The Equation of State in Astrophysics: IAU Colloquium 147 (International Astronomical Uni)

What do we understand of the properties of the dense ionised matter found in the interiors of low-mass stars and giant ...

Author: Gilles Chabrier | Evry Schatzman

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What do we understand of the properties of the dense ionised matter found in the interiors of low-mass stars and giant planets? The 147th IAU Colloquium gathered together international experts to address this question, and their answers are provided in this opportune proceedings. In this volume, reviews by world experts in plasma and dense matter physics and in stellar astrophysics cover everything from the cooling theory of white dwarfs and their accretion-induced collapse through to the internal structure of low-mass stars and giant planets. They cover a wide range of topics related to the equation of state in dense matter, from the fundamental basis of the iV-body problem to astrophysical applications. Together these articles provide an essential review of the most recent achievements in the field and give direction for future research, for graduate students and researchers

The Equation of State in Astrophysics

IAU Astronomical Union Union Astronomique International The following Colloquia of the International Astronomical Union are published for the Union by Cambridge University Press.

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The Equation of State in Astrophysics Proceedings of IAU Colloquium No. 147 Saint-Malo, France 14-18 June 1993

Edited by Gilles Chabrier Ecole Normale Superieure de Lyon and Evry Schatzman Observatoire de Meudon

Hi CAMBRIDGE UNIVERSITY PRESS

Published by the Press Syndicate of the University of Cambridge The Pitt Building, Trumpington Street, Cambridge CB2 1RP 40 West 20th Street, New York, NY 10011^1211, USA 10 Stamford Road, Oakleigh, Melbourne 3166, Australia © Cambridge University Press 1994 First published 1994 Printed in Great Britain at the University Press, Cambridge A catalogue record for this book is available from the British Library Library of Congress cataloguing in publication data available ISBN 0 521 47260 1 hardback

Contents

Group photograph List of participants Preface

X

xi xvi

Reviews 1 Equations of state in stellar structure and evolution H.M. Van Horn

page 1

2 Equation of state of stellar plasmas F.J. Rogers

page 16

3 Statistical mechanics of quantum plasmas. Path integral formalism A. Alastuey

page 43

4 Onsager-molecule approach to screening potentials in strongly coupled plasmas Y. Rosenfeld

page 78

5 Astrophysical consequences of the screening of nuclear reactions J. hern and M. Hernanz

page 106

6 Crystallization of dense binary ionic mixtures. Application to white dwarf cooling theory R. Mochkovitch and L. Segretain page 126 7 Non crystallized regions of White dwarfs. Thermodynamics. Opacity. Turbulent convection /. Mazzilelli page 144 8 White dwarf crystallization E. Garcia-Berro and M. Hernanz

page 161

9 Gravitational collapse versus thermonuclear explosion of degenerate stellar cores J. Isern and R. Canal

page 186

10 Neutron star crusts with magnetic fields D.G. Yakovlev and A.D. Kaminker

page 214

11 High pressure experiments for astrophysics P. Loubeyre

page 239

12 Equation of state of dense hydrogen and the plasma phase transition; A microscopic calculational model for complex fluids F. Perrot and C. Dharma-wardana

page 272

13 The equation of state of fluid hydrogen at high density G. Chabrier

page 287

14 A comparative study of hydrogen equations of state D. Sanmon

page 306

15 Strongly coupled ionic mixtures and the H/He equation of state H.M. DeWitt

page 330

16 White dwarf seismology: Influence of the constitutive physics on the period spectra G. Fontaine and P. Brassard page 347 17 Helioseismology: the Sun as a strongly-constrained, weakly-coupled plasma W. Da'ppen

page 368

18 Transport processes in dense stellar plasmas N. Itoh

page 394

Vll

Vlll

Contents

19 Cataclysmic variables: structure and evolution J.-M. Hameury

page 420

20 Giant planet, brown dwarf, and low-mass star interiors W.B. Hubbard

page 443

21 Searches for brown dwarfs J. Liebert

page 463

22 Jovian seismology B. Mosser

page 481

Observational projects 23 EVRIS: first space experiment devoted to stellar seismology A. Baglin

page 512

24 The HIPPARCOS mission and tests for the equation of state A. Baglin and Joao Fernandes

page 517

25 Ground based heliosismology: IRIS and GONG F.-X. Schmider

page 525

26 The spatial GOLF project S. Turck-Chieze

page 532

27 The DENIS survey T. Forveille

page 537

28 PRISMA: A mission to study interior and surface of stars P. Lemaire

page 540

Posters 29 Towards a helioseismic calibration of the equation of state in the solar convective envelope S. V. Vorontsov, V. A. Baturin, D. 0 . Gough, W. Dappen page 545 30 Thermal cyclotron and annihilation radiation in strong magnetic fields V.G. Bezchastnov and A.D. Kaminker

page 550

31 Modified adiabatic approximation for a hydrogen atom moving in a magnetic field V.G. Bezchastnov and A.Y. Potekhin

page 555

32 Computations of static white dwarf models: A must for asteroseismological studies P. Brassard and G. Fontaine

page 560

33 The Chandrasekhar mass of a gravitating electron crystal D.Engelhardt and I. Bues

page 565

34 Coulomb corrections in the nuclear statistical equilibrium regime D. Garcia and E. Bravo

page 571

35 Molecular Opacities: Application to the Giant Planets T. Guillot, D. Gautier and G. Chabrier

page 576

36 On Radiative Transfer Near the Plasma Frequency at Strong Coupling Yu. K. Kurilenkov and H.M. Van Horn

page 581

37 Effects of Superfluidity on Spheroidal Oscillations of Neutron Stars Umin Lee, T.J.B. Collins, R.I. Epstein and H.M. Van Horn

page 586

38 Magnetic Field Decay in the Non-superfluid Regions of Neutron Star Cores A. G. Muslimov and H. M. Van Horn

page 591

Contents

ix

39 On the equation of state in Jovian seismology J. Provost, B. Mosser and G. Chabrier

page 596

40 Analysis of the screening formalisms in solar and stellar conditions H. Dzitko, S. Turck-Chieze, P. Delbourgo-Salvador and Ch. Lagrange

page 601

41 Theoretical Description of the Coulomb Interaction by Pade-Jacobi Approximants W. Stolzmann and T. B locker

page 606

42 New Model Sequences from the White Dwarf Evolution Code M. Wood

page 612

43 Low temperature opacities C. Neuforge

page 618

LIST OF PARTICIPANTS

Numbers in parentheses refer to the group photograph

A. ALASTUEY, (58)

J. M. APARICIO, (37)

W. APPEL, (14)

A. BAGLIN, (2)

I. BARAFFE, (31)

V. BEZCHASTNOV, (61)

T. BLOCKER,

H. BOFFIN, (22)

P. BRASSARD, (46)

Laboratoire de Physique, Ecole Normale Superieure de Lyon, 69364 Lyon Cedex 07 - FRANCE [email protected] C.E.A. de Blanes, 17300 Blanes - SPAIN aparicio@ceab. es Laboratoire de Physique, Ecole Normale Superieure de Lyon, 69364 Lyon Cedex 07 - FRANCE wappel@physique. ens-lyon.fr DASGAL, Observatoire de Meudon, 92195 Meudon Cedex - FRANCE [email protected] Laboratoire de Physique, Ecole Normale Superieure de Lyon, 69364 Lyon Cedex 07 - FRANCE ibaraffe@physique. ens-lyon.fr Ioffe Institute of Physics and Technology, 194021 St Petersburg - RUSSIA [email protected] Institiit fiir Theoretische Physik und Sternwarte, Universitat Kiel, 2300 Kiel - GERMANY pas96 @rz. uni-kiel. dbp. de Institut d'Astrophysique, Universite Libre de Bruxelles, 1050 Brussels - BELGIUM [email protected] Departement de Physique, Universite de Montreal, H3C 3J7 Montreal, Quebec - CANADA brassard@astro. umontreal. ca

XI

List of participants I. BUES, (56)

G. CHABRIER, (11)

C. CHARBONNEL, (13) F. D'ANTONA, (40)

W. DAPPEN, (27)

H. E. DE WITT, (20)

W. EBELING,

D. ENGELHARDT,(60)

J. FERNANDES, (55)

G. FONTAINE, (45)

A. FORSTER, (17)

E. GARCIA-BERRO, (8)

Remeis-Sternwarte Bamberg, D-8600 Bamberg - GERMANY [email protected] Laboratoire de Physique, Ecole Normale Superieure de Lyon, 69364 Lyon Cedex 07 - FRANCE [email protected] Observatoire de Geneve, CH-1290 Sauverny - SWITZERLAND Osservatorio Astronomico di Roma, 00040 Monte Porzio - ITALIA [email protected] and [email protected] Department of Physics and Astronomy, University of Southern California, 90089-1342 Los Angeles, California - USA [email protected] Lawrence Livermore Laboratory, 94550 Livermore, California - USA [email protected] Hiimboldt Universitdt, Sektion Physik, 1040 Berlin - GERMANY [email protected] Dr. Remeis Sternwarte, 8600 Bamberg - GERMANY [email protected] DASGAL, Observatoire de Meudon, 92195 Meudon Cedex - FRANCE [email protected] Departement de Physique, Universite de Montreal, H3C 3J7 Montreal, Quebec - CANADA [email protected] Institiit fur Theoretische Physik, Hiimboldt Universitat, 1040 Berlin - GERMANY [email protected] Universidad Politecnica de Barcelona, Dept Fisica, 08031 Barcelona - SPAIN [email protected]

Xll

List of participants Institut d'Estudis Catalans, Laboratori d'Astrofisica, Barcelona - SPAIN dgarcia@fen. upc. es Centre d'Etudes de Limeil-Valenton, D. GILLES, (36) 94195 Villeneuve-Saint-Georges - FRANCE gilles@limeil. cea.fr J. F. GONZALEZ, (12) Laboratoire de Physique, Ecole Normale Superieure de Lyon, 69364 Lyon Cedex 07 - FRANCE [email protected] DASGAL, Observatoire de Meudon, M. J. GOUPIL, (43) 92195 Meudon Cedex - FRANCE goupil@mesiob. obspm. circe.fr Institute Physics of the Earth, T. V. GUDKOVA, (6) Russian Academy of Sciences, 123810 Moscow - RUSSIA [email protected] T. GUILLOT, (51) Observatoire de Nice, 06304 Nice Cedex 4 - FRANCE [email protected] nice.fr J. GUTIERREZ-CABELLO, Facultad de Fisica, Dipartamento di Astronomia, 08028 Barcelona - SPAIN [email protected] Observatoire de Strasbourg, J. M. HAMEURY, (34) 67000 Strasbourg - FRANCE [email protected] L.P.L., University of Tucson, W. B. HUBBARD, (44) 85721 Tucson, Arizona - USA [email protected] C.S.I.C., cami de Santa Barbara, J. ISERN, (5) 17300 Blanes - SPAIN [email protected] N. ITOH, (4), Mrs ITOH, (3) Department of Physics, Sophia University, 102 Tokyo - JAPAN [email protected] D. GARCIA-SENZ,

Xlll

List of participants T. KAHLBAUM, (49)

A. KOVETZ, (41)

P. 0 . LAG AGE, U. LEE, (25)

P. LEMAIRE, J. LIEBERT, (23)

P. LOUBEYRE, (15)

I. LOPEZ, (35)

G. MASSACRIER, (32)

J. MATIAS, (42)

I. MAZZITELLI, (57)

R. MOCHKOVITCH,

WIP, Arbeitsgruppe Nieder Temperaturplasmaphysik, 2200 Greifswald - GERMANY Department of Geophysics and Planetary, Sciences, Tel Aviv University, 69978 Tel Aviv - ISRAEL attay@etoile. tau.ac. il CEA - CEN Saclay, 91191 Gif-sur-Yvette Cedex - FRANCE Laboratoire de Physique, Ecole Normale Superieure de Lyon, 69364 Lyon Cedex 07 - FRANCE [email protected] IAS, Universite Paris 11, 91405 Orsay Cedex - FRANCE Steward Observatory, University of Tucson, 85721 Tucson, Arizona - USA [email protected] Universite Paris 6 Laboratoire de Physique des Milieux Condenses 75252 Paris - FRANCE DAPNIA, CEN Saclay, 91191 Gif-sur-Yvette Cedex - FRANCE [email protected] Laboratoire de Physique, Ecole Normale Superieure de Lyon, 69364 Lyon Cedex 07 - FRANCE [email protected] Observatoire de Toulouse, 31400 Toulouse - FRANCE [email protected] CNR, Instituto di Astrofisica Spaziale, 00044 Frascati (Rm) - ITALIA italo@irmias. ias.fra. cnr. it LA.P., 75014 Paris - FRANCE [email protected]

XIV

List of participants B. MOSSER, (33)

N. MOWLAVI,

C. NEUFORGE, (21)

A. PEREZ, (59)

F. ROGERS, (50)

Y. ROSENFELD, (48)

D. SAUMON, (47)

E. SCHATZMAN, (10)

M. SCHLANGES, (19)

F. X. SCHMIDER,

L. SEGRETAIN, (16)

G. SHAVIV, (38)

I.A.P., 75014 Paris - FRANCE [email protected] Universite Libre de Bruxelles, 1050 Brussels - BELGIUM nmowlavi@astro. ulb. ac. be Universite de Liege, Institut d'Astrophysique de Liege, 4000 Liege - BELGIUM Laboratoire de Physique, Ecole Normale Superieure de Lyon, 69364 Lyon Cedex 07 - FRANCE [email protected] Lawrence Livermore Laboratory, 94550 Livermore, California - USA [email protected] Physics Department, Nuclear Research Center-Negev, 84190 Beer-Sheva - ISRAEL L.P.L., University of Tucson, 85721 Tucson, Arizona - USA [email protected] DASGAL, Observatoire de Meudon, 92195 Meudon Cedex - FRANCE [email protected] Universitat Greifswald, Fachrichtung Physik, 2200 Greifswald - GERMANY Universite de Nice, Departement d'Astrophysique, 06108 Nice Cedex 2 - FRANCE Laboratoire de Physique, Ecole Normale Superieure de Lyon, 69364 Lyon Cedex 07 - FRANCE [email protected] Technion-Israel, Institute of Technology 32000 Haifa - ISRAEL [email protected]

XV

List of participants C. STEHLE, (30)

W. STOLZMANN, (39)

J. TULLY, (1)

S. TURCK-CHIEZE,

H. M. VAN HORN, (53)

G. VAUCLAIR, (52)

S. VORONTSOV, (26)

A. WEISS, M. A. WOOD, (9)

G. WUCHTERL, (62)

D. YAKOVLEV, (54)

V. N. ZHARKOV, (7)

DAMAP, Observatoire de Meudon, 92195 Meudon Cedex - FRANCE [email protected] Instittit fiir Theoretische Physik und Sternwaxte, Universitat Kiel, 2300 Kiel - GERMANY pas96@rz. uni-kiel. dbp. de Observatoire de Nice, 06304 Nice Cedex 4 - FRANCE [email protected] CEA - CEN Saclay, 91191 Gif-sur-Yvette Cedex - FRANCE turck@frsacl 1 .bitnet Department of Physics and Astronomy, University of Rochester, 14627 Rochester, New York - USA hmvh@uordbv. bitnet Observatoire de Toulouse, 31400 Toulouse - FRANCE vauclair@fromp 51. bitnet Astronomy Unit, Queen Mary and Westfield College, E l 4NS London - ENGLAND [email protected] Max Planck Institute fiir Astrophysik, 8046 Garching - GERMANY Florida Institute of Technology, 32901-6988 Melbourne, Florida - USA [email protected]. fit. edu Universitat Heidelberg, Instittit fiir Theoretische Astrophysik, 6900 Heidelberg - GERMANY [email protected] Ioffe Institute of Physics and Technology, 194021 St Petersburg - RUSSIA [email protected] Institute Physics of the Earth, Russian Academy of Sciences, 123810 Moscow - RUSSIA [email protected] (att. Prof. Zharkov)

XVI

PREFACE In the Internal Constitution of the Stars, published in 1926, Eddington gave a central temperature of white dwarfs of several billions degrees. The Fermi-Dirac statistics appeared just one year later, in 1927, and the new equation of state for degenerate matter provided the explanation of white dwarfs. The solution of the problems we have to consider presently are probably not relevant of the same kind of intellectual jump. But who knows! Anyhow, the time of the perfect gas law is definitely over. It is possible, in many cases, to get astrophysical orders of magnitude, using simple or oversimple relations betwen physical quantities. However, modeling correctly observational results has become, nowaday, more and more difficult. Data are of a better precision and provide more information, would it be chemical abundances, evolutionary tracks or those wonderful helioseismological data. An elementary statement is that, in order to look Inside the Stars (the title of a recent colloquium), we need more accurate descriptions of basic physical laws: equation of state, opacities, thermonuclear reactions. We are still facing many difficulties in the field, and we can give a few examples: we do not have a theory which decribes consistently both the equation of state of a plasma and the level population of the atoms; we still have to improve the theory of screening effects in dense plasmas; we have now a description of cold, dense, weakly ionized matter of brown dwarfs, but it is still incomplete. And this is not the end of the list! There has not been many meetings devoted to the equation of state. The 1978 meeting on dense matter in Paris dealt with several aspects of the equation of state, but the present meeting represents an important new landmark. It provides the link between the astrophysical problems related to the equation of state and the underlying most recent physical theories. It is not necessary to give here the list of the reviews: they can be found in the table of contents. Let just say that most regions in the pressure-temperature diagram have been covered by the present review papers. From main sequence stars to white dwarfs, from brown dwarfs to giant planets, from supernovae to neutron stars (ultra high density being excluded), most aspects of the equation of state have been considered. The connections with new observational results (HIPPARCOS, GOLF, PRISMA) have also been presented. Our knowledge of the equation of state will certainly have to be improved within the next few years in order to obtain the best description of a large amount of new observational results. We hope the present conference will help reaching this goal. We are very endebted to the different speakers for their excellent reviews, as well in the oral as in the written presentations. We are grateful to the Commission of the European Communities, the 'Centre National de la Recherche Scientifique (CNRS)', the 'Ministire de la Recherche el de la Technologie' and the International Astronomical Union for granting financial support for the organization of the conference and the edition of the present proceedings. We would like to express our special thanks to Marie-Pierre Fuchs (who unfortunately could not come to St-Malo) for carrying efficiently the practical organization and to the whole staff of the Palais du Grand Large for their efficiency and their warm hospitality. Finally we express our warmest thanks to all the participants who made this a fruitful, stimulating and enjoyable meeting.

Evry Schatzman and Gilles Chabrier February

XV11

1994

PREFACE Dans Internal Constitution of the Stars, publid en 1926, Eddington donnait une temperature centrale pour les naines blanches de plusieurs milliards de degree. La statistique de Fermi-Dirac apparut un an plus tard, en 1927, et la nouvelle Equation d'dtat pour la matiere ddgdndrde fournit 1'explication des naines blanches. La solution des problemes que Ton considere aujourd'hui ne relive probablement pas du meme genre de revolution intellectuelle. Mais qui sait! II est possible, dans la plupart des cas, d'obtenir des ordres de grandeur astrophysiques en utilisant des relations simples entre grandeurs physiques. Une modelisation correcte des rdsultats observationnels, cependant, est devenue de nos jours de plus en plus difficile. Les donndes sont de plus en plus prdcises et fournissent de plus en plus d'informations, que ce soit les abondances chimiques, les trace's devolution ou ces superbes donndes d'hdliosismologie. II est trivial de constater que, afin de voir Inside the Stars (titre d'un congres recent), nous avons besoin de descriptions de plus en plus correctes des outils physiques de base : Equation d'dtat, opacity's, reactions thermonucldaires. II reste encore beaucoup de difficulty's dans le domaine, et nous pouvons en donner quelques exemples : il n'y a toujours pas de thdorie qui ddcrive de facon cohdrente a la fois l'dquation d'dtat d'un plasma et la population des niveaux atomiques; il nous faut encore amdliorer la thdorie des effets d'dcran dans un plasma; nous avons ddsormais une description de la matiere froide, dense, partiellement ionisde dans les naines brunes, mais elle est encore incomplete. Et ce n'est pas une liste exhaustive! II n'y a pas eu beaucoup de congres ayant pour theme l'dquation d'dtat. La conference de 1978 sur la mati£re dense a Paris traitait de plusieurs aspects de l'dquation d'dtat, mais la prdsente conference reprdsente une marque importante, car elle fournit le lien entre les problemes astrophysiques lids a l'dquation d'dtat et les theories physiques sous-jacentes les plus rdcentes. II n'est pas ndcessaire de donner la liste des revues: elles sont donndes dans la table des matieres. Disons simplement que la plupart des rdgions du diagramme densitd-tempdrature sont couvertes par les revues de ce recueil. Des dtoiles de la Sdquence Principale aux naines blanches, des naines brunes aux planetes gdantes, des supernovae aux dtoiles a neutrons (en excluant la matiere ultra-dense), la plupart des aspects de l'dquation d'dtat ont dtd considdrds. Les liaisons avec les nouveaus rdsultats observationnels (HIPPARCOS, GOLF, PRISMA) sont dgalement prdsentdes. Notre connaissance de l'dquation d'dtat devra certainement Stre amdliorde dans les prochaines anndes afin d'obtenir une description correcte des nombreux nouveaux rdsultats observationnels. Puisse le prdsent recueil etre utile a un tel but. Nous sommes extremement reconnaissants aux diffdrents confdrenciers pour leurs excellentes revues, tant dans la prdsentation orale qu'dcrite. Nous remercions la Commission des Communautis Europiennes, le Centre National de la Recherche Scientifique, VUnion Astronomique Internationale et le Ministere de la Recherche et de la Technologie pour leur soutien financier dans l'organisation de la confdrence et l'ddition du prdsent recueil. Nous voudrions exprimer plus particulierement nos remerciements a Marie-Pierre Fuchs (qui malheureusement n'a pas pu venir a St-Malo) pour son efficacitd dans ('organisation pratique de la confdrence, ainsi qu'a tout le personnel du Palais du Grand Large pour son efficacitd et sa cordiale hospitalitd. Enfin nous exprimons nos plus sinceres remerciements a tous les participants, qui ont fait de cette confdrence une rdunion fructueuse, stimulante et sympathique.

Evry Schatzman et Gilles Chabrier Fdvrier 1994

XV111

Equations of State in Stellar Structure and Evolution H. M. VAN HORN Department of Physics and Astronomy, C. E. Kenneth Mees Observatory, and Laboratory for Laser Energetics, University of Rochester, Rochester, NY 14627-0011. Current address: Division of Astronomical Sciences, Room 1045, National Science Foundation, 4201 Wilson Boulevard, Arlington, VA 22230

Abstract In this paper I summarize some of the recent advances in studies of dense matter. Research on phase separation in the binary ionic mixtures (BIMs) that constitute the matter in white dwarfs has been motivated by the need to obtain accurate estimates for the ages of the faintest white dwarfs and thus of the disk of our Galaxy. Substantial age increases appear possible, but it is not yet clear whether such large increases occur in real white dwarfs. A second advance is the prediction, based on state-of-the-art physical calculations, that ionization of H at low temperatures and increasing densities may occur via a first-order "plasma phase transition" (PPT). Astrophysical consequences of this result are still being explored in an effort to test this prediction. Related to these equation-of-state calculations are calculations of the enhancement of nuclear reaction rates at high densities. New thermonuclear rates have been computed for C+C reactions in BIMs, although there is currently some controversy about results at the highest densities. New pycnonuclear reaction rates have also been calculated for BIMs, and it has been suggested that He-burning at T — 0 may occur through a first-order phase transition. Finally, calculations of the equation of state of matter in strong magnetic fields and of radiative opacities at high densities have undergone very recent and substantial improvements, which are just beginning to be utilized in astrophysical calculations. Dans cette revue, je resumme les travaux les plus recents dans l'etude

2

Van Horn: Stellar structure and evolution

de la matiere dense. Les recherches sur la separation de phase dans les melanges ioniques binaires qui constituent l'interieur des naines blanches ont ete motivees par la necessite d'obtenir une estimation correcte de Page des naines blanches les moins brillantes et done du disque de notre Galaxie. II apparait possible d'accroitre l'age substantiellement, mais il n'est pas encore certain qu'un tel accroissement ait reellement lieu dsans les naines blanches. Une seconde percee est la prediction, basee sur des calculs trs detailles, que l'ionisation de H a basse temperature et haute pression ait lieu au travers d'une "transition de phase plasma (PPT) ". Les consequences astrophysiques de ce resultat sont actuellement encore en cours d'etude, afin de tester cette prediction. En relation avec ces equations d'etat, ont lieu des calculs d'accroissement de taux de reactions nucleaires a haute densite. De nouveaux taux termonucleaires ont ete calcules pour la reaction C+C dans les melanges binaires, bien au'il y ait actuellement une controverse quant a ces resultats. De nouveaux taux pycnonucleaires ont ete calcules pour les melanges, et il a ete suggere que le brulage de He a T = 0 ait lieu au travers d'une transition de phase du premier ordre. Enfin, des calculs de l'equation d'etat de la matiere dans les champs magnetiques forts et des opacites radiatives a haute densite ont ete effectues recemment.

1.1 Introduction The purpose of this paper is to summarize various problems related to the equations of state (EOS) used in stellar structure and evolution calculations. I shall be concerned with quantities such as the EOS itself, various transport coefficients, nuclear reaction rates, and so forth. I shall endeavor to summarize some of the most recent advances in these areas, but it is not possible here to cover all topics of interest. In particular, there have been some interesting recent calculations of the EOS of matter in strong magnetic fields which are mentioned only briefly at the end of this paper. Let me begin with a reminder of the parameters needed for stellar evolution calculations. The four equations describing the time-dependent structure of a star are (c/. Clayton 1968, pp. 436 fF.): dr GMr

dP dr

P r2 '

dT

3 up

dr

3

(2) Lr

4acT 4irr2'


and , OS x

/. v

lr)

(4)

In order to solve these equations, we need to know the following properties of stellar matter: the pressure P = P(p,T), the entropy 5 = S(p,T), the opacity K = n(p,T), and the net nuclear energy generation rate e = e(p,T). The relations giving the dependence of the pressure and entropy upon the thermodynamic state variables p, the mass density, and T , the temperature, are collectively called the "equation of state." Quantities such as the radiative opacity K and the so-called "conductive opacity" /c,., proportional to the inverse of the thermal conductivity, are termed transport coefficients; they determine the rate of transport of some physical quantity, such as heat. For some purposes, we also need to know the viscosity (which controls the rate of transport of momentum) or the ambipolar diffusion coefficients (which govern the rate of gravitational separation of elements in a star); for the most part I shall neglect these quantities in this brief review. The thermonuclear and pycnonuclear (density-induced) reaction rates e and the neutrino loss rates ev round out the list of physical quantities needed. It is also useful to have some idea of the values of the dimensionless physical parameters that characterize the state of matter at different temperatures and densities. Van Horn (1991) has illustrated this for plasmas composed of H or of Fe. Here we briefly describe the changes in the physical conditions, beginning at low p and at thermal energies kT much less than the ionization potentials of the atoms. Under these conditions, matter consists of an ideal gas of neutral atoms or molecules. As the temperature increases, but still at low densities, the molecules first dissociate into neutral atoms, and the atoms subsequently ionize. At sufficiently high T, matter consists of a fully ionized plasma of bare atomic nuclei and electrons. Under these conditions, matter can still be described as a mixture of ideal gases (the ions and electrons). If the density is now increased along some isotherm, new, density-dependent effects begin to appear. As the electrons in the plasma become crowded increasingly close together, degeneracy first becomes important. This occurs when kT ~ ep, where €p is the Fermi energy. At still higher densities, the Coulomb interaction energy (Ze)2/a between the electrons and ions becomes comparable to kT. Here a is a measure of the average separation of electrons and ions; it is of the order of the size of the "Wigner-Seitz sphere" containing just enough electrons to neutralize the ionic charge. The quantity F = (Ze)2/akT is the dimension-

4


less parameter generally used to characterize the strength of the Coulomb interaction. The condition T ^*^

(9)

Rogers: Stellar plasmas

21

In equations (7, 9) i and j are the particle types, X^ = (h212^kT)1'2, n and I, are quantum numbers, the Eni are bound state energies, 5/ is the phase shift, p is the relative momentum, and |iy is the reduced mass. The connection between bound and scattering states can readily be established by integrating equation (9) by parts and using the fact that the phase shift at zero energy is just nn (Levinson 1949). Adding the zero energy term to equation 8 (Rogers, Graboske, and DeWitt 1971) gives B

- -1)

(10)

If now each exponential (Boltzmann) factor is expanded in powers of PE n / it is apparent that the first term in the expansion just removes the negative term in equation (10). In other words, an analytical term having the form of the most divergent term in Zjm is not actually present in By; at least at temperatures where the Boltzmann factors are near unity. The sum over the PE n / terms in the expansion of BJ£ is also divergent, but with a second integration by parts of equation (9) it too can be shown to be analytically missing from the total By at high temperature (Rogers 1979; 1977; Bolle 1989; 1987; Pisano and McKellar 1989). There are divergences in By, but they are all in the final form for B^s after the two integrations by parts. These two integrations by parts have effectively redefined the continuum such that it begins at -kT, rather than the usual zero of energy. The effective internal partition function is thus,

(ID It is worthwhile noting that the value of By has not changed in this series of manipulations, rather it has been shown that the analytic properties of the complete set of states dictates a specific separation of By into effective bound and scattering state parts. This separation will play an important role in the many-body statistical mechanical methods to be described in the next section. 2.4 Classical Density Expansion for Plasmas The natural way to treat reacting, multicomponent plasmas is in the grand canonical ensemble (Hill 1956). In this approach one views the system in terms of its fundamental constituents, so that bound complexes arise naturally from the theory. As a result, this approach is commonly referred to as the physical picture method. The standard procedure is to

22


expand the pressure in terms of two body, three body clusters, etc., i.e., a cluster expansion. The same is true for plasmas, but the long range of the Coulomb potential introduces substantial complications. In addition, the quantum nature of electrons introduces degeneracy and exchange corrections. The attractive electron-ion interaction leads to short distance divergences in classical cluster coefficients, so that the use of quantum mechanical methods is essential. Graphical resummation procedures are required to remove the long-range divergences occurring in all cluster coefficients of plasmas. The divergences in B^js, mentioned in the previous section, are only the simplest examples. A detailed description of the procedure is given elsewhere (Dashen, Ma, and Bernstein 1969; Kraeft, Kremp, Ebeling and R6pke 1986; Rogers 1981). For illustrative purposes it is much easier to consider the related procedure in the canonical ensemble, i.e., an expansion of the free energy in the density. For simplicity we also focus on classical procedures where possible. In early graphical analysis of the classical one-component plasma, ie., heavy ions emmersed in a continuous neutralizing background of electrons, Mayer (1950)referredto the non-ideal free energy (aside from a factor VkT) as -S, so that, (F-F0)/VkT = -S

(12)

where S = 2LBj-i^

(13)

Bj is the jth virial coefficient. The classical second virial coefficient for the one component plasma (OCP) is given by B2 = -—\d~rx'd~ri(e-pu{ra)-\) v 2VJ

= -2n\"{e-MT) -l)r2dr / J o

(14)

where rl2 = r = The divergence in B2 is easily seen by expanding the classical Boltzmann factors in powers of |5u. The leading term diverges as r2, the second term as r, and the third term logarithmically as r-»°o. The leading divergence posses no problem, since it is canceled by an opposing term provided by the neutralizing electron background; leaving the (Pu)2 as the leading


23

divergence. In general the lowest order term from each of the virial coefficients is given by \ \

i

) . . . u { r

j x

)

(15)

The sum over the most divergent terms from each of the higher B j ; i.e., those that have just one power of the potential turned on between each pair of particles, can be represented graphically as shown in Figure 1.

Fig. 1 Graphic illustration of the classical ring sum for the OCP. Solid circles represent ions. A diagram involving n ions comes from B n .

Due to their topology, they are known as ring diagrams. The result of the summation over ring diagrams is t

^

j

(16)

y

where A = Z2e2/kTXD

(17)

A D =(*774/rZVp) 1/2

(18)

and

The pressure for a v component plasma, corresponding to multicomponent generalization of equation (12), is (19)

24


Using equation (16) this gives in the Debye-Huckel approximation for the OCP

P/kT=p(l-±) o

(20)

The Debye-Hiickel correction is just the leading term in an infinite expansion analogous to the virial expansion for an ordinary gas. However due to the Coulomb modifications it involves p 3 # rather than p 2 (see equation 16). The next higher term in the expansion is obtained by summing over the next most divergent diagrams, the three rung ladder diagrams; so named by their ladder-like quantum mechanical form. Figure 2 shows (in the classical limit) how one rung of the three rung ladder diagram is screened by summing over chains, of ever increasing length, that have one power of the potential, Bu, turned on between each pair of ions. Similar summations produce screening in the remaining two rungs of the diagram (often called watermelon diagrams in classical mechanics).

0 0 l-B,(T,XD)-27tjdrr2(q,

25

-

(24)

_2 where _

Z Z

a ft
0.2. The S2 correction, analogous to the second virial coefficient for the Debye-Hiickel potential, is somewhat of an improvement The important point is that the Abe series offers a systematic way of calculating the EOS of the OCP. It was shown in Rogers (1981) that, if large numbers of Abe nodal terms are included, this convergence persists to large values of the coupling parameter, T=Z2e2/&. This shows that, contrary to some claims in the literature, that, when properly implemented, the Debye-Huckel potential is a valid plasma potential. Similar studies have been done for multicomponent mixtures of heavy ions (TCP) in a neutralizing background. Our interest here is limited mainly to weak to moderately coupled plasmas, but this example shows that the Abe procedure could be applied to plasmas occurring in white dwarf stars. Related examples of the convergence of the activity expansion method (see Section 7) to the OCP are given in Rogers and DeWitt (1973). They found that the convergence is somewhat faster in the

26


activity expansion. Similar results have also been reported for electrolytic solutions (Wood, Lilley, and Thompson 1978). Since electrons in real plasmas are always quantum mechanical, direct use of classical methods is not possible. Nevertheless, the main complications in real plasmas have to do with the highly classical long range correlations. The divergences associated with short range quantum interactions are easily handled by replacing the classical virial coefficients occurring in the Sj with their quantum-mechanical equivalents (Rogers 1991; 1986; 1981; 1974; Rogers and DeWitt 1973). Quantum mechanical calculations for real reacting plasmas follow similar steps and the convergence should be similar to the example in Figure 3. In the current implementation the ion-ion correlations are included in all orders, but the electron-electron and electron-ion correlations are included only through 5/2 order in the density (activity). Approximate methods for including electron-electron and electron-ion terms when even the electron-ion coupling is strong are given in Rogers 1979. 1

1.0

0.5

-

3

2 1

4 1

1

\ -

OCP

0 —

\ \

i

N . . .

8

12

16

A Fig. 3 Comparison of the Abe nodal expansion with Monte Carlo simulations of the OCP by Slattery, Doolen, and Dewitt (1982). Labels indicate the free energy terms used to calculate the pressure.

Elaborating on the remarks of the previous paragraph, we note that for a real plasma, involving electrons and ions, the diagrammatic sum that gives the Debye-Hiickel correction equivalent to equation 16, involves additional terms corresponding to electron-electron and electron-ion interactions.

27


o
0.5 . For Boltzmann statistics the quantum corrected Debye-Huckel term can be written in the form (30)

28


where f(0,0)=l. There are also some additional corrections due to exchange effects. The ion-ion diffraction parameter, "fa, does not appear in equation (30), since 7 i i « l due to the mass of the ion. Figure 5 displays the variation of f with Yee for the simple case of a non-degenerate electron gas for the "exact" calculations of Graboske and DeWitt (1974) and the approximate effective potential results of Rogers (1979).

100

Fig. 5 Diffraction corrections to the electron-electron ring sum.

2 5 EOS of a Hydrogen Plasma We have chosen the simple example of ionization equilibrium in a nondegenerate hydrogen plasma to compare the free energy obtained from a many body quantum statistical approach with a commonly used free energy minimization model, i.eM

e+pjtH The typical free energy minimization method would have

where the first three terms on the right correspond to the translational free energies for electrons, protons, and hydrogen atoms, respectively, Zmt is the sum over states (internal partition function), and SoH is gi v e n by


29

equation (28). As described in Section 2, some model must be introduced to make the sum in Zjm finite. In practice this can be one of a number of possibilities such as the use of the energy levels of the Debye potential. Since Zjnt changes discontinuously at conditions where a state moves into the continuum and is no longer counted, equation (31) is not physically consistent The free energy that results from the many body diagrammatic approach is (Rogers 1991; 1989; 1986) (32) \PA.)

\PPA>'p)

KPH^'H

)

where (33) which is closely related to Z^ t of equation (11), is the so called PlanckLarkin partition function. The sum in Zg}t ranges over the states in a screened potential which approaches the Debye-Hiickel potential at very low density. As described in Rogers (1986; 1981) the energy levels appearing in 3m are unscreened except for high lying states near the plasma continuum. The states that are screened change with plasma conditions. As a result Zg{t is both finite and a continuous function of temperature and density; although the density dependence is very slight for normal stellar conditions. The MHD EOS displays a similar property through the use of the occupation probability formalism. Figure 6 is an attempt to explain the result in equation (32). It shows (schematically) the contribution to the partition function for hydrogen as a function of principal quantum number at temperatures of 1-2 eV. Due to the large value of the Boltzmann factor at these temperatures, the ground state contribution is large. Because of the wide energy separation between n=l and n=2, the contribution from n=2 has already dropped close to unity. For higher states the n2 degeneracy causes the contribution to again increase. For the pure Coulomb potential the area under the curve becomes infinite and leads to the well known divergence of the atomic partition function. The high lying states are classical (closely spaced), which means that sums over quantum numbers can be replaced with integrals; while the low lying states are quantum mechanical. This suggests that the partition function can be separated into two parts; one requiring the explicit use of quantum numbers, the other only involving

30


integrals as shown in the figure. The natural way to do this seems to be to use the analytic properties (see Section 3). The boundary between the two regions is of course fuzzy and not abrupt as actually shown. The discussion leading to equation (28) showed that, even though quantum mechanics is required to remove the divergence in the electron-ion terms, the ring sum is highly classical. This suggests that the electron-ion part of the ring summation leading to equation (28) has in some sense included the classical parts of Zint. As a result, using the full Zjnt in equation (31) is incorrect, since it would double count much of the excited state part.

8 Fig. 6 Contributions to the hydrogenic bound state partition function. Lower shaded region corresponds to the quantum mechanical part; upper shaded region to the (approximate) classical part. The total is shown as a smoothed curve to emphasize the classical nature of the high n contribution.

The pressure for a partially ionized hydrogen plasma corresponding to equation (32), but also including degeneracy and exchange corrections, has the form (34)

where the In/2 functions are the usual Fermi Functions, a e =Me/kT is the degeneracy parameter,

31


(35)

is the first order electron exchange, and gee is the electron-electron distribution function for an ideal Fermi gas, the Debye length corrected for electron degeneracy is 1/2

kT

(36)

and fp(Yee.Yei) is the diffraction correction to the pressure similar to Figure 5. There are some additional exchange corrections to equation (34), which are not explicitly shown. The first order exchange correction (equation (35)) is frequently omitted in astrophysical EOS calculations, but in view of the current need for high precision that is no longer acceptable. The origin of the exchange term is easily seen from the electronelectron distribution function for an ideal gas, as shown in Figure 7. 1.0

Fig. 7 Schematic representation of the ideal Fermi gas electron-electron distribution function.

For an ideal Boltzmann gas, gee is everywhere unity, as is gpp ang gep (not shown), so that, the first order contribution to the pressure, i.e., the average over ZiZje2/r, is exactly zero. However, when the electron

32


distribution is allowed to redistribute itself, due to the Paulirepulsion,gee is forced to have the value 1/2 at r=0 and gradually increases to unity as r > ao. This produces a small, negative, non-cancellation in the average of ZiZje2/r. In other words the quantum statistical effects that produce the degeneracy corrections are always partially reduced by the Coulomb interactions. The presence of the Coulomb interaction also affects the electronic charge distribution, which produces additional corrections to the first order average over ZiZje2/r. This leads to what are known as second order (or higher) exchange corrections. The Fermi function ratios appearing in equation (34) are shown schematically in Figure (8) as a function of the degeneracy parameter. The ratio of I3/2/I1/2 increases with increasing degeneracy, indicating that degeneracy increases the electron contribution to the pressure, while the ratio L1/2/I1/2 decreases, indicating that highly degenerate electrons do not contribute to the screening.

Fig. 8 Fermi function ratios appearing in equation (34).

2.6 The Activity Expansion Method A many body Quantum statistical procedure for calculating the EOS of reacting multi-component plasmas is described in Rogers (1991). A somewhat different procedure, specialized to hydrogen plasmas is given in Bartsch and Ebeling (1971). These approaches treat the plasma in terms of its electrons and nuclei and are thus "physical picture" methods.


33

Composite particles, i.e., ions, atoms, and molecules, arise naturally in the physical picture, such that, plasma screening effects on the bound states are determined from theory. This is a definite advantage over chemical picture methods in current use; all of which introduce models to obtain these effects. For a discussion of the activity expansion method, as developed for multi-component plasmas, the reader is referred to Rogers (1991). The main steps in the method can be summarized as follows I.

Start from the classical multicomponent canonical ensemble.

n.

Carry out the Abe reorganization in powers of |3u (see Section 4) to obtain a finite expression for S(T, {pi}, XD). The reason for this step is that the analogous summation in the grand-canonical ensemble gives S(T, {z{}, X), i.e., exactly the same analytic function except density is replaced by the activity; where (37)

is the activity, i={e, 0Cj}, otj is an atomic nucleus of charge Zj, and A. is a screening length that approaches XD when the Coulomb coupling is weak. HI.

Construct a generating function method for developing the corresponding grand canonical ensemble from functional derivatives of S(T, [z[), X). The advantage of this procedure is that, as a result of the resummation in step n, S(T, {zj}, X) is divergence free and therefore so is the grand canonical expression.

IV.

Recollect terms to introduce screened cluster coefficients; i.e. construct the grand canonical ensemble analogs to equations (2127). At this point all states are screened by the plasma.

V.

Replace all the classical Boltzmann factors in the classical grand canonical ensemble with Tr exp-PH.

VI.

Introduce an augmented set of activity variables to account for the formation of ions, atoms, and molecules; i.e., construct composite particle activities from products of ze and za. This renormalization removes the screening of low lying, bound states that were initially present after step IV.

34

VII.


Reorganize the resulting activity expressions to take advantage of the charge asymmetry of high Z ions. Since the ion-ion terms are classical, it is possible to include them in all orders. Limitations on the range of validity are set by the electron-ion diagrams, which require quantum mechanics and are only included to 5/2 order in the coupling parameter.

This procedure will only recover the linear correction in Yij in the expansion of f(Yee>Yei)- However, it is possible to introduce additional corrections using known results from other sources (Rogers 1981). Since the activity expansion approach is based on a systematic method, it is possible to determine its range of validity. This is shown in Figure 9 for an argon plasma which shows contours of constant A=< Z^>e^/kTA.D* As already shown in Figure 3 for the classical OCP, the contour A=0.2 is the approximate limit of validity of the Debye-Hiickel theory. The current version of the code includes all orders of ion-ion interactions, but only terms through 5/2 order in the activity for electron-electron and electronion interactions. Consequently, in the shaded region on the right side of the figure the errors in the pressure are estimated to exceed 5%. The limitation at the low temperature end of this region is due to the inclusion of only two particle neutral-neutral and ion-neutral interactions. 1000

10"3 1(T 2 Density (fl/cm3)

1CT1

Fig. 9 Contours of constant A= e2/kTXD for an argon plasma.

35


The short dashed line in the lower right hand part of Figure 9 is the approximate location of some recent shock wave EOS measurements by Erskine, Rosznyai, and Ross (1994). Figure 10 compares this experimental work with the Saha equation and the activity expansion method. The coupling parameter, T, is around unity and the temperature is 2.5 eV at the densest point. The activity expansion method is in good agreement with the experiment. Additional experimental work is in progress.

-

OPAL-H-

0.16 —

^0.12

£

/y> -.

a °"0.08 — ; lonlzatlon onset v 0.04 1

> ^^£*

• i

0

6.004

i

i

0.008 0.012 Density (g/cm3)

i i

i

i i

i

i

i

i~

0.016

Fig. 10 Comparison of the activity expansion method (curve labeled OPAL) with the shock wave experiments of Erskin, Rosznyai, and Ross (1994). To demonstrate the importance of Coulomb interations, the simple Saha equation is also shown.

2.7 Equation of State Comparisons The EOS properties frequently needed in stellar modeling are the first order thermodynamic quantities and the following second order quantities (Cox and Guili 1968) XT =

{dinp)T

(37)

36


(38)

(39) Comparisons of the several EOS methods described above have been carried out by Da'ppen (1992). A few selected examples are repeated here. Figure 11 compares results for XT. Figure lla shows that for stellar envelope conditions, that there is only a few percent difference between the EFF and MHD approaches. Figure lib shows that the differences between the MHD and OPAL equations of state are much smaller than their differences with EFF. Similar results were found for the other second order quantities. The seismic modes are determined only by the deviations of Fi from the ideal gas value 5/3. Consequently, differences similar to those shown in Figure lla for XT substantially affect the agreement with the p-mode data. The MHD and OPAL equations of state are in somewhat better agreement with observational data than is the simple EFF model (Christensen-Dalsgaard and Dappen 1993). At higher densities, where the Coulomb coupling is outside the range of validity of the Debye-Hiickel theory, differences between MHD and OPAL become significant. Figure 12 shows a comparison of Fi along an isochore having density 0.1 gm/cm^- The region around two million degrees corresponds to conditions near the bottom of the solar convection zone. Differences in the two EOS methods are fairly small in this important solar region. However, substantial differences occur at temperatures of a few hundred thousand degrees. These differences could effect the modeling of very low mass stars. In a recent paper Dziembowski, Pamyatnykh, and Sienkiewicz (1992) used helioseismological data to test the MHD equation of state. They found evidence that this approach is inadequate for conditions that exist in the fractional solar radius range r//?=0.85 to 0.95. We have used the activity expansion method (Section 7) to study two points in this region (Rogers and Iglesias 1993). One point is located at r//?=0.88, T=744,000 K and p=0.0348 g/cm3; the other is at r/R=0.90, T=612,000 K and p=0.0259 g/cm3. We were able to draw several conclusions (see Figure 1 of Rogers and Iglesias 1993). 1) The activity expansion result is appreciably different than the MHD result for Fi t on the fine sensitivity

37


I

1

1

1

1

1

2.0 1.8 -

/

— —

.1.6

\

i

1.4 1.2 1.0 "~

V

|

1

— — 1

1

1

1 I

0.1 _

/

n

U

CO

i —

V—— —

-0.1

-0.2 -

\

1

4.2

/

—

i 4.4

i i i 4.6 4.8 5.0 log T (Kelvin)

i 5.2

I

5.4

Fig. 11 Comparison of %p on an isochore with p=10*5-5. Part a compares the EFF and MHD equations of state; solid line EFF and the dashed line MHD. The mixture is 90% H and 10% He by number abundance. OPAL is indistinquishable from MHD on this scale. Part b compares the MHD and OPAL equations of state by looking at the relative differences with EFF: i. e., (X-rOpal . X j E F F ^ E F F ^ 4 line) ^ j (ftMHD _x T EFF)/x T EFF (dashed line). [From Dappen (1992) with permission].

38


I

U"1.60

I

I

I

I

I

6.0

6.2

-

1.55 -

1.50

5.4

5.6 5.8 log T (Kelvin)

Fig. 12 Ti for p=0.1 g/cm3 for the mixture of Fig. 11; solid line CEFF; dashed line MHD: dotted line OPAL equation of state. [From Dappen (1992) with permission ]

scale of the experimental data; 2) the Coulomb corrections are important; 3) the metallicity has a surprisingly large effect on I*i; and 4) the composition of Z is important. For example, we found that if we assume Z is composed entirely of Si, that a close match of the inversion data is obtained with just solar metallicity (0.0193). 2.8 OPAL Equation of State Tables The inability to resolve a number of long standing discrepancies between theory and observation through improved modeling led to the speculation that the widely used Los Alamos opacities were missing important sources of opacity in the lO^-lO** K temperature range (Simon 1982). Due to this speculation and the need for the opacity of low Z materials to model laser produced plasmas, the OPAL opacity effort was undertaken (Iglesias, Rogers, and Wilson 1987; Rogers and Iglesias 1992; Iglesias and Rogers 1993). While it was necessary to calculate the occupation numbers as part of this effort, the much greater accuracy and table density required to calculate derivatives of the equation of state,

39


made it necessary to defer these calculations. The success of the OPAL opacities in helping to improve theoretical models (Guenther 1992) has made it essential to also provide EOS data that is consistent with the opacity tables. We now have available EOS tables for the following conditions: Grevesse 1991 mixture (Rogers and Iglesias 1992) 0.005>T6")*a(*?. "') = 0 while two operators associated to different species always commute. The second-quantized expressions of the operators of interest read

52

Alastuey: Statistical mechanics of quantum plasmas

*«(->*)

(15)

(16) x e a e,t> c (|f(17) (now JVa is the number-operator which counts the particles of species a). The averages like (14 ) appearing in the V-expansions then involve traces over the Fock space of the products of ( ^ ) T . and ($^) r . weighted by the free-particles measure

(18)

Since the measure (18) is Gaussian in the $'s, and since all the \P's commute or anticommute except for c-numbers, the above traces can be calculated with help of Wick's theorem. Similarly to the formula which relates any moment for a Gaussian distribution of real variables to the covariances, the free-particle average of any product of $'s is identically equal to the sum over the products of all the possible corresponding two-operators contractions. The sole non-vanishing contractions are those which conserve the number of particles of each species. They reduce to the finite-temperature one-particle Green's functions of So, i.e.

The second-quantization analysis of the V-expansions provides well-defined rules for interpreting each term as a Feynman graph similar to those which appear in field theory. Roughly speaking, the graphs are made of oneparticle fermionic or bosonic oriented loops connected by two-body interaction lines. The two points linked by a given interaction line are affected of the same "time" r,-. They may belong either to the same loop or to different


53

loops. Each loop contains an arbitrary number nj_, of points f,- at time r w n i c ^ o n ly depends on the pairs of permutations The Slater-sum representation (35) of HA alows a natural identification of the exchange effects. Indeed, the "square" terms (Va = V'a for any a), where the diagonal matrix elements of exp(—(3H]v) in configuration space appear, obviously correspond to MB statistics. A "rectangle" term (Va / "P'a f° r a * least one species) involves the exchange of n particles (n > 2). The corresponding matrix elements of exp(—(3HN) are off-diagonal with respect to the positions of the exchanged particles. The structure of the FK representation of these off-diagonal matrix elements can be interpreted in two ways which lead to different treatments of the exchange contributions. A first possible interpretation consists in introducing opened filaments J-î associated to the exchange of a particle a from position rjt to position fi. The shape of J7^ is parametrized by

w&(*) = (1 - s)fk + sft + Xa({s)

(36)

which describes a path of the exchanged particle in the genuine Feynman path integral (uki (0) = Ffc and Uki (1) = ft)- The above closed filaments £ are again associated to the non-exchanged particles. For instance, if one considers the off-diagonal matrix element fN\ exp(-PHN)\rir2r3

...fN>

(37)

which corresponds to the exchange of two particles, there appear two opened filaments T"2 and T2\ a n ^ (N-2) closed filaments S3,..., £N- This situation is illustrated in Figure 6. By collecting together all the contributions with the same finite number n of exchanged particles, we are then left with a problem of impurities, the n opened filaments, immersed in the bath S* of closed

62


Fig. 1.6 Two opened filaments F°2 and T%x surrounded by closed filaments of S*.

filaments. This inhomogeneous situation can be dealt with along standard perturbative techniques where the reference system is the homogeneous bath S* described in section 3.2. The second interpretation of the exchange contributions is due to Brydges (private communication). Any permutation of n objects, which characterizes the exchange of n particles, is the product of p cycles with p < n. Therefore, the corresponding n opened filaments may be always viewed as a set of p closed filaments. Each of these new closed filaments is made of q opened filaments, q > 1, and will be noted £(*). It can be associated to a closed path described in a "time" qflh. For instance, in the above example the union of T^ and T%± gives raise to £^2K Therefore, the whole Slater representation (35) of 3\ is identified as the grand-partition function of a mixture of classical closed filaments £^ with q = 1,2,3...oo (the S^'s are the closed filaments £ introduced in section 3.2). The typical size of £^ depends on the "time" q(3h. Its activity incorporates a self-energy term arising from two-body interactions of the type (30) between the opened filaments which constitute £^q\ In the present approach, the MB and exchange effects are treated on an equal footing, while the previous interpretation leads to a perturbative treatment of the exchange contributions.

3.3.4 Possible

applications

Within the above equivalences which follow from the FK representation, the equilibrium properties of the quantum system S can be studied by applying the usual methods of classical statistical mechanics to S*. Indeed, the system of closed filaments is isomorphic to an ordinary classical system


63

of point objects with two-body interactions. In «S*, the position Fis replaced by the generalized coordinate £. The familiar calculation rules remain unchanged, apart from this simple substitution, because all the quantities of S* behave as commuting c-numbers (the operatorial structure of quantum mechanics "disappears" in the FK representation). The familiar Mayer series can be extended to S*. For systems with shortrange forces, Ginibre (1971) proved the convergence of the activity expansions by exploiting the classical structure of the Mayer-like graphs. For the present Coulomb systems, the Mayer-like series for S* constitute a powerful tool in the systematic derivation of density-expansions (see Section 4). Aside from these exact calculations, one might introduce approximate methods by extending well-known integral equations (likeHNC) to the correlations of S* (see e.g. Chandler (1981) for a review relative to systems with short-range forces). Although such extensions do not cause any trouble at a formal level, we stress that the explicit calculations might be rather difficult because of the functional integrations over the shapes of the filaments.

3.4 Virial-Like Expansions Like in the classical case, all the above Mayer-like graphs diverge because of the long-range Coulombic nature of the filament-filament potential. Alastuey, Cornu and Perez (1993) have shown that the corresponding series can be reorganized in series of finite resummed graphs. In this section, we just sketch the main steps of their method. First of all, they consider only MB statistics and the resummation procedure is applied to the Ursell function h(Sa, £(>) of S*. The density expansions of the MB thermodynamic functions of SMB are then evaluated via standard identities. The exchange effects are included perturbatively within the impurities approach exposed in Section 3.3.

3.4-1 Diagrammatic resummations The two-point Ursell function h(£a, £b) of S* is defined as usual by

p(£a)p(£b)h(£a,£b) = z(£a)z(£b)]im

f^fA

(38)

It can be represented by series of Mayer graphs F in terms of the filament density

64


Fig. 1.7 A typical graph T which contributes to the Mayer-like densityexpansion of h(£a,£b). The closed filaments are drawn as in Figure 5, with the sole difference that the positions of the root filaments £a and £\, are representated by white circles. The tubes connecting the filaments are the Mayer bonds /(£ l ea d to three resummed bonds FD, (Af. V-^b) a n d Fdip. All the other structures, involving several chains and/or the bonds fa and /c/2, give raise to the fourth bond FR. All these cases are illustrated in Figures 8 and 9. The resummation procedure automatically excludes the convolutions FD *

66


(a)

-o

Fig. 1.8 a, 1.8b, 1.8c : Three f-diagrams which belong to the same resummation class. For clarity, the shapes of the filaments are not represented. The big black circles are filaments V which remain fixed through the resummation process. The small black circles are filaments C which are "eaten" by the resummation "machinery". Solid lines : bonds fe ; lines with one arrow : bonds A£.v/ C (the arrow indicates the point with respect to which acts the gradient) ; double solid lines : bonds / c 2 /2 ; dashed lines : bonds

Fig. 1.9 The H-graph generated by the f-class illustrated in Figures 8a, 8b and 8c. Strings : bonds FQ ; strings with one arrow : bonds X^.^FD ; strings with two opposite arrows : bonds Fdip ; hatched bubbles : bonds FR.

*i ft- Vi^b * FD, FD* \j(j. S7jFD and A,-£. between j Ijtwo filaments (Vi,Vj) in any graph II. The bonds F can be calculated explicitely in terms of the MB particle densities. Indeed, since the /c-bonds are shape-independent, the functional


67

integrations over the shapes of the intermediate filaments C in the chains lead to the replacement of each p(£) = pa( £) by the MB particle density

(43)

Thus the summation of all the convolution chains can be performed in terms of the familiar Debye potential 0o( r ) = exp (—KT)/T with K = (47r/?2_,e^p£ffl) . In particular, one finds a

FD(Vi,Vj) = -PeaieajD(\?i ~ ?j\)

(44)

We stress that , contrarily to FD and A £. V^D which decay exponentially fast (like 4>D essentially), the bonds Fap and FR are found to decay algebraically as 1/r3 when r —• oo. These behaviours are related to the efficiency of the screening of the multipole-like interactions, which appear in the expansion of the bare filament-filament potential (33) in powers of f and £'. The charge-charge and dipole-charge interactions are perfectly screened via the usual classical process while the screening of the higher order multipole interactions is inhibited by quantum fluctuations. The above slow decays should ultimately pollute the correlations with algebraic tails, in accord with the absence of exponential clustering predicted by Brydges-Seiler (1986), Alastuey-Martin (1988-1989) and Cornu-Martin (1991). Although the screening mechanisms are less efficient than in the classical case, they do eliminate the long-range Coulomb divergencies, i.e., each graph II does converge, as expected. The resummed diagrammatic expansion of the two-point correlations of SMB is immediately obtained by inserting the previous Il-representation of h(£a,£b) i n the identity

P%B(0, fifyaepvc{r) (46) A a,0

which expresses the free-energy per unit of volume in terms of the two-point correlations. In (46), Pxg 1S calculated for a fictitious system Sg where all the interactions are multiplied by the dimensionless coupling parameter g. Moreover the temperature and the MB particle densities of Sg are identical to those of S. The insertion of the II-representation of p"* in (46) provides a diagrammatic expansion of the free-energy. At this level, the Il-series for (0FMB/A) do not constitute an explicit expansion with respect to the particle densities, because of the presence of the shape-dependent statistical weights pg(£). In fact, the functional pg{€) can be itself expanded in powers of the p%B 's as follows. One starts from the familiar Mayer expansion of pg{€) in terms of the fugacities z{£) = z*. The corresponding set of divergent Mayer graphs G is then transformed into a set of convergent graphs P via a resummation process similar to the one described in Section 4.1. Now there appear five resummed bonds which can be expressed in terms of functions entirely scaled by KDfZ = (47r/?£ a e£z a ) and of / r scaled by the Landau and de Broglie lengths (these lengths depend only on the temperature). A scaling analysis with respect to KQ,Z of the spatial integrals in the graphs P allows to express pg(£) as a double integer series in z*1/2 and Inz*. The half-integer powers come from KD,« ~ z*1?2 while the logarithms arise from the l/r 3 -tail in / j . Eventually, the fugacities are eliminated in favor of the particle densities by and using (43), and pg(£) is rewritten as a double integer series in (pMB) MB \np , the first terms of which read

/ dsvc(\f + A7fi(s) - A«f()|)) Jo

'

(47)

- JX>(|))exp(-/3eae7 J dsvc(\r + X^(i(s) - Xa(t(s)\))\

After the replacement of the statistical weights pg(£) by their particle-


69

densities expansions, it remains to determine the density dependence of the spatial integrals in the II-graphs. This analysis is carried out via a scaling method similar to the one used for the P-graphs with KD in place of KD,ZIndeed, all the resummed bonds F can be expressed in terms of KD-scaled functions and fa. The final form of the density expansion of (/?F Mfl /A) is a double integer series in (pMB)1/2 and lnpMB. The corresponding expansion of the pressure @PMB derived from the identity

has the same structure. We stress that the macroscopic instability of SMB does not cause any divergency in the virial coefficients of the above series. However, it should prevent their convergence.

3.4.3 Exchange contributions The exchange effects are taken into account by inserting the Slater expansion (35) of EA in the grand-canonical expression (3) of /?P. This gives

= (3PMB + }2En

(49)

n=2

where En is the contribution of n exchanged particles. According to the impurities point of view, En may be expressed in terms of the density p{£a\Fki) of S* in presence of n opened filaments J-ki- For instance, Ei reads

J x exp [ - / ? £ dg J dSaP(Sa\g; J? 2 , Jft)(t»(f., Jfi) + v(€a, In (50), the first exponential represents the exchange contribution in the vacuum while the second one describes the many-body effects on the twoparticle exchange. The structure of all the other En's is similar to the one displayed in (50). The diagrammatical method exposed in Section 4.1 can be extended to

70


the inhomogeneous system S* in presence of the .F's. This provides a representation of p (€a\^Fki) in terms of graphs made of the n opened filaments Tki and closed filaments. The statistical weight of a closed filament is the density p(£) of the homogeneous system. Two closed filaments are linked by at most one resummed bond F. A closed filament £ and an opened filament T are linked by at most one "external" Mayer bond (exp [—f3v(£,Jr)] — 1). The density expansion of En follows from use of the previous diagrammatic representation of p{£a\Fki)- The integrals over the positions of the opened and closed filaments can be evaluated via a scaling analysis with respect to nD. The resulting expression for En takes the general form zn multiplied by a double integer series in (pMB) and lnpMB. 3-4-4 General structure of the density

expansions

The present method gives access first to the pressure. Use of the expansions of f)PMB and En in (52), allows to rewrite f3P as a triple integer and \npMB. The fugacities and MB densities are series in z, (pMB) then eliminated in favor of the real densities pa by combining the identities p%B = zad(/3PMB)/dza and pa = zad((}P) / dza (p%B differs from pa at given fugacities). The resulting virial expansion of the pressure is a double integer series in p1/2 and Inp. The density expansions of the other thermodynamic functions, obtained via the usual identities, have the same structure. According to the above scaling analysis, the contributions of a fully resummed graph of the II-type to a given virial coefficient, reduce to subgraphs built with bonds / j , FD, fc Af. V^b> A£. v/c- The physical nature of these contributions can then be identified by inspection of the bonds and filaments which appear in the subgraphs according to the correspondance rules, - long-range classical interactions : bonds fc and - quantum diffraction : bonds A^. v / c and Af. - bound and scattering states : bonds fa - exchange : opened filaments In general, all these physical effects are coupled at high orders in p. We stress that, in the present formalism, all the corresponding contributions are treated simultaneously in a systematic and coherent way. Of course, the exact density expansions can be also derived in the framework of the formalisms described in Section 2. However, the FK diagrammatic method is well-suited for this purpose. In particular, all the long-


71

range Coulomb divergencies are automatically eliminated at any order in the density via the introduction of the generic resummed bonds F. In the effective-potential method, it remains to treat the long-range part of the many-body effective interactions in a systematic way. Furthermore, the density is the natural expansion-parameter in the FK diagrammatics. Indeed only a finite number of graphs II contributes to a given virial coefficient (roughly speaking the order in the density of II increases with the number of filaments and the number of bonds). On the contrary, the calculation of such a coefficient from many-body perturbation requires the collection of infinite sets of Feynman graphs with Ladder structures. Moreover, in this approach, the elimination of the fugacities in favor of the densities is more cumbersome than in the FK method where part of this transformation is automatically done. Although the FK representation is quite efficient for performing infinite resummations at a formal level, one has to keep in mind that, for practical calculations, the difficulty relies in the functional integration over the shapes of the filaments. The purely diffraction contributions, which only involve moments of the Gaussian measure T>(£), are evaluated from the covariance (28) by using Wick's theorem. The contributions of bound and scattering states, which involve one or more bonds /x, cannot be so easily computed in the space of filaments. In fact, it is more convenient to express them in terms of matrix elements of exp (-(3HN) by applying backwards the FK formula (26). The corresponding explicit calculations are then limited by the absence of exact solutions for the N-body quantum-mechanical problem, as soon as N is larger than two. Beside the complexity of the graphs, this problem intrinsic to quantum mechanics and which appear in any formalism, prevent a detailed knowledge of the virial coefficients at orders higher than

3.5 Low-Density Equations of State 3.5.1 the exact EOS at the order phl2 The truncation of the above virial expansion of the pressure at the order

72


p5/2 gives (Alastuey and Perez (1992)) K3

a

a,fi

3

X)

l l l n («*«/

ID

C2 1/2

with Kx, = (47ry9 ^ e2apQ)

, / a/3 = ^CaC/3, map = mampl{ma+mp),\ap

=

a

(Ph2/mQl3)1/2, C\ = 15.205 ± .001, C2 = -14.733 ± .001 and EulerMascheroni's constant C = .577216... Moreover, Q{-y/2lQp/\ap) is the so-called quantum second-virial coefficient first introduced by Ebeling

(52)

while E(—y/2laa/Xaa)

is the exchange integral,


l dr< -f\e-

73

ph

""\r>

(53)

In (52) and (53), hap is the one-body Hamiltonian of the relative particle with mass map submitted to the Coulomb potential e a e^/r. The functions Q and E only depend on the temperature via the single dimensionless parameter {—y/2l/X). All the physical effects mentioned is Subsection 4.4 give contributions to (51), the structures of which require the following comments. First, the classical contributions of the long-range part of the interactions are polynomials in the inverse temperature, the charges and the densities, which do not involve Planck's constant. The involved coefficients are evaluated analytically or, like C\ and C2, by numerical computations of dimensionless integrals. The lowest-order contribution is nothing but the familiar Debye-Hiickel term in p3?2, and constitutes the leading correction to the MB ideal pressure in (51). By specifying (51) to the case of the classical OCP, it can be checked that the classical terms found here, up to the order p 5 / 2 , do coincide with those calculated by Cohen and Murphy (1971), as it should be. The contribution of quantum diffraction at large distances appears only at the order p 5 / 2 and reduces to

The occurence of this term shows that the long-range part of the interactions cannot be entirely treated at a classical level. This diffraction term is missing in the expressions obtained by the effective-potential method. In this formalism, it arises from the three-body interactions w-3' which are longranged. Indeed, for large-triangular configurations, u\23 typically behaves as (f33 h e\ ei e$ / (12mi))ViVc( r i2)«Vi u e( r i3) ( a P ar * from a sum over the permutations of 1, 2, 3). These slow l/r 2 -decays lead to screened contributions which are of order phl2 instead of p3 ( a similar mechanism makes the Debye correction be of order p3!2 instead of p2). The diffraction correction (54) is merely proportional to h2, because it arises from large distances where the quantum effects can be treated perturbatively "a la WignerKirkwood". By the way, the presence of this term is crucial for recovering from (51) the well-known ^-correction (Hansen and Pollock (1973))

74


(55) 6m to the classical pressure of the OCP calculated by the WK method. In fact, the sole contributions to (55) do arise only from the />2-term in (51), while (54) exactly cancels the 7i2-term which comes from p2KDQ. The term papp â/jQ(—\/2/ajg/Aajg) is the total contribution from both bound and scattering states of two charges ea and ep. The truncation of < f\ exp (—(3hap) | r > in the integral (52) defining Q ensures that this contribution is finite. This regularization in not an arbitrary mathematical artefact. It is directly related to the truncated stucture of the bond /y, and reflects the screening of the Coulomb interaction at large distances. For opposite charges such that eQep < 0, one may extract from Q a contribution of the bound states which reduces to the familiar Planck-Brillouin-Larkin (PBL) sum

f ) n2 [exp(-/fc?») - 1 + pef]

(56)

n = ll

where €%& = — e2 e% map/(2 h2n2) are the energy levels of the hydrogenoid atom with Hamiltonian hap. However, other definitions of the bound states contributions can be introduced from (52) by using the basic properties of the trace. For instance, as shown by Bolle (1987), there exists an infinite set of arbitrary decompositions in terms of bound and scattering contributions of the PBL sum itself. So, as far as thermodynamic quantities are concerned, only the total contribution of both bound and scattering states is an unambiguous quantity. The />5'2-contribution from bound and scattering states merely reduces to its p2 counterpart multiplied by /3eaepKD. This multiplicative factor arises from many-body effects which induce a constant shift —eaepKD on the energy levels of the two-particles states. The contribution pl,E(—y/2laa/Xaa) arises from the exchange of two charges ea in the vacuum. It is finite, independently of any screening effect, because the off-diagonal matrix elements < —f\ exp (—fihaa) \ r > are short-ranged. The magnitude of this contribution is smaller than the one relative to free particles because the repulsive potential e^/r inhibits the exchange. Similarly to what happens for the contributions of bound and scattering states, at the order p 5 ' 2 , the many-body effects on the twoparticles exchange amount to lower the repulsive barrier e2 / r by the constant —e\n D .


75

Eventually, the high-temperature series can be recovered from (51) by expanding the virial-coefficients in powers of /?. Since //A is proportional to /3 1 / 2 , the /3-expansions of Q (-y/2l/X) and E (-V21/X) coincide with their Taylor series in powers of //A. In agreement with DeWitt's observations, such series do involve singular powers of h because //A is also proportional to 1/h. In fact, the expression (23) up to the order /3 5 / 2 is exactly recovered by inserting the /^-expansions of Q and E in (51). The calculation of the higher order terms from the virial expansion requires a suitable reorganization of the density series, similar to the one relative to the e2-series evocated in Section 2.2. 3.5.2 Application to the Sun For practical applications, the range of validity of the truncated EOS (51) is determined by the conditions I < a and X < a (with a ~ p" 1 / 3 ), which mean respectively weak coupling and weak degeneracy. It turns out that these conditions are met in the inner regions of the Sun which can be viewed, in a first approximation, as a three-component plasma made of electrons, protons and Helium nuclei. Perez (private communication) has evaluated the various physical corrections appearing in (51) within a typical temperature- and density-profile. Moreover, he used representations of Q (Kraeft et al (1986)) and E (Jancovici (1978)) which allow simple and accurate numerical computations. His results are sketched in Table 1. As expected, the inner regions almost behave as an ideal MB gas, since the relative magnitudes of the above corrections do not exceed a few percent. In the core, the correction to the ideal MB pressure is positive and mainly due to the electronic exchange. In the outer layers, this correction becomes negative because the classical Debye term then dominates. In the intermediate regions, the partial cancellation between the exchange and Debye terms increases the relative importance of the other contributions in (51), in particular those taking into account the recombinations between one nuclei and one electron in hydrogen atoms and He+ ions. The previous results illustrate the usefulness of the truncated EOS (51) for helioseismology. The actual observations lead to very accurate determinations of the speed of the sound and of adiabatic coefficients. For a proper interpretation of these data, one needs a reliable theoretical description of the small deviations from the ideal behaviour. The EOS (51) fullfills this requirement since it comes out from an exact expansion. Furthermore, its analytical character allows simple and consistent calculations of any thermodynamic coefficient via partial differentiations with respect to the density

76


Table 1.1. Relative deviations from the MB ideal pressure for the inner regions of the Sun, calculated from the truncated EOS (51). R represents the distance to the center. The magnitudes of all the physical corrections arising in (51) are also indicated R

log/)

logT

PP/P-I

Class.

Bound/

Exch.

Diff.

[106] km

g/cc

(K)

[io-2]

[io-2]

Scatt.St. 2 2 [io-2] [io- ] [io- ]

0.62 0.15 0.00

-2.2 1.5 2.2

5.82 6.95 7.19

-0.698 0.110 1.446

-0.710 -1.000 -0.971

0.006 0.089 0.101

Int.

0.005 1.006 2.294

0.000 0.014 0.022

or the temperature. This EOS should be quite efficient for describing the regions close to the core, where all the chemical species are almost fully ionized (the inclusion of the contributions of heavier nuclei, like F%6+, is straightforward). Near the surface, the validity of the p5/2-truncated EOS is limited by the presence of atoms (like He) and ions which result from recombinations between one nuclei and two or more electrons. In the virial expansions, the contributions of these entities enter in terms of order f? (at least) which are not included in (51). Acknowledgment I am indebted to Marie-Pierre Fuchs for typing this manuscript, and to Asher Perez for communicating unpublished data.

References Abe R., Progr. Theor. Phys. 22, 213, (1959) Alastuey A., Cornu F. and Perez A., to be published in Phys. Rev. E, (1994) Alastuey A. and Martin Ph. A., Phys. Rev. A 40, 6485, (1989) Alastuey A. and Perez A., Europhys. Lett. 20, 19, (1992) Bolle D., Phys. Rev. A 36, 3259, (1987) Brydges D. and Seiler E., /. Stat. Phys. 42, 405, (1986) Chandler D., "Studies in Statistical Mechanics", edited by E. W. Montroll and J. L. Lebowitz (North Holland, Amsterdam, 1981) Cohen E. G. D. and Murphy T. J., Phys. Fluids 12, 1404, (1969) Cornu F. and Martin Ph. A., Phys. Rev. A 44, 4893, (1991) De Witt H. E., J. Math. Phys. 3, 1216, (1962) De Witt H. E., J. Math. Phys. 7, 616, (1966) Dyson F. J. and Lenard A., J. Math. Phys. 8, 423, (1967) Ebeling W., Ann. Phys. (Leipzig) 19, 104, (1967) Fetter A. and Walecka J. D., "Quantum Theory of Many-Particle Systems" (Me Graw-Hill, New York, 1971)

Alastuey: Statistical mechanics of quantum plasmas Feynman R. P. and Hibbs A. R., "Quantum Mechanics and Path Integrals" (Me Graw-Hill, New York, 1965) Gell-Mann M. and Brueckner K. A., Phys. Rev. 106, 364, (1957) Ginibre J., "Statistical Mechanics and Quantum Field Theory " (Les Houches Lectures, ed. by C. De Witt and R. Stora (Gordon and Breach, New York, 1971) Jancovici B., Physica 91 A, 152, (1978) Kraeft W. D., Kremp D., Ebeling W. and Ropke G., "Quantum Statistics of Charged Particle Systems" (Plenum Press, New York, 1986) Lenard A. and Dyson F. J., J. Math. Phys. 9, 698, (1968) Lieb E. H., Rev. Mod. Phys. 48, 553, (1976) Lieb E. H. and Lebowitz J. L.,Adv. Math. 9, 316, (1972) Mayer J. E., J. Chem. Phys. 18, 1426, (1950) Meeron E., J. Chem Phys. 28, 630, (1958) Montroll E. W. and Ward J. C., Phys. Fluids 1, 55, (1958) Morita T., Prog. Theor. Phys. (Japan) 22, 757, (1959) Pollock E. L. and Hansen J. P., Phys. Rev. A 8, 3110, (1973) Rogers F. J., Phys. Rev. A10, 2441, (1974) Salpeter E. E., Ann. Phys.(New York) 5, 183, (1958) Simon B., "Functional Integration and Quantum Physics" (Academic, New York, 1979)

11

Onsager-molecule approach to screening potentials in strongly coupled plasmas YAAKOV ROSENFELD Nuclear Research Center- Negev, P.O.Box 9001, Beer-Sheva, Israel

Abstract The solution of the exact integral equation for the liquid pair-structure in the asymptotic strong coupling limit for the plasma, as mapped on the Onsager charge-smearing optimization for the energy lower bound, features "Onsager atoms" and "Onsager molecules". The universal properties of this asymptotic limit make it a natural reference starting point for an asymptotic strong coupling expansion for the fluid structure and thermodynamics, playing the role of an "ideal liquid" state. In particular, the leading strong coupling terms for the potential energy, direct correlation functions, and screening potentials for the Coulomb and Yukawa mixtures (corresponding to classical plasmas and electron screened classical plasmas), with full thermodynamic consistency, are presented. These are in complete agreement with the Alastuey-Jancovici analysis of early simulations data by Hansen in strong coupling, and with recent highly accurate simulations data of Ogata, Iyetomi, and Ichimaru. Data analysis errors lead Ogata, Iyetomi, and Ichimaru to incorrect results for the short range screening potentials in strong coupling. Their calculations for the short range screening potentials, bridge functions, and enhancement factors for nuclear reaction rates in strongly coupled plasmas, should be revised. La solution de l'equation integrate exacte pour la structure du liquide dans la limite asymptotique de couplage fort pour le plasma, calquee sur 1'optimisation de charge d'Onsager pour la limite inferieure de l'energie, met en evidence des "atomes d'Onsager" et des "molecules d'Onsager". Les proprietes universelles de cette limite asymptotique en font un point de reference naturel pour un developpement asymptotique en couplage fort pour la structure et la thermodynamique du fluide. En particulier, les ter78

Rosenfeld: Onsager-molecule approach

79

mes dominants en couplage fort pour l'energie potentielle, les fonctions de correlation directes, et les potentiels d'ecran des melanges de Coulomb et de Yukawa (correspondant aux plasmas dassiques et aux plasmas dassiques ecrantes), sont pr'esentes dans cette revue. Ces resultats sont en parfait accord avec l'analyse de Alastuey et Jancovici des premieres simulations de Hansen en couplage fort, et avec les recentes simulations splus precises de Ogata, Iyetomi et Ichimaru. Des erreurs d'analyse de donnees ont conduit Ogata, Iyetomi et Ichimaru a publier des resultats incorrects pour la partie a courte portee des potentials d'ecran en couplage fort. Leurs calculs pour les potentiels d'ecran, les fonctions bridge, et les facteurs d'accroissement des taux de reactions nudeaires dans les plasmas fortement couples doivent etre revus. 4.1 Introduction : screening potentials as related to the inverse scattering problem for the fluid pair structure Classical plasmas composed of classical positive ions in a uniform neutralizing background charge density of degenerate electrons, are basic models for dense stellar materials and provide important reference systems in condensed matter physics (Rogers & DeWitt (eds.), 1987; Ichimaru (ed.), 1990). These systems are referred to as "one component plasma" (OCP), "binary ionic mixture" (BIM), or "multi ionic mixture", depending on the number of different ionic species. Considerable effort has been invested over the years (Hansen & Baus, 1980; Ichimaru 1982; Ichimaru et al., 1987; Isern, these proceedings) in increasing the accuracy of the calculations of the short range screening potentials in strongly coupled plasmas because of the following two main reasons: (1) The enhancement factors for the thermonuclear reaction rates which are important for stellar evolution, particularly for Carbon ignition in degenerate cores, are essentially controlled by the short range part of the screening potential, (the zero-separation value , H(0) , in particular). The ions inside dense stars (such as white dwarfs and neutron stars) are strongly coupled ,r ~(potential energy)/(kinetic energy) ~100, the enhancement factor is roughly proportional toexp[H(0)], and H(0) ~ T, so that an error of 2% in the value of H(0) may yield a reaction rate which is off by an order of magnitude. (2) The screening potentials play a key role in the study of the short range behavior of the bridge functions, notably their universal properties, which proved seminal for developing an accurate theory of liquid structure (Rosenfeld k Ashcroft, 1979; Rosenfeld 1980a,b). The zero separation theorem (Hoover & Poirer, 1962; Widom, 1963; Rowlinson & Widom, 1982) provides an important test for the accuracy of the

80


equation of state of mixtures, and offers a consistency check (Rosenfeld, 1990a) for predictions based on closure approximations in integral equation theories for the fluid pair structure. The screening potentials H{J{T) of the classical plasma are defined in terms of the bare Coulomb interaction between the two ions of charges Z, and Zj , and the pair correlation function gij(r):

Hair) = ^

+ hfe(r)]

(1)

It is convenient to measure all distances in units of the Wigner-Seitz radius, a = ( j ^ ) 1 ^ 3 , where n is the total number density of the ions, and to define the Coulomb coupling parameter T = a£ y. The pair correlation function can be expressed through the free energy change upon fixing the positions of the pair of fluid particles in the appropriate configuration to form an interaction-site molecule (Hoover & Poirer, 1962; Widom, 1963; Jancovici, 1977; Rosenfeld, 1987a,b) H.(A_

WOO ~ FS

Here FQX is the configuration^ (excess over ideal gas) free energy of the Nparticle system (in a uniform neutralizing background) and F{X(T) is that of the same system but with the pair of particles kept at fixed separation , r , forming a two-site charge cluster. Ffx(r) does contain the intramolecular interaction ' r ; , so that Hij(r) is finite as r —• 0. The simulation data is limited to r < Rmin where Rmin ~ 1 for F ~ 160 and Rmin ~ 0.5 for T ~ 10. It is impossible to get H(r ~ 0) by simulations directly from eq.(l) because of the essentially zero probability for very close encounters. It is possible to try and extrapolate these results down to r = 0, but early attempts were not very successful (DeWitt, Graboske, and Cooper, 1973; Itoh, Totsiji, Ichimaru, and DeWitt,1978,1979) as pointed out by Rosenfeld (1980a, 1987a,b). On the basis of eq.(2), however, the calculation of H(0) can be carried out via the equation of state of the plasma mixture, if it is available. Salpeter (1954) and Salpeter & Van Horn (1969) pioneered the calculation of eq.(2) near r = 0 by means of the ion-sphere model. Using eq.(2) and assuming the validity of the linear mixing rule (Hansen et al., 1977; Brami et al., 1979) for the free energy of the BIM and by employing an accurate OCP equation of state (DeWitt, 1976), Jancovici (1977) obtained a good estimate of H(0) for the OCP. He also derived the exact leading r 2 term in the expansion around r = 0, which was used by


81

Alastuey and Jancovici (1978) to extrapolate the simulation data for H(r) towards the origin r = 0. The first direct (structural, via eq.l) theoretical calculations of H{j{r) for the OCP and the BIM were performed by the assumption of universality of the bridge functions in the modified-hypernetted-chain (MHNC) theory (Rosenfeld, 1980a,b). These results are in agreement with Jancovici's calculations as well as with the ion-sphere scaling for mixtures. The classical direct ((r), can be reduced by exact diagrammatic analysis to the solution of the hypernetted-chain (HNC) integral equation for an effective potential (denote: 0 = ^

*(r) =ttr)+ ^p-

(3)

The "exact" HNC equation is composed of the Ornstein-Zernike relation between the direct correlation function, C(T), and the radial distribution function g(r) = h(r) — 1, with the A;-space form (also defining the structure factor S(k)) h(k) = c(k) + nh(k)c(k) = S(k)c(k)

(4)

and the HNC-dosure for $(r):

H(r) - /ty(r) = lnfaf(r)] = -0#(r) + h(r) - c(r)

(5)

n — N/V is the number density . The heart of the problem is B(r) - the bridge function - which may be expanded in diagrams with the "dressed" /i(r)-bond. B(r) = 0, i.e. $(r) = (r), defines the HNC-approximation. The diagrammatic low density expansion represents a very slowly convergent route for obtaining meaningful results for a highly correlated system like a dense fluid, while, on the other hand , even the HNC-approximation provides an excellent point of departure for describing liquid pair structure. Along this alternative route, a first order improvement on the HNCapproximation , the ansatz of the universality (Rosenfeld & Ashcroft, 1979) of the repulsive short range structure of B(r) was found empirically to be very accurate. It provides the key (Rosenfeld, 1986a; Aers & Dharmawardana, 1984) to consistent solutions of the direct and inverse "scattering problems". Heuristic arguments were given in favor of "universality", but

82


the computational intractability of the bridge diagrams prohibited its direct assessment. More recently, the "Onsager molecule" method (Rosenfeld, 1987a,b) was advanced for the evaluation of eq.(2). It is based on a general result in liquid state theory (Rosenfeld 1985,1986b), which holds under broad assumptions, that the leading term in the asymptotic strong coupling expansion of the configurational free energy is also its Onsager-type exact lower bound, e.g. the ion-sphere result for the classical plasmas. Using the Onsager-Molecule approach we can calculate the leading strong coupling expansion term of 77(r) (Rosenfeld, 1988, 1991a; Rosenfeld, Levesque, and Weis, 1989) and from it derive the asymptotic B(T), closing a fully self consistent cycle. The results using this method (Rosenfeld, 1992a) are in agreement with Jancovici's expression, with the MHNC calculations, and with the simulation results, as will be described below. Very recently, using an extrapolation of very accurate "extra long" computer simulations for strongly coupled classical plasmas, new results for the short range screening potential H(r) were obtained by Ogata, Iyetomi and Ichimaru (1991). These results are reported to be of 0.1% accuracy and to differ significantly from previous calculations (e.g. Jancovici's expression (1977) for H(0)). Careful analysis of the same data finds (Rosenfeld, 1992a), however, that these new simulations data are in excellent agreement with the previous calculations. Moreover, these high accuracy simulations provide an additional strong support to the asymptotic expansion ("Onsager molecule") analysis (Rosenfeld, 1987a,b) of short range screening potentials in multi-ionic plasmas. It is mainly their incorrect data analysis, which led Ogata, Iyetomi and Ichimaru (1991) to wrong conclusions. The analysis by Ogata, Iyetomi and Ichimaru (1991) is also used as the lounching platform for similar calculations of electronic-response effects by Ichimaru and Ogata (1991) and for general multi ionic mixtures by Ogata, Ichimaru and Van Horn (1991). The errors in these later calculations were also pointed out (Rosenfeld, 1992a). A very recent review by Ichimaru (1993), summarizes the many papers on screening potentials and enhancement factors for nuclear reaction rates by Ichimaru and coworkers, which perpetuate these errors. Although mentioning this criticism, the review by Ichimaru does not present any serious attempt to address it. In turn, the difference of results from the Ichimaru group vs the older Jancovici-Alastuey work could make a difference of a massive accreting white dwarf exploding into a Type I supernova or collapsing into a neutron star (Isern, these proceedings). The present review was invited in order to clarify the situation. This review describes the properties of strongly coupled plasmas as obtained analytically from the Onsager-molecule theory in complete agreement


83

with the numerical simulations data. The Onsager molecule theory provides the asymptotic strong coupling properties of classical plasmas in a simple physically intuitive way, and is an essential tool for analysing numerical simulation results for strongly coupled plasmas. Ogata, Iyetomi and Ichimaru completely ignore the asymptotic analysis, which could have provided them an early worning about their wrong assumption. Along with revealing the errors in the analysis by Ogata, Iyetomi and Ichimaru, this review proposes a revised calculation of the short range screening potentials from their accurate simulations data. 4.2 Asymptotic strong coupling limit for plasmas (Onsager atoms and molecules) as a paradigm for the "ideal liquid" state Unlike the solid, on one hand, and the ideal gas, on the other hand, the dense fluid near freezing does not contain a "natural" small parameter. The solid features the crystal structure as the "ideal" reference state; the Lindemann melting rule identifies the ratio of particle's vibration to the nearest neighbor distance as a natural small parameter; the "elementary excitations" are phonons, providing the basis for the harmonic expansion. The low density fluid features the uniform distribution "ideal gas"as the reference state, with a natural small parameter given by the particle's size to the WignerSeitz radius. The "elementary excitations" are pair-collisions, and the virial power series in density is the natural expansion (for plasmas consider the Debye theory and the Abe-Meeron cluster expansion). There is, however, no natural "ideal liquid" available, but it can be theoretically constructed by using the asymptotic high density limit (bypassing the solid) of the exact diagramatic theory of liquids, to have all the needed properties in order to serve its purpose (Rosenfeld, 1985,1986b,1991a). This ideal liquid features universal structure-factor peaks, e.g. at tg(kia) = A;,o for D — 3. In parallel it features "Onsager atoms" with self energy that provides an exact energy lower bound. The small parameter is a generalized free-volume, the "elementary excitations" are "Onsager molecules", and the "basis functions" are the fundamental geometric measures of the sphere. The "ideal liquid" is defined through mapping of the asymptotic high density or strong coupling limit of the exact liquid state theory for pair-correlations onto the Onsager exact lower bound for the potential energy (the corresponding variational problems coincide). It is extended by recursive definitions for higher order correlations and provides basis functions for fluid structure and free energy functional (Rosenfeld, 1993a).

84


Exact liquid state theory for pair correlations can be reduced to an HNC equation for some potential and thus general properties of the solution of this equation for different potentials are of central importance. For any non-singular potential $(r) with strong repulsion at short distances , the asymptotic T -*• oo solution of the HNC equation has the following universal features (Rosenfeld, 1988) ( use a (= (yf^) 1 / 3 in 3-D) as the unit of length, and /3$(r) = £ for the OCP in 3D as the specific example) : (1) Madelung behavior of the potential energy featuring: the Onsager exact lower bound for the potential energy = sum of self energy of individual dressed particles (Onsager atoms and molecules), e.g. for the OCP

5 £ = PUOA = -0.9I\

(6)

is the self-energy of an "Onsager atom" consisting of a point charge at the center of a neutralizing unit sphere having the background charge density. It is equal to the energy integral with a universal pair correlation function

J

D

[9D(r) -

(in our units n=ft^ 1 , where ftjrj is the volume of a unit D-dimensional sphere). The universal functions <jf£>(r) denote the limit r} —* \ of the solution of the PY equation for D-dimensional hard spheres of packing fraction t). (2) Saturation of the direct correlation functions: —^p- — ^ ( r ) = electrostatic interaction between the uniformly smeared charges (spheres with the ionsphere radius), for the OCP, given in ZD by

* ( r ) = | ^ 5 2 $(r) = -

r

+

16

r r 160

2

T

(3) Ewald identity and Mean Spherical Approximation boundary conditions:

N

\

J

= -{-nj

,

(9) D

[c(r) + P4>(r)]d r + c(r = 0)}


85

(a) The direct correlation function is a short range Ewald function: c(r > 2) = — /3$(r > 2). (b) There exists a positive definite structure factor, S(k) > 0 : thus also c(k) < 0. (4) Pair exclusion: g(r < 2) = 0 , i.e. c(r < 2) + /3$(r < 2) > 0. The ideal liquid limit is associated with an effective packing fraction rj = 1. e = 1 — rj serves as the small parameter. (5) Universal structure factor peaks: Let ft(r) denote the overlap volume of two D-dimensional unit spferes at separation r (ft(0) = fto)» a n d define

u(r) = §£J. For D = 3 obtain « ( r ) = l - | r + ^

r2

The zeroes c(fcj) = 0 are identical to the zeroes v(ki) = 0 , e.g. £,- = tan(ki) in 3Z). For the Coulomb potential c(k) ex. — T J T > ^ or the Yukawa potential c(fc) a —fcaVa^• (6) Correlation functions and screening potentials feature the Onsager molecules naturally and by recursive definition: In the limit

(11)

nx{r)

= (N-

2)UQA + UOM(T)

(12)

and thus from eq.(2)

HOM{T)

= - - PuOM{r) + 2(3UOA

(13)

Similarly to the Onsager atom, UOM{T) 1S * n e self-energy of an "Onsager molecule" consisting of a pair of ions separated by a distance r in a uniform neutralizing charge cloud of background charge density. The shape of this molecule is determined by the surface on which the electrostatic field vanishes. HOM{T) < 7 by virtue of the antibonding of the Thomas-Fermi molecules. Onsager molecules have the property to "dissociate" whenever the distance between the two point charges is larger than 2a, i.e. = 2UOA

T i.e.

HOM(T) = —

(14a) for

r

>2a

86


Since the Onsager molecule is achieved by optimization, < »(r)

(146)

Since (14.) then the asymptotic functions obey H(r) = -c(r) - B(r)

(15)

(7) The asymptotic "energetics" is exact: recall that the energy integral is equal to that with a universal pair correlation function, and change in B(r) does not affect the asymptotic (Madelung) energy \HOM(T) ls * n e exact asymptotic limit for the screening potential1

4.3 Exact solution of the classical inverse scattering problem in the strong coupling limit, and asymptotic expansion for the correlations We now obtain (Rosenfeld, 1988) the solution of the inverse scattering problem in the ideal liquid limit in 3D. Consider the asymptotic HNC limit for $(r) = <j>(r) + kBTB(r), given the Onsager-molecule screening potential BoM(f)- Our purpose is to find Ac(r) = c(r) — co(r) and B(r) , where co(r) is the asymptotic HNC limit for ^(r). (1) Note that as part of the general "ideal liquid" properties: Ac(r > 2) = B(r > 2) = 0

(16a)

(2) Because c(r) is an Ewald function corresponding to the Onsager atom we obtain (recall eq.9) Ac(fc) < 0 3 f

, Ac(r = 0) = f- / Ac(r)d3 47T J

(3) There is only one solution: Ac(r) is proportional to the overlap volume function


Ac(r) = -Au{r) < 0

87

(16c)

(4) The coefficient A > 0 is obtained by requiring thermodynamic consistency between the compressibility and energy equations of state, knowing that the latter is the exact asymptotic limit. For the OCP , A = 0.2F. (5) Finally, the bridge function is obtained from eq.(15): B(r) = - Ac(r) - co(r) - HOM{r)

(17)

(6) Because of (14a) the asymptotic bridge functions satisfy

BOM{T)

> -Ac(r)

(18a)

In the range relevant to the scattering problem of strongly coupled plasmas, r > ~ 1 ( as g(r < ~ 1) = 0, in strong coupling), the deviations between HOM(T) and \t(r) is less than 1%. However, the corresponding deviations between BOM(^) and — Ac(r) are much larger and may exceed 50%! The bridge function as obtained from the right hand side of (18a) is universal , i.e. it is of the same shape for all potentials, and equals

Buniver,ai(r) = -Ac(r) = ATu(r)

(186)

It thus represents the universal component of the strong coupling B(T). Buniversai(r) always provides an exact lower bound to the asymptotic bridge function. Here is a summary of the asymptotic strong coupling limit results for the OCP (Rosenfeld 1988, 1991a): (a) Energy:

^

-0.9r

(19)

(b) Structure: For the short range part (before "dissociation"), r < 2 , we have

(20)


H(r,T) r

=

HOM(r,T) 9

r

1

(21)

^ Wr) = ^(2 5 / 3 - 2) - Y + h2(r)r* h2(r) ^ 0.038 - 0.0026r2 => h2{r = 2) * 0.0277

T

r

j

(22)

(23)

= 6(r) = ¥(r) + i * ( r ) - W(r) /i2(r) w a s obtained by fitting the Onsager molecule numerical data (Stein, Shalitin and Rosenfeld, 1988) ,while its value at r = 2 is determined by H{r = 2) = 0.5. In the longer range part (after "dissociation"),r > 2, we have

r

1 c(r,r) . ^(r,T) " r "" rr ' r ~ u

^

J

The Onsager molecule functions for mixtures, namely 1r»j(r) and w»y(r) are easily obtained analytically as generalizations of the OCP. HOM,%J{T < 2) is still needed to be calculated numerically (see, however, the next section). Finally, the ideal liquid Onsager limit and the liquid pair structure belong to the same basin of attraction with respect to the diagramatic iterative map as implied by eqs.3-5. This iterative map exhibits an instability of the pair function which correlates well with the simulation data for the freezing of simple fluids and plasmas. The availability of the Onsager limit result is seminal for these instability calculations (Rosenfeld, 1991b, 1992b).

4.4 Ion-sphere and Onsager molecule scaling properties for multi-ionic mixtures

The leading term in the asymptotic strong coupling expansion of the configurational free energy of the fluid is an exact lower bound (Rosenfeld, 1985, 1991a), e.g. the ion-sphere result for the classical plasmas (Lieb and Narnhofer, 1975). The two free energies in eq.2 are bounded from below by these leading terms. In particular, the excess free energy of the given mixture is bounded by the sum of Onsager-atom self energies:


89

(25)

NkBT

i*OA,i = ~0-9~ft^ = —0.9F, is the self-energy of an "Onsager atom" consisting of a point charge Z{ at the center of a neutralizing sphere of radius Ri having the background charge density, R{ = ( is the average charge per ion, then the linear-law reads (Hansen, Torrie, and Vieillefosse, 1979; Brami, Hansen, and Joly, 1979): pex

M^

= Z*>MT ^î^TÔ./tn.reap.l. 0 1 -^"/ t

/.in (.410)

*

corresponding to replacing uo(t) in (39) by

= Mt)-±-l = ~-£t>

+ ....

(42)

It should be noted that the functions Q(t) and tio,/m.rc»p.(') aê even functions of their argument so that the Onsager bound for (41a) is an even function of a. Thus, for small values of a the Onsager bound predicts the following expansion for the linear-response excess free energy:

It is interesting to note here that the best fits (Hansen, 1975; Young, Corey, and DeWitt, 1991) to simulations data yield F(T,a)

8.992

18.165 ,

=

°

+

(AA,

(44)

-

for large values of F, in excellent agreement with the bound (43). The weak screening expansion for the Onsager bound for the mixture (eqs.(37)-(40)) is given by ttmi.,lin.raip. = -0.9 < Z 5 / 3 >< Z

+ ANon.Lin.^3

+ O(aA)

where the first term is the standard linear-mixing (ion-sphere) result for unscreened plasmas, and

94


< Z A

N o

n.Lin.-

, 18 i +

< Z5/3 >2

3 l

Z5/3 >2

1

2/3

l

J

+

100 l < Z 7 / 3 This expansion is in agreement only with general trends among the coefficients in a fit (Hubbard & DeWitt, 1985) to simulations data which, as it stands, can not be compared in full detail with the present result. To obtain we solve eqs.(40) to leading order :

Zi

/3

The Onsager bound is the exact leading term in the strong coupling expansion for the fluid potential energy (Rosenfeld, 1985, 1986b, 1988). To the extent that its scaling properties apply also for arbitrary values of the coupling parameter, this strong coupling limit suggests the following general scaling approximation ("non-linear" mixing rule, because of eq.40) for repulsive-Yukawa fluids:

^

fmix =

(48a)

2 2 f ( T ) , a, a, = aRi

(486)

where u m , r and u refere to the mixture and to the one component energies (per particle, in temperature units), respectively, and fmix and / refere similarly to the corresponding excess free energies, and where eqs.(40) are used in (48b). The Onsager exact energy bound for charged Yukawa mixtures and the corresponding non-linear mixing rule have a simple physical meaning which is revealed by treating the "Yukawa" problem in "Coulomb" language. In the standard linear-response (Firey & Ashcroft, 1977; Chabrier & Ashcroft, 1990) treatment for point ions in (e.g.) a nearly degenerate Fermi sea of electrons, it is assumed that the electron number density varies in space according to (let e = 1)


pe(?)

=< pe > +Ap c (7*) = n

95

a2

+^*(~H

(49)

where $(~r*) is the electrostatic Coulomb potential at the point r in space, and (Dharma-wardana & Taylor, 1981; Dandrea, Ashcroft, and Carlsson, 1986) a is related to the Fermi function I\fi. Let Up be the total electrostatic potential energy (relative to the infinite self energy of the point charges) of this electro-neutral system ,and let UK be the non-ideal kinetic energy of the electrons in the linear response leading order. Their sum,

(50) satisfies the following electrostatic inequality (Rosenfeld, 1991a,b, 1993b): UP

+ UK > £ NiUOA,i = UOTF

(51)

i

where

UOA,i =

^

(52)

The integral / ...dv{ is over the volumes ,«i, of individual, confined, isolated, spherical, and neutral "Onsager-atoms" composed of a central point charge Z{ and an electron cloud of number density (49) and radius R{. Using the definition of Q(t) in Eqs.35 the electron density at the surface can be written as

Pe{Ri)

~ [4«R>\

(53)

The optimized bound (51) subject to the condition of total charge neutrality, J2ixiR^ = 1> yields the bound (41b). Comparison of (40) with (53) reveals its true physical meaning: it is the condition of constant (independent of t) electron surface density, i.e. constant surface electrostatic potential. This is of course the expected result on physical grounds, and which is missing in the linear rule. In view of the dominance of the electrons contribution to the total pressure of the plasma, which is determined for Thomas-Fermi theory

96


by their density at the surface of the confined atom, the condition (40) also corresponds to the well known "volume additivity rule" for Thomas -Fermi mixing of elements (More, Warren, Young, and Zimmerman, 1988),

t>(P,T) = J>t»,-(P,r)

(54)

i

namely combining specific-volumes , v=]^ , Vj=$- ,at same pressure and temperature. Using the non-linear mixing rule (48) obtain the following result for the zero separation screening potential of the Yukawa system ,-,a0 + f(Tj,aj) - f(Ti+i,ai+j)

(55)

rz?

a«+j = = aRi+j aRi+j ,, and where Z,+ - = Z{ + Zj . In the ,, a«+j 7 asymptotic strong coupling limit obtain

where Ti+j — îJf*

,ij(0) T

=

9 JZ}

Z?

t

10 \R{

Rj

ZlA Ri+j)

175 i.e.

>»/» (Z,5/3 + Z,5/3 - (Z,

~ / 5/3

ioo*/3i

z
~ 70, with results in about 0.3% agreement with the original Alastuey-Jancovici extrapolation for H(Q). This also

100


shows that the functional form of / ( r ) as chosen by Ogata, Iyetomi, and Ichimaru (1991) ,with 4 free coefficients, is inferior to the Alastuey-Jancovici form with 3 coefficients. Increasing the number of free coefficients to 5 almost does not affect the result of the extrapolation for these relatively large values of F. The resulting function /*o(r) decreases with F towords the asymptotic limit of 1.05732, while the function ^{T) gradually increases towards the predicted Onsager molecule result. For F = 10, however, only with 5 coefficients it is possible to fit the function / ( r ) to better than 0.1%, and the result features a negative coefficient h^. This last result, and the gradual change in the value of /12, from its asymptotic value of about 0.038 to zero (and even negative values) are in qualitative agreement with the direct simulation results of Ogata, Iyetomi, and Ichimaru (1991) for the coefficient /12, as also discussed below.

4.6.2 Concerning the data analysis of Ogata, Iyetomi, and Ichimaru Ogata, Iyetomi, and Ichimaru (1991) and Ichimaru & Ogata (1990) employ the following methodology: they estimated the coefficient /12 by the Monte Carlo method. They find that the computed values (which are smaller in magnitude than the extent of errors) are an order of magnitude smaller than h\. As a result they decide to truncate the expansion at the second term with hi = 0.25 , and obtain the extrapolated screening potential in the follwing form :

—±-L = ho — h\ r 2 for r < ro

p. = f(r)

for r>r0

The remaining two unknown parameters ho,ro are determined from the requirement that the function and its first derivative are continuous at TQ:

-0.5r 0 = . ho - 0.25rg = /(ro) Since / ( r ) is dominated by the linear part at ro , Ogata, Iyetomi, and Ichimaru (1991) obtain


r 0 = IB = 0.912 - 0.026 ln(r) ho = A - B2 = 1.148 - 0.0094 ln(r) - 0.00017[ln(r)]2

101

(

'

Near freezing, for example at F = 160 , corresponding to their Figure lb, they obtain r 0 = 0.780 , h0 = 1.095. Jancovici's result (1977) , ho = 1.071, is 2.2% smaller. Ogata, Iyetomi, and Ichimaru (1991) estimate the error , in their H(Q) = ho so extrapolated, to stay on the order of 0.1%. This analysis by Ogata, Iyetomi, and Ichimaru (1991) is wrong. It is based on incorrect logic and incorrect methodology : (1) In strong coupling, e.g. F = 160 , the value of ro (e.g. TQ — 0.78) is much smaller than Rmin (e.g. Rmin ~ 1.15). Thus, the connection point for the extrapolation is beyond the actual data points, making the procedure dependent on the choice of fitting function for the data. The obvious next step is to check what happens with the addition of extra terms. Indeed, keeping the term /i2, and repeating the procedure it is found (Rosenfeld, 1992a) that the connection point is now inside the data range, and the results are in 0.3% agreement with Alastuey and Jancovici. (2) Even if the coefficient h2 is an order of magnitude smaller than h\ its effect on the extrapolated ho can be of order of 1% (e.g. with ro ~ 1 and hi ~ hi/25 ~ 0.01). More specifically, observe their equations A1,A2 and Table 4 . They write h2 in the general form Th2 = A — F 2 /32 , so that if Th2 is close to zero then A is close to F 2 /32. Since, for large F , A is a large number for which one needs high accuracy, then it is indeed a good idea to do, as they did, and monitor the quantity nu" in their eq.A2, which should provide the indication for the accuracy of the calculation for A. Now observe their Table 4 to find (e.g.) for T = 80 : u = 0.9910 ± 0.000089. in other words, the statistical accuracy is indeed very high, but the result for "«" is 1% off the exact value "u"= 1 in all the cases in Table 4. Thus, maybe the statistical error for A is high and indicates ±0.01 for /12, but on the basis of the "litmus test" via the quantity "u" (as indeed intended and presented by them), one expects at least a similar error of about 1% for A, which means an error of about T/3200 for h2 , i.e. 0.05 for F = 160 . Thus their claim h2 = 0.00 ± 0.01 is may be correct for F = 10 but not for F = 160. In addition to this expected systematic error, their data point for F = 160 is: h2 = 0.063 ± 0.133, which at its maximum is of the order expected by the Onsager molecule theory. Such a possible gradual transition in the behavior of h2 is in accord with our analysis ,above, of the simulation data as embodied by their function

f(r). Ogata, Iyetomi, Ichimaru, and Van Horn (1992,1993) find deviations from

102


the linear mixing rule which seem to justify their results for H(0) based on /12 = 0. A closer look at their data reveals however that again, as above, they project F ~ 10 results to much stronger couplings. The significant data for the purpose of calculating H(0) is for very dilute ((12 ~ 0.01) mixture of larger charge (Z2 > 1)) in a solvent of unit charge. The effective coupling for these relevant mixtures in the simulations of Ogata, Iyetomi, Ichimaru, and Van Horn (1992,1993) are relatively weak (i.e. F ~ 10). Moreover, their results for a "negative" deviation from the linear mixing rule lead to unphysical crystalization diagrams (Mochkovitch, these Proceedings), and contradict more accurate simulations by DeWitt & Slattery (1993) which find a small positive deviation from the linear mixing rule. In their analysis of their simulation data for mixtures, Ogata, Iyetomi, and Ichimaru (1991) do not consider the prefactor G, and thus effectively use the scaling (32a) with G = 1. On the basis of the high accuracy of the linear law it is expected that scaling relations between the OCP and mixtures, which are similar to the linear law and are likewise based on the "Onsager molecule" (ion-sphere) limit, to be of similar high accuracy ( e.g. better than the 0.3% which is the uncertainty in the OCP results). The additional systematic deviation introduced for unlike ions (t ^ j) by the assumption that G = 1 , which is the assumption made by Ogata, Iyetomi, and Ichimaru can be estimated using (32b) by the deviations of Go from unity. Thus the results Go - 1 = 0.999,0.993 for s = 4/3,2 , respectively, justify the scaling relation used by them for Carbon/Oxygen and (somewhat less) Hydrogen/Helium mixtures. For larger charge ratios, however, significant deviations from the assumption G — 1 will occur as, e.g. Go — 1 ~ 0.03,0.06 for s = 5,10 , respectively. In later work Ogata, Ichimaru, and Van Horn (1991, 1993) and Ogata, Iyetomi, Ichimaru, and Van Horn (1992) they use for the mixture the same incorrect extrapolation method as for the OCP, assuming the ion-sphere scaling via d{j for the function / ( r ) , again without regard to the short range correcting function G(r). The raw simulations data of Ogata, Iyetomi, Ichimaru, and Van Horn agrees very well, however, with our Onsager molecule analysis. For the unscreened OCP at F = 44.9 Ichimaru and Ogata (1991) obtain H(0)/T = 1.110 in contrast to the Jancovici type extrapolation (Rosenfeld, 1992a) H(0)/T = 1.088. Considering the contribution of the electron response then the Ichimaru-Ogata result for a = 0.3125, namely 8 = 1.110 — 0.825 = 0.285 , is similar to the corresponding alternative extrapolation result (Rosenfeld, 1992a) 6 = 1.088 - 0.800 = 0.288, because the value of /12 changes little, for this small value of a = 0.3125 , from the


103

a = 0 case. Ichimaru and Ogata compare their value of S with another estimate ,which they proposed earlier, given by (their eq.ll) 6 = 1.057(1 - e~Q) = 1.057a - 0.5285a 2 + ...

(72)

which gives 6 = 0.278 for a = 0.3125. This value also compares well with the Onsager molecule estimate (eq.58) 6 = a - 0.36860a2 + .... = 0.277. It is not clear what is the origin of eq.72, but it seems to be a fitting function. In any case, it should be noted that eq.(72) is analytically different (and for larger values of a also numerically different) from the Onsager molecule prediction eq.58 which is derived from basic physics.

4.7 Conclusion The present analysis finds excellent agreement between the simulations data for the equation of state and for the screening potentials of strongly coupled plasmas (F > ~ 70), in accordance with the "Onsager molecule" asymptotic limit. The Alastuey-Jancovici-type extrapolation of the highly accurate simulations data of Ogata, Iyetomi, and Ichimaru (1991) for the OCP screening potential, together with the correct "Onsager molecule" scaling (Eq.32) provide the short range screening potentials of strongly coupled plasmas to an accuracy of about 0.3%, in agreement with the linear law and equation of state data. The extended Jancovici formula (eq.55) for the Columb and Yukawa OCP agrees very well with the present extrapolation of the data of Ichimaru and Ogata (1991). The inevitable conclusion from the present analysis is that the calculations of the enhancement factors for nuclear reaction rates of Ogata, Iyetomi, and Ichimaru (1991) and the calculations of the short range bridge functions of Iyetomi, Ogata, and Ichimaru (1992), which are all based on incorrect analysis of the short range screening potential for strongly coupled plasmas, should be revised. In particular, the new data analysis should take into account the gradual change of the coefficient /^(F) from its Onsager F -+ oo limit value of about 0.038, to values near zero for F ~ 20 and to possibly negative values for F < ~ 10. Acknowledgements I thank Hugh DeWitt for bringing to my attention the papers by Ichimaru and coworkers, and for sending valueable material. Interesting discussions with Gilles Chabrier, Hugh DeWitt, Jordi Isern and Hugh Van Horn are acknowledged with gratitude. This work was supported by The Basic Re-

104


search Foundation administered by The Israel Academy of Sciences and Humanities under Grant No.492/92.

References Aers G.C. and Dharma- wardana M.W.C., Phys.Rev.A29, 2734, (1984) Alastuey A. and Jancovici B., Ap.J. 226, 1034 , (1978) Brami B., Hansen J.P., and Joly F., Physica 905A, 505, (1979) Chabrier G. and Ashcroft N.W., PAya.iiew.i442 , 2284, (1990) Dandrea R.G., Ashcroft N.W., and Carlsson A.E., Phys.Rev.B 34 , 2097, (1986) Dharma-wardana M.W.C. and Taylor R. , J.Phys.Cl4, 629, (1981) DeWitt H.E., Graboske H.C., and Cooper M.S., Ap.J. 181 , 439, (1973) DeWitt H.E., Phy8.Rev.Al4,1290, (1976) DeWitt H.E., Slattery W.L., and Stringfellow G.S., in "Strongly Coupled Plasma Physics", edited by S.Ichimaru, North-Holland/Yamada Science Foundation, Amsterdam, 1990, page 635. DeWitt H.E. and Slattery W.L., privat communication, (1993) Firey B. and Ashcroft N.W., Phys.Rev.Al5 , 2072, (1977) Hansen J.P., J.Physique.36, L-133, (1975) Hansen J.P., Torrie G.M., and Vieillefosse P., PAys.iicw.yil6, 2153, (1977) Hansen J.P. and Baus M., Phys.Rep.59, 1, (1980) Hoover W.G. and Poirer J.C., J.Chem.Phys.37 ,1041, (1962) Hubbard W.B. and DeWitt H.E., Astrophys. J. 290, 388, (1985) Jancovici B., J.Stat.Phys.17, 357, (1977) Ichimaru S., Rev.Mod.Phys. 54, 1017, (1982) Ichimaru S. .Iyetomi H., and Tanaka S., Phys.Rep. 149, 91, (1987) Ichimaru S., Ed., Strongly Coupled Plasma Physics", North- Holland/Yamada Science Foundation, Amsterdam, 1990 Ichimaru S. and Ogata S., in "Strongly Coupled Plasma Physics", edited by S.Ichimaru, North-Holland/Yamada Science Foundation, Amsterdam, 1990, page 101. Ichimaru S. and Ogata S., Ap.7.374 ,647, (1991) Ichimaru S., Rev.Mod.Phys.65, 255, (1993) Itoh N., Totsuji H., Ichimaru S., Ap.J.21S, 477, (1978) Itoh N., Totsuji H., Ichimaru S., and DeWitt H.E., vlp.7.234, 1079, (1979) Iyetomi H., Ogata S., and Ichimaru S., Ap.J.334, L17, (1988) Iyetomi H., Ogata S., and Ichimaru S., in US-Japan Workshop on Nuclear Fusion in Dense Plasmas, Institute for Fusion Studies, The University of Texas, Austin. Preliminary Abstracts, Edited by S.Ichimaru and T.Tajima, Oct. 1991 Iyetomi H., Ogata S., and Ichimaru S., Phys.Rev.A46 ,1051, (1992) Lieb E.L. and Narnhofer H., J.Stat.Phys.12, 291, (1975) More R.M. , Warren K.H., Young D.A., and Zimmerman G.B., Phys.Fluids 31, 3059,(1988) Ogata S., Iyetomi H., and Ichimaru S., in "Strongly Coupled Plasma Physics", edited by S.Ichimaru, North-Holland/Yamada Science Foundation, Amsterdam, 1990, page 59 Ogata S., Iyetomi H., and Ichimaru S., Ap.7.372, 259, (1991)


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Ogata S., Ichimaru S., and Van Horn H.M., in US-Japan Workshop on Nuclear Fusion in Dense Plasmas, Institute for Fusion Studies, The University of Texas, Austin. Preliminary Abstracts, Edited by S.Ichimaru and T.Tajima, Oct. 1991 Ogata S., Iyetomi H., Ichimaru S., and Van Horn H.M. , in proceedings of the International Conference on the Physics of Strongly Coupled Plasmas, Rochester, New-York, USA, August 17 - 21,1992 (University of Rochester Press, in print, edited by H. VanHorn and S. Ichimaru); Ogata S., Ichimaru S., and Van Horn H.M. , preprint , 1993 Onsager L. , J.Phys.Chem. 43, 189, (1939) Poll P.D., Ashcroft N.W., and DeWitt H.E., Phys.Rev.A37 , 1672, (1988) Rogers F.J.and DeWitt H.E., Eds., Strongly Coupled Plasma Physics", Plenum, New York, 1987 Rosenfeld Y. and Ashcroft N.W., Phys.Rev.A20 ,1208, (1979) Rosenfeld Y., Phys. Rev. Letters, 44, 146, (1980a) Rosenfeld Y., /. Physique. 41 , C2 -77, (1980b) Rosenfeld Y. ( Phys.Rev.A25 ,1206(1982a) Rosenfeld Y., Phys.Rev.A26, 3622, (1982b) Rosenfeld Y. and Gelbart W.M., J. Chem. Phys., 81, 4574, (1984) Rosenfeld Y., Phys.Rev.A32, 1834(1985) Rosenfeld Y., J. Stat. Phys. 42, 437, (1986a) Rosenfeld Y., Phys.Rev.A 33, 2025, (1986b) Rosenfeld Y. and Blum L., J. Chem.Phys. 85 , 1556, (1986) Rosenfeld Y., Phys.Rev.A35, 938, (1987a) Rosenfeld Y., in "Strongly Coupled Plasma Physics", edited by F.J.Rogers and H.E.DeWitt, Plenum, New York, 1987b. Rosenfeld Y., Phys.Rev.A37, 3403, (1988) Rosenfeld Y., Levesque D., and Weis J.J., Phys.Rev.A 39 , 3079, (1989) Rosenfeld Y., Phys.Rev.440,1137, (1989) Rosenfeld Y., J.Chem.Phys. 93, 4305, (1990) Rosenfeld Y.,in High-Pressure Equations of State : Theory and Applications, edited by S. Eliezer and R. Rici, Italian Physical Society International School of Physics Enrico Fermi, Course CXIII ,1989, North- Holland, Amsterdam, (1991a) Rosenfeld Y., Phys.Rev.A43, 6526, (1991b) Rosenfeld Y., Phys.Rev.A46, 1059, (1992a) Rosenfeld Y., A46, 4922, (1992b) Rosenfeld Y., J.Chem.Phys.98,%126, (1993a) Rosenfeld Y., Phys.Rev.E47, 2676, (1993b) Rowlinson J.S. and Widom B., Molecular Theory of Capilarity (Clarendon, Oxford, 1982) Rowlinson J.S. , Physica 4 1 5 6 , 15, (1989) Salpeter E.E., Aust.J.Phys.7, 373, (1954) Salpeter E.E. and Van Horn EM.,Ap.J. 155, 183, (1969) Stein J., Shalitin D., and Rosenfeld Y., Phys.Rev.A37 , 4854, (1988) Widom B., J.Chem.Phys.39, 2808, (1963) Young D.A., Corey E.M., and DeWitt H.E., Phys.Rev.A44, 6508, (1991)

Astrophysical consequences of the screening of nuclear reactions J. ISERN AND M.HERNANZ Centre d'Estudis AvanqaU Blanes (CSIC), Camt dt Santa Barbara an, 17S00 Blancs, Spain.

Abstract The rate of nuclear reactions depends on the influence of the surrounding particles that compose the plasma. At high densities the situation is far from being satisfactory and the influence of electron polarization has not been completely elucidated. In particular, it is shown that the possibility of an accretion induced collapse of a carbon-oxygen white dwarf instead of a supernova explosion completely depends on the screening factors and pycnonuclear rates that are adopted. Similarly, the possibility of detecting isolated neutron stars that accrete matter from the interstellar medium depends on the adopted pycnonuclear rates. Low rates allow the formation of a metastable layer that can release energy explosively and produce a7-ray burst. Nevertheless, current rates seem to prevent such a situation. Le taux des reactions nucleaires depend de l'influence exercee par les particules voisines qui composent le plasma. A haute densite, la situation est loin d'etre satisfaisante et l'influence de la polarization electronique n'est pas sufisamment claire. En particulier, on montre que la possibility d'obtenir un collapse non explosif d'une naine blanche de carbone oxygene depend des facteurs d'ecrantage et des taux pycnonucleaires adoptes. Egalement, la possibility de detecter des etoiles a neutrons isolees depend des taux pycnonucleaires adoptes. Des petites valeurs favorisent la formation d'une couche metastable qui peut liberer de l'energie explosivement et produire 106

Isern & Hernanz: Screening of nuclear reactions

107

une eruption gamma. Quand- meme, les taux actuels semblent empecher cette situation

5.1 Introduction A nuclear reaction happens when two nuclei approach to a distance of the order of 10~13 cm after tunneling the Coulomb barrier. In the case of two isolated nuclei, this barrier is described by: Vu(r) = — - —

(5.1)

but in the case of a plasma it must be modified to account for the influence of the environment. It is possible to write: Vl2(r) = ^ ^ - + w(r)

(5.2)

where w(r) represents the average contribution of the surrounding particles to the interaction potential. Since the effect of this term is to lower the barrier, the transmission coefficient increases and the reaction rate can drastically differ from the case of isolated nuclei. For instance, in the case of a pure carbon plasma with a density p — 109 g/cm 3 and a temperature T = 109 K, the rate of the reaction 12 C+ 12 C increases by a factor of 1016 due to the influence of the surrounding particles. This effect was noticed for the first time by Schatzman (1948) and was later developed by several authors (Salpeter, 1954; Salpeter and Van Horn, 1969; DeWitt et al, 1973; Graboske et al, 1973; Jancovici, 1977; Alastuey and Jancovici, 1978; Itoh et al, 1978, 1979, 1990; Yakovlev and Shalybkov, 1988; Ogata et al 1991). Usually, the screening factor is written as

E=j-

(5.3)

where R and Ro are the rate of thermonuclear reactions with and without allowance for screening. If | E - 1 |> 1, it is called strong. In the first case, the theory developed by DeWitt et al (1973), Graboske et al (1973) or Yakovlev and Shalybkov (1988) is completely satisfactory from the astrophysical point of view since corrections are usually smaller than the current uncertainties in the astrophysical factors. This regime almost covers the evolution of stars in the main sequence (except the low luminosity end) but not the advanced stages of the evolution of intermediate stars. Once H is exhausted in the central regions, intermediate and low mass

108


stars develop an electron degenerate core. The typical densities of these cores do not allow to neglect the screening effects on nuclear reactions, which turn out to be dominant. This regime is not yet completely understood (see Rosenfeld , this volume), specially at high densities and improvements can change the chemical composition of degenerate cores or even the loci of the ignition curve in the cold, high density regions of the p-T plane. If densities are high enough, the barrier penetration is driven by the energy of the ground state of nuclei in a lattice instead of by the thermal energy and nuclear fusion happens in the so called pycnonuclear regime. This implies that, even at zero temperature, nuclear fusion can happen and that it is not possible to reach arbitrarily high densities without a nuclear rearrangement of matter. The transition from the strong screening to the pycnonuclear regime is not well known and all the work done up to now relies on interpolations that try to avoid discontinuities. A correct understanding of both regions, pycnonuclear and strong screening, is crucial in order to elucidate if CO white dwarfs can make a silent collapse to a neutron star or they will always explode as a Type la supernova when they accrete enough matter from a companion star. Both regions, specially the pycnonuclear one, are also relevant to elucidate if the long term evolution of slowly accreting neutron stars can result into a 7-ray burst, a thermonuclear explosion or just into a peaceful growth of their mass. Here in this paper, we will only address to the fate of accreting CO white dwarfs and accreting neutron stars. 5.2 The physical problem Dense plasmas can be characterized by several dimensionless parameters which facilitate the classification of physical properties in each problem. In order to simplify, it is possible to consider a plasma consisting of only one chemical species with atomic and mass numbers Z and A respectively, with a ion number density n< (or, quivalently, a density p = AHfii, where H is the atomic mass number) and a temperature T. The ion system can be described through the following parameters: The strength of the Coulomb coupling of ions is described by the so called Coulomb coupling constant, F,

where a is the ion-sphere or Wigner-Seitz radius, a = (3/47rn,)1/3. A weakly coupled plasma satisfies the relationship F 1. When T > 180, the ion plasma suffers a phase transition and solidifies into a bcc crystal (Slattery, Doolen, DeWitt, 1980, 1982; Heifer, McCrory, Van Horn 1984; Ogata and Ichimaru, 1987) The ratio between the thermal de Broglie wavelength of an ion, Ay, and the ionic spacing, a, describes the importance of the quantum effects in the description of the fluid properties. A = — = —.

(5.5)

A classical fluid satisfies A 1 imply that the contribution of the quantum plasma effects to the reaction rates cannot be longer neglected. The electron component can be characterized through its Wigner-Seitz radius, usually expressed in units of the Bohr radius (Pines and Nozieres, 1966): r. = - ^

(5.8)

OB

where ae = (3/47rn c ) 1/3 and aB = H2/me2 = 5.292 10~9. The Fermi momentum of electrons in units of me is given by

X3A.) me

, ijOljg r,

while the Fermi energy, in units of me2, is given by EF = mc 2 (\/l-|-1.9610- 4 7-r 2 - 1)

(5.10)

The degree of degeneracy can be expressed through the well known parameter * kT~ kT where fi is the chemical potential of electrons.

110


The coupling strength between electrons and nuclei is given by the rate between the Thomas-Fermi screening length and the radius of the electronic sphere ae

For densities higher than 106 g/cm3, the electrons are weakly polarized and they can be treated as an ideal Fermi gas (Ajjr/a e >> 1). The temperature is usually referred to the Gamow energy, EQ. In the case of nuclei interacting via the bare Coulomb potential (case of a tenuous plasma), the rate, Ro, of the reaction is Ro oc exp(-3EG/kT)

(5.13)

and the parameter r, that only depends on the temperature, is denned as %jJ-/(J

r, £i % 7l

i £iaJL £J\ f^Q C

I F ~ ^~4~^ kTh

2

* 1 /O

J

*

(5

v

' ^

where y, is the reduced mass of the colliding particles. The radius, ro, of the classical turning point for colliding pairs with an energy equal to the Gamow energy is (Alastuey and Jancovici 1978) ro _ 3F _ a

T

TT

5.2.1 Strong screening regime

5.2.1.1 Nonresonant reactions In the case of nonresonant reactions, the reaction rate can be written as:

Rex jTS(E)P(E)exp(-^)dE

(5.16)

where P(E) is the Coulomb barrier-penetration factor, the exponential term is the Boltzmann factor and S(E) is the astrophysical factor. Since S(E) is a smooth function of the energy and P(E) exp(—E/kT) displays a pronounced peak around the Gamow energy, EG, it is possible to write: R oc S(EG) JQ°° P(E) exp (~)dE

(5.17)

and the integral can be interpreted as the pair correlation function at a distance of the order of the nuclear radius, g(r n ) (DeWitt, Graboske and


111

Cooper, 1973; Graboske et al, 1973; Alastuey and Jancovici, 1978; Yakovlev and Shalybkov, 1988): 9(r)

ex exp | -

^

|

(5.18)

Therefore, the enhancement factor can be written as (5.19)

)

where EG 0 and go are the Gamow energy and the pair correlation function for nuclei in an infinitely diluted plasma. Despite g(r n ) is a strongly varying function in the neighborhood of r n = 0 , the ratio g(r n )/go(rn) is a slowly varying function of r n and it is possible to write 9o(0) It has been shown (Jancovici, 1977; Yakovlev and Shalybkov, 1988) that AF

(T,n))

(5-21)

where the first term, which is dominant, corresponds to the purely classical contribution and is equal to the difference between the Coulomb excess of free energies after and before the interaction, and the second term contains the quantum corrections. This term, which is not negligible at all at high densities (Figure 1), has been computed by several authors using different methods: the WKB approximation (Itoh et al, 1977, 1979), the path integral (Alastuey and Jancovici, 1978) or directly (Ogata et al, 1991). A critical discussion of the consistency of the approximations involved in these calculations can be found in Yakovlev and Shalybkov (1989) and in Rosenfeld (this volume). Figure 2 displays the values of In E as a function of F for two values of 3F/r, and in section 3 we discuss the astrophysical consequences of using different approximations. 5.2.1.2 Resonant reactions In the case of resonant reactions, their rate can be written as: Ttot(Er)

6XP(

kT>

(5 22)

'

where E r is the resonant energy and Tin, Tout are the width of the direct and reverse reactions respectively, and Ttot — r,-n + Tout is the total width. In the case of a strongly coupled plasma, the resonant energy is shifted to £j. =

112


6.

8.5

9.5

10.5

log(RHO) Fig. 5.1 Ignition line for a pure carbon plasma. The continuous liie has been computed using the full expression of Alastuey and Jancovici (1978). The dashed line corresponds to the case where only the purely classical contribution is used. The dotted line, 3F/r = 1.6, outlines the limit of validity of the expression Er + AJP and the enhancement factor becomes (Mitler 1977, Mochkovitch and Nomoto, 1986):

AF Tin(Br)T(E'r)

(5.23)

5.2.1.3 Photodesintegration reactions The photodesintegration rate of a nucleus is given by

A oc< av > exv(-Q0/kT)

(5.24)

where < av > is the rate per pair of interacting particles of the forward reaction:

avxx. f

Jo

a(E)exp(-E/kT)dE

(5.25)

113


200

Fig. 5.2 Comparison of the screening factors obtained by Alastuey and Jancovici (1978), updated with the values proposed by Rosenfeld (this volume), H(RAIJ), and those obtained by Ogata et al (1991), H(OII). The continuous line corresponds to 3T/r = 1 and the dashed line to ZF/r = 0.5

E is the kinetic energy in the center of mass system, and increases by a factor E = ex P [-

, 17)] = e x p [ | £ + (T, r,)]

(5.26)

but since the energy threshold must include the contribution of the Coulomb interactions to the chemical,potential: Q = Qo + A/*, the photodesintegration rate is reduced by a factor exp(—Afi/kT) which cancels the dominant term of the screening factor (Mochkovitch, 1983; Mochkovitch and Nomoto, 1986). Therefore, the reaction rate becomes:

A = A(r =

(5.27)

and since (T, rf) is negative, the photodesintegration rate is reduced instead of enhanced.

114


5.2.1-4 Electron polarization In all the previous calculations it has been assumed that the degenerate electrons form a uniform, perfectly rigid background. This approximation is justified since ATf/a e , the ratio of the Thomas-Fermi screening length to the radius of the electronic sphere is larger than one in the regions of interest. Nevertheless, the polarization of electrons is not zero and this effect must be taken into account. If the polarization of electrons is small, the Coulomb excess of free energy per ion, or the chemical potential, can be written in the form of a term computed assuming a uniform electron background plus a small correction due to the electron polarization (Mochkovitch and Hernanz 1986; Yakovlev and Shalybkov 1988): inE = Ht + Hpot where Hpoi = -AFpoi/kT and AFPoi/kT is the difference of polarization energies of electrons. Notice that despite this term is usually called electron screening, it is not the screening due to electrons but the perturbation induced by the polarization of electrons to the screening computed assuming a uniform electron background. The contribution of electron polarization can be computed in the following way (Mochkovitch and Hernanz 1986): Hpoi = - 4 F r [ 2 4 / 3 / ( 2 5 / 3 r , x) - / ( r , *)]

(5.28)

where >S0(q)F(x,y)

-dq (5.29) q2 So(q) is the structure factor corresponding to the pure Coulomb unperturbed potential, F(x,y) is related to the dielectric function of relativistic electrons and x = hkF/mec, y = q/qF, 9 F = (97TZ/4)1/3, qrF = a/^TF, q = ak. In the case of a pure carbon plasma, Z=6, p = 109 g/cm3 and T= 108 K, H p0 | = 0.55 while H; = 37.2. Therefore, due to the uncertainties involved in the nuclear reactions and in the screening factor itself, it is usually possible to neglect the contribution of this term. These results, however, strongly differ from those of Ichimaru (1993) who found enhancements much more important, by one or two orders of magnitude. This issue is very important and must be clearly elucidated (see Rosenfeld, this volume) since these corrections can completely modify the results discussed below about the accretion induced collapse of white dwarfs. Recent independent calculations by Sahrling (1994) confirm the results of Mochkovitch and Hernanz, and Yakovlev and Shalybkov. The effect of the electron polarisability in the afore-mentioned example is found to be negligible (of the order of 1%).


115

5.2.2 Pycnonuclear reactions

The evaluation of the reaction rate in the pycnonuclear regime is quite complicated since the energy distribution of nuclei is no longer given by the Boltzmann distribution and it is necessary to solve the Schrodinger equation to obtain the wave function. The first discussion of this process was given by Cameron (1959) and was later elaborated by Wolf (1965), Salpeter and Van Horn (1969), Schramm and Koonin (1990) and Ogata et al (1991), in the case of one component plasma. The case of binary ionic mixtures has been recently treated by Ichimaru et al (1992), while the case of the 3a process has been treated by Salpeter and Van Horn (1969), Fushiki and Lamb (1987), Schramm et al (1992) and Langanke and Mtiller 1993). 5.2.2.1 Fusion rates in one component plasma In the pycnonuclear regime, no independent particle model provides an adequate description of the fusion process and there are not simple approximations to the potential of the interacting particles. It is possible, however, to find two limiting cases for the potential (Salpeter and Van Horn, 1969): The "static" and the "relaxed" approximations. The first one assumes that all the nuclei as well as the center of masses of the reacting pair are frozen at the equilibrium positions, while the second one assumes that the position of the center of masses is fixed and the remaining lattice points polarize into the positions determined by the separation of the two reacting nuclei. Wolf (1965), using a simplified study of the dynamics of the crystal, obtained an intermediate potential between these two limiting cases, but nearer to the static case. Lately, Ogata et al (1991) determined the interparticle potential using a MonteCarlo method and found a result that is near to the relaxed approximation. The result of Ogata et al (1991) can be easily understood in the following terms: Let At be the time spent by a reacting pair with a typical energy hup into the barrier and let u~x be the characteristic response time of the crystal. Since up At ~ 140Z5/2p^"1/4 (Schramm and Koonin 1991) is bigger than one for all the situations of physical interest, the lattice will respond adiabatically and will polarize under the influence of the reacting pair. Nevertheless, since the kinetic energy associated to the polarization must be taken into account during the calculation of the tunneling probability, it turns out that both effects of the polarization almost cancel and the final fusion rate is very near to the static one (Schramm and Koonin, 1991). The inclusion of dynamic terms has been contested by Ichimaru (1993) who argued that all the dynamic effects are already included in the

116


5.00

3.00 -

1.00 -

-1.00

RHO/1E9 Fig. 5.3 Comparison of different fusion rates normalized to the static rates,log(R/R,«o«,c), versus A, where A is the ratio of the nuclear Bohr radius and the lattice spacing (A a p 1 ' 3 ). Solid line is from Schramm and Koonin (1991). Dashed line is from Ogata et al (1993). Dotted line corresponds to the relaxed approximation of Salpeter and Van Horn (1969). pair correlation function through the dynamic structure factors since nuclear reactions are very rare events as compared with all the other collision processes. Figure 3 displays the comparison between the different fusion rates quoted here.

5.2.2.2 Fusion rates in a binary ionic mixture In the case of a plasma with several components the situation is rather complicated because of the uncertainties about the structure of the crystal. Usually, the fusion rate has been taken as c(x) = c(x = 1) x 2 , where x is the molar fraction. The validity of this expression relies on the assumption that the reacting nuclei are randomly distributed and that the lattice potential is uniform. The existence of lattice imperfections can be handled in the following way


117

(Salpeter and Van Horn 1969): The presence of a default produces a local increase of the density that translates into an increase of the local rate by a factor exp(1.3A~1/2 6X/X), where SX is the local excess of the parameter A. Because of the extreme sensitivity of the pycnonuclear reactions to the density, the nuclear fusion will start at these defaults and, if the relaxation of the reaction products produces new defaults, the reaction rate will be strongly enhanced. The presence of other chemical species can have two different consequences. If the reacting and non-reacting nuclei have similar charges (case of C O mixtures) the heavier species blocks the reaction and its rate is strongly reduced. If they have very different values, the heaviest one induces a local increase of the density and the reaction rate is strongly enhanced. The critical value that separates both behaviors is Z2/Z1 = 2.3, where Zi > Z\. Figure 4 shows the influence the presence of oxygen and iron on the pycnonuclear fusion of carbon. 5.2.2.3 The 3a fusion rate The 3a reaction in the pycnonuclear regime was considered for the first time by Cameron (1959) and later on by Salpeter and Van Horn (1969), Fushiki and Lamb (1987) and Schramm et al (1992). Besides the problem of its three-body nature, the pycnonuclear 3a reaction is characterized by the fact that temperatures are too low to allow the resonance of 8Be and by the fact that in the region of interest helium is a quantum fluid rather than a Coulomb lattice. The present results are quite satisfactory since different approaches give the same results. Nevertheless, the effect of the presence of impurities remains to be studied. It is interesting to notice that above p ~ 2 — 3 109 g/cm3 a lattice made of 8Be has a lower energy than a 4He fluid (Schramm et al 1992) and 8Be becomes stable. However, the pycnonuclear 3a reaction is able to convert helium into carbon before reaching this density and the possibility of the existence of 8Be matter seems to be prevented (Langanke and Miiller, 1993). 5.3 The astrophysical problem 5.3.1 Collapse or explosion of mass accreting white dwarfs More than fifteen years ago, Schatzman suggested that besides the standard mechanism for neutron star formation (collapse of the fuel exhausted core of a massive star), a second mechanism was possible: gravitational collapse of a white dwarf due to accretion of matter in a close binary system. In

118


2.00

IMII

0.00 -

-1.00

RHO/1E9 Fig. 5.4 Blocking and catalyzing effects of impurities in the pycnonuclear fusion of carbon. The vertical axis displays log[R(x £ 0)/R(x = 0)] for the 1 2 C+ 1 2 C reaction in the presence of oxygen (continuous line) and iron (dashed line). In both cases the molar abundance of the heavier species is x=0.5

both cases, collapse would be induced by growth above the effective Chandrasekhar's mass. The second mechanism would account for the presence of neutron stars in systems such as the low-mass X-ray binaries, where survival to a Type I supernova explosion appears unlikely. Accretion induced collapse of white dwarfs has later been incorporated in evolutionary scenarios for the origin of several kinds of binary systems containing neutron stars (Van den Heuvel 1983, 1989). Even if white dwarfs might grow in mass at any arbitrary rate, which is not the case (Isern et al 1983; Hernanz et al 1988; Canal et al 1990), accretion induced collapse would in any case encounter the difficulty that explosive ignition of the nuclear fuel in the electron degenerate cores always precedes reaching the Chandrasekhar's mass. For white dwarfs made of carbon-oxygen or oxygen-neon-magnesium, ignition can be delayed until losses due to electron captures overcome the energy released by ther-


119

monuclear reactions (Isern and Canal, this volume). CO white dwarfs were proposed by Canal and Schatzman (1976) while ONeMg white dwarfs were advocated by Miyaji et al (1980). Concerning CO white dwarfs, it was soon realized that central ignition in a partially solid core provided the most favorable conditions for accretion induced collapse (Canal and Schatzman 1976; Canal and Isern 1979). The behavior of a cool white dwarf interior during the accretion phase depends on the competition between physical processes that increase the temperature of the material (compression, nuclear reactions in the inner core and thermonuclear burning of the accreted matter) and those that cool down the star (photon and neutrino losses). Obviously, heat conduction plays an important role, as being the main mechanism for heat transport in the stellar interior. Conduction can be characterized by the time required for a thermal signal to travel from the center to the surface of the star. This time is given by Henyey and L'Ecuyer (1969) as: •

(5-30)

where / is the linear extent of the region considered and all the remaining symbols have their usual meanings. The time scale for increasing central density, when mass approaches the Chandrasekhar's limit, can be expressed as (Canal and Schatzman 1976):

where M is the mass accretion rate and yo = -Jl + x2F, where x/r is the dimensionless Fermi momentum. The effects of the compression induced by the accreted matter can be divided in two terms (Nomoto 1982). The first one is due to the increase in density at a fixed mass fraction as the stellar mass increases, and its effects are quite uniform throughout the whole star. The second term corresponds to compression as matter moves inwards in the mass-fraction space. It is negligible in the inner strongly degenerate regions of the star, but it is very large in the semi degenerate external layers. This means that a thermal wave is generated in those layers which diffuses inwards. A rough estimate of the compressional luminosity of the external layers is (Nomoto 1982): LNH -?— = 1.4 lO- 3 r 7 Mio I©

(5.32)

120


T7 being the temperature in units of 107K and M\Q the accretion rate in units of 1O"10 M©/yr. The effects of this thermal wave on the temperature profile, and thus on the physical state of the white dwarf interior, will depend on the time needed for the thermal wave to reach the center of the star as compared to the time required for the star to reach the Chandrasekhar's limit and also on the efficiency of thermal cooling of the white dwarf (Hernanz et al 1988). For low accretion rates (10~12M©/yr< M < 3 10"10M©/yr), the thermal wave generated by the accretion process in the external layers can reach the center of the star before any instability starts, but compression is so slow that radiative cooling through the star surface dominates. The white dwarf, in this case, evolves keeping an isothermal profile and the internal temperature is determined by the balance between compression and cooling through the envelope. For high accretion rates (M > 5 10~8 M©/yr), compressional luminosity is much higher. However, in this case, the thermal wave has no time to reach the center for high enough masses, and only the outermost solid layers are affected. In the central solid layers TTH » TP and they evolve quasiadiabatically. The slope for the evolutionary path in the log p — log T diagram is given by (Mochkovitch 1983) 0.815 + 0.215F1/4 (5 33) T3 1= ~ 0.945 + 0.6461^ ' F3 — 1 being the adiabatic index. For F values in the range 100 - 200, the adiabatic index is ~ 0.5, For intermediate accretion rates, (310- 10 M©/yr< M < 510~8M©/yr), the thermal wave has again enough time to reach the center of the star. Compressional heating dominates the normal cooling and matter heats up. The outcome, collapse or explosion, depends on the density at which the thermonuclear runaway starts and this critical density depends on the velocity at which the burning front propagates. The minimum density necessary to get a collapse is 8.5 109g/cm3 (Isern and Canal, this volume). Garcia and Bravo, also this volume, propose an even smaller value, 7109 g/cm3. However, if all the modes of propagation of the burning front are taken into account, this critical density adopts a value in the range 9.2 to 9.5 109 g/cm 3 , although these limits are rather uncertain. It is thus interesting to examine the dependence of the ignition density on the adopted value of the nuclear reaction rate. Table 1 displays the ignition density for different accretion rates. All the models have been constructed in the same way as in Hernanz et al (1988). Models labeled with a capital letter


121

Table 5.1. Ignition densities (in units of 109 g/cm3) for different pycnonuclear reaction rates M(M0/yr)

A

B

a

b

c

io- 6 io- 97

10.60 9.05 7.87 7.98 7.73

12.40 10.10 9.10 9.92 9.66

10.40 6.63 7.82 7.98 7.71

12.10 6.63 8.97 9.89 9.63

10.85 6.51 7.99 8.26

lO"

io- 10 io- 11

started at M^D ~ 1-2M®, those with a small letter started at 1.1. The initial temperature was To = 4 IO6 K for all of them. Pycnonuclear reaction rates from Ogata et al (1991) were used in models A and a. Models B and b were obtained using the rates of Schramm and Koonin (1990), while model c was obtained with the pycnonuclear rates of Ichimaru et al (1992), which include the blocking effects due to the presence of oxygen nuclei. Columns A and B show the importance of including the dynamics of the crystal in the reaction rates. In the first case, almost all of them collapse. Comparison of model c with a shows that the blocking effects of oxygen effectively delay the runaway but do not introduce any qualitative change in the above picture. The initial mass is also a critical factor. Below 1.05 M®, the accretion induced collapse is impossible. Since the outer thermal wave has less time to reach the centre, collapse is favored by increasing the initial mass. This implies that only the tail of most massive CO white dwarf can collapse to form a neutron star. The strong screening factors adopted and the interpolation algorithm between strong screening and pycnonuclear regimes are also critical. Table 2 displays the ignition density in the case of a white dwarf with an initial mass MWD = 1.2 M®, initial temperature T = 410 6 K, which accretes matter at a rate M = 10~ 7 M^/yr and is made of carbon and oxygen. The screening factors have been obtained from Ogata et al (1991), Oil model, and from Rosenfeld (1993) and Alastuey and Jancovici (1979), RAJ model. The pycnonuclear rate was obtained from Ichimaru et al (1993). The gap between pycnonuclear and strong screening reactions was covered with the interpolation algorithm described by Bravo et al (1983). In model A, the rates were obtained interpolating between the line 3F/r = 1 (strong screening regime) and 3 r / r = 8.1 (pycnonuclear region), Model B by interpolating between 3T/T = 1 and T = 180 and model C between ZT/T = 1.5 and T = 180. As it

122


Table 5.2. Ignition densities (in units of 109 g/cm3) for different strong screening factors B

OH RAJ

9.23 8.88

9.90 9.70

off 9.88

can be seen from table 2, the results are spectacular and even an off center ignition is obtained in case C model OIL 5.3.2 Slowly accreting neutron stars Neutron stars are rather abundant in the Galaxy. Their estimated density in the solar neighborhood is n ~ 104/(300pc)3. Few of them can be detected as pulsars, if they are young enough or/and they rotate fast enough, or as bright X-ray sources, if they are members of a close binary system (in this case, they accrete matter at a rate M ~ lO^Mo/yr and they emit Lx — 1038 erg/s in X-rays). The question is: what happens with the remaining, isolated and old, neutron stars that accrete at a typical rate of 1010 g/s?. The evolution of neutron stars that accrete mass at a very low rate, 10~ 16 < M < 10~llMQ/yr, has been studied many times (Hameury et al 1983; Blaes et al 1989, 1990, 1992; Miralda-Escude et al 1990; Zdunik et al 1992) In all of them the pycnonuclear rates have proved to play a critical role since the mass-accretion is so slow that the temperature is too low to allow thermonuclear rearrangement of nuclei. Their evolution can be summarized in the following way: a) Few years after the explosion, the neutron star captures the bound debris and burns them to 56 Fe. Because of the weight of the newly accreted matter, this material sinks and starts capturing electrons: 56Fe—>56Cr—^56Ti. b) After the pulsar phase, the neutron star can start accreting matter from the interstellar medium, M ~ 1010 V^QUISM g/s> where U40 is the speed of the star relative to the interstellar medium in units of 40 km/s and niSM is the hydrogen number density of the interstellar gas in cm" 3 (Hoyle and Lyttleton, 1939), which imply accretion rates of the order of 1010 g/s. The newly accreted matter compresses the crust and hydrogen is burnt to helium due to pycnonuclear reactions or electron captures, depending on the accretion rate. Since spallation reactions at the surface of the star destroy most of the metals (Bildstein et al 1991), burning is stable as far


123

as M < 2 1013 g/s and the crust can be considered isothermal, Ts = M^ (Blaes et al 1992, for such low rates. c) The helium newly formed in this way solidifies at a temperature T9 ~ 3.3 lOV 1 / 3 . For an isothermal crust of the kind mentioned in b), this will happen at a density pa = 2.8 107Mfflu (Blaes et al 1992). As He behaves like a quantum liquid for densities higher than 310 8 g/cm3, (Mochkovitch and Hansen 1979, Chabrier 1993), a layer of solid He surrounded by a classical liquid in the top and a quantum liquid in the bottom will form. When the density is high enough, helium ignites due to pycnonuclear reactions. The characteristic time scales for the a-captures are: r[ 12 C(a, 7 ) 16 O] ~ 10"9s r[ 16 0(a,7) 2O JVe]~lO 15 3 Thus, if the 3o>reaction were slow, the freshly synthesized oxygen would have time to capture an a-particle and approach to Si. Since the 3a rate is very high (Schramm, Langanke and Koonin, 1991) only 12C has time to capture an a-particle and an 16O-rich mixture forms. As oxygen crystallizes at higher temperatures than helium (pc a 1.3 105M10 ' g/cm 3 ), it immediately solidifies and sinks, accumulating at the bottom of the quantum He-ocean. d) As the accretion proceeds, the density at the bottom of the accreted material increases and finally at p = 1.921010 g/cm 3 , 1 6 0 undergoes a twostage electron capture to 16 C. If this 16C accumulates or fuses depends on the pycnonuclear rate of the reaction 16 C + 1 6 C. If it accumulates, it can undergo an elastic Rayleigh-Taylor instability with the old underlying crust resulting in a starquake which releases some 1040 ergs of gravitational and nuclear energy (Blaes et al 1990, 1992). The fusion of 16C was studied by Bravo et al (1983). It has the following characteristics: 16

C + 1 6 C -^ 28 Mg + An (90%) Q = l

i e c + i 6 c _^29 Mg

+ 3n

( 1Q% ) Q __ 1

with So = 5.5 108 MeV-barn. At the density at which it is produced, its characteristic fusion time scale is 1000 yrs, quantity that can be much shorter if some heavy elements coming from the original crust diffuse and mix with the 16C layers. Therefore, the composition will change to 28Mg. During all this process, the underlying original crust makes a transition to

124


56

Ti, 62Cr or 62Fe because of electron captures and, since its mean molecular weight per electron is equal or higher than that corresponding to 28Mg, the accreted layer will be stable. 5.4 Conclusions Carbon-oxygen white dwarfs can either collapse or explode depending on their initial properties. If they are massive and cool enough, they can nonexplosively collapse and form a neutron star. The frequency and relevance of this phenomenon, as compared with the accretion induced collapse of ONeMg white dwarfs, critically depends on the adopted rates and on the way as the transition between strong screening and pycnonudear regimes is handled. The effects due to the polarization of electrons can be critical. If they increase the reaction rates by one or two orders of magnitude as claimed by Ichimaru (1983), the collapse to a neutron star will be prevented in the majority of cases. The possibility of observing isolated neutron not detectable as pulsars also depends on the adopted pycnonudear rates. If the rates are high enough, due to the presence of defaults or heavy impurities, the slowly accreted material is gradually transformed and neutronized without forming a metastable layer that could release its energy in a burst. Therefore, due to the current uncertainties on the screening factors it is not possible to reach a definite conclusion in both cases. It should be of the highest interest to definitively elucidate the behavior of nuclear reaction rates at high densities (3F/r > 1) and in the pycnonudear region (F > 180). Acknowledgements This work has been partially financed by the DGICYT grant PB91-060, the Spanish-French Action "Physics of White Dwarfs and Brown dwarfs" and the CESCA grants "Structure and Evolution of Galaxies" and "Accretion onto White Dwarfs". References Alastuey A., Jancovici B.,Astrophys. J. 226, 1034 (1978) Bravo E., Isern J., Canal R., Labay J.,Astron. Astrophys.124, 39 (1983) Bildsten L., Salpeter E.E., Wasserman l.,Astroph. 7.384, 143 (1992) Blaes O., Blandford R., Goldreich P. and Madan P.,Astrophys. 7.343, 839 (1989) Blaes O., Blandford R., Madan P. and Koonin S.,Astrophys. 7363, 612 (1990) Blaes O., Blandford R., Madan P. and Yan L., Astrophys. 7.399, 634 (1992) Cameron A.G.W., Astrophys. J. 130, 916 (1959) Canal R., Isern., in White Dwarfs and Variable Degenerate Stars IAU Colloq N o l 5 3 , ed H.M. Van Horn, V. Weidemann, p.52, Rochester NY: Univ. Rochester Press (1979)


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Canal R., Isern J., Labay J., Ann. Rev. Astron. Astrophys. 28, 183 (1990) Canal R., Schatzman E., Astron. Astrophys. 46, 229 (1976) Chabrier G., Astrophys. J. 414, 695 (1993) DeWitt H.E., Graboske H.C., Cooper M.S.,Astrophys. J. 181, 439 (1973) Graboske H.C., DeWitt H.E., Grossman A.S., Cooper U.S.,Astrophys. 7. 181, 457 (1973) Fushiki I., Lamb D.Q., Astrophys. J. 317, 368 (1987) Hameury J.M., Heyvaerts J. and Bonazzola S.,Astron. Astrophys.121, 259 (1983) Heifer H.L., McCrory R., Van Horn H.M., 7. Stat. Phys 37, 577 (1984) Henyey L., L'Ecuyer J ., Astrophys. J. 156, 549 (1969) Hernanz M., Isern J., Canal R., Labay J., Mochkovitch R., Astrophys. 7. 324, 331 (1988) Hoyle F. and Lyttleton R.A.,Proc. Cambridge Phil. Soc.35, 592 (1939) Ichimaru S.,Rev. Mod. Phys. 65, 255 (1973) Ichimaru S., Ogata S., Van Horn H.M., Astrophys. J. Letters 401, L35 (1992) Isern J., Labay J., Hernanz M., Canal R., Astrophys. 7. 273, 320 (1983) Itoh N., Kuwashima F., Munakata E.,Astrophys. 7. 362, 620 (1990) Itoh N., Totsuji H., Ichimaru S.,Astrophys. 7. 220, 742 (1978) Itoh N., Totsuji H., Ichimaru S., DeWitt H.E.,Astrophys. J. 1079, (1979) Jancovici B.,7. Stat. Phys. 17, 357 (1977) Miralda-Escude J., Haensel P. and Paczynski B.,Astrophys. 7.362, 572 (1990) Miyaji S., Nomoto K., Yokoi K., Sugimoto D., Publ. Astron. soc. Japan 32, 303 (1980) Mochkovitch R., Astron. Astrophys. 122, 212 (1983) Mochkovitch R., Hansen J.?.,Phys. lett.73A, 35 (1979) Mochkovitch R., Hernanz, M., in Nucleosynthesis and its implications on nuclear and particle physics, eds. J. Audouze and J. Tran Thanh Van, p. 109, Reidel: Dordretch (1986) Nomoto K., Astrophys. 7. 257, 780 (1982) Ogata S., Ichimaru S., Phys. Rev A36, 5451 (1987) Ogata S., Iyetomi H., Ichimaru S.,Astrophys. 7. 372, 259 (1991) Pines D, Nozieres P., The theory of quantum liquids (Benjamin: New York) (1966) Sahrling M., to appear in A&A, (1994) Salpeter E.E., Aust. J. Phys. 7, 373 (1954) Salpeter E.E., Van Horn H.M., Astrophys. J 155, 183 (1969) Schatzman E., presented at InT. Sch. Cosmol. Gravit. Erice, Italy (1974) Schramm S., Koonin S.E., Astrophys. 7. 365, 296 (1990) Schramm S., Langanke K and Koonin S.E.,Astrophys. 7.397, 579 (1992) Slattery W.L., Doolen G.D., DeWitt H.E., Phys Rev A21, 2087 (1980) Slattery W.L., Doolen G.D., DeWitt H.E., Phys Rev A26, 2355 (1982) Van den Heuvel E.P.J., in Accretion-Driven Stellar X-Ray Sources Ed. W.H.G. Lewin, E.P.J. Van den Heuvel, p.303, Cambridge Univ. Press () Van den Heuvel E.P.J., in Timing Neutron Stars, ed H.O. Ogelman E.P.J. Van den Heuvel, Dordrecht: Kluwer, p.523 (1989) Yakovlev D.G., Shalybkov D.A., Adv. Space. Res. 8, (2)707 (1988) Zdunik J.L., Haensel P., Paczynski B. and Miralda-Escude J.,Astrophys. 7.384, 129 (1992)

6 Crystallization of dense binary ionic mixtures. Application to white dwarf cooling theory. R. MOCHKOVITCH Institut d'Astrophysique it Paris, 75014 Paris, France

L. SEGRETAIN Laboratoire de Physique. Ecole Normale Superieure de Lyon, 69364 Lyon Cedex 07. France

Abstract This paper is organized in two parts. First, phase diagrams for dense binary mixtures are computed with the density functional theory (DFT). The method of calculation is reviewed and the different approximations which are used are clearly stated. The DFT is then applied to several mixtures of astrophysical interest. A comparison is made between several existing phase diagrams and the origin of some discrepancies among them is discussed. In a second part, the consequences of these phase diagrams on the cooling of white dwarfs are presented in a pedagogical way starting from the simple Mestel theory. The importance of the partial separation of carbon and oxygen at crystallisation is emphasized and the possible effect of minor species such as 22Ne or Fe is also considered. The separation of carbon and oxygen adds 1 - 2 Gyr to age of the galactic disk estimated from the white dwarf luminosity function while the delay resulting from the presence of minor species is probably negligible when the chemical evolution of the Galaxy is properly taken into account. Cet article est organise en deux parties. Tout d'abord, les diagrammes de phase des melanges binaires denses sont calcules a l'aide de la theorie de la fonctionnelle de densite. La methode de calcul est detaillee et les differentes approximations utilisees sont clairement expliquees. Le theorie est ensuite appliquee a plusieurs melanges d'interet astrophysique. Une comparaison est faite entre plusieurs diagrammes de phase publies et l'origine de certains 126

Mochkovitch & Segretain: Crystallization of dense binary ionic mixtures

127

disaccords entre eux est discutee. Dans la seconde partie, les consequences de ces diagrammes de phase sur le refroidissement des naines blanches sont presentees de maniere pedagogique avec pour point de depart la theorie de Mestel. L'importance de la separation partielle du carbone et de l'oxygene durant la cristallisation est soulignee et l'effet possible des especes mineures, comme le 22Ne et le fer, est aussi considere. La separation du carbone et de l'oxygene ajoute 1 a 2 milliards d'annees a l'age du disque galactique obtenu a partir de la fonction de luminosite des naines blanches alors que le delai du a la presence des especes mineures est probablement negligeable quand revolution chimique de la Galaxie est convenablement prise en compte. 6.1 Calculation of the phase diagrams 6.1.1 Introduction The importance of Coulomb interactions in white dwarf interiors has been recognized long ago (Kirshnitz 1960; Abrikosov 1960; Salpeter 1961), and was shown to lead eventually to crystallization of the core of the star (Van Horn 1968; Lamb and Van Horn 1975). The importance of the alloying behavior of dense Coulomb lattices to astrophysics was first pointed out by Dyson (1971), and the importance of the crystallization diagram of C/O mixtures on the evolution of white dwarfs was first considered by Stevenson (1980). In order to study the crystallization of the white dwarf core, we consider this one as a multi-component plasma, an extension of the one component plasma model (OCP) (Brush, Sahlin, Teller 1966; Hansen 1973). The OCP is characterized universally by the dimensionless parameter T = Zs/3e2/kTae = Z 5 / 3 r c where Z is the charge of the ion and ae — (3/(4n-ne))1^3 is the electronic sphere radius with n e the number density of electrons. The present critical value of the paramater T at crystallization is Tc « 172 - 180 (Ogata, Ichimaru 1987; Strinfellow et al. 1990; Dubin 1990; Farouki and Hamaguchi 1993) and the release of the latent heat is about Ar^T'/'particle. However, besides carbon and oxygen, minor elements like neon and iron are present in the core of white dwarfs and the crystallization of this mixture is more difficult to study than for the one component system. In the next section we examine the different methods used to build a phase diagram and we present all existing results. 6.1.2 Methods of calculation The determination of a phase diagram for a multi-component plasma requires the determination of the free energy in the liquid (FL) and the solid

128

Mochkovitch & Segretain: Crystallization of dense binary ionic mixtures l.U

1.4 =

>

: p=cst

:

'

A

1.3 /

1 1 1 1

1.2

9

_

—

TI

—

/

. • '

|

1.1

n

1 0 . 2 0 4 0.6 0.8

—

1 0.8 0.6 0.4 0.2 0

Fig. 6.1 Construction of phase diagram (Fs) phase, as shown infig.1.1. We compute free energies in the two phases at a given temperature for all concentrations and we determine the equilibrium concentrations in the solid (is) and in the liquid (z£,) phase with a double-tangent method. We iterate this process for several temperatures to build the phase diagram. Two methods can be used to compute JFJ and Fa. The first one is the Monte-Carlo simulation. The advantage of this method is to be an "exact" calculation since the energy is calculated directly from the canonical average of the interaction , but a reliable result requires a large number of particles since the accuracy of the method is proportional to 1/N. This requirement becomes really restricting for a mixture when the concentration of one of the species is small ; in this case the total number of particle becomes prohibitive, and the simulation requires a tremendously large computation time. The second method is based on the density fonctional theory of freezing, and is described below. 6.1.3 The density functional theory of freezing The density functional theory (DFT) is based on the theorem (Honenberg and Kohn 1964) that any thermodynamic function can be written as a functional of the density. It has been applied to phase diagrams for the first time by Ramakrishnan and Yussouf (1979) for the crystallization of a hard


129

sphere system. Since then, the DFT has been used for different short range repulsive potentials (V(r) oc r ~ n , n = oo,12,6,4) (Barrat 1987) and was shown to be in good agreement with the simulations. Within the framework of the DFT, we can calculate directly A F = Fa—F[ by considering the solid near melting as a non-uniform liquid and by calculating the related correlation functions from the Taylor expansion around the liquid correlation functions. In order to calculate AF, several approximations are necessary. i) The first one is the truncation of the Taylor expansion at a given order, usually the second order. That yields, for the OCP, to : A F = AFid + AFex

(1)

with

NkT ~ Jv

PL

l

PL

(2)

and

53^ represents the sum over all non zero reciproqual lattive vectors (RLV), C(G) denotes the Fourier transform of the direct correlation function (DCF) of the liquid and the one-particule density in the solid is written as :

p(r) = pL(l + Y1P(G) exp(ig.f))

(4)

However a problem appears for Coulomb systems if the Taylor expansion is truncated at the second order. In this case, Rovere and Tosi (1985) showed that the solid phase is never stable, its energy being overestimated because of the missing compensating negative contribution of the higher order terms. Two solutions can be used to solve this problem. The first one is to include higher orders terms (third order or more). This is tremendously complicated and has been done only for the OCP (Barrat 1987; Iyetomi and Ichimaru 1988; Likos and Aschroft 1992). Then a second, more phenomenological, approach is used. The effect of higher order terms is equivalent to a reduction of the contribution of the second RLV (i.e. the term C^p2^ ) (Likos and Aschroft 1992). Then the solution is to reduce arbitrarily this contribution in such a way that the crystallization of the OCP is recovered

130


for Tc = 178 (Barrat et al. 1988; Ichimaru et al. 1988; Segretain and Chabrier 1993). ii) The second approximation concerns the description of the solid phase, namely the choice of a description of the solid density fluctuation (i.e. pg), of a crystal lattice (i.e. {G}), and of a type of alloy (for a BIM). For the description of the solid density fluctuation, a gaussian approximation is usually adopted (i.e pg = exp(—G2/a)). The accuracy of such an approximation has been assessed in all situations studied so far (Tarazona 1984; Baus and Colot 1985; Baus 1987; Barrat et al. 1987,1988; DeWitt et al. 1993). With respect to the choice of the crystal lattice, the BCC (body centered cubic) lattice has been shown to be the most stable configuration for the OCP (Stringfellow et al. 1990). Finally the type of alloy can be either ordered (like a CsCl crystal) or random, i.e. each species occupies a node of the lattice with a probability equals to xu so that :

XvM*)3'2 £ex P (-a,,(f- Rtf)

(5)

Although an ordered crystal is more stable at T = 0, a random alloy is favored at finite temperature because of the entropic contribution (Segretain and Chabrier 1993; Ogata et al. 1993). iii) The third and last approximation concerns the calculation of the correlation functions (Cg). Two N-body theories are used. The first one is the so-called mean spherical approximation (MSA) (Hansen and MacDonald 1976) which reads (for one component) : f g(r) = 0,

r F\m, which leads to an eutectic diagram. The calculations of Barrat et al. (1988) (fig. 1.2) are based on the DFT formalism, and use the MSA theory to calculate the correlation functions between the particles, but with the same diameter a for the two ionic species. Moreover, Barrat et al. used the old value F c = 168 for the cristallization of

132


Fig. 6.2 Phase diagram for C/O mixture. Full line : Segretain and Chabrier; dotted line : Barrat et al.; dashed line : Ichimaru et al. . Note that for the diagram of Barrat et al. Tc = 168 instead of Tc = 178 for the other two. the pure phase. The diagram is found to be of a spindle form (dotted line). More recently, Segretain and Chabrier (1993) extended these calculations by considering different diameters for the two ionic species and by using Tc = 180 for the crystallization value for the OCP. The diagram (full line) confirms essentially the results of Barrat et al. The calculations of Ichimaru et al. (fig. 1.2) are also based on the DFT, but the correlation functions are calculated within the framework of the IHNC. They find an azeotropic diagram with an azeotropic point characterized by xc = 0.16, with XQ the concentration of carbon. As long as they are used for astrophysical applications, it is interesting to note that these diagrams are very similar, and lead essentially to the same results. 6.1.4-2 Arbitrary ionic mixtures Recently, Segretain and Chabrier (1993) (fig. 1.2 and 1.3) extended the former calculations to examine the evolution of the crystallization diagram of arbitrary ionic mixtures, as a function of the charge ratio. The calculations were performed within the framework of the DFT, and the correlation functions were calculated within the MSA theory, with different diameters for the two ionic species. The width of the gaussian, a,,, in the ion distribution function of the solid (eq. (5)), was found to agree perfectly (< 0.5%),


133

with the value obtained by molecular dynamics simulations, for the OCP (Slattery and DeWitt 1993). They find that the phase diagram evolves from a spindle form for 0.72 < C = Zi/%2 < 10, to an azeotropic form for 0.58 < C < 0-72 and to an eutectic form for ( < 0.58. This bears important consequences for the cooling of white dwarfs, as will be detailed below. Some recent calculations, by Ogata et al. (1993) use MC simulations to calculate the free energies of the solid and the liquid ionic mixture. These authors claim a negative departure from the linear mixing rule (LMR) in the liquid phase for the mixture at small concentration of high charges. This negative departure in the free energy of the liquid of course favors the liquid phase, so that, when computing the phase diagrams, they obtain an azeotropic form for 0.71 < C < 1.0 and an eutectic form for ( < 0.58. For 0.58 < C < 0.71 they get the superposition of an eutectic and an azeotropic diagram. These calculations, however, are very dubious for two reasons. First, as we already pointed out in § 2, for the statistical error to be negligible in MC simulations, the number of particles must be large enough. In Ogata et al. calculations, the departure from the LMR is found to occur for X2 < 10%, i.e. JV2 < 10, where %i and N2 denote respectively the concentration and the number of particles of species 2. Second, recent similar MC simulations, involving a larger number of particles, by Slattery and DeWitt (1993), confirm a positive, though small, departure from the LMR in the liquid phase. Since the results by Ogata et al. rely entirely on their claim of a negative departure, this has to be confirmed with no ambiguity for their results to be trust.

6.2 Astrophysical consequences 6.2.1 Historical overview We now discuss the consequences on the cooling of white dwarfs of the phase diagrams computed above. We shall naturally focus on the delay introduced by the crystallization process which directly affects any estimate of the age of the galactic disk obtained from the white dwarf luminosity function. Before describing the results of detailed calculations we present an historical and very simple overview of the problem, starting from the original Mestel theory (1952). We first write down the energy equation of the cooling white dwarf L + Lu-

Lnuc = - / CvTdm T — Jo Jo \oT) = ~Eth — Egrav

p-^dm p

(11)

134


i—i

1—|—i—i—i

Z./Z2=0.75

Fig. 6.3 Phase diagram for an arbitrary mixture. (Segretain and Chabrier)

where L, Lv and Lnuc are respectively the photon and neutrino luminosities and the nuclear energy release in the envelope; Eth and Egrav are the thermal and gravitational energy sources of the white dwarf (all other symbols have their usual meanings). Mestel theory follows from the energy equation after a series of approximations, some of them being excellent and others more questionable: (i) The interior temperature is assumed to be uniform due to the high thermal conductivity of degenerate electrons. (ii) Only the photon luminosity is included, the radiation transfer through the envelope being computed with a Kramers opacity. (iii) The thermal energy source is dominant in the white dwarf interior. (iv) The heat capacity of the dense plasma inside the white dwarf is


135

approximated by the ideal gas contribution of the ions. There is no phase transition at any stage of the cooling. All these simplifications being made eq. (11) reduces to 3 M , dT k

where fi and ma are the mean atomic weight and the atomic mass unit. Together with L oc T3S from (i) eq. (12) can be integrated to give L oc t~7^ and the luminosity function oc dt/dLog L oc L~hf7. Since Mestel's pioneering work the theory of cooling white dwarfs has received a continuous interest and has been improved in many different ways. While approximation (i) has proved to be valid after a few 105 years of cooling, major changes have been incorporated to (ii), (iii) and (iv). (ii) Nuclear burning at the end of the planetary nebula phase still affect the first ~ 104 years of white dwarf evolution (Iben and Tutukov, 1984). For approximately 107 years, neutrino losses dominates the cooling until the photon luminosity takes over. Naturally, the luminosity is now obtained from detailed atmosphere models instead of the simple result deduced from the Kramers law (Fontaine and Van Horn, 1976; D'Antona and Mazzitelli, 1979; Wood, 1992). However, in spite of the progress which have been made, the determination of an accurate relationship between the internal temperature and the luminosity remains one of the major challenge in white dwarf cooling theory. The source of the difficulties comes from uncertainties in the opacity which is still poorly known in some regions of the envelope where the material is partially degenerate (see D'Antona and Mazzitelli, 1990). (iii) The assumption that the gravitational energy source is negligible was first discussed by Mestel and Ruderman (1967). The relative importance of the different terms entering the cooling equation was then evaluated from the virial theorem in a very elegant and simple way by Lamb and Van Horn (1975). In a pure carbon white dwarf, where the separation of elements at crystallization discussed below does not occur, it can be shown that (13) where ep is the typical Fermi energy of the electrons in the white dwarf interior. (iv) The ions certainly do not behave like a perfect gas as can be seen from the large value of the plasma coupling constant T. The thermodynamics of the ions is controlled by Coulomb effects and is now known with

136


a high degree of accuracy thanks to the work of a number of people since twenty years (see Baus and Hansen, 1980 and Ichimaru et al., 1987 for good reviews). The most spectacular consequence of the Coulomb effects is probably the existence of the fluid-solid transition at F c ~ 180. The related release of latent heat (of the order offcjgTper particle) introduces a delay in the cooling of white dwarfs which was first discussed by Van Horn (1968). At low temperature, when fcgT < hup {up being the plasma frequency), quantum effects become important and the heat capacity of the ions decreases according to the Debye law, leading to an acceleration of cooling in very faint white dwarfs, known as "Debye cooling". Since white dwarfs are made of carbon and oxygen (or of oxygen, neon and magnesium for the most massive ones) plus several other minor species such as 22Ne and Fe, it was clear that calculations ignoring a possible separation of elements at crystallization were uncomplete. The problem was considered by Stevenson (1980) who proposed an eutectic phase diagram for the C/O mixture. The implied total separation of carbon and oxygen had a drastic effect on the time needed to reach the cut-off of the luminosity function which was increased by about 5 109 years (Mochkovitch, 1983). The Stevenson phase diagram was the result of a very simple analytical model for the free energies in both liquid and solid phases. Much more reliable results have been obtained recently using the density functional approach described above. Instead of an eutectic all the new phase diagrams for carbon and oxygen have a spindle (or azeotropic) shape. The delays introduced in the cooling of white dwarfs are less spectacular but remain important.

6.2.2 The crystallization of carbon-oxygen white dwarfs The solid being more oxygen rich than the liquid, it is also slightly denser to maintain the continuity of total pressure at the phase transition. An estimate of the density change between the solid and liquid phases is given by ^

^

^

(14)

where P{ and Pe are respectively the ionic and electronic pressures, 7 is the adiabatic index for the electron gas and Ye is the number of electrons per nucleon. Inside a 0.6 M© white dwarf ^ « 10~4. The solid therefore settles down at the center while the lighter liquid left behind is redistributed with great efficiency by convective transport (Mochkovitch, 1983). The result of the crystallization process is then an enrichment in oxygen in the central

137


1

1.60.8 _

0.6 \

: .

>:

0.4 -

\

:

0.2

0 0.2 0.4 0.6 0.8 1

n

........... r

'0 0.2 0.4 0.6 0.8 1

Fig. 6.4 Redistribution of carbon and oxygen in a crystallizing white dwarf; a spindle phase diagram leads to the formation of a composition gradient in a white dwarf where Xe = X0= 0.5 before crystallization. In the example shown a constant enrichment in oxygen, a = 0.3, has been adopted.

regions of the white dwarf and a depletion in the outer parts. This has been illustrated in fig. 1.4 in the case of a spindle phase diagram and for carbon and oxygen with initially equal mass fractions throughout the white dwarf. Making the rough approximation that the enrichment in oxygen is a constant a ~ 0.3 X'o = (1 + a)Xl0 ,

(15)

1

(X'' being respectively the oxygen mass fractions in solid and liquid phases) and assuming a perfect mixing outside the solid core it becomes possible to obtain the composition gradient analytically (Barrat et al., 1988). When the solid core has grown to a mass M» the remaining oxygen mass fraction in the liquid mantle is

' XI dm — MB

(16)

where Mwd is the total mass of the white dwarf. Taking the derivative of eq. (16) with respect to Ma and introducing (15) yields

dxi

_

,17)

138


which after integration gives X'o= 0.5(1 + a ) ( l - - ^ ) « .

(18)

An order of magnitude of the release of gravitational energy produced by the redistribution of elements is simply e9rav ~—gRZ

1013 erg.g-1 ,

(19)

where we have adopted ^ „ i o " 4 , g - 1.2 108 cm.s" 2 and R = 8300 km, which are typical for the gravity and radius of a 0.6 M© white dwarf. The value of egrav is comparable to the latent heat and cannot be neglected in cooling calculations. The resulting time delay Atgrav

~

Mv

*

e

*™ Z 109 yr ,

(20)

where < Z/crya > ~ 10~4 L© is the average luminosity of the white dwarf during the crystallization process. After having shown from simple estimates the importance of the redistribution of carbon and oxygen at crystallization, we summarize the results from detailed calculations such as those discussed extensively by GarciaBerro et al. in this volume. An additional complication which has been included in these calculations comes from the possibility of an initial composition gradient in the white dwarf core. Mazzitelli and D'Anton a (1986) who followed the evolution of intermediate mass stars from the main sequence to the white dwarf stage found a high oxygen concentration at the center of 0.6 M© white dwarfs (Xo ~ 0.8 from the center to M/M©=0.4 and Xo ~ 0.4 in the outer parts). Only massive white dwarfs (M~ 1 M©) show a nearly uniform composition with Xc — Xo = 0.5. Table 1 gives the time needed to reach the cut-off of the luminosity function at Log(L j LQ) « —4.5 as a function of the white dwarf mass in models which either include or neglect the redistribution of carbon and oxygen and the initial stratification. The results listed in Table 1 have been obtained with the spindle phase diagram of Segretain and Chabrier (1993). We have checked that the azeotropic phase diagram of Ichimaru et al. (1988) gives more or less the same results. This is not surprising since the crystallization of the white dwarf occurs far from the azeotropic point. It can be seen from Table 1 that the typical delay introduced by the redistribution of carbon and oxygen is 1 - 2 Gyr. It is smaller when the initial stratification is taken into account. For massive white dwarfs however there is no stratification and cases 1-3 and 2-4 are respectively identical. The delay directly affects any age determination of


139

Table 6.1. Values of the cooling time (in Gyr) at Log(L/Lq)) = —4.5 for different white dwarf masses, in the following four cases: (1) no initial stratification, no CO redistribution at crystallization; (2) no stratification, CO redistribution; (3) initial stratification, no redistribution; (4) initial stratification, CO redistribution. The delays due to CO redistribution are given for the cases 2-1 and 4-3. M/M0

0.5

(1)

8.73 9.24 9.39 9.27 8.96 8.53 8.10 8.07

(2)

10.78 11.45 11.53 11.27 10.78 10.13 9.45 9.13

A r ( 2 - 1)

2.05 2.21 2.14 2.00

(3)

8.10 8.76 9.24 9.28 8.97 8.53 8.10 8.07

(4)

9.18 9.96

10.76 11.12 10.73 10.13 9.45 9.13

A r ( 4 - 3)

1.08

1.52

0.6

1.20

0.7

0.8

1.84

0.9

1.82

1.76

1.0

1.60

1.60

1.1

1.35

1.35

1.2

1.06

1.06

the galactic disk coming from the white dwarf luminosity function. A calculation which would neglect the redistribution of carbon and oxygen would then yield an age underestimated by at least 1 Gyr. 6.2.3 The effect of minor species 6.2.3.1 -22Ne From the different minor species which can play a role in the cooling of white dwarfs 22Ne is certainly the most important. It is relatively abundant since it is produced from 14N during He burning by the chain of reactions 14 N(a,7) 18 O(a,7) 22 Ne. Its abundance is directly related to the initial concentration of CNO elements, so that one can expect a mass fraction X(22Ne) ~ 1 - 2% in population I white dwarfs. 22Ne is a neutron rich nucleus (Ve(22Ne)=10/22=0.45) which can produce a large release of gravitational energy if it accumulates at the center during crystallization. The physics of 22 Ne deposition should in principle be described by a 3-component (C-ONe) phase diagram which is unfortunately not presently available. One has to rely on N-Ne binary phase diagrams where nitrogen mimics the behavior of the CO mixture. As shown in Sect. 1.1.4, for a charge ratio less than 0.72


140

0.98 Fig. 6.5 Neon poor side of the N-Ne phase diagram. The neon concentration is smaller in the solid. The liquid gets more and more neon rich until the azeotropic point is reached.

there is an azeotropic point. For the N-Ne mixture, the neon mass fraction of the azeotrope is Xa(Ne)=0.16 which means that with X(22Ne) ~ 1 - 2%, the crystallizing white dwarf is on the neon poor side of the phase diagram (see fig. 1.5). This has many interesting consequences. The solid which freezes out from the liquid has a smaller neon concentration and is therefore lighter due to a smaller Ye (eq. (14)). It will rise and melt in lower density regions so that the liquid at the center will become more and more neon rich until the azeotropic point is reached. Crystallization at the composition of the azeotrope then takes place until the whole 22Ne has been collected in a central sphere of mass

X(Ne)

< 0.1Mwd ,

(21)

This "neon distillery" releases a large amount of gravitational energy, three times more than CO redistribution for X(Ne)=0.0l! Translated into delays on the cooling times this implies that a 0.6 M© CO white dwarf with 1% of 22 Ne will take ~ 14 Gyr to reach the cut-off of the luminosity function.


141

Does it mean that the age of the galactic disk is 14 Gyr? Probably not, for at least two independent reasons. First, the white dwarfs at the cutoff come from stars formed very early in the history of the disk and then have a low metallicity. Since the delay is proportional to the abundance of 22Ne, it follows that the faint end of the luminosity function and the estimated age of the disk are only weakly affected by neon deposition. A detailed calculation, where the luminosity function is obtained consistently with a model of galactic evolution is presented by Garcia-Berro et al. in this volume. Another more fundamental reason is simply that Ne may very well not deposit in the realistic case of a ternary mixture C-O-Ne. As mentioned above the phase diagram is not known for more than two components. Nevertheless, one can try to use the results for the CO and N-Ne cases to predict that the solid which freezes out from the ternary mixture will be oxygen rich and neon poor. The higher oxygen concentration and the depletion in neon have opposite effects on the density difference between the solid and the liquid. Since the "neon distillery" works only if the solid is lighter, this leads to a maximum allowed enrichment in oxygen for a given depletion in neon. Let us assume that in liquid phase Xj. = Xl0 = 0.495 and Xl(Ne) = 0.01. In the solid X*(Ne) = aXl(Ne) with a < 1. We have represented in fig. 1.6 the maximum oxygen mass fraction in the solid as a function of a which keeps Apti < 0. If we adopt the value a « 0.5 as indicated from the study of N-Ne mixtures, X™ax « 0.65, close to the oxygen mass fraction in the solid obtained for a CO mixture with Xj. = Xl0 = 0.5. The chance that the solid remains lighter are naturally increased if a w 0 as proposed by Ogata et al. (1993).

6.2.3.2 - Fe The possibility of iron deposition has been discussed by Xu and Van Horn (1992) and the problem appears very similar to that of neon. The phase diagram of the N-Fe mixture has an eutectic shape and the solid can then be expected to be iron free and lighter than the liquid. In principle an "iron distillery" could work and bring all the iron at the center. For an iron mass fraction of 10~3, the resulting additional delay would reach 0.8 Gyr. However, exactly as for neon, iron deposition is uncertain in a multicomponent plasma and even if it occurs, it does not affect the cooling of the old, low metallicity white dwarfs which make the faint end of the luminosity function.


142

0.9

0.8

no Ne deposition

0.6

Ne deposition 0.5,

0.25

0.5 a

0.75

Fig. 6.6 Maximum oxygen mass fraction in the solid as a function of a (the neon depletion factor) to have Ap,i < 0 and neon deposition.

6.3 Conclusion During the past twenty years our understanding of the physics of the interior of white dwarfs has been transformed. The thermodynamics of dense plasmas is now known with a high degree of accuracy and the importance of Coulomb effects on the stucture and evolution of white dwarfs has been fully recognized. For one of the most difficult problem - the shape of the phase diagram for mixtures - the density functional theory provides for the first time a reliable solution (at least for binary mixtures) and cooling sequences of white dwarfs with the best input physics for the interior can now be calculated. The partial separation of carbon and oxygen at crystallization adds 1 - 2 Gyr to the time needed to reach Log(L/LQ) = -4.5 and hence to the age of the galactic disk deduced from the white dwarf luminosity function. The main source of error in white dwarf cooling theory now probably comes from the envelope, where the opacity is still very uncertain in some regions of the p — T plane.

References Abrikosov A.A., Zh. Eksp. i Teor. Fiz. 39, 1798, (Soviet Phys. JETP 12, 1254), (1960)


143

Barrat J.L., Thesis, university of Paris VI (1987) Barrat J.L., Europhys. Lett. 3, 523, (1987) Barrat J.L., Baiis M., Hansen J.P.,, J. Phys. C 2 0 , 1413, (1987) Barrat J.L., Hansen J.P., Mochkovitch R., A&A 199, L15, (1988) Baus M., Colot J.L , Mol. Phys. 55, 653, (1985); Baus M., /. Stat. Phys., 48, 1129,(1987) Baus M., Hansen J.P., Phys. Rep. 59, 1, (1980) Brami F., Hansen J.P., Joly B., Physica A 95, 505, (1979) Brush S.G., Sahlin H.L., Teller E., J. Chem. Phys. 45, 2102, (1966) Chabrier G., Ashcroft N.W., Phys. Rev. A 42, 2284, (1990) D'Antona F., Mazzitelli I., A&A , 74, 161, (1979) D'Antona F., Mazzitelli I., Ann. Rev. Astron. Astrophys. , 28, 139, (1990) DeWitt H.E, Slaterry W.L., (1993) Private communication DeWitt H.E, Slattery W.L., Yang J., The international conference on the physics of strongly coupled plasmas, (1993), in press Dubin D.H.E., Phys. Rev. A 42, 4972, (1990) Dyson F., Ann. Phys. 63, 1, (1971) Farouki R.T., Hamaguchi S., Phys. Rev. E47, 4330, (1993) Fontaine G., Van Horn H.M., Ap.J. Suppl. , 31, 467, (1976) Hansen J.P., Phys. Rev. A 8, 3096, (1973) Hansen J.P., MacDonald I.R., Theory of simple liquids, (Academic Press), (1976,1989) Honenberg P., Kohn W., Phys. Rev. , 136, B 864, (1964) Iben I.Jr., Tutukov A., Ap.J. , 282, 615, (1984) Ichimaru S., Iyetomi H., Tanaka S., Phys. Rep. 149, 91, (1987) Ichimaru S., Iyetomi H., Ogata S., y i p / 3 3 4 , L17, (1988) Iyetomi H., Ichimaru S., Phys. Rev. B 38, 6761, (1988) Kirshnitz D.A., Zh. Eksp. i Teor. Fiz. 38, 503, (Soviet Phys. J E T P 11, 365), (1960) Lamb D.Q., Van Horn H.M., Ap.J. 200, 306, (1975) Landau L.D. , Lifshitz E.M., Statistical Physics, (London: Pergamon) (1958) Likos C.N., Aschroft N.W., Phys. Rev. Lett. 69, 316, (1992) Mestel L., M.N.R.A.S. , 112, 583, (1952) Mestel L., Ruderman M.A., M.N.R.A.S. , 136, 27, (1967) Mochkovitch R., A&A , 122 , 212, (1983) Mazzitelli I., D'Antona F., Ap.J. , 308, 706, (1986) Ogata S., Ichimaru S., Phys. Rev. A 36, 5451, (1987) Ogata S., Iyetomi H., Ichimaru S., Van Horn H.M. Phys. Rev. E 48, 1344, (1993) Parrinello M., Tosi M.P., Chem. Phys. Lett. 64, 579, (1979) Ramakrisnan T.V., Yussouf M., Phys. Rev. B 19, 2775, (1979) Rovere M., Tosi M.J.Phys. C, 18, 3345, (1985) Salpeter E.E., Ap.J. 134, 669, (1961) Segretain L., Chabrier G., A&A 271, L13, (1993) Singh Y., Phys. Rep. 207, 351, (1991) Stevenson D.J., J. Physique 41, C2-61, (1980) Stringfellow G.S., DeWitt H.E., Slaterry W.L., Phys. Rev. A 4 1 , 1105, (1990) Tarazona P., Mol. Phys. 52, 81, (1984) Van Horn H.M., Ap.J., 151, 227, (1968) Wood M., Ap.J. , 386, 539, (1992) Xu , Van Horn H.M., Ap.J. , 387, 662, (1992)

Non crystallized regions of White Dwarfs. Thermodynamics. Opacity. Turbulent convection. I. MAZZITELLI Istituto di Astrofisica Spaziak. Consiglio Nazionalc dtlk Ricerche, C.P. 67, 00044 Frascati. Italy

Abstract The evolution of White Dwarf stars along their cooling sequences is governed not only by their thermal content, but also by the rate at which heat flows through the external, partially degenerate and non-isothermal layers. In particular, cooling is found to be largely influenced both by the optical atmosphere, and by the convective envelope. The first one, in fact, determines the internal density stratification, down to the point at which electron degeneracy takes over, while the second one affects the temperature stratification in the same layers. The reliability of the present generation of models of White Dwarf envelopes is discussed, on the grounds of the main physical inputs (thermodynamics, opacity, convection theory), for both H-rich and He-rich surface chemical compositions. The conclusion is that, below LogL/LQ < —3, we can build little more than test models. L'evolution des naines blanches le long de leur sequence de refroidissement est gouvernee non seulement pas leur contenu thermique, mais aussi par la vitesse a laquelle la chaleur s'echappe a travers les couches externes, nonisothermes et partiellement degenerees. En particulier, le refroidissement est largement influence a la fois par l'atmosphere optique et par l'enveloppe convective. La premiere determine la stratification interne en densite jusqu'a ce que la degenerescence electronique prenne le dessus, alors que la seconde affecte la stratification en temperature dans les memes couches. Nous discutons la validite de la generation actuelle de modeles d'enveloppes de 144

Mazzitelli: Non crystallized regions of White dwarfs

145

naines blanches, sur la base des ingredients physiques (thermodynamique, opacite, theorie de la convection), pour a la fois des compositions chimiques de surface riches en hydrogene et en helium, respectivement. La conclusion est que, en dessous de LogL/L@ < —3, il n'est guere possible de proposer autre chose que des modeles test.

7.1 Introduction Observations seem to definitely show that the luminosity function of the nearby White Dwarfs (WDs) display an abrupt break-down in the luminosity range LogL/LQ= -4.0 4- -4.5 (Liebert et al. 1989). The obvious interpretation of this feature is the finiteness of the age of the galactic disk —in the vicinity of the Sun— such that even the first born among the WDs are not yet old enough to have cooled below this luminosity (D'Antona and Mazzitelli 1978). The alternative explanation of WDs being already in the fast Debye cooling phase (D'Antona and Mazzitelli 1989), should in fact lead to a milder decrease of the luminosity function, than apparently suggested by the observations. In this framework, it should be legitimate to assume WDs as powerful probes for evaluating the age of the galactic disk (Winget et al. 1987). However, if this is the goal, we have first to make sure about the reliability of the present generation of stellar models, also because the external layers of the WDs —which determine the cooling rate— are at such (relatively) high densities and (relatively) low temperatures, that matter is definitely far from ideal gas conditions. Also, in old (and cold) WDs, the largest fraction of the temperature difference between surface and centre is found in the convective region. Although the convective gradient is in any case very close to the adiabatic gradient —due to the large density— the Mixing Length Theory (MLT), tuned on the sun, is hardly expected to be the more realistic model for dealing with turbulent convection in dense, partially ionized and partially degenerate conditions. In the following, WDs models of different luminosities and surface chemical compositions will be discussed showing that, especially in the case of He-rich stars, the p-T region where our present physical understanding of the thermodynamics and of the heat transport properties of a partially ionized real gas is still rather poor, is met relatively soon during the evolution.

146


7.2 The main ingredients There is by now wide agreement about some general features of WDs. The masses of the most of WDs, for instance, seem to lie in the range 0.5<M -3. Below this limit, we can certainly build test models but, basing upon these last, we can scarcely draw sound quantitative conclusions about galactic evolution.


150 1

1

•

.

I

1

I

i

-

glycerine 0 -

H-rich WD

olive oil

/

—

o - 2 --

water -

/

Log L/L o = - 2.5

' - 4 --

air

/ L L

-16

.

-14

.

.

i

-12

.

.

.

i

.

-10

•

.

i

-8

. . .

-6

Log M/Mtot Fig. 7.3 The dynamical viscosity rj as a function of the depth, in the subatmospheric region of a H-rich WD at LogL/Lo= -2.5. The abscissa shows the Log of the fractional mass starting from the surface.

7.3.3 Turbulent convection Up today, the MLT has been widely applied to the modeling of WDs convective envelopes since, in any case, the rate of overadiabaticity in the dense external layers of these objects is expected to be negligible or quite so. This seems in fact to be the case as long as the thermal properties of WDs are concerned, even if other properties of WDs, as the pulsational ones, are instead much more sensitive to the treatment of convection (Pelletier et al. 1986). However, the problem is perhaps not so settled, as will be shown below. The MLT is usually tuned on the sun, and one can legitimately ask which can be its predictive power, when applied to conditions in which the density is up to six or seven orders of magnitude larger than in the solar subatmosphere. Let us examine this point more in detail.

151

Mazzitelli: Non crystallized regions of White dwarfs I

1

I

i

i

T

H-rich WD

-2

SO

o -4

-6 -16

•14

-12

-10

-8

Log M/Mtot Fig. 7.4 The behavior of the Prandtl number as a function of depth, as in Fig. 1.3, at various luminosities, for a H-rich WD. For comparison, remember that the Prandtl number in the solar subatmosphere is a ~ 10~9. Much larger densities, could imply for instance larger viscosities than in the case of the almost inviscid sun. The coefficient of dynamical ion viscosity r) can be easily computed for a coulombic plasma, according to Wallenborn and Baus (1978). The results for the case of the subatmosphere of a H-rich WD, are shown in Fig. 1.3. Viscosity can be quite large, so large as to equal that of very viscous terrestrial fluids. Actually, the knowledge of viscosity alone is not sufficient, since also for a relatively large value of 77, matter in a star can still have room enough to generate a wide spectrum of viscous eddies, and full developed turbulence. The Prandtl number a = v/x, where v is the kinematical viscosity coefficient and x the thermometric conductivity, is in this respect, much more significant, since it is directly connected to the amplitude of the eddies spectrum.

152


As can be seen from Fig. 1.4, during the cooling of the WD, the value of a in the convective region steadily increases, reaching ~1 below LogLjZ©= -4. This leads to a consistent shrinking of the turbulent eddies spectrum with respect to an inviscid case, and the efficiency of the convective heat transfer decreases. In particular, from Canuto and Mazzitelli (1991) one can see that the integral of the energy spectrum of the turbulent eddies versus the vawenumber does not change significantly when a decreases below 10—3, but this is not the case for larger values of being LQ the white dwarf luminosity at birth (assuming that all white dwarfs were born with the same luminosity), 1/1(1, m,Z) is ls proportional to the characteristic cooling time. Bwd(totm^) * n e so ~ called source function for white dwarfs at time to, denned as the total num-

Garcia-Berro & Hernanz: White dwarf crystallization 11

o

-3 log(VU)

Fig. 8.7 Characteristic cooling times vs luminosity, for a 0.6 M© C/O white dwarf with XnNe - 0% (solid line), 1% and 3% respectively. ber of white dwarfs with mass m and metallicity Z born at time to, per mass, metallicity and time units. Thus

Bwd(tQ,m,Z)

=

tPN

- Z{tb))

Here M is the mass of the progenitor of a white dwarf of mass m, (M) the initial mass function, generally assumed to be constant in time, TJ){t) the star formation rate, expressed as mass of stars formed per unit volume (pc 3 ), to is the birth time of the white dwarf (with a luminosity LQ which we will consider to be large compared with the actual luminosity, LQ > L), and tf, is the birth time of the progenitor star, obtained from: h + tm,(M) + tcool(L, M, Z(tb)) = tiiak

(8)

where tdiak is the age of the disk and tms is the time spent on the main sequence. Note that by adopting this expression for the source function we are implicitly assuming that the metallicity of white dwarfs is that of the interstellar medium when their progenitors were born. The integration of (7) gives the white dwarf luminosity function n(L), i.e.

Garcia-Berro & Hernanz: White dwarf crystallization

174 -3

-3.5

-4

no Carbon—Oxygen separation Carbon-Oxygen separation Carbon-Oxygen separation + MD no Carbon-Oxygen separation + MD

-4.5

-5,

9.5

log(t)

10

Fig. 8.8 Effect of initial stratification on the cooling time of a C/O 0.6 MQ white dwarf. Full line: no separation, no stratification; dotted line: separation, no stratification; dotted-dashed line: no separation, stratification; dashed line: separation, stratification. MD stands for Mazzitelli and D'Antona (1986).

the number of white dwarfs per unit of bolometric magnitude, per cubic parsec, with luminosity L: rM,

n(L) = / JM

tip

TC00,(L,

M,

, M, Z(tb))-

tma(M))(M)dM where M9up and M, n / denote respectively the maximum and the minimum mass of the white dwarf progenitors which contribute at luminosity L. M, n / is obtained by setting tt, = 0 in expression (8):

teooi(L, Minf, ZQ) =

(10)

where Z o is the initial (t = 0) metallicity of the interstellar medium, usually taken as 0. If we focus on pure C/O white dwarfs, then eqn. (9) reads


rM,Up

n(L) = / r c o o / ( i , M)il>(tdisk - tcool(L, M) JMini(L)

175

tma(M))

(11)

4>{M)dM where Af,n/ satisfies now: tma{Minj) + tcooi{L, Af,n/) = Conversely, if we focus on the influence of minor chemical species on the white dwarf luminosity function, the cooling rates depend on the mass of the white dwarf, its luminosity and its metal content. For a sake of simplicity, the dependence of cooling times and characteristic cooling times on the mass of the white dwarf will be ignored and calculations will be performed for a typical M* = 0.6 M© white dwarf only. This is justified by the weak mass-dependence of tcoo/ and rcoo/ when compared with their strong metallicity-dependence. In this case, the luminosity function reads:

n(L)= / Tcoo^M^ZWtdisktcooiiLtM^Z) JMinf(f(L) tms(M))<j>(M)dM

(12)

where Minf is now given by: tdiak = tcooi(L, Af*,Z0) + tms(Minf). A close look at equation (12) reveals that, in fact, the effect of different progenitor masses is taken into account correctly. As can be seen from the previous expressions, the computation of the white dwarf luminosity function requires not only accurate cooling sequences, but also a model of galactic evolution. A good standard model for our purposes is that of Clayton (1984), which takes into account the infall of matter over the galaxy for a certain amount of time. The reason of this choice is basically its simplicity. Of course, there are more elaborate models of galactic evolution (see for instance Abia et al, 1991, and Bravo et al, 1993) but for our purposes - that is, showing the importance of a proper treatment of crystallization - this SFR is enough. Concerning the initial mass function, IMF, the one proposed by Salpeter (1961) has been adopted. We have computed luminosity functions for the cooling sequences described in the previous section. Figures 9 and 10 show the luminosity function (11) for our reference model and case (iii), i.e. no separation at crystallization without and with initial stratification respectively, computed for two different ages of the galactic disk in each case, namely 8.8 and 105 Gyr and 8.5 and 10.3 Gyr, respectively. These values best reproduce the position of the observed cut-off, obtained with two different approximations for

176


O

Fig. 8.9 Theoretical luminosity functions for C/O white dwarfs with no fractionation (reference model), for ages of the disk t Po)(Vi < VQ) or (Pi < Po)(Vi > VQ). The first solution corresponds to a detonation and the second one to a deflagration. From the equation of conservation of energy it is possible to write eo + q - «i + \{PQ + Pi)(Vb - Vi) = 0

(9.13)

which is called the detonation adiabat (the case q=0 is called the shock adiabat). This equation, together with that defining the mass flux determines the final state once the characteristics of the burning front have been specified. The physical meaning of the intersection between the adiabat and the mass flux is clear. A shock heats and compresses the material to a state (P 3 , Vs) given by the intersection of the shock adiabat with the line defined by j 2 . Because of the increase in temperature, material burns and reaches the state (Pi,Vi) defined by the detonation adiabat and the jf2-line intersection. Since q > 0, Pi < Ps and V\ > Vs which implies that a rarefaction is associated to the postchock burning. The family of solutions obtained in this way, with j as a free parameter,

Isern & Canal: Degenerate stellar cores

191

has an extremum for which j and the front velocity are minima. This solution, called Chapman-Jouguet detonation, corresponds to the case where the j-line is tangent to the detonation adiabat. It has the important properties of being univoquely determined by the thermodynamic properties of the material (including q) and having a propagation velocity that is equal to the sound velocity of the burned material. All the remaining solutions, called strong detonations, move subsonically with respect to the burned material. In stars, due to the spherical simmetry, material must be at rest at the centre. Therefore, the velocity has to decrease from some positive value behind the front to zero at the centre. This means that a rarefaction wave, moving at the sound velocity, must follow the detonation. Since strong detonations are subsonic respect the burned material, they are overtaken by the rarefaction wave and only the Chapman-Jouguet one can survive. Although it has not been completely elucidated, it is generally accepted that due to the high densities existing in the central regions of CO white dwarfs, the overpressures induced by the burning front cannot give rise to a detonation (Mazurek, Meier and Wheeler 1977). However, as the subsonic flame propagates into regions of progressively decreasing density, it accelerates and it can eventually become a detonation (Blinnikov and Khokhlov 1986; Woosley 1986). It is interesting to notice here that a detonation does not necessarily produces the complete incineration of the material into iron. If the density is smaller than ~ 107 g/cm3, nuclear statistical equilibrium cannot be reached and, as a consequence, elements of masses intermediate between C-0 and Fe are produced. If the density is smaller than 106 g/cm3 even the fuel, the C-0 mixture, has no time to be exhausted (Khokhlov 1989). In the case that the fuel is He this is not true and Fe is synthesized. c) Deflagration. Deflagrations are the solution to the conservation laws that fulfill the condition (Pi < P2) {Vx > V2). The main dificulty with these solutions arises from the fact that matter is subsonic at both sides of the front and boundary conditions behind it can affect the front as well as matter ahead. In the case of a star with a subsonic burning front propagating outwards with a velocity D, the condition of matter being at rest at the center demands the existence of a shock precursor that boosts matter outwards (Mazurek and Wheeler 1980) and causes the expansion of the star. There are two modes whereby a nuclear deflagration can propagate inside a degenerate core: The laminar mode and the turbulent mode. In the laminar case (often known as conductive front), electrons transport the energy released in the burning regions to the surroundings, inducing their ignition. The velocity can be estimated in the following way (Landau and Lifchitz 1959): The width of the front is given by 6 ~ y/xr, where x is

192


the thermometric conductivity and r is the time that the burning lasts. The velocity of the burning front is thus given by D ~ 6T = y/xjr. This velocity is always of the order or smaller than 10~2Cg, where ca is the sound velocity. Useful approximations are provided by (Timmes and Woosley 1992):

valid in the range 0.01 < />g < 10 in the case of a C-0 mixture and:

valid in the range 1 < pg < 14 in the case of an ONeMg mixture. In the turbulent case, a hot and less dense layer is formed below a cold dense layer. Because of gravity, the interface is Rayleigh-Taylor unstable and both layers are mixed, that noticeably increasing the propagation velocity due to the increase in the efficiency of the conductive transfer. Although at present there is not a satisfactory theory of turbulent flames in stellar interiors, it is possible to make an estimate of the velocity of a turbulent flame in the central regions of a star. Nomoto et al (1984) proposed from mixing length arguments, a turbulent velocity vt c± \fgeffl/2 where geff = GMr2/Sp/p is the effective acceleration, 6p is the difference of densities between both sides of the front and / is the mixing length taken as / = min(r,a.flp) where a is an adjustable parameter of the order of unity. The use of the mixing length theory has been questioned not only because of the jump in density across the front but also because all other characteristicsof the burning strongly violate the basic hypothesis of the theory itself. Woosley(1990) proposed a fractal description of the burning front to take into account its wrinkling and Livne and Arnett (1993) proposed to treat the turbulent deflagration in terms of an ablative front in order to correctly handle the growth of the different unstable modes. At present, the question is completely open and in fact the mixing- length model, despite being physically incorrect, provides for a = 0.7 the best agreement with observations. In practice, numerical treatments like those of Sutherland and Wheeler (1984) and Unno (1967) are used. It must be also taken into account that electron captures behind the burning front can completely inhibit the development of the Rayleigh-Taylor instability (Timmes and Woosley 1992).


193

9.3 The collapse of degenerate cores

The evolution of the primary star in a close binary system can give rise to either a helium white dwarf, a carbon-oxygen white dwarf, or an oxygenneon-magnesium white dwarf, depending on the initial parameters of the system. However, not all compositions can be involved in the accretion induced collapse. Helium white dwarfs can immediately be discarded. Their mass growth would lead to explosive helium ignition at the center of the star for a central density pc < 4 108 g/cm3, no matter how low is the temperature (Sugimoto and Nomoto 1980). After incineration, electron captures would be too slow, the overpressures would be large enough to start a detonation and the star would thus be completely disrupted (Woosley and Weaver 1986). This leaves only carbon-oxygen and oxygen-neon-magnesium white dwarfs as possible candidates for a core collapse. These groups have to be considered separately as accretion induced collapse poses different problems for each of them. Both cooling in the stage between the formation of the white dwarf and onset of mass accretion and reheating by mass accretion are specially relevant in the CO case, whereas semiconvection associated with electron captures plays an important role in the ONeMg case. As it was stated in the Introduction, the collapse/ explosive behavior alternative for a degenerate core depends on the density at which the burning front starts, for a fixed velocity of the flame. Since there is not yet a theory giving the velocity of the burning front, it is not possible to decide which is the value of the density beyond which the collapse ensues. It is possible, however, to establish two firm bounds and a guess as to this critical value. The upper limit is determined by the maximum velocity of the flame: the sound velocity. In this case, the minimum density necessary to guarantee the collapse is p « 2 1010 g/cm3. The lower limit is determined by the minimum velocity of the flame: the conductive velocity. In this case, the minimum density necessary to get a collapse is 8.5 109 g/cm3. In a previous calculation by Canal et al (1992), this limit was set to 9.5 109 g/cm3 because the Coulomb corrections to the equation of state of the ions were not correctly included. The guess can be obtained by using everal recepies appeared in the literature and making some kind of average. For densities higher than 9 109 g/cm3 some of them predict a collapse, and for 9.5 109 g/cm3 all of them do.

194


9.3.1 The CO case Carbon-oxygen white dwarfs can form in binary systems either by Roche lobe overflow just before or just after ignition of He in initially close binaries or by Roche lobe overflow during the early or the thermally pulsing asymptotic giant branch phases in initially wide binaries. In the latter case, common envelope evolution should allow enough orbital angular momentum to be lost so that the wide binary evolves into a close binary. An important question is the upper mass limit for CO white dwarfs formed in this way. Observations of classical novae give average masses of 1.23 M©, although this figure is not truly representative of the average since several selection effects favor the detection of massive white dwarfs, and models for the recurrent nova U Sco give a mass M~ 1.38 M© (Starrfield et al 1989). However, the most massive white dwarfs found in these systems may well be ONeMg and not CO white dwarfs. Theoretically, CO cores of single stars should ignite C non explosively when M core > 1.1 — 1.2M© (Iben and Tutukov 1986). The behavior of the central layers of a growing CO core is determined not only by the local balance among the nuclear energy released, the neutrino losses and the compressional work, but also by the properties of the outer layers. If such CO core is part of an isolated white dwarf star, it will cool down because of the photospheric losses. If this core is growing, as is the case of an accreting white dwarf in a binary system, it is heated not only by the H and He burning shells but also by the compression of the outer, partially degenerate layers. An important ingredient ofthe problem of determining the density at which central ignition happens comes from the fact that nonperturbed white dwarfs cool down and eventually solidify. For instance, a CO white dwarf with M > 1M© starts to solidify after 1 Gyr since its formation. That ~ 0.5, that implies that the adiabatic coefficient is small (d\nT/d\np)s the heating by compression is gentle (Hernanz et al 1988) and that, in the absence of the influence of the outer layers, the ignition of the carbon-oxygen mixture is entirely controlled by pycnonuclear reactions (Canal and Isern 1979). In that case, the density at which the runaway starts is pc ~ 10io g/cm3 instead of 4 109 g/cm3 typical of the fluid phase. The energy released by the compression of the outer layers is, by far, the most important heating mechanism. It can roughly be approximated by (Nomoto 1982): Lg/LQ = 1.4 10- 3 (T/10 7 /O(M/10- 10 M©/yr)

(9.16)

195


9.00 B

C

A

8.50

8.00

— 7.50

7.00

6.5

I

.50

I

I

I

I

8.50

1 I

I

1

I

I

I

I

I

9.50

10.50

log(RHO) Fig. 9.1 Path followed by the center of a mass-accreting white dwarf with an initial mass of 1.15 M, T o = 4 106 K and z c = x 0 = 0.5. Cases A, B and C correspond tho M = 10~ 6 ,5 10~ 8 and 1O~10 MO/yr respectively. The dashed line is the ignition line

If the accreted rate is smaller than 3 10" 10 M 0 /yr, energy losses through the photosphere are dominant and the star cools down. If the accretion rate is higher, compressional heating is dominant and a thermal wave propagating inwards is generated. The consequence is that the inner layers are heated up, they cross the ignition line, denned by the condition that neutrino losses exactly balance the energy released by the carbon burning, in the (/), T) plane and a thermonuclear runaway starts. Nevertheless, if the initial mass of the white dwarf is M > 1.2MQ and the accretion rate is M > 5 10~8M©/3/r, the thermal wave coming from the surface has no time to reach the center and the thermonuclear runaway is entirely determined by the local properties of matter (Hernanz et al 1988). Figure 1 displays the evolution of the center of an accreting carbon-oxygen white dwarf. The three different behaviors already mentioned are clearly displayed: In case A, the thermal wave has no time to reach the central

196


-10

-9

-6

log(M)

Fig. 9.2 Ignition density as a function of the mass accretion rate in the case of a white dwarf with the same characteristics as in Figure 1. The dashed line represents the minimum density necessary to have a gravitational collapse instead of a thermonuclear explosion

regions and the center follows a trajectory of slope 0.5 in the temperaturedensity plane. In case B, a strong thermal wave coming from the surface heats the material and finally induces its thermonuclear runaway. In case C, heating and surface cooling exactly balance each other after a transient phase and the white dwarf evolves isothermally. Figure 2 displays the ignition density as a function of the mass accretion rate for a 1.15 M© white dwarf. Only those accreting either at a very small rate or at a high rate have a chance to collapse. Therefore, if we take into account that some of them can be initially hotter, it can be concluded that the majority of those white white dwarfs will explode and only a small fration will collapse. Notice also that if more massive white dwarfs were considered, the ignition density would increase and the collapse would be favored.


197

9.3.2 The ONeMg case White dwarfs made of ONeMg would result, in close binary systems from loss of the helium layer when it expands to read giant size during C shell burning (Iben and Tutukov 1984; Nomoto 1984) and they are not expected to form from single-star evolution (Habets 1985). In contrast, their presence in close binary systems might be indicated by the observation of Ne-novae (Starrfield 1990; Truran and Livio 1986). They should be, on average, more massive and less frequent, by a factor ~ 104, than CO white dwarfs (Iben and Tutukov 1984). When the ONeMg white dwarfs are compressed, the Fermi energy increases and nuclei undergo electron captures. The behavior of the temperature depends on the relationship: ^<X(EF

+ ET-EV-

Etht0)

(9.17)

where EF is the Fermi energy of electrons, Ey is the energy of the excited states produced during the capture process, Eu is the energy of the neutrino emiited and Eth,o is the threshold energy for the electron capture to the ground state (Miyaji et al 1980). If the Fermi energy is high enough, the derivative is positive and the temperature increases in such a way that electron captures can trigger by themselves the thermonuclear runaway. The first element starting to capture electrons is 24Mg (p ~ 4 109 g/cm 3 ), followed by 20Ne (p ~ 9.1109 g/cm 3 ) and 16O (p ~ 1.9 109 g/cm 3 ). Since electron captures produce an excess of entropy that tends to induce convection and, at the same time, a chemical gradient that tends to suppress it, the density at which the runaway actually starts is extremely sensitive to the way in which convection is handled (Mochkovitch 1984). If the Schwarzschild criterion is applied, which does not take into account the presence of chemical gradients (| V |>| Vad | where V is the actual gradient and Vad is the adiabatic gradient), the entropy generated by electron captures induces the formation of a convective zone that transports very efficiently the excess of entropy. As the captures proceed, the star gradually contracts until a density of 2 1010 g/cm3 is reached. When this happens, electron captures on 0 trigger the ignition of this element and matter is completely incinerated to 56Ni. Electron captures on 56Ni are so fast that, independently of the speed of the burning front, the white dwarf collapses to a neutron star. The Ledoux criterion takes into account the existence of chemical inhomogeneities. According to this criterion, the condition for the onset of convection becomes | V |>| VL, |, where V^, is equal to the adiabatic gradient plus a stabilizing term that depends on the gradient of the chemical

198


composition (see, for instance, Cox and Giuli 1968). Consequently, convection is inhibited and strong local heating is produced. 24Mg is exhausted and the temperature drops due to thermal neutrino emission before the temperature for explosive ignition is reached, but the laster happens at the onset of electron captures on 20Ne. In this case, the ignition takes place at 9.2 109 g/cm 3 when the Takahara et al (1989) electron capture rates are adopted and the influence of the chemical potential on the electron threshold is properly taken into account. 9.3.3 The nonexplosive collapse of white dwarfs Once nuclear reactions start at the center, the burning propagates through all the star. The one-dimensional calculations made up to now assume that the flame propagates with a velocity determined by the fastest mode of burning: either spontaneous, conductive or turbulent. The detonation mode is not considered since the central density, pc > 8 109 g/cm3, of the models considered here is always very high. Except for minor differences, the behavior of the CO and ONeMg cores is always the same: burning propagates outwards, the electron captures reduce the mean electron mole number and induce the contraction of the star, the Chandrasekhar limit falls below the actual mass of the star, and finally the star collapses homologously. It should be stressed here that in the case of CO white dwarfs the ignition happens in the interior of a solid. The strength of this solid is enough (Hernanz et al 1988) to prevent or at least strongly delay the developpement of convection. However, the neutrinos emitted by the electron captures on the burned material deposit enough energy to melt the crystal (the latent heat per nucleon is / ~ kTm, where Tm is the melting temperature). The energy deposited by neutrinos in their interaction with relativistic electrons is given by (Gehrstein et al 1976; Chechetkin et al 1980):

where the units are erg/g/s, L51 is the neutrino luminosity in units of 1051 erg/s and 17 is the radius in units of 107 cm, x = E^/Ep, E,, being the energy of neutrinos and Ep the Fermi energy in MeV. To quantify this situation, Isern et al (1990) considered a burning front placed at 107 cm from the center and assumed that L51 = 0.1, Ep = 10 MeV and E,, = 9 MeV. They found that the crystal melted in less than 1 second, a quantity that is of the order of the time necessary to develop the Rayleigh-Taylor instability.


199

The ejection of matter due to neutrino deposition following the collapse can be treated as a neutrino-driven wind (Hernanz et al 1993). A general description of these winds can be found in Duncan, Shapiro and Wasserman (1986) and in its relativistic form in Paczynski (1990). The equations that define the wind in its stationary form are: =M

HYM + LQO + LV = E

v2/c2)Y2

Y = yJ{\-TalT)lyl(\-v*l#)

(9.19)

(9.20)

(9.22)

(9.23)

with, T9 = 2GM/c2, H = c2 + (P + U)/p0, where Eo and Mo are the respective rates at which energy and mass are injected at the base of the wind, H is the enthalpy, po is the rest mass density, Loo is the photon luminosity measured by an observer at infinity, L is the photon luminosity in the comoving frame, Lj, is the neutrino luminosity and k is the opacity. The equation of state is that of a gas composed by radiation, nuclei and e~e+ pairs, and the energy deposited by neutrinos takes into account the captures by protons and neutrons, the scattering by electrons and the creation of e~e+. Figure 3 displays an example that is in remarkably agreement with the numerical models of Woosley and Baron (1992). The recent observations of 7-ray bursts by BATSE at the Gamma Ray Observatory have shown that these events are distributed isotropically but not uniformly in radius (Fishman et al 1991, Meegan et al 1991). These observations have opened the possibility of a cosmological origin. In this case, they must be placed at zw 1 on average and emit ~ 1051 ergs in ~ 15 s (Paczynski 1991). One of the scenarios that have been proposed is the accretion induced collapse of a white dwarf (Dar et al 1992). We have solved equations (19-23) for a set of reasonable values of the temperature and radius of the neutrinosphere. In all cases we have obtained a heavy wind characterized by: M ~ E{GM/RU)~X > E/c2, where M and R,, are the mass of the compact object and the radius of the neutrinosphere respectively. Since 7-rays can only emerge if the condition M < 10~2 (E/c2) is fulfilled (Paczynski 1990), we must conclude that the collapse of white dwarfs cannot explain the existence of 7-ray bursts (Hernanz et al 1993).


-

2

-

T

-

1

o

sS, GO

o

log(r)

Fig. 9.3 Characteristics of a neutrino driven wind when Tv = 4.961010 K and Rv = 30 km. At the sonic point p, = 9.95105g/cm3, T, = 5.9109 K and R, = 210.4 km. The intensity of the wind is M = 1.161031 g/s.

9.4 The explosion of degenerate cores For a long time it has been assumed that the observational constraints that Type la supernovae should satisfy were the following ones: 1) The surfaces of their progenitors should be devoided of H at the time of explosion in order to explain the absence of Balmer lines in the spectra. 2) The progenitors should be long-lived stars in order to account for their occurrence in all types of galaxies, even the elliptical ones. 3) Their explosion should produce at least ~ 0.5 M© of 56Ni in order to account for the light curve and to explain the late time spectra. 4) Intermediate mass elements should be present in the outer layers in order to explain their spectra at maximum light. 5) The explosion should produce events with very homogeneous peak magnitudes (Miller and Branch 1990; Branch and Tamman 1992) whereas the light curve shapes and the photospheric expansion velocities might show some degree of variability.


201

6) The abundance ratios of Fe and Ni isotopes should agree with the Solar System values after combining SNIa yields with those from gravitational collapse supernovae. 7) The death rates of the progenitors should agree with the observational estimates of the frequency of SNIa events. Mainly due to points 1), 2) and 3) it is thought that SNIa are due to the explosion of CO white dwarfs in a binary system. Point 4) strongly suggests that the burning front propagates subsonically in the inner parts of the star and that only in the very outer layers, where p < 107 g/cm3, it could propagate supersonically. The observational situation mentioned in point 5) is very puzzling. It was claimed for some time that the rate of fading and the magnitude at the peak of the light curves as well as the expansion velocities of the photosphere of Type la supernova outbursts displayed a continuous behavior, i.e. the most luminous supernovae were declining more slowly and expanding more quickly than the less luminous ones (deVaucouleurs and Pence 1976; Pskovskii 1977; Branch 1981, 1982). This was challenged by Cadonau et al (1985), who examined the shape of twelve SNI light curves in a sample of elliptical galaxies and reached the conclusion that the dispersion of the light curves was smaller than 0.3m when only photoelectric photometry was taken into account. Concerning the maximum of the light curve, Miller and Branch (1990) examined the Pskovskii's sample and found that the dispersion in maximum brightness is smaller than 0.4m if the inclinations of the galaxies are taken into account. Branch and Tammann (1992) proposed an absolute blue magnitude at maximum of MB = -19.6±0.4. Concerning the postpeak decline, it has been shown that contamination by the light of the background galaxy and the K-corrections (Boisseau and Wheeler 1991; Leibundgut et al 1991) might account for the dispersion. Nevertheless, there are clear evidences of SNIa displaying peculiar behaviors. SN1885A, in M31, was very fast (deVaucouleurs and Corwin 1985). SN1986G in NGC128 (Phillips et al 1987) seems to have been intrinsically dim, to have a low expansion velocity and a fast light curve decline. SN 1991T (Filippenko et al 1992; Ruiz-Lapuente at al 1992) seems to have been overluminous and extremely peculiar in several aspects. Finally, the existence of different expansion velocities near the maximum of the light curve has been confirmed in a number of cases (Branch 1987; Branch et al 1988; Schneider et al 1988; Philips et al 1987; Barbon et al 1990). These velocities range from 10,000 km/s in SN1986G, SN1986A and SN1989B to at least 15,000 km/s in SN1983G and SN 19841 (Branch and Tammann 1992). The most extreme case of peculiar behavior

202


has been provided by SN1991bg in NGC 4374 (an elliptical galaxy in the Virgo cluster) which was clearly underluminous: at maximum light its B magnitude was ~ 2.5 m fainter and its V magnitude ~ 1,6m fainter than a normal SNIa in the same galaxy, it declined very fast after maximum and entered the nebular phase sooner than other SNIa (Filippenko et al 1992; Leibundgut et al 1993). Therefore, the question to elucidate is the following one: is there a bulk of very homogeneous events, with some "dissidents" which can be explained just by allowing some minor changes in the main parameters of the drflagration/detonation model or is it necessary to build a new scenario to account for the existence of the "anomalous cases"?. The discovery of SN1991bg seems to point towards the second alternative. Point 6) has been recently analyzed by Bravo et al (1993) and the combined yields of SNI and SNII seem to account fairly well for the observed abundances. Point 7) has recently turned out to be critical. A popular scenario for SNIa explosions involves the merging of two CO white dwarfs in a binary system due to the emission of gravitational radiation (Iben and Tutukov 1984). However, the negative results of searches for progenitor systems have raised serious objections to this scenario (Munari and Renzini 1992). This has renewed the interest in the single degenerate scenario (Wheelan and Iben 1973) where a white dwarf grows to the point of explosive ignition by accreting matter from a nondegenerate companion, typically a red giant or a supergiant, probably forming a symbiotic star. New estimates of the space density of symbiotic stars have increased their inferred numbers by a factor ~ 10 — 100. That means that the fraction of such stars that should reach the Chandrasekhar mass is just ~ 4% (Munari and Renzini 1992) or ~ 40% (Kenyon et al 1993) according to the higher and lower estimates respectively. The observation of transient hydrogen lines in at least two SNIa, provides aditional support to the symbiotic scenario. In this context, it is especially relevant the discovery of H-lines in the nebular spectrum of SNl991bg (RuizLapuente et al 1993), which can be interpreted as due to H-rich material, stripped from the companion by the kinematic interaction with the supernova ejecta, that appears as low velocity material in the late type spectra (Chugai 1986). Nevertheless, it is necessary to keep in mind the peculiar behavior of this supernova before generalizing this observational evidence. The symbiotic scenario also opens up a new and interesting possibility. After burning, the accreted hydrogen is converted into heliun and accumulates on the surface of the star. Depending on the accretion rate and on


203

the initial mass of the star, helium eventualy detonates and produces an inwards shock wave whose strenght increases due to geometric effects. Two dimensional hydrodynamic simulations (Livne and Glasner 1991) shows that this shock wave ignites a detonation in the center of the CO core that completely disrupts the star. This has lead the suggestion (Woosley and Weaver 1993, Canal 1993) that the helium detonation of CO white dwarfs with initial masses in the range 0.5 - 1.3 MQ after accreting ~ 0.1 - 0.2 M© of hydrogen-rich material could be at the origin of SNIa. Therefore, the question is whether both models, central ignition or He-induced detonation of CO white dwarfs, can coexist or one of them must be eliminated. Notice that the symbiotic scenario is compatible with the central ignition model and that one of the criteria to discriminate among them is their ability to reproduce the emergent variety of SNIa events. 9.4.1 Models igniting carbon at the center This family of models assumes a mass accreting CO white dwarf in a binary system that approaches to the Chandrasekhar's mass and is partially incinerated by a subsonic burning front. A close examination of the preexplosion evolution reveals that the thermal runaway can happen at a density p in the range 2 109 < p < 1.3 1010 g/cm3, that depending on the history of the binary system (Hernanz et al 1988). The shape of the light curve and the luminosity at maximum depend on the kinetic energy as well as on the total amount of 56Ni newly synthesized and on its distribution accross the star, since 7-rays emitted by radioactive nuclei must be thermalized before escaping to contribute to the optical light curve. Both properties are affected by the total amount of electron captures undergone by the incinerated material, which depends on the ignition density, and by the total amount of matter that is completely incinerated (Graham 1987; Canal et al 1988). For a given chemical composition, there are two properties that can modify the characteristics of the light curve. One is the velocity of the burning front and the other is the density at which the thermal runaway starts. If the burning front propagates as a Chapman-Jouguet detonation, the velocity of the burning front is completely determined by the thermodynamic properties of the burned material or, equivalently, by the density of the white dwarf. If the front propagates as a deflagration, the flame velocity is not uniquely determined by the density and it can be very different from one event to another, even if all the objects have a similar structure. In this section we examine the dependence on the density of the explosion characteristics.

204


The initial phases of the explosion, those encompassing from the ignition at the center to homologous expansion, have been modelled by several groups. The most critical, point as mentioned above, is the treatment of the burning front. A typical model is, for instance, that of Bravo et al (1993), who used an explicit difference scheme similar to that of Colgate and White (1966), an equation of state for the ion component taken from Ichimaru et al (1988) and for the electron component an ideal Fermi gas plus electronpositron pairs. Radiation was also included. The electron capture rates were taken from the compilation of Fuller et al (1982) and when not available there they were computed from the gross theory of /?—decay (Kodama and Takahashi 1975). The equation of state for the nuclear statistical equilibrium (NSE) material has been computed using a set of 722 nuclei with 0.39 56Ni and the rates were taken from Caughlan and Fowler (1988). NSE was assumed when T> 2 109 K or T> 5.5 109 K, for p > 710 7 g/cm3 and p > 106 g/cm3, respectively. It is also possible to obtain a rough bolometric light curve using the diffusion approximation and a finite difference scheme similar to that proposed by Falk and Arnett (1977). A flux limiter, of the form proposed by Alme and Wilson (1974), has to be used to avoid overestimate of the radiative flux in the outer layers. The equation of state is that of an ionized ideal gas plus radiation and the degree of ionization can be computed from the Saha equation (see Hoflich et al 1992 for details). The justification of the procedure only simplicity. Below 3500 K, the degree of ionization is very low. However, since the space is pervaded by energetic 7-photons, Ey ~ 1 MeV, coming from the radioactive nuclei, the degree of ionization of matter is higher than that corresponding to such temperature and the opacity is increased by several orders of magnitude. The resulting opacity is conveniently modelled by (Swartz 1991): KC = max[/csc, 1.4 lO" 1 0 ^) 1 ' 3 ] (9.24) P where KSC is the opacity obtained assuming local thermodynamic equilibrium, e is the radioactive energy locally deposed and p is the density. The sources of opacity that have to be be considered are scattering Thomson by free electrons, bound-free and free-free transitions, and the contribution from the lines modified by the expansion effects (Karp et al 1977). The total average Rosseland opacity obtained in this way lies in the range of 0.05 to


205

0.1 cm 2 /g. Finally, the energy deposited by 7-photons can be handled in several ways, the simplest one being to treat them as a simple absorption process (Sutherland and Wheeler 1984) assuming a 7 opacity K~, = 0.03 cm 2 /g, which is accurate enough for the majority of purposes. The development of the Rayleigh-Taylor instability associated with the deflagration can be handled in several ways, no one being completely satisfactory. Table 1 displays the main characteristics of several models. The columns have the following meanings: />g is the density at which the central runaway starts, in units of 109 g/cm3; Mj, is the total burned mass, incinerated plus partially burned; M M is the ejected mass of radioactive Ni; K51 is the kinetic energy in units of 1051 ergs, and Mpe is the mass of 56Fe synthesized. In the models labelled R, the development of the Rayleigh-Taylor instability associated with the deflagration was computed in the way proposed by Sutherland and Wheeler (1984), taking a = 0.7. Models labelled J were designed to allow the burning front to propagate as a deflagration in the central regions and as a detonation in the outer layers. The development of the Rayleigh-Taylor instability was handled, in this case, with Unno's theory of time dependent convection (Unno, 1967). Concerning the two free parameters of the theory, the excess of temperature and the initial velocity of the "bubbles", ATo and VQ respectively, it was assumed that VQ = 0, and l r ,dT x dP

dT.

where / is the mixing length. The last choices implies that the transition regime is very short and that the steady state is attained almost instantaneously. The characteristic mixing length scale was taken to be equal to the density length scale. In all cases, when the density was p = 3 — 4 107 g/cm3 respectively, the deflagration spontaneously turned into a detonation that partially incinerated the material. The models show that despite the increase in the total burned mass, which monotonically increases with the ignition density, the total mass of Ni produced during the explosion decreases and is strongly reduced in model R8. This is due to the electron captures on the incinerated material near the center, which are more important at high densities. Models J, however, propagate the burning faster, thus reducing the time available for electron captures, and so the final amount of 56Ni is similar in all of them. The total kinetic energy dos not appreciably change with density because the increase of the burned mass is compensated by the energy losses due to electron captures and the initially greater binding energy. In the models J the total

206

Isern & Canal: Degenerate stellar cores Table 9.1. General characteristics

of the computed

models

Model

P9

M6(M0)

Mm{MQ)

K 51

M F e (M 0 )

R2 R4 R8

2.5 4.0 8.0

0.86 0.89 0.96

0.56 0.52 0.34

0.85 0.91 0.86

0.58 0.59 0.46

J2 J4 J8

2.5 4.0 8.0

1.00 1.06 1.19

0.63 0.63 0.51

1.42 1.42 1.40

0.68 0.73 0.66

Table 9.2. General characteristics of the computed light curves Model

tbol

Mio/

\ph

0o

Pi

R2 R4 R8

13 13 11

-19.23 -19.20 -18.81

10400 11000 11200

6.5 6.5 7.0

3.9 4.0 4.0

J2 J4 J8

12 12 10

-19.56 -19.54 -19.54

12424 12700 13800

7.1 7.7 8.5

3.3 3.4 3.4

amount of burned matter is very large and, consequently, the kinetic energy is also very large. Table 2 displays the parameters that characterize the models. M^,/ is the magnitude at maximum; tj, 0 / is the time in days, from the explosion to the maximum; vphot is the velocity of the photospheric layer, defined as the layer of optical depth unit, in km/s; /3o and /?i are the slopes in magnitudes per 100 days of the light curve 10 days after maximum and during the exponential tail. The most noticeable fact in Table 2 is that the magnitude at maximum is the same in all the families of models and only the case R8 displays a significant deviation from the average. The reason is twofold: in the models where the runaway starts at high densities, the destruction of 56 Ni by electron captures in the central layers is counterbalanced in part by the production of this element in the outer layers and the proximity of the radioactive material to the surface, which facilitates the scape of photons. The large amount of 56 Ni newly synthesized and the small opacities of the models translate into very bright light curves, with the exception of the R8 one. Leigbungut et al (1991) have determined that the average apparent


207

magnitude, free from extinction, of six SNIa in the Virgo cluster is 11.92 ± 0.11. If the distance to that cluster is assumed to be 20.03 ± 3 Mpc, their absolute magnitude would be MB = -19.6 ± 0.4 and only models J, i.e. the delayed detonation models, would be fully consistent with this value. However, if a distance of 16.5 Mpc is adopted (Jacoby et al 1992), the magnitude at maximum would be MB — —19.2 and models R would be completely acceptable. Concerning the shape of the light curves and leaving aside the problem of their absolute calibration, models R perfectly fit the shape of the bolometric light curve since, on average, the observed rate of decline of the bolometric light curves is /3Q = 6 ± 0.5 and /? ~ 3.3, although their exponential tail is too step. On the contrary, models J display a first decline after maximum and an exponential tail with a slower slope than in the case R. In any case, however, the light curves of each family are very similar and differ by less than 0.5 m , which implies that the ignition density does not seem to be able, just by itself, to introduce significant changes in the shape of the light curves. The expansion velocity changes from model to model. In the family labelled R, the velocity differences are modest, less than 800 km/s. In the J models, this difference reaches 1500 km/s. In any case, these values define a range of variation that is smaller that the observed one. Due to the low opacities of the models, K ~ 0.05 — 0.1 cm 2 /g on average, the photospheres of models J are rather deep and the velocity rather low, 11,000 to 12,000 km/s during maximum. These quantities are in agreement with the observations (Branch et 1 1985), but there is a noticeable amount of matter, 0.01 M©, moving at high velocities. Because of the lack of ultraviolet observations of supernovae, it is hard to obtain any conclusion from these figures. On one hand, Harkness and Wheeler (1990) obtained from UV observations that the maximum expansion velocity was 25,000 km/s for SN1981B. On the other hand, there are several supernovae like SN1984A (Wegner and McMahan 1987), SN1983G (McCall et al 1984), and SN1990N (Leibungudt et al 1991) that display material moving at very high velocities. The nucleosynthesis can be computed with a post/processing code that uses the previously computed time evolution of the temperature and density for each shell. The most noticeable feature is the extraordinary increase of neutronized elements when the ignition density increases. This implies that the number of exploding high density white dwarfs which have contributed to the building of Solar System abundances, must had been very low in order to account for the observed values. It is also important to notice that even the models that ignite carbon at low densities produce an excess of

208


50

Ti, 54 Cr, 54-58Fe, and 56Ni. The first two nuclei are essentially produced in the central region and their overabundances cannot be avoided if, as it actually happens, the velocity of the burning front is very small at the center. Nevertheless, the degree of acceptability of these excesses can only be ascertained in the context of a model of the chemical evolution of the galaxy and taking into account the uncertainties that characterize our knowledge of isotopic and elementary abundances in stars and in the solar system. A simplified model (Bravo et al 1993) shows that the constraints introduced by the cosmic abundances of the elements are much less restrictive than previously thought. 9.4.2 Models igniting He off-center The symbiotic scenario demands a non-violent burning of the freshly accreted hydrogen in order to effectively incorporate the newly synthesized helium onto the white dwarf. The usual limits for the behavior of hydrogen are as follow: Low rates, M < 10~9 MQyr"1, lead to a nova outburst that not only removes all the accreted matter but even erodes the original white dwarf. High accretion rates, M > 10~6 MQyr"1, lead to the formation of a red giant envelope, while intermediate rates in between these two limits lead to the formation of a common envelope. The nova limit is based on calculations that assume spherically symmetric and soft accretion. Soft means here that material is deposited at the surface at rest and with the same entropy as the underlying material. It is thus cold and becomes strongly degenerate before igniting. Most likely, however, this material will form a disk around the compact star, and the accretion process will thus include angular momentum and kinetic energy dissipation. When those effects are taken into account, the actual range of MH producing nova explosions becomes ill defined (Shaviv and Starrfield 1987, Sparks and Kutter 1987) and further research on this point is thus needed. Anyway, even in the most favorable cases, nova outbursts would limit the actual mass growth to at most 10% of MHThe development of a common envelope (either due to accretion above the Eddington limit or to formation of a red-giant envelope should induce mass-loss by the system as a whole and thus inhibit further growth of the white dwarf. Nonetheless, accretion rates in the range 10~9 < Mfj < 10~6 M©yr~x allows to convert H into He trough steady combustion or weak flashes. A further constraint is that the He layer resulting from the burning of the accreted H will explosively ignite only if it is accumulated at a rate 10~9
pd, where Pd « 4 x 1011 g cm" 3 is the neutron drip density. Matter of the outer crust consists of electrons and atoms. The electrons can be nondegenerate in a very thin surface layer, and they are strongly degenerate deeper in the star. The atoms are fully ionized by the electron pressure (they are actually bare nuclei) everywhere except near the very surface. In the inner crust, free neutrons appear (in addition to the electrons and nuclei) owing to the drip from the nuclei. The neutrons are degenerate and superfluid. The superfluidity is caused by the Cooper pairing of the neutrons due to nuclear forces; corresponding critical temperature is about 108 - 1010 K (e.g., Wambach et al., 1991). The rotation of the superfluid component of matter in rotating neutron stars is realized in the form of quantized vortices which are parallel to the rotational axis. The interaction of the vortices and the nuclei (pinning and depinning) in the inner crust is thought to be responsible for pulsar glitches (e.g., Pines 1991). The properties of nuclei and free neutrons are model dependent (Baym et al 1971, Negele and Vautherin 1973) and rather uncertain. At the bottom

Yakovlev & Kaminker: Neutron star crusts with magnetic fields

216

9 8

:

T -"Fl 6 -- N

N

y'

/

?e•

N

y \

i

>

LOGt ^> , G/CM" Fig. 10.1 p-T diagram of Fe matter for B = 1012 G. 7> is the electron degeneracy temperature, 7} corresponds to F = 1, Tm is the melting temperature, Tp is the ion plasma temperature; TB is given by (8); pu is explained in Sec. 3. Lines N restrict the low-T low-/) domain of incomplete ionization and electron gas non-ideality. Dashes show the curves for B = 0 of the inner crust, the nuclei may form clusters and droplets with strongly nonspherical shapes (Lorenz et al. 1993). Below we shall mainly consider the outer crusts. The state of electrons is determined by the electron Fermi momentum PFO and 'relativistic parameter' x:

(

\ —)

1/3 .

(!)

where ne is the electron number density, p% is density in units of 106 g cm" 3 , and fie is the number of nucleons per one electron. For x •< 1 (p$ < 1) the electron gas is non-relativistic, while for x >• 1 it is relativistic. The electron degeneracy temperature Tp is (Fig. 1) TF = To(\/1 + x2 — 1), where To = mec2/ks « 5.930 X 109 K, ks is the Boltzmann constant, and m e the electron mass.


217

The state of the ions is characterized by the ion-coupling parameter

akBT

a 0.2275^ 1 ^ 1 ,

(2)

where a = [3/(47rn,)]1/3 is the radius of ion sphere (the charge of electrons within the sphere compensates the ion charge), n,- is the ion number density, and Tg = T/(10 8 K). For simplicity, we consider one component plasma of ions. Some properties of multi-component ion mixtures in neutron star crusts are reviewed by Yakovlev and Shalybkov (1989). At sufficiently high temperatures the ions form a classical Boltzmann gas. With decreasing T, the gas gradually (without any phase transition) becomes a Coulomb liquid, and then (with a phase transition) a Coulomb crystal. The gaseous regime occurs (Hansen 1973) for F < 1 (T > T/, Fig. 1). The classical Coulomb crystal melts (Nagara et al. 1987) at F « 172 (T = T m ). The thermodynamics of strongly coupled Coulomb systems has been studied extensively by Monte Carlo and other methods (see Hansen 1973, Pollock and Hansen 1973, Hansen et al. 1977, Slattery et al. 1980, 1982, and references therein). At low T the quantum effects in ion motion (zero-point ion vibrations) become important. These effects are especially pronounced if T < Tp, where

T p =^«7.832xl0 6 (^)

K, 1 (electrons suffer many Larmor rotations between successive collisions). The p — T domain where the electrons are magnetized (770 ]> 1) is commonly wide. For instance the field B = 1012 G magnetizes the electrons almost for all p and T shown in Fig. 1. The quantum effects are associated with quantization of electron motion transverse to the magnetic field. If the Landau gauge of the magnetic field vector potential (A = (—By, 0,0)) is used, an electron state can be characterized (Klepikov 1954, Kaminker and Yakovlev 1981) by four quantum numbers, pz, n, s and px. In this case pz is an electron momentum along B, n = 0,1,... enumerates the Landau levels, s is the sign of the projection of the electron spin onto the momentum (5 = ±1 for n > 0; 5 = —sign(pz) for n = 0) and px determines an y-coordinate of the electron Larmor guiding center. Then the electron energy is D

£ = Jm\c* + c2pl + 2nmec2hijJB, h^B = mec2b, b = — , v &

(5)

where ug = \e\B/(mec) is the electron cyclotron frequency, and Bc = 4.414 x 1013 G is the 'relativistic' magnetic field {huge = fnec2). The number density of the free electron gas is related to the electron chemical potential \i by the equation

where / is the Fermi-Dirac distribution. The electron degeneracy temperature is TF = To(\/l + xB — 1), where the 'relativistic parameter' XQ — PFJ(mec) depends generally on B. Using (6) one can easily show that the strongly degenerate electrons (T 1. If, in particular, X]Q > 1 then the transverse conductivities are greatly suppressed (K± W «||/^) due to rapid Larmor rotation of the electrons. For instance, in the nonrelativistic electron gas at B ~ 1012 G and T > Tm the Hall parameter can be as high as 770 ~ 10 2 -10 3 . Then the suppression reaches 4 - 6 orders of magnitude.

10.5.3

Quantizing magnetic fields

The theory of longitudinal and transverse transport properties of a degenerate electron gas in quantizing magnetic fields is based on the linearized relativistic kinetic equation which describes relaxation of the electron Landau states (Sec. 3) due to electron collisions. The longitudinal conductivities Ky or TB (P < 1), the field is nonquantizing and its effect is weak, «|| « K± « KQ. If T < TB (/? > 1), the magnetic field is quantizing, and the thermal conductivity is anisotropic and strongly enhanced by the magnetic field, K|| J_ ~ KO/32. The enhancement occurs because the main heat carriers at (3 ^> 1 are extraordinary mode photons with frequencies u> •< u g . The mean free path of these photons is a factor of (U;B/U>)2 larger than at B = 0. The enhancement K a /?2 for (3 > 1 was first outlined by Tsuruta et al. (1972) and Lodenquai et al. (1974) using a simplified approach.


229

An important fact is that K± becomes larger than «|| in high magnetic fields. For instance, in the case of the Thomson scattering at /? > 1 one obtains K± « 2«|| « (32/(2n2). These asymptotes are valid for large /? while for intermediate ft Silant'ev and Yakovlev (1980) got analytic fits

where «o = 7.5 x 1020T^p/fie erg cm" 1 s" 1 K" 1 , and p is expressed in g cm" 3 . For example, consider a helium plasma with T = 4.5 X 10s K, and B = 2 x 1012 G. The Thomson scattering dominates at p < 1 g cm" 3 . In the latter case K|j « 100«o and n± « 180KO- If P > 1 g cm" 3 , free-free transitions dominate, and the conductivity enhancement is weaker, KJJ « 31/co and KJ. w 21KO (now with respect to the free-free conductivity «o for the same T and p). At very low p the radiative thermal conductivity is affected by electron - positron vacuum polarization in strong magnetic fields (Pavlov and Yakovlev 1982). 10.6 Neutrino energy losses in magnetized crusts Neutron stars become transparent to neutrinos in half a minute after their birth. Neutrino generation produces powerful energy losses in the cores and crusts of young neutron stars. The main neutrino production mechanisms in the crusts are: electron bremsstrahlung on atomic nuclei (e + Z —• e + Z + v + F), electron-positron annihilation (e~ + e"*" —• u + V), plasmon decay (hu>pe —• v + 17), and photon decay (7 + e —• e + v + V). These

processes have been studied in detail for 5 = 0 since the classic works of Beaudet et al. (1967) and Festa and Ruderman (1969) (see Itoh et al. 1989 for the references to further works). All processes can be affected by magnetic fields. Moreover magneticfieldslead to a new process, synchrotron radiation of neutrino pairs by electrons (e —• e + v +F), which is forbidden by momentum and energy conservation for B = 0. Neutrino synchrotron radiation was first studied by Landstreet (1967) but rather qualitatively (see Kaminker et al. 1992a, for critical remarks). So far two processes have been investigated in strong magnetic fields: the neutrino synchrotron and pair-annihilation radiations. These processes are described by similar matrix elements. According to the Weinberg - Salam theory both processes proceed via charged and neutral currents, and hence all neutrino flavors can be generated. For temperatures T 1013 G, and p ~ 107 - 108 g cm" 3 . Further work is required for studying other neutrino generation mechanisms in strong magnetic fields.

231


To

20

o Or

;

T=10 K — —'

oi \5

•

-

PL_

_

-£^—

. In the "average neutral-pseudoatom" approach the Z + 1 ionic species are replaced by one ionic species with "average" charge Z* and the EOS is determined in that simplified model. The name "neutral pseudo-atom" (NPA) refers to the neutral object consisting of an ion in a suitable profile (e.g., a Wigner-Seitz cavity) plus its cloud of bound and free electrons that form a neutral object (a more rigorous definition is given in terms of a sum rule on the phase shifts). The bound and free-electron distribution etc., at the NPA are determined by the self-consistent solution of Kohn-Sham equations for the electrons and ions in the plasma (Dharma-wardana and Perrot, 1982, and 1987). In the average-NPA approach the individual species concentrations i{ are not evaluated and hence Z* is fixed by other considerations, e.g., as in the studies of the EOS of dense Al (Perrot 1990) and Be (Perrot 1993). Recently we have implemented a DFT calculation of the EOS of the Z + 2 component mixture without appealing to the so called "chemical pictures", hard-sphere models etc. In this approach we construct Z + 1 different neutral pseudoatoms (each self-consistently determined for the given plasma conditions) and use their interactions with one another and with electrons, instead of a single average-NPA which interacts with the electrons.

274

Perrot & Dharma-wardana: Equation of state of dense hydrogen

The above discussion of metallic plasmas takes on a new dimension of difficulty at lower densities and temperatures when molecule formation becomes important. Alkali metal vapors and Hydrogen plasmas have some similarities in this regard. Thus, a hydrogen plasma at 0.25 Mbar and leV could contain H2, H, H + , e~ and possibly H~, H*, and small amounts of other clusters. As the density is increased the identity of these" chemical" objects becomes "blurred". Thus an excited H2 molecule having a "bond length" R begins to resemble two unbound hydrogen atoms separated by a distance R. In effect, if mean distances become comparable to bond lengths the simple "chemical" picture breaks down. Even if chemical species like H2, H+, etc., could be identified, their electronic energy levels, vibrational spectra etc., have to be calculated self-consistently, including the interactions with the medium, i.e., the field-particles (FP) that surround a given "molecule-like" entity. Progress in this type of problem is easier for vapors of simple metals for which pseudopotential theory is applicable. The problem is more difficult for protons where the full non-linear consequences of a point-charge (with no moderation effect arising from a finite core size) have to be taken into account and no meaningful construction of pseudopotentials is possible. Thus in our study of ionized hydrogen plasmas (Dharma-wardana and Perrot 1982) we retained electron- and ion- coordinates at every stage of the calculation, and avoided pseudopotentials or linear response. The objective of this paper is to examine a tractable microscopic approach to EOS of hydrogen fluid in the proposed plasma phase transition (PPT) regime (Saumon and Chabrier 1992) where molecular species exist. It is useful to examine not just the thermodynamics, but also the dynamical process of ionization. Such a study sharpens our concepts about the PTT and leads the way for a future numerical study of this difficult regime of hydrogen fluids. 12.2 The mechanism of ionization Wigner and Huntington (1935) noted the similarities between hydrogen and the alkalis and sowed the seeds leading to the concept of a plasma phase transition (Stevenson and Salpeter 1977, Ebeling and Richert 1985). The most elaborate study of the PPT is due to Saumon and Chabrier (1992). They considered a hydrogen fluid containing the four species H2, H, H + , e~ and found that a fluid phase which was predominantly H2, with very low ionization (xfj+ < 0.5%) and low monatomic H concentration undergoes a first order phase transition to a significantly ionized (xH+ « 25%)


275

fluid phase, still containing H2, H, H+, e~, at Tc = 1.3 eV and Pc = 0.614 Mbar corresponding to a critical density of 0.347 g/cc. Since a sharp increase in ionization is proposed, it is important to understand the process of ionization. Ionization of a single atom in the vacuum is the promotion of an electron from some bound state v = n,l,m of energy — |£KI to a continuum state k,l, m with energy k2/2 (here we use atomic units: h = 1, |e| = 1, me = 1). In Hummer and Mihalas (1988), and in Mihalas, Hummer and Dappen (MHD, 1988) the ionization process is modeled by assuming that the field particles (FP) create a microfield which Stark ionizes the atom. However, if an effectively spherically symmetric potential were applied to the atom (this could happen from a cubic packing of a coordination shell of FP around the atom) the microfield is zero but a very large destabilization of the atomic bound state could arise. Thus the MHD model is incorrect from the outset. The MHD-estimates of the "critical fields" for ionization of atoms do not contain effects of possibly large potential fluctuations and electron-exchange effects which nevertheless have small microfields (note that the fields are vectors, while the potentials are scalars). Ionization from metal clusters, and the workfunction of metal surfaces are known to be strongly determined by the exchange energy of the ionizing electron, i.e., a quantum effect unrelated to the Stark effect. The discussion given below suggests that the ionization process involves cluster states of the FP in a fundamental way. Consider an atom in a plasma. For simplicity, let the inner shells of the atom be full and let the last occupied electron (in the ground state) be ns where n is a principle quantum number. The "radius" of the atom is about n2 atomic units (a.u.). Consider a dilute plasma of atom density pa where the mean separation rws = (3/)a/47r)1/3 is large, say 100 n2 a.u. An ionizing electron acquires energy from the FP via thermal and random fluctuations of the particle distribution. Most collisions are long-range, weak, multiple collisions which slowly raise the electron to higher energy levels. When the electron reaches the excited state with nj = lOn the electron is still in a bound state but the "size" of the atom is comparable to the inter-atomic distance rwa. From then on the electron gains energy by hoping to orbitals which span several atomic centers (i.e, clusters) prior to ionization. That is, the atomic electron becomes a "hopping electron" before it passes into the continuum to become a fully delocalized ("free") electron. Thus the ionization process involves three types of electrons, viz., bound, hopping and free electrons. The discussions in terms of only bound electrons and free electrons apply only in situations where hopping electron concentrations are negligible. Unfortunately, hopping electrons are important except

276


in fully ordered solids at zero temperature, or in very hot strongly ionized plasmas. The theory of hopping electrons, and the relationship of the ionization process to the concept of the "mobility edge" transition (see Davis and Mott 1975) present in disordered materials apply to plasma situations as well (Dharma-wardana and Perrot 1992). The same ideas are relevant for an understanding of the PPT. The cluster ionization picture can be restated as a local-band theory of disordered materials. Consider a Hydrogen fluid in the weakly ionized prePPT phase of Saumon and Chabrier (to be denoted the prephase, while the phase with a higher ionization will be called the postphase). The prephase is mostly H2 molecules. If we sit on an H2 molecule (the origin), then there is local order as defined by the range Rc beyond which particle correlations die-off and the pair function g(r) becomes essentially unity. This is effectively the "cluster" to consider. We could either think of cluster states or of a local bandstructure in the region r < Rc around the molecule at the center. Electron hopping will be determined by a matrix element T(ry) linking atomic centers i and j , and determines the local bandwidth. In H2 this band is fully occupied at T=0K and hopping occurs by transferring to the unoccupied conduction band, or via an upper "Hubbard band" if that is energetically more favorable (the upper Hubbard band of H2 is approximately the valence band of HJ). However, the situation becomes radically different if even a small amount of H*, H + , etc., and electrons are also present, as in the prephase. Then we have a small number of holes in the H2 local-valence band, a few electrons in the local-conduction band, as well as "impurity gap states" of H^", H*, H + , H and H~ structures (the H and H~ bandstructures are essentially like the lower and upper Hubbard bands of H). The gap states provide a degree of hopping conductivity via overlapping localized states, but the mean free path of the hopping electron remains smaller than the cluster size Rc- However, when the percolation threshold (e.g. Stauffer 1979) is reached the hopping paths "percolate" through the whole volume and provide at least one electronic state which carries the electron out of the cluster. This is the onset of the so-called mobility edge and amounts to a phase transition in the sense that localized carriers have now become delocalized. The idea of the mobility edge was first presented by Mott and further developed by many authors (see Davis and Mott 1975). It is very likely that if the proposed PPT exists, then it is essentially a transition across the mobility edge, and its existence depends on the sharpness of the mobility edge. Note that crossing the mobility edge to reach the more conductive postphase does not require that the carriers have reached the conduction band - they have merely crossed into the per-


277

colating impurity states in the local band gap. The postphase need not to be a fully ionized phase, as has been assumed by some authors. The cluster picture of ionization involves a gradual delocalization of a bound electron on a given atomic site to occupy increasingly larger-sized bound states defined on a local transient cluster (hopping states) and finally in the whole plasma (full ionization). This picture is more convenient than the local-band picture with gap states. However, the two models complement and clarify the nature of the ionization process and the PPT. In an earlier study (1982) we considered a hydrogen plasma where each proton supported only one bound state. A significant feature of the bound ls-state reported in that study was that its average radius was larger than the mean proton-proton separation i.e., a hopping state. The bound state was calculated with the proton interacting with an "average" cluster of field ions (FI) and electrons around a proton. The Fl-cluster is modeled by the ion distribution pg(r), where p is the average (bulk) nuclear density. Although such an "average cluster" is incorrect at short time scales, it is valid for thermodynamics which depend only on space and time averaged quantities. To treat the molecular species present in the prephase and the postphase, a multi-center model is needed and will be taken up next.

12.3 Construction of pseudoatoms and pseudomolecules In the so called "chemical" picture for EOS calculations one assumes (by "chemical intuition") that certain well defined chemical species, e.g., H2, H, etc., exist in a given fluid and that their "internal" electronic coordinates do not appear in the discussion. Instead, energy spectra and weights of isolated molecules appear in "internal" partition functions. The internal partition functions of isolated atoms or molecules contain divergencies which are removed by various prescriptions. In a more fundamental analysis (sometimes called the "physical picture") electrons and nuclei interact via the Coulomb interaction, and the interplay between the electron coordinates and the nuclear coordinates is retained right up to the final stage of free energy minimization which determines the mixture composition, particle distributions and the thermodynamics. Even in the physical picture we can talk of molecular species noting that these are "pseudomolecules" (PM) which are "coupled to each other" since the internal structure of a PM depends on the environment around it, i.e., the self-consistent distribution of field particles around it. In the low density limit these PM reduce to the molecules of the chemical picture. Also, the pair-interactions of the PM will

278


be a function of the particle distributions in the medium which have to be generated in situ while the free-energy is minimized. If we consider the prephase and the postphase of the PPT, then we need to construct pseudomolecules like H2, H, H*, H + , H~, H^" in the presence of the equilibrium concentration of electrons, at a given temperature and total density. Here an H + is really a proton carrying no bound electrons, plus a distribution of electrons, ions, and other molecular species (the field-particles), while H and H" carry a singly or doubly occupied bound state which spills into the respective self-consistent FP-distribution (FPD). These field-ion distributions are given by the classical form of the Kohn-Sham equations, while the boundstates and field-electron distributions are given by the (quantum) Kohn-Sham equations which are coupled to the classical equations. The electron coordinates and the ion coordinates are retained through out the energy minimization. Unfortunately, this procedure is numerically too arduous since we need to self-consistently resolve a multi-center, multi-species problem as well as quantum mechanical problems for bound and continuous spectra. Hence we consider a less ambitious procedure based on using single-center DFT calculations to construct a tight-binding-type model to account for the molecular states of the pseudomolecules.

12.3.1 Tight binding model A hydrogen atom placed in a given field-particle distribution (FPD) will have a finite set of bound states u , v = n,ltm and energy €u as well as a spectrum of scattering states v = k,l,m with eu = k2/2. From our previous work (single center calculations) we know how to determine this Kohn-Sham energy spectrum of bound and free states for a given FPD. To simplify the discussion we assume that there is only one bound state, «i 9 , but several bound states would be needed in many applications. We

fit the uu(r) function to the form u(Z*,r) = (Z**/n)l/2exp(-Z*r). This simple form where Z* is the only fit-parameter is probably adequate for the plasma-problem. Although we are concerned with H-plasmas, we also carry out single-center calculations for the He + + and He+ ions in the given FPD and determine their Kohn-Sham spectra. He + + and He+ correspond to the "zero bond length" limit of two protons, or a proton and an H atom. This "united-atom" limit of the molecular species is required to correctly recover the interaction energies at smaller particle separations. Now we consider the construction of pseudomolecules.


279

12.3.2 H2 pseudomolecule The electron Kohn-Sham equation for the H* molecule in the fluid is :

(1)

1 + Vxc{r, n(f), p(r)} ^(r, R) = E^{r, R)

Here R is the internuclear separation between the protons labeled A and B; the electron coordinate f referred to the midpoint, and to the centers A and B are related by rA = r + R/2 and ?B = f— R/2. Also Vp and Vxc are the Poisson potential and the exchange-correlation potential due to the FPD. The xc-potential is a density-functional potential which brings in the many-body effects of exchange and correlation with electrons, and correlations with ions. Their formulation has been discussed in our earlier work (e.g., see Perrot, Furutani and Dharma-wardana, 1990). An approximate construction of Vp and Vxc for the two-center problem is given below. Consider the isolated molecule where Vp and Vxc = 0. In the limit where R is large we have two (gerade and ungerade) solutions : 4>(r,Z*,R)g,u = [u(Z*,rA)±u(Z*,rB)]/21/2

large R limit

(2)

with Z* = 1 and the energy

_x [(1 + R)e~2R ± (1 - 2r 2 /3)e- fi ] Eg,u = f Is + R

f, I /I i .

However, this is a poor solution for small R. In the R « 0 limit we use a value of Z* = ZQ consistent with the solution of the He+ problem, i.e., the "united atom" obtained from H*. That is, we find Z*(R) variationally for each R for the isolated H* problem so that

interpolates from R = 0 to R = 00, with Zoo = 1- The minimum of Eg(R) a,t R = Ro is the binding energy with the optimal Z*(RQ). This approach provides an excellent solution to the H^ problem (i.e., to the accuracy we needed for the plasma-fluid problem) since the experimental (Herzberg 1954) binding energy Eb = Eg(R0) - ei3 = -2.79 eV and Ro = 2.003 a.u.,

280


while this method (Bransden et al 1983) gives Eb = -2.25 eV and # 0 = 2.0 a.u. with the optimal Z*(Ro) = 1.23. The vibration- and rotation- spectra can be calculated as usual from Eb(R). Now consider the pseudomolecule H* in the presence of the FP (e.g., H2, H, H*, H + , H~, HJ etc). A given FPD can be rewritten as a density p(r) of protons and a density n(f) of electrons. The p(f) and n(f) interacting with the pseudomolecule whose mid-point is at the origin of coordinates produce the potentials Vp(f) and Vxc(r). These potentials are constructed from the one-center functions Vp(f,H),Vp(f,'H.e+) and V xc (^H),V rc (f,He + ) which are known from the single center calculation for H and He + . In effect we define Vp(?) = Vp(rA,Re+)F(R) + ~{VP(rA, H) + Vp(rB, E)}{Vp(rA, H+) + Vp(fB, H+)}

(5)

x {1 - F(R)} where the superposition takes place in the independent atom limit for very large R, while for small R we recover the united-atom limit of He + . The interpolation function F(R) could be modeled by f(R) of Eq. (4) or a better form can be constructed. The same interpolation scheme can be used for constructing Vxc(f) from the results of the one-center calculation and hence all the potentials need in the two-center Kohn-Sham equation, i.e., Eq.(l) are known. Now in variationally determining Z* appearing in ij)(r, Z*) we do not use just Eq.(3) but include also the FP-energy contribution, i.e. < V(r, Z*)\Vp(r) + Vxc(r)\j>(r, Z*) >. This will give us a new Z*(R,p,n) to replace Eq.(4), where the limiting forms ZQ and ZQ© correspond to the He"*" and H atom wavefunction-exponents from the one-center calculations with the field particles self-consistently included. A new binding energy curve Ef,(R,p, n) dependent on the FP-distributions p(r), and n(f) and a new intermolecular equilibrium distance RQ will result from this calculation. If there are no H* molecules in the system under the assumed conditions, then there will be no stabilizing minimum and E\,{R) > 0. Thus the method by itself determines the pseudomolecules found in the fluid, independently of any "chemical intuition". The binding energy curve E\,(R,r,n) of H^ calculated here is basically the interaction potential between an H atom and an H + ion separated by a distance R in the fluid, inclusive of all the fluid effects. Given Ef,(R,r,n), the vibrational and rotational spectra and the "internal partition function" can be calculated as usual. The concept of an internal partition function has a meaning only if an


281

"internal part" and an "external part" can be defined. Since the KohnSham calculation includes the whole "correlation sphere" or cluster defined by the FPD, this issue is non-trivial. In the usual neutral-pseudoatom calculations (e.g. Perrot 1993) for simple metallic fluids, it is sufficient to use a spherical Wigner-Seitz cavity to represent the FPD at the pseudo-ion. The effect of the cavity is corrected for using linear response theory applied to the electron gas. In more complicated pseudomolecular systems the simplifications available for "simple metallic" fluids are not present. However, the effect of the FPD can be allowed for and factored out consistently by using a response function constructed from the Kohn-Sham basis of the pseudomolecule, instead of the response function constructed with plane wave states. Then the numerical work is more demanding. Simplified approaches using Wigner-Seitz volumes for each pseudomolecule and projecting out the contributions from neighboring cells according to some physical scheme etc may also be used. Since the electronic spectra contain only finite numbers of bound states, there are no spurious degeneracies appearing in the partition function summation. The electron density n(f) will be adjusted to self-consistence but the 2center FPD is not reevaluated except in the singe-center step. That is, the electron density results, bond lengths etc., of the two-center step are inputs to recalculate the self-consistent FPD of the one-center step. This approach is valid to second order in the density corrections, and has given good results in other problems (e.g., see Harris 1987). 12.3.3 H2 pseudomolecule Here again we begin with a simple one-parameter model of the isolated H2 molecule and re-optimize this parameter as a function of the field-particle potentials Vp(f) and Vrc(f). For constructing the H2 pseudomolecule we use as our tight-binding basis a generalization of the solutions ^(fj Z*, R) of the H* system. The simplest symmetric form for the spatial part of the H2 wavefunction (singlet spin state) is :

This can be rewritten in terms of the atomic ls-functions as $(ri,r 2 ) = $Cov(ri,r2) + $ion(ri,r2) where

(6)

282


Instead of Eq.(6) we use the trial form

$(n,r 2,Z*,A) = (1 - A)*cot,(n,r2) + (1 + A)$ ion (n,r 2 )

(7)

where A is the new variational parameter. That is, Z*(R) is the value determined to be optimal for the H* problem and A is specific to the H2 problem. We may also optimize both Z* and A, with improved results. Using this approach for isolated H2 a binding energy Eb(Ro) = —4.00 eV and an equilibrium bond distance Ro = 1.5 a.u, are obtained (cf. experimental values (Herzberg 1954) Eb = -4.7 eV, Ro = 1.4 a.u.). In the pseudo-H2 system inclusive of FPD we have to construct the twocenter potentials Vp(f) and Vxc(r). We interpolate between the infiniteR limit where Vp(r*) is a superposition of two single-center Vp(r) H-atom potentials calculated using the single-center DFT equations, and the R = 0 united-atom limit which is the He-atom. The interpolation function F(R) is similar to that of Eq.(5), and constructed using results of trial calculations. Once the potentials Vp(f) and Vxc(f), and the limiting Z*(R,f, n) values for R = 0 and R = 00, are prepared from the one-center calculations for H and He pseudoatoms, A is now minimized (for each R) including the FPterm, i.e. < {rx,r2,Z*,\)\Vp{r) + Vxc(f)\(n,r2, Z*, A) >. The resulting binding energy curve E(,(R,n,p) of pseudo-H 2 in the fluid can be used to obtain the vibrational and rotational spectra and the "internal partition function" inclusive of the effects of the fluid environment. If the binding energy function Ef,(R, n, p) has no minimum at some ifo then the fluid does not support the existence of H 2 molecules.

12.3.4 Other pseudo-molecules and clusters These methods can be applied to almost any simple hydrogenic cluster. We begin with a simple tight-binding model for the cluster in isolation and then include the FPD via the optimization of a relevant parameter in the Kohn-Sham functions used in the tight-binding model. The method also yields interaction potentials between pseudomolecules without resorting to perturbation theory. Thus we may consider the interaction between


283

an H2-pseudomolecule with an H + ion in the fluid. The H + ion has a FPD around it. The construction of the potential around the pseudo-H2 molecule was already discussed. Hence if we have to consider the interaction potential between a pseudo-H2 molecule and a pseudo-H+ ion, we use the $(ri,f2,Z*,A) solutions of the previous problem and taking a linear combination involving $ABi $BC, a n d $CA where A, B, and C are the three nuclei of the H2 and H + interacting system. This includes the binding due to electrons hopping among the three nuclei and goes beyond the usual polarization potential models for the H2-H+ interaction. A three-body interaction-potential (which is what the H2-H"*" system is) can be reduced to a 2-body potential either by imposing an equilibrium bond distance RQ to one of the three inter-nuclear separations (valid if the other two distances are >> RQ), or by averaging with pair-distributions functions, as in Aers and Dharma-wardana (1984). The method proposed here is free of hard-sphere models, cutoffs in the partition function etc. It provides a microscopic approach which is capable of (i) determining the stability of a given pseudo-molecular cluster in the fluid and hence deciding whether we need to include it in the thermodynamics, (ii) providing environment-dependent binding energy functions and interaction energy functions, (iii) providing the "internal partition function" of each species taking account of interactions with the plasma, (iv) obtaining triplet and singlet spin states, £ , II momentum states etc., and their correct thermal averages. 12.4 Minimization of the free energy function It is convenient to rewrite the free-energy minimization step outside the multispecies density functional calculation which involves the self-consistent resolution of a set of coupled Kohn-Sham equations for each species, as well as electron spectra for each species. Then the proposed procedure is as follows: (i) We prepare a trial "mixture" of species like H2, H, H*, H + , H~, HJ, e~ at mean nuclear density pn and compositions z;. The electron density is constrained by charge neutrality. (ii) Single-center DFT calculations are done for the united-atom and single atom limits, viz., for He, He+, H+, H, H", and He". The FPD can be simplified if desired, by replacing the FP-molecules by simpler distributions. Thus an H2 molecule is replaced by a distribution of electrons centered at the mid-point of the H2-bond and calculated from |$(ri,r2,Z*, A)|2. The electrons contribute U(T)HI to the total electron distribution n(r). We also

284


include the two nuclei which contribute />(r)#2 to the proton distribution p(r). These contributions TI{T)H2, P(T)H2 have no effect on the Poisson and xc-potentials (in the local density approximation) unless n(r)H2 n a s some overlap with the pseudomolecule at the origin whose detailed structure is being calculated. The existence of any overlap leads to hopping electronic states which cause modifications in the spectrum of the central particle. (iii) The output from the single center calculations will be electron distributions and FP-distributions. (iv) These are now fed into the multi-center calculations for pseudomolecules. We do not directly solve multicenter Kohn-Sham equations, but use the tight-binding approach where one or two carefully chosen parameters determining the bound state wavefunctions are optimized. The FPD and the continuous electron spectrum are interpolated from the single- and united-atom limits as discussed before, in the spirit of the Harris functionals. (v) The new electron distributions are introduced into step number (ii) above. (vi) Step (iv) is repeated and interaction potentials, binding energy.curves and the internal partition functions are calculated. (vii) A more refined evaluation of the FPD, using the interaction potentials obtained at this stage may be carried out using MD or integrals equations (N.B. The DFT equation for classical particles are formulated as an effective HNC equation, coupled to the electronic Kohn-Sham equation). (viii) A total free energy minimization is carried out to obtain the mixture composition x,-. (ix) These new X{ are fed into step (ii) and the procedure is repeated until self-consistency is attained. (x) At this stage the mixture composition, particle distribution functions, interaction potentials etc.. are completely converged. The equation of state, electrical conductivity (Perrot and Dharma-wardana 1987), thermal conductivity and other properties which could serve as a "diagnostic" of a phase transition can be calculated at this stage. Thus the free-energy minimization step (i.e., step viii) is carried out within the large iterative loop (steps ii to viii) where the adjustment of the internal electronic coordinate is carried out in consort with the evaluation of FPD and the structure and spectra of the pseudomolecules. This is exactly the "physical picture" of a microscopic theory. In this discussion we have not addressed a number of issues. We assume that the protons can be treated as classical particles. Also, when an ion is immersed self-consistently in a fluid containing continuum ("free") electrons, the continuum density of states becomes modified. The effec-


285

tive masses m* of the electrons near each pseudomolecule change from the usual value of m*/me = 1, and the existence of finite sized bound cores on each ion also affect the compressibility of the electron gas. All these subtle effects contribute to the electron partition function and have to be consistently treated. These issues are basically simpler than the pseudomolecular problems discussed here, and have been adequately treated elsewhere by us, in a recent (unpublished) study of Al-plasmas. 12.5 Conclusion We have presented a tractable microscopic approach to the study of complex fluids containing neutral and charged molecular species and electrons. The intuitive "chemical picture" based on data relevant to isolated molecules, hard spheres etc., is no longer needed. The model is able to take into account the electronic interactions between molecular species and the fluid environment. This allows room for three types of electrons, viz., (i) electrons fully localized on the molecular species, (ii) electrons which hop between the molecular species and the neighboring molecules in the fluid , and (iii) fully delocalized (i.e. "free") electrons. It is suggested that the prephase of the PPT is a system with a high population of hopping electrons, while the postphase is rich in delocalized electrons. The PPT is really a transition across the "mobility edge" and its physical reality depends on the sharpness of the associated percolation threshold. Acknowledgment We thank Gilles Chabrier for requesting us to contribute to this volume and for stimulating our interest in the plasma phase transition. References G. C.Aers and M.W.C. Dharma-wardana, Phys. Rev. A 29, 2734 (1984) B. H. Bransden and C. J. Joachain, Physics of atoms and molecules, Longman, London (1983) J. Davis and N. F. Mott, Theory of Disordered Solids Oxford, U.K. (1975) M. W. C. Dharma-wardana and F. Perrot, Phys. Rev. A 26, 2096 (1982); Phys. Rev. A 45, 5883 (1992) W. Ebeling and W. Richert, Phys. Lett, 108A, 80 (1985) E. K. U. Gross and R. M. Dreizler, Density Functional methods in Physics NATO ASI, II Ciocco, Italy (Plenum 1994) J. Harris, Phys. Rev. 5 31,1770 (1987) G. Herzberg, Molecular Spectra and Molecular Structure, van Norstand, New Jersey (1950); also K. P.Huber and Herzberg, Constants of diatomic molecules (van Norstand, 1979) P. Hohenberg and W. Kohn, Phys. Rev. B 136, 864 (1964)

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D. J. Hummer and D. Mihalas, Astrophys. J. 331, 794 (1988) W. Kohn and L. J. Sham, Phys. Rev. A 140, 1133 (1965). S. Lindqvist and N. March, ed\, Theory of the inhomogeneous Electron Gas (Plenum, New-York 1983). N. D. Mermin, Phys. Rev. A 137, 1141 (1965). D. Mihalas, D. J. Hummer, and W. v. Dappen, Astrophys. J. 331, 815 (1988) F. Perrot, Phys. Rev. £ 4 7 , 570 (1993); Phys. Rev. A 42, 4871 (1990) F. Perrot and M. W. C. Dharma-wardana, Phys. Rev. A 30, 2619 (1984); Phys. Rev. A 36, 238 (1987) F. Perrot, Y. Furutani and M. W. C. Dharma-wardana, Phys. Rev. A 41,1096 (1990) D. Saumon and G. Chabrier, Phys. Rev. A 46, 2084 (1992) D. Stauffer, Physics Reports, Scaling theory of percolation clusters 54, 1 (1979) D. J. Stevensen and E. E. Salpeter, Astrophys. J. Suppl. Ser. 35, 221 (1977) E. B. Wigner and H. B. Huntington, J. Chem. Phys. 3, 764 (1935)

13 The equation of state of fluid hydrogen at high density G. CHABRIER Laboratoire ie Physique, Ecolt Normak Superieure it Lyon, 69864 Lyon CtitxOl, France

Abstract We present a free energy model for fluid hydrogen at high-density and hightemperature. This model aims at describing pressure dissociation and ionization, which occur in partially ionized plasmas encountered in the interiors of giant planets and low-mass stars. The model describes an interacting mixture of H2,H,H+ and e~ in chemical equilibrium. The concentrations of H 2+ and H~ ions are found to be negligible for equation of state purposes. Our model relies on the so-called chemical picture approach, based on the factorization of the partition function into translational, internal and cohfigurational degrees of freedom. The present model is found to be unstable in the pressure-ionization regime and predicts the existence of a first-order plasma phase transition (PPT) which ends up at a critical point given by Tc = 15300 K, Pc = 0.614 Mbar, and pc = 0.35 gem" 3 . The transition occurs between a weakly ionized phase and a partially ionized (~ 50%) phase. Nous presentons un modele d'energie libre pour l'hydrogene fluide a haute densite et haute temperature. Le but de ce modele est de decrire la dissociation et l'ionisation en pression, telles qu'elles se produisent dans les plasmas partiellement ionises rencontres a l'interieur des planetes geantes et des etoiles de faible masse. Le modele decrit un fluide en interaction compose de H2,H,H+ et e~ en equilibre chimique. Les concentrations de H 2+ et H~ sont negligeables pour les calculs d'equation d'etat. Notre modele repose sur 287

288

Chabrier: Fluid hydrogen at high density

l'approche chimique, basee sur la factorisation de la fonction de partition en degres de liberte translationnel, interne et configurationnel. Le modele presente une zone d'instabilite dans le domaine d'ionisation en pression et predit l'existence d'une transition de phase plasma (PPT) du premier ordre, se terminant en un point critique donne par Tc = 15300 K, Pc = 0.614 Mbar, et pc = 0.35 gcm~3. La transition a lieu entre une phase faiblement ionisee et une phase partiellement ionisee (~ 50%). 13.1 Introduction The similarities between hydrogen and alkali metals lead Wigner and Huntington (1935) to suggest that pressure-ionized hydrogen would behave like a monovalent 'metal' even at zero-temperature, and it has often been argued that first-order phase transition must occur between the two states, given the large difference between the molecular and the metallic states (Stevenson and Salpeter 1977; Ebeling and Richert 1985). Most of the recent investigations have focused on the zero-temperature or the room-temperature, where static compression experiments are now available above the megabar domain. These results indicate that a transition between a molecular and a semi-metal state occurs around 2 Mbar, but true metallization has not been observed unambiguously yet. This state is believed to occur around 2-3 Mbar. Few models exist however at high-temperature, in the fluid range, where shock-wave experiments have clearly establish the stability of the fluid molecular phase up to 0.8 Mbar (Nellis et al. 1984). Fluid hydrogen is the main component of stellar interiors and atmospheres. In most of astrophysical objects, hydrogen is ionized by temperature, but pressure ionization occurs in the interior of giant planets, brown dwarf stars and partially in the outermost layers of white dwarfs and low-masss stars. The recent discovery of global oscillations in Jupiter (see Mosser, these proceedings), as well as the recent achievements of helio- and astero-seismology (see Dappen and Fontaine, these proceedings) give us new information on the structure of these objects, and then on the properties of matter under extreme thermodynamic conditions. This stress the need for a correct calculation of fluid hydrogen at high-density, including a proper treatment of pressure dissociation and ionization. A simplified phase diagram of hydrogen at high temperature is shown on Figure 1, with the internal temperature profiles of a few dense astrophysical objects. In section II, we give a short presentation of our free energy model, which has been presented extensively elsewhere (Saumon and Chabrier 1991,

J

289


6

5

iog,oT[K] Fig. 13.1 Simplified (/>,T) phase diagram for hydrogen. A few physical regimes are identified : above the line F = 1, where F is the usual coupling parameter, the classical ionic plasma is strongly coupled whereas correlations are dominant in the quantum electron plasma below the line r, — 1, where r, is the mean inter-electronic distance in unit of Bohr radius. Electrons are degenerate above the line 0 = kT/ej = 1, where ep is the electron Fermi energy. Protons are classical below the line A, = 1, where A,- is the De Broglie wavelength in unit of the mean inter-ionic distance a. The curve F = 178 denotes the crystallization line of the H + plasma. The various temperature profiles are characterise of the interior of Jupiter (J), a brown dwarf (BD), a ZZ-Ceti white dwarf (WD) and the Sun (S).

1992). In section III, we discuss in detail the PPT predicted by our model whereas a comparison with other models is discussed in Section IV. Section V is devoted to the astrophysical applications and Section VI to the conclusion. 13.2 Description of the free energy model The model relies on the so-called chemical picture, in the sense that we assume the existence of independent, bound configurations such as H atoms, H2 molecules, interacting with pair potentials. At densities corresponding to pressure ionization, such a scheme is erroneous and the concept

290


of individual pair potential fails, requiring the use of quantum-statistical many-body theory, i.e. a physical picture where only fundamental particles (electrons and nuclei) exist (see the reviews by Rogers, Alastuey and Perrot and Dharma-wardana). In particular, our approach does not take into account the possibility of excitonic states, i.e. clusters or pseudo-atoms and pseudo-molecules, as described in Perrot and Dharma-wardana (these proceedings). Although formally exact, the physical picture, however, involves either diagramatic expansion which converge only at low-density or hightemperature (see the reviews by Rogers and Alastuey), or, when extended to higher-density, involve a coupled treatment of classical and quantummechanical N-body theories, which renders practical applications for the calculation of astrophysical EOS nearly impossible. In view of these difficulties, the chemical picture, and its inherent factorization of the partiction function in terms of different particle interactions, remains a very powerful method. It can be view as the best compromise between the rigorous treatment and the practical application. It is why it is important to compare the results obtained with "chemical" models with the ones derived from "physical" ones. Our EOS consists of a general free energy model which applies in the regime of partial temperature- or pressure-ionization, and which reduces to a so-called "neutral" model and "plasma" model respectively at lowdensity /low-temperature and at high-density and/or high-temperature. 13.2.1 Model for neutral hydrogen At low-density (p £ 1 gem" 3 ), low temperature (T^ 104 K), hydrogen is adequately described as a mixture of H atoms and H2 molecules. The concentrations of H~ and H 2+ ions are found to be negligible (< 10~3) for EOS purposes. Because all particles are very nearly classical in this regime, we can factorize the partition function and treat the small quantum effects with a semi-classical approximation. If we make the additional assumption that the internal levels of atoms and molecules are only weakly affected by the presence of nearby particles, as suggested by available experimental results up to electronic densities as large as Ne/V ~ 1021 cm" 3 (Weise et al. 1972, Grabowski et al. 1987, Hashimoto and Yamaguchi 1983), the Helmoltz free energy separates into ideal, configurational, internal, and quantum contributions : F(NH, NH2 ,V,T) = Fid + Fconf + Fint + Fqm

(1)


291

13.2.1.1 the configuration term This term arises from the interaction between the different particles in their ground state. Computation of these interactions requires the knowledge of three interaction potentials 4>H2-H2, H-H and <j>H2-H- For H2-H2 use an effective potential derived from shock tube experiments (Nellis et al. 1984; Ross et al. 1983) which implicitly includes many-body effects. Since no similar experimental data exist for 4>H-H and 4>H2-HI we have used ab-initio potentials (Kolos and Wolniewicz 1965; Porter and Karplus 1964). We treat the spin dependence of the H-H interaction by averaging the interaction potentials of the singlet and triplet states; the resulting H-H potential has no bound states. The three potentials have been fitted by generalized Morse potentials. The configuration free energy FCOnf is derived from theua interaction potentials in the framework of the WCA fluid perturbation expansion (Weeks, Chandler and Andersen 1971). In this theory, the interaction potential is split into a repulsive reference potential 4>re*(r) and a weak, attractive perturbation potential 4?ert{r). We approximate the free energy of the reference system by that of a hard sphere fluid, which is known analytically (Mansoori et al. 1975), whereas the contribution of the perturbation potential is given by the first term of the free energy expansion (High Temperature Approximation) :

Fconf(N,V,T) = f 4%p{r)g5${r)d?

J

(2)

Here the ga,p(f) are the hard sphere pair correlation functions (Griindke and Henderson 1972) and a\ and o-i are the density and temperature-dependent hard sphere diameters determined thermodynamically by the WCA criterion (Weeks, Chandler and Andersen 1971). The standard WCA scheme, derived originally for liquid state theory, has been extended for this particular approach to high-density and high-temperature (Saumon, Chabrier and Weis 1989). The excess internal energy and pressure derived from this expansion scheme agree within less than 3% with MC simulations for the density and temperature range of interest (Saumon, Chabrier and Weis 1989). This assesses the validity of the configuration energy (2) for the H-H2 mixture.

we

292


13.2.1.2 The Internal Free Energy The effect of near-neighbor interactions on the internal structure of bound species is essential to a correct description of pressure dissociation and ionization. We have used in our model the so-called occupation probability formalism (OPF) derived by Hummer and Mihalas (1988). In this formalism, the internal free energy reads :

Fint = -kTLn J2 Na^2uaigQie-e^kT a=l

(3)

i

where i runs over all internal states of species a and u)ai,gai and €ai are respectively the occupation probability, the multiplicity and the unperturbed energy of state i. The density dependent uai are computed from the configuration term Fconf in the free energy. This ensures consistency of both the interactions and their effects on the IPF. It also provides a smooth cut-off of the IPF, and therefore a plausible pressure dissociation/ionization effect. Moreover the present method uses unperturbed energy eigenvalues and does not invoke hypothetical energy level shifts of doubtful validity. As a matter of fact, such shifts have been shown to be too small to be significant, as mentioned above (Wiese et al. 1972, Grabowski et al. 1987, Hashimoto and Yamaguchi 1983). In practice, however, one must resort to a linearization of Fconf to compute the occupation probability (see Hummer and Mihalas (1988) for details). Calculations including terms beyond the density-linear term have been computed recently for helium at high-density (Aparicio and Chabrier 1994). In addition our occupation probabilities include neutral particle interaction only. The effect of charged particles, i.e. Stark ionization, requires a knowledge of the plasma microfield distribution at high-density, which complicates tremendously the calculations. The effect of the microfield will be discussed in detail later in the paper. In our treatment of the IPF of H2, we have included all vibrational and rotational levels of each bound state of the molecule (Hiiber and Herzberg 1979). The term Fqm in Eq.(l) is the quantum contribution to the free energy, which is always a weak perturbation of the classical free energy in the domain of interest for the hydrogen EOS, and then has been calculated to the first non-vanishing order in the Wigner-Kirk wood h2 expansion.


293

13.2.2 Model for fully ionized hydrogen For kT £ lRyd or p £ 2gcm~3 (corresponding to ra ~ 1, where rs is the mean inter-electronic spacing in units of Bohr radius a 0 ), the fluid is a fully ionized electron-proton plasma. At these densities, the mean electron-ion potential energy E{e = e2 /rsao is smaller than the electron Fermi energy Ep so that the plasma can be described as a superposition of an electronscreened ionic fluid and a rigid electron background (Ashcroft and Stroud 1978). Under these conditions, the free energy of the plasma reads : F = Fid - NkTIn I e/3U —1, so that it is not possible to estimate the chemical equilibrium in this dense fluid with simple theories. At even higher densities, near log/) = 0 for hydrogen, the mean distance between H atoms becomes comparable to twice the value of the Bohr radius and the electronic wave functions of neighboring atoms overlap. The electrons are forced into unbound states and the fluid becomes a pressure ionized plasma. A calculation of pressure ionization by Saumon and Chabrier (1992) reveals that pressure ionization t Throughout this work, log T is the logarithm of the temperature in K, and log p is the logarithm of the mass density in g/cm 3 .

310

Saumon: Hydrogen equations of state

£ o 00

O

-10

-

Fig. 14.1 Phase diagram for hydrogen. Heavy solid lines separate various physical regimes. Below the Praa = Pga, line, the pressure of the photon gas exceeds that of the matter (H+ and e). Electrons are degenerate above the 0 = 1 line, and protons form a strongly coupled plasma above the T = 1 line. The thick curve labeled PPT shows the metastable region of the Plasma Phase Transition. The abundance of atomic hydrogen, H, is 50% along the dashed curve which indicates regimes of partial dissociation and ionization. The dotted curves are interior models for a) Jupiter, b) 0.3 MQ main sequence star, c) the Sun, d) the outer hydrogen layer of a T = 12 500 K DA white dwarf and e) a 15 MQ main sequence star.

of hydrogen may not be a gradual process at all temperatures but could occur discontinuously through a first order phase transition, the so-called plasma phase transition (PPT). The metastable region of this transition is shown by the curved labeled "PPT" and ends at a critical temperature of logTc = 4.185. Two important issues pertaining to the plasma are the degree of electron degeneracy and the strength of the Coulomb coupling between the charged particles. Above the solid line labeled 9 = 1, where 9 = eF/kT, the Fermi energy of the electrons €F is larger than kT, and they are therefore degenerate. Protons, on the other hand, remain classical over most of this diagram and in all astrophysical conditions (except in neutron stars). Above the line T = 1, non-ideal Coulomb effects play an important role as the electrostatic potential energy between two protons, e2/a, where e is the


311

charge quantum and a is the mean interparticle distance, becomes larger than their kinetic energy, kT. At intermediate temperatures (logT « 5) and densities of log p « 0, temperature and pressure ionization are of comparable importance. In this regime, thermal excitation of hydrogen atoms is significant and they are immersed in a moderately coupled plasma (F « 1) where electrons are partially degenerate (6 « 1). This regime is particularly difficult to treat as the internal levels of the atoms are strongly perturbed by the surrounding plasma. For most elements, this is, along with pressure ionization, the regime where equations of state are most unreliable. At low densities and high temperatures, radiation pressure Prad becomes larger than the gas pressure, PgM- Finally, the upper left part of Fig. 1 represents conditions which are not realized in astrophysical contexts, where hydrogen is a high-T molecular solid or possibly forms a Coulomb lattice. Interior models of various hydrogen-rich objects are shown by dotted curves in the density-temperature plane of Fig. 1. The gaseous envelope of Jupiter is shown by the curve labeled 'a.' The envelope is dominated by molecular hydrogen and it goes through the region of pressure ionization. If the PPT calculated by Saumon and Chabrier (1992) occurs in nature, it should also be found in the envelope of Jupiter. Just below the PPT, the dense molecular fluid becomes strongly non-ideal due to the strongly repulsive intermolecular forces. Curves 'b', 'c', and 'e' represent main sequence stars with masses of 0.3, 1 and 15 Af©, respectively, where M© is the mass of the Sun. The 15 M© star has the simplest EOS physics. It is fully ionized throughout its interior and the plasma is very weakly coupled (F 3.7. This is somewhat surprising if we consider that the SESAME 5251 hydrogen EOS is actually a deuterium EOS scaled in density. Differences in pressure reach a maximum of 25% in the regime of pressure ionization. In the regime of temperature ionization differences are as high as 40%. At the lower temperatures where molecules dominate the EOS, systematic differences of « 6% are found in U (Saumon and Van Horn 1987). This arises from the density scaling procedure which is not appropriate in the molecular phase. The energy levels of the molecule depends on the moment of inertia and the reduced mass of D2 which are twice as large as for H2. Rogers (1981) has developed an EOS with an approach entirely different from FMIN, using an activity expansion which considers only protons and electrons interacting with the Coulomb potential. Bound states (atoms) arise naturally in this approach and are not treated as a separate chemical species, as in the FMIN method. This approach is very rigorous and fundamental (Rogers 1994). Over the (/>, T) domain where this complex method can be presently solved, it leads to a most accurate EOS. While we have not yet compared it with the SC EOS, it has been compared with the MHD EOS under the conditions found in the solar envelope (see Dappen 1994 and Rogers 1994). The two equations of state are in extremely good agreement, with differences of less than 0.1% in the second derivatives of the free energy. While such differences are important when comparing the computed solar oscillation spectrum to the wealth of extremely precise data, they are completely negligible in all other astrophysical situations. It is very satisfying that two equations of state based on entirely different approaches should agree so well. This indicates that our understanding of the EOS of normal stellar material is now excellent, at least over some parts of the phase diagram.

316


14.5 About the Saumon-Chabrier EOS The free energy model underlying the SC EOS is described in details in Saumon and Chabrier (1991, 1992) but the EOS itself is not yet available (Saumon, Chabrier and Van Horn 1993). The model is summarized in the review by Chabrier (1994) and a few additional remarks relevant to the present comparison follow. Pressure ionization received particular attention in the SC EOS calculation. It was found that the adopted free energy model (and a number of variants) becomes thermodynamically unstable and predicts the existence of a first order phase transition between a mostly molecular phase and a dense, partially ionized phase. This plasma phase transition (PPT) is shown in Fig. 1. It terminates at high temperature at a critical point located at Tc = 15300K, Pc = 6.14 x 10 u dyn/cm 2 , and pc = 0.35g/cm3. Pressure ionization is a most difficult problem in EOS calculations and much remains to be said on this challenging topic. There is currently no experimental result which bears on the existence of the PPT. To allow for the possibility that the PPT is not realized in nature, there is an "interpolated" version of the SC EOS where the discontinuities associated with the PPT have been smoothed by interpolation. It is otherwise identical to the SC EOS with PPT. This interpolated version used for the comparisons in § 6. The interpolated region has an irregular shape but it extends roughly over 3.50 < logT < 4.78 and -0.5 < logp < 0.5. Ideally, the interpolation of P and the entropy S (or P and U) should be constrained by the requirement of thermodynamic consistency, which reflects that P and S are not independent quantities but derive from the same thermodynamic potential, in this case the Helmholtz free energy. Fontaine, Graboske and Van Horn (1977) applied this constraint when interpolating across the regime of pressure ionization. In the case of the SC EOS, however, it was found that the requirements of 1) continuity of P, S and their derivatives at the boundaries of the interpolation region and of 2) thermodynamic consistency overconstrain the interpolation. This difficulty can be avoided by widening the density range of the interpolation but only to an unacceptable degree where parts of the EOS table believed to be reliable (based on experimental data and an assessment of the model) would be replaced by less accurate, interpolated values. This suggests that while there may not be a PPT in hydrogen, pressure ionization probably occurs rather suddenly. Reliable EOS values were preserved at the cost of losing thermodynamic consistency and P and S were interpolated separately along isotherms over as narrow a density range as possible.


317

The free energy model underlying the SC EOS can be improved upon and ameliorations are being considered. The most important of which involves the effect of charged particles on the bound states of hydrogen atoms. Interactions with neighboring particles, charged and neutral, affect the number of bound states, or the internal partition function, of atoms and molecules. A proper treatment of this effect is essential for an accurate description of partial dissociation and ionization, particularly at larger densities (logp > —3). While the activity expansion of Rogers (1981) accounts for this naturally and rigorously, the FMIN method is only weakly constrained in this respect. In its current form, the free energy model developed by Saumon and Chabrier (1992) accounts only for the effect of neutral particles on bound states (by an excluded volume effect). In reality, neighboring charged particles also affect the bound states by inelastic collisions with bound electrons and also through the fluctuation micro-electric field induced by their thermal motion. This microfield has the effect of a time dependent perturbation on the Coulomb potential of the nucleus and can induce Stark ionization of the upper levels of an atom. Collisions and microfield effects on hydrogenic atoms are discussed in great details in Hummer and Mihalas (1988) who conclude that for log/> < —1.5, the microfield is the dominant mechanism. Being caused by random thermal motions, the fluctuating microfield is described by a statistical distribution. Hummer and Mihalas have adopted the F = 0 Holtzmark distribution. We have found that this distribution, which does not account for the correlations which arise between charged particles at T > 0, has much too strong an effect on the IPF and leads to spurious results for F « 1. Generating microfield distributions for finite F is computationally involved and a suitable, parametrized form was not available when the SC EOS was computed. As a consequence, the effect of the microfield is ignored altogether until an adequate distribution function becomes available. The net effect of this omission is that as the gas becomes mostly ionized by temperature, the IPF is less affected by the neighboring particles than when it was surrounded by neutral particles. This creates a long tail of residual atoms in the partial ionization zone. We will return to this point in § 6. Two thermodynamic surfaces from the SC EOS are show in Figs. 3 and 4. Only second derivatives of the free energy are shown because they display the various physical regimes more clearly than first derivatives such as P and 5". They also amplify defects in F they are very useful to reveal flaws in an EOS as well as its degree of smoothness. Note that the MHD free energy model is sufficiently simple to allow for analytic differentiation of F and their EOS is consequently very smooth. As emphasized in the reviews

318


Fig. 14.3 Inverse compressibility, xP = 01og.P/dlogp|x, for the (p,T) range covered by the Saumon-Chabrier EOS (interpolated version). The (p, T) grid shown is that of the original tabular data and no smoothing has been applied.

by Fontaine (1994) and Dappen (1994), smoothness of the EOS can be more desirable than accuracy in the context of non-adiabatic stellar pulsations. Figure 3 shows Xp — dlogP/dlogpfr, which measures the stiffness of the EOS. For an ideal gas, Xp = 1> as can be seen over most of the lowdensity part of the figure. The photon gas pressure depends only on T and Xp = 0 in that limit. The degenerate electron gas is less compressible than the Maxwell-Boltzmann ideal gas and Xp Tlses t o a plateau at 5/3. The two shallow "valleys" seen on either side of logT = 4 are due to molecular dissociation and ionization. They clearly separate the regions dominated by H2, H and H + , respectively. The very steep rise seen at low-T and highp is caused by the repulsive core of the H2-H2 interaction potential. Like

319


log Fig. 14.4 Adiabatic gradient, Vad = dlog7'/01og.P|.s for the (p,T) range covered by the Saumon-Chabrier EOS (interpolated version). The (p,T) grid shown is that of the original tabular data and no smoothing has been applied. all liquids, dense fluid H2 is relatively incompressible, a property reflected by the high value of Xp- There are a few spurious features caused by the interpolation procedure at intermediate T and high p. As indicated in Fig. 2, the EOS does not extend to the low-T and high-p limit where Xp — 0. Figure 3 is best interpreted in reference to the phase diagram shown in Fig. 1. Most of the physical regimes discussed above can be identified in Fig. 4 which shows the adiabatic temperature gradient, V»d = d\ogT/dlogP\s. When expressed in terms of p and T, V ^ is a function of the four second

derivatives of F: dP/dp\T, dP/dT\p, dS/dp\T and dS/dT\p. It therefore combines all the defects and noise found in the second derivatives. This is

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the origin of the several spikes seen in the interpolation region. Other peaks and oscillations found along the high-/) border of the EOS are caused by edge effects in the table. 14.6 The EOS comparison Even by limiting the comparison to four EOS, it is not possible to do justice to the great efforts which went in their development or to review the merits of each one of them. The discussion will be limited to the areas were the largest differences arise. The introduction of spurious errors was kept to a minimum by avoiding numerical interpolation in the tables as much as possible. For that purpose, six isotherms common to all four EOS were selected: logT = 3.70, 4.10, 4.50, 5.30, 6.10, and 6.90 (Fig. 2). The last three isotherms are not tabulated by MM and the necessary T-interpolation was performed with a program provided with the table. The figures show the density points of the original tables connected by a straight line. The EOS were not "smoothed", however, it has been remarked before that the FGVH EOS has a number of "bad points" where the second derivatives of the free energy show anomalous behavior. Since these points are isolated, they must not arise from deficiencies in the underlying thermodynamic description but represent some localized numerical quirk. A few of these points are found in the six isotherms under consideration and the discordant values were corrected by a simple interpolation in density. These points are located at (logr,logp): (4.10, -5.667), (4.50, -3.667), (5.30, -2.333) for Vad. The quantities compared are logP, logf/ and Vad for all six isotherms. Exceptions are the MHD EOS which is not shown for logT = 6.10 and 6.90, the table available being limited to logT < 6, and the MM EOS which gives only logP, Vad and C p , the specific heat. This last quantity is not used in the present comparison. Note that all quantities shown here are taken directly from the EOS tables and are not constructed from other quantities by using thermodynamic identities, for example. This avoids introducing potential errors due to thermodynamic inconsistency in the EOS or numerical inaccuracies in the procedure. In all four cases, the zero of energy is chosen as the ground state of the H2 molecule and the contribution of the photon gas is included. The four EOS are compared in Figures 5-9 where it is readily apparent that the differences can be substantial. As a point of reference, the SC EOS indicates that P and U are within 1% of their ideal gas value for log/9 < —2. It is easy to verify that the non-ideal terms are very small at this density


1

321

1

14 -

——"

a

12 -

W /

0,

on O

10 -

8

-4

/

• sc - FGVH MHD - MM

^

1

1

1

-3

-2

-1

_

1 0

3

log p (g/cra ) Fig. 14.5 Comparison of pressure isotherms from the four equations of state. The isotherms are (from top to bottom): logT = 6.90, 6.10, 5.30, 4.50, 4.10 and 3.70. by comparing the volume occupied by atoms (or molecules) to the total volume. Surprisingly, significant differences are found even for densities below logp = — 2.

14-6.1 The pressure Figure 5 shows the pressure from the four EOS along the six isotherms. At lower and at higher densities than shown in this figure, the agreement is satisfactory. The two hottest isotherms correspond to a fully ionized gas of H + and e interacting weakly in the Debye-Hiickel limit. The photon pressure dominates gas pressure when P becomes independent of p along the log T = 6.90 curve. The agreement is excellent in this relatively simple

322


regime but severe divergences are found at lower temperatures. Curiously, the MM EOS systematically overestimates P at low densities, where the gas is ideal for all practical purposes. This is most likely due to an overestimate of the degree of dissociation and perhaps ionization which arises from their treatment of the IPF of H and of H2. Their IPF for H2 has been corrected since we obtained the MM table in 1987 (Mazzitelli 1993). For the three lowest isotherms, the MHD and FGVH EOS predict much higher pressures than either the SC or the MM EOS at moderate densities. For temperatures up to logT = 3.6, the SC EOS reproduces experimental results and can be considered as a reference for this comparison. The high pressures of FGVH and MHD are caused by the hard sphere potential used to model the interactions between neutral particles. This potential qualitatively models the strongly repulsive cores of the actual potentials, but being infinitely repulsive, it fails to describe the softness of the repulsion. This feature of neutral-neutral interactions becomes important at high densities. The hard sphere potential is too repulsive at high densities and leads to overestimated pressures even in a regime where the gas should be nearly ideal. The authors of the MHD EOS point out that their EOS should be used for log/> < —2, a safe limit at low-T in view of the above observation. In the regime of pressure ionization (—0.5 < log/9 < 0.5), the SC, MM and FGVH EOS can differ by up to a factor of 2. In all three cases, thermodynamic quantities were smoothly interpolated between a low-density and a high-density regime where the authors felt that their respective EOS were reliable.

1^.6.2 The internal energy Most of the features discussed above can also be seen in the internal energy U, shown in Fig. 6. Again, we see that for logT < 4.50 and \ogp > —2, the hard sphere model used by FGVH and MHD leads to an overestimate of U. The logp = — 2 limit recommended by MHD is a sensible choice for U as well as for P. The two intermediate isotherms illustrate the importance of a careful treatment of the influence of neighboring particles on the IPF. For the logT = 4.50 isotherm, SC lies above MHD and the reverse is true for logT = 5.30. At these low densities, characteristic of the ideal gas, this arises from differences in the degree of ionization, which is directly affected by the IPF of atomic hydrogen. At these temperatures, thermal excitation of H becomes significant and the chemical equilibrium depends on how

323


cm u

(90

o

13

-

12 —

-2

0 log p (g/cm 3 )

Fig. 14.6 Comparison of internal energy isotherms. The isotherms are (from top to bottom): logT = 6.90, 6.10, 5.30, 4.50, 4.10 and 3.70. many states are allowed in the IPF sum.f This effect was not visible in the pressure because it is relatively insensitive to excitation energies of bound species. At logT = 4.50, the degree of ionization is sufficiently low for the finite "size" of atoms to be the main non-ideal contribution. This, in effect, is an excluded volume interaction which removes the upper levels of the IPF to ensure that the atoms do not "overlap." MHD adopted a fixed and somewhat arbitrary diameter for the H atom in its ground state (1.06 A) while SC use a thermodynamic criterion (Saumon and Chabrier 1991) to compute a temperature and density dependent value ranging from 1.1 to t The SC and MHD EOS do not use a cut off in the IPF sum but a gradual removal of bound states based on the occu pation probability formalism presented in Hummer and Mihalas (1988). It is nevertheless useful to think in terms of a sharp cut off in the present context.

324


about 1.6 A. Fewer states are retained in the IPF when the hard sphere diameter is larger, favoring a higher degree of ionization in the SC EOS and a larger U. Because the SC EOS uses more realistic interaction potentials between neutral particles and a thermodynamic criterion to obtain the hard sphere diameters of H and H2, it is more reliable in this regime than the MHD EOS. The situation is quite different along the log T = 5.30 isotherm where the degree of ionization is high and atoms are surrounded mostly by charged particles. As discussed in § 5, the motion of the ions and electrons induces a fluctuating micro-electric field which can cause Stark ionization of the upper levels of the atom, thereby removing them from the IPF. Since this effect is missing in the SC EOS, the IPF retains too many states and the degree of ionization as well as the internal energy are underestimated. According to Fig. 6, this effect is not very large, but the MHD EOS is nevertheless more accurate in this regime. Along the logT = 5.30 isotherm, the MHD and the SC EOS differ most notably for —1 < logp < 1. Under these conditions, pressure ionization occurs where thermal excitation of the atoms is large. We discussed this regime in § 2. While the MHD model is well beyond its limit of validity (log/* < — 2), none of the EOS presented here can be considered reliable in this difficult regime. 14-6.3 The adiabatic gradient As discussed above, the second derivatives of the free energy are very sensitive to the choice of thermodynamic model and display itsflawsprominently. The adiabatic temperature gradient is particularly interesting since it forms the basis of the Schwarzschild criterion for convective instability in stars. The six isotherms for the adiabatic gradient are shown on Figs. 7-9. Except in a few well known limits, figures of Vad are particularly difficult to interpret physically. We will limit the analysis to listing the failures and problems with each EOS. Figure 7 shows the two lower isotherms over a wide density range. The overall wavy structure is caused by partial dissociation and ionization. A number of features are immediately apparent: • Even at very low densities where the gas is ideal, the agreement is not perfect. Differences of 10% are commonplace. • The FGVH EOS can be very noisy. • The MHD EOS shows pathological behavior for log/) > —2, once again reinforcing their warning about not using their EOS above this limit.

325


0.50

0.40

0.30

>

a

0.20

-

0.10

log P (g/cm ) Fig. 14.7 Comparison of adiabatic gradient isotherms showing the logT : 3.70 and 4.10 isotherms. • The SC EOS is not very smooth in the regime of the fully ionized plasma (log/) > 0.5). The next two isotherms are displayed on Fig. 8. Again, the wavy structure seen for logT = 4.50 is due to partial ionization. Hydrogen is nearly fully ionized everywhere along the logT = 5.30 isotherm and the drop to = 0.25 at very low densities is due to the photon gas. We find that: • There are still differences in the ideal gas regime, but they are below the 10% level. • The FGVH EOS appears smoother in this regime • Above log/7 = —1.5, the MHD EOS shows pathological behavior along both isotherms.


326

0.50

0.40

-

0.30

-

0.20

-

0.10

-

-8

-6

-4

-2

log P (g/cm 3 ) Fig. 14.8 Comparison of adiabatic gradient isotherms showing the log T — 4.50 and 5.30 isotherms. • For the logT = 4.50 isotherm, the MM EOS shows a "phase lag." This indicates an ionization zone which is displaced to comparatively higher densities. This originates in their treatment of the IPF. • At the high-density end of these isotherms, T > 10 and 0 < 1, conditions under which the Coulomb interactions are strong. In FGVH and MM, these are described with a Thomas-Fermi-Dirac model and both show Vnd rising as the density is increased. On the other hand, SC use a screened onecomponent plasma model (SOCP, Chabrier 1994), a much more accurate description of the plasma, and find that V«i decreases along the isotherm. The SC EOS remains rather noisy in this regime. Finally, Figure 9 shows the two hottest isotherms. For a pure photon gas, Vad = 0.25 and it approaches 0.4, the value for a non-interacting (ideal), classical, monoatomic gas, as the pressure of the plasma comes into play.

327


0.50

0.40

1

T

1

-

6. 10

/ /

0.30 —

/

1 '

1 '

1

1

/*"" / f // / / / / 6 .90 / /

0.20 — -

0.10

0.0 -8

SC FGVH MM

-

, 1

1 ,

1 .

•

1

- 6 - 4 - 2 0 log p (g/cm 3 )

Fig. 14.9 Comparison of adiabatic gradient isotherms showing the logT = 6.10 and 6.90 isotherms. Both of these limits are readily apparent on this figure. The adiabatic gradient of a mixture of photons and non-interacting protons and electrons can be calculated analytically (Cox and Giuli 1968, § 9.17), a result accurately reproduced by FGVH and SC. The divergence of the MM curves from the analytic expression cannot be explained on physical grounds. At high densities, Vad drops below 0.4 due to relatively weak to moderate Coulomb interactions (r < 1). Both the FGVH and SC EOS show a downward trend in Vad and agree quite well while the MM EOS displays an increase similar to that observed in Fig. 8. Because of its strong connection with convective instability, Vad plays an important role in models of stellar interiors and envelopes. Figures 7-9 show differences of the order of 10% in the ideal gas regime of partial dissociation and ionization, underscoring the sensitivity of Vad to the treatment of the

328


states in the IPF. When strong non-ideal effects come into play, it appears that the adiabatic gradient remains a rather poorly determined quantity. 14.7 Concluding remarks This exercise of comparing several equations of state developed for applications to astrophysical problems (mainly stellar interiors) reveals that the situation is not as satisfactory as is commonly assumed. Much progress has been accomplished over the time span represented by these EOS: the SC and the MHD EOS represent considerable improvement over the older FGVH and MM EOS. This is due in part to new high-pressure experiments which probe the H2-H2 potential to smaller interparticle separations, the development of a solid knowledge of dense plasmas through numerical simulation and a more acute awareness of the importance of consistency between the treatment of the internal partition function and the interactions between particles. It also shows that each of these EOS has flaws or limitations, most of which can be addressed in the near future. The most challenging areas remain associated with partial dissociation and ionization for hydrogen. The treatment of temperature ionization with the FMIN method has improved considerably in the last few years but we have seen that none of the EOS presented here is truly satisfactory in this respect. The more rigorous activity expansion technique may provide a definitive treatment of temperature ionization. On the other hand, pressure ionization remains by far the most poorly understood phenomenon and maintains a shroud of uncertainty over a part of the phase diagram which is important for low-mass stars, brown dwarfs and most critically, the jovian planets. The calculation of such equations of state is a complex problem. Those presented here each required a few man-years of effort, and still they display flaws and problems of various importance. This raises strong doubts about the validity of the much simpler and often crude equations of state used in many astrophysical problems. Any problem calling for an equation of state should first be cast in a phase diagram (Fig. 1) to determine the relevant physical regimes, the magnitude of non-ideal effects and whether partial ionization and dissociation are expected. In numerous cases, the EOS is sufficiently simple for semi-analytic treatments to be adequate. However, if non-ideal effects are expected and the accuracy of the final result is important, there is no justification for not using the appropriate tabular equation of state. In conclusion, astrophysicists should be more critical of the equations of state they use.


329

I am very grateful to F. D'Antona, and to D. G. Hummer, who kindly provided the MM and the pure hydrogen MHD EOS tables, respectively. I thank G. Fontaine who generated Figs. 3 and 4 and provided the white dwarf model shown in Fig. 1, and F. J. Swenson for sending me the 0.3 MQ main sequence star model (Fig. 1.). This research was supported in part by NSF grant AST-8910780 and by NASA grant HF-1051.01-93A from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS526555.

References Chabrier, G., these proceedings (1994) Chabrier, G., Saumon, D., Hubbard, W.B., and Lunine, J. Ap. J., 391, 817 (1992) Clayton, D. D., Principles of Stellar Evolution and Nucleosynthesis, 2 nd Ed., (Chicago: University of Chicago Press) (1983). Cox, J.P., and Giuli, R.T. Principles of Stellar Structure, Vol. 1, (Gordon and Breach: New York) (1968) Dappen, W., these proceedings (1994) Dappen, W., Mihalas, D., Hummer, D.G., Mihalas, B.W. Ap. J., 332, 261 (1988) Fontaine, G. these proceedings (1994) Fontaine, G., Graboske, H.C., Jr., and Van Horn, H.M. Ap. J. Supp., 35, 293 (1977) Graboske, H.C., Jr., Harwood, D. J., and Rogers, F. J. Phys. Rev., 186, 210 (1969) Hubbard, W.B., and DeWitt, H.E. Ap. J., 290, 388 (1985) Hummer, D.G., and Mihalas, D. Ap. J., 331, 794 (1988) Kerley, G.I. Phys. Earth Planet. Inter., 6, 78 (1972) Lamb, D.Q. PhD Thesis, University of Rochester (1974) Lamb, D.Q., and Van Horn, H.M. Ap. J., 200, 306 (1975) Magni, G., and Mazzitelli, I. Astron. Astrophys., 72, 134 (1979) Marley, M.S., and Hubbard, W.B. Icarus, 73, 536 (1988) Mazzitelli, I. private communication (1993) Mihalas, D., Dappen, W., and Hummer, D.G. Ap. J., 331, 815 (1988) Rogers, F.J., Phys. Rev., A24, 1531 (1981) Rogers, F.J., these proceedings (1994) Saumon, D., and Chabrier, G. Phys. Rev. A, 44, 5122 (1991) Saumon, D., and Chabrier, G. Phys. Rev. A, 46, 2084 (1992) Saumon, D., Chabrier, G., and Van Horn, H.M. in preparation for Ap. J. (1993) Saumon, D., and Van Horn, H.M. in Strongly Coupled Plasma Physics, F.J.. Rogers and H. E. DeWitt, Eds. (Plenum: New York), p. 173 (1987) Stevenson, D.J. Phys. Rev. B, 12, 3999 (1975) Stevenson, D.J., and Salpeter, E.E. Ap. J. Suppl, 35, 229 (1977)

15 Strongly Coupled Ionic Mixtures and the H/He EOS Hugh E. DcWitt Lawrence Livermore National Laboratory, Livermore, CA 94550 Abstract This paper summarizes recent work on the strongly coupled OCP and Binary Ionic Mixture equation of state and other thermodynamic quantities in white dwarf interior conditions for both fluid and solid phases with the assumption of a uniform background. Conditions for phase separation of different elements in fluid or solid phases is strongly dependent on deviations from the linear mixing rule which gives the equation of state as an additive function of the OCP equation of state. These deviations turn out to be small (a few parts in 105) and always positive including the case where the fraction of the higher Z component approaches 0. Also the equation of state of strongly coupled light elements (H and He particularly) obtained from simulations with a linear response description of the electrons is given for conditions appropriate to brown dwarf star interiors. Recent Livermore work on a band structure calculation of the enthalpy of H and He mixtures under jovian conditions is discussed. This work leads to a prediction of a high temperature (15000 °K) for miscibility of He in ionized H at 10 Mb.

Resume Ce papier resume l'ouvrage recent sur le OCP a fort couplage et sur l'equation d'etat et d'autres quantites thermodynamiques pour le melange binarire ionique aux conditions interieure des nains blancs. Les conditions pour seperation de phae dan les elements divers dans l'6tat solide ou fluide sont tres sensible mix deviations de regie lineaire qui donne l'equation d'e"tat comme function additive sur celui du OCP. Cettes deviations sont tres petities (quelques parts dans 105) et toujours positive meme guard la fraction du composant a Z superieur va vers O. L'equation d'&at pour les elements leger (surtout H et He) a fort couplage obtenu par simulations avec response lineaire des electrons est donne p;our les conditions des nains brun. Nous discutons aussi calculs recent, fait a Livermore, predisent que le He dans H cst immiscible a haut temperature (15000 K) a la pression de 10 Mb. 330

DeWitt: Strongly coupled ionic mixtures

33\

15.1 Introduction Hydrogen and helium mixtures form the main components of jovian planets, brown dwarf stars, and ordinary main sequence stars. In these various conditions the state of the mixtures ranges from molecular and atomic fluids in the outer layers of the jovian planets, to ionized H and neutral He deeper in jovian planets, to fully ionized low Z elements in brown dwarfs and main sequence stars. Hydrogen and helium are found on the surfaces of white dwarf stars but most of the interiors are composed of heavier elements beginning with C, O, Mg, and up to Fe that are fully ionized. The relativistically degenerate electron gas provides a nearly uniform density neutralizing background and the positive pressure that balances the gravitational contraction force and thus determines the size of the white dwarf. For applications to these various astrophysical objects we need to know the equation of state of mixtures of light elements from the molecular-atomic region at low temperature and high density on up to the extreme high densities where the ions are bare nuclei. One can distinguish four density regions: i) Ionic fluid in a (nearly) uniform background of degenerate electrons. Ts -> 0. An example is the C and O fluid (bare nuclei, usually assumed to be classical) above the crystallizing core of a white dwarf star • Monte Carlo simulations give very accurate results for the ionic interactions in both the fluid and the solid state. This system is well understood classically, though quantum effects are now known to play a strong role in real white dwarfs^. ii) Ions in a responding background of degenerate electrons fa < 1). This is the situation in brown dwarf star interiors with pressures above 80 Mb so that both H and He are fully ionized. The equation of state is given with fairly good accuracy by using Monte Carlo simulations with the Coulomb potential screened by using an appropriate dielectric function in linear response theory. Also fairly accurate results can be obtained by solving coupled Hypernetted Chain (HNC) equations for the ionic mixtures^. At the moderate temperature ( a few eV) but density high enough that 0.1 < rs < 1) the electrons are highly degenerate (kT « Ep) but still polarizable. iii) Partially Ionized Mixtures (rs >) with the ions still strongly coupled. For P > 3 Mb as in the interior of most of Jupiter one finds H + along with neutral He, and consequently both free and bound electrons. In this density and temperature region both MC and HNC calculations are of questionable

332


accuracy, and there is still considerable uncertainty. A plasma phase transition 4 is possible for the hydrogen at roughly kT = 1 eV. Such a transition may be modified by the presence of neutral helium5'6*7. This region is unfortunately still outside the possible region of experimental measurements. iv) Neutral H? and He as in jovian atmospheres, 0 < P < 3 Mb. This region has been studied experimentally to the 1 Mb region for H2 and is accessible to 3 Mb in future experiments. Most of this paper will deal with region i), the extremely high density region where one has some hope of obtaining some nearly exact results from very long numerical simulations, either Monte Carlo or molecular dynamics. Yakovlev and Shalybkov1 have given an excellent review of most known results up until 1988 for the OCP and ionic mixtures in the strong coupling region, and discussed the applications to very dense stars, particularly white dwarf interiors. Problems that remain to be addressed with greater quantitative precision include the questions of the value of the coupling parameter at which the fluid solid transition occurs, and the possibility of phase separation, i.e. separation of heavy from light elements in the freezing process or even in the fluid region. Associated with the phase separation process is the need for an accurate representation of the equation of state of the ionic mixtures in both fluid and solid phases. The well known linear mixing rule gives an excellent first approximation to the mixture equation of state in terms of the OCP results for the energy and the Helmholtz free energy. Small deviations from the linear mixing rule determine the phase diagram of mixtures, for example the possible separation of Fe from the C and O in the fluid region of the white dwarf interior. These questions need to be addressed by the generation of very accurate Monte Carlo energy data for ionic mixtures using several hundred to a thousand particles and averaging over as many as a few hundred million configurations. Some new results for mixtures will be given here. 15.2 Strongly Coupled Ionic Mixtures in a Uniform Background We will use the now standard definitions of parameters. For the OCP with number density n = N/V , temperature p = 1/kT, and classical point charges Ze moving in a uniform background the coupling parameter is V — (Ze)2/akT with a = (—n)"1^3, the Wigner-Zeitz or ion sphere radius. At the extreme densities id white dwarf stars the deviation from a uniform


333

background due to ionic polarization of the relativistic electrons is measured by r s = ae/ae with OQ = a/Z1/3 the electron sphere radius. rs from about 0.01 for p = 106 gm/cc down to 0.001 for p = 109 gm/cc, so that to a good approximation electron screening effects on the ion-ion contribution to the equation of state is quite small. The interior of a white dwarf if it were a single element is believed to be crystallized with an energy of the form: U/NkT = (Uo + Ulh)/NkT = a M r + (3/2 + A i/T + A2/T2 + ...)

(1)

where aM = - 0.89592926 is the Madelung constant for the bcc lattice, 3/2 is the classical thermal energy of the harmonic lattice, and the first anharmonic energy 8 term has the coefficient Ai = 10.84. The second anharmonic energy term9 is known only from MC simulations and is A2 « 600. The Helmholtz free energy for the OCP bcc lattice can be obtained by temperature integration: F/NkT = aMF + J l n r + C - S H - Ai/T - - ^ +

(2)

where C =1 + ln(2(3/47i)1/3) and S H I S the entropy constant ( , averaged over lattice phonon frequencies which for the OCP bcc lattice is

2.4939. The OCP fluid energy, U/NkT, obtained from long Monte Carlo simulations 10 in the strongly coupled region, 1 < F < 200 again has the remarkable property of splitting to a good approximation into a static piece and a thermal piece, but with the thermal energy governed by a power law: U/NkT = (U0(p) + Uui(p, P))/NkT = a F r + b P + c

(3)

where s is a number ranging from 1/4 to 2/5 depending on the number of terms used in the fitting form in Eq. 3; the best estimate is s » 1/3. Integration over the temperature from p to 0, gives the OCP fluid free energy as; F/NkT = a F T + (1 /s)br s +clnr + D

(4)

Where D is an integration constant. The coefficient of T in Eqs 3 and 4 is the 'fluid Madelung coefficient' which Rosenfeld11 has shown to have an exact limiting value of a F = - 9/10, which is the value in the F - * «»for the fluid. If a F is allowed to be a fitting parameter for the OCP fluid data the

334


coefficients for the fit to the best available OCP MC data is aF = - 0.89921, b = 0.596, c = 0.268 and s = 0.3253. Other equally accurate fits to the OCP fluid data with ap fixed at -9/10 are given in Ref. 10. Generally the OCP energy data for large F is known to about ± 0.0005 (which is a few parts in 10 6 ), and simple fitting functions like Eq. 3 can reproduce the known OCP data to about ± 0.001. This kind of accuracy is ultimately needed also for the ionic mixtures in order to determine the freezing line for ionic mixtures and the conditions for phase separation in the freezing process or in the fluid. For the OCP the fluid and solid Helmholtz free energies cross at F = 172, which should be regarded as the best available estimate of the OCP freezing transition F. The inclusion of the first order anharmonic energy in Eq. 1 has changed the estimate from the earlier value 12 of 178 to 172. Since the fluid and bcc solid free energies have so nearly the same slope, a very small change in one of the free energies can send the crossing point up or down very much.. This fact will be even more true for ionic mixtures which means that current estimates1^ of the phase diagrams for ionic mixtures must be regarded with some skepticism. It should also be noted that the classical OCP and the classical ionic mixture is a serious approximation for white dwarf star interiors since in fact the ions in these stars have serious quantum diffraction effects2. The measure of QM diffraction effects is r\ = hC0p/kT which ranges from 3 to 8 in white dwarf interiors. However, the location of the freezing transition is only slightly affected by quantum effects. The transition temperature is lowered (F increased), but even for Tj = 8 the transition F is changed to only abut 200. Rosenfeld has demonstrated a number of other exact limiting results for the OCP fluid and the ionic mixtures for large F. The screening function, H(x) with x = r/a, in the pair correlation function: g(x) = exp{-F/x + H(x)} is needed for calculations of the screening enhancement of thermonuclear reactions in very dense stellar interiors. H(0) gives the lowering of the Coulomb barrier for two rapidly approaching ions. The Onsager molecule method used by Rosenfeld gives a result for the screening function for finite F: H(x)/F = ho(F) - hix 2 + h2(F)x4 - h3(F)x6 +

(5)


335

In this expansion around the x = 0 only the coefficient hi is a constant independent of F, namely hi = 1/4.112 and 113 must be determined from the best available MC data for g(x). For large F ( 100 to 200) h2 - 0.038 which is large enough to influence the determination of ho. H(0) = hoF is given by the difference of the Helmholtz free energy for N charges Z and the free energy of N - 2 charges Z and on charge of 2Z. The result can be shown to be: H(0) = 2f 0C p(F) - fOCP(25/3F) - @/ax2)Afocp(F)

(6)

= (9/10)(25/3 - 2) = 1.0573F as F -» where X2 = N2AN1 + N2). The first line of Eq. 6 is the linear mixing rule result for H(0) with AfocP = ^mixture - ^LM > and f = F/NkT. Rosenfeld has shown that by inclusion of the x 4 term in Eq. 5 , that H(0) can be obtained very accurately without using the linear mixing rule 14 , and that the results agree remarkably well with the linear mixing result. The sign of Af is quite important for this discussion. The general statement of the linear mixing rule for binary ionic mixtures is: U(Zi, xi, Z 2) x2)/NkT = - ^ " x 1 foCP(F 1) + x2foCP(F2)

(7)

with Fi = Zi 5 / 3 F C . Linear mixing applies exactly in the large F limit but is at best a good approximation for the fluid mixture thermal energy. Thus the usual statement of the linear mixing rule in Eq. 7 is indicated as an approximation. It is the deviations from linear mixing in the binary ionic thermal energy that result in the possibility of phase separation, i.e. Z2 ions separating from the Zi ions, and similarly the possible separation of large Z ions from smaller Z ions in the freezing process. Ogata, et a l ] 3 reported conditions for phase separation of high Z ions in the fluid phase and phase diagrams for binary ionic mixtures upon freezing which were very much influenced by their MC observations of some negative deviations from linear mixing in the limit of a small fraction of the large Z component, i.e. X2—»0. Our recent results (Slattery and DeWitt 15 ) indicate that the deviation from linear mixing is only positive.

336


Brami, Joli, and Hansen 16 did a detailed study of the conditions for phase separation in binary ionic fluids using the pure HNC mixture equations (no bridge function correction) and found always positive deviations with a magnitude of never more than 0.027 for X2 = 0.05 and Z2/Z2 = 8. Ogata et al 13 reported negative deviations for MC binary fluid runs with Ni = 990, Zi = 1, and N2 = 10, Z 2 = 3 and 5 for I*i = 20, thus X2 = 0.01. We first tried the Z2 = 3 case with HNC mixture equations and found that Umix/NkT = -17.4614 and that ULM/NkT = -17.4621 (also from HNC). The difference is AUHNC/NICT = + 0.0007. All mixture runs done with the HNC equations give positive deviations from linear mixing. Since the HNC equation is an approximation, the above results may be questioned with the presumably more accurate MC fluid mixture simulations. We did a few MC mixture runs for X2= 0.01 (Ni = 990 and N2 = 10) but with 150 million configurations which is 20 times the number of configurations reported by Ogata et al. For I*i = 20 our MC mixture result was Umjx/NkT = -17.7231 and the linear mixing result using the best available fit to the OCP fluid energy data, Eq. 3, is ULM/NkT = -17.7241. Thus we obtain a positive deviation of AUMC/NkT = + 0.0010, whereas Ogata et al report a value of - 0.002. A possible source of error in the Ogata et al results is their fitting function for the OCP data which is inaccurate by about 0.003 for some values of T. The deviations from linear mixing are obviously very small and the reported results depend very much on the accuracy of the MC energy results and the accuracy of the fit to the OCP fluid data. Our conclusion is that the MC mixture results are always positive. A positive deviation from linear mixing has a number of consequences. Ogata, Iyetomi, and Ichimaru16 evaluated H(0) from their MC data for g(r) on the assumption that Ii2 in Eq. 5 was 0, whereas Rosenfeld14 finds h2 = 0.038 in agreement with Alastuey and Jancovici 17 from their work on enhancement of thermonuclear reactions in 1978. The Ogata et al 16 results for H(0)/T are about 2% larger than the results of Rosenfeld and of Alastuey and Jancovici which are also in close agreement with the linear mixing rule. Ogata et al 13 (in their report to the Rochester Conference) cite the correction term in Eq. 6 to explain this difference. However, their agreement with linear mixing plus the correction requires that their deviation from linear mixing be negative, but our results indicate that Af is positive. The 2% difference in estimates of h 0 = H(0)/T makes a factor of 10 difference in the final enhancement rate for thermonuclear reactions at large F. As Isern has pointed out at this Conference18 a very small change in the enhancement of thermonuclear reactions in a white dwarf star at the Chandrasekhar limit can make the difference in the star becoming a supernova or to collapse into a


337

neutron star. For reasons given above we think the Alastuey-Jancovici estimates17 of the screening enhancement of thermonuclear reactions are more accurate and reliable than the results of Ogata, Iyetomi, and

Ichimaru16.

The crystallization of binary ionic mixtures provides another example for which the sign of the deviation from liner mixing can make a qualitative difference. Using a density functional theory of freezing Segretain and Chabrier19 estimated the domains of three types of phase diagrams depending upon the ratio of the charges, Z1/Z2. For 0.72 < Z1/Z2 < 1 the phase diagram for solidification is a spindle type; for 0.58 < Z1/Z2 < 0.72 an azeotropic phase diagram results; and for an even greater charge disparity, Z1/Z2 < 0.58, a eutectic phase diagram results. This is the same progression of phase diagrams as is found for mixtures of hard sphere diameters with different diameters. Ogata et al 13 , however, based on their MC simulations that lead to negative values of Af, find that the azeotropic form can persist for any value of Z1/Z2 near 1 when the fraction of the higher Z component is very small, i.e. X2=s0.01. The positive values of Af that we find support the spindle shaped phase diagram for all values of X2 for Z1/Z2 near 1. the type of phase diagram for binary ionic mixtures in white dwarf stars, C and O, and C and Fe, has major astrophysical consequences for the possible separation and crystallization of trace elements19, and the resulting effect on the cooling time for white dwarfs. It is clear that a lot more very accurate numerical simulation of binary ionic mixture energies is needed to finally settle these questions. 15.3 Strongly Coupled Ionic Mixtures in a Responding Background For applications to brown dwarf stars and the jovian planets the light elements, especially H and He, may be partially to fully ionized. Although the free electrons are largely degenerate the r s value is typically in the vicinity of 0.5 to 1. Thus the electrons can screen the protons and alpha particles appreciably. The electrons may also have finite temperature effects measured by the parameter 6 = kT/EF which leads to complete degeneracy when 9 « 1, but in some stars we may have the worst of all possibilities, namely 9 = 1 . Both the density dependence (rs) and the temperature dependence (0) can be dealt with by using an appropriate density and temperature electron dielectric function in linear response theory. This has been developed by Chabrier and Ashcroft20 for solution in the HNC ion mixture equations. The energy and pressure of the ionic mixture in linear response (from their paper) is:

338


N

1 ^^(^w*^)] 2{2n) :

(8)

2 (2/r)3

(9)

In Eqs. 8 and 9 E is the total ion-ion and ion-electron interaction energy and P e x is the corresponding pressure contribution. v(k) is the Fourier transform of the Coulomb potential, 4rce2/k2, Sz(k) is the structure factor, and e(k) is the density and temperature dependent dielectric function. The first term in u cx and p c x is the usual BIM functions appropriate for a uniform background, rs = 0. The second term is the linear response density contribution, and third term is the result of the temperature dependence in the dielectric function. The screened potential, v(k)/e(k), is used in the


339

solution of the coupled HNC equations for the BIM in order to calculate Sz(k) which is then used in Eqs. 8 and 9 to evaluate the energy and pressure of the mixture. Chabrier and Ashcroft presented results for a variety of mixtures, H and He, H and O, with temperatures and densities that went from weak coupling to strong coupling and with X2 = 0, .25, .75, and 1. They checked the linear mixing rule and found that deviations from linear mixing ranged from 0 % to at most 3% and were positive. The largest deviations from linear mixing occurred for 9 = 1, the most difficult temperature region. For the very high temperatures in the center of the sun Iyetomi and Ichimaru21 have solved the HNC equations for H and Fe mixtures in order to construct a phase diagram for H and Fe. They find that the central temperature of the sun is too hot by a factor three to allow for thepossibility of any separation of Fe from the remaining solar plasma. Their results suggest that heavy elements, such as Fe, might be able to separate from small cool stars with interior temperatures of only a few million °K. Eqs. 8 and 9 could in principle be used for evaluation by Monte Carlo simulation, though this has not been done yet with a temperature dependent dielectric function. Hubbard and DeWitt have done extensive MC calculations on H and He mixtures22, assumed to be fully ionized, with the density dependent RPA dielectric function (the Lindhard function) and 8 = 0. 45 MC simulations were done on pure H, pure He, and mixtures of HHe with values of rs ranging from 0.2 up 1.4 and values of F e ranging from 7 to 155. Three mixtures were computed: 80%H and 20% He, 50% and 50%, and 20% and 80%. Unfortunately because of limited computer time the number of particles used was only Ni + N2 = 50, and the energies were obtained after thermalization with 105 configurations. By today's standards these simulations are far too short and too few particles to give accurate results for the energy and pressure. Nevertheless, enough data was obtained that a Helmholtz free energy model could be constructed: f = F m i x /NkT = -ar c + blV/4 - clnr e + d

(10)

which is similar to Eq. 4 except the exponent s was chosen to be 1/4 and the coefficients in Eq. 10 are all functions of rs. Specifically, the coefficient of F e was expanded to the second power in rs: a(rs) = ai[(l-x)Zi5/3 + xZ 2 5 / 3 ] + a 2 (l-x)r s + a3xrs + [a4(l-x)

(11)


340

The coefficients b, c, and d were modeled with only the linear dependence in rs# All together there were 10 numerical coefficients (ai , a2 ,...) that were obtained numerically by a least squares fit to the 45 MC data points. These coefficients used in Eq. 10 gave a useful numerical model for the free energy mixture of H + , He + + , and the screening electrons. In order to obtain the equilibrium curves for the coexistence of two liquid phases at a pressure of 8 Mb, the equilibrium equations were solved:

^(r.N, / V,N21V) = (dF I dNx\yjlt

(12) ^(T,NXa I V,N2a IV) = li2(T,Nlb I V,N2b I V\ P(T,Nla I V,N2,1V) = P(J,Nlh I V,N2b IV), P(T,NU/V,N2JV)

= PS,

using Eq. 10. The resulting equilibrium curve for the coexistence of to liquid phases is shown in Fig. 1 (from the Hubbard-DeWitt paper) Stevenson toooo

5000

0.5

i.o

Fig. 1 - Kqnilibrium curve for coexistence of two liquid phases of H iind lie at S Mb, compared with earlier work by Stevenson23-


™l

At x = 0.07, the He fraction in Jupiter, the demixing temperature is 8000 °K. This estimate of the demixing temperature in Jupiter is open to criticism because the helium atoms in Jupiter from 3 Mb where hydrogen is presumed to pressure ionize on the 35 Mb pressure at the center it is probable that the helium atoms are neutral, i.e. with two bound electrons. Strictly speaking the Hubbard-DeWitt simulation of H and He applies only to astronomical objects with pressures above 50 to 80 Mb in order to assure the pressure ionization of the He. The free energy model for the H - He mixture can probably be considerably improved with more accurate MC simulations involving up to 1000 charges and up to several million configurations for each value of F e and rs. It should also be mentioned that the ten fitting coefficients used in Eq. 10 can be somewhat reduced in number by using the scaling obtained in recent work by Rosenfeld on charged Yukawa mixtures24. Thus the coefficients a3 and 04 can be replaced by a single coefficient weighted with ?/3 and

7/3

15.4 Partially Ionized H and He The obvious problem for theoretical calculations of the H-He equation of state and the possibility of phase separation of the He in jovian planets is the fact that the hydrogen whether in molecular or atomic form become pressure ionized at approximately 3 Mb while the tightly bound helium electrons remain attached to their nuclei in the H-He mixture until some much larger pressure, ~ 50 Mb. Chabrier and Saumon4 have given a detailed model for the equation of state of hydrogen from the molecular region to the fully ionized region that exhibits a plasma phase transition with a critical point at T c = 15300 K, P c =0.614 Mb, and p c = 0.35 gm/cc. The Chabrier-Saumon EOS is probably the best theoretically based and most quantitative result available for use in modeling the jovian planets. Whether there is a true phase transition or simply a sharply defined region where pressure ionization occurs does not affect the accuracy of the EOS very much. Unfortunately this possible phase transition is out of range of current experimental measurements. Pure He is believed to be pressure ionized at some far higher pressure, perhaps as much as 80 Mb. He atoms mixed in high pressure ionized hydrogen will probably ionize at some lower pressure which may be greater than the interior pressure of Jupiter. To address this problem of the H- He mixture in the 10 to 30 Mb range it is essential to have a detailed understanding of the electronic structure of the mixture. Klepeis, Schafer, Barbee, and Ross at Livermore have approached this problem of the


342

electronic structure with total energy calculational methods of condensed matter physics at T = 0 K In this approach the atoms are placed in a lattice, fee or bec, and electronic enthalpy, H(x) = E(x) + P(x)V with x as the He fraction, is calculated with the local density approximation. A brief description of the method and band structure results for the H-He mixture is given in Ref. 5. A complete report is available from Livermore. The main thrust of this work is an accurate complete T = 0 K calculation with the ions on a lattice from which one obtains the enthalpy of mixing from: (13)

AH(x) = H(x) - xH(x =1) - (l-x)H(x =0) Results for 10.5 Mb are shown in Fig. 2

1

I ' '

I

" P = 10.5 Mbar

s o

-

a • — \ A

/

^\

1 -

u

/

P.

/

CtO

\

a

:

0.5

/ /

0

a bee lattice (sc) A bec lattice (dia) o bee lattice (rhom) • fee lattice (sc) . I , , • I

0

•

\

\ \ \ -

ill

0.2 0.4 0.6 0.8 x (Atomic Percent Helium)

1

Fig. 2 The enthalpy of mixing per atom in ev from Eq. 13 obtained from first principles total energy calculations for four different lattices at T = 0 K and P = 10.5 M b .


343

The different lattices used have only a small effect on the mixing enthalpy. Calculations were also done at different pressures to find AH(x = 1/2). This peak enthalpy was nearly constant between 5 and 20 Mb and dropped only by 35 % at 1000 Mb. The large values of the mixing enthalpies are a direct consequence of the fact the helium-derived electrons are more tightly bound to the nucleus than the loosely bound hydrogen-derived electrons. To go to finite T one needs a Gibbs free energy of mixing. This was done by adding a simple result for the ideal gas mixing entropy to obtain: (14)

AG(x) = AH(x) + kT [ x In x + (l-x)ln(l-x)]

and from Eq. 14 the Gibbs mixing free energy was obtained for several temperatures as shown in Fig. 3.

. 1 ' . . 1 .

1

1

1

1

•

1

i

• .

I • •

• 1

-p == 10.5 Ml)ar>

I u A

I

^ -

/ 10 000 K

/

0.5 |--

15,000

yy

^

—

-V

20,000

30,000

-0.5 -

•

y^

O

t

Nv

*yk

-it

0]

i . . .

0.2

0.4

0.6

i . . .

\ -

0.8

x (Atomic Percent Helium) Fig. 3 Gibbs free energy of mixing from Eq. 14 for several temperatures at 10.5 Mb. The double tangent construction is Shown lor T = 15000 K

344


Finally using the double tangent construction at various temperatures the demixing temperature is obtained and shown in Fig. 4

I

I

I

I

I

I

I

I

I

l

l

)

P = 10.5 Mbar -

r u fl)

I «

This work OCP-LM Iindhard Ion—sphere

9

o 3 CO

I

0

Fig. 4

I

I

I

I

I

I

I

I

I

I

I

I

0.2 0.4 0.6 0.8 x (Atomic Percent Helium)

1

Immisibility temperature limit for H-He mixtures.

The demixing temperatures obtained by this procedure are very much larger than those obtained by Stevenson and by Hubbard and DeWitt. At x = 0.07 corresponding to the helium fraction in Jupiter Fig. 4 gives the demixing temperature as close to 15000 K vs. the 8000 K from MC with Lindhard dielectric function. The error estimate on this calculation is ± 3000 K. However, it is clear that the approach needs serious improvement for finite temperature since i) the ions are assumed to be in a lattice rather than in a fluid state, and ii) the Gibbs thermal energy of the mixture is not included. Clearly more work needs to be done with this approach. The earlier estimate of demixing at 8000 K is consistent with the belief that H and He in Jupiter remain mixed through out much of the planet. If the 15000 K


345

estimate given by the Livermore work is correct, it would have serious astrophysical consequences since it would mean that substantial separation of the He in Jupiter has already happened. Work performed under the auspices of the U.S. Department of Energy by the Lawrence Livermore National Laboratory under contract number W7405-ENG-48. References 1. 2. 3. 4. 5. 6. 7. 8. 9.

10. 11. 12. 13.

14. 15. 16. 17.

D.G. Yakovlev and D. A. Shalybkov, Sov. Sci. Rev. E. Astrophys. &Space Phys. Vol.7, 311-386 (1989) G. Chabrier, N.W. Ashcroft, H.E. DeWitt, Nature 360,48 (5 Nov. 1992) W.B. Hubbard and H.E. DeWitt, Astrophysical J. 290, 388 (1985) D. Saumon and G. Chabrier, Phys.Rev. A46, 2084 (1992): Phys.Rev.A44. 5122(1991) J.E.KIepeis, K.J. Schafer, T.W. Barbee III, M. Ross, Science 254, 986 (15 Nov., 1991) D. Saumon, W.B. Hubbard, G. Chabrier, H.M. VanHorn, Astrophysical J. 391, 327 (1992) G. Chabrier, D. Saumon. W.B. Hubbard, andJ.I. Lunine, Astrophysical J. 391, 317 (1992) D.H.E. Dubin, Phys. Rev. A42, 4972 (1990) H.E. DeWitt, W.L. Slattery, J. Yang, Strongly Coupled Plasma Physics, ed.H.M. Van Horn and S. Ichimaru, University of Rochester Press: Rochester, NY, p.425 (1993) G.S.Stringfellow, H.E. DeWitt, W.L. Slattery, Phys. Rev.A41, 1105(1990) Y. Rosenfeld, Proceedings of this Conference; Phys. Rev. A33, 2025 (1986); Phys. Rev. A37, 3403 (1988) W.L. Slattery, G.D. Doolen, H.E.DeWitt, Phys. Rev.A26, 2255 (1982) S. Ogata,H.Iyetomi,S. Ichimaru, H.M.Van Horn, Strongly Coupled Plasma Physics, ed. H.M. Van Horn and S. Ichimaru, University of Rochester Press: Rochester, NY, p.53 (1993) Y. Rosenfeld, Phys. Rev. A46, 1059 (1992) W.L. Slattery and H.E. DeWitt, recent MC results at Los Alamos S. Ogata, H. Iyetomi, and S. Ichimaru, Astrophys. J. 372, 259 (1991) A. Alastuey and B. Jancovici, Astrophys.J 226,1034 (1978)

346

18. 19. 20. 21. 22. 23. 24. 25.


J. Isern,proceedings of this Conference L. Segretain and G. Chabrier, Astron. Astrophys. 271, L13 (1993) G. Chabrier and N.W. Ashcroft, Phys. Rev. A42, 2284 (1990) H. Iyetomi and S. Ichimaru, Phys. Rev. A34, 3203 (1986) W. B. Hubbard and H.E. DeWitt, Astrophys. J. 290, 388 (1985) DJ. Stevenson, phys. Rev. B12, 3999 (1975) Y Rosenfeld, Phys. Rev. E47.2676 (1993) J. E. Klepeis, KJ. Schafer, T. W. Barbee III, M. Ross, Phase Separation in Mixtures of Hydrogen and Helium at Megabar Pressures, UCRL-JC-107995, July 1991. See also the brief report by the same authors in Strongly Coupled Plasma Physics, ed. H.M. Van Horn and S. Ichimaru, University of Rochester Press: Rochester,NY, p.73 (1993)

16 White dwarf seismology: Influence of the constitutive physics on the period spectra. G. FONTAINE Departement ie Physique, Universite it Montreal, C.P. 6128, succursale A, Montreal, Quebec, HSC 5/7, Canaia.

P. BRASSARD Departement ie Physique, Universite de Montreal, C.P. 6128, succursale A, Montreal, Quebec, HSC 5/7, Canaia.

Abstract We present the results of numerical experiments aimed at demonstrating how the 1, the expected period difference for the same mode (same k) between the two models should be

"Jrrh^-^

(4)

We have used this last equation to obtain the results listed in Table 2 which apply to I = 2 modes with a period P ~ 1000 s (k ~ 31 for the standard model). These expected period differences can be contrasted with the results of the detailed calculations. We discuss each of our numerical experiments in what follows. Figure 4 summarizes part of our results for a first experiment in which the envelope equation of state has been varied. This is a plot of the period difference for the same mode (same value offc)computed, on the one hand, for a model in which the recent Saumon-Chabrier (SC; Saumon & Chabrier 1989, 1991, 1992) equation of state for the H/He envelope has been used and, on the other hand, for our standard model with the FGV equation

Fontaine & Brassard: White dwarf seismology

357

Table 16.2. Expected period differences for £ = 2 modes at P=1000 s Models compared

AP(s)

SC-STD IDEAL-STD NC-STD 2K-STD 2K-HOT OPAL-STD CM-STD

0.5 -25.2 22.2 -92.1 -1.3 12.1 -0.1

of state, all other things being the same. The period difference is plotted in terms of the period of the standard model for the sequence of modes belonging to £ = 2. These results are quite typical of the other values of £ (the periods and period differences must be scaled by a factor \/£(£+ 1)). What the diagram shows is that the periods of the two models are quite similar with maximum relative deviations of ~ 3%. Typically, the differences are much less than that. The largest deviations occur for the low-order modes which are formed relatively deep in the star, and which are most sensitive to the thermodynamics of the helium plasma. The largest differences between the SC and the FGV data indeed arise for helium in the regime of interest for pulsating DA white dwarfs. Note that, even for hydrogen, there are substantial local differences in thermodynamic quantities between the two generations of equation of state tables (see the paper by Saumon in these Proceedings). However, as discussed above, pulsation modes are mostly sensitive to integrated properties, so that these local differences tend to average out. The very similar values of the integrals no(SC)=75.78 s and II0(STD)=75.74 s (see Table 1) indeed reflect the fact that the pulsation properties of the two models must be very similar. The expected period difference of 0.5 s for the £ = 2, 1000 s mode (Table 2) is quite consistent with the actual difference shown in Fig. 4. A happy consequence here, at least from our point of view, is that the newer SC equation of state data do not change significantly the period spectra of pulsating DA white dwarfs. The model parameters we have used in our experiments are typical of these stars, and varying the parameters within acceptable ranges will not change this conclusion. This means, in particular, that past calculations based on the FGV tables remain qualitatively and quantitatively reliable (see, e.g., Brassard et al. 1992a). The reader should not be left with the impression that any envelope

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envelope_eos 30.0

i

m

10.0 -

o

s u V

£

•g -IO.O •c

s. -30.0

100.

300.

500.

700.

900.

Period (STD) Fig. 16.4 Comparison of the pulsation periods of the SC and STD models. The figure shows the period difference of the two models for the same mode (same value of Jb) in terms of the period of the STD model. The results refer to g modes belonging to the I = 2 sequence.

equation of state is adequate to describe pulsating white dwarfs, however. In this context, we show, in Figure 5, some of the results of an experiment in which we have removed all nonideal terms and computed an envelope equation of state for pure H, He, and C on the basis of Saha equations only (IDEAL). Of course, as is well-known, this approach leads to complete recombination at sufficiently high densities, which is unphysical. In a format identical to that of Fig. 4, the diagram shows period differences which are large compared to the previous results and which, moreover, are systematic. Indeed, the period difference grows systematically larger (in an absolute sense) with increasing radial order. The periods of the model based on the ideal equation of state become smaller than the periods of the standard model. For the £ = 2, 1000 s mode, the expected period difference


359

envelope.eos 30.0

i

i

i

i

i

10.0 Q>

O 0 U V

•o -10.0 o

•c »

-30.0

100.

300.

500.

700.

900.

Period (STD)

Fig. 16.5 Same as Fig. 4, but for the IDEAL and STD models. is -25.2 s according to Table 2, which is comparable to the exact result shown in Fig. 5. Note also that, to a large extent, the jagged appearance of the curve is caused by mode trapping/confinement effects in the region where we rather crudely switch from our neutral, recombined ideal gas in the envelope to our totally ionized dense Coulomb fluid in the core. We found it also interesting to investigate the effects of nonideal terms in the core equation of state. In the next experiment, we have thus removed all the nonideal terms in the pure C equation of state of Lamb (1974; see also Lamb & Van Horn 1975). This could be easily done because Lamb's code, among many nice features, is modular in its construction, so only the kinetic energy terms, for example, can be retained. Figure 6 summarizes our results for the £ = 2 modes. The figure is on the same vertical scale as the previous ones, but there is a shift of the zero point. We can observe a systematic increase of the period with increasing radial order for the modified model (NC, which stands for "no Coulomb" terms)

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core.eos

50.0

30.0 -

g o

a 0>

•o

1 -10.0 100.

300.

500.

700.

900.

Period (STD) Fig. 16.6 Same as Fig. 4, but for the NC and STD models.

as compared to the reference model. The accumulated period difference at 1000 s is comparable to the expected value of 22.2 s (Table 2). We note that by keeping the surface gravity constant, the modified model can accommodate a larger mass than the standard model because the ideal NC core equation of state is harder. The upshot is that the altered model is slightly more degenerate in its interior, \N\ is smaller, II0 is larger, and the periods are larger than for our reference model. We have carried out similar experiments with the opacity. For example, we have computed the period distributions of models in which the radiative opacity, the conductive opacity, and then the total opacity have been arbitrarily multiplied by a factor of 2. The results for the experiment with the total opacity are illustrated in Figure 7 (modes with £ = 2). Again, the period difference is plotted in terms of the period of the standard model; the symbol "2K" here stands for the modified, more opaque model. Note that

361


total-opacity

-90.0 100.

300.

500.

700.

900.

Period (STD)

Fig. 16.7 Same as Fig. 4, but for the 2K and STD models. the vertical scale has been expanded to make allowance for the relatively large period differences. We can observe that the g-mode periods of the opaque model are systematically smaller than those of the standard model. This is easily understood since an increase of the global opacity leads, for a model with a fixed surface temperature, to a larger core temperature. This, in turn, implies that the overall degeneracy of the core is less, the Brunt-Vaisala, frequency is larger, and, consequently, the characteristic period spacing Tl0 is smaller in the modified model. We find indeed that the expected period difference of -92.1 s (Table 2) based on the n o arguments match rather well the exact result for a mode with a period of 1000 s. It is interesting to point out that the more opaque model mimics very closely a hotter model. In other words, we can recover almost exactly the period spectrum of the more opaque model by considering a model with standard input physics, but with a higher effective temperature. Such a

362


equivalent_hot_model

I

-10.0 -

I

~-30.0 o

g I -50.0

I

s. -70.0 -

-90.0 100.

300.

500.

700.

900.

Period (HOT) Fig. 16.8 Same as Fig. 4, but for the 2K and HOT models. model is referred to here as "HOT", and is the only equilibrium model for which we have changed a parameter (Tefj in the present case). By trial and error, we have determined that a model with Teff = 14,280 K has nearly the same central temperature as the opaque model at Teff = 12,500 K, and a very similar value of the characteristic period spacing n o (see Table 1). Accordingly, it is not surprising that the period spectra of the two models are nearly the same as illustrated in Figure 8 (plotted on the same scale as Fig. 7). A better agreement could have been found by fine tuning even more the effective temperature of the hot model, but we believe this is not necessary to make our point. In another experiment, we have taken advantage of the recent availability of the OPAL Rosseland radiative opacity tables (Rogers & Iglesias 1992) to compute a model (OPAL) incorporating the new data for pure H, pure He, and pure C. The periods of this model (for I = 2 modes) are contrasted to those of the standard model in Figure 9. We find that the OPAL opacities


363

radiative.opacity 30.0

-30.0

100.

300.

500. 700. Period (STD)

900.

Fig. 16.9 Same as Fig. 4, but for the OPAL and STD models. lead to systematically larger periods than the case based on LAO opacities. Again, the expected period difference of 12.1 s at 1000 s based on the integrals II 0 is consistent with the result of Fig. 9. On the whole, we find that the OPAL opacities are slightly smaller than the LAO opacities in white dwarf envelopes, leading to models with slightly lower central temperatures and increased values of II 0 . We think that, in a large part, the differences are mainly caused by a difference in chemical composition: our older LAO tables refer to mixtures which contain small traces of heavy elements (Z= 10~3), whereas we used the OPAL data for pure elements. We believe that the contribution of the Z elements to the LAO results makes the opacity somewhat larger, on the average, than the OPAL opacity (since the differences in opacity for pure H or pure He, in particular, are not expected to be very significant between the two generations of tables). The test should probably be redone with radiative opacity tables referring to exactly the same chemical compositions.

364


The effects of varying the convective efficiency using the ML1, ML2, and ML3 versions of the mixing-length theory have been thoroughly investigated by Brassard et al. (1992a), and will not be repeated here. We simply recall that an increase of the convective efficicency leads to a systematic increase of the y-mode periods. The effect is negligible for low-order modes but increases with increasing radial order. We have recently incorporated in our model building code the parameterfree convection theory of Canuto & Mazzitelli (1991, 1992; CM). At Teff = 12,500 K, the CM model shows a hydrogen convection zone which is slightly less deep than the ML2 zone of the standard model. Since, at that effective temperature, the convection zone is quite high in the envelope (see Fig. 1), it does not contribute very much to the II0 integral. Accordingly, there is very little difference in the values of II 0 for the CM and STD models (Table 1). Figure 10 illustrates that period differences only appear for the high-order modes, i.e., for those which have nodes reaching out into the high envelope and interacting with the convection zone. In the present case, the period differences are quite small. They would actually increase for cooler models (which develop deeper convection zones), but low-order modes would remain unaffected by convection. Finally, we again refer the reader to Brassard et al. (1992a) for a discussion of the effects of varying the physical treatment used to describe the composition transition zones. In a nutshell, the main effect of, for example, decreasing the thickness of a composition transition zone is to increase the nonuniformities in the period spectrum; the average period spacing IIO remains largely unchanged. 16.5 Discussion We have presented the results of a number of numerical experiments to illustrate how theflf-modeperiod spectra of pulsating DA white dwarfs depend on the various components of the constitutive physics. In this exercise, we used stellar models with typical values for their parameters. In this connection, it should be reminded that, for reasons of distance, it is more complicated to read the asteroseismological records of pulsating stars than that of the Sun. For instance, when observing pulsating white dwarfs, we do not dispose a priori of independent and accurate estimates of basic stellar parameters such as total mass, total radius, surface gravity, effective temperature, and age. (We do not dispose either of thousands of well-identified modes!). Because these parameters are well known for the Sun, helioseismology can potentially be used to test the equation of state of the solar

365


convective_efficiency

100.

300.

500.

700.

900.

Period (STD) Fig. 16.10 Same as Fig. 4, but for the CM and STD models. plasma as discussed by Dappen (these Proceedings). This is not a realistic claim to make for pulsating white dwarfs. In fact, what white dwarf seismology can really bring us are the means to infer the stellar parameters through the signatures they leave on the period distributions. In this approach, the stellar parameters are derived under the implicit assumption that the set of constitutive physics used in the model building phase is a good representation of the true physical conditions in the actual stars. The method we have used in this paper remains the only avenue for estimating the "internal errors" due to the constitutive physics. We note that Brassard et al. (1993) used a similar approach in their analysis of G117-B15A, and were able to show that their error budget is dominated by observational uncertainties (and not by shortcomings in the constitutive physics). We conclude by pointing out that the constitutive physics bears on two more aspects of white dwarf seismology, which could not be discussed in this paper due to space shortage. While we have focussed here on the influence

366


of the input physics on the period spectra, the question of rates of period change is also quite important. These rates depend, of course, directly on the cooling rate, a problem which has received considerable new attention in recent years, and which has been reviewed by Chabrier, Garcia-Berro, and Mazzitelli at this Colloquium. We note that in the range of effective temperature where the pulsating white dwarfs are found, no "exotic" mechanism (neutrino cooling, crystallization, phase separation in the core, etc.) are expected to dominate, so the computed cooling rates across the narrow instability strip should be quite reliable. Finally, there is the question of mode excitation in pulsating white dwarfs. Nonadiabatic pulsation calculations by several independent groups all indicate that the driving region is located near the base of the hydrogen convection zone in pulsating DA white dwarfs. The question of mode driving and, in particular, the question of the location of the theoretical blue edge of the instability strip boil down to how deep is the hydrogen convection zone in a given model. Clearly, the most important component of the constitutive physics in the present context is the physical model used to describe convection. (The envelope equation of state as well as the radiative opacity also bear on the problem). A nonadiabatic survey to assess the influence of different convection models on the question of the location of the theoretical blue edge is a timely project. We have undergone such a project (Brassard, Fontaine, k Wesemael 1993). Acknowledgement We wish to thank friends and collaborators for providing invaluable support, input, and insight during the course of our ongoing investigations of pulsating white dwarfs. Special thanks are due to those who have provided the basic data used in this paper: Gilles Chabrier, Hal Graboske, Bill Hubbard, Walter Huebner, Naoki Itoh, Don Lamb, Italo Mazzitelli, Forrest Rogers, Didier Saumon, and Hugh Van Horn. This work was supported in part by the NSERC Canada and by the Fund FCAR (Quebec). References Brassard P., Fontaine G., Wesemael F., k Tassoul M. Ap. J. Suppl. 81, 747, (1992a) Brassard P., Pelletier C , Fontaine G., k Wesemael F. Ap. J. Suppl. 80, 725, (19926)

Brassard P., Fontaine G., k Wesemael F., in preparation, (1993) Brassard P., Fontaine G., Bergeron P., Wesemael F. k Vauclair G., in preparation, (1993)


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Canuto V., k Mazzitelli I. Ap. J. 370, 295, (1991) Canuto V., k Mazzitelli I. Ap. J. 389, 724, (1992) Fontaine G., Graboske H.C. Jr. k Van Horn H.M. Ap. J. Suppl. 35, 293, (1977) Fontaine G., Villeneuve B., k Wilson J. Ap. J. 243, 550, (1981) Hubbard W.B. k Lampe M. Ap. J. 163, 297, (1969) Huebner W.F., private communication, (1980) Iben I. Jr., k Tutukov A.V. Ap. J. 282, 615, (1984) Itoh N., Mitake S., Iyetomi H. k Ichimaru S. Ap. J. 273, 774, (1983) Itoh N., Kohyama Y., Matsumoto N. k Seki M. Ap. J. 285, 758, (1984) Kawaler S.D., k Hansen C.J., IAU Colloq. 114, White Dwarfs, ed. G. Wegner (New York:Springer-Verlag), 97, (1989) Koester D., k Schonberner D. Astr. Ap. 154, 125, (1986) Lamb D.Q., Ph. D. thesis, University of Rochester, (1974) Lamb D.Q. k Van Horn H.M. Ap. J. 200, 306, (1975) McGraw J.T., Ph. D. thesis, University of Texas at Austin, (1977) Robinson E.L., IAU Colloq. 53, White Dwarfs and Variable Degenerate Stars, eds. H.M. Van Horn k V. Weidemann (Rochester: University of Rochester Press), 343, (1979) Rogers F.J. k Iglesias, C.A. Ap. J. Suppl. 79, 507, (1992) Saumon D. k Chabrier G. Phys. Rev. Letters 62, 2397, (1989) Saumon D. k Chabrier G. Phys. Rev. A 44, 5122, (1991) Saumon D. k Chabrier G. Phys. Rev. A 46, 2084, (1992) Tassoul M. Ap. J. Suppl. 43, 469, (1980) Van Horn H.M., Proc. 25th Liege Astrophysical Colloquium, Theoretical Problems in Stellar Stability and Oscillations, eds. A. Noels k M. Gabriel (Liege:Universite de Liege), 307, (1984) Wesemael F., Bergeron P., Fontaine G. k Lamontagne R.L., Proc. Seventh European Workshop on White Dwarfs, eds. G. Vauclair k E.M. Sion (NATO ASI Series), 159, (1991) Winget D.E., IAU Symp. 123, Advances in Helio- and Asteroseismology, eds. J. Christensen-Dalsgaard k S. Frandsen (Dordrecht:Reidel), 305, (1988) Winget D.E. k Fontaine G., Pulsations in Classical and Cataclysmic Variable Stars, eds. J.P. Cox k C.J. Hansen (Boulder: University of Colorado Press), 46, (1982) Winget D.E., Hansen C.J. k Van Horn H.M., Nature 303, 781, (1983)

17 Helioseismology: the Sun as a stronglyconstrained, weakly-coupled plasma W. DAPPEN Department of Physics and Astronomy, University of Southern California, Los Angeles, CA 90089-1342, USA

Abstract Accurate measurements of observed frequencies of solar oscillations are providing a wealth of data on the properties of the solar interior. The frequencies depend on the solar structure, and on the properties of the plasma in the Sun. Except in the very outer layers, the stratification of the convection zone is almost adiabatic. There, the sound-speed profile is governed principally by the specific entropy, the (homogenous) chemical composition and the equation of state. It is therefore essentially independent of the uncertainties in the radiative opacities. The sensitivity of the observed frequencies is such that it enables to distinguish rather subtle features of the equation of state. An example is the signature of the heavy elements in the equation of state. This opens the possibility to use the Sun as a laboratory for thermodynamic properties. Les frequences observees des oscillations solaires constituent une base de donnees extremement riche qui nous permet d'etudier les proprietes de l'interieur du soleil. Les frequences dependent de la structure solaire et des proprietes locales du plasma (surtout de la vitesse du son). Sauf dans les couches tres exterieures, la structure de la zone convective du soleil est essentiellement adiabatique. Le profil de la vitesse du son est done donne par l'entropie specifique, la composition chimique (homogene) et l'equation d'etat. L'opacite radiative ne joue pas de role. Grace a la grande precision des frequences observees on arrive a distinguer des phenomenes assez sub368

Dappen: Helioseismology

369

tiles dans l'equation d'etat, comme la signature faible des elements lourds. Le soleil est devenu un laboratoire de physique des plasmas stellaires.

17.1 Introduction Solar acoustic oscillations have opened a new window into the Sun. By their nature they link the local sound speed in the interior with the observed oscillation frequencies. The spatial resolution of the solar disk allows the identification of a large number of individual oscillation modes, which are classified in terms of spherical harmonics. Modes in a large range of angular degrees, between / = 0 and a few thousand, are observed. The frequencies of these modes are centered around 3 mHz, which corresponds to periods around 5 minutes. They have been determined with high precision: typical relative errors are of the order of 10~4. The modes are confined to a cavity, which extends, broadly speaking, from the surface of the Sun, where the waves lose their material support, to the inner turning point which lies deeper the lower the angular degree / is. Radial modes, with / = 0, have no inner turning point and their cavity is the entire Sun. The observed solar oscillation modes are standing acoustic waves; hence the quantity most obviously probed is sound speed. Since the oscillations are largely adiabatic (except very near the surface), the frequencies are determined predominantly by the local adiabatic sound speed, which is a thermodynamic quantity. In addition, the frequencies depend on the density distribution in the Sun. Therefore, these helioseismic frequencies can be used as a diagnosis of the plasma of the solar interior. A high-quality thermodynamic potential is needed for the pressure-density relation (i.e. the equation of state, which is essential for determining the hydrostatic equilibrium between pressure gradient and gravity) and for thermodynamic quantities (mainly adiabatic sound speed). Introductions to helioseismology are, for example, the reviews by Deubner & Gough (1984), Christensen-Dalsgaard, Gough & Toomre (1985), Bahcall & Ulrich (1988), Christensen-Dalsgaard (1988), Libbrecht (1988), Vorontsov & Zharkov (1989), Gough & Toomre (1991), Libbrecht & Woodard (1991), Christensen-Dalsgaard & Berthomieu (1991), Gough (1992), and TurckChieze et al. (1993). The reviews by Christensen-Dalsgaard (1991) and Christensen-Dalsgaard & Dappen (1992) specifically address the helioseismic determination of the equation of state.

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Fig. 17.1 Observed p-mode frequencies obtained from a 20-day sequence obtained at the 60-Foot Solar Tower of Mount Wilson Observatory.

17.2 Helioseismology: observations After the discovery of the solar five-minute oscillations by Leighton et al. (1962), it took 15 years before they were recognized as global oscillations. Figure 1 shows a typical display of the helioseismic data. While early data looked extremely noisy, the observational progress made since has been tremendous, resulting in very clean data. In a typical representation of helioseismic data, the frequencies of all observed oscillation modes are plotted against their angular degree /. In general, for a given angular degree one observes more than one frequency. They belong to modes of different numbers n of radial nodes. If one plots the observed frequencies, those belonging to modes with the same number of radial nodes can be connected with smooth lines; this is true for any vibrating gas sphere. Figure 1 shows such a v — I diagram obtained from current observations. Since modes with the same radial order n lie on the same ridge, one can therefore identify the radial order n with the different ridges of the diagram. Such an identification is possible up to an unkown global constant no. Duvall (1982) found a technique to resolve this remaining ambiguity and to identify the radial order uniquely. In the mid-seventies, the ridges in the v — l diagram began to emerge from the noise (Deubner, 1975); once they were seen, they definitely established the solar nature of the five-minute oscillations as a superposition of global oscillation modes, a suggestion made earlier by Ulrich (1969, 1970).


371

If the Sun were spherically symmetric, then each mode frequency vn\ would be 2/ + 1 times degenerate. The solar rotation (like any other nonspherical perturbation, such as, e.g., magnetic fields) breaks this symmetry, thus each frequency is split into a multiplet. The splitting is small, since it is of the order of the angular frequency of the solar rotation, which has a period of a little less than a month. Therefore the rotational splitting is too small to be visible in a plot of absolute frequencies such as Figure 1. However, thanks to observational series of weeks and months, the splittings can be well observed for a wide range of / (see, e.g., Harvey, 1988; Rhodes et a/., 1990). Why did it take some 15 years before the oscillations were properly identified? The reason is that the oscillation velocities are tiny, less than 1™. And yet, such velocities are observed using the Doppler effect of light. From each wing of a given spectral line, a narrow piece is cut out and sent through an interferometer into a comparator. The intensity difference of the two parts then becomes a measure of the Doppler line shift, and thus radial velocity. Since the solar disk can be well resolved, such measurements can nowadays be made typically for 1024 X 1024 pixels simultaneously, and this at a rate of a few times per minute. Using the Doppler effect of light, velocities of the order of 1-jp are only marginally detectable. One might therefore wonder why one can obtain so clean a picture as in Figure 1. This question is even more in order if one considers the seemingly chaotic motion on the Sun, granulation, supergranulation, flares, rotation, and so on. The answer lies in the extreme regularity and the surprisingly long life time of the modes, which allow the observers to follow an individual oscillation mode for days and weeks. Therefore, the strict periodicity of the signal is exploited, so that in the end the frequencies can be determined very accurately against all initial odds. The data like those of Figure 1 allow a high precision analysis of the structure of the solar interior. Tabulated frequencies are given in the article by Libbrecht et al., 1990. The relative precision, with which each of the observed mode frequencies vn\ is determined, now attains 10~4, which is at least one order of magnitude better than the uncertainties of any current theoretical predictions. The reason for this inadequacy of the theoretical models is that they are not (yet) sufficiently sophisticated, because the usual simplifying assumptions on convection, opacity, equation of state, nuclear physics, internal rotation, and other physical ingredients are not good enough to explain all the details encountered in the seismological data.

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17.3 Helioseismology: theory Broadly speaking, theoretical inferences from the observed helioseismic frequencies can be made in two ways. In the forward approach, we build a solar model and compute its normal modes. Then the "best" model is the one that satisfies all observational constraints. Should there be more than one "best" model, an aesthetic principle such as Occam's razor is invoked to select the simplest of them. In the inverse approach, we try to make as few theoretical assumptions as possible to infer the physical state of the solar interior directly from the oscillation frequencies.

17.3.1 The general equations for evolution and oscillations For tutorial purposes I will go somewhat off the beaten track and discuss the evolution and oscillations of the star at the hand of the same set of hydrodynamic equations. Of course, the time scales of evolution and oscillations are so much different that in practical calculations one always separates the two parts. Making here as many simplifications as I dare, I refer the interested reader to the superb book by Unno et al. (1989). The solar case is extensively dealt with, for instance, in the reviews by ChristensenDalsgaard and Berthomieu (1990) and Turck-Chieze et al. (1993). I neglect viscosity, and assume that any treatment of turbulent motion, or convective heat transfer, is done in terms of a mean-field approach. This means that state variables are averaged over time-scales of turbulent motion. Such an approach is justified except in a thin layer beneath the solar photosphere. Under the assumptions of inviscid motion and mean-field variables, the resulting 9 equations are

^ + div(pv)] = 0

(2) (3)

= AirGp

(4)


373

F = K(VT;7>,X)

(5)

enuc = €nuc(T,p,X)

(6)

K=K(T,P,X)

(7)

p = p(T,p,X)

(8)

s = s(T,p,X)

(9)

Here, v is the (Eulerian) velocity field, p and p are pressure and density, respectively, 4> is the (self-) gravitational potential, s is specific entropy (per unit mass), enue is the nuclear energy generation rate, F is the energy flux (in the mean-field sense) through the star, the vector X = (X\,X2,...Xn) symbolizes the chemical composition, with X{ being the mass fractions of element t, and K is the opacity. Note the square brackets in Eqs. (1-3). They are there for the discussion of the separate issues of evolution and oscillation (see below). Taking into account the vector nature of Eq. (1) and (5), Eqs. (1-9) are 13 equations for the 13 hydrodynamic fields v, p, p, , T, s, F, € nuc , «• Eqs. (1-4) are partial differential equations, Eq. (5-9) are "material" equations, and it is no surprise that they are the hard part of the overall problem. The toughest among them is the expression of the "conductivity" for energy (Eq. 5), because it is the result of wholly different physical processes according to the physical conditions, given by T, p, VT. Energy transport by radiation, convection and electron conduction are the most familiar ones. As long as stellar matter is optically thick (which it is except near the stellar surface), Eq. (5) can be simplified with the help of the diffusion approximation

but when matter is optically thin, strictly speaking even the form of Eq. (5) is inappropriate, because then radiative transport becomes intrinsically nonlocal, and radiation hydrodynamic will have to be brought into the game (see the book by Mihalas and Mihalas, 1984).

374


Eq. (5) also needs some "switch" to change to convective form of energy transport when the local conditions do not warrant a stable radiative stratification. For the equilibrium model, one usually assumes a stability criterion d la Schwarzschild or Ledoux plus some mixing-length formalism (see e.g. Cox & Giuli, 1968; Gough k Weiss, 1976; Unno et a/., 1989). For the oscillation part, the interplay between convective and oscillatory motion can become very complicated. Compared with the question of energy transport (Eq. 5), the rest of the material equations (6-9) look relatively harmless. The most difficult among them is opacity (7), which appears in the diffusion approximation (10).

17.3.2 Evolution Formally speaking, the problem of stellar evolution is the one of Eqs. (1-9) without the parts in the big square brackets, that is without the inertia term of Eq. (1) and the thermal term of Eq. (3). Just to illustrate with a familiar equation, note that in the approximation of a spherically symmetric configuration Eqs. (1) and (4) allow elimination of and become the wellknown equation of stellar structure (see, e.g., Schwarzschild, 1958). dp _ GMrp ( } dr~ r* * Here, as usual, Mr denotes the mass of the sphere of radius r and G the constant of gravitation. Similarly, in the spherical approximation, Eq. (3) becomes (with Lr = 4irr2F) — = 4wr2pcnuc, (12) ar and Eq. (5) [in the form of the special case of Eq. (10)] becomes the equally familiar equation of the radiative temperature gradient

dr 647TKo and their partial derivatives evaluated at the equilibrium values. Under these assumptions (static and spherical equilibrium and linearization), there are special solutions that are products of a radial amplitude with an exponential time dependence and a (possibly vector-) spherical harmonic function. If we consider the example of pressure we write

376


(^0)-

(14)

There are analogous expressions for all other variables. The general solution is a superposition of such particular solutions. If we assume that thermal heat losses are negligible small during the fast oscillations, then we deal with adiabatic oscillations. Drastic simplifications become possible. The whole equation (3) disappears, because the left-hand side vanishes identically, showing that the equilibrium condition is exactly preserved (this statement is true even in the nonlinear case. In the absence of equation (3), temperature, energy flux and opacity do not participate in the oscillation equation, though they are of course important in the equilibrium part. Thus equations (5-7) are also gone. Thermodynamics becomes ultra simple, especially in the linear case, where the co-moving Lagrangian pressure and density fluctuations (Sp and Sp) are simply related through the equilibrium adiabatic gradient Fi = (dlnp/d\np)a

.

(15)

P

In the nonlinear adiabatic case, this simple relation would have to be replaced by the function p(p) that follows from the integral of motion s, i.e. from the implicit equation s = s(p,p,X.) = const. A further simplification of the adiabatic problem is that the equation of continuity permits expressing the tangential component of the displacement field in terms of the pressure fluctuation (for details see Unno et a/., 1989). We thus arrive at the famous adiabatic eigenvalue problem of stellar oscillations, which plays a central role in helioseismology. The result is, for each angular degree /, an eigenvalue problem, which consists of coupled equations for the radial amplitudes of the displacement vector £ r , the fluctuation of pressure pf and of the gravitational field , and of the usual boundary conditions at the center and the surface of the star. For radial oscillations or in the so-called Cowling approximation for nonradial oscillations (where one neglects changes of the gravitational field during the oscillatory motion), the equations become especially simple. Their formal type is (again, see Unno et al., 1989, for details)


377

f = Dp + Et + F

(16)

dr The Coefficients A,B,C,D,E,F are not constant but functions of the radius. They contain the properties of the equilibrium model and, most importantly, the eigenvalue u\. As mentioned, boundary conditions complement the equations (16). Thus the problem of adiabatic stellar oscillations is, for each /, completely analogous to an inhomogeneous vibrating string. For each / there is a set of solutions with different radial nodes n and frequencies uni. It should be clear by now that the solution of the eigenvalue problem is much easier than finding the equilibrium model through stellar evolution. Of course one can make things more complicated here as well. By considering nonadiabatic motion, the energy equation (3) is coining back, and with it temperature, which forces us to bring in £nuc» K» a n ^ convection. Again, I refer to the book by Unno et al. (1989). Near the solar surface there are nonadiabatic effects that have to be treated properly before the theoretical data will match the observations shown in Figure 2. 17.3.4 Inverse analysis If one writes Eq. (16) very formally, the oscillation frequencies un\ can be written as functionals r), p(r),...]

(17)

of the structure of the Sun. So far we have discussed how to obtain the frequencies, given the structure. With the ability to do so, one can compare observed frequencies with computations based on different models, and in this way obtain some information about the solar structure. However, it is evidently desirable to attempt to invert the process, to obtain more extensive information about the properties of the solar interior from the observed frequencies. Such inverse analyses are, in a certain sense, implicit in any type of scientific measurement, since a raw measurement rarely supplies the quantity that one is interested in. However, in the present case the relation between the desired properties of the Sun, e.g. p(r), and the observed quantities is more complex, since each frequency is sensitive to the structure of a substantial part of the Sun; thus the inverse problem is correspondingly more difficult. Similar problems are encountered in other

378


branches of science, such as geophysics and radiation theory, and there is a substantial literature dealing with them {e.g. Parker 1977; Deepak 1977; Craig k Brown 1986; Tarantola 1987). An alternative method of inversion is based on asymptotic theory, where local propagation properties for acoustic waves are approximately examined in the spirit of a JWKB analysis. The need for such an approximate discussion comes from the fact that, although the numerical solution of the equations of adiabatic oscillations is relatively simple, it does not immediately provide an understanding of the properties of the oscillations. Such a direct understanding can come from the approximate asymptotic analysis It was shown by Gough (cf. Deubner & Gough 1984; Gough 1986) how to write down an approximate form of the oscillation equations, from which it is straightforward to obtain the asymptotic behavior of the solution. It turned out immediately that this asymptotic approach also opens the door for elegant asymptotic inversion methods. I refer the reader to the papers by Gough (1985), Thompson (1991), Gough & Thompson (1991), Brodsky & Vorontsov (1993), and Gough & Vorontsov (1993). The last two papers deal with a nonlinear asymptotic inversion. The power of such inversions for the equation of state is illustrated in the article by Vorontsov et al. (tfiese proceedings).

17.4 Comparison of theory with observations The most direct way to compare theory and observation is to compute the analogue of Fig. 1 with the forward techniques mentioned above, so that the difference between each observed and computed frequency can be taken. Figure 2 shows four such diagrams of frequency differences, each for a different theoretical model. Two equations of state and two different opacity tables were used in the models. The two equations of state were (i) the Eggleton, Faulkner & Flannery (1973) (EFF) equation of state and (ii) the CEFF equation of state, which is, as explained below, an EFF plus a Coulomb term (Christensen-Dalsgaard, 1991; Christensen-Dalsgaard & Dappen, 1992). The opacities used were the Cox and Tabor (CT) (1976) and Los Alamos Opacity Library (LAOL) tables. Since I merely want to illustrate the sensitivity of the helioseismic method, it doesn't matter that these opacities are not the most current ones. A recent calculation based on Livermore opacitites can be found in Berthomieu et al. (1993a). I remark in passing that in such comparisons of observed with computed data ("0-C diagrams"), it is useful if an appropriate scale factor is taken out (see, e.g., Christensen-Dalsgaard 1988; Christensen-Dalsgaard

379


I £

-30

a)

-10

-30

2000

3000 4000 »„ (/iHz)

2000

3000

4000

2000

3000 4000 V- (MHz)

2000

3000

4000

-30

Fig. 17.2 Frequency differences, scaled by the factor Qn\ (see text), between observed frequencies in the compilation by Libbrecht et al. (1990) and four sets of computed frequencies, in the sense (observation) - (theory). The abscissa is cyclic frequency i/n/. The points have been connected with lines according to the value of the degree /: / — 20,30: ; / = 40,50,60,80,100: ; / = 120,150,200,300,400: ; and / = 500,600,700,800,900,1000: . The models are distinguished by their equation of state and opacity (a) EFF equation of state, CT opacity; (b) EFF equation of state, LAOL opacity; (c) CEFF equation of state, CT opacity; (d) CEFF equation of state, LAOL opacity (from Christensen-Dalsgaard k. Dappen, 1992).

380


& Berthomieu 1991). This scale factor Qni, which is essentially the inertia or kinetic energy - of the mode with quantum numbers nl (normalized to the same surface amplitude), contains the principal / and frequency dependence of the individual mode frequencies vn\. The purpose of the illustration in Fig. 2 is to show the sensitivity of the helioseismic analysis with respect to changes in the physics of the model. A perfect model would yield a horizontal line corresponding to all f>vn\ = 0. Note that the discrepancies between theory and observation are huge compared to the observational errors which are nowadays significantly below 1 fiUz. Such a combination of quantity and quality of astrophysical data is truly exceptional. 17.5 The equation of state As we have seen, the three basic material properties required in stellar models are the equation of state, opacity, and the nuclear-energy generation rate. At this meeting, the focus is on the equation of state. I shall use the term equation of state in a slightly broader sense than usual, so that it encompasses not only pressure as a function of temperature and density, but also all thermodynamic quantities. These quantities must be consistent with each other, that is, their appropriate Maxwell relations have to be satisfied. Such formal consistency is always achieved if the equation of state and the thermodynamic quantities stem from a single thermodynamic potential. In trivial models (e.g. in a plasma assumed to be fully ionized everywhere) it is possible to write down a consistent equation of state and thermodynamic quantities independently. However, in more realistic cases, modeling a thermodynamic potential is the only practical way to obtain the equation of state and thermodynamic quantities. A quick glance at Fig. 2 reveals that solar observations are indeed very sensitive to details of the equation of state. One might go further and conclude that the Sun prefers the CEFF to the EFF equation of state. However, such conclusions are fraught with danger, although probably not in this clear-cut case. The reason why one has to be prudent is that there are too many uncertainties in the solar model, coming, e.g., from convection or opacity, so that one has to be alert to the possibility that by changing the equation of state one could trigger changes in the other physical parameters. An illustration for this is found at each railroad crossing in France, where a sign warns: "un train peut en cacher un autre" (which, applied to our situation, means: proceed with caution, watch out for a hidden train of thought). If, say, the opacity is bad, one can not rule out that a worse equation of


381

state could cause an overall better agreement with observations. Only when simultaneous progress with the other physical quantities is made (that is, if someone is watching the other track, to use the train metaphor), we will learn how to disentangle the different effects. However, for a sensitivity analysis, Figure 2 is already sufficient. The transitions from panels a to c and 6 to d, respectively, are obtained by putting some additional nonideal effects (the Coulomb pressure) into the equation of state with everything else unchanged. The response of the Sun, as seen through the "eyes" of helioseismology, is huge. I will not elaborate how the equation of state is modeled. Several authors of these proceedings do it (Rogers, Alastuey, Saumon, and Chabrier). My message is different: I intend to show why there is still a long way to go before rigorous theories (for instance that presented by Alastuey, these proceedings) can be used in solar and stellar models. I will begin with requirements for any solar or stellar equation of state. I insist that formal aspects (such as consistency and smoothness) play a crucial role. As a consequence, I would like to raise sympathy for the many home-grown formalisms that stellar modelers have been constantly developing. Then I will discuss the nonideal plasma effects that have to be included in realistic solar equations of state. Finally I will present a few selected results from equation of state comparisons. In the absence of a perfect equation of state, the comparisons can give us at least important information about the amount of the current uncertainty in the equation of state. Also, it will tell us at which temperatures and densities the uncertainty is most noticeable and to what degree solar observations can discriminate between various models. 17.5.1 Requirements on an equation of state for stellar models A stellar equation of state has to satisfy four conditions: (i) a large domain of applicability (in p, T), (ii) a high precision of its numerical realization, (iii) consistency between the thermodynamic quantities, and (iv) the possibility to take into account relatively complex mixtures with at least several of the more abundant chemical elements. More specifically, the first condition demands that the formalism can be used from the stellar surface (the photosphere), where T is typically a few 103 K and p some 10~7 g/cm3, to the center of a star where T is, again typically, about 107 K and p some 102 g/cm3. The second condition demands that a given formalism can be cast in an algorithm that converges without ambiguity and with sufficient precision, so that all required thermodynamic derivatives (such as adiabatic

382


gradients) can be computed. Note, that for this only formal precision is required: reality of the physical description is a different issue. The third condition, consistency, states that all thermodynamic quantities stem from a single thermodynamic potential. This condition is often violated in twoor more-zone formalisms, which contains a different physical theory in different parts of a star. An example is the ad hoc imposition of full ionization in the central region, in order to mimic a pressure-ionization device, in combination with a conventional Saha equation in the envelope of the star. Such a formalism leads to a discontinuous thermodynamic potential and a violation of thermodynamic identities. Such violations of thermodynamic identities are inadmissible in calculations of stellar structure and oscillations. As we have seen, calculations of stellar oscillation frequencies often exploit thermodynamic quantities to transform one variable into another. Equation (15) shows such a transformation. There the adiabatic gradient Fi is used to establish a connection between density and pressure changes, and it is an absolute necessity that the Fi is consistent with the equation of state and other thermodynamic variables of the model. This example illustrates the necessity of formal consistency. Finally, the third and last condition, i.e. the possibility to describe rather realistic chemical compositions, is a bit less important for the equation of state itself. However, for opacity, heavy elements are very important, and a good equation of state plays an important role in any opacity calculation. 17.5.2 The role of the solar convection zone Energy transport by radiation is treated adequately in the solar interior in the diffusion approximation; on the other hand, energy transport by convection is usually treated in a rather crude way, with an a priori unknown parameter, the so-called mixing length (see, e.g., Cox and Giuli, 1968). Near the surface, convection is probably sufficiently vigorous to cause dynamic effects on the average hydrostatic equilibrium, yet such effects are often ignored. At the lower boundary of the convection zone, motion is normally supposed to stop at the point where convective instability ceases; there is no doubt, however, that motion extends into the convectively stable region through convective overshoot, although the extent of the overshoot is uncertain (see, e.g. Berthomieu et al., 1993b). Despite the complications it introduces, in a certain sense convection simplifies the structure of the outer parts of the Sun. Regardless of the uncertain details of convective energy transport, there is no doubt that


383

except in a thin boundary layer near its top the convection zone is very nearly adiabatically stratified {e.g. Gough k Weiss 1976). One can show (Christensen-Dalsgaard, 1986) that the structure of the almost adiabatically stratified convection zone only depends on the equation of state, the composition and the constant value of the specific entropy, which in turn is essentially fixed by the value of the mixing-length parameter; particular, the convection zone structure is insensitive to the opacity. Another simplification of convection is that it makes the chemical composition homogeneous in the convection zone, although there is of course the possibility of gravitational settling (for a recent calculation, see Christensen-Dalsgaard et a/., 1993). Beneath the convection zone, the stratification becomes highly dependent on radiative opacity. It is difficult to disentangle the helioseismic effects of equation of state and opacity, but if opacity can be nailed down relatively accurately, an equation of state diagnosis can also become possible. An examlpe of an equation of state issure is the possibility of partial recombination of He+ ions in the solar center (see Christensen-Dalsgaard k Dappen, 1992). 17.6 Equation of state comparisons The most direct way to test the equation of state would be laboratory experiments. However, so far they have not yet helped to check realistic stellar equations of state. For instance, attempts to use constraints from a highprecision optical emission spectrum (e.g. Wiese, Paquette k Kelleher, 1973) have failed, because line-broadening effects were overshadowing the subtle details of statistical mechanics. It is therefore no wonder that - despite their difference in statistical mechanics - several of the currently popular equations of state have been able to reproduce that optical experiment (Dappen, Anderson and Mihalas, 1987; Seaton, 1990; Iglesias k Rogers, 1992). An alternative "experimental" approach is to use solar oscillation data. As the comparisons between observed and theoretical solar oscillation frequencies (Fig. 2) demonstrate, one can use the Sun to test the equation of state (for more details, see Christensen-Dalsgaard, Dappen k Lebreton, 1988; Christensen-Dalsgaard, 1991; Christensen-Dalsgaard k Dappen, 1992). Inversions of solar oscillation frequencies, such as those presented by Vorontsov et al. (these proceedings), have also demonstrated a high diagnostic potential for subtle effects, such as the location of the pressure-ionization region of helium and the influence of heavy elements in the equation of state. The disadvantage of a solar diagnosis is of course that we cannot vary the

384


parameters: we have to accept solar conditions as they are. Only asteroseismology carries the promise to overcome this handicap (for a recent review, see e.g., Christensen-Dalsgaard, 1993). In the absence of a rigorous computation of the equation of state (to the needed accuracy), one can make comparisons between different models of the equation of state. Such comparisons will give us information about the overall uncertainty in the equation of state. But they also allow solar physicists to determine how uncertainties in the equation of state propagate into theoretically predicted oscillation frequencies. In this way, a "map" of the T—p plane can be drawn, showing localized "interesting" regions, where nonideal effects of one or another kind are important. I will briefly present the equations of state used in the comparisons. More details about them (and further references) can be found in the article by Christensen-Dalsgaard & Dappen (1992). I just recall that all currently used stellar equations of state can be classified in terms of the so-called "chemical picture" and the "physical picture" (Krasnikov, 1977). While in the more conventional chemical picture bound configurations (atoms, ions and molecules) are introduced and treated as new and independent species, only fundamental particles (electrons and nuclei) appear in the physical picture. In the chemical picture, reactions between the various species occur, and thus the thermodynamic equilibrium must be sought among the stoichiometrically allowed set of concentration variables by means of a maximum entropy (or minimum free-energy) principle. In contrast, the physical picture has the aesthetic advantage that there is no need for a minimax principle; the question of bound states is dealt with implicitly through the Hamiltonian describing the interaction between the fundamental particles. For exhaustive treatments of these issues, consult the three books by Ebeling, Kraeft & Kremp (1976), Kraeft et al. (1986), Ebeling et al. (1991).

17.6.1 EFF Eggleton, Faulkner & Flannery (1973) developed a simple equation of state in the chemical picture (EFF) that is formally consistent and includes an ad hoc pressure ionization device that works at least qualitatively correctly. The device is not based on a physical model (e.g. a description of an atom and its surrounding particles), but is imposed by forcing the anticipated result, i.e., full ionization at high densities. In addition, the EFF equation of state incorporates a correct treatment of the partially degenerate electrons according to Fermi-Dirac statistics. Bound systems (atoms and ions) are


385

always assumed to be in their ground state; the ground-state energy is constant and equal to the free-particle value.

17.6.2 CEFF To overcome the lack of a Coulomb term in the EFF equation of state, J0rgen Christensen-Dalsgaard and I have added a Coulomb configuration^ term in the Debye-Hiickel approximation (taken from the MHD equation of state). Such an upgrade of the EFF equation of state was motivated by the fact that adding a Coulomb term to the EFF equation of state makes a significant contribution towards a more realistic equation of state (see below and the papers by Christensen-Dalsgaard, 1991; Christensen-Dalsgaard & Dappen, 1992). Of course the remaining disadvantages of the EFF equation of state still point to the need of more complete formalisms. However, the successful application of the CEFF equation of state to solar physics makes it very well suited as a reference equation of state.

17.6.3 MHD The Mihalas-Hummer-Dappen (MHD) equation of state (Hummer & Mihalas, 1988; Mihalas et al., 1988; Dappen et a/., 1988) is realized in the chemical picture with the free-energy minimization method. Occupation probabilities are introduced on the one hand to avoid the famous (or rather notorious) discontinuities that come along with simple cut-off recipes for internal partition functions. On the other hand they represent a result that should come from quantum mechanics, namely the fraction of atoms or ions for which a given state can exist (given the constraints of the surrounding particles). Only then, these "available" states are populated according to statistical mechanics. It is dear that such an approach is largely intuitive. However, its advantage is that complicated plasmas can be modeled, with detailed internal partition functions for a large number of atomic, ionic, and molecular species. All particles are allowed to interact with each other. Also, full thermodynamic consistency is assured by analytical expressions of the free energy and its first- and second-order derivatives. This not only allows an efficient Newton-Raphson minimization, but, in addition, the ensuing thermodynamic quantities are of analytical precision and can therefore be differentiated once more, this time numerically. Reliable third-order thermodynamic quantities are thus calculated. The MHD equation of state was realized for the international "Opacity Project" (see Seaton, 1987).

386


17.6.4 OPAL The OPAL equation of state is realized in the physical picture. A detailed presentation is given by Rogers (these proceedings). In the physical picture, the concept of a perturbed atom in a plasma is not needed at all. Therefore, no assumptions about energy-level shifts or the convergence of internal partition functions have to be made. On the contrary, properties of energy levels and the partition functions come out from the formalism. The OPAL equation of state was developed by a group at Livermore as part of their opacity project (Rogers, 1986; Iglesias, Rogers & Wilson, 1987; Rogers, these proceedings). This equation of state does satisfy the requirements from stellar modelling that I mentioned above; however, a systematic application application of the OPAL equation of state to helioseimology is still awaiting. 17.6.5 Results from the comparisons Early comparisons showed a striking agreement between the MHD and OPAL equation of state for conditions as found in the hydrogen-helium ionization zones of the Sun (Dappen, Lebreton & Rogers, 1990; Dappen, 1990). For convenience, a representative result from this early comparison is shown in Figure 3, which compares the MHD and OPAL results with that of the simple EFF formalism (which is essentially a consistent ground-stateonly Saha equation of state under these conditions). The absolute curves of part a of Figure 3 are merely able to show the difference between MHD (or OPAL) and the simple EFF results. To see the difference between the MHD and OPAL results, one needs the magnified part 6, which shows the relative differences between MHD and EFF, and between OPAL and EFF values, respectively. This relative plot now not only allows one to see the difference between MHD and OPAL results, but also their striking similarity. Later, it turned out that this agreement was nearly accidental. The reason for this was found by varying the parameters of the MHD equation of state. It followed that on the chosen isochore, all thermodynamic quantities are dominated by the Coulomb pressure correction (Dappen, 1990; ChristensenDalsgaard, 1991; Christensen-Dalsgaard k Dappen, 1992). The Coulomb correction overshadows the effect of the excited states (which are of course treated differently in the MHD and OPAL approach). Note that the Coulomb term acts directly and indirectly, at least in the language of the chemical picture, because it is not mainly the free-energy of the Debye-Hiickel term itself, but rather also the Coulomb-induced shift in the ionization equilibrium, which is responsible for the deviation from the unperturbed EFF result.

387


2.5

0.02 i

-^

0.00

i !

—«^^

f

|

\ \

1 1 1 !

0.02

N

f

I k

t f

•

,

b)

0.04 4.0

4.5

5.0

5.5

log T

Fig. 17.3 Comparison of XT = (dlnp/dlnT)fi for /> = 10"6 5 g cm"3. Absolute quantities (a) and relative differences (with respect to EFF) (b) are shown. See text for more details. Of course, solar physicists were happy that two completely different formalisms delivered the same equation of state, but, by the same token, a first attempt to use the Sun as an equation-of-state test was also thwarted. This discovery suggested to upgrade the simple EFF equation of state with the help of the Coulomb interaction term. The resulting equation of state (called CEFF) has become a useful tool for solar physics (Christensen-Dalsgaard, 1991; Christensen-Dalsgaard & Dappen, 1992); at the same time, however, it became also clear that a helioseismic test of the important issue of chemical versus physical picture would be more difficult than first thought. For reasons not yet fully understood it seems that in the chemical picture, the signature of internal partition functions, such as those employed in the MHD equation of state, is much less visible in the thermodynamic quantities than a naive estimation of the shift in the ionization equilibrium would predict. It is likely that there are accidental cancellations in the derivatives of the free energy. The cancellations of partition-function effects in the chemical picture seem to be greatest for the ionization zone of hydrogen and somewhat less for those of helium. A more recent comparison of MHD


388

1.6B0

1.670

1.660

5.0

5.5

6.0

6.5

7.0

Fig. 17.4 Fi for p = 5.00 x 10~3g cm" 3 and a representative solar mixture of H, He, and O. Parts (a) and (b) as in Fig.3, but here with CEFF instead of EFF. See text for more details.

and OPAL values (Dappen, 1992) has examined selected cases of higher densities (where sizeable discrepancies appear) and a first case of a mixture involving a representative solar heavy element (oxygen). It appears that for the heavier elements, the internal partition functions finally lead to the intuitively expected consequences for the thermodynamic quantities. Figure 4 shows the result of this comparison with oxygen for the quantity Ti. Density was chosen as p = 0.005 g cm" 3 , a value suggested by a helioseismic study of the solar helium abundance (Kosovichev et al., 1992). Here, not only do the large MHD partition functions cause shifts in the ionization balance but these shifts also significantly propagate into the thermodynamic quantities. The effect is large enough so that it appears, despite the small relative number of the heavy elements in the mixture, to be within reach of helioseismology (for more details see Christensen-Dalsgaard & Dappen, 1992; Dappen et al. (1993)). To examine the MHD ionization fractions, a single case was examined (T = 2.10 X 10s K,p = 5.00 X 10~3g cm" 3 ), once with the full MHD equation of state, once with a "stripped-down" version of MHD, which does not


389

contain any excited states (but is otherwise identical). The resulting ionization fractions of 0 3 + , 0 4 + , 0 5 + were, respectively, 0.314, 0.248, 0.364 for the stripped-down MHD (without excited states), and 0.304, 0.476, 0.182 for the full MHD. (The result for the stripped-down very closely reflects the ground-state weights of the ions). Not unexpectedly in view of the Planck-Larkin partition function (see Rogers, these proceedings), the OPAL equation of state predicts ionization fractions close to those of the strippeddown MHD equation of state (Rogers, private communication). This comparison for the first time establishes a clear case of disagreement between the MHD and OPAL results. Clearly, the origin of the discrepancy in the ionization degrees is due to the treatment of the excited states. Of course, only some 2 percent of the matter in the Sun consist of elements heavier that H and He, and therefore the signature of the MHD-OPAL discrepancy in Fi (Figure 4) is small (of the order of 10~3). Nevertheless, as has been demonstrated by Christensen-Dalsgaard & Dappen (1992), even the resulting tiny sound-speed differences are within reach of a helioseismic diagnosis.

17.7 Conclusions Even weakly-coupled plasmas can pose tough problems if high accuracy is demanded. Solar oscillations are an example of a case where the present observational material is much better than the theoretical models. The solar convection zone is especially well suited for a study of the equation of state. It was suggested in a number of early papers {e.g. Berthomieu et a/., 1980; Ulrich, 1982; Shibahashi et a/., 1983, 1984) that improvements in the equation of state can reduce discrepancies between theory and observations. Later, Christensen-Dalsgaard, Dappen & Lebreton (1988) showed that the MHD equation of state significantly reduced these discrepancies for a large range of oscillation modes. Since the MHD equation of state simultaneously incorporates several different types of non-ideal corrections, it did not become immediately clear which one of these corrections was mainly contributing to this success. iFrom selected comparisons of the MHD with the OPAL equation of state, it turned out, rather surprisingly, that the net effect of the hydrogen and helium bound states on thermodynamic quantities was to a large degree eclipsed beneath the influence of the Coulomb term, which was thus recognized as the dominant non-ideal correction in the hydrogen and helium ionization zones. This discovery led to an upgrade of the simple EFF

390


equation of state through the inclusion of the Coulomb interaction term (CEFF). However, for the heavier elements it appears that, in the chemical picture, the internal partition functions finally lead to the expected consequences for the thermodynamic quantities. The heavy elements can thus become the ideal testing ground for the effects of bound states in partially ionized plasmas. The small abundance of heavy elements in the Sun will make a diagnosis difficult and stretch the power of helioseismology to its limits, but as the study by Vorontsov et al. (these proceedings) shows, there are encouraging signs that the difficulties can be overcome. Acknowledgement: I would like to thank Angel Alastuey, Vladimir Baturin, Gilles Chabrier, J0rgen Christensen-Dalsgaard, Werner Ebeling, Andreas Forster, Douglas Gough, Asher Perez, Ed Rhodes, Forrest Rogers, and Sergei Vorontsov for stimulating discussions. I specifically want to thank J0rgen Christensen-Dalsgaard for the results displayed in Figure 2, Forrest Rogers for the OPAL results contained in Figures 3 and 4, and Ed Rhodes for the v — I diagram shown in Figure 1.

References Alastuey, A., these proceedings Bahcall, J.N. & Ulrich, R.K., Rev. Mod. Phys. 60, 297-372 (1988) Berthomieu, G., Cooper, A.J., Gough, D.O., Osaki, Y., Provost, J. & Rocca, A., in Lecture Notes in Physics, Vol. 125: Nonradial and Nonlinear Stellar Pulsation, eds Hill, H.A. & Dziembowski, W., Springer, Berlin, 307-312 (1980) Berthomieu, G., Provost, J., Morel, P., Lebreton, Y., Astron. Astrophys. 268, 775-791 (1993a) Berthomieu, G., Morel, P., Provost, J., Zahn, J.-P., in Proc. IAU Symposium No 137: Inside the Stars, eds. Werner W. Weiss and Annie Baglin, ASP Conference Series, Vol. 40, 60-62 (1993b) Brodsky, M. k Vorontsov, S.V., Astrophys. J., 409, 455 (1993) Chabrier, G., these proceedings Christensen-Dalsgaard, J., in Seismology of the Sun and the distant stars, ed D.O. Gough, Reidel, Dordrecht), 23-53 (1986) Christensen-Dalsgaard, J., in Seismology of the Sun & Sun-like Stars, eds Domingo, V. & Rolfe, E.J., ESA SP-286, Noordwijk, The Netherlands, 431-450 (1988) Christensen-Dalsgaard, J., In Lecture Notes in Physics, Vol. 388: Challenges to Theories of the Structure of Moderate-mass Stars, eds Gough, D.O. k. Toomre, J., Springer, Heidelberg, 11-36 (1991) Christensen-Dalsgaard, J., in Proc. IAU Symposium No 137: Inside the Stars, eds. Werner W. Weiss and Annie Baglin, ASP Conference Series, Vol. 40, 483-496 (1993).


391

Christensen-Dalsgaard, J. k Berthomieu, G., in Solar Interior and Atmosphere, eds Cox, A.N., Livingston, W.C. k Matthews, M., Space Science Series, University of Arizona Press, Tucson, 401-478 (1991) Christensen-Dalsgaard, J. k Dappen, W., Astron. Astrophys. Review 4, 267-361 (1992) Christensen-Dalsgaard, J., Dappen, W. k Lebreton, Y., Nature 336, 634-638 (1988) Christensen-Dalsgaard, J., Gough, D.O. k Toomre, J., Science 229, 923-931 (1985) Christensen-Dalsgaard, J., Profitt, C.R. k Thompson, M.J., Astrophys. J. 403, L75 (1993) Cox, A.N. k Tabor, J.E.: 1976, Astrophys. J. Suppl. 31, 271-312 (1985) Cox, J.P., and Giuli, R.T., Principles of Stellar Structure, Gordon and Breach, New York, 1968 Craig, I.J.D. k Brown, J.C., Inverse Problems in Astronomy: a Guide to Inversion Strategies for Remotely Sensed Data, Adam Hilger, Bristol (1986) Dappen, W., in Lecture Notes in Physics, Vol. 367: Progress of Seismology of the Sun and Stars, eds. Osaki, Y. k Shibahashi, H., Springer, Berlin, 33-40 (1990) Dappen, W., in Astrophysical Opacities, eds. C. Mendoza k C. Zeippen (Revista Mexicana de Astronomia y Astrofisica) 141-149 (1992) Dappen, W., Anderson, L.S. k Mihalas, D., Astrophys. J. 319, 195-206 (1987) Dappen, W., Lebreton, Y. k Rogers, F., Solar Physics 128, 35-47 (1990) Dappen, W., Mihalas, D., Hummer, D.G. k Mihalas, B.W., Astrophys. J. 332, 261-270 (1988) Dappen, W., Gough, D.O., Kosovichev, A.G. k Rhodes, E.J., Jr., in Proc. IAU Symposium No 137: Inside the Stars, eds. Werner W. Weiss and Annie Baglin, ASP Conference Series, Vol. 40, 304-306 (1993). Deepak, A., Inversion Methods in Atmospheric Remote Sounding, Academic Press, New York (1977) Deubner, F.-L., Astron. Astrophys. 44, 371 (1975) Deubner, F.-L. k Gough, D.O., ylnn. Rev. Astron. Astrophys. 22, 593-619 (1984) Duvall, T.L., Nature 300, 242-243 (1982) Ebeling, W., Kraeft, W.D. k Kremp, D., Theory of Bound States and Ionization Equilibrium in Plasmas and Solids, Akademie Verlag, Berlin, DDR (1976) Ebeling, W., Forster, A., Fortov, V.E., Gryaznov, V.K. k Polishchuk, A.Ya., Thermodynamic Properties of Hot Dense Plasmas, Teubner, Stuttgart, Germany (1991) Eggleton, P.P., Faulkner, J. k Flannery, B.P., Astron. Astrophys. 23, 325-330 (1973) Gough, D.O., Solar Phys. 100, 65-99 (1985) Gough, D.O., in Hydrodynamic and Magnetohydrodynamic Problems in the Sun and Stars, ed. Osaki, Y., University of Tokyo Press, 117-143 (1986) Gough, D.O., in Astrophysical Fluid Dynamics, eds Zahn, J.-P. k Zinn-Justin, J., North-Holland, Amsterdam (1992) Gough, D.O. k Thompson, M.J., in Solar Interior and Atmosphere, eds Cox, A.N., Livingston, W.C. k Matthews, M., Space Science Series, University of Arizona Press, 519-561 (1991) Gough, D.O. k Toomre, J., Ann. Rev. Astron. Astrophys. 29, 627-685 (1991) Gough, D.O. k Vorontsov, S.V., Mon. Not. R. astr. Soc, (1993) submitted Gough, D.O. k Weiss, N.O., Mon. Not. R. astr. Soc. 176, 589-607 (1976)

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Harvey, J., in Seismology of the Sun & Sun-like Stars, eds Domingo, V. k Rolfe, E.J., ESA SP-286, Noordwijk, The Netherlands, 55-66 (1988) Hummer, D.G. k Mihalas, D., Astrophys. J. 331, 794-814 (1988) Iglesias, C.A. k Rogers, F.J., in Astrophysical Opacities, eds. C. Mendoza k C. Zeippen (Revista Mexicana de Astronomia y Astrofisica) 161-170 (1992) Iglesias, C.A., Rogers, F.J. k Wilson, B.G., Astrophys. J. 322, L45 (1987) Kosovichev, A.G., Christensen-Dalsgaard, J., Dappen, W., Dziembowski, W.A., Gough, D.O., k Thompson, M.J., Mon. Not. R. astr. Soc. 259, 536-558 (1992) Kraeft W.D., Kremp D., Ebeling W. k Ropke G., Quantum Statistics of Charged Particle Systems, Plenum, New York (1986) Krasnikov Yu.G., Zh. Eksper. teoret. Fiz. 73, 516 (1977) (English translation: Soviet Phys. - JETP 46, 270-274; author's name mis-spelt as "Karsnikov") Leighton, R.B., Noyes, R.W., Simon, G.W., Astrophys. J. 135, 474 (1962) Libbrecht, K.G., Space Sci. Rev. 47, 275-301 (1988) Libbrecht, K.G. k Woodard, M.F., Science 253, 152-157 (1991) Libbrecht, K.G., Woodard, M.F. k Kaufman, J.M., Astrophys. J. Suppl. 74, 1129-1149 (1990) Mihalas, D.M. k Mihalas, B.W., Foundations of radiation hydrodynamics (Oxford University Press) (1984) Mihalas, D., Dappen W. k Hummer, D.G., Astrophys. J. 331, 815-825 (1988) Noether, E., Nachr. Konig. Gesell. Wissen. Gottingen, Math.-Phys. Kl., 235-257 (1918) [for an English translation see Transport Theory and Stat. Phys. 1 (1971), 186-207] Parker, R.L., Ann. Rev. Earth Planet. Sci. 5, 35-64 (1977) Rhodes, E.J., Jr., Cacciani, A., Korzennik, S., Tomczyk, S., Ulrich, R.K., Woodard, M.F., Astrophys. J., 351, 687-700 (1990) Rogers, F.J., Astrophys. J. 310, 723-728 (1986) Rogers, F.J., these proceedings Saumon, D., these proceedings Schwarzschild, M., Structure and evolution of the stars (Princeton University Press, Princeton, New Jersey) (1958) Seaton, M., /. Phys. B: Atom. Molec. Phys. 20, 6363-6378 (1987) Seaton, M., J. Phys. B: Atom. Molec. Phys. 23, 3255-3296 (1990) Shibahashi, H., Noels, A. k Gabriel, M., Astron. Astrophys. 123, 283-288 (1983) Shibahashi, H., Noels, A. k Gabriel, M., Mem. Soc. Astron. Hal. 55, 163-168 (1984) Tarantola, A., Inverse Problem Theory, Elsevier, Amsterdam (1987) Thompson, M.J., in Lecture Notes in Physics Vol. 388: Challenges to Theories of the Structure of Moderate-mass Stars, eds Gough, D.O. k Toomre, J., Springer, Berlin, 61-80 (1991) Turck-Chize, S., Dappen, W., Fossat, E., Provost, J., Schatzman, E., Vignaud, D., Physics Report 230, 57-235 (1992) Ulrich, R.K., it Astrophys. J, 158, 427 (1969) Ulrich, R.K., it Astrophys. J, 162, 993 (1970) Ulrich, R.K., Astrophys. J. 258, 404-413 (1982) Unno, W., Osaki, Y., Ando, H. k Shibahashi, H., Nonradial Oscillations of Stars, 2nd Edition, University of Tokyo Press, Tokyo (1989) Vorontsov, S.V., Baturin, V.A., Gough, D.O., Dappen, W., these proceedings


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Vorontsov, S.V. & Zharkov, V.N., Sov. Set. Rev. E. Astrophys. Space Phys. 7, 1-103 (1989) Wiese, W.L., Kelleher, D.E. & Paquette, D.R., Phys. Rev. A6 1132-1153 (1972)

18 Transport processes in dense stellar plasmas NAOKIITOH Deparimant of Physics, Sophia University, 7-1, Kioi-cho, Chiyoda-ku, Tokyo 102, Japan

Abstract Transport processes in dense stellar plasmas which are relevant to the interiors of white dwarfs and neutron stars are reviewed. The emphasis is placed on the accuracy of the numerical results. In this review we report on the electrical conductivity and the thermal conductivity of dense matter. The methods of the calculations are different for the liquid metal phase and the crystalline lattice phase. We will broadly review the current status of the calculations of the transport properties of dense matter, and try to give the best instructions available at the present time to the readers. Nous presentons une revue des propoietes de transport dans les plasmas denses stellaires caracteristiques des interieurs de naines blanches et d'etoiles a neutrons. L'accent est mis sur la precision des resultats numeriques. Nous presentons la conductivite electrique et la conductivite thermique dans la matiere dense. Les methodes de calcul sont differentes dans la phase liquide et dans la phase cristalline. Nous donnons une revue generale des calculs des proprietes de transport dans la matiere dense, et nous essayons de donner les meilleures instructions quant aux donnees disponibles actuellement.

18.1 Introduction In recent years white dwarf asteroseismology opened up a new fertile land of astrophysics (Bradley k Winget 1991; Bradley, Winget, k Wood 1992). Consequently, the basic physics data which go into white dwarf models need 394

Itoh: Transport processes in dense stellar plasmas

395

to be sufficiently accurate that they should live up to the standard required by the asteroseismological data. In order to match the accuracy required by white dwarf asteroseismology, tremendous effort has been devoted in the recent years to improve the numerical accuracy of the calculations of the transport properties of dense matter. For the liquid metal phase the papers of Itoh et al. (1983) and Mitake, Ichimaru, k Itoh (1984) have given significantly improved results of the conductivities upon some of the previous results reported in the papers of Flowers & Itoh (1976), Flowers k Itoh (1981), and Yakovlev k Urpin (1981). The paper of Itoh et al. (1984) has significantly improved the results of the conductivities in the cyrstalline lattice phase reported in the papers of Flowers k Itoh (1976), Flowers k Itoh (1981), Yakovlev k Urpin (1981), and Raikh k Yakovlev (1982). Recently, Itoh, Hayashi, k Kohyama (1993) have included the lower densities 10° — 104gcm~3 for the calculation of the electrical and thermal conductivities of dense matter in the crystalline lattice phase. The impurity scattering contributions to the electrical and thermal conductivities of dense matter in the crystalline lattice phase have been recently recalculated by Itoh k Kohyama (1993). The viscosity of dense matter in the liquid metal phase has been calculated by Itoh , Kohyama , k Takeuchi (1987).

18.2 Electrical and Thermal Conductivities in the Liquid Metal Phase In this section we review the calculations of the electrical and thermal conductivities of dense matter in the liquid metal phase following the paper of Itoh et al. (1983). We consider the case that the atoms are completely pressure-ionized. The corresponding condition is expressed by EF(p)>Z2

Ry,

(1)

where Ep{p) is the electron Fermi level at zero temperature and at a given mass density p. Numerically, the condition (1) can be rewritten as p > 0.378AZ2

gem" 3 ,

(2)

where Z and A are the atomic number and mass number of the atom considered, respectively. Thus we have p > 6.05gcm~3 for 4He,/> > 1.63 x 102gcm~3 for 12 C,p > 3.87xl0 2 gcm" 3 for 16 O, and p > 1.43xl04gcm~3for

396


56

Fe. We further restrict ourselves to the density-temperature region in which electrons are strongly degenerate. This condition is expressed as

T < TF = 5.930 x 109 [[l + 1.018 (Z/A)2/3 pl/3] ^ - ll K, (3) where Tp is the Fermi temperature and ^6 is the mass density in units of 106gcm"3. For the ionic system , we consider in this section the case that it is in the liquid state. The criterion for that is a subject of the recent debate (see the review paper of H.E. DeWitt in the present Proceedings); but we follow in this review the criterion given by Ogata & Ichimaru (1987)

where a = [3/ (47rnj)] ' is the ion-sphere radius and T& is the temperature in units of 108K. The analytic fitting formulae presented in the paper of Itoh et al. (1983) are valid for 2 < T < F m , irrespective of the precise value ofF m . The low-temperature quantum corrections for the ions can be neglected unless the parameter

where kp is the Fermi wavenumber of the electrons and M is the mass of the ion, becomes appreciable compared with unity. The prescriptions to take into account the low-temperature quantum corrections for the ions have been given in detail by Mitake, Ichimaru, & Itoh (1984). However, these prescriptions are not complete, because they are powerless for the case y > 1. This situation is quite ironical, as the correct prescrptions for low-temperature quantum corrections are badly needed for y > 1. When we can calculate the low-temperature quantum corrections accurately, they are small anyway. Therefore, if one is not worried by the small corrections, it would be probably wise to disregard altogether the low-temperature quantum corrections and use simply the results of Itoh et al. (1983) for the whole range of 2 < T < Tm. Then one would not be too wrong in getting the right behavior of the conductivities. On the other hand, if one uses the results of Mitake, Ichimaru, & Itoh (1984) for the cases y > 1, one would be led to unphysical results, because the calculation of Mitake Ichimaru,


397

4h »——

^-r=i7i

i

i

2

3

i

. 4 log/ 7

i 3

5

6

(g-cm- )

Fig. 18.1 The density-temperature diagram for the XH matter.

& Itoh (1984) is based on the lowest-order expansions with respect to the small values of y. Thus the method completely breaks down for y > 1. The expressions for the electrical conductivity a and the thermal conductivity K in the liquid metal phase are a = 8.693 x 1 0 2 1 ^ A

[1 + 1.018 (Z/A)2/3 p26/3\

K = 2.363 x 10 17

< S >

1.018(Z/A)2/3

ergscm~ ~ 11s~ s~11K~ 1 , <S>

= I'd (±.\ (± Jo

(6)

\2kFj\2kF

1.018 (Z/A)2/3p26/3 1 + 1.018 (Z/A)2/3pl/3

(7)

S(k/2kF)

398


logP

6

,

8

-3v

3 (gem ) 4

Fig. 18.2 The density-temperature diagram for the He matter.

log) 0

8

-x (g-cm3)

Fig. 18.3 The density-temperature diagram for the 12C matter.

10

399


6

8 log/?

10 (gem )

12

3

Fig. 18.4 The density-temperature diagram for the 56Fe matter.

S(k/2kF) 2kFJ \2kFJ =<S-i

[(k/2kF)2e(k/2kFyQ)\

1.018 (Z/A)2/3pl/3 >— 1 + 1.018 (Z/A)2/3pl/3 *

>,

(8)

where hk is the momentum transferred from the ionic system to the electron, S(k/2kF) is the ionic liquid structure factor, and e(k/2kF, 0) is the static dielectric screening function due to the degenerate electrons. The first term in equation (8) corresponds to the ordinary Coulomb logarithmic term, and the second term is a relativistic correction term. In Itoh et al. (1983) explicit calculations have been carried out for Z = 1,2,6,8,10,12,14,16,20,26, and asymptotic formulae have been presented for Z > 27. Some of the results are shown in Figures 5-8. The difference between the results of Yakovlev and Urpin (1981) and those of Itoh et al. (1983) is the following: Itoh et al. (1983) carried out the integrations in equation (8) by using the IHNC structure factor of the classical one-component plasma calculated by Iyetomi & Ichimaru (1982) and Jancovici's (1962) relativistic dielectric function for the degenerate electrons, whereas Yakovlev & Urpin (1981) used an approximate form of the srtuc-

400


2

„ 3 4 log/0 (g-cnr3)

Fig. 18.5 Comparison of Yakovlev and Urpin's results (dashed curves) with those of Itoh et al. (solid curves) for the lYL matter. ture factor and neglected the screening due to the degenerate electrons by setting e(k/2kF,0) = 1. Yakovlev and Urpin's (1981) approximations are quite satisfactory at high densities as can be seen from Figures 5-8, but bring about significant deviations from the correct results at low densities where the screening due to the degenerate electrons is appreciable. Accurate analytical fitting formulae which summarize the numerical results have been presented by Itoh et al. (1983) 18.3 Electrical and Thermal Conductivities in the Crystalline Lattice Phase In this section we review the calculations of the electrical and thermal conductivities of dense matter in the crystalline latttice phase (F > r m = 180) following the papers of Itoh et al. (1984a) and Itoh,Hayashi,&Kohyama (1993). The relativistic extension of the expressions for the electrical and thermal conductivities due to degenerate electrons has been given by Flowers & Itoh (1976). The electrical conductivity a and thermal conductivity K are related to the effective electron collision frequencies ua and vK by

401


1.0 -

Fig. 18.6 Comparison of Yakovlev and Urpin's results (dashed curves) with those of Itoh et al. (solid curves) for the 4He matter.

a = - ^ - = 1.525 x 10 20 -/» 6 m*V(,

x 1 + 1.018 (f

A

*)'

101I88s. - 1 ergs cm x s lK

(9)

l

,

(10)

where ne is the number density of eletrons and m* is the relativistic effective mass of an electron at the Fermi surface. In this section we are interested in the scattering of electrons by phonons. The collision frequencies va and uK due to one-phonon processes can be calculated by the variational method (Flowers k Itoh 1976; Yakovlev & Urpin 1981; Raikh & Yakovlev 1982) as


402

4

6. log/ 0

, 8 % (gem3)

10

Fig. 18.7 Comparison of Yakovlev and Urpin's results (dashed curves) with those of Itoh et al. (solid curves) for the 12 C matter.

kBT = xZx + (1 - x)Z2 ,

(47)

< A>=xA1+(l-x)A2

(48)

.

As the unperturbed state we take a regular lattice whose lattice sites are

414


0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

LogT Fig. 18.18 Thermal conductivity of the "pure" 12 C matter as a function of the parameter T for the densities 10 4 gcm~ 3 ,10 6 gcm~ 3 ,10 8 gcm~ 3 ,10 10 gcm" 3 .

occupied by (< Z >,< A >) ions. Then we suppose that impurity ions are added at lattice sites to make up the compositionally disordered lattice. Thus at a site occupied by a (Zi,Ai) ion, the added impurity charge is Zx- < Z >= (1 - x)(Zi - Z2) = (1 - x)AZ

.

(49)

At a site occupied by a {Z2,A2) ion, the added impurity charge is

Z2-

-

-xAZ

(50)

Next we consider these charge impurities as static, and calculate their contributions to electrical and thermal resistivities. This is exactly the same as that has been done by Itoh et al. (1983) when they calculated the electrical and thermal conductivities of dense matter in the liquid metal phase. We consider the case that the atoms are completely pressure-ionized. We further restrict ourselves to the density-temperature region in which electrons are strongly degenerate. This condition is expressed as = 5.930 x 109

[[1


415

where TF is the electron Fermi temperature, X\ and X2 are the mass fractions (Xi + X2 = 1) of the ions (Z\,A\) and {Z2,A2), a n d ps is the mass density in units of 106gcm~3. For the ionic system we consider the case that it is in the crystalline lattice state. The latest criterion corresponding to this condition for the case of the one-component system is given by (Ogata & Ichimaru 1987)

where a = [3/(47rnj)]1/3 is the ion-sphere radius, and T% is the temperature in units of \Q%K. Extending the work of Itoh et al. (1983) to the case of charged impurities, and assuming that the correlations between the two kinds of impurity charges are the same, we obtain the expressions for the electrical conductivity a and the thermal conductivity « limited by impurity scattering: a = 8.693 x 102Ve (%• + %

K = 2.363 X 1017T8P6 ( £ + %

\M

_ \$ \( JJoJo 1.018 1.018

k

\( ( k V )\2kF$\2kFJJ

Ai

S(k/2kF)

416

Itoh: Transport processes in dense stellar plasmas U

\ 5

[(k/2kFye(k/2kF,0)}2

2kF)) \2kF) 1.018

(z

Z V 3 p 2 /3 2

6

1 + 1.018 (fj-Xx + %X2) p6 where S(k/2kF) is the structure factor for the impurity charges, and c(k/2kF, 0) is the static dielectric function due to degenerate electrons, kF being the electron Fermi wavenumber. The results (53) and (54) nicely reproduce the experimentally verified rules. The property that the resistivity is proportional to x(l — x) is Nordheim's rule; the property that the resistivity is proportional to (AZ) 2 is Linde's rule (Ziman 1960). In order to calculate the structure factor S(k) for the impurity charges, we employ a simple model. We adopt the ion-sphere model, and assume the following radial distribution function for the impurity charges :

where a is the ion-sphere radius a = {3/[47r(n1 + n 2 )]} 1 / 3 .

(57)

Then the structure factor S(k) is obtained as

S(k) =l + (ni+n2)J

d3r[g(r) - 1] exp(-tk • r)

3

= 1-

(sinfca — kacoska)

.

(58)

{kay

We notice that ^(Jfca) 2 ,asfc-»0 .

(59)

Since the screening effect due to electrons is negligible compared with the effect of the impurity charge correlation expressed by the structure factor (58), we set e(k/2kF,0) = 1 in equation (55). We further replace the structure factor (58) by a simplified form


417

which satisfies the condition (59). Insertion of this into equation (55) gives

\

\f

,

(61)

The electron Fermi wavenumber kp is written as kF = 2.613 X 1010 (^rXx + ^-X2) ) \Ai A2 The ion-sphere radius a is written as

pl/3cm-1

a = 0.7346xl0- lo f4 1 + 4 i> )" 1/ %6 1/3 cm \Ai

.

.

(63)

(64)

A2J

Thus we have

f) l / 3 (^ f)- I/3 .

(^

(65)

It is straightforward to generalize the results in the above to higher component systems. Let the mass fraction of the (£j,Aj) component be Xi(52iXi = 1). The result is a = 8.693 X 1021ps \Y

-r1}

^s"1 '

(66)

Y, ^f )

K = 2.363 x IO17TSP6 \ , ^ f ) 1

*+1.018

(%)2/%1 is given by equations (55),(61),(62) with kFa=

/

y.\ 1 / 3 /

1.920 ( j > * )

x\~1/3

(Ef:)

•

(71)

18.5 Concluding remarks We have reviewed the recent developments in the field of the transport properties of dense matter. The modern calculations of the transport properties of dense matter make full use of the recent developments in statistical physics, plasma physics, liquid state physics, and solid state physics. Interparticle correlations play essential roles in the quantitative evaluation of the transport properties of dense matter. Now the time is ripe for the close examinations of our understanding of the physics of dense matter by using the observational data of the asteroseismology of white dwarfs.

18.6 Appendix In this Appendix we directly insert equation (58) into eqation (55) and carry out the integration. The result is

(72)

The differences between the < S > values calculated from eqations (61) and (62) and those calculated from equations (72) and (73) are typically 10 % - 20 % (the latter is generally larger). The structure factor (58) based on the ion-sphere model is an approximation of the real ionic correction, and its oscillatory behavior does not have a rigorous meaning. Thus, in this paper, we will adopt the smooth structure factor (60) which leads to a simpler expression for < 5_i >. In any case, we should bear in mind that the differences in the < S > values caused by the use of the different structure factors indicate the accuracy of the present calculation.


419

18.7 References Bradley, P.A., k Winget D.E. 1991, ApJS, 75, 463 Bradley, P.A., Winget, D.E., k Wood, M.A. 1992, ApJ, 391, L33 Coldwell-Horsfall, R.A., k Maradudin, A.A. 1960, J. Math. Phys., 1, 395 Flowers, E., k Itoh, N. 1976, ApJ, 206, 218 Flowers, E., k Itoh, N. 1981, ApJ, 250, 750 Itoh, N., Hayashi, H., k Kohyama, Y. 1993, ApJ, in press Itoh, N., k Kohyama, Y. 1993, ApJ, 404, 268 Itoh, N., Kohyama, Y., Matsumoto, N., k Seki, M. 1984a, ApJ, 285, 758; erratum 404, 418 Itoh, N., Kohyama, Y., k Takeuchi, H. 1987, ApJ, 317, 733 Itoh, N., Matsumoto, N., Seki, M., k Kohyama, Y. 1984b, ApJ, 279 , 413 Itoh, N., Mitake, S., Iyetomi, H., k Ichimaru, S. 1983, ApJ, 273, 774 Iyetomi, H., k Ichimaru, S. 1982, Phys. Rev. A, 25. 2434 Jancovici, B. 1962, Nuovo Cimento, 25, 428 Ogata, S., k Ichimaru, S. 1987, Phys. Rev. A, 36, 5451 Pollock, E.L., k Hansen, J.P. 1973, Phys. Rev. A, 8, 3110 Raikh, M.E., k Yakovlev, D.G. 1982, Ap. Space Sci., 87, 193 Yakovlev, D.G., k Urpin, V.A. 1981, Soviet Astr., 24, 303 Ziman, J. 1960, Electrons and Phonons (Oxford Univ. Press)

19 Cataclysmic variables: structure and evolution J.-M. HAMEURY Observatoire it Strasbourg, 11 rut it I'Universite, 67000 Strasbourg. France (present address) DAEC, Observatoire it Paris, F-92195 Mtudon ceiez, France

Abstract I discuss the structure and evolution of cataclysmic variables, with a particular emphasis on the influence of the physics used in calculating the internal structure of the secondary. The available observational data is very rich, and can, in principle, be used to constrain the stellar physics. It is found that, in order to explain the lack of systems with periods in the range 2 3 hr, it is required that main sequence star become convective for masses below 0.3 M0. This has little consequences on the equation of state, but constrains the opacities and the treatment of subphotospheric layers. On discute la structure et revolution des variables cataclysmiques, en s'attachant plus particulierement a l'infiuence de la physique de l'etoile secondaire. Les donnees observationelles, tres abondantes, peuvent en principe etre utilisees pour contraindre la physique stellaire. On trouve que, pour expliquer l'absence de systemes entre 2 et 3 heures, il faut que les etoiles de la sequence principale deviennent convectives lorsque leur masse atteint 0.3 M©. Ceci a peu de consequences sur 1'equation d'etat, mais contraint les opacites et le traitement des couches sub-photospheriques.

19.1 Introduction Cataclysmic variables (CV's) are binary systems containing a white dwarf and a normal star, which fills its Roche and transfers mass onto the compact 420

421

Hameury: Cataclysmic variables

Accretion Disk

| Low Mass jSecondary

Hot Spot

White Dwarf Primary

Fig. 19.1 Schematic view of a cataclysmic variable (from Ritter, 1985)

object. The orbital period is in the range 1 to 10 hr, although we know a few cases in which it is longer than one day or shorter than 1 hr. These systems are quite similar to low-mass X-ray binaries, with the difference that the primary star is a neutron star or a black hole. Figure 1 shows a schematic view of a cataclysmic variable. Because the orbital angular momentum of the accreting matter is high, matter cannot flow directly onto the white dwarf, and an accretion disc forms, except in the particular case of strongly magnetic white dwarfs. In the disc, mass flows inwards, whereas angular momentum is transported outwards to the outer edge of the disc, where it is transferred to the orbit via tidal effects. The disc therefore extends up to a significant fraction of the Roche radius of the primary (typically 0.8 or 0.9 times the Roche radius). The mass transfer rate in these systems is of the order of 10~10 - 10~9 M© yr" 1 , and is strongly variable on time scales ranging from days to years or more, giving rise to bolometric luminosities in the range 1032 - 1034 erg s" 1 . Most of the optical light is emitted by the accretion disc itself, while infrared is emitted by the secondary (usually a K or M dwarf); the white dwarf and the boundary layer between the disc and the surface of the white dwarf are responsible for XUV emission.

422


We know several hundreds of these systems; their typical distance is of the order of 100 pc to a few hundred pc, so that the total number of cataclysmic variables in the galaxy is estimated to be about 200,000. We therefore have a wealth of available observational data that could be used to constraint models of low mass stars, which are normally quite difficult to observe. An interesting characteristic of these system is that, because the secondary fills its Roche lobe, there is a relation between the secondary radius and its mass, as we shall see later. This is particularly valuable for low mass stars, since the mass and radius determination is quite a difficult task in isolated low mass stars, and is model dependant. Another advantage of cataclysmic variables as compared to low mass X-ray binaries is, apart from the fact that they are much more numerous and therefore closer, that the effect of illumination of the secondary by radiation emitted by the primary is almost negligible, although some effect may still be present (Sarna, 1990). This is not the case for LMXB's, which may have an evolution rather different from that of CV's, due precisely to the illumination effect (see e.g. Ruderman et al., 1989; Podsiadlowski, 1991; Hameury et al., 1993) 19.2 Structure of cataclysmic variables 19.2.1 Orbital period distribution Figure 2 shows the orbital period distribution of cataclysmic variables. It is seen that most systems have orbital periods below a few hours, with a peak at 1.5-4 hr. There are only 4 systems having a period shorter than 80 min, which is the so-called "minimum period", whereas there is a lack of systems between 2 and 3 hr; this interval is called the "period gap", although there are indeed a few systems in it. As we shall see later, the orbital period distribution is an essential tool for the understanding the evolution of cataclysmic variables. 19.2.2 Primary and secondary masses The masses of both stars are difficult to determine, since, in order to get a reliable information on the orbital velocity of both components, one requires double spectroscopic systems. In order to determine the masses, one moreover has to know the inclination i of the orbital plane of the system. This is for example the case in eclipsing systems, for which i has to be larger than 70 - 80°. Finally, one must also know where the observed lines are emitted from: this may be either from the whole surface of the secondary (this is usually the case of absorption lines as in normal stars), but lines

423


20

6

15

_

5

_

J

o 6

0

2

4 6 Orbital Period (hr)

8

10

Fig. 19.2 Orbital period distribution of cataclysmic variables. Data from Ritter (1985) may also originate from the vicinity of the L\ region (in the case of emission lines due to heating of the secondary by hard radiation from the white dwarf). There are also emission lines from the accretion disc; in order to obtain the velocity of the white dwarf, one must assume circular symmetry for the accretion disc, which is certainly not true for its outer part, because of distortion due to tidal forces. With all these caveats, it is found that the average mass of the white dwarf in CVs is about 1 M©, significantly larger than that of isolated white dwarfs, which is about 0.6 M©. This difference, as has been shown by Ritter and Burkert (1986) is not due to differences in the formation mechanism of isolated and non-isolated white dwarfs, nor to the steady mass increase of the primary as a result of accretion (in fact as much mass is removed during novae explosions as is accreted in between), nor to systematic errors. This difference can be simply explained by selection effects, that tend to favour the observation of systems with a massive white dwarf, that has smaller radius, so that the luminosity for a given mass transfer rate is higher. After correction of the selection effects, the intrinsic average white dwarf mass turns out to be of the order of about 0.6 M©, identical to that of isolated stars. The secondary mass is always found to be in the range 0.1 - 1.0 M©

424


19.2.3 Mass-radius relation Kepler's third law relates the orbital separation a and period i^,r measured in hours via:

-f- = 0.50(Afi + M 2 ) 1 / 3 P h 2 / 3

(1)

ft©

where Mi and M2 are respectively the primary and the secondary masses measured in solar masses. Because the secondary fills its Roche lobe, the secondary radius J22 is determined by the Roche geometry, so that:

(Paczynski, 1971) for M2 < Mi (see also Eggleton, 1983 for a more accurate and more general fit of R^. Combining Eqs. (1) and (2) gives the massradius relation:

= 0.23J defined as M2/(4/37riZf) is 110 P ^ 2 g cm" 3 , and depends only on the orbital period of the system. For orbital periods of a few hours, < p > is of the order of a few g cm" 3 , typical of main sequence stars. If one makes the further assumption that the secondary is indeed on the main sequence, then R2 = M2R0 and the secondary mass is uniquely determined by the orbital period of the system via: M2 = O.llPhr

(4)

As we will see later, this assumption is not quite justified, but, at least for orbital periods larger than 3 hr, is not too extreme. A first important consequence can be deduced from this approximate relation: because mass is lost from the secondary, the orbital period of the system must decrease (again only for systems which have main sequence secondaries). Ritter (1985) has shown that the secondary in many CVs is not too far from the main sequence. Fig. 2 shows the mass and radius of several cataclysmic variables, as compared with theoretical predictions. One can see first that, because of the mass-radius relation resulting from the geometry of the system, the error box in the plane M 2 , R2 is reduced to a line segment, and, more important, that the secondary in those systems is quite close to the main sequence, the 10 - 20 % difference in radii can be either intrinsic, or due to a systematic error in theoretical models.

425

Hameury: Cataclysmic variables 1

-1.0

-OS

,

,

0.0

tog(M/Mo) Fig. 19.3 Comparison of determined masses and radii of the secondary in 13 systems with the prediction of theoretical models of main sequence stars.

19.3 Evolution of cataclysmic variables 19.3.1 A preliminary

approach

As we have seen, the typical mass transfer rate in cataclysmic variables is about 10~ 9 M© y r ~ l , so that for a 1 M© secondary, the mass transfer time scale tjfr = M2/M is about 109 yr, much shorter than the Hubble time. On the other hand, the nuclear time W is longer or much longer than the Hubble time for stars less massive than 1 M©. This means that nuclear evolution is negligible, and that systems evolve under the effect of mass transfer. The Kelvin-Helmholtz time fo-H ~ 107M2~3 becomes larger than tjfr when the mass of the secondary reaches 0.3 - 0.4 M©. This means that for larger masses, corresponding to periods larger than 3 - 4 hr, the secondary has time to adjust thermally to its new structure, modified by mass transfer, and remains on the main sequence. At shorter periods, it must deviate from main sequence, and it turns out that the secondary becomes degenerate. In these stars, the mass-radius relation is quite different from that of main sequence stars, and is R% oc M% , so that applying the same

426


argument as in section 1.2.3, one finds that M2 oc P ^ 1 , so that the orbital period of short period systems must increase. The transition main-sequence - degenerate therefore corresponds to a minimum in the orbital period, as has been shown by Paczyriski and Sienkiewicz (1981) and Rappaport et al. (1982). 19.3.2 Mass transfer; stability The evolution of a binary system is governed by the variation of three important quantities: the total mass M\ + M2, the secondary radius R2 equal to the Roche radius RL, and the total angular momentum of the system J given by: Ga

J = MXM2 '

Since the secondary must fill its Roche lobe at any time, R2 = RL, where the dot denotes time derivative. If one furthermore assumes for simplicity that the total mass of the system remains constant, then differentiation of Eq. (5) yields: a

R

~J

L

nM2

/

M2

J M2/5 2 Z

U

In order to get the mass transfer rate, one needs to know the relation between M2 and R2; for a main sequence star, M2/M2 = R2/R2- Departure from thermal equilibrium must however be taken into account, which requires a mode detailed description analysis of the secondary. The surface of a star is however not a well defined concept; matter extends above the photosphere, defined as the point where the optical depth is 2/3. Thus the prescription R2 = R\, can only be a first order approximation, that is accurate to a few photospheric pressure scale height Hp. It turns out that in the lower main sequence, Hp is a very small fraction (typically 10~4) of the radius, so that the prescription R2 = RL is a very good approximation. This is not the case for systems in which the secondary is an evolved star; one would then require a more refined description of mass transfer. This is also the case if one is interested in short episodes of the secular evolution, during which the secondary radius does not vary by much more than a few scaleheight, as for example the turn on of mass transfer. The hydrodynamics


427

of mass transfer have been studied in quite some details by Lubow and Shu (1975) who showed that the flow is approximately isothermal and reaches the sound velocity at the lagrangian point L\. This enabled Ritter (1988) to determine the mass transfer rate as: -M2 = Moe-(R*-R^/H*>

(8)

where Mo is a quantity that depends of the photospheric temperature and density, as well as on the binary parameters. 19.3.2.1 Dynamical stability A star subject to mass loss reacts on a short time scale according to:

where £ 10 hr) systems: evolved secondaries

There are three systems with an orbital period longer than 15 hr. GK Per (48 hr), U Sco (30 hr), and V394 Cra (18.2 hr). These cannot obviously contain main sequence stars, since they would have a mass by far exceeding that of the white dwarf, and mass transfer would be unstable. The secondaries in long period systems are subgiant whose nuclear evolution drive the evolution of the system. Their evolution has been discussed by Webbink et al. (1983). The mass-period relation for these systems is given by (King,


437

1988):

(

7.65 ^

where Mc is the mass of the helium core. It is seen that the period depends very sensitively upon Mc, and that the orbital period of these systems increase with time, as the helium core grows; long periods (up to a few days) can be obtained. This is confirmed by numerical models of the evolution of binaries containing an evolved secondary (Pylyser and Savonije, 1988a,b). 19.4-4-2 Ultrashort systems (Porb < 1 hr) systems: He secondaries Two systems have confirmed periods below 80 min: GP Com (46 min) and AM Cvn (18 min). These systems cannot have evolved through the sequence described above; they are too compact to contain H-rich main sequence or degenerate secondaries, but instead contain He stars that may or may not be degenerate. These systems have been studied in quite some details (see e.g. Rappaport and Joss, 1984; Tutukov et al., 1985; Nelson et al., 1986; Fedorova and Ergma, 1989). He rich secondaries have a much smaller radius than H-rich secondaries, and hence have shorter orbital periods. The formation of such systems is not well understood; they are clearly very rare. 19.4-4-3 Systems in the period gap ^From the period histogram shown in Fig. 1.3, it can be seen that there are some systems inside the period gap. This is not really unexpected, since depending on the initial secondary mass, systems may form at periods between 2 and 3 hr. Except in very special cases, they are fully convective, and evolve down to the minimum period. It seems however that the period gap il less pronounced for a subclass of cataclysmic variables, the AM Her systems, that contain a strongly magnetic white dwarf rotating synchronously with the orbit. This will be further discussed in the next section. 19.4-4-4 The case of magnetic systems Cataclysmic variables containing a magnetized white dwarf are divided into two subclasses: systems in which the interaction between the dipoles of the primary and secondary is strong enough to force co-rotation of the white dwarf with the orbit, the so-called AM Her systems or polars, and systems in which the white dwarf spin period is shorter than the orbital period (the DQ Her systems, or intermediate polars). Magnetic systems have received much attention, still many questions are left unanswered (see King, 1993 for a recent review),

438


The fact that the average orbital period of AM Her systems is shorter than that of DQ Hers lead Chanmugam and Ray (1984) and King et al. (1985) to propose that intermediate polars must synchronize when the orbital separation has become small enough and evolve into polars. The issue is still controversial (see e.g. Lamb and Melia, 1987; Hameury et al., 1987), but it now appears that, although most AM Her systems must have been born as intermediate polars, most observed intermediate polars will never become synchronous. The period distribution of AM Her systems have two interesting features: (1) there is an accumulation of systems at a period of 114 min, and (2) the period gap is much less pronounced for these systems. An explanation of (1) has been suggested by Hameury et al. (1988): as systems emerge from the gap, their mass transfer is high and they spend a long time at about that period (see Fig. 1.4). This imposes tight constraints on the white dwarf mass distribution in these systems, as well as on angular momentum losses (Hameury et al., 1988; Ritter and Kolb, 1992). It is however not clear whether the spike is real, or will disappear as the number of detected systems increases. Point (2) may indicate that there is mass transfer while AM Her systems evolve through the gap, and that magnetic braking is either not interrupted or never effective. Wu and Wickramasinghe (1993) and Wickramasinghe (1993) suggested that the strong white dwarf field would prevent open field lines from the secondary, and consequently magnetic braking would be impossible. On the other hand, Schmidt et al. (1986) and Frank et al. (1993) argued that the interaction of the secondary wind with the white dwarf magnetic field would lead to enhanced magnetic braking that would persist even after the secondary has become fully convective. 19.4'5 Predicted orbital distribution As mentioned earlier, the evolutionary tracks obtained for different binary parameters must be convolved by the distribution of initial masses and periods to obtain the intrinsic orbital period distribution of cataclysmic variables. This has been done by Hameury et al. (1990) for AM Her systems only, and more recently by Kolb (1993) and Shafter (1992) for all systems. Both Kolb (1993) and Shafter (1992) used Politano's (1988,1990) and/or de Kool (1992) differential formation rates of cataclysmic variables, but Shafter assumption that the secondary lies on the main sequence is extremely crude, whereas Kolb used a bi-polytrope code. Kolb (1993) main results are: (1) only 1% of CVs are located below the period gap; (2) the intrinsic period distribution depends weakly on the details of magnetic braking; realistic


439

10

CU

o oo

Fig. 19.5 Predicted intrinsic (dotted line) and observed (solid line) period distribution of cataclysmic variables (from Kolb, 1993).

braking laws from Mestel and Spruit (1987) or Verbunt and Zwaan (1981) lead to essentially the same results and (3) there is a much more significant dependence on the details of the stellar structure. Once the intrinsic population of CVs is determined, it must be corrected for selection effects. These are quite difficult to estimate (see e.g. a discussion by Ritter and Burkert, 1986); in particular, they are different for each sub-class of cataclysmic variables, depending on how they are detected. Figure 1.5 shows one of Kolb's (1993) result in the case of a bolometric luminosity limited sample (i.e. the detection probability is proportional to the total luminosity to the power 3/2), which is not a very good approximation (most of the luminosity emitted by a CV is in the UV range, and hence undetectable). It is seen that this is in disagreement with observations, as far too many systems are found above the period gap, and that the predicted excess at the minimum period does not show up in the observed distribution. Assuming a visual magnitude limited sample would solve the first problem, whereas it is quite difficult to explain why one does not observe an accumulation of systems at the minimum period. It is possible that this is due to some unforeseen selection effect; it is nevertheless worth noting that the secondary mass is of the order of 0.05 M© at the minimum period, and that the stellar structure for those low masses is not well understood.

440


19.5 Conclusion Models for the evolution of cataclysmic variables are relatively sensitive to the assumed stellar physics, the strongest dependance being that due to the opacities and the treatment of the photospheric layers. If one were to know with reasonable accuracy the angular momentum losses from these systems, one would be able to test stellar models for masses in the range 0.05 0.5 M©. The presence of a more massive compact companion certainly helps in determining the characteristics of the secondary. In order to test the stellar models, one would also need to know the white dwarf mass in a larger number of systems; observational selection effects should also be better understood. The subclass of magnetic systems appear very promising from that point of view, even though the presence of strong magnetic field from the white dwarf introduces a further complication. The disagreement of the observed and predicted orbital period distribution for systems close to the minimum period is also quite interesting in that it is unlikely to be solely due to selection effects, and might therefore tell us something on the structure of very low mass stars.

References Alexander D.R., Astrophys. J. Suppl. Ser. 29, 363 (1975) Chanmugam G., Ray A., Astrophys. J. 285, 252 (1984) Cox A.N., Stewart J.N., Nauchn. Informatsii, 15, 1 (1969) Cox A.N., Tabor J.E., Astrophys. J. Suppl. Ser. 31, 271 (1976) D'Antona F., Mazzitelli I., Astrophys. J. 260, 722 (1982) D'Antona F., Mazzitelli I., Ritter H., Astron. Astrophys. 225, 391 (1989) de Kool M., Astron. Astrophys. 261, 188 (1992) Dorman B., Nelson L.A., Chau W.Y., Astrophys. J. 342, 1003 (1989) Eggleton P.P., Astrophys. J. 268, 368 (1983) Fedorova A.V., Ergma E.V., Astrophys. Sp. Sci., 151, 125 (1989) Fontaine G., Graboske H.C.Jr., Van Horn H.M., Astrophys. J. Suppl. Ser. 35, 293 (1977) Fowler W.A., Caughlan G.R., Zimmerman B.A., Ann. Rev. Astron. Astrophys. 13, 69 (1975) Frank J., et al., in preparation (1993) Graboske H.C.Jr, De Witt H.E., Grossman A.S., Cooper M.S., Asirophys. J. 181, 457 (1973) Hameury J.M., Astron. Astrophys. 243, 419 (1991) Hameury J.M., King A.R., Lasota J.P., Ritter H., Astrophys. Sp. Sci. 131, 583 (1987) Hameury J.M., King A.R., Lasota J.P., Livio M., Mon. Not. R. astr. Soc. 237, 835 (1989) Hameury J.M., King A.R., Lasota J.P., Mon. Not. R. astr. Soc. 242, 141 (1990) Hameury J.M., King A.R., Lasota J.P., Raison F., Astron. Astrophys. 227, 81 (1993)


441

Hameury J.M., King A.R., Lasota J.P., Ritter, H., Astrophys. J. 316, 275 (1987) Hameury J.M., King A.R., Lasota J.P., Ritter, H., Mon. Not. R. astr. Soc. 231, 535 (1988) Harris M.J., Fowler W.A., Caughlan G.R., Zimmerman B.A., Ann. Rev. Astron. Astrophys. 21, 185 (1983) Joss P.C., Rappaport S., Astrophys. J. 270, L73 (1983) King A.R., Q. Jl R. astr. Soc, 29, 1 (1988) King A.R. in the proceedings of the Monte-Porzio conference Evolutionary links in the zoo of interacting binaries, in press (1993) King A.R., Frank J., Ritter H., Mon. Not. R. astr. Soc. 213, 181 (1985) Kolb U., Astron. Astrophys. 271, 149 (1993) Kolb U., Ritter H., Astron. Astrophys. 254, 213 (1992) Kovetz A., Prialnik D., Astrophys. J. 291, 812 (19*) Kraft R.P., Matthews J., Greenstein J.L., Astrophys. J. 136, 312 (1962) Lamb D.Q., Melia F., Astrophys. Sp. Sci. 131, 511 (1987) Livio M., Govarie A., Ritter H., Astron. Astrophys. 246, 84 (1991) Livio M., Shankar A., Burkert A., Truran J.W., Astrophys. J. 356, 250 (1990) Lubow S.H., Shu F.H., Astrophys. J. 198, 383 (1975) McDermott P.N. Taam R.E., Astrophys. J. 342, 1019 (1989) McDonald J., Astrophys. J. 305, 251 (1986) Mestel L. Spruit H.C., Mon. Not. R. astr. Soc. 226, 57 (1987) Nelson L.A., Chau W.Y., Rosenblum A., Astrophys. J. 299, 658 (1985) Nelson L.A., Rappaport S.A., Joss P.C., Astrophys. J. 304, 231 (1986) Paczyriski B., Ada Astron., 19, 1 (1969) Paczyriski B., Ann. Rev. Astron. Astrophys. 9, 183 (1971) Paczynski B., Sienkiewicz R., Astrophys. J. 248, L27 (1981) Paczynski B., Sienkiewicz R., Astrophys. J. 268, 825 (1983) Podsiadlowski P., Nature 350, 136 (1991) Politano M., PhD thesis, University of Illinois, Urbana-Champaign (1988) Politano M., in Accretion-powered compact binaries, ed. Mauche C.W., Cambridge University Press, p. 421 (1990) Pylyser E.H.P., Savonije G.J., Astron. Astrophys. 191, 57 (1988a) Pylyser E.H.P., Savonije G.J., Astron. Astrophys. 208, 52 (1988b) Rappaport S., Joss P.C., Astrophys. J. 283, 232 (1984) Rappaport S., Joss P.C., Webbink R.F., Astrophys. J. 254, 616 (1982) Rappaport S., Verbunt F., Joss P.C., Astrophys. J. 275, 713 (1983) Ritter H. 1985, in High energy astrophysics and cosmology, eds. Yang J. and Zhu, C , Gordon and Breach Science Publ. Inc., New York, p. 207 (1985) Ritter H., Astron. Astrophys. 202, 93 (1988) Ritter H., Astron. Astrophys. Suppl. Ser. 85, 1179 (1990) Ritter H., Burkert, Astron. Astrophys. 158, 161 (1986) Ritter H., Kolb, U., Astron. Astrophys. 259, 159 (1992) Ruderman M., Shaham J., Tavani M., Eichler D., Astrophys. J. 342, 292 (1989) Sarna M., Astron. Astrophys. 239, 163 (1990) Saumon D., Chabrier, G. Phys. Rev. A, 44, 5122 (1991) Schatzman E., Ann. Astr., 25, 18 (1962) Schatzman E., in IAU Symp. No. 22, ed. Lust R., Reidel, Dordrecht, p. 153 (1965) Schmidt G.D., Stockman H.S., Grandi S.A., Astrophys. J. 300, 804 (1986) Shafter A.W., Astrophys. J. 394, 268 (1992) Shara M.M., Livio M., Moffat A.F.J., Orio M., Astrophys. J. 311, 163 (1986)

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Skumanich A., Astrophys. J. 171, 565 (1972) Spruit H.C., Ritter H., Astron. Astrophys. 124, 267 (1983) Truran J.W., Livio, M., Astrophys. J. 308, 721 (1986) Tutukov A.V., Fedorova A.V., Ergma E.V., Yungelson L.R., Pis'ma Astr. Zh., 11, 123 (1985) VandenBerg D.A., Hartwick F.D.A., Dawson P., Alexander D.R., Astrophys. J. 266, 747 (1983) Vardya M.S., Astrophys. J. Suppl. Ser. 42, 281 (1960) Verbunt F., Mon. Not. R. astr. Soc. 209, 227 (1984) Verbunt F., Zwaan C , Astron. Astrophys. 100, L7 (1981) Webbink R.F., in White dwarfs and variable degenerate stars, IAU Colloq. 53, p. 426, eds. Van Horn H.M. and Weidemann V., University of Rochester (1979) Webbink R.F., Rappaport S., Savonije G.J., Astrophys. J. 270, 678 (1983) Wickramasinghe D.T., in Cataclysmic variables and related physics, Annals of the Israel Physical Society No. 10, p. 208, eds. Regev O. and Shaviv G. (1993) Wu K., Wickramasinghe D.T., in Cataclysmic variables and related physics, Annals of the Israel Physical Society No. 10, p. 336, eds. Regev O. and Shaviv G. (1993)

20 Giant planet, brown dwarf, and low-mass star interiors W.B. HUBBARD Department of Planetary Sciences, Lunar and Planetary Laboratory, University of Arizona, Tucson, AZ 85721, USA

Abstract Astrophysical objects of low mass, ranging from giant planets to extreme dwarf main-sequence stars, have a number of physical characteristics in common due to properties of their equations of state. Their luminosities are low (much less than the solar luminosity X©) and their evolutionary timescales are typically measured in Gyr. So far there are few observational examples of these objects, although they are undoubtedly numerous in the galaxy. The lower mass limit is set by the object's ability to retain hydrogen during accumulation (about the mass of Saturn), while the upper mass limit is set by the lifting of electron degeneracy by high internal temperature. Objects confined within this broad range, which extends up to about 0.1 M©, are governed by the thermodynamics of liquid metallic hydrogen. In this paper, we discuss the implications of this feature of their interior structure for their radii, interior temperatures, thermonuclear energy generation rates, and luminosities. We conclude with a brief assessment of the confrontation between observations and theory in galactic clusters and in the solar system. L'equation d'etat des corps celestes de faible masse, qui vont des planetes geantes aux etoiles naines qui sont a la limite de la sequence principale, est a l'origine d'un ensemble commun de proprietes physiques. Leur luminosite est de beaucoup inferieure a celle du Soleil et leur temps caracteristique devolution se mesure en milliards d'annees. A ce jour, nous ne connaissons que quelques exemples de ces objets malgre la conviction qu'ils sont nom443

444

Hubbard: Giant planet, brown dwarf, and low-mass star interiors

breux dans la Galaxie. La limite inferieure en masse (approximativement la masse de Saturne) est fixee par la capadte de retenir l'hydrogene au cours du processus d'accumulation de la matiere. La limite superieure est atteinte lorsque la temperature interne est suffisamment elevee pour que les electrons ne soient plus degeneres (environ 0.1 M©). Les proprietes des corps qui se retrouvent dans ce domaine etendu en masse sont principalement determinees par la thermodynamique de l'hydrogene metallique liquide. Cet article presente les effets de ce point commun de leur structure interne sur leurs rayons, leurs temperature internes, leurs taux de generation d'energie thermonucleaire et leurs luminosites. Nous concluons par une breve discussion de la confrontation entre les observations et la theorie dans le cas des amas galactiques ainsi que dans le systeme solaire.

20.1 Introduction In this chapter, we shall discuss the implications of the equation of state (mainly that of hydrogen) for giant planets, brown dwarfs, and very lowmass stars. Although the mass range covered by these seemingly disparate objects is moderately large (about two orders of magnitude), the physics of the equation of state is basically the same, and leads to certain common characteristics. By giant planet, we mean the four largest planets of the solar system, Jupiter, Saturn, Uranus, and Neptune, as well as their so far hypothetical counterparts in other solar systems. The equation of state of hydrogen is particularly relevant to the largest two giant planets, Jupiter and Saturn. Giant planets have masses M which lie in the range 5 X 10~5M© < M < 1 x 10~3M© (M© = mass of the sun). The term brown dwarf (BD) has become standard usage for designating a class of hydrogen-rich objects with the following characteristics: (a) composition similar to the sun, i.e. dominated by hydrogen; (b) masses about ten times larger than that of Jupiter; (c) masses smaller than the critical mass for sustained thermonuclear fusion of hydrogen. Masses of brown dwarfs lie roughly in the range 1 x 10~2Af© < M < 1 X 10~1M©. At present there is some uncertainty about the possible modes of origin of objects in the mass range 1 x 10~3M© < M < 1 X 10"~2M©; such objects could be considered either very large giant planets or extremely small BD's. According to Boss (1986), the minimum mass for direct formation of a BD from collapse of an interstellar cloud of H and He is ~ 0.02M©, and objects of lower mass form through a different sequence of events, which begins with coagulation of planetesimals from solid particles. As we shall discuss, these lower mass


445

objects lie below the critical mass for fusion of deuterium and it is therefore convenient to classify them as giant planets (GP's) rather than BD's. Objects which are more massive than ~ 10~1M© but still substantially less massive than the sun are termed very low mass stars (VLM), or extreme M dwarfs. In contrast to the situation for GP's and BD's, many observational examples of VLM's exist. All of the objects under discussion here, GP's, BD's, and VLM's, have intrinsic luminosities L which are small compared with the luminosity of the sun £©. Their luminosity and associated interior thermal state change very slowly with time, typically over time scales measured in Gyr. At the same time, depending on the relative efficiency with which these objects are formed, they may comprise an appreciable fraction of the mass of the Galaxy. This mass could thus be largely hidden in objects which are difficult to detect. In modeling objects across the indicated mass range, we assume that the composition is similar to that of the sun, i.e., predominantly hydrogen. Although there is some uncertainty about the precise composition and indeed there may be some variation with mass and age, for our purposes it is sufficient to take a uniform composition with a helium mass fraction Y = 0.25, and a hydrogen mass fraction X = 1 - Y — Z, where Z is the mass fraction of all elements heavier than helium (the so-called metals). The value of Z plays little or no role in the equation of state as it does not exceed 0.02 for solar composition. However, the value of Z affects photon opacities in the outermost layers of these hydrogen-rich objects, and hence has a large impact on their interior thermal state. In the lowest mass range, M ~ < 3 X 10~4M©, significant amounts of hydrogen are lost during accumulation of the object, and Z becomes large enough to play a significant role in the equation of state. The similarity of the physics of the equation of state in all of these objects ultimately arises from the fact that their interiors lie for the most part within the following limits: 6 = eF/kT > 1

(1)

T = e2/akT > 1

(2)

and

Here 8 is the electron degeneracy parameter, the ratio of the electron fermi energy ep to a typical thermal excitation energy kT. Similarly, the ion

446


coupling parameter (which for hydrogen is the same as the electron coupling parameter) T measures the ratio of a typical ion coulomb energy e2/a to kT. Here e is the ion charge and a is the average distance between ions. Because both of these parameters are large, the object's equation of state is mainly governed by the physics of metallic hydrogen. And, because F typically lies in the range 1 < T < 100, the metallic hydrogen is in a strongly coupled liquid phase. A third dimensionless parameter of relevance to the equation of state is the density parameter rs = a/ao, where ao is the Bohr radius. This parameter ranges from r , « l for a giant planet such as Jupiter to ra « 0.1 for the most massive BD's. Under these circumstances, the relation between pressure P and mass density p is largely independent of temperature T, and can be expressed in the form P ~ P°

(3)

with 1.6 < a < 2. This result is universally true for hydrogen-rich objects in the relevant mass range, and leads to the remarkable result that such objects have very similar radii R, regardless of their mass. However, this similarity does not extend to quantities related to the object's interior thermal state, such as its total intrinsic luminosity L, its effective temperature Te, and its central temperature T centra i, which in general depend sensitively on both the mass and age of the object. Some of these points are illustrated in Table 1.1, in which we compare Jupiter, a well-studied metallic-hydrogen object with an age of about 5 Gyr, with a hypothetical BD of the same age and gross chemical composition. Since Jupiter rotates rapidly and is therefore nonspherical, the radius R which is given is the equatorial radius at the 1-bar pressure level. For Jupiter, the quantity /9Centrai 1S actually the highest density of the metallic hydrogen zone in the planet's interior, and does not refer to the density of a Z-rich central core. 20.2 Radius vs. mass; relation to equation of state The overall behavior of the R(M) curve for giant planets, BD's, and VLM's is extremely diagnostic of the equation of state of metallic hydrogen-helium mixtures, and its general shape is shown in Figs. 1.1 and 1.2. Figure 1.1 shows a maximum radius for solar composition of 75740 km at M = 0.004M© (about four Jupiter masses). For pure hydrogen, the maximum in R(M) moves to 0.006MQ and 87200 km. The maximum radius exists as a direct consequence of a competition between electrostatic con-


447

Table 20.1. Comparison between Jupiter and a typical brown dwarf

M(0) /2(km) age (Gyr) L/LQ •^'nuclear/-'-'

T (10 bar) Pcentral -* central

16

(

I

I

14 -

1

I

I

Jupiter

brown dwarf

0.001 71492 ± 4 5 0.9 x lO" 9 0 124 K 337 K 4 g cm" 3 22600 K

0.070 55870 5 1.5 x 10" 5 0.28 1262 K 550 K 979 g cm" 3 1.683 x 106 K

1

1

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age = 5 Gyr

1

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M/M0 Fig. 20.1 Solid line (—): radius vs. mass for solar-composition objects at a cooling age of 5 Gyr. Objects to the left of the short break at 0.01M o are considered giant planets. BD's extend from the break to 0.08A/ o . •: GP's, shown in more detail in Fig. 1.2.

tributions to the equation of state which contribute a negative component to the pressure, and the electron fermi pressure, which contributes positively. For M < 0.004AfQ (r, ~ < 0.6) the electrostatic contributions to the pressure are sufficiently important (pressure increases sufficiently rapidly with increasing density) that there is a positive slope to the M(R) curve in this mass range. For masses greater than 0.004M©, the degenerate electron fermi pressure begins to dominate the equation of state, causing a decrease in R with increasing M. As Fig. 1.1 shows, there is a minimum in the R(M) curve at about 0.07M©. The minimum is not especially related to the equation of state, but is instead produced by the onset of thermonuclear energy

448

Hubbard: Giant planet, brown dwarf, and low-mass star interiors i i i I i i i I i i i I i i i I i i i I i i i I i i i I

O

o o o

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-u I I I .0002 .0004 .0006 .0008

.001

.0012 .0014

M/Mo Fig. 20.2 Solid line (—): radius vs. mass for spherically symmetric solarcomposition objects at a finite interior temperature. Dashed line ( ): same, but for pure H. •: giant planets Jupiter, Saturn, Uranus, Neptune.

generation in the BD's core, which tends to raise the interior temperature and reduces the electron degeneracy. According to Burrows et al. (1993), the minimum mass for sustained thermonuclear energy generation in BD's is 0.0767M©; objects which are more massive are not BD's but are instead VLM's, and settle eventually on the main sequence. The minimum in R(M) lies slightly below this minimum mass because Fig. 1.1 is computed for a finite age of 5 Gyr, and the most massive BD's are able to burn hydrogen for several Gyr before they fail to settle on the main sequence. Figure 1.2 shows an expanded view of the extreme left-hand corner of Fig. 1.1. The solid curve again shows R(M) for solar-composition material, calculated for temperatures along an adiabatic compression curve corresponding to the interiors of Jupiter and Saturn, while the dashed curve is the same, but for pure hydrogen. At masses slightly below the mass of Saturn (0.0003 MQ), the metallic-hydrogen core vanishes, and R(M) is determined by the equation of state of dense molecular hydrogen. At masses slightly below the mass of Uranus or Neptune (5 x 10~5M©), the molecular hydrogen adiabat falls entirely in the ideal-gas region, and as a result R(M) begins to increase with falling mass. However, the latter behavior cannot be realized in nature. As Fig. 1.2 makes clear, giant planets less massive than Saturn cannot capture significant amounts of hydrogen when they are formed, and their interior equations of state are dominated by heavier nuclei. More detailed analysis of the interior compositions of Jupiter and Saturn (Chabrier et al., 1992) shows that both are more enriched in the Z-component than


449

solar composition, with Saturn more enriched than Jupiter. In Fig. 1.2, Jupiter plots slightly above the solar-composition curve because correction for rotation of Jupiter has not been included in the solid curve; when this correction is included, Jupiter plots slightly below the theoretical curve for solar composition. 20.3 Variation of luminosity with mass The heat flow which corresponds to the observed luminosity L is presumed to be derived from two sources: (a) heat release from the object due to work done on the object's interior and due to changes in internal energy E; (b) heat release due to nuclear reactions in the object's interior. Both of the sources can be combined into the equation rM P An L = / —±dm-Jo

pL at

A rM rM / Edm+ / €N(p,T)dm at Jo

(3)

Jo

where dm is an element of mass, t is the time, and ejv is the rate of release of energy by nuclear reactions. For the giant planets, ejq is effectively zero because temperatures and densities are too low for fusion reactions to proceed, and the abundances of radioactive high-Z elements such as 40 K, 232 Th, and 238 U are too low for radioactive decay to be significant. In the giant planets, only the first two terms in Eq. 3 are important, although they must be carefully evaluated generalized in the case of Saturn to account for immiscibility of helium in liquid metallic hydrogen and consequent formation of a helium-enriched core (Stevenson, 1975). As masses increase into the BD range, one must consider contributions to CN from the following reactions (Burrows et al., 1993): p + p->d + e+ + ve p + d -* 3He + 7

(1.442 MeV) (5.494 MeV)

(4) (5)

In the BD mass range, these reactions stop at 3He. In addition, the following reaction does not contribute significantly to cN but does serve as a useful tracer of BD evolution: p + 7U^2a

(6)

Basically, the division between VLM's and BD's is defined by reaction (4). For masses above the critical mass of O.O767M0, central densities exceed about 103 g cm" 3 , and central temperatures exceed about 2x 10^ K. Under these circumstances, enough heat is liberated by reactions (4) and (5) to


450

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log t (Gyr) Fig. 20.3 Curves of luminosity vs. time for VLM's and BD's of various mass. The upper curve is for a VLM of O.2OOM0, while the lowest curve is for O.OIOMQ. A smaller mass interval has been used for objects near the critical mass for hydrogen burning. • shows the 0.070M© model presented in Table 1.1. balance the heat radiated by the VLM's atmosphere. For objects below the critical mass, reaction (5) can still proceed but reaction (4) does not. As a result, the BD has a relatively brief phase during which it burns primordial deuterium, but once the deuterium is gone, only heat release from the first two terms of eq. (3) plays a role in the BD's luminosity. At the lower end of the BD mass range, even deuterium does not burn. This second critical mass lies at about 0.015MQ and corresponds to central densities about ~ 20 g cm" 3 , and central temperatures ~ 0.5 X 106 K. Objects below this second critical mass can be considered GP's. Figure 1.3 illustrates the above-described effects, showing the evolution of luminosity with time on a double-logarithmic scale. The bifurcation of objects into VLM's and BD's at the right side of the figure is apparent. The model with the prolonged curve is just subcritical at 0.0765M©, and shows a gradual decline of luminosity with time over time periods longer than the age of the universe. Other features in Fig. 1.3 are worthy of note. A "ripple" in L(t) is apparent for —3 < log* < —1.5, with the onset of the "ripple" progressively later for smaller masses. This "ripple" is caused by thermonuclear fusion of


451

primordial deuterium via reaction (5), assuming that the primordial deuterium mass fraction is Yd = 2 x 10~5. The "ripple" vanishes for masses between O.OlOAf© (lowest curve in Fig. 1.3) and 0.020MQ (second lowest curve in Fig. 1.3). Thus the critical mass for deuterium burning lies at M « 0.015M©, and we conclude that GP's, such as Jupiter, will contain the primordial deuterium abundance, but BD's older than ~ 0.1 Gyr will be depleted in deuterium. Another, fainter, "ripple" in Fig. 1.3 can be discerned at L ~ 10~ 4 .£Q. This "ripple" is not related to the equation of state or to nuclear reaction rates, but is caused by temporarily increased atmospheric opacity at this luminosity level, as a consequence of the formation of dust grains in the BD's atmosphere at levels where most photons are just able to escape to space (Burrows et al., 1989). This increase in opacity causes the BD luminosity to temporarily drop and the cooling age of the BD to be correspondingly extended. As Fig. 1.3 makes clear, L is a function of both M and t for BD's and VLM's, and thus identification of an object as a possible BD requires knowledge of its age as well as its luminosity. But there is, theoretically, an independent test which can be used to determine whether an object lies within the BD mass range. Fragile nuclei such as deuterium and lithium are destroyed via reactions (5) and (6) respectively, at significantly lower temperatures than those required for reaction (4) to proceed. We have already discussed the destruction of an initial deuterium component in a BD. The corresponding destruction of initial lithium requires a mass equal or greater than about O.O65M0 (Magazzu, Martin, and Rebolo, 1993, D'Antona and Mazzitelli, 1993), and occurs at t ~ < 0.1 Gyr. Because the initial lithium abundance is very small, reaction (6) does not contribute appreciably to 6;v- An object older than 0.1 Gyr with detectable lithium in its atmosphere could be safely considered to be a BD. However, detection of such a lowabundance atom in such intrinsically faint objects presents a formidable observational challenge which has not yet been overcome. 20.4 Observational tests 20.4.1 Cluster luminosity functions One of the best methods available at present to investigate properties of putative BD's in the solar vicinity is to examine low-luminosity objects in nearby galactic clusters. Such clusters are relatively young and have a known t as established by the evolution of their more massive members. One may use the function L(M,t) as displayed in Fig. 1.3 to calculate the so-

Hubbard: Giant planet, brown dwarf, and low-mass star interiors 1 . , , , , , , . , , ,

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log L/L, Fig. 20.4 Luminosity functions for p Oph for various values of a, compared with data from Comeron et al. (1993).

called luminosity function N(L, t), where N is the number of objects within a given interval in logX at a given time t. To calculate N, one must also know the initial number of objects created within the cluster as function of M; this is given by the so-called initial mass function (IMF):

Z(M)dM = CM~adM

(7)

where £(M) is the number of initial objects created within M and M + dM. The exponential dependence of £ on M was discovered by Salpeter (1955), who also showed that a ss 2.35 for stars in the galaxy. It is at present uncertain whether a power law of the form of eq. (7) applies to BD's. Clearly, the proportion of the mass of the Galaxy comprised of BD's depends on the relevant value of a for the BD mass range. For VLM's (and possibly BD's) in the Hyades galactic cluster (t = 0.6 Gyr), Hubbard et al. (1990) found that the theory matched low-luminosity star counts best for a w 0. As is clear from Fig. 1.3, BD's are best detected in young clusters, when their luminosities are relatively elevated. Fig. 1.4 shows a comparison of star counts in the young galactic association p Oph (t = 0.003 Gyr; Comeron et al., 1993) with the theory of Burrows et al. (1993). In such a young cluster, deuterium burning is still important in the BD's. While the number of detected low-luminosity objects is still quite small, a = 1.14 (Comeron et al., 1993) is a reasonable mean value for the BD-VLM mass range. This result, when added to f(M) for objects more massive than VLM's, implies that at most 1/3 of the mass of the Galaxy is composed of BD's. Figure 1.4 shows a very indirect test of the equation of state of BD's and VLM's. The primary influence of the equation of state on this plot is via its effect on the radii of the metallic-hydrogen objects, and on their


453

thermal properties such as heat capacities and thermonuclear reaction rates. However, of equal importance are the atmospheric opacities which regulate the escape of interior heat, and the value of a.

20.4-2 Luminosity of giant planets With the exception of Uranus, the solar system's giant planets have measurable intrinsic luminosities. These were determined with considerable precision by experiments on the Voyager spacecraft after encounters with all four bodies during the previous decade (Pearl et al., 1991). Values of both the intrinsic luminosity L and the specific luminosity (L/M) are given in Table 1.2. The theory of giant-planet luminosity is developed by integrating eq. (3) over t (after setting c^ = 0), thus determining the time interval required for the planet's luminosity to decline to the present observed value. This can be compared with the known ages of these planets (t = 4.6 Gyr). In the case of Jupiter and Saturn, the luminosities are governed by the thermal properties of hydrogen, and the most recent calculation of L(t) for these objects (Saumon et al, 1992) has made use of the Plasma Phase Transition (PPT) theory of Saumon and Chabrier (1989, 1991, 1992). The calculation finds that interior adiabats in both Jupiter and Saturn cross the PPT during their evolution, at an interior point where P « 1 Mbar. The evolutionary age of the planet t£ is defined by the value of t for which the planet's luminosity drops to the value given in Table 1.2. For Jupiter, the theory gives *# = 5 Gyr, but for Saturn %E = 2.5 Gyr. Thus Jupiter's heat flow is in accordance with the latest equation of state of hydrogen and eq. (3), but Saturn's is not. The traditional explanation of the discrepancy for Saturn is that additional gravitational energy is liberated in Saturn's interior if helium becomes immiscible in hydrogen over part of the pressuretemperature range traversed in Saturn's interior (Stevenson and Salpeter, 1977). In this case helium droplets would form and sink to deeper layers in Saturn, liberating gravitational energy. Eq. (3) must be modified to take this mechanism into account (Hubbard and Stevenson, 1984). While the theory of helium-hydrogen phase separation in Saturn is as yet not fully quantitative, it does predict the substantial depletion of helium which is observed in Saturn's atmosphere (Conrath et al., 1984). In the case of Uranus and Neptune, the equation of state is dominated by heavier materials than hydrogen or helium, as we discuss below. The most relevant quantity for the evolution of a giant planet's luminosity is its interior specific heat at constant volume per unit mass, Cy. Neglecting eyy

454


Table 20.2. Measured luminosity and specific luminosity of GP's

Jupiter Saturn Uranus Neptune

L/LQ

L/M ( 1 0 - U W/kg)

0.9 x 10~ 9 0.2 x 10- 9 < 0.002 x 10" 9 0.01 x 10" 9

18 15

)

Fig. 20.10 Theoretical adiabats for mixtures of H and He in solar proportions, mixed with ice. Diamonds show shock compression data on ice.

10-*

10-'

p(g/cm3) Fig. 20.11 Interior models of Uranus and Neptune. interior models of Neptune calculated by Zharkov and Gudkova (1991); in the latter models, the hydrogen-helium envelope extends to about 0.2 Mbar.

20.5 Conclusion The properties of a broad range of astrophysical objects, ranging from VLM's (masses ~ 0.1M©) to giant planets (masses ~ O.OOIMQ) can be investigated within the framework of a general equation of state for a mixture of hydrogen and helium. For the more massive objects, confrontation between theory and data comes primarily from a comparison of the predicted spectral properties of photons emitted from BD atmospheres, as a function of their interior thermal properties and age, with the (sparse) observational data set.


461

For the giant planets within our solar system, it is possible to study not only the relation between intrinsic luminosities and interior thermal state, but also the relation between the gravitational potential coefficients Jit and the interior P(p) relation. A further constraint on the interior equation of state comes from the M(R) relation. A primary result of this study is that, commencing at masses comparable to that of Jupiter, giant planets form in a process which tends to lose some of the hydrogen-helium component as the planet accretes. As the object's mass decreases, ever more hydrogenhelium is lost, such that objects in the mass range of Uranus and Neptune contain only a small fraction (~ few % by mass) of hydrogen-helium. In the mass range of Uranus and Neptune, typical central pressures reach only a few Mbar. Likely interior material (ice) is susceptible to experimental determination of its equation of state in this pressure range. Thus, theory can be at least partially replaced with experiment when equations of state are tested for the lowest mass giant planets. This work was supported in part by NASA Grant NAGW-1555 and by NSF Grant INT-8907133.

References Boss A.P., in Astrophysics of Brown Dwarfs (M.C. Kafatos, R.S. Harrington, S.P. Maran, eds.), Cambridge Univ. Press, pp. 206-211, (1986) Burrows A., Hubbard W.B., Lunine J.I., Astrophys. J. 345, 939, (1989) Burrows A., Hubbard W.B., Saumon D., Lunine J.I., Astrophys. J. 406, 158, (1993) Chabrier G., Saumon D., Hubbard W.B., Lunine J.I., Astrophys. J. 391, 817, (1992) Comeron F., Rieke G.H., Burrows A., Rieke M.J., Astrophys. J. 416, 185, (1993) Conrath B.J., Gautier D., Hanel R.A., Hornstein J.S., Astrophys. J. 282, 807, (1984) D'Antona F., Mazzitelli I., Astr. J. Suppl., in press, (1993) Dyson F., Ann. Phys. 63, 1, (1971) Hubbard W.B., Burrows A., Lunine J.I., Astrophys. J. 358, L53, (1990) Hubbard W.B., Nellis W.J., Mitchell A.C., Holmes N.C., Limaye S.S., McCandless P.C., Science 253, 648, (1991) Hubbard W.B., Pearl J.C., Podolak M., Stevenson D.J., in Neptune and Triton (D. Cruikshank, ed.) Univ. of Arizona Press, in press, (1994) Hubbard W.B., Stevenson D.J., in Saturn (T. Gehrels, M.S. Matthews, eds.), Univ. of Arizona Press, pp. 47-87, (1984) Magazzu A., Martin E.L., Rebolo R., Asirophys. J. 404, L17, (1993) Nellis W.J., Hamilton D.C., Holmes N.C., Radousky H.B., Ree F.H., Mitchell A.C., Nicol M., Science 240, 779, (1988) Pearl J.C., Conrath B.J., J. Geophys. Res. 96, 18921, (1991) Salpeter E.E., Astrophys. J. 121, 161, (1955) Saumon D., Chabrier G., Phys. Rev. Lett. 62, 2397, (1989)

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Saumon D., Chabrier G., Phys. Rev. A 44, 5122, (1991) Saumon D., Chabrier G., Phys. Rev. A 46, 2084, (1992) Saumon D., Hubbard W.B., Chabrier G., Van Horn H.M., Astrophys. 3. 391, 827, (1992) Stevenson D.3.,Phys. Rev. 12B, 3999, (1975) Stevenson D.J., Salpeter E.E., Astrophys. J. Suppl. 35, 221, (1977) Zharkov V.N., GudkovaT.V., Ann Geophysicae 9, 357, (1991)

21 Searches for brown dwarfs JAMES LIEBERT Steward Observatory and Department of Astronomy, University of Arizona, Tucson, AZ 85721, USA

Abstract This review attempts a brief summary of the numerous and diverse searches for the so-called brown dwarfs, substellar objects having masses between giant planets and the lowest mass M dwarf stars. Cette revue donne un bref apergu de l'etat actuel des diverses recherches de naines brunes, objects substellaires ayant des masses comprises entre les planetes geantes et les naines M de faible masse.

21.1 Introduction Between the giant planets such as Jupiter ( 10~ 3 MQ) and stars at the bottom of the hydrogen-burning main sequence (< 0.1M©) - spanning more than two orders of magnitude in mass - the sequence of brown dwarfs has yet to be discovered and analyzed in detail. The previous sentence carries the positive bias of this author that - despite the current lack of a single, unambiguous example for me to discuss at this meeting - the flurry of searches now underway by a variety of techniques will identify at least some genuine brown dwarfs during the present decade. Our motivation for thinking and speaking positively is to encourage advances in the theory of both the interiors and atmospheres of such gaseous objects, in order to make possible positive identifications among the candidates found by observers. Indeed, numerous candidates exist of different kinds, some with measured masses, 463

464

Liebert: Searches for brown dwarfs

luminosities and temperatures which straddle the stellar mass limit (SML) near 0.08 M©. This paper is the observational complement to Bill Hubbard's, in which recent theoretical modelling of objects near and below the SML mass limit is discussed. One area of rapid and important theoretical progress not addressed by Dr. Hubbard is the application of new stellar atmosphere analyses to fit the infrared and optical spectra of very low mass stars and, potentially, the more luminous brown dwarfs. I will review this work briefly in Section 2. The remaining sections are devoted to overviews of the different techniques used in current searches, and the recent results. Most of this material is discussed much more extensively in Burrows and Liebert (1993), but there are some updated references and new results. 21.2 Low Mass Stars or Brown Dwarfs? In the solar neighborhood, the most accurately determinable stellar parameter is often the luminosity. This is because a large, accurate trigonometric parallax combined with measurements of multiple colors yields a good estimate of the bolometric flux even if the temperature is poorly known. To look at the situation in simpler, observational terms: if the magnitude at K(2.2/x) is known, the bolometric correction is small, and the luminosity may be estimated. It might therefore seem appropriate to establish the relationship between mass and luminosity near the bottom of the main sequence, in order to determine whether a given candidate is below the SML. We shall see, however, that this is not possible. 21.2.1 The Mass - Luminosity Relation On the hydrogen-burning main sequence, the luminosity is of course a monotonic and steeply-increasing function of mass. Note that a low mass star has a hydrogen-burning lifetime that is much longer than the age of the Galaxy. Hence, the so-called "zero age" main sequence (ZAMS) position at a given mass and chemical composition remains unchanged for time scales of interest to us. The mass below which hydrogen-burning cannot continue indefinitely is near 0.075-0.08 M©for the solar composition. The main sequence luminosity is often expressed as a power law of the mass with the exponent varying from approximately three for massive stars to nearly five below the solar mass. Such a relationship is also valid on the low mass main sequence (with smaller slope) until the approach to the stellar mass limit. The predictions of stellar interiors models have been tested


465

empirically by comparing stars with measured masses (and luminosities) in astrometric binary systems. Henry and McCarthy (1993) have added an impressive number of new binary components using their technique of infrared speckle interferometry. Using the absolute K magnitude (M#) which, as we pointed out, is closely related to the log of the luminosity, they found a simple power law fit for the mass, o = -0.166M* + 0.560 which is an excellent fit to the 0.1-1 M©range. The problem comes in extending the fit below 0.1 M©towards the stellar mass limit, where several complications arise. First, the theory predicts that, as the stellar mass limit is approached, the M-L function will steepen radically - that is there will be a much larger decrease in luminosity over a given interval decrease in mass. Second, the pre-main sequence phase prior to the ZAMS lasts longer with decreasing mass, and below 0.1 of solar the objects require over 109 years to reach the ZAMS. Thus, the M-L relation becomes a substantial function of the age of a star. Moreover, below the SML the (substellar) objects never reach the ZAMS. Again, their phase of gravitational contraction brings them slowly through the same luminosities as the dimmest ZAMS stars. Finally, there are the so-called "transition" objects predicted to have masses of 0.07-0.075 M© for solar composition, which undergo limited hydrogen-burning for up to a few Gyr, though this energy release is unable to halt the contraction and growing degeneracy of the core. Nonetheless, before entering the brown dwarf cooling sequence, they may linger for these relatively long times in the luminosity range of the faintest ZAMS objects. Thus, we must know the age of a stellar object at a given low luminosity, before a mass can be assigned to it from its luminosity.

21.2.2 Temperature Estimates from Spectra and Model Atmosphere Fits The effective temperature (Te) estimates for very low mass stars have been very poor up to now and, until recently, based almost entirely on fits of observed colors to blackbodies. Now it is well known that the blackbody shape is a very poor approximation to the energy distributions of these objects. However, there had been relatively little attention to studying the spectra and atmospheres of stars on the M dwarf sequence, the hydrogenburning main sequence stars of lowest mass. This neglect has been due

466


primarily to the complexity of especially the molecular opacity sources in such cool stars. In recent years there has been impressive progress on some of the most relevant of the molecular band systems, such as CO, TiO and HjO. The two PhD dissertations of Allard (1990) at the University of Heidelberg and Ruan (1991) at the Australian National Observatory have changed this bleak situation. These resulted in the first model atmosphere grids reaching down to temperatures appropriate to the SML, and both demonstrated fair success in matching infrared and optical spectra of M dwarfs. The result of the analysis using the Allard model atmospheres was a set of temperatures for low mass stellar "standards" - well-studied, bright stars in the solar neighborhood. When spectra extending from 0.6 to 1.55 \i were fitted with synthetic spectra from models with solar composition and log g = 5, the first real attempt at defining a temperature scale for low mass stars based on model atmospheres (Kirkpatrick et al. 1993). Now these can be combined with the more accurate luminosities to place the objects in the astronomers' favorite diagram, and compare these with the predictions of theory as a function of mass and chemical composition. 21.2.3 The Hertzsprung-Russell Diagram Fig. 1 is a Hertzsprung-Russell (HR) Diagram, a plot of log L/Z©vs. log Te, in which the stars from Kirkpatrick et al. (1993) are shown asfilledcircles. Examples of the blackbody temperature determinations are shown as open circles in the diagram. Also shown for comparison are theoretical interiors calculations for masses approaching the stellar mass limit of 0.08 M©. It is perhaps not too much of a surprise that the model fits give Te values closer to the predicted locations than the blackbody fits, except for the coolest stars of lowest luminosities. However, the spectroscopic fits are far from perfect over the wavelength range covered. Moreover, Tinney, Mould and Reid (1993) show that Allard models give a poor fit to the spectral energy distributions observed at longer infrared wavelengths. There is clearly much work to be done in refining the temperature and HR diagram determinations for stars of the lowest masses. Also shown in Fig. 1, however, are a few tracks showing the very real evolution in this diagram of substellar objects. These brown dwarfs fall towards the main sequence in their phase of gravitational contraction, and limited nuclear burning for those with "transition" masses. Finally, they enter the cooling sequence at a fixed radius with the onset of degeneracy. During these phases the figure shows how closely their evolution parallels the


467

main sequence, although the brown dwarf always reaches a given luminosity in a shorter time. Thus, the ambiguity of the stellar luminosity is not resolved - at least not easily - by using the effective temperature. It is nearly impossible, therefore, to distinguish a brown dwarf fairly near the SML from a stable star, based on position in the HR Diagram alone. The temperature of the brown dwarf is typically only a few hundred degrees lower than the star at a given luminosity. We have already seen that the Te assignable to a low mass star is at least that uncertain. Either of two additional stellar parameters may provide enough information to resolve the ambiguity - that is, determine whether a given low luminosity object is a star or a brown dwarf. These are the mass or the age. We have already mentioned the growing sample of nearby stars where the masses are estimated from the solutions to the binary orbit. Likewise, in young clusters or associations where the age may be known, single objects may be analyzed.

21.3 Searching for Field Stars 21.3.1 Proper Motion Surveys Selection by motion on the sky has been the traditional way of finding the Sun's nearest neighbors. The most important survey to date was that of Luyten (1963) using plates taken with the Palomar 1.2-meter Schmidt telescope at two different epochs generally some 12 years apart; the first epoch was the original Palomar Sky Survey of the early 1950s. The sample one assembles has a kinematic bias - for example, the stars which happen to have the smallest tangential velocities with respect to the Sun might be missed. However, this has been a very efficient way of finding solar neighbors over the entire sky. The nearest, low luminosity stars to the Sun are those amenable to the most accurate followup observations - trigonometric parallaxes, searches for companions, etc. The least luminous of the solar neighbors currently are two stars cataloged in Luyten (1979) as LHS 2924 and 2065, each with absolute visual magnitudes (My) fainter than +19, implying luminosities of a few x 10~*Z/©. Such luminosities place them very close to the bottom of the ZAMS (Fig. 1), but also in the realm of the young brown dwarfs. 21.3.2 Optical and Infrared Color Surveys The red colors of the lowest-mass stellar objects provide a method of selection free of kinematical bias. With the availability of sensitive emulsions,

468


-2

A

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Fig. 21.1 The HR Diagram, showing the near juxtaposition of stellar and substellar masses of differing ages (from Burrows and Liebert 1993, their Fig. 19). automated measuring machines, and sophisticated computers, large photographic color surveys have been carried out over large areas of the sky at high galactic latitude (where contamination by red giants and heavilyreddened background stars is not a problem). Gilmore and Reid (1983) presented the first large samples, as did Hawkins (1986) and his collaborators. Still, the surface area of the sky covered by the color surveys does not yet approach that of the Luyten proper motion surveys. Hence, using redder colors they must go to larger distances than the Palomar Sky Survey and necessarily find objects that are more difficult to study in detail. In particular, the measurement of trigonometric parallaxes might be more difficult, so that the luminosities of objects which may be below 0.1M© may have to be assigned from photometric colors. Using the new Palomar Sky Survey of the 1980s, Tinney, Reid and Mould


469

(1993) have combined both proper motion and color selection into a more comprehensive attempt at finding low mass stars. Their project includes the measurement of trigonometric parallaxes as well. This sample promises to be the best collection of "field" stars. Already they have identified stellar objects somewhat fainter than the LHS stars mentioned previously. Tinney (1993) argues that the faintest known stars may not be stably supported by nuclear burning. An example of a more specialized color survey is that of Kirkpatrick (1992) using photometric CCD data rather than photographic magnitudes. He presents evidence in agreement with Tinney's (1993) conclusion about the faintest field stars. Finally, we note that one of the coolest known objects of this type - called PC 0025+0447 - was found as part of a survey for highredshift QSOs (Schneider et al. 1991). This object is characterized by a color somewhat redder than the benchmark LHS stars, as well as extremely strong Ha emission. Unfortunately, its luminosity is not measured, nor is the survey characterized to find a complete sample of very red stellar objects which might lack strong emission.

21.4 Searching in Young Clusters The advantages of confining the search to members of a young stellar aggregation are obvious. First, the objects will be young and relatively luminous. Secondly, they should have at least approximately the same ages, so that it will be possible to assign masses based on an estimate of the luminosity alone. Furthermore, all candidates lie at the same known distance - that is, if they really are members of the cluster or group. Finally, there are several interesting aggregations close enough for observation - ranging from molecular clouds forming stars of order 106"7 years to clusters nearly as old at 109 years. The development of large format CCD and infrared arrays has made observation of accurate colors over large regions of a cluster possible. The disadvantages are also formidable. Star-forming regions may be in the galactic plane, so that heavily reddened background stars may be confused with genuinely cool members of an association. Newly formed stars may possess a remnant accretion disk and/or strong chromospheric activity, which may distort the spectral energy distribution from that of a simple photosphere. Furthermore, the youngest groups may have a significant spread in age. All of these problems are generally magnified the younger the aggregation is. Finally, the candidates themselves are relatively far away and faint compared to field objects found in the solar neighborhood or in

470


the color surveys, such that the opportunities for followup observations are limited. 21.4.1 Star-Forming, Molecular Clouds The youngest star-forming regions where it might be worth looking for luminous substellar objects are of the order of 106 years old, and range in density from the loose Taurus-Auriga clouds to Rho Ophiuchus, a giant molecular cloud and newly-forming star cluster. Both happen to be approximately 150 pc away. No clear success has been achieved with the former, which is characterized by a treacherous, heavily-reddened stellar background. Rho Ophiuchus differs in having a very high internal extinction of Av > 50 magnitudes. The work has been carried out exclusively in the infrared, especially the K band, using wide-format arrays. An exhaustive review of the work on Rho Oph is beyond the scope of this paper, but I will mention a recent highlight. Comeron, Rieke, Burrows and Rieke (1993) have now completed a survey of 200 square arc minutes to a completeness of K = 15.5. They have found 91 faint sources, all with multiple observations in H(1.6//) and K. They used the color information to estimate the (highly variable) extinction to each object and hence the luminosity. Then, comparison with a theoretical isochrone for an assumed age of (up to) 2 x 106 years yields a unique mass for each luminosity. Perhaps seven objects have indicated masses at or below 0.05M©- well below the SML! The method is simple and straightforward, since the amount of information is quite limited. In order to pursue these candidates further, it may be necessary to have a better understanding of the infrared spectra and energy distributions of low mass stellar and substellar objects. 21.4.2 The Pleiades and Hyades These clusters have attracted the most attention in the search for substellar objects of known age - that is, 6-7 x 107 years for the Pleiades and 6 x 108 for the Hyades. Since the latter is closer to the Sun (44 vs. 125 pc) it turns out that the predicted apparent brightnesses of their respective brown dwarf sequences would be very similar. We have space here to discuss only a few of the many searches in the fields of these clusters. Stauffer et al. (1989) surveyed some 900 sq. arcmin at V and / using a CCD detector. They found several good candidates which are likely to be members based on radial velocities and Ho emission line activity (Stauffer


471

et al. 1994). However, the first paper concluded that the mass function peaked near 0.2MQ and hence it was unlikely that the cluster had a high density of substellar objects. A deeper CCD survey has been published by Simons and Becklin (1992), which can penetrate well into the brown dwarf luminosity regime. At the moment, the implication of this work is not yet clear to this author. The most comprehensive and complete Pleiades survey is that of Hambly & Jameson (1991) and Hambly, Hawkins and Jameson (1991; HHB), using Schmidt photographic plates at R and / to study a three degree diameter field - the core of the cluster. The second citation above includes proper motion measurements to determine with fairly high probability the stars which are astrometric members of the cluster - of those which have appropriate magnitudes and colors to fit the predicted Pleiades low mass pre-main sequence. To illustrate how this double-selection method works, we show in Fig. 2 the plot of proper motions measured in the Pleiades field by HHB. The vast majority of background stars form a huge "core" near zero velocity. But an excess of stars clearly appears at the Pleiades velocity (lower southeast circle), which fortunately is well offset from zero. Still, some allowance for a background having this same velocity is necessary, and a "control field" (eastern circle) helped HHB estimate that number. Secondly, they plotted the candidates surviving astrometric selection into an I-R / color magnitude diagram (Fig. 3), for both the Pleiades and astrometric control field. The diagonal line is the expected locus of the main sequence at the Pleiades distance. Clearly there is an excess of points in the set with Pleiades motions corresponding to possible pre-main sequence objects above the diagonal line, with very few such points in the control field. Nonetheless, HHB could not expect that 100% of the candidates surviving both tests are actual Pleiades low mass stars and brown dwarfs. Further confirmation can be achieved with radial velocity measurements: Stauffer, Liebert, Giampapa and Hambly (1994, in preparation) indicate that this survey has selected brown dwarf and very low mass stellar members with high efficiency. The Hyades is sufficiently old that the luminosity function (LF) should have separated into distinct lower main-sequence and brown dwarf components. Due to the size or relative looseness of the cluster - and, ironically, its proximity to the Sun - it is difficult to establish membership reliably from proper-motion measurements alone. Moreover, a new astrometric study (Reid 1992) shows evidence for mass segregation; thus the lower-mass objects may be less centrally concentrated and hence even more difficult to identify as cluster members against a difficult background field.

472


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Leggett and Hawkins (1988, 1989) selected field stars with large R —I colors for followup infrared (JHK) photometry and derived infrared LFs. Again, there was evidence of a peak in the LF near MK ~ +6.7 (or ~ 0 . 2 M Q ) . Bryja et al. (1992) identified several faint, red objects on multiple epochs of red Palomar Sky Survey plates as brown dwarf candidate members. The visual-infrared color measurements were somewhat puzzling, though spectroscopic followup work has strengthened the case for at least a few of these stars. This work is still in progress, as is an / and K band imaging project by Macintosh et al. (1992).

21.5 Discovery and Analysis of Substellar Companions For stellar objects in close binary systems, the masses may be determined directly by analysis of the binary orbits. The observational techniques include direct photography of visual binaries near the Sun, astrometric perturbation analyses, and speckle interferometry especially at infrared wavelengths. But wider pairs are even easier to find and at least the luminosity of a faint companion can be established.


473

Fig. 21.3 The color magnitude diagram for the Pleiades candidates surviving astrometric selection (top) and from a control field, as discussed in the text, (from HHB)

21.5.1

Wide Optical

Companions

Photography of thefieldsaround nearby stars has been employed for decades to discover resolved companions sharing the space motion of the primary. Several early "benchmark" stars of low luminosity were found by van Biesbroeck (1961); vB8 and vBlO are among the best studied of these. The latter, at Mv ~ +18.7, was the least luminous known star until the 1980s.

474


Fig. 21.4 Infrared images of the 10,000 K white dwarf GD 165 (top) in 7(1.2/i), H(l.6fi), and AT(2.2/J) from left to right, taken from Becklin and Zuckerman (1988).

21.5.2 Infrared Imaging and

Photometry

Continuation of the search for fainter, resolved companions in recent years has employed new infrared array detectors to search at friendlier wavelengths. Skrutskie, Forrest and Shure (1989) found only one new very low luminosity companion in their survey of known, nearby stars, and this Gliese 569 (Forrest, Skrutskie, and Shure 1988) is luminous enough to be a main sequence star. An even deeper survey to K = 15.5 of all northern stars to distances out to 8 pc (G.H and M.J. Rieke, 1992 private communication) found no plausible brown dwarf candidates. The most exciting discoveries by this method are infrared detections around white dwarf stars, by E.E. Becklin and B. Zuckerman. The white dwarf GD 165 has a companion at least 120 a.u. away, with a color temperature of 2100 K and luminosity of 8 x 10~s£Q(Becklin and Zuckerman 1988), substantially cooler and fainter than any well-studied field star. The infrared images (Fig. 4) show a companion star dramatically cooler than its white dwarf primary at a separation of 4.3 arcsec. The companion is not detected at 1.2/x (left frame), but is brighter than the primary at 2.2/x (right). GD 165B also exhibits a very late type spectrum (Kirkpatrick, Henry and Liebert 1993). Nonetheless, various authors have shown that GD 165B may fit tracks of marginally stellar mass. A more puzzling but potentially more decisive case is the unresolved infrared excess of the white dwarf G 29-38 (Zuckerman and Becklin 1988). If due to a brown dwarf companion, as these authors originally suggested, the separation on the plane of the sky cannot exceed several a.u. However, detections of flux longward of 2/x out to 10/x led Telesco, Joy and Sisk (1990)

475


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Fig. 21.5 Unresolved excess far-infrared emission from G 29-38, with an 800 K blackbody fit. Shortward of 2p thefluxof the white dwarf primary star dominates, (from Tokunaga, Becklin and Zuckerman 1990). and Tokunaga, Becklin and Zuckerman (1990) to attribute the infrared flux to some kind of cooler dust shell around the white dwarf. The far-infrared energy distribution presented in the latter reference is shown in Fig. 5. The fit suggests that the dust has a temperature near 800 K. It was logical to ask how a white dwarf with a cooling age of 109 years could retain such a warm dust shell? However, the age of the white dwarf poses a problem for a brown dwarf interpretation as well, since, if the infrared luminosity were attributed primarily to a companion, it could not be very old. Finally, Barnbaum and Zuckerman (1992) report that G 29-38 is probably a small amplitude (5-10 km s"1) radial velocity variable with a possible period near 11 months; this could be consistent with a substellar

476


mass companion less than an a.u. away. The study of this fascinating object must continue. 21.5.3 Infrared Speckle Interferometry Henry and McCarthy (1990, 1992) have used two-dimensional speckle interferometric observations to survey a complete sample of nearby stars out to 8 pc for faint companions most easily detectable at 2.2 \i. Multiple observations of the separation and position angle of newly-discovered and previously-known binary systems lead to improved determinations of the masses (and luminosities) of low mass stars and brown dwarf candidates. These serve as the data points for their mass-luminosity function discussed earlier. The Henry and McCarthy (1992) LF declines sharply at MK ~ +10. In most cases, if companions two magnitudes fainter existed, they would have been found over a wide range of separations. The Zuckerman and Becklin (1992) search for companions to white dwarfs produced a very similar LF and sharp decline at the faint end - despite the discovery of GD 165B. 21.5.4 Radial Velocity Surveys A complementary technique to the various imaging approaches for finding unresolved companions to nearby stars is to search for radial velocity variations due to the orbital motion of a visible component. The stars must be bright enough for precise, high-resolution line profiles to be measured. The most comprehensive search to date (Marcy and Benitz 1989) covered 70 low mass M dwarfs, some 80% of allo known single stars later than dM2 and brighter than V = 10.5 accessible from the Northern Hemisphere. Since the brightness constraint requires the stars also to be within 10 pc, there is considerable overlap with the speckle sample, but greater sensitivity to smaller orbital separation. There is also the advantage that the companion need not emit any radiation - it need not be a young brown dwarf. As we shall see shortly, however, the inability to study directly the companion is also a disadvantage. Marcy and Benitz (1989) uncovered only one companion that is possibly substellar - Gliese 623B (Marcy and Moore 1988) - which was discovered independently in the infrared speckle work. Again, this star has a possible mass range (0.067-0.087 M©) that straddles the SML cutoff. Several ongoing studies are sensitive to even smaller velocity variations. Campbell, Walker and Yang (1988) and McMillan (1992, private commu-


477

ideation) have relative accuracy near 10 meters per second. The latter continues a multi-year monitoring of 16 bright solar-type stars. The most exciting discovery to date from this method happened somewhat circumstantially. Repeated observations of the G dwarf HD 114762 by Latham et al. (1989) were intended to establish this star as a radial velocity standard under an International Astronomical Union program. Instead, they found that the star is variable with a period of 84 days and an amplitude of 0.55 km s" 1 . The unseen companion has a mass of eleven Jupiters - divided by the sine of the unknown orbital inclination. Thus, if it were viewed less than 8 degrees from pole-on, the companion could still be stellar. A new analysis by Cochran, Hatzes and Hancock (1991) finds an upper limit of only 1 km s" 1 for the projected rotation rate - suggesting indeed that a pole-on orientation is possible. Now the Harvard-Smithsonian group are engaged in a systematic monitoring program, with a target list including 24 nearby M dwarfs (Mazeh et al. 1990). 21.6 Halo stars, brown dwarfs and MACHOS The stellar mass limit for a halo star of 1/100 solar metallicity is close to 0.1M©(D'Antona 1987); from stellar interiors calculations, we would expect such a boundary star to be substantially more luminous and hotter than its counterpart at solar metallicity. Indeed, a Pop II main sequence is observed to "cut off' near My ~ +14, as expected, some five magnitudes brighter than the end of the disk sequence (Monet et al. 1992). The Population II main sequence is up to three magnitudes subluminous in comparison to disk stars of the same color (but much higher mass) - hence the low mass halo stars are called Msubdwarfs. Any existing halo brown dwarfs must be far too low in luminosity to be detectable. Information on the halo LF and mass function at the faint end is sketchy. Several globular clusters have been observed down to near the mass limit using CCD detectors (Richer et al. 1991), without evidence of a flattening or turnover. However, this ground-based work has necessarily focussed on stars in outer regions of the cluster. Yet there is evidence that the dynamical relaxation would cause the lower mass objects to be less centrally concentrated than more massive stars, so that the mass function would be biased to a steeper slope in such studies. A repaired Hubble Space Telescope could perform an unbiased determination of the average cluster mass function. Likewise, Richer and Fahlman (1992) have now attempted a corresponding mass function for the field halo population from one large-format CCD field. There are formidable problems with this kind of study: even if the

478


stellar needles can be successfully separated from the extragalactic haystack, it is difficult to estimate their space density because the metallicities and distances are poorly determined. The conclusion of these authors that the halo has a steeply-rising halo mass function accounting for the "missing mass" must be regarded with caution, especially since it appears to conflict with other studies. For example, there was no evidence for such cool stars in the comparable study of Tyson (1988, and private communication). Moreover, proper motion studies should have found very efficiently most representatives of this population in the immediate solar neighborhood, yet the ratio of low mass M stars of Pop II to Pop I does not appear to differ from that for more massive stars (Hartwick et al. 1984; C.C. Dahn and Liebert, unpublished). Nonetheless, the halo mass and luminosity functions remain poorly determined. I suppose that there is also the possibility that virtually ali the unseen halo mass is below the SML. To account for all of our galaxy's alleged massive halo, however, requires a density of something like 0.5 brown dwarf per cubic parsed It is worth asking if there are any other ways of detecting invisible or very faint stellar objects? The proposal of Paczynski (1986) to observe gravitational lensing events of more distant stars by compact, unseen objects in the foreground of our Galaxy generated much excitement as a possible answer to this question. The search for what are commonly called MACHOS - massive, compact, halo objects, a term coined by Kim Griest - has been undertaken by several groups. Since the St. Malo meeting, two groups have reported the first, probable microlensing events of background Large Magellanic Cloud giants by halo objects - the EROS group (Aubourg et al. 1993) and the Livermore group (Alcock et al. 1993). A Polish project tied more directly to Paczynski has also reported a similar detection from the Galactic bulge (Udalski, A. et al. 1993). What are these microlenses likely to be? It is too early to draw conclusions. A perfectly satisfactory explanation at the moment for the joint EROS-Livermore event with an estimated mass in the 0.03-0.3 M@ range, of course, is that this is a low mass star. Time will tell how many such events are found in the well-defined search programs. 21.7 The Bottom Line What is the net result of the application of all of these attempts to find brown dwarfs in the galactic field populations, in young stellar associations, and as companions to nearby stars? First, it has to be said that not a single.


479

unambiguous case of a brown dwarf can be pointed to with confidence. Yet, the pessimists must also acknowledge that literally dozens of interesting candidates have been found with this variety of search techniques, and more are being added by the month. The lack of proven examples may be blamed on the great difficulty of establishing a substellar mass. There is a need to sharpen our theoretical tools - through more accurate equations of state and opacities - so that the candidates may be unveiled for what they are. Better theoretical atmosphere, envelope and interior models are needed in order to tell the difference between a brown dwarf and a very low mass star. The author wishes to acknowledge support from the National Science Foundation through grant AST 92-17961. I thank Adam Burrows for many discussions that contributed greatly to this work, and George Rieke and Fernando Comeron for results in advance of publication.

References Alcock, C , et al., Nature, 365, 621, (1993) Allard, F. PhD dissertation, University of Heidelberg, (1990) Aubourg, E. et al., Nature, 365, 623, (1993) Barnbaum, C , and Zuckerman, B., Ap.J.Let, 396, L31, (1992) Becklin, E.E., and Zuckerman, B., Nature, 336, 656, (1988) Bryja, C., Jones, T.J., Humphreys, R.M., Lawrence, G., Pennington, R.L., and Zumach, W. Ap.J.Let,, 388, L23, (1992) Burrows, A. and Liebert, J. Rev. Mod. Phys. 65, 301, (1993) Campbell, B., Walker, G.A.H., and Yang, S., Ap.J., 331, 902, (1988) Cochran, W.D., Hatzes, A.P., and Hancock, T.J., Ap.J.Let, 380, L35, (1991) Comeron, F., Rieke, G.H., Burrows, A., and Rieke, M.J., Ap.J., in press, (1993) D'Antona, F., Ap.J., 320, 653, (1987) Forrest, W.J., Skrutskie, M.F., and Shure, M. Ap.J.Let., 330, L119, (1988) Gilmore, G., and Reid, I.N., Mon.Not.R.A.S., 202, 1025, (1983) Hambly, N.C., Hawkins, M.R.S., and Jameson, R.F., Mon.Not.R.A.S., 253, 1, (1991) Hambly, N.C., and Jameson, R.F. Mon.Not.R.A.S., 249, 137, (1991) Hartwick, F.D.A., Cowley, A.P., and Mould, J.R., Ap.J., 286, 269, (1984) Hawkins, M.R.S., Mon.Not.R.A.S., 223, 845, (1986) Henry, T.J., and McCarthy, Jr., D.W. Ap. J. 350, 334, (1990) Henry, T.J., and McCarthy, Jr., D.W., in Complementary Approaches to Double and Multiple Star Research, Proc. IAU Coll. 135, eds. H.A. McAlister and W.I. Hartkopf, ASP Conf. Series, p. 10 (1992) Henry, T.J., and McCarthy, Jr., D.W., Astron. J. 106, 773, (1993) Kirkpatrick, J.D., PhD dissertation, University of Arizona, (1992) Kirkpatrick, J.D., Henry, T.J., and Liebert, J., Ap.J., 406, 701, (1993) Kirkpatrick, J.D., Kelly, E.M., Rieke, G.H., Liebert, J., Allard, F., and Wehrse, R. Ap.J. 402, 643, (1993) Latham, D.W., Mazeh, T., Stefanik, R.P., Mayor, M., and Burki, G., Nature, 339, 38,(1989)

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Leggett, S.K., and Hawkins, M.R.S., Mon.Not.R.A.S., 234, 1065, (1988) Leggett, S.K., and Hawkins, M.R.S., Mon.NoLR.A.S., 238, 145, (1989) Luyten, W.J., Proper Motion Survey with the 48-inch Schmidt Telescope, No. 1 (University of Minnesota: Minneapolis), (1963) Luyten, W.J., it The LHS Catalogue, (University of Minnesota: Minneapolis), (1979) Macintosh, B., Zuckerman, B., Becklin, E.E., and McLean, I.S., Bull. Am. Astr. Soc, 24, 773, (1992) Marcy, G.W., and Benitz, K.J., Ap.J., 344, 441, (1989) Marcy, G.W., and Moore, D., Ap.J., 241, 961, (1988) Mazeh, T., Latham, D.W., Mathieu, R.D., and Carney, B.W., in Active Close Binaries, ed. C. Ibanoglu, (Kluwer Academic: Dordrecht), p. 145, (1990) Monet, D.G., Dahn, C.C., Vrba, F.J., Harris, H.C., Pier, J.R., Luginbuhl, C.B., and Abies, H.D., Astron.J., 103, 638, (1992) Paczynski, B., Ap.J., 304, 1, (1986) Reid, I.N. 1992, Mon.Not.R.A.S., 257, 257. Richer, H.B., and Fahlman, G.G., Nature, 358, 383, (1992) Richer, H.B., Fahlman, G.G., Buonanno, R., Fusi Pecci, F., Searle, L., and Thompson, I.B., Ap.J., 381, 147, (1991) Ruan, Phd dissertation, Australian National University, (1991) Schneider, D.P., Greenstein, J.L., Schmidt, M., and Gunn, J.E., Astron. J., 102, 1180,(1991) Simons, D.A., and Becklin, E.E., Ap.J., 390, 431, (1992) Skrutskie, M.F., Forrest, W.J., and Shure, M. Astron. J., 98, 1409, (1989) Stauffer, J.R., Hamilton, D., Probst, R., Rieke, G.H., and Mateo, M., Ap.J.Let, 344, L21, (1989) Stauffer,J.R., Giampapa, M., Liebert, J., Hamilton, D., Macintosh, B. and Reid, N., Ap.J., in press, (1994) Telesco, CM., Joy, M., and Sisk, C , Ap.J.Let., 358, L17, (1990) Tinney, C.G., Astron. J., 105, 1169. (1993) Tinney, C.G., Mould, J.R., and Reid, I.N., Astron. J. 105, 1045, (1993) Tinney, C.G., Reid, I.N., and Mould, J.R., Ap.J., 414, 254, (1993) Tokunaga, A.T., Becklin, E.E., and Zuckerman, B., Ap.J.Let., 358, L21, (1990) Tyson, A., Astron.J., 96, 1, (1988) Udalski, A., et al. Ada Astron., in press, (1993) van Biesbroeck, G., Astron. J., 51, 61, (1961) Zuckerman, B., and Becklin, E.E., Nature, Nature, 330, 138, (1988) Zuckerman, B., and Becklin, E.E., Ap.J., 386, 260, (1992)

22 Jovian seismology BENOIT MOSSER Institut d'Astrophysique de Paris, 98bis, bd Arago, 75014 Paris, France ; [email protected]

Abstract This paper reviews a new astrophysical subject: seismology of the giant planets. Seismology is dedicated to the sounding of the interior structure of any object; on the other hand, the interiors of the Jovian planets need to be constrained, in order to improve our knowledge of their structure and of their evolution, as well as the thermodynamical laws involved at high pressures and low temperatures. The relationship between Jovian seismology and, first, Jovian internal structure, and second, high pressure physics, is examined, in order to determine the task of "dioseismology" f in the next years. We present then the seismological theoretical approaches developped since the pionnering work of Vorontsov et ai. (1976), who calculated the frequencies of the Jovian eigenmodes. We report the first observational attempts for the detection of the oscillations of Jupiter. We discuss the observational results and examine what can be done in the future. La sismologie des planetes geantes apparait comme un centre d'interet astrophysique d'avenir. Elle doit permettre en effet - et il s'agit en fait du seul outil dont l'on dispose - de sonder les interieurs de ces planetes, actuellement mal connus, mais dont la determination represente un interet majeur. Cet article recapitule aussi bien les diverses approches theoriques developpees depuis Particle precurseur de Vorontsov et ai. (1976) que les dit This neologism, constructed in the same manner as the substantive helioseismology, should represent the seismology of all four giant, or Jovian, planets. 481

482

Mosser: Jovian seismology

verses experiences menees pour detecter les oscillations de la planete Jupiter. L'accent est mis sur les liens reliant l'etude sismologique des planetes geantes avec d'une part leur structure interne, d'autre part la physique hyperbare gerant les equations d'etat utilisees pour decrire le comportement de l'enveloppe fluide. II apparait que les deux problematiques, que l'on souhaiterait distinctes, s'enchevetrent a l'envi. L'enjeu de la "diosismologie" consiste en parvenir, essentiellement par un developpement dans un premier temps des observations, a demeler l'imbroglio actuel. Des pistes sont proposees, qui permettent d'etablir, a partir de resultats sismologiques, des resultats univoques en terme de structure interne.

22.1 22.1.1

Introduction The interior structure of giant planets

The image we have from any astrophysics! object is two-dimensional. What we see from the giant planets is in fact a very thin layer from where the photons escape or are reflected. In Jupiter for instance, the upper atmosphere is sounded by spectroscopy from a few microbars down to the 10-bar level. Some physical data, as pressure, temperature or density, cannot be measured in the deep interior. However, the values of the mass, the gravitational moments J2, J\ and J&, the rotation period of the planetary core and the luminosity constrain interior models. But the density profiles obtained from these integral quantities are strongly non unique. In addition, the following points must be noted: • The giant planet interiors correspond to pressure and density ranges where the equation of state (EOS) of hydrogen, helium and heavier elements are very far from the perfect gas law (Chabrier, these proceedings). The determination of the pressure-density profiles in the planetary interior would be a unique tool for determining the EOS of a hydrogen-helium gas mixing at very high pressure. • The precise determination of the actual state of the giant planet interior is a clue for their former evolution. • The measurement of the concentrations of helium or other elements in the whole planet and not only in the upper atmosphere, as well as the determination of the structure discontinuities in all four giant planets are key points for planetology.


483

22.1.2 Seismology Seismology is a very powerful tool for the investigation of the interior structure of any object. A simple seismological experience consists of sounding a wall by knocking it: the sound it makes helps determining its consistency. In the same way, we need to "listen" to the Jovian resonances in order to determine of what the planetary interior is made. Because of its fluid interior, Jupiter looks like a star or the Sun, and the Jovian seismological study is a priori very similar to the one of any spherical fluid object. The acoustic modes (e.g. sound waves) which are favored in a sphere are expressed very crudely by: „

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(22.1) v

'

where n is the radial order of the mode. The degree I is related to the first index of the spherical harmonics Y™ associated to the mode, and VQ is the acoustic characteristic frequency. The pattern described by Eqt. 22.1 is approximately followed, for example, by the solar modes, the one of a-Cen (Pottasch et ai. 1992) as well as Procyon (Gelly et al. 1986) and the Jovian modes (Schmider et al. 1991, Mosser et al. 1993). The signature of each object appears in the smaller terms which are not expressed by Eqt. 22.1. The calculation, measurement and interpretation of these terms are the task of dioseismology. 22.1.3 Historical review The first paper about the seismology of giant planets is from Vorontsov, Zharkov & Lubimov (1976). The authors present the two basic ideas of dioseismology: the measurements of the oscillation periods is a uniqua tool for investigating the planetary interiors; intensive energetics can lead to the excitation of such oscillations. Low degree and low order pressure modes of Jupiter and Saturn are calculated, as well as discontinuity modes due to the core. This paper was followed by four other articles (Vorontsov & Zharkov 1981; Vorontsov 1981, 1984a and b), which represent a very complete approach of the specific problems of Jovian seismology. Structure discontinuities, oblateness and differential rotation are considered as perturbations of a spherical and continuous state. Bercovici & Schubert (1987) have introduced a more simple approach which is based on the ray tracing theory. They propose some possible excitation mechanisms for the modes, and are the first to give an estimate of the Jovian mode amplitude, expressed by the observable velocity in the troposphere: about 0.5 m.s" 1 . The first

484


Jovian echelle diagrams (see Table 22.7) are due to Mosser et ai. (1988) and Vorontsov et ai. (1989); they show how strong is the influence of the core on the oscillation pattern. The first attempt for detecting the Jovian oscillations came 13 years after the first theoretical paper. It is due to Deming et ai. (1989). The long delay between the first theoretical development of Jovian seismology and the first observations is surprising, when considering the importance of the subject as well as the efforts which have been made at the same time in helioseismology. It can be understood only because of the real difficulty of the observation. In order to achieve the necessary resolution, the detection has to be made, with a large telescope, continuously over several nights, and therefore needs a stable detector. The first tentative detection was negative. Due to the geometry of the IR detector, only "high" degree modes (£ > 10) with azimuthal order \m\ = l\ were searched. According to the conditions required for the detection of Jovian pulsations (Schmider et ai. 1991, Mosser 1993), it seems that the observation conditions were not favorable.

22.1.4

Summary

A review on planetary seismology by Lognonne & Mosser (1993) has presented general theoretical and observational results on Jovian seismology. In this review, we propose a more detailed approach of the interconnection of the giant planets seismology and their interior structure. The main questions concerning the interior of the giant planets and their seismological consequences are presented in Section 22.2. Seismological calculations and observations of pressure modes are developped in Section 22.3. Finally, the discussion proposed in Section 22.4 gives clues to what can be in the future the efficiency of seismology for disentangling the planetary interiors. We will restrict our attention principally to Jupiter, the only giant planet whose oscillations have been likely detected. There is no conceptual difference between Saturn and Jupiter. The case of Uranus and Neptune is somewhat different, since they have a greater core. Their theoretical oscillations spectra are depicted in Section 22.3. But, due to their low luminosity, it will be a long time before they can be observed. t Such degrees cannot be considered as high in helioseismology, since the Sun appears 40 times greater than Jupiter in the s Icy!


485

22.2 Giant planets structure The aim of this chapter is twofold. It wants first to establish the link between the construction of giant planets models and high pressure physics; secondly between the models and seismology. In order to examine these two points, we will focus on the method currently used for the construction of the giant planets interior models.

22.2.1

Current

constraints

22.2.1.1 The gravitational moments The current parameters constraining the structure are the gravitational moments J2n (Table 22.1). These parameters express the decomposition of the non spherical gravitational potential of the planet. Jo is simply the mass of the planet. J2, J4 and Js of Jupiter have been measured by the Voyager spacecrafts (Campbell & Synnot 1985). J2 indicates the presence of a more dense core, which is supposed to be made of heavy materials, ices and rocks (Hubbard & Marley 1989). The uncertainty on J& is too high to permit any constraint. 22.2.1.2 Convection and composition The internal flux radiated by Jupiter is supposed to be transported everywhere in the planet by convection (Hubbard 1968). The radiative opacity of hydrogen and helium is too high to allow energy to be carried out by radiation. The presence of convection implies that the planet is adiabatically stratified (in fact, a very little of superadiabaticity is needed to evacuate the internal energy, but at such a low level that it is usually neglected). Jovian models are then supposed to be fully adiabatic. Since an adiabatic temperature profile has been measured under the 1-bar pressure level by Voyager 2 (Lindal et ad. 1981), the adiabat starts at the 1-bar pressure level, with a tropospheric composition as given by the observations (Gautier & Owen 1989), and a temperature of 165 K. Models follow the same adiabat from the 1-bar level to the center of the planet. The gradient in the core is surely not adiabatic, since the planetary flux goes to zero at the center. Furthermore, the adiabat surely changes with the composition gradients. However, the EOS of the heavy materials used in the core, extraplolated from lower pressure theoretical estimations (Hubbard & Marley 1989) are also assumed to be adiabatic.

486


1 12 -

-0.5

0.0 log p (g/cm )

Fig 22.1 Pressure-density relation along a pure hydrogen adiabats (from Chabner et al. 1992, Fig. 2). Differences between the EOS (solid curvebaumon et al. EOS with plasma phase transition (PPT), dashed curvewithout PPT (interploation), dot-dashed: Marley & Hubbard EOS) are very small, except in the vicinity of the PPT.

22.2.1.3 The plasma phase transition (PPT) The influence of the plasma phase transition of hydrogen (PPT) on the Jovian structure has been presented by Chabrier et al. (1992) and Saumon et al. (1992). The unique role of the PPT in the Jovian thermal balance, if really a first order transition, must be emphasized. Even if, according to Saumon et ai. (1992), the actual latent heat release contributes to only about 1% of the radiated power, the hypothetical transformation of all the metallic hydrogen into molecular hydrogen should provide the actual Jovian flux during not less than 1010 years! The role of the latent heat in the planetary evolution has to be taken very carefully into account.

22.2.1.4 Saturn, Uranus and Neptune Standard models of Saturn do not differ qualitatively from Jovian models Because of the lower mass, the PPT occurs deeper in the planet. The low helium abundance measured in the troposphere is explained by the unmiscibiUty of helium in the metallic region, hence its depletion in the envelope. The pressure in the fluid envelope of Uranus and Neptune is not high enough to permit PPT. These two giant planets present a large ice shell surrounding a rock core, and a thin fluid envelope (Fig. 22.2, Table 22.3).


22.2.2

487

Construction of the interior models

The models take into account the oblate shape of the planet (see Table 22.1) due to the high rotation velocity. To be consistent, the calculations must be two-dimensionally developped. Variables are functions of the inner radius r and of the colatitude 9 (axisymmetry is assumed). The theory offigures,as exposed in Zharkov & Trubitsyn (1978), shows how it is possible to reduce, through some coefficients describing the oblate shape of the planet, the twodimensional dependence to a pure radial dependence. All models assume a solid body rotation, with a rotation period equal to that of the magnetic field (system III). Then, two equations govern the evolution of density and pressure: hydrostatic equilibrium : EOS = adiabat :

— = —p g dr p = p(p)

(22.2)

The gravitational field g expresses simply in function of the density: g = 4wGfQpu2du/r2. The hydrostatic equilibrium equation carries no essential information, but the interior is fluid. What governs the model is the EOS which is used. The Jovian adiabat results p(p) from the ideal additive volume law: 1

Y y Z 773- + p(p)z THT P(p)x + p(p)y

(22-3)

where X, Y and Z are respectively the mass fraction of hydrogen, helium and heavier elements; p(p)i is the EOS of species i following the planetary adiabat. Y and Z in the outer envelope are in agreement with the observational results, whereas their mean value is inferred from the solar composition (YjUp =YQ, enrichment in Z, cf. Table 22.2). This implies a composition discontinuity, whose exact location remains undetermined (Gudkova et al. 1989, Zharkov & Gudkova 1991, 1992). Fig. 22.1 represents three pure hydrogen adiabats, which have been used for the construction of Jovian standard models (Fig. 22.2). It can be remarked that the difference in the pressure-density relations (as high as 20 % at the location of the PPT) leads to difference in the density profiles (Fig. 22.3) less than 2%, except at the core level. The main characteristics of a Jovian standard model (Table 22.3) are finally: • the rotation and the resulting oblateness of the planet. • the core, with a discontinuity constrast of about 4 with respect to the fluid envelope.

488


Table 22.1. Jupiter: primary constraints Mass Equatorial radius (1 bar) Oblateness Rotation period

h

u

1.899xl027kg 71492±4 km

6.48% 9h55min33s

} xlO

6

h

14697± 1 1± 5 &84± 5 31±20

Table 22.2. Jupiter: composition and structure envelope PPT core

Y ~ 0.20 and Z ~ 0.02 ; adiabatic gradient transition H2-Hmetai around the 1.2Mbar level rocks and ices, mixed or separated

Table 22.3. Giant planet models Model JUPl JUP2 JUP4 JUP5 JUP6 JUP7, JUP8

SATl URAl NEPl

Authors

Core

Hubbard & Marley 1989

mixed ice and rock

Chabrier & Saumon 1992 rock core + ice shell

Envelope

AZÔ no PPT PPT PPT, AYÔ

Gudkova & Zharkov 1989 mixed ice and rock Hubbard & Marley 1989 Hubbard & Marley 1989 Podolak 1991

rock core + ice shell rock core + ice shell

• the transition from molecular to metallic hydrogen, occuring around the 1.2 Mbar level in the Jovian interior. • convection everywhere in the planet. Finally, in the standard model frame, the use of a given EOS, associated with a given set of secondary hypothesis, corresponds to a given interior model (Table 22.3). Uniqueness of the model is not insured, so that new constraints, stronger than the gravitational constraints, are needed.


489

JuplUr

-2.10* K - 5 0 Mbor

Uraaui

- 0 0 8 RV

Ncptuao

Saturn

1 bor

1 %

1 Mbor

0.79 I

H, + H.

IIP S Mbor 12.1CTK 12 Mbor

0.25

0.13 R.

Fig. 22.2 Standard model of the four giant planets. Models of Jupiter and Saturn have three layers: the core, and the two fluid envelopes. The pressure in the fluid envelope of Uranus and Neptune is not high enough to permit the transition to metallic hydrogen. The greatest density contrast is at the rock core for Jupiter and Saturn, and the ice core for Uranus and Neptune.

Fig. 22.3 Comparison of the density profile of current Jovian models (see Table 22.3). Differences in the fluid hydrogen-helium envelope are not perceptible, but the core structures are very different.

490


Fig. 22.4 Comparison of the sound speed profile of current Jovian models (see Table 22.3). Differences in the fluid hydrogen-helium envelope are as high as 10%, despite the similarity between the density profiles.

22.2.3 iFrom the interior structure to seismology An important parameter for studying pressure oscillations is the sound speed. Its expression is easily derived from the interior structure: 1 def / Op

dpJs

adiabaticity

(22.4)

where p and p represent the pressure and density perturbation of the wave and po and po the non perturbed terms. The definition of the sound speed - a second order derivative of the free energy - first shows the capability of seismology to distinguish between different EOS. In fact, even if different adiabatic EOS used for Jupiter give very similar density profiles in the fluid envelope (Fig. 22.3), they lead to sound speed profiles that differ by about 10% (Fig. 22.4). Secondly, the calculation of the sound speed puts in evidence the importance of the hypothesis of adiabaticity. A non adiabatic gradient would lead to a different sound speed profile. Finally, one must emphasize the influence of the core. The high density constrast at the core frontier induces a strong sound speed contrast, which drastically affects the oscillations pattern (Section 22.3.). Furthermore, according to the manner the models are constructed, the core plays a very important role: its mass, even if very small in current models (about 1% of the mass of Jupiter) is adjusted so that the mass of the model fits the planetary mass. Therefore, the core mass and size - and therefore its seismological signature - depend crucially on the EOS used in the fluid envelope.


491

22.3 Giant planets seismology The oscillations of the giant planets correspond to perturbations of the pressure, density and gravitational potential. What perturbs the static equilibrium state will be examined in subsection 3.4. We first present the linear analysis whose object is the determination of the eigenfrequency pattern. The evolution of the velocity field of the oscillations and of the perturbed pressure, density and gravity terms associated to the oscillations is governed by the following equations (the analytical expressions and the complete resolution are given in Unno et al. 1979): • the equation of the movement, which exhibits three restoring forces: pressure perturbation, buoyancy, and perturbation of the gravitational potential. The Cowling approximation considers that the last term is negligible. The pressure term dominates for sound waves. • the equation of continuity, which expresses the mass conservation. • the Poisson equation (in fact useless when neglecting the perturbation of the gravitational potential). • the equation governing the evolution of the perturbation. Since the perturbation evolves much more rapidly than the characteristic evolution time of the unperturbed state, the waves propagate adiabatically. The most simple solution is the ray tracing theory, which only considers the propagation of the wave vector. It is sustained by the assumption that the propagative wave is a pure plane wave. Mosser et a/. (1988) use the the helioseismologic analysis proposed by Gough (1986) with this method, but include the structure discontinuity of Jupiter. The theory is valid for the determination of Eqt. 22.1, but inadequate for a further description of the eigenfrequencies pattern. In the following, we will focus on the asymptotic method, which proved to be powerful for the understanding of the planetary oscillations (Provost et al. 1993), as well as on the numerical calculations developped for precisely taking into account all the features of the Jovian interior structure, namely the core, the PPT and the rapid rotation. 22.3.1 Asymptotic approach An asymptotic method for calculating the Jovian oscillation spectrum has been proposed by Provost et al. (1993). It follows the asymptotic development of Tassoul (1980), but includes the discontinuity of the Jovian core. The exact eigensolutions describing the movements are asymptotically developped in Bessel's functions to the second order in frequency, near the surface (where the evolution is dominated by the adiabatic index n e of the upper atmosphere) and near the center (where the evolution is dominated

492


by the degree t of the mode), and finally connected by the discontinuity. The main advantage of this analytical procedure is to obtain seismological parameters which can be related to the internal structure parameters. On the other hand, the method does not permit a precise description of more than one discontinuity. 22.3.1.1 The modulation due to the core Asymptotic eigenfrequencies accounting for the discontinuity in the sound speed profile are given by:

e2N-2 - — ^

e . - -smanit

. sin2an>/

with L2 = £(£ + 1) and

The first part of Eqt. 22.5 reproduces the solar asymptotic expression, while the second expresses the influence of the core. • The characteristic frequency UQ measures the travel time of the sound along a planetary diameter: *b =

2 / -

(22.7)

R is the planetary radius, commonly defined at the 1-bar pressure level. As shown by Mosser (1990), the upper level which should be considered when integrating VQ is the tropopause. Eqt. 223 recapitulates the different steps for the exact calculation of UQ, taking into account the tropospheric contribution and also the effect of the oblate planetary shape: Interior

•*-

Troposphere /•tropo P Buse ( J r \1-l

L

T)

Rotation

X

r

g-|

he

(228)

The cavity where the modes propagate ends at the tropopause (a level which is not included in current interior models). The contribution of the planetary oblateness e is due to the fact that the planetary seismological mean radius R(l — e/3) differs from the geometrical mean radius R(l — 2e/9). The modes favor in fact the equatorial regions (Mosser 1992).


493

• The second order characteristic frequency is also a simple function of the sound speed, accounting for all the variations of c:

Vi = fRJO

(22.9) T

whereas the other V{ are complex integrals of interior parameters. • The core modulation has a period N and a relative amplitude e. The integer N measures the acoustic radius of the core compared to the whole planetary acoustic radius:

* - / T/I T •/planet

c

/

./core

c

< 2210 >

• e measures the discontinuity at the core frontier (Table 22.4). Finally, the pertinence of the seismological parameters shown by the previous equations must be noted (Provost et a/. 1993). Their determination, extrapolated from a non-asymptotic but numerical oscillation pattern, agrees with their theoretical values. 22.3.1.2 The rotation The comparison of Jupiter and the Sun shows the importance of rotation for Jovian seismology. Both objects present similar freefall frequencies, but their rotational frequencies are respectively 28.2 and 0.4/xHz. The removal of degeneracy due to the Jovian rotation cannot be neglected. Mosser (1990) describes the four different effects of rotation: the non-galilean planetary referential, the oblate shape of the planet, the Coriolis and centrifugal forces. The two last terms become negligible for high overtones (vn UQ). But the influence of rotation is severe (Fig. 22.6). The non-degenerated eigenfrequencies vn,e,m express as a function of the degenerated frequencies vn,i (Mosser 1990): »n,t,m = Vn,l [l + £(m 2 )] - 771 l/rot

(22.11)

where the azimuthal order m varies from —I to I (m is the second index of the spherical harmonics Yf1 associated with the mode). E(m2) is a function related to the oblateness and with the same order of magnitude. 22.3.1.3 Oscillation spectra of the giant planets The asymptotic data which describe the modulation due to the core are given in Table 22.5. What differs principally between Jupiter and Saturn on one side and Uranus and Neptune on the other side is the period N. Jupiter and Saturn have a small rock core, which implies a large JV, whereas Uranus

494


N

0

0.5 n=13

'.

*

* *° o *

o

m

1

a

*

•

o

o

°

*• *

0

m

°n-2

o

m

• a c? o o a* o a* * o a*

o

'

*

*

i

0

0.5

1

1

0

•

*

*

"•

i•

• in

,

o

o

0

*

1

aa a o a a a a

*o

N

x

d

*

o * *

o * «

2 a)

(73

,

1

•

:

°

n= 10

o

a

CO

a a)

. -

X

•

E

u

°n- 2 °

i

2 m '

0.5

1

i

n-10 o * o o* o o*

• *

•

*

o

1

*

a a

* _

a a a

n •O

• O

i

-

CM

a

•

n-2 o

Lune

m

0

°

as

:

1

0.5

Fig. 22.5 Asymptotic echelle diagrams for giant planets p-modes with degree t = 0 —• 3: v/v0 is plotted as a function of the reduced frequency hv = u/vQ - [n + int(£/2) + n e /2 + 1/4]. The modulation is function of the frequency, namely of the radial order n: its period is about N. The amplitude of the modulation is related to the coefficient t. For each planet, the radial order n varies from 2 to the maximal order of the trapped modes (cf. Table 22.5). (£ = 0: o; 1= 1: n ; £ = 2: • ; £ = 3: *)


495

Table 22.4. Jovian sound speed discontinuities Discontinuity e Core 0.13 — 0.44 PPT 0 — 0.05

-0.5

Ac/r (mrad.s"1) 3 - 0 . 1 — 0.1

0 0.5 1 f0 - |n+int(je/2)+n#/2+6/4l

o 1.5

Fig. 22.6 Jovian echelle diagram including rotation, according to the perturbation theory (Mosser 1990). Only modes with (t + m) even are shown. (* = 0: O ; / = 1 : D ; / = 2 :

« ; / = 3: *)

and Neptune have a large ice shell and N ~ 2. Table 22.6 summarizes the various frequencies governing the qualitative aspect of the planetary oscillation spectrum. • up = \JGMlR3/2ir measures the mean density of the planet and varies as i/o- The ratios vp/vo are similar for the four giant planets. • i/rot: the rotation frequency, compared to VQ, measures the complexity of the spectrum due to the rotation. For Saturn, 1 — 1 modes already overlap (^n,2,2 — ^n-1,2,-2); this overlap occurs for I = 3 Jovian modes and I = 5 for Uranus et Neptune, as indicated by the ratio i/o/2frof • vc; the cutoff frequency at the tropopause represents the highest possible eigenfrequency. It is related mainly to the temperature. The ratio i/e/i/o gives an estimate of the number of modes effectively trapped; modes with n greater than this ratio cannot be reflected at the tropopause level. Finally, the echelle diagrams of all four giant planets are presented on Fig. 22.5. The echelle diagram representation exhibits the meaningful difference between the eigenfrequencies developped to the second order in frequency and Eqt. 22.1. The method to build the echelle diagram is explained

496


Table 22.5. Asymptotic seismological data Planet

V7

N (/iHz)


155 111 164 195

V> =

12.6 10.0 2.0 2.2

0.40 0.1 0.41 -0.3 0.21 -0.4 0.13 -0.4

2.3 1.5 0.1 1.9

-0.8 -0.7 -0.1 -0.1

-1.3 -0.3 0.6 -3.9

Table 22.6. Characteristic frequencies Planet

vp

j/ 0 i/rot

ve

vo/vp

. . . . ( / i H z ) . . . . (mHz)


99 70 94 105

155 111 165 190

28 26 16 18

3.0 1.6 1.8 2.2

1.59 1.59 1.74 1.80

2.8 2.1 5.1 5.3

18 13 10 10

the Sun

99

136 0.4

7.0

1.37

165

50

Table 22.7. Principle of the echelle diagram frequencies

eigenfrequency v equ

asymptotic relation

equidistance Ai> ~ 6u\

= P

I

i echelle diagram

vertical axis

P€1N

horizontal axis

in Table 22.7f. The echelle diagram of Fig. 22.6 includes the rotational removal of degeneracy. 22.3.2 Numerical approach Different numerical codes have been developped for the calculations of Jovian pressure modes (p-modes), fundamental modes (f-modes) or surface modes (these one have significant amplitudes in the vicinity of the discontinuities). Numerical calculations (Fig. 22.7, 22.8, 22.9 and 22.10) can explore the whole [n, t] domain, contrary to asymptotic calculations. The precision is limited by the absence of the tropospheric contribution, except for the very low frequency modes (o + Sv (p is an integer).

497


8

/"c

- |n+lnt(V2)l

Fig. 22.7 Jovian numerical echelle diagram: same as Fig. 22.5, but using numerically computed eigenfrequencies (Provost et a/. 1993). (I = 0: o; / = 1 : n;£ = 2: » ; * = 3: *)

mo MO 2020

-tO

-20

0

20

60

80

6v

Fig. 22.8 Jovian numerical echelle diagram according to Vorontsov (from Vorontsov et a/. 1989, Fig. 5). This diagram represents the same low degree modes as the one of Fig. 22.7. Differences between the two figures are mainly related to differences between the interior models.

22.3.2.1 Normal mode theory Most of the approaches (Vorontsov et ai. 1989, Marley 1991, Provost et aJ. 1993) are derived from helioseismology, following the normal mode theory, but including density and sound speed discontinuities. Rotation (and differential rotation) are considered as perturbations (Vorontsov 1981,1984a and b, Vorontsov & Zharkov 1981). On the other hand, Lee (1993) introduces the rotation at the zeroth order of the calculations. Because of rotation,

498

Mosser: Jovian seismology 1.0

a

tc ZJ

1

-[

a • &

a a ra > a

A

a

A A A A

u a IX)

A A A

o

0J -

D O O A

1—

a o a o a o a

« «

A A

A A A A A

*7±4 ••• ( t n e index m remains the same because of axisymmetry). The main result of the couplage between the modes is that, contrary to perturbation results, crossings between modes of different degrees are avoided (Fig. 22.9). However, the principal property of the rotational splitting remains true = 2m

(22.12)

22.3.2.2 The Saturnian ring system as a seismometer Marley (1991) focusses his interest on low frequency Saturnian f-modes. These low frequency oscillations perturb the gravitational potential of the planet and can open gaps in the rings. Therefore, the Saturnian ring system acts as a seismometer. The great advantage of ring seismometry consists in the possibility of detecting very low frequencies £ modes, which are not detectable with a ground based detector. However, such a detection can only be applied to Saturn! An important observational result reported by Marley is the fact that low frequency Saturnian modes do not open large unexplained gaps in the ring system, which implies a surface amplitude lower than about 1 meter for very low frequency modes (a few tens of /iHz).

499


Oscillation period (min) Fig. 22.10 Saturnian low frequency £ = m f-modes (from Marley 1991, Fig. 10). £modes are modes without radial nodes. For each degree £, the different dashes correspond to different Saturnian interior models. As for Jupiter, the differences between interior models express by huge differences in the eigenfrequency pattern.

22.3.3 Observations 22.3.3.1 Three observations Three positive observation runs of Jovian oscillations have been conducted in 1987, 1990 and 1991, using two different seismometric techniques. The principle of the detection is as follows. The oscillations induce in the upper troposphere a vertical velocity field (Fig. 22.11). The spectral lines, reflected or formed at these levels, are Doppler shifted. The Doppler shift of the solar sodium line reflected by the planet has been observed with a sodium resonance cell (Schmider et ai. 1991), and the Fourier transform spectrometry method has analysed the Doppler shift in the interferogram of the Jovian methane lines at 1.1 /xm (Mosser et ai. 1993). The Doppler signal is recorded over several consecutive nights. The resulting temporal series is cleaned and its Fourier transform is calculated in order to search for the planetary eigenfrequencies. 22.3.3.2 Analysis The most obvious signature which appears in the Fourier spectra is the non continuity of the observations. Observations conducted in a single


500

2.1

MM

•m 3.2

4.2

Ifl

Fig. 22.11 Oscillation velocity field in the upper troposphere, corresponding to the projection of the spherical harmonics Y™ on the view axis. site are limited to about 8 hours. The resulting effect (Fig. 22.12) on the oscillation pattern is desastrous, as seen on a theoretical oscillation spectrum (Fig. 22.13). The very high mode density precludes any treatment of the window effect; the solution which consists of separately considering the nights to obtain a continuous series is also inoperative, since the resolution after one single night is not sufficient to resolve the modes. 22.3.3.3 Detection of the oscillations A commonly used criterion for the detection of oscillations (Gelly et ai. 1986, Pottasch et ai. 1992) is the building of an echelle diagram which should show regular patterns aligned vertically. This criterion cannot be extensively used for Jovian modes, since the modulation excludes vertical alignments in an entire echelle diagram (Provost et ai. 1993). In fact, the signature of the oscillations has been given by the signature of the rotation. The rotation affects the signal in two ways: first, it modulates the photometric signal, when some atmospheric features pass through the field of view; secondly, it removes the degeneracy of the modes, with the relation indicated by Eqt. 22.11. The first effect cannot be responsible for the signal at frequencies much higher than the rotational frequency (Schmider et ai. 1991). The second rotational signature is a consequence of Eqt. 22.12. The signature of the rotational removal of degeneracy is represented in the

501


-40

Frequency

G*Hz)

o

Time

(night)

Fig. 22.12 Fourier spectrum of the window function of the observation made at the CFHT in 1990 with the Fourier transform spectrometer (below: the shaded zone corresponds to effective observation). The diurnal signature appears in the spectrum as aliases at ±11.6/iHz. The duration of the observation determines the sharpness of the peaks in the Fourier spectrum. The extension of the spectrum is given by the duty cycle: less than 8 hours daily observation imply the dilution of one peak of the spectrum over about 50/iHz, with only 1/4 of the power in the central peak.

m

•

6

•d

1= I cqd

,1,

1, ,1. JlL

li.

d d d

o 500

1... LhJ. Him 1000

1500

2000

2500

Frequency G*Hz)

Fig. 22.13 Theoretical spectra, as continuously observed, or with a window of 8 hours observation per night. Amplitudes have been arbitrarily chosen: Gaussian envelopes; cutoff above the cutoff frequency at the tropopause; decreasing amplitude with increasing £ value; same amplitude for multiplets

Fourier spectrum of the observed Fourier spectrum by half of the rotation period of Jupiter, and by overtones (Fig. 22.14). Finally, this second effect

502


1

7A 7 I

CO

• 10

Times (day)

Fig. 22.14 Fourier spectrum of the Fourier spectrum observed in 1991 at the CFH telescope. The signature of the removal of degeneracy due to rotation appears at periods multiple of half the Jovian period of rotation. The empty zones correspond to the absence of observations during daytime. 1

.

.

• •• 77 "" 0HP87 • ». J'J ir" 1 41 II1 I 11 IIi m 1IBWUTH T

i,

'

_

s

I31 1

1

ill1 J:ir III II llll

4 1 1 • 3

4 1

i 1 1 lid l i • 1 a iWli.ll til III

LfUllJl 500

i:

i

1 1 Lit • -

i L • 1

1000

••

0 0

RIJ J i l v 'oiH4

is 0 Illi ^ '

lill 1 M l i Ah?|"0*-0 1500

2000

17

0

0

A.

2500

CFH90

500

1000

1500

2000

2500

Frequency G*Hz)

Fig. 22.15 Power spectra obtained in 1987 at the OHP, with the sodium cell resonance technique (Schmider et ad. 1991), and at the CFHT in 1990, with the Fourier Transform Spectrometer (Mosser et sd. 1993). A possible identification of the modes [n, (, m] is proposed (± is for m = ±1).

has proved the detection of a propagative signal. Because the other possible propagative signals - tropospheric or stratospheric waves - have very low frequencies ( < 1 mHz), the signal has been identified with Jovian global oscillations.


503

22.3.4 Observational results The Jovian oscillation spectra are presented on Fig. 22.15. As reported in Mosser et ai. (1991, 1993) the unambiguous identification of all individual eigenfrequencies is not possible. However, a plausible identification may be proposed, that agrees for all three spectra, and is based on the identification of the signature of the rotational removal of degeneracy, namely the [£ = 1 or 2, m = ±1] doublets. 22.3.4-1 Discrepancies In fact, what is more interesting is the extrapolation from the three oscillations patterns of some characteristic asymptotic parameters. We will focus on the characteristic frequency VQ, revealed by a possible equidistance frequency Av. The theoretical value of VQ calculated from current interior models lies, according to Eqt. 22.8, between 155 and 160 //Hz. The observed value of the equidistance Av is about 136 //Hz. It can be related to a characteristic frequency VQ in the frequency range [Av(l — 2e/N), Af(l + 2e/N)]. Typically, Av and VQ may differ from about 10//Hz. At least, the discrepancy between the theoretical and observed values of VQ is about 10 //Hz. This discrepancy should have a manifold explanation. It can be related to problems encountered in physics (the determination of the EOS), or in seismology (the calculation of the sound speed and of vo), or in planetology (the validity of the hypothesis of the standard model: adiabaticity and homogeneous composition). • EOS: The relationship between the sound speed c and the EOS has already been emphasized. The precision on c largely depends on the one of the EOS (1% inaccuracy on the EOS leads to a much bigger inaccuracy on c). If accuracy is now insured for the EOS of molecular and metallic hydrogen, that is not the case for helium and other heavier elements. • Calculation of the sound speed: The calculation of c supposes the structure to be adiabatic in the whole planet, what is only an approximation. The influence of the superadiabaticity needed for evacuating the internal flux is limited, but still exists. Furthermore, the continuity of the adiabat from the 1-bar level to the deep interior supposes no compositional gradient and convection everywhere. Any change in the value of the adiabat would substantially modify the temperature profile at all the deeper levels, and then modify significantly the sound speed profile. • Composition gradient: A composition gradient must exist somewhere in the planet to reconcile the observed abundances in the outer envelope with the values assumed to occur in the metallic region on the basis of cosmogonical arguments. Such a gradient modifies inevitably the adiabat.

504


• Radiative gradient: As calculated in Guillot et ai. (1993), a radiative window should exist in Jupiter near the 3000 K level. This window would change the entire planetary temperature profile, conducting to lower inner temperature, and maybe to lower sound speeds. • Inhomogeneities: Let us consider in Jupiter a layer dr, composed of hydrogen and heavier elements, with volumic fractions respectively (1—a) et a. The acoustic radius of the layer 6r measures the inverse of the mean sound speed c (dr = dr/c) and depends on the constitution: [ Homogeneous mixing: H

I 2 distinct levels:

dr2

^

1+

(22.13)

dr

Cz

/.

Since the sound speed in hydrogen, CH, is much higher than the one in the heavier elements, c z , the acoustic radius in case of heterogeneity is larger, which implies a lower characteristic frequency. • Perturbation of the adiabat by clouds: The planetary adiabat is for many elements over the critical point, implying no phase separation. However, the presence of water clouds introduces a discontinuity in the adiabat, and leads to lower interior temperatures, and subsequently to smaller sound speed. Finally, the hypotheses supporting the standard model should be revised. 22.3.4-2 Excitation mechanism The velocity of the observed low degree modes has been estimated about a few m.s" 1 at the 0.5-bar pressure level. The dependence of the velocity of the wave with altitude is very important (Mosser et a/. 1992). A velocity of about 1 m.s" 1 at the 0.5-bar level corresponds to only a few mm.s" 1 at the PPT level. Depending on the frequency u of the mode compared with the cutoff frequency, this velocity v corresponds to the amplitude a of an isobar level (Mosser, in preparation): a =

PQCV

, or a = v/2v u

(22.14)

That leads to an amplitude of the 1-bar level of about 100 m for v=l m.s" 1 . The excitation mechanism proposed by Goldreich et aJ. (1988) for the solar oscillations gives an order of magnitude of the expected Jovian velocity, which is too low by about a factor 100 or 1000 when considering the convection in the upper Jovian atmosphere (parallely to what is done for the Sun). Then, it seems inadequate for Jupiter, and no other mechanism is able to quantitatively explain the high amplitudes detected. On the other hand, Mosser (1991) has shown that:


505

• The estimated energy in the oscillations is compatible with the flux radiated by the planetf. • The PPT is a possible region of excitation of the modes. Since the entropy shift per proton at the PPT is very high (about k^/2 per proton), the PPT, if it is a first order transition, acts as an impermeable surface. Surface waves can then develop. They have exponentially decreasing amplitudes and are therefore not detectable, but can be coupled with sound waves. • The couplage is efficient, because it occurs between phenomena with similar periods, and the excitation too, because it appears in the deep planet, contrary to convection, which is too slow in the deep region or inefficient in the upper atmosphere. 22.4 Discussion 22.4-1 The seismological test Helioseismology was an unknown solar field 30 years ago, but has now provided measurements which are among the most precise in astrophysics, and have permitted to reconstruct the solar interior profile (ChristensenDalsgaard et ai. 1985). Can we expect the same improvements for the giant planets? Bearing in mind that the precise measurement of many eigenfrequencies will not be obtained rapidly, we focus the discussion on global asymptotic criteria. Pure hydrogen adiabats, as well as density and sound speed profiles resulting from the different EOS have already been compared, showing how the seismological parameter c is sensitive to the EOS. Measuring the sound speed profile would require what we have exluded, the measurement of numerous eigenfrequencies. However, as shown by Eqt. 22.7 and Eqt. 22.9, most of the asymptotic parameters are related to the sound speed and may be translated into interior structure parameters. 22.4.2

The signature of the PPT

224-2.1

Low degree modes

In current models, the influence of the PPT is hidden by that of the core. This is not only due to the fact that the core discontinuity is much stronger; this is also conceptually related to the way the models are made. Any f This estimation is problematic, since it needs the determination of a characteristic damping time of the oscillations! We can guess that this time is not short according to the following informations: the viscosity of molecular hydrogen is very low; there is no radiative zone in Jupiter, and therefore no radiative viscosity; the quality factor of the planet is very high.

506


change of the EOS (for example, with or without PPT) leads to a change of the size of the core, which implies a huge change in the seismological modulation. This appears very clearly on Fig. 3 of Provost et ai. (these proceedings), where eigenfrequendes of different models based on different EOS are compared. The EOS modification affects principally the low degree modes modulation. The second effect is the change of the temperature profile, corresponding then to a small change of the characteristic frequency VQ, which causes the small mean slope in the diagram, visible for all degree modes. But this can be considered only as an indirect consequence of the PPT. In fact the frontier at the PPT induces qualitatively the same kind of modulation as the one due to the core, but with a much lower amplitude (Table 22.4). This modulation already appears on the numerical echelle diagram (Fig. 22.7). Due to the expected location of the transition, the period iVPPT is about 2. The amplitude is very small, so that finally the direct signature of the PPT corresponds to a slight and rapid modulation. The detection of this modulation requires a very high frequency resolution. 22.4.2.2

High degree modes

High degree modes, which propagate in the upper envelope, are less sensitive to the core. We have compared different models based on different equations of state in order to test the capability of high degree modes to sound the PPT. Results are presented on Fig. 22.16, an histogram of the number of modes refracted at a given level. The direct signature of the PPT appears with evidence on modes with degrees higher than 8. The number of modes refracted at the PPT strongly depends on the sound speed discontinuity. If it is negative, modes penetrate deeper in the planet; if positive, the sound speed jump acts as a wall where the propagation towards the planet is stopped. 22.4.3 How to disentangle interior structure and EOS ? The previous chapter has put in evidence a major problem: how to discriminate between the influence of interior structure and that of the EOS? An answer is given by the seismological parameters. The work of Provost et aJ. (1993) has first shown the dependence between seismological parameters and interior structure parameters, and second proved that the seismological parameters are observable. The results are recalled by Fig. 22.17 and Fig. 22.18, and demonstrate that the measurement of asymptotic parameters corresponds to the measurement of interior structure parameters.

507


0 -

20

40

I

1

o

PPT : Ac >

60 '

1

ESS"

o o — CM

ft _r

o o

-^£

-i—i

n

J

" -f~i=- ~l

^ 1 r-i_

Fig. 22.16 Histograms counting the modes refracted at a given level in the planet, depending on the degree range. Three models have been used, which differ by the description of the PPT. The number of high degree modes refracted at the PPT strongly depends on the sound speed discontinuity. If Ac < 0, modes are refracted deeper in the planet; if Ac > 0, total reflection may occur. The last model has no PPT.

22.4.4

future

It appears now very important to define the next steps which must be done in dioseismology. The improvement of the description of the Jovian standard model is first required. The connection between the twofields- seismology and interior structure modelling - is already operative and fruitful. New


508 2.3

•

'

i

i

•

2.2

JUP2+

i

juP7+

i

:

+ jup6

-

j"P8+

ci :

t

-

ju P 5 + .

154

156

i

.

i

.

i

162

160

158

G*Hz) Fig. 22.17 [i/0, r o m ] diagram, indicating how the measure of the frequency i/0 can be related to the interior structure. T0<m represents the adiabatic coefficient which fits the inner (metallic) region of the fluid envelope. Models with a real PPT (JUP6 and JUP7) are separated from the others because of a higher I/Q value.

J"PB?-\

oo

1 £

jiJp^-^ +

• sups

to -

JUp1

t

-2

.

0

^~~^-H.

1

.

|

2 V, (mHz)

Fig. 22.18 [Vi, re] diagram, where rc is the core radius. The quasi-linear dependence proves that the observation of the characteristic frequency Vi provides the direct measurement of the core radius.

observations of Jovian oscillations are urgently needed. In order to improve the quality of the observed spectra, three solutions might be considered: • The window effect can be reduced only with ground-based observations made with a network of site, or with space observation. • In order to disentangle the spectrum, imaging must be developped (IR photometry at 10 /nn). An image of the Jovian oscillationfieldpermits the calculation of different spectra corresponding to different degree ranges and with a much lower mode density.


509

• The cruise flight of an interplanetary spacecraft is a unique solution for observing continuously stellar or planetary oscillations. This solution will be used during the cruise flight of Mars94 (Baglin 1991), for asteroseismology. Observing Saturn in the last years was far from favorable, with the ring system vignetting the major part of the Southern hemisphere. However, Saturn has been observed in July 1993 with the Fourier transform spectrometer at CFHT. The data reduction is under progress. During the opposition in August 1995, the rings will be seen edge on, providing a unique opportunity to obtain a regular and symmetric visibility of the modes.

22.5 Conclusion In this paper, we have reviewed the first development of a new domain: the seismological study of giant planets, or dioseismology. Theoretical results, derived from helioseismology, are much in advance compared to the observations, except for the understanding of the modes excitation. The observations tend to prove the detection of the oscillations on Jupiter, with pretty large amplitudes, but they cannot yet be translated precisely in terms of interior structure. Even if theoretical results have shown the complexity of the oscillation spectrum due to the structure discontinuities, dioseismology is able to interpret their signature. One current problem is to distinguish between the different signatures. The one due to the core is obvious, and will provide information on the core size and structure. The direct observation of the PPT is more difficult, being hidden by the core. Finally, the new seismological constraints imply the development of a new generation of Jovian interior models.

Acknowledgements I thank very much the organizers of the colloquium, first for having given the opportunity of a first review about dioseismolgy, second for having run the risk of inviting a novice in research. Je tiens egalement a remercier Daniel Gautier, pour ses directives toujours judicieuses ainsi que son aide precieuse. This work has been supported by the Programme National de Planetologie from the Institut National des Sciences de l'Univers (INSU) and by the Groupement de Recherches 'Structure interne des etoiles et des planetes geantes' from the Centre National de la Recherche Scientifique (CNRS).

510


References Baglin A. 1991. Adv. Space Res., 11, 4, (4)133-(4)140. Bercovici D. and G. Schubert 1987. Icarus 69, 557-565. Campbell J. K. and S. P. Synnot 1985. Astron. J. 90, 364-372. Chabrier G., D. Saumon, W.B. Hubbard and J.I. Lunine 1992. ApJ 391, 817-826. Christensen-Dalsgaard J., T.L. Duvall Jr., D.O. Gough, J.W. Harvey, and E.J. Rhodes 1985. Nature 315, 378-382. Deming D., M. J. Mumma, F. Espenak, D. E. Jennings, Th. Kostiuk, G. Wiedemann, R. Loewenstein, and J. Piscitelli 1989. ApJ 343, 456-467. Gautier D. and T. Owen 1989. In Origin and Evolution of Planetary and Satellites Atmospheres, (S.K. Atreya, J.B. Pollack and M.S. Matthews, Eds.) University of Arizona Press, Tucson. Guillot T., D. Gautier, Chabrier G., B.Mosser. Submitted to AkA Gelly B., G. Grec and E. Fossat 1986. AkA 164, 383-394. Goldreich P. and P. Kumar 1988. Astrophys. Journal 326, 462-478. Gough D.O. 1986. In Hydrodynamics and MHD Problems in the Sun and Stars (Y.Osaki, Ed.), 117-143. University of Tokyo Press. GudkovaT.V., V.N. Zharkov and V.V. Leont'ev 1989. [Astron. Vestnik, 22,3 252-261], Solar Sys. Res 22, 159-166 Hubbard W.B. 1968. ApJ 152, 745-753. Hubbard W.B. and M.S. Marley 1989. Icarus 78, 102-118. Lee U. 1993. ApJ. 405, 359-374. Lindal G.F., G.E. Wood, G.S. Levy, J.D. Anderson, D.N. Sweetnam, H.B. Hotz, B.J. Buckles, D.P. Holmes, P.E. Doms, V.R. Eshelman, G.L. Tyler and T.A. Croft 1981. J. Geophys. Res. 86, 8721-8727. Marley M.S. 1991. Icarus 94, 420-435. Mosser B., D. Gautier and Ph. Delache 1988. In Seismology of the Sun and Sun like-stars. Tenerife, Spain, Sept. 1988. Proc. ESA, SP-286, 593-594. Mosser B. 1990. Icarus 87, 198-209. Mosser B., F.-X. Schmider, Ph. Delache and D. Gautier 1991. ApJ 251, 356-364. Mosser B., D. Gautier and Th. Kostiuk 1992. Icarus 96, 15-26 Mosser B. 1992. Etude des oscillations globales de Jupiter et des planetes geantes. PhD thesis, Universite Paris-XI, Orsay. Mosser B., D. Mekarnia, J.-P. Maillard, J. Gay, D. Gautier and Ph. Delache 1993. AkA 267, 604-622. Pottasch 1992. AkA 264, 138. Provost J., B. Mosser and G. Berthomieu 1993. AkA 274, 595-611. Saumon D. and G. Chabrier 1989. Phys. Rev. Letters 62, 2397-2400. Saumon D., W.B. Hubbard, G. Chabrier and H.M. Van Horn 1992. ApJ 391, 827-831. Schmider F.-X., B. Mosser and E. Fossat 1991. AkA 248, 281-291. Stevenson D.J. 1985. Icarus 62, 4-15. Tassoul M. 1980. ApJ Suppl 43, 469-490. Unno W. ,Y. Osaki, H. Ando and H. Shibashi 1979. Nonradial oscillation of stars, (W. Unno, Ed.), 149-159. University of Tokyo press. Vorontsov S.V., V.N. Zharkov and V.M. Lubimov 1976. Icarus 27, 109-118. Vorontsov S.V. and V.N. Zharkov 1981. Astron. Zh. 58, 1101-1114, [Sov. Astron. 25, 627-634, 1982]. Vorontsov S.V. 1981. Astron. Zh. 58, 1275-1285, [Sov. Astron. 25, 724-729, 1982]. Vorontsov S.V. 1984a. Astron. Zh. 61, 700-707, [Sov. Astron. 28, 410-414, 1984].


511

Vorontsov S.V. 1984b. Astron. Zh. 61, 854-859, [Sov. Astron. 28, 500-503, 1984]. Vorontsov S.V., T.V. Gudkova and V.N. Zharkov 1989. Pis. Astron. Zur. 15, 646-653. Zharkov V.N. and V.P. Trubitsyn 1978. In Physics of the planetary interiors (W.B. Hubbard, Ed.), 221-284. Pachart publishing house, Tucson. Zharkov V.N. and T.V. Gudkova 1991. Ann. Geophysicae 9, 357-366. Zharkov V.N. and T.V. Gudkova 1992. In High-Pressure research: Application to Earth and Planetary Sciences, Y. Syono, M.H. Manghnani Eds, 357-366.

23 EVRIS A. BAGLIN DASGAL, Observatoirc de Paris. URA CNRS 335. 92125 Afeudon Ctdtx. France

Abstract The EVRIS experiment is an exploratory mission devoted to stellar seismology. It will observe approximately ten bright stars, for 20 days each, during the cruise of the Russian MARS 94 mission. The photometer will be able to detect amplitudes of modes as small as a few 10~6 magnitude. Some objects of masses lower than the solar one will allow to test the thermodynamics. EVRIS est la premiere experience devouee a la sismologie des etoiles. Elle sera lancee par la mission Russe MARS 94. Elle observera une dizaine d'objets, chacun pendant une vingtaine de jours, avec un seuil de detection de quelques 10~6 magnitude. Plusieurs etoiles de masse plus faible que celle du Soleil devraient permettre des tests significatifs de leur thermodynamique. 23.1 An exploratory instrument for asteroseismology. 23.1.1 Scientific objectives and strategy. After the success of detection, measurements and interpretation of the eigemodes of the Sun, it is tempting to try to achieve similar progress on other stars and to allow for a comparative and differential study of the seismical stellar behavior. The major difficulties when going from the Sun to stars is the lack of photons and the lack of spatial resolution. The rationale for such an aim 512

Baglin: EVRIS

513

has already been developed several times (i.e. Hudson et al. 1986, Praderie et al. 1988). The need to go to space has also been extensively documented (see i.e. Mangeney and Praderie 1984, Hudson et al. 1986, Baglin 1990, Weiss 1992).As an important and costy mission is difficult to defend and to launch, it was decided to propose first a small mission with a quite restricted program, but exploring the field. After 10 years with various attemps and versions, we have finally succeeded in being selected on the Russian MARS 94 mission. The EVRIS experiment will work during the 300 days cruise to MARS, then stop when the spacecraft enters the MARS neighbourhood and starts orbiting the planet.

23.1.2

The instrument.

The instrument is a classical photometer, working in photon counting mode, in a large bandpass (3000 A) centered on the visible region, to collect as many photons as possible. Due to the limited weight and space (and also to the small budget!) available, we had to restrict ourselves to a very small telescope; the diameter of the entrance pupil is 70mm. Associated to a stellar sensor, it is placed on a pointing platform (PAIS) designed and manufactured by IKI (Russia), and situated in the shadow of the solar panels of the spacecraft. The pointing accuracy is 30 arcsec.

23.2 Detection threshold for a coherent signal. The condition to detect a coherent signal in the Fourier space, if the noise level is limited by the Poisson photon noise is:

Np inf(T,t)

> A~2

(1)

A, amplitude of the coherent signal; Np, number of photons collected in Is.; T, duration of the observing run (in s.); t, time of coherence of the signal. EVRIS is able to detect amplitudes of a few 10~6 in 5 days for stars brighter than mv « 3.5 in the frequency range 0.1 to 10 mHz. At lower frequencies, the level of detection increases due to the unavoidable rise of instrumental and environment sources of noise (Fig 1).

514

Baglin: EVRIS

Power spectrum Hz" 1

coherent signal ot amplitude A

s - 0.5 A 2 0

b~ 2.3/N

confidence level 90 %

1/N

(20J)

mn

1

s

1

frequency

Fig. 23.1 Detection threshold of a coherent signal as a function of frequency.The power spectrum of a coherent signal is compared to the spectrum of a representative noise (white noise plus instrumental noise at low frequency).

23.3 Choice of the observing program. The observing program is not yet completely fixed. It is submitted to many constraints due to instrumental conditions, like visibility from the space craft, stellar sensor operations, level of instrumental noise and to the scientific specifications.

23.3.1 Possible Targets. Approximately 180 stars fullfill the preliminary conditions. Excluding giants and OB stars for which the characterisitic timescales are too long, 46 objects remain up to now. Their position in a luminosity/surface temperature diagram (the so-called HR diagram) is shown on Fig.2. A final choice will be done to improve the scientific return. Some stars lie in the lower part of the main sequence, and their structure is essentially controlled by thermodynamics. Their envelope is convective, then independent of the radiative transfer but governed by the equation of state. They have been observed by Hipparcos (See Baglin (1993), this

515

Baglin: EVRIS -2

4.05

4

3.95 3.9 3.85 3.8 3.75

3.7 3.65

Fig. 23.2 HR diagram of possible tar gets. Absolute visual magnitudes and effective temperature are computed through the Genava calibration (Hauck, 1973, 1985).

conference), and their modelisation will then be strongly constrained. If pulsations are detected, they will permit to test precisely the stratification in the convective zone.

23.3.2

Duration of the runs.

It will be fixed during the mission depending on the first results, but never shorter than 5 days. The term inf(T,t) in eq.l will be optimised.

516

Baglin: EVRIS

23.4 Post EVRIS projects. - COROT, a small mission proposed to CNES, to follow a few objects, detected by EVRIS as multimode pulsators. The runs on each object will be longer to study the statistic of the amplitude and frequency of the modes. If accepted, the launch is scheduled around 98. - STARS, a M3 ESA proposal to beflownaround 2005 (see Lemaire(1993), this conference), is now selected for an assessment study.

Acknowledgments. We want to thank all the Co-investigators of Evris and all the technical team who is now manufacturing the experiment and its pointing platform. References Baglin, A. : 1993, this conference Baglin, A. : 1990, Solar Phys., 133, 155. Baglin, A.: 1991, Adv. Space Res., 11, 133. Harvey, C. : 1988, in Advances in Helio-and Asteroseismology (J. Christensen-Dalsgaard and S. Frandsen eds.), p. 497. Hudson, H.S., Brown, T.M., Christensen-Dalsgaard, J., Cox, A.N., Demarque, P., Harvey, J.W., Me Graw, J.T., Noyes, R.W.: 1986, A concept for an Asteroseimology Explorer. Proposal submitted to NASA. Lemaire Ph. : 1993, this conference. Praderie, F., Mangeney, A., Lemaire, P., Puget, P., Bisnovatyi-Kogan, G. : 1988, in Advances in Helio-and Asteroseismology (J. Christensen-Dalsgaard and S. Frandsen eds.), p. 549. Mangeney, A., Praderie, F. : 1984, Proceedings of the workshop on Space Prospects in Stellar Activity and Variability, Paris Observatory Press. Weiss W. : 1992, in Inside the stars (A. Baglin and W. Weiss eds.) ASP Conferences Series 40, 708.

24 HIPPARCOS A. BAGLIN AND J. FERNANDES DASGAL, Observatoire de Paris. URA CKRS 335. 92125 Meudon Cedex. France

Abstract The HIPPARCOS mission will permit a decisive step forward in the comparison between observed and predicted global properties of stars, in producing distances and apparent magitudes with accuracies more than one order of magnitude higher than before. Nearby stars of intermediate and low mass will allow for statistical tests on the validity of the equation of state, like for instance the steepness of the main sequence. La mission HIPPARCOS va permettre un pas en avant fondamental dans les tests des propriees thermodynamiques des etoiles de masse intermediaire en fournissant des distances et des magnitudes apparentes beaucoup plus precises que celles obtenues au sol. 24.1 Introduction. Tests of the physical description of stellar interiors rely on a theory vs observation comparison. The stellar evolution theory predicts the variation with time of the state of the interior of a star and, also, of its fundamental, observable parameters, i.e. luminosity, surface temperature, for a given mass. The HIPPARCOS (High Precision PARallax Collecting Satellite) mission will permit a decisive step forward in this confrontation by producing distances and apparent magnitudes with accuracies more than one order of magnitude higher than before (Baglin, 1988). 517

518

Baglin & Fernandes: The HIPPARCOS mission

For a description of the mission see for instance Perryman et al., 1992, and the 'Hipparcos Input Catalogue' (Turon et al., 1992). 24.2 Distances measurements. HIPPARCOS measures parallaxes i.e. distances, and proper motions. Aproximately 120 000 stars brighter than mv w 12.5 are observed; the survey is complete up to the apparent magnitude 7.5. The accuracy is 2 milliarcseconds or better, i.e. 10 to 50 times better than from the ground. This high accuracy is reached by accumulation of individual measurements, i.e. it increases with the duration of the mission. After slightly more than 3 years of observations, the nominal accuracy is obtained despite the poor quality of the orbit, far from the expected one (Perryman, 1993). The results obtained from only one year of data is given in Figure 1. 24.3 Apparent H magnitudes. Hipparcos has proven to be an excellent photometric instrument, working in only one large H band. It will provide H apparent magnitudes with an accuracy of few milli-magnitudes. In addition, BT and UT Tycho magnitudes will be obtained for star brighter than about 11. 24.4 Luminosities. Accurate distances provide accurate luminosities for the closest objects, if an apparent magnitude is known also accurately, and if the so-called 'bolometric correction' is also well known. L == lH d2 BC

(1)

L total flux radiated in one second; Ifj flux received on earth in the H band in one second; d distance, expressed in 10 parsecs; BC bolometric correction associated to the H magnitude, which corrects the energy radiated in the electromagnetic spectrum outside the H band. At 5 parsecs the relative precision on the distance is 1 %; it generates an uncertainty of 2% on luminosities, i.e. 10 to 50 times better than presently. The progress of spectrophotometry and of the modelisation of stellar atmosphere have also reduced drastically the uncertainties on the effective temperature and on the bolometric correction.

519

Baglin & Fernandes: The HIPPARCOS mission

DISTRIBUTION OF PARALLLAX ERRORS 25000

C/>

20000

15000

-

1 0000

-

5000

-

I I I I I I' I I I I I I I I I I ~ 8

10

12

14

16

18

20

22

24

mas Fig. 24.1 The accuracy of the parallax measurements after only one year of observations (2000 stars observed more than 5 times) is already of the order of 2 milliarcseconds.

24.5

Low m a s s - s t a r s .

One of the most fundamental reasons to study low-mass stars (M) of the particles at u < uc(v,n), while the annihilation ones are caused by the reactions \n + v) + |n) —»• \UJ) between e~ and e + at u> > u>a(is, n) > wa(0, 0) = 2. At u ^> 6 ^> t, one ma,y keejj the only terms n — 0 in Eq. (1) and replace the sum over v by the integral. Then

These spectra are in good agreement with the exact spectra (1). Averaging Eq. (2) over energy interval 2 < u < 1 + \/l + 26 between the first and the second annihilation peaks we obtain (Ja)

N-N+ ~

exp (-^)

.

(7)

552

Bezchastnov & Kaminker: Thermal cyclotron and annihilation radiation

a) j / N for N=102,2 cm"3 b) 10 xj/N for N=1Cr° cm"3 c) K T ' V ' N for N=10'8 cm"3

0.1 1 Photon energy. Mev Fig. 30.1 Spectra of differential emissivity j/N as a sum of the cyclotron and one-photon pair annihilation radiations in e~e+— plasma with T = 0.05mc- and B = 0.2Bcr for different number densities JV_ = N+ = N. Smooth solid lines show the averaged cyclotron spectra (6).

The ratio of Eq. (7) to Eq. (6) at u> = 2 gives the critical number density, H

;— exp I —; . (8) b / \ b t) For N > Ncr, the annihilation spectral component forms a jump at the main threshold (E = 1.022 Mev) followed by a sequence of the fine structure peaks (Fig. 3). The annihilation features are more pronounced on the cyclotron background, when the condition N > Ncr is stronger. Ncr(T,B) shows sharp growth with increasing T arid/or with decreasing B (Fig. 4).

Bezchastnov & Kaminker: Thermal cyclotron and annihilation radiation -.22

aj)

553

-3

j'/N for N = 1CT c m . )) 10~lw xj/N for N=1Cr cm"J 3 to

° 10"'°xj/N

b) 10" xj/N

c) 10 I5|

for N = 1

°

° c_... m

b=0.2 t=0.07

f (

10 i

/

%| Nj,b)

*

/

/

< iJ , >

IT—jf

r

!/

I'°- / / /

7

NJ,C)

V\\i

10-4 / io-"[/ ir,

-»L_.

Tttl

0.1 . I Photon energy, Mev

Fig. 30.2 Same as in Fig. 1 for T = 0.07mc2. The spectrum inside the rectangle is presented separately in Fig. 3.

30.3 Discussion Soft thermal-like spectra have been observed in the X-ray pulsar state of the 5 March 1979 event, and in the soft 7-ray repeaters (Mazets et al., 1982, Golenetskii et a/., 1984). The fits yield T ~ (3 - 5) x 10 s K. Thus one can expect to observe the one-photon annihilation features in the spectra of these soft sources under quite realistic conditions (B > 1012 G, TV < 102" cm~' J ). The work was partly supported by the ESO C&EE Grant A-01-068.

554

Bezchastnov & Kaminker: Thermal cyclotron and annihilation radiation 10 * E u 10

Photon energy, Mev

Fig. 30.3 Spectrum of differential emissivity j near the annihilation jump for T = 0.07mc2, B = 0.2£ cr and N = 1020 cm" 3 . The bar between the first and the second annihilation peaks corresponds to the averaged value obtained from Eq. (7).

0.05 0.07 0.09 0.11 0.130.15

Fig. 30.4 Critical particle number density Ncr as a function of t = T/mc* for three values of h — B/Bcr (numbers near the curves). References Bezchastnov V.G. and Pavlov G.G., Astrophys.Sp.Sci. 178, 1, (1991) Golenetskii S.V. tt a/., Nature 307, 41, (1984) Harding A., AsUvphys.J. 300, 167, (1986) Harding A., Phys.Reports 206, 327, (1991) Mazets E.P. tt a/., Astrophys.Sp.Sci. 84, 173, (1982)

31 Modified adiabatic approximation for a hydrogen atom moving in a magnetic field V.G. BEZCHASTNOV loffe Institute of Physics and Technology, 194021, St.Petersburg, Russia

A.Y. POTEKHIN loffe Institute of Physics and Technology, 194021, St.Petersburg, Russia

Abstract Motion of a hydrogen atom across the magnetic field shifts center of election density distribution. For strong magnetic fields, the radiative transitions can be considered in the modified adiabatic approximation in which the shifts are taken into account. The method is illustrated by calculating the photoionization cross sections.

31.1 Introduction The presence of hydrogen atoms affects radiative transfer in cool atmospheres of neutron stars with strong magnetic fields B — 10 11 — 1013 G (Shibanov et al., 1993). If 7 = 5/(2.35 x 109 G) > 1, the atomic structure and radiative transitions have been considered by many authors for non-moving atoms (see, e.g., Potekhin and Pavlov, 1993 and the references therein). The effects of atomic motion across B have been studied using the adiabatic approximation (Gorkov and Dzyaloshinsky, 1968), variational calculations (Vincke et ul., 1992), and the perturbation theory (Pavlov and Meszaros, 1993). For strong fields, it is natural to use the basis of the states of free charged particles (Landau functions). In the adiabatic approximation, one keeps the main term of the wave function expansion. In the traditional approach, the transverse part of the wa.ve function is localized in a magnetic, well. However, the real shift of the electron density from the Coulomb center cannot coincide with the shift of the magnetic well. We 555

556

Bezchastnov & Potekhin: A hydrogen atom moving in a magnetic field

consider matrix elements for radiative transitions assuming that the adiabatic wave functions for initial and final states are arbitrary shifted. This approach is applied to calculating the photoionization cross sections.

31.2 Basic equations The structure of the hydrogen atom with generalized momentum hK in the magnetic field B = (0,0,5) is determined by the Schrodinger equation

(/?

s

I)

f

\r + i)ro\

E

K

(f) = 0,

(1)

where E is the atomic energy, M — mc + m p , n — (mcmp)/M; mf: and m p are electron and proton masses, respectively, f = fe — rv — qra is the shifted relative co-ordinate, FQ = -(ch/eB2) K± X B and pz = -ihVz. The Hamiltonian

(ae = (nip— rnc)/2M and p± = — ihV±) corresponds to harmonic motion in the "x-y"-plane; ;/ is an arbitrary parameter. Each atomic state is described by the family of eigenfunctions with various ?;,

fa } ^{f- f.),

(3)

where F* = (77 — r]')ro. If 7 > 1, one may construct the adiabatic solution

^r)

= yg]^nMr±)-

(4)

Here $n,Af is an eigenfunction of H±, n = 0 , 1 , 2 , . . . and N = 0 , 1 , 2 , . . . enumerate the electron and proton Landau levels, respectively. Then the. full set of atomic quantum numbers is |A', i) = \K,n, iV,f), where t (positive or negative) determines the "longitudinal" energy. The motion across B shifts the electron density distribution in the direction of Fo - see Vincke ,N>(r± + r»)|exp {iae(r? - i/) whose direct calculation is complicated because F* ^ 0. However, using the shift transformation *n',N'(r± + ?*) = exp ( ( r)f *

n',N'(r±),

(6)

we may rewrite Eq. (5) in the form I± = (n',N'\ exp | i ( i ; ' - //)

\n,N),

(7)

where the initial and final states belong to the same Landau basis. For this basis, p± and f± can be expressed through the annihilation and creation operators, which have the following properties: a | n,N) A\TI,N)

= y/Ti\ n - 1,7V),

A+

n, N-l),

= T/N\

a+ | n,N)

= y/n + 1 | n+

\ n , N) = VWTT

[d,a+] = [A,A+] = 1 .

\n,N+

1,/V), 1),

(8)

Expanding the exponent in Eq. (7) over powers of the operators a, A, u+ and A + , we obtain '2u) , (9)

I± = where 4> = TT/2 + arctan(A', / /A', ; ),

u = (q - i/)2(aMK±)2/2,

(10)

— \/ch/eB and F^ ^ is a normalized Laguerre function. This yields the cyclic components of .D/,;: D+1 = (Dx + iDy)/V2 ^TlFn, 0.1. It is remarkable, that the maximum polarization curve approaches the usual scalar Fermi function inside the crystal. The central density of the crystal approaches 10 8 kg/m 3 .

of the virial theorem. The gravitational energy is balanced by the pressure in matrix form. The additional pressure dependent on the magnetic, field is written according to Cox et al. (1968) in terms of the gravitational energy with magnetic flux conservation inside and outside of the crystal.

570

Engelhardt & Bues: A gravitating electron crystal

Fig. 33.5 The mass in units of 1.44Mo versus macroscopic, magnetization M for different polarizations P = 0.17,0.36,0.54 of the nondegenerate plasma. The phase transition enhances the pressure at M = 0, whereas the binding energy of the crystal is lower than the scalar Fermi energy. This picture avoids the singularity which occurs, when the pressure density relation is written in a polytropic form.

33.4 Conclusion From the analysis within the combined Chandrasekhar - Heisenberg model we conclude, that a white, dwarf with a magnetic field, which is beginning to crystallize has a larger Chandrasekhar mass of 1.4 * 1.44M0 for the plasma regime above the crystallization zone which is polarized at half maximum. Along the cooling sequence a larger magnetic field is needed to stabilize the white dwarf. References Cox, J.P., Giuli, R.T.,stellar structun, Science, New York, (1968) Engelhardt, D., Bues, I.,H7tt*c Dwarfs, Kluwer, Dordrecht, 229, (1993) Giosse,R,,Models in Statistical Physics and Quantum Fitld, Springer Berlin (1988) ' ' Jones, R.C., J.Opt.Soc.Am., 31,488, (1941) Lieb.E.H., Cornmun. Math. Phy.s.,112, 147, (1987) Sakurai 1.1.,Advanced Quantum Mechanics, Addison- Wesley, Reading, (1967)

34

Coulomb corrections in the nuclear statistical equilibrium regime D. GARCi'A Dpi de Fistca i Enginytria Nuclear, UPC, Barcelona, Spuin, and Laboratori d'Astrofisica, Insiitut d'Estudis Catalans.

E. BRAVO Dpt de Fisica t Enginyeria Nuclear, UPC, Barcelona, Spain. Laboratori d'Astrofisica, Institut d'Estudis Catalans and Centre d'Estudis Avancats, C.S.I.C, Blanes, Spain

Abstract The ionic contribution to the Equation of State (EOS) of a multicomponent plasma of nuclei in the nuclear statistical equilibrium regime is studied, and a method to compute the coulombic corrections within the framework of the linear mixing hyphothesis is proposed. Some consequences of including these corrections in the EOS are briefly analysed in relation with two concrete astrophysical scenarios, the Supernovae la explosion and the Accretion Induced Collapse of a massive white dwarf.

34.1 Introduction Since the pioneering work of Salpeter (1961) dealing with the corrections to an ideal plasma at zero temperature non ideal effects and, specially, the Coulomb corrections have been incorporated to the EOS of stella.r evolutionary codes in order to get a better understanding of the stellar evolution. The late stages of stellar evolution are specially sensitive to the non-ideal effects: the negative Coulomb pressure contributes to bound the cores of red giant stars and in white dwarfs the Coulomb and quantum corrections can drastically alter its cooling time. In the case of the progenitors of supernovae and neutron stars, the Coulomb pressure diminishes the Chandrasekhar mass limit below its nominal value of MQU = 5.8ye2 MQ. The importance of Coulomb corrections in a. finite temperature plasma, is measured by the plasma coupling constant, T, = (Ze) /r\k&T, where r\ 571

572

Garcia & Bravo: Coulomb corrections

is the mean interionic distance. Because of the relative independence of the Coulomb terms with respect to the temperature, traditionally this correction has been applied to moderately cold material at high density, like that found in white dwarfs or in the cores of giant stars, being important to determine the presupernova structure. Nevertheless, the Coulomb correction has usually been neglected once the explosive thermonuclear runaway starts in the cores of the progenitors of supernovae. In view of the Z 5 / 3 dependence of Fj, the Coulomb correction to the equation of state could be important at high temperature when matter is in the nuclear statistical equilibrium (NSE).

34.2 Coulomb corrections in NSE For a MCP, in the linear mixing approximation and for T\ > 1, the correction due to the electric interaction of the ions with an uniform electron background is given by a. mean over the correction each element would ha.ve if it was the only species present (one-component plasma, OCP). For Fj < 1 this assumption could not be true but in this case the corrections are not so relevant and a. calculation by using an interpolation formulae (Yakovlev &: Shalybkov 1987) probably suffice. Adopting the expression obtained by Ogata & Ichimaru (1987) for the OCP, we have calculated the coulombic correction to the free energy per nuclei in NSE, / c , from:

in units of k%T, being F e = e2/aek^T, a e the mean interelectronic distance, a = -0.898004, b = 0.96786, c = 0.220703, d = -0.86097, e = 2.5269. The '0i are averages over the chemical composition,

where Xu is the number fraction of the species v. In NSE the photodisintegration reactions a.re so fast that maintain the chemical abundances in equilibrium for densities greater tha.n ~ 10s g/cuv J , once a. high enough temperature (> 5 109 K) has been achieved. Thus, the chemical composition can be obtained a.s a function ofp, T and Ye only, and the 4>i can be evaluated a.s well. We have computed the NSE abundances for a set of values of p, T, and Ye (10 8 < p < 1012 g/cin 3 , 2 109 < T < 4 1O10 K, and 0.40 < Ye < 0.50), taking into account the nuclear binding energy


573

and the chemical potential of each nuclei (Mochkovitch & Nomoto 1986) corrected by the electrostatic interaction. In Fig. 1 there is plotted the pressure correction relative to the ideal gas pressure, as a function of the temperature for different densities, and for Ye = 0.5 mol/g.

34.3 Specific scenarios 34.3.1 Supernovae la

(SNIa)

The ultimate effect of the coulombic corrections in NSE must be evaluated following a complete hydrodynamical calculation of a explosion model. Nevertheless, we can by now outline the main consequences on the dynamics of the explosion and the nucleosynthesis. As can be seen in Fig. 1 the corrections in NSE are specially important at densities near 1O10 g/cm3. In the standard model of SNIa a carbon deflagration sets in at a central density p ~ 3 — 4 109 g/cm3. Therefore, although no negligible, coulombic corrections will probably not alter too much the dynamics of the explosion. The composition of the ejecta of SNIa is altered by the inclusion of the Coulomb chemical potentials in the Saha, equation that governs the abundances of the nuclides in NSE. The negative contribution of the electrostatic, interaction to the chemical potential is more important for high Z nuclei, so its abundance is increased. In Fig. 2 there are compared the NSE abundances calculated with and without coulombic corrections for typical conditions at freeze-out (/> = 108 g/cm 3 , T = 5 109 K). The abundance of alpha particles is the most affected, being reduced by about 15% for Ye = 0.5 mol/g. This should reduce the amount of alpha-rich freeze-out in the. expanding material of the SNIa, reducing the abundance of nuclei such as 58Ni and 62 Ni, and allowing more 54Fe to be ejected to the interstellar medium. The abundance of neutronized nuclei can be affected in other way by reducing the net rate of electron capture in NSE. At p = 2 10'* g/cnr* and T = 8 109 K (typical conditions of the material when most of the electron captures take place) the main contribution to the neutronization comes from the Co isotopes. In Fig. 3 there are shown the variation of the Co isotopes abundances, due to the inclusion of the Coulomb chemical potentials. The differences can reach ~ 40%, and a. similar increase in the neutronization rate can be expected (except for the important contribution of free protons, whose abundance is nearly unaffected).

574


34.3.2 Accretion Induced Collapse

(AIC)

For certain cases, the central ignition of a massive white dwarf destabilized by matter accretion from a. companion can take place at densities as high as 10 10 g/cm 3 (Hernanz et al. 1988). At such a high density the electron captures onto the nuclei present in the NSE regime can be fast enough to induce the collapse, rather than the explosion, of the white dwarf, forming a neutron star. Then, for a. fixed initial chemical composition of the white dwarf (Carbon and Oxygen, or Oxygen-Neon-Magnesium) there does exist a. critical density which separates the explosive and collapsing outcome. The value of this density is a function of the velocity of the conductive burning front that determines the advance of the flame near to the center. As a general rule, slow fronts lead to collapse and fast fronts lead to explosion. Recently, Timmes & Woosley (1992) have calculated the velocity of a. conductive burning front a.s a. function of the chemical composition prior to ignition, and of the ignition density. When this expression for the conductive velocity is used to follow the evolution of a massive white dwarf without taking into account the coulombic corrections, a value of pc ~ 8.5 109 g/cm 1 is obtained for the critical density. However: - Coulombic effects alter the velocity of the conductive combustion front itself. In order to obtain an idea of the importance of non-ideal corrections on the conductive velocity, the hydrodynamic evolution of a. conductive flame in a medium composed of C and 0 ha.s been followed by means of an hydrocode, in the form described in Garcia et al. (1990), and using a, small nuclear network to follow the changes in the elemental abundances. Although a, larger network is needed for a. suitable representation of the flame velocity (Timmes & Woosley 1992), our approach is sufficient to coiupaxe the effect of including the coulombic correction. Our calculations show that an overestimation of the flame velocity in about 10% results when the specific heat cv is not corrected from non ideal terms. - Near p = 1010 g/cm'1 the overpressure caused by the almost instantaneous rise in temperature is about 10%, and it is greatly due to the ionic contribution. In these conditions a coulombic correction of about 35% (see curve c in Fig. 1) represents a. substantial reduction in the total overpressure. In Fig. 4 there is shown the evolution of a. white dwarf after the explosive carbon ignition at />c = 7 10'' g/c.nrJ. Curve (1) represents the evolution obtained using a.n EOS without coulombic corrections in the NSE regime, and with the conductive burning velocity given by Timmes & Woosley. Curve (2) is the same, but including coulombic corrections in NSE, and with the

575


front velocity multiplied by a factor 0.9, according with the above mentioned results. As can be seen, the track of the central density is substantially altered, changing from explosion to collapse for the same ignition density.

34.4 Conclusions A simple method to compute the Coulomb corrections in a complex system composed of several hundred of nuclei which are in nuclear statistical equilibrium has been presented. The main implications for SNIa models are of nucleosynthetic nature: the rate of neutronization at a fixed temperature can be increased by ~ 40% in the heavy nuclei contribution at Ye = 0.50 mol/g, and the amount of alpha-rich freeze-out reduced by about 15%. The AIC is favoured by the pressure decrease and specific heat increase due to the Coulomb terms. This fact, besides a slight reduction on the conductive velocity of the thermonuclear front, allows the minimum density at which a C-0 white dwarf could collapse be as low as 7 109 g/cm 3 . This work has been supported by the D.G.I.C.Y.T. grant PB00-0912, and the C.E.S.C.A. project Hydrodynamical Evolution of Compact Stars.

References Garcia, D., Labay, J., Canal, R. and Isern, J., Proceedings of the Symposium "Nuclei In The Cosmos", Baden (Austria), Eds H. Oberhummer and W. Hillebrandt, P 97, 1990. Herriariz, M., Isern, J., Canal, R. and Labay, 3.,ApJ, 324, p331, 1988. Mochkovitch, R. and Nomoto, K., Astronomy and Astrophysics,l54, pi 15, 1986. Ogata, S. and Ichimaru, S., Physical R.eview, A36, p5451> 1987. Salpeter, E., ApJ, 134, p669, 1961. Timmes, F.K. and Woosley, S.E., ApJ, 396, p649, 1992. Yakovlev, D.G. and Shalybkov D.A...SW 5c?'. Rev. E. Astrophysics and Space Physics,!, p311, 1989.

0.47

0.4a

0.49

Time (sj

Y. (mol/g) ~* Figure 1. Pressure correction relative to the ideal ion pressure. Densities: a) 10* , b) 1 0 . c) 10" g/cm

~* Figure 2. Abundances in NSE at freew out. With Ceulomo chemical ootentiols (solid line) and »ithout (dashed line)

j Abundances of the main n . „ , i n ,) o n < j «ith out n J c l , i , w i t h r,M

e-capture

(dasned line) Coulomo chemical potential

Hours 4 Evolution of a for o C™ .hue udwarf stortina at 7 10'a/crr ' w u " ="J™n« •"• '"!/»

35 Molecular Opacities: Application to the Giant Planets T. GUILLOT Observatotre de la Cote d'Azur, BP229, 06304 Ar«ce Cedex 4. France

D. GAUTIER Observatotre de Paris, 5 pi J.Janssen, 92195 Meudon Cedex. France,

G. CHABRIER Laboratoire de Physique, E.N.S. Lyon, 69364 Lyon Cedex 07. France

Abstract Present available interior models of giant planets assume that the internal transport of energy is entirely convective and, accordingly, rule out any possibility of radiative transport. New opacity calculations at temperatures and densities occurring within the giant planets, taking into account H2H2 and H2-He collision-induced absorption as well as infrared and visible absorption due to hydrogen, water, methane and ammonia are presented. These opacities are not high enough to exclude the presence of a. radiative zone in the molecular Hj envelope of Jupiter, Saturn and Uranus.

Abstract Les modeles de structure interne des planetes geantes developpes actuellement supposent que le transport de l'energie s'effectiie entierement, par convection, ce qui elimine toute possibility de transport radiatif. Des nouveaux calculs d'opacite aux temperatures et densites caracteristiques des planetes etudiees, tenant compte de l'absorption induite par collisions H2H2 et H/2-He ainsi que de l'absorption dans l'infrarouge et. dans le visible de l'hydrogene, l'eau, le methane et l'ammoniaque, sont presentees. Ces opacites ne sont pas suffisainment elevees pour exclure la presence d'une zone radiative dans l'enveloppe (l'hydrogene moleculaire de Jupiter, Saturne et Uranus. 576

Guillot et al.: Molecular opacities

577

35.1 Introduction Since the estimations of the conductive and radiative opacities in Jupiter by Hubbard (1968) and Stevenson (1976) all the interior models of the four giant planets have been calculated under the assumption that the energy is transferred by convection through the entire hydrogen-helium envelope. Consequently, the thermal profile is assumed to be adiabatic at all depths. This hypothesis is based on the fact that these conductive and radiative opacities are high and that at least Jupiter, Saturn and Neptune have a substantial intrinsic luminosity. New facts prompt us to reexamine the question. Firstly, new calculations have permitted to improve substantially the hydrogen-helium opacity. Secondly, progress in molecular spectroscopy allows one to take into account the opacity due to the most abundant minor atmospheric, components. Thirdly, Voyager measurements have provided a new upper limit of the intrinsic luminosity of Uranus which is significantly weaker than that previously thought. In the next Section, we present the method used to determine the presence of a radiative zone. Then we calculate radiative opacities. In the last section we comment our results.

35.2 Method Neglecting rotation and compositional gradients, we use the Schwarzschild criterion: the medium is convective when V a j < V r a j, and radiative otherwise. V a j = (d In T/d In P)s is the adiabatic gradient and Vra,j the radiative gradient. This latter is proportional to the intrinsic luminosity of the planet (taken from Pearl and Conrath, 1991) and to the Rosseland mean opacity:

where KU is the monochromatic absorption and Bv is the Planck function. Therefore, we have to calculate Rosseland opacity tables for each planet, with chemical abundances compatible with the infrared observations of their atmospheres (Gautier & Owen, 1989). In particular, the abundances of the CNO compounds and hea,vier elements are set to 2, 4 and 50 times the solar value for Jupiter, Saturn and Uranus, respectively. We use for this comparison the interior models of Chabrier ct al. (1992) for Jupiter and Saturn, and those of Hubbard & Marley (1989) for Uranus.

578


CM

U

c o o (0
Vrad- The dashed lines correspond to the radiative gradient calculated with the opacity of hydrogen and helium alone. The various dot-dashed lines correspond to values of the cut-off equal to 100, 500 and 1000 cm"1. The vertical bars show the uncertainties on the measured intrinsic, luminosities.

Non-idea] effects for the CIA of H2-H2 and H2-He are taken into account, following the method described by Lenzuni & Saumon (1992). Figure35.1 shows two synthetic spectra at r=300K and T=3000K. One can see that the contribution of H2O, CH4, NH3 to the total opacity is significant at low temperatures, as these molecules are strong absorbers in the infrared, region which has then the most important weight in the Rosseland opacity. At higher temperatures and larger densities, the CIA, proportional to p2, and H<J free-free and H~ bound-free absorptions dominate the spectrum.

35.4 Results We compare in Fig 35.2 the radiative and adiabatic gradients for Jupiter and Uranus. Saturn and Jupiter have similar internal structures. Therefore all the results for Jupiter are also true for Saturn, at least qualitatively. The case of Neptune has not been treated here. The dashed lines in Fig35.2 correspond to radiative gradients calculated with hydrogen and helium only. A comparison of these curves to the adiabatic gradients (plain lines) shows that hydrogen and helium cannot ensure convection in the entire molecular hydrogen-rich envelopes of Jupiter and

580


Uranus. In both planets, the presence of H2O, CH4 and NH3 (dot-dashed lines) restore convection at temperatures below 1200 K (corresponding to pressures below 1 and lOkbar for Jupiter and Uranus, respectively). According to our calculations, a deep radiative window is therefore present in Jupiter, Saturn and Uranus. In the case of Jupiter, convection appears again at temperatures above 2900 K for which the absorption of H~ and H^~ becomes preponderant. At these levels the contribution of metals is significant. In the case of Uranus the comparison ends at 2000K, as our calculations cannot apply to the "ices"-f "rocks" core of this planet. We emphasis that our opacity calculations are uncertain, due to the lack of knowledge on both the chemical composition and absorption of the medium considered. However the predicted radiative zones will vanish only if these opacities are underestimated by more than one order of magnitude. Moreover, the observations do not preclude a very low value for Uranus' intrinsic luminosity. This would lead to a still larger radiative zone in this planet. Non-adiabatic models of the giant planets are needed for studying the consequences of the presence of a radiative zone on their internal structure. This is the subject of our next communication. References Barshay S.S., Lewis J.S., Icarus 33, 593-611, (1978). Bell K.L., J. Phys. B 13, 1859-1865, (1980). BirnbaumG., J. Quant. Spectrosc. Radial. Transfer 21, 597-607, (1979). Borysow A., Frommhold L., Astrvphys. J. 341, 549-555, (1989). Borysow A., Frommhold L., Astrophys. J. 348, L41-L43, (1990). Chabrier G., Saumon D., Hubbard W.B., Lunine J.I., Astrophys. J. 391, 817-826, (1992). Dalgarno A., Williams D.A., Proc. Phys. Soc. 85, 585-589, (1965). Gautier D., Owen T., In Origin and Evolution of Planetary and Satellite. Atmospheres (eds. S.K. At.reya, J.B. Pollack, and M.S. Matthews), University of Arizona Press, Tucson, pp. 487-512, (1989). Hubbard W.B., Astrophys. J. 152, 745-753, (1968). Hubbard W.B., Marley M.S., Icarus 78, 102-118, (1989). Husson N., Bonnet, B., Scott N.A., Chedin A., J. Quant. Spectrosc. Radiat. Transfer 48, 509-518, (1992). John T.L., Astron. Astrophys. 193, 189-192, (1988). Kurucz R.L., Smithsonian Obs. Spec. Rep. 309, 1-291, (1970). Lenzuni P., Chernoff D.F., Salpeter E.E., Astrophys. J. Suppl. 76, 759-801, (1991). Lenzuni P., Saumon D., Rev. Mex. Astron. Astrofis. 23, 223-230, (1992). Pearl J.C., Conrath B.J., J. Geophys. Res. Suppl. 96, 18921-18930, (1991) . Stevenson D.J., Ph.D. thesis, Cornell University, (1976).

36 On Radiative Transfer Near the Plasma Frequency at Strong Coupling YU. K. KURILENKOV Institute for High Temperatures, Russian Academy of Science,, Moscow 1211,12

H.M. VAN HORN Department of Physics and Astronomy and C. E. Kenneth Mees Observatory, University of Rochester, Rochester, NY 14627-0011, U.S.A.

Abstract The effects of strong coupling on the frequency-averaged optical characteristics of plasmas, such as the Rosseland mean-free-path, are considered. The general expression for the Rosseland mean opacity has been analyzed in terms of the transverse dielectric function of a dense plasma, and the frequency-dependent effective collision frequency. The corresponding values of the absorption coefficient and the refractive index for a dense plasma are presented at u < u)p in obvious forms.

36.1 Introduction We are concerned with rather "cold," classical, strongly coupled plasmas (SCPs), where the energy hu>p of a plasmon is comparable to or higher than the thermal energy of the elections. The electron plasma frequency is up = (47rn e e 2 /m c ) 1 / 2 . The plasma under consideration is specified by two parameters: (1) the ion coupling parameter T = (Ze)2/3/a > 1,

(1)

where /? = l/A.-gT' is the inverse temperature, Z is the ionic charge, and a = (3/47rn{)1/3 is the ion sphere radius, and (2) the electron degeneracy parameter

0 = 1//3EF = 2(4/9n)2^Z^:irs/T

> 1,

(2)

where Ep - (3n2nc)2/9Ti2/2mc is the Fermi energy of electrons at the temperature T = 0, and rs = (m c e 2 /ri 2 )(3/47rn e ) 1 / 3 is the electron density parameter. Under these plasma, conditions, nonideality effects in the optical properties are expected to be important. On the other hand, crystallization — which occurs for T > 180 — does not yet take place in this modest 581

582

Kurilenkov & Van Horn: Radiative transfer near the plasma frequency

F range. We are thus dealing with an intermediate state between ideal and solid-density plasmas. Such plasmas occur frequently both in postmain-sequence phases of stelar evolution (stellar interiors, core-envelope boundaries, and surface areas) and in laboratory experiments with high local concentrations of energy (see Van Horn and Ichimaru 1993). SCP effects manifest themselves differently in the total absorption coefficient for different frequency domains. We restrict ourselves here mainly to the important, but partial, case when the frequency is u> < up. This frequency range is critical for radiative transfer, especially under conditions where hup ~ kT. The collective modes are important in spite of the collision-dominated character of the SCP, so we attempt to consider the Rosseland mean absorption in terms of the collision frequency and the dielectric function.

36.2 Dielectric Function and Dynamic Collision Frequency In fact, all the information on the optical properties of a plasma is included in the transverse dielectric function £tr(u). In turn, the dielectric function depends upon the dynamic, electron-ion collision frequency v(u): u2 etr(u)=l-

p

(3)

u[u + ii/(u)\ for v(u) < u. Explicit theoretical results for v{u) are well-known in two limiting cases: in classical dilute plasmas (F < < 1 and 0 > > 1) and in quantum cases (0 < < 1), when the electron-ion interaction is weak enough. The Born approximation, which is acceptable for u> < u>pc (Ichimaru, Iyetomi, and Tanaka 1987; Berkovsky, Kurilenkov, and Milchberg 1992) gives the following expression for the dynamic collision frequency: 2

2

* M?K

2(

unSij) + CwuSij,

(1)

and dafij = I<swltbij + CuHbi:n

(2)

where the quantities u\j and W{j are the strain tensors of the lattice and fluid (it// = un + «22 + U33)i I1 is the shear modulus of the solid lattice, the A's are bulk moduli, and C is the coupling coefficient between the two media. The superscripts ,$' and N denote the quantities associated with the superfluid and normal matter, respectively. The stress tensors defined above lead to coupled momentum equations, which govern the oscillations of the two media. In this study, nonradial oscillations of neutron stars are investigated within the framework of the Newtonian theory of stellar oscillations. Oscillations are assumed to be a,diaba.tic, and no thermal effects associated with

Lee et al: Effects of superfluidity

588

1 14 -

34 =

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32

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00 30 -

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V \ 1

1.0

Fig. 37.1 The densities and pressures as functions of the fractional radius. The solid lines denote the total density p(= pN + ps) and pressure p = (pN + PS)- The dash-dotted lines denote ps and ps in the inner crust, while the dotted lines correspond to pN and pN in the inner crust. oscillations are included. We neglect the effects of magnetic fields and rotation on the oscillations. We employ the Cowling approximation and ignore the Eulerian perturbation of the gravitational potential.

37.2 Numerical Results The physical parameters of the unperturbed neutron star model used in the present calculations are M* = 0.503M©, /?.* = 9.839km, Tc = 1.03 x 107K, and pc = 9.44 x 1014g cm"'. We have computed the shear modulus for this model from the expression /i = 0.2945Z2e2n^3 (Baym and Pines 1971), using the the relation between the number density n/v of nuclei with charge Z and the mass density /> as given by Negele and Vautherin (1973). Above the neutron-drip density (4.3 x 10 n g cm" 3 ), neutrons emerge from the nuclei to form a free neutron superfliiid. This density marks the boundary between the inner and outer crust. The quantities that describe the neutron superfliiid in the inner crust are calculated following the prescription given by Epstein (1988). The densities and pressures in and around the inner crust are shown in Fig. 1. In Table 1, we present results for several low-order / =• 2 spheroidal modes including the effects of superfluidity, as well as results for those without superfluidity, where a = Os/KUGM*. The results are quite dramatic. The increased speed of the transverse waves, ct = (/z/^yv)1/2, greatly changes the frequency spectrum of the shear modes trapped in the crust. It changes the frequency spectrum of acoustic modes as well.

Lee et al.\ Effects of superfluidity

589

Table 37.1. The dimensionless frequency a and the period IT for surface g-modes, i-modes, s-modes and p-modes for 1 = 2.

nx

Mode 92

9\ «2 »i

*2 *3 *4

F

f

V\ Vi P3 P4

n2

2.037(-3) 3.230(-3)

368.43 232.38

2.037(-3) 3.230(-3)

368.43 232.38

8.282(-3) 1.028(-l)

90.626 7.2956

8.282(-3) 1.716(-1)

90.626 4.3738

3.093(-l) 5.556(-l) 7.483(-l) 8.571(-1)

2.4266 1.3509 1.0030 0.8740

7.015(-l) 1.183(+0) 1.437(+0) 1.728(+0)

1.0700 0.6344 0.5224 0.4344

— 1.886(+0) 4.038(+0) 4.736(+0) 6.062(+0) 7.129(+0)

— 0.3979 0.1859 0.1585 0.1238 0.1053

2.065(+0) 1.482(+0) 4.893(+0) 6.231(+0) 7.837(+0) 9.755(+0)

0.3635 0.5064 0.1534 0.1205 0.0958 0.0768

The quantities d\ and IIi are without superfluidity, while a-z and II2 are with superfluidity in the inner crust. The periods II are given in ms. Fig. 2 shows the radial component of the displacement vector, £,/r, for /-modes (modes with no nodes of £r/r) with and without superfluidity in the inner crust. We have two /-modes in the case with superfluidity. One of the two /-modes, denoted by F in Table 1, has a large amplitude £r/r for the supeifluid component, which is comparable with that of the lattice. Note that the oscillation amplitudes of the supeifluid component are usually quite small compared with those of the lattice. In particular, this is true for the conventional /-mode, denoted by / in Table 1. As shown by Table 1, the surface g modes are not affected at all by superfluidity in the inner crust.

37.3 Conclusions We have computed several low-order, global / = 2 spheroidal oscillation modes of neutron stars with M = 0.503M©, including the effects of superfluidity. The presence of supeifluid neutrons in the inner crust increases the local transverse wave speed substantially. This decreases the global oscillation periods of those modes which have significant amplitudes in the

590

Lee et al: Effects of superfluidity

0.0

0.2

0.4

0.6

0.8

1.0

0.4

0.6

rlR Fig. 37.2 The eigenfunctions £,/r of the F- and / - modes. In each panel, the solid and dotted lines denote the eigenfunctions associated with the normal matter and the supeifluid, respectively. In the right-hand panel, the dash-dotted line is the /-mode for the case without superfluidity in the inner crust. inner crust, often by very large factors, thus effectively restricting motions associated with the pulsations to the outer crust. Conversely, modes which do not have significant amplitudes in the inner crust, such as the surface (/-modes, are completely unaffected. Acknowledgements This work has been supported in part by the National Science Foundation under grant AST 91-15132, in part by NASA grant NAGW-2444, both through the University of Rochester, and in part by the U.S. Department of Energy.

References Baym, G., and Pines, D., Annals of Phy*., 66, 816, (1971) Epstein, R. I., Ap. ./., 333, 880, (1988) McDermott, P. N., Van Horn, H. M., and Hansen, C. J., Ap. J., 325, 725, (1988) Mendell, G., Ap. J., 380, 515, (1991) Negele, J. W., and Vautherin, D., Nucl. Phys., A207, 298, (1973) Van Horn, H. M., ami Epstein, R. I., Bull. A. A. S., 22, 748, (1990)

38 Magnetic Field Decay in the Non-superfluid Regions of Neutron Star Cores A.G. MUSLIMOV AND H.M. VAN HORN Department of Physics and Astronomy and C.E. Kenneth Mees Observatory, University of Rochester, Rochester NY 14627-0011 USA

Abstract We consider a simple model for the evolution of a poloidal magnetic field initally trapped in a. region containing normal npe matter within the outer liquid core of a neutron star. We have performed numerical computations for neutron stars with masses of 1.4, 1.6, and 1.7 MQ that undergo very rapid cooling due to the direct Urea process. Because the timescale for the magnetic field deca.y is directly proportional to T2, such a cooling history produces a rapid decline in the magnetic-field strength B, even for B as low as ~ 1012 G. In particular, we show that an initially quasi-homogeneous magnetic field of strength B = 1012 G declines during the first ~ 1 Myr. 38.1 Introduction The calculations of Ba.ym, Pethick, and Pines (1969a) have shown that the electrical conductivity of matter in the core of a neutron sta.r is too large to permit ohmic decay of the magnetic field within the age of the Universe. Recently, Haensel, Urpin, and Ya.kovlev (1990; hereafter HUY) have pointed out. that the magnetic-field strength |B| ~ 1012 G typical of pulsars is sufficiently strong that the anisotropy of the transport coefficients cannot be neglected and that the "resistivity" for current flow perpendicular to B is many orders of magnitude larger than that for current flow parallel to B. Using a simple "toy" model, they found that internal fields B > 101'1 G can decline to ~ 1012 G in times ~ 107 years, but tha.t fields < 1012 G remain practically unchanged on this timescale. In this paper, we derive a.n equation for the. evolution of the dipole magnetic field, taking into account the anisotropy and nonlinearity of the conductivity tensor produced by the strong field. We solve this equation numerically for the case in which the field is entirely confined to the outer, 591

592

Muslimov & Van Horn: Non-super fluid neutron star cores

non-superfluid, normal-ytpe-matter core of the neutron star. We regard these calculations as a step toward the construction of more complex models of field evolution in neutron stars. We have used the new neutron-star cooling histories obtained by Page and Applegate (1992; hereafter PA) which allow the possibility of the direct Urea process in the neutron star cores. For our present calculations, we have chosen their 1.4, 1.6, and 1.7 MQ models with maximal cooling.

38.2 Basic Equations and Numerical Results In this section, we sketch briefly the derivation of an equation describing the evolution of the magnetic field confined to that part of the core of a neutron star containing normal (z.e., non-superfluid) npe matter. Explicit expressions for the components of the resistivity tensor 7£ under conditions appropriate to neutron star interiors with normal npe matter are presented in Haensel, Urpin, and Yakovlev (1990). For strong fields, they find TZ± = 7£|| + A.B2, where TZ± and 1Z\\ are the components of V, associated with current flow perpendicular arid parallel to B, respectively, and A is a coefficient given below. For this case, an appropriately generalized form of Ohm's Law can be rewritten as E = ft||j + ft//j x B + AB x (j x B),

(1)

where the first term is responsible for ohmic dissipation, and the second represents the Hall drift. ^Frorn the strong-field relations given by HUY, the coefficients 1Zy and A are ^

5

A

(

2

)

where in* ss Tikf(c)/c. is the effective mass of the election, including relativistic effects, m* is the effective mass of the proton (~ 0.8 mp, c.f. Frima.11 and Maxwell 1979), nc is the number density of electrons, r c p is the electron relaxation time (due to Coulomb scattering from protons), and rpn is the proton relaxation time due to scattering from neutrons. According to Baym, Pethick, and Pines (1969a,b), the relaxation times rep and r pil are, respectively, 1/2

s and Tpn « 10" 1 9 Ty2 s.(3) Here To = T/10 5 ' K; po = 2.8 x 1014 g • cm" 3 is the nuclear-matter density; and other symbols have their usual meanings.

Muslimov & Van Horn: Non-superfluid neutron star cores

593

We now consider the evolution of the dipole component of a purely poloidal magnetic field confined to the neutron star core. Note that the Hall effect disappears in this case. For a purely poloidal field, we can express B in the form [in spherical coordinates (r, 6,)]: / S B = Bo\2^rcos0e rV r1

1 OS -—sineeo r Or

\ , )

(4)

where Bo is some normalization magnitude of the field, and e r and e# are unit vectors in the radial and transverse directions. To obtain the partial differential equation satisfied by the stream function S, we first note tha.t the form assumed for B yields a purely azimuthal (and solenoidal) current density j , from Ampere's Law. With the Hall term neglected, eq. (1) yields E from B and j . For this case, the r- and ^-components of Faraday's Law can be shown to contain the same physical information. Using Fa.rada.y's La.w we arrive at the following nonlinear equation for evolution of the function S(r,t): dt ~ The main problem with eq. (5) is that even if the field were initially dipolar, higher-order multipoles may be generated due to the third term in expression (1), which we have not taken into account. While the presence of higher-order multipoles can, in principle, modify the evolution of the dipole magnetic field, this problem is beyond the scope of the present paper. To solve equation (5) we need to specify appropriate boundary conditions. Here, we consider a simplified model in which the internal magnetic field is entirely confined to the core liquid region with normal npe matter (i.e., without superfluidity or superconductivity). For the 1.4 M© neutron star model we have employed in these calculations (Page and Applegate 1992), the stellar radius is B* = 11.3 km, the radius of the core is ~ 10 km, with central density pc = 4.76po! aiul the normal ripe-matter fluid occupies the outer layer of the core, with thickness ~ 4 km. We take the magnetic field to be confined entirely to the layer of normal npe matter. We assume that both the function S\r, t) and its derivative OS/Or vanish at the outer and inner boundaries of that region. We also adopt the normalization value Bo = 0.5 x 1012 G (see eq. [4]) corresponding to an initial, quasi-homogeneous magnetic field with field strength 1012 G at the magnetic poles. For the 1.4 MQ model, the internal temperature remains relatively constant, at T ~ 108 K, during the first ~ 102 yr; then it suddenly drops to ~ 3 x 10' K. Thereafter, the temperature decreases monotonically until it

594

Muslimov & Van Horn: Non-superfluid neutron star cores

1.0

o

s 2

3

4 5 LOG t (yr)

6

Fig. 30.1 Lett, panel: The temporal evolution of the magnetic energy, £jif (in p / a s m a / < av >,, acuum (1). 40.2.1 Analytic form of the screening factor.

The screening factor / can be expressed as / = exp[H(0)/kT] if the internuclear spacing is much larger than the classical turning point. For stellar evolution applications, the. analytic forms of the screening factors which have generally been used are those developed by Salpeter (1954, hereafter S54) arid Graboske et al (1973, GWCD), namely: /,„ = expA = exp(ZlZ2e1)/{B.DkT),

B.D =

R,D is the Debye-Huckel radius, £ = \ZJ2i(î(î + &c))Xi/Ai the rms charge of the plasma and 0c the electronic degeneracy factor. A is the natural screening parameter, ie the ratio of the Coulomb energy interaction to the kinetic, energy of the reacting particles. /„, is only valid when A < 1, that is, in the weak screening (WS) regime. Later GWGC presented a formalism recovering the limits of WS and SS (strong screening) regimes: log fG = %//(,A£[(Z1 + Z 2 ) 1 + 6 -Z 1 1 + 6 -Z,| + ( > ], A O =

Dzitko et al.\ Screening formalisms in solar and stellar conditions Ao is a charge independent variable characterizing the plasma, b lies between 1 and 2/3 for WS or SS and 0.860 for IS (intermediate screening), value obtained with the cluster expansion theory, r/j, depends on the appropriate charge plasma average £. For WS 7/j, = £, kb — 1/2 leading to the classical Salpeter's formula. Thus, 3 screening regimes are defined: A < 0.1 -> WS, O . K A < 5 -» IS, and 5 < A -> SS. In the previous standard assumptions, it was written l o g / = H(0)/kT = AU/kT where All is the electrostatic energy difference of the 2 reacting nuclei between infinite separation and fusion, but Mitler showed that AF, the difference in Helmholtz free energy, is better adapted than AU (Mitler 1977, hereafter M77). In fact, the first two terms of the expansion of the effective interaction energy are: ZiZ2e

AF(0) \AF(Q)\ —^, hence log/MO = —TTJTr kg! kT This calculation is carried out in the 2 fluid approximation, within the statistical equilibrium assumption, and with a better charge density where 2 spherical domains around the reacting ions are denned and limited by some radius r\ (roughly, the radius of the Wigner-Seitz sphere): if r < T\ the electron density is taken uniform (as in S54), and if r > ri the density distribution goes like e~Kr/r, as in the Debye theory. This leads to a prescription which recovers the previous S54 and GWGC results in WS, and which is also valid for all regimes, but without any discontinuities over the A range, as it is the case for the GWGC formalism (see figure 1.1). U(r) =

40.2.2

Numerical computation with radial dependence of the screened potential function H(r)

In this case, one checks the validity of the expression (1). H(0) is then replaced by H(r) which accounts for the distorted ion-electron distribution around the 2 ions in interaction. At short distance, their common cloud is assumed to be uniformly distributed following an ellipsoid of revolution, and a.t large distance H(r) turns into an usual Yukawa potential. This radial dependence involves a. recalculation of the screened reaction rates < ov >screened- The enhancement factor / is thereby deduced as follows: JMr

=
-T loci for models with the indicated effective temperatures.

42.3 Neutrino Rates The neutrino rates of Itoh and collaborators are comparable to the rates of Beaudet et al. (1967) and De Zotti (1972). The largest difference between the old and new rates is for the bremsstrahlung process, which is roughly equivalent at high temperatures, but a factor of three or more weaker at low temperatures.

42.4 Results In order to explore the importance of surface metallicity, I computed computed parallel 0.4, 0.6, and 0.8 M 0 C-core DA sequences with Z = 0.000 and 0.001. The evolutionary summary listings for the Z - 0.000 sequences are given in Tables 1-3, sampled every 0.5 dex in l o g ( Z / I 0 ) before crystallization onset, and every 0.2 dex thereafter!. Comparing, we find that t Tin? listings of these anil other sequences are available in electronic form. F'lease sen

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