Introduction to the Physics of Stellar Interiors

127 Pages·1973·4.431 MB·English

by  V. Kourganoff (auth.)

INTRODUCTION TO THE
PHYSICS OF STELLAR INTERIORS
ASTROPHYSICS AND
SPACE SCIENCE LIBRARY
A SERIES OF BOOKS ON THE RECENT DEVELOPMENTS
OF SPACE SCIENCE AND OF GENERAL GEOPHYSICS AND ASTROPHYSICS
PUBLISHED IN CONNECTION WITH THE JOURNAL
SPACE SCIENCE REVIEWS
Editorial Board
J. E. BLAMONT, Laboratoire d'Aeronomie, Verrieres, France
R. L. F. BOYD, University College, London, England
L. GOLDBERG, Harvard College Observatory, Cambridge, Mass., U.S.A.
C. DE JAG ER, University of Utrecht, Holland
Z. KOPAL, University of Manchester, Manchester, England
G. H. LUDWIG, NASA, Goddard Space Flight Center, Greenbelt, Md., U.S.A.
R. L liST, Institut fur Extraterrestrische Physik, Garschin!{-Munchen, Germany
B. M. MCCORMAC, Lockheed Palo Alto Research Laboratory, Palo Alto, Cali[., U.S.A.
H. E. NEWELL, NASA, Washington, D.C., U.S.A.
L.1. SEDOV, Academy of Sciences of the U.S.S.R., Moscow, U.S.S.R.
Z. S v ES TK A, Fraunhofer Institute, Freiburg im Breisgau, Germany
Secretary of the Editorial Board
W. DE GRAAFF, Sterrewacht 'Sonnenborgh', University of Utrecht, Utrecht, Holland
VOLUME 34
INTRODUCTION TO
THE PHYSICS OF
STELLAR INTERIORS
by
v. KOURGANOFF
University of Paris-Sud, Orsay
D. REIDEL PUBLISHI G COMPANY
DORDRECHT-HOLLA D / BOSTO -U.S.A.
INTRODUCTION A LA PHYSIQUE DES INTERIEURS STELLAIRES
First published in 1969 by Dunod, Paris
Translated from the French by Janet Rountree Lesh
Library of Congress Catalog Card Number 72-86104
ISBN-13: 978-94-0 I 0-2541-6 e-ISBN-13: 978-94-0 I 0-2539-3
DOT: 10.1007/978-94-0 I 0-2539-3
Published by D. Reidel Publishing Company,
P.O. Box 17, Dordrecht, Holland
Sold and distributed in the U.S.A., Canada, and Mexico
by D. Reidel Publishing Company, Inc.
306 Dartmouth Street, Boston,
Mass. 02116, U.S.A.
All Rights Reserved
Copyright © 1973 by D. Reidel Publishing Company Dordrecht, Holland
No part of this book may be reproduced in any form, by print, photoprint, microfilm,
or any other means, without written permission from the publisher
PREFACE
All astrophysicists are acquainted with the fundamental works ofS. Chandrasekhar [6]
and M. Schwarzschild [1] concerning the internal structure of stars. Although both
of these works accentuate the principal mathematical devices of the theory (and use,
for this reason, notations that are rather perplexing for the non-specialist), the work
of Schwarzschild is distinguished by care in demonstrating the physical meaning of
the principal equations, while that of Chandrasekhar makes every effort not to skip
a single step in the calculations. On the other hand, Schwarz schild , who considers
his two introductory chapters as simple reviews of results which are already known,
passes a bit rapidly over certain difficult arguments, and Chandrasekhar never goes
far enough in the analysis of the physical mechanisms involved.
From another point of view, the excellent review articles published in the Ency
clopedia of Physics [5] by M. H. Wrubel, P. Ledoux, and others, and those published
in Stars and Stellar Systems [4] by H. Reeves, B. Stromgren, R. L. Sears and
R. R. Brownlee, and others, are principally intended for research workers who are
already initiated into the theory of internal structure. These monographs are on a level
that is clearly too high for the general physicist who is approaching these astrophysical
questions for the first time, and more particularly for the post-graduate student.
The many readers in this category require both the continuity of a Chandrasekhar
(logical presentation of the arguments and the calculations) and the 'physical intuition'
of a Schwarz schild or a Stromgren.
For this reason we have been led to reconsider this set of treatises and monographs
from the pedagogical point of view, and to try to clarify the principal physical concepts
used in 'formulating' the theory; for experience in teaching has shown us that these
concepts are generally poorly understood by most beginners. The systematic use of
numerical integration (in an approximation which makes it possible to follow the
details of the calculations) makes the reader aware of the orders of magnitude of the
parameters involved. This is all the more reasonable in that we are faced with a prob
lem in which the data and the results are almost always defined by numerical tables,
and not in terms of functions with known mathematical properties.
All this leads us, after a brief review of some classical results, to emphasize the
derivation of the equation of mechanical equilibrium for the (gaseous) stellar material
- which is often likened, on the basis of a purely formal analogy, to the equation of
equilibrium for a liquid at rest.
Next we take as a starting point, a sort of working hypothesis, the density distribu
tion e(r) (a function of the distance r from the center of the star) obtained by
VI PREFACE
Schwarzschild (at the end of his book) for a model of the Sun in an evolutionary
state close to its present state. Then we dissect, so to speak, the physical mechanism
by which the density distribution (! (r ) determines the mass, pressure, and temperature
distributions in the interior of the star.
We then come to the problem of the energy equilibrium of the star, which leads
us to introduce the principal nuclear reactions involved (the p-p chain and the C-N
cycle). We specify the composition of the nuclei by appropriate schematic represen
tations which enable the beginner to follow more easily the details of the reactions
(which he is all too inclined to consider as a simple play on notation). The use of the
recent monograph by Fowler et al. [2] enables us to specify the numerical values of
the atomic weights and the cross-sections, on which the actual calculation of the
energy output from nuclear reactions depends.
We pass somewhat rapidly over the 'tunnel' effect, which is very well treated in
most of the classical works on quantum mechanics, but we emphasize the specifically
thermonuclear nature of the opposition between the favourable effect of an increase in
temperature on the overcoming of the electrostatic repulsion of the nuclei, and the
unfavourable effect of an excessive increase in temperature on the phenomenon of
'fusion' as such.
We likewise devote special attention to finding the physical meaning of certain ideas
which are generally - and wrongly - considered to be 'obvious', such as the convergence
towards an equilibrium state of a set of cyclical reactions, or the 'mean duration of
cycle'.
After a brief review of the empirical representation of the energy outputs e and
pp
eCN as a function of various parameters and of the temperature T, we proceed to a
'final test' which shows how taking into account the energy production and the
opacity* of the stellar material 'justifies' (and consequently determines) the distri
bution (! (r) used as a starting point.
We conclude with a presentation of the great discovery of M. Schwarzschild: the
necessity of considering the evolution of stellar models if one wants to understand
the structure of the static models.
A section of the last chapter summarizes the essentials of the fundamental mathe
tical concepts used in the construction of models, leaving out everything that makes
the more 'advanced' accounts cumbersome for a beginner.
Finally, since this book is intended not only for the general physicist but also - and
especially - for the student, we give at the end of each chapter a large number of
exercises, mostly drawn from recent original publications. These problems are given
with 'answers' **, which enable the student to check himself; but no detailed solution
is given, as this might make the student too passive. The solution of such exercises
* We assume that the reader is familiar with the theory of the mechanical effects of radiation (radia
tion pressure), with the notion of opacity, and with their application to the theory of stellar structure.
These concepts are explained, in great detail and in a completely elementary fashion, in Chapter VII
of our book Introduction to the General Theory of Particle Transfer [9].
** Except for Exercises l' and 2', which are especially easy.
PREFACE VII
constitutes an excellent introduction to theoretical research for all young physicists,
whatever their ultimate field of specialization.
It goes without saying that the present work, whose ambitions are purposely limited,
constitutes only an introduction to the theory of internal structure and of thermonu
clear reactions. The reader who wishes to pursue the subject can turn to the more
'advanced' works already mentioned above; and since the bibliography in these works
generally stops around 1963 (except for Fowler's monograph), we place at his disposal
a list of references to more recent publications (up to the end of 1971).
We are happy to take this opportunity to thank Dr L. Bottinelli, maitre-assistant
in Astrophysics at the University of Paris-Sud (Orsay), who drew up some of the
questions and most of the 'answers' for the exercises, under our direction.
TABLE OF CONTENTS
v
PREFACE
CHAPTER I. GENERAL CONSIDERA nONS CONCERNING THE ENERGY
RADIATED BY STARS
1. The Energy Output and Its 'Spectral Composition' 1
2. The Observational Data 1
3. Generalities Concerning the Energy Sources 2
CHAPTER II. MECHANICAL EQUILIBRIUM: THE EQUILIBRIUM BETWEEN
THE GRAVITATIONAL FORCE PER UNIT VOLUME AND
THE GRADIENT OF THE TOTAL PRESSURE
1. Introduction 3
2. The Equilibrium between the Gradient of the Total Pressure and the Gravi-
tational Force per Unit Volume 4
2.1. Newton's Theorem. The Gravitational Force per Unit Volume 4
2.2. The Force per Unit Volume Produced by the Pressure Gradient 8
2.3. The Equation Expressing the Mechanical Equilibrium of (d V) 9
3. The Relation between My and the Density Q at a Distance r from the
Center 10
4. The Expression for div g as a Function of the Local Density Q.
Poisson's Equation 10
5. The Calculation of the Gas Pressure P The Concept of the Mean Mass
gas•
J1 of a Particle of the Mixture in Units of m (where m is the Mass in Grams
H H
of a 'Real' Microscopic Hydrogen Atom) 12
6. A Model of the Sun at 'Constant Density' Q = Q 15
7. The 'Homologous' Model. Expressions for Pc and Tc in Terms of M
and R 18
Exercises 20
CHAPTER III. THE DETERMINATION OF THE INTERNAL STRUCTURE
BY THE DENSITY DISTRIBUTION Q(r)
1. Introduction 24
2. The Determination of the Distribution of the Mass Mr Contained in a
Sphere of Radius r 25
x TABLE OF CONTENTS
3. The Determination of the Distribution of the Total Pressure P as a Func-
tion of r 28
4. The Determination of the Distribution of the Temperature T as a Function
of r 31
5. Summary. The Empirical Representation of the Functions g (r '), e( r '),
per'), and T(r'). The Polytropic Index n 32
6. The (Superficial) 'Convective Zone' of the Sun 35
Exercises 37
CHAPTER IV. ENERGY EQUILIBRIUM AND NUCLEAR REACTIONS
1. The Equation of Energy Equilibrium 44
2. The p-p Chain and the C-N Cycle 45
2.1. Introduction 45
2.2. An Explicit Schematic Representation of the Composition of Nuclei 45
2.3. The Details of the Reactions in the p-p Chain (Bethe, 1938) 45
2.4. The Details of the C-N Cycle (Bethe, 1938) 49
3. Calculation of the Energy B. Generalities 52
3.1. Calculation of R12 for a Given Reaction 52
4. The 'Mean Lifetime' of a Given Nucleus with Respect to an Isolated
Reaction (R) 57
4.1. Generalities 57
4.2. The Physical Meaning of !p(e) 59
4.3. !p(e) as the 'Mean Duration of an Isolated Reaction (R)' 60
4.4. !p(e) as an 'Exponential Decrement' 61
4.5. The Transition Probability Pea per Reaction (R) 61
5. The Convergence of Cyclic Reactions to a Stationary (,Equilibrium') State 61
6. The 'Mean Duration of a Cycle'. The Calculation of the Energy B when
Cyclic Reactions Are Present 68
7. The Empirical Representation of Bpp and BCN 70
8. Application to the Sun. The 'Final Test' 73
8.1. Review of the Principal Results 73
8.2. The Region in which B Is Negligible 74
8.3. The 'Central' Region (r' < 0.40), where L; and X Vary 76
8.4. The 'Final Test' 77
Exercises 78
CHAPTER V. EVOLUTIONARY MODELS. THE ACTUAL DETERMINATION
OF STRUCTURE
I. Introduction 84
1.1. The Advantage of Studying 'Evolutionary Sequences' 85
1.2. The 'Fossilized' Composition 85
2. The Evolution of the Distributions X( r) and Y( r) 87
3. Discussion 89
TABLE OF CONTENTS XI
4. The Mathematical Structure of the Problem. Principles of the Integration
Methods 93
5. The Age of a Star 97
6. The Relations between P, T, L, R, and Parameters such as M, ko, GO, and
f1 for 'Homologous' Models. The 'Mass-Luminosity' and 'Mass-Radius'
Relations 98
CONCLUSION 101
SOLUTIONS FOR THE EXERCISES 104
BIBLIOGRAPHY 110
INDEX OF SUBJECTS 114