The Electromagnetic Spectrum of Neutron Stars (2007)(en)(386s)

The Electromagnetic Spectrum of Neutron Stars NATO Science Series A Series presenting the results of scientific meeti...

Author: Altan Baykal | Sinan K. Yerli | Sitki C. Inam | Sergei Grebenev

12 downloads 668 Views 9MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

The Electromagnetic Spectrum of Neutron Stars

NATO Science Series A Series presenting the results of scientific meetings supported under the NATO Science Programme. The Series is published by IOS Press, Amsterdam, and Springer (formerly Kluwer Academic Publishers) in conjunction with the NATO Public Diplomacy Division.

Sub-Series I. II. III. IV.

Life and Behavioural Sciences Mathematics, Physics and Chemistry Computer and Systems Science Earth and Environmental Sciences

IOS Press Springer (formerly Kluwer Academic Publishers) IOS Press Springer (formerly Kluwer Academic Publishers)

The NATO Science Series continues the series of books published formerly as the NATO ASI Series.

The NATO Science Programme offers support for collaboration in civil science between scientists of countries of the Euro-Atlantic Partnership Council. The types of scientific meeting generally supported are “Advanced Study Institutes” and “Advanced Research Workshops”, and the NATO Science Series collects together the results of these meetings. The meetings are co-organized by scientists from , NATO countries and scientists from NATO s Partner countries – countries of the CIS and Central and Eastern Europe. Advanced Study Institutes are high-level tutorial courses offering in-depth study of latest advances in a field. Advanced Research Workshops are expert meetings aimed at critical assessment of a field, and identification of directions for future action. As a consequence of the restructuring of the NATO Science Programme in 1999, the NATO Science Series was re-organized to the four sub-series noted above. Please consult the following web sites for information on previous volumes published in the Series. http://www.nato.int/science http://www.springeronline.com http://www.iospress.nl

Series II: Mathematics, Physics and Chemistry – Vol. 210

The Electromagnetic Spectrum of Neutron Stars edited by

Altan Baykal Middle East Technical University, Ankara, Turkey

Sinan K.Yerli Middle East Technical University, Ankara, Turkey

Sitki C. Inam Baskent University, Ankara, Turkey and

Sergei Grebenev Space Research Institute (IKI), Moscow, Russia

Published in cooperation with NATO Public Diplomacy Division

Proceedings of the NATO Advanced Study Institute on The Electromagnetic Spectrum of Neutron Stars Marmaris, Turkey 7-18 June 2004 A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN-10 ISBN-13 ISBN-10 ISBN-13 ISBN-10 ISBN-13

1-4020-3860-7 (PB) 978-1-4020-3860-7 (PB) 1-4020-3859-3 (HB) 978-1-4020-3859-4 (HB) 1-4020-3861-5 (e-book) 978-1-4020-3861-5 (e-book)

Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands.

www.springeronline.com

Printed on acid-free paper

All Rights Reserved © 2005 Springer No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed in the Netherlands.

Contents

List of participants, speakers, and guests

ix

Preface

xv

Part I Formation of Neutron Stars Neutron Star Formation Chris L. Fryer, Aimee Hungerford

3

Young Neutron Stars and Their Wind Nebulae Patrick Slane Part II

15

Neutron Star Structure and Magnetic Field

Neutron Star Superfluidity, Dynamics and Precession M. Ali Alpar

33

A Biography of the Magnetic Field of a Neutron Star Malvin Ruderman

47

Elementary Processes in Strong Magnetic Fields Joseph Ventura

63

Drifting subpulses in PSR B0826-34 J. Gil, Y Gupta, J. Kijak, M. Sendyk

75

Temperature Distribution in Magnetized Neutron Star Crusts U. Geppert, M. Kueker, D. Page

79

Coupling of Radial and Non-radial Oscillations of Neutron Stars Andrea Passamonti, Marco Bruni, Leonardo Gualtieri, Carlos F. Sopuerta

83

Part III Neutrino Astrophysics High-Energy Neutrino Astronomy Francis Halzen

89

Gravitational Wave Interactions with Magnetized Plasmas

v

103

vi

EM SPECTRUM OF NEUTRON STARS

Joachim Moortgat, Jan Kuijpers The Double-Neutron-Star Inspiral Rate and Expectations for GW Detection Chunglee Kim, Vassiliki Kalogera, Duncan R. Lorimer, Mia Ihm, Krzysztof Belczynski

107

Proto-Neutron Star Neutrino Emission Aimee L. Hungerford, Christopher L. Fryer, Aristotle Socrates, Omer Blaes

111

Part IV

Cooling and Atmospheres of Neutron Stars

Observations of Cooling Neutron Stars J.E. Trümper

117

Optical Studies of Isolated Neutron Stars and Their Environments Roberto P. Mignani

133

Disks Around Radio Pulsars K. Yavuz Eksi, M. Ali Alpar

137

Radiation from Condensed Surface of Magnetic Neutron Stars Matthew van Adelsberg, Dong Lai, Alexander Y. Potekhin

141

Part V

Pulsars

Observations of Radio Pulsars Nichi D’Amico

147

Radio Pulsar Statistics D. R. Lorimer

161

Emission Mechanisms and Broad Band Spectra of Pulsars K.S. Cheng

173

Modeling Spectral Features from Isolated Neutron Stars Wynn C.G. Ho, Dong Lai, Alexander Y. Potekhin, Gilles Chabrier

185

Part VI

Binaries, accretion

Formation and Evolution of Neutron Stars in Binary Systems E.P.J. van den Heuvel

191

Accretion Disks H.C. Spruit

203

High mass X-ray binaries in LMC P. Shtykovskiy, M. Gilfanov

249

vii

Contents Astrophysics with Lobster Telescopes Rene Hudec, Libor Sveda, Ladislav Pina, Adolf Inneman

253

The Stability of Magnetized Rotating Plasmas with Strong Toroidal Fields Martin E. Pessah, Dimitrios Psaltis

257

Part VII X-ray binaries Accretion Powered X-ray Pulsars Altan Baykal

263

Delayed Iron Lines in GX 1+4 Takayoshi Kohmura, Shunji Kitamoto

279

Timing Neutron Stars M. van der Klis

283

Millisecond X-Ray Pulsars and QPOs Frederick K. Lamb

311

Part VIII

Gamma Ray Bursts and Magnetars

The Electromagnetic Spectrum of AXPs GianLuca Israel, Sandro Mereghetti, Nanda Rea, Luigi Stella

329

The distribution of kHz QPO frequencies in Sco X-1 Tomaso Belloni, Mariano Mendez, Jeroen Homan

339

Part IX Science with current and future X/ gamma-ray missions The Chandra X-Ray Observatory Martin C. Weisskopf

345

The INTEGRAL mission Sergei A. Grebenev

357

Part X

CONCLUDING REMARKS

Concluding Remarks E.P.J.van den Heuvel

373

Object Index

379

Subject Index

383


Akgün, Taner ([email protected]) Cornell University, USA Aksaker, Nazım ([email protected]) Çukurova Üniversitesi, Turkey Akyüz, Aysun ([email protected]) Çukurova Üniversitesi, Turkey Alpar, M. Ali ([email protected]) Sabanci University, Turkey Altamirano, Diego ([email protected]) University of Amsterdam, The Netherlands Ankay, A¸skın ([email protected]) TÜB˙ITAK Feza Gürsey Institute, Turkey Aslan, Zeki ([email protected]) TUG, Turkey Balman, Sölen ¸ ([email protected]) METU, Turkey Baykal, Altan ([email protected]) METU, Turkey Becker, Werner ([email protected]) MPE, Germany Beklen, Elif ([email protected]) METU, Turkey Belloni, Tomaso ([email protected]) INAF - Brera Astronomical Observatory, Italy Bhattacharya, Dipankar ([email protected]) RRI, India Bogdanov, Slavko ([email protected]) Harvard University, USA Cackett, Edward ([email protected]) University of St. Andrews, UK Casella, Piergiorgio ([email protected]) Third University of Rome & Rome Astronomical Observatory, Italy Chachoua, Elie ([email protected]) Laboratoire Univers Theorique, Observatoire de Paris, France Chan, Chi-Kwan ([email protected]) University of Arizona, USA Chang, Philip ([email protected]) University of California Santa Barbara, USA Chelovekov, Ivan ([email protected]) Space Research Institute, Russion Academy of Science, Russia Cheng, Kwong Sang ([email protected]) HKU, China D’Amico, Nichi ([email protected]) INAF, Italy Den Hartog, Peter R. ([email protected]) SRON, National Institute for Space Research, The Netherlands Dönmez, Orhan ([email protected]) Nigde University, Turkey Dönmez, Burçin ([email protected]) University of New Hampshire, USA Dupuis, Rejean ([email protected]) University of Glasgow, UK Ek¸si, K. Yavuz ([email protected]) Sabanci University, Turkey Erkoca, Arif Emre ([email protected]) METU, Turkey Erkut, Mehmet Hakan ([email protected]) Sabanci University, Turkey Ertan, Ünal ([email protected]) Sabanci University, Turkey Frackowiak, Michal ([email protected]) Nicolas Copernicus Astronomical Center, PAN, Poland Fryer, Christopher L. ([email protected]) LANL, USA

ix

x Gainutdinova, Razia ([email protected]) Institute of Physics, National Academy of Sciences, Kyrgyzstan Rep. Gelfand, Joseph ([email protected]) Harvard University, USA Gil, Janusz ([email protected]) Institute of Astronomy, University of Zielona Gora, Poland Gilfanov, Marat ([email protected]) MPA, Germany Gö˘gü¸s, Ersin ([email protected]) Sabanci University, Turkey Grebenev, Sergei ([email protected]) SRI, Russia Grindlay, Jonathan E. ([email protected]) Harvard-Smithsonian CfA, USA Gürkan, Mehmet Atakan ([email protected]) Northwestern University, Department of Physics and Astronomy, USA Güver, Tolga ([email protected]) Istanbul University Department of Astronomy & Space Sciences, Turkey Hakano˘glu, Ba¸sak ([email protected]) Bogazici University, Turkey Halzen, Francis ([email protected]) University of Wisconsin, Madison, USA Ho, Wynn ([email protected]) KIPAC/Stanford, USA Hudec, Rene ([email protected]) Astronomical Institute Ondrejov, Czech Rep. Hungerford, Aimee ([email protected]) Los Alamos National Laboratory, USA Hüseyin, Oktay ([email protected]) TÜB˙ITAK Feza Gursey Institute, Turkey Hüdaverdi, Murat ([email protected]) Bogazici University, Turkey Ibragimov, Askar ([email protected]) Kazan State University, Department of Astronomy , Russia Ibrahim, Alaa ([email protected]) NASA GSFC & GWU, USA Ilfan, Bikmaev ([email protected]) Kazan State University, Department of Astronomy, Russia ˙ Inam, S. Ça˘gda¸s ([email protected]) METU, Turkey Israel, Gianluca ([email protected]) INAF, Italy Kalemci, Emrah ([email protected]) Space Sciences Laboratory / UC Berkeley, USA Kaneko, Yuki ([email protected]) University of Alabama in Huntsville, USA Karimov, Kazimir ([email protected]) Institute of Physics, National Academy of Sciences, Kyrgyzstan Rep. Kim, Chunglee ([email protected]) Northwestern University, USA Kızılo˘glu, Nilgün ([email protected]) METU, Turkey Kızılo˘glu, Ümit ([email protected]) METU, Turkey Kohmura, Takayoshi ([email protected]) Kogakuin University, Japan Kouveliotou, Chryssa ([email protected]) NSSTC, USA Küpçü Yolda¸s, Aybüke ([email protected]) Max-Planck-Institut für extraterrestrische Physik, Germany Lamb, Frederick K. ([email protected]) UIUC, USA Lommen, Dave ([email protected]) University of Amsterdam, The Netherlands Lorimer, Dunc ([email protected]) University of Manchester, UK Mangano, Vanessa ([email protected]) Astronomical observatory of Rome, Italy Marco, Bruni ([email protected]) Institute of Cosmology and Gravitation (University of Portsmouth), UK Medin, Zach ([email protected]) Cornell University, USA Migliari, Simone ([email protected]) University of Amsterdam, The Netherlands Mignani, Roberto ([email protected]) European Southern Observatory, Germany Moortgat, Joachim ([email protected]) Department of Astrophysics, University of Nijmegen, The Netherlands Özel, Nesibe ([email protected]) METU, Turkey Ögelman, Hakki B. ([email protected]) University of Wisconsin, Madison, USA


xi

Özkan, Safiye ([email protected]) Akdeniz Üniversitesi, Turkey Parolin, Ilaria ([email protected]) INAF - Brera Astronomical Observatory, Italy Passamonti, Andrea ([email protected]) Institute of Cosmology and Gravitation (University of Portsmouth), UK Patel, Sandeep K. ([email protected]) NSSTC/USRA, USA Pavlinsky, Mikhail ([email protected]) Space Research Institute, Russia Pessah, Martin ([email protected]) The University of Arizona, USA Posselt, Bettina ([email protected]) Max-Planck-Institut für extraterrestrische Physik, Germany Poutanen, Juri (juri.poutanen@oulu.fi) University of Oulu, Finland Psaltis, Dimitrios ([email protected]) University of Arizona, USA Rea, Nanda ([email protected]) University of Rome 2 / INAF-Observatory of Rome, Italy Ruderman, Malvin A. ([email protected]) Columbia University, USA Shtykovskiy, Pavel (pav [email protected]) Space Research Institute, Russia Slane, Patrick O. ([email protected]) CFA, USA Slowikowska, Agnieszka ([email protected]) Nicolaus Copernicus Astronomical Center, Poland Sonbas, Eda (eda [email protected]) Çukurova Üniversitesi, Turkey Spruit, Henk ([email protected]) MPA, Germany Sunyaev, Rashid ([email protected]) MPA, Germany Swank, Jean ([email protected]) NASA GSFC, USA Tagieva, Sevinç ([email protected]) Academy of Science - Physics Institute, Azerbaijan Republic Ta¸skın, Mustafa Özgür ([email protected]) METU, Turkey Tepedelenlioglu, Emre ([email protected]) University of Wisconsin at Madison, USA Thompson, Chris ([email protected]) University of Toronto, Canada Trümper, Joachim ([email protected]) MPE, Germany Tsygankov, Sergey ([email protected]) Space Research Institute, Russia Ulrich R.M.E., Geppert ([email protected]) Astrophysikalisches Institut Potsdam, Germany van Adelsberg, Matthew ([email protected]) Cornell University, USA van den Heuvel, Ed ([email protected]) University of Amsterdam, The Netherlands van der Klis, Michiel ([email protected]) University of Amsterdam, The Netherlands Ventura, Joseph ([email protected]) University of Crete, Greece Voss, Rasmus ([email protected]) Max-Planck Institut fur Astrophysik, Germany, Germany Vurm, Indrek ([email protected]) Tartu University, Estonia Weisskopf, Martin ([email protected]) NASA MSFC, USA Wijers, Ralph A. M. J. ([email protected]) University of Amsterdam, The Netherlands Woods, Peter ([email protected]) USRA/NSSTC, USA Yerli, Sinan Kaan ([email protected]) METU, Turkey Yilmaz, Arda ([email protected]) METU, Turkey Zavlin, Slava ([email protected]) University of Starsbourgh, France


xiii

Preface

Neutron stars hold a central place in astrophysics for two broad reasons. First, for the physics, neutron stars are interesting because they are made of the most extreme states of condensed matter in bulk quantities. This is indeed the most condensed directly observable state of matter since black holes are not directly observable in principle. The structure of a neutron star, namely its equation of state, involves many important fundamental problems of physics such as the strong interactions of nuclear physics, condensed matter physics including superconductivity and superfluidity, and exotic states of matter. The other reason attracting the astronomers’ interest is that neutron stars, along with white dwarfs and black holes, represent one of the stable configurations that stars reach at the end of stellar evolution. The population and distribution of neutron stars in the Galaxy are central topics of interest in Astronomy. Being extreme compact objects, neutron stars posses the highest rotation rates and strongest magnetic fields among all stars. They radiate prolifically, in high energy electromagnetic radiation and in the radio band. Radio pulsars make up the majority of observed neutron stars. For all these reasons, the study of neutron stars holds an important place in graduate study in astrophysics. This is an area that requires, and is ideally suited for, the learning of theoretical physics, astrophysics, techniques of observation and data analysis of optical, radio, X-ray and Gamma-ray astronomy. With advances in theoretical work and with the improved observational facilities of each successive observatories, our knowledge of neutron stars expands continuously. This volume is devoted to the selected lectures presented in the 6th NATO ASI series entitled "The Electromagnetic Spectrum of Neutron Stars" in Marmaris, Turkey, on 7-18 June 2004. Altogether about 600 participants have attended the tutorial sessions of this school since its inception in 1988. The Advanced Study Institutes in the neutron star series have all addressed the main properties, structure, dynamics and evolution of neutron stars, and each school has in addition focused on one particular aspect of neutron star research. The pace of development in the field is such that at each school, at average intervals of three years, there have been new developments to be included in the basic tutorial lectures addressed to each new generation of graduate stu-

xv

xvi dents, while new observations have multiplied with each new major space observatory, including GRANAT, ROSAT, GRO, RXTE, BeppoSAX, Chandra, XMM-Newton and most recently INTEGRAL, along with increasingly sensitive surveys of radio pulsars. The first ASI in the series had concentrated on the timing properties of neutron stars. This ASI is devoted to the spectral properties of neutron stars. Relating the radiation patterns from the surface of the neutron star to its structure (mass, radius, magnetic field), to its cooling history, its age and evolution is a key avenue in neutron star research. The continuum as well as emission and absorption lines in the spectra must be observed and understood to uncover this information. This has led to models for neutron star atmospheres with different compositions and magnetic fields. The magnetospheric emission processes of isolated radio pulsars and the emission processes of accreting neutron stars can be better understood with the help of observations of their spectra. These spectra emerge not only from the surface but also from and through the interactions of radiation with matter surrounding the neutron star. Observations by recent space observatories Chandra and XMM-Newton yield higher resolution X-ray spectra and clues of red-shifted iron lines, electron cyclotron lines and possibly ion cyclotron lines indicating magnetar fields. Thermal and power law components of the spectra still challenge atmospheric and circumstellar models. Data in hard Xrays and gamma rays are starting to come in from the INTEGRAL mission. This ASI included spectral information from the neutron stars in the broadest sense, namely neutrinos and gravitational radiation along with the electromagnetic spectrum. The formation of a neutron star in a supernova explosion gives most of its energy output in the form of neutrinos, which were detected from the supernova 1987A in the Large Magellanic Cloud. The existing and developing neutrino detectors are capable of detecting neutrinos from a nearby supernova. As for gravitational radiation, neutron star binaries are known to be important sources: observations of the famous binary pulsar PSR 1913+165 constituted the best test of general relativity. These observations yielded indirect evidence for the emission of gravitational waves. Detectors such as LIGO will be available to detect the gravitational waves, for which neutron star binaries are believed to be among the promising sources, particularly in the last stages of evolution leading to a merger that also yields the gamma ray bursts. From the tutorial nature of lectures, we believe that this volume can serve as graduate level of text including the broad range of properties of neutron stars. We are grateful to the NATO Scientific Affairs Division for its funding of this Advanced Study Institute. We thank M.A. Alpar and M. Gilfanov for helping to prepare NATO ASI proposal in the early stages of this meeting. The Scientific Organizing Committee consisted of A. Baykal (codirector), S. A. Grebenev (codirector), A.M. Alpar, N. d’Amico, M. Gilfanov, Ü. Kızılo˘glu, C. Kouveliotou, H. Ögelman, R. Sunyaev, J. Trümper, E. van den Heuvel, M.

PREFACE

xvii

van der Klis, and J. Ventura. The Local Organizing Committee consisted of S. ¸ Balman, S.K. Yerli, E. Gö˘gü¸s, Ü. Ertan, and S.Ç. ˙Inam. S.K. Yerli and S.Ç. ˙Inam deserve special thanks for their assistance in all phases of the organization as well as in editing of the proceeding. We thank Sölen ¸ Balman for her efforts during the talks and local organisations. In addition to this we thank to our students Arif Erkoca, Elif Beklen, Arda Yılmaz and Baybars Külebi for their efforts during the meeting. ALTAN BAYKAL

I

FORMATION OF NEUTRON STARS

NEUTRON STAR FORMATION Constraints from/on Neutron Star Populations Chris L. Fryer Theoretical Astrophysics, T-6 Los Alamos National Laboratory, Los Alamos, NM 87545 and Physics Department University of Arizona, Tucson, AZ 85721 [email protected]

Aimee Hungerford Transport Methods, CCS-4 Los Alamos National Laboratory, Los Alamos, NM 87545 and Astronomy Department University of Arizona, Tucson, AZ 85721 [email protected]

Abstract The nature of a neutron star differs based on whether the neutron star formed from the collapse of a white dwarf, the collapse of a ∼ 8 − 11M star, or the collapse of a more massive star (∼ 11 − 20M ). Although the details of the supernova mechanism are not understood completely, the intuition we have gained from modeling stellar collapse does allow us to understand many of the qualitative features of stellar collapse. Here we review the basic neutron star formation scenarios and apply our understanding of these formation processes to constrain neutron star kick mechanisms and their relation to supernova asymmetries as well as the relation between stellar rotation and neutron star magnetic fields. We conclude with a discussion of the constraints placed by observed populations of pulsars and magnetars.

Keywords:

1.

stars: supernovae – stars: neutron – stars: formation

Neutron Star Formation

A number of reviews of both stellar evolution (Woosley et al. 2002; Young & Arnett 2004) and the supernova mechanism (Bethe 1990; Burrows 2000; Mezzacappa & Bruenn 2000; Janka 2001; Burrows & Thompson 2003; Janka 3 A. Baykal et al. (eds.), The Electromagnetic Spectrum of Neutron Stars, 3–14. © 2005 Springer. Printed in the Netherlands.

4


Figure 1. Left: Diagram of the convection-enhanced, neutrino-driven supernova mechanism. Neutrinos from the cooling proto-neutron star heat the base of a convectively unstable regions. The heated material rises, converting its neutrino-heated thermal energy to kinetic energy which pushes against the ram-pressure of the infalling star. If it can overcome this ram-pressure, a supernova is born. The accretion rate decreases most dramatically at composition boundaries, and it is likely that an explosion will be launched when such a boundary hits the edge of the convective cell. Right: Slice from a 3-dimensional simulation showing a single snapshot in time of this convective process. The grayscale denotes entropy, and the vectors denote the velocity magnitude (length) and direction (vector).

et al. 2004, Fryer 2003) exist. Some of these reviews are chapters in a book on stellar collapse (Fryer 2004) which includes detailed chapters on much of the important physics and numerics involved in stellar collapse. In this proceedings, we will build upon the intuition already known on stellar collapse which coincides most with the review by Fryer (2003) and we direct the reader to that review for more details on the explosion mechanism. For the purposes of distinguishing between neutron star formation scenarios, one need keep the following basic picture in mind that will hold for all neutron star forming scenarios. The inner core of each progenitor will collapse down to nuclear densities and then bounce. In all 3 scenarios, the bounce shock stalls before driving an explosion. If neutrinos are able to deposit enough energy to blow off the infalling material, an explosion is launched (Fig. 1). The ram pressure of the infalling material forms a “lid” on top of this neutrino driven pressure cooker. The pressure in this lid when an explosion occurs determines the energy in the explosion. For the collapse of a white dwarf, whether or not an explosion occurs, the collapsed object will ultimately cool to form a neutron

5


10 10

1

1 0.1

0.01 0.1

1.5

2

2.5

3

0.5

1

1.5

2

Time (s)

Figure 2. Left: Accretion rate as a function of enclosed infalling mass for 11 (solid), 15 (dotted), 25 (dashed), 40 (dot-dashed) M stars. The thick solid line corresponds to an accretioninduced-collapse white dwarf. Right: Accretion rate versus time for these same models. The more massive progenitors have higher accretion rates (and hence higher ram pressures) and hence are harder to explode. Both the 11 and 15 M stars have a sharp decrease in the accretion rate (dropping below 1 M s−1 ) in a few hundred milliseconds. In the accretion-induced collapse model, this decrease is even more dramatic.

star. But for the other two scenarios, an explosion must occur to prevent the accretion of the entire star and the formation of a black hole. The structure of the inner ∼ 1M for all of these stars is very similar and so one might expect that the core bounce for all of these stars shouldn’t be too different. But they do differ in their structure beyond 1 M . It is this structure that determines the accretion rate, and hence the ram pressure on top of the star, as a function of time. Figure 2 shows the accretion rates as a function of enclosed mass and of time for a series of stars from the accretion induced collapse of a white dwarf (model from Fryer et al. 1999) to a series of

6


massive stars range from 11 to 40 M (see Rauscher et al. 2003). The larger accretion rates for more massive stars means that it is more difficult to make such stars explode. Fryer (1999) found that even when the 25 and 40 M stars did manage to explode, the resulting shock was weak and a lot of fallback occured, ultimately forcing the proto-neutron star to collapse to form a black hole. Just using this mass accretion as a guide (more physical effects can play a role - see Janka 2001), we can differentiate 3 different formation paths of neutron stars: Accretion Induced Collapse: The collapse of a white dwarf that exceeds the Chandrasekhar limit by accreting in a binary system. Because of the lack of a massive envelope, these stars will explode fairly rapidly. The total fraction of neutron stars formed in this manner are at least 1001000 times lower than the other two formation scenarios (Fryer et al. 1999), but they may still play an important role in forming neutron stars in globular clusters. Collapse of stars between ∼8-∼11 M : These stars are likely to have very diffuse envelopes, not too different than the accretion induced collapse case. They will explode quickly with very little convection. Depending upon the initial mass function of stars, this scenario will make 49-52% of all neutron stars. The uncertainty in the actual limits of these stars will also make a huge difference in the relative fraction of these stars. Collapse of stars between ∼11-20 M : Stars above ∼11 M have much more dense layers beyond the iron core and will take longer to explode. They will convect much longer. The rate for these objects is roughly 48-51%, but bear in mind that the total fraction depends sensitively on the lower mass limit. With this basic picture of neutron star formation in mind, we can now gain some intuition on the effects of rotation, magnetic fields and asymmetries.

2. 2.1

Modifications to the Basic Picture The interplay between Magnetic Fields and Rotation

It is generally believed that the magnetic field strength of the star prior to collapse has little bearing on the magnetic field strength of the resulting neutron star because an efficient dynamo can easily drive any arbitrary seed magnetic field up close to the saturation limit (e.g. Thompson & Duncan 1993; Akiyama et al. 2003). It is the star’s rotation speed (or the magnitude of its differential

Modifications to the Basic Picture

7

Figure 3. Left: Angular velocity vs. radius (top) and mass (bottom) for three stellar progenitors: SN15A, SN15B (models E15A and E15B from Heger et al. 2000), and SN15C (15 M model from A. Heger et al. 2004, in preparation). The angular velocity remains relatively constant in burning shells because of the convection that efficiently transports angular momentum. However, at the boundaries of these layers, the spins can decouple, causing jumps in the angular velocity. These jumps persist, although with much smaller magnitudes, in the progenitor (SN15C), which includes magnetic fields (which can transport angular momentum across these boundaries). Right: Angular momentum vs. mass zone as a function of time. The solid line shows the angular momentum profile 90 ms before bounce, the dotted line 40 ms after bounce, and the dashed line 140 ms after bounce. Note that in the protoneutron star interior, the star quickly loses 80% of its total angular momentum. This angular momentum is transported to the surface of the protoneutron star (note the rise in angular momentum beyond 1 M 140 ms after bounce). See Fryer & Warren (2004) for more details.

rotation), then, that is the major determiner of the magnetic field strength in the neutron star. The left side of figure 3 shows the angular velocity in the cores of 3 different stars. Two of these stars were formed with different initial spin rates. The cores have angular velocity that take on step-like profiles. This is because convection drives the spin rate to equilibrate within a burning shell. Between burning shells, friction tries to equilibrate the spins, but is less successful, allowing discrete steps at the shell (composition) boundaries. The third star (SN15C) began with a spin rate roughly equal to the fastest spinning star. But for this star, Heger et al. (2004 - in preparation) included a recipe for magnetic field

8


generation. The magnetic field adds an additional friction term between shell boundaries, leading to more equilibration, and ultimately a slower spinning core. The larger the magnetic field in the progenitor of the core, the slower the core’s spin will be. But recall that, for the current dynamo mechanisms (e.g. Duncan & Thompson 1993), it is the spin of the core, not the initial magnetic field, that ultimately determines the strength of the magnetic field in the neutron star. Thus we are led to the seemingly contradictory conclusion that the highest magnetic field stars (at least the stars with the largest magnetic fields in their interiors) produce the lowest magnetic-field neutron stars. We now know about the effect of rotation on magnetic fields in the pulsar itself, but what about the effect of magnetic fields on rotation of the progenitor star. As the star evolves, this “equilibration” of angular velocity leads to a decrease in the angular momentum of the core. Even the fastest rotating cores shown in figure 3 have lost roughly 99% of their angular momenta during the course of the star’s evolution to collapse. Magnetic fields can cause this angular momentum to drop an additional ∼ 2 orders of magnitude (Fig. 3). A common mistake by the community is to take the rotation periods from the pre-collapse progenitors from Figure 3 and assume that the angular momentum is conserved through collapse and supernova explosion. If this were true, even the slowest star in figure 3 would produce a neutron star spinning faster than 10 ms. But it isn’t true. Figure 3 (right) shows the angular momentum of the core of the collapse before and after bounce. As the core compresses, the high angular momentum material is left in a disk above the neutron star. At the launch of the explosion, the inner core has lost 80% of its angular momentum and the high angular momentum material remains in a thermally+centrifugally supported disk (Fig. 4). It is likely that this disk will lose most of its angular momentum through a wind prior to accreting onto the neutron star. At this time, even the fastest spinning cores have periods just below 100 ms. When the neutron star cools, it will spin up. The fastest cores will contract to pulsars with spin-periods of a few ms (Fryer & Warren 2004). The slower star from figure 3 will end up as a >20 ms pulsar. During this contraction, potential energy is converted into rotational energy (a 1 ms pulsar has over 1051 ergs of energy!). But what happens if a strong magnetic field develops in the hot protoneutron star. This magnetic field will couple the proto-neutron star to the wind, fallback, and ejecta of the supernova. If it is strong enough, it can slow the proto-neutron star even before energy conversion can occur. As far as we know, this process has not been calculated in detail (the details of the supernova explosion must be first understood). But it is likely that any reasonably high magnetic field proto-neutron star will become a slow (> 100 ms) pulsar.

Modifications to the Basic Picture

9

Figure 4. Radial velocity distribution of a rotating 15 star 1.6 s after bounce (Fryer & Heger 2000). Note the strong jet being driven in the polar region. The velocity is a factor of 2 times stronger along the poles than in the equator. Note the disk that has formed around the protoneutron star. This is the bulk of the high-angular momentum material in the collapsing core.

The only way to obtain a fast spinning compact neutron star is to wait until after the proto-neutron star has become compact to produce the magnetic field (e.g. through the merger of a low-mass helium star - Chris Thompson, private communication).

2.2

Asymmetries and Kicks

Observations of a range of neutron star systems from X-ray binaries to pulsar proper motion studies suggest that neutron stars receive “kicks” leading to velocities, in some cases, in excess of 1000 km s−1 (see, for example, Fryer & Kalogera 1997; Lai et al. 2001). Although the consensus is that these kicks

10


Figure 5. Left: Slice of the exploding core of an asymmetric collapse (see Fryer 2004 for details). Shading denotes entropy, and vectors give velocity direction (vector length denotes velocity magnitude). The filled circle denotes the core or protoneutron star (defined by that material with densities above 1013 g cm−3 ). The crossed lines show the x- and y-axes. The core of the star was initially centered at x = y = z = 0. Note that the core has already moved 20 km but that the downflows are strongest on the leading edge of the neutron star. Right: Isosurface of material moving outward with a radial velocity of 3000 km s−1 (Fryer & Warren 2004). In this rotating model, the convective bubbles ultimately developed into to outflows along the rotation axis. The outward motion in the positive z axis was much stronger. It is this sort of convection that Scheck et al. (2004) have argued will produce neutron star kicks.

are imparted during the formation of the neutron star, the exact mechanism behind neutron star kicks remains unknown. At the same time, the evidence for asymmetries in the supernova explosion has also grown (see Hoflich 1991; Hungerford et al. 2003 for reviews). From figure 4, we see that rotation can produce large asymmetries in the supernova explosion. In this case, the convection that drives the explosion was inhibited by the angular momentum gradient, leading to a bipolar explosion. In most cases, the bimodal explosion is too symmetric to impart a significant kick onto the nascent neutron star. But there are a number of instabilities in the supernova explosion that will produce kicks. We review these here. A few mechanisms have been proposed that use magnetic fields to alter the effective opacity of neutrinos, leading to asymmetries. These proposals range from asymmetric opacities (Arras & Lai 1999) to resonant transitions

Observational Constraints

11

to sterile neutrinos (Fuller et al. 2003) to magnetic instabilities on neutron starspots (Socrates et al. 2004 - in preparation; see also Hungerford et al. in this proceedings). Although all of these mechanisms produce larger kicks with larger magnetic fields, it is difficult to compare these necessarily to observations. Some invoke magnetic starspots which may not have any bearing on the observed dipole field. In any event, it is not clear that the magnetic field in the proto-neutron star has anything to do with the observed magnetic field 1,000-1 million years later. The other main class of kick mechanisms is driven by asymmetries in the convective engine. One such class invokes perturbations in the stellar core just prior to collapse. Although preliminary 2-dimensional results (Burrows & Hayes 1996) suggested that these asymmetries would not only produce kicks, but also large explosion asymmetries in the supernova explosion, Fryer (2004) found not only that the explosion asymmetries were much more mild than Burrows & Hayes (1996) predicted, but that kicks would only occur in quick explosions such as those expected from stars in the ∼ 8 − 11 M range (Fig. 5 left panel). The kick mechanism that has been attracting a lot of attention in the last year simply argues that the convective cells in the convection engine of figure 1 will merge, ultimately driving one upflowing bubble and one downflow. Based on the low-mode convection seen in simulations from Herant et al. (1994), Herant (1995) argued that if this convection could develop into single modes, large kicks could be imparted onto the neutron star (see Fig. 5 - right panel). Now, most of the major supernova groups have argued strongly that their simulations show single mode convection (Blondin et al. 2003; Scheck et al. 2004). Although their remains some debate as to the exact mechanism driving the convection to these low modes, this mechanism appears to be a moderately successful mechanism for stars above 11 M .

3.


The deviations away from the spherical picture of supernova explosions make predictions for a number of observations. Neutron star observations, in turn, can be used to place constraints on the supernova mechanism. We discuss both the predictions and constraints here.

3.1

Ejecta Remnants

Both the rotationally-induced bimodal explosions and the single-mode convective explosions can produce asymmetries in the supernova ejecta. These asymmetries have profound effects on the mixing that occurs in the supernova explosion, and ultimately can lead to observational features in the gamma-ray signature (Hungerford et al. 2003, 2004), iron lines, polarization, and the ap-

12


Figure 6. Left: Nickel distribution (isosurface) and density distribution (shading) for a bipolar explosion such as might be expected from a rotating progenitor (Hungerford et al. 2003). Right: Nickel distribution (isosurface) and density distribution (shading) for a single-lobe explosion such as might be expected from convection (Hungerford et al. 2004). This Nickel distribution will have decided signatures in the gamma-rays, polarization, and remnants of these supernovae.

pearance of the supernova remnant. Figure 6 shows the nickel distribution for both a bimodal and single-lobe explosion. The sorts of asymmetries in the element abundances are likely to fit such remnants like Cas A. Comparing detailed simulations with remnant observations such as the recent megasecond observation of Cas A (Hwang et al. 2004), we can derive constraints on the explosion asymmetries.

3.2

Pulsars

As this chapter focuses on the formation of neutron stars, let’s focus our discussion on the constraints neutron-star observations can place on the supernova calculations. In particular, the pulsar velocity distribution can help to home in on the correct neutron star kick mechanism and this, in turn, will tell us much about the cause of supernova explosions. It is now generally accepted that the pulsar velocity distribution has two components: a fast component with a mean velocity of roughly 500−600 km s−1 , and a slow component below ∼ 200 km s−1 (Fryer et al. 1998; Arzoumanian et al. 2002). Both components make up roughly 50% each of the total pulsar


13

population. Any kick mechanism must be able to explain both populations. Recall that the main two supernova progenitor groups (stars between ∼ 8 − 11 and ∼ 11 − 20 M ) each make up roughly 50% of all neutron stars. One natural way of explaining the two component pulsar velocity distribution is to have a mechanism that works in one progenitor population (to explain the high-velocity peak), but not in the other (allowing small asymmetries to explain the low-velocity peak). Both the asymmetric collapse and single-mode convection mechanisms have such a feature. But a better understanding of the pulsar velocity distribution is required to determine whether these two stellar populations provide the correct dividing line for the two-component pulsar velocity distribution. Many of the kick mechanisms predict trends, increasing either with increasing magnetic field strength or possibly even pulsar rotation. Although no such correlation has yet been found, the magnetic field mechanisms can not yet be ruled out. Better information on these trends could well be the easiest constraint on the current set of proposed kick mechanisms.

3.3

Magnetars

One of the biggest constraints on the supernova explosion is the actual rate of magnetars. Are magnetars, as C. Thompson (private communication) believes, a rare occurrence in nature (∼ 1% of all neutron stars). If so, we can explain them in some binary interaction (e.g. the merger of a low-mass helium star with a neutron star suggested by Thompson) and the magnetar population will not have much bearing on our understanding of the supernova mechanism. Or do magnetars make up a sizable fraction of the neutron star population (20-50%) as suggested at this meeting? If this is the case, our supernova mechanism must be able to produce these objects, and that means that a sizable fraction of progenitors must be spinning rapidly. (Recall that the spin of a star is the dominant factor in making high magnetic fields). This would suggest that the latest stellar models with magnetic field viscosities are wrong (at least in the stars that make magnetars), so determining the magnetar fraction not only tells us about the supernova mechanism, but also stellar evolution. We are reaching the point in supernova theory that we have pushed the limits on what theory can teach us about neutron star populations. But we are now in the era where we have developed enough intuition that we can use what we have observed about neutron star populations to constrain both supernova and stellar theory. By taking advantage of these constraints, the next decade shows exciting promise.

14


Acknowledgments This work was funded under the auspices of the U.S. Dept. of Energy and the University of Arizona, and supported by its contract W-7405-ENG-36 to Los Alamos National Laboratory, by a DOE SciDAC grant number DE-FC0201ER41176 and by NASA Grant SWIF03-0047-0037. The simulations were conducted on the Space Simulator at Los Alamos National Laboratory.

References Akiyama, S., Wheeler, J. C., Meier, D. L., Lichtenstadt, I. (2003), ApJ, 584, 954. Arras, P., & Lai, D. (1999) ApJ, 519, 745. Arzoumanian, Z., Chernoff, D. F., & Cordes, J. M. (2002) ApJ, 568, 289. Bethe, H. A. (1990), RvMP, 62, 801. Blondin, J. M., Mezzacappa, a., DeMarino, C. (2003), ApJ, 584, 971. Burrows, A. S. & Hayes, J. (1996) PRL, 76, 352. Burrows, A. S. (2000) Nature, 403, 727. Burrows, A. S. and Thompson, T. A. From Twilight to Highlight: The Physics of Supernovae. Proceedings of the ESO/MPA/MPE Workshop held in Garching, Germany., 53, (2003). Fryer, C. L. & Kalogera, V. (1999), ApJ, 489, 244 Fryer, C. L., Burrows, A., Benz, W. (1998), ApJ, 496, 333 Fryer, C. L. (1999), ApJ, 522, 413 Fryer, C. L., Benz, W., Herant, M., Colgate, S. A. (1999) ApJ, 516, 892 Fryer, C. L., & Heger, A. (2000) ApJ, 541, 1033 Fryer, C. L. (2003) IJMPD, 12, 1795. Fryer, C. L. (2004) Stellar Collapse, ed. C. L. Fryer, Kluwer publishers, 5 (2004). Fryer, C. L. (2004) ApJ, 601, L175 Fryer, C. L., & Warren, M. S. (2004) ApJ, 601, 391 Fuller, G. M., Kusenko, A., Mocioiu, I., & Pascoli, S. (2003) Phys. Rev. D, 68, 103002 Heger, A., Langer, N., & Woosley, S. E. (2000) ApJ, 528, 368 Herant, M., Benz, W., Hix, W. R., Fryer, C. L., & Colgate, S. A. (1994), ApJ, 435, 339. Herant, M. (1995), Phys. Rep., 256, 117. Hoflich, P. (1991), A&A, 246, 481. Hungerford, A. L., Fryer, C. L., & Warren, M. S. (2003) ApJ, 594, 390 Hungerford, A. L., Fryer, C. L., & Rockefeller, G. (2004) in preparation Hwang, U. et al. (2004) submitted to ApJL. Janka, H.-T. (2001) A&A, 368, 527. Janka, H.-T., Buras, R., Kifonidis, K., Rampp, M. & Plewa, T. (2004) in Stellar Collapse, ed. C. L. Fryer, Kluwer publishers, 69. Lai, D., Chernoff, D. F., & Cordes, J. M. (2001), ApJ, 549, 1111 Mezzacappa, A. and Bruenn, S. W. (2000) Mem. Soc. Astron. Ital., 71, 515. Rauscher, T., Heger, A., Hoffman, R. D., & Woosley, S. E. (2003) Nuc. Phys. A, 718, 463. Scheck, L., Plewa, T., Janka, H.-Th., Kifonidis, K., & Muller, E. (2004), PRL, 92, 011103. Socrates, A. et al. in preparation for ApJ Thompson, C., & Duncan, R. C. (1993) ApJ, 408, 194 Woosley, S. E., Heger, A. & Weaver, T. A. (2002), RvMP, 74, 1015. Young, P. A. and Arnett, W. D. Stellar Collapse, ed. C. L. Fryer, Kluwer publishers, 5 (2004).

YOUNG NEUTRON STARS AND THEIR WIND NEBULAE Patrick Slane Harvard-Smithsonian Center for Astrophysics [email protected]

Abstract

With Tera gauss magnetic fields, surface gravity sufficiently strong to significantly modify light paths, central densities higher than that of a standard nucleus, and rotation periods of only hundredths of a second, young neutron stars are sites of some of the most extreme physical conditions known in the Universe. They generate magnetic winds with particles that are accelerated to energies in excess of a TeV. These winds form synchrotron-emitting bubbles as the particle stream is eventually decelerated to match the general expansion caused by the explosion that formed the neutron stars. The structure of these pulsar wind nebulae allow us to infer properties of the winds and the pulsating neutron stars themselves. The surfaces of the the stars radiate energy from the rapidly cooling interiors where the physical structure is basically unknown because of our imprecise knowledge of the strong interaction at ultrahigh densities. Here I present a summary of recent measurements that allow us to infer the birth properties of neutron stars and to probe the nature of their winds, the physics of their atmospheres, and the structure of their interiors.

Keywords:

stars: neutron – stars: pulsar – stars: circumstellar matter

1.

Introduction

Young neutron stars (NSs) probe some of the most extreme physical environments in the Universe. Their rapid rotations and large magnetic fields combine to accelerate particles to extremely high energies, producing energetic winds that result in the slow spin-down of the stars and generate nebulae of synchrotron-emitting particles spiraling in a wound-up magnetic field. The structure of these nebulae is determined by the energy input from the central pulsars as well as the structure and content of the medium into which they expand. In the centermost regions, relativistic outflows in the form of rings and jets are formed; the geometry of these emission regions reveals the orientation of the pulsar spin axes and can provide information on the formation of kicks imparted in the moments following their formation. Their large-scale struc15 A. Baykal et al. (eds.), The Electromagnetic Spectrum of Neutron Stars, 15–30. © 2005 Springer. Printed in the Netherlands.

16

Young Neutron Stars and Their Wind Nebulae

tures reveal details of the magnetic field and signatures of interaction with the ejecta from the explosions that gave them birth. The stellar interiors are characterized by conditions and physical processes otherwise observed only within the nuclei of atoms. They are born hot, but cool rapidly due to neutrino production in their interiors. However, details of the interior structure of such stars remain poorly understood owing to our incomplete understanding of the strong interaction at ultrahigh densities and, since the neutrino production rate is critically dependent on the structure of the interior, the cooling rate is highly uncertain. In the standard cooling scenario, neutrino production proceeds primarily via the modified Urca process. Residual heat diffuses from the core to the surface, manifesting itself as blackbody-like emission – modified by effects of any residual atmosphere – which peaks in the soft X-ray band. The rate at which the surface temperature declines depends critically upon the neutrino emission rate; thus, its measurement provides constraints on hadronic physics at high densities. Perhaps the most stunning thing about NSs is the fact that we can actually make measurements that, directly or indirectly, probe the above properties. Particularly with the advent of sensitive high-resolution X-ray observations, we can now image jets and outflows from the wind termination shocks, identify magnetic filaments in the nebular interiors, detect the thermal emission from shock-heated ejecta, and measure directly the pulsations from the rotating stars and the emission from their ultra-hot surfaces. Here I describe the basic properties of young NSs and their nebulae, and summarize recent observational work that has begun a revolution in our understanding of how these stars work. Brevity precludes a thorough review, and the reader is referred to recent articles by Kaspi et al., 2004 and Yakovlev and Pethick, 2004 for additional information and references.

2.

Pulsar Wind Nebulae

Our basic understanding of PWNe stems from the picture presented by Rees and Gunn, 1974, and expanded upon by Kennel and Coroniti, 1984, in which an energetic axisymmetric wind is injected from a pulsar into its surroundings. As illustrated schematically in Figure 1, the structure of a PWN is regulated by the input power from the pulsar and the density of the medium into which the nebula expands; the pulsar wind and wound-up toroidal magnetic field inflates a bubble which is confined in the outer regions by the expanding shell of ejecta or interstellar material swept up by the SNR blast wave. The boundary condition established by the expansion at the nebula radius rN results in the formation of a wind termination shock at which the highly relativistic pulsar wind is decelerated to merge with the particle flow in the nebula. The shock forms at the radius rw at which the ram pressure of the wind is balanced by the

17

Pulsar Wind Nebulae

Figure 1. Schematic view of a pulsar and its wind nebula. See the text for a complete description. (Note the logarithmic size scaling in the PWN figure when comparing with images shown elsewhere in the text.)

internal pressure of the PWN: 2 ˙ rw = E/(4πηcp),

(1)

where E˙ is the rate at which the pulsar injects energy into the wind, η is the fraction of a spherical surface covered by the wind, and p is the total pressure outside the shock. Ultimately, the pressure in the nebula is believed to reach the equipartition value; a reasonable pressure estimate can be obtained by integrating the radio spectrum of the nebula, using standard synchrotron emission expressions, and assuming equipartition between particles and the magnetic field. Typical values yield termination shock radii of order 0.1 pc, which yields an angular size of several arcsec at distances of a few kpc. As the relativistic fluid comprising the PWN encounters the freely-expanding ejecta, Rayleigh-Taylor instabilities result in the formation of a network of dense, optical line-emitting filaments (Jun, 1998). The density and magnetic field strength becomes enhanced in regions where the PWN encounters these filaments, producing enhanced synchrotron emission observed as radio filaments. Due to the pinching effect of the global toroidal magnetic field, the overall morphology of a young PWN is often elongated along the pulsar spin axis (Begelman & Li, 1992; van der Swaluw et al., 2004). Along the rotation axis the flow becomes collimated, producing jets. Pinch instabilities may disrupt the toroidal structure, however, changing the structure of the magnetic field in the outer nebula regions and relaxing the collimation of the jets far from the pulsar (Begelman, 1998). The overall geometry of the PWN, as well as that of the emission from jets or ring-like structures near the termination shock, thus provides a direct indication of the pulsar geometry. The details of the jet morphology and the emission structure in the postshock region provide the strongest constraints available on

18


Figure 2. Chandra images of the Crab Nebula (left), G54.1+0.3 (center), and PSR B1509−58 (right) showing the complex emission from these PWNe, including jet outflows and toroidal structures.

wind composition and particle acceleration in PWNe. For cases in which the pulsar proper motion is also known, constraints on the kick velocity mechanism can be derived based on the degree of alignment between the velocity vector and the pulsar spin axis. In later stages the PWN interacts with the reverse shock formed in the SNR in which the NS was born. This interaction causes the disruption of the PWN, often leading to composite SNRs with complicated PWN structures in their interiors.

2.1

Jets and Tori

In the inner portions of the Crab Nebula, optical wisps mark the position of the wind termination shock, at a distance of ∼ 0.1 pc from the pulsar. The brightness and position of these wisps varies in time, with inferred outflow speeds up to 0.7c (Hester, 1998). As shown in Figure 2 (left), high resolution X-ray images reveal a ring of emission at the position of the wisps (Weisskopf et al., 2000), providing a direct connection between the unshocked pulsar wind and the bulk properties of the nebula. Material from the inner ring forms a series of toroidal X-ray wisps that are variable with time (Hester et al., 2002). The geometry of these X-ray features imply a tilted torus, and a jet of material flows perpendicular to the plane of the toroid, extending some 0.25 pc from the pulsar. A faint counterjet is also observed, along with significantly enhanced X-ray emission from the leading portion of the toroid, presumably the result of Doppler beaming. One troubling aspect of this suggestion is that the brightness distribution around the inner ring does not match that of the outer toroid; indeed, the brightness is rather uniform except for some small clump-like structures that vary in position and brightness with time. A handful of other PWNe display X-ray features that suggest the presence of extended ring-like structures and narrow collimated components. The size of the ring-like features places the emission region near the pulsar wind ter-

Pulsar Wind Nebulae

19

mination shock. The spectral and temporal properties of the collimated structures argue that they are focused jets of high speed material, as observed in the Crab. Such observations have already begun to inspire new axisymmetric MHD models that predict similar features (e.g., Komissarov & Lyubarsky 2004), and ongoing observational studies promise to further constrain and refine such models. In particular, the confining mechanism for jets is not wellunderstood; many jets display some amount of curvature, with the Vela pulsar jet being an extreme example in which the morphology is observed to change on timescales of months (Pavlov et al., 2003). This may be the result of pinch instabilities disrupting the toroidal structure of the confining magnetic field (Begelman 1998), or could be indicative of an interaction of the jet material with the ambient medium. There also appears to be a wide variation in the fraction of spin-down energy channeled into the jets, ranging from roughly 2.5 × 10−5 for PSR J0205+6449 in 3C 58 to nearly 10−3 for PSR B1509−58. And, while Doppler beaming is invoked to explain the large brightness variations in jets and the associated counterjets, as well as around the observed toroidal structures, it is not clear that this alone is sufficient to explain the observations. Chandra observations of G54.1+0.3 (Lu et al., 2002) reveal a central 136 ms pulsar (Camilo et al., 2002a) embedded in a diffuse 1. 5 × 1. 2 nebula (Figure 2, center). The pulsar is surrounded by an X-ray ring for which the X-ray emission is brightest along the eastern limb. When interpreted as the result of Doppler boosting, this implies a post-shock velocity of ∼ 0.6c (Lu et al. 2002; Romani & Ng 2003). Faint bipolar elongations running roughly east-west, perpendicular to the long axis of the ring, are also observed. These apparent outflows, which presumably lie along the pulsar rotation axis, are more diffuse than the jets in the Crab Nebula, yet appear to carry away a considerably larger fraction of the energy; they comprise roughly the same luminosity as the central ring, which is in stark contrast to the Crab where the torus outshines the jets by a large factor. Chandra observations of PSR B1509−58 (Gaensler et al., 2002) demonstrate that this young and energetic pulsar associated with G320.4–1.2 powers an extended and extremely complicated PWN, with structures on scales from ∼ 10 down to the spatial resolution limit (Figure 2, right). The elongated PWN has a clear axis of symmetry centered on the pulsar, presumably representing the projected orientation of the pulsar spin axis. To the southeast of the pulsar, the nebula is dominated by a narrow jet-like feature approximately 6 pc in length. The lack of a similar feature to the north can be explained by Dop< 30◦ pler boosting if the pulsar’s spin axis is inclined to the line-of-sight by ∼ (Gaensler et al., 2002). In the central regions of the PWN, a pair of semicircular arcs lie ∼ 0.5 and ∼ 1 pc to the north of the pulsar. Gaensler et al. (2002) note that if the inner region of these arcs represents the position of the

20


pulsar wind termination shock, then the flow time to the arcs is much shorter than the synchrotron lifetime of the emitting particles based on equipartition estimates of the magnetic field. Thus, unlike for the Crab torus, where these timescales are similar, the emission from the arcs is not the result of large synchrotron cooling at this position. Instead, the arcs appear to resemble the series of concentric wisps seen for the Crab which are interpreted as sites of electron compression in an ion-dominated flow (Gallant and Arons, 1994, Gaensler et al., 2002). The innermost region of 3C 58 (see Figure 3) consists of a bright, elongated compact structure centered on the pulsar J0205+6449. This inner nebulosity is bounded along the western edge by a radio wisp (Frail and Moffett, 1993), and is suggestive of a toroidal structure that is tilted about a north-south axis, with the pulsar at its center. The eastern side of the toroid is slightly brighter than the western side, suggesting that the eastern side is beamed toward us. If interpreted as a circular termination shock zone, the inferred inclination angle in the plane of the sky is roughly 70 degrees (Slane et al., 2002). The elongated structure extending westward from the position of the pulsar has the appearance of a jet (Figure 3, right). Its orientation is consistent, in projection, with the pulsar rotation axis inferred from the wind termination shock region discussed above, and also the east-west elongation of the entire PWN (Figure 3, left). The structure shows considerable curvature, similar to that seen in the Crab Pulsar jet. A faint structure that may be a counterjet is observed to the east of the pulsar. The observed luminosity is nearly a factor of 10 smaller than that for the torus. For the Crab Nebula, the torus is nearly 20 times more luminous than the jet in X-rays, while for PSR B1509–58 the jet is brighter than the extended inner emission (Gaensler et al. 2002). The jet/torus morphology observed in these PWNe provides the geometry of the pulsar system, yielding both the projected direction of the spin axis and the inclination angle. Modeling of such emission in other PWNe holds promise for

Figure 3. Left: Chandra image of 3C 58. Complex filamentary loops fill the interior region. Right: The innermost region of 3C 58 showing the NS embedded in an elongated structure. A curved jet extends to the west, with a hint of a counterjet component in the east.

Pulsar Wind Nebulae

21

understanding the kicks that give pulsars their large space velocities (Ng and Romani, 2004). The jets observed in the Crab and Vela pulsars, for example, are aligned with their proper motion vectors (Aschenbach & Brinkman 1975; Helfand et al. 2001). If the kick that gave these pulsars their proper motion was generated in the supernova explosion by some asymmetric mass ejection, then this alignment requires an initial pulsar spin period that is short relative to the kick timescale, so that the impulse of the kick is averaged over many rotations of the star (Lai et al., 2001). Romani and Ng, 2003 reach similar conclusions for PSR J0538+2817 in the supernova remnant (SNR) S147. By modeling the faint extended PWN emission as a jet and torus, they derive a spin axis direction that is aligned with the vector from the SNR center to the current pulsar position. For some pulsars [e.g. J0205+6449 in 3C 58 (Murray et al., 2002) and J1811–1925 in G11.2–0.3 (Kaspi et al., 2001)], we believe that the initial spin period was much longer than typical pulsar kick timescales. This would suggest that their proper motions should not necessarily be aligned with the jet direction. Future radio timing observations of these pulsars will ultimately lead to such proper motion measurements.

2.2

Filaments in PWNe

Extensive filamentary structure is observed in Hα, [OIII], and other optical line images of the Crab Nebula. Based on their observed velocities, these filaments form an expanding shell of ejecta that surrounds the nonthermal optical emission from the nebula. High resolution images with HST reveal detailed morphology and ionization structure suggesting that the filaments form from Rayleigh-Taylor instabilities as the expanding relativistic bubble encounters slower moving ejecta (Hester et al., 1996), a picture supported by MHD simulations that show that 60-75% of the swept-up mass ends up concentrated in such filaments (Jun 1998, Bucciantini et al. 2004). Radio observations reveal filaments that coincide with these optical filaments, presumably corresponding to synchrotron emission from regions of enhanced density and magnetic field in the form of magnetic sheaths that form as the pulsar-injected energy encounters the thermal filaments (Reynolds, 1988). Such filamentary structure is not observed in X-rays, however, suggesting that the electrons with sufficient energy to radiate X-rays do not reach the shell of filaments. This is consistent with the observed smaller extent of the X-ray emission in the Crab nebula relative to its radio size, and indicates a larger magnetic field than is observed in 3C 58 and PSR B1509−58. Recent Chandra observations of 3C 58 reveal a complex of loop-like filaments most prominent near the central regions of the PWN (Figure 3, left), but evident throughout the nebula (Slane et al., 2004a). These structures, whose X-ray spectra are nonthermal, are very well correlated with features observed

22


in the radio band (Reynolds and Aller, 1988). Optical observations reveal faint thermal filaments as well (van den Bergh, 1978), which presumably have an origin similar to that of the Crab filaments. The velocities of these optical filaments in 3C 58 are ∼ ±900 km s−1 (Fesen, 1983), sufficiently high to indicate that the PWN is young, but too small to account for the current size of 3C 58 if the historical age is assumed – one of several standing problems with regard to its evolution (Chevalier, 2004). A detailed comparison of the X-ray and optical images shows that most of the X-ray filaments do not have corresponding optical structures, however. While comparisons with deeper optical images are clearly needed, the fact that many of the X-ray features without optical counterparts are brighter than average in X-rays suggests that these may actually arise from a different mechanism. Slane et al. (2004) propose that the bulk of the discrete structures seen in the X-ray and radio images of 3C 58 are magnetic loops torn from the toroidal field by kink instabilities. In the inner nebula, the loop sizes are similar to the size of the termination shock radius, as suggested by Begelman (1998). As the structures expand, they enlarge slightly as a consequence of the decreasing pressure in the nebula. Some of the observed X-ray structure in the outermost regions may be the result of thermal filaments produced by Rayleigh-Taylor instabilities, similar to the filaments in the Crab Nebula. A shell of thermal X-ray emission demonstrates the presence of ejecta in these outer regions (Bocchino et al., 2001, Slane et al., 2004a. It is worth noting that considerable loop-like filamentary structure is evident in Chandra observations of the Crab Nebula as well (Weisskopf et al., 2000). These features are primarily observed encircling the bright Crab torus, perpendicular to the toroidal plane, and may result from currents within the torus itself. It is at least conceivable that such currents are signatures of the kink instabilities suggested above.

2.3

Large-Scale Structure of PWNe

The large-scale elongated shape of 3C 58 is similar to that found (particularly in the radio band) for a number of other PWNe including the Crab Nebula and G54.1+0.3. Magnetohydrodynamical calculations by Begelman & Li, 1992, and van der Swaluw, 2004, show that such an elongation can result from the pinching effect of a toroidal magnetic field for which the projected axis lies along the long axis of the PWN. The pinching effect results in a low pressure at the edge of the bubble along the major axis with respect to the (higher) pressure at the edge of the minor axis, which yields the elongated structure. The elongation thus marks the projection of the spin axis of the pulsar producing the wound-up field. In 3C 58 this is consistent with the inference of an east-west direction for the projected spin axis based on the interpretation of the extended

Pulsar Wind Nebulae

23

Figure 4. Left: Chandra image of G292.0+1.8. The inset shows the central region at energies above 4 keV, where the pulsar and its wind nebula dominate. Right: Schematic diagram of the evolutionary state of G292.0+1.8. The lack of iron observed in the spectrum indicates that the reverse shock has not yet made its way to the center of the remnant, where the PWN resides.

structure in the inner nebula as being associated with a tilted ring-shaped wind termination shock zone (Slane et al., 2002). The structure of a PWN can be altered significantly through interaction with the reverse shock from the SNR in which it resides. In its early evolution the PWN is basically freely-expanding, encountering only small amounts of slowmoving ejecta in the SNR interior. As the SNR blast wave sweeps up sufficient amounts of circumstellar/interstellar material, a reverse shock is driven back through the ejecta. As this reverse shock propagates, heating the ejecta, it will eventually reach the PWN. Chandra studies of the oxygen-rich remnant G292.0+1.8 reveal an SNR in the intermediate stages of this process. The 0.510 keV X-ray image is presented in Figure 4 (left), and shows the complex structure associated with the shock-heated ejecta and CSM (Park et al., 2002). The inset shows the central image at energies above 4 keV, and reveals a compact pulsar surrounded by a wind nebula (Hughes et al., 2001, Camilo et al., 2002b, Hughes et al., 2003). X-ray spectra of the SNR show metal-rich ejecta with strong lines of oxygen and neon, but a distinct shortage of iron emission (Park et al., 2004), indicating that the reverse shock has not yet propagated sufficiently far toward the center to heat the iron-rich material that was formed closest to the core of the progenitor (Figure 4, right). The morphology of the PWNe in G327.1−1.1 presents a rather different picture. The radio image (Figure 5, left) reveals a well-defined SNR shell with a bright PWN in its interior, distinctly offset from the geometric center. A finger of emission extends to the northwest of the radio PWN, and Chandra

24


observations reveal a compact X-ray source at this location (Figure 5, right). The source is slightly extended (see inset) suggesting that we are seeing material near the wind termination shock. The compact source resides at the tip of a trail of emission that leads back to the bulk of the radio nebula, rather than at the center of the PWN (as in G292.0+1.8, for example), suggesting that the PWN morphology results from a combination of the pulsar motion and the passage of the reverse shock which has apparently disrupted the western side of the nebula (Slane et al., in preparation).

3.

Neutron Star Cooling

The cooling rate of isolated NSs has been a subject of considerable theoretical work predating even the discovery of the first pulsars (e.g., Bahcall & Wolf 1965). The poorly understood properties of the strong nuclear potential at the densities found in NS interiors make these calculations difficult, and lead to a wide range of predictions based on different assumptions for the equation of state, composition, and details of superconductivity (see, e.g., reviews by Tsuruta, 1998, and Yakovlev & Pethick, 2004). While there is a clear consensus that the early cooling proceeds via neutrino emission from the NS core, the timescale over which this dominates depends critically on the neutrino production rate which, in turn, can vary by orders of magnitude depending upon the state of matter in the interior. Broadly speaking, models can be divided into “standard” and “non-standard” cooling scenarios based on the rate of neutrino production in the NS interior.

Figure 5. Left: MOST image of the composite SNR G327.1−1.1. Right: Chandra image of the PWN in G327.1−1.1 (with radio contours), showing diffuse emission surrounding a compact, but resolved, X-ray source (inset).

Neutron Star Cooling

25

Figure 6. Left: Representative neutron star cooling curves for neutrino rates corresponding to different interior conditions. Data from Page, 1998. Right: Temperature vs. Age measurements for neutron stars, along with plots for standard cooling (bold), “minimal” cooling, and massdependent cooling invoking dUrca (see text).

At the high densities and low proton fractions expected in NS interiors, the direct Urca reactions (n → p + e− + ν¯e and p + e− → n + νe ) cannot conserve both energy and momentum. Instead, a bystander baryon is required for each interaction to absorb momentum. The neutrino rate for this so-called modified Urca (mUrca) process is considerably lower than in the direct (dUrca) process because of the extra interaction required, and is the basis for standard cooling models. neutrino bremsstrahlung, and plasmon neutrino processes). In Figure 6 (left), we plot cooling curves for different models of the NS interior and its properties using representative neutrino rates (see Page, 1998 and references therein). The solid curve corresponds to “standard” cooling using an equation of state of moderate stiffness. Non-standard cooling models incorporate neutrino emissivities associated with other processes that may operate in NS interiors, such as those arising from the presence of pion condensates which may form at sufficiently high densities. The resulting pion-induced beta decay leads to very a high neutrino emissivity and a correspondingly shorter cooling time for the NS interior. Similar processes involving kaon condensates or quark matter may operate as well. Alternatively, equations of state that allow a high proton fraction in the interior may allow the dUrca process to proceed; this also leads to extremely high neutrino production rates (e.g., Kaminker et al., 2002). These nonstandard cooling mechanisms modify the NS cooling curves substantially. The dashed curves in Figure 6 represent approximations for several nonstandard cooling models and illustrate the associated rapid cooling (Page 1998). The effects of superfluidity can substantially moderate such rapid cooling because the significantly reduced heat capacity of the superfluid particles re-

26


duces the neutrino rate considerably. Thus, slow cooling can occur from a combination of slow neutrino rates and a high degree of superfluidity, while rapid cooling places strong constraints on both, and thus on the structure and physics of the NS interior, as well as on the microphysics of superfluidity. In particular, as the temperature approaches the critical temperature for the superfluid state, the formation and breaking of Cooper pairs opens another channel for neutrino emission that can lead to more rapid cooling. Page et al., 2004 have calculated a “minimal cooling” scenario which extends the standard mUrca cooling scenario to include the contributions from this Cooper pair process. As discussed below, for some superfluidity models they find sufficiently rapid cooling to explain most observations of young NSs. Yakovlev and Pethick, 2004, on the other hand, have considered both standard and enhanced cooling models. Using the NS mass as a free parameter, they find models for which the dUrca process becomes active for sufficiently high masses, thus leading to a picture in which the young, cool NSs correspond to those with higher mass (see below).

3.1

Measuring Neutron Star Temperatures

Treating the emission from the NS surface as a blackbody, X-ray spectral fitting provides a measure of the gravitationally redshifted temperature and luminosity (assuming the distance is known): 2 2GM 2GM ∞ ∞ Ts = 1 − Ts ; L = 1 − L (2) Rc2 Rc2

where the quantities on the left are observed at infinity, and those on the right are at the NS surface; M and R are the mass and radius of the NS. From this we calculate the effective radius, which can be compared directly with 2 ∞ ∞ 4 predications for different equations of state: Ref f = L /[4πσ(Ts ) ] (where σ is the Stefan-Boltzmann constant). Alternatively, if only an upper limit on the source luminosity is determined, a temperature upper limit can be derived by assuming a value for the NS radius. As with all stars, the emission from the surface of a NS is not a blackbody; rather, it is modified by the presence of whatever atmosphere might exist. One expects the surface of the NS to be covered with Fe, but an atmosphere consisting of H, He, and/or intermediate-mass elements acquired either from ejecta fallback following the neutron star’s formation, or from material accreted from the ISM, is also a possibility. From models of nonmagnetic atmospheres, the primary effect of H or He atmospheres is a considerable deviation of the high energy end of the spectrum relative to the Wien tail of a pure blackbody. The result is that attempts to fit the observed emission with a blackbody model will overestimate the effective temperature – typically by as much as a factor of

27

Summary

two. The inferred size of the NS would, in turn, be underestimated in order to yield the same flux. For atmospheres dominated by heavier elements the effect is considerably reduced, and the blackbody fit gives a good approximation to the temperature.

3.2

Confronting Cooling Models

In Figure 6 (right) we plot the measured temperatures, or upper limits, for the pulsars and compact objects in SNRs for which these values are best determined. Values for known pulsars are plotted with closed circles. For comparison, curves are shown for standard cooling as well as the “minimal” cooling model of Page et al., 2004 (dashed curve). The latter model assumes no enhanced cooling mechanisms in the interior. The primary mechanism which results in faster cooling is neutrino emission from the Cooper pair breaking and formation process, which is heavily dependent upon the assumed superfluidity model (here we have plotted their model “a” for the neutron 3 P2 gap); the predicted cooling rate is sufficiently rapid to explain the measurements for most pulsars, although the temperatures for the Vela Pulsar and J0205+6449 in 3C 58 fall appear to require more rapid cooling. The inferred temperature for RX J0007.0+7302 (plotted as an open box in Figure 6), a compact X-ray source in the SNR CTA 1, also falls below this minimal cooling scenario (Slane et al., 2004). While pulsations have yet to be detected from this source, Chandra observations by Halpern et al., 2004 reveal a slightly extended source with a distinct jet-like tail that is the signature of the pulsar outflows described in Section 1.2. For comparison, cooling curves illustrating the effects of increasing mass are plotted as solid curves, using the model 1p for proton superfluidity from Yakovlev and Pethick, 2004. Also plotted in the Figure, as dashed arrows, are upper limits based on a neutron star search in nearby SNRs (Kaplan et al., 2004). In this study, deep X-ray observations were used to identify, sources within the field of the SNRs, and follow-up optical and IR observations were used to screen for non-NS counterparts. While the lack of detection of a NS in these remnants (G127.1+0.5, G84.2-0.8, G93.3+6.9, and G315.4-2.3 in increasing order of temperature upper limit) may indicate that these SNRs originated in Type Ia events, or all formed black holes, this is statistically unlikely. Future measurements from this study will solidify this picture, and perhaps provide further evidence for rapid cooling in young neutron stars.

4.

Summary

Due in large part to the availability of high resolution X-ray measurements, the study of young NSs and their PWNe have yielded dramatic new information on the nature of the stellar interiors and the structure of their winds. It

28


is now clear that the axisymmetric wind from a pulsar goes through a termination shock as it joins the slower flow of its extended nebula, and that jets and toroidal structures characterizing the inner structure can be used to infer the orientation of the pulsar spin axis. The brightness variations in the inner nebula and jets, as well as the spectral and spatial structure of these regions, yield information on the outflow geometry as well as the fraction of spin-down energy being channeled into these regions. With broad application to particle acceleration and jet formation in astrophysical settings, these observations are providing constraints on theoretical models of considerable importance for a wide range of problems. Similarly, new observations are providing unprecedented capabilities for detecting young neutron stars in SNRs, and for characterizing their emission. These have forced a revised look at models for the structure of NS interiors and provide the best opportunity for addressing the possibility that exotic states of matter reside in NS cores. Through additional and more sensitive observations of these systems, we anticipate significant refinements, and undoubtedly new surprises, in broad picture of their structure that is currently unfolding.

Acknowledgments The author wish to thank Bryan Gaensler, David Helfand, Jack Hughes, and Fred Seward for their particular contributions to this work. Informative discussions on NS cooling with Dima Yakovlev and Dany Page are also gratefully acknowledged. This work was supported in part by NASA contract NAS839073 and grants GO0-1117A, NAG5-9281, and GO1-2054X.

References Aschenbach, B. and Brinkmann, W. (1975). A model of the x-ray structure of the Crab Nebula. A&A, 41:147. Bahcall, J. N. and Wolf, R. A. (1965). An Observational Test of Theories of Neutron-Star Cooling. ApJ, 142:1254–1256. Begelman, M. C. (1998). Instability of Toroidal Magnetic Field in Jets and Plerions. ApJ, 493:291. Begelman, M. C. and Li, Z. (1992). An axisymmetric magnetohydrodynamic model for the Crab pulsar wind bubble. ApJ, 397:187–195. Bocchino, F., Warwick, R. S., Marty, P., Lumb, D., Becker, W., and Pigot, C. (2001). The x-ray nebula of the filled center supernova remnant 3C 58 and its interaction with the environment. A&A, 369:1078–1087. Bucciantini, N., Amato, E., Bandiera, R., Blondin, J. M., and Del Zanna, L. (2004). Magnetic Rayleigh-Taylor instability for Pulsar Wind Nebulae in expanding Supernova Remnants. A&A, 423:253–265. Camilo, F., Lorimer, D. R., Bhat, N. D. R., Gotthelf, E. V., Halpern, J. P., Wang, Q. D., Lu, F. J., and Mirabal, N. (2002a). Discovery of a 136 millisecond radio and x-ray pulsar in supernova remnant G54.1+0.3. ApJ, 574:L71–L74.

Summary

29

Camilo, F. , Manchester, R. N., Gaensler, B. M., Lorimer, D. L., and Sarkissian, J. (2002b). PSR J1124–5916: A young and energetic pulsar in supernova remnant G292.0+1.8. ApJ, 567:L71–L75. Chevalier, R. A. (2004). Pulsar wind nebulae: theoretical aspects and observational constraints. Advances in Space Research, 33:456–460. Fesen, R. A. (1983). Discovery of large radial velocities in the supernova remnant 3C 58. ApJL, 270:L53–L57. Frail, D. A. and Moffett, D. A. (1993). Deep VLA imaging of pulsar-powered nebulae and the beaming fraction of young pulsars. ApJ, 408:637–645. Gaensler, B. M., Arons, J., Kaspi, V. M., Pivovaroff, M. J., Kawai, N., and Tamura, K. (2002). High-resolution X-ray imaging of pulsar B1509–58 and its environment. ApJ, 569:878–893. Gallant, Y. A. and Arons, J. (1994). Structure of relativistic shocks in pulsar winds: A model of the wisps in the Crab Nebula. ApJ, 435:230. Halpern, J. P., Gotthelf, E. V., Camilo, F., Helfand, D. J., and Ransom, S. M. (2004). X-Ray, Radio, and Optical Observations of the Putative Pulsar in the Supernova Remnant CTA 1. ApJ, 612:398–407. Helfand, D. J., Gotthelf, E. V., and Halpern, J. P. (2001). Vela Pulsar and Its Synchrotron Nebula. ApJ, 556:380–391. Hester, J. J. (1998). Wind interactions in the Crab Nebula. Mem. della Soc. Ast. It., 69:883. Hester, J. J., Mori, K., Burrows, D., Gallagher, J. S., Graham, J. R., Halverson, M., Kader, A., Michel, F. C., and Scowen, P. (2002). Hubble Space Telescope and Chandra Monitoring of the Crab Synchrotron Nebula. ApJL, 577:L49–L52. Hester, J. J., Stone, J. M., Scowen, P. A., Jun, B., Gallagher, J. S., Norman, M. L., Ballester, G. E., Burrows, C. J., Casertano, S., Clarke, J. T., Crisp, D., Griffiths, R. E., Hoessel, J. G., Holtzman, J. A., Krist, J., Mould, J. R., Sankrit, R., Stapelfeldt, K. R., Trauger, J. T., Watson, A., and Westphal, J. A. (1996). WFPC2 Studies of the Crab Nebula. III. Magnetic RayleighTaylor Instabilities and the Origin of the Filaments. ApJ, 456:225–233. Hughes, J. P., Slane, P. O., Burrows, D. N., Garmire, G., Nousek, J. A., Olbert, C. M., and Keohane, J. W. (2001). A pulsar wind nebula in the oxygen-rich supernova remnant G292.0+1.8. ApJ, 559:L153–L156. Hughes, J. P., Slane, P. O., Park, S., Roming, P. W. A., and Burrows, D. N. (2003). An x-ray pulsar in the oxygen-rich supernova remnant G292.0+1.8. ApJ, 591:L139–L142. Jun, B. (1998). Interaction of a Pulsar Wind with the Expanding Supernova Remnant. ApJ, 499:282. Kaminker, A. D., Yakovlev, D. G., and Gnedin, O. Y. (2002). Three types of cooling superfluid neutron stars: Theory and observations. A&A, 383:1076–1087. Kaplan, D. L., Frail, D. A., Gaensler, B. M., Gotthelf, E. V., Kulkarni, S. R., Slane, P. O., and Nechita, A. (2004). An X-Ray Search for Compact Central Sources in Supernova Remnants. I. SNRS G093.3+6.9, G315.4-2.3, G084.2+0.8, and G127.1+0.5. ApJS, 153:269–315. Kaspi, V. M., Roberts, M. S. E., and Harding, A. K. (2004). Isolated neutron stars. In Lewin, W. H. G. and van der Klis, M., editors, Compact Stellar X-ray Sources, Cambridge. CUP. in press (astro-ph/0402136). Kaspi, V. M. , Roberts, M. S. E., Vasisht, G., Gotthelf, E. V., Pivovaroff, M., and Kawai, N. (2001). Chandra observations of G11.2-0.3: Implications for pulsar ages. ApJ, 560:371–377. Kennel, C. F. and Coroniti, F. V. (1984). Magnetohydrodynamic model of Crab Nebula radiation. ApJ, 283:710–730. Komissarov, S. S. and Lyubarsky, Y. E. (2004). Synchrotron nebulae created by anisotropic magnetized pulsar winds. MNRAS, 349:779–792.

30


Lai, D., Chernoff, D. F., and Cordes, J. M. (2001). Pulsar jets: implications for neutron star kicks and initial spins. ApJ, 549:1111–1118. Lu, F. J., Wang, Q. D., Aschenbach, B., Durouchoux, P., and Song, L. M. (2002). Chandra observation of supernova remnant G54.1+0.3: A close cousin of the Crab Nebula. ApJ, 568:L49– L52. Murray, S. S., Slane, P. O., Seward, F. D., Ransom, S. M., and Gaensler, B. M. (2002). Discovery of x-ray pulsations from the compact central source in the supernova remnant 3C 58. ApJ, 568:226–231. Ng, C.-Y. and Romani, R. W. (2004). Fitting pulsar wind tori. ApJ, 601:479–484. Page, D. (1998). Thermal evolution of isolated neutron stars. In The Many Faces of Neutron Stars. Edited by R. Buccheri, J. van Paradijs, and M. A. Alpar. Dordrecht ; Boston : Kluwer Academic Publishers, 1998., p.539, pages 539. Page, D., Lattimer, J. M., Prakash, M., and Steiner, A. W. (2004). Minimal Cooling of Neutron Stars: A New Paradigm. in press (astro-ph/0403657). Park, S., Hughes, J. P., Slane, P. O., Burrows, D. N., Roming, P. W. A., Nousek, J. A., and Garmire, G. P. (2004). Nucleosynthesis in the Oxygen-rich Supernova Remnant G292.0+1.8 from Chandra X-Ray Spectroscopy. ApJ, 602:L33–L36. Park, S. , Roming, P. W. A., Hughes, J. P., Slane, P. O., Burrows, D. N., Garmire, G. P., and Nousek, J. A. (2002). The structure of the oxygen-rich supernova remnant G292.0+1.8 from Chandra x-ray images: shocked ejecta and circumstellar medium. ApJ, 564:L39–L44. Pavlov, G. G., Teter, M. A., Kargaltsev, O., and Sanwal, D. (2003). The Variable Jet of the Vela Pulsar. ApJ, 591:1157–1171. Rees, M. J. and Gunn, J. E. (1974). The origin of the magnetic field and relativistic particles in the Crab Nebula. MNRAS, 167:1–12. Reynolds, S. P. (1988). Filamentary structure in Crab-like supernova remnants. ApJ, 327:853– 858. Reynolds, S. P. and Aller, H. D. (1988). Radio observations of the Crab-like supernova remnant 3C 58. I - Total intensity observations. ApJ, 327:845–852. Romani, R. W. and Ng, C.-Y. (2003). The Pulsar Wind Nebula Torus of PSR J0538+2817 and the Origin of Pulsar Velocities. ApJL, 585:L41–L44. Slane, P., Helfand, D. J., van der Swaluw, E., and Murray, S. S. (2004a). New constraints on the structure and evolution of the pulsar wind nebula 3C 58. ApJ. in press (astro-ph/0405380). Slane, P., Zimmerman, E. R., Hughes, J. P., Seward, F. D., Gaensler, B. M., and Clarke, M. J. (2004b). X-Ray Observations of the Compact Source in CTA 1. ApJ, 601:1045–1049. Slane, P. O., Helfand, D. J., and Murray, S. S. (2002). New constraints on neutron star cooling from Chandra observations of 3C 58. ApJ, 571:L45–L49. Tsuruta, S. (1998). Thermal properties and detectability of neutron stars. II. Thermal evolution of rotation-powered neutron stars. Phys Rep, 292:1–130. van den Bergh, S. (1978). Observations of the Optical Remnant of SN 1181 = 3C 58. ApJL, 220:L9. van der Swaluw, E., Downes, T. P., and Keegan, R. (2004). An evolutionary model for pulsardriven supernova remnants. A hydrodynamical model. A&A, 420:937–944. Weisskopf, M. C., Hester, J. J., Tennant, A. F., Elsner, R. F., Schulz, N. S., Marshall, H. L., Karovska, M., Nichols, J. S., Swartz, D. A., Kolodziejczak, J. J., and O’Dell, S. L. (2000). Discovery of spatial and spectral structure in the x-ray emission from the Crab Nebula. ApJ, 536:L81–L84. Yakovlev, D. G. and Pethick, C. J. (2004). Neutron Star Cooling. Ann. Rev. Astr. Ap., 42:169– 210.

II

NEUTRON STAR STRUCTURE AND MAGNETIC FIELD

A BIOGRAPHY OF THE MAGNETIC FIELD OF A NEUTRON STAR Malvin Ruderman Department of Physics and Columbia Astrophysics Laboratory Columbia University [email protected]

Abstract

After some post-natal cooling, a spinning, magnetized, canonical neutron-star (NS) has a core of superconducting protons, superfluid neutrons, and degenerate extreme relativistic electrons, all surrounded by a thin highly conducting solid crust. The quantum fluids are threaded by a dense array of quantized vortexlines which can interact strongly with a denser and much less uniform one of quantized magnetic flux-tubes. The physics of such a core predicts the evolution of a NS’s surface magnetic field and related phenomena as the star’s spin changes. Predictions include changes in NS magnetic dipole moments, anomalously small polar caps in millisecond pulsars, properties of two different families of spin-period “glitches", and spin-down ages much greater than true ages for some pulsars. Quantitative model-based estimates for all of these are given. None are in conflict with observations.

Keywords:

stars:pulsar – stars:neutron – stars:magnetic fields

1.

Introduction

There is abundant observational evidence that the magnetic dipole-moment of a rapidly spinning neutron star (NS) evolves during a pulsar’s lifetime. Most of the relevant data comes from observed NS-periods (P) and spin-down rates (P). With simple plausible models of NS-magnetospheres the spin-down torque on a (presently) solitary NS depends only on the NS spin (Ω = 2π/P ) and dipole moment (µ): µ2 Ω3 (1), I Ω˙ ∼ − 3 c where I is the NS moment of inertia. Fig. 1 shows NS-surface-dipole magnetic fields, B ≡ µR−3 , over a large range of pulsar spin-periods. (All pulsars are assumed to have I = 1045 gcm2 and radii R = 106 cm.) Also shown is the typical evolution of B with P from the microphysics inside a canonical, 47 A. Baykal et al. (eds.), The Electromagnetic Spectrum of Neutron Stars, 47–61. © 2005 Springer. Printed in the Netherlands.

48

A Biography of the Magnetic Field of a Neutron Star

rapidly spinning, strongly magnetized NS. In the model considered here a NS consists mainly of a sea of superfluid neutrons together with less dense components of superconducting protons and very relativistic degenerate electrons. (Effects from possible, much smaller, central volumes of more exotic particles are ignored.) Surrounding this sea of quantum fluids is a thin solid conducting crust of thickness ∆ ∼ 10−1 R in which the NS protons clump into conventional, but extremely neutron-rich, nuclei. In such an object B is expected to evolve mainly because of changes in Ω. Fig. 1 shows evolutionary segments a → b → c → d → e predicted from this canonical NS model. The slopes of B(P ) in the segments a → b, b → c are not significantly different from those determined directly from certain observations of P, P˙ , and P¨ (Sect. 4), and from inferred ages of radio pulsars observed in other B − P regions. Fig. 1 segments a → a; a → b − c; c → d; d → h → g; d → e → f ; d → h → q give more explicit B(P ) predictions than what can be inferred directly from B − P data, but can be compared with other kinds of observations (Sects. 3, 7). How this same model leads to two different kinds of “glitches" in pulsars is discussed in Sect. 8. In all comparisons of observations with model-based predictions no disagreements have yet become apparent. In almost all biographies, and certainly including those of NS B-fields, authors and readers have good reason to be much less confident about descriptions of earliest life (conception, infancy, and childhood) than of adolescence, maturity, and old age. This is even more the case here where there is not yet a consensus about what is happening in any one of these stages. We shall begin our account of the development of a NS magnetic field with this caveat very much in mind.

2.

Conception, Birth, and Infancy

Neutron stars are believed to be born in violent implosions of much less dense ancestors. There is no consensus about the origin of a newly formed NS’s B(1012 − 1015 G?). Speculations include (a) conservation during a NS’s violent birth of flux already inside its ancestor (NS fluxes may be comparable to those in magnetic white dwarfs, the toroidal field within the sun, and cores of red and blue giants); (b) short-lived post-partem dynamos[1]; (c) field amplification in asymmetric supernova explosions; (d) toroidal field breakout after wind-up from differential rotation imparted at birth [2]; (e) thermoelectric generation[3]; (f) exterior field reduction from burial by fall-back of some of the initially exploded matter.

Conception, Birth, and Infancy

49

Figure 1. Dipole fields (B) of pulsars inferred from observed P, P˙ , together with the B(P ) predicted for a typical pulsar from the model discussed in the text. The point c is about where coherent radio emission is no longer observable. Point d is for P and B of a solitary NS after 1010 yrs. The millisecond pulsar (MSP) population in the lower left corner is presumed to be populated by the evolution of some NSs off the bcd segment from accretion-induced spin-up by companions (many of which have been evaporated by the MSPs). After accretion stops the MSP’s again spin-down like solitary radio pulsars. Open circles are pulsars in binaries which are not presently interacting with their companions. (The P , P˙ of many more radio pulsars are shown in reference [28].) The diagonal dashed line is the canonical accretion-driven spin-up line.

Because of a NS’s violent, unstable birth, the initial distribution of B within a very rapidly spinning NS is probably magneto-hydrodynamically (MHD) unstable for a time[4] tMHD ∼ 106 ( 10ms P ) sec, if the initial P ∼ 10ms. This MHD relaxation time may exceed the “freezing time" (tf ∼ 10 sec) for neutrino emission to cool a new-born and initially very hot (T > 1010 K) NS-crust to below the temperature where crust-solidification begins. Some relevant expected solid crust properties are shown in Table 1. The main uncertainties in it are the maximum sustainable shear strain (θmax ) and ∆θ, the size of sudden strain-relaxation (“crust-breaking") if Ymax is slowly exceeded. If tf > tMHD

50


Table 1. Estimates of some properties of NS-crusts crust thickness shear modulus maximum sustainable shear strain maximum averaged shear strength strain relaxation when Ymax is exceeded Eddy current decay time

∆ ∼ 105 cm κ ∼ 1030 dyne cm−2 θmax ∼ 10−3 Ymax ∼ κθmax ∆/R ∼ 1026 dyne cm−2 ∆θ ∼ 10−4 − 10−3 ∼ 107 yrs

the NS begins its childhood (t ∼ 1 yr) with relatively small magnetic stress in its crust, but if tf < tMHD , j × B forces in the crust may sustain a surfacedipole B up to (8πYmax )1/2 ∼ 5 × 1013 G despite MHD relaxation to well below this for the dipole field at the surface of the NS’s core.

3.

Childhood: t ∼ 1 yr – 10 yrs (Fig. 1, a → a; a → b − c)

About a year or so after its birth an initially very hot NS will cool its interior below the transition temperature (T ∼ 3 × 109 K?) at which it becomes a proton-superconductor (p-sc). Any magnetic field within the Type II p-sc expected within the NS core organizes itself on a submicroscopic scale into a dense array of quantized flux-tubes (flux = 2 × 10−7 Gcm2 ) each of which has a radius Λ ∼ 10−11 cm and an interior magnetic field Bc ∼ 1015 G. Local flux-tube area densities are huge: nφ ∼ 5B12 × 1018 cm−2 . (If B > Bc , p-sc is quenched.) Submicroscopically, the B-field structure now becomes −1/2 extremely inhomogeneous (B ∼ 0 between flux tubes 5 × 10−1 B12 cm apart, and B = Bc ∼ 1015 G within them). On much larger scales the p-sc flux-tube array also varies greatly with initial densities and twists in direction reflecting the complicated combination of poloidal and toroidal B in the cooling NS core just before it began its transition to a p-sc. Just after the transition the previously stabilized core B would be strongly out of equilibrium. The initially MHD-stabilized configuration is based upon a compromise between minimizing the sum of tension-energy (B 2 /8π) along B and a similar B 2 /8π contribution from repulsion between field lines, After the transition into quantized flux-tubes the effective tensile pull per unit area jumps greatly, B 2 /8π → BBc /8π, while the repulsion between flux tubes almost vanishes ¯ ∼ Bc . unless flux-tubes are squeezed to separation distances ∼ Λ so that B Therefore, after the p-sc transition, magnetic forces act to pull the NS-core’s magnetic field toward a new equilibrium configuration. It is generally a much smaller one as flux-tubes try to minimize their length until they touch each

Adolescence

51

other. If this is achieved, the µ of the core-field at the surface of the p-sc sea (its interface with the base of the NS-crust) would be greatly diminished from its value before the transition1 . It is difficult to calculate the time scale-needed for this new configuration to be achieved. (Some movement of flux-tubes is by co-advection together with their e-p-n embedding fluids. This may involve induced flux-tube bunching and backflow between bunches[7]. Some flux-tube movement is, instead, through those fluids at a velocity allowed by electron-current ohmic dissipation from the random part of the electron scattering on a partly disordered flux-tube array[6]. Composition gradients give important constraints.) The NS’s solid, strongly conducting crust can prevent changes in magnetic field through it for up to 107 yrs as long as any shearing stress on it at the core-crust interface < Ymax . However, after the p-sc transition, that stress may jump from B 2 /8π to BBc /8π, giving a reduction in the maximum possible crust-stabilized B by almost 2 × 102 , from a pre-childhood B ∼ 5 × 1013 G to 3 × 1012 G. This suggests that many pulsars may begin their childhoods with very large dipolar Bs which may not survive this stage of their lives. Although there is no reliable estimate of the time-scale for this survival based upon the microphysics of all the possible contributions to it which should be considered, an interpretation of observed build-up of one the Crab pulsar’s “glitches" suggests about 10 years (Sect. 8 footnote). Then a a pulsar entering childhood with B ∼ 5 × 1013 G would spin-down to P ∼ 0.5 sec as its previously crust-stabilized B drops to about 3 × 1012 G. Crucial evidence for such pulsars would be the observation of spin-down ages (tsd ≡ P/2P˙ ) about 2 × 102 times greater than their true ages, those inferred from the ages (tsnr ) of the supernova remnants in which these NS’s are still embedded. This large reduction would, of course, become smaller if B before the p-sc transition were smaller, and could disappear if that B < 3 × 1012 G (a in Fig. 1). Fig. 2 compares spin-down and supernova-remnant ages of pulsars for which both have been reported. The two pulsars with the largest differences, 1E 1207.4-5209 and PSR J1952+3252, have present magnetic dipole fields of about 3 × 1012 G and present spin-down times about 2 × 102 times longer than their true ages. (One alternative, and presently more common, explanation of this discrepancy is that many pulsars are born with such very long spin-periods that they have not spun-down much from those periods in their first 104 yrs.)

1 If,

unexpectedly, the p-sc transition is to a type I superconductor[5], proton-superconductivity is quenched by a B > Bc everywhere inside discrete regions very much larger than flux¯ c when tubes and vanishes throughout the superconducting volume outside them. Then B 2 > BB averaged over large areas passing through both kinds of proton-phases. The new force distribution and its consequences would then be very similar to those following a type II transition.

52


4.

Adolescence: t ∼ 103 − 104 yrs; Crab-like pulsars (Fig. 1 a → b)

About 103 yrs after their violent births, NSs will have cooled below the transition temperatures (several ×108 K) to neutron superfluidity (n − sf ). An initially nearly uniformly rotating NS neutron sea then changes its fluid rotation pattern by establishing a nearly parallel array of quantized vortex lines aligned along the spin-direction. These vortex-lines, with an area number density nv ∼ 105 (Ω/ΩCrab )cm−2 , pass through the the hugely more abundant, curved, twisting, magnetic flux-tubes already formed in the p-sc during early NS childhood. When the neutrons in a NS core spin-down(-up) these corotating n-vortex-lines move away from(toward) the NS spin-axis. Vortex lines, parallel to the spin-axis but displaced from it by r⊥ , move out with an average ˙ ⊥ /2Ω ∼ 10−5 × (NS age/103 yrs) cm s−1 . Because of the velocity v = −Ωr strong velocity-dependence of the neutron-proton interaction, segments of nvortex-lines and p-flux-tubes also interact strongly with each other when they are within a distance Λ. Therefore the moving n-vortex-lines of a NS whose Ω is changing must either cut through or carry along with themselves the geometrically complicated flux-tube array which they thread. Which occurs, depends on the magnitude of the resistance of various regions of the flux-tube array to being moved through the embedding seas[6], MHD instabilities which can result in flux-tube bunching[7], possibilities for advective movements of flux-tubes together with their embedding electrons and protons, the force just before cut-through of the vortex-flux-tube interaction, and the possibility of

Figure 2. Observed spin-down ages (P/2P˙ ) of some young NSs together with estimates of their true ages from those of the supernova remnants in which they are embedded, or from historical records of associated supernova explosions.

53

Maturity

temporary anchoring of the core’s flux-tubes by the solid crust. Both frictional drag and inelastic cut-through generate heat within the core, Therefore thermal x-ray emission bounds for spinning-down NSs (especially where they exceed neutrino-emission in NS cooling) give an empirically determined limit to both. Consideration of the above supports the following comments. a) Characteristic n-vortex velocities and the induced flux-tube array velocities are small enough in the cores of adolescent pulsars (spin-down age > 103 yrs) that cut-through of flux-tubes by moving vortex-lines is expected to be unimportant in them. b) This conclusion is more firmly based for older pulsars and quite compelling for those whose spin-up (-down) ages exceed 105 yrs. (Even if the p-sc is type I in some of the core region, there will be such strong forces there between n-vortex lines and the field boundary regions of thickness Λ, in which B goes from ∼ 0 to > Bc , that slowly moving vortex-lines resist passage through such boundaries and move the whole B-field regions with themselves.) c) The crust is not strong enough to prevent the surface B of a NS from following vortex-array controlled flux-tube movement near the base of the crust as long as B > 1012 G and P < 1 sec. Outside this B −P range the conducting crust could delay this for about 107 yrs, but not prevent it (cf. the discussion on very short delays in Sect. 8). We consider next observational consequences of a very simple evolutionary model expected to have validity for magnetic field evolution in adolescent NSs and, with even more confidence, for such evolution in more slowly spinningdown (-up) older NSs: the magnetic dipole field on the surface of such NSs follows that of the core’s p-sc flux-tube array near the base of the crust. The movement of that array is, in turn, controlled by the expanding (contracting) n–sf vortex-array of the spinning-down(-up) core n–sf. From here on our discussion follows much of what is in the published literature, supplemented by new supporting x-ray data from the XMM and Chandra satellites. It will therefore be rather abbreviated with more detail and references available elsewhere[6,7]. In the adolescent (Crab-like) pulsars r⊥ ∝ P 1/2 until r⊥ reaches the NS radius R. The predicted evolution of µ⊥ , the component of µ perpendicular to Ω, is then particularly simple. Models which attribute spin-down mainly to the Maxwell ¨Ω ˙ −2 = 3 − 2µ˙ ⊥ Ω = 2 torque, µ2⊥ Ω3 /c3 , have a “spin-down index" n ≡ −ΩΩ ˙ µ Ω ⊥

as long as all r⊥ ∝ P 1/2 . As r⊥ reaches R for a significant fraction of the vortex lines, n grows from 2 toward 3. It reaches 3 when flux-tubes cannot move out further. This behavior is shown in the upward moving a → b segment of Fig 1. and n values observed in some adolescent pulsars (for which P has been measured): n = 2.5, Crab[8]; n = 2.8, PSR B1509-58[9]; n = 1.8, PSR B0540-69[10]; n = 2.9, PSR J1119-7936[11]. (For a different model for 2 < n < 3 in which the effective I varies cf. ref. 30).

54

5.


Maturity: ages 104 − 106 yrs; Vela-like and older pulsars (Fig.1 b → c)

As flux-tubes are pushed out of the p-sc core by the core’s expanding n-sf vortex-array, North and South poles at the surface will ultimately reconnect. (cf the discussion in Sect.8 on “giant glitches".) Thereafter, µ⊥ typically decreases as P −1 , which gives an average spin-down index n = 5. (This is because the part of the core’s flux, which has not yet been pushed out to r ∼ R where it reconnects, is the source of the NS’s remaining surface dipole B. That remaining flux is proportional to the remaining number of core n-sf vortex-lines.) Differences between spin-down ages and kinematically determined ones of observed pulsars around the b → c implies n = 4.5 ± 0.8 The n at a particular P for any one pulsar cannot be predicted a priori without detailed knowledge of the NS core’s magnetic field structure2 .

6.

Old age (Fig.1, c → c → d; c → d → d)

The point c in Fig. 1 is near where the maximum expected magnetic shear stress on the crust’s base no longer exceeds the crust’s yield strength (BBc /8π < Ymax ). The evolution of surface B beyond c depends on time scales. The core’s surface B should follow the trajectory c → c → d, but the crust’s surface B would now follow it only after ten or so million years, when crustal Eddy currents have died out (or perhaps earlier if plastic flow in the crust has allowed sufficient reduction of its j × B stresses). This time-lag is plausible as the reason why some x-ray pulsars, NSs spun-down relatively rapidly by binary companions (e.g. to P ∼ 103 sec for Vela X-1), can temporarily maintain surface dipole B ∼ 1011 G. The death line of a solitary NS as an observable radio pulsar is expected at P ∼ several seconds and B ∼ 1011 G, but spin-down would continue.

7.

Resurrection of some 108 − 1010 yr old NSs (Fig. 1 d → h → g ; d → h → e → f ; d → h → q ; d → c)

Some dead pulsars in binaries will be spun-up by accretion from companions which have evolved, orbitally or in size, to fill their Roche lobes. (Before or during this phase, the interaction with the companion may first have given

2 For

example, N and S surface polar caps might be connected by pushed core-flux just below the surface. This pulls them toward each other until they ultimately reconnect (n > 3), or first pulls them apart (n < 3) until they are on opposite sides of the star. In the latter case further pulling will then bring them closer (n > 3) until reconnection is finally achieved. Alternatively, an initially complicated surface field could have many N and S pole regions. Then the vector sum which gives the net dipole moment may be either decreased (n < 3) or increased (n > 3) by reconnection of any one pair. Only after long evolution would n ∼ 5 be realized (cf the reported n ∼ 1.4 for Vela[13].)

Resurrection of some 108 − 1010 yr old NSs

55

the NS larger additional spin-down with accompanying reduction in B than would have been the case if it was solitary.) Such a genesis by accretion from a low mass companion (LMXB) is widely proposed as the origin of the millisecond radio pulsars (MSPs) in the lower left corner of Fig.1. The superfluidvortices’s radial velocities within a NS being spun-up to a millisecond period in an LMXB (∼ 10−9 cm sec−1 ) are so small that flux-tube movement which follows it seems inescapable. The very slow inward movement of the n-sf vortices squeezes all magnetic flux inward with it toward the NS spin-axis. Evolutionary tracks for surface-dipole B in Fig. 1 depend upon the initial B-field configuration at the beginning of the long, slow spin-up[6,14,15,16]: (d → c) - N and S polar caps are in opposite spin-hemispheres. The final dipole moment is almost aligned and somewhat bigger than it was initially. (d → g) - N and S polar caps are in the same spin-hemisphere. They are then squeezed together near the spin-axis to form an orthogonal rotator (µ ⊥ Ω) whose µ is reduced by a factor (Pg /Pd )1/2 ∼ 10−2 . Spin-up alone, by bringing the N and S magnetic poles so close together, reduces µ to the small value essential for accretion to spin-up the NS to a MSP (by approaching the limiting accretion spin-up line of Fig. 1). (d → h → q; d → e → f ) - most, but not all, flux from either spinhemisphere returns to the NS surface in the same hemisphere from which it came. Continued spin-up reduces the orthogonal component, µ⊥ ∝ (P )1/2 , together with a slightly increased, aligned µ . When these two components of the total dipole approach equal magnitudes at, say, h (or e), the total dipole moment (and surface dipole B) can no longer be strongly reduced by further spin-up since it diminishes only the orthogonal component. The initial conditions needed for accretion-induced spin-up into the extreme lower left corner of Fig. 1 from such d → h → e strongly suggest that the exceptionally fast MSPs they evolve into should usually have a surface magnetic field resembling that from an orthogonal dipole positioned on the NS spin-axis at the interface between the NS’s perfectly diamagnetic p-sc core and the bottom of its crust (Fig. 3b). A second large MSP family there should consist of almost aligned rotators (less frequently observed since their radio beams are directed so close to the NS spin-axes) whose µ⊥ have been “spin-up squeezed" to negligible strength. Their surface-field configuration should then closely resemble that from a N(S) pole on the spin-axis where magnetic field leaves the diamagnetic core in the upper spin-hemisphere and an equal strength S(N) pole where it re-enters the core in the lower spin-hemisphere (Fig. 3a). In both MSP families the size of the polar caps at the crust core interface is expected to be small. [Polar cap radii there ∼ (P/Pd )1/2 R ∼ 104 cm.] There is strong observational support for both of these two MSP families. (a) An exceptionally large fraction of the most rapidly spinning MSPs in the Galactic disk fit orthogonal rotator criteria of two sub-pulses of comparable

56


Figure 3. Magnetic field configurations of strongly spun-up millisecond pulsars (MSPs): (b) field configuration after prolonged spin-up when magnetic poles were in the same spinhemisphere; (a) final configuration when N and S poles were in opposite spin-hemispheres or had a more complicated distribution with many poles but the spin-up was extremely great; (c) a magnified view of (a) around the North polar cap. Indicated in (c) is a gamma-ray of the curvature radiation from an extreme relativistic lepton moving in from an accelerator along an open field line. Where it hits the surface there is a hot polar cap (very black in the figures). Energetic curvature radiation disappears near the polar cap because of the disappearing curvature of open field-lines near the essentially isolated pole.

strength separated by around 180◦ in phase[15,17]. Of the 6 MSPs reported to have x-ray pulses as well as radio-emission[18], 5 are consistent with orthogonal rotators. This same orthogonality criterion is met in only about 10−2 of the rest of the radio pulsar population. (b) The very special predicted B-field structure in the near-magnetosphere from “spin-up flux-squeezing" into an orthogonal rotator gives just the radiosubpulse polarization properties, and their frequency dependence[15], observed in the first, fastest MSP, PSR B1937+21 (P = 1.6 ms). (c) The sixth x-ray identified radio-MSP, PSR J0437-4715, has a huge radiopulsewidth (∼ 270◦ ). This and the structure of its radiopulse-polarization[19] strongly support its categorization as a nearly aligned rotator. (d) Just below the accretion spin-up line of Fig. 1 is where a large fraction of nearly aligned pulsar candidates are observed. (e) The aligned MSP, PSR J0437-4715 (P ∼ 6 ms), would have a surface polar cap area Apc ∼ πΩR3 /c ∼ 1011 cm2 for a conventional central dipole field or that from a uniformly magnetized core. This Apc should also be the emission area of blackbody radiation x-rays sustained by backflow of extreme relativistic particles down onto the polar cap from the pulsar’s open field line particle accelerator(s). The expected blackbody radiation is indeed observed, but its emission area is only 4 × 108 cm2 , and it is surrounded by a comparably luminous but cooler blackbody annulus 2 × 105 cm away[20]. These two observed features are hard to understand with conventional models of a NS’s

Pulsar Glitches

57

surface magnetic field. They are, however, just what are predicted on the surface of a strongly spun-up, flux-squeezed, aligned MSP[14]. In that model open field line bundles span very much smaller polar cap areas on the crust surface (just above each of the two core poles) than polar cap areas in models with more conventional B-field configurations. The new predicted polar cap area ∼ πΩR∆2 /c ∼ 109 cm2 ∼ 10−2 × that from central dipole models. For PSR 0437 the predicted polar cap radius, 0.17 km, is a tenth the canonically estimated one and consistent with the 0.12 km deduced from observations. Curvature radiation from extreme relativistic particle inflow onto such polar caps is strong along almost all of an incoming particles’s trajectory and would heat up a large surface area extending far from the polar caps. However, with the special squeezed flux geometry of Figs. 3 a,c such strong curvature radiation sources should disappear above the polar cap where the local B-field lines lose their curvature. Strong curvature radiation heating of the NS surface outside the polar cap itself should then only be important beyond about 2 km away, in agreement with the inner radius of the reported hot annulus. Up to this point all of the agreement between the simple spinning-NS model predictions and related observations have involved relatively slow, time averaged, changes in B at the surface of a NS crust as it responds to changing spin of a NS core’s quantum fluids. We turn now to a consideration of other observations which test expectations of the detailed way in which such crustal field changes are accomplished.

8.

Small sudden changes in B through overstressed crusts: pulsar “glitches"

Moving core flux-tubes continually build up stress in surrounding conducting crust which anchors the B-field that traverses it. If this stress grows to exceed the crust’s yield strength (Ymax ), subsequent relaxation may, at least partly, be through relatively sudden crustal readjustments (“crust breaking"). Such events would cause very small spin-ups(-downs) in spinning-down(-up) NSs (spin-period “glitches"). The above model for the evolution of a core’s flux-tube array in adolescent and mature pulsars suggests glitch details in such pulsars similar to those of the two observed glitch families: Crab-like glitches (C) and the very much larger giant Vela-like ones (V).

a) Crab-like (C) glitches. In both adolescent and mature pulsars an expanding quasi-uniform n-sf vortex-array carries a p-sc flux-tube array outward with it. If growing flux-tube-induced stress on the crust is partly relaxed by “sudden" outward crust movements of magnitude s where the stress is strongest (with density preserving backflow elsewhere in the stratified crust) the following consequences are expected:

58


˙ : ∆Ω˙ ∼ (1) a “sudden" permanent increase in µ⊥ , spin-down torque, and |Ω| ˙ Ω s/R ∼ ∆θ (strain relaxation) < θmax ∼ 10−3 . (This is the largest nontransient fractional change in any of the pulsar observables expected from breaking the crust.) A permanent glitch-associated jump in high density (ρ) n-sf of this sign and magnitude (∼ 3 × 10−4 ) is indeed observed in the larger Crab glitches[21,22]. (2) a “sudden" reduction in shear stress on the crust by the flux-tubes below it. Its estimated magnitude is (BBc /8π)(s/R). This is also the reduction in pull-back on the core’s expanding n-sf vortex array by the core’s p-sc fluxtube array which it tries to drag with it. The n-vortices therefore move out to a new equilibrium position where the Magnus force on them is reduced by just this amount. The n-sf, therefore, spins down a bit. All the (less dense) charged components of the NS (crust, core-p and-e) together with the n- vortex array must spin-up much more. (The total angular momentum of the NS does not significantly change in the brief time for development of the glitch.) A new equilibrium is established in which the charged components (all that is observed, of course, is P of the crust’s surface) have acquired a ˙ BBc ∆Ω ∆Ω −4 ∼ . (2) (s/R) ∼ 10 ˙ Ω 8πρR2 Ω2 Ω ˙

−8 and ∆Ω ∼ 3×10−4 are observed. Crab-glitch ∆Ω ˙ Ω with magnitudes ∼ 3×10 Ω So are many much smaller glitches, and the proportionality factor of Eqn. 2 holds approximately for them as well[22]3 . b) Giant Vela-like (V)glitches. A second V-family of glitches differs from that of Crab-like ones (C) in several ways[29]. (1) (∆Ω/Ω)V −102 ×(∆Ω/Ω)C . (2) V-glitches develop their ∆Ω in less than 102 sec.: the spin-period of a V-glitch is already decreasing in magnitude when first resolved[24], while C-glitches are still rising toward their full ∆Ω for almost 105 sec[23]. (3) V-glitches are observed in mature pulsars (mainly, but not always in Fig. 1 along b → c) while C -glitches are observed in both adolescent and mature pulsars. (4) Eq. 2 for ˙ Ω) ˙ for V-glitches. The existence C-glitches would greatly overestimate (∆Ω/ of a second glitch family, with V-properties, should result from a different kind

3 One

of the larger Crab glitches has been observed throughout its early development[23]. It took a time 8 × 104 sec for the observed ∆Ω to rise to its full glitch value. One possible interpretation of this delay is that this is the time it takes for the suddenly unbalanced force on the core’s n-vortex array to drag this array to its new equilibrium position. The main drag retarding such a repositioning would be from moving the flux-tube array which is forced to co-move with the n-sf vortices. If the very small speed with which the combined vortex-line - flux-tube arrays responds (∼ 10−8 cm/sec) is assumed proportional to the small unbalanced force exerted on them just after the crust-breaking, this would scale to about 103 yrs for fluxtubes to move to their new very distant equilibrium positions when subject to the much stronger forces on them discussed in Sect. 3.

Pulsar Glitches

59

of vortex-driven flux-tube movement in a NS core. If there were no very dense, comoving, flux-tube environment around them, outward moving core-vortices could smoothly shorten and then disappear entirely as they reached the core’s surface at its spin-equator. However, the strongly conducting crust there resists entry of the flux-tube array which the vortices also bring with them to the crust’s base. This causes a pile-up of pushed flux-tubes in a small equatorial annulus, which opposes the final vortex-line disappearance. The final vortexline movement in which they vanish occurs either in vortex-line flux-tube cutthrough events, or, more likely, in a sudden breaking of the crust which has been overstressed by the increasing shear-stress on it from the growing annulus. Giant V-glitches are proposed as such events, allowing a sudden reduction of part of this otherwise growing annulus of excess angular momentum and also some of the magnetic flux trapped within it. These would not begin until enough vortex-lines, initially spread almost uniformly throughout the core, have piled up in the annulus to support the needed stress. This happens when adolescence is completed, i.e. when crust B-field reconnection (maturity) begins. The P at which this occurs may vary very considerably among pulsars, depending on their childhood B and its history. If crust-breaking displacements in such events involve crust movements of about the same size as those in the largest C-glitches (|s| ∼ 3 × 102 cm), and these crust movements (with their associated expulsion of flux and its ultimate reconnection) are responsible for the average decrease in magnetic moment of mature pulsars (Fig. 1 b → c), then the interval between V-glitches −4 Ω |n − 3|−1 . (3) τg ∼ 3 × 10 ˙ Ω For the Vela pulsar this gives an average interval between giant glitches of about 4 years, less than twice that observed. [Depending upon the sign of s, Vela might have n ∼ 3 + 2 = 5 or 3 − 2 = 1 (cf. the footnote in Sect 5.). The τg from Eq.3 is the same.] If the “Big Glitcher"[25], PSR J0537, with its observed n ∼ 7 and tsd = 5 × 103 yrs, is already a “mature" pulsar, its predicted τg = 0.7 yrs. This is in reasonable agreement with its observed τg = 0.4 yrs. The density of excess angular momentum and flux-tubes stored in the annulus is limited by the strong repulsion among flux-tubes when they become so closely packed that average B among them approaches Bc . The annulus builds up to a volume VA where the growing stress from its coupling to the NS crust reaches the crust’s yield-strength. This new crust-breaking epoch is reached after very substantial spin-down, from early adolescence where vortexlines first develop and start their outward movement to the onset of “maturity" where giant glitches and n ∼ 5 begin. Subsequent spin-down continues forced entry into this annulus of small stretches of vortex-lines together with

60


those parts of accompanying flux-tubes they bring into the annulus with them. But new inflow is now balanced by glitch-events which give a comparable loss of flux and angular momentum from the annulus. (The annulus does not grow larger after the crust’s yield-strength is reached.) For a constant density neutron star core and tsd /τg glitches in a spin-down time, each glitch would give a relative jump in observed pulsar spin τg 5VA ∆Ω ∼ × . (4) Ω 2VN S tsd with VA /VN S the ratio of the small annulus volume (VA ) to the volume of the whole NS core (VN S ). Unfortunately the present value of VA /VN S depends on unobserved features in the development of a NS’s core B-field during infancy and childhood. If, for example, that field at the beginning of adolescence is near the surface dipole field B(a) ∼ 2 × 1012 G, and the total volume of the core’s flux-tubes is conserved as they are pushed outward and compacted into VA , then VA /VN S ∼ B(a)/Bc ∼ 2 × 10−3 . Probably more realistically, as discussed in Sect.3, B-evolution in childhood suggests a final core B typically about an order of magnitude larger than the surface dipole when childhood ends. But, during subsequent outward movement and compactification of the core’s flux- tubes, it is likely to be their total number rather than their total volume that is conserved. If so, VA /VN S ∼ [10B(a)/Bc ] ∼ 3 × 10−3 . It is fortuitous that both estimates for the VA /VN S ratio in Eq. 3 happen to agree. With it and Vela pulsar parameters τg = 3yrs, tsd = 104 yrs, the predicted (∆Ω/Ω) ∼ 2 × 10−6 , typical of that of a giant glitch in Vela. For the “Big Glitcher" the predicted (∆Ω/Ω) = 7 × 10−7 , comparable to the 4 × 10−7 of its observed ones[25]. This completes our brief biography of a canonical (usually solitary) pulsar’s magnetic field through all phases of its life. Different kinds of observations, many of which would otherwise seem puzzling, all give considerable support for a very simple model in which the biography of a NS’s magnetic field is closely and simply tied to the history of it’s spin4 .

Acknowledgments I am happy to thank E. Gotthelf, J. Halpern, P. Jones, J. Sauls, J. Trumper, and colleagues at the Institute of Astronomy (Cambridge) for helpful discussions. 4 This glitch model is quite different from the presently widely applied one in which some of the n − sf vortices which are located inside the NS crust and are normally pinned to crust nuclei, collectively un-pin and move outward[26,31]. This sudden movement of crust-vortices reduces the crust’s n-sf angular velocity and spins-up the rest of the NS (cf. ref. 27 for a criticism of this as a basis for a model for giant glitches). In the model proposed here these crustal n − sf vortices do not play an important role in triggering a glitch, but may have observable consequences in post-glitch healing.

Pulsar Glitches

61

References [1] Thompson, R. & Duncan, R. 1996, ApJ, 473, 322. [2] Ruderman, M., Tao, L., & Kluzniak, W., 2000, ApJ, 542, 243. [3] Applegate, J., Blandford, R., & Hernquist, L. 1983, MNRAS, 204, 1025. [4] Flowers, E. & Ruderman, M. 1977, ApJ, 215, 302. [5] Link, B. 2003, Phys. Rev. Lett., 91, 101101. [6] Ruderman, M., Zhu, T., & Chen, K. 1998, ApJ, 492, 267. [7] Ruderman M. 2009 in From X-Ray Binaries to Gamma-Ray Bursts, ed. E. van den Heuvel, L. Kapper, E. Rol, & R. Wijers, ASP conf. Series 308, 251. [8] Lyne, A., Pritcherd, R., & Graham-Smith, F. 1988, MNRAS 233, 267. [9] Kaspi, V., Manchester, R., Siegman, B., Johnston, S., & Lyne, A. 1994, ApJ, 422, L83. [10] Zhang, W., Marshall, F., Gotthelf, E., Middleditch, J., & Wang, Q. 2001, ApJ, 544, L177. [11] Camilo, F. et al. 2000, ApJ, 541, 367. [12] Cordes, J. & Chernoff, D. 1998, ApJ, 505, 315. [13] Lyne, A., Pritchard, R., Graham-Smith, F., & Camilo, F. 1996, Nature, 381, 497. [14] Cheng, K., Ruderman, M., & Zhu, T. 1998, ApJ, 493, 397. [15] Cheng, K., & Ruderman, M. 1993, ApJ, 408, 179 [16] Ruderman, M. 2004 in X-ray and γ-ray Astrophysics of Galactic Sources, Proc. 4th Agile Science Workshop, 2003, ed. M. Tavani, A. Pellizoni, & S. Varcellone, IASF. [17] Jayawardhana, R. & Grindlay, J. 1995, J. Astron. Soc. Pac. Conf. Ser., 105, 231. [18] Becker, W. & Aschenbach, B. 2002, “X-ray Observations of Neutron Stars", Proc. 270 WEHeraeus Seminar, eds. W. Becker, H. Lesch, J. Trumper, MPE Rpt. 278, astro-ph/0208466. [19] Gil, J., & Krawczyk, A. 1997, MNRAS, 285, 561 [20] Pavlov, G., Zavlin, v., & Sanwal, D. 2002, “Thermal Radiation from Neutron Stars," Proc. of ref. [18]. [21] Gullahorn, G., Isaacman, R., Rankin, J., & Payne, R. 1977, AJ, 81, 309.; Demianski, M. & Prószyński, M. 1983, MNRAS, 202, 437. [22] Wong, J., Backer, D., & Lyne, A., 2001, ApJ, 548, 477. [23] Lyne, A., Graham-Smith, F., & Pritchard, R. 1992, Nature, 359, 706. [24] Flanagan, C. 1990, Nature 345, 416; McCulloch, P., Hamilton, P., McConnel, D., & King, E., 1990 Nature, 346, 822. [25] Marshall, F., Gotthelf, E., Middleditch, J., Wang, Q., & Zhang, W. 2004, ApJ, 603, 572. [26] Alpar, A., Chau, H., Cheng, K.S., & Pines D. 1993, ApJ, 109, 345. [27] Jones, P. 1998, MNRAS, 296, 217. [28] Manchester, R. 2004, Science, 304, 489. [29] Lyne, A., Shemar, S., & Graham-Smith, F. 2000, MNRAS, 315, 534. [30] Alpar, A., Chau, H., Cheng, K.S., & Pines D. 1996, ApJ, 459, 706. [31] Anderson, P. & Itoh, N. 1975, Nature, 265, 25.

COUPLING OF RADIAL AND NON-RADIAL OSCILLATIONS OF NEUTRON STARS Andrea Passamonti,1 Marco Bruni,1 Leonardo Gualtieri,2 and Carlos F. Sopuerta1,3 1 ICG, University of Portsmouth, Portsmouth PO1 2EG, Britain 2 Dipartimento di Fisica “G. Marconi”, Università di Roma “La Sapienza” and Sezione INFN ROMA 1, piazzale Aldo Moro 2, I–00185 Roma, Italy 3 IGPG and CGWP, The Pennsylvania State University, University Park, PA 16802, USA

Abstract

This is a progress report on our study of the coupling of first-order radial and non-radial relativistic perturbations of a static spherical star. Our goal is to investigate the effects of this coupling on the gravitational wave signal of neutron stars. In particular, we are looking for the existence of resonances and parametric amplifications, changes in the damping time of non-radial oscillations, etc. To that end, we have developed a formalism that introduces gauge invariant quantities to describe the coupling. Their equations have the same structure as the equations for first-order non-radial perturbations plus some source terms, which makes them very appealing for time domain studies.

Keywords:

stars:neutron – stars:oscillations

1.

Introduction

Neutron stars are important gravitational wave sources both as isolated objects or in binary systems. The analysis of their gravitational radiation could open up a direct window on their interior, possibly revealing details on the equation of state of nuclear matter, the dynamics of the crust-mantle interaction and the inner superfluid/superconducting core. An accurate modeling of sources is crucial to the final end of providing templates of the wave form and spectrum of the signal for the analysis of data that comes from the new generation of gravitational wave detectors. Linear perturbations and instabilities of neutron stars have been studied for a long time [1] but relatively little is known of non-linear dynamical effects (see [2] for references) and therefore second order studies may help to understand known problems and even reveal a new phenomenology.

83 A. Baykal et al. (eds.), The Electromagnetic Spectrum of Neutron Stars, 83–86. © 2005 Springer. Printed in the Netherlands.

84

Coupling of Radial and Non-radial Oscillations of Neutron Stars

Here we outline our work in [2], where we introduced a framework to study the coupling of radial and non-radial first-order relativistic perturbations of static spherical stars. The goal is to study effects of this coupling (possible resonances, parametric amplification, etc.) that could make a significant impact in the gravitational wave signal of neutron stars.

2.

Perturbative Framework

Using 2-parameter relativistic perturbation theory [3]; [4] we parametrize separately the radial and non-radial perturbations. This allows us to split second order perturbations into three differentiated types: second-order radial and non-radial and the coupling between them. We are interested in the third type. The basic ingredients of our perturbative framework are: (i) The background model for the star. We use a Tolman-Oppenheimer-Volkov model, that is, an equilibrium perfect-fluid spherically symmetric configuration. (ii) First-order radial perturbations (see [5] and references therein). These are perturbations preserving the spherical symmetry of the background model. They can be described by just three functions: two matter variables (in our case we choose the enthalpy and the fluid velocity radial component perturbations) and a metric variable. They obey a system of evolution equations containing only first-order time derivatives and subject to a constraint, the Hamiltonian constraint, which can be used to monitor the accuracy of a time-domain integration. (iii) Firstorder non-radial perturbations. Due to the symmetry of the background, these perturbations can be expanded in terms of (tensor) harmonics, so that the angular dependence is explicitly separated. Depending on how they behave under parity transformations they are called polar (even) or axial (odd) perturbations. Every harmonic (the monopole part would correspond to the radial perturbations, so it is not consider here) can be described in terms of gauge-invariant variables by using the formalism developed in [6]; [7]; [8]. These perturbations have been extensively studied in the literature due their interest in relation with the gravitational radiation produced by star oscillations. Most of the works in the past were done in the frequency domain [9]; [10]; [11]. However, recent works have studied them by using a time-domain approach [12]; [13]; [14]. (iv) Coupling of radial and non-radial perturbations. As we have mentioned above, these perturbations are a part of the second-order perturbations of our background. The part that is generated by the coupling of the radial and nonradial first-order perturbations. This is the sector of the second-order perturbations that we have to study in order to look for the physical phenomena we described in the abstract and introduction, and it is the subject of [2]. The structure of these perturbations is very particular. By pure inspection of Einstein’s equations one can see that they are generated by source terms that can be expressed as a sum of products of radial and non-radial first order perturba-

85

Perturbative Framework

tions. This means that they can be also expanded in (tensor) harmonics, which makes the analysis much simpler than the analysis of the whole set of secondorder perturbations. Moreover, in [2] we were able to show that one can also have a gauge-invariant description just by extending in an appropriate way (in particular, by fixing the gauge for the radial perturbations) the formalism for non-radial perturbations introduced in [6]; [7]; [8]. Going into more detail about the structure of the equations for the coupling perturbations, we have seen in [2] that these perturbations, once decomposed in harmonics, obey the same equations as non-radial perturbations do, with the only difference that for the coupling terms we have source terms which, as we mentioned before, can be written as the sum of products of radial and nonradial perturbations. This structure has very important consequences for practical purposes, in particular for the numerical integration of the equations. Indeed, given a numerical code capable of evolving the non-radial perturbations we can construct a code evolving the coupling perturbations just by adding the sources. Having this interesting property in mind, an appropriate way of formulating (choice of variables and equations) the equations for non-radial perturbations would be the one chosen in [14], where the Hamiltonian constraint (an elliptic-type equation) is used to solve for one of the perturbative variables instead of using an evolution equation. Apart from the obvious interest that this procedure has (we make sure that constraints are preserved during the evolution), it has an extra interest thinking on solving also for the coupling perturbations: If we do not solve for the Hamiltonian constraint the errors produced by its violation would increase since we would accumulate the ones coming from the integration of the non-radial perturbations with the ones after solving for the coupling. Therefore, solving for the Hamiltonian constraint can improve substantially the accuracy of the calculations. As it has been shown for the case of non-radial perturbations it can estimate damping times and mode frequencies with an accuracy comparable to frequency domain calculations [14]. Hence, the structure of the system of equations governing the stellar interior is given by a gravitational wave equation for the non conformal-flat metric perturbation S, a sound wave equation for a fluid perturbation H (coincident with the enthalpy perturbation in some particular gauges), and finally the Hamiltonian constraint mentioned above, used to update the value of the conformal-flat metric perturbation k at every time-step, −S,tt + e2(Φ−Λ) S,rr + ...... = e2Φ SS . −H,tt + k,rr +

+ ...... = e SH , 2 (Λ,r + Φ,r ) H + ...... = SHamil , r¯ c2s

c¯2s e2(Φ−Λ) H,rr

2Φ

(1) (2) (3)

where SS , SH , SHamil are the source terms containing the product of first order radial/nonradial perturbations [2].

86

Coupling of Radial and Non-radial Oscillations of Neutron Stars

To sum up, our framework to study the coupling of radial and non-radial oscillations leads to a hierarchy of equations (from the background to the coupling terms). In order to solve them in the time domain we have to pay attention to a number of other important issues. Of particular relevance are the boundary conditions: We need to impose the regularity of the perturbations at the origin, the vanishing of the Lagrangian perturbation of the pressure at the star surface, and the continuity of metric perturbations at the surface (junction conditions), connecting with the exterior, which can be described by the corresponding Zerilli equation, which will propagate the gravitational wave signal. Then, we can use the well-known black-hole perturbation machinery to compute the energy and angular momentum that has been radiated away. A discussion of all these issues can also be found in our first work [2].

3.

Future work

We are presently working in the construction of numerical codes for the time domain integration of the equations derived in [2]. In this work we focused on polar perturbations, which are the most relevant for stars, but we are also exploring the axial case. On the other hand, this work can also be considered as a step towards a more comprehensive study of second order perturbations of compact stars and mode coupling.

Acknowledgments This work has been partially supported by the EU (Research Training Network contract HPRN-CT-2000-00137). CFS was supported by EPSRC and presently by NSF grants PHY-9800973 and PHY-0114375.

References [1] N. Andersson, Class. Quant. Grav. 20, R105 (2003). [2] A. Passamonti, M. Bruni, L. Gualtieri, and C.F. Sopuerta, (2004), gr-qc/0407108. [3] M. Bruni, L. Gualtieri, and C.F. Sopuerta, Class. Quant. Grav. 20, 535 (2003). [4] C.F. Sopuerta, M. Bruni, and L. Gualtieri, Phys. Rev. D 70, 064002 (2004). [5] K.D. Kokkotas and J. Ruoff (2002), gr-qc/0212105. [6] U.H. Gerlach and U.K. Sengupta, Phys. Rev. D 19, 2268, (1979). [7] C. Gundlach and J.M. Martin-Garcia, Phys. Rev. D 61, 084024 (2000). [8] J.M. Martin-Garcia and C. Gundlach, Phys. Rev. D 64, 024012 (2001). [9] K.S. Thorne and A. Campolattaro, Astrophys.J. 149, 591 (1967). [10] S. Detweiler and L. Lindblom, Astrophys.J. 292, 12 (1985). [11] S. Chandrasekhar and V. Ferrari, Proc. Roy. Soc. (London) A 432, 247 (1991). [12] G. Allen, N. Andersson, K.D. Kokkotas, and B.F. Schutz, Phys. Rev.D 58, 124012 (1998). [13] J. Ruoff, Phys. Rev.D 63, 064018 (2001). [14] A. Nagar, G. Diaz, J.A. Pons, and J.A. Font, Phys. Rev. D 69, 124028 (2004); A. Nagar and G. Diaz, (2004), gr-qc/0408041.

III

NEUTRINO ASTROPHYSICS

DRIFTING SUBPULSES IN PSR B0826-34 J. Gil,1 Y. Gupta,2 , J. Kijak,1 and M. Sendyk1 1 Institute of Astronomy

University of Zielona Góra, Poland [email protected] 2 National Centre for Radio Astrophysics, TIFR

Pune University Campus, Pune 411007 India

Abstract

We analyze single pulse data from PSR B0826-34 showing apparent changes of subpulse drift direction, inconsistent with basic pulsar electrodynamics. We demonstrate that this unusual behaviour is not genuine, and results from aliasing in sampling of the intensity fluctuations. Aliasing resolved, drifting subpulses in PSR B0826-34 provide first direct evidence of a system of sparks circulating on the polar cap.

Keywords:

stars:neutron, stars:pulsars:individual PSR B0826-34

1.

PSR B0826-34

The emission in PSR B0826-34 occurs in the entire pulsar period (left panel in Fig. 1), which indicates an almost aligned rotator. Therefore, the observer’s line-of-sight stays in the emission beam for most of the rotational period P1 . This provides a unique opportunity of scanning the polar cap along its circumference and, therefore, detecting radiation from a relatively large number N of subpulse-associated beams of radio emission. These beams are believed to rotate around the magnetic axis in the time interval P4 = N P3 , where P3 is the usual drift periodicity (vertical separation between drift-bands in pulsar periods P1 ). Although the horizontal drift-band separation is known (P2 ≈ 25◦ ), the real value of P3 (and therefore the value of the actual drift rate D0 = P2 /P3 ) cannot be determined without aliasing resolving. However, we can first determine the number of involved subpulse subbeams, using the fact that the magnetic and the spin axes of the pulsar are nearly aligned. In such case, the angular separation between adjacent sparks on the polar cap is almost the same as observed P2 and hence, N ≈ 360◦ /P2 . This yields a value of 14 sparks circulating on the polar cap. 75 A. Baykal et al. (eds.), The Electromagnetic Spectrum of Neutron Stars, 75–78. © 2005 Springer. Printed in the Netherlands.

76

Drifting subpulses in PSR B0826-34

Figure 1. The observed (left panel) and simulated (right panel) subpulse patterns (see text for explanations).

In order to find the value of P4 we have to determined P3 . This requires aliasing resolving, which can be achieved by realistic simulations of radiation associated with these 14 subbeams. However, this in turn requires the knowledge of geometrical parameters, such as: the inclination angle α (between the pulsar’s rotation and magnetic axes), the impact angle β (of the closest approach of the line-of-sight to the magnetic axis) the circulation distance d of sparks from the pole and the emission altitude r at which the subpulse radiation is emitted tangently to dipolar magnetic field lines connected with the sparks. These geometrical parameters can be determined from measurements of the pulse width W and variations of the mean position angle Ψ along the pulse longitude ϕ, as these quantities depend on different combinations of α, β and d. Using all the available observational information, we obtained α = 2.5◦ ±0.2◦ , β = 1.0◦ ± 0.1◦ and d = 0.45rp = 33 m for PSR 0826-34.

Simulations

2.

77

Simulations

We can now simulate the radiation of PSR 0826-34, assuming that its single pulse structure reflects the circumferential motion of 14 sparks at a distance of about 33 meters from the pole. The spark-associated subpulses are emitted tangently to dipolar field lines at an altitude r determined by the conventional radius-to-frequency mapping (Kijak & Gil 1998). The number of subpulses and their phases in a single pulse are determined by the angles α and β, as well as by the angular drift rate Dr = 360◦ /P4 . Since the latter is not known a priori, we performed an experiment with time varying drift rates, starting with Dr = 0 and incrementing it by ∆Dr = 0.03◦ every pulsar period P1 . Such value of ∆Dr was chosen to make sure that cycles of gradual variations of the drift rate have duration of about 100P1 , as observed. The idea is to find a sequence of about 100 pulses with varying Dr = n∆Dr in a range appropriate to produce curved drift-bands similar to those visible in Fig. 1 (left panel). The sample result of our simulations is presented in Fig. 1 (right panel) with all important information written in the top panel and on both sides of the pulse window. The last column indicates the actual value of Dr = n0.03◦ /P1 , where n is the sequential pulse number indicated on the vertical axis. Next to Dr we show values of P4 /P1 = 360◦ /Dr . On the other side of the pulse window we show values of P3 /P1 = P4 /(14P1 ) and the column just next to it shows the fluctuation frequency f3 P1 = P1 /P3 . What we can learn from simulated patterns presented in Fig. 1. First of all, pulse n = 1 shows that were there no drift (Dr = 0), the observer (α = 2.4◦ and β = 1.1◦ ) would clearly see 7 out of 14 sparks in the form of longitude stationary subpulses. As the drift rate increases with the increasing pulse number, the subpulse drift with time varying rate becomes more and more apparent. However, up to about pulse number n = 100, the subpulse drift is relatively slow, non-aliased and proceeds from the leading to the trailing edge of the profile. This is the real drift direction and the observed drift-bands are formed by the same sparks/subbeams. This is, however, not true in the region well above pulse n = 100, where all kinds of stroboscopic effects become visible. We have marked regions where the apparent drift-bands are formed by subpulses appearing at approximately the same phase every m-th pulse period P4 , where m = 5, 4, 3, 2 and so on. It is worth nothing that the number of apparent driftbands is about 7 m. The drift-bands change the apparent drift direction due to the aliasing effect, every time f3 crosses a multiple of the Nyquist frequency. Obviously, the region below pulse n = 800 does not correspond to drifting subpulses in PSR 0826-34, because it shows alternating, longitude stationary intensity modulations, which are not observed in this pulsar. It seems, however, that its drifting subpulses patterns are well modeled by the region between pulses n = 800 and n = 900, which represents just one cycle of the observed

78

Drifting subpulses in PSR B0826-34

multiple curved drift-bands. A clear pattern of seven drift-bands is visible, moving in an aliased direction from the trailing to the leading edge in the first half of the cycle, and in the true direction from the leading to the trailing edge in the second half of the cycle. The drift direction change occurs at f3 = 1/P1 (or P3 = P1 ), which is twice the conventional Nyquist frequency. At this stage the carousel advances exactly by one subbeam per one pulsar period P1 , and the apparent drift-bands are formed by successive adjacent subbeams. The corresponding values of P4 /P1 = 14 and Dr = 25.7◦ /P1 . However, 50 pulses earlier, at the beginning of a cycle P4 /P1 = 14.4 and Dr = 25◦ /P1 , while 50 pulses later, at the end of a cycle, P4 /P1 = 13.3 and Dr = 27.1◦ /P1 . This means that the carousel speeds up along each cycle, increasing Dr by about 8%. This can be converted into drift velocities of sparks circulating at a distance d = 33 m from the pole. Since P1 = 1.84 s and P4 = 2πd/vd , we obtain vd = 7.8, 8.0 and 8.5 m/s at the beginning, at the reversal phase and at the end of a cycle, respectively.

3.

Ruderman & Sutherland model

It is desirable to check whether the derived velocity vd = 8 m/s is consistent with the Ruderman & Sutherland model, in which vd = c∆V /rp Bs ,

(1)

where c is the speed of light, rp = 73 m is the polar cap radius, Bs = 2.7 × 1012 G is the surface magnetic field at the pole, and ∆V = (2π/cP1 )Bs h2

(2)

is the potential drop across the vacuum gap of height h. Thus, the actual drift velocity is vd = (2π/P1 )(h2 /rp ). The height of the gap is approximately equal to the distance between the adjacent sparks. Since at P3 = P1 the sparks cover this distance in exactly one pulsar period P1 = P , the sparks cover this distance in exactly one pulsar period P1 = 1.84 s, moving with vd = 8 m/s, we can reasonably adopt that the effective gap height h = vd P1 = 14.7 m in PSR 0826-34. Thus, the estimated drift velocity vd = 9.8 m/s, in very good agreement with the observationally derived value vd = 8 m/s. The perfect agreement requires h = 13.3 m, which implies the effective potential drop above the polar cap ∆V = 2.6 × 1011 V. To explain the curved subpulse driftbands this potential drop has to vary systematically by several percent during a 100 period cycles. The mechanism of these quasi-periodic variations remains to be understood.

ELEMENTARY PROCESSES IN STRONG MAGNETIC FIELDS Joseph Ventura Department of Physics, University of Crete, and Institute of Electronic Structure and Laser 71003 Heraklion, Crete, Greece [email protected]

Abstract

The magnetic constriction of electronic orbits in strong magnetic fields (SMF) drastically modifies the properties of electronic matter, while SMF will even modify the properties of electromagnetic radiation in vacuum through such processes as polarization, pair creation, and photon splitting. We review the bulk properties of matter in SMF with emphasis on radiative opacities and transport in external magnetic fields appropriate for application to pulsars and magnetars. SMF changes in atomic matter and condensed matter at the surface layers of neutron stars are also touched upon.

Keywords:

stars:neutron – stars:pulsars – stars:magnetic fields

1.

Introduction

Neutron stars were first discovered thirty five years ago as radio pulsars, strongly magnetized, rotating dense stars with surface magnetic fields of the order of a few times 1012 G. The actual presence of such huge fields, first implied by the pulsar spin-down rates, has since been documented by the discovery of electron cyclotron lines at photon energy ωc = 11.6B/B12 keV in the X-ray spectra of about half of the known accreting X-ray pulsars (Trümper et al. 1978, Makishima et al. 1999, Dal Fiume et al. 2000, Ventura 2003), where ωc = eB/me c is the electron cyclotron frequency. In some cases cyclotron harmonics have also been detected (see also Santangelo et al. 1999, Heindl et al. 1999). Candidate magnetars, or neutron stars with magnetic fields two or three orders of magnitude higher than the above, made their observational debut in astrophysics more recently (Kouveliotou et al. 1998), while the presence of such fields in soft gamma repeaters (SGRs) and anomalous X-ray pulsars (AXP) had been theorized several years earlier by Duncan and Thompson 1992, and Paczyński 1992 (see reviews by Woods and Thompson 2004, Kouveliotou 2003, 2005). 63 A. Baykal et al. (eds.), The Electromagnetic Spectrum of Neutron Stars, 63–74. © 2005 Springer. Printed in the Netherlands.

64

Elementary Processes in Strong Magnetic Fields

In pulsar magnetic fields the electron motion transverse to the external field is quantized giving rise to the Landau energy levels, 2 + c2 p2z )1/2 , E = (E⊥

E⊥ = me c2 (1 + 2nb)1/2 ,

(1)

where n = 0, 1, 2, ... is the Landau quantum number, and b = B/Bcr , with Bcr =

m2e c3 = 4.414 × 1013 G e

the electrodynamic critical field. In the non-relativistic limit this reduces to the well-known harmonic oscillator type spectrum, E = me c2 + nωc +

p2z . 2me

> Bcr , the transverse motion of the electron becomes relativistic (i.e. If B ∼ > 1) and the sequence of the Landau levels is un-harmonic. nb ∼ In the presence of these very strong magnetic fields, only the ground state, n = 0, is occupied, and this turns out to have a determining impact on the properties of electronic matter because of the magnetic constriction of the − b−1/2 , electron orbital to a very small gyro-radius am = (c/eB)1/2 = λ C − where λ C = /me c – see reviews in Mészáros 1992 [M92 in the following], Lai 2001 [L01], Canuto and Ventura 1977 [CV].– This brings in turn dramatic changes in the dynamic and radiative properties of plasma, as well as in atomic matter. Atoms become elongated and compact with a drastic increase in their binding energy, and a qualitative change in their energy spectrum. All these changes in atomic and plasma properties are imprinted in the spectra emitted or absorbed. The reduced atomic size also has immediate implications on the condensed magnetic matter which is characterized by much higher densities than the B = 0 case.

2.

Radiative Processes

Radiation processes in nature are usually led by the motion of the electrons, which are the lightest charged particles. Electrons in excited Landau levels spontaneously de-excite in a typical electromagnetic timescale, τc =

3 me c3 −2 = 2.576 × 10−16 B12 s, 4 e2 ωc2

emitting synchrotron/ cyclotron radiation. In a strong magnetic field, the reduced mobility of electrons transverse to the external field results in profound changes in photon polarizations and opacities. Low frequency photons, ω

ωc , having their E-polarization vector perpendicular to B, are thus found to

65

Radiative Processes

interact less, and have a much longer mean free path than in the field free case, and also as compared to photons with their E-vector along the magnetic field (e.g. Canuto et al. 1971, Ventura 1979 [V79]). Even the vacuum acquires drastically altered properties in the presence of intense B fields. The pulsar magnetosphere is thus optically thick to e− e+ pair creation, for photons with energy greater than the pair creation threshold, 2me c2 . For lower energy photons the magnetic vacuum is birefringent due to the action of virtual pairs (e.g., Adler et al. 1970). Energetic (≥ 50 keV) X-ray photons may also not freely escape the magnetosphere, however, due to photon splitting, giving rise to polarization and spectral changes in the escaping radiation (Adler et al. 1970, Thompson and Duncan 1995).

2.1

Photon Propagation and Opacities

Vacuum Birefringence. As mentioned above, the magnetic vacuum is birefringent, and the normal modes propagating therein are linearly polarized with their e-vector parallel or perpendicular to the B-k plane. The corresponding refractive index at low frequencies, ω 2me c2 is independent of the photon frequency: q 2 2 α 2 , (2) b sin θ, with δ = n ,⊥ = 1 + 45π h where α = 1/137, and the quantities q and h take the limiting expressions q 7δ, h 4δ valid at low fields b 1 (Adler et al. 1970, Meszaros and Ventura 1978, 1979 [MV], Gnedin, et al. 1978a, b), while for higher fields b > 1,

2 α 2 α −1 −1 − b + 1.27 − b (0.85 + ln b) , h + b (0.145 − ln b) q 2π 3 2π 3 (Lai and Ho 2002, Heyl and Hernquist 1997).

Magnetoactive Plasma. The photon normal modes propagating in a magnetized plasma are elliptically polarized, with the axis of the polarization ellipse oriented to the field B – ordinary ( ) mode – or ⊥ to the field – extraordinary (⊥) mode (e.g., V79, M92). The refractive index is simply expressed in terms of the polarization tensor, with the electric field polarization vector eiα (i = 1, 2) appearing in cyclic coordinates α = ±, z (in which the polarization tensor is diagonal): n2i

ω2 =1− e ω

|ei− |2 |eiz |2 |e+ i|2 + + ω ω + ωc ω − ωc

.

(3)

66


Figure 1. Angle averaged scattering (σ) and free-free (κ) opacities for the two normal modes in a cold plasma (left), and including vacuum corrections (right). Mode exchange scattering σ12 plays a pivotal role in the radiation transport through birefringent media. (ωH = ωc )

The electron plasma frequency is given by ωe2 = 4πe2 ne /me , and dissipation effects are included through ω = ω + iγr , with the radiation damping rate, γr =

2 e2 ω 2 , 3 me c3

accounting for radiation losses via Thomson scattering. The ion component contribution has been omitted in the above expression for simplicity, but is easily accounted for, becoming important at the ion cyclotron resonance, see e.g. Ho and Lai (2001). It is interesting to note that the normal modes and, therefore, the dispersion relation have a strong dependence on frequency and on the direction of propagation.Thus, if θ is the inclination angle of propagation with respect to the external field B, the normal modes always have circular polarization at θ = 0 ( propagation), while at θ = π/2 (⊥ propagation) the polarization modes are linear, oriented and ⊥ to B – the polarization ellipse having degenerated into a straight line in this case. Between these two limits the normal mode ellipticity varies in a continuous way. At low frequencies, ω ωc , the polarization ellipse tends to be very elongated, and the normal modes are almost linear at all, except the very small, angles. In this case the dispersion relation is largely dominated by the z component in (3), whereas the ± components are of the order O(ω/ωc ).

67

Radiative Processes

The (photon scattering) absorption coefficient is Absorption Coefficient. directly related to the imaginary part of the refractive index (Eq. 3) through the optical theorem, from which we obtain (V79, M92): (i) ω 2 |ei+ |2 ω 2 |ei− |2 σes = |eiz |2 + + σT (ω + ωc )2 (ω − ωc )2 + γr2

(4)

in terms of the usual Thomson absorption coefficient σT = (8π/3)r02 ne , where r0 = me c2 /e2 is the electron classical radius, and ne is the electron number density. Since γr ω, ωc at X-ray frequencies, the quantity γr in the denominators is usually neglected, except in the last term where it is needed to ensure a finite value at resonance. Our previous remarks following Eq. (3) are applicable here as well. The magnetic Thomson opacity has a sensitive dependence on polarization, frequency and angle of propagation. At low frequencies ω ωc , in particular, the normal modes are almost linearly polarized, while Eq. (4) is dominated by the leading term:

= |eiz |2 + O ω 2 /ωc2 implying long mean free paths for the ⊥-mode, and also for propagation almost aligned with B (θ ω/ωc ) for either mode. It is clear

that the opacity of ⊥photons is greatly reduced in the average, O ω 2 /ωc2 , as compared to that of

-photons (σ1 σ2 in Fig.1), and that the magnetized plasma is a strongly polarizing, anisotropic medium. If, on the other hand, the damping rate γr in (3) is replaced by the Coulomb collision rate, Ze4 νC = 2π 2 3 ne ΛC , me vT the imaginary part of (3) will yield an estimate of the free-free absorption coefficient, κ, for a magnetized plasma. A more careful ab initio calculation yields for a thermal plasma at temperature kT (Nagel and Ventura 1983) νC kT κi 1 − e−ω/kT , σi γr ω

(5)

where the index i designates the polarization normal mode (e.g. Gnedin and Sunyaev 1974, MV78). Since γr is proportional to ω 2 , one notes that the plasma is absorption dominated (σ κ) at low frequencies, and scattering dominated (σ κ) at high frequencies. A high frequency photon will thus scatter several times prior to being thermalized with implications on the emitted spectrum. The angle averaged free-free and electron scattering absorption coefficients are shown in Figure 1. The plasma parameters assumed are the same as in Fig. 2.

68

2.2


Radiation from a Plasma Slab

The ratio κi /σi given in the previous paragraph is an important parameter in radiation transport. If κ/σ 1, a photon is very unlikely to scatter before being absorbed and, due to Coulomb collisions, the medium is in local thermodynamic equilibrium (LTE). The medium in this case is said to be absorption dominated, and the local emissivity is given by the Planck spectrum (e.g. Rybicki and Lightman 1979). Photon scattering can be neglected in analyzing this case. If κ/σ 1, on the other hand, a photon will typically scatter many times prior to being absorbed, and the medium is scattering dominated. The medium is then not in LTE and, in the optically thick case, the radiated intensity can strongly deviate from blackbody. The radiated spectral intensity emitted by a uniform, 1 km thick, thermal plasma slab is shown in Fig. 2 (see VNM). For the parameters assumed in Figs.1 and 2, the plasma is found to be scattering dominated above the frequency of 0.1 keV, and absorption dominated for ω < 0.1 keV. At frequencies below 0.1 keV then photon scattering can be neglected. Yet the emission of mode-1 falls well below the blackbody values because the slab is optically thin to mode-1 photons. Above 0.1 kev the medium is scattering dominated, and

Figure 2. Emitted spectra from plasma slab of linear dimension R = 1 km compared to the Planck spectrum (top curve). Spectral intensities are shown for mode 1, mode 2, and total (solid curve). The plasma parameters assumed are ne = 1022 cm−3 , kT = 10 keV and B = 4.41 × 1012 G.

69

Radiative Processes

the emitted spectrum deviates substantially from blackbody. Whereas mode 2 ( ) photons dominate at low frequencies and at resonance, the polarization is reversed at intermediate frequencies. Photons at frequencies above 0.1 keV are typically produced in the short mean free path -mode, and escape after converting to the long mean free path ⊥-mode via mode exchange scattering (σ12 in Fig. 1). Note that Comptonization was not included in the above radiative transfer calculation. Its inclusion would dramatically alter the appearance of the spectral features (e.g. Nagel 1981, Meszaros and Nagel 1985).

Vacuum-Induced Spectral Features. Comparing Eqs. (2) and (3) we see that the plasma polarization term, |n2 − 1|, decreases with photon frequency, while vacuum polarizability is independent of frequency. Thus, depending on the values of B and ne , there is always a frequency ω0 above which vacuum polarization dominates, while plasma polarization dominates below this frequency. With increasing frequency the polarization normal modes are found to change their characteristics smoothly at the transition frequency ω0 with the ⊥ to B mode changing to and vice versa, while passing through circular at ω0 . According to Eqs. (4) and (5) this change in the polarization modes results in accordingly altered free-free and scattering photon coefficients, as seen in Fig. 1b. This is again reflected on the emitted spectra of Fig. 2b (e.g. VNM, M92, Lai and Ho 2002). Note that a vacuum-induced spectral feature appears at ω0 in this uncomptonized simple model calculation. In both the high and low field limits ω0 is found to be proportional to the electron plasma frequency with the limiting values, (15π/α)1/2 b−1 ω0 ={ ωe (3π/α)1/2 b−1/2

for b 1 for b 1.

The polarization of the radiation observed at infinity is also modified at and around ω0 . This is important because the observed X-ray radiation from magnetic neutron stars is expected to be highly polarized, and its polarization is measurable in principle (Gnedin et al. 1978a, Novick et al. 1977, Meszaros et al. 1988). While the emitted radiation is dominated by the ⊥ mode (mode 1 in Fig. 2) at frequencies both below and above ω0 , VNM found that this polarization direction would be preserved for ω > ω0 while, at lower frequencies, it would adiabatically switch into the mode as the radiation propagates through lower plasma densities in the NS magnetosphere. This happens because at high frequencies the emitted radiation is vacuum dominated and remains so all the way to infinity, while at ω < ω0 the initially plasma dominated normal modes will adiabatically change into the vacuum dominated mode while retaining their helicity as the photon propagates through lower densities. X-ray polarimetry has been technically rather limited in the past yielding results only for the very bright Crab Nebula. New technology proposed re-

70


cently (Costa et al. 2001), however, offers much improved expectations in this important field. Polarization propagation through a density gradient has been taken up recently in a much more systematic way, in the context of the neutron star atmosphere density gradient, yielding important new results (see Lai and Ho 2002, 2003). These authors have found that that the quiescent emission from a magnetar atmosphere tends to become spectrally softer due to mode conversion, while spectral features tend to be suppressed (Ho and Lai 2004, Ho et al. 2003). They find that the adiabatic mode conversion is effective in the NS surface layers only for higher energy photons.

3.

Atomic Matter

Atomic matter is dramatically altered in strong magnetic fields, with atoms being constricted to one-dimensional, needle-shaped, and very tightly bound entities aligned with the external field (Ruderman 1974, CV, L01, Ruder et al. 1994). At such high fields the electron occupies the nL = 0 Landau level, its motion limited along the z axis (the direction of the B-field), and may be viewed as one-dimensional. In its ground state, the atom is tiny, being confined within a transverse radius r⊥ = am = αb−1/2 a0 , and a length lz 2a0 / ln (b/α2 ). The quantities a0 , and α denote the Bohr radius, and the fine structure constant respectively. The binding energy of the hydrogen atom at rest is thus found to reach ∼ 250 eV in a 4.7 × 1012 G field, and 710 eV in 4.7 × 1014 G (see e.g. Potekhin 1994 [P94], 1998, L01). As the magnetic field increases, the atomic volume decreases in inverse proportion to ∼ b ln (b/α2 ) with immediate consequences on the atomic density, and the density of condensed matter (see Table 1). Table 1. Magnetic hydrogen atom. Properties of the ground state b = B/Bcr 10−1 1 10

E0 (Ryd)

E0 (eV)

lz /a0

ρa (g/cm3 )

18.02 32.27 52.15

245 439 709

0.198 0.142 0.109

∼ 104 1.3 × 105 1.7 × 106

In obtaining the atomic excitation energy spectrum for the hydrogen atom, it is crucial to include the finite mass of the proton, whose transverse motion is quantized in multiples of the ion cyclotron quantum, ωi = (me /mp )ωc , contributing a significant fraction of the atomic excitation energy – see Fig. 3 (Herold et.al 1981, Wunner et al. 1980, Ventura et al. 1992). A further important complication is the effect of the atom’s motion perpendicular to B. Thermal motion ⊥ to B induces opposing Lorentz forces on the electron and ion which reduce the binding energy, and tend to destabilize the

Condensed State

71

Figure 3. Energy spectrum of hydrogen atom. Bound states as well as continuum bands are shown for several values of the angular momentum Lz number s.

atom (Wunner et al. 1980, P94). Giving, for instance, the atom a transverse kinetic energy of the order of the atomic binding energy will grossly deform the atomic shape making it non-axisymmetric, and reducing its binding energy by a factor of about two (Fig. 3). At typical surface temperatures of a keV for a magnetar, and for a young radio pulsar, one thus expects to have strongly deformed hydrogen atoms in the neutron star atmosphere. Analytic fitting formulas for these atomic properties have been obtained by Potekhin (1998). These deformations result in asymmetric and very strongly broadened atomic lines, and ionization edges (Pavlov and Potekhin 1995, Potekhin and Pavlov 1993, 1997).

4.

Condensed State

Apart from the atomic structure, the bulk properties of condensed matter are also affected by the presence of strong magnetic fields in the NS surface layers. The condensed state generally requires low temperatures, below the Fermi temperature TF . These changes are best illustrated on a free electron gas (e.g. L01, VP03).

72

4.1


Electron Gas in Magnetic Field

The Fermi momentum, pF is given as usual in terms of the electron number density, ne . Under conditions of strong degeneracy a Fermi momentum, pFn , is defined for each Landau level occupied, n∗

1 gn pFn , ne = 2 2 2π am n=0

> 1), where gn denotes the electron spin degeneracy (g0 = 1, gn = 2 for n ∼

1/2 pFn = me c γF2 − 1 − 2nb . Here γF = 1 + χ2 , and χ = pF0 /me c2 denote the dimensionless energy, and momentum while n∗ is the maximum Landau level occupied. The familiar Fermi surface is thus substituted by a discrete sequence of Fermi momenta, pFn , while the Fermi temperature is kTF = me c2 (γF − 1). Numerically one finds TF = 5.93 × 109 χ2 /(1 + γF ) K.

(6)

For a degenerate electron gas the thermodynamic quantities, and those of the bulk properties of matter, such as energy density, magnetization and conductivity, exhibit quantum oscillations of the de Haas – van Alfven type as functions of density, whenever √ the dimensionless Fermi momentum reaches the characteristic values χ = 2nb, which signify the occupation of a new Landau level. Typically in these oscillations the quantities take values not deviating far from their corresponding classical B = 0 values, except in the limit of a strongly quantizing field (n∗ = 0) where one finds substantial deviations. This limit occurs when the typical energies are less than the Landau level energy quantum kT, kTF < ωc , i.e. at T TB and ρ ρB , where ρB = (21/2 π 2 λ3C )−1 mp µe b3/2 2.08 × 106 µe b3/2 gcm3 , TB = ωc /kγF ≈ 1.343 × 108 (B12 /γF ) K, and µe = A/Z is the molecular weight per electron. This limit is of particular interest in defining the properties of the surface layers of magnetic neutron stars.

Strongly quantizing field. When the electron’s transverse motion is frozen in the ground state Landau level, n = 0, the phase space is effectively onedimensional. Then 2 − −1 − )χ λ C ) χ = (b/2π 2 λ ne = (2π 2 αm C 3

− n , so we have as compared to the isotropic B = 0 case, where χ30 = 3π 2 λ C e 3

− 3 n /b = 2 χ3 /b = (0.6846/b)ρ /µ . χ = 2π 2 λ 6 e C e 3 0

(7)

Condensed State

73

Figure 4. Temperature-density diagram illustrating the magnetic field effects on the properties of a free electron gas in strong magnetic fields. Solid curves, B = 1012 G; short dashed lines, 1013 G; long dashed lines, 1014 G. [L01]

The Fermi temperature given by Eq. (6) is also applicable in the B = 0 case, but with χ0 replacing χ and, since χ/χ0 = (4/3)1/3 (ρ/ρB )2/3 , one can easily see that TF is strongly reduced at ρ ρB . An initially degenerate electron gas at a given density and temperature can thus become nondegenerate when a quantizing magnetic field is switched on. Conversely, for a given temperature T < TB degeneracy will take hold at an increased density 1/2 ρ = 0.19 × 105 b T8 g/cm3 increasing proportionally with B. This is nicely demonstrated in Fig. 4, where the vertical long-dashed line gives for B = 1014 G the density ρB at about 7 × 106 g/cm3 , and degeneracy at 108 K setting in at density of ∼ 106 g/cm3 . Once again we note that the condensed matter state in strong magnetic fields is characterized by very high densities. This has led some authors to investigate the rates at which cold fusion would take place in magnetic molecular chains at the surface layers of quiescent magnetars (Lai and Salpeter 1997).

References S.L. Adler, J.N. Bahcall, C.G. Callan, M.N. Rosenbluth 1970, Phys. Rev. Lett. 25, 1061 S.L. Adler 1971, Ann. Phys. 67, 599 V. Canuto, J. Lodenquai and M. Ruderman 1971, Phys. Rev. D 3, 2303 V. Canuto and J. Ventura 1977, Fundam. Cosm. Physics, 2, 203 [CV] E. Costa, P. Soffita, R. Bellazzini, A. Brez, G. Spandre 2001, Nature 411, 662 D. Dal Fiume, M. Orlandini et al. 2000, Adv. Sp. Res. 25, 399 R. Duncan and C. Thompson 1992, ApJ 392, L9

74


Yu.N. Gnedin and R.A. Sunyaev 1974, A&A 36, 379 Yu.N. Gnedin, G.G. Pavlov and Yu.A. Shibanov 1978, JETP Lett. 27, 325 Yu.N. Gnedin, G.G. Pavlov and Yu.A. Shibanov 1978a, Sov. Astron. Lett. 4, 117 J.S. Heyl and L. Hernquist 1997, J. Phys. A 30, 6485 W.A. Heindl et al. 1999, ApJ 521, L49 H. Herold, H. Ruder and G. Wunner 1981, J. Phys. B 14, 751 W. Ho and D. Lai 2002, ApJ 607, 420 W. Ho, D. Lai, A.Y. Potekhin, G. Chabrier 2003, ApJ 599, 1293 W. Ho and D. Lai 2004, ApJ 607, 420 C. Kouveliotou, et al. 1998, Nature 393, 235 C. Kouveliotou 2003, in From X-ray Binaries to Gamma-ray Bursts, edt. E.P.J. van den Heuvel, et al., ASP Conference Series, Vol. 308 C. Kouveliotou 2005, These proceedings D. Lai 2001, Revs. Mod. Phys. 73, 629 [L01] D. Lai and E.E. Salpeter 1997, ApJ 491, 270 D. Lai and W. Ho 2002, ApJ 566, 376 D. Lai and W. Ho 2003, ApJ 588, 962 K. Makishima, T. Mihara, F. Nagase, Y. Tanaka; 1999, ApJ 525, 978 P. Mëszäros 1992, High Energy Radiation from Neutron Stars, U. of Chicago Press [M92] P. Mëszäros and J. Ventura 1978, Phys. Rev. Lett. 41, 1544 [MV] P. Mëszäros and J. Ventura 1979, Phys. Rev. D 19, 3565 P. Mëszäros, W. Nagel and J. Ventura 1980, ApJ 238, 1066 [MNV] P. Mëszäros, R. Novick, A. Szentgyorgyi, G.A. Chanan, M.C. Weisskopf 1988, ApJ 324, 1056 W. Nagel 1981, ApJ 251, 278 W. Nagel and J. Ventura 1983, A&A 118, 66 R. Novick, M.C. Weisskopf, J.R.P. Angel, P.G. Sutherland 1977, ApJ 215, L117 B. Paczyński 1992, Acta Astron. 42, 145 G.G. Pavlov and A.Y. Potekhin 1995 ApJ 450, 883 A.Y. Potekhin 1994, J. Phys. B 27, 1073 A.Y. Potekhin 1998, J. Phys. B 31, 49 A.Y. Potekhin and G.G. Pavlov 1993, ApJ 407, 330 [P94] H. Ruder, G. Wunner, H. Herold and F. Geyer 1994, Atoms in Strong Magnetic Fields (SpringerVerlag, Berlin) G.B.Rybicki and A.P. Lightman (1979), Radiative Processes in Astrophysics, Wiley, New York. M. Ruderman 1974, in Physics of Dense Matter, IAU Symposium No. 53, edt. C.J. Hansen (Dordrecht), p. 117 A. Santangelo et al. 1999 ApJ 523, L85 C. Thompson and R. Duncan 1995, MNRAS 275, 255 J. Trümper et al. 1978, ApJ 219, L105 J. Ventura 1979, Phys. Rev. D 19, 1684 [V79] J. Ventura, W. Nagel and P. Mëszäros 1979, ApJ 233, L125 [VNM] J. Ventura, H. Herold, H. Ruder, F. Geyer 1992, A&A 261, 235 J. Ventura and A.Y. Potekhin 2001, in The Neutron Star – Black Hole Connection, edt. C. Kouveliotou et al., Kluwer Academic Publishers [VP03] J. Ventura 2003, in From X-ray Binaries to Gamma Ray Bursts, edt E.P.J. van den Heuvel et al., ASP Conference Series, Vol. 308 P.M. Woods and C. Thompson 2004, astroph/0406133 G. Wunner, H. Ruder and H. Herold 1980, Phys. Lett. 79A, 159

NEUTRON STAR SUPERFLUIDITY, DYNAMICS AND PRECESSION M. Ali Alpar Sabancı University Orhanlı, Tuzla, 34956 Istanbul, Turkey [email protected]

Abstract

Basic rotational and magnetic properties of neutron superfluids and proton superconductors in neutron stars are reviewed. The modes of precession of the neutron superfluid are discussed in detail. We emphasize that at finite temperature, pinning of superfluid vortices does not offer any constraint on the precession. Any pinning energies can be surmounted by thermal activation and there exists a dynamical steady state in which the superfluid follows the precession of the crust at a small lag angle between the crust and superfluid rotation velocity vectors. At this small lag the system is far from the critical conditions for unpinning, even if the observed precession of the crust may entail a large angle between the figure axis and the crust’s rotation velocity vector. We conclude that if long period modulations of pulse arrival times and pulse shapes observed in a pulsar like the PSR B1828-11 are due to the precession of the neutron star, this does not have any binding implications about the existence of pinning by flux lines or the existence of Type II superconductivity in the neutron star.

Keywords:

stars:neutron – stars:pulsars – superfluidity – superconductivity – precession

1.

Introduction

This lecture consists of two parts. The first part is a general review of neutron superfluidity and proton superconductivity, and the determining effects of superfluidity and superconductivity on neutron star dynamics. The second part is a discussion of neutron star precession and possible constraints imposed by superfluidity on precession, with a critical review of recent inferences on the nature of proton superconductivity in neutron stars based on the occurrence of precession. In Marmaris I devoted roughly half of the lecture to the first part. In the written version here I will summarize this general discussion, and supplement it with references to lectures on neutron star superfluidity and superconductivity in previous ASI of this series (Sauls, 1989; Pines, 1991; Alpar, 1991, 1995, 1998, 2001). Taken together, those lectures form a fairly compre33 A. Baykal et al. (eds.), The Electromagnetic Spectrum of Neutron Stars, 33–46. © 2005 Springer. Printed in the Netherlands.

34

Neutron Star Superfluidity, Dynamics and Precession

hensive introduction to the subject. The bulk of the exposition here will be devoted to a discussion of precession, on the lines and level of detail of the original lecture.

2.

Superfluidity and Superconductivity in Neutron Stars

Like all interacting fermion systems at low temperature, neutrons and protons in the neutron star interior are expected to be superfluid. Neutron superfluidity exists in two different regimes. In the inner crust regions of the neutron star, neutrons are found in bound states in the nuclei that form a solid lattice in the crust, as well as the continuum Bloch states of the crust lattice. The neutrons in the continuum states form a superfluid that coexists with the lattice. This coexistence determines the particular dynamical properties of the neutron star. The neutron and proton superfluids in the core of the neutron star exist in a homogeneous medium where their dynamical properties are very different from those of the crust superfluid. The minimum free energy state of any rotating system is the state of rigid body rotation. A charged superfluid - a superconductor - like the protons in the neutron star core achieves the state of rigid body rotation by setting up a magnetic field called the London field, of magnitude BLondon = (2mc/e)Ω sustained by supercurrents at the boundaries of the superconducting region. The energy cost of the London field is negligible compared to the free energy gain of the rigid rotation state. A London field of only 10−4 G is needed for neutron star rotation rates Ω ∼ 100 rad s−1 . The neutron superfluid achieves a minimum energy state very close to the state of rigid body rotation by forming an array of quantized vortices, each carrying a quantum of vorticity κ = h/2mn where h is the Planck constant and mn is the neutron mass. A uniform area density of vortices n0 = 2Ω/κ sets up a macroscopic rotation state of the superfluid that is the state of rigid rotation almost everywhere, except for the vicinity of individual vortex lines. The cores of the vortex lines, within a distance ξ ∼ EF /(kF ∆) from the vortex axis, are populated with neutrons in the normal phase. Here EF and kF are the Fermi energy and wavenumber of the neutrons and ∆ is the neutron superfluid’s energy gap. It is the interaction of these vortex cores with the normal matter around them that conveys to the superfluid any differential rotation with respect to the normal matter components of the star, including the outer crust. The motion of the vortices in response changes the rotational state of the superfluid to follow the spinup or spindown of the outer crust and normal matter under the external torques on the star. In the core superfluid the neutron vortices interact with a continuous, homogeneous system of electrons and protons. The dynamics of the neutron superfluid in the core of the neutron star is determined by the vortex-electron interaction, which is expected to

Superfluidity and Superconductivity in Neutron Stars

35

be very efficient (Alpar & Sauls 1988) on both observational and theoretical grounds. This strong coupling results from a spontaneous magnetization of the neutron vortex lines (Alpar, Langer & Sauls 1984) and is expected to achieve rotational equilibrium between the neutron superfluid and the normal matter including the outer crust of the star with a time lag τ ∼ 400 P only, where P is the rotation period of the neutron star. In the inner crust vortex lines can get pinned to the nuclei forming the crust lattice. Pinning will constrain the rotation rate of the superfluid, and leads to a lag between the rotation rate of the superfluid and the normal matter outer crust of the star as the normal matter spins down (or up) in response to external torques. The vortex lines will then have an average flow rate through thermal activation against the pinning energy barriers (Alpar et al. 1984a,b, Alpar, Cheng & Pines 1989). By this thermal creep mechanism the pinned crustal superfluid can also follow the spindown (or up) of the star in response to external torques. In fact, in the rotational equilibrium state for the pinned crust superfluid, as well as for the homogeneous core superfluid, the superfluid and the normal matter components spin down (or up) at the same rate. Observations of glitching pulsars indicate that these neutron stars are always close to their rotational equilibrium state. Magnetic free energy is minimized in the state with a uniform magnetic field distributing the total flux throughout the body of the system. For most observed stars the required magnetic field that corresponds to the dipole magnetic moment of the star is of the order of 1012 G, in the direction of the magnetic moment. (The magnetic moment constraint is distinct from the rotational constraint which, for the protons, is met by the tiny London field.) The superconducting protons can get very close to the free energy minimum by sustaining a uniform array of flux lines each carrying a quantum of flux Φ0 = hc/2e at an area density nf corresponding to the required uniform field B. This density of flux lines is nf = B / Φ0 . The central structure of the flux lines contain normal matter within a length scale ξ and magnetic field within a length scale λ ≡ c(4πnp e2 /mp )−1/2 called the ”London penetration depth”. Here c is the velocity of light, np , mp and e are the proton number density, mass and charge respectively. As is the case with the neutron vortex lines, there is an energy cost associated with the core of each flux line. Wherever a flux line and a vortex line intersect, the junction formed will represent an energy gain because of the overlap of the flux line core with the vortex line core. This means the two lines are pinned to each other at the junction. Any change in the rotational state of the neutron superfluid has to involve a motion of the vortex lines, which is not possible, topologically, without encountering junctions with flux lines and therefore being hindered by pinning energy barriers. This should have important consequences for the short term rotational behaviour as well for the rotational and magnetic evolution of the star. Ruderman, Zhu & Chen

36


(1998) have argued that the pulsar spindown will proceed with the motion of the vortex lines either carrying pinned magnetic flux lines with them, in the case of pulsars of the age of the Vela pulsar (∼ 104 yrs) or older, or just cutting through the flux lines in the case of younger pulsars like the Crab pulsar.

3.

Precession

The precession of neutron stars first attracted attention in connection with the 35 d cycle of Her X-1 which is likely to involve the precession of the accretion disk which might be coupled to the precession of the neutron star. The involvement in precession of the pinned superfluid in the neutron star crust was discussed in some detail. Shaham (1977) pointed out that pinning will keep the angular momentum of the pinned superfluid fixed in the body frame (the rotating and precessing frame of the neutron star’s solid crust), and that such a constraint will require ωpr /Ω ∼ Ip /I. In the case of pinning the precession frequency would be determined by the ratio of pinned moment of inertia Ip to total moment of inertia. Alpar & Ögelman (1987) showed that this constraint will be avoided in practice because at finite temperature vortex creep can provide a steady state in which the rotation velocity vector Ωp of the pinned superfluid follows the precession of the rotation velocity vector Ωc of the crust. The vector lag Ωc − Ωp that drives the precession by creep is perpendicular to Ωp , and has a small magnitude. Alpar & Ögelman (1987) also showed that precession by vortex creep has a steady state only in the linear regime of vortex creep, with a lag small compared to ωcr for unpinning. (The nonlinear regime of vortex creep in a spinning down pulsar, in which the lag in steady state is close to but slightly smaller than ωcr , does not have an analogue for precession by creep.) Periods and amplitudes that are consistent with precession driven by the expected effective triaxialities for neutron stars do not necessarily rule out the presence of pinning. In short, at finite temperatures pinning constraints are not absolute. Furthermore, when steady state creep allows the pinned superfluid components to take part in precession, the steady state lag is much smaller than the lag needed for unpinning, and spontaneous unpinning is unlikely. These early discussions investigated the pinned crust superfluid, and possibility of pinning in the core superfluid at vortex line-flux line junctions was ignored. However, these arguments are equally applicable to any site of pinning. In recent years the issue of neutron star precession has attracted renewed attention as a result of observations of what seems to be free precession in the radio pulsar PSR B1828-11 (Stairs, Lyne & Shemar 2000, Hobbs et al. 2004). The evidence comes as correlated periodic changes in the pulse shape and arrival times of the pulses at the rotation period of the radio pulsar. This is naturally explained in terms of the pulsar beam nodding about the line of

Precession

37

sight as the neutron star precesses. Another line of explanation could invoke the periodic migrations of magnetic field patterns rooted on the neutron star surface, roughly in analogy with the model for the bursting, accreting pulsar GRO J1744-28 (Miller 1996). Here we will discuss neutron star precession, and application of precession models to the case of PSR B1828-11. An important pinning constraint would arise from the pinning of neutron vortex lines in the core of the neutron star by their junctions with the proton flux lines. The importance of this interaction for neutron star dynamics was first pointed out by Sauls (1989). In view of the fact that the potentially pinned core neutron superfluid comprises most of the neutron star’s moment of inertia, Shaham’s absolute pinning argument would require a precession period that is comparable to the rotation period, which is not the case observed. The amplitude angles of precession required to fit the observed wandering of pulse arrival times are larger than the critical angle between the rotation axis of the pinned superfluid and that of the crust and normal matter under absolute pinning. Vortex lines would unpin, but this would not free the superfluid from the absolute pinning constraint as the vortex lines would encounter new pinning junctions with flux lines. This process is highly dissipative. The observed amplitudes of precession would not be sustained by an absolutely pinned superfluid if the angle or rotation rate difference between the crust, together with its pinned lines, and the superfluid is of the order of Ω times the observed precession amplitude right from the beginning of the precession of the crust. This argument, relying on absolute pinning, was invoked to conclude that the core neutron superfluid is not pinned, and therefore that the proton superconductor does not contain quantized flux tubes that would pin the neutron superfluid’s vortex lines. Thus it is claimed that the proton superconductor in neutron stars is Type I and not Type II (Link 2003). In a Type I superconductor magnetic free energy is minimized by alternating macroscopic layers of magnetized normal matter and diamagnetic regions where the protons are superconducting. This conclusion, if definite, would have important consequences. Calculations of the proton superconducting energy gap at the high density neutron star interior are highly uncertain because of our incomplete knowledge of the strong interaction and because of the difficulties of the many body calculations at these supernuclear densities. Evidence for Type I rather than Type II superconductivity would reverse the constraint on the √ proton superconductivity gap ∆p (ρ) provided by the condition λ < (>) ξ/ 2 for Type I (Type II) superconductivity. Thus ∆p (M eV ), the value of the proton energy gap in units of MeV, would be less than ( greater than) 0.09 (x/0.05)5/6 (ρ14 )5/6 for Type I (Type II) superconductivity - here x is the ratio of the number density of protons to the number density of nucleons, and ρ14 is the density in units of 1014 gm cm−3 . More importantly for astrophysics, the existence of pinning between vortex and flux lines has as yet not fully explored consequences for

38


neutron star dynamics and evolution. Particularly interesting is the idea that flux vortex pinning would induce magnetic field decay coupled to the spindown of the neutron star on evolutionary timescales, providing an intriguing qualitative explanation for the prevalence of low magnetic fields among old neutron stars in low mass X-ray binaries and millisecond pulsars (Srinivasan et al. 1990). This attractive possibility is not viable if neutron stars have Type I proton superconductors. But the entire argument leading to these sweeping conclusions relies on the prevalence of absolute pinning, as would hold only at absolute zero temperature. The allowance of precession by vortex creep means a loophole in the argument. Jones (2004) has already commented that consideration of pinning and energy dissipation in the crust superfluid shows that any precession would be damped. A careful discussion by Alpar & Ögelman (1987) showed that vortex creep in the crust superfluid and dissipative vortex electron coupling in the core superfluid make precession possible, but also provide strong damping. Here we extend the discussion of Alpar & Ögelman (1987) to the consequences of pinning of neutron vortices in the core superfluid at junctions with the proton flux lines. Let us now examine the conditions for precession by creep more carefully. We shall take a two component model for the neutron star. The component ext consists of the crust, norwhich is coupled directly to the external torque N mal matter including the electrons in the core and the proton superconductor together with its flux lines. The other component, which contains most of the moment of inertia, is the neutron superfluid in the core of the star. The two int . We label the two components are coupled through an internal torque N components with c for crust and n for the neutron superfluid in the core. The Euler equations describing the motion of the rotation vectors Ωc and Ωn are: c dΩ c ×L c = N ext + N int +Ω dt n dΩ c ×L n = −N int +Ω In dt Ic

(1) (2)

c and L n are the angular momenta of the two components. where L Let us note, as a generality, that the precession of any fluid components of a body are driven by internal torques which depend on rotational velocity lags between the solid and the fluid. (We shall discuss specific models of this velocity dependence below.) With velocity dependent torques, energy will be dissipated and the precession of the fluid will inevitably be damped. In the case of strict pinning of the vortex lines the angular momentum of the superfluid in the body (crust, c ) would remain constant. Adding the Euler

39

Precession

equations for the two components, with the condition In Ic

n dΩ dt

=

n dL = 0, dt

c dΩ c × (L c + L n) = N ext +Ω dt

(3) (4)

is obtained as the equation governing the precession of the neutron star with ext = 0. pinned superfluid. The case of free precession is obtained by setting N As suggested by the form of Eq.(4), the precession frequency under pinning is of the order of (In + Ic )/Ic times the rotation frequency Ω. The solution of Eq.(4) gives precession at the frequency: ωpr = Ω +

In Ω Ic

(5)

for any precession angle. The distinction between Ωc and Ωn is negligible here. If the core neutron superfluid is absolutely pinned to the crust system, which includes the charged proton superconductor, as a result of the absolute pinning of the neutron vortex lines to the proton flux lines, Eqs. (4) and (5) would apply. As the ratio of the core neutron superfluid and the crust (including all normal matter and charged components ) moments of inertia is of the order of 10 or maybe more the inferred precession of PSR B 1828-11, with ωpr ∼ = 7.7 × 10−8 clearly cannot be supported by neutron stars with absolutely pinned core superfluids, or for that matter, with pinned crust superfluids (Alpar and Ögelman 1987). At finite temperature a superfluid can change its rotational state by vortex creep even in the presence of pinning. As noted above the theory of vortex creep was developed first in the context of pulsar spindown, glitches and postglitch response. The work of Alpar and Ögelman (1987) applied the vortex creep model to the precession of the inner crust superfluid in which the vortices are pinned to the nuclei that form the crust lattice. The core superfluid was treated in terms of the dynamical coupling that arises from the scattering electrons ( and charge-coupled superconducting protons) by the spontaneously magnetized neutron vortex cores. The magnitude of the damping torque for the core superfluid, and therefore the magnitude of the external counter-torque that would be needed to keep the apparently free precession behaviour of Her X-1 in its 35 d cycle were estimated, and found to be commensurate with the torques available from the accretion disk, though the vector matching, and locking in, of the external torque at the right phase to keep driving precession might be difficult. It is now considered more likely that the 35 d cycle of Her X-1 is actually due to the modulation of the accretion column and the emerging beam of radiation by the precession of the accretion disk, not necessarily

40


entailing any precession of the neutron star. The contribution of the Alpar and Ögelman paper is the demonstration that analogously with the situation for spindown (or spinup) vortex creep enables a pinned superfluid to take part in precession too. Let us now write the Euler equations for our two component model, taking into account only the core superfluid, which contains most of the moment of inertia of the star and modeling the internal torque that couples the pinned superfluid to the crust as due to the angular momentum transfer through vortex creep: n dΩ c ×Ω n = − Nint = Ωc − Ωn +Ω (6) dt In τ Here we have taken the moment of inertia In to be isotropic. Alpar and Ögelman have shown that steady state creep for precession has to be in the linear n , is perpendicular to the c ×Ω regime: The second term on the left hand side, Ω c ×Ω n Ωc Ωn plane, while the torque Nint has to be in that plane. Therefore Ω must be balanced by a component of the first term dΩn /dt. In the nonlinear n | ∼ ωcr Ω, which is much c × Ω creep regime steady state the magnitude |Ω n /dt| ∼ ωpr Ω. Hence Eq.(6) larger than the magnitude of the first term, |dΩ does not have a steady state creep solution in the nonlinear regime. The form of the internal torque is therefore given as appropriate for the linear regime in n )/τ c − Ω the last equality in Eq. (6). Thus the torque is proportional to (Ω where τ is the linear creep regime relaxation time. The component of this n describes spindown or spinup, while the component equation parallel to Ω perpendicular to Ωn is of interest here, as it describes precession: Ωn

ˆn dΩ c ×Ω n = (Ωc − Ωn )⊥ . +Ω dt τ

(7)

n ", and Ω ˆ n is the unit vector for Ω n. The subscript ⊥ means "perpendicular to Ω In a steady state the rotation vector Ωn (t) for the pinned core superfluid would c (t) of the crust in the body follow the precessing rotation velocity vector (Ω frame with a fixed angular relation. We try the solutions c (t) = Ωc [sin α cos(ωpr t), sin α sin(ωpr t), cos α] Ω n (t) = Ωn [sin β cos(ωpr t − φn ), sin β sin(ωpr t − φn ), cos β] Ω

(8) (9)

In these trial solutions we have taken the precession of the crust as given. The precession frequency ωpr of the crust is set by the crust’s effective triaxiality. c and Ω n precess with respective cone angles α and The rotation vectors Ω β ≡ α − around the crust figure axis, as shown in Fig. 1. The angle φn is the lag with which the pinned superfluid’s angular velocity follows behind that of the crust in precession. We find that a solution for steady

41

Precession

Figure 1. Precession of PSR B1828-11. In the corotating frame of the crust precession is around the figure axis zˆ. In the inertial frame precession takes place around the Precession axis, c. which is coplanar with zˆ and Ω

state precession of the fluid component exists, with

∼ = φn ∼ =

ωpr sin α Ω ωpr ⊥ r c −Ω n )⊥ | Ep |(Ω v0 = ) exp (−Ep /kT ) 2 sinh( r kT ωcr c −Ω n )⊥ | v0 Ep |(Ω ∼ exp (−Ep /kT ). = 2 r kT ωcr

ωpr =

(13)

Here r denotes the distance from the rotation axis, Ep denotes the pinning energy at each vortex line-flux line junction and v0 an average microscopic fluctuation velocity of the vortex lines (Alpar et al. 1984a). The geometry is as shown in Fig. 2b of Alpar and Ögelman (1987). The last equality is valid in the linear regime. Using Eqns (12 ) and ( 13), we write ˙ n,⊥ = (Ωc − Ωn )⊥ Ω τ⊥

(14)

where the timescale for the linear response of precession to the lag (Ωc − Ωn )⊥ is τ⊥ =

r kT ωcr kT exp(Ep /kT ) = exp(Ep /kT ). 2v0 Ep Ωn,⊥ 2v0 ρκλf Ωn,⊥

(15)

Here the ratio between the critical frequency for unpinning ωcr and the pinning energy Ep has been expressed in terms of the superfluid density ρ, the vortex quantum κ, the length scale across a pinning junction, which in the present case is the London length λ and the distance between successive pinning junctions along a vortex line, which is simply the distance f between flux lines. In steady state precession, using Eqs. (13) and (14), n )⊥ | = ωpr Ω sin α τ⊥ . c −Ω |(Ω

(16)

The requirement that the creep by precession is actually in the linear regime means that the argument of the sinh in Eq. (14) must be less than 1, c −Ω n )⊥ | Ep ωpr Ω sin α τ⊥ Ep |(Ω = < 1. kT ωcr kT ωcr

(17)

43

Precession

From Eqs. (16) and (18) we obtain Ep < kT ln(

2v0 ) rωpr

(18)

To estimate the temperature kT in the neutron star core we start with an estimate of the surface temperature Ts . The dominant contribution to energy dissipation in the neutron star is due to the energy dissipation in vortex creep for the spindown of the pinned crust superfluid. Energy dissipation rates in all other dynamical couplings between the normal matter and the pinned crust and core superfluids, including spindown or precession by creep are negligible in comparison. This can be easily verified by calculating the energy dissipation rate E˙ diss for each process. For neutron stars past their initial cooling stage, the surface temperature can be estimated by equating the energy dissipation rate from the spindown of the pinned crust superfluid to the blackbody luminosity of the neutron star surface (Alpar et al. 1984a), ˙ = 4πR2 σTs 4 . E˙ diss = Ip ωcr |Ω|

(19)

Here Ip ∼ 1043 gm cm2 is the moment of inertia of the pinned crust superfluid. ωcr ≤ 1 rad s−1 is the estimated steady state lag for spindown by creep, which should be prevalently in the nonlinear regime for this pulsar (Alpar, Cheng & Pines 1989). On the right hand side, R is the neutron star radius and σ is the Stefan-Boltzmann constant. For PSR B1828-11, we estimate the surface temperature as Ts = 5 × 104 K. Using the Gudmunsson, Pethick and Epstein (1982) relation between the surface temperature and the core temperature of a neutron star, we estimate the core temperature as kT ∼ = 2.3 keV. Using v0 ∼ 106 cm s−1 , and r∼106 cm in Eq.(19), we find that the typical pinning energy Ep at each vortex line-flux line junction must be less than about 40 keV in order for precession by vortex creep, which, as mentioned above, has to be in the linear regime to be possible at the estimated temperature of PSR B1828-11. This is substantially less than earlier estimates of Ep ∼ 1 MeV. However, there are many uncertainties in Ep . There is no real calculation, but qualitatively, effects of tension in vortex and flux lines, of small junction angles achieved by bending and of collective effects involving small angle pinning of a vortex line to many flux lines making up a very weakly defined junction, analogous to superweak pinning in the case of vortex-lattice interactions in the crust (Alpar et al. 1984b), all point towards weaker pinning. A particularly important consideration is the fact that the core neutron superfluid is dynamically coupled to the crust and normal matter electrons with a short coupling time of the order of only 400 times the rotation period. The proton superconductor and the flux lines anchored in it must also follow the motion of the electrons and therefore the crust because of very tight electromagnetic coupling. The flux lines and vortex lines will therefore have reduced relative velocities and

44


effective pinning energies - the electromagnetic forces tend to enforce corotation of the two line systems and will therefore reduce the pinning energies. We therefore conclude that a steady state of precession in the presence of pinning might well be possible for the core superfluid in the neutron star. The linear creep regime has a steady state lag that is much less than the critical lag for unpinning, so that if the steady state precession by creep can be attained the pinned superfluid will be far from unpinning conditions. The precession of the crust may have been triggered by a rare event, like a glitch, that offset the pinned superfluid and the crust rotation rates to a level comparable to unpinning conditions. In that case the initial response of creep in the pinned superfluid will be in the nonlinear regime, which does not have a steady state. This initial response will be in the form of rapid vortex creep which will bring the lag down, closer to the steady state in the linear regime. The timescale to relax into steady state will be of the order of the linear regime relaxation time τ⊥ . For the steady state to be reached within one period of the crust precession, we have the condition: τ⊥ ≤ Ppr ∼ 1000days,

(20)

for PSR B1828-11. This translates into the requirement Ep ∼ = 47keV + 2.3keV (1/2 ln(ρ14 /B12 ))

(21) s−1

at the estimated core temperature of 2.3 keV, using Ω 16 rad for PSR ∼ B1828-11 and appropriate values of λ = and of f normalized for ρ = 1014 g cm−3 and B = 1012 G. We already found that steady state precession by creep is possible if typical pinning energies Ep are less than about 38 keV. The condition for the existence of steady state precession by creep satisfies the condition to reach such a steady state within one precession period. Once precession is set up in an isolated neutron star like PSR B1828-11, for how long can it survive? Jones (2004) has noted that it would take only a small fraction of the pinning energy per junction to be dissipated in each precession period for precession to be damped. Thus, the observed timing and pulse shape excursions might be due to overdamped precession in the presence of pinning, and the observation of such a transient would not imply that pinning and Type II superconductivity do not exist. Let us now take a careful look at the damping of precession in terms of the coupling mechanisms between the pinned core superfluid and the crust. Fluids will damp precession as they can freely adjuct their shape. The timescale τpr for damping precession is expected to satisfy the Bondi-Gold relation ωpr τpr ∼ Ωτrot

(22)

where τrot is the dynamical coupling or damping time between the rotation rates of the solid and fluid components of the star. For the core neutron superfluid, the shortest rotational coupling/damping time is not due to the vortex

45

Precession

line-flux line interactions, but rather due to the vortex line-electron interactions. As we mentioned in Section 1, the core neutron superfluid is dynamically coupled to the crust and normal matter through the interactions of the spontaneously magnetized neutron vortex lines and the charged particles (electrons and superconducting protons which are electromagnetically very tightly coupled with the electrons). This coupling time is expected to be τrot ∼ 400 rotation periods from theory (Alpar & Sauls 1988). This is supported by observational bounds from the Vela pulsar glitches (see Dodson, McCulloch & Lewis for the Vela glitch with the tightest observational resolution). The associated precession damping time is τpr ∼ 400Ppr ∼ 400yrs

(23)

for PSR B1828-11. While this is only a small fraction of the pulsar’s lifetime, given the total amount of observation time that can resolve precessional excursions devoted to the entire pulsar sample, and if occasional fraction of the glitches that are expected to occur in every pulsar do trigger precession, the probability of our observing precession from some (one or two) single pulsars might not be all that small. Using models developed for the Vela pulsar glitches in a manner consistent with the statistics of all pulsar glitches, the time between glitches in PSR B1828-11 is estimated to be about 80 yrs. Thus if one in five glitches of PSR B1828-11 resets the effective triaxiality of the solid crust to trigger precession observable by us, the pulsar would be sustained in extended precession for intervals of more than one damping time.

Summary In summary, we find that neutron stars can precess in the presence of vortex line-flux line pinning constraints, so that precession does not necessarily show that pinning and Type II superconductivity do not exist in neutron stars. Furthermore, even the tightest damping mechanism we know, based on observational constraints from pulsar spindown and glitches, do not make it completely unlikely that we observe some single pulsars in precession.

Acknowledgments This work was supported by Sabancı University Astrophysics and Space Forum and by the Turkish Academy of Sciences. I thank Altan Baykal, Sinan Kaan Yerli and the members of the LOC for their help, and Ünal Ertan for the figure.

46


References Alpar, M.A. (1991), in Neutron Stars: Theory and Observation, Proc. NATO ASI - Agia Pelagia, Greece, 1990; eds. J. Ventura & D. Pines, (Dordrecht, Kluwer), 49 Alpar, M.A. (1995), in The Lives of the Neutron Stars, Proc. NATO ASI - Kemer, Turkey, 1993; eds. M.A. Alpar, Ü. Kızılo˘glu & J. van Paradijs, (Dordrecht, Kluwer), 185 Alpar, M.A. (1998), in The Many Faces of Neutron Stars, Proc. NATO ASI - Lipari, Italy, 1996; eds. R. Buccheri, J. van Paradijs & M.A. Alpar, (Dordrecht, Kluwer), 59 Alpar, M.A. (2001), in The Neutron Star - Black Hole Connection, Proc.NATO ASI- Elounda, Greece, 1999; eds. C. Kouveliotou, J. Ventura & E.P.J. van den Heuvel,(Dordrecht, Kluwer),57 Alpar, M.A., Anderson, P.W., Pines, D. & Shaham, J. (1984a), ApJ 276, 325 Alpar, M.A., Anderson, P.W., Pines, D. & Shaham, J. (1984b), ApJ 278, 791 Alpar, M.A., Cheng, K.S. & Pines, D. (1989), ApJ 346, 823 Alpar, M.A., Langer, S.A. & Sauls, J.A. (1984), ApJ 282, 533 Alpar, M.A. & Sauls, J.A. (1988), ApJ 327, 723 Alpar, M.A. & Ögelman, H. (1987), A & A 185, 196 Gudmunsson, E.H., Pethick, C.J. & Epstein, R.I. (1982), ApJL259, L19 Hobbs, G., Lyne, A. G., Kramer, M., Martin, C. E. & Jordan, C. A. (2004), MNRAS, 353, 1311 Jones, P.B. (2004), Phys. Rev Lett. 92, 149001 Link, B. (2003), Phys. Rev Lett. 91, 101101 Miller, G.S. (1996), ApJL468, L29 Pines, D. (1991), in Neutron Stars: Theory and Observation, Proc. NATO ASI-Agia Pelagia, Greece, 1990; eds. J. Ventura & D. Pines, (Dordrecht, Kluwer), 57 Sauls, J.A. (1989), in Timing Neutron Stars, Proc. NATO ASI - Çe¸sme, Turkey, eds. H. Ögelman & E.P.J. van den Heuvel, (Dordrecht, Kluwer), 457 Shaham, J. (1977), ApJ 214, 251 Srinivasan, G., Bhattacharya, D., Muslimov, A. & Tsygan, A. (1990), Curr. Sci. 59, 31 Stairs, I.H., Lyne, A.G. & Shemar, S. (2000) Nat 406, 484.

TEMPERATURE DISTRIBUTION IN MAGNETIZED NEUTRON STAR CRUSTS U. Geppert Astrophysikalisches Institut Potsdam D-14482 Potsdam, Germany [email protected]

M. Kueker Astrophysikalisches Institut Potsdam D-14482 Potsdam, Germany [email protected]

D. Page Instituto de Astronomia, UNAM, 04510 Mexico D.F., Mexico [email protected]

Abstract

The effect of axisymmetric dipolar fields which are either restricted to the stellar crust, “crustal fields”, or allowed to penetrate the core, “core fields”, on the temperature distribution in neutron star crusts is investigated. While core fields result in practically isothermal crusts unless the surface field exceeds 1015 G, crustal fields with surface strength above a few times 1012 G cause significant deviations from crustal isothermality if the core temperatures is ≤ 108 K. At the stellar surface, the cold equatorial region produced by the suppression of heat transport perpendicular to the field by the Larmor rotation of the electrons in the envelope, present for both core and crustal fields, is significantly extended by that classical suppression at higher densities in the case of crustal fields. Then, two small warm polar regions appear which have observational consequences: the neutron star has a small effective thermally emitting area and the X-ray pulse profiles are expected to have a distinctively different shape compared to the case of a core field. These features, when compared with X-ray data on thermal emission of young cooling neutron stars, provide a first step toward a new way to study the magnetic flux distribution within a neutron star.

Keywords:

stars:neutron star – stars:magnetic fields – conduction – dense matter


80

Temperature Distribution in Magnetized Neutron Star Crusts

All the heat stored in the core of the neutron star and eventually irradiated away from its surface by photons has to be transported through the crust. In the absence of rotation and magnetic field, this transport in the stably stratified layers of the crust is spherically symmetric. While the effects of rotation are quite small even for millisecond pulsars, the presence of magnetic fields may cause significant deviations from the spherical symmetry of the transport processes, even for quite “standard” field strength of ∼ 1012 G. Magnetic fields penetrating the shallow layer below the surface, the envelope, causes a non-uniform surface temperature distribution (Greenstein & Hartke (1983), Potekhin & Yakovlev (2001)).

Figure 1. Representation of both field lines and temperature distribution in the crust whose radial scale is stretched by a factor of 5, assuming B0 = 3 · 1012 G and Tcore = 106 K. Left panel corresponds to a crustal field, right panel to a star-centered core field. Bars show the temperature scales in units of Tcore .

However, the non-isothermality of the subjacent crust depends strongly on the geometry of the field. While outside the star the magnetic field may be well described by a dipolar one, its internal structure can be qualitatively very different. A star centered dipolar field which is in wide regions of the crust almost radial (“core field”) causes only small deviations from isothermality even for extreme field strengths. A field maintained by currents circulating exclusively in the crust (“crustal field”) has strong meridional components, suppresses the radial heat transport and channels the flux toward the magnetic poles. Any

Ruderman & Sutherland model

81

Figure 2. The surface temperature Ts as a function of the polar angle θ and for Tcore = 106 K (left panels), Tcore = 107 K (mid panels), or Tcore = 108 K (right panels) The dashed lines show the surface temperature distribution when an isothermal crust is assumed. The full lines represent the surface temperatures when the crust temperature distributions take into account the anisotropy of heat transport induced by a crustal magnetic field (the temperature at the crustcore interface being fixed at the Tcore ). Almost indistinguishable from the isothermal crust model is the Ts -distribution for a star-centered core field. It is shown by dot-dashed lines for Tcore = 108 K; for lower Tcore the differences are even smaller. The assumed polar surface field strengths B0 are 1012 G (upper panels), 3 · 1012 G (mid panels) and 1013 G (lower panels).

presence of a magnetic field causes anisotropy of the heat flux since the heat conductivity becomes a tensor whose components perpendicular to the field lines are strongly decreased in comparison to the component parallel to them. In Fig. 1 both the temperature distribution and the field lines in the crust for the two qualitatively different field configurations are shown. These results are obtained by solving the stationary two–dimensional energy balance equation assuming the field structures fixed by the choice of its polar surface strength B0 , the temperature at the crust-core interface Tcore fixed and isotropic, and applying the field dependent outer boundary condition (for details of the model calculations see Geppert et al. (2004)). The drastic difference in the crustal temperature distribution for the different field structures which are characterized by the same dipolar field structure and strength outside the neutron star, causes significant differences in the surface temperature distribution as shown in Fig. 2

82

Temperature Distribution in Magnetized Neutron Star Crusts

The magnetic field permeating the envelope induces a non-uniform surface temperature distribution, mostly due to quantizing effects of the field at low densities, even in the case of a uniform crustal temperature (Page (1995)). However, the non-isothermality of the crust produced by a crustal field will result in an even more pronounced non-uniformity of the surface temperature. An obvious result is that the different field structures do not only affect the relation between polar and equatorial surface temperature but also the setup and the extension of the warm polar regions. This will have several, perhaps observational, consequences: 1. For a crustal field a larger pulsed fraction can be expected since the stronger corresponding meridional temperature gradient reduces the warm polar cap area in comparison with a core field. 2. This may open a new way to distinguish between crustal and core magnetic fields: A strong crustal magnetic field implies a smaller effective area for thermally emitting cooling neutron stars. The small effective emitting area inferred from the blackbody spectra of PSR 0656+14, PSR 1055-52 and Geminga: Becker & Trümper (1997) can be explained by the existence of a relatively small warm polar region, created by a strong crustal field and emitting almost all the thermal radiation. 3. The differences in the photon luminosities for a core or a crustal field will also affect the long term cooling of neutron stars. Due to its lower photon luminosity, a neutron star having a magnetic field confined to its crust will stay warmer for a longer time than a neutron star with a field penetrating its core. 4. The non-isothermality of the crust may have consequences for the crustal field itself. The electric conductivity is smaller in the hot polar regions than in the cooler equatorial layer. Therefore, currents circulating closer to the poles will decay faster than elsewhere and the crustal field structure may differ from that of a field maintained by currents in an isothermal crust.

References Becker, W., & Truemper, J. 1997 A&A, 326, 682 Geppert, U., Kueker, M., & Page, D. 2004, A&A, in press, see astro-ph/0403441 Greenstein, G. & Hartke, G. J. 1983, ApJ, 271, 283 Page, D. 1995, ApJ 442, 273 Potekhin, A. Y., & Yakovlev, D. G. 2001, A&A, 374, 213

GRAVITATIONAL WAVE INTERACTIONS WITH MAGNETIZED PLASMAS Joachim Moortgat & Jan Kuijpers Department of Astrophysics, University of Nijmegen PO Box 9010, 6500 GL Nijmegen, The Netherlands [email protected], [email protected]

Abstract

Gravitational waves (GWs) propagating through a uniformly magnetized plasma interact directly with the magnetic field and excite magnetohydrodynamic (MHD) waves with both electromagnetic and matter components. We study this process for arbitrary geometry in the MHD approximation and find that all three fundamental MHD modes – slow and fast magnetosonic, and Alfvén – are excited depending on both the polarization of the GW and the orientation of the ambient magnetic field. The latter two modes can interact coherently with the GW resulting in damping of the GW and linear growth of the plasma waves.

Keywords:

gravitational waves – magnetohydrodynamics – plasmas – stars:neutron – stars: pulsars – stars:magnetic fields

1.

Introduction

This decade is expected to witness the historical first direct detection of gravitational waves with detectors such as (Advanced) LIGO, VIRGO, TAMA and others. Gravitational waves are emitted by highly energetic events that occur at relatively large distances. Because the GW amplitude falls off with distance from its source, the signal that reaches Earth is exceedingly weak and can only be filtered from a noise signal with some theoretical knowledge of the expected waveforms. To identify a detection of a GW burst, any additional electromagnetic signature of such an event would be extremely useful. It so happens that many of the proposed GW sources are embedded in a strong magnetic field. Examples are rapidly spinning neutron stars with a small oblateness that precess, accrete, or have an r-mode instability, supernovae core collapse and bounce, newly born ‘boiling’ and oscillating neutron stars and magnetars (high frequency GW: Andersson and Kokkotas, 2004), magnetars with crust fracturing (low frequency GW: Messios et al., 2001) and coalescing compact binaries in which at least one component is a magnetic neutron star. 103 A. Baykal et al. (eds.), The Electromagnetic Spectrum of Neutron Stars, 103–106. © 2005 Springer. Printed in the Netherlands.

104


In the last case, maximum GW luminosities of the order of 1055 erg/s (Janka et al., 1999) are released into a wound-up magnetic field of field strength up to 1012 – 1015 Gauss (Ibrahim et al., 2003). We investigate whether these extreme space-time distortions perturb the ambient magnetic field sufficiently to produce an observable electromagnetic counterpart of the GW burst. In these proceedings we focus our attention on a discussion of the relevant physics and leave the mathematical details to Moortgat and Kuijpers, 2004. Gaussian geometrized units are used throughout this discussion (G = c = 1) and Latin indices are used for time-space components (a = 0 . . . 3).

2.

Coupling of the GW to the magnetic field

The interaction of dynamical space-time with matter and energy is described by Einstein’s field equations (EFE). For gravitational waves interacting with a magnetofluid the EFE can be linearized in the GW amplitude h and linear perturbations in the energy and momentum density δT ab : 1 (1) Gab − 2hab = 8πδT ab . 2 In the transverse-traceless gauge, a GW propagating in the z direction only has two independent components h+ (z, t) and h× (z, t), corresponding to different polarizations. The only components of δT ab , in the rest frame of a perfect magnetofluid, that couple to the GW and can not be removed by a gauge transformation depend on the magnetic field. Explicitly, Eq. (1) reduces to: 2h+ (z, t) = 4Bx0 δBx (z, t),

2h× (z, t) = 4Bx0 δBy (z, t),

(2)

=B 0 + where the ambient magnetic field is chosen to lie in the x-z plane: B 0 0 and B = B (sin θ, 0, cos θ). Eq. (2) are evolution equations for the GW. δB Similarly, we have derived evolution equations for the magnetic field by solving a closed set of magnetohydrodynamic (MHD) equations in an unspecified GW metric. Together with Eq. (2) these can be solved to find a self-consistent dispersion relation for the coupled gravitational-plasma waves (Moortgat and Kuijpers, 2004). However, we can approximate the GW as a driving wave propagating at the speed of light in the limit 8π 0 2 (B ) < ω∆k, µ0 x

3.

(∆k = k − ω).

(3)

Alfvén, slow and fast magneto-acoustic waves

In the approximation of a GW driver, the evolution equations for the magnetic field can be solved and look like: 1 δBx ∝ h+ Bx0 , 2

1 δBy ∝ h× Bx0 . 2

(4)

GWs propagating through a relativistic wind

105

These results are reminiscent of the spatial deviations of test masses in interferometers such as LIGO [δx = 12 (h+ x0 + h× y0 ) and δy = 12 (h× x0 − h+ y0 )]. The solution for δBx corresponds to a compressional fast magnetosonic wave (MSW) with both electromagnetic and gas properties. Coherent interaction with the GW is possible when the phase velocity of the MSW approaches that of the GW. In a Poynting flux dominated plasma where the Alfvén velocity uA is relativistic and much larger than the sound velocity, this limit is satisfied and the perturbations are allowed to grow linearly with distance: h+ 0 B sin θ ωz eiω(z−t) . δBx (z, t) (5) 2 The phase velocity of the slow MSW is always much smaller than the fast mode, so it can never interact coherently with the GW. The second expression in Eq. (4) corresponds to non-compressional shear Alfvén waves. The condition for coherent interaction with the GW is more stringent because its phase velocity uA = uA cos θ has to approach the velocity of light, but at the same time its amplitude is ∝ Bx0 ∝ uA sin θ. Therefore, in the case of coherent interaction the amplitude of the Alfvén waves is suppressed by a small factor θ 1. δBy (z, t)

h× 0 B θ ωz [eiω(z−t) ] + O[θ2 ]. 2

(6)

As was mentioned in the previous section, the GW only interacts directly with the magnetic field, and in particular the plasma motion in a GW is generally non-compressional. However, in a perfectly conducting plasma the particles are ‘glued’ to the magnetic field lines and the electromagnetic Maxwell equations couple to the matter conservation laws through the current density. Consequently, pressure, density and magnetic field gradients, currents and a drift velocity are also excited in the MSW, whereas the non-compressional Alfvén waves cause a divergence of the electric field and a corresponding charge density fluctuation.

4.

GWs propagating through a relativistic wind

In many of the GW sources mentioned in Sect 10.1, the spinning matter winds up the magnetic field and causes a collimated relativistic outflow of charged particles in a magnetized plasma wind or jet. This wind is already present before the cataclysmic event such as a binary merger. Therefore the wind has had time to expand over large distances before it is overtaken by the GW. To study the interaction between the GW and the wind we can simply Lorentz boost the results of the previous sections to the frame of an observer looking at the relativistic outflow. The exact expressions for all the wave components can be found in Moortgat and Kuijpers, 2004. The general result is that

106


the plasma wave amplitudes are typically suppressed by a factor Γ−2 , where Γ is the Lorentz factor of the wind. This suppression is due to the red-shifted GW frequency and magnetic field in the wind frame. However, this is compensated by the large interaction length scale in the extended wind.

5.

Conclusions

We have found that GWs propagating through a magnetized plasma excite all three fundamental MHD modes. Furthermore, if the plasma is magnetically dominated, coherent interaction is possible with the Alfvén and fast MSW, allowing these waves to grow. The total energy transferred from the GWs to the plasma is proportional to the square of the ambient magnetic field, the interaction length scale and the GW frequency and amplitude. The most favorable astrophysical sources are therefore merging (magnetic) neutron star binaries and young vibrating magnetars, that have the strongest known magnetic fields and emit GWs with the highest frequencies. However, even in the most extreme GW sources, it is not yet clear whether this will in fact produce an observable electromagnetic signature directly related to the GW emission (for some numerical estimates see Moortgat and Kuijpers, 2003). At present, we are investigating whether nonlinear effects in a dilute plasma can lead to radio emission which can be detected by LOFAR.

References Andersson, N. and Kokkotas, K. D (2004). Gravitational-wave astronomy: the high-frequency window. ArXiv General Relativity and Quantum Cosmology e-prints. Ibrahim, A. I., Swank, J. H., and Parke, W. (2003). New Evidence of Proton-Cyclotron Resonance in a Magnetar Strength Field from SGR 1806-20. Astrophysical Journal Letters, 584:L17–L21. Janka, H.-T., Eberl, T., Ruffert, M., and Fryer, C. L. (1999). Black Hole-Neutron Star Mergers as Central Engines of Gamma-Ray Bursts. Astrophysical Journal Letters, 527:L39–L42. Messios, N., Papadopoulos, D. B., and Stergioulas, N. (2001). Torsional oscillations of magnetized relativistic stars. MNRAS, 328:1161–1168. Moortgat, J. and Kuijpers, J. (2003). Gravitational and magnetosonic waves in gamma-ray bursts. Astronomy & Astrophysics, 402:905–911. Moortgat, J. and Kuijpers, J. (2004). Gravitational waves in magnetized relativistic plasmas. Physical Review D, 70(2):023001–+.

HIGH-ENERGY NEUTRINO ASTRONOMY Francis Halzen Department of Physics, University of Wisconsin, Madison, WI, 53706, USA

Abstract

Kilometer-scale neutrino detectors such as IceCube are discovery instruments covering nuclear and particle physics, cosmology and astronomy. Examples of their multidisciplinary missions include the search for the particle nature of dark matter and for additional small dimensions of space. In the end, their conceptual design is very much anchored to the observational fact that Nature accelerates protons and photons to energies in excess of 1020 and 1013 eV, respectively. The cosmic ray connection sets the scale of cosmic neutrino fluxes. In this context, we discuss the first results of the completed AMANDA detector and the reach of its extension, IceCube. Similar experiments are under construction in the Mediterranean. Neutrino astronomy is also expanding in new directions with efforts to detect air showers, acoustic and radio signals initiated by super-EeV neutrinos.

Keywords:

neutrinos – instrumentation:detectors – telescopes

1.

Neutrinos Associated with the Highest Energy Cosmic Rays

The flux of cosmic rays is summarized in Fig. 1a,b[1]. The energy spectrum follows a broken power law. The two power laws are separated by a feature dubbed the “knee"; see Fig. 1a. Evidence has been accumulating that cosmic rays, up to EeV energy, originate in galactic supernova remnants.Any association with our galaxy disappears however in the vicinity of a second feature in the spectrum referred to as the “ankle". Above the ankle, the gyroradius of a proton exceeds the size of the galaxy and it is generally assumed that we are witnessing the onset of an extragalactic component in the spectrum that extends to energies beyond 100 EeV. Experiments indicate that the highest energy cosmic rays are predominantly protons. Above a threshold of 50 EeV these protons interact with cosmic microwave photons and lose their energy to pions before reaching our detectors. This is the Greissen-Zatsepin-Kuzmin cutoff hat limits the sources to the local cluster of galaxies. Models for the origin of the highest energy cosmic rays fall into two categories, top-down and bottom-up. In top-down models it is assumed that the 89 A. Baykal et al. (eds.), The Electromagnetic Spectrum of Neutron Stars, 89–101. © 2005 Springer. Printed in the Netherlands.

90

High-Energy Neutrino Astronomy

Figure 1. At the energies of interest here, the cosmic ray spectrum consists of a sequence of 3 power laws. The first two are separated by the “knee" (left panel), the second and third by the “ankle". There is evidence that the cosmic rays beyond the ankle are a new population of particles produced in extragalactic sources; see right panel.

cosmic rays are the decay products of cosmological remnants with Grand Unified energy scale MGU T ∼ 1024 eV. These models predict neutrino fluxes most likely within reach of first-generation telescopes such as AMANDA, and certainly detectable by future kilometer-scale neutrino observatories[2]. In bottom-up scenarios it is assumed that cosmic rays originate in cosmic accelerators. Accelerating particles to TeV energy and above requires massive bulk flows of relativistic charged particles. These are likely to originate from the exceptional gravitational forces in the vicinity of black holes. Examples include the dense cores of exploding stars, inflows onto supermassive black holes at the centers of active galaxies and annihilating black holes or neutron stars. Before leaving the source, accelerated particles pass through intense radiation fields or dense clouds of gas surrounding the black hole. This results in interactions producing pions decaying into secondary photons and neutrinos that accompany the primary cosmic ray beam as illustrated in Fig. 2. How many neutrinos are produced in association with the cosmic ray beam? The answer to this question, among many others[2], provides the rationale for building kilometer-scale neutrino detectors. We first consider a neutrino beam produced at an accelerator laboratory; see Fig. 2. Here the target absorbs all

Neutrinos Associated with the Highest Energy Cosmic Rays

Figure 2.

91

Diagram of cosmic ray accelerator producing photons and neutrinos.

parent protons as well as the secondary electromagnetic and hadronic showers. Only neutrinos exit the dump. If nature constructed such a “hidden source" in the heavens, conventional astronomy will not reveal it. It cannot be the source of the cosmic rays, however, because in this case the dump must be transparent to protons. A more generic “transparent" source can be imagined as follows: protons are accelerated in a region of high magnetic fields where they interact with photons via the processes p + γ → ∆ → π 0 + p, p + γ → ∆ → π + + n. While the protons may remain trapped in the acceleration region, equal numbers of neutrons, neutral and charged pions escape. The energy escaping the source is therefore equally distributed between cosmic ray, gamma rays and neutrinos produced by the decay of neutrons and neutral and charged pions, respectively. The neutrino flux from a generic transparent cosmic ray source is often referred to as the Waxman-Bahcall flux[3]. It is easy to calculate and the derivation is revealing.

92


Figure 1b shows a fit to the observed spectrum above the “ankle" that can be used to derive the total energy in extragalactic cosmic rays. The energy content of this component is ∼ 3×10−19 erg cm−3 , assuming an E −2 energy spectrum with a GZK cutoff. The power required for a population of sources to generate this energy density over the Hubble time of 1010 years is ∼ 3 × 1037 erg s−1 per (Mpc)3 or, as often quoted in the literature, ∼ 5 × 1044 TeV per year per (Mpc)3 . This works out to[4] ∼ 3 × 1039 erg s−1 per galaxy, ∼ 3 × 1042 erg s−1 per cluster of galaxies, ∼ 2 × 1044 erg s−1 per active galaxy, or ∼ 2 × 1052 erg per cosmological gamma ray burst. The coincidence between these numbers and the observed output in electromagnetic energy of these sources explains why they have emerged as the leading candidates for the cosmic ray accelerators. The coincidence is consistent with the relationship between cosmic rays and photons built into the “transparent" source. In the photoproduction processes roughly equal energy goes into the secondary neutrons, neutral and charged pions whose energy ends up in cosmic rays, gamma rays and neutrinos, respectively. We therefore assume that the same energy density of ρE ∼ 3 × 10−19 erg cm−3 , observed in cosmic rays and electromagnetic energy, ends up in neutrinos with a spectrum Eν dN/dEν ∼ E −γ cm−2 s−1 sr−1 that continues up to a maximum energy Emax . The neutrino flux follows from the relation Eν dN/dEν = cρE /4π . For γ = 1 and Emax = 108 GeV, the generic source of the highest energy cosmic rays produces a flux of Eν 2 dN/dEν ∼ 6 × 10−8 GeV cm−2 s−1 sr−1 of all flavors. There are several ways to modify this simple prediction: The derivation fails to take into account the fact that there are more cosmic rays in the universe producing neutrinos than observed at earth because of the GZK-effect and evolution of the sources with redshift. This increases the neutrino flux by a factor 3. Only 1/3 of the neutrino energy at earth will be in the form of neutrinos of muon flavor. For proton-γ interactions the pions receive only 1/4 of the energy of the neutrons; in p-p interactions this fraction is close to unity.

Kilometer-Scale Detectors

93

We therefore estimate the muon-neutrino flux associated with the sources of the highest energy cosmic rays in the range Eν 2 dN/dEν = 1 ∼ 5 × 10−8 GeV cm−2 s−1 sr−1 yielding 50 ∼ 250 detected muon neutrinos per km2 per year. The event rate is obtained by folding the predicted flux with the probability that the neutrino is actually detected in a high energy neutrino telescope; the latter is given by[2] the ratio of the muon and neutrino interaction lengths in the detector medium, λµ /λν . The number depends weakly on Emax and the spectral slope γ. This flux has to be compared with the sensitivity of ∼10−7 GeV cm−2 s−1 sr−1 reached with the first 4 years of data collected with the AMANDA detector[5]. The analysis has not been completed but a preliminary limit of 2.9 × 10−7 GeV cm−2 s−1 sr−1 has been obtained with a single year of data[6]. On the other hand, after three years of operation IceCube will reach a diffuse flux limit of Eν2 dN/dEν = 1 ∼ 3 × 10−9 GeV cm−2 s−1 sr−1 depending on the magnitude of the dominant high energy atmospheric neutrino background from the prompt decay of atmospheric charmed particles[7]. The level of this background is small but difficult to predict.

2.

Kilometer-Scale Detectors

Arguing that a generic cosmic accelerator produces equal energies in cosmic ray, photons and neutrinos, we derived the “Waxman-Bahcall" flux. A kilometer-scale detector is required to detect the roughly ∼125 events per km2 year. Model calculations assuming that active galaxies or gamma-ray bursts are the actual sources of cosmic rays yield similar, or even smaller event rates. The case for kilometer-scale detectors also emerges from the consideration of “guaranteed" cosmic fluxes. Neutrino fluxes are guaranteed when both the accelerator and the pion production target can be identified: The extragalactic cosmic rays produce 0.1 ∼ a few events per km2 year in interactions with cosmic microwave photons. Furthermore, these cosmic rays are magnetically trapped in galaxy clusters and may produce additional neutrinos on the X-ray emitting gas in the cluster. Galactic cosmic rays interact with hydrogen in the disk producing an observable neutrino flux in a kilometer-scale detector. Air shower arrays have observed a “directional" flux of cosmic rays from the galactic plane, unlikely to be protons whose directions are scrambled in the magnetic field. The flux appears only in a narrow energy range from 1 ∼ 3 EeV, the energy where neutrons reach typical galactic kiloparsec distances within their lifetime of minutes. Both the directionality

94


and the characteristic energy make a compelling case for neutron primaries. For every neutron reaching earth, a calculable number decays into electron antineutrinos before reaching us. Their flux should be observable in neutrino telescopes[8]: from the Cygnus region at the South Pole and from the galactic center for a Mediterranean detector. In conclusion, “guaranteed" sources also require kilometer-size neutrino detectors, preferably operated over many years. Finally and most importantly, thanks to recent observations[9] of the supernova remnant RX J1713.7-3946 with the Cangaroo and Hess atmospheric Cherenkov telescopes, neutrino astronomy can claim a truly guaranteed source [10]. The observations of TeV-gamma rays from the supernova remnant RX J1713.7-3946 have most likely identified the first site where protons are accelerated to energies typical of the main component of the galactic cosmic rays. The signal appears to be produced by the interaction of shock accelerated protons in a supernova remnant with a molecular cloud. The high statistics Hess data for the flux are power-law behaved over a large range of energies without any signature of a cutoff. It looks unlikely that a synchrotron or inverse Compton spectrum generated by electrons can reproduce it. Furthermore, already after the Cangaroo observation a search for such electrons with the VLA and Chandra turned out to be negative. The most natural explanation is that the signal is of pion origin. From the fact that equal numbers of neutral and charged pions are produced, it is straightforward to calculate the high-energy neutrino flux associated with this source to be at least 20 muon-type neutrinos per kilometer-squared per year. From a variety of such sources we can expect event rates of cosmic neutrinos of galactic origin similar to those estimated for extragalactic neutrinos in the previous section. Supernovae associated with molecular clouds are a common feature of the OB associations that exist throughout the galactic plane. They have been suspected to be the sources of the galactic cosmic rays. It is important to realize that the relation between the neutrino and gamma flux is robust. The νµ + ν¯µ neutrino flux (dNν /dEν ) produced by the decay of charged pions in the source can be derived from the observed gamma ray flux by imposing energy conservation:

Eγmax Eγmin

dNγ Eγ dEγ = K dEγ

Eνmax

Eνmin

Eν

dNν dEν dEν

(1)

where Eγmin (Eγmax ) is the minimum (maximum) energy of the photons that have a hadronic origin. Eνmin and Eνmax are the corresponding minimum and maximum energy of the neutrinos. The factor K depends on whether the π 0 ’s are of pp or pγ origin. Its value can be obtained from routine particle physics. In pp interactions 1/3 of the proton energy goes into each pion flavor on

Mediterranean Telescopes

95

average. In the pion-to-muon-to-electron decay chain 2 muon-neutrinos are produced with energy Eπ /4 for every photon with energy Eπ /2 (on average). Therefore the energy in neutrinos matches the energy in photons and K = 1. This flux has to be reduced by a factor 2 because of oscillations. The estimate should be considered a lower limit because the photon flux to which the calculation is normalized, may be partially absorbed in the source or in the interstellar medium.

3.

Neutrino Telescopes: First “Light"

While it has been realized for many decades that the case for neutrino astronomy is compelling, the challenge has been to develop a reliable, expandable and affordable detector technology to build the kilometer-scale telescopes required to do the science. Conceptually, the technique is simple. In the case of a high-energy muon neutrino, for instance, the neutrino interacts with a hydrogen or oxygen nucleus in deep ocean water and produces a muon traveling in nearly the same direction as the neutrino. The Cherenkov light emitted along the muon’s kilometer-long trajectory is detected by a lattice of photomultiplier tubes deployed on strings at depth shielded from radiation. The orientation of the Cherenkov cone reveals the roughly collinear muon and neutrino direction. The AMANDA detector, using natural 1 mile-deep Antarctic ice as a Cherenkov detector, has operated for more than 4 years in its final configuration of 667 optical modules on 19 strings. The detector is in steady operation collecting roughly 7 ∼ 10 neutrinos per day using fast on-line analysis software. The lower number will yield a background-free sample all the way to the horizon. AMANDA’s performance has been calibrated by reconstructing muons produced by atmospheric muon neutrinos in the 50 GeV to 500 TeV energy range[11]. Using the first 4 years of AMANDA II data, the AMANDA collaboration is performing a search for the emission of muon neutrinos from spatially localized directions in the northern sky. Only the year 2000 data have been published [12]. The skyplot is shown in Fig. 3. A 90% upper limit on the neutrino fluency of point sources is at the level of 6 × 10−8 GeV cm−2 s−1 or 10−10 erg cm−2 s−1 , averaged over declination. This corresponds to a flux of 6 × 10−9 cm−2 s−1 integrated above 10 GeV assuming an E −2 energy spectrum typical for shock acceleration of particles in high energy sources. The most significant excess is 3.4 σ from the Crab with a probability of close to 10% given the trial factor for 33 sources searched. IceCube is needed to make conclusive observations of sources. The AMANDA II detector has reached a high-energy effective telescope area of 25,000 ∼ 40,000 m2 , depending on declination. This represents an interesting milestone: known TeV gamma ray sources, such as the active galax-

96


Figure 3. Skymap showing declination and right ascension of neutrinos detected by the AMANDA II detector during four Antarctic winters of operation in 2000–2003.

ies Markarian 501 and 421, should be observed in neutrinos if the number of gamma rays and neutrinos emitted are roughly equal as expected from cosmic ray accelerators producing pions[10]. Therefore AMANDA must detect the observed TeV photon sources soon, or, its observations will exclude them as the sources of cosmic rays.

4.

Mediterranean Telescopes

Below PeV energy, South Pole neutrino telescopes do not cover the Southern sky, which is obscured by the large flux of cosmic ray muons and neutrinos. This and the obvious need for more than one telescope — accelerator physics has clearly demonstrated the value of multiple detectors — provide compelling arguments for deploying northern detectors. With the first observation of neutrinos by a detector in Lake Baikal with a telescope area of 2500 m2 for TeV muons[13] and after extensive R&D efforts by both the ANTARES[14] and NESTOR[15] collaborations in the Mediterranean, there is optimism that the technological challenges to build neutrino telescopes in deep sea water have been met. Both Mediterranean collaborations have demonstrated their capability to deploy and retrieve optical sensors, and have reconstructed down-going muons with optical modules deployed for R&D tests. The ANTARES neutrino telescope is under construction at a 2400 m deep Mediterranean site off Toulon, France. It will consist of 12 strings, each equipped with 75 optical sensors mounted in 25 triplets. The detector per-

Kilometer-scale Neutrino Observatories

97

formance has been fully simulated[14] with the following results: a sensitivity after one year to point sources of 0.4 − 5 × 10−15 cm−2 s−1 (note that this is the flux of secondary muons, not neutrinos) and to a diffuse flux of 0.9 × 10−7 GeV cm−2 s−1 above 50 TeV. As usual, an E −2 spectrum has been assumed for the signal. AMANDA II data have reached similar point source limits (0.6 × 10−15 cm−2 s−1 sr−1 ) using 4 Antarctic winters of data, or about 1000 days[12]); this value depends weakly on declination. Also the diffuse limits reached in the absence of a signal are comparable[5]. We have summarized the sensitivity of both experiments in Table 1, where they are also compared to the sensitivity of IceCube. Given that AMANDA and ANTARES operate at similar depths and have similar total photocathode area (AMANDA II is actually a factor of 2 smaller with 667 8-inch versus 900 10-inch photomultipliers for Antares) the above comparison provides us with a first glimpse at the complex question regarding the relative merits of water and ice as a Cherenkov detector. The conclusion seems to be that, despite many differences in optics of the medium and the background counting rates of the photomultipliers, the telescope sensitivity is approximately the same for equal photocathode area. The comparison is summarized in Table 1 where the point source limits are tabulated for secondary muon rather than neutrino flux. In the same context, the NEMO collaboration has done the interesting exercise of simulating the IceCube detector (augmen-

98


ted from 4800 to 5600 optical modules; see next section) in water rather than ice. While the effective area is only slightly reduced at PeV energy, it is smaller by a factor 6 at TeV energy in water. This most likely is a result of the larger spacings of the strings and the lower absorption length of blue Cherenkov light in water[16].

5.


The baseline design of kilometer-scale neutrino detectors maximizes sensitivity to νµ -induced muons with energy above hundreds of GeV, where the acceptance is enhanced by the increasing neutrino cross section and muon range but the Earth is still largely transparent to neutrinos. The mean-free path of a νµ becomes smaller than the diameter of the earth above 70 TeV — above this energy neutrinos can only reach the detector from angles closer to the horizon. Good identification of other neutrino flavors becomes a priority, especially because ντ are not absorbed by the earth. Good angular resolution is required to distinguish possible point sources from background, while energy resolution is needed to enhance the signal from astrophysical sources, which are expected to have flatter energy spectra than the background atmospheric neutrinos. Overall, AMANDA represents a proof of concept for the kilometer-scale neutrino observatory, IceCube[7], now under construction. IceCube will consist of 80 kilometer-length strings, each instrumented with 60 10-inch photomultipliers spaced by 17 m. The deepest module is 2.4 km below the surface. The strings are arranged at the apexes of equilateral triangles 125 m on a side. The instrumented (not effective!) detector volume is a cubic kilometer. A surface air shower detector, IceTop, consisting of 160 Auger-style Cherenkov detectors deployed over 1 km2 above IceCube, augments the deep-ice component by providing a tool for calibration, background rejection and air-shower physics, as illustrated in Fig. 4. The transmission of analogue photomultiplier signals from the deep ice to the surface, used in AMANDA, has been abandoned. The photomultiplier signals will be captured and digitized inside the optical module. The digitized signals are given a global time stamp with a precision of < 10 ns and transmitted to the surface. The digital messages are sent to a string processor, a global event trigger and an event builder. Construction of the detector is expected to commence in the Austral summer of 2004/2005 and continue for 6 years, possibly less. The growing detector will take data during construction, with each string coming online within days of deployment. The data streams of IceCube, and AMANDA II, embedded inside IceCube, will be merged off-line using GPS timestamps. IceCube will offer advantages over AMANDA II beyond its larger size: it will have a higher efficiency and superior angular resolution in reconstructing


99

Figure 4. Relative sizes of the IceCube, AMANDA, and Superkamiokande neutrino detectors. AMANDA will be operated as a lower threshold subsystem of IceCube. As the size of the detector grows, so does the threshold energy of neutrinos detected.

tracks, map showers from electron- and tau-neutrinos (events where both the production and decay of a τ produced by a ντ can be identified) and, most importantly, measure neutrino energy. Simulations, benchmarked by AMANDA data, indicate that the direction of muons can be determined with sub-degree accuracy and their energy measured to better than 30% in the logarithm of the energy. The direction of showers will be reconstructed to better than 10◦ above 10 TeV and the response in energy is linear and better than 20%. Energy resolution is critical because, once one establishes that the energy exceeds 1 PeV, there is no atmospheric muon or neutrino background in a kilometer-square detector and full sky coverage of the telescope is achieved. The background counting rate of IceCube signals is expected to be less than 0.5 kHz per optical sensor. In this low background environment, IceCube can detect the excess of anti-νe events from a galactic supernova. NEMO, an INFN R&D project in Italy, has been mapping Mediterranean sites and studying novel mechanical structures, data transfer systems as well as low power electronics with the goal to deploy a next-generation detector similar to IceCube. A concept has been developed with 81 strings spaced by 140 m. Each consists of 18 bars that are 20 m long and spaced by 40 m. A bar holds a pair of photomultipliers at each end, one looking down and one

100


horizontally. As already mentioned, the simulated performance[17] is, not unexpectedly, similar to that of IceCube with a similar total photocathode area as the NEMO concept. Recently, a wide array of projects have been initiated to detect neutrinos of the highest energies, typically above a threshold of 10 EeV, exploring other experimental signatures: horizontal air showers and acoustic or radio emission from neutrino-induced showers. Some of these experiments, such as the Radio Ice Cherenkov Experiment[18] and an acoustic array in the Caribbean[19], have taken data; others are under construction, such as the Antarctic Impulsive Transient Antenna[20]. The more ambitious EUSO/OWL project aims to detect the fluorescence of high energy cosmic rays and neutrinos from a detector attached to the International Space Stations.

Acknowledgments I thank my AMANDA/IceCube collaborators and Teresa Montaruli for discussions. This research was supported in part by the National Science Foundation under Grant No. OPP-0236449, in part by the U.S. Department of Energy under Grant No. DE-FG02-95ER40896, and in part by the University of Wisconsin Research Committee with funds granted by the Wisconsin Alumni Research Foundation.

References [1] T. K. Gaisser, Proceedings of the 31st International Conference on High Energy Physics, Amsterdam, The Netherlands, July 2002. [2] T. K. Gaisser, F. Halzen, and T. Stanev, Phys. Rept. 258, 173 (1995) [Erratum 271, 355 (1995)], hep-ph/9410384; J.G. Learned and K. Mannheim, Ann. Rev. Nucl. Part. Science 50, 679 (2000); F. Halzen and D. Hooper, Rept. Prog. Phys. 65, 1025 (2002), astro-ph/0204527. [3] J. N. Bahcall and E. Waxman, Phys. Rev. D 64, 023002 (2001). [4] T. K. Gaisser, OECD Megascience Forum, Taormina, Italy, 1997, astro-ph/9707283. [5] G. C. Hill et al. (AMANDA collaboration), Proceedings of the 28th International Cosmic Ray Conference, Tsukuba, Japan, 2003. [6] M. Ackermann et al. (AMANDA Collaboration), astro-ph/0405218. [7] J. Ahrens et al. (IceCube Collaboration), Particle Astrophysics, in print, astro-ph/0305196 and http://icecube.wisc.edu/science/sci-tech-docs/ [8] L. A. Anchordoqui et al., Phys. Lett. B 593, 42 (2004), astro-ph/0310417. [9] Talks at Gamma 2004, Heidelberg, Germany, 2004. [10] J. Alvarez-Muniz and F. Halzen, Ap. J. 576, L33 (2002). [11] E. Andres et al. (AMANDA Collaboration), Nature 410, 441 (2001); Phys. Rev. D 66, 012005 (2002), astro-ph/0205109. [12] J. Ahrens et al. (AMANDA Collaboration), Phys. Rev. Lett. 92, 171102 (2004), astroph/0309585. [13] V. A. Balkanov et al. (Baikal Collaboration), Nucl. Phys. Proc. Suppl. 118, 363 (2003).


101

[14] T. Montaruli et al. (ANTARES Collaboration), Proceedings of the 28th International Cosmic Ray Conference, Tsukuba, Japan, 2003. [15] http://www.nestor.org.gr/ [16] E. Migneco, CRIS04 Workshop, Catania, Italy, 2004. [17] R. Coniglione et al. (NEMO Collaboration), http://nemoweb.lns.infn.it/publication.htm [18] G. M. Frichter et al., Phys. Rev. D 53, 1684 (1996 ), astro-ph/9507078 [19] N. G. Lehtinen et al, Astropart Phys. 17, 272 (2002), astro-ph/0104033. [20] P. Gorham, 2002 Aspen Winter Conference on Ultra High Energy Particles from Space, http://astro.uchicago.edu/home/web/olinto/aspen/astroweb

PROTO-NEUTRON STAR NEUTRINO EMISSION Aimee L. Hungerford, Christopher L. Fryer Los Alamos National Laboratory and The University of Arizona [email protected], [email protected]

Aristotle Socrates, Omer Blaes University of California - Santa Barbara Physics and Astronomy Department [email protected], [email protected]

Abstract Neutrino spectra during the cooling phase of neutron stars provide a relatively direct probe of the physics in the hot, dense interiors of these compact objects. Details of the equation of state and instabilities in these extreme physical regimes have a significant affect on the properties of the emitted neutrino spectrum. In addition to the direct ramifications for the cooling time, the natal kicks observed in young pulsars may be the result of asymmetric neutrino emission due to opacity variations within the neutron star itself. Furthermore, such asymmetries in the emission are also reflected in neutrino deposition, resulting in asymmetrically driven winds above the proto-neutron stars where r-process elements are thought to be synthesized. In this proceedings, we present preliminary results from a study of the neutrino emission from neutron stars, including asymmetries arising from magnetoacoustic instabilities and neutron star rotation. Keywords:

1.

stars:neutron – neutrinos – instabilities – stars:kinematics

Introduction

High space velocities of pulsars have convincingly demonstrated that corecollapse supernovae are asymmetric. The departure from spherical symmetry is likely imprinted in the explosion mechanism itself and will lead to asymmetries in the remnant neutron star structure. This may result in altered neutrino opacity profiles, giving rise to asymmetric neutrino emission, an important consequence of which is that a net momentum (NS kick) can be imparted to the proto-neutron star (Lai & Qian 1998). Additionally, asymmetries in the neutrino deposition could give rise to asymmetric wind structures. Neutron 111 A. Baykal et al. (eds.), The Electromagnetic Spectrum of Neutron Stars, 111–114. © 2005 Springer. Printed in the Netherlands.

112

Proto-Neutron Star Neutrino Emission

star winds are the prime site for r-process nucleosynthesis. To date, spherically symmetric wind calculations of the r-process yields in supernovae (see Hoffman, Woosley, & Qian 1997; Thompson, Burrows, & Meyer 2001 and references therein) seem unable to produce the observed r-process yields. To investigate the range of kicks and wind asymmetry possible from asymmetric neutrino fluxes, the transport of neutrinos must be followed through aspherical models of proto-neutron stars. Using a neutrino Monte Carlo transport code, we investigate neutrino emission and deposition profiles from neutron star asymmetries which result from the presence of magnetoacoustic instabilities (MAI) and stellar rotation. In a stratified medium with a background radiative flux, acoustic waves can be driven unstable by periodic radiative forcing. Neutrino fluxes through magnetic spots in young proto-neutron stars seem sufficiently large and diffusive to drive such instabilities. Ultimately, these MAI’s result in density fluctuations that may serve to increase the neutrino mean free path over the solid angle subtended by the magnetic spot. This will lead to asymmetric neutrino emission concentrated where the magnetic field is strongest (i.e., in starspots). Stellar rotation is also capable of generating asymmetries in neutrino emission. The current paradigm in stellar collapse requires convective motions above the proto-neutron star to drive the supernova explosion. Rotation in the massive star progenitors of core-collapse supernovae inhibit convection in the equatorial plane giving rise to asymmetric explosions with rapidly spinning neutron stars remaining (Fryer & Heger 2000). This creates a neutron star with an accretion disk-like structure in the equator where the angular momentum is highest, which will give rise to asymmetries in the angular neutrino emission.

2.

Simulation Details

Asymmetric neutrino emission, arising from either the MAI or the rotationally deformed proto-neutron star, has ramifications for potentially observable quantities such as neutron star winds, gravitational wave emission and pulsar kicks. Using Monte Carlo neutrino transport we track the spatial neutrino energy deposition (important for driving NS winds), the angular distribution of neutrino luminosity (important for gravitational wave emission) and net momentum of escaping neutrinos (important for pulsar kick values) against background neutron star structures. The proto-neutron star structure used to investigate the effect of a MAI is constructed from the collapsed core of a one-dimensional supernova simulation (using the 1D collapse code described in Herant et al. 1994; Fryer et al. 1996). We artificially perturb the 1D proto-neutron star structure (taken at t = 130 ms post-bounce) to mimick the effects of a MAI. We assume the instability exists in a single dominant starspot with covering area C. Throughout this starspot,

113

Results

we lower the opacity by a factor f from a radius Rin to Rout . Table 1 shows the values we have adopted for the simulations presented here. The ranges explored are guided by the linear instability analysis in Socrates et al. (2004, in preparation). The altered proto-neutron star model at 130 ms is then mapped onto a 3-dimensional (1763 ) grid used by the Monte Carlo transport code. C

f

0.1 0.1 0.1 0.1 0.05 0.05 0.01 0.01

2 2 1.33 1.33 2 2 2 2

Rout (km) 120 120 120 120 120 120 120 120

Rin (km) 50 60 50 60 50 60 50 60

Vkick (km/s) 2350 1500 950 700 1250 800 250 200

< hT T >e ( 10 kpc) 5.2E-21 4.7E-21 4.7E-21 4.7E-21 5.2E-21 4.8E-21

Table 1. Kick velocities and mean gravitational wave magnitudes which result from our suite of MAI model asymmetries. This assumes a 1 second duration for the instability.

The proto-neutron star structure used to investigate the effect of rotational asymmetry is taken from the inner 122 km of the rotating SPH core-collapse simulation from Fryer & Heger (2000). The model is taken at t = 1.9 s postbounce and the SPH particle properties (e.g. density, electron fraction) are mapped onto a 25×50 2D grid. This 2D grid is, in turn, mapped to a 3D (1763 zone) grid used in the Monte Carlo calculations. These star models are taken as the material background through which we simulate the transport of electron neutrinos using a version of the Monte-Carlo γ-ray transport code Maverick (Hungerford et al. 2003). This code has been altered to follow the transport of neutrinos by including functional forms for the absorptive and scattering opacities as given by Janka (2001). Neutrino annihilation is not included. Neutrino emission rates (electron capture and electron-positron annihilations) are taken from Herant, Benz & Colgate (1992) for the MAI models. Neutrino emissivities, and mean neutrino energy, for the rotating model were taken directly from the flux limited diffusion results included in the collapse simulation itself. The energy distribution for the emitted neutrinos was taken from Keil, Raffelt & Janka (2003).

3.

Results

With our proto-neutron star models, we can now study the various effects these asymmetries have on our 3 observables. For these asymmetries to affect the r-process or the supernova explosion, they need not alter the total energy deposition of neutrinos dramatically, but must induce an asymmetry in the deposition. Figure 1 shows the neutrino energy deposition profiles for the rotational asymmetry and the most extreme MAI asymmetry in our parameter study. It is clear that the MAI produces only a small asymmetry, and the total

114

Proto-Neutron Star Neutrino Emission

Figure 1. Left Panel: Neutrino energy deposition contours for C = 0.1, f = 2, Rout = 120 km, Rin = 50 km. Note that the contours move inward over the cone of lower opacity (cone opening angle is roughly 37◦ for C = 0.1). Right Panel: Neutrino energy deposition contours for the 2D neutron star model taken from the simulations of Fryer & Heger 2000.

heating from the proto-neutron star is essentially spherically symmetric. The rotational asymmetry, however, significantly alters the neutrino energy deposition from spherical symmetry, though one must keep in mind that neutrino annihilation may alter this profile. The small asymmetries from the MAI do make a large difference in the gravitational wave spectrum. For our suite of MAI models, the gravitational wave amplitudes (listed in Table 1 and assuming the asymmetry persists for 1 s1 ) are sufficiently high to be detectable by advanced LIGO observing a Galactic supernova. The most dramatic effect of the MAI arises from the net linear momentum carried away by neutrinos. In order to conserve momentum, the neutron star must receive a kick in the opposite direction. For our suite of models, and assuming a total energy loss of 5 × 1053 erg, kick amplitudes can range from 100 to 2000 km s−1 !

References Fryer, C., Benz, W., & Herant, M. (1996), ApJ, 460, 801 Fryer, C., & Heger, A. (2000), ApJ, 541, 1033 Herant, M., Benz, W., & Colgate, S. (1992), ApJ, 395, 642 Herant, M., Benz, W., Hicks, W. R., Fryer, C., & Colgate, S. (1994), ApJ, 435, 339 Hoffman, R., Woosley, S., & Qian, Y. (1997), ApJ, 482 , 951 Hungerford, A., Fryer, C., & Warren, M. (2003), ApJ, 594, 390 Janka, H.-Th. (2001), A&A, 368, 527 Keil, W., Raffelt, G., & Janka, H.-Th. (2003), ApJ, 590, 971 Lai, D., & Qian, Y. (1998), ApJ, 505, 844 Thompson, T., Burrows, A., & Meyer, B. (2001), ApJ, 562, 887

1 Depending

on explosion epoch, neutrino luminosity can vary considerably over this 1 s time period, but this value gives a rough estimate of the expected signal.

IV

COOLING AND ATMOSPHERES OF NEUTRON STARS

THE DOUBLE-NEUTRON-STAR INSPIRAL RATE AND EXPECTATIONS FOR GRAVITATIONAL-WAVE DETECTION Chunglee Kim,1 Vassiliki Kalogera,1 Duncan R. Lorimer,2 Mia Ihm,1 Krzysztof Belczynski,1,3 (1) Northwestern University, Department of Physics and Astronomy, 2145 Sheridan Rd., Evanston, IL, 60201, USA (2) University of Manchester, Jodrell Bank Observatory, Macclesfield, Cheshire, SK11 9DL, UK (3) Lindheimer Postdoctoral Fellow [email protected], [email protected], [email protected], [email protected], [email protected]

Abstract

We present the most up-to-date inspiral rate estimates of double-neutron-star (DNS) systems in our Galaxy. We find that the DNS inspiral rate (R) is most likely to be ∼83 Myr−1 for our reference model. Incorporating the systematic effects of the pulsar luminosity function, we also calculate a global probability distribution of the rate estimates and compare it with the empirical supernova (SN) rate estimates. We conclude that our empirical rate estimates are consistent with SN rates, although the uncertainty in the contribution of Type Ib/c SNe in the formation of DNS system is presumably an order of magnitude.

Keywords:

stars:neutron – stars:binaries – gravitational waves – methods:statistical

1.

Introduction

Soon after the discovery of the highly relativistic pulsar J0737−3039 (Burgay et al. 2004) we applied our analysis method for pulsar populations and updated the inspiral rate estimates for the current sample of Galactic close DNS (Kalogera et al. 2004a; 2004b). Our main conclusion was that this new, remarkably relativistic system dominates the Galactic DNS inspiral rate and leads to a rate increase by a factor of 6 − 7. This implies a correspondingly significant increase in DNS inspiral event rates for gravitational-wave (GW) interferometers like LIGO (Abramovici et al. 1992). In what follows, we summarize our recent results and present new results on the calculation of a global probability density function (PDF) of rate estimates. We also discuss constraints from the Type Ib/c SN rates on the rate estimates. 107 A. Baykal et al. (eds.), The Electromagnetic Spectrum of Neutron Stars, 107–110. © 2005 Springer. Printed in the Netherlands.

108

The Double-Neutron-Star Inspiral Rate and Expectations for GW Detection

Figure 1. P (R) of the DNS inspiral rate estimates is shown on a log scale. The thick solid line is for the Galactic rate estimate overlapped with results for individual systems (dashed lines). Dotted lines indicate confidence intervals for the rate estimates. The same results are shown on a linear scale in the small inset. All results shown are for our reference model.

2.

The Galactic DNS Inspiral Rate

In Fig. 1, we show the PDF of the rate estimates, P (R), for our reference model (Model 6 in Kim et al. 2003). The most likely value of R turns out to be ∼ 83 Myr−1 , larger by a factor of 6.4 than the rate estimated before the discovery of J0737−3039. For all models we consider, the peak values of the DNS inspiral rate are found in the range ∼ 4−220 Myr−1 . The revised inspiral rate implies an increase in the detection rate of DNS inspirals for ground-based GW interferometers. We obtain the most probable event rates to be one per 30 yr and one per 2 days, for initial and advanced LIGO, respectively. At the 95% confidence interval, the most optimistic predictions for the reference model are one event per 8 yrs and two events per day for initial and advanced LIGO, respectively. For more details, see Kalogera et al. (2004a; 2004b).

3.

Global Probability Distribution Of The Rate Estimates

In Kim et al. (2003), we showed that estimated Galactic DNS inspiral rates are strongly correlated with the assumed pulsar luminosity function. Here, we describe how we can incorporate systematic uncertainties from the pulsar

Rate Constraints From Type Ib/c Supernovae

109

luminosity function and calculate, Pg (R), a global PDF of rate estimates. We note that specific quantitative results would change when constraints on the luminosity function are derived from the the recent discoveries of faint pulsars (Camilo 2003). In order to calculate Pg (R), we first derive the prior distributions of two model parameters for the pulsar luminosity function, the cut-off luminosity Lmin and power-index p, by fitting the marginal PDFs presented by Cordes & Chernoff (1997). We note, however, that their results does not reflect the current pulsar sample. Then we calculate Pg (R) by incorporating those priors with P (R). We find that Pg (R) is strongly peaked at only around 15 Myr−1 (Fig. 2). This is a factor 5.5 smaller than the revised rate from the reference model (R 83 Myr−1 ). At the 95% confidence interval, we obtain R ∼ 1 − 170 Myr−1 and the corresponding LIGO event rates are in the range ∼ (0.4 − 70) × 10−3 yr−1 (initial) and ∼ 2 − 380 yr−1 (advanced).

4.

Rate Constraints From Type Ib/c Supernovae

Based on our current understanding of DNS formation, the progenitor of the second neutron star is expected to form during a Type Ib/c supernova. There-

Figure 2. Pg (R) on a linear scale (bottom panel). Dotted lines represent the lower (SNL ) and upper (SNU ) bounds on the observed SN Ib/c rate (600–1700 Myr−1 ) scaled by 1/10 and 1/100. In the top panel, we show the assumed prior functions for Lmin (left) and p (right).

110

The Double-Neutron-Star Inspiral Rate and Expectations for GW Detection

fore, empirical estimates for the Type Ib/c SN rate in our Galaxy provide constraints on the DNS inspiral rate estimates as an upper limit. In order to compare the observed SN rate and the DNS inspiral rate estimates, it is necessary to know the fraction of SN Ib/c actually involved in the formation of DNS. We adopt the empirical rate RSN Ib/c 1100 ± 500 Myr−1 (Cappellaro et al. 1999) and, as a simple trial, scale them by 1/10 and 1/100. The result is shown in Fig. 2. Within an order of magnitude of uncertainty, we can conclude that the empirical rate estimation for DNS is consistent with the observed SN rate.

5.

Conclusion

We calculate the Galactic DNS inspiral rate considering currently observed systems. Having a PDF allows us to give statistical confidence to the rate estimates. We also derive the global PDF of the rate estimates. If constraints on the pulsar luminosity distribution for the current pulsar sample become available, it is possible to establish more realistic distribution for the rate estimates. With a simple analysis, we show that the empirical rate estimates are roughly consistent with the observed SN Ib/c rates. Finally, we note that only the discovery of a significantly different system from the three already known, i.e. in pulsar and/or binary properties, may appreciably change the Galactic inspiral rate.

Acknowledgments We would like to thank Kip Thorne for suggesting incorporating the systematics into a single PDF. This research is partially supported by NSF Grant 0121420, and a Packard Foundation Fellowship in Science and Engineering to VK. DRL is a University Research Fellow supported by the Royal Society. He also thanks the Theoretical Astrophysics Group at Northwestern University for support. KB is a Lindheimer Fellow at Northwestern University and also acknowledges support from grant PBZ-KBN-054/p03/2001.

References Abramovici, A., et al. 1992, Science 256, 325 Burgay, M., et al. 2004, Nature, 426, 531 Camilo F. 2003, Radio Pulsars, ASP Conference Series, eds. M. Bailes, D.J.Nice, and S.E. Thorsett, 145 Cappellaro, E., Evans, R., & Turatto, M. 1999, ApJ, 351, 459 Cordes, J.M., & Chernoff, D.F. 1997, ApJ, 482, 971 Kalogera, V., et al. 2004a, ApJ, 601, L179 Kalogera, V., et al. 2004b, ApJ, 614, L137 Kim, C., Kalogera, V., & Lorimer, D.R. 2003, ApJ, 584, 985

DISKS AROUND RADIO PULSARS Can the Disk Survive the Radiation Pressure? K. Yavuz Eksi ˙ Sabancı University, 34956, Orhanlı–Tuzla, Istanbul, Turkey Present address: Harvard Smithsonian CFA, 60 Garden St. Cambridge, MA 02138 [email protected], [email protected]

M. Ali Alpar ˙ Sabancı University, 34956, Orhanlı–Tuzla, Istanbul, Turkey [email protected]

Abstract

The magnetic field around a pulsar varies from the near zone dipole magnetic field (B ∝ r−3 ) dominating the electric field, to the radiation zone field B ∼ E ∝ r−1 , across the light cylinder radius RL = c/Ω∗ . We argue that the transition from near zone to the radiation zone can be very broad if the inclination angle between the rotation and magnetic axis is small. This allows us to define stable equilibrium between the magnetosphere of the neutron star and a surrounding disk beyond the light cylinder radius. We conclude that radio pulsar activity and disks are not mutually exclusive and can exist together. We discuss implications for fallback disks and accreting millisecond pulsars.

Keywords:

accretion disks— stars: neutron—X-rays:binaries—stars: pulsars

1.

Introduction

Neutron stars with accretion disks can manifest themselves as X-ray pulsars. Although they are highly magnetized rotating objects, accreting pulsars are not observed simultaneously as radio pulsars. The presence of the disk inside the light cylinder radius, RL = c/Ω∗ , suppresses the radio pulsar action. The radio pulsar could turn on if the inner radius of the disk goes beyond RL corresponding the innermost open field line. Indirect evidence of a turned on radio pulsar has been claimed for the quiescent state of the soft X-ray transient Aquila X–1 [1] and for the transient millisecond pulsar SAX J1808.4–3658 [2]; [3]. Inside RL , the inner radius of the disk is estimated by the Alfvén radius [4], RA at which kinetic energy density of the disk EK ∝ r−5/2 is balanced by the 137 A. Baykal et al. (eds.), The Electromagnetic Spectrum of Neutron Stars, 137–140. © 2005 Springer. Printed in the Netherlands.

138

Disks Around Radio Pulsars

magnetic energy density Em ∝ r−6 of the dipole field of the star. For r < RA , Em would be greater than EK justifying the disruption of the disk and formation of an inner boundary. In the radiative zone electromagnetic energy density produced by the neutron star scales as Eem ∝ r−2 , i.e. decreases less rapidly than EK beyond the equilibrium point and hence sweeps away the disk. Noting this, Shvartsman [5] concluded that a stable equilibrium beyond RL would only be possible beyond the gravitational capture radius [6] . In this work we employ a model with a transition zone for the fields where they gradually change from the near zone dipole fields to the radiation fields at infinity. We show that the disk need not be disrupted once it goes beyond RL as there can be a domain where the electromagnetic energy density drops more rapidly (with increasing r) than EK ∝ r−5/2 , allowing stable equilibrium configurations.

2.

Transition From the Near Zone to the Radiation Zone

In order to describe the transition from the near zone to the radiation zone, a global solution for the electromagnetic field structure of a neutron star is required. A global solution for a perfectly conducting, rigidly rotating star in vacuum is given by Deutsch (1955) [7]. In the approximation R∗ RL where R∗ is the radius of the neutron star, the Deutsch solutions for the electromagnetic energy density Eem = (E 2 + B 2 )/8π give (see [8] for further discussion)

4 2 µ2 −6 5 2 4 2 2 1 + x cos ξ + x + 2x + sin ξ x (1) Eem = 9 2 8πRL6 where x = r/RL and ξ is the inclination angle between the rotation and magnetic axis. Note that this reduces to Eem ∝ r−6 for r RL and Eem ∝ r−2 for r RL , as expected. The local power-law index of the electromagnetic energy density, γ ≡ d ln Eem /d ln x, calculated from equation (1) is shown for a variety of ξ in Figure (1). It is seen that the transition from the near zone to the radiation zone is broader for small inclination angles. The inner radius can be estimated as the electromagnetic radius Rem , determined by Eem = EK which gives a generalization of the Alfvén radius. The critical radius at which γ = −5/2 will separate the stable and unstable equilibrium domains is shown in Figure 2. It is seen that the inner radius will be stable beyond the light cylinder, for distances up to many RL , practically for all radii for aligned (ξ = 0o) rotators and up to Rcrit ∼ = 2.85RL for orthogonal (ξ = 90o) rotators. The value of γ at the light cylinder, γL ≡ γ(x = 1), varies between −5.38 to −4.55. For spherical accretion, the kinetic energy density scales with x−5/2 . Note that for all inclination angles γL < −5/2. For thin disks, the density of the gas in the disk scales as ρ ∝ r−15/8 [9] implying EK ∝ ρvK2 ∝ ρr−1 ∝ r−23/8 where vK is the Keplerian velocity. This scaling is slightly more steep than the spherical accretion case, but again

Discussion

Figure 1. Power-law index of the electromagnetic energy density for a variety of inclination angles. The power-law index changes between -6 and -2. The transition becomes broader for small inclination angles.

139

Figure 2. The critical radius, in terms of light cylinder radius, at which γ = −5/2. For small inclination angles, a stable inner radius for the disk can be found for distances up to many RL .

γL < −23/8 i.e. the electromagnetic energy density is steep enough to balance the kinetic energy density of the disk at the light cylinder. For a thin disk, the minimum critical inner radius will be Rcrit,min = 2.13RL .

3.

Discussion

We presented a generalization of the Alfvén radius that is stable beyond the light cylinder up to a critical radius Rcrit determined by the inclination angle between the rotation and magnetic axis. Within the model, a disk can survive beyond the light cylinder, even if the radio pulsar activity commences. As long as Rem < Rcrit , the disk will not be ejected, and the radio pulsar may turn off again when M˙ increases such that Rem < RL , switching back to the propeller phase. Even when the pulsar activity is turned on, the presence of the disk may effect the coherent radio emission. The magnetic dipole radiation torque will act on the neutron star even if a radio pulsar is not observed. Sources on their evolutionary path to higher or lower mass inflow rates will make a transition from or to a stage with a disk inner radius stably placed in the radiation zone, and with possible rotation powered activity. Accretion driven millisecond Xray pulsars are likely examples of late stages in the evolution of LMXBs into millisecond radio pulsars through spin-up by accretion [10]; [11]. Such sources might hover around the transition, exhibiting transient behavior. Burderi et al. [2] argued that the irradiation of the companion by the switched on magneto-dipole rotator, in the quiescent stage, can explain the modulation of the flux in the optical (see also [3]). If the source is in the propeller stage [12]

140

Disks Around Radio Pulsars

accretion of only a small fraction of the disk inflow through a limited bunch of field lines will probably allow the magnetospheric gaps and pulsar radiation to survive. This would be easier for the outer gaps. Magnetospheric voltages in millisecond pulsars are of the order of those in the Vela pulsar, so optical, X and gamma ray pulsar activity may be possible if the outer gap can survive. For small inclination angles, accretion near the magnetic polar caps would also be near the rotational pole and therefore avoid the centrifugal barrier. For paths avoiding the inner gap even the radio pulsar activity might survive in the propeller phase, with the disk protruding inside the light cylinder. Depending on the beaming geometry, searches for radio and high energy pulsar activity might yield very interesting results. Menou et al. (2001) [13] and Alpar et al. (2001) [14] assumed that the inner radius of fallback disks [15] tracks the light cylinder radius. In such a model the disk can assist the magnetic dipole radiation torque without quenching the radio pulsar mechanism. These models would not work [16] if the disruption of the disk takes place [5] as soon as its inner edge reaches the light cylinder from within the near zone. The present work shows that such a disk need not be necessarily disrupted till its inner radius moves to a few times the light cylinder radius.

Acknowledgments This work was supported by Sabancı University Astrophysics and Space Forum, by the High Energy Astrophysics Working Group of TÜB˙ITAK and by the Turkish Academy of Sciences for MAA.

References [1] Campana, S. et al. 1998, ApJ, 499, L65 [2] Burderi, L., Di Salvo, T, D’Antona, F., Robba, N.R., & Testa, V., 2003, A&A, 404, L43 [3] Campana, S. et al. 2002, ApJ Lett., accepted, astro-ph/0408584 [4] Davidson, K., Ostriker, J.P., 1973, ApJ, 179, 585 [5] Shvartsman, V. F., 1970, Astron. Zh., 47, 660 [6] Lipunov, V.M., 1992, Astrophysics of Neutron Stars (Berlin: Springer) [7] Deutsch, A.J., 1955, Annales D’Astrophysique, 18, 1. [8] Ek¸si, K.Y., Alpar, M.A., 2005, ApJ, in print [9] Shakura, N.I., Sunyaev, R.A., 1973, A&A, 24, 337 [10] Alpar, M. A., Cheng, A.F., Ruderman, M.A. & Shaham, J., 1982, Nature, 300, 728 [11] Radhakrishnan, V. & Srinivasan, G., 1982, Curr. Sci., 51, 1096 [12] Illarionov, A.F., Sunyaev, R.A., 1975, A&A, 39, 185. [13] Menou, K., Perna R., & Hernquist, L. 2001a, ApJ, 554,L63. [14] Alpar, M. A., Ankay, A., and Yazgan, E. 2001, ApJ, 557, L61 [15] Michel, F.C., Dessler, A.J., 1981, ApJ, 251, 654 [16] Li, X.-D., 2002, ApJ Lett., 579, L37.

OBSERVATIONS OF COOLING NEUTRON STARS J.E. Trümper Max-Planck-Institut für extraterrestrische Physik 85741 Garching Germany

Abstract

Observations of cooling neutron stars allow to measure photospheric radii and to constrain the equation of state of nuclear matter at high densities. In this paper we concentrate on neutron stars, which show thermal (photospheric) Xray emission and have measured distances. After a short summary of the radio pulsars falling into this category we review the observational data of the 7 radio quiet isolated neutron stars discovered by ROSAT which have been studied in detail by Chandra, XMM-Newton and optical observations. Their spectra show blackbody temperatures between 0.5 and 1 million Kelvin and an optical excess of a factor of 5-10 over the extrapolation of the X-ray spectrum. Four of these sources show periodicities between 3.45 and 11.37 sec indicating slow rotation. The pulsed fractions are small, between 6 and 18 %. The magnetic fields derived from spin down and/or possible proton cyclotron lines are of the order 1013 -1014 G. We then discuss RX J1856.5–3754 in detail and suggest that the remarkable absence of any line features in its X-ray spectrum is due to effects of strong magnetic fields (∼1013 G). Assuming blackbody emission to fit the optical and X-ray spectrum we derive a conservative lower limit of the “apparent” neutron star radius of 16.5 km × (d/117 pc). This corresponds to the radius for the “true” radius of 14 km for a 1.4 M neutron star, indicating a stiff equation of state at high densities. A comparison of the result with mass-radius relations shows that in this case a quark star or a neutron star with a quark matter core can be ruled out with high confidence.

Keywords:

stars:neutron – radiation mechanisms:thermal – equation of state

1.

Introduction and History

1960’s and 1970’s: The rocket experiments and the early satellites like Uhuru and Ariel-5 were not sensitive enough to detect the weak and soft thermal emission of neutron stars. A speculation by Chiu (1964) that the X-rays from the Crab nebula were due to a hot neutron star with kT∼4 keV was soon disproved by the famous NRL lunar occultation experiment which found only an extended source (Bowyer et al. 1964). 117 A. Baykal et al. (eds.), The Electromagnetic Spectrum of Neutron Stars, 117–131. © 2005 Springer. Printed in the Netherlands.

118

Observations of Cooling Neutron Stars

1980’s: The Einstein observatory gave the first sensitive upper limit for the temperature of the Crab pulsar, kT24 m50CCD = 28.6 B>27 V=25.7 R>23

pr. mot. (mas/y) 97 145 332 -

distance

Four of the seven sources show X-ray pulsations with periods of typically 10 sec and pulsed fractions of typically 10% (c.f. Table 1), suggesting that the neutron stars have an inhomogeneous temperature distribution. The slow down rate measured for RX J0720–3125 (henceforth RX J0720) leads to estimates of the magnetic field of ∼3 × 1012 G and of the age of 106 years. A few of these objects exhibit small but significant changes of the spectra with pulse phase which may be explained by the anisotropic emission of strongly magnetized plasmas. Four of the sources show broad absorption line features which have been attributed to proton cyclotron absorption/scattering in magnetic fields of a few times 1013 G (Table 2). In RX J0720 long term spectral changes have been found (de Vries et al. 2004), which have been interpreted in terms of neutron star precession. Table 2. Magnetic field estimates for radio quiet isolated neutron stars (Haberl 2004) Object (name) RX J0420.0-5022 RX J0720.4-3125 RX J0806.4-4123 1RXS J130848.6+212708 RX J1605.3+3249 RX J1856.5-3754 1RXS J214303.7+065419

P (s) 3.45 8.39 11.37 10.31 -

dP/dt (10−13 ss−1 ) < 92 (1.4±0.6) < 18 -

Ecyc (eV) 329 262 100-300 450-480 -

Bdb (10−13 G) < 18 2.8-4.2 < 14 ∼1 -

Bcyc (10−13 G) 6.6 5.2 2-6 9.1-9.7 -

RX J1856–3754

123

In summary, these findings strongly suggest that these “magnificent seven” are strongly magnetized (1013 -1014 G), slowly rotating neutron stars having an inhomogeneous temperature distribution over the stellar surface. Their main energy source must be heat loss from the hot interior (cooling), since accretion of matter from the interstellar medium is too inefficient due to the high stellar velocities. These sources do not show radio emission, probably because either they are evolved beyond the pulsar death line or because their radio beam is too narrow due to their large light cylinder radius.

4. 4.1

RX J1856–3754 General Properties

Among the radio quiet isolated neutron stars RX J1856.5-3754 (henceforth RX J1856) is the brightest and the only one with a known distance. Therefore it is best qualified for detailed studies aiming at a determination of its radius, and in the rest of this paper we concentrate primarily on this object. RX J1856 was discovered serendipitously in a ROSAT PSPC field by Walter et al. (1996). Using the (HST) Walter & Matthews (1997) identified the X-ray source with a faint blue star (V ∼ 26 mag). Its distance and proper motion were determined with the HST by Walter & Lattimer (2002), to be (117±12) pc and 0.33 arcsec/year, respectively. With the VLT van Kerkwijk & Kulkarni (2001) found a faint nebula surrounding the point source which has a cometary-like geometry with a 25? tail extending along the direction of motion. None of the X-ray observations revealed any variability on time scales up to ten years. The so far best upper limit of 1.3% (2σ) on periodic variations in the range 10−3 –50 Hz has been established by Burwitz et al. (2003) using a XMMNewton EPIC-pn observation. Chandra LETG observations with high spectral resolution show a spectrum that can be fit by a Planckian spectrum with a temperature of 63±3 eV, (c.f. Fig. 3). Despite the excellent photon statistics and the good energy resolution of the LETG this spectrum is devoid of any spectral features. Compared with the optical spectrum which shows a Rayleigh-Jeans slope (∼ ν 2 ), the X-ray spectrum is reduced by a factor of ∼6. Therefore, the overall spectrum of the source has often been described by a two-temperature blackbody model (e.g. Pons et al. 2002, Burwitz et al. 2003, Pavlov & Zavlin 2003, Trümper et al. 2004). A large number of papers have been dealing with the questions concerning the nature of this compact object and the proposed answers include everything from “normal” neutron stars with stiff or soft equations of state over neutron stars having a quarks core to bare (strange) quark stars, P-stars etc (for references c.f. Turolla et al. 2003). Before coming back to this topic we want to summarize some more observational data and their immediate consequences in somewhat more detail.

124


Figure 3. The Chandra LETG X-ray spectrum of RX J1856 fitted with (non-magnetic) photospheric models assuming pure iron and solar composition. The best fit is obtained with a Planck spectrum (Burwitz et al. 2003).

4.2

The Magnetic Field Strength of RX J1856.5–3754

The impressive lack of any significant spectral features in the LETG spectrum excludes magnetic fields of (1.3 - 7) × 1011 G (electron cyclotron lines) and (2 - 13) × 1013 G (proton cyclotron lines), see Burwitz et al. (2003). This leaves the possibility open of a low magnetic field characteristic for millisecond pulsars or a high magnetic field typical for normal pulsars. Unfortunately, due to the absence of a periodicity the usual estimate of the magnetic field of RX J1856 based on the rotating dipole model is not possible. Using phenomenological arguments based on the very small pulsed fraction in Xrays and on a comparison with other objects van Kerkwijk & Kulkarni (2001) have argued that the star has a relatively low magnetic field of a few 1011 G which may be marginally consistent with the absence of proton cyclotron lines. But this is not the only possibility. We estimate the magnetic field using the spin-down luminosity dE/dt ∼ 4 × 1032 erg/s required for powering the cometary-like emission nebula (Kerkwijk & Kulkarni 2001) and the age of the

RX J1856–3754

125

star (t ∼ 5 × 105 years) inferred from its proper motion and the distance to its likely birthplace in the Upper Sco OB association (Walter & Lattimer 2002). Applying the model of magnetic dipole braking we find a period of a ∼1.8 sec and a magnetic field strength of ∼1.1 × 1013 G. We emphasize that these figures are very similar to those of the second brightest object of this kind, the pulsating source RX J0720 whose spectral characteristics are very similar to those of RX J1856. While the estimate of dE/dt may be considered as rather reliable, the age derived from the birthplace argument is not so certain. However, an age of t ∼ 5 × 105 years (with an uncertainty of a factor of two) is fully consistent with what we know empirically about the cooling of neutron stars. We therefore conclude that the magnetic field of RX J1856 is probably large, i.e. of the order of >1013 G. To confirm this, it is necessary to exclude the alternative hypothesis of a millisecond pulsar (van Kerkwijk & Kulkarni 2001, Pavlov & Zavlin 2003). To this end a high time resolution observation with XMM-Newton has already been scheduled.

4.3

The Featureless X-ray Spectrum of RX J1856.5–3754

The main puzzle of RX J1856 is the observational fact that its X-ray spectrum (Fig. 3) is completely featureless. It has been pointed out by Burwitz et al. (2001, 2003) that nonmagnetic photospheric spectra assuming a pure iron composition are incompatible with the measured spectrum because the predicted Fe-L features are not detected with high significance. Even a solar composition model with its small abundance of metals leads to unacceptable spectral fits. Doppler smearing of the spectral lines due to fast rotation does not wash away completely the strongest spectral features (Braje & Romani 2002, Pavlov et al. 2002)]. On the other hand hydrogen or helium photospheres can be excluded, because they would over-predict the optical flux by a very large factor (Pavlov et al. 1996). Therefore any nonmagnetic photosphere can be firmly excluded. This argument can be extended to magnetized hydrogen and helium photospheres (Zavlin & Pavlov 2002). Iron photospheric models have been calculated by Rajagopal et al. 1997 for B=1012.5 and 1013 G.Unfortunately they suffer from the fact that the radiative properties of iron atoms/ions in super strong magnetic fields are not known exactly, but only in Hartree-Fock approximation (work of Neuhauser et al. (1986). The resulting spectra contain a lot of lines having spacings of 50-100 eV, which could be easily resolved by the LETG (resolution 10000 sec by the propeller effect. This requires an extremely strong magnetic field (∼1015 G) and a relatively low velocity (Mori & Ruderman 2003). The simplest explanation is that the rotational axis of the neutron star is closely aligned with the line of sight or with the magnetic axis. This may look unlikely in view of the low pulsed fraction of 16.5 km (3σ). As an alternative we use a model with a continuous temperature distribution (c.f. Fig. 4b) of the form T = Th × {1 + (θ/θ0 )γ }−1

(1)

128


Figure 4. Blackbody fits to the optical and X-ray spectra of RX J1856.5-3754 for a twocomponent model (a) and a model with a continuous temperature distribution (b), see text.

The best fit to the overall spectrum yields a central temperature of the hot spot Th = 82 eV, an angular size of the hot spot θ0 = 400 and γ = 2.1. In this case the neutron star radius turns out to be 16.8 km (>3σ), not much different from that of the simpler model. These apparent radii R measured by a distant observer are related to the “true” stellar radius R0 by R = R0 (1 − Rs /R0 )−1/2

(2)

where Rs = 2GM/c2 is the Schwarzschild radius. The corresponding bound in the M – Ro diagram is shown in Fig. 4. For a standard neutron star of 1.4 solar masses the true radii are R0 = 14.0 km (Fig. 4a) and R0 = 14.1 km (Fig. 4b), respectively, and thus considerably larger than the canonical radius of 10 km.

RX J1856–3754

129

Figure 5. The mass-radius relations for various equations of state for the nuclear matter according to [26]. The thick dashed curve represents the apparent minimum neutron star radius derived from both the two-component and continuous temperature blackbody models and a source distance of 117 pc.

This implies a rather stiff equation of state. We note, that the same conclusion was reached by Braje & Romani (2002) using a two-component model and similar arguments. In order to compare our results with the predictions of theoretical neutron star models in more detail we use the mass-radius diagram given by Pons et al. (2002) This diagram is shown in Fig. 5 to which we have added a curve corresponding to the apparent radius of R = 16.5 km. It is evident that the result of our analysis excludes the quark star models discussed by Pons et al (2002) and by Schertler et al. (1998). Also the neutron star models with quark matter cores discussed in the latter paper are rejected. We conclude that for a source distance of 117 pc this neutron star must have a very stiff equation of state. Recent improvements of the RX J1856–3754 parallax, which use additional HST observations at four different epochs (Kaplan 2004) yield an even larger distance of 160 pc. This result considerably sharpens our conclusion. In this context one may speculate that a too large radius could imply that this neutron star has an anomalously low mass (14 km instead of the canonical size of 10 km. This result gets support from the observations of the three radio pulsars discussed in section 2. This has consequences for pulsar astrophysics: E.g. the magnetic field strengths estimated from pulsar spin down observations have to be lowered by at least a factor of two since B ∼R−2 o , and the moment of inertia and therefore the estimate of the pulsar rotational energy of a pulsar increases by a similar factor. However, the most important result of our analysis is that the behavior of nuclear matter at very high densities is governed by a very stiff equation of state.

5.

Outlook

We have learned a lot about cooling neutron stars from X-ray and optical observations since 1990, and we can hope to learn a lot more from Astro-E, XEUS and Constellation-X in the future. However, an all-sky survey in the soft X-ray band which is at least ten times more sensitive than ROSAT would be most important, to find more and fainter sources of this type. At the same time, it would be necessary to improve the sensitivity of optical observations with instruments of the 30-100 m class in order to measure the faint optical spectra. Finally, it seems essential to develop further our understanding of atomic and condensed matter physics under the conditions of very strong magnetic fields.

Acknowledgments The continuous support and cooperation of Werner Becker, Vadim Burwitz, Frank Haberl and Slava Zavlin is gratefully acknowledged.

References Akiyama, S., Wheeler, J. C., Meier, D. L., Lichtenstadt, I. (2003), ApJ, 584, 954. Becker, W. 2004, private communication. Bowyer, C.S., et al. 1964, Science 146, 912. Braje, T.M. & Romani, R.W. 2002, ApJ 580, 1043. Brinkmann, W. 1980, A&A 82, 352. Brisken, W.F., et al. 2003, Apj 593, L89. Burwitz, V., et al. 2001, A&A 379, L35. Burwitz, V., et al. 2003, Astron. & Astrophys. 399, 1109. Caraveo, P.A., et al. 1996, ApJ 461, L91. Chiu, H.-Y. 1964, Ann. Phys. 2, 364. Dodson, R., et al. 2003, MNRAS 343, 116. De Vries, C.P., et al. 2004, to be published in A&A. Haberl, F. 2004, Adv. Space Res. 33, 638. Harnden, F.R. & Seward, F.D. 1984, ApJ 283, 279. Kaplan, D. 2004, private communication. Lai, D. 2001, Rev.Mod.Phys. 73, 629.

Outlook

131

Mc Gowan, K.E., et al. 2003, ApJ 591, 380. Mori, K. & Ruderman, M.A. 2003,ApJ 592, L75. Neuhauser, D., et al. 1987, Phys.Rev. A36, 4163. Pavlov, G.G., et al. 2002, in Proc. of the 270th Haereus Seminar on Neutron Stars and Supernova Remnants, W. Becker, H. Lesch, & J. Trümper (eds.) MPE Report 278, 273 astroph/0206024). Pavlov, G.G. & Zavlin, V.E. 2003, in Proceedings of the XXI Texas Symposium on relativistic Astrophysics, B. Rino, R. Maiolino, & M. Filippo (eds.), astro-ph/ 0305435). Pavlov, G.G., et al. 1996, ApJ 472, L33. Pons, J.A., et al. 2002, ApJ 564, 981. Rajagopal, M., et al. 1997, ApJ 479, 347. Schertler, K., et al. 1998, Nucl. Phys. A637, 451. Trümper, J. & Lenzen, R. 1978, Nat 271, 216. Trümper, J. 1983, Adv. Space Res. 2, 142. Trümper, J., et al. 2004, Nucl. Phys., Vol 132C, 560. Turolla, R., et al. 2003, ApJ 603, 265. Van Kerkwijk, M.H. & Kulkarni, S.R. 2001, A&A 380, 221. Walter, F.M., et al. 1996, Nat 379, 233. Walter, F.M. & Lattimer, J. 2002, ApJ 576, L145. Walter, F.M. & Matthews, L.D. 1997, Nat 389, 358. Weisskopf, M. 2005, (this volume). Zavlin, V.E. & Pavlov, G.G. 2002, in Proc. of the 270th Haereus Seminar on Neutron Stars, Pulsars and Supernova Remnants, W. Becker, H. Lesch & J. Trümper (eds.), MPE Report 278, 263.

OPTICAL STUDIES OF ISOLATED NEUTRON STARS AND THEIR ENVIRONMENTS Roberto P. Mignani European Southern Observatory [email protected]

Abstract

The results of optical studies of Isolated Neutron Stars (INSs), their Pulsar-Wind Nebulae (PWNe) and Pulsar Bow Shocks are reviewed and discussed.

Keywords:

stars:pulsars – stars:neutron – stars:circumstellar matter

1. 1.1

Isolated Neutron Stars The identification record

The Isolated Neutron Stars (INSs) with an associated optical counterpart, including both rotation-powered pulsars and the so-called X-ray Dim INSs

Table 1. INSs identification status. The columns give the name, the year of the proposed identification, the used telescope and its aperture, the magnitude (V -band when available), the distance, the interstellar absorption AV and the identification evidence. Name Crab Vela B0540-69 Geminga B0656+14 B0950+08 B1929+10 B1055-52 RXJ1856-3754 J0720-3125 B1509-58 RXJ1308.6+2127 RXJ1605.3+3249 J0437-4715

Year 1969 1976 1984 1987 1994 1996 1996 1997 1997 1998 2000 2002 2003 2004

Tel Steward CTIO CTIO CFHT NTT HST HST HST HST Keck VLT HST HST HST

∗ http://rsd-www.nrl.navy.mil/7213/lazio/ne

Size 0.9m 4m 4m 3.6m 3.5m 2.4m 2.4m 2.4m 2.4m 10m 8.2m 2.4m 2.4m 2.4m model/

mag 16.6 23.6 22 25.5 25 27.1 25.6 24.9 25.7 26.7 25.7 28.6 26.8 ∗∗ PM:

d(kpc)∗ 1.73 0.23 49.4 0.16 0.29 0.26 0.33 0.72 0.14 4.18

0.14 Proper Motion,

AV 1.6 0.2 0.6 0.07 0.09 0.03 0.15 0.22 0.12 0.30 5.2 0.14 0.06 0.11

∗∗∗ Ph:

Identification Pulsations Pulsations Pulsations PM∗∗ /Pulsations Pulsations/PM Position/Ph∗∗∗ PM Position PM PM Position Position PM Spectroscopy Photometry


134

Optical Studies of Isolated Neutron Stars and Their Environments

(XDINSs), are now 14 (Table1; see also Mignani et al. 2004a) i.e. about as many as those detected in X-rays in the pre-ROSAT era. Mostly thanks to the HST and the high UV sensitivity of the FOC and the STIS, the initial identification score of two objects per decade has increased to almost one per year. As a matter of fact, in the last 10 years HST has detected all the INSs it was targeted to. On the other hand, large ground-based telescopes like the Kecks and the VLT so far played only a marginal role. Recent HST observations might have identified also the counterpart of the young 16 ms pulsar PSR J0537-6910 (Mignani et al. 2004b). Owing to their intrinsic faintness, most INSs have been detected only because of their close distance and small interstellar absorption. This made also possible to use proper motion as an alternative, and indeed very efficient, identification technique to optical timing. Table 2. Optical INS database grouped by age decades. The columns give the name, spectroscopy, photometry, the spectral index α and temperature (in units of 105 K) of the power-law and blackbody components (P Lo ;BBo ) and the comparison with the optical extrapolation of the X-rays ones (P Lx ;BBx ). Polarization measures are indicated in the last column. Name Crab

Spec.(ρA) Phot. 1100-9000 UV,UBVRI,JHK

α -0.11

T -

Comments P Lo < P L x

B1509-58 B0540-69 Vela B0656+14

R 2500-5500 UBVRI 4500-8600 UV,UBVRI,JH UV,UBVRI,JHK

+0.2 +0.12 +0.45

8.5

Geminga

3700-8000 UV,UBVRI,JH

+0.8

4.5

P Lo < P L x P Lo ∼ P L x P Lo ∼ P L x BBo ∼ BBx P Lo < P L x BBo ∼ BBx

+0.5 +0.65 -1.4 -

1.0 4 2.3

B1055-52 B1929+10 B0950+08 J0437-4715 RXJ0720-3125 RXJ1856-3754 RXJ1605.3+3249 RXJ1308.6+2127

U UV,U U,BVI 1150-1700 UV,UBVR 3600-9000 UV,UBV VR V

∗∗ IP=Inter Pulse; OP=Off Pulse; TI=Time Integrated

1.2

Pol.∗∗ 20% (IP) 40% (OP) 10% (TI) 5 % (TI) 8.5% (TI) 100% (IP)

P Lo < P L x P Lo ∼ P Lx BBo > BBx BBo > BBx BBo > BBx BBo > BBx BBo > BBx

Photometry and Spectroscopy

Table 2 summarizes the optical INSs database (see also Mignani et al. 2004a and references therein). Only for six of them optical/UV spectroscopy is available and only for four photometry spans all the way from the IR to the UV. This is crucial to identify thermal and non-thermal spectral components whose contributions are expected to be markedly different in the IR and in the UV. As

Pulsar Bow-Shocks

135

a general trend, the spectrum grows in complexity with the age from a single power-law (PL) dominated to a composite one featuring both PL and blackbody (BB) components. While in some cases the optical PL/BB components do match the extrapolation of the X-ray ones, apparently this is not a general rule, which suggests that the optical and X-ray emission mechanisms are not always related to each other. In particular, for XDINs the optical BB spectrum appears to be systematically above the extrapolation of the X-ray one. Apart from the decrease of the temperature of the BB component, which follows from the cooling of the neutron star surface, there is no clear indication for an evolution of the spectral parameters with the neutron star’s age.

1.3

Timing

After the “historical” optical pulsars Crab, Vela and PSR B0540-69, pulsations have been clearly detected from Geminga (Romani & Pavlov, in preparation) and PSR B0656+14 (Gull et al., 2004) thanks to recent HST/STIS observations, confirming and improving the earlier results of Shearer et al. (1997) and Shearer et al. (1998). In all cases but PSR B0540-60 the lightcurves are double-peaked and for both the Crab and Geminga the peaks are aligned in phase with the γ-ray ones.

1.4

Polarimetry

Till recently, the only INS with measured optical polarization was the Crab (see also Kamback et al., these proceedings). The breakthrough came with the VLT which measured the time-integrated polarization for PSR B0540-69, Vela and PSR B1509-58 (Wagner & Seifert 2000). More recently, time-resolved polarization was measured for PSR B0656+14 (Kern et al. 2003).

2.

Pulsar-Wind Nebulae

So far, Pulsar Wind Nebulae (PWNe) have been detected in the optical only for two young pulsars: the Crab (e.g., Hester et al. 2002) and PSR B0540-69 (Caraveo et al. 2000). In both cases, HSTobservations have clearly resolved the counterparts of the X-ray structures detected by Chandra. The PWN around the Vela pulsar originally claimed by Ögelman et al. (1989) was not confirmed by HST observations (Mignani et al. 2003) which put 3σ upper limits of ≈ 27.9 and ≈ 28.3–27.8 mag arcsec−2 on the brightness of the inner and outer X-ray PWN, respectively, i.e. close the extrapolation of the X-ray/radio data. Comparable deep upper limits were set on the optical emission of the PWN around PSR J0537-6910 (Mignani et al. in preparation).

136

3.

Optical Studies of Isolated Neutron Stars and Their Environments

Pulsar Bow-Shocks

The interaction between the pulsar’s relativistic wind and the ISM compressed by the pulsar supersonic motion originates a shock which ionizes the ISM and produce emission in Hα . The Hα luminosity (LHα ) depends on the ˙ on the neutron star’s velocity (vN S ) and on pulsar’s rotational energy loss (E), the fraction X of neutral Hydrogen in the ISM. The bow-shock shape tends to be symmetric wrt the pulsar proper motion (µ) direction, with deviations determined by the local ISM density distribution, featuring either arc-like or bullet-like structures according to the perspective. So far, optical bow-shocks have been clearly identified around 6 INSs (see Table 3). By imposing pressure balance between the (radial) pulsar wind and the ISM and assuming a geometrical model for the bow-shock one can derive the local ISM density (ρISM ) and the angle of the pulsar’s velocity vector wrt the line of sight. From the 3-D velocity, the distance and the age one can then trace back the galactic orbital motion of the pulsar, given a galactic potential model, and localize its birth place, hence identify its progenitor stellar population. Table 3. INSs with Hα bow-shocks. Luminosities and E˙ are in units of erg cm−2 s−1 . Name B0740-28 B2224+65 B1957+20 J0437-4715 J2124-3358 RXJ1856-3754

Log(Age) (yrs) 5.20 6.05 9.18 9.2 9.8 ?

˙ Log(E) 35.14 33.08 35.20 34.07 33.63 ?

µ (mas/yr) 29 182 30.4 141 52.6 333

d (kpc) 1.9 2 1.53 0.14 0.27 0.14

vN S (km/s) 204 1700 225 98 72 220

Log(LHa ) ∼29 ∼30 ∼31 ∼28 ∼27 ∼26

Comment

ms,binary ms,binary ms,isolated

References Caraveo, P.A. et al., 2000, Proc. of A decade of HST science, Eds. M. Livio, K. Noll, and M. Stiavelli, p.9 Gull, T. et al. 2004, Proc. IAU Symposium 218 "Young Neutron Stars and Their Environments", eds F. Camilo and B. M. Gaensler Hester, J. J. et al. 2002, ApJ, 577, L49 Kern, B. et al., 2003 ApJ 597, 1049 Mignani, R.P. et al. 2004, Proc. IAU Symp. 218 "Young Neutron Stars and Their Environments", eds F. Camilo and B. M. Gaensler (astro-ph/0311468) Mignani, R.P., et al. 2004a, submitted to A&A Ogelman, H. B, Koch-Miramond, L., Aurieére, M. 1989, ApJ, 342, 83 Shearer, A. et al. 1997, A&A, 487, L181 Shearer, A. et al. 1998, ApJ 335, L21 Wagner, S.J. & Seifert, W., 2000, Proc. of IAU Coll. 177 "Pulsar Astronomy: 2000 and Beyond", ASP Conference Series, Vol. 202, p. 315, Eds. M. Kramer, N. Wex, and N. Wielebinski

RADIATION FROM CONDENSED SURFACE OF MAGNETIC NEUTRON STARS Matthew van Adelsberg,1 Dong Lai,1 and Alexander Y. Potekhin2,3 1 Center for Radiophysics and Space Research, Department of Astronomy, Cornell University, Ithaca, NY 14853; [email protected], [email protected] 2 Ioffe Physico-

Technical Institute, Politekhnicheskaya 26 194021 St. Petersburg, Russia; [email protected], 3 Isaac Newton Institute of Chile, St. Petersburg Branch, Russia

Abstract

Recent observations show that thermal X-ray spectra of many isolated neutron stars are featureless and in some cases well fit by a blackbody. Such perfect blackbody spectra are puzzling as radiative transport through typical neutron star atmospheres causes noticeable deviation from blackbody. Previous studies have shown that in a strong magnetic field, the outermost layer of the neutron star may be in a condensed solid or liquid form because of the enhanced cohesive energy of the condensed matter. Thus, thermal radiation can emerge directly from the degenerate metallic condensed surface, without going through a gaseous atmosphere. Here we discuss emission properties of condensed Fe and H surfaces of magnetic neutron stars in the regimes where such condensation may be possible.

Keywords:

stars:magnetic fields – stars:neutron – radiation mechanisms:thermal – X-rays:stars

1.

Introduction

So far about 20 NSs have been detected in thermal emission (for a review see Pavlov & Zavlin 2003). With the exception of 3-4 sources, the thermal spectra of isolated NSs are featureless and sometimes well fit by a blackbody. For example, deep observations with Chandra and XMM-Newton show that the soft X-ray (0.15-1 keV) spectrum of RX J1856.5−3754 (Walter et al. 1996) can be fit with an almost perfect blackbody at kT = 64 eV (e.g., Drake et al. 2002; Burwitz et al. 2003). The featureless, and in some cases “perfect” blackbody spectra observed in isolated NSs are puzzling. This is because a NS atmosphere is not a perfect blackbody emitter due to nongrey opacities. Recently, several groups have suggested that the spectrum of RX J1856.5−3754 might be explained if the NS has a condensed surface with no atmosphere above it (Burwitz et al. 2001,Burwitz et al. 2003,Mori & Ruderman 2003,Turolla et al. 2004). Lai & Salpeter 141 A. Baykal et al. (eds.), The Electromagnetic Spectrum of Neutron Stars, 141–144. © 2005 Springer. Printed in the Netherlands.

142

Radiation from Condensed Surface of Magnetic Neutron Stars

1997 studied the phase diagram of the H surface layer of a NS and showed that for strong magnetic fields, if the star surface temperature is below a critical value, the atmosphere can undergo a phase transition into a condensed state (see also Lai 2001). For B ≥ 1014 G, this may occur even for temperatures as high as 106 K. In this paper, motivated by recent observations of dim isolated NSs, we present calculations of the emissivity of condensed Fe or H surface of magnetic NSs in the regime where we expect condensation might be possible.

2.

Properties of Condensed Surface Layer

It is well known that strong magnetic fields can qualitatively change the properties of atoms, molecules and condensed matter. For B B0 (B0 = Z 2 e3 m2e c/3 = 2.35 Z 2 × 109 G, where Z is the nuclear charge number), the electrons in an atom are confined to the ground Landau level, and the atom is elongated, with greatly enhanced binding energy. Covalent bonding between atoms leads to linear molecular chains, and interactions between molecular chains can lead to the formation of three-dimensional condensed matter (see Lai 2001 for a recent review). For the case of hydrogen, Lai & Salpeter 1997 showed that in strong magnetic fields, there exists a critical temperature Tcrit below which a phase transition from gaseous to condensed state occurs, with kTcrit about 10% of the cohesive energy of the condensed hydrogen. Thus, Tcrit ∼ 8 × 104 , 5 × 105 , 106 K for B = 1013 , 1014 , 5 × 1014 G (Lai 2001). For heavy elements such as Fe, our current knowledge of the various forms of matter in strong magnetic fields is quite incomplete. Numerical results of Jones 1986, together with approximate scaling relations suggest an upper limit 2/5 of the cohesive energy (for Z ≥ 10) of Qs ≤ Z 9/5 B12 eV, where B12 = B/(1012 G). Thus for Fe, the critical temperature for phase transition Tcrit ≤ 2/5 0.1Qs /k ≤ 105.5 B12 K (Lai 2001). The emissivity of the condensed surface layer is related to the reflectivity (e) R of the surface through Kirchoff’s Law Iν = (1 − R)Bν (T ). As a first approximation, we use the free electron gas model for our calculations. The dielectric tensor for the condensed layer takes the form of Ginzburg 1970, with the collisional damping rates calculated as described by Potekhin 1999. For more details, see van Adelsberg et al. 2004.

3.

Emission from Condensed Surface Layer

We present results of surface emission for several cases: Figs. 1a and 1b show the dimensionless emissivity J = 1 − R as a function of photon energy E for Fe surface at 1013 G, and H surface at 1014 G, respectively. T = 106 K

143

Emission from Condensed Surface Layer

(a)

(b)

Figure 1. Dimensionless emissivity for Fe surface at B = 1013 G (a) and H surface at B = 1014 G (b).

in all our calculations, and the B field is assumed to be normal to the surface (the features in Figs. 1a and 1b are representative of our results for more complicated geometries). In both cases, the emissivity is reduced from blackbody at low energies, approaching unity for energies a few times the electron plasma energy Epe , and there is feature associated with the ion cyclotron energy EBi . For H, Epe is too high to be of observational interest. Figure 2 depicts specific flux as a function of photon energy for Fe surface composition at B = 1012 , 1013 G, and H surface for B = 1014 G. For the case of Fe, there is a reduced emission (by a factor of 2 or so) around EBi ≤ E ≤ Ec compared to the blackbody at the same temperature. For the H surface at B = 1014 G, the flux is close to blackbody at all energies except for a broad feature around EBi . Our calculations show that the emission spectrum resembles that of a diluted blackbody, with the reduction factor in the range of J = 0.4 − 1 depending on the photon energy (see Figs. 1 & 2). This would increase the inferred emission radius by a factor of J −1/2 . We note that the results presented in this paper correspond to a local patch of the NS; combining the emission from different surface elements will likely result in smoothing the absorption features of Figs. 1& 2. Furthermore, we have assumed a perfectly smooth surface. This is valid if the condensed matter is in a liquid state, as is likely to be the case for H condensate (see van Adelsberg et al. 2004). For Fe, the condensed surface is most likely a solid and we may expect a rough surface. If this is the case, the surface may be much less reflective than the results shown in Figs. 1 & 2, and the emission will be closer to the blackbody spectrum. The emission from a condensed NS surface is distinct from atmospheric emission in several aspects: (i) Atmospheric emission generally possesses a hard spectral

144

Radiation from Condensed Surface of Magnetic Neutron Stars

Figure 2.

Specific flux for Fe surface at B = 1012 , 1013 G and H at B = 1014 G.

tail whereas the condensed surface emission does not; (ii) The spectrum of a cool NS atmosphere can have both cyclotron and atomic absorption features (iii) The polarization signature of condensed matter emission is qualitatively different from that of atmospheric emission.

References Brinkmann, W. 1980, A&A, 82, 352 Burwitz, V., Zavlin, V.E., Neuhäuser, R., Predehl, P., Trümper, & J., Brinkman, A.C. 2001, A&A, 379, L35 Burwitz, V., Haberl, F., Neuhäuser, R., Predehl, P., Trümper, J., & Zavlin, V.E. 2003, A&A, 399, 1109 Drake, J., et al. 2002, ApJ, 572, 996 Ginzburg, V.L. 1970, Propagation of Electromagnetic Waves in Plasmas (2d ed.; Oxford: Pergamon Press) Jones, P.B. 1986, MNRAS, 218, 477 Lai, D. 2001, Review of Modern Physics, 73, 629 Lai, D., & Salpeter, E.E. 1997, ApJ, 491, 270 Mori, K., & Ruderman, M. 2003, ApJ, 592, L75 Pavlov, G. G., & Zavlin, V. E. 2003, in XXI Texas Symposium on Relativistic Astrophysics, ed. R. Bandiera et al. (Singapore: World Scientific), 319 Pons, J.A., Walter, F.M., Lattimer, J.M., Prakash, M., Neuhäuser, R., & An, P. 2002, ApJ, 564, 981 Potekhin, A.Y. 1999, A&A, 351, 787 Turolla, R., Zane, S., & Drake, J.J. 2004, ApJ, 603, 265 van Adelsberg, M., Lai, D., Potekhin, A. 2004, ApJ, submitted Walter, F., Wolk, S.J., & Neuhauser, R. 1996, Nature, 379, 233

V

PULSARS

EMISSION MECHANISMS AND BROAD BAND SPECTRA OF PULSARS K.S. Cheng Department of Physics, University of Hong Kong, Pokfulam Road, Hong Kong, China [email protected]

Abstract

We review some theoretical models, which are used to explain some X-ray and gamma-ray data from rotation powered pulsars emitted from regions within and beyond the light cylinder. In particular, the high energy emission resulting from the interactions between pulsar wind and its surrounding medium are considered.

Keywords:

stars:pulsars – stars:neutron – radiation mechanisms:non-thermal

1.

Introduction

Pulsars were first accidentally detected in radio band by Cambridge astronomers (Hewish et al. 1968). Soon after they are also detected in optical, X-rays and gamma-rays. Currently there are nearly two thousand pulsars have been discovered. Their periods (P ) and magnetic field strengths (B) range from 1.55ms (Backer et al. 1982) to 8.5s (Young et al. 1999) and 108 G to 1015 G, respectively. In order to understand the nature of pulsars, multiwavelengths approach becomes a consensus. In particular in the past two decades there has been tremendous progress of X-ray and gamma-ray observations to rotation-powered pulsars by ROSAT, ASCA, RXTE, BeppoSAX, CGRO, Chandra, XMM-Newton, INTEGRAL etc. The data from these satellites provide very important information to constraint the local properties of emission regions including local charged density, electric field, magnetic field, geometry etc. These constrains provide strong restrictions for the acceleration mechanisms and radiation processes involved. In this paper,we will first review some acceleration mechanisms and radiation processes inside and outside the light cylinder. Then we select some X-ray and gamma-ray data, which can be explained in terms of these simple theories. We can see that some X-ray data of rotation powered pulsars from different satellites, which seem to be insistent with each other, but it turns out that these discrepancies can be resolved 173 A. Baykal et al. (eds.), The Electromagnetic Spectrum of Neutron Stars, 173–184. © 2005 Springer. Printed in the Netherlands.

174

Emission Mechanisms and Broad Band Spectra of Pulsars

from the characteristic of individual satellites. Some model predictions will be proposed. In particular, we would like to suggest that high energy radiation from the interactions between pulsar wind and its surrounding medium play a crucial role to identify some unidentified high energy point sources as pulsars.

2.

Theories of Gamma-ray emission from regions within the light cylinder - outer gap model

Pulsars are rapidly rotating, strongly magnetized, neutron stars surrounded by corotating plasma up to the light cylinder (RL , where the corotating speed is c). Detailed discussions of the structure of pulsar magnetospheres can be found, for example, in Michel (1991) and Beskin, Gurevich & Istomin (1992). It is generally believed that pulsed gamma-rays are emitted within the light cylinder. There are two popular classes of charged accelerator models, i.e. polar cap models (e.g. Daugherty & Harding 1996;Harding & Zhang 2001) and outer gap models. Here we will focus on the outer gap models. Cheng, Ho and Ruderman (1986, hereafter CHR) proposed a two dimensional outer gap model to explain the observed data of the Crab and Vela pulsars. Their model assumed that the radiation regions are thin in the longitudinal direction. Their double peak γ-ray structure was from two topologically disconnected outer gaps, each of which is associated with different magnetic poles. However, Romani and co-workers (Chiang & Romani 1994; Romani and Yadigaroglu 1995) have shown that only one outer gap with only outgoing current can already produce a broad, irregularly-shaped emission beam of which is particularly dense near the edge, so that two γ-ray peaks would be observed when the line of sight from the Earth crosses these enhanced γ-ray beam regions; the inner region of the beam provided a significant amount of emission between the peaks. Cheng, Ruderman & Zhang (2000) (hereafter CRZ) have re-considered the three dimensional magnetosphere by introducing various physical processes (including pair production which depends sensitively on the local electric field and the local radius of curvature, surface field structure, reflection of e± pairs because of mirroring and resonant scattering) to determine the three-dimensional geometry of the outer gap. They have shown that two outer gaps and both outgoing and incoming currents are in principle allowed, but it turns out that outgoing currents dominate the emitted radiation intensities. According to CRZ, the azimuthal extension of the outer gap (∆Φ ∼ 150◦ for the Crab pulsar) is finite and is determined by the local pair production condition. For the Crab-like pulsars, the fractional size of the outer gap is given by

f0 ≈ 5.5P

26/21

−4/7 B12

∆Φ 2π

1/7 (1)

Pulsar Wind Nebula

175

˙ where E˙ is the and the total gamma-ray luminosity is given by Lγ = f03 E, spin-down power of pulsar. Inside the light cylinder, high energy photons will be emitted nearly tangent to the magnetic field lines in the corotating frame because of the relativistic 1/γ beaming inherent in high energy processes unless |E × B| ∼ B2 . Then the propagation direction of each emitted photons by relativistic charged particles can be expressed as (ζ,Φ), where ζ is the polar angle from the rotation axis and Φ is the phase of rotation of the star. Effects of the time of flight and aberration are taken into account. A photon with velocity u = (ux , uy , uz ) along a magnetic field line with a relativistic addition of velocity along the azimuthal angle gives an aberrated emission direction u = (ux , uy , uz ). The time of flight gives a change of the phase of the rotation of the star. Combining these two effects, and choosing Φ = 0 for radiation in the (x,z) plane from the center of the star, ζ and Φ are given by cos ζ = uz and Φ = −φu − r · uˆ , where φu is the azimuthal angle of uˆ and r is the emitting location in units of RL . In panel A of Fig. 1, the emission morphology in the (ζ, Φ) plane is shown. For a given observer with a fixed viewing angle ζ, a double-pulsed structure is observed because photons are clustered near two edges of the emission pattern due to the relativistic effects (cf. panel B of Fig. 1).

3.

Gamma-ray emission from regions beyond the light cylinder - pulsar wind nebula

The non-thermal radiation can be contributed by the shock wave produced between the pulsar wind and the outflow matter of the companion star (Stella et al. 1994, Tavani & Arons 1997) or the surrounding nebula (Chevalier 2000). In pulsar binaries, the nebula non-thermal emission may be released from the shock front between the pulsar wind and the stellar wind of the companion and interstellar medium (ISM). In this section we introduce simple models to calculate the high energy spectral properties produced from pulsar wind. The following calculations will be made according to Chevalier (2000). In general, a pulsar is moving in the interstellar medium with a large velocity. When the velocity is larger than the sound speed in the medium, a bow shock will be formed. The typical sound speed of interstellar medium is about cs ∼ (P/ρ)1/2 ∼ (10 − 100)km s−1 . Most pulsars have a typical kick velocity of 300 − 500km s−1 which is larger than the sound speed. The termination radius of the shock waves by the pulsar wind is given as Rs = (E˙ sd /4πρvp2 c)1/2 ∼ 1/2 −1 1016 E˙ 34 n−1/2 vp,300 cm, where vp is the pulsar’s proper velocity, n is the num−3 ber density 1cm and E˙ 34 is the spin-down power in units of 1034 erg/s. Unless the source is located very near the Earth, most of these sources are regarded as point sources for EGRET. Generally, assuming the equipartition between the electron energy and magnetic field, i.e. e ∼ B ∼ 0.5, the

176


Figure 1. Emission projection onto the (ζ,Φ) plane and pulse profile. The emission consists of the emission outwards from both outer gaps and inwards only from pair creation regions of both outer gaps. The outer gaps are limited along the azimuthal direction by pair production. (a) The emission projection and (b) corresponding pulse profile. Crab parameters, α = 65◦ , ζ = 82◦ and δφ = 150◦ are used.

˙ s2 c)1/2 . For the strong magnetic field in the emitting region is B = (6 B E/R relativistic shock, the postshock electron energy distribution is N (γ) ∝ γ −p for γm < γ < γmax , where γm = p−2 p−1 e γw , γw is the Lorentz factor of the relativistic pulsar wind. γmax can be estimated by the equivalence between the electron acceleration time-scale and the synchrotron cooling time-scale. The electron synchrotron cooling time is tsyn = 6πme c/σT γB 2 , the acceleration time is given by tacc = γme c/eB, then γmax = (6πe/σT B)1/2 , where σT is the Thompson cross section. So the maximum photon energy emitted by the 2 relativistic electrons is about Emax ∼ 3eB 2me c γmax ∼ 250 MeV. The number of radiating particles at γ, N (γ), is determined by a balance between the rate at which particles are injected at the shock front N˙ (γ) and synchrotron losses. The synchrotron power of an electron with γ is P (γ) = p−1 4 2 B2 2 −1 ˙ −p ˙ 3 σT cγ 8π and N (γ) = (p−1)γm (γw me c ) Esd γ . The balance between

177

Applications

(γ) injection and synchrotron losses can be expressed as me1c2 ∂N (γ)P = N˙ (γ). ∂γ p−1 ˙ −(p−1) . The luminosity of raThe solution is N (γ) = γm (γw P (γ))−1 Eγ diating particle in the range from γ to γ + dγ is P (γ)N (γ), leading to the luminosity per unit frequency p−2 −(p−2)/2 ˙ (p+2)/4 −p/2 E Rs ν , Lν = k(p)γw 2

(2)

(p−2)/4

p−1 ( 6e )(p−2)/4 ep−1 B . The luminosity at a where k(p) = 12 ( p−2 p−1 ) 4π 2 me c3 particular frequency ν can be estimated as νLν . Since p = 2 ∼ 3, we expect that the photon spectral index of high energy spectrum produced by pulsar wind should be 1.5 ∼ 2.5.

4. 4.1

Applications Lx vs E˙ for non-accreting pulsars

In order to understand how the loss of rotational energy is converted into high energy radiation, many observational and theoretical studies have sought to determine the relationship between the X-ray luminosity, Lx , and the rate of ˙ Indeed, a correlation of the form rotational energy loss or spin down power, E. 1.39 ˙ was found in Einstein data by Seward & Wang (1988). Subsequent Lx ∝ E studies using a larger sample of pulsars led to a relation of the form, Lx ∝ E˙ based on ROSAT data (see Becker & Trümper 1997) and Lx ∝ E˙ 1.5 based on ASCA data (see Saito 1998). Recently, a reanalysis of 39 pulsars based on data obtained from several X-ray satellites by Possenti et al. (2002) led to an intermediate relation Lx ∝ E˙ 1.34 , similar to that of Seward & Wang (1988). The discrepancy of these different correlations seems difficult to understand. However, the deduced existence of a correlation between Lx and E˙ suggests that the observed X-rays are produced by a process which taps the rotational energy of the neutron star, a detailed description of the mechanism remains elusive. This is, in part, a result of the fact that the data from different satellites are obtained in different energy ranges. The results can be affected, for example, by interstellar absorption especially for those pulsars studied in the soft X-ray regime of ROSAT (0.1- 2.4 keV). Different satellites also have different detection characteristic as well. For example ROSAT has better angular resolution than ASCA. In addition, the total X-ray luminosity is composed of contributions from both the pulsed and non-pulsed components, and these components are likely to reflect physical conditions in diverse spatial environments. X-rays radiated by spin powered pulsars could include 5 components with different origins: 1. Non-thermal non-pulsed diffuse radiation from pulsar wind nebulae. This is the major contribution to the non-pulsed X-rays in ASCA observations because of its poor angular resolution.

178


2. Non-thermal non-pulsed radiation from the pulsar magnetosphere: this component could be important as suggested by Becker et al. (2004) when the angular resolution of X-ray detector is sufficiently high to separate this contribution from that of the nebula. 3. Non-thermal pulsed radiation from the pulsar magnetosphere: the pulsed nonthermal emission may be produced in the vicinity of the polar cap as a result of inverse Compton scattering of higher order generation pairs of particles on soft photons emitted by the neutron star (Zhang & Harding 2000) or in the outer magnetosphere as a result of synchrotron radiation of downward cascades from the outer gap electron/positron particles (Cheng & Zhang 1999). Although ASCA does not have enough spatial resolution, its timingresolution can extract this component. 4. Thermal non-pulsed radiation from the pulsar surface. This may result from neutron star cooling. 5. Thermal pulsed radiation from the pulsar surface. the pulsed thermal emission likely originates in a polar cap on the neutron star surface and can contribute to hard X-rays (Cheng & Zhang 1999). Cheng et al. (2004b) have suggested that the X-ray data of ASCA are mainly contributed by the components 1, 3 and 5; but X-rays by ROSAT include the components 3, 4, and 5 as suggested by Becker & Trümper 1997). They separate ASCA (2-10 keV) data into two components, i.e. pulsed and non-pulsed ˙ 1.2±0.08 whereas components, and they show that the former satisfies Lpul x ∝E npul 1.4±0.1 the latter satisfies Lx ∝ E˙ . Fig. 2 and Fig. 3 compare the model predictions with ASCA data.

4.2

Phase resolved spectrum of young pulsars

In this subsection, we describe how to calculate the phase-dependent spectra of pulsars. We will use the parameters of the Crab pulsar as example. Because the Crab pulsar outer gaps are thin, we use the electric field of the 3/2 2 ΩB(r)f 2 (r)RL r , where f (r) ∼ f . The charged CHR model: E|| (r) = 0 RL cs(r) particles inside the gap will be accelerated to extremely relativistic and radiate curvature photons with characteristic energy γe (r)3 c/s(r), where γe (r) = 1/4 2 3 s eE (r)c is the local Lorentz factor of the accelerated electrons / 2 || 2e c positrons in the outer gap and s(r) ∼ (rRL )1/2 . Because of the high soft photon density, the high energy emission from the Crab pulsar is described by synchrotron self-Compton process. In order to calculate the phase-resolved spectrum, the inclination angle and the viewing angle must be given. We have chosen ζ = 82◦ and α = 65◦ . The emission trajectories is shown in Fig. 9 of CRZ and the phase-resolved spectrum can be obtained by identifying the emission regions in the corresponding phase bins. Fig. 10 of CRZ shows the

179

Applications

Figure 2. The pulsed X-ray luminosity (2-10 keV) from ASCA observations versus spin-down power of 23 X-ray pulsars. The solid line is LX = 10−21 E˙ 3/2 , and ˙ the dashed line represents LX = 10−3 E. The relation between the pulsed component and spin-down power cannot be described by both the two formulae. The best fitting function is shown as the dotted line, LX,pul = 10−11 E˙ 1.2 as suggested by Cheng and Zhang (1999).

Figure 3. The non-pulsed X-ray luminosity (2-10 keV) from ASCA observations versus spin-down power of 23 Xray pulsars. The solid line is LX = 10−21 E˙ 3/2 , and the dashed line represents ˙ The best fitting function is LX = 10−3 E. 1.4 ˙ LX ∝ E and is constant with p = 2.8.

phase-resolved spectrum of the Crab pulsar. The phase-resolved spectrum of the Vela pulsar and Geminga can be found in Romani (1996) and Zhang & Cheng (2001) respectively.

4.3

Some faint X-ray sources in galactic center

ecently, deep X-ray surveys of this region were conducted using the Chandra Observatory which revealed a multitude of point X-ray sources ranging in luminosities from ∼ 1032 − 1035 ergs s−1 (Wang, Gotthelf, & Lang 2002) over a field covering a 2 × 0.8 square degree band and from ∼ 3 × 1030 − 2 × 1033 ergs s−1 in a deeper, but smaller field of 17 × 17 (Muno et al. 2003). At present, four bright elongated X-ray sources have been discovered (Wang, Lu, & Lang 2002; Lu, Wang, & Lang 2003; Sakano et al. 2003). These are characterized by non thermal spectra in the energy range 2-10 keV and luminosities ∼ 1033 − 1034 ergs s−1 . A number of lower luminosity elongated sources have also been detected (Park et al. 2004). Among models involving pulsars, young pulsars are not likely to be a major contributor to the X-ray source population since few supernova remnants are presently observed in the Galactic center region. This viewpoint is also supported by pulsar birth rate estimates. Specifically, the birth rate of young pulsars in

180


the Milky Waybirth rate of young pulsars in is about 1/150 yr (Arzoumanian, Chernoff, & Cordes 2002). As the mass in the inner 20 pc of the Galactic center is ∼ 108 , the birth rate of young pulsars in this region is only 10−3 of that in the entire Milky Way, or ∼ 1/150 000 yr. Few young pulsars are likely to remain in the Galactic center region since only a fraction (∼ 40%) of young pulsars in the low velocity component of the pulsar birth velocity distribution would remain within the 20 pc region of the Galactic center studied by Muno et al. (2003) on timescales of > 105 yrs. To produce the observed ˙ of the pulsars should be > 1033 X-ray luminosities, the spin-down power, E, −1 31 2 −4 ˙ ergs s . Given that E ∼ 10 B12 P ergs s−1 where B12 and P are the 12 magnetic field strength in units of 10 Gauss and the spin period respectively, and assuming that E˙ ∼ 1033 ergs s−1 and B12 ∼ 1, the pulsar period would correspond to P ∼ 0.3 s, requiring the age of normal pulsars to be ∼ 106 yr. Hence, the total available number of normal pulsars in this region of the Galactic center is ∼ 3 − 30. On the other hand, there may exist a population of old neutron stars with low space velocities which have not escaped the Galactic center (see Belczynski & Taam 2004b). Such neutron stars could have been members of binary systems and been recycled to millisecond periods, having formed from low mass X-ray binaries in which the neutron stars accreted sufficient matter from either white dwarf, evolved main sequence star or giant donor companions (e.g., Belczynski & Taam 2004a). The current population of these rapidly rotating neutron stars may either be single (having evaporated its companion) or have remained in a binary system. Since the elongated X-ray emission from the pulsar B1957+20 (see Stappers et al 2003) provides evidence that millisecond pulsars emit relativistic winds, the wind nebulae surrounding these pulsars likely contribute more significantly to the faint non thermal X-ray source emission from the Galactic center region than young pulsars. Cheng et al. (2004c) explore the X-ray properties of the millisecond pulsar population as potential candidates for a subset of the point-like and tail-like X-ray sources in the Galactic center region. They argue that based on binary population synthesis investigations, a significant fraction of the neutron stars in this region may have been recycled as millisecond pulsars. A simple model for the wind nebula surrounding such pulsars (cf. section 3) produces sources that can emit at X-ray luminosities in the range of 1031 − 1033 ergs s−1 with a spectrum characterized by a photon index, Γ ∼ 1.5 − 2.5. The emission from pulsars moving at low velocities would not appear spatially extended in Chandra surveys, however those pulsars with high spatial velocities (> 100 km s−1 ) may have a tail-like structure, and could contribute to the filamentary X-ray source population.

Discussion

4.4

181

TeV emission from unidentified EGRET sources

There are about 170 unidentified γ-ray sources in the third EGRET catalog, and nearly one third of these sources lie close to the Galactic plane |b| < 5◦ (Hartman et al. 1999). Most of those unidentified sources in the Galactic plane can be identified as γ-ray pulsars, possibly Geminga-like pulsars which are radio quiet (Cheng & Zhang 1998; Zhang, Zhang & Cheng 2000). For the medium and high latitude sources, it has been suggested that some of them are associated with the supernova remnants in the nearby Gould Belt (Gehrels et al. 2000; Grenier 2000). If these unidentified sources are indeed pulsars, which can produce pulsar wind nebulae through the interactions between relativistic wind particles with the interstellar medium (ISM) . The pulsar wind nebulae will contribute to the production of non-pulsed X-ray emission by synchrotron processes (Chevalier 2000), and TeV photons through inverse Compton scattering (ICS, Aharonian, Atoyan & Kifune 1997). These excessive TeV photons have been detected in some known pulsar wind nebulae, such as Crab and Vela(Kifune et al. 1995; Yoshikoshi et al. 1997). Therefore, if γ-ray pulsars contribute to the unidentified EGRET sources, it is expected that possible TeV signals could be detected in these EGRET sources. Several groups have searched for TeV signals in the error boxes of unidentified EGRET sources, for example, with the HEGRA AIROBICC array (Aharonian et al. 2002), and the Whipple 10m Gamma-Ray Telescope. No TeV source detection has been confirmed at the Whipple, with only an upper limit TeV flux of about 20 EGRET sources determined at ∼ (3 − 6) × 10−11 photon cm−2 s−1 (Fegan & Weekes 2004). However, the next generation of Cherenkov arrays can have a much better sensitivity down to ∼ 10−13 photon cm−2 s−1 (Weekes 2004). We can estimate the TeV luminosity produced by inverse Compton scatterIC , where Lsyn can be estimated by equation 2 in section 3. If we ing by LLsyn take some typical parameters of pulsars and nebulae to find the TeV luminosity: νLν∼1018 Hz ∼ 1034 erg s−1 , e ∼ 0.5, B ∼ 0.01, γw ∼ 106 , Rs ∼ 1016 cm, the electron energy spectral index p = 2.2, then we obtain the TeV luminosity by ICS processes LIC (TeV) ∼ 1031 erg s−1 . If we follow the Monte Carlo simulation results of Cheng et al. (2004a) for the Unidentified Sources and assuming that they are pulsars, we can obtain the correlation between GeV flux and TeV flux shown in Fig. 4 as well as the distribution of TeV fluxes. In Fig. 5, we can see that if all Unidentified EGRET Sources are indeed pulsars there will be ∼20 sources in the galactic plane and ∼30 sources in high galactic latitude detected by the next generation Cherenkov telescopes respectively.

5.

Discussion

Finally, we have the following remarks. (1)The phase resolved high radiation spectrum must be explained in terms of three dimensional pulsar mag-

182


Figure 4. The GeV gamma-ray flux versus the TeV flux for the simulated γray pulsars for the high latitude |b| > 5◦ (solid) and the Galactic latitude |b| ≤ 5◦ (circle) sample.

Figure 5. The distribution of the TeV flux from the wind nebulae of the simulated γ-ray pulsars which could be the unidentified EGRET sources. The distributions of the pulsars in the high latitude (|b| > 5◦ ) (solid), and in the Galactic disk (|b| ≤ 5◦ ) (dashed) are shown.

netospheric model. Although some phase resolved data between two pulses for young pulsars can be successfully explained, the origin of the off-pulsed component still required further study(Weisskopf, 2004). (2)It is likely that ASCA data consists of X-rays from inside magnetosphere as well as contribution from the pulsar wind nebula. However, the original of non-thermal non-pulsed component is not known. In explaining the relation between Lx vs E˙ for ASCA data, we have assumed that the contribution of this component is small. Recently Becker et al. (2004) showed that this component is extremely important for old pulsars. (3)We predict that if the Unidentified gamma-ray sources are associated with pulsars they should be detectable in TeV range for future generation ground based Cherenkov detectors. However, the Monte Carlo predictions can only provide the correlation between the GeV and TeV fluxes. It is not possible to give definite predictions for individual sources. (4)We suggest that there is a possible population of millisecond pulsars clustered in the galactic center. They can exhibit themselves as faint X-ray point sources detected by Chandra and also contribute to diffuse background gamma-rays in this region. Perhaps part of e± lines from the galactic center may be also contributed from this population. This provides a great challenge for radio astronomers to identify them.

Discussion

183

Acknowledgments We thank M. Ruderman and L. Zhang for useful discussion on the outer gap models, and R. Taam and W. Wang on pulsar wind models. This work is supported by a RGC grant of Hong Kong Government.

References Aharonian, F. A., Atoyan, A. M., & Kifune, T. 1997, A&A, 291, 162 Aharonian, F. A., et al. 2002, A&A, 390, 39 Arzoumanian, Z., Chernoff, D. F., & Cordes,J. M. 2002, ApJ, 568, 289 Backer, D.C., Kulkaini, S.R., Heiles, C. et al. 1982, Nature, 300, 615 Becker, W. and Trümper, J. 1997, A&A, 326, 682 Becker, W. et al. 2004, astro-ph/0405180 Beskin, V.S., Gurevich, A.S. and Istomin, Ya.N. 1992, “Physics of the Pulsar", Cambridge University Press, Cambridge Belczynski, K. and Taam, Ronald E. 2004a, ApJ,603, 690. Belczynski, K. and Taam, Ronald E. 2004b, ApJ, in press Cheng, K.S., Ho, C. and Ruderman, M.A. 1986, ApJ, 300, 500 (CHR) Cheng, K.S., Ruderman, M. and Zhang, L. 2000, ApJ, 537, 964 (CRZ) Cheng, K.S. and Zhang, L. 1998, ApJ, 498, 327 Cheng, K.S. and Zhang, L. 1999, ApJ, 515, 337 Cheng, K.S., Zhang, L., Leung, P. and Jiang, Z. 2004a, ApJ, 608, 418 Cheng, K.S., Taam, R.E. and Wang, W. 2004b, ApJ, in press Cheng, K.S., Taam, R.E. and Wang, W. 2004c, ApJ, submitted Chevalier, R.A. 2000, ApJ, 539, L45 Chiang, J. and Romani, R.W. 1994, ApJ, 436, 754 Daugherty, J.K. and Harding, A.K. 1996, ApJ, 458, 278 Fegan, S.J. & Weekes, T.C. 2004, in Cosmic Gamma-Ray Sources, ed. K. S. Cheng & G. E. Romero, (Kluwer Academic Publishers), in press Gehrels, N., Macomb, D.J., Bertsch, D.L. et al. 2000, Nature, 404, 363 Grenier, I.A. 2000, A&A, 364, L93 Harding, A.K. and Zhang, B. 2001, ApJ, 548, L37 Hartman, R.C., Bertsch, D.L., Bloom, S.D. et al. 1999, ApJS, 123, 79 Hewish, A., Bell, S.J., Pilkington, J.D.H., Scott, P.F. and Collins, R.A. 1968, Nature, 217, 709 Kifune, T., et al. 1995, ApJ, 438, L91 Lu, F. Wang, Q.D. & Lang, C. 2003, AJ, 126, 319 Michel, F.C. 1991, “Theory of Neutron Star Magnetospheres", University of Chicago Press, Chicago Muno, M.P., Baganoff, F.K., Bautz, M.W. et al. 2003, ApJ, 589, 225 Park, S., Muno, M. P., Baganoff, F. K., Maeda, Y., Morris, M., Howard, C., Bautz, M. W., & Garmire, G. P. 2004, ApJ, 603, 548 Possenti, A. et al. 2002, A&A, 387, 993 Romani, R.W. 1996, ApJ, 470, 469 Romani, R.W. and Yadigaroglu, I.A. 1995, ApJ, 438, 314 Saito, Y. 1998, Ph.D. Thesis, Univ. of Tokyo Sakano, M. Warwick, R. S., Decourchelle, A., & Predehl, P. 2003, MNRAS, 340, 747 Seward, F. D., & Wang, Z. 1988, ApJ, 332, 199

184


Stappers, B. W., Gaensler, B. M., Kaspi, V. M., van der Klis, M., & Lewin, W. H. G. 2003, Science, 299, 1372 Stella, L., Campana, S., Colpi, M. et al. 1994, ApJ, 423, L47 Tavani, M. and Arons, J. 1997, ApJ, 477, 439 Wang, Q.D., Gotthelf, E.V. and Lang, C.C. 2002, Nature, 415, 148 Wang, Q.D., Lu, F. & Lang, C. 2002, ApJ, 581, 1148 submitted Weekes, T. C. 2004, in Cosmic Gamma-Ray Sources, ed. K. S. Cheng & G. E. Romero, (Kluwer Academic Publishers), 345 Weisskopf, M. C. et al. 2004, ApJ, 601,1050 Yoshikoshi, T., et al. 1997, ApJ, 487, L65 Young, M.D., Manchester, R.N. and Johnston, S. 1999, Nature, 400, 848 Zhang, L. and Cheng, K.S. 2001, MNRAS, 320, 477 Zhang, B. and Harding, A.K. 2000, ApJ, 532, 1150 Zhang, L., Zhang, Y.J. and Cheng, K.S. 2000, A&A, 357, 957

MODELING SPECTRAL FEATURES FROM ISOLATED NEUTRON STARS Wynn C.G. Ho,1 Dong Lai,2 Alexander Y. Potekhin,3 and Gilles Chabrier4 1 Hubble Fellow; Kavli Institute for Particle Astrophysics and Cosmology, Stanford University,

PO Box 20450, Mail Stop 29, Stanford, CA 94309, USA 2 Center for Radiophysics and Space Research, Department of Astronomy, Cornell University,

Ithaca, NY 14853, USA 3 Ioffe Physico-Technical Institute, Politekhnicheskaya 26, 194021 St. Petersburg, Russia 4 Ecole Normale Supérieure de Lyon, Centre de Recherche Astrophysique de Lyon (UMR CNRS No. 5574), 69364 Lyon Cedex 07, France

Abstract

We study several effects that influence the strength of the proton cyclotron and atomic features in the thermal spectra of magnetic neutron stars. Magnetic field variations over the neutron star surface leads to broadening of features. Vacuum 14 polarization can strongly suppress spectral lines when B > ∼ 10 G. The surface 13 spectrum is unaffected by vacuum polarization when B < ∼ 7 × 10 G; thus the proton cyclotron absorption line (and atomic lines) can have a large equivalent width, possibly explaining the features seen in some isolated neutron stars.

Keywords:

stars:atmospheres – stars:magnetic fields – stars:neutron – X-rays:stars

1.

Observations

Considerable observational resources have been devoted to the study of thermal emission from isolated neutron stars (NSs) and, in particular, to the search for spectral features in the radiation. Thermal radiation from the surface of isolated NSs can provide invaluable information on the physical properties and evolution of NSs (Zavlin 2005). For many NSs, the spectra are found to be featureless and often well fit by a blackbody (see Pavlov et al. 2002 for a review). However, absorption features have been found in the thermal emission of several isolated NSs. For example, the spectrum of the young NS 1E 1207.4 − 5209 shows features at 0.7 and 1.4 keV (Sanwal et al. 2002; Mori et al. 2004) and possibly at 2.1 and 2.8 keV (De Luca et al. 2004). Several of the dim, radio-quiet isolated NSs (Haberl 2004; Trümper 2005) have been observed to possess broad absorption features (Table 1). It is particularly striking 185 A. Baykal et al. (eds.), The Electromagnetic Spectrum of Neutron Stars, 185–188. © 2005 Springer. Printed in the Netherlands.

186

Modeling Spectral Features from Isolated Neutron Stars

that, although four of these have similar effective temperatures, the equivalent widths (EW) of their lines are very different.

Table 1. Spectral Features in Dim Isolated Neutron Stars DINS RX J1308.6 + 2127 RX J1605.3 + 3249 RX J0720.4 − 3125 RX J0806.4 − 4123 RX J0420.0 − 5022

Period (s)

kT (eV)

Eline (keV)

EW (keV)

Ref

10.3 ? 8.4 11.4 3.45 or 22.7

86 95 85a 96 45

0.2−0.3 0.45 0.27 0.41−0.46 0.33

0.15 0.08 0.04 0.03−0.06 0.045

1 2 3 4 4

References—(1) Haberl et al. 2003; (2) van Kerkwijk et al. 2004; (3) Haberl et al. 2004a; (4) Haberl et al. 2004b a see de Vries et al. 2004 and Vink et al. 2004 for long-term spectral changes

2.

Partially Ionized Atmospheres and Vacuum Polarization Effect

Because the strong magnetic field significantly increases the binding energies of atoms, molecules, and other bound states (see Lai 2001 for a review), these bound states may have abundances appreciable enough to contribute to the opacity in the atmosphere. Recently, thermodynamically consistent equation of state (EOS) and opacities for a magnetized, partially ionized H plasma have been obtained by Potekhin & Chabrier (2003, 2004). These EOS and opacities have been implemented by Ho et al. (2003) and Potekhin et al. (2004, where we have included the effect of bound species on the polarization vectors of the photon modes) for modeling NS atmospheres. For models with “ordi13 nary” magnetic field strengths (1012 G< ∼B< ∼ 7 × 10 G), the spectral lines associated with bound species lie in the extreme UV to very soft X-ray energy bands and are difficult to observe. However, the opacities are sufficiently different from the fully ionized opacities that they can change the atmosphere structure and continuum flux, which can affect fitting of the observed spectra (e.g., Ho et al. 2004). In a magnetized NS atmosphere, both the plasma and vacuum polarizations contribute to the dielectric property of the medium. A “vacuum resonance” arises when these two “compensate” each other (Ventura 2005). Away from the vacuum resonance, the photon modes (for E EBe = 1.16 B14 MeV, the electron cyclotron energy) are almost linearly polarized. Near the vacuum resonance, the normal modes become circularly polarized. When a photon propagates outward in the NS atmosphere, its polarization state will evolve adiabatically if the density variation is sufficiently gentle; thus, a photon of

Partially Ionized Atmospheres and Vacuum Polarization Effect

O−mode X−mode

187

O−mode X−mode

Figure 1. Diagram illustrating how vacuum polarization-induced mode conversion affects the Vacuum O−mode emergent radiation from a Photosphere Resonance magnetized NS atmosphere. The photosphere is defined by where the optical depth O−mode Vacuum (measured from the surface) Photosphere Resonance is 2/3 and is where the photon decouples from the matter. Left: In the “normal” field regime, the X−mode X−mode resonance lies Photosphere vacuum Photosphere outside the photospheres of the two modes. Right: In the “superstrong” field regime, the vacuum resoX−mode O−mode X−mode O−mode nance lies between the two 13 13 X10 G X10 G > photospheres. B< 7 B 7 ~ ~

one mode will be converted into the other mode as it traverses the vacuum resonance (Fig. 1; Lai & Ho 2002, 2003). 14 At B > ∼ 10 G, vacuum polarization can significantly affect the radiation spectrum from magnetized NS atmospheres: it softens the high-energy tail and suppresses the proton cyclotron feature and features due to bound species (Lai & Ho 2002; Ho & Lai 2003; Ho et al. 2003). The latter could provide an explanation for the non-detection thus far of lines in the observed thermal spectra of several magnetars (Israel 2004), which are thought to possess B > ∼ 1014 G. We note here that there have been spectral features seen in the nonthermal emission from magnetars (Rea et al. 2003; Ibrahim 2005). 14 At B < ∼ 10 G, vacuum polarization has little effect on the atmosphere emission spectra. Therefore, strong proton cyclotron or other atomic features may be present in the thermal spectrum. Our calculations of NS synthetic spectra, taking into account the line broadening effect due to magnetic field variation over the NS surface, show that the observed broad absorption features in the dim isolated NSs (Table 1) could be explained naturally as the proton cyclotron line, with possible blending from atomic lines of neutral hydrogen (Ho & Lai 2004). The variation in the strength of the observed spectral features in these sources is then due to different fractions of the surface with B < ∼ 1014 G.

188

Modeling Spectral Features from Isolated Neutron Stars

Acknowledgments W.H. is supported by NASA through Hubble Fellowship grant HF-01161.01A awarded by STScI, which is operated by AURA, Inc., for NASA, under contract NAS 5-26555. D.L. is supported in part by NSF grant AST 0307252 and NASA grant NAG 5-12034. The work of A.P. is supported in part by RFBR grants 02-02-17668 and 03-07-90200, and RLSS grant 1115.2003.2.

References De Luca, A., Mereghetti, S., Caraveo, P.A., Moroni, M., Mignani, R.P., & Bignami, G.F. 2004, A&A, 418, 625 de Vries, C.P., Vink, J., Méndez, M., & Verbunt, F. 2004, A&A, 415, L31 Haberl, F. 2004, Adv. Sp. Res., 33, 638 Haberl, F., Schwope, A.D., Hambaryan, V., Hasinger, G., & Motch, C. 2003, A&A, 403, L19 Haberl, F., Zavlin, V.E., Trümper, J., & Burwitz, V. 2004a, A&A, 419, 1077 Haberl, F., et al. 2004b, A&A, 424, 635 Ho, W.C.G. & Lai, D. 2003, MNRAS, 338, 233 Ho, W.C.G. & Lai, D. 2004, ApJ, 607, 420 Ho, W.C.G., Lai, D., Potekhin, A.Y., & Chabrier, G. 2003, ApJ, 599, 1293 Ho, W.C.G., et al. 2004, in preparation Ibrahim, A.I. 2005, this volume Israel, G.L. 2005, this volume Kulkarni, S.R., Kaplan, D.L., Marshall, H.L., Frail, D.A., Murakami, T., & Yonetoku, D. 2003, ApJ, 585, 948 Lai, D. 2001, Rev. Mod. Phys., 73, 629 Lai, D. & Ho, W.C.G. 2002, ApJ, 566, 373 Lai, D. & Ho, W.C.G. 2003, ApJ, 588, 962 Mori, K., Chonko, J.C., & Hailey, C.J. 2004, ApJ, submitted (astro-ph/0407369) Pavlov, G.G., Zavlin, V.E., & Sanwal, D. 2002, in Proc. 270 WE-Heraeus Seminar on Neutron Stars, Pulsars, and Supernova Remnants, eds. Becker, W., Lesch, H., & Trümper, J., (MPE Rep. 278; Garching: MPI), p.273 Potekhin, A.Y., & Chabrier, G. 2003, ApJ, 585, 955 Potekhin, A.Y., & Chabrier, G. 2004, ApJ, 600, 317 Potekhin, A.Y., Lai, D., Chabrier, G., & Ho, W.C.G. 2004, ApJ, 612, 1034 Rea, N., Israel, G.L., Stella, L., Oosterbroek, T., Mereghetti, S., Angelini, L., Campana, S., & Covino, S. 2003, ApJL, 586, L65 Sanwal, D., Pavlov, G.G., Zavlin, V.E., & Teter, M.A. 2002, ApJL, 574, L61 Trümper, J. 2005, this volume van Kerkwijk, M.H., Kaplan, D.L., Durant, M., Kulkarni, S.R., Paerels, F. 2004, ApJ, 608, 432 Ventura, J. 2005, this volume Vink, J., de Vries, C.P., Méndez, M., & Verbunt, F. 2004, ApJ, 609, L75 Zavlin, V.E. 2005, this volume

VI

BINARIES, ACCRETION

OBSERVATIONS OF RADIO PULSARS Nichi D’Amico Universita degli Studi di Cagliari, Dipartimento di Fisica, SP Monserrato-Sestu km 0,700, 09042 Monserrato (Ca), Italy

Abstract

Pulsars, rapidly rotating highly magnetized neutron stars, have many exciting applications in physics and astronomy. After nearly 40 years since the original discovery, pulsar research has great vitality, making major contributions to fields ranging from ultra-dense matter physics to relativistic gravity, cosmology and stellar evolution. In the last few years, with a series of successful pulsar search experiments carried using the Parkes 64m radio telescope, we have produced an unprecedented boom of radio pulsar discoveries, including the first ever known double-pulsar. We review the main results achieved recently, with particular emphasis to the discovery of the first double pulsar.

Keywords:

stars:pulsars – stars:neutron – stars:magnetic fields – dense matter – surveys

1.

Introduction

Pulsars are relatively weak radio sources. Successful pulsar surveys therefore require a large radio telescope, low-noise receivers, a relatively wide bandwidth and long observation times. Pulsars have steep spectra, typically S(ν) ∝ ν −1.7 , and low-frequency cut-off occur usually below 200-300 MHz. So, in principle, we would better observe them at relatively low frequency around 400 MHz. However, pulsar signals suffer dispersion due to the presence of charged particles in the interstellar medium. To retain sensitivity, especially for shortperiod, high-dispersion pulsars, the observing bandwidth must be sub-divided into many channels. In most pulsar searches, this has been achieved using a filterbank system. The sensitivity of pulsar searches is also limited by the Galactic radio continuum background and by interstellar scattering, especially for low radio frequencies and at low Galactic latitudes. Interstellar scattering results in a onesided broadening of the observed pulse profile with a frequency dependence ∼ ν −4.4 [1] which cannot be removed by using narrow bandwidths. While relatively young pulsars tend to be located at low Galactic latitudes, old pulsars and in particular recycled binary pulsars and millisecond pulsars, can be found at high Galactic latitudes where dispersion is rather low, and 147 A. Baykal et al. (eds.), The Electromagnetic Spectrum of Neutron Stars, 147–160. © 2005 Springer. Printed in the Netherlands.

148

Observations of Radio Pulsars

multipath scattering is negligible. It has been proved that large scale surveys carried out at low frequency are prone to discover many millisecond pulsars. While the larger telescope beams available at low frequency make such surveys relatively fast, they are not very efficient to probe the inner regions of the Galactic disk, because the high background temperature, strong dispersion and scattering reduce significantly the sensitivity. In general, there is no ideal radio frequency to be adopted for pulsar observations, and a given frequency choice, coupled with the other system parameters, simply results in a better sampling of a given volume of the pulsar parameters space. It results that observations at low frequency are ideal to observe the local sample of millisecond pulsar, at high Galactic latitude, while relatively high frequency, around 1400 MHz, are more efficient to probe the inner part of the Galaxy, at low Galactic latitude. Most pulsar searches along the Galactic plane therefore have been carried at higher radio frequencies, often around 1400 MHz [2]; [3]. One of the main limitation of searches at relatively high frequencies is the narrow size of radio telescope beams, which make large scale surveys relatively slow. This limitation has been solved only recently with the availability of multibeam receivers.

2.

Sensitivity considerations

The minimum detectable flux density for a pulsar of period P can be described by the following equation [4]: Tsys + Tsky We mJy (1) Smin = nσ G Np ∆t∆νM Hz P − We where nσ is the minimum signal-to-noise ratio S/N considered, Tsys and Tsky the system noise temperature and the sky temperature in K respectively, G the gain of the radio telescope (in K/Jy), ∆t the integration time in seconds, Np the number of polarizations and ∆νM Hz the bandwidth in MHz. is a factor ∼ 1.5 accounting for sensitivity reduction due to digitization and other losses. Finally, We is the effective width of the pulse: We = W 2 + δt2 + δt2DM + δt2scatt (2) its value depends on the intrinsic pulse width W , on the time resolution δt of the detection apparatus and on the broadening of the pulse introduced both by the dispersion of the signal in each frequency channel (δtDM ) and by the scattering induced by inhomogeneities in the ISM (δtscatt ). The time taken by a signal with a frequency ν to reach the observer is given by: d

d 1 νp 2 d 1 e2 1 dl 1 1+ dl = + ∼ DM (3) t= 2 ν c 2 πme c ν 2 0 vg 0 c

149

Sensitivity considerations

d where DM = 0 ne dl is the dispersion measure and d is the distance to the observer. Since the radio pulsar signal is broad band, the dependence of t on the frequency produces a broadening ∆tDM of the pulse profile: ∆tDM

e2 = 2πme c

1 1 − 2 2 ν1 ν2

DM ∼ 8.3·103

∆νM Hz DM 3 νM Hz

s

(4)

where νM Hz and ∆νM Hz are the central observing frequency and the total bandwidth in MHz respectively and DM is given in pc cm−3 .

Figure 1. Effect of the dispersion on the pulse of PSR J1644-4559. In this observation a total bandwidth ∆ν = 288 MHz, is split into 96 × 3 MHz channels, and the central frequency is ν = 1372.5 MHz. The bottom integrated profile is obtained by dedispersing the signal.

One method for minimizing the effects of the dispersion is to split the total bandwidth into several channels having a frequency width δν. In this way, in each frequency channel, the pulse is poorly affected by the dispersion in the ISM and the signal in the different channels appears as in figure 1, with the pulses arriving in each channel at a different time. Knowing the dispersion measure allows to correct for this time shift integrating the signal along the

150


line connecting the edges of the pulses in each channel (dedispersion of the signal) and obtaining a sharp and high signal-to-noise profile. Better results can be obtained applying the so-called coherent de-dispersion method, in which the incoming signals are de-dispersed over the whole bandwidth using a filter which has the inverse transfer function to that of the interstellar medium [5]. As a result, the pulse profile is perfectly aligned in frequency, without any residual dispersive smearing caused by finite channel bandwidths. The inhomogeneities in the ISM act like center of scattering onto the signal emitted from a pulsar. Because of this interaction with clumps of matter, the radio waves follow different paths and arrive at the observer at different times thus producing a broadening δtscatt of the pulse. According to the simple "thin screen" model δtscatt scales as: δtscatt ∝

DM 2 ν4

(5)

Unlike the case of the dispersion, no technical skills can be adopted to mitigate the effects of interstellar scattering. The only way to reduce its effect is to observe at higher frequencies.

3.

Recent results achieved at Parkes

In this section we review the main results of some pulsar survey programs carried out at Parkes in the last few years adopting a new sensitive multibeam receiver.

3.1

The survey of the Galactic plane

This survey covered a strip with |b| < 5o along the Galactic plane between Galactic longitudes of 260o and 50o . The data were sampled every 250 µs, and the observation time per pointing was 35 min, giving a very high sensitivity, about seven times better than previous similar surveys [2]; [3], at least for pulsars not in short-period binary systems. The survey has been outstandingly successful, with over 700 pulsars discovered so far. Fig. 2 shows the locations of the new pulsars in the P-P˙ diagram, showing that a substantial number of relatively young pulsars are now available in the sample. Several reports on the multibeam survey and its results are available [6]; [11]; [10]; [7]; [12]; [9]; [8]. Also, papers on the discovery of several pulsars of particular interest have been published. In term of pulsar counting, this is the most successful pulsar survey carried out so far. Fig. 3 shows the distribution of the new pulsar in the Galaxy, and clearly indicates that much deeper regions of the Galaxy were probed by this survey


151

Figure 2. P -P˙ diagram containing the pulsars discovered by the PM survey (black dots),for which timing parameters have been already published, overlaid on the previous known population.

3.2

The search of the Globular Cluster system

About 60 clusters among the ones visible at Parkes were observed in this search, according to their optical central density and satisfying the requirement DMexp ≤ 300 cm−3 pc (where DMexp is the DM expected for the cluster according to a model for the Galactic distribution of the ionized gas [13]). So far, 12 new pulsars have been discovered in 6 globular clusters, none of which had previously known associated pulsars. Seven of these pulsars are members of binary systems, and 6 of them have relatively high DM values. One pulsar follows a highly eccentric orbit and another one is eclipsed for a large fraction of the orbital period. This is a rather peculiar system and it is associated with NGC 6397. This cluster was one of the most promising targets for searching MSPs: in fact, it is one of the closest clusters, at a distance

152


Figure 3. Schematic diagram of the Galactic Coordinate system viewed from above, showing the distribution of the new discovered pulsars overlaid on the previous known sample

of 2.6 kpc ± 6% [14], and probably has a collapsed core with hints of mass segregation [15]. It lies in fourth place in the list of GCs ranked according to central luminosity [16]. Moreover, it contains ∼ 20 X-ray sources detected with Chandra within 2 arcmin of the cluster center, 8 of which probably identified with CVs [17]. The aforementioned peculiar object found in this cluster is, a binary MSP with a spin period of 3.65 ms [18]. It displays eclipses at 1.4 GHz for more than 40% of the orbit. This phenomenon is not uncommon (more than 10 eclipsing systems containing a MSP are known [19]; [21]; [22]; [24]; [23]; [20]) and in the case of PSR B1744−24A the eclipses show duration and irregularities similar to those of PSR J1740−5340. However this new system is 2 − 3 times wider (with an orbital separation of ∼ 6.5 R ) than any other known


153

eclipsing pulsar binary, and has a heavier minimum mass for the companion (> 0.19 M ) than any known eclipsing system. In addition, the radio signal exhibits striking irregularities (delays and intensity variations) over a wide range of orbital phases, indicating that the MSP is orbiting within a large envelope of matter released from the companion with a high mass loss rate. These characteristics challenge the evaporation models from a degenerate companion, suggesting more likely that the companion is a bloated main-sequence star or the remnant of the star that spun up the MSP[20].

3.3

The Parkes high latitude survey

The analysis of the data of another survey, the Parkes High-Latitude Pulsar Survey (PH survey), collected in three years of observations starting from November 2000, resulted in the discovery of 18 new pulsars, four of which belonging to the class of millisecond, or recycled, pulsars. Of these, three are in a binary system, and one is an isolated mildly recycled object. The PH survey more than doubled the number of MSPs in the selected region, fulfilling one of the main aim of this project, that is increasing the statistics on these objects. The spatial distribution of millisecond pulsars, in previous large area surveys, appeared isotropic; in the present work, as well as in the intermediate and high latitude Swinburne surveys [25]; [26], which have the same observational parameters as the PH survey, the millisecond pulsar 0.4 GHz. Pulsar

Emitted Pulse

Telescope

Detected Pulse

Figure 4. Left: Pulse scattering by irregularities in the ISM. Right: A simulation showing the fraction of pulsars undetectable due to scattering as a function of observing frequency.

Propagation effects in the interstellar medium (ISM). Dispersion and scatterbroadening of the pulses in the ISM hamper detection of short period and/or distant objects. The effects of scattering are shown in Figure 4. Fortunately, like Tsky , the scatter-broadening time τscatt has a strong frequency dependence, scaling roughly as ν −4 . Figure 4 shows that for survey frequencies below 1 GHz, scattering ‘hides’ a large fraction of the population. Additionally, scintillation, the diffractive and refractive modulation of apparent flux densities by turbulences in the ISM (Rickett 1970) affects pulsar detection. For example, two northern sky surveys carried out 20 years apart with comparable sensitivity (Damashek et al. 1978; Sayer et al. 1997) detected a number of pulsars above and below the nominal search thresholds of one experiment but not the other. Surveying the sky multiple times minimizes the effects of scintillation and enhances the detection of faint pulsars through favorable scintillation. Finite size of the emission beam. The fact that pulsars do not beam to 4π sr means that we see only a fraction f of the total active population. For a circular beam, Gunn & Ostriker (1970) estimated f ∼ 1/6. A consensus on the precise shape of the emission beam has yet to be reached. Narayan & Vivekanand (1983) argued that the beams are elongated in the meridional direction. Lyne & Manchester (1988), on the other hand, favour a circular beam. Using the same database, Biggs (1990) presented evidence in favor of meridional compression! All these studies do agree that the beam size is period

Techniques to correct for observational selection

167

dependent, with shorter period pulsars having larger beaming fractions. Tauris & Manchester (1998) found that f = 0.09 [log(P/s) − 1]2 + 0.03, where P is the period. A complete model for f needs to account for other factors, such as evolution of the inclination angle between the spin and magnetic axes. Pulse nulling. The abrupt cessation of the pulsed emission for many pulse periods, was first identified by Backer (1970). Ritchings (1976) presented evidence that the incidence of nulling became more frequent in older long-period pulsars, suggesting that it signified the onset of the final stages of the neutron star’s life as an active radio pulsar. Since most pulsar surveys have short (< few min) integration times, there is an obvious selection effect against nulling objects. Means of overcoming this effect are to look for individual pulses in search data (Nice 1999), survey the sky many times, or use longer integrations.

4.

Techniques to correct for observational selection

From an observationally-biased sample, we seek to characterize the underlying population accounting for the aforementioned selection effects. For a given survey of integration time, τ , and bandwidth, ∆ν, the quantity Tsys W/P (1) Smin ∼ G ∆ντ is the limiting sensitivity to pulsars of a certain period, P , and pulse width, W , given an antenna with gain, G, and system temperature, Tsys . For further details, see the review by D’Amico in this volume. For a given Smin , then, there is a maximum detection volume Vmax = (L/Smin )3/2 to pulsars of luminosity, L. This idea is used to correct the sample in two ways described below.

4.1

Population inversion techniques

The first method, originally developed by Large (1971), is of particular interest to determine the spatial distribution of the parent population. Given the observed distribution N (P, z, R, L) in terms of period, P , distance from the Galactic plane, z, Galactocentric radius, R, and luminosity, L, we may write dN (P, z, R, L) = V (P, z, R, L) ρ(P, z, R, L) dP dz dR dL,

(2)

where V is the volume of the Galaxy effectively searched and ρ is the underlying (true) distribution of the population. Since we know N and can estimate V on the basis of pulsar survey sensitivities, we can invert equation (2) to solve for ρ. The only simplification required to do this is to assume that P , z, R and L are independent quantities. Fortunately, apart from a very weak coupling between P and z, there are no significant relationships between any of these parameters. The problem then reduces to four equations which can be solved for the underlying distributions: ρP (P ), ρz (z), ρR (R) and ρL (L).

168

Radio Pulsar Statistics

Galactocentric radius (kpc)

Galactocentric radius (kpc)

Figure 5. Left: the observed radial distribution and corrected Galactic radial density function ρ(R) as derived by Lyne, Manchester & Taylor (1985). Right: corrected radial density functions proposed by Narayan (1987), Lorimer (2004) and Yusifov & Kucuk (2004).

Of particular interest is ρR , the underlying radial pulsar density. For many years, the standard reference for ρR was Lyne, Manchester & Taylor (1985) As can be seen from the results of this work shown in Figure 5, the form of ρR at small R is poorly constrained. These results were approximated in most subsequent work using a Gaussian distribution for ρR (e.g. Narayan 1987). As the left panel of Figure 5 clearly shows, there is no reason to prefer a Gaussian over a function which tends to zero at small R. Using the results of the Parkes multibeam survey, which has discovered many more pulsars in the inner Galaxy, I revisited this method recently (Lorimer 2004) and found strong evidence in favor of a non-Gaussian radial distribution. This result substantiates earlier work by Johnston (1994) and more recently Yusifov & Kucuk (2004). Whether the deficit in the inner Galaxy is a real effect is not yet clear.

4.2

Scale factors and pulsar current

If the form of the Galactic distribution is known, a related approach, pioneered by Phinney & Blandford (1981) and Vivekanand & Narayan (1981), can be used to estimate the pulsar birth rate. The method involves binning the population in period and computing the flow or ‘current’ of pulsars J(P ) =

nbin ˙ Pi ξi 1 . ∆P fi

(3)

i=1

Here, nbin is the number of pulsars in a period bin of width ∆P , ξi is the ‘scale factor’ and fi is the beaming fraction of the ith pulsar. As discussed earlier, fi is based on some beaming model. For a given pulsar, its scale factor ξi represents the number of pulsars with similar parameters in the Galaxy and is computed using a Monte Carlo simulation of N pulsars with identical periods

Some of the many outstanding problems

169

and luminosities. Using accurate models for the various pulsar surveys, it is relatively straightforward to calculate the number of pulsars n that are detectable from that population. As a result, ξi = N/n. The beauty of the pulsar-current analysis is that it makes just two fundamental assumptions about pulsars: (i) they are a steady-state population; (ii) they are spinning down steadily from short to long periods. The first of these is justified since the ages of pulsars, while not well known (107−8 yr), are certainly less than the age of the Galaxy, 1010 yr. The second is, of course, well in accord with timing observations. The birth-rate can be computed from this analysis by simply plotting J as a function of P . In the standard model where pulsars are born spinning rapidly, there should be a peak in the current at short periods followed by a decline in the current as pulsars end their life with longer periods. The birth rate is then just the height of this peak. Somewhat controversially, the first such analysis by Vivekanand & Narayan (1981) found a step function at P = 0.5 s in their distribution of J. This was claimed as evidence for an ‘injection’ of pulsars into the population with P ∼ 0.5 s The most recent analysis of this kind was carried out by Vranesevic et al. (2005) using a sample of 815 normal pulsars from the Parkes multibeam survey. The total birthrate of the population was found to lie between 1–2 pulsars per century for 1400-MHz luminosities above 1 mJy kpc2 . Dividing the population into groups according to magnetic field strength, Vranesevic et al. found that over half of the total birthrate is contributed by pulsars with fields > 2.5 × 1012 G. This is in spite of the fact that such pulsars make up less than 30% of the observed sample and, based on their scale factors, only about 5–10% of the total population. While no evidence was found for a significant population of pulsars injected into the population with intermediate spin periods, the observed distribution of pulsar current is consistent with up to 40% of all pulsars being born with periods in the range 0.1–0.5 s.

5.

Some of the many outstanding problems

In view of the difficulties in correcting for these selection effects, and the inherent problem of small-number statistics, many controversies have pervaded pulsar statistics over the years. I review here only two topics: the period evolution of normal pulsars and the apparent paucity of isolated recycled pulsars.

5.1

Period evolution and field decay of isolated pulsars

The classic model for spin-down of an isolated pulsar is to write the braking torque as a generalized power law. For an angular velocity Ω = 2π/P , the ˙ = KΩn , where K is proportional to the equation of motion is given by Ω braking torque and n is the so-called braking index. For a constant value of K

170


and pure magnetic dipole braking n = 3, the equation of motion on the P –P˙ diagram is such that pulsars follow a slope of –1 in a log-log plot like Figure 2. The dipolar braking hypothesis can be tested for a handful of young pulsars, where timing measurements provide n. So far, all 6 measured values of n are consistent with a flat distribution in the range 1.4–2.9. In other words, all of the pulsars with measured values of n are moving along lines with slopes greater than –1 on the P –P˙ diagram. When these vectors are plotted (see, for example Lyne 2004) one sees that the directions these young pulsars are moving would place them above the pulsar island! So the conundrum is, either the pulsars in the island have a different set of progenitors than the young objects, or there is some evolution in the braking index as a function of time. The evolution in braking index can either be provided by integrating the equation of motion assuming that n is genuinely a function of time, or that K decays with time. In all simulations of the P –P˙ plane that I am aware of to date, the shape of the diagram is reproduced by modeling the evolution of K with time. Excellent fits to the observed diagrams (see, for example, Figure 8 in Gonthier et al. 2004) can be obtained by decay laws of the form K(t) ∝ exp(−t/tD ) for decay times tD of a few million years. This is usually interpreted as exponential decay of the magnetic moment of the neutron star on a timescale of a few million years. While earlier versions of these simulations were criticized by van Leeuwen (2004) as not taking into account period dependent beaming, the work of Gonthier et al. (2004) does, I believe, account for this effect and still prefers a short magnetic field decay time. Despite the good agreement on the P –P˙ plane, there are a number of vexing issues: (a) spontaneous decay of the magnetic field on such short timescales is inconsistent with the observations of millisecond pulsars which have Gyr ages and yet field strengths at the level of 108 G; (b) the exponential model is inconsistent with all braking index measurements, since it always predicts an effective n ≥ 3; (c) in principle, the same behavior could be reproduced by modeling the evolution of n rather than field decay; (d) what is the ultimate fate of low-braking-index pulsars? For example, the Vela pulsar has n = 1.4 (Lyne et al. 1996) and is moving towards the magnetars on the P − P˙ diagram, rather than the pulsar island. Lyne (2004) proposed that such objects might be the progenitors of the magnetars. This idea requires further investigation.

5.2

Where are all the isolated ‘recycled’ pulsars?

The discovery of new pulsars often sheds light on previously unseen areas of the neutron star ‘zoo’ which likely represent quite rare evolutionary processes. One example is the discovery of two isolated pulsars J2235+1506 (Camilo et al. 1993) and J0609+2130 (Lorimer et al. 2004) with spin properties similar to the double neutron star binaries. Camilo et al. suggested that J2235+1506

Current and future pulsar search projects

171

might be the remains of a high-mass binary system that disrupted during the second supernova explosion. Is this hypothesis consistent with the observations? One way to test this is to consider the fraction, η, of binary systems that remain bound after the second supernova explosion. Numerous authors have followed the orbital evolution of a wide variety of binary systems containing neutron stars using detailed Monte Carlo simulations. For example, Portegies Zwart & Yungelson (1998) find η ∼ 4%. We therefore expect for each double neutron star system we observe to find of order 20 systems which disrupted. Currently we know of 8 double neutron star binaries. Why, then, do we not see of order 160 pulsars like J0609+2130 or J2235+1506? This currently outstanding problem may indicate a different evolutionary scenario for these objects and warrants further study.

6.

Current and future pulsar search projects

Pulsar astronomy is currently enjoying the most productive phase of its history, with applications providing a wealth of new information about compactobject astrophysics, general relativity, the Galactic magnetic field, the interstellar medium, binary evolution, planetary physics and even cosmology. Our understanding of the Galactic pulsar population has improved dramatically thanks largely to the success of the Parkes multibeam survey. Like our colleagues in other parts of the electromagnetic spectrum, radio astronomers are highly active in a number of areas which will bring new advances in sensitivity. The first of these is the Arecibo L-band feed array project (ALFA), a sevenbeam system which is currently in active use in large-scale surveys for pulsars and neutral hydrogen. The excellent sensitivity of Arecibo means that ALFA will be able to probe much deeper into the Galaxy than was possible in the Parkes surveys. A simulation of a survey with ALFA that is currently in progress along a narrow strip of the Galactic plane (|b| < 1◦ ) shows that it should discover over 200 pulsars. Over the next 5–10 years, as surveys with ALFA extend to higher latitudes, we can reasonably expect to detect of order 1000 pulsars. This sample the population will provide a further quantum leap to pulsar statistical analysis and sample to the edge of the Galaxy. The ALFA system, however, is only a precursor for what might be possible with the Square Kilometer Array (SKA), an ambitious world-wide collaboration currently planned for the year 2020 (see skatelescope.org). Simulations for pulsar surveys with this instrument demonstrate that the increase in sensitivity of the SKA (around two orders of magnitude over current radio telescopes!) would mean that essentially every Galactic pulsar beaming towards us (of order 30,000 objects!) could be detectable. Perhaps by the year 2030, the sample of radio pulsars will be finally free of selection effects.

172


Acknowledgments I wish to thank the Royal Society and NATO for supporting my attendance at this meeting, and to the organizers for putting together an excellent program.

References Arzoumanian, Z., Cordes, J. M. & Wasserman, I., 1999, ApJ, 520, 696 Backer D. C., 1970, Nature, 228, 42 Bhattacharya, D. & van den Heuvel, E. P. J., 1991, Phys. Rep., 203, 1 Biggs J. D., 1990, MNRAS, 245, 514 Bisnovatyi-Kogan, G. S. & Komberg, B. V., 1974, Sov. Astron., 18, 217 Cordes, J. M. & Lazio, T. J. W., 2002, astro-ph/0207156 Damashek M., Taylor J. H., & Hulse R. A., 1978, MNRAS, 225, L31 Ghosh, P. & Lamb, F. K., 1979, ApJ, 234, 296 Gunn J. E. & Ostriker J. P., 1970, ApJ, 160, 979 Gonthier, P. L., Van Guilder, R. & Harding, A. K., 2004, ApJ, 604, 775 Hewish A. et al., 1968, Nature, 217, 709 Johnston S., 1994, MNRAS, 268, 595 Lamb, F. K. & Yu, W., in Binary & Millisecond pulsars, eds: Rasio & Stairs, PASP, in press Large M. I., 1971. IAU Symposium No. 46, Dordrecht, Reidel, p. 165 Lorimer, D. R., 2004, in IAU Symposium no. 218, Eds: Camilo & Gaensler, PASP, 105 Lorimer, D. R., et al. 2004, MNRAS, 347, L21 Lyne, A. G. et al., 1996, Nature, 381, 497 Lyne, A. G., 2004, in IAU Symposium no. 218, Eds: Camilo & Gaensler, PASP, 257 Lyne A. G. & Manchester R. N., 1988, MNRAS, 234, 477 Lyne A. G., Manchester R. N., & Taylor J. H., 1985, MNRAS, 213, 613 Lyne A. G. et al., 1998, MNRAS, 295, 743 Manchester, R. N. et al. 2001, MNRAS, 328, 17 Narayan R. & Vivekanand M., 1983, A&A, 122, 45 Narayan R., 1987, ApJ, 319, 162 Nice D. J., 1999, ApJ, 513, 927 Phinney E. S. & Blandford R. D., 1981, MNRAS, 194, 137 Portegies Zwart S. F., Yungelson L. R., 1998, A&A, 332, 173 Radhakrishnan, V. & Shukre, C. S., 1985, In: Supernovae, Their Progenitors and Remnants, p. 155, eds Srinivasan, G. & Radhakrishnan, V., Indian Academy of Sciences, Bangalore Rickett B. J., 1970, MNRAS, 150, 67 Ritchings R. T., 1976, MNRAS, 176, 249 Sayer R. W., Nice D. J., & Taylor J. H., 1997, ApJ, 474, 426 Tauris T. M. & Bailes M., 1996, A&A, 315, 432 Tauris T. M. & Manchester R. N., 1998, MNRAS, 298, 625 van Kerkwijk, M. H., 1996, In: IAU Colloquium 160, p. 489, eds Johnston, Walker & Bailes van Leeuwen, J., 2004, PhD Thesis, Utrecht University. Vivekanand M. & Narayan R., 1981, JA&A, 2, 315 Weisberg, J. M., 1996. Vranesevic, N. et al. 2004, ApJ, in press Weisberg, J. M., 1996, In: IAU Colloquium 160, p. 447, eds Johnston, Walker & Bailes Yusifov, I. & Kucuk, I., 2004, A&A, 422, 545

ACCRETION DISKS H.C. Spruit Max-Planck-Institut für Astrophysik Postfach 1523, D-85740 Garching, Germany

Abstract

In this lecture the basic theory of accretion disks is reviewed, with emphasis on aspects relevant for X-ray binaries and Cataclysmic Variables. This is followed in the second part of the lecture with an introduction to the theory of radiatively inefficient accretion, including its possible association with outflows and jets, and mechanisms for the transition from a cool optically thick disk to ion supported accretion flow near black holes and neutron stars.

Keywords:

stars:neutron – black hole physics – stars:white dwarfs – stars:variables – – accretion – accretion disks

1.

Introduction

Accretion disks are inferred to exist in objects of very different scales: km to millions of km in low Mass X-ray Binaries (LMXB) and Cataclysmic Variables (CV), solar radius-to-AU scale in protostellar disks, and AU-to-parsec scales for the disks in Active Galactic Nuclei (AGN). An interesting observational connection exists between accretion disks and jets (such as the spectacular jets from AGN and protostars), and outflows (the ‘CO-outflows’ from protostars and possibly the ‘broad-line-regions’ in AGN). Lacking direct (i.e. spatially resolved) observations of disks, theory has tried to provide models, with varying degrees of success. Uncertainty still exists with respect to some basic questions. In this situation, progress made by observations or modeling of a particular class of objects is likely to have direct impact for the understanding of other objects, including the enigmatic connection with jets. In this lecture I concentrate on the more basic aspects of accretion disks, but an attempt is made to mention topics of current interest, such as magnetic viscosity, as well. Emphasis is on those aspects of accretion disk theory that connect to the observations of LMXB and CV’s. For other reviews on the basics of accretion disks, see Pringle (1981), Treves et al. (1988). For a more in-depth treatment, see the textbook by Frank et al. (2002). More recent developments are reviewed in sections xxx. 203 A. Baykal et al. (eds.), The Electromagnetic Spectrum of Neutron Stars, 203–247. © 2005 Springer. Printed in the Netherlands.

204

2.

Accretion Disks

Accretion: General Gas falling into a point mass potential Φ=−

GM r

from a distance r0 to a distance r converts gravitational into kinetic energy, by an amount ∆Φ = GM (1/r − 1/r0 ). For simplicity, assuming that the starting distance is large, ∆Φ = GM/r. If the gas is then brought to rest, for example at the surface of a star, the amount of energy e dissipated per unit mass is e=

GM r

(rest)

or, if it goes into a circular Kepler orbit at distance r: e=

1 GM 2 r

(orbit).

The dissipated energy goes into internal energy of the gas, and into radiation which escapes to infinity (usually in the form of photons, but neutrino losses can also play a role in some cases).

2.1

Adiabatic Accretion

Consider first the case when radiation losses are neglected. This is adiabatic accretion. For an ideal gas with constant ratio of specific heats γ, the internal energy per unit mass is P . e= (γ − 1)ρ With the equation of state P = RρT /µ

(1)

where R is the gas constant, µ the mean atomic weight per particle, we find the temperature of the gas after the dissipation has taken place (assuming that the gas goes into a circular orbit): 1 T = (γ − 1)Tvir , 2

(2)

where Tvir , the virial temperature is given by Tvir =

GM µ . Rr

In an atmosphere with temperature near Tvir , the sound speed ss = (γRT /µ)1/2 is close to the escape speed from the system, the hydrostatic pressure scale

205

Accretion: General

height is of the order of r. Such an atmosphere may evaporate on a relatively short time scale in the form of a stellar wind. A simple example is spherical adiabatic accretion (Bondi, 1952). An important result is that such accretion is possible only if γ ≤ 5/3. The larger γ, the larger the temperature in the accreted gas (eq. 2), and beyond a critical value the temperature is too high for the gas to stay bound in the potential. A classical situation where adiabatic and roughly spherical accretion takes place is a supernova implosion: when the central temperature becomes high enough for the radiation field to start desintegrating nuclei, γ drops and the envelope collapses onto the forming neutron star via a nearly static accretion shock. Another case are Thorne-Zytkow objects (e.g. Cannon et al. 1992), where γ can drop to low values due to pair creation, initiating an adiabatic accretion onto the black hole. Adiabatic spherical accretion is fast, taking place on the dynamical or free fall time scale τd = r/vK = (r3 /GM )1/2 ,

(3)

where vK is the Kepler orbital velocity. When radiative loss becomes important, the accreting gas can stay cool irrespective of the value of γ, and Bondi’s critical value γ = 5/3 plays no role. With such losses, the temperatures of accretion disks are usually much lower than the virial temperature. The optical depth of the accreting flow increases with the accretion rate M˙ . When the optical depth becomes large enough so that the photons are ‘trapped’ in the flow, the accretion just carries them in, together with the gas (Rees 1978, Begelman 1979). Above a certain critical rate M˙ c , accretion is therefore adiabatic.

2.2

The Eddington Limit

Objects of high luminosity have a tendency to blow their atmospheres away due to the radiative force exerted when the outward traveling photons are scattered or absorbed. Consider a volume of gas on which a flux of photons is incident from one side. Per gram of matter, the gas presents a scattering (or absorbing) surface area of κ cm2 to the escaping radiation. The force exerted by the radiative flux F on one gram is F κ/c. The force of gravity pulling back on this one gram of mass is GM/r 2 . The critical flux at which the two forces balance is c GM (4) FE = κ r2 Assuming that the flux is spherically symmetric, this can be converted into a critical luminosity LE = 4πGM c/κ,

(5)

206

Accretion Disks

the Eddington luminosity (e.g. Rybicki and Lightman, 1979). If the gas is fully ionized, its opacity is dominated by electron scattering, and for solar composition κ is then of the order 0.3 cm2 /g (about a factor 2 lower for fully ionized helium, a factor up to 103 higher for partially ionized gases of solar composition). With these assumptions, LE ≈ 1.7 1038

M M erg/s ≈ 4 104 L M M

If this luminosity results from accretion, it corresponds to the Eddington accretion rate M˙ E : GM ˙ (6) ME = LE → M˙ E = 4πrc/κ. r Whereas LE is a true limit that can not be exceeded by a static radiating object except by geometrical factors of order unity (see chapter 10 in Frank et al, 1992), no maximum exists on the accretion rate. For M˙ > M˙ E the plasma is just swallowed whole, including the radiation energy in it (cf. discussion in the preceding section). With κ = 0.3: M˙ E ≈ 1.3 1018 r6 g/s ≈ 2 10−8 r6 M yr−1 , where r6 is the radius of the accreting object in units of 10 km.

3.

Accretion with Angular Momentum

When the accreting gas has a zonzero angular momentum with respect to the accreting object, it can not accrete directly. A new time scale appears, the time scale for outward transport of angular momentum. Since this is in general much longer than the dynamical time scale, much of what was said about spherical accretion needs modification for accretion with angular momentum. Consider the accretion in a close binary consisting of a compact (white dwarf, neutron star or black hole) primary of mass M1 and a main sequence companion of mass M2 . The mass ratio is defined as q = M2 /M1 (note: in the literature q is just as often defined the other way around). If M1 and M2 orbit each other in a circular orbit and their separation is a, the orbital frequency Ω is Ω2 = G(M1 + M2 )/a3 . The accretion process is most easily described in a coordinate frame that corotates with this orbit, and with its origin in the center of mass. Matter that is stationary in this frame experiences an effective potential, the Roche potential (Ch. 4 in Frank, King and Raine, 1992), given by φR (r) = −

GM GM 1 − − Ω2 r2 r1 r2 2

(7)

Accretion with Angular Momentum

207

where r1,2 are the distances of point r to stars 1,2. Matter that does not corotate experiences a very different force (due to the Coriolis force). The Roche potential is therefore useful only in a rather limited sense. For non-corotating gas intuition based on the Roche geometry is usually confusing. Keeping in mind this limitation, consider the equipotential surfaces of (7). The surfaces of stars M1,2 , assumed to corotate with the orbit, are equipotential surfaces of (7). Near the centers of mass (at low values of φR ) they are unaffected by the other star, at higher Φ they are distorted and at a critical value Φ1 the two parts of the surface touch. This is the critical Roche surface S1 whose two parts are called the Roche lobes. Binaries lose angular momentum through gravitational radiation and a magnetic wind from the secondary (if it has a convective envelope). Through this loss the separation between the components decreases and both Roche lobes decrease in size. Mass transfer starts when M2 fills its Roche lobe, and continues as long as the angular momentum loss from the system lasts. A stream of gas then flows through the point of contact of the two parts of S1 , the inner Lagrange point L1 . If the force acting on it were derivable entirely from (7) the gas would just fall in radially onto M1 . As soon as it moves however, it does not corotate any more and its orbit under the influence of the Coriolis force is different (Fig. 1).

Figure 1. Roche geometry for q = 0.2, with free particle orbit from L1 (as seen in a frame corotating with the orbit). Dashed: circularization radius.

Since the gas at L1 is usually very cold compared with the virial temperature, its sound speed is small compared with the velocity it gets after only a small distance from L1 . The flow into the Roche lobe of M1 is therefore highly supersonic. Such hypersonic flow is essentially ballistic, that is, the stream flows approximately along the path taken by free particles.

208

Accretion Disks

Though the gas stream on the whole follows an orbit close to that of a free particle, a strong shock develops at the point where the orbit intersects itself. [In practice shocks already develop shortly after passing the pericenter at M1 , when the gas is decelerated again. Supersonic flows that are decelerated, by whatever means, in general develop shocks (e.g. Courant and Friedrichs 1948, Massey, 1968). The effect can be seen in action in the movie published in Ró˙zyczka and Spruit, 1993]. After this, the gas settles into a ring, into which the stream continues to feed mass. If the mass ratio q is not too small this ring forms fairly close to M1 . An approximate value for its radius is found by noting that near M1 the tidal force due to the secondary is small, so that the angular momentum of the gas with respect to M1 is approximately conserved. If the gas continues to conserve angular momentum while dissipating energy, it settles into the minimum energy orbit with the specific angular momentum j of the incoming stream. The radius of this orbit, the circularization radius rc is determined from (GM1 rc )1/2 = j. The value of j is found by a simple integration of the orbit starting at L1 and measuring j at some point near pericenter. In units of the orbital separation a, rc and the distance rL1 from M1 to L1 are functions of the mass ratio only. As an example for q = 0.2, rL1 ≈ 0.66a and the circularization radius rc ≈ 0.16a. In practice the ring forms somewhat outside rc , because there is some angular momentum redistribution in the shocks that form at the impact of the stream on the ring. The evolution of the ring depends critically on nature and strength of the angular momentum transport processes. If sufficient ‘viscosity’ is present it spreads inward and outward to form a disk. At the point of impact of the stream on the disk the energy dissipated is a significant fraction of the orbital kinetic energy, hence the gas heats up to a significant fraction of the virial temperature. For a typical system with M1 = 1M , M2 = 0.2M having an orbital period of 2 hrs, the observed size of the disk (e.g. Wood et al. 1989b, Rutten et al. 1992) rd /a ≈ 0.3, the orbital velocity at rd about 900 km/s, the virial temperature at rd is 108 K. The actual temperatures at the impact point are much lower, due to rapid cooling of the shocked gas. Nevertheless the impact gives rise to a prominent ‘hot spot’ in many systems, and an overall heating of the outermost part of the disk.

209

Thin Disks: Properties

4.


4.1

Flow in a Cool Disk is Supersonic

Ignoring viscosity, the equation of motion in the potential of a point mass is ∂v 1 GM + v · ∇v = − ∇P − 2 ˆ r, ∂t ρ r

(8)

where ˆ r is a unit vector in the spherical radial direction r. To compare the order of magnitude of the terms, choose a position r0 in the disk, and choose 3 1/2 as typical time and velocity scales the orbital time scale Ω−1 0 = (r0 /GM ) and velocity Ω0 r0 . The pressure gradient term is R 1 ∇P = T ∇ ln P. ρ µ In terms of the dimensionless quantities r˜ = r/r0 , t˜ = Ω0 t,

v˜ = v/(Ω0 r0 ), ˜ = r0 ∇, ∇

the equation of motion is then ∂˜ v ˜ ln P − 1 ˆ ˜v = − T ∇ +v ˜ · ∇˜ r. Tvir r˜2 ∂ t˜

(9)

All terms and quantities in this equation are of order unity by the assumptions made, except the pressure gradient term which has the coefficient T /Tvir . If cooling is important, so that T /Tvir 1, the pressure term is negligible to first approximation, and vice versa. Equivalent statements are also that the gas moves hypersonically on nearly Keplerian orbits, and that the disk is thin, as is shown next.

4.2

Disk Thickness

The thickness of the disk is found by considering its equilibrium in the direction perpendicular to the disk plane. In an axisymmetric disk, using cylindrical coordinates (, φ, z), measure the forces at a point r0 (, φ, 0) in the midplane, in a frame rotating with the Kepler rate Ω0 at that point. The gravr balances the centrifugal acceleration Ω20 at itational acceleration −GM/r 2 ˆ this point, but not at some distance z above it because gravity and centrifugal acceleration work in different directions. Expanding both accelerations near r0 , one finds a residual acceleration toward the midplane of magnitude gz = −Ω20 z.

210

Accretion Disks

Assuming again an isothermal gas, the condition for equilibrium in the z direction under this acceleration yields a hydrostatic density distribution z2 . ρ = ρ0 () exp − 2H 2 H(), the scale height of the disk, is given in terms of the isothermal sound speed ci = (RT /µ)1/2 by H = ci /Ω0 . We define δ ≡ H/r, the aspect ratio of the disk, and find that it can be expressed in several equivalent ways: T 1/2 ci H −1 = =M = , (10) δ= r Ωr Tvir where M is the Mach number of the orbital motion.

4.3

Viscous Spreading

The shear flow between neighboring Kepler orbits in the disk causes friction due to viscosity. The frictional torque is equivalent to exchange of angular momentum between these orbits. But since the orbits are close to Keplerian, a change in angular momentum of a ring of gas also means it must change its disctance from the central mass. If the angular momentum is increased, the ring moves to a larger radius. In a thin disk angular momentum transport (more precisely a nonzero divergence of the angular momentum flux) therefore automatically implies redistribution of mass in the disk. A simple example (Lüst 1952, see also Lynden-Bell and Pringle 1974) is a narrow ring of gas at some distance r0 . If at t = 0 this ring is released to evolve under the viscous torques, one finds that it first spreads into an asymmetric hump with a long tail to large distances. As t → ∞ the hump flattens in such a way that almost all the mass of the ring is accreted onto the center, while a vanishingly small fraction of the gas carries almost all the angular momentum to infinity. As a result of this asymmetric behavior essentially all the mass of a disk can accrete, even if there is no external torque to remove the angular momentum.

4.4

Observations of Disk Viscosity

Evidence for the strength of the angular momentum transport processes in disks comes from observations of variability time scales. This evidence is not good enough to determine whether the processes really have the same effect as a viscosity, but if this is assumed, estimates can be made of the magnitude of the viscosity.

211


Cataclysmic Variables give the most detailed information. These are binaries with white dwarf (WD) primaries and (usually) main sequence companions (for reviews see Meyer-Hofmeister and Ritter 1993, Cordova 1995, Warner 1995). A subclass of these systems, the Dwarf Novae, show semiregular outbursts. In the currently most developed theory, these outbursts are due to an instability in the disk (Smak 1971, Meyer and Meyer-Hofmeister 1981, for recent references see, King 1995, Hameury et al. 1998). The outbursts are episodes of enhanced mass transfer of the disk onto the primary, involving a significant part of the whole disk. The decay time of the burst is thus a measure of the viscous time scale of the disk (the quantitative details depend on the model, see Cannizzo et al. 1988, Hameury et al. 1998): tvisc = rd2 /ν, where rd is the size of the disk. With decay times on the order of days, this yields viscosities of the order 1015 cm2 /s, about 14 orders of magnitude above the microscopic viscosity of the gas. Other evidence comes from the inferred time scale on which disks around protostars disappear, which is of the order of 107 years (Strom et al, 1993).

4.5

α-parametrization

The process responsible for such a large viscosity has not been identified with certainty yet. Many processes have been proposed, some of which demonstrably work, though often not with an efficiency as high as the observations of CV outbursts seem to indicate. Other ideas, such as certain turbulence models, do not have much predictive power and are based on ad-hoc assumptions about hydrodynamic instabilities in disks. In order to compare the viscosities in disks under different conditions, one introduces a dimensionless vsicosity α: ν=α

c2i , Ω

(11)

where ci is the isothermal sound speed as before. The quantity α was introduced by Shakura and Sunyaev (1973), as a way of parametrizing our ignorance of the angular momentum transport process (their definition is based on a different formula however, and differs by a constant of order unity). How large can the value of α be, on theoretical grounds? As a simple model, let’s assume that the shear flow between Kepler orbits is unstable to the same kind of shear instabilities found for flows in tubes, channels, near walls and in jets. These instabilities occur so ubiquitously that the fluid mechanics community considers them a natural and automatic consequence of a high Reynolds number: LV Re = ν

212

Accretion Disks

where L and V are characteristic length and velocity scales of the flow. If this number exceeds about 1000 (for some forms of instability much less), instability and turbulence are generally observed. It has been argued (e.g. Zel’dovich 1981) that for this reason hydrodynamic turbulence is the cause of disk viscosity. Let’s look at the consequences of this assumption. If an eddy of radial length scale l develops due to shear instability, it will rotate at a rate given by the rate of shear, σ, in the flow, here σ=r

3 ∂Ω ≈ − Ω. ∂r 2

The velocity amplitude of the eddy is V = σl, and a field of such eddies produces a turbulent viscosity of the order (leaving out numerical factors of order unity) νturb = l2 Ω.

(12)

In compressible flows, there is a maximum to the size of the eddy set by causality considerations. The force that allows an instability to form an overturning eddy is the pressure, which transports information about the flow at the sound speed. The eddies formed by a shear instability can therefore not rotate faster than ci , hence their size does not exceed ci /σ ≈ H (eq. 10. At the same time, the largest eddies formed also have the largest contribution to the turbulent viscosity. Thus we should expect that the turbulent viscosity is given by eddies with size of the order H: ν ∼ H 2 Ω, or α ∼ 1. Does hydrodynamical turbulence along these lines exist in disks? Unfortunately, this question is still open, but current opinion is leaning toward the view that the angular momentum transport in sufficiently ionized disks is due a a small scale magnetic field (Shakura and Sunyaev 1973). This is discussed in section 8.

5.

Thin Disks: Equations

Consider a thin (= cool, nearly Keplerian, cf. section 4.2) disk, axisymmetric but not stationary. Using cylindrical coordinates (r, φ, z), (note that we have changed notation from to r compared with section 4.2) we define the surface density Σ of the disk as ∞ ρdz ≈ 2H0 ρ0 , (13) Σ= −∞

213


where ρ0 , H0 are the density and scaleheight at the midplane. The approximate sign is used to indicate that the coefficient in front of H in the last expression actually depends on details of the vertical structure of the disk. Conservation of mass, in terms of Σ is given by ∂ ∂ (rΣ) + (rΣvr ) = 0. ∂t ∂r

(14)

(derived by integrating the continuity equation over z). Since the disk is axisymmetric and nearly Keplerian, the radial equation of motion reduces to vφ2 = GM/r.

(15)

∂vφ ∂vφ vr vφ + vr + = Fφ , ∂t ∂r r

(16)

The φ-equation of motion is

where Fφ is the azimuthal component of the viscous force. By integrating this over height z and using (14), one gets an equation for the angular momentum balance: ∂ ∂ ∂Ω ∂ (rΣΩr2 ) + (rΣvr Ωr2 ) = (Sr3 ), ∂t ∂r ∂r ∂r

(17)

where Ω = vφ /r, and

∞

S= −∞

ρνdz ≈ Σν.

(18)

The second approximate equality in (18) holds if ν can be considered independent of z. The right hand side of (17) is the divergence of the viscous angular momentum flux, and is derived most easily with a physical argument, as described in, e.g. Pringle (1981) or Frank et al. (1992)1 . Assume now that ν can be taken constant with height. For an isothermal disk (T independent of z), this is equivalent to taking the viscosity parameter 1 If

you prefer a more formal derivation, the fastest way is to consult Landau and Lifshitz (1959) chapter 15 (hereafter LL). Noting that the divergence of the flow vanishes for a thin axisymmetric disk, the viscous stress σ becomes (LL eq. 15.3) ∂vi ∂vk , σik = η + ∂xk ∂xi where η = ρν. This can be written in cylindrical or spherical coordinates using LL eqs. (15.15-15.18). The viscous force is ∂σik 1 ∂η Fi = = σik + η∇2 vi , ∂xk η ∂xk Writing the Laplacian in cylindrical coordinates, the viscous torque is then computed from the φ-component of the viscous force by multiplying by r, and is then integrated over z.

214

Accretion Disks

α independent of z. As long as we are not sure what causes the viscosity this is a reasonable simplification. Note, however, that recent numerical simulations of magnetic turbulence suggest that the effective α, and the rate of viscous dissipation per unit mass, are higher near the disk surface than near the midplane. See the discussion in section 22.8. While eq (17) is still valid for rotation rates Ω deviating from Keplerian (only the integration over disk thickness must be justifiable), we now use the fact that Ω ∼ r−3/2 . Then Eqs. (14-17) can be combined into a single equation for Σ: r

∂ ∂ ∂Σ = 3 [r1/2 (νΣr1/2 )]. ∂t ∂r ∂r

(19)

Under the same assumptions, eq. (16) yields the mass flux M˙ at any point in the disk: ∂ M˙ = −2πrΣvr = 6πr1/2 (νΣr1/2 ). ∂r

(20)

Eq. (19) is the standard form of the thin disk diffusion equation. An important conclusion from this equation is: in the thin disk limit, all the physics which determines the time dependent behavior of the disk enters through one quantitity only, the viscosity ν. This is the main attraction of the thin disk approximation.

5.1

Steady Thin Disks

In a steady disk (∂/∂t = 0) the mass flux M˙ is constant through the disk and equal to the accretion rate onto the central object. From (20) we get the surface density distribution:

r 1/2 1 ˙ i M 1−β , (21) νΣ = 3π r where ri is the inner radius of the disk and β is a parameter appearing through the integration constant. It is related to the flux of angular momentum FJ through the disk: FJ = −M˙ βΩi ri2 ,

(22)

where Ωi is the Kepler rotation rate at the inner edge of the disk. If the disk accretes onto an object with a rotation rate Ω∗ less than Ωi , one finds (Shakura and Sunyaev, 1973, Lynden-Bell and Pringle, 1974) that β = 1, independent of Ω∗ . Thus the angular momentum flux (torque on the accreting star) is inward (spin-up) and equal to the accretion rate times the specific angular momentum at the inner edge of the disk. For stars rotating near their maximum rate (Ω∗ ≈ Ωi ) and for accretion onto magnetospheres, which can rotate faster

215


than the disk, the situation is different (Sunyaev and Shakura 1977, Popham and Narayan 1991, Paczyński 1991, Bisnovatyi-Kogan 1993, Rappaport et al. 2004). Accreting magnetospheres, for example, can spin down by interaction with the disk. This case has a surface density distribution (21) with β < 1 (see also Spruit and Taam 1993, Rappaport et al. 2004). The angular momentum flux is then outward, and the accreting star spins down. This is possible even when the interaction between the disk and the magetosphere takes place only at the inner edge of the disk. Magnetic torques due interaction with the magetosphere may exist at larger distances in the disk as well, but are not necessary for creating an outward angular momentum flux. Numerical simulations of disk-magnetosphere interaction (Miller and Stone 1997) give an interesting new view of how such interaction may take place, and suggests it happens very differently from what is assumed in the previous ‘standard’ models.

5.2

Disk Temperature

n this section I assume accretion onto not-too-rapidly rotating objects, so that β = 1 (eq. 21). The surface temperature of the disk, which determines how much energy it loses by radiation, is governed primarily by the energy dissipation rate in the disk, which in turn is given by the accretion rate. From the first law of thermodynamics we have ρT

dS = −divF + Qv , dt

(23)

where S the entropy per unit mass, F the heat flux (including radiation and any form of ‘turbulent’ heat transport), and Qv the viscous dissipation rate. For changes which happen on time scales longer than the dynamical time Ω−1 , the left hand side is small compared with the terms on the right hand side. Integrating over z, the divergence turns into a surface term and we get ∞ 4 Qv dz, (24) 2σr Ts = −∞

where Ts is the surface temperature of the disk, σr is Stefan-Boltzmann’s radiation constant σr = ar c/4, and the factor 2 comes about because the disk has 2 radiating surfaces (assumed to radiate like black bodies). Thus the energy balance is local (for such slow changes): what is generated by viscous dissipation inside the disk at any radius r is also radiated away from the surface at that position. The viscous dissipation rate is equal to Qv = σij ∂vi /∂xj , where σij is the viscous stress tensor (see footnote in section 5), and this works out2 to 2 using,

e.g. LL eq. 16.3

216

Accretion Disks

be Qv = 9/4 Ω2 νρ.

(25)

Eq. (24), using (21) then gives the surface temperature in terms of the accretion rate:

r 1/2 9 2 GM 3M˙ i 4 1− . (26) σr Ts = Ω νΣ = 3 8 r 8π r This shows that the surface temperature of the disk, at a given distance r from a steady accreter, depends only on the product M M˙ , and not on the highly uncertain value of the viscosity. For r ri we have Ts ∼ r−3/4 .

(27)

These considerations only tells us about the surface temperature. The internal temperature in the disk is quite different, and depends on the mechanism transporting energy to the surface. Because it is the internal temperature that determines the disk thickness H (and probably also the viscosity), this transport needs to be considered in some detail for realistic disk models. This involves a calculation of the vertical structure of the disk. Because of the local (in r) nature of the balance between dissipation and energy loss, such calculations can be done as a grid of models in r, without having to deal with exchange of energy between neighboring models. Schemes borrowed from stellar structure computations are used (e.g. Meyer and Meyer-Hofmeister 1982, Pringle et al. 1986, Cannizzo et al. 1988). An approximation to the temperature in the disk can be found when a number of additional assumptions is made. As in stellar interiors, the energy transport is radiative rather than convective at high temperatures. Assuming local thermodynamic equilibrium (LTE, e.g. Rybicki and Lightman 1979), the temperature structure of a radiative atmosphere is given, in the Eddington approximation by: 3 d σr T 4 = F. dτ 4

(28)

The boundary condition that there is no incident flux from outside the atmosphere yields the approximate condition σr T 4 (τ = 2/3) = F,

(29)

∞ where τ = z κρdz is the optical depth at geometrical depth z, and F the energy flux through the atmosphere. Assuming that most of heat is generated

217


near the midplane (which is the case if ν is constant with height), F is approximately constant with height and equal to σr Ts4 , given by (26). Eq (22.2.2) then yields 2 3 (30) σr T 4 = (τ + )F. 4 3 Approximating the opacity κ as constant with z, the optical depth at the midplane is τ = κΣ/2. If τ 1, the temperature at the midplane is then: 27 T 4 = σr−1 Ω2 νΣ2 κ. (31) 64 With the equation of state (1), valid when radiation pressure is small, we find for the disk thickness, using (21): 1/10 H = (R/µ)2/5 64π32 σr (κ/α)1/10 (GM )−7/20 r1/20 (f M˙ )1/5 r 1/20 = 5 10−3 α−1/10 r6 (M/M )−7/20 (f M˙ 16 )1/5 , (Pr P ) (32) where r6 = r/(106 cm), M˙ 16 = M˙ /(1016 g/s), and f = 1 − (ri /r)1/2 . From this we conclude that: i) the disk is thin in X-ray binaries, H/r < 0.01, ii) the disk thickness is relatively insensitive to the parameters, especially α, κ and r. It must be stressed, however, that this depends fairly strongly on the assumption that the energy is dissipated in the disk interior. If the dissipation takes place close to the surface [such as in some magnetic reconnection models (Haardt et al. 1994, Di Matteo et al. 1999a and references therein)], the internal disk temperature will be much closer to the surface temperature. The midplane temperature and H are even smaller in such disks than calculated from (32). The viscous dissipation rate per unit area of the disk, Wv = (9/4)Ω2 νΣ [cf. eq. 26)] can be compared with the local rate WG at which gravitational energy is liberated in the accretion flow. Since half the gravitational energy stays in the flow as orbital motion, we have 1 GM M˙ , (33) WG = 2πr 2r2 so that Wv /WG = 3f = 3[1 − (ri /r)1/2 ].

(34)

At large distances from the inner edge, the dissipation rate is 3 times larger than the rate of gravitational energy release. This may seem odd, but becomes understandable when it is realized that there is a significant flux of energy through the disk associated with the viscous stress3 . Integrating the viscous 3 See

LL section 16

218

Accretion Disks

energy dissipation over the whole disk, one finds ∞ GM M˙ 2πrWv dr = , 2ri ri

(35)

as expected. That is, globally, but not locally, half of the gravitational energy is radiated from the disk while the other half remains in the orbital kinetic energy of the accreted material. What happens to this remaining orbital energy depends on the nature of the accreting object. If the object is a slowly rotating black hole, the orbital energy is just swallowed by the hole. If it has a solid surface, the orbiting gas slows down until it corotates with the surface, dissipating the orbital energy into heat in a boundary layer. Unless the surface rotates close to the orbital rate (‘breakup’), the energy released in this way is of the same order as the total energy released in the accretion disk. The properties of this boundary layer are therefore crucial for accretion onto neutron stars and white dwarfs. See also section 22.9.1 and Inogamov and Sunyaev (1999).

5.3

Radiation Pressure Dominated Disks

In the inner regions of disks in XRB, the radiation pressure can dominate over the gas pressure, which results in a different expression for the disk thickness. The total pressure P is 1 (36) P = Pr + Pg = aT 4 + Pg . 3 Defining a ‘total sound speed’ by c2t = P/ρ the relation ct = ΩH still holds. For Pr Pg we get from (31), with (26) and τ 1: 3 κf M˙ , 8π (where the rather approximate relation Σ = 2Hρ0 has been used). Thus, cH =

3 κ 3 M˙ H ≈ f M˙ = f , (37) R 8π cR 2 M˙ E where R is the stellar radius and M˙ E the Eddington rate for this radius. It follows that the disk becomes thick near the star, if the accretion rate is near Eddington (though this is mitigated somewhat by the decrease of the factor f ). Accretion near the Eddington limit is evidently not geometrically thin any more. In addition, other processes such as angular momentum loss by ‘photon drag’ have to be taken into account. Three locally defined time scales play a role in thin disks. The dynamical time scale td is the orbital time scale: td = Ω−1 = (GM/r 3 )−1/2 .

(38)

219

Comparison with CV Observations

The time scale for radial drift through the disk over a distance of order r is the viscous time scale: tv = r/(−vr ) =

2f r 2 2 rf = ( ) , 3 ν 3αΩ H

(39)

(using (20 and (21), valid for steady accretion). Finally, there are thermal time scales. If Et is the thermal energy content (enthalpy) of the disk per unit of surface area, and Wv = (9/4)Ω2 νΣ the heating rate by viscous dissipation, we can define a heating time scale: th = Et /Wv .

(40)

In the same way, a cooling time scale is defined by the energy content and the radiative loss rate: tc = Et /(2σr Ts4 ).

(41)

For a thin disk, the two are equal since the viscous energy dissipation is locally balanced by radiation from the two disk surfaces. [In thick disks (ADAFs), this balance does not hold, since the advection of heat with the accretion flow is not negligible. In ADAFs, tc > th (see section 22.12]. Thus, we can replace both time scales by a single thermal time scale tt , and find, with (25): ∞ 1 γP dz, (42) tt = Wv −∞ γ − 1 where the enthalpy of an ideal gas of constant ratio of specific heats γ has been used. Leaving out numerical factors of order unity, this yields tt ≈

1 . αΩ

(43)

That is, the thermal time scale of the disk is independent of most of the disk properties and of the order 1/α times longer than the dynamical time scale. This independence is a consequence of the α-parametrization used. If α is not a constant, but dependent on disk temperature for example, the dependence of the thermal time scale on disk properties will become apparent again. If, as seems likely from observations, α is generally < 1, we have in thin disks the ordering of time scales: tv tt > td .

6.

(44)

Comparison with CV Observations

The number of meaningful quantitative tests between the theory of disks and observations is somewhat limited since in the absence of a theory for ν, it is a

220

Accretion Disks

bit meagre on predictive power. The most detailed information perhaps comes from modeling of CV outbursts. Two simple tests are possible (nearly) independently of ν. These are the prediction that the disk is geometrically quite thin (eq. 32) and the prediction that the surface temperature Ts ∼ r−3/4 in a steady disk. The latter can be tested in a subclass of the CV’s that do not show outbursts, the nova-like systems, which are believed to be approximately steady accreters. If the system is also eclipsing, eclipse mapping techniques can be used to derive the brightness distribution with r in the disk (Horne, 1985, 1993). If this is done in a number of colors so that bolometric corrections can be made, the results (e.g. Rutten et al. 1992) show in general a fair agreement with the r−3/4 prediction. Two deviations occur: i) a few systems show significantly flatter distributions than predicted, and ii) most systems show a ‘hump’ near the outer edge of the disk. The latter deviation is easily explained, since we have not taken into account that the impact of the stream heats the outer edge of the disk. Though not important for the total light from the disk, it is an important local contribution near the edge. Eclipse mapping of Dwarf Novae in quiescence gives a quite different picture. Here, the inferred surface temperature profile is often nearly flat (e.g. Wood et al. 1989a, 1992). This is understandable however since in quiescence the mass flux depends strongly on r. In the inner parts of the disk it is small, near the outer edge it is close to its average value. With eq. (26), this yields a flatter Ts (r). The lack of light from the inner disk is compensated during the outburst, when the accretion rate in the inner disk is higher than average (see Mineshige and Wood 1989 for a more detailed comparison). The effect is also seen in the 2-dimensional hydrodynamic simulations of accretion in a binary by Ró˙zyczka and Spruit (1993). These simulations show an outburst during which the accretion in the inner disk is enhanced, between two episodes in which mass accumulates in the outer disk.

7.

Comparison with LMXB Observations: Irradiated Disks

In low mass X-ray binaries a complication arises because of the much higher luminosity of the accreting object. Since a neutron star is roughly 1000 times smaller than a white dwarf, it produces 1000 times more luminosity for a given accretion rate. Irradiation of the disk by the central source leads to a different surface temperature than predicted by (26). The central source (star plus inner disk) radiates the total accretion luminosity GM M˙ /R (assuming sub-Eddington accretion, see section 2). If the disk is concave, it will intercept some of this luminosity. If the central source is approximated as a point source the irradiat-

Comparison with LMXB Observations: Irradiated Disks

221

ing flux on the disk surface is GM M˙ , (45) 4πRr2 where is the angle between the disk surface and the direction from a point on the disk surface to the central source: Firr =

= dH/dr − H/r.

(46)

The disk is concave if is positive. We have 2 r Firr = , F 3f R where F is the flux generated internally in the disk, given by (26). On average, the angle is of the order of the aspect ratio δ = H/r. With f ≈ 1, and our fiducial value δ ≈ 5 10−3 , we find that irradiation in LMXB dominates for r > 109 cm. This is compatible with observations (for reviews see van Paradijs and McClintock 1993), which show that the optical and UV are dominated by reprocessed radiation. When irradiation by an external source is included in the thin disk model, the surface boundary condition of the radiative transfer problem, equation (29) becomes σr Ts4 = F + (1 − a)Firr ,

(47)

where a is the X-ray albedo of the surface, i.e. 1 − a is the fraction of the incident flux that is absorbed in the optically thick layers of the disk (photons absorbed higher up only serve to heat up the corona of the disk). The surface temperature Ts increases in order to compensate for the additional incident heat flux. The magnitude of the incident flux is sensitive to the assumed disk shape H(r), as well as on the assumed shape (plane or spherical, for example) of the central X-ray emitting region. The disk thickness depends on temperature, and thereby also on the irradiation. It turns out, however, that this dependence on the irradiating flux is small, if the disk is optically thick, and the energy transport is by radiation (Lyutyi and Sunyaev 1976). To see this, integrate (28) with the modified boundary condition (47). This yields 2 (1 − a)Firr 3 . σr T 4 = F (τ + ) + 4 3 F

(48)

Thus the irradiation adds an additive constant to T 4 (z). At the midplane, this constant has much less effect than at the surface. For the midplane temperature and the disk thickness to be affected significantly, it is necessary that Firr /F > ∼ τ.

(49)

222

Accretion Disks

The reason for this weak dependence of the midplane conditions on irradiation is the same as in radiative envelopes of stars, which are also insensitive to the surface boundary condition. The situation is very different for convective disks. As in fully convective stars, the adiabatic stratification then causes the conditions at the midplane to depend much more directly on the surface temparture. The outer parts of the disks in LMXB with wide orbits may be convective, and their thickness affected by irradiation. In the reprocessing region of the disks of LMXB, the conditions are such that F ∼ 10 K.

Radiation Supported Radiatively Inefficient Accretion

231

In order for the flow to be radiation pressure and advection dominated, the optical depth has to be sufficiently large so the radiation does not leak out. The energy density in the flow, vertically integrated at a distance r, is of the order E ≈ aT 4 H,

(65)

and the energy loss rate per cm2 of disk surface is given by (59). The cooling time is therefore, tc = E/F = 3τ H/c.

(66)

This is to be compared with the accretion time, which can be written in terms of the mass in the disk at radius r, of the order 2πr2 Σ, and the accretion rate: tacc = 2πr2 Σ/M˙ .

(67)

This yields tc /tacc ≈

4 R κ ˙ M= m ˙ , πrc η r

(68)

(where a factor 3/2 H/r ∼ O(1) has been neglected). Since r > R, this shows that accretion has to be of around the Eddington rate or larger in order to be both radiation- and advection-dominated. This condition can also be expressed in terms of the so-called trapping radius rt (e.g. Rees 1978). Equating tacc and tc yields rt /R ≈ 4m. ˙

(69)

Inside rt , the flow is advection dominated: the radiation field produced by viscous dissipation stays trapped inside the flow, instead of being radiated from the disk as happens in a standard thin disk. Outside the trapping radius, the radiation field can not be sufficiently strong to maintain a disk with H/r ∼ 1, it must be a thin form of disk instead. Such a thin disk can still be radiationsupported (i.e. Pr Pg ), but it can not be advection dominated. Flows of this kind are called ‘radiation supported accretion tori’ (or radiation tori, for short) by Rees et al. 1982. They must accrete at a rate above the Eddington value to exist. The converse is not quite true: a flow accreting above Eddington is an advection dominated flow, but it need not necessarily be radiation dominated. Advection dominated optically thick acretion flows exist in which radiation does not play a major role (see section 22.12.1). That an accretion flow above M˙ E is advection dominated, not a thin disk, follows from the fact that in a thin disk the energy dissipated must be radiated away locally. Since the local radiative flux can not exceed the Eddington energy flux FE , the mass accretion rate in a thin disk can not significantly exceed the Eddington value (61).

232

Accretion Disks

The gravitational energy, dissipated by viscous stress in differential rotation and advected with the flow, ends up on the central object. If this is a black hole, the photons, particles and their thermal energy are conveniently swallowed at the horizon, and do not react back on the flow. Radiation tori are therefore mostly relevant for accretion onto black holes. They are convectively unstable (Bisnovatyi-Kogan and Blinnikov 1977): the way in which energy is dissipated, in the standard α-prescription, is such that the entropy (∼ T 3 /ρ) decreases with height in the disk. Recent numerical simulations (see section 22.14) show the effects of this convection.

11.1

Super-Eddington Accretion onto Black Holes

As the accretion rate onto a black hole is increased above M˙ E , the trapping radius moves out. The total luminosity increases only slowly, and remains of the order of the Eddington luminosity. Such supercritical accretion has been considered by Begelman and Meier (1982, see also Wang and Zhou 1999); they show that the flow has a radially self-similar structure. Abramowicz et al. (1988, 1989) studied accretion onto black holes at rates near M˙ E . They used a vertically-integrated approximation for the disk, but included the advection terms. The resulting solutions were called ‘slim disks’. These models show how with increasing accretion rate, a standard thin ShakuraSunyaev disk turns into a radiation-supported advection flow. The nature of the transition depends on the viscosity prescription used, and can show a nonmonotonic dependence of M˙ on surface density Σ (Honma et al. 1991). This suggests the possibility of instability and cyclic behavior of the inner disk near a black hole, at accretion rates near and above M˙ E (for an application to GRS 1915+105 see Nayakshin et al., 1999).

11.2

Super-Eddington Accretion onto Neutron Stars

In the case of accretion onto a neutron star, the energy trapped in the flow, plus the remaining orbital energy, settles onto its surface. If the accretion rate is below M˙ E , the energy can be radiated away by the surface, and steady accretion is possible. A secondary star providing the mass may, under some circumstances, transfer more than M˙ E , since it does not know about the neutron star’s Eddington value. The outcome of this case is still somewhat uncertain; it is generally believed on intuitive grounds that the ‘surplus’ (the amount above M˙ E ) somehow gets expelled from the system. One possibility is that, as the transfer rate is increased, the accreting hot gas forms an extended atmosphere around the neutron star like the envelope of a giant. If it is large enough, the outer parts of this envelope are partially ionized. The opacity in these layers, due to lines of the CNO and heavier elements, is then much higher than the electron scattering opacity. The Eddington lumi-

233

ADAF Hydrodynamics

nosity based on the local value of the opacity is then smaller than it is near the neutron star surface. Once an extended atmosphere with a cool surface forms, the accretion luminosity is thus large enough to drive a wind from the envelope (see Kato 1997, where the importance of this effect is demonstrated in the context of Novae). This scenario is somewhat dubious however, since it assumes that the mass transferred from the secondary continues to reach the neutron star and generate a high luminosity there. This is not at all obvious, since the mass transfering stream may instead dissipate inside the growing envelope of the neutron star. The result of this could be a giant (more precisely, a Thorne-Zytkow star), with a steadily increasing envelope mass. Such an envelope is likely to be large enough to engulf the entire binary system, which then develops into a commonenvelope (CE) system. The envelope mass is then expected to be ejected by CE hydrodynamics (for reviews see Taam 1994, Taam and Sandquist 2000). A more speculative proposal, suggested by the properties of SS 433, is that the ‘surplus mass’ is ejected in the form of jets. The binary parameters of Cyg X-2 are observational evidence for mass ejection in super-Eddington mass transfer phases (King and Ritter 1999, Rappaport and Podsiadlowski 1999, King and Begelman 1999).

12.

ADAF Hydrodynamics

The hydrodynamics of radiatively inefficient flows can be studied by starting, at a very simple level, with a generalization of the thin disk equations. Making the assumption that quantities integrated over the height z of the disk give a fair representation (though this is justifiable only for thin disks), and assuming axisymmetry, the problem reduces to a one-dimensional time-dependent one. Further simplifying this by restriction to a steady flow yields the equations 2πrΣvr = M˙ = cst,

(70)

rΣvr ∂r (Ωr2 ) = ∂r (νΣr3 ∂r Ω),

(71)

1 vr ∂r vr − (Ω2 − Ω2K )r = − ∂r p, ρ

(72)

Σvr T ∂r S = q + − q − ,

(73)

where S is the specific entropy of the gas, Ω the local rotation rate, now different from the Keplerian rate ΩK = (GM/r 3 )1/2 , while q − = divFr dz (74) q + = Qv dz

234

Accretion Disks

are the height-integrated viscous dissipation rate and radiative loss rate, respectively. In the case of thin disks, equations (70) and (71) are unchanged, but (72) simplifies to Ω2 = Ω2K , i.e. the rotation is Keplerian, while (73) simplifies to q + = q − , expressing local balance between viscous dissipation and cooling. The left hand side of (73) describes the radial advection of heat, and is perhaps the most important deviation from the thin disk equations at this level of approximation (hence the name advection dominated flows). The characteristic properties are seen most cearly when radiative loss is neglected altogether, q − = 0. The equations are supplemented with expressions for ν and q + : ν = αc2s /ΩK ;

q + = (r∂r Ω)2 νΣ.

(75)

If α is taken constant, q − = 0, and an ideal gas is assumed with constant ratio of specific heats, so that the entropy is given by S = cv ln(p/ργ ),

(76)

then equations (70)-(73) have no explicit length scale in them. This means that a special so-called self-similar solution exists, in which all quantities are powers of r. Such self-similar solutions have apparently first been described by Gilham (1981), but have since then been re-invented several times (Spruit et al. 1987; Narayan and Yi, 1994). The dependences on r are Ω ∼ r−3/2 ; H ∼ r;

ρ ∼ r−3/2 , T ∼ r−1 .

(77) (78)

In the limit α 1, one finds

γ−1 , vr = −αΩK r 9 5−γ

(79)

5 − 3γ 1/2 , Ω = ΩK 2 5−γ

(80)

c2s = Ω2K r2

H = r

γ−1 , 5−γ

γ−1 5−γ

(81)

1/2 .

(82)

The precise from of these expressions depends somewhat on the way in which vertical integrations such as in (74) are done (which are only approximate).

ADAF Hydrodynamics

235

The self-similar solution can be compared with numerical solutions of eqs. (70)–(73) with appropriate conditions applied at inner (ri ) and outer (ro ) boundaries (Nakamura et al. 1996, Narayan et al. 1997). The results show that the self-similar solution is valid in an intermediate regime ri r ro . That is, the solutions of (70)–(73) approach the self-similar solution far from the boundaries, as is characteristic of self-similar solutions. The solution exists only if 1 < γ ≤ 5/3, a condition satisfied by all ideal gases. As γ ↓ 1, the disk temperature and thickness vanish. This is understandable, since a γ close to 1 means that the particles making up a gas with gas have a large number of internal degrees of freedom. In thermal equilibrium the accretion energy is shared between all degrees of freedom, so that for a low γ less is available for the kinetic energy (temperature) of the particles. Second, the rotation rate vanishes for γ → 5/3. As in the case of spherical accretion no accreting solutions exist for γ > 5/3 (cf. section 22.2). Since a fully ionized gas has γ = 5/3, it is the most relevant value for optically thin accretion near a black hole or neutron star. Apparently, steady advection dominated accretion can not have angular momentum in this case. The question then arises how an adiabatic flow with γ = 5/3 will behave if one starts it as a rotating torus around a black hole. In the literature, this problem has been circumvented by arguing that real flows would have magnetic fields in them, which would change the effective compressibility of the gas. Even if a magnetic field of sufficient strength is present, however, (energy density comparable to the gas pressure) the effective γ is not automatically lowered. If the field is compressed mainly perpendicular to the field lines, for example, the effective γ is closer to 2. Also, this does not solve the conceptual problem what would happen to a rotating accretion flow consisting of a more weakly magnetized ionized gas. This conceptual problem has been solved by Ogilvie (1999), who showed how a gas cloud initially rotating around a point mass settles to the slowly rotating self-similar solutions of the steady problem discussed above. He found a similarity solution to the time dependent version of eqs (70)–(73), in which distance and time occur in the combination r/t2/3 ). This solution describes the asymptotic behavior (in time) of a viscously spreading disk, analogous to the viscous spreading of thin disks (see section 22.4.3). As in the thin disk case, all the mass accretes asymptotically onto the central mass, while all the angular momentum travels to infinity together with a vanishing amount of mass. For all γ < 5/3, the rotation rate at a fixed r tends to a finite value as t → ∞, but for γ = 5/3 it tends to zero. The size of the slowly-rotating region expands as r ∼ t2/3 . It thus seems likely that the typical slow rotation of ADAFs at γ near 5/3 is a real physical property. In such a flow the angular momentum gets expelled from the inner regions almost completely.

236

12.1

Accretion Disks

Other Optically Thick Accretion Flows

The radiation-dominated flows discussed in section 22.11 are not the only possible optically thick advection dominated flows. From the discussion of the hydrodynamics, it is clear that disk-like (i.e. rotating) accretion is possible whenever the ratio of specific heats is less than 5/3. A radiation supported flow satisfies this requirement with γ = 4/3, but it can also happen in the absence of radiation if energy is taken up in the gas by internal degrees of freedom of the particles. Examples are the rotational and vibrational degrees of freedom in molecules, and the energy associated with dissociation and ionization. If the accreting object has a gravitational potential not much exceeding the 2.3 + 13.6 eV per proton for dissociation plus ionization, a gas initially consisting of molecular hydrogen can stay bound at arbitrary accretion rates. This translates into a limit M/M R /R < 0.01. This is satisfied approximately by the giant planets, which are believed to have gone through a phase of rapid adiabatic gas accretion (e.g. Podolak et al. 1993). A more remotely related example is the core-collapse supernova. The accretion energy of the envelope mass falling onto the proto-neutron star is lost mostly through photodisintegration of nuclei, causing the well known problem of explaining how a shock is produced of sufficient energy to unbind the envelope. If the pre-collapse core rotates sufficiently rapidly, the collapse will form an accretion torus (inside the supernova envelope), with properties similar to advection dominated accretion flows (but at extreme densities and accretion rates, by X-ray binary standards). Such objects have been invoked as sources of Gamma-ray bursts (Popham et al. 1999). A final possibility for optically thick accretion is through neutrino losses. If the temperature and density near an accreting neutron star become large enough, additional cooling takes place through neutrinos (as in the cores of giants). This is relevant for the physics of Thorne-Zytkow stars (neutron stars or black holes in massive supergiant envelopes, cf. Bisnovatyi-Kogan and Lamzin 1984, Cannon et al. 1992), and perhaps for the spiral-in of neutron stars into giants (Chevalier 1993, see however Taam 2000).

13.

Optically Thin Radiatively Inefficient Flows (ISAFs)

The optically thin case has received most attention, because of the promise it holds for explaining the (radio to X-ray) spectra of X-ray binaries and the central black holes in galaxies, including our own. For a review see Yi (1999). This kind of flow occurs if the gas is optically thin, and radiation processes sufficiently weak. The gas then heats up to near the virial temperature. Near the last stable orbit of a black hole, this is of the order 100 MeV, or 1012 K. At such temperatures, a gas in thermal equilibrium would radiate at a fantastic rate, even if it were optically thin, because the interaction between electrons and


237

photons becomes very strong already near the electron rest mass of 0.5MeV. In a remarkable early paper, Shapiro Lightman and Eardley (1976) noted that this, however, is not what will happen in an optically thin accreting plasma. They showed that, instead, thermal equilibrium between ions and electrons breaks down and a two-temperature plasma forms. We call such a flow an ion supported accretion flow (ISAF), following the nomenclature suggested by Rees et al. (1982). The argument is as follows. Suppose that the energy released by viscous dissipation is distributed equally among the carriers of mass, i.e. mostly to the ions and ∼ 1/2000 to the electrons. Most of the energy then resides in the ions, which radiate very inefficiently (their high mass prevents the rapid accelerations that are needed to produce electromagnetic radiation). Their energy is transfered to the electrons by Coulomb interactions. These interactions are slow, however, under the conditions mentioned. They are slow because of the low density (on account of the assumed optical tickness), and because they decrease with increasing temperature. The electric forces that transfer energy from an ion to an electron act only as long as the ion is within the electron’s Debye sphere (e.g. Spitzer, 1965). The interaction time between proton and electron, and thus the momen−1/2 where Tp is the proton tum transfered, therefore decrease as 1/vp ∼ Tp temperature. In this way, an optically thin plasma near a compact object can be in a twotemperature state, with the ions being near the virial temperature, and the electrons, which are doing the radiating, at a much lower temperature around 50– 200 keV. The energy transfer from the gravitational field to the ions is fast (by some form of viscous or magnetic dissipation, say), from the ions to the electrons slow, and finally the energy losses of the electrons fast (by synchrotron radiation in a magnetic field or by inverse Compton scattering of soft photons). Such a flow would be radiatively inefficient since the receivers of the accretion energy, the ions, get swallowed by the hole before getting a chance to transfer their energy to the electrons. Most of the accretion energy thus gets lost into the hole, and the radiative efficiency η (section 22.2.2) is much less than the value ≈ 0.1 for a cool disk. The first disk models which take into account this physics of advection and a two-temperature plasma were developed by Ichimaru (1977). It is clear from this description that both the physics of such flows and the radiation spectrum to be expected depend crucially on the details of the ionelectron interaction and radiation processes assumed. This is unlike the case of the optically thick advection dominated flows, where gas and radiation are in approximate thermodynamic equilibrium. This is a source of uncertainty in the application of ISAFs to observed systems, since their radiative properties depend on poorly known quantities such as the strength of the magnetic field in the flow.

238

Accretion Disks

· · M/ME 10

slim disk α=2

1 α=0.1 0.1

AF AD

α=2

SLE

0.01

α=0.1

standard disk

0.001 0.1

1

10

100

1000

τ

Figure 2. Branches of advection-dominated and thin disks for two values or the viscosity parameter α, as functions of accretion rate and (vertical) optical depth of the flow (schematic, after Chen et al. 1995, Zdziarski 1998). Optically thin branches are the ISAF and SLE (ShapiroLightman-Eardley) solutions, optically thick ones the radiation dominated (‘slim disk’ or ‘radiation torus’) and SS (Shakura-Sunyaev or standard thin disk). Advection dominated are the ISAF and the radiation torus, geometrically thin are the SLE and SS. The SLE solution is a thermally unstable branch.

The various branches of optically thin and thick accretion flows are summarized in figure 1. Each defines a relation between surface density Σ (or optical depth τ = κΣ) and accretion rate. ISAFs require low densities. These can result either because of low accretion rates or large values of the viscosity parameter. The condition that the cooling time of the ions by energy transfer to the electrons is longer than the accretion time yields a maximum accretion rate (Rees et al. 1982), 2 m ˙ < ∼α .

(83)

If α ≈ 0.05 as suggested by current simulations of magnetic turbulence, the maximum accretion rate would be a few 10−3 . If ISAFs are to be applicable to systems with higher accretion rates, such as Cyg X-1 for example, the viscosity parameter must be larger, on the order of 0.3.


13.1

239

Application: Hard Spectra in X-ray Binaries

In the hard state, the X-ray spectrum of black hole and neutron star accreters is characterized by a peak in the energy distribution (νFν or E F (E)) at photon energies around 100 keV. This is to be compared with the typical photon energy of ∼ 1 keV expected from a standard optically thick thin disk accreting near the Eddington limit. The standard, and by far most likely explanation is that the observed hard photons are softer photons (around 1 keV) that have been up-scattered through inverse Compton scattering on hot electrons. Fits of such Comptonized spectra (e.g. Sunyaev and Titarchuk 1980, Zdziarski 1998 and references therein) yield an electron scattering optical depth around unity and an electron temperature of 50–100 keV. The scatter in these parameters is rather small between different sources. The reason may lie in part in the physics of Comptonization, but is not fully understood either. Something in the physics of the accretion flow keeps the Comptonization parameters constant as long as it is in the hard state. ISAFs have been applied with some success in interpreting XRB. They can produce reasonable X-ray spectra, and have been used in interpretations of the spectral-state transitions in sources like Cyg X-1 (Esin et al. 1998 and references therein). An alternative to the ISAF model for the hard state in sources like Cyg X1 and the black hole X-ray transients is the ‘corona’ model. A hot corona (Bisnovatyi-Kogan and Blinnikov 1976), heated perhaps by magnetic fields as in the case of the Sun (Galeev et al. 1979) could be the medium that Comptonizes soft photons radiated from the cool disk underneath. The energy balance in such a model produces a Comptonized spectrum within the observed range (Haardt and Maraschi 1993). This model has received further momentum, especially as a model for AGN, with the discovery of broadened X-ray lines indicative of the presence of a cool disk close to the last stable orbit around a black hole (Fabian et al. 2002 and references therein). The very rapid X-ray variability seen in some of these sources is interpreted as due magnetic flaring in the corona, like in the solar corona (e.g. Di Matteo et al. 1999a).

13.2

Transition from Cool Disk to ISAF

One of the difficulties in applying ISAFs to specific observed systems is the transition from a standard geometrically thin, optically thick disk, which must be the mode of mass transfer at large distances, to an ISAF at closer range. This is shown by figure 1, which illustrates the situation at some distance close to the central object. The standard disk and the optically thin branches are separated from each other for all values of the viscosity parameter. This separation of the optically thin solutions also holds at larger distances. Thus, there is no plausible continuous path from one to the other, and the transition between

240

Accretion Disks

the two must be due to additional physics that is not included in diagrams like figure 1. A promising possibility is that the transition takes place through evaporation. Two distinct mechanisms have been elaborated for such evaporation. In the first (Meyer and Meyer-Hofmeister 1994, Liu et al. 2002), the evaporation starts at a relatively large distance from the hole, where the virial temperature is of the order of 106 − 107 K. As in the solar corona, the strong decrease of radiative efficiency of gas with temperature in this range produces a hot optically thin corona in contact with the cool disk below, and exchange of mass can take place through evaporation and condensation, and the process is mediated by electron heat conduction. In this scenario, a corona flow at ∼ 107 K at a distance of several hundred Schwarzschild radii transforms into a twotemperature ISAF further in. Observations indicate that cool disks can also coexist with a hot, hard X-ray producing plasma quite close to the hole (for references see Dullemond and Spruit 2004). At these close distances, evaporation must behave differently from the coronal evaporation model, since the interaction of a two-temperature plasma with a cool disk is very different from that of a plasma at coronal temperatures (Spruit 1997). The energy in an ISAF is in the ions, and electron conduction of heat unimportant. Moreover, the ions penetrate a substantial distance into the cool disk, and loose their energy long before they can heat up the disk to virial temperatures. Nevertheless, evaporation can still take place in this case, since it turns out that the interaction of the ions with the cool disk produces a layer of intermediate temperature (around 100 keV) that becomes thermally unstable in the presence of viscous dissipation, and heats up ISAF temperatures (Deufel et al. 2000, 2001, 2002, see also Spruit and Deufel 2002). This model explains both the hard spectra of typical black hole accreters and the coexistence of cool and hot plasma indicated by the observations (Dullemond and Spruit 2004).

13.3

Quiescent Galactic Nuclei

For very low accretion rates, such as inferred for the black hole in the center of our galaxy (identified with the radio source Sgr A*), the broad band spectral energy distribution of an ISAF is predicted to have two humps (Narayan et al. 1995, Quataert et al. 1999). In the X-ray range, the emission is due to bremsstrahlung. In the radio range, the flow emits synchrotron radiation, provided that the magnetic field in the flow has an energy density order of the gas pressure (‘equipartition’). Synthetic ISAF spectra can be fitted to the observed radio and X-ray emission from Sgr A*. In other galaxies where a massive central black hole is inferred, and the center is populated by an X-ray emitting gas of known density, ISAFs would also be natural, and might ex-


241

plain why the observed luminosities are so low compared with the accretion rate expected for a hole embedded in a gas of the measured density. In some of these galaxies, however, the peak in the radio-to-mm range predicted by analogy with Sgr A* is not observed (Di Matteo et al. 1999b). This requires an additional hypothesis, for example that the magnetic field in these cases is much lower, or that the accretion energy is carried away by an outflow.

13.4

Transients in Quiescence

X-ray transients in quiescence (i.e. after an outburst) usually show a very low X-ray luminosity. The mass transfer rate from the secondary in quiescence can be inferred from the optical emission. This shows the characteristic ‘hot spot’, known from other systems to be the location where the mass transfering stream impacts on the edge of an accretion disk (e.g. van Paradijs and McClintock 1995). These observations thus show that a disk is present in quiescence, while the mass transfer rate can be measured from the brightness of the hot spot. If this disk were to extend to the neutron star with constant mass flux, the predicted X-ray luminosity would be much higher than observed. This has traditionally been interpreted as a consequence of the fact that in transient systems, the accretion is not steady. Mass is stored in the outer parts and released by a disk instability (e.g. King 1995, Meyer-Hofmeister and Meyer 1999) producing the X-ray outburst. During quiescence, the accretion rate onto the compact object is much smaller than the mass transfer from the secondary to the disk. ISAFs have been invoked as an alternative explanation. The quiescent accretion rate onto the central object is proposed to be higher than in the diskinstability explanation, the greater energy release being hidden on account of the low radiative efficiency of the ISAF. Some transient systems have neutron star primaries, with a hard surface at which the energy accreted by the ISAF must somehow be radiated away. A neutron star, with or without ISAFs, can not accrete in a radiatively inefficient way. In order to make ISAFs applicable, it has been proposed that the neutron stars in these systems have a modest magnetic dipole moment, such that in quiescence the gas in the accretion disk is prevented, by the ‘propeller effect’ (Illarionov and Sunyaev 1975, Sunyaev and Shakura 1977) from accreting onto the star.

13.5

ISAF-disk Interaction: Lithium

One of the strong predictions of ISAF models, whether for black holes or neutron stars, is that the accreting plasma in the inner regions has an ion temperature of 10–100 MeV. Nearby is a cool and dense accretion disk feeding this plasma. If only a small fraction of the hot ion plasma gets in contact with the disk, the intense irradiation by ions will produce nuclear reactions (Aharonian

242

Accretion Disks

and Sunyaev 1984, Martín et al. 1992). The main effects would be spallation of CNO elements into Li and Be, and the release of neutrons by various reactions. In this context, it is intriguing that the secondaries of both neutron star and black hole accreters have high overabundances of Li compared with other stars of their spectral types (Martín et al. 1992, 1994a). If a fraction of the disk material is carried to the secondary by a disk wind, the observed Li abundances may be accounted for (Martín et al. 1994b).

14.

Outflows?

The energy density in an advection dominated accretion flow is of the same order as the gravitational binding energy density GM/r, since a significant fraction of that energy went into internal energy of the gas by viscous dissipation, and little of it got lost by radiation. The gas is therefore only marginally bound in the gravitational potential. This suggests that perhaps a part of the accreting gas can escape, producing an outflow or wind. In the case of the ion supported ISAFs, this wind would be thermally driven by the temperature of the ions, like an ‘evaporation’ from the accretion torus. In the case of the radiation supported tori, which exist only at a luminosity near the Eddington value, but with much lower temperatures than the ion tori, winds driven by radiation pressure could exist. The possibility of outflows is enhanced by the viscous energy transport through the disk. In the case of thin accretion disks (not quite appropriate in the present case, but sufficient to demonstrate the effect), the local rate of gravitational energy release (erg cm−2 s−1 ) is W = Σvr ∂r (GM/r). The local viscous dissipation rate is (9/4)νΣΩ2 . They are related by ri Qv = 3[1 − ( )1/2 ]W, r

(84)

where ri is the inner edge of the disk (see ‘accretion disks’ elsewhere in this volume). The viscous dissipation rate is less than the gravitational energy release for r < (4/9)ri , and larger outside this radius. Part of the gravitational energy released in the inner disk is transported outward by the viscous stresses, so that the energy deposited in the gas is up to three times larger than expected from a local energy balance argument. The temperatures in an ADAF would be correspondingly larger. Begelman and Blandford (1999) have appealed to this effect to argue that in an ADAF most of the accreting mass of a disk might be expelled through a wind, the energy needed for this being supplied by the viscous energy transport associated with the small amount of mass that actually accretes. These suggestions are in principle testable, since the arguments are about two-dimensional time dependent flows (axisymmetric), which can be studied fairly well by numerical simulation. Igumenshchev et al. (1996), and Igu-

Outflows?

243

menshchev and Abramowicz (1999) present results of such simulations, but unfortunately these give a somewhat ambiguous answer to the question. For large viscosity (α ∼ 0.3) no outflow is seen, but for small viscosity time dependent flows are seen with outflows in some regions. Some of these flows may be a form of convection and unrelated to systematic outflows.

References Abramowicz, M.A., Czerny B., Lasota J.-P., Szuszkiewicz E., 1988, Astrophys. J. 332, 646 Abramowicz M.A, Kato S., Matsumoto R., 1989, Publ. Astr. Soc. Japan 41, 1215 Aharonian F.A., Sunyaev, R.A., 1984 Mon. Not. R. Astron. Soc. 210, 257 Armitage P.J., 1998, Astrophys. J. 501, L189 Armitage P.J., and Livio M., 1998, Astrophys. J. 493, 898 Balbus S.A. and Hawley J.F., 1991, Astrophys. J. 376, 214 Balbus S.A. and Hawley J.F., 1992, Astrophys. J. 400, 610 Balbus S.A. and Hawley J.F., 1996, Astrophys. J. 464, 690 Begelman M.C., 1979, Mon. Not. R. Astron. Soc. 187, 237 Begelman M.C., Meier D.L., 1982, Astrophys. J. 253, 873 Begelman M.C., McKee C.F., Shields G.A., 1983, Astrophys. J. 271, 70 Begelman M.C., Blandford R.D., 1999, Mon. Not. R. Astron. Soc. 303, L1 Bisnovatyi-Kogan G., 1993, Astron. Astrophys. 274, 796 Bisnovatyi-Kogan G., and Ruzmaikin A.A. 1976, Ap. Space Sci. 42, 401 Bisnovatyi-Kogan G.S., Blinnikov S.I., 1976, Pi’sma Astron. Zh 2, 489 (translation 1977, Soviet Astron.L 2, 191) Bisnovatyi-Kogan G.S., Blinnikov S.I., 1977, Astron. Astrophys. 59, 111 Bisnovatyi-Kogan G.S., Lamzin S.A., 1984, Astron. Zh 61, 323 (translation Soviet Astron. 28, 187) Blaes O. and Hawley J.F., 1988, Astrophys. J. 326, 277 Blandford R.D., 1976, Mon. Not. R. Astron. Soc. 176, 465 Blandford R.D. and Payne D.G., 1982 Mon. Not. R. Astron. Soc. 199, 883 Blandford R.D., 1989, in Theory of Accretion Disks, eds. F. Meyer, W. Duschl, J. Frank and E. Meyer-Hofmeister, Kluwer, Dordrecht, p.35 Blandford R.D., Rees M.J., 1992, In: Testing the AGN paradigm, Holt, S. Neff, S. Urry, C. Megan (eds.), Univ. of Maryland, p. 3-19. Bondi H., 1952, Mon. Not. R. Astron. Soc. 112, 195 Bradshaw P., 1969, J. Fluid Mech. 36, 177 Brandenburg A., Nordlund ρA., Stein R.F. and Torkelsson U., 1995, Astrophys. J. 446, 741 Cannon R.C., Eggleton P.P., Zytkow A.N., Podsiadlowski P., 1992, Astrophys. J. 386, 206 Cannizzo J.K., Shafter A.W., Wheeler J.C., 1988, Astrophys. J. 333,227 Chen X.M., Abramowicz M.A., Lasota J.-P., Narayan R., Yi I., 1995, Astrophys. J. 443, L61 Chevalier R.A., 1993, Astrophys. J. 411, 33 Cordova F.A., 1995, in W.H.G. Lewin et al. eds. X-ray Binaries, Cambridge University Press, Cambridge, p.331 Courant R. and Friedrichs K.O., 1948, Supersonic Flow and Shock Waves, Interscience, New York, and Springer, Berlin (1976) Chandrasekhar C., 1961, Hydrodynamic and Hydromagnetic Stability, Oxford Univ. Press, Oxford, and Dover Publications Inc., 1981 De Villiers, J., & Hawley, J.F. 2003, Astrophys. J. 592 1060 Di Matteo T., Celotti A., Fabian A.C., 1999a, Mon. Not. R. Astron. Soc. 304, 809

244

Accretion Disks

Di Matteo T., Fabian A. C., Rees M.J., Carilli C.L., Ivison R.J.,1999b, Mon. Not. R. Astron. Soc. 305, 492 Deufel, B., & Spruit, H. C. 2000, Astron. Astrophys. , 362, 1 Deufel, B., Dullemond, C.P., & Spruit, H.C. 2001, Astron. Astrophys. , 377, 955 Deufel, B., Dullemond, C.P., & Spruit, H.C. 2002, Astron. Astrophys. , 387, 907 Dullemond, C.P. & Spruit, H.C., 2004, Astron. Astrophys. submitted Eardley D., and Lightman A., 1975, Astrophys. J. 200, 187 Emmering R.T., Blandford R.D., Shlosman I., 1992, Astrophys. J. 385, 460 Esin A.A., Narayan R., Cui W., Grove J.E., Zhang S.N., 1998, Astrophys. J. 505, 854 Fabian, A.C., et al. 2002, Mon. Not. R. Astron. Soc. , 335, L1 Frank J., King A.R., and Raine D.J., 2002, Accretion Power in Astrophysics (3rd edition), Cambridge University Press. Galeev A.A., Rosner R., Vaiana G.S., 1979, Astrophys. J. 229, 318 Gilham S., 1981, Mon. Not. R. Astron. Soc. 195, 755 Haardt F., Maraschi L., Ghisellini G., 1994, Astrophys. J. 432, 95 Hameury J.-M., Menou K., Dubus G., Lasota J.-P., Hure J.-M., 1998, Mon. Not. R. Astron. Soc. 298, 1048 Hawley J.E., 1991 Astrophys. J. 381, 496 Hawley J.F., Gammie C.F. and Balbus S.A., 1995, Astrophys. J. 440, 742 Hawley J.F., Balbus S.A., Winters W.F. 1999, Astrophys. J. 518, 394 Hirose, S., Krolik, J.H., De Villiers, J., & Hawley, J.F. 2004, Astrophys. J. 606, 1083 Honma F., Kato S., Matsumoto R., Abramowicz M., 1991, Publ. Astr. Soc. Japan 43, 261 Horne K., 1985, Mon. Not. R. Astron. Soc. 213, 129 Horne K., Marsh T.R., Cheng F.H., Hubeny I., Lanz T., 1994, Astrophys. J. 426, 294 Horne, K., 1993, in J.C. Wheeler, ed., Accretion Disks in Compact Stellar Systems, Singapore: World Scientific Publishing H¯oshi R., 1979, Prog. Theor. Phys. 61, 1307 Ichimaru S., 1977, Astrophys. J. 241, 840 Igumenshchev I.V., Chen X.M., Abramowicz A., 1996 Mon. Not. R. Astron. Soc. 278, 236 Igumenshchev I.V., Abramowicz M.A., 1999, Mon. Not. R. Astron. Soc. 303, 309 Illarionov A.F., Sunyaev R.A., 1975, Astron. Astrophys. 39, 185 Inogamov N.A., Sunyaev R.A., 1999, Astron. Lett., 1999, 25, 269 (=astro-ph/9904333) Kato M., 1997, Astrophys. J. Suppl. 113, 121 Kippenhahn R. and Thomas H.-C., 1981, in D. Sugimoto, D. Lamb and D.N. Schramm, Fundamental Problems in the Theory of Stellar Evolution (IAU Symp. 93), Reidel, Dordrecht, p.237 King A.R., 1995 in Lewin, W.H.G., van Paradijs, J., van den Heuvel, E.P.J., eds., X-ray Binaries, Cambridge University press King A.R. and Ritter H., 1999 Mon. Not. R. Astron. Soc. 309, 253 King A.R., Begelman M.C., 1999, Astrophys. J. 519, L169 Königl A., 1989 Astrophys. J. 342, 208 Königl A., Kartje J.F.,1994 Astrophys. J. 434, 446 Königl A. and Ruden S.P., 1993, in Protostars and Planets III, eds. E.H. Levy, J.I. Lunine, Univ. Arizona Press, Tucson, p.641 Kley W., 1991, Astron. Astrophys. 247, 95 Kley,W., Papaloizou J.C.B., Lin D.N.C, 1993, Astrophys. J. 416, 679 Kuulkers E., Howell S.B., van Paradijs J., 1996, ApJ 462, 87 Landau L.D. and Lifshitz E.M., 1959 Fluid Mechanics, Pergamon Press, Oxford Lee J.C., Fabian W.N., Brandt W.N., Reynolds C.S., Iwasawa K., 1999, Mon. Not. R. Astron. Soc. 310, 973

Outflows?

245

Lewin W.H.G., van Paradijs J., van den Heuvel E.P.J., 1995, X-ray Binaries, Cambridge University press Lightman A.P., Eardley D.M., 1974, Astrophys. J. 187, L1 Lin D.N.C., and Taam R.E., 1984, in S.E. Woosley, ed., High Energy Transients in Astrophysics, AIP Conference Procs. 115, p.83 Liu, B.F., Mineshige, S., Meyer, F., Meyer-Hofmeister, E., & Kawaguchi, T. 2002, Astrophys. J. , 575, 117 Loeb A., Laor A., 1992, Astrophys. J. 384, 115 Lovelace R.V.E., 1976, Nature 262, 649 Lüst R., 1952, Z. Naturforsch. 7a, 87 Lynden-Bell D. and Pringle J.E., 1974, Mon. Not. R. Astron. Soc. 168, 603 Lyutyi V.M. and Sunyaev R.A., 1976, Astron. Zh. 53, 511, translation in Sov. astron. 20, 290 (1976) Martín E L., Rebolo R., Casares J., Charles P.A., 1992, Nature 358, 129 Martín E.L., Rebolo R., Casares J., Charles P.A., 1994a Astrophys. J. 435, 791 Martín E., Spruit H.C., van Paradijs J., 1994b, Astron. Astrophys. 291, L43 Massey B.S., 1968, Mechanics of Fluids, Chapman and Hall, London (6th Ed. 1989) Meyer F., 1990, Rev. Mod. Astron. 3, 1 Meyer F. and Meyer-Hofmeister E., 1981, Astron. Astrophys. 104, L10 Meyer F. and Meyer-Hofmeister E., 1982, Astron. Astrophys. 106, 34 Meyer-Hofmeister, E. and Ritter, H., 1993, in The Realm of Interacting Binary Stars, ed. J. Sahade, Kluwer, Dordrecht, p.143 Meyer F., Meyer-Hofmeister E., 1994, Astron. Astrophys. 288, 175 Meyer-Hofmeister E., Schandl S., and Meyer F.,1997 Astron. Astrophys. 318, 73 Meyer-Hofmeister E., Meyer F., 1999, astro-ph/9906305 Miller, K.A., and Stone, J.M., 1997, Astrophys. J. 489, 890 Mineshige S. and Wheeler J.C., 1989 Astrophys. J. 343, 241 Mineshige S., and Wood J.A., 1989, in Theory of Accretion Disks, eds. F. Meyer, W. Duschl, J. Frank and E. Meyer-Hofmeister, Kluwer, Dordrecht, p.221 Nakamura K.E., Matsumoto R., Kusunose M., Kato S., 1996, Publ. Astr. Soc. Japan 48, 769 Narayan R., Goldreich P., Goodman,J., 1987, Mon. Not. R. Astron. Soc. 228, 1 Narayan R. and Popham R., 1993, in Theory of Accretion Disks, II, eds. W. Duschl et al., Kluwer Dordrecht, p293 Narayan R., Yi I., 1994, Astrophys. J. 428, L13 Narayan R., Yi I., Mahadevan R., 1995, Nature 374, 623 Narayan R., Kato S., Honma F., 1997, Astrophys. J. 476, 49 O’Donoghue D.O., Chen A., Marang F., Mittaz P.D., Winkler H., Warner B. 1991, Mon. Not. R. Astron. Soc. 250, 363 Ogilvie G.I., 1999, Mon. Not. R. Astron. Soc. 306, L9 Osaki Y., 1974, Publ. Astr. Soc. Japan 26, 429 Ostriker E.C., Gammie C.F., Stone J.M., 1999, Astrophys. J. 513, 259 Osaki Y., 1993 in Theory of Accretion Disks, II, eds. W. Duschl et al., Kluwer Dordrecht, p93 Paczyński B., 1978, Acta Astron. 28, 91 Paczyński B., 1991, Astrophys. J. 370, 597 Piran T., 1978, Astrophys. J. 221, 652 Podolak M., Hubbard W.B. Pollack J.B., 1993, in Protostars and Planets II, University of Arizona Press, p.1109 Popham R. and Narayan R., 1991, Astrophys. J. 370, 604 Popham R., Woosley S.E., Fryer C., 1999, Astrophys. J. 518, 356

246

Accretion Disks

Popham R., 1997, in Accretion Phenomena and related Outflows (IAU Colloquium 163), ASP Conference Series 121, ed. D. T. Wickramasinghe; G. V. Bicknell; and L. Ferrario, p.230 Pringle J.E., 1981 Ann. Rev. Astron. Astrophys. 19, 137 Pringle J.E. and Savonije, G.J., 1979, Mon. Not. R. Astron. Soc. 187, 777 Pudritz R., 1981, Mon. Not. R. Astron. Soc. 195, 881 Quataert E., Narayan R., 1999 Astrophys. J. 517, 101 Rappaport S., Podsiadlowski P., 1999, astro-ph/9906045 Rappaport, S.A., Fregeau, J.M., Spruit, H.C., 2004, Astrophys. J. 606, 436 Ró˙zyczka M. and Spruit H.C., 1993, Astrophys. J. 417, 677 (with video) Rees M.J., 1978, Physica Scripta 17, 193 Rees M.J., Phinney E.S., Begelman M.C., Blandford R.D., 1982, Nature 295, 17 Regev, O., Hougerat, A. 1988, Mon. Not. R. Astron. Soc. 232, 81 Rybicki G.R. and Lightman A.P., 1979, Radiative Processes in Astrophysics, Wiley, New York, Ch 1.5 Rutten R., van Paradijs J., Tinbergen J., 1992, Astron. Astrophys. 260, 213 Sakimoto P., Coroniti F.V., 1989, Astrophys. J. 342, 49 Sawada K., Matsuda T., Inoue M. & Hachisu 1987, Mon. Not. R. Astron. Soc. 224, 307 Schandl S., Meyer F., 1994, Astron. Astrophys. 289, 149 Schandl S., Meyer F., 1997, Astron. Astrophys. 321, 245 Schultz A.L., Price R.H., 1985, Astrophys. J. 291, 1 Shakura N.I. and Sunyaev R.A., 1973, Astron. Astrophys. 24, 337 Sunyaev R.A., Shakura N.I., 1975, Pi’sma Astron. Zh. 3, 216 (translation Soviet Astron. L. 3, 114) Shapiro S.L., Lightman A.P., Eardley D.M., 1976 Astrophys. J. 204, 187 Smak J., 1971, Acta Astron. 21, 15 Smak J., 1984, Publ. Astr. Soc. Pac. 96, 54 Spitzer L., 1965, Physics of fully ionized gases, Interscience Tracts on Physics and Astronomy, New York: Interscience Publication, 1965, 2nd rev. Spruit H.C., 1987, Astron. Astrophys. 184, 173 Spruit H.C. 1996, in Evolutionary Processes in Binary Stars, R.A.M.J. Wijers et al. (eds.), NATO ASI C 477, Kluwer Dordrecht, p249 Spruit H.C., Rutten R.G.M., 1998, Mon. Not. R. Astron. Soc. 299, 768 Spruit H.C., Matsuda T., Inoue M., Sawada K., 1987, Mon. Not. R. Astron. Soc. 229, 517 Spruit H.C., Taam R.E., 1993, Astrophys. J. 402, 593 Spruit H.C., 1997, in Accretion disks-new aspects, eds. E. Meyer-Hofmeister and H.C. Spruit, Lecture Notes in Physics 487, Springer, p. 67 Spruit, H.C., & Deufel, B. 2002, Astron. Astrophys. , 387, 918 Strom S.E., Edwards S., Skrutskie M.F., 1993, in Protostars and Planets III, eds. E.H. Levy, J.I. Lunine, Univ. Arizona Press, Tucson, p.837 Sunyaev R.A., Shakura N.I., 1977, PAZh 3, 262 it Soviet Astron. Letters 3,138) Sunyaev, R. A., & Titarchuk, L. G. 1980, Astron. Astrophys. , 86, 121 Taam R.E., 1994, in Compact stars in Binaries (IAU Symp 165), J. van Paradijs et al. eds., Kluwer, p.3 Taam, R. E., & Sandquist, E. L. 2000, Ann. Rev. Astron. Astrophys. , 38, 113 Treves A., Maraschi L., and Abramowicz M., 1988 Publ. Astr. Soc. Pac. 100, 427 Tritton D.J., 1993, J. Fluid Mech. 241, 503 Turner, N. J. 2004, Astrophys. J. Letters , 605, L45 van Paradijs J. and McClintock J.E., 1995, in Lewin, W.H.G., van Paradijs, J. and van den Heuvel, E.P.J., eds., X-ray Binaries, Cambridge Univ. Press, Cambridge, p.58

Outflows?

247

van Paradijs J, Verbunt F. 1984, in S.E. Woosley, ed., High Energy Transients in Astrophysics, AIP Conference Procs. 115, p.49 Velikhov, 1959, J. Expl. Theoret. Phys. (USSR), 36, 1398 Verbunt F., 1982, Space Sci. Rev. 32, 379 Wang J.M., Zhou Y.Y., 1999 Astrophys. J. 516, 420 Wardle M., and Königl A., 1993, Astrophys. J. 410, 218 Warner B. 1987, Mon. Not. R. Astron. Soc. 227, 23 Warner B. 1995, Cataclysmic Variable Stars, Cambridge: CUP Wood J.A., Horne K., Berriman G, Wade R.A. 1989a Astrophys. J. 341, 974 Wood J.A., Marsh T.R., Robinson E.L. et al., 1989b, Mon. Not. R. Astron. Soc. 239, 809 Wood J.H., Horne K., Vennes S., 1992, Astrophys. J. 385, 294 Yi I., 1999, in Astrophysical Discs, eds. J.A. Sellwood and J. Goodman, Astronomical Society of the Pacific Conference series 160, 279. (astro-ph/9905215) Zdziarski A.A., 1998, Mon. Not. R. Astron. Soc. 296, L51 Zel’dovich Ya. B., 1981, Proc. Roy. Soc. London A374, 299

ASTROPHYSICS WITH LOBSTER TELESCOPES Rene Hudec Astronomical Institute of the Academy of Sciences of the Czech Republic, Fricova 298, CZ–251 65 Ondrejov, Czech Republic [email protected]

Libor Sveda and Ladislav Pina Czech Technical University, Faculty of Nuclear Sciences and Physical Engineering, V Holesovickach 2, CZ–180 00 Praha 8, Czech Republic

Adolf Inneman Centre for Advanced X-ray Technologies, Reflex, Novodvorska 994, 142 00 Praha 4, Czech Republic

Abstract

We refer on novel X–ray telescopes with high sensitivity as well as large field of view. The novel telescopes will monitor the sky with unprecedented sensitivity and angular resolution of order of 1 arcmin. They are expected to contribute essentialy to study and to understand various astrophysical objects such as AGN, SNe, GRBs, X–ray flashes, galactic binary sources, stars, CVs, X-ray novae, various transient sources, etc. We further refer on the project of a Lobster X– ray All Sky Monitor (ASM). The related scientific issues are also presented and discussed.

Keywords:

telescopes – methods:observational – X-rays:general

1.

Introduction

Wide field X-ray telescopes with imaging optics are expected to represent an important tool in future space astronomy projects, especially those for deep monitoring and surveys in X-rays over a wide energy range. The Lobster– Eye wide field X-ray optics has been suggested in 70ies by Schmidt ( [7], orthogonal stacks of reflectors) and by Angel ( [8], array of square cells) but has not been constructed until recently. This novel X-ray optics offers an excellent opportunity to achieve very wide fields of view (FOV, 1 000 square degrees 253 A. Baykal et al. (eds.), The Electromagnetic Spectrum of Neutron Stars, 253–256. © 2005 Springer. Printed in the Netherlands.

254

Astrophysics with Lobster Telescopes

and more) while the widely used classical Wolter grazing incidence mirrors are limited to roughly 1 deg FOV.

2.

Lobster Eye X–ray Telescopes

The Lobster Eye Wide Field X–ray telescopes in Schmidt arrangements are based on perpendicular arrays of double–sided X–ray reflecting flats. In the first prototypes developed and tested, double–sided reflecting flats produced by epoxy sandwich technology as well as gold coated glass foils have been used [2]. Recently, micro Schmidt lobster eye arrays with foils thickness as low as 30 microns have been developed and tested in order to confirm the capability of these systems to achieve fine angular resolutions of order of a few arcmin. The thin foils are separated by 70 microns gaps in these prototypes. On the other hand, large lobster eye systems with Schmidt geometry have been designed and constructed, achieving dimensions up to 300 × 300 × 600 mm. Their optical tests have confirmed the expected performance according to calculations (computer ray–tracing). The calculations and the measurement results indicate that the lobster eye telescope based on multiarray of modules with thin and closely spaced glass foils (analogous to those already assembled and tested) can meet the requirements e.g. of the ESA ISS Lobster mission (including the angular resolution and with better transmission) and can hence represent an alternative to the recently suggested MCP technique [1]. For the Angel geometry, numerous square cells of very small size (about 1 × 1 mm or less at lengths of order of tens of mm, i.e. with the size/length ratio of 30 and more) are to be produced. This demand can be also solved by modified innovative replication technology. Test modules with LE Angel cells have been succesfully produced. Linear test module has 47 cells 2.5×2.5 mm, 120 mm long (i.e. size/length ratio of almost 50), surface microroughness 0.8 nm, f = 1300 mm. Another test module is represented by a L–shaped array of 2 × 18 = 36 cells of analogous dimension. The surface microroughness of the replicated reflecting surfaces is better than 1 nm.

3.

The Lobster All Sky Monitor

We propose a focusing wide–field X–ray optics for the All-Sky Monitor (ASM), such as a Multi–Foil optic (MFO) [4][5][6] in Schmidt [7] arrangement (Figure 1). Multi–foil optic (MFO) in general is an X–ray optic based on the thin glass foils. The dimensions of the plates can be optimized for a given focal length, plate spacing, photon energy, and surface quality. The spacing between the reflecting plates changes the angular resolution substantially. If a wide energy range response is needed, the gain in the whole energy range should be optimized. We have simulated various LE MFO samples designed for various energies. The most interesting design suitable for an orbital scanning experi-

Scientific objectives

Figure 1. Schematic view of the MFO sample in Schmidt arrangement. Two perpendicular sets of reflecting surfaces are clearly visible

255

Figure 2. Example of LE module assembly strategy. The modules are divided into three groups each covering 60 × 6 deg. Total FOV of 180 × 6 deg is covered.

ment, based on current simulations, seems to be: 78.0 × 11.5 × 0.1 mm3 gold coated plates, 0.3 mm spacing between plates, and focal length f = 375 mm. Field of view of such an optic is approximately 6 × 6 deg (FWHM). The optic, together with the planar detector 4 × 4 cm2 large with 150 × 150 µm2 pixel size, necessary electronic, and the casing will create a single LE module. A number of modules suitably arranged will create a whole ASM. An example of such an arrangement is plotted in Figure 2. The ASM built from the modules will have a Field of View (FOV) 180 × 6 deg2 . It will scan the sky once per orbit (∼ 90 min) and hence will cover the whole sky several times per day. The limiting detectable flux after a one–day operations depends on the position on the sky relative to the orbit and can be ∼ 10−12 erg/s/cm2 near the orbit and can reach ∼ 4 × 10−13 erg/s/cm2 near the orbit poles. The angular resolution will be ∼ 3 − 4 arc min.

4.

Scientific objectives

Soft X–Ray ASM scientific targets can be obviously divided into two work modes. The first work mode is the fast discovery of new X–Ray sources and/or discovery of sudden flux changes of known sources. Prompt emission study, precise positioning, and alert system for narrow field instruments will be an important output in this work mode. Gamma Ray Bursts study (20 − 60 triggers per year), X-Ray flashes (> 8 triggers per year), supernovae prompt X–Ray emission (10 − 20 triggers per year), X–Ray binaries and cataclysmic variables sudden flux/spectra changes, and stellar events at the nearby stars will be observable targets. All these sources are relatively bright and are above the daily detection limit, hence can be detected much faster. The second work mode is the long–term monitoring of large number of X– Ray sources with sampling rate from hours to days (depending on the actual source flux). Light curves and rough spectra will be gathered during the whole

256

Astrophysics with Lobster Telescopes

mission lifetime for X–Ray binaries (∼ 700 in the Milky Way galaxy), cataclysmic variables (∼ 200), nearby stars (∼ 600), and AGN (∼ 4000).

5.

Conclusions

Results of analyses and simulations of lobster–eye X–ray telescopes have indicated that they will be able to monitor the X–ray sky at an unprecedented level of sensitivity, an order of magnitude better than any previous X–ray all– sky monitor. Limits as faint as 10−12 erg cm−2 s−1 for daily scanning observation as well as the angular resolution < 4 arcmin in soft X–ray range are expected to be achieved allowing monitoring of all classes of X–ray sources, not only X–ray binaries, but also fainter classes such as AGNs, coronal sources, cataclysmic variables, as well as fast X–ray transients including gamma–ray bursts and the nearby type II supernovae.

Acknowledgments We acknowledge the support from the Ministry of Industry and Trade of the Czech Republic, projects FB–C3/29/00 and FD–K3/052.

References [1] Fraser, G. W. et al.: LOBSTER–ISS: an imaging X– ray all–sky monitor for the International Space Station, Proc. SPIE, 2002, Vol. 4497, p. 115–126 [2] Inneman A., et al., 1999, SPIE Vol. 3766, 72. [3] Schmidt, W. K. H.: a proposed X– ray focusing device with wide field of view for use in X–ray astronomy, 1975, NucIM, vol. 127, p. 285–292 [4] Hudec, R. et al., Wide-field x-ray optics, Proc. SPIE 4012, p. 432.–441., 2000 [5] Hudec, R. et al., Lightweight x-ray optics for future space missions, Proc. SPIE 4851, p. 578– 586, 2003 [6] Inneman, A. et al., Progress in lobster-eye x-ray optics development, Proc. SPIE 4138, p. 94– 104, 2000 [7] Schmidt, W. K. H., A proposed X–ray focusing device with wide field of view for use in X–ray astronomy, NucIM 127, p. 285-292, 1975 [8] Angel J. R. P. Astroph. J. 364, 233 (1979).

FORMATION AND EVOLUTION OF NEUTRON STARS IN BINARY SYSTEMS E.P.J. van den Heuvel Astronomical Institute “Anton Pannekoek” and Center for High Energy Astrophysics, University of Amsterdam, The Netherlands [email protected]

Keywords:

1.

stars:neutron – binaries:symbiotic – stars:formation – stars:evolution

Introduction and Summary

A large part of this lecture is similar to the lecture “Formation and Evolution of Neutron Stars and Black Holes in Binary Systems” presented in the 1999 NATO Advanced Study Institute “The Neutron Star – Black Hole Connection” (Kouveliotou, Ventura and van den Heuvel 2001) and the reader is referred the to the corresponding paper in the Proceedings of that Advanced Study Institute (van den Heuvel 2001). Here I just mention some important new developments in this field that happened since 1999. This concerns in particular the discovery of a separate class of low kick velocity neutron stars and the implications of the existence of this new class for the formation of double neutron stars. It is pointed out that five of the seven known double neutron stars in the galactic disk have low orbital eccentricities, indicating that the secondborn neutron stars in these systems received hardly any velocity kicks at their birth. The masses of the second-born neutron stars in these systems are quite low (between 1.18 and 1.30 M : on average 1.24 M ). It is suggested that these low-mass, low-kick neutron stars were formed by the electron-capture collapse of the degenerate O-Ne-Mg cores of helium stars of relatively low mass (< 3.5 M ), whereas the higher -mass, higher-kick-velocity neutron stars were formed by the collapse of the iron cores of stars of higher initial mass. This “no-kick” model for the second-born neutron stars in the double neutron star systems can also explain – at least qualitatively – the correlation between the orbital eccentricities and the spin periods of the recycled components of the double neutron stars discovered by Faulkner et al. (2004).


192

2.

Formation and Evolution of Neutron Stars in Binary Systems

The bimodal distribution of the birth kick velocities of neutron stars

Pfahl et al. (2002) discovered the existence of a separate class of B-emission X-ray binaries (abbreviated here as Be/X-ray binaries) with wide orbits of low eccentricity (< 0.25). The systems in this class tend to have relatively low Xray luminosities (< 1034 ergs/s). A well-known example is X-Per, in which the neutron star has an almost circular orbit with a period of 250 days. About half of all Be/X-ray binaries with known orbits appear to belong to this class and the relatively low X-ray luminosities of these sources imply that these systems are on average considerably nearer to us than the high-eccentricity Be/X-ray binaries (which during outbursts can reach a luminosity of 1038 ergs/s). Therefore, as Pfahl et al. (2002) pointed out, the systems in the low-eccentricity class probably form the bulk of the Be/X-ray binary population, since the known numbers of sources in both classes are about the same. These authors pointed out that the neutron stars in the low-eccentricity systems cannot have received a kick velocity at their birth exceeding 50 km/s. Until the discovery of this class of X-ray binaries it was generally thought that all neutron stars receive a high kick velocity at their birth, of order at least a few hundred km/s (see e.g.: Lyne and Lorimer 1994; Hansen and Phinney 1997). Often a Maxwellian distribution is used to represent the observed distribution of pulsar velocities, and the characteristic velocity of these Maxwellians is typically around 200 – 300 km/s. Hansen and Phinney (1997) pointed out that from their analysis there is hardly any room for low-velocity neutron stars, while there are several neutron stars with space velocities exceeding 800 km/s. On the other hand, other authors, such as Hartmann (1997) argued that there is room for a few tens of per cents of low-velocity pulsars and Cordes and Chernoff (1998) argued that the pulsar velocity distribution is bi-modal, with one peak representing a lower-velocity population. Pfahl et al. (2002) showed, by means of population synthesis calculations that include the evolution of binaries and the presence of birth kicks imparted to neutron stars, that with the assumption of only one Maxwellian with a high characteristic velocity (100 to 200 km/s) one can reproduce the higheccentricity population of the Be/X-ray binaries, but one totally fails to reproduce the presence of a large population of systems with low eccentricities. They convincingly showed that the only way in which both the observed highe and the low-e populations of the Be/X-ray binaries can be reproduced is: by assuming that there are two distinct populations of neutron stars: one population that receives hardly any kick velocity at birth (vk < 50 km/s) and another which receives the “canonical” high velocity kick of order 200 km/s or more at birth.

Double neutron stars and the low kick velocity neutron star population

3.

193

Double neutron stars and the low kick velocity neutron star population

At present 8 double neutron stars are known, 7 of them in the galactic disk and one in a globular cluster (see Stairs 2004). The seven systems in the galactic disk are listed in table 1. As the table shows, the double neutron stars tend to have very narrow orbits. They are the later evolutionary products of wide high-mass X-ray binary systems with orbital periods > 100 days (van den Heuvel and Taam 1984; cf. van den Heuvel 2001), mostly B-emission X-ray binaries. When the massive star in such a system has expanded to become a red giant, its envelope engulfs the neutron star, causing this star to spiral down into this envelope, reducing its orbital separation by several orders of magnitude. The large energy release due to friction and accretion during this spiral-in process is expected to cause the hydrogen-rich envelope of the giant to be expelled such that a very close binary remains, consisting of the helium core of the giant together with the neutron star (van den Heuvel and Taam 1984; Dewi and Pols 2003). (Depending on the orbital separation at the onset of spiral in, the helium core itself may already be (somewhat) evolved and possibly contain already some C and O in its core). [In Be/X-ray systems that started out with orbital periods < 100 days the neutron stars spirals in so deeply that it most probably merges with the core of the giant, and so no binary will be left; e.g. see Taam 1996]. Due to the large frictional and tidal effects during spiral in the orbit of the system is expected to be perfectly circular. The helium star generates its luminosity by helium burning, which produces C and O, and subsequently by carbon burning, producing Ne and Mg. If the helium star has a mass in the range 1.6 to 3.5 M (corresponding to a main-sequence progenitor in the range 8 to 12 – 14 M , the precise limits of this mass range depending on metallicity and on the assumed model for convective energy transport (Sugimoto and Nomoto 1980, Mijaji et al. 1980, Podsiadlowski et al. 2004)) it will during carbon burning develop a degenerate O-Ne-Mg core, surrounded by episodic C-and He-burning shells (e.g. Nomoto et al. 1982ab, Habets 1986ab). When such a degenerate core develops, the envelope of the helium star begins to expand, causing in a binary the onset of mass transfer by Roche-lobe overflow (Habets 1986ab; Dewi and Pols 2003). Roche-lobe overflow leads to the formation of an accretion disk around the neutron star and accretion of matter with angular momentum from this disk will cause the spin frequency of the neutron star to increase. Therefore one expects that during the later evolution of these helium stars of relatively low mass the first-born neutron star in the system will be “spun up” to a short spin period. This neutron star had already a long history of accretion: first when it was in a wide binary with an early-type (presumably Be) companion; subsequently during the spiral-in phase into the envelope of its companion and now as com-

194


panion of a Roche-lobe overflowing helium star. Since all binary pulsars which had a history of mass accretion (so-called “recycled” pulsars (Radhakrishnan and Srinivasan 1982)) tend to have much weaker magnetic fields than normal single pulsars, it is thought that accretion in some way causes a weakening of the surface dipole magnetic field of neutron stars (Taam and van den Heuvel 1986) and several theories have been put forward to explain this accretioninduced field decay (Bisnovatyi-Kogan and Komberg 1975; see Bhattacharya and Srinivasan 1995 for a review and Cumming 2005). With a field weakened to about 1010 Gauss (as observed in the recycled components of the double neutron stars (see table 1)), and an Eddington-limited accretion rate of helium (rate 4.10−8 M /yr) a neutron star can be spun-up to a shortest possible spin period of a few tens of milliseconds (Smarr and Blandford 1976, Srinivasan and van den Heuvel 1982). When the helium star finally explodes as a supernova, the second neutron star in the system is born. This is a newborn neutron star without a history of accretion and is therefore expected to resemble the “normal” strong-magnetic field single radio pulsars (Srinivasan and van den Heuvel 1982), which have typical surface dipole magnetic fields strengths of 1012 – 1013 Gauss. This theoretical expectation has been beautifully confirmed by the discovery of the double pulsar systems PSRJ0737-3039AB, which consists of a recycled pulsar (star A) with a very rapid spin (P = 23 ms) and a weak magnetic field (7.109 G) and a normal strong-magnetic-field (6.1012 G) pulsar (star B) with a “normal” pulse period of 2.8 sec (Burgay et al, 2003, Lyne et al. 2004; see table 1). The explosive mass loss in the second supernova has made the orbit eccentric and since the two neutron stars are basically point masses, tidal effects in double neutron star systems will be negligible and there will be no tidal circularization of the orbit. (On timescales of tens of millions of years the orbits may be circularize by a few percent due to the emission of gravitational waves in the shortest-period system of PSRJ0737-3039, but in all the other double neutron stars this is a negligible effect, except in the final stages of spiraling together (see e.g. Shapiro and Teukolsky 1983)). In case of spherically symmetric mass ejection in the supernova there is a simple relation between the orbital eccentricity and the amount of mass ∆Msn ejected in the supernova: (1) e = ∆Msn /(Mns1 + Mns2 ) where Mns1 and Mns2 are the masses of the first- and the second-born neutron stars. The “conventional” kick velocities of neutron stars of about 200 – 400 km/s (Hansen and Phinney 1987) are quite similar to the orbital velocities of the neutron stars in close double neutron stars such as the Hulse-Taylor binary pulsar PSRB1913+16 (Porb = 7.75 hours). Therefore, a kick velocity of this order produces a major disturbance of the orbit and – unless it is imparted in a very specific direction – will in general impart a large eccentricity to the orbit, of order 0.5 or more. Indeed, the Hulse-Taylor binary pulsar has a large

Masses of the 2nd-born NSs in the double NS sys. & in PSRJ1145-6545

195

eccentricity e = 0.617 and the same is true for the system PSRJ1811-1736 (e = 0.828). However, as table 1 shows, very surprisingly all of the other 5 double neutron stars in the galactic disk have very small orbital eccentricities, in the range 0.088 to 0.27. Such eccentricities are the ones which one expects from the pure sudden mass loss effects in the supernova explosion, given by equation (1), but not in case a randomly directed kick velocity of order 200 – 400 km/s is imparted to the second-born neutron star at birth. [In particular, the small orbital eccentricities of the two relatively wide double neutron stars PSRJ1518+4909 and PSRJ1829+2456 are impossible to reconcile with high kick velocities imparted at birth]. It thus appears that the second-born neutron stars in these 5 systems belong to the same “kick-less” class as the neutron stars in the low-eccentricity class of Be/X-ray binaries (van den Heuvel 2004). The same holds for the young strong-magnetic-field pulsar in the eccentric radio-pulsar binary PSRJ1145-6545 which has a massive white dwarf as a companion (Bailes 2005). The orbital eccentricity of 0.172 of this binary shows that the neutron star was the last-born object in the system (Tauris and Sennels 2000; formation of a white dwarf cannot introduce an orbital eccentricity). The low value of its eccentricity would be hard to understand if the neutron star received the canonical 200 – 400 km/s kick at its birth.

4.

The masses of the second-born neutron stars in the double neutron star systems and in PSRJ1145-6545

In the eccentric white-dwarf/neutron-star system of PSRJ1145-6545 the mass of the neutron star is known from the measurement if relativistic effects to be 1.28(2) M (Bailes 2005) (the number within parentheses indicates the 95% confidence uncertainty; the total mass of the system is 2.30 M and the mass of the white dwarf is at least one solar mass). Also in two of the low-eccentricity double neutron stars the masses of both stars are accurately known from measured relativistic effects (see Stairs 2004): (i) in PSRJ0737-3039 the second-born neutron star has MB = 1.250(3) M and the first-born one has MA = 1.330(3) M (Lyne et al. 2004). (ii) in PSRJ1756-2251 the second-born neutron star has a mass of 1.18(3) M and the first-born one a mass of 1.40(3) M (Faulkner et al. 2004). In most of the other double neutron stars the masses of the stars are not yet accurately known, but in 3 of these the second-born neutron stars must be less massive than 1.30 M for the following reasons. In all double neutron star systems the relativistic parameter that can be measured most easily is the General Relativistic rate of periastron advance, which directly yields the sum of the masses of the two neutron stars (e.g. see Stairs 2004). In the systems of PSRJ1518+4904, PSRJ1811-1736 and PSRJ1829+2456 the resulting sum of the masses turns out to be 2.62, 2.60 and 2.53 M , respectively. The in-

196


Table 1. Double neutron star binaries and the eccentric-orbit white-dwarf neutron star system J1145-6545. Pulsar Name

Spin Per. (ms)

Porb e

Compan. Mass (M )

Pulsar Mass (M )

Sum of masses (M )

(1010 G)

J07373039A

22.7

0.10

0.088

1.250(5)

1.337(5)

2.588(3)

0.7

(1)

J07373039B

2770

0.10

0.088

1.337(5)

1.250(5)

2.588(3)

6.102

(1)

40.9

8.63

0.249

1.05 (+0.45) (-0.11)

1.56 (+0.13) (-0.45)

2.62(7)

0.1

(2)

B1534+ 12

37.9

0.42

0.274

1.3452(10)

1.3332(10)

2.678(1)

1

(3)

J17562251

28.5

0.32

0.18

1.18(3)

1.40(3)

2.574(3)

0.54

(4)

J18111736

104

18.8

0.828

1.11 (+0.53) (-0.15)

1.62 (+0.22) (-0.55)

2.60

1.3

(3)

J1829+ 2456

41.0

1.18

0.139

1.27 (+0.11) (-0.07)

1.30 (+0.05) (-0.05)

2.53(10)

∼1

(5)

B1913+ 16

59

0.33

0.617

1.3873(3)

1.4408(3)

2.8281(1)

2

(3)

J11456545

394

0.20

0.172

1.00(2)

1.28(2)

2.288(3)

102

(6)

J1518+ 4904

(d)

Bs Ref

References: (1) Lyne et al. (2004); (2) Nice et al. (1996); (3) Stairs (2004); (4) Faulkner et al. (2004); (5) Champion et al. (2004); (6) Bailes (2005).

dividual masses of the neutron stars in these systems are still rather poorly determined, but in all these systems the already crudely determined other relativistic parameters indicate that the second-born neutron star has the lowest mass of the two (see references in van den Heuvel 2004). As in all these systems the sum of the masses is around 2.60 M , the second-born neutron stars in these systems cannot be more massive than 1.30 M .

Formation mechanisms of neutron stars and possible resulting kicks

197

Thus we find that in these six systems the second-born neutron star has a low mass, in the range 1.18 to 1.30 M and belongs to the low-kick category. This is strong evidence that no (or a low) kick velocity is correlated with a low neutron star mass of around 1.24 (±0.06) M . A neutron star of 1.24 M corresponds to a pre-collapse mass of about 1.44 M , as during the collapse the gravitational binding energy of the neutron star of about 0.20 M (slightly depending on the assumed equation of state of neutronized matter) is lost in the form of neutrinos. So apparently the cores, which collapsed to these second-born neutron stars, had a mass very close to the Chandrasekhar mass.

5.

Formation mechanisms of neutron stars and possible resulting kicks

It is long known (Mijaji et al. 1980, Sugimoto and Nomoto 1980) that there are two basically different ways in which neutron stars are expected to form, i.e.: (i) In stars which originated in the main-sequence mass range between 8 and about 12 – 14 M , which in binaries produce helium stars in the mass range 1.6 to 3.5 M (Habets 1986ab, Dewi and Pols 2003), the O-Ne-Mg core which forms during carbon burning becomes degenerate and when its mass approaches the Chandrasekhar mass, electron captures on Mg and Ne cause the core to collapse to a neutron star. Since these stars did not reach Oxygenand Silicon burning, the baryonic mass of the neutron star, which forms in this way, is expected to be purely determined by the mass of the collapsing degenerate core, which is the Chandrasekhar mass. The gravitational mass of this neutron star is then the Chandrasekhar mass minus the gravitational binding energy of the neutron star, which is about 0.20 M . Thus a neutron star with a mass of about 1.24 M is expected to result. (ii) In stars initially more massive than 12 – 14 M , the O-Ne-Mg core does not become degenerate and these cores proceed through Oxygen and Silicon burning to form an iron core. When the mass of this iron core exceeds a critical value it collapses to form a neutron star. The precise way in which here neutrino transport during core bounce and shock formation results in a supernova explosion is not yet fully understood. It appears that first the shock stalls and then several hundreds of milliseconds later, is revitalized. Some fall back of matter from the layers surrounding the proto neutron star is expected to occur (see Fryer, this volume) such that the neutron star that forms may be substantially more massive than the mass of the collapsing Fe-core. In fact there are two expected mass regimes for the resulting neutron stars: for stars with initial main-sequence masses in the range 12 – 14 M to 19 M the collapsing cores are expected to be about 1.3 M , whereas for stars more massive than 19 M

198


the collapsing iron core is expected to have a mass > 1.8 M (Timmes et al. 1996), leading to the formation of neutron stars with (gravitational) masses > 1.6 M . Taking some fall-back of matter into account, the neutron stars formed from these types of iron cores may be expected to have gravitational masses > 1.3 M and > 1.7 M , respectively. The fact that the pre-collapse masses of the low-mass, low-kick neutron stars were very close to the Chandrasekhar limit suggests that these neutron stars are the result of the electron-capture collapse of the degenerate O-Ne-Mg cores of helium stars that originated in the mass range 1.6 to 3.5 M (initial main-sequence mass in the range 8 to 12 – 14 M ). Can one understand why such neutron stars would not receive a birth kick whereas those formed by the collapse of an iron core would? Burrows and Hayes (1996) suggest that the violent large-scale convective motions in the core during O- and Si-burning just prior to the formation of the Fe-core may produce considerable large-scale density inhomogeneities in the mantle of the proto-neutron star. They showed that this may lead to asymmetric neutrino transport and escape, which may easily impart enough momentum to the neutron star to produce a space velocity of 500 km/s. Recent 3-D numerical hydrodynamic core collapse and neutrino transport calculations by Scheck et al. (2004) confirm this expectation. As no O- and Si-burning occur prior to the e-capture collapse of a degenerate O-NeMg core, neutrino transport in this case may be close to spherically symmetric, leading to no (or a very small) kick velocity imparted during collapse.

6.

The eccentricity spin-period relation for the recycled components of double neutron stars

Faulkner et al. (2004) pointed out that there is a relation between the orbital eccentricities and the spin periods of the recycled components of the double neutron star systems. In terms of the above-described model, in which the second-born neutron stars in the systems received no velocity kick at birth, this relation can, at least qualitatively, be understood by using equation (1), as follows. According to equation (1) the orbital eccentricity e induced by the second Supernova explosion is determined by the amount of mass ∆Msn ejected in the formation of the second-born neutron star. The progenitor of this neutron star was a helium star. The lifetime of a helium star decreases with increasing stellar mass (roughly inversely proportional to the helium star mass to the power 2.5) and the same is true for the duration of the phase of Rochelobe overflow from the helium star, which produces the spin-up of the rotation of the recycled first-born neutron star. Hence, the more massive the helium star, the shorter the duration of the spin-up phase of the recycled neutron star. A shorter duration of the spin-up phase implies that this star cannot be spun-up to a very short period, thus: the longer the spin period of the recycled first-born

199

Conclusions

pulsar. But also: the larger the mass of the helium star, the larger the mass of this star will be at the time of the second Supernova explosion and thus: the larger the amount of mass ∆Msn ejected in the second supernova explosion (the helium star has a larger mass at the time of the explosion for two reasons: it had a larger mass to start with and it lost less mass by Roche-lobe overflow than a lower-mass helium star). Because of the larger amount of mass ejected in the supernova, the orbital eccentricity of the resulting double neutron star will be larger, in case of spherically symmetric mass ejection (no birth kick imparted to the neutron star). In view the above one expects, if the mass ejection in the second supernova explosion is symmetric (no kicks), that there will be a positive correlation between the orbital eccentricities and the spin periods of the recycled components of the double neutron stars, as observed. The observed e − P relation (Faulkner et al.2004) is rather “noisy”, the points spreading over quite a large range. The best-fit straight line through the 7 points gives – roughly – the linear relation: e = 0.01(P − 17)

(2)

where P is the pulse period of the recycled neutron star in milliseconds. Using equation (1) this relation gives: ∆Msn = 0.01(Mns1 + Mns2 )(P − 17)

(3)

Inserting the longest and the shortest pulse periods observed for the recycled components of double neutron stars: P = 104 for PSRJ1811-1736 and P = 23 for PSRJ0737-3039A, respectively, one obtains, using (Mns1 + Mns2 ) = 2.60 M : ∆Mns = 2.26 M and 0.16 M , respectively. With a neutron star mass of 1.30 M this would imply a mass of the progenitor helium stars of 3.56 M and 1.46 M , respectively. The calculations of Roche-lobe overflow of helium stars by Dewi and Pols (2003) show that the duration of Roche-lobe overflow of a 3.5 M helium star is very short, probably less than 5000 years, while the calculations of Lommen et al. (2005) for Roche lobe overflow from helium stars with a mass around 1.6 M show that here the Roche-lobe overflow lasts an order of magnitude or more longer. Therefore the low-mass helium star progenitor of PSRJ07373039B can have spun up PSRJ0737-3039A to a very short spin period, whereas the companion of PSRJ1811-1736 had hardly any time to spin up its first-born neutron star companion.

7.

Conclusions

The most important new development in the field of formation and evolution of neutron stars in binaries is the discovery by Pfahl et al. (2002) - thanks to

200


the study of Be/X-ray binaries - of the existence of two distinct populations of neutron stars, one population that does not receive a velocity kick at birth and another population that receives a quite large velocity kick at birth, on average of order several hundreds of km/s. The double neutron star systems show that the “low-kick” neutron stars have low masses, around 1.24 M , which is consistent with their formation by electron-capture collapse of a degenerate O-Ne-Mg core in stars that originated in the main-sequence mass range 8 to about 12 – 14 M (helium-star mass range 1.6 to 3.5 M ). It is suggested here that the “high-kick” neutron stars originate from more massive stars, in which the neutron star was formed by the iron-core collapse. This may result in two regimes of neutron star masses: if the initial main-sequence mass was < 19 M , a neutron star with a mass around 1.3 to 1.4 M will result, whereas for M > 19 M a neutron star of mass > 1.7 M may result or a black hole.

References Bailes, M. 2005, in “Binary radio Pulsars” (ed. F.A.Rasio and I.H.Stairs), ASP Conf.Series 328, 33. Bhattacharya, D. and Srinivasan, G., 1995, in : “X-ray Binaries” (ed. W.H.G.Lewin, J.A.van Paradijs and E.P.J.van den Heuvel), (Cambridge, Cambridge Univ. Press) 495. Bisnovatyi-Kogan, G.S. and Komberg, B.V., 1975, Soviet Astron. 18, 217. Burgay, M., D’Amico, N., Possenti, A., et al. 2003, Nature 426, 531. Burrows, A. and Hayes, J., 1996, Phys.Rev.Letters, 76, 352. Cordes, J. and Chernoff, 1998, Ap.J. 505, 315. Cumming, A., 2005, in: “Binary Pulsars” (editors: F.A.Rasio and I.H.Stairs), ASP Conf. Series, Vol.328, 311. Dewi, J.D.M and Pols, O.R. 2003, MNRAS 344, 629. Faulkner, A.J., Kramer, M., Lyne, A.G., Manchester, R.N., McLaughlin, M.A., Stairs, I.H., Possenti, A., Lorimer, D.R., D’Amico, N., Camilo, F., Burgay, M.,2004, Astro-Ph/0411796v1. Habets, G.M.H.J., 1986a, Astron.Ap. 165, 95. Habets, G.M.H.J., 1986b, Astron.Ap. 167, 61. Hansen, B.M.S. and Phinney, E.S. 1997, MNRAS 291, 569. Hartmann, J.W. 1997, Astron.Ap. 322, 127. Kouveliotou, C., Ventura, J. and van den Heuvel, E.P.J. (editors), 2001, “The Neutron Star Black Hole Connection”, NATO Science Series Vol.567, Kluwer Acad. Publ., Dordrecht, 522pp. Lommen, D., Yungelson, L., van den Heuvel, E.P.J., Nelemans, G. and Portegies Zwart, S., 2005, Astron.Ap. (submitted). Lyne, A.G. and Lorimer, D.R., 1994, Nature 369, 127. Lyne, A.G. et al., 2004, Science, 303, 1153. Mijaji, S., Nomoto, K., Yokoi, K., Sugimoto, D., 1980, P.A.S.J. 32, 303. Nomoto, K., 1982a, Ap.J. 253, 798. Nomoto, K.,1982b, Ap.J. 257, 780. Pfahl, E, Rappaport, S., Podsiadlowski, P., and Spruit, H. 2002, Ap.J. 574, 364. Podsiadlowski, P., Langer, N., Poelarends, A.J.T., Rappaport, S., Heger, A. and Pfahl, E., 2004, Ap.J. 612, 1044. Radhakrishnan, V. and Srinivasan, G., 1982, Current Science 51, 1096.

Conclusions

201

Scheck, L., Plewa, T., Janka, H.-T., Mueller, E., 2004, Phys.Rev.Letters 92(1),Jan. Shapiro, S.L. and Teukolsky, S.A., 1983, “Black Holes, White Dwarfs and Neutron Stars”, (New York, Wiley-Interscience), 645pp. Smarr, L.L., and Blandford, R.D., 1976, Ap.J. 207, 574. Srinivasan, G. and van den Heuvel, E.P.J., 1982, Astron.Ap. 108, 143. Stairs, I.H., 2004, Science 304, 547. Sugimoto, D. and Nomoto, K. 1980, Space Sci. Rev. 25, 155. Taam, R.E. 1996, in “Compact Stars in Binaries” (editors J.van Paradijs, E.P.J.van den Heuvel and E.Kuulkers), Proc.IAU Symp. 165, Kluwer Acad. Publ., Dordrecht, 3-15. Taam, R.E. and van den Heuvel, E.P.J. 1986, Ap.J. 305, 235. Tauris, T. M., and Sennels, T., 2000, Astron.Ap. 355, 236. Timmes, F.X., Woosley, S.E., Weaver, T.A. 1996, Ap.J. 457, 834. Van den Heuvel, E.P.J. 2001 in: “The Neutron Star Black Hole Connection” (eds. C.Kouveliotou, J.Ventura and E.P.J.van den Heuvel), Kluwer Acad. Publ., Dordrecht, 173-243. Van den Heuvel, E.P.J., 2004, in: Proc. 5th INTEGRAL Workshop, (eds. V.Schoenfelder, G.Lichti and C.Winkler), ESA SP-552, (Noordwijk, ESA Publ. Div. ESTEC), 185-194. Van den Heuvel, E.P.J. and Taam, R.E., 1984, Nature 309, 235.

HIGH MASS X-RAY BINARIES IN LMC P. Shtykovskiy Space Research Institute, Russian Academy of Sciences, Profsoyuznaya 84/32, 117997 Moscow, Russia pav [email protected]

M. Gilfanov Max-Planck-Institute für Astrophysik, Karl-Schwarzschild-Str. 1, D-85740 Garching bei Muenchen, Germany Space Research Institute, Russian Academy of Sciences, Profsoyuznaya 84/32, 117997 Moscow, Russia [email protected]

Abstract

We study population of compact X-ray sources in the Large Magellanic Cloud using the archival data of XMM-Newton observatory. Based on the properties of the optical and near-infrared counterparts of the detected sources we identified HMXB candidates in LMC. The bright end of this distribution in general is consistent within statistical and systematic uncertainties with extrapolation of the universal luminosity function of HMXBs. However, there seems to be fewer low luminosity sources, log(LX ) < 35.5, than predicted. We consider the impact of the “propeller effect” on the HMXB luminosity distribution and show that it can qualitatively explain the observed deficit of low luminosity sources.

Keywords:

X-rays: galaxies – X-rays: binaries – stars: neutron – galaxies: individual: LMC

Introduction As has been shown by Grimm et al. (2003), the X-ray luminosity function (XLF) of HMXBs obeys, to the first approximation, the universal power law distribution with the differential slope of ≈ 1.6, whose normalization is proportional to the star formation rate of the host galaxy. Validity of this universal HMXB XLF has been established in the luminosity range log(LX ) > 35.5 − 36. LMC owing to its proximity offers unique possibility to study its lowluminosity part. Indeed, the sensitivity of a typical Chandra or XMM/Newton observation, ∼ 10−14 erg/s/cm2 corresponds to the luminosity of ∼ 3 · 1033 erg/s at the LMC distance. 249 A. Baykal et al. (eds.), The Electromagnetic Spectrum of Neutron Stars, 249–252. © 2005 Springer. Printed in the Netherlands.

250

High mass X-ray binaries in LMC

Figure 1. The incompleteness-corrected XLF of HMXB candidates in LMC (left) and the log(N)-log(S) distribution of X-ray sources in the LMC field (right). The upper histogram on the left figure shows sources of uncertain nature together with likely HMXB candidates, the lower one shows only likely HMXB candidates. The upper grey line and shaded area on the same figure show the luminosity distribution predicted from the “universal” XLF of Grimm et al. (2003) extrapolated towards low luminosities and its uncertainty. The lower solid and dashed lines show the same XLF modified by the “propeller effect”(section 23.3) assuming black hole fraction 0% and 30% respectively. The different histograms on the right figure show distribution of all sources, high mass X-ray binaries and CXB sources.

1.

Nature of X-ray sources in the field of LMC

We have selected 23 XMM-Newton archival observations with the pointing direction towards LMC and performed source detection procedure with standard SAS tasks in 2–8 keV energy band. The 2–8 keV source counts were converted to the 2–10 keV energy flux assuming a power law spectrum with the photon index 1.7 and NH =6·1020 cm−2 . The total corrected for incompleteness number of sources with the flux FX [2 − 10 kev] > 3.34 · 10−14 erg/s/cm2 equals 214. From the comparison of this number with number of CXB sources NCXB (> S) ≈ 218 (Moretti et al., 2003) it is obvious that the majority of the detected sources are background AGNs (see Fig. 1). To filter out contaminating background and foreground sources, we use the fact, that optical emission from HMXBs is dominated by the OB optical companion, whose properties, such as absolute magnitudes and intrinsic colors are sufficiently well known. Combining positions of possible optical counterparts on the Hertzsprung-Russel diagram with the distance modulus of LMC and reddening towards LMC, we created a number of selection criteria for search of HMXB candidates. We applied these criteria to optical counterparts of X-ray

251

Propeller effect and HMXBs XLF

sources in the field of LMC (obtained from optical and near-infrared catalogs USNO-B (Monet et al., 2003), GSC2.2.1 (Morrison & McLean, 2001), the CCD survey of the Magellanic Clouds (Massey et al., 2002), 2MASS (Cutri et al., 2003)) and identified a number of HMXB candidates.

2.

The luminosity function of HMXB candidates in LMC

The incompleteness-corrected luminosity distribution of HMXB candidates is shown in Fig. 1. The upper and lower histograms correspond to sources of uncertain nature plus likely HMXB candidates and to the likely HMXB candidates only respectively. These two histograms provide upper and lower limits for the true X-ray luminosity function of HMXBs in the observed part of LMC. +0.26 , the norWe obtain best fit value for the differential slope α = 1.28−0.23 malization corresponds to N (> 1035 erg/s) ≈ 5 HMXBs. As is evident from Fig.1, the slope of the luminosity distribution appears to be somewhat flatter and its normalization smaller than predicted from extrapolation of the “universal” HMXB luminosity function of Grimm et al. (2003) – α ≈ 1.6 and N (> 1035 erg/s/cm2 ) ≈ 11 ± 5. However, the XLF flattening is not statistically significant in the LX ≥ 2.5 · 1034 erg/s luminosity range – the Kolmogorov-Smirnov test do not allows us to reject universal model. Apart from effects of statistics the low-luminosity flattening of the luminosity distribution should be expected due to the “propeller effect”, we consider it in the following section.

3.

Propeller effect and HMXBs XLF

As suggested by Illarionov & Sunyaev (1975), at low mass accretion rate, the spin frequency of the neutron star can exceed the Keplerian frequency at the magnetospheric radius. In this case, corresponding to ω = Ω∗ /ΩK (Rm ) > 1, the flow of the matter towards the neutron star will be inhibited by the centrifugal force exerted by the rotating magnetosphere and the matter can be expelled from the system due to the “propeller effect”. The corresponding value of the critical luminosity at which the “propeller effect” occurs: −2/3

−7/3

−7/3 5 2 R6 M1.4 B12 P100 LX,prop = 3.4 · 1033 ωprop

erg/s

(1)

where P100 is the NS (neutron star) spin period in units of 100 sec, R6 is the NS radius in units of 106 cm, M1.4 is its mass divided by 1.4M , B12 is strength of the magnetic filed on the NS surface in units of 1012 Gauss,√ωprop is value of ω at which the propeller effect occurs, we assume ωprop = 2. The value of LX,prop defines the lower limit on the possible X-ray luminosity of an X-ray binary with given parameters of the neutron star. The existence of the lower limit on the luminosity of an accreting neutron star will result in the deficit of

252

High mass X-ray binaries in LMC

low luminosity sources. The example of modified luminosity distribution is shown on Fig.1. Here we used typical NS parameters and distributions of the HMXBs over the NS spin periods and surface magnetic fields observed in the Milky Way and Small Magellanic Cloud. This behavior is qualitatively similar to the observed XLF (Fig.1). However, due to lack of distinct features of the “propeller effect” in the XLF at log(LX ) > 33 and large uncertainty in the observed HMXB XLF at low luminosities, it is premature to draw any definite conclusion regarding its influence on the HMXBs XLF. The detailed analysis of properties of HMXBs population in LMC can be found in Shtykovskiy & Gilfanov (2004).

References Cutri, R. M., Skrutskie, M. F., van Dyk, S., Beichman, C. A., Carpenter, J. M., Chester, T., Cambresy, L. et al. 2003, yCat, 2246, 0 Grimm, H.-J., Gilfanov, M.R., Sunyaev, R.A. 2003, MNRAS, 339, 793 Illarionov, A. F., Sunyaev, R. A. 1975, A&A, 39, 185 Massey, P., 2002, yCat, 2236, 0 Monet D.G., Levine S.E., Canzian B., Ables H.D., Bird A.R., Dahn C.C., Guetter H.H. et al. 2003, AJ, 125, 984 Moretti, A., Campana, S., Lazzati, D., Tagliaferri, G. 2003, ApJ, 588, 696 Morrison, J. E., McLean, B., GSC-Catalog Construction Team, II, 2001, DDA, 32.0603 Shtykovskiy P. & Gilfanov M. 2004, astro-ph/0404300

THE STABILITY OF MAGNETIZED ROTATING PLASMAS WITH STRONG TOROIDAL FIELDS Martin E. Pessah,1,2 and Dimitrios Psaltis,2,1

1 Astronomy Department, 2 Physics Department, University of Arizona

Abstract

It is widely accepted that the magnetorotational instability offers a viable mechanism for enhanced angular momentum transport in weakly magnetized accretion disks around compact objects. Here, we show that, contrary to the results of most previous studies, the presence of a strong azimuthal component in the magnetic field plays a crucial role not only in the growth rates of the unstable modes but also in determining which modes are subject to instabilities.

Keywords:

accretion disks – magnetohydrodynamics – instabilities – plasmas

1.

Introduction

Over the last decade, several local linear mode analysis have pointed out the relevance of the magnetorotational instability (MRI) as a promising mechanism for effective angular momentum transport and the subsequent accretion of matter onto the central objects (Balbus & Hawley 1998, Balbus 2003). In most early studies, it was found that the only effect of a toroidal field component is to quench the growth rates of the unstable modes already present when only a weak field perpendicular to the disk is considered (Balbus & Hawley 1991; Blaes & Balbus 1994). These treatments, usually invoked a number of approximations appropriate for the study of short wavelength perturbations in the presence of weak fields. An important simplification consisted on neglecting the curvature of the background magnetic field, thereby putting the magnetic field in a different footing than the background velocity field considered to be circular. This approximation is indeed well justified when the Alfven speed is negligible when compared to the sound speed. However, there is plenty of theoretical (Pringle 1989; Pariev et al. 2003) and numerical (Kudoh et al. 2002; Machida et al. 2000, Miller & Stone 2000) evidence supporting the existence of regions where the plasma is subject to strong magnetic fields. Examples of settings with non-negligible toroidal fields include the innermost region of an accretion disk around a magnetic neutron star, the rarefied coronae above accretion disks surrounding compact objects and magnetically 257 A. Baykal et al. (eds.), The Electromagnetic Spectrum of Neutron Stars, 257–260. © 2005 Springer. Printed in the Netherlands.

258

The Stability of Magnetized Rotating Plasmas with Strong Toroidal Fields

supported tori in AGN. In these environments, the conditions under which it is safe to neglect the field curvature might no longer be satisfied. Therefore, extrapolation of the results obtained in the weak field limit will no longer properly describe the evolution of modes that are allowed in the plasma.

2.

Dispersion relation including field curvature

In order to perform the linear mode analysis, let us assume an axisymmetric steady background with no vertical stratification characterized by a cylindrical rotational profile Ω = Ω(r) and threaded by a background magnetic field of the form B = [0, Bφ , Bz ]. Deriving the dispersion relation involves a lengthly but otherwise straightforward process. For the sake of brevity, we briefly outline the steps to be followed and present here only the result. Starting with the set of equations that govern the behavior of a polytropic MHD fluid we perturb all the flow variables A to A + δA with δA = δAk ei(kz−ωt) . Considering only vertical waves is physically motivated (since these correspond to the most unstable modes in the well studied MRI) and more tractable mathematically. Neglecting second order terms in the perturbed quantities we obtain a homogeneous linear system in δB, δv, δρ, and δP (i.e., the perturbations in the magnetic field, the velocity field, the density, and the pressure respectively). Setting its determinant equal to zero we find a 6th degree dispersion relation of the form ω 6 + a4 ω 4 + a3 ω 3 + a2 ω 2 + a1 ω + a0 = 0 with coefficients 2 2 2 + 2vAz )κ2 + vAφ a4 = k 2 (c2s + vAφ a3 = 6 kvAφ vAz

d ln Ω 2 4 2 2 2 2 2 2 2 2 v a2 = k vAz (2cs + vAφ + vAz ) + k κ (cs + vAφ ) + 2 d ln r Az

a1 a0

2 2 2 + k 2 vAφ (2c2s + vAφ − vAz ) 2

3 2 = 4 k vAφ vAz 2cs + vAφ 2 2 2 d ln Ω 2 2 2 = −k 4 vAz (2c2s + vAφ ) , cs k vAz + 2 − vAφ d ln r

(1)

where cs , vAz , vAφ and κ stand for the sound speed, the vertical and toroidal components of the Alfven speed and the epicyclic frequency respectively. Note that we have defined dimensionless variables by scaling all the frequencies with the rotational frequency Ω and all speeds with the circular velocity Ωr. We have also defined a dimensionless wavenumber by multiplying the physical wavenumber by the radial coordinate r. The parameter is a dummy variable that we have introduced in order to keep track of the curvature terms that are usually neglected. Setting = 1 provides our full treatment while setting = 0 provides the dispersion relation where only the curvature of the velocity field is considered (see, e.g., Balbus & Blaes 1994).

Results and Discussion

259

Figure 1. Solutions of the dispersion relation (1) for two different strengths of the toroidal field for a Keplerian rotation profile (κ = 3/2) with vAz = 0.01 and cs = 0.05. Left panel: only the curvature of the velocity field is considered ( = 0). Right panel: all curvature terms are considered ( = 1).

3.

Results and Discussion

Numerical Solutions.— The solutions ω(k) for two different values of the toroidal Alfven speed are shown in Figure 1. In the case where only the curvature of the velocity field is considered (left panels) and when all the curvature terms are accounted for (right panels). Note that the mode structure is very similar in both cases for vAφ = 0.1 but it is remarkably different in the case of vAφ = 0.4. Figure 2 shows the range of modes that are unstable (black dots) as a function of the toroidal field strength. Three different regions are clearly distinguishable. In particular, the MRI is stabilized (Region I) and two distinct instabilities appear for stronger toroidal fields (Regions II and III). Importance of Curvature Terms.— In previous local linear mode analysis of the MRI, the forces due to the curvature of the toroidal field were usually neglected when compared against other forces produced by the bending of vertical field lines. The argument being that the ratio of the former to the latter is order 1/k. To illustrate the risks of this reasoning when considering strong toroidal fields let us consider the ratio R1 of the magnetic tension produced by the bending of a vertical field line in the radial direction to the ratio of the radial force produced by a density variation when the finite curvature or the toroidal field is considered. In this case, we can write R1 = Bφ2 δρ/ikρBz δBr .

260

The Stability of Magnetized Rotating Plasmas with Strong Toroidal Fields

Figure 2. Unstable modes (black dots) as a function of toroidal field strength.

Figure 3. Importance of toroidal field lines. R1 is clearly not ∝ 1/k

Although it is true that R1 ∝ 1/k, it is also proportional to the ratio of perturbed quantities, which we do not know a priori. Therefore, it is only after having found the eigenfrequencies ω(k) by taking into account all curvature terms that we can properly find R1 (k). The ratio R1 for the unstable modes in regions II and III in Figure 2 are shown in Figure 3. We can now understand why the mode structure in the lower right panel of Figure 1 is so different with respect to its counterpart on the lower left. For vAφ = 0.4, the functional form of R1 (k) not only differs significantly from the assumed 1/k but its magnitude is of order unity. Summary.— In this paper we have investigated the role of strong toroidal fields on the stability of local axisymmetric perturbations in compressible, differentially rotating, MHD flows when the geometrical curvature of the background is taken into account. We have shown that the MRI is stabilized and two distinct instabilities appear for strong toroidal fields. In a future paper, (Pessah & Psaltis 2004), we will present a more detailed analysis of the different instabilities in Figure 2 and the potential implications of our findings for shearing box models of strongly magnetized accretion disks.

References Balbus, S. A. 2003, ARA & A, 41, 555 Balbus, S. A., & Hawley, J. F. 1991, ApJ, 376, 214 ———. 1998, Rev. Mod. Phys., 70, 1 Blaes, O. M., & Balbus, S. A. 1994, ApJ, 421, 163 Kudoh, T., Matsumoto, R., & Shibata, K. 2002, PASJ, 54, 121 Machida, M., Hayashi, M. R., & Matsumoto R. 2000, ApJ, 532, L67 Miller, K. A., & Stone, J. M. 2000, ApJ, 534, 398 Pariev, V. I., Blackman, E. G., & Boldyrev, S. A. 2003, A&A, 407, 403 Pessah, M. E., & Psaltis, D. ApJ, submitted (astro-ph/0406071) Pringle, J. E., 1989, MNRAS, 236, 107

VII

X-RAY BINARIES

ACCRETION POWERED X-RAY PULSARS Altan Baykal Middle East Technical University, Physics Department, 06531, Ankara, Turkey [email protected]

Abstract

I will review the class of accreting powered neutron stars that are in close binary systems. I will summarize the recent developments and the main properties of the neutron stars. These include the formation of X-ray spectra, formation of pulse profiles, quasi periodic oscillations, pulse frequency fluctuations, long term variations and orbital period decay of X-ray binaries.

Keywords:

stars:neutron – stars:pulsar – stars:binaries:symbiotic – stars:winds, outflows – X-rays:binaries

1.

Introduction

Accretion powered X-ray pulsars were discovered over 30 years ago (Giacconi et al., 1971). After the discovery of an X-ray pulsar in the Crab Nebula, X-ray pulsations were detected by the UHURU satellite from two X-ray binaries. These sources were Centaurus X-3 (Giacconi et al., 1971, Schreier et al., 1972) and Hercules X-1 (Tananbaum et al., 1972). The discovery of radio and X-ray pulsars confirmed the evidence of the existence of the neutron stars which had been predicted theoretically in the 1930’s. The physical mechanism of pulsed emission was understood as X-ray pulsars which are rotating and strongly magnetized (B>1011 Gauss) neutron stars that accrete plasma from stellar companion (Pringle and Rees 1972, Davidson and Ostriker 1973, Lamb, Pethick and Pines 1973). When the accreting material approaches the neutron star, the plasma is funneled to the magnetic poles. Then it releases its gravitational energy as X-ray and gamma-ray radiation. If the magnetic and rotation axes are misaligned, and if the beamed emission from the magnetic poles rotates through the line of sight the pulsations can be observed (Meszaros, Nagel and Ventura 1980). However, if the magnetic field of the neutron star is relatively weak (B 1037 ergs−1 or pencil beam if Lx < 1037 ergs−1 . Figure 1 shows X-ray profiles in three energy bands for six pulsars (White, Swank and Holt, 1983). For X-ray luminosities less then 1037 erg sec−1 pulse profiles agree each other in all energy bands. For X-ray luminosities greater then 1037 erg sec−1 pulse profiles change as function of energy. A pulse reversal at low energies and in the highest energy band is clearly seen for 4U 1626-67. Her X-1 has shown phase shift at lowest energy. Possible explanation of this phase shift is the magnetospheric emission rather then polar cap emission of soft black body component (Endo, Nagase and Mihara 2000). Generally, pulse profiles of accretion powered X-ray binaries have shown single or double peaks at high energies above 10 keV. Modeling of pulse profiles leads an upper limit between the magnetic axis and the rotation axis as 50o (Bulik, Gondek-Rosinka, Santangelo et al., 2003). In the pulse profiles of 42 seconds transient X-ray pulsar EXO 2030+375, the relative strength of the main pulse and interpulse changes when the outburst

X-ray Spectra

265

Figure 1. The pulse profiles of 6 X-ray pulsars (White, Swank and Holt 1983). At the top right log of the luminosity in the 0.5-60 keV is given. The pulse periods in seconds are shown at the top center.

decays (Parmar, White and Stella 1989). This can occur if the beam pattern has changed phase by 180◦ and interprets a transition from fan beam to pencil beam (see Figure 2). Accretion induced bursting pulsar GRO J1744-28 was initially discovered by BATSE (Kouveliotou., et al., 1996). The pulse profile of 426 msec pulse advanced ∼ 25 msec during the burst (Stark, Baykal, Strohmayer and Swank 1996). Either in quiescent or burst states X-ray luminosity of GRO J1744-28 was greater then 1037 erg sec−1 and the pulse profile was very similar in both states. Possible interpretation of phase shifts in pulse arrival times is the shift of accretion footprint during the burst. Therefore the fan beam is shifted with respect to the observer line of sight during the burst (see Figure 3 and 4).

266

3.

Accretion Powered X-ray Pulsars

X-ray Spectra

The ionized plasma follows the field lines and accretes to the magnetic polar caps. The accretion X-ray luminosity is L=

GM M˙ R

(1)

where M˙ is the instantaneous mass accretion rate, and M and R are the neutron star mass and radius. In the simplest model, blackbody temperature can be obtained from accretion X-ray luminosity L = Acap σSB T 4 ,

(2)

as ∼ 3 keV for a polar cap area Acap ∼ km2 . Even though this value is comparable with the peak of X-ray spectra of accretion powered X-ray pulsars, the observed X-ray spectra are much harder than a blackbody and have been

Figure 2. Pulse profile of EXO 2030+375 as function of X-ray luminosity (Parmar, White and Stella 1989)

267

X-ray Spectra

Figure 3. Light curve (top) and pulse phase lag (below) of GRO J1744-28 (Stark, Baykal, Strohmayer and Swank 1996).

Figure 4. The geometries of magnetospheric flow in quiescence (top left) and during a burst (top right) are shown, along with their corresponding accretion footprints (Miller, 1996).

represented by a various models. Most commonly X-ray spectra have been represented by a power law with an exponential cutoff or a broken law (White, Swank and Holt 1983, Mihara 1995, Coburn, Heindl, Rothschild et al., 2002). In these models, overall continuum can be represented as, f (E) = AE −Γ , f (E) = AE

−Γ (Ecut −E)/Ef old

e

E ≤ Ecut

(3)

, E > Ecut ,

(4)

where Γ is power law index varying between 0.0 and 1.0, and Ecut is the high energy cut off varying between 10 and 20 keV. The spectrum above the high energy cutoff Ecut can be approximated by the function e(Ecut −E)/Ef old where Ef old is e folding energy. In some of accretion powered X-ray binaries, soft blackbody component have been observed. These soft components can be interpreted either as emission from polar cap or emission from accretion disk and in some cases as reprocessed emission of the surrounding material (see table 1.) Measurements of the magnetic field of the pulsar can be obtained from the cyclotron resonance scattering features (CRSF). This is commonly called as "cyclotron lines". These line like spectral features arise as a result of the resonant scattering of photons by electrons whose energies are quantized into Landau levels by strong magnetic field (Meszaros 1992). The fundamental energy at which CRSF observed is given as Ec = 11.6

B (1 + z)−1 keV, 1012 G

(5)

268


Table 1. Soft Blackbody components of Accretion Powered X-ray Binaries Name

kT (keV)

Em.Rad. (km)

References

∼ 1.1-1.3 ∼0.5-0.8 ∼ 0.7 ∼1.9

∼1-6 ∼3 ∼ 5-8 ∼ 0.4

Reig and Coe 1999 Gierlinski, Done, Barret 2002 Juett, Galloway, Chakrabarty 2003 Inam, Baykal, Swank, Stark 2004

Polar Cap Interpretation EXO 2030+375 SAX J1808.4-3658 XTE J0920-314 SAX J2103.5+4545 Emission from accretion disk or the surrounding material Her X-1 Cen X-3 SMC X-1 LMC X-4

0.16-0.17 ∼ 0.1 0.13-0.15 ∼ 0.17

Endo, Nagase, Mihara 2000 Burderi, et al., 2000 Paul, Nagase, Endo, et al., 2002 Paul, Nagase, Endo, et al., 2002

where B is the magnetic field in the scattering region and z is the gravitational red shift. The quantized energy levels of the electrons are harmonically spaced at 2Ec , 3Ec , etc.. At sufficiently high magnetic fields (Ec > 35 keV), relativistic effects become important. The first detection of CRSF was made by Trumper et al., (1978) from Her X-1 at ∼ 40 keV. Basic CRSF models are given as H(E) =

τ W 2 (E/Ec )2 , (E − Ec )2 + W 2

H(E) = τ e−(E−Ec )

2 /2σ 2

,

Lorentzian

(6)

Gaussian

(7)

where W and σ are the width of cyclotron energy for Lorentzian and Gaussian models respectively, and τ is the optical depth. Using the power law with high energy cutoff models given at eqs. 3 and 4 complete continuum model can be represented at the form F (E) = f (E)e−H(E) .

(8)

As indicated by dotted line in Figure 5, which represents Ecut ∝ Ec0.7 (Makishima et al., 1999) correlation between spectral cutoff energy and CRSF is valid below ∼35 keV. This trend saturates above ∼35 keV, where the relativistic effects become more important. Coburn et al. (2002) have shown that cyclotron line width (σ) is proportional to cyclotron line energies (Ec ) and that fractional line width (σ/Ec ) scales with optical depth of the line (τ ). The correlation between σ ∝ Ec is expected from the Doppler broadening scattering cross section for electrons in the effectively

Pulse Frequency Fluctuations

269

Figure 5. Spectral cutoff energy versus CRSF is plotted is from RXTE data (Coburn et al., 2002). The dotted line indicates the power law correlation inferred from GINGA data by Makishima et al., (1999).

one dimensional gas along the magnetic field lines. The correlation between σ/Ec and τ indicates that as CRSFs increase in depth, the width of the feature as a percentage of the resonance energy also increases. Therefore as CRSFs become deeper σ/Ec becomes broader.

4.


The pulse frequency fluctuations in accreting neutron stars are produced by torques originating outside and inside the object. The external torque depends on the angular momentum carried with accretion flow; the internal torque depends on the coupling between the superfluid interior and the solid outer crust. In general, external fluctuations of the torque are filtered by the coupling between the crust and superfluid interior to produce output represented by observed changes in the angular velocity (Lamb, Pines and Shaham 1978a,b, Deeter and Boynton 1982). In order to estimate the type of noise processes and core superfluidity, techniques were developed by Deeter (1984) and Cordes (1980) for the estimation of red noise power density and associated random walk noise strengths (see also Scott, Finger and Wilson 2003, Erkoca 2004). If we assume that the noise process consists of individual torque events followed by neutron star’s response by δΩ(t − ti ) and that these events occur at random times ti and at a rate R such that time during interval T, the number of

270


events obeys a Poisson distribution with mean value RT, and the times ti are uniformly distributed for 0 ≤ ti ≤ T then angular velocity fluctuations of the crust can be written in terms of individual events, ∆Ω =

K

δΩ(t − ti )

(9)

i=1

in the form of shot noise (Rice 1954). Then the power spectrum of the noise in the ∆Ω, (10) P(∆ω) = R|g(ω)|2 , where g(ω) = δΩ(t)eiωt dt is the Fourier transform of the individual shot. For the r=1,2...th time integral of white noise time series (or rth order red noise) with strength Sr , the mean square residual for data spanning an interval T is given by (Rice 1954), < ∆Ω2 >= Sr T 2r−1 ,

(11)

where Ω is the variation in angular velocity of the neutron star and the noise strength Sr = R < δΩ2 > can be expressed in terms of rate of the events R times RMS step size of angular velocities < δΩ2 >. Noise strengths Sr in time domain are related by power spectra in the following form (12) PΩ = Sr (2πf )−2r . For r=1 case, power spectra of angular acceleration fluctuations and angular velocity fluctuations can be related as follows, PΩ˙ = (2πf )2 PΩ = Sr=1 .

(13)

In Figure 6 and figure 7, angular velocity time series history of 38 seconds pulsar OAO 1657-415 and its power density spectrum is represented respectively. Figure 8, presents the rotational noise strengths of accreting powered pulsars as a function of their X-ray luminosities. In order to compare magnitudes of noise strengths with other class of sources, the noise strengths of two magnetars (SGR 1900+14, SGR 1806-20) and two AXPs (1E 1048.1-5937, 1E 2259+59) and one rotation powered pulsar (Crab) are included in the plot. The short term pulse frequency fluctuations can be approximated with a two component neutron star model (Baym et al., 1969). In this model, one component is the crust charge particle system, which consists of protons, electrons and the crust with inertia Ic which rotates with angular velocity Ωc . The second component is the core neutron superfluid, with moment of inertia Is , which rotates with angular velocity Ωs . Any external torque on the crust creates a lag

271


between Ωc and Ωs . The two components are coupled by crust core coupling time τ , ˙ c = N (t) − Ic (Ωc − Ωs ), Ic Ω (14) τ ˙ s = Ic (Ωc − Ωs ), Is Ω (15) τ where N(t) is the external torque exerted on the neutron star. For OAO 1657415 (P= 38 sec), either the crust core coupling time τ 600 Hz, and that production of gravitational radiation by uneven heating of the crust or excitation of r-waves is not as easy as was originally thought (Ushomirsky et al. 2000; Lindblom & Owen 2002). At present there is no unambiguous evidence that the spin rates of neutron stars in LMXBs is affected by gravitational radiation. Production of millisecond rotation-powered pulsars.—Soon after rotationpowered radio-emitting MSPs were discovered, it was proposed that they have been spun up to millisecond periods by steady accretion in LMXBs at rates ∼ M˙ E (see Bhattacharya & van den Heuvel 1991), with the implicit assumption that accretion then ends suddenly; otherwise the stars would track νeq to low spin rates as the accretion phase ends. This simplified picture is sometimes still used (see, e.g., Arzoumanian, Cordes, & Wasserman 1999), but—as noted above—most neutron stars in LMXBs accrete at rates M˙ E , many accrete only episodically, and the accretion rates of others dwindle as their binary systems evolve. The real situation is therefore more complex. The initial spins of rotation-powered MSPs recycled in LMXBs are the spins of their progenitors when they stopped accreting. These spins depend sensitively on the magnetic fields and the appropriately averaged accretion rates of the progenitors when accretion ends. Comparison of the equilibrium spinperiod curves for a range of accretion rates with the P –P˙ distribution of known rotation-powered MSPs (Fig. 3) suggests three important conclusions:

320

Millisecond X-Ray Pulsars and QPOs

(1) The hypothesis that the accretion torque vanishes at a spin frequency close to the calculated νeq predicts that MSPs should not be found above the spin-equilibrium line for M˙ = M˙ E , because this is a bounding case. The observed P –P˙ distribution is consistent with this requirement for the RPD model of the inner disk that was used for M˙ greater that ∼ 0.1M˙ E , except for two pulsars recently discovered in globular clusters: B1821−24 and B1820−30A (Hobbs et al. 2004). Either the intrinsic P˙ ’s of these pulsars are lower than shown or the RPD model of the inner disk does not accurately describe the accretion flow that spun up these stars. (2) The accretion spin-equilibrium hypothesis predicts that MSPs should be rare or absent below the spin-equilibrium line for M˙ = 10−4 M˙ E , because stars accreting at such low rates generally will not achieve millisecond spin periods during their accretion phase. The observed P –P˙ distribution is consistent with this prediction. (3) The MSPs near the 15 Gyr spin-down line were produced in situ by final accretion rates less than ∼ 3 × 10−3 M˙ E rather than by spin-up to shorter periods by accretion at rates greater than ∼ 3 × 10−3 M˙ E followed by magnetic braking, because braking would take too long. This result accords with the expectation (see above) that most neutron stars in LMXBs accrete at rates

M˙ E toward the end of their accretion phase.

3.

Generation of Kilohertz QPOs

The observations of kilohertz QPO pairs summarized in § 1 provide strong hints about the mechanisms that generate them (see Lamb & Miller 2005): 1. It appears highly likely that the frequency of one of the two kilohertz QPOs reflects the orbital frequency of gas in the inner disk. The frequencies of the kilohertz QPOs are similar to those of orbital motion near neutron stars. They also vary by hundreds of Hertz on time scales as short as minutes (see, e.g., Méndez et al. 1999; van der Klis 2000; Wijnands 2005). Such large, rapid variations are possible if they are related to orbital motion at a radius that varies (Lamb 2003). 2. The star’s spin is somehow involved in producing the frequency separation of the two kilohertz QPOs in a pair. This involvement is clear in XTE J1807−294, where ∆νQPO ≈ νspin , and in SAX J1808.4−3658, where ∆νQPO ≈ νspin /2. It is strongly indicated in the other kilohertz QPO sources, because in all cases where both ∆νQPO and νburst have been measured, the largest value of ∆νQPO is consistent or approximately consistent with either νspin or νspin /2 (see van der Klis 2000; Lamb 2003; Lamb & Miller 2005). 3. A mechanism that produces a single sideband is indicated. Most mechanisms that modulate the X-ray brightness at two frequencies (such as amplitude modulation) would generate at least two strong sidebands. Although weak


321

Figure 5. Left: Twin kilohertz QPOs and other variability of the X-ray flux of Sco X-1. The total variation (power density times frequency) is dominated by the kilohertz QPOs, which in this observation are at ∼ 600 Hz and ∼ 900 Hz. The horizontal-branch oscillation (HBO) at ∼ 50 Hz and its second harmonic at ∼ 100 Hz are also visible, as are several broad-band noise components. From Wijnands & van der Klis (1997). Right: Correlation between lower and upper kHz QPO frequency for the Sco X-1 from A03 [black circles], a sample of atoll sources [open circles] and Z sources [stars] from the literature. The line represents a fixed 3:2 ratio. In the inset, the distributions of ratios for the atoll (black) and Z sources (gray) are shown. From Belloni et al. (2005).

sidebands have been detected close to the frequency of the lower kilohertz QPO (Jonker, Méndez, & van der Klis 2000), at most two strong kilohertz QPOs are observed in a given system (van der Klis 2000; Méndez & van der Klis 2000). This suggests that the frequency of one QPO is the primary frequency while the other is generated by a single-sideband mechanism. Beat-frequency mechanisms naturally produce a single sideband. Because one QPO frequency is almost certainly an orbital frequency, the most natural mechanism would be one in which the second frequency is generated by a beat with the star’s spin frequency or with another orbital frequency. 4. Mechanisms for generating kilohertz QPO pairs (see, e.g., Fig. 5) like the mechanism proposed by Abramowicz & Kluzniak (2001) to explain the high-frequency QPOs observed in black hole candidates and by Kluzniak et al. (2003) to explain the kilohertz QPOs observed in SAX J1808.4−3658 are excluded for the kilohertz QPOs seen in neutron stars, because these mechanisms require a low-order resonance between the geodesic frequencies of test particles orbiting at a fixed radius, which disappears when the two frequencies change substantially. As noted above, the separation frequency is approximately constant, which implies a roughly linear relation between the two QPO frequencies (see Fig. 5), which is incompatible with a fixed frequency ratio (Belloni et al. 2005). This type of mechanism also cannot explain the commensurability of ∆νQPO and the spin frequency (see, e.g., Lamb 2004).

322


Lamb & Miller (2005) have proposed a modification of the original sonicpoint beat-frequency model (MLP98) that potentially can explain within a single framework why the frequency separation is close to νspin in some stars but close to νspin /2 in others. In this “sonic-point and spin-resonance” (SPSR) beat-frequency model, gas from perturbations orbiting at the sonic-point radius rsp produces a radiation pattern rotating with a frequency νQPO2 close to the orbital frequency νorb at rsp , as in the original model, and that this rotating pattern is responsible for the upper kilohertz QPO. This mechanism for generating the upper kilohertz QPO is supported by the observed anticorrelation of the upper kilohertz QPO frequency with the normal branch oscillation flux in Sco X-1 (Yu, van der Klis, & Jonker 2001) and the anticorrelation of the kilohertz QPO frequency with the mHz QPO flux in 4U 1608−52 (Yu & van der Klis 2002). A new ingredient in the modified model is preferential excitation of vertical motions in the disk at the “spin-resonance” radius rsr , where νspin − νorb is equal to the vertical epicyclic frequency νψ , by the magnetic and radiation fields rotating with the neutron star. Preliminary numerical simulations show that the resulting vertical displacement of the gas in the disk is much greater at the resonant radius than at any other radius. In a Newtonian 1/r gravitational potential, νψ (r) = νorb (r). Although νψ (r) is not exactly equal to νorb (r) in general relativity, the difference is < 2 Hz at the radii of interest (where νorb < 300 Hz). Consequently, at the resonance radius where vertical motion is preferentially excited, νorb ≈ νψ ≈ νspin /2, i.e., at this radius the orbital and vertical frequencies are both approximately half the star’s spin frequency. Lamb & Miller (2005) argue that interaction of the radiation pattern rotating with frequency νQPO2 with the vertical motion of gas excited at the spinresonance radius produces a second QPO at νorb (rsp ) − νspin /2, if the gas at rsr is highly clumped, or at νorb (rsp ) − νspin , if the flow at rsr is relatively smooth. They interpret this second QPO as the lower kilohertz QPO. To see how flux modulations are possible, suppose first that the gas in the disk near the spin-resonance radius is highly clumped. When illuminated, each clump orbiting at rsr scatters radiation in all directions. In effect, each clump redirects the radiation propagating outward from the sonic radius in the modest solid angle that it subtends (as seen from the sonic radius) into all directions. From the point of view of a distant observer, each individual clump looks like a light bulb that is blinking on and off with a frequency equal to νorb (rsp ) − νorb (rsr ) ≈ νorb (rsp ) − νspin /2. If there are only a modest number of clumps at rsr , the scattering from the individual clumps dominates the time variation of the X-ray flux. The left-hand panel of Figure 6 shows the power spectrum of the flux variation generated in a simulation in which five randomly-positioned clumps scatter the radiation pattern coming from the sonic radius. The peak at νorb (rsp ) − νspin /2 is clearly dominant. Because the radiation is scattered in


323

Figure 6. Power spectra of the X-ray flux modulation produced by simulations of a disk with a small number of clumps near the spin-resonance radius (lefthand panel) and a large number of clumps (right-hand panel). The star’s spin frequency is 400 Hz while the orbital frequency at the sonic point is 1,000 Hz. These power spectra demonstrate that if the flow near the spin-resonance radius is clumpy, the effect of individual clumps dominates and the dominant frequency is νorb (rsp )− ν spin /2 . If instead the flow is relatively smooth, the effect of the clump pattern dominates and the dominant frequency is νorb (rsp )− ν spin. This simulation did not include any signal with the orbital frequency of the gas at the sonic radius.

all directions, an observer does not have to be close to the disk plane to see the X-ray flux modulation. Suppose instead that the gas in the disk near the spin-resonance radius is less highly clumped. There may be a larger number of smaller clumps or the flow may even be relatively smooth. As before, each element of gas is oscillating vertically with frequency νspin /2. Together they form a pattern of raised fluid elements that rotates around the star with frequency νspin . Because a large number of fluid elements are scattering radiation to the observer at any given moment, their individual contributions blend together, so the dominant time variation has frequency νorb (rsp ) − νspin . In this case the brightness variation produced by the pattern of scattering clumps dominates the brightness variation produced by the individual clumps. The right-hand panel of Figure 6 shows the power spectrum of the flux variation generated in a simulation in which 500 randomly-positioned clumps scatter the radiation pattern coming from the sonic radius. The peak at νorb (rsp ) − νspin is clearly dominant. Magnetic forces may cause the gas in the accretion disk to become more clumped as it approaches the neutron star (MLP98; Lamb & Miller 2001; Lamb & Miller 2005). Consequently, the parameters that may be most im-

324


portant in determining whether the flow at the spin resonance radius rsr is clumpy or smooth are the star’s spin frequency and magnetic field. For a given stellar magnetic field, the flow is likely to be more clumpy if the star is spinning rapidly and rsr is therefore close to the star. For a given spin rate, the flow is likely to be more clumpy if the star’s magnetic field is stronger. The four sources with νspin > 400 Hz and measurable frequency separations have ∆νQPO ≈ νspin /2 whereas the three sources with νspin < 400 Hz have ∆νQPO ≈ νspin (see Muno et al. 2001). With such a small sample, one cannot make any definite statements, but the apparent trend is consistent with the sonic-point and spin-resonance beat-frequency model. These trends suggest that if kilohertz QPOs are detected in the recently-discovered 185 Hz and 314 Hz accretion-powered X-ray pulsars XTE J0929−314 (Galloway et al. 2002) and XTE J1814−338 (Strohmayer et al. 2003), their frequency separations should be approximately equal to their respective spin frequencies. The 435 Hz spin frequency of XTE J1751−305 (Markwardt et al. 2002) is high enough that ∆νQPO could be either approximately 435 Hz or approximately 217 Hz; QPOs at both frequencies might even be detectable. Finally, we note that there is no known reason why the mechanism for producing a lower kilohertz QPO proposed in the original sonic-point beatfrequency model would not operate. Apparently this mechanism does not produce a strong QPO in the fast rotators, but it might produce a weak QPO in these sources. If it operates in the slow rotators, it would produce a QPO near νorb (rsp ) − νspin that might appear as a sideband to the lower kilohertz QPO. Although the sonic-point and spin-resonance beat-frequency model appears qualitatively consistent with the basic properties of the kilohertz QPOs, many aspects of the model require further exploration and development.

Acknowledgments It is a pleasure to thank L. Bildsten, D. Chakrabarty, P. Kaaret, C.B. Markwardt, D. Marković, M. Méndez, M.C. Miller, M. Nowak, D. Psaltis, R. Remillard, T. Strohmayer, J. Swank, M. van der Klis, and W. Zhang for useful discussions. This research was supported in part by NSF grant AST 0098399, NASA grant NAG5-12030, and the Fortner Endowed Chair at the University of Illinois.

References Abramowicz, M.A., & Kluzniak, W. 2001, A&A, 374, L19. Alpar, M.A., Cheng, A.F., Ruderman, M.A., & Shaham, J. 1982, Nature, 300, 178. Arzoumanian, Z., Cordes, J.M., & Wasserman, I. 1999, ApJ 520, 696. Bhattacharya, D., & Srinivasan, G. 1995, in X-Ray Binaries, ed. W.H.G. Lewin, J. van Paradijs & E.P.J van den Heuvel (Cambridge, Cambridge University Press), 495. Bhattacharya, D., & van den Heuvel, E.P.J. 1991, Phys. Rep., 203, 1.


325

Bildsten, L. 1998, ApJ, 501, L89. Belloni, T., Méndez, M., and Homan, J. 2005, this volume. Chakrabarty, D. 2005, in Binary Radio Pulsars, ed. F.A Rasio and I.H. Stairs (ASP Conf. Series, Vol. 328), 279. Chakrabarty, D., & Morgan, E.H. 1998, Nature, 394, 346. Chakrabarty, D., Morgan, E.H., Muno, M.P., Galloway, D.K., Wijnands, R., van der Klis, M., & Markwardt, C.B. 2003, Nature, 424, 42. Elsner, R.F., Ghosh, P., & Lamb, F.K. 1980, ApJ, 241, L55. Galloway, D.K., Chakrabarty, D., Morgan, E.H., & Remillard, R.A. 2002, ApJ, 576, L137. Galloway, D.K., Chakrabarty, D., Muno, M.P., & Savov, P. 2001, ApJ, 549, L85. Ghosh, P., & Lamb, F.K. 1979, ApJ, 234, 296. Ghosh, P., & Lamb, F.K. 1992, in X-Ray Binaries and Recycled Pulsars, ed. E.P.J. van den Heuvel & S. Rappaport (Dordrecht: Kluwer), 487. Hartman, J.M., Chakrabarty, D., Galloway, D.K., Muno, M.P., Savov, P., Mendez, M., van Straaten, S., Di Salvo, T., 2003, American Astronomical Society, HEAD Meeting No. 35, abstract 17.38. Hasinger, G., & van der Klis, M. 1989, A&A., 225, 79. Hobbs, G.B., Lyne, A.G., Kramer, M., Martin, C.E., & Jordan, C.A. 2004, M.N.R.A.S., submitted. Hobbs, G.B., & Manchester, R.N. 2004, ATNF Pulsar Catalogue, v1.2, http://www.atnf.csiro.au/research/pulsar/psrcat/psrcat help.html . Jonker, P.G., Méndez, M., & van der Klis, M. 2000, ApJ, 540, L29. Joss, P.C., & Li, F.K. 1980, ApJ, 238, 287. Kaaret, P., Zand, J.J.M. in’t., Heise, J., & Tomsick, J.A., 2002, ApJ, 575, 1018. Kaaret, P., Zand, J.J.M. in’t., Heise, J., & Tomsick, J.A. 2003, ApJ, 598, 481. Kluzniak, W., Abramowicz, M.A., Kato, S., Lee, W.H., & Stergioulas, N. 2003, ApJ, 603, L89. Lamb, F.K. 1989, in Proc. 23rd ESLAB Symp. on X-ray Astronomy, ed. J. Hunt & B. Battrick (ESA SP-296), 215. Lamb, F.K. 2003, in X-Ray Binaries and Gamma-Ray Bursts, ed. E.P.J. van den Heuvel, L. Kaper, E. Rol & R.A.M.J. Wijers (San Francisco: Astron. Soc. Pacific), 221. Lamb, F.K. 2004, in X-Ray Timing 2003: Rossi and Beyond, ed. P. Kaaret, F.K. Lamb, & J.H. Swank (AIP Conf. Proc., Vol. 714), 3. Lamb, F.K., & Miller, M.C. 2001, ApJ, 554, 1210. Lamb, F.K., & Miller, M.C. 2005, ApJ, in press, astro-ph/0308179. Lamb, F.K., and Yu, W. 2005a, in Binary Radio Pulsars, ed. F.A Rasio and I.H. Stairs (ASP Conf. Series, Vol. 328), 299. Lamb, F.K., and Yu, W. 2005b, in preparation. Lewin, W.H.G., van Paradijs, J., & Taam, R. 1995, in X-Ray Binaries, ed. W.H.G. Lewin, E.P.J. van den Heuvel & J. van Paradijs (Cambridge Univ. Press), 175. Lindblom, L., & Owen, B. 2002, Phys. Rev. D, 65, 063006. Markwardt, C.B., Smith, E., & Swank, J.H. 2003a, IAU Circ. 8080. Markwardt, C.B., Strohmayer, T.E. & Swank, J.H. 1999, ApJ, 512, L125. Markwardt, C.B., & Swank, J.H. 2003b, IAU Circ., 8144, 1. Markwardt, C.B., Swank, J.H., Strohmayer, T.E., in ’t Zand, J.J.M., & Marshall, F.E. 2002, ApJ, 575, L21. Markwardt, C.B., Swank, J.H., & Strohmayer, T.E. 2004, The Astronomer’s Telegram, 353, 1. Méndez, M., & van der Klis, M. 2000, MNRAS, 318, 938. Méndez, M., van der Klis, M., Ford, E.C., Wijnands, R.A.D., & van Paradijs, J. 1999, ApJ, 511, L49.

326


Miller, M.C., Lamb, F.K., & Psaltis, D. 1998, ApJ, 508, 791 (MLP98). Muno, M.P., Chakrabarty, D., & Galloway, D.K., & Psaltis, D. 2002, ApJ, 580, 1048. Muno, M.P., Chakrabarty, D., Galloway, D.K., & Savov, P. 2001, ApJ, 553, L157. Muno, M.P., Fox, D.W., Morgan, E.P. 2000, ApJ, 542, 1016. Psaltis, D., & Chakrabarty, D. 1999, ApJ, 521, 332. Psaltis, D., & Lamb, F. K. 1998, in Neutron Stars and Pulsars, ed. N. Shibazaki, N. Kawai, S. Shibata, & T. Kifune (Tokyo: Univ. Acad. Press), 179. Radhakrishnan, V., & Srinivasan, G. 1982, Curr. Sci., 51, 1096. Ritter H., & King A.R. 2001, in ASP Conf. Ser. Vol. 229, Evolution of Binary and Multiple Star Systems, ed. Podsiadlowski, P., Rappaport S.A., King A.R., D’Antona F. & Burderi L., San Francisco, Astron. Soc. Pac., 423. Shibazaki, N., et al. 1989, Nature, 342, 656. Smith, D.A., Morgan, E.H., & Bradt, H. 1997, ApJ, 482, L65. Strohmayer, T.E., Jahoda, K., Giles, A.B., & Lee, U. 1997, ApJ, 486, 355. Strohmayer, T.E., & Markwardt, C.M. 2002, ApJ, 577, 337. Strohmayer, T.E., et al. 1998, ApJ, 498, L135. Strohmayer, T.E., Markwardt, C.M., Swank, J.H., and in’t Zand, J.J.M., 2003, ApJ, 596, L67. Strohmayer, T.E., Zhang, W., Swank, J.H., Smale, A., Titarchuk, L., Day, C., & Lee, U. 1996, ApJ, 469, L9. Strohmayer, T.E., Zhang, W., Swank, J.H., White, N.E., & Lapidus, I. 1998, ApJ, 498, L135. Ushomirsky, G., Cutler, C., & Bildsten, L. 2000, M.N.R.A.S., 319, 902. van den Heuvel, E.P.J. 1992, in X-Ray Binaries and Recycled Pulsars, ed. E.P.J. van den Heuvel & S.A. Rappaport (Dordrecht: Kluwer), 233. van der Klis, M. 2000, ARA&A, 38, 717. Villarreal, A.R., & Strohmayer, T.E. 2004, ApJ, 614, L121. White, N., & Stella, L. 1987, M.N.R.A.S., 231, 325. Wijnands, R.A.D. 2005, in Pulsars New Research (New York: Nova Science Publishers), in press. Wijnands, R., Strohmayer, T., Franco, L.M., 2001, ApJ, 554, L59. Wijnands, R., & van der Klis, M. 1998, Nature, 394, 344. Wijnands, R.A.D., van der Klis, M., Homan, J., Chakrabarty, D., Markwardt, C.B., & Morgan, E.H. 2003, Nature, 424, 44. Yu, W., & van der Klis, M. 2002, ApJ, 567, 67. Yu, W., van der Klis, M., & Jonker, P.G. 2001, ApJ, 559, L29. Zhang, W., Jahoda, K., Kelley, R.L., Strohmayer, T.E., Swank, J.H. & Zhang, S. N. 1998, ApJ, 495, L9. Zhang, W., Lapidus, I., Swank, J.H., White, N. E., & Titarchuk, L. 1996, IAU Circ. 6541.

VIII

GAMMA RAY BURSTS AND MAGNETARS

TIMING NEUTRON STARS and the link with black-hole timing M. van der Klis Astronomical Institute “Anton Pannekoek” University of Amsterdam [email protected]

Abstract

This lecture deals with aspects of the rapid X-ray variability of low-magnetic neutron stars in low-mass X-ray binaries. After summarizing some basic considerations related to the physics of orbital motion in strong field gravity and introducing the main types of system, the relations of the kilohertz quasi-periodic oscillations with orbital motion, neutron star spin, luminosity and accretion rate and other, slower variability components are reviewed. Finally, possible relations with the variability of black holes are discussed.

Keywords:

stars:neutron – stars:pulsar – X-rays:binaries – X-rays:stars – black hole physics – stars:oscillations

1.

Introduction

X-ray binaries are interesting in their own right and from the point of view of stellar evolution, but perhaps their main interest is in their role as cosmic laboratories to study strong gravity and dense matter. Two prominent aims of X-ray binary studies are the direct determination of the properties of the strong gravitational fields near black holes and neutron stars, and of the supranuclear density matter in the interior of neutron stars. In the inner few kilometers of the accretion flow onto a low-magnetic-field neutron star or stellar-mass black hole of mass M the accreting matter is moving close to the Schwarzschild radius at RS = 2GM/c2 . This means we are looking at the motion of matter in strongly curved spacetime and can learn about gravitation in the strong-field regime, a regime where classical physics fails and general relativity has not yet been tested to any degree of confidence by means other than extrapolation from observations in weak gravity. In the case of a neutron star, the properties of the accretion flow constrain fundamental neutron star parameters such as mass and radius, and thereby (via stellar structure theory) the equation of state 283 A. Baykal et al. (eds.), The Electromagnetic Spectrum of Neutron Stars, 283–310. © 2005 Springer. Printed in the Netherlands.

284

Timing Neutron Stars

(EOS) of the ultradense matter at the star’s core, whose properties are the subject of much speculation involving, among others, pion and kaon condensates, hyperons, and even strange matter. Measurements of the varying emission of hot spots on the surface of a spinning neutron star can likewise constrain both gravitation theory and neutron star parameters. X-ray timing as well as spectroscopy can be used to study the dynamics of the inner accretion flows and the neutron star spins; for best results one wants to combine these techniques. Both X-ray binaries and AGNs are useful to study the accretion flow in the strong-field region; both types of system have advantages, and again for best results one wants to study both, and describe the results in one coherent picture. An important aspect of compact object studies that is unique to X-ray binaries is the possibility to compare systems that are similar except in the type of compact object they contain. A particularly interesting comparison is that between low-magnetic field neutron stars and black holes, which are both expected to allow gravity-dominated accretion flows down to well into the strong-field region. This comparison, at least in principle, allows to determine which of the observed properties are caused by the unique characteristics of a black hole, and which are in common with other accreting compact objects. Future additional techniques will allow to address these issues in new ways. One is direct imaging of the strong-field region, which becomes possible (in principle) initially in radio to IR/sub-mm wavelengths for the galactic center and AGNs, and only later in X-rays, as the latter requires the development of large baseline X-ray interferometers, which is very challenging indeed. Another is gravitational wave detection of accretion and merging events involving compact objects; ground based gravitational wave experiments such as LIGO and later space based ones such as LISA are directed towards this aim. My charge for this review is to concentrate on neutron-star X-ray binary timing. I shall put particular emphasis on the millisecond time variability of the X-rays emitted by the accreting low-magnetic field neutron stars in low mass X-ray binaries, as it is in these neutron stars that a plasma flow not dominated by the neutron star magnetic field extends down into the strong gravity region. The link of the timing phenomena observed in these neutron stars with those in stellar mass black holes, where the same basic physical situation applies, is also addressed. The emphasis on millisecond variability comes from the consideration that the dynamical time scales of stellar mass compact objects are of this order; hence the main characteristic time scales in the strong-field region, such as the orbital time scale, as well as the spins of weakly magnetic neutron stars spun up by accretion, tend to be of the order of milliseconds as well. I shall be very brief in this write-up on the topic of the data analysis of rapid X-ray variability and just mention that, certainly in the millisecond domain the analysis method of choice for detecting the variability and measuring

285

Some basic physical considerations

its characteristic frequencies is the calculation of power spectra using Fourier technique (see, e.g., van der Klis 1989, van der Klis 2004) — examples of such power spectra are shown below. Narrow features in such power spectra are called quasi-periodic oscillations (QPOs) and broad ones are called noise components. It is interesting to note that a number of millisecond phenomena that were predicted during the 1970’s and 80’s have all been discovered with the Rossi Xray Timing Explorer (RXTE) satellite within a few years following its launch on December 30, 1995. Millisecond wavetrains due to clumps orbiting near the innermost stable circular orbit from general relativity were predicted by Sunyaev (1973) and it is likely that the kilohertz QPO signals discovered with RXTE in 1996 are caused by orbital motion effects of this kind. Short-lived periodic signals at the neutron star spin frequency during X-ray bursts were predicted by Livio & Bath (1982) and discovered with RXTE as burst oscillations also in 1996. Millisecond accreting pulsars were also predicted in the 1980’s (Alpar at al. 1982, Radhakrishnan & Srinivasan 1984) shortly after the discovery of the millisecond radio pulsars and finally found with RXTE in 1998. The pulsations and most likely the burst oscillations are diagnostic of the millisecond neutron star spins; the various high frequency QPOs nearly certainly probe the accretion flow, likely, the inner disk. The RXTE Proportional Counter Array (PCA) remains the only instrument that can study these fundamental phenomena.

2.


The gravitational setting in which the plasma moves through the the strongfield region is best introduced in terms of the classic results describing orbital motion in general relativity (see also van der Klis 2004). The time scales of orbital motion relate to the Keplerian frequency from Newtonian physics: νK =

GM/r 3 /2π ≈ 1184 Hz

r −3/2 m1.4 1/2 , 15 km

where m1.4 is the central mass M in units of 1.4 M , and r is the orbital radius. In general relativity the frequencies of the azimuthal, radial and vertical components of the orbital motion differ, so that in general orbits are not closed. In addition to the azimuthal motion at the general-relativistic orbital frequency νφ , there are the radial and vertical epicyclic frequencies νr and νθ . Due to this, eccentric orbits waltz at the periastron precession frequency νperi = νφ − νr and orbits tilted relative to the equatorial plane of a spinning central mass wobble at the nodal precession frequency νnodal = νφ − νθ : the periastron, and the equatorial nodes, of the orbit, respectively, themselves circle around the center with uniform speed.

286


Figure 1.

Orbital (νK ) and epicyclic (νr , νθ ) motions.

For infinitesimally tilted and eccentric orbits in Kerr spacetime GM/r 3 /2π = νK (1 + j(rg /r)3/2 )−1 ; νφ = 3/2 1 + j(rg /r) 1/2 ; νr = νφ 1 − 6(rg /r) + 8j(rg /r)3/2 − 3j 2 (rg /r)2 1/2 νθ = νφ 1 − 4j(rg /r)3/2 + 3j 2 (rg /r)2 , where j ≡ Jc/GM 2 is the Kerr angular-momentum parameter which is 0 < j < 1 for prograde orbits and −1 < j < 0 for retrograde ones; J is the angular momentum of the central point mass and rg ≡ GM/c2 . Close to a compact object no stable orbital motion is possible. In a Schwarzschild geometry (j = 0) the innermost stable circular orbit (ISCO) or marginally stable orbit has a radius rms = 6rg = 6GM/c2 ≈ 12.5m1.4 km, and a corresponding orbital frequency νms = c3 /2π63/2 GM ≈ 1566 Hz m1.4 −1 . For prograde equatorial orbital motion in a Kerr geometry the ISCO is smaller. Spinning neutron stars have an exterior spacetime that is Kerr only to first order in j; to higher order the metric, and hence the precise frequencies, depend on the mass distribution. Depending on mass and internal structure, spinning neutron stars could have appreciable angular momentum (e.g., j∼0.2 and ∼0.5

287


280

νθ

Frequency (Hz)

240

j=0 3:1

200 160

2:1

120

3:2

80

νr

40 0

5

6

7

8

9

10

11

r / rg

Figure 2. Radial and vertical epicyclic frequencies vs. r/rg in Schwarzschild geometry. Three resonant radii are indicated. After Abramowicz et al. (2004).

for 500 and 1000 Hz spins, respectively, Miller et al. 1998). To first order in j (Kluzniak et al. 1990, Miller et al. 1998) rms ≈ (6GM/c2 )(1−0.54j)

and

νms ≈ (c3 /2π63/2 GM )(1+0.75j).

Some disk flows can penetrate down to inside the ISCO before the matter plunges in (e.g., Abramowicz et al. 2004), but not beyond the marginally bound orbit at rmb = rg (2 − j) + 2rg (1 − j)1/2 , which is inside the ISCO (at 4rg in Schwarzschild geometry). As νr and νθ are both less than νφ , periastron and nodal precession are both prograde. Periastron precession is a consequence of the non-1/r2 nature of gravity in general relativity; the classic example is Mercury’s generalrelativistic perihelion precession. Nodal or Lense-Thirring precession is due to the frame dragging caused by the central object’s spin and does not occur if j = 0. In the weak-field (rg /r 1) slow-rotation (j 1) limit νnodal = (GM )2 j/πc3 r3 = 8π 2 νφ2 Iνspin /M c2 , so: νφ 2 νspin I45 ; νnodal = 13.2 Hz m 1000 Hz 300 Hz here νspin is the neutron star spin frequency, I its moment of inertia, I45 denotes I in units of 1045 g cm2 and m is M in units of M : by measuring νnodal , νφ and νspin the neutron star structure dependent quantity I/M can be constrained. For acceptable theoretical equations of state, values of I45 /m between 0.5 and 2 are expected (Stella & Vietri 1998). Of course, the motions in an accretion disk, while occurring in the gravitational potential producing the free-particle orbits described above are not

288


GX wLMXB

Atoll NS transients

0.001

0.01

0.1 Lx / L Edd NS

Z

BH transients 1.0

10

Figure 3. Luminosities attained by Z sources, GX atoll sources, ordinary atoll sources and weak LMXBs, respectively, as well as by neutron-star and black-hole transients. The extent of the Lx overlaps between these source types is undecided in detail, but those shown here are likely.

in fact free-particle motions. Hydrodynamic, electromagnetic and radiative stresses will also affect them. Yet in nearly all models at least some of the orbital and epicyclic frequencies show up in the collective particle motions that characterize the plasma as a whole. However, other frequencies can occur as well, such as those related to disturbances propagating at sound speed. Moreover, due to hydrodynamic effects these frequencies, and the orbital and epicyclic ones can all be coupled, so that various combination frequencies are in general produced as well.

3.

Source types

There are a number of sub-types of low-magnetic field neutron star systems. The three main subtypes are the Z sources, the atoll sources (Hasinger & van der Klis 1989) and the ’weak LMXBs’. Z sources are the most luminous, and accrete at an appreciable fraction of the Eddington critical rate (perhaps 0.5–1 LEdd ). Atoll sources, many of which are X-ray burst sources, cover a much wider range in luminosities, from less than 0.001 LEdd all the way up to the range of the Z sources. Ordinary atoll sources are usually in the 0.01– 0.2 LEdd range, while the ’GX’ atoll sources in the galactic bulge usually hover at the upper end (perhaps 0.2–0.5 LEdd ), and the weak LMXBs at the lower end (

The Electromagnetic Spectrum of Neutron Stars (2007)(en)(386s)

The Electromagnetic Spectrum of Neutron Stars

The Cooling of Neutron Stars

Neutron Stars and Pulsars

High-energy radiation from magnetized neutron stars

Black holes, white dwarfs, and neutron stars

The Plane Wave Spectrum Representation of Electromagnetic Fields (IEEE OUP Series on Electromagnetic Wave Theory)

Isolated Neutron Stars: from the Surface to the Interior

Alien Vision: Exploring the Electromagnetic Spectrum with Imaging Technology

Neutron Stars 1: Equation of State and Structure

The Spectrum of Responsibility

Handbook of Neutron Optics

Handbook of Neutron Optics

The Neutron Bomb

Neutron Star

Neutron Star

Neutron Star

Neutron Solstice

Neutron Star

The Spark Spectrum of Neon

Neutron Star

Neutron Star

Neutron Star

Neutron Star

neutron star

neutron solstice

Neutron Solstice

Neutron Soltice

On the Spectrum of Argon

The Electromagnetic Spectrum of Neutron Stars (2007)(en)(386s)