A Quest for Perspectives: Selected Works of S. Chandrasekhar: With Commentary (Volume 1)

A Quest for Perspectives Selected Works of S. Chandrasekhar With Commentary Volume 1

Editor: Kameshwar C. Wali Imperial College Press

A Quest for Perspectives Selected Works of S. Chandrasekhar With Commentary Volume 1

A Quest for Perspectives Selected Works of S. Chandrasekhar With Commentary Volume 1

Editor

Kameshwar C. Wali Syracuse University

Imperial College Press

Published by Imperial College Press 57 Shelton Street Covent Garden London WC2H 9HE Distributed by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

The editor and the publisher would like to thank the following organisations and publishers for their assistance and their permission to reproduce the articles found in this volume: Academic Press (J. Math. Anal. Appi), American Academy of Arts and Sciences, American Mathematical Society, American Philosophical Society, American Physical Society (Phys. Rev. Lett., Rev. Mod. Phys.), Blackwell Science (Mon. Not. R. Astron. Soc), Current Science Association, Elsevier Science Publishers B. V., Graduate Institute for Applied Mathematics (J. Rat. Mech. Anal.), Kluwer Academic Publishers, Macmillan Magazines (Nature), New York Academy of Sciences, Plenum Press, Royal Society, Springer-Verlag (Z Astrophys.), University of Chicago Press (Astrophys. J.)

Cover picture: An Individual's View of the Individual (Man on the Ladder), by Piero Borello. Courtesy of the artist. A QUEST FOR PERSPECTIVES Selected Works of S. Chandrasekhar (with Commentary) Volume 1 Copyright © 2001 by Imperial College Press All rights reserved. This book, or parts thereof may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN ISBN ISBN ISBN

1-86094-201-6 (set) 1-86094-208-3 (pbk) (set) 1-86094-283-0 1-86094-284-9 (pbk)

Printed in Singapore by Uto-Print

To Lalitha Chandrasekhar

Vll

Contents Preface

ix

Volume 1 I.

Early Years; The Theory of White Dwarfs and Stellar Interiors

II.

Radiative Transfer, the Polarization of the Sunlit Sky, and the

1

Negative Ion of Hydrogen

143

III.

Stochastic and Statistical Problems in Astronomy

307

IV

Turbulence and Hydromagnetic Problems in Astrophysics

457

V. Hydrodynamic and Hydromagnetic Stability

531

Volume 2 VI.

Ellipsoidal Figures of Equilibrium

651

VII.

Relativistic Instabilities and Post-Newtonian Approximations

743

VIII.

The Mathematical Theory of Black Holes; Colliding Waves

877

The Nonradial Oscillations of Stars in General Relativity

997

IX. X.

Miscellaneous Writings

Publications by S. Chandrasekhar

1089 1411

IX

Preface It is well known that S. Chandrasekhar (known simply as Chandra in the scientific world) followed a unique style of research. This is best described in his own words in the autobiographical account published along with his Nobel Lecture in 1983: After the early preparatory years, my scientific work has followed a certain pattern motivated principally by a quest after perspectives. In practice, this quest has consisted in my choosing (after some trials and tribulations) a certain area, which appears amenable to cultivation and compatible with my taste, ability, and temperament. And when after some years of study, I feel that I have accumulated a sufficient body of knowledge and achieved a view of my own, I have an urge to present my point of view ab initio, in a coherent account with order, form, and structure. Thus, Chandra's research, motivated principally by a quest after perspectives, covered a wide range of investigations that comprised: (1) stellar structure, including the theory of white dwarfs (1929-39); (2) stellar dynamics, including the theory of Brownian motion (1938^43); (3) the theory of radiative transfer, the theory of the illumination and the polarization of the sunlit sky, and the quantum theory of the negative ion of hydrogen (1943-50); (4) hydrodynamic and hydromagnetic stability (1952-61); (5) the equilibrium and the stability of ellipsoidal figures of equilibrium (1961-68); (6) the general theory of relativity and relativistic astrophysics (1962— 71); (7) the mathematical theory of black holes (1974-83); (8) colliding wave, the nonradial oscillations of stars in general relativity and the writing of Newton's Principia (1983-95). During each of these periods lasting several years, Chandra produced a series of papers, and in most cases the series ended with a monograph on the subject that represented his "matured outlook, and the subject matter organized in a coherent framework." Because of these books and monographs, one may question the need for a volume or volumes of collected original papers. Indeed, Chandra himself, when approached by the University of Chicago Press, with the idea of publishing a collection of his papers, thought it would be superfluous to do so. Eventually, when he was persuaded, he sought, in the selection process, the help of Martin Schwarzschild, Robert Mullikin, H.C. van de Hulst, Norman Lebovitz, Kip Thorne, and Basilis Xanthopoulos, who were familiar with one or another major area of research in which he had worked. The selection of the papers was based on two criteria: first, none of the selected papers should have already been included in any of his published books, and second, the papers of possible historical interest should be given preference. The University of Chicago

X

Press has published six volumes (and a seventh one after his death) containing a major body of Chandra's original papers and articles. These volumes containing published papers in their original form are of extreme importance to researchers and historians of science. They are an invaluable source for studying the subject as it developed, uninfluenced by Chandra's own perspective and his attempt to provide an integrated and coherent account in the monographs. Bearing this in mind, I have collected together in this single volume a subset of original papers from each of the aforementioned periods of Chandra's research. The first nine parts contain technical articles, while the last part is devoted to some articles of a popular and historical nature. In selecting articles for this anthology, my principal aim has been to include those papers that are to a great extent self-contained and laid the groundwork for subsequent detailed investigations in a given area and a given problem. I have also made it a point to include many of Chandra's articles that are based on his talks at various conferences. They are masterly, complete-with-historical-background, concurrent work, and his own work put in a proper perspective. While the seven volumes of Selected Papers published by the University of Chicago Press are indispensable to libraries and research institutions, I hope the present anthology of selected papers will make Chandra's original papers more affordable and accessible to individual researchers. It gives only a glimpse, perhaps, at the vast arena of theoretical physics and astrophysics that Chandra dominated with his monumental contributions. The readers may benefit by two other books that will supplement this anthology. These are: (1) Black Holes and Relativistic Stars (edited by Robert M. Wald, University of Chicago Press, 1998), the proceedings of a "working scientific symposium" on the theory of black holes and relativistic stars that was the preoccupation of Chandra during the last phase of his life; (2) From White Dwarfs to Black Holes: The Legacy of S. Chandrasekhar (edited by G. Srinivasan, University of Chicago Press, 1999). The articles in these two volumes provide a grand tour of the colossal scientific edifice Chandra has left behind. While presenting a summary of the important contributions to a particular area, the articles describe also the impact of Chandra's work on the further development of the subject. Finally, I would like to thank Ms Emily Davis for her help in preparing this anthology. My thanks are also due to Senior Editor H.T. Leong and his colleagues at World Scientific for the excellence they have brought to this book.

Kameshwar C. Wall November 30, 2000

1

I.

Early Years; The Theory of White Dwarfs and Stellar Interiors

The papers in this section span the years 1929-40, beginning with the very first paper Chandra published, when he was still an undergraduate at the Presidency College, Madras, India. This paper deals with the effect of the then new Fermi-Dirac quantum statistics on Compton scattering. Chandra was introduced to the new quantum mechanics by Arnold Sommerfeld during the latter's visit to Madras in the fall of 1928. The new quantum mechanics, which had stunned Europe, had not yet made its way to India. Sommerfeld was invited to speak to the science students and Chandra, who was among them, made arrangements to see him the following day in his hotel room. Chandra had mastered the atomic theory as laid out in Sommerfeld's classic book on the old quantum theory, Atomic Structure and Spectral Lines. He approached him with the brash confidence of a young undergraduate to impress the master with his knowledge as well as his intense desire to pursue a research career in physics. But Sommerfeld shocked him by telling him that the old quantum theory in his book was no longer of any use. It had been replaced by the revolutionary new quantum mechanics due to Schrodinger, Heisenberg, Dirac, and others. While Chandra had also studied on his own the classical Maxwell-Boltzmann statistics, Sommerfeld told him that too had undergone a fundamental change in the light of the new quantum mechanics. Seeing a crestfallen young student in front of him, Sommerfeld offered him the galley proofs of his as-yet-unpublished paper that contained an account of the new FermiDirac quantum statistics and its application to the electron theory of metals. Chandra would later characterize this encounter as the "single most important event" in his scientific career. For someone with less determination and passion for study than Chandra, such an encounter would have had disastrous consequences. But Chandra immediately launched into a serious study of the new developments in atomic theory. From Sommerfeld's paper, he learned enough about Fermi-Dirac statistics to write, within a few months, his first paper, titled "The Compton Scattering and the New Statistics." He sent it to Ralph H. Fowler in Cambridge, England, requesting him to communicate it for publication in the Proceedings of the Royal Society. Chandra had chosen Fowler since he had come across, by pure chance, soon after his encounter with Sommerfeld, Fowler's paper on the theory of collapsed configuration of stars, namely the white dwarfs. Published in the Monthly Notices of the Royal Astronomical Society, it contained still another application of Fermi-Dirac statistics to the stellar matter in the form of degenerate electrons in white dwarfs. So for Chandra, at the time, Fowler was someone who knew Fermi-Dirac statistics and consequently someone who could understand his paper and help its publication. The paper was indeed published in the Proceedings. However, this chance

2

circumstance was to have a profound influence on Chandra's future scientific career. The following year, when he was offered unexpectedly a Government of India scholarship to continue his research in England after his graduation, he didn't have to think hard before choosing Cambridge University, and Fowler as his doctoral adviser. He began to study the theory of white dwarfs and was able immediately to extend Fowler's work by combining Fowler's ideas with Eddington's polytropic considerations for a star. Papers 2 to 5 contain Chandra's most significant work in his early years, leading to the celebrated discovery of the Chandrasekhar limit on the mass of a star that could become a white dwarf and its broader implications for the problem of stellar evolution. Paper 2 contains a brief account of the discovery of the limiting mass he made on his long voyage from India to England in 1930. A point of historical interest: Paper 3, written soon thereafter, contains a more detailed account. Chandra had withheld from submitting it for publication in the British journals because of E.A. Milne's dissent and his influence on what was published and what was not. It is this paper that concludes with the often quoted statement "Great progress in the analysis of stellar structure is not possible before we can answer the following fundamental question: Given an enclosure containing electrons and atomic nuclei (total charge zero), what happens if we go on compressing the material indefinitely?" In order to avoid confrontation with Milne, Chandra waited till he was in Copenhagen at the Niels Bohr Institute, before sending it for publication in Zeitschrift fuer Astrophysik at Potsdam, Germany. But, as luck would have it, Milne was in Potsdam and he was asked to referee the paper. Milne advised against publishing the paper and wrote to Chandra saying, "Unfortunately I have been unable to recommend acceptance, as the paper contains a mistake in principle, and in any case it would only do harm to your reputation if it were published." Papers 4, 5 and 7 are the seminal papers on the theory of white dwarfs. During the fall of 1934, Chandra worked out the complete theory using the exact relativistic equation of state describing degenerate matter. With extensive numerical work, he established beyond any doubt the validity of the limiting mass condition, reiterating the question he had posed in his earlier paper. A star with mass greater than the limiting mass would not reach an equilibrium state of a white dwarf in the course of its evolution. It will continue to collapse. Chandra presented his results at the January 1935 meeting of the Royal Astronomical Society. His findings raised challenging, fundamental questions. What happens to the more massive stars as they continue to collapse? Are there terminal stages of stars other than that of white dwarfs? Paper 6 contains what transpired after Chandra presented his paper (no. 5). Instead of getting appreciation and recognition for a fundamental discovery, Chandra unexpectedly faced what amounted to a public humiliation. Because no sooner had he presented his paper than Sir Arthur Eddington tore apart Chandra's dramatically stated conclusions by attacking the very concept of relativistic degeneracy on which Chandra's work depended. Characterizing it as leading to stellar buffoonery and reductio ad absurdum behavior of a star, he made it look as though Chandra had made a simple conceptual error and gotten it all wrong. While he himself missed the great opportunity of being the first one to recognize the possibility of the existence of black holes and other terminal stages of a star, Eddington's attack had a profound influence on Chandra's personal life as well as the progress of astronomy. The irony of this encounter and the consequences of this unexpected occurrence are more fully described elsewhere.

3

After this historic meeting and the unexpected encounter, Chandra put aside his work on white dwarfs and went on to study stellar dynamics and other problems. But, four years later, in 1939, he was invited to an international meeting in Paris devoted to the special topics of novae and white dwarfs. Chandra and Eddington both gave talks. Paper 8 contains Chandra's paper and the discussion afterward at this meeting. Chandra took the opportunity to state again his conclusions regarding the limiting mass based on relativistic degeneracy. Included in this paper is some work he had done on rotating white dwarfs. He had withheld it from publication because of the controversy. Paper 9 is drawn from the Proceedings of the American Philosophical Society. It contains a description of the general methods to determine the physical conditions in stellar interiors and the theory of white dwarfs at the end of the decade of the 1930s. The last paper in this part is of special interest, in that it is totally out of line with the general body of Chandra's work. It is a very brief communication published in Nature on certain numerical coincidences based on purely dimensional arguments. Saying that he was hesitating to publish this on account of the conviction that "purely dimensional arguments" do not lead one very far, he presents a formula in terms of the well-known fundamental constants, h (Planck's constant), c (velocity of light), G (Newton's constant), and H (mass of the proton). The formula, for certain powers of the combination of these constants, yields cosmological constants such as masses of stellar magnitude, the number of particles in the universe, and the mass of the Milky Way.

5

I.

Early Years; The Theory of White Dwarfs and Stellar Interiors

1.

The Compton Scattering and the New Statistics Proceedings of the Royal Society A125 (1929): 231-37

2.

The Maximum Mass of Ideal White Dwarfs The Astrophysical Journal 74, no. 1 (1931): 81-82

13

3.

Some Remarks on the State of Matter in the Interior of Stars Zeitschrift fuer Astrophysik 5, no. 5 (1932): 321-27

15

4.

The Highly Collapsed Configurations of a Stellar Mass (second paper) Monthly Notices of the Royal Astronomical Society 95, no. 3 (1935): 207-25

22

5.

Stellar Configurations with Degenerate Cores Monthly Notices of the Royal Astronomical Society 95, no. 3 (1935): 226-60

41

6.

Discussion of Papers 4 and 5, A.S. Eddington and E.A. Milne The Observatory 58, no. 729 (1935): 37-39, 52

76

7.

Stellar Configurations with Degenerate Cores (second paper) Monthly Notices of the Royal Astronomical Society 95, no. 8 (1935): 676-93

80

8.

The White Dwarfs and Their Importance for Theories of Stellar Evolution Conference du College de France, Colloque International d'Astrophysique, 17-23 Juillet 1939 (Hermann, Paris, 1941)

98

9.

The Internal Constitution of the Stars Proceedings of the American Philosophical Society 81, no. 2 (1939): 153-86

10. The Cosmological Constants Nature 139 (1937): 757

6

108

142

The Compton Scattering By S.

CHANDRASEKHAR,

and the New

Statistics.

The Presidency College, Madras.

(Communicated by R. H. Fowler, F.R.S.—Received June 20, 1929.) 1. Introduction. Great success has been achieved by Sommerfeld in the electron theory of metals by assuming that there are free electrons in them which obey the Fermi-Dirac statistics. I t has been assumed in the case of univalent metals that on the average one electron per atom is free. In general, however, the valency electrons can be considered as free.* These free electrons will take part in the Compton scattering. The analysis of such a Compton effect reduces to the analysis of the collisions between radiation quanta and an electron gas. The general features of such a scattering was first considered by Dirac.f But he has assumed a Maxwellian distribution for the electrons which will not be applicable to the case under consideration, because the electrons in a conductor being degenerate do not obey the Maxwell's law, but the Fermian distribution. In considering such a process we take it that the conservation of momentum and energy principles are satisfied for each particular collision just as in Compton's theory—only we are here dealing with moving electrons instead of stationary electrons which Compton considers. Thus electrons of different momenta components will produce different Compton shifts, and the intensity of any particular shift will depend on the number of electrons in that state. Thus we have to average for the radiation falling on an assembly of electrons whose momenta are distributed according to the Fermi-Dirac law. The above is just a natural extension of Compton's theory. In this connection mention should be made of Jauncey'sJ theory of bound electrons whose arguments are essentially what we have put forward in the previous paragraph. But his theory has not been quite satisfactory because he has not assumed any definite distribution of the electrons.

* Roaenfeld,' Naturw.,' p. 49 (1929). t ' M.N.R.A.S.,' vol. 85, p. 825 (1925). % ' Phys. Rev.,' vol. 25, p. 723 (1925).

[231]

2. Compton scattering ivith Moving Electrons. Let nij., mv, in. be the momentum of the scattering electron and gx. gy, gt those of the quantum, g,, int represent the masses of the electron and the quantum multiplied by the velocity of light c.

If we take polar co-ordinates

gx — /tv cos 0/c ; g„ = /*v sin 0 cos /c ; g. = sin 6 sin (f> gt = Av/c.

(1)

Then the conservation of momentum and energy gives (mtt, gu) — (mu, gu') = (gu, g„').

(2)

The above equation gives the frequency of the scattered quantum in terms of the initial momentum of the electron and the incident quantum, and the directions of the incident and scattered quanta. Equation (2) reduces to m, — mx cos 6' — m„ sin 0' = — (mt — ma) v

(1 — cos 0'),

(3)

c

if we assume that the directions of the incident quantum is along the x axis and that of the scattered quantum in the xy plane. Here 0' is simply the angle of scattering. 3. The Spectral-intensity Distribution

Function.

Before considering the case of scattering of monochromatic X-radiation, we will consider first the more general case when the incident radiation is continuous. Suppose we have such a pencil of radiation confined to a small solid angle dm and let I„ be the intensity per unit frequency range. Let this radiation be incident on an assembly of dn electrons of momentum mx, mv> m2. Let the intensity of radiation scattered in the solid angle do' and frequency range v' and v' + dv' be given by R(\>')dVdo>'.

(4)

Then it has been shown by Dirac (he. cit., equation (8)) that h -at >\ * A T J v'F (o, b) K (v') = —— .dn.ly da s-!—'-. (5) mrcr vwit Here v' is to be regarded as a function of gx', gv', g„' and mx, mv, m„ being that frequency of the incident quantum which will be scattered by an (mx, mv, mz) electron into the frequency range v' to v' -f- dv'. In the above equation F (a, b) is a function which depends on the scattering law adopted and a and b the two invariants connected with the scattering

[232]

process which as well as the initial momentum mx, mv, mz of the electron and ffz> 9v> 9' °f *^ e < l u a i i t u m specify the collision. Now for dn in equation (5), we have to put the Fermi-expression ,

= n

V e h3'

_ dmr dmv dm^ ' e x p (Smx2/2mJfcf )/A + 1 '

(6)

and integrate with respect to mx, mv, mz. In the above equation A is the constant appearing in the Fermi-Dirac statistics. I t has different values according as we consider a degenerate or a non-degenerate gas. When the Bystem is non-degenerate A is a small positive quantity and then has the value A = nh3. (2nmkT)-z'ijG. (7) A degenerate system corresponds to A being a large quantity and in that case , . / 3n \ 2/3 Jfi ia.

logA =

(8)

te) -2SSf-

Then by equation (6) m2c3 h? W rfc3 Where

JJ J

I„dcov'F dm- dm,, dm. vm, exp (Zmx*l2mkT)IA + 1

V .Gf"l r dfi)(Kv, v')iv. h3 Jo J J-oo

(9) (10)

F dmv dmz VOT vm, exp (2m x 2 /2m£T)/A + 1/ 8m,

(11)

Where mx and 8 v/3mx are to be evaluated in terms of mv, mz and v by means of equation (3) mc h (6) ^ = (12) w ev c' v ' (1 - cos 6') and 2 2 r K s i— n S 'Jl 2 +, K (13) mx2 +L me22 = ^P m - (/ ,l sin 6' v where P = 1 — 2v cos 6'/v' + (v/v')2, y = v/v* - cos 0', K = - » n e ( v / v ' —1) + Av(l —cos6')/c F

P A P|"dk_: ex

.'..2

1 v(l — cos 6') KsinG'"

L

Y2 2w»AT

hV fa, im, mc2v.sin2 0' 1

P

-1- ».

(18)

If we introduce the new variables y = m'v2 {3/y2 . 2mkT, z = m^flrnkT. Then V^>

>

°U(l-cos0')

mc2vJ

p*

J J0

e"+2/B + l

where U0 is the special case of the general Sommerfeld integral

u,=—-L-.r rhich gives for p = 0*

uPdu

,

( 20)

r( P + i) Jo C«/B + I

U0 = 7ilog(B + l). Hence we get our intensity distribution function

^•^^[w^¥)-^-y-^-^(B+i)-

(21)



(2)

x = the opacity coefficient, L = luminosity in ergs cm -3 , M = mass in grams, k = BOLTZMANN'S Constant, /( = molecular weight = OC/KJJ (say), nig = mass of the hydrogen atom. Since for the standard model the gas pressure p is given by = we have V P > \ ) where

*= ^ ("/ k x* 8 1 - Wl* 2.682 • 10" r 1 - jjl 1 /.

') Refered to as I.e. Copyright 1932 by Springer-Verlag. Reprinted by permission.

[321]

®

The equation of state in the degenerate zone is v = -£i