Physics of Gravitating Systems. Equilibrium and Stability

Transcription

1 Physics of Gravitating Systems I Equilibrium and Stability

2 A. M. Fridman V. L. Polyachenko Physics of Gravitating Systems I Equilibrium and Stability Translated by A. B. Aries and Igor N. Poliakoff With 85 Illustrations Springer-Verlag New York Berlin Heidelberg Tokyo

3 A. M. Fridman V. L. Polyachenko Astrosovet UI. Pyatnitskaya Moscow Sh-17 U.S.S.R. Translators A. B. Aries 4 Park A venue New York, NY U.S.A. Igor N. Poliakoff 3 Linden A venue Spring Valley, NY U.S.A. Library of Congress Cataloging in Publieation Data Fridman, A. M. (AlekseI Maksimovieh) Physies of gravitating systems. Includes bibliography and index. Contents: 1. Equilibrium and stability -2. Nonlinear eolleetive proeesses. Astrophysieal applieations. 1. Gravitation. 2. Equilibrium. 3. Astrophysies. 1. Polfaehenko, V. L. (Valerii L'vovieh) II. Title. QC178.F This is a revised and expanded English edition of: Ravnovesie gravitiru{ushchikh sistem. Moseow, Nauka, ustolchivost' 1984 by Springer-Verlag New York Ine. AII rights reserved. No part of this book may be translated or reprodueed in any form without written permission from Springer-Verlag, 175 Fifth Avenue, New York, New York 10010, D.S.A. Typeset by Composition House, Ltd., Salisbury, England. Printed and bound by R. R. Donnelley & Sons, Harrisonburg, Virginia ISBN ISBN (ebook) DOI /

4 Dedication This book, which has been written by physicists, originated from the ancient problems of astronomy which are related to the shapes and evolution of celestial bodies and their systems. It has become a tradition among physicists to study individual fields of astrophysics. About 30 years ago this would have been a great event. At that time Professor D. A. Frank-Kamenetsky was one of the first Soviet physicists to begin to develop consistently the fundamental problems of astrophysics. Being a very erudite scientist he succeeded in influencing, by his passionate inquisitiveness, quite a number of young physicists who chose astrophysics as their speciality. We dedicate this book to the memory of D. A. Frank-Kamenetsky-a prominent scientist and a man of pure soul. v

5 Preface l It would seem that any specialist in plasma physics studying a medium in which the interaction between particles is as distance-dependent as the interaction between stars and other gravitating masses would assert that the role of collective effects in the dynamics of gravitating systems must be decisive. However, among astronomers this point of view has been recognized only very recently. So, comparatively recently, serious consideration has been devoted to theories of galactic spiral structure in which the dominant role is played by the orbital properties of individual stars rather than collective effects. In this connection we would like to draw the reader's attention to a difference in the scientific traditions of plasma physicists and astronomers, whereby the former have explained the delay of the onset of controlled thermonuclear fusion by the "intrigues" of collective processes in the plasma, while many a generation of astronomers were calculating star motions, solar and lunar eclipses, and a number of other fine effects for many years ahead by making excellent use of only the laws of Newtonian mechanics. Therefore, for an astronomer, it is perhaps not easy to agree with the fact that the evolution of stellar systems is controlled mainly by collective effects, and the habitual methods of theoretical mechanics III astronomy must make way for the method of self-consistent fields. [ This extended preface to the Russian edition (\976) reflects the achievements in all the topics referred to therein over the last eight years. It especially concerns Part II, specifically the theory of multi-component systems and the problems of the nonlinear theory, the foundations of which were described in the Russian edition (for instance, the nonlinear theory of a gra vitating disk, and the nonlinear waves and solitons traveling in such a disk). VII

6 viii Preface Small oscillations of the medium can be considered as the simplest phenomenon in which collective effects are essential. The main purpose of this book is to treat systematically the theory of small oscillations, equilibrium and stability of gravitating systems. The theory to be presented below already has a variety of applications. Especially widespread recognition has been gained by its application to the problem of galactic spiral structure. Application of the results of the stability theory of gravitating disks with a central body to the problem of the law of planetary distances or of the rings of Saturn has gained a widespread reputation. In this book the first priority is assigned to systems composed of a large number of gravitating masses not colliding with each other. Stellar systems of various sizes (for example, galaxies or globular star clusters), first of all, belong to such collision less gravitating systems. The foundations of the statistical mechanics of stellar systems (physical statistics of particles interacting in accordance with Newton's law) were laid down in the 1930s by Ambartsumian [128 ad ]. One of the fundamental results in stellar dynamics (the proof of the "short-time scale" [129 ad ]) had been derived while examining the evolution of stellar systems which makes use of the distribution function in phase space while depending on the fundamental integrals of motion. The approach of statistical mechanics of stellar systems is largely used in this book. Theoretical investigations in stellar dynamics, particularly the problem of the stability of star systems, have risen to a completely new level upon penetration into this field of the research methods developed earlier in plasma physics. It is not out of place to mention here the pioneering investigations of collective effects in a gravitating medium carried out by Antonov [4], Lynden-Bell [281], Sweet [330], and others. They introduced the idea of instability as the spontaneous excitation of collective modes of oscillations of the medium, widely used in the theory of plasma instabilities. Thus, obsolete works devoted to the stability of orbits of individual particles were discarded from the theory of stability of gravitating systems, and the" one-particle" description of Newtonian mechanics was replaced by the statistical method for describing systems of many particles, which makes use of the distribution function concept that satisfies the kinetic equation. The dispersion equation method, usually applied in plasma theory, became the main method for studying stability. Also, many methods of solving the kinetic equation were taken from plasma theory-the method of "integration over trajectories," widely used in this book, may be cited as an example. Finally, in the theory of gravitational instability, different approximated research methods, such as the WKB method (or the method of local analysis), also became habitual. The period of general passion for plasma methods and their somewhat one-sided application to gravitation was followed by a period of more attention to the possibilities of the theory, which have also revealed deep differences between plasma and the gravitating medium, are due to the difference in sign of interaction of electrically charged particles and gravi-

7 Preface IX tating masses and to the absence of screening of gravitational attraction. We treat the question of the analogy between the plasma and gravitating media in the Introduction, where we also give the basic equations of the theory, and where the problem of gravitational (Jeans) instability of a homogeneous medium is discussed. In Chapter I, which also has an introductory character, the stability theory of the simplest gravitating system-a homogeneous collisionless fiat layer-is set forth. The methods used later can be readily illustrated, in particular, by this example. Chapters II-IV deal with a study of equilibrium and stability of four "classical" figures of equilibrium: cylinder, sphere, ellipsoid, and disk. In the first paragraphs of each chapter a review of results of the equilibrium theory of relevant systems is provided. It should be noted that the equilibrium theory of collisionless gravitating systems started to develop much earlier than stability theory. The works of Eddington [196,197], Jeans [241,242], Camm [179, 180], Freeman [ ], and others played the main role in the formation of this theory. Primarily, works by Kuz'min [58-62], Weltmann [32], Bisnovatyi-Kogan and Zel'dovich [21-23] should be distinguished among the works of Soviet investigators. Small perturbations of cylindrical configurations are considered in Chapter II. The investigation of stability of a collisionless cylinder with respect to arbitrary perturbations has been carried out by the authors and Mikhailovskii; also the beam instability in a gravitating medium was investigated in collaboration with Mikhailovskii [88]. "Flute" -type oscillations have been discussed by the authors (together with Shukhman) [112, lis] as well as by Antonov [14]. Chapter III is devoted to spherically symmetric systems. In this chapter we present the results which were obtained primarily by our group and by Antonov. Chapter IV gives the analysis of stability of ellipsoidal gravitating systems, carried out by the authors together with Morozov and Shukhman. The problem of the stability of disk systems (Chapter V) drew the attention of a large number of investigators. As far as applications of the theory of stability of a gravitating disk to the problem of spiral structure formation of galaxies are concerned, the basic ideas and conclusions belong to Lin and Shu [ ], Kalnajs [250, 289], Toomre [333, 334], and Lynden-Bell [289]. In spite of the undoubtedly interesting results achieved in these areas, we have arrived at the opinion that even the qualitative picture of the formation of spiral structure of galaxies depicted now is the only possible one. 2 For instance, in the process of the airal structure formation a distinct role can be played by instabilities of the non-jeans type, which are discussed in Chapter VI of this book. These instabilities have been investigated by the authors together with Moiozov, 2 This opinion is shared by many authors (see, e.g., [II, 303, 304]). Reviews related to the problem of the spiral structure of galaxies also point out difficulties of the existing theory (e.g., [84]).

8 x Preface Fainstein, and Shukhman as well as by Bisnovatyi-Kogan and Mikhailovskii and Mark and Kulsrud. Chapter VII is notable for the discussion of some questions of nonlinear theory. It presents the basic problems, which have now been solved Gointly with Mikhailovskii, Petviashvili, Frenkel, Churilov, and Shukhman). One section of this chapter is dedicated to the results of the "pancake" theory as derived by Zel'dovich and his co-workers. Some astrophysical applications of the theory of equilibrium and stability of gravitating systems are treated in Part II. We decided that it might be useful to have, for reference, tables of all the instabilities studied hitherto which can develop in various gravitating systems. There are two such tables in the book. The first one characterizes the Jeans instability of a multicomponent homogeneous medium. In commentaries to the table, among other things, a theorem is formulated of the Jeans instability in a multi component medium at rest. (The theorem of the number of instabilities for the general case of moving components is formulated in l of Chapter VI.) The second table characterizes all the non-jeans instabilities and is placed in the last section ( 12) of the Appendix. It contains: the geometry of the system considered, instability conditions, instability growth rates, typical frequencies of perturbed waves, and references to works in which this or that instability was first described. In commentaries to the table we describe briefly the physical mechanisms responsible for non-jeans instabilities. At the end of most of the chapters is a series of problems (and their solutions) which throw some light on details of mechanisms either operating in gravitating systems or somewhat removed from the basic class of questions. It seemed reasonable to number the formulae separately within each section (and even in subsections), because references to formulae from other sections are comparatively rare. Of course, we were unable to present the results of all authors on the subject of interest. When selecting any given work, we proceeded, first of all, from the reliability of results obtained in it. Therefore, we have omitted papers written on an almost purely" intuitive" level and those not supported by strict mathematical analysis. There is also considerable duplication of results. In such cases we selected, as a rule, the original work. It is possible, however, that taking into account the reservations made above, not all the papers which deserve to be mentioned have been cited in the book. We express our sincere apologies to the authors of those papers. In order to read this book a general knowledge of the kinetic equation and of the basic concepts of potential theory is sufficient. Of course, for the reader familiar with the principles of plasma instability theory it will be easier to orientate himself in the subject of the book. Nevertheless, when writing the book, we have not supposed the reader to have any special knowledge of the physics of plasma instabilities (or of gravitational stability theory). We acknowledge with gratitude our joint work with A. B. Mikhailovskii,

9 Preface xi R. Z. Sagdeev, and Ya. B. Zel'dovich without whose support and advice this book would not have come into being. The general plan of the book was discussed with A. B. Mikhailovskii, L. M. Ozernoy, and Ya. B. Zel'dovich, who made a number of suggestions which we tried to take into account in the course of our work on the manuscript. We would especially like to thank our colleagues and friends I. G. Shukhman, A. G. Morozov, S. M. Churilov, and V. S. Synakh. Their valuable contribution is not restricted to providing the above-mentioned results but extends to the actual writing of some sections of Chapter VII. I. G. Shukhman together with A. G. Morozov wrote Section 2.2 and, together with S. M. Churilov, 4 and the supplement to it. Section 3 of this chapter is devoted to the results of the" pancake" theory obtained by Ya. B. Zel'dovich and his colleagues. This section was written at the authors' request by A. G. Doroshkevich. V. S. Synakh co-authored our first papers with the use of a computer. We wish to convey our sincere gratitude to all of them. We are also grateful to G. I. Marchuk and A. G. Massevitch for their support during the preparation of the English edition. We thank L. A. Chujanova for her great help in the design of the manuscript. Moscow April, 1984 A. FRIDMAN V. POLYACHENKO

10 Contents (Volume I) Introduction I I. Basic Concepts and Equations of Theory 2 2. Equilibrium States of Collisionless Gravitating Systems 6 ~ 3. Small Oscillations and Stability 9 4. Jeans Instability of a One-Component Uniform Medium 10 ~ 5. Jeans Instability of a Multicomponent Uniform Medium Basic Theorem (on the Stability of a Multicomponent System with Components at Rest) Four Limiting Cases for a Two-Component Medium Table of Jeans Instabilities of a Uniform Two-Component Medium General Case of n Components Non-Jeans Instabilities Qualitative Discussion of the Stability of Spherical, Cylindrical (and Disk-Shaped) Systems with Respect to Radial Perturbations 21 PART I Theory CHAPTER I Equilibrium and Stability of a Nonrotating Flat Gravitating Layer Equilibrium States of a Collisionless Flat Layer Gravitational (Jeans) Instability of the Layer 31 XIll

11 xiv Contents (Volume I) 3. Anisotropic (Fire-Hose) Instability of a Collisionless Flat Layer Qualitative Considerations Derivation of the Dispersion Equation for Bending Perturbations of a Thin Layer Fire-Hose Instability of a Highly Anisotropic Flat Layer Analysis of the Dispersion Equation Additional Remarks Derivation of Integro-Differential Equations for Normal Modes of a Flat Gravitating Layer Symmetrical Perturbations of a Flat Layer with an Isotropic Distribution Function Near the Stability Boundary Perpendicular Oscillations of a Homogeneous Collisionless Layer Derivation of the Characteristic Equation for Eigenfrequencies Stability of the Model Permutational Modes Time-Independent Perturba.tions (co = 0) 69 Pro~~ ~ CHAPTER II Equilibrium and Stability of a Collisionless Cylinder Equilibrium Cylindrical Configurations Jeans Instability of a Cylinder with Finite Radius Dispersion Equation for Eigenfrequencies of Axial-Symmetrical Perturbations of a Cylinder with Circular Orbits of Particles Branches of Axial-Symmetrical Oscillations of a Rotating Cylinder with Maxwellian Distribution of Particles in Longitudinal Velocities Oscillative Branches of the Rotating Cylinder with a Jackson Distribution Function (in Longitudinal Velocities) Axial-Symmetrical Perturbations of Cylindrical Models of a More General Type Nonaxial Perturbations of a Collisionless Cylinder The Long-Wave Fire-Hose Instability Nonaxial Perturbations of a Cylinder with Circular Particle Orbits Stability of a Cylinder with Respect to Flute-like Perturbations Local Analysis of the Stability of Cylinders (Flute-like Perturbations) Dispersion Equation for Model (2), I Maxwellian Distribution Function Comparison with Oscillations of an Incompressible Cylinder Flute-like Perturbations (k z = 0) Flute-like Oscillations of a Nonuniform Cylinder with Circular Orbits of Particles 119 Problems 125 CHAPTER III Equilibrium and Stability of Collisionless Spherically Symmetrical Systems Equilibrium Distribution Functions Stability of Systems with an Isotropic Particle Velocity Distribution The General Variational Principle for Gravitating Systems with the Isotropic Distribution of Particles in Velocities U~ = j~(e), f~ = dfo/de :-:; 0) 152

12 Contents (Volume I) xv 2.2. Sufficient Condition of Stability Other Theorems about Stability. Stability with Respect to Nonradial Perturbations Variational Principle for Radial Perturbations Hydrodynamical Analogy On the Stability of Systems with Distribution Functions That Do Not Satisfy the Condition f~(e) S; Stability of Systems of Gravitating Particles Moving On Circular Trajectories Stability of a Uniform Sphere Stability of a Homogeneous System of Particles with Nearly Circular Orbits Stability of a Homogeneous Sphere with Finite Angular Momentum Stability of Inhomogeneous Systems Stability of Systems of Gravitating Particles Moving in Elliptical Orbits Stability of a Sphere with Arbitrary Elliptical Particle Orbits Instability of a Rotating Freeman Sphere Stability of Systems with Radial Trajectories of Particles Linear Stability Theory Simulation of a Nonlinear Stage of Evolution Stability of Spherically Symmetrical Systems of General Form Series of the Idlis Distribution Functions First Series of Camm Distribution Functions (Generalized Poly tropes) Shuster's Model in the Phase Description Discussion of the Results 235 Problems 238 CHAPTER IV Equilibrium and Stability of Collisionless Ellipsoidal Systems 246 I. Equilibrium Distribution Functions Freeman's Ellipsoidal Models "Hot" Models of Collision less Ellipsoids of Revolution Stability of a Three-Axial Ellipsoid and an Elliptical Disk Stability of a Three-Axial Ellipsoid Stability of Freeman Elliptical Disks Stability of Two-Axial Collisionless Ellipsoidal Systems Stability of Freeman's Spheroids Peebles-Ostriker Stability Criterion. Stability of Uniform Ellipsoids, "Hot" in the Plane of Rotation The Fire-Hose Instability of Ellipsoidal Stellar Systems Secular and Dynamical Instability. Characteristic Equation for Eigenfrequencies of Oscillations of Maclaurin Ellipsoids 294 Problems 296 CHAPTER V Equilibrium and Stability of Flat Gravitating Systems 323 I. Equilibrium States of Flat Gaseous and Collisionless Systems Systems with Circular Particle Orbits Plasma Systems with a Magnetic Field 334

13 XVI Contents (Volume I) 1.3. Gaseous Systems "Hot" Collision less Systems Stability of a "Cold" Rotating Disk Membrane Oscillations of the Disk Oscillations in the Plane of the Disk Stability of a Plasma Disk with a Magnetic Field Qualitative Derivation of the Stability Condition Variational Principle Short-Wave Approximation Numerical Analysis of a Specific Model Stability of a "Hot" Rotating Disk Oscillations in the Plane of the Disk Bending Perturbations Methods of the Stability Investigation of General Collisionless Disk Systems Exact Spectra of Small Perturbations Global Instabilities of Gaseous Disks. Comparison of Stability Properties of Gaseous and Stellar Disks 428 Problems 434 References 445 Additional References 459 Ind~ %5

14 Contents (Volume II) CHAPTER VI Non-Jeans Instabilities of Gravitating Systems CHAPTER VII Problems of Nonlinear Theory PART II Astrophysical Applications CHAPTER VIII General Remarks CHAPTER IX Spherical Systems CHAPTER X Ellipsoidal Systems CHAPTER XI Disk-Like Systems. Spiral Structure CHAPTER XII Other Applications Appendix References Additional References Index XVII