Table Of ContentAn Introduction to
Astrophysical
Hydrodynamics
Steven N. Shore
GHRS Science Team
Computer Sciences Corporation
Goddard Space Flight Center
Greenbelt, Maryland
ACADEMIC PRESS, INC.
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Library of Congress Cataloging-in-Publication Data
Shore, Steven N.
An introduction to astrophysical hydrodynamics / Steven N. Shore,
p. cm.
Includes bibliographical references and index.
ISBN 0-12-640670-7
1. Hydrodynamics. 2. Astrophysics. I. Title.
QB461.S446 1992
523.01~dc20 91-28920
CIP
PRINTED IN THE UNITED STATES OF AMERICA
92 93 94 95 96 97 MB 9 8 7 6 5 4 3 2 1
To Lys Ann
There's beauty in the bellow of the blast,
there's a grandeur in the growling of the gale.
Preface
For he who'd make his fellow creatures wise should always
gild the philosophic pill!
W. S. Gilbert, Yeomen of the Guard
The universe is the most wonderful hydrodynamics laboratory imaginable. Every
kind off low we observe in terrestrial environments occurs in the cosmos on a truly
grand scale. Supersonic motion, magnetohydrodynamic flows, turbulence of every
variety, and instabilities of perplexing richness all present themselves to the alert
eye. It is the purpose of this book to introduce these processes to astronomers in
the hope that they will findt he tools necessary for understanding both their obser
vations and the related theoretical literature.
Over the past decade the dramatic improvements in detectors and the instru
ments to which they are mated have opened a vast range of new phenomena to the
scrutiny of observations. Not only can we determine velocities and observe struc
ture, we can also get true flux measurements for objects fainter than ever before
accessible. The resolution of spectrographs has always been high for optical obser
vations; but now spacecraft, especially the Gbddard High Resolution Spectrograph
(GHRS) on the Hubble Space Telescope, have made it possible to obtain resolu
tions of order a few kilometers per second for the entire satellite ultraviolet range
between 1200 and 3000 A. Millimeter detectors have dramatically improved in
sensitivity, and with new interferometers as well as large single dishes now avail
able, both galactic and extragalactic observations of molecular clouds are possible
at resolutions of a few hundreds of meters per second. In short, we aire passing into
a heroic epoch, one in which many of the hydrodynamic phenomena of the universe
will lie open to observation.
In light of these advances, it appears to me to be a serious deficiency in the
curriculum that so few courses on astrophysical hydrodynamics are offered. It is
clear that astronomers know how intimately their work is related to fluid dynamics,
but they have not had many resources to turn to when looking for an introduction
to the subject written in their language and reflecting their concerns. I hope that
xi
xii Preface
this book helps to fill this lacuna. It is intended for the student with a physics
background, with the usual complement of mathematics courses of a senior under
graduate or graduate level. However, no prior experience, other than mundane
observation of the world at large, has been assumed. The astronomy is introduced
in context. The material can be taught in a single semester, and the instructor
and student should find the references sufficient so that a second semester semi
nar course could include deeper discussion of some of the topics in the various
chapters.
The topics treated here are a selection of those frequently encountered both in
the literature and in observations. Chapter 1 examines the basic conservation equa
tions. It includes a section on the Bernoulli equation as an example of the simplest
kind of solution for a flow. The effects of viscosity and boundary layers are intro
duced in Chapter 2. Chapter 3 addresses vorticity and rotation, including the math
ematical devices and physical concepts required for magnetohydrodynamics
(MHD) and turbulence theory. Chapter 4 deals with shock phenomena. Chapter 5
treats similarity solutions. In order to connect such solutions with a more familiar
problem normally discussed in courses on stellar interiors, the polytropic sphere is
presented as an example of a scaling solution. Chapter 6 addresses MHD and
magnetic field generation. It is meant to serve as a general overview of the subject
and for this reason does not deal explicitly with dilute plasmas. Chapter 7 treats
turbulence, including the effects of self-gravitation and supersonic flows. Chapter
8 is an introduction to outflows and accretion phenomena. This chapter is, frankly,
an experiment. I am trying here to show that many of the phenomena usually
treated separately are really quite similar. Specifically, it is the theme of this chapter
that in many ways accretion is just a wind with the direction of flow reversed.
Chapter 9 deals with a range of astrophysically interesting instabilities. These are
chosen to be representative rather than complete and are also intended to serve as
a guide to the techniques for assessing the nature of instabilities in both the linear
and nonlinear regimes. Finally, Chapter 10 presents some examples of the obser
vational diagnosis of astrophysical flows. It introduces most of the tools needed for
simple radiative transfer (the Sobolev approximation) and contains examples of
some recent observational results for stellar winds and novae. While specific ref
erences have been included in each chapter, the general bibliography of mono
graphs and conferences at the end of the book is intended as a supplement. I have
tried to include as much as reasonable of the standard fluid dynamics corpus,
especially from the engineering literature, but I have heavily spiced the selection
with works on specific astrophysical applications.
A word now about the problems that follow Chapter 10. They are loosely
grouped in the order of chapters, but there is crossover between sections. I hope
this will help integrate some of the ideas and also lead to further study of specific
astrophysical problems. I have found that the best way to learn the material is to
re-derive the results and apply them to real situations rather than to just solve
artificially posed problems.
Preface xiii
I first formed the desire to write this book several years ago while teaching a
course in astrophysics at Case Western Reserve University. The students in that
course were expecting a traditional discussion of stellar physics and got a version
of these lectures instead. Part of the MHD chapter was used in lectures in a grad
uate course on plasma physics at New Mexico Institute of Mining and Technology.
Parts of the instabilities chapter and the accretion disk section of the wind-accre
tion chapter were delivered as a series of lectures in seminar series at the Very
Large Array. And finally, the bulk of the winds and accretion sections of the book
formed the basis of two GHRS seminar series on stellar winds at the Laboratory
for Astronomy and Solar Physics at Goddard Space Flight Center.
And so now having given you, the reader, a view of my motives for producing
this book and some background on its pedigree, I ask you only to turn a page or
two and join me in what I hope will be some happy hours of contemplation of
astrophysical hydrodynamics.
Steve Shore
Acknowledgments
While I have, by Heaven, I'll give, so damn your economy.
Sheridan, School for Scandal
There are many students and colleagues I feel under a deep obligation to acknowl
edge here. First, I want to thank Pierre Encrenaz and Fran9oise Praderie for their
continuing kindness and interest in inviting me for many years to spend extended
periods at the Ecole Normale Superieure (Paris) and the Observatoire de Meudon.
My continuing association with the group DEMIRM at Meudon has brought many
stimulating hours of discussion, and I want especially to thank Patrick Boisse\
Xavier Desert, Christophe Dupraz, Edith Falgarone, Michel Perault, and Jean-
Loup Puget for their kindness and interest in different aspects of this work. Their
comments have been invaluable, as has been their friendship. Alain Omont and
Bernard Lazareff kindly invited me to stay for some months at the Groupe d'Astro-
physique at Grenoble, and I thank them for their hospitality. Federico Ferrini ar
ranged several stays at Pisa and in addition has been a valued collaborator and close
friend for many years. He also provided numerous incisive critiques for most of
the chapters in this book. Marcello Rodono and Santo Catalano were kind enough
to arrange a visiting appointment at Catania, where some of the lectures in this
book were delivered as a graduate short course.
I want to especially thank Ted LaRosa, my former student and deeply valued
friend, for his indispensable help. He has been a constant source of discussion and
encouragement and has provided detailed criticisms and suggestions for all of the
chapters in this book, often forcing considerable rewriting and clarification of ob
scure points. I also thank Bob Hjellming, Jean Eilek, and Aileen O'Donoghue for
discussions during VLA seminars on topics related to material in this book. Doug
Brown, George Sonneborn, Sumner Starrfield, Tom Ake, Bruce Altner, and Sally
Heap have been the finest co-workers anyone could ever hope for; they have borne
my distractions as a result of this book like true and understanding friends. Albert
Petschek and Dave Raymond, two of the finest physicists I have ever known or
xv
xvi Acknowledgments
worked with, have been friends, colleagues, and wise guides when I was getting
lost in the deeper waters of the subject. I also thank Xavier D£sert, Christophe
Dupraz, Ivan Hubeny, Richard Robinson, Ron Polidan, and Eli Dwek for com
ments on various chapters, Keith Feggans for his generous advice and assistance
with software development and systems, and Dara Norman for introducing me to
Superpaint, the program with which I have drawn most of the illustrations in this
book.
I reserve special thanks for Tom Bolton, my advisor many years ago at Toronto
and a constant source of scientific insight and personal counsel, and Kevin Pren-
dergast, whose inspiration sparked my love of hydrodynamics.
My thanks also to the staff at Academic Press, especially Marvin Yelles, who
first encouraged me to submit a prospectus of this work and has been a valued
guide to the publishing process. My thanks also to Dean Irey and Steven Martin
for their skillful and sensitive editorial treatment of this work.
Three acknowledgments are painful to record, but I trust the reader will un
derstand. One is to my father, Frederick Shore, who died in 1988, just as this book
was getting started. From him I learned the meaning of education, of service, and
of hard work. I deeply regret that he never had a chance to hold in his hands the
work you are now reading. The second is to Johannes Hardorp, my first teacher
and colleague in astrophysics, who died in 1988. He welcomed me as a novice into
his work and his life, and it was from him that I learned what riches lay in the field.
Finally, I record here my great debt to Nick Sanduleak, whose death in 1990 cut
short a long collaboration and golden friendship. He was the gentlest person I have
ever known and the one most responsible for making me understand the deepest
motivations and methods of a true observer of nature.
The most important acknowledgment of all is saved for the dedication. This
book would never have happened without the prodding, wisdom, skill, and unfail
ing encouragement of Lys Ann Shore. There is not a portion that she has not read
and not a slip of writing that her keen eye has not made some comment on. What
ever flaws still remain in this work are not for her lack of trying to make it a good
read and a coherent and logical argument. There is no way that I'll ever be able to
record how much I owe to her, but I hope she, and you, the reader, will be pleased
with the final product.
I have received support from a number of organizations while writing portions
of this work. Visiting appointments have been supported by grants from the CNRS
(France), the CNR (Italy), and the French and Italian ministries of education. Some
support has come from NASA and from the GHRS Science Team.
The final draft of this book was written during a visiting appointment at the
ENS in February 1991.
C H A P T ER 1
The Equations of
Fluid Motion
Upon the word, accoutered as I was, I plunged
in and bade him follow. So indeed he did.
Shakespeare, Julius Caesar
We live in an incredibly dynamical universe. Any chance observation bears
witness to the enormous diversity of motions and interactions that domi
nate the structure of matter in astrophysical environments. It is the aim
of this book to put some of this in context. We see that gravitation, the
ultimate structuring force on the scales typical in astrophysical problems,
plays a key role in understanding how matter evolves. But we also must
reckon with magnetic fields, turbulence, and many of the same processes
that dominate laboratory studies.
1.1 Introductory Remarks
When you think of a fluid, the idea of a structureless deformable con
tinuous medium comes to mind. Other properties, ingrained in childhood,
are also likely to occur to you; things like incompressibility, viscosity (if
you were really precocious), and perhaps even the statement that a fluid
will not support shear and will yield freely in the presence of an applied
force. So what have these to do with an astrophysical context? More than
these definitions and properties might lead you to suspect. Of course, you
will likely think, since stars are gaseous (something we've known since the
1930s), that we should be dealing with kinetic theory and gas dynamics.
The processes of rarefied media are likely to dominate our understanding
1
2 1. The Equations of Fluid Motion
of cosmic bodies. This isn't quite true. Since a star is composed of gas
which is homogeneous (in most cases) and which acts collectively to create
its own gravitational field, it mimics rather well the behavior of a fluid mov
ing (or sitting) under gravity. The collision times are so short (or, put another
way, the mean free paths are so short compared with any scale lengths in
the medium) in the interior of the star that any disturbances can be
washed out and the structure can be described as continuous. Naturally,
this is, for the moment, only an assertion. We shall prove it in due time.
The most crucial point is that stars and all other cosmic matter can be
treated as an ensemble object or system when we have carefully chosen
some scales of length and time. In a nutshell, the reason for this book is that
we can always, at some magnification of scale or some rate of clock ticking,
apply a fluid approximation to the problems at hand. This book is meant to
provide the machinery, both computational and conceptual, with which to
begin treating dynamical and static problems posed by fluids in a nonter-
restrial environment.
1.2 Equations of Motion
1.2.1 Distribution Functions
To begin with, we shall treat the case of a homogeneous medium of
identical particles. Forget for the moment that this may be too restrictive an
assumption. Imagine that this group of particles is characterized by a global
velocity distribution. Also, assume that we can know this distribution
function and that the positional information can eventually be derived for
the particles as well. Let us start with a gas that consists of a collection of
myriads of particles, all of identical mass. If we assume that these particles
execute collective motions, we will be able to take ensemble averages and
treat them as if they were a continuous medium. This is what we mean by a
fluid in an astrophysical context. But before we can reach the stage of
describing the matter as a classical substance, we need to consider the
microscale phenomena and how to incorporate them into a macroscopic
description of the motion and thermal properties. To do this, we begin with
a statistical mechanics treatment and then generalize from there.
Let us say that there exists a possibly time-dependent distribution
function /, which is a function of x, the particle positions, and v, their
velocities, and which provides a complete description of the chance of any
single particle having a specific position and velocity at any time. We
assume that the particle motions are individually governed by any forces
imposed externally by the medium and also by any interactions that the
