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\noindent
{\twelveb Chapter 3}
\vskip .25truein
\noindent
{\twelveb Hydrodynamics}
\vskip 1.44truein plus .06truein
\noindent
{\bf 3.1 Equations}
\vskip \baselineskip
\noindent
Most astronomical objects are fluid. In fact, those which are not
are exceptional---portions of the terrestrial planets, various
small bodies in the solar system, interstellar grains, neutron
stars, and perhaps the cores of the oldest white dwarves. Just about
everything else the astronomer deals with is fluid---stars, the
interstellar medium, flows of gas away from stars, onto stars, and
between stars, the clouds which will become stars, and the debris
left over after the deaths of stars. Even the stars themselves
may be sometimes regarded as the particles making up a fluid.
The equations of hydrodynamics, sound, and shocks are discussed in many
texts; one of the most elegant is that by Landau and Lifshitz (1959).
Some of the fluids an astronomer deals with are far from thermodynamic
equilibrium, and relaxation to equilibrium is very slow; good examples
of this are the relativistic particles in the cosmic rays and in
nonthermal radio sources, and the stars in a galaxy. But in many cases,
particularly for ionized plasmas, local
thermodynamic equilibrium applies. This means that at any point in the
fluid it may be completely described by a single density, velocity, and
temperature. The velocity distribution function of the individual
particles is Maxwellian (if they are nondegenerate), and the distribution
of their ionization and excitation states is described by the Saha
equation. Any processes of
excitation, dissociation, ionization, or chemical reaction are either
negligibly slow or so rapid that they may be assumed to be in
equilibrium. Then, for many purposes, the fact that the
particles are charged is inessential, and the ordinary equations of
hydrodynamics may be used. We need to remember that we are
dealing with a plasma when considering deviations from
equilibrium---transport processes, acceleration of particles to
high energy, and macroscopic currents and magnetic fields, but
often (particularly at high densities) these effects are
insignificant.
The first equation of hydrodynamics is that of continuity, or the
conservation of mass. Stated verbally, the rate at which mass
accumulates in an element of volume is equal to the net rate at
which it flows in through that element's boundaries. The rate
of mass flow through any unit element of area is
$\rho \vec u \cdot \hat n$, where
$\rho$ is the density,
$\vec u$ is the velocity, and
$\hat n$ is the unit normal to the surface. Using vector calculus,
the net mass flow from an infinitesimal element of unit volume is
given by $\nabla \cdot ( \rho \vec u)$, and the contained mass is
$\rho$, so that the equation of continuity is
$${\partial \rho \over \partial t} + \nabla \cdot \left( \rho \vec u
\right) = 0 . \eqno(3.1.1)$$
Although here
$\rho$ is the mass density, this equation is quite general; it
applies to any conserved scalar quantity. Further, if a quantity
is conserved except for known sources or sinks, a similar equation
applies if these are added to the right hand side. Sources need not be
anything as exotic as matter creation in a steady state cosmology, but
can include gas \lq\lq created\rq\rq\ by the evaporation of a solid,
produced in a chemical reaction, or introduced through a narrow pipe.
Finally, if an equation like (3.1.1) applies to the individual
components of a vector or tensor quantity, another similar equation
applies to that entire quantity.
\headline={\ifodd\pageno\rightheadline \else\leftheadline\fi}
\def\leftheadline{\vbox{\vskip 0.125truein\line{\tenbf\folio\qquad
Hydrodynamics\hfil}}}
\def\rightheadline{\vbox{\vskip 0.125truein\line{\tenbf\hfil Equations
\qquad\folio}}}
\nopagenumbers
It is useful to consider a total (or convective) derivative,
which measures the change in a quantity per unit time seen by an
observer borne along with the fluid flow. This is defined
$${{\cal D} \over {\cal D}t} \equiv {\partial \over \partial t} +
\vec u \cdot \nabla . \eqno(3.1.2)$$
The continuity equation may be rewritten
$${{\cal D} \rho \over {\cal D}t} + \rho \nabla \cdot \vec u = 0 .
\eqno(3.1.3)$$
If $\rho$ is constant the fluid is called incompressible, and
$\nabla \cdot \vec u = 0$. This condition leads to a drastic
simplification of the equations of hydrodynamics, and to a great
body of mathematical results, but to very few astrophysical ones.
Changes in density are important in astrophysics.
The second equation of hydrodynamics is equivalent to
$F = ma$, or the conservation of momentum. An element of fluid
feels a force which is given by the difference between the
pressures on its surfaces. For an infinitesimal element the
resulting equation is
$$ \rho {{\cal D} \vec u \over {\cal D}t} = - \nabla P ,\eqno(3.1.4)$$
where
$P$ is the pressure. Note that the convective derivative which
follows the motion of a fluid element is used---at a fixed point
in space
$\vec u$ might change because fluid with different velocity is
swept through the point (\lq\lq is advected by the flow,\rq\rq\ in the
jargon of fluid mechanics), so the partial time derivative of
$\vec u$ contains contributions in addition to those of
$\nabla P$. Equation (3.1.4) may be rewritten in terms of the
partial derivatives, using (3.1.2):
$${\partial \vec u \over \partial t} + (\vec u \cdot \nabla )\vec u
= - {1 \over \rho} \nabla P . \eqno(3.1.5)$$
In this form it is called Euler's equation. Viscous stresses and
gravity and other body forces may be readily added,
if present---they are \lq\lq source terms\rq\rq\ for momentum, in
analogy with the sources of mass one might add to the continuity
equation. Equation (3.1.5) may be rewritten, after some manipulation
and the use of (3.1.1), in the form
$${\partial \over \partial t} (\rho \vec u) + \nabla \cdot \left( \rho
\vec u \vec u + \hbox{\bf P} \right) = 0 , \eqno(3.1.6)$$
where
$\vec u \vec u$ is a dyad and $\hbox{\bf P}$
is the stress tensor. For inviscid fluids $\hbox{\bf P}$
is the scalar pressure $P$ multiplied by the unit tensor. Equation
(3.1.6) is written in a form analogous to (3.1.1), so that it is
obviously a conservation law for the momentum density $\rho \vec u$;
$(\rho \vec u \vec u +\hbox{\bf P})$ is the momentum flux density tensor.
There generally is no independent hydrodynamic equation derivable from
the conservation of angular momentum. Taking the cross product of a
radius vector with the momentum equation would give an equation
for the conservation of angular momentum, but this would contain
no new information. An exception to this would occur for
a fluid having internal stores of angular momentum other than its
bulk motion (for example, particle spin), in which case there would be
an additional equation relating the internal angular
momentum to the angular momentum of the bulk motion.
The third equation of hydrodynamics is obtained from the
conservation of energy. The first law of thermodynamics for a
fluid element is
$${{\cal D}{\cal U} \over {\cal D}t} + P {{\cal D} V \over {\cal D}
t} = Q , \eqno (3.1.7)$$
where
${\cal U} = {\cal E}/\rho$ is the internal energy per gram,
$V \equiv 1/\rho$ is the specific volume, and
$Q$ is the external power supplied per gram (the source term); we now
assume a scalar pressure $P$.
$Q$ includes such effects as thermal conduction, viscous
frictional heating, radiation emission and absorption, and heat
produced by chemical or nuclear reactions. Equation (3.1.7) may be
rewritten, after some manipulation and the use of the continuity
and momentum equations, in the form
$${\partial \over \partial t} \biggl\lbrack {\cal E} + {1 \over 2}
\rho u^{2} \biggr\rbrack + \nabla \cdot \biggl\lbrack \vec u \left(
{\cal E} + {1 \over 2} \rho u^{2} \right) + P \vec u \biggr\rbrack
= \rho Q , \eqno(3.1.8)$$
paralleling equations (3.1.1) and (3.1.6) in explicit conservation
form. It is apparent that
${\cal E} + \rho u^{2}/2$ is the energy density, and
$P \vec u$ is the flux of mechanical work. The momentum and energy
equations are formally identical to (3.1.1), with suitable
definitions of the fluxes. This is as it should be, for they
are all based on the conservation of quantities carried with the
moving fluid.
These equations may be rewritten in a coordinate system which
moves with the fluid, so that the independent space variables are
replaced by masses. These coordinates, called Lagrangian (in
contrast to the Eulerian coordinates of equations (3.1.1,5,6,8),
are very useful in numerical computation, because the
advective terms
$\vec u \cdot \nabla$ are difficult to compute accurately.
Lagrangian coordinates are particularly powerful in
\lq\lq one-dimensional\rq\rq\ calculations, in which all material
quantities depend on a single spatial coordinate. In these calculations,
fluid cells or zones are generally infinitely broad flat slabs,
infinitely long cylinders, or spherical shells. Two- and
three-dimensional Lagrangian meshes (in which quantities depend on two
or three spatial coordinates) are also used, but may
tangle when a fluid flow is heavily sheared.
In Lagrangian coordinates the time derivative is
${\cal D} / {\cal D}t$, so the
$\vec u \cdot \nabla$ terms do not appear explicitly and do not
have to be computed. In one dimensional slab symmetry the
spatial variable is the mass, with
$dm \equiv \rho dx$. The equations become
$$\eqalignno{{{\cal D} V \over {\cal D} t}&= {\partial u \over \partial
m},&(3.1.9)\cr
{{\cal D} u \over {\cal D} t}&= - {\partial P \over \partial m},&
(3.1.10)\cr
{{\cal D}{\cal U} \over {\cal D} t} + P {{\cal D} V \over {\cal D} t}
&= Q. &(3.1.11)\cr}$$
The variables
$u$, $V$, and ${\cal U}$ give the velocity, specific volume, and
internal energy (per gram) of a specified fluid or mass element as a
function of time. Eulerian coordinates may be obtained from
$$x(m,t)=\int_{0}^{t} u(m,t^{\prime})\,dt^{\prime}+x(m,0)=\int_{0}^{m}
V(m^{\prime},t)\,dm^{\prime}+x(0,t) . \eqno(3.1.12)$$
Similar Lagrangian equations may be obtained for cylindrical and
spherical geometry. Spherical geometry is of most astrophysical
interest. The equations are:
$$\eqalignno{dm&= 4 \pi r^{2} \rho \,dr&(3.1.13)\cr
{{\cal D} V \over {\cal D} t}&= {\partial (4 \pi r^{2} u) \over \partial
m}&(3.1.14)\cr
{{\cal D} u \over {\cal D} t}&= -4 \pi r^{2} {\partial P \over \partial
m}&(3.1.15)\cr
{{\cal D}{\cal U} \over {\cal D} t} + P {{\cal D} V \over {\cal D} t}
&= Q.&(3.1.16)\cr}$$
The radius $r$ is obtained from
$$r(m,t) = \int_{0}^{t} u(m,t^{\prime})\,dt^{\prime} + r(m,0)=\int_{0}^{m
}{V(m^{\prime},t) \over 4 \pi r^{2}}\ dm^{\prime} + r(0,t) .
\eqno(3.1.17)$$
A striking feature of the equations of hydrodynamics is their
nonlinearity. This is apparent even in (3.1.1) in the product of
$\vec u$ and $\rho$, but is true even for incompressible fluids
because of the
$(\vec u \cdot \nabla )\vec u$ term in (3.1.5). The practical
consequence of this is that exact solutions are scarce. The most
commonly used tools of the theoretical hydrodynamicist's trade are
linearization of the equations for infinitesimal disturbances
about a static equilibrium, and numerical computation of the full
nonlinear equations.
The alert reader will have noticed that the first equation (3.1.1)
involves two variables,
$\rho$ and $\vec u$. The number of scalar variables exceeds the
number of equations by the number of spatial dimensions. Adding
the momentum equation (3.1.4) or (3.1.5) adds another variable
$P$. This is a vector equation, with as many components as
spatial dimensions, so now there is one more scalar variable than
equations. Adding the energy equation (3.1.7) adds still another
variable, ${\cal U}$ or ${\cal E}$ (assuming the source term is known).
The excess of one variable remains. This is reminiscent of the closure
problem encountered in radiation transport theory ({\bf 1.7}) and in
kinetic theory ({\bf 2.1}). Its origin is similar, because
in kinetic theory the hydrodynamic equations are obtained by taking
velocity moments of the Boltzmann equation (2.1.9).
In order to close the system of equations an additional constraint
is required. This is generally a constitutive relation, or
\lq\lq equation of state\rq\rq\
$$P = P({\cal E},V) . \eqno(3.1.18)$$
The justification for the use of (3.1.18) is the assumption of
thermodynamic equilibrium.
It may be possible to define the variables $P$ and ${\cal E}$ in
disequilibrium fluids, but only in equilibrium is there a unique relation
among them. This relation is closely analogous to the Eddington
approximation (1.7.12) of radiative transport theory, which has the form
of an equation of state (3.1.18) for the radiation field.
In equilibrium any two of the variables
$P$, ${\cal E}$, and $V$ are a complete description of the state of
the fluid in its rest frame. Permeable or dielectric fluids in
external fields, and intrinsically anisotropic fluids, require
generalizations which are, in principle, straightforward; such
complications are generally irrelevant to the astrophysicist.
To make thermodynamic equilibrium a valid approximation,
it is necessary that the processes which bring it about occur
either very rapidly, or very slowly, compared to the hydrodynamic
processes of interest. A good example is a 100 Hz sound wave in
air. Collisional relaxation between the molecules occurs in
$\sim 10^{-9}$ seconds, and relaxation of the populations of
most of the important rotational states is nearly that rapid, so the
instantaneous achievement of equilibrium thermodynamic properties may
be assumed. On the other hand, the nuclei are far from equilibrium;
their lowest energy state is as
$^{56}$Fe nuclei. But nuclear reactions at room temperature
are very slow
($\gg 10^{100}$ years), so the fact that
$^{14}$N and
$^{16}$O nuclei have other, more energetically favorable,
states available may be ignored. Then thermodynamic equilibrium
is well justified. But if the air contains a molecule whose
rotational or vibrational relaxation time is
$10^{-3}$ sec, then equilibrium is not obtained, and more
complex equations than those of hydrodynamics must be solved.
Such intermediate time scale processes do occur, and produce an
observable excess attentuation of sound in air.
The equations of hydrodynamics have their ultimate derivation from
the more complex \lq\lq kinetic\rq\rq\ equations, which deal with
the complete distribution functions of the particle states. In
attempting to solve these kinetic equations one generally takes
their velocity moments, multiplying and integrating them
$\int d^{3}\vec u\ (\vec u \vec u \cdots \vec u\,),$ where
the $n$-th moment equation is obtained if there are $n$ factors
in the parentheses. Each moment
gives a new equation, but also an additional unknown moment of
the distribution function. There is always an excess unknown, as
we saw with the equations of hydrodynamics. In order to close this
hierarchy of equations some additional assumption must be made. The
section of this book {\bf 1.7} on radiation transport presents another
example. In hydrodynamics the added assumption (3.1.18) is that of
thermodynamic equilibrium. It is essential, and the term
{\it hydrodynamics} is often taken implicitly to mean thermodynamic
equilibrium. When similar equations arise there sometimes
is no good way to close the hierarchy, as in the theory of turbulence,
and uncertain approximations and assumptions must be made.
\goodbreak
\vskip 2\baselineskip plus2\baselineskip
\goodbreak
\noindent
{\bf 3.2 Sound Waves and Jeans Instability}
\vskip \baselineskip
\noindent
Sound waves are one of the most important applications of the
equations of hydrodynamics. In order to derive their properties,
consider a one-dimensional disturbance of small amplitude in an infinite
fluid which is otherwise uniform and at rest. Then we may write
$$\eqalignno{\rho&= \rho_{0} + \delta \rho (x,t)&(3.2.1)\cr
P&= P_{0} + \delta P(x,t) &(3.2.2)\cr
u&= \delta u(x,t). &(3.2.3)\cr}$$
Substitute these expressions into eqs. (3.1.1) and (3.1.5), and keep
only terms of the first power in small quantities, to obtain a system
of linear equations for small disturbances:
$$\eqalignno{{\partial \delta \rho \over \partial t} + \rho_{0}
{\partial \delta u \over \partial x} &= 0 &(3.2.4)\cr
{\partial \delta u \over \partial t} + {1 \over \rho_{0}} {\partial
\delta P \over \partial x}&= 0 . &(3.2.5)\cr}$$
\def\rightheadline{\vbox{\vskip 0.125truein\line{\tenbf\hfil
Sound Waves and Jeans Instability\qquad\folio}}}
To eliminate one of the three variables, write
$$\delta P = \left( {\partial P \over \partial \rho} \right) \delta \rho
. \eqno(3.2.6)$$
This relation plays a role analogous to that of the constitutive
relation (3.1.18) in closing the hierarchy of equations. Equation
(3.2.6) may be regarded as a constitutive relation for the changes
in pressure and density a fluid element undergoes in the sound
wave, and may be derived from (3.1.18) if an additional assumption is
made---for example, that processes in the sound wave are
adiabatic.
In order to eliminate
$\delta u$ from (3.2.4) and (3.2.5) take the partial derivative with
respect to time of (3.2.4) and that with respect to space of (3.2.5),
and algebraically eliminate the cross-derivative term. The
result is
$${\partial^{2} \delta \rho \over \partial t^{2}} = \left( {\partial P
\over \partial \rho} \right) {\partial^{2} \delta \rho \over \partial
x^{2}} . \eqno(3.2.7)$$
If we define
$$c_{s} \equiv \left( {\partial P \over \partial \rho} \right)^{1/2} ,
\eqno(3.2.8)$$
the solutions to (3.2.7) are of the form
$$\delta \rho = f(x-c_{s}t) + g(x+c_{s}t), \eqno(3.2.9)$$
where
$f$ and $g$ are arbitrary functions. For a periodic function
$$\delta \rho = \delta \rho_{0} \exp (ikx - i \omega t),\eqno(3.2.10)$$
eq. (3.2.7) leads to
$$\omega^{2} = c_{s}^{2}k^{2} . \eqno(3.2.11)$$
Thus
$c_{s}$ is the phase velocity (and also the group velocity
$\partial \omega \over \partial k$) of sound waves. (3.2.11) is called
the dispersion relation (even though the waves are not dispersive).
Most commonly, the derivative in (3.2.8) is evaluated at
constant entropy, because the fluid motions in sound waves are usually
nearly adiabatic. Because we are concerned with infinitesimal
disturbances, it is evaluated at the conditions of the undisturbed fluid.
Then $c_{s}$ is called the adiabatic sound speed.
Sound waves are not always adiabatic. For example, for short wavelength
waves in a fluid of high thermal conductivity it may be a better
approximation to evaluate (3.2.8) at constant temperature. Then
$c_{s}$ is an isothermal sound speed. For conditions intermediate
between adiabatic and isothermal, sound waves are both dispersive and
damped.
If the fluid satisfies the power law equation of state
$P \propto \rho^{\gamma}$ for adiabatic changes, as is frequently a
good approximation (for perfect monatomic gases
\hbox{$\gamma = 5/3$} for nonrelativistic point particles and
$\gamma = 4/3$ if the particles are relativistic; see {\bf 1.9}), then
$c_{s} = \sqrt{\gamma P /\rho}$. If, in addition,
$P = \rho N_{A} k_{B} T/ \mu$, where
$\mu$ is the molecular weight (as is the case for gases of
noninteracting particles), then
$$c_{s} = \sqrt{{\gamma N_{A} k_{B} T \over \mu }} . \eqno(3.2.12)$$
The most important qualitative implication of (3.2.12) is the
increase of
$c_{s}$ with $T$.
For the diatomic molecules in air at room temperature the
vibrational degrees of freedom are not significantly excited (the
first excited state has an energy corresponding to 3340 $^{\circ}$K in
N$_{2}$ and 2230 $^{\circ}$K in O$_{2}$), while the rotational
degrees of freedom are excited to such high quantum numbers that
they may be considered classical oscillators (the first excited
states are at energies corresponding to
6 $^{\circ}$K and 4 $^{\circ}$K respectively). Then there are effectively
5 degrees of freedom per molecule, and (3.2.12) with
$\gamma = 1.40$ (1.9.8) is an excellent approximation to the sound speed
in air.
We now consider sound waves in a self-gravitating fluid. To the
momentum equation we must add the force of gravity, which we express
in terms of the gravitational potential $\phi$:
$${\partial \vec u \over \partial t} + (\vec u \cdot \nabla )\vec u =
-{1 \over \rho} \nabla P - \nabla \phi . \eqno(3.2.13)$$
The potential is given by Poisson's equation
$$\nabla^{2} \phi = 4 \pi G \rho . \eqno(3.2.14)$$
As before, we consider small perturbations about a uniform infinite
fluid at rest.
This procedure is incorrect for a self-gravitating fluid, because the
assumed initial uniform state is not a solution of the equations;
(3.2.14) cannot be solved for such an infinite fluid without having
$\phi \rightarrow \infty$ and $\nabla \phi \rightarrow \infty$,
implying an infinite gravitational acceleration! In fact, (3.2.13)
cannot have a uniform static equilibrium solution even for a finite
self-gravitating fluid, because a uniform fluid possesses no
pressure gradient to oppose the acceleration of gravity. Although
technically improper, the calculation is still informative.
For small disturbances $\delta \rho (x,t)$, $\delta P(x,t)$, $\delta
u(x,t)$, and $\delta \phi (x,t)$, (3.2.13) and (3.2.14) become
$$\eqalignno{{\partial \delta \rho \over \partial t} + {1 \over
\rho_{0}} {\partial \delta P \over \partial x} + {\partial \delta \phi
\over \partial x} &= 0 &(3.2.15)\cr
{\partial^{2} \delta \phi \over \partial x^{2}} - 4 \pi G \delta \rho
&= 0 &(3.2.16)\cr}$$
(3.2.15) replaces (3.2.5). As before, we assume that all the small
quantities have spatial and temporal variability $\propto \exp (ikx-
i\omega t)$, and eliminate $\delta \rho$, $\delta P$, and $\delta \phi$
from equations (3.2.4), (3.2.6), (3.2.15) and (3.2.16). The
result (if all the small quantities are nonzero) is an algebraic
dispersion relation:
$$\omega^{2} = c_{s}^{2} k^{2} - 4 \pi G \rho_{0} . \eqno(3.2.17)$$
The important property of (3.2.17) is the existence of a critical
wavevector $k_{J}$ and wavelength $\lambda_{J}$:
$$\eqalignno{k_{J}&=\sqrt{4 \pi G \rho_{0}} / c_{s}&(3.2.18a)\cr
\lambda_{J}&=2 \pi c_{s} / \sqrt{4 \pi G \rho_{0}}&(3.2.18b)\cr}$$
For $k \gg k_{J}$ the second term on the right hand side of (3.2.17)
is negligible, and it closely approximates (3.2.11);
these disturbances are essentially ordinary sound waves, and
self-gravity is unimportant. For $k < k_{J}$ we have $\omega^{2} < 0$,
so there are both exponentially growing (in time) and exponentially
damped solutions. The growing solutions describe Jeans instability.
Their growth rate is $\sim \sqrt{4 \pi G \rho_{0}}$.
If this derivation had begun from consistent initial conditions, the
Jeans instability criterion would set an upper bound $\lambda_{J}$
on the size of a stable self-gravitating gas cloud. Because it did
not, it is better to regard $\lambda_{J}$ as a parameter dividing length
scales for which self-gravity is important from those for which
it is not. A cloud of radius $r < \lambda_{J}$ will, if unconfined,
expand under the influence of its own pressure forces. If it is
to survive for a time $\gapp r/c_{s}$ it must be confined by an
external pressure; its self-gravity is inadequate. This is believed
to be a qualitatively correct description of most interstellar gas
clouds, which have $r \ll \lambda_{J}$, and which are believed to be
immersed in a dilute but hot medium with which they are in pressure
balance.
In a cloud of radius $r > \lambda_{J}$ the force of gravity exceeds that
of pressure, and the Jeans analysis predicts gravitational collapse in a
time $\sim (4 \pi G \rho_{0})^{-1/2}$. If such a collapse is followed
into the nonlinear regime, the pressure forces are found (if $\gamma >
4/3$) to increase more rapidly that the gravitational forces, and the
cloud settles into (or oscillates around) a state of hydrostatic
equilibrium. In this state, which resembles the hydrostatic equilibrium
of a star, $r \sim \lambda_{J}$ provided the new, increased, $\rho$ and
$c_{s}$ are used in the calculation of $\lambda_{J}$. The frequencies
of the low modes of oscillation are $\sim \sqrt{4 \pi G \rho} \sim
t_{h}^{-1}$, where $t_{h}$ is the hydrodynamic time scale (1.6.1).
Because of the improper assumptions made in the derivation of $\lambda_{
J}$ it is best to regard it only as a criterion for the importance of
self-gravity. In objects with $r < \lambda_{J}$ self-gravity is
negligible, while in those with $r \sim \lambda_{J}$ it is important.
For example, an interstellar cloud which grows to this size will then
collapse.
The analysis is more complicated if the fluid has internal motions in
its initial state. Then it is qualitatively
correct to replace $c_{s}$ by a characteristic fluid velocity $v$. We
may smoothly interpolate between the $c_{s} \gg v$ and $c_{s} \ll v$
limits with an expression like
$$\lambda_{J} = 2 \pi \left({c_{s}^{2} + v^{2} \over 4\pi G \rho_{0}}
\right)^{1/2} ; \eqno(3.2.19)$$
the particular form of the numerator is suggested by
the addition of the hydrodynamic stress $\rho v^{2}$ to the gas
pressure, but is not quantitatively justified. More importantly, the
appropriate value of $v$ will almost always depend on the length scale
being considered, because it is only the {\it differential} velocity
across a distance $r$ which can prevent collapse (it is always possible
to consider the problem in the center-of-mass frame of the region under
consideration).
For a turbulent flow the differential $v(r)$ depends on the statistical
properties of the turbulence, and will generally be an increasing
function of $r$. If the turbulent velocities are subsonic, then $v$
makes a minor contribution to (3.2.19), whatever its dependence on
$r$. In supersonic turbulence fluid elements collide and strong shocks
form ({\bf 3.3}), increasing the sound speed so that $c_{s} \sim v$.
An important case of differential motion is that of rotation, such as
is found in a disc of gas orbiting a central object. Examples include
the interstellar gas orbiting the centers of galaxies, and accretion
discs surrounding stars and collapsed objects. In these cases $v =
\Omega r$, where $\Omega$ is the local angular velocity of rotation.
Substituting this into (3.2.19), neglecting $c_{s}$, and using
$\Omega = \sqrt{GM/R^{3}}$, where $M$ and $R$ are the radius
and mass of the central object (this holds approximately even if
the mass is distributed in a non-spherically symmetric way throughout
a volume of radius $R$, as is the mass of galaxies), we find that
$r > \lambda_{J}$ requires
$$\rho_{0} > \pi {M \over R^{3}} . \eqno(3.2.20)$$
The numerical coefficient is rather approximate (because it depends on
the boundary conditions on the disturbances, which we have ignored).
The qualitative conclusion is that discs much denser than the mean
density of their enclosed mass may be unstable. Because
$c_{s} > 0$ it is also necessary that $r > \lambda_{J}$ where
$\lambda_{J}$ is obtained from (3.2.18b); to be unstable a rotating
cloud must have both a minimum size and a minimum density.
For the mean interstellar medium near us $\rho_{0} \sim 10^{-24}$
gm/cm$^{3}$, while $M/R^{3} \sim 10^{-23}$ gm/cm$^{3}$, so that
instability is not expected. In much denser clouds of sufficient size
instability will occur; this is probably how stars and star clusters
begin to form. During their subsequent collapse
angular momentum is conserved, so that $v \propto r^{-1}$ and $\rho_{0}
\propto r^{-3}$; $r$ decreases faster than $\lambda_{J} \propto r^{1/2}$
(3.2.19), halting the collapse. Further contraction is limited by
the rate at which other processes (probably magnetic torques) can
remove angular momentum from this protocluster or protostar. These
processes are incalculable, and may be very slow.
Accretion discs are usually (but uncertainly) estimated to have very
little mass and low density, and therefore not to satisfy the
instability criterion (3.2.20).
A closely related problem is the Jeans instability of a gas of
collisionless gravitating particles, such as the stars in a galaxy,
or the denser clouds (which may be protostars) formed by the Jeans
instability in a more dilute gaseous medium. The analysis is more
complicated because the equations of hydrodynamics cannot be used
to describe the motions of collisionless particles, but the
instability criterion is similar, if $c_{s}$ is now taken to be the
velocity dispersion of the particles. The stars in the disc of
our Galaxy have $\rho_{0} \sim 10^{-23}$ gm/cm$^{3}$, and are thus
closer to instability than the gas alone. Because the particles of a
collisionless gas pass by each other freely, without exchanging
momentum or \lq\lq sticking,\rq\rq\ one consequence of Jeans
instability is an increase in their velocity dispersion as $\delta u$
is added to the pre-existing velocities. It may be
that this process maintains the velocity dispersion, and limits the
density, of the stars in a galactic disc. It is also possible that
spiral arms are a consequence of density clumping produced by
Jeans instability of the disc, stretched into a spiral pattern by
galactic differential rotation.
In a roughly spherical object, whether collisional (a star), or
collisionless (an elliptical galaxy or star cluster), $\rho_{0} \sim
M/R^{3}$ and $c_{s}$ (or $v$) $\sim \sqrt{GM/R}$, so that $R < \lambda_{
J}$, and there is no instability.
\goodbreak
\vskip 2\baselineskip plus2\baselineskip
\goodbreak
\noindent
{\bf 3.3 Shocks}
\def\rightheadline{\vbox{\vskip 0.125truein\line{\tenbf\hfil Shocks\qquad
\folio}}}
\vskip \baselineskip
\noindent
A shock is a propagating irreversible discontinuity in the thermodynamic
state of a body of material. For astrophysical purposes we
are interested only in shocks in fluids, though they can occur in
solids. We will assume that everywhere in the fluid, except in
the infinitesimally thin sheet which we call the shock,
thermodynamic equilibrium holds, and the fluid is locally
characterized by its velocity, density, pressure, and internal
energy. As discussed in the previous section, in thermodynamic
equilibrium these are a complete description of the fluid, and an
additional constitutive relation makes one of the variables redundant.
We are interested in finding relations among the parameters of the
unshocked fluid (\lq\lq ahead\rq\rq\ of the shock), those of the shocked
fluid (\lq\lq behind\rq\rq\ the shock), and the velocity of propagation
of the shock itself. These relations, often referred to as the
\lq\lq jump conditions,\rq\rq\ are determined by the conservation laws.
It is not {\it a priori}
obvious how a shock may actually be produced. It is even less
obvious (and quite a difficult problem to determine) how
unshocked fluid is transformed to its shocked state. The
relations among the parameters of the shocked and unshocked fluid are
only a restatement of the conservation laws, and do not answer these
questions.
Imagine a fluid initially at rest with uniform density
$\rho_{0}$ and pressure
$P_{0}$. At
$t = 0$ we consider a shock propagating in the
$+x$ direction from the origin with a velocity
$U$. The gas behind the shock moves with a velocity
$u$, and has a uniform density
$\rho_{1}$ and pressure
$P_{1}$ throughout its volume. The
equations of hydrodynamics are satisfied everywhere but at the
shock front, for everywhere else
$\partial /\partial t = \partial /\partial x = 0$.
This configuration (Figure 3.1) may be produced by a
piston beginning to move at
$t = 0$ from the origin with speed
$u$, so that the shocked gas is at rest with respect to the
piston.
\topinsert
\vskip 10.5truecm
\ctrline{{\bf Figure 3.1.} Shock produced by a piston.}
\endinsert
Conservation of mass gives the equation
$$\rho_{1} (U-u) t = \rho_{0} U t , \eqno(3.3.1)$$
where the right hand side is the rate of mass flow into the shock
front, and the left hand side is the rate of mass flow out of it.
No mass can accumulate in the infinitesimally thin shock.
Conservation of momentum gives
$$\rho_{0}Utu = (P_{1}-P_{0})t, \eqno(3.3.2)$$
where
$P_{1}-P_{0}$ is the pressure jump across the shock.
Without specifying the internal workings of the shock, we can say
that matter can acquire momentum only by the application of a
pressure gradient or pressure drop. Alternatively, we may note
that the piston at the left must apply a pressure
$P_{1}$, balancing the fluid pressure upon its face, and a
hypothetical piston at
$x = +\infty$ must apply a pressure
$P_{0}$. These imply a change in momentum per unit area per
unit time
$P_{1}-P_{0}$, which appears in the momentum matter
acquires by changing its velocity as it crosses the shock.
Conservation of energy gives
$$\rho_{0}Ut \left( {\cal U}_{1} - {\cal U}_{0} +{1 \over 2} u^{2}
\right) = P_{1} ut , \eqno(3.3.3)$$
which is the rate at which the piston at the left does work on
the fluid, measured in the laboratory frame. In order to keep the
shock propagating at a constant rate, it is necessary to supply
continually energy and momentum from the outside. If this
is not done then our simple solution, uniform everywhere but at
the shock, will be replaced by a more complex solution in which
the shock gradually weakens.
Now make a Galilean transformation to a frame moving to the right at a
speed $U$, so that the shock is stationary. In this frame the
unshocked matter has a velocity
$$u_{0} = -U , \eqno(3.3.4)$$
and the shocked matter has a velocity
$$u_{1} = u - U . \eqno(3.3.5)$$
Equations (3.3.1)--(3.3.3) may then be rewritten,
after a little algebra:
$$\eqalignno{\rho_{1}u_{1}&= \rho_{0}u_{0} &(3.3.6)\cr
P_{1}+\rho_{1}u_{1}^{2}&= P_{0}+\rho_{0}u_{0}^{2} &(3.3.7)\cr
{\cal U}_{1}+{P_{1} \over \rho_{1}}+{1 \over 2}u_{1}^{2}&= {\cal U}_{0}
+{P_{0} \over \rho_{0}}+{1 \over 2}u_{0}^{2}.&(3.3.8)\cr}$$
These equations bear a striking resemblance to the equations of
hydrodynamics in conservation law form, (3.1.1), (3.1.6), and (3.1.8).
This should be no surprise, because both sets of equations only state
the basic conservation laws. To derive (3.3.6)--(3.3.8) from
the hydrodynamic equations, integrate each of the latter over a
small spatial region including the shock front, working in the frame
in which the shock is stationary. As the thickness of this region
becomes infinitesimal, $\int dx {\partial \over \partial t}$ goes to
zero, because $\partial /\partial t$ is finite everywhere. At the
shock $\partial /\partial x$ is a $\delta$-function, but $\partial /
\partial t$, although not defined, is not a $\delta$-function because
no variable exhibits a discontinuous change in time. (For flows
uniform on each side of the shock $\partial / \partial t = 0$;
integrating over
an infinitesimal thickness generalizes the conclusion to flows
whose properties vary smoothly away from the shock.) Then
$\int dx {\partial f(x) \over \partial x}$ across the shock
reduces to
$f(x_{1}) - f(x_{0})$, and if there are no
$\delta$-function sources at the shock the result is
$f(x_{1}) = f(x_{0})$ for the three functions
$f$ of (3.1.1), (3.1.6), and (3.1.8). This gives
(3.3.6)--(3.3.8); to obtain the last of these divide the equation
for the third $f$ by that for the first one.
>From (3.3.6) and (3.3.7) we readily obtain
$$u_{0}^{2} = {\rho_{1} \over \rho_{0}} \left( { P_{1} - P_{0} \over
\rho_{1} - \rho_{0} } \right) . \eqno(3.3.9)$$
This has a simple but interesting interpretation. We know that
the speed of an infinitesimal sound wave in a fluid is
$\sqrt{\partial P / \partial \rho}$.
The ratio
$(P_{1}-P_{0})/(\rho_{1}-\rho_{0})$
is an approximation to
$\partial P / \partial \rho$, and should lie between the values of
$\partial P / \partial \rho$ for the unshocked and shocked fluid.
The sound speed in the heated, compressed, shocked fluid is larger
than in the unshocked fluid, so we may generally take
$c_{1}^{2} > (P_{1}-P_{0})/(\rho_{1}-\rho_{0}) > c_{0}^{2}$,
where $c$ denotes the sound speed. We also have
$\rho_{1} > \rho_{0}$, so that (3.3.9) gives
$$u_{0}^{2} > c_{0}^{2} . \eqno(3.3.10)$$
The shock advances into the unshocked fluid at a speed in excess
of its sound speed (supersonically); shocks have Mach numbers
greater than 1.
Elementary algebra (or an interchange of indices in 3.3.9, for the
original equations are symmetric in them) leads to
$$u_{1}^{2} = {\rho_{0} \over \rho_{1}} \left( {P_{1}-P_{0} \over
\rho_{1}-\rho_{0}} \right) . \eqno(3.3.11)$$
By essentially the same argument as before this leads to
$$u_{1}^{2} < c_{1}^{2} . \eqno(3.3.12)$$
The shocked matter moves away from the shock at a speed less than
its sound speed (shocks are subsonic with respect to the matter
behind them).
Equations (3.3.6)-(3.3.8) are three equations relating the four
\lq\lq unknowns\rq\rq\
$\rho_{1}$, $u_{1}$, $P_{1}$, ${\cal U}_{1}$
to the \lq\lq knowns\rq\rq\
$\rho_{0}$, $u_{0}$, $P_{0}$, ${\cal U}_{0}$.
If there is an additional constraint, it is possible to solve
explicitly for the unknowns in terms of the knowns. Because
we have assumed that the fluid is in thermodynamic equilibrium on each
side of the shock, we are assured that such an additional constraint
exists in the form of an equation of state, eq. (3.1.18).
Take the perfect gas equation of state
$${\cal E} = {1 \over \gamma -1} P , \eqno(3.3.13)$$
where $\gamma$ is a constant. For
reversible adiabatic transformations we showed in {\bf 1.9} that such a
gas follows the law
$$P \propto \rho^{\gamma} . \eqno(3.3.14)$$
This is a convenient and often useful law, but we must remember
that shocks are not reversible, and do not follow (3.3.14), even if
the gas satisfies (3.3.13). In this section we use (3.3.13), and
only mention (3.3.14) to remind the reader of the significance of
$\gamma$.
The jump conditions give
$${\cal U}_{1} - {\cal U}_{0} = {1 \over 2} (P_{1}+P_{0})(V_{0}-V_{1}) ,
\eqno(3.3.15)$$
which neatly expresses the change in
${\cal U}$ as a result of the mean \lq\lq$PdV$\rq\rq\ work. Using the
equation of state (3.3.13) we obtain
$$\eqalignno{
{P_{1} \over P_{0}}&= {(\gamma +1)V_{0}-(\gamma -1)V_{1} \over (\gamma
+1)V_{1}-(\gamma -1)V_{0}}&(3.3.16a)\cr \noalign{\hbox{or}}
{V_{1} \over V_{0}}&=
{(\gamma -1)P_{1}+(\gamma +1)P_{0} \over (\gamma +1)P_{1}+(\gamma -1)
P_{0}}.&(3.3.16b)\cr}$$
Equation (3.3.16) has a striking and important consequence. In the
limit
$P_{1}/P_{0} \rightarrow \infty$ we have
$${V_{1}\over V_{0}} \rightarrow {\gamma -1 \over \gamma +1} .
\eqno(3.3.17)$$
No single shock, no matter how strong, can compress matter by a
factor of more than
$(\gamma + 1)/(\gamma - 1)$, which is equal to 4 for a simple
monatomic gas with
$\gamma = 5/3$. The pressure and temperature of the shocked
matter may increase without bound, but the density cannot. The
only exception to this rule is obtained if
$\gamma = 1$. This describes an isothermal gas, for which the
temperature is fixed, but whose density may be increased
arbitrarily by a single shock.
A practical realization of an isothermal gas is one which cools
radiatively, and for which the radiative cooling rate is a very
steeply increasing function of temperature. This is, in fact, often
a good approximation for the interstellar gas, because its
cooling depends on collisional excitation of energy levels with
excitation energy
$\chi \gg k_{B}T$. The cooling rate then varies approximately as
$\exp (-\chi /k_{B}T)$,
with a very large coefficient. The consequence of this is that
strong shocks in the interstellar medium (such as those produced
by supernova explosions) produce sheets or shells of very high
density, at least in calculations.
These dense shells may be sites of star formation. This suggestion is
controversial, in part because the compression is in one dimension only.
The Jeans length of a flattened cloud is approximately determined by its
mean density averaged over a spherical volume $\lambda_{J}^{3}$; it is
easily seen that the flattening to a pancake of a cloud initially
smaller than $\lambda_{J}$ will not, by itself, make it gravitationally
unstable. The increased confining pressure behind the shock will
contribute to instability. The interstellar medium has a very
heterogeneous density distribution, so that different parts of the shell
will travel at different speeds, depending on the density of the material
they encounter. Transverse density gradients will also alter their
direction of motion (refracting the shock). Fragments of the dense
shell may acquire a significant velocity dispersion, which interferes
with gravitational collapse ({\bf 3.2}).
Sonic booms are familiar examples of weak shocks. The pressure jump
$P_{1}-P_{0}$ is orders of magnitude less than $P$. Writing
$P_{1} = P_{0}(1+\alpha )$, and taking
$\alpha \ll 1$ in (3.3.16) leads to
$\rho_{1} \approx \rho_{0}(1+\alpha / \gamma)$. Weak shocks therefore
satisfy
$${d \ln P \over d \ln \rho} = \gamma , \eqno(3.3.18)$$
the same equation that applies to infinitesimal sound waves in a gas
with equation of state (3.3.13). By (3.3.9) and (3.3.11),
a weak shock moves at the sound speed with respect to both the shocked
and unshocked gas, and in all respects resembles a sound wave of
impulsive form. The frequency spectrum is given by the Fourier
transform of its step function pressure profile.
Shocks may be produced in a wide variety of circumstances. The
piston discussed at the beginning of this section is only the
simplest possibility. If a fluid contains a propagating pressure
profile resembling Figure 3.2.a,
the sound speed will be higher in the adiabatically compressed
and heated matter near the crest of the wave than elsewhere. For
this reason, and because the nonlinear $(\vec u \cdot \nabla )\vec u$
term has a similar effect, the peak of the pressure profile will
gradually overtake its leading edge, and its shape will steepen, as
shown in Figure 3.2.b. Eventually a shock will form, as shown in
Figure 3.2.c. This qualitative description is confirmed by numerical
calculations.
\topinsert
\vskip 10.5truecm
\ctrline{{\bf Figure 3.2.} Shock formation.}
\endinsert
Sound waves of infinitesimal amplitude propagate at constant
speed with a stationary profile. The amplitude of any real
disturbance is not infinitesimal, so it might be expected to steepen and
form a shock, as sketched in Figure 3.2. This usually does not happen.
The steepening is slow if the amplitude is small. For example, the
sounds of ordinary speech have an intensity of about 75 dB in the jargon
of the acoustic engineer, who defines the intensity of sound as
$20 \log_{10} (\delta P/2 \times 10^{-10}$ atmospheres) dB. They would
require propagation through a distance $D \sim 10^{6}$ wavelengths to
steepen substantially, even if there were no spherical divergence.
Unless a sound wave
is initially quite strong, its steepening is usually overwhelmed
by losses resulting from viscosity, heat conduction, and delayed
relaxation to thermodynamic equilibrium (bulk viscosity).
Sound waves are also significantly attenuated at boundaries between
fluid media and solids, or between two fluids, because there are usually
dissipative heat flows or viscous forces at interfaces.
Sound waves share with shallow water waves the property that the
velocity of propagation increases at the crest of the wave. As a
result, shallow water waves also show a steepening process
qualitatively resembling that of Figure 3.2. The \lq\lq shock\rq\rq\
is called a hydrodynamic bore, and is observed on beaches as water runs
up them after a wave breaks, in some tidal estuaries, and in other
circumstances where a large surge of water is suddenly released.
The resemblance is only qualitative, for the dispersion relation
for infinitesimal shallow water waves is dispersive. As a result, there
exist stationary nondissipative propagating structures of finite
amplitude, called solitons, in which the dispersion is balanced
by the nonlinearity. Acoustic solitons are not possible because
there is no dispersion to balance their tendency to steepen into a shock.
We have said nothing about what actually goes on in a shock;
that is, about the nonequilibrium processes by which the material is
transformed from its equilibrium unshocked state to its equilibrium
shocked state. Weak shocks may, in some circumstances, be calculated as
if the matter were always close to equilibrium, so that ordinary
coefficients of viscosity and thermal conductivity (whose derivation
assumes small deviations from equilibrium) may be used.
A strong shock is a region of large velocity and thermal gradients,
about a mean free path thick, in which deviations from thermodynamic
equilibrium are large, and the material undergoes an irreversible change
of state. The hydrodynamic description of a fluid is not valid there,
and the transport coefficients (viscosity and thermal conductivity) are
not well defined. To compute the structure of a strong shock
quantitatively requires consideration of the full nonequilibrium
particle distribution function, and is quite difficult.
Various additional complications are possible, including
\lq\lq collisionless\rq\rq\ plasma shocks, in which dissipation is
provided by plasma instabilities (in effect, macro-collisions)
rather than by microscopic particle collisions. Fortunately, for most
purposes it is possible to ignore the detailed structure of the shock
itself. The power of the assumption of thermodynamic equilibrium
outside of the shock is that it permits calculation of all the
properties of the fluid from the conservation laws alone, without any
consideration of the way in which the equilibrium states are achieved.
\goodbreak
\vskip 2\baselineskip plus2\baselineskip
\goodbreak
\noindent
{\bf 3.4 Blast Waves}
\vskip \baselineskip
\noindent
An explosion or other instantaneous point release of energy within a
fluid produces an outward travelling shock. This is called a blast
wave. A full description of the flow is very complex, but if certain
assumptions can be justified a simple theory is both accurate and
informative. The theory was first developed in order to describe
man-made explosions in air, and is known as the Sedov-Taylor solution,
but also has astrophysical applications, particularly to supernova
explosions in the interstellar medium. Zeldovich and Raizer (1966)
give a thorough discussion with references to the original literature.
We first assume that the fluid medium is uniform and at rest. This is
well justified for most atmospheric explosions (so long as the radius
of the shock is much less than the scale height of the atmosphere, and
the shock has not touched the ground), but is more questionable for
interstellar explosions. The interstellar medium is strongly
heterogeneous, with its density believed to vary by a factor of at
least $\sim 10^{3}$ between the abundant ordinary clouds and the hot
intercloud medium (rarer dense clouds may be orders of magnitude denser
still). It is thus necessary to assume either that an average may be
taken over the heterogeneities (combining high and low density regions
to form an effective mean medium), or that the presence of the clouds
may be ignored (if they
do not affect the flow around them, or if they are sparsely enough
distributed that a typical blast wave does not encounter them). The
validity of any of these assumptions of the interstellar medium has not
been established. Many interstellar blast waves produced by supernovae
(called supernova remnants) are observed, but tests of the theory are
indirect and uncertain because its basic parameters---the age and energy
of the explosion and the density of the medium---are generally not
directly observable or quantitatively known.
\def\rightheadline{\vbox{\vskip 0.125truein\line{\tenbf\hfil
Blast Waves \qquad\folio}}}
This problem is described by several parameters: the energy $Y$ and mass
$M$ of the exploding matter, and the density $\rho_{0}$ and pressure
$P_{0}$ of the medium. We always assume $Y/M \gg P_{0}/\rho_{0}$;
if this were not the case the explosion would never produce a strong
shock. Three regimes may be distinguished on the basis of the radius
$R$ of the spherical blast wave:
In the early regime I, $R\ll (M/\rho_{0})^{1/3}$, and the mass contained
within the radius $R$ is almost entirely that of the exploding object.
The surrounding medium has had little influence on its motion.
Supernova remnants typically remain in this regime for the first
$\sim 300 - 3000$ years of their lives, depending on $M$, $\rho_{0}$,
and their initial velocity of explosion. The Crab Nebula supernova
remnant, the relic of a supernova of the year 1054, is believed to be in
this regime, in large part because for it $\rho_{0}$ and the initial
expansion velocity were unusually low.
In the late regime III, $R \gg (Y/P_{0})^{1/3}$. Most of the energy
contained within the blast wave was internal energy of the fluid prior
to the explosion, with $Y$ contributing only a small fraction. The
shock is now weak and propagates at nearly the sound speed.
The middle regime II is the most interesting one; here $(Y/P_{0})^{1/3}
\gg R \gg (M/\rho_{0})^{1/3}$. In this case most of the mass within
the blast wave was that of the fluid medium, but most of the energy was
that of the explosion. If these inequalities hold to high accuracy
then $P_{0}$ and $M$ may be ignored, and the problem is described by
only two parameters, $Y$ and $\rho_{0}$, which permits great
simplification. We first discuss the early part IIa of this middle
regime, in which radiative losses are negligible, the total energy
within the blast wave is essentially constant (and equal to $Y$), and
the fluid is described by an adiabatic exponent $\gamma > 1$.
All physical quantities are products of powers of mass, length, and time.
If a problem has three intrinsic dimensional parameters of independent
dimensionality (meaning that none is proportional to a product of
powers of the other two; equivalently, the matrix formed by the powers
of mass, length, and time in the three quantities is not singular),
then it is possible to define a characteristic length (or any other
dimensional quantity) as a product of appropriate powers of the
dimensional parameters. The estimates of {\bf 1.4} were examples of this
procedure; a characteristic density $\rho$, pressure $P$, and thermal
energy $k_{B}T$\footnote*{The use of the temperature $T$ in place
of $k_{B}T$ is only a redefinition of the temperature scale
for the sake of convenience; $^{\circ}$K is not a physical unit
independent of energy.} were found for a star as a function of its
parameters $G$, $M$, and $R$.
If only two independent dimensional parameters exist then, in general,
no characteristic length or time may be defined (some characteristic
dimensional parameters may be formed from the two given parameters,
but in our case not the important ones of length, time, or velocity).
The solutions are of the form
$$\hskip 1.23truein\eqalign{P(r,t)&=P_{1}(t){\tilde P}\bigl(r/R(t)\bigr)
\cr u(r,t)&=u_{1}(t){\tilde u}\bigl(r/R(t)\bigr)\cr
\rho (r,t)&=\rho_{1}(t){\tilde \rho}\bigl(r/R(t)\bigr),\cr}
\eqno(3.4.1)$$
where the quantities $P_{1}$, $u_{1}$, and $\rho_{1}$ are the
pressure, velocity, and density immediately behind the shock,
${\tilde P}$, ${\tilde u}$, and ${\tilde \rho}$ are dimensionless
functions of dimensionless arguments, and $R(t)$ gives the blast wave
radius as a function of time. The form (3.4.1) is called a similarity
solution because the solution at any one time looks like that at any
other time if all lengths are multiplied by a single scale factor;
both solutions have the same shape. Such a solution is possible only
if the intrinsic parameters of the problem define no characteristic
lengths or times, for if they did the solutions when $R$ or $t$ were
near these characteristic lengths or times could be of a different
form than those when $R$ or $t$ were much greater or much less.
Our assumption that there are no dimensional parameters other than $Y$
and $\rho_{0}$ requires that the constitutive relations be of the form
(3.3.13) and (3.3.14), because any other form would contain
additional dimensional parameters. Because $P_{0}$ is negligible
the strong shock jump conditions ({\bf 3.3}) may be used to express
$P_{1}(t)$, $u_{1}(t)$, and $\rho_{1}(t)$ in terms of the shock
velocity; $\rho_{1}(t)$ is the constant density $\rho_{0}(\gamma +1)/
(\gamma -1) \geq 4 \rho_{0}$. Because $\rho_{1} \gg \rho_{0}$ most of
the swept up mass is concentrated in a dense shell just inside the blast
wave.
Equations (3.4.1) may be substituted in the hydrodynamic equations
in spherical geometry in order to obtain ordinary differential equations
for $\tilde P$, $\tilde u$, and $\tilde \rho$. It is more interesting
and important to find the function $R(t)$, and a simple dimensional
argument suffices. If we consider the blast wave at a specific time
$t$, this provides a third dimensional parameter. From $Y$, $\rho_{0}$,
and $t$ we can construct one quantity with the dimensions of length.
Because there are also no dimensionless parameters (pure numbers) in
the problem other than $\gamma$, $R(t)$ must be proportional to this one
characteristic length:
$$R(t) = \xi_{0} (\gamma )\left({Y t^{2}\over\rho_{0}}\right)^{1/5}.
\eqno(3.4.2)$$
The dimensionless function $\xi_{0}( \gamma )$ is the constant of
proportionality, and is found by integration of the equations for
$\tilde P$, $\tilde u$, and $\tilde \rho$; it is fairly close to unity.
>From (3.4.2) we obtain the shock velocity
$${dR(t) \over dt}={2 \over 5} \xi_{0} ( \gamma ) \left({Y \over \rho_{0}
t^{3}} \right)^{1/5} . \eqno(3.4.3)$$
The functions $P_{1}(t)$ and $u_{1}(t)$ may be obtained from (3.4.3)
and the strong shock jump conditions; on dimensional grounds alone we
have
$$\eqalignno{u_{1}(t)&\sim {dR(t) \over dt} \propto \left({Y \over
\rho_{0}} \right)^{1/5} t^{-3/5} \propto \left( {Y \over \rho_{0}}\right)
^{1/2} R^{-3/2}&(3.4.4a)\cr P_{1}(t)&\sim \rho_{0} \left( {dR(t) \over
dt}\right)^{2} \propto Y^{2/5}\rho_{0}^{3/5}t^{-6/5} \propto YR^{-3}
&(3.4.4b)\cr}$$
Equations (3.4.2)--(3.4.4) are used in theories of supernova remnants.
Because they are believed to be typically $10{,}000-30{,}000$ years old,
their ages and the time-dependence of their properties have not been
directly observed, and comparisons with the theory are only statistical.
The numerical values of $Y$ and $\rho_{0}$ are also poorly known, and
perhaps the most important application of this theory is to
determine the ratio $Y/\rho_{0}$ from (3.4.2). Typical values
obtained from statistical analyses are $\sim 3 \times 10^{75} \
{\rm cm}^{5}/{\rm sec}^{2}$ (Clark and Caswell 1976). There are at
least two distinct types of supernovae and the interstellar $\rho_{0}$
is known to be very heterogeneous, so it is not clear how a mean
$Y/\rho_{0}$ should be interpreted.
In the early part of regime II, just described, the temperature of
the shocked matter (which is proportional to \hbox{$P_{1}/\rho_{1}
\propto t^{-6/5} \propto R^{-3}$)} exceeds $10^{6\ \circ}$K,
and radiative cooling is unimportant. However, the radiative
cooling rate of interstellar matter (Spitzer 1978) rises steeply when
$T$ drops below $10^{6\ \circ}$K.
We now consider regime IIb, in which $(Y/P_{0})^{1/3} \gg R$ still
holds, but radiative energy losses are rapid, and the matter may be
approximately described as an isothermal fluid (the adiabatic exponent
$\gamma \rightarrow 1$). Now $\rho_{1} \rightarrow \infty$, and the
swept up mass forms a very thin and dense shell just behind the blast
wave. A simple \lq\lq snowplow\rq\rq\ model is useful.
Because radiative cooling is effective, the pressure in the low density
region inside the dense shell is very low. Each
element of solid angle of the shell now moves independently of all
other elements, and is slowed as it sweeps up further interstellar
material. Its collision with the interstellar matter is completely
inelastic because of the rapidity of the radiative losses; momentum
is conserved but all the kinetic energy (in the center-of-mass frame
of each element) is radiated. If we enter regime IIb at time $t_{0}$
with a shell of radius $R_{0}$, velocity $u_{0}$, and mass per unit
area $\sigma_{0} = R_{0}\rho_{0}/3$, its further slowing is described by
$$\eqalignno{u&=u_{0}\left({R_{0} \over R}\right)^{3}&(3.4.5a)\cr R&=
\left(R_{0}^{4} + 4(t-t_{0})u_{0}R_{0}^{3}\right)^{1/4}.&(3.4.5b)\cr}$$
When $u$ calculated from (3.4.5a) becomes comparable to $c_{s}$, the
blast wave enters the weak shock regime III. This is
discussed in Appendix {\bf A.2}.
\goodbreak
\vskip 2\baselineskip plus2\baselineskip
\goodbreak
\noindent
{\bf 3.5 Accretion}
\def\rightheadline{\vbox{\vskip 0.125truein\line{\tenbf\hfil Accretion
\qquad\folio}}}
\vskip \baselineskip
\noindent
Accretion is the name given the process by which an object increases
its mass by the capture of surrounding matter. We are here concerned
with the gravitational capture of gas by stars, and not with the
agglomeration of solid particles (for example, in the early Solar
System), in which surface forces are important. In this section we
consider only the processes by which gravitationally unbound matter
is captured, and not its subsequent flow onto the massive object which
attracted it; these flows are described in {\bf 3.6} and {\bf 3.7}.
Accretion was first investigated by astronomers who were concerned
with the possibility that the masses of stars might grow appreciably
during their lifetimes by accretion of interstellar material, or that
the potential energy released by infall onto the Sun might affect the
climate of the Earth. The development of a quantitative theory of
accretion and better estimates of the density of the interstellar
medium demonstrated that accretion is almost always insufficient to
produce these effects,
although between a protostar forming from a dense interstellar cloud and
a star is a state which may be described as a rapidly accreting star.
Modern interest in accretion focuses on two distinct problems. The
first concerns stars (including neutron stars
and black holes) in close binaries, which may accrete significant
amounts of mass from their companions. In some cases this mass flows
smoothly from one star to the other, and is always gravitationally
bound, but in others it must be captured from a high velocity wind
(like those discussed in {\bf 1.15}); in the latter case the
dynamics of accretion resemble that of accretion from the interstellar
medium. Accretion onto collapsed stars is of interest because a great
deal of gravitational potential energy is released. The second problem
concerns peculiar stars, particularly white dwarves, whose surface
composition and spectrum may be observably affected by the accretion
of very small amounts of interstellar material.
Consider accretion by a star of mass $M$ and radius $R$ moving at a
velocity $v$ through an initially uniform medium of density $\rho_{0}$.
Assume
$$v \ll \sqrt{GM/R} , \eqno(3.5.1)$$
as is usually the case, and first take the particles of the medium to
be collisionless. Then each particle follows the hyperbolic path of a
free test particle, unless it actually collides with the stellar surface.
It is simple to calculate the limiting impact parameter $b_{1}$,
within which particles collide with the star and are accreted.
Particles with this impact parameter have orbits just tangent to the
stellar surface. Because of (3.5.1) their orbits are nearly
parabolic, and by conservation of angular momentum we have
$$b_{1}v = \sqrt{2GMR} . \eqno(3.5.2)$$
The accretion rate $A$ is then given by
$$\eqalign{A&=\pi b_{1}^{2} \rho_{0} v \cr
&={2 \pi GMR\rho_{0} \over v} . \cr}\eqno(3.5.3)$$
For a star like the Sun moving at $v=20$ km/sec through a medium with
$\rho_{0}=10^{-24}$ gm/cm$^{3}$ we have $A \approx 3 \times 10^{7}
{\rm gm/sec} \approx 5 \times 10^{-19}\ M_{\odot}$/yr, a negligible
value.
The actual accretion rate of a collisional fluid is expected to be
much larger than that given by (3.5.3), and was calculated by Hoyle
and Lyttleton (1939). Figure 3.3 shows a flow pattern stationary in
the frame of the star, which is moving to the right. Fluid inside the
nearly parabolic hyperbola AB (actually a hyperboloid of revolution
whose axis is the star's direction of motion) falls directly onto the
stellar surface, with the accretion rate (3.5.3). Material in paths
with larger impact parameters, like CD and ED, converges on the axis of
motion behind the star and undergoes particle collisions there. The
accretion wake (qualitatively sketched as a cross-hatched area) contains
particles which have collided; if mean free paths are short this region
is sharply bounded by a thin shock.
\topinsert
\vskip 10.5truecm
\ctrline{{\bf Figure 3.3.} Accretion flow.}
\endinsert
All the fluid is on hyperbolic orbits about the star until collisions
occur. They contribute to accretion in two ways. Collisions among
atoms, ions with bound electrons, and molecules are often inelastic,
leaving the collision products in excited states whose energy is usually
promptly radiated, or produce dissociation and ionization; these
processes reduce the particle kinetic energy. When collisions are
frequent the matter follows the equations of hydrodynamics. Even if
its total (kinetic plus internal) energy exceeds its gravitational
binding energy, as it did before collisions began, there will be a flow
in the accretion wake towards the attracting star. Matter close to the
star will be in supersonic free-fall (this is an assumed boundary
condition, which is justified if matter radiates and cools rapidly
upon impact with the stellar surface, as is usually the case). No
acoustic signal can propagate supersonically and reach the outer parts
of the wake; there is no pressure support from the stellar surface, and
the pressure and internal energy of the wake do not effectively
oppose gravity. This is very different from a stellar wind, in which
subsonically moving matter near the stellar surface supports and
accelerates the outflow.
Roughly, we may say that matter in the wake whose
kinetic energy alone is less than its gravitational energy will be
accreted. The importance of the shock at the surface of the accretion
wake is that it converts a portion of the kinetic energy of the matter
to internal energy, and thus reduces its velocity below escape velocity.
To calculate the accretion rate we need to determine the impact
parameter $b$ for which the matter becomes bound after entering the
accretion wake. On the wake axis the tangential
component of velocity $v_{\theta}$ is zero (it drops by a large factor
at the boundary shock, and is then reduced to zero by a pressure
gradient within the wake). In order that the lost kinetic energy per
unit mass equal that at infinity, $v_{\theta}$ of the matter entering
the accretion wake must satisfy
$${1 \over 2} v_{\theta}^{2} = {1 \over 2} v^{2} . \eqno(3.5.4)$$
\vskip7pt plus7pt minus7pt
\endpage
The matter follows hyperbolic orbits, which are given by
$$\eqalignno{r&={b^{2}v^{2}/GM \over 1 + e \cos \theta}&(3.5.5a)\cr
\noalign{\hbox{where the eccentricity}}
e&=\sqrt{{b^{2}v^{4} \over G^{2} M^{2}} +1 } , &(3.5.5b)\cr}$$
and the angle $\theta$ is measured from the point of closest approach.
The asymptote is at the angle $\theta_{a} = \cos^{-1} (-1/e)$, and the
point D is at $\theta_{D} = \theta_{a} \pm \pi$, so that $\cos \theta_{D}
= 1/e$. Then the distance $r_{D}$ from D to the attracting star is
$$r_{D} = {b^{2}v^{2} \over 2GM} . \eqno(3.5.6)$$
By conservation of angular momentum
$$r_{D}v_{\theta} = bv \eqno(3.5.7)$$
so that at D
$$v_{\theta} = {2GM \over bv} . \eqno(3.5.8)$$
Now (3.5.4) is satisfied if
$$b = {2GM \over v^{2}}, \eqno(3.5.9)$$
implying an accretion rate
$$\eqalign{A&= \pi b^{2} \rho_{0} v \cr
&= {4 \pi G^{2} M^{2} \rho_{0} \over v^{3}}.}\eqno(3.5.10)$$
This result is larger than (3.5.3) by a factor $2GM/(v^{2}R)$, which
is $\sim 10^{3}$ for typical stellar parameters. The accretion rate is
still negligible under ordinary interstellar conditions ($\sim 10^{-15}
M_{\odot}$/yr for the Sun), but may be significant in other problems.
The quadratic dependence of $A$ on $M$ is noteworthy; $A$ is
proportionally more important for very massive objects (large
interstellar clouds, star clusters, and galaxies) than for ordinary
stars.
The apparent divergence of $A$ as $v \rightarrow 0$ is also noteworthy.
We have so far assumed that the fluid is initially pressureless and has
zero sound speed. Calculations by Bondi (1952) have shown that if
$v=0$ (3.5.10) gives the correct answer, if $v$ is
replaced by the sound speed $c_{s}$, and an uncertain (but $\approx 1$)
coefficient is introduced. A plausible interpolation formula between
the $v \ll c_{s}$ and $v \gg c_{s}$ regimes is
$$A = {4 \pi G^{2} M^{2} \rho_{0} \over (v^{2} + c_{s}^{2})^{3/2}} .
\eqno(3.5.11)$$
Numerical calculations by Hunt (1971) have confirmed this expression.
Perhaps his most important result is the applicability of (3.5.11)
even when there are no radiative losses; this justifies the assertion
than only kinetic, and not internal, energy contributes to the escape
of fluid from the gravitational potential well.
For collisionless particles (3.5.3) should still be used; under
typical interstellar conditions neutral atoms and molecules may be
considered collisionless, and their accretion rate is very low.
Thus the ionization state of the fluid
determines its accretion rate. Careful calculations of interstellar
accretion must consider photoionization by the radiation of the
accreting object and the quantitative cross-sections
for ion-neutral charge exchange and other collisional processes.
If the fluid is heterogeneous the accretion rate will be reduced below
that of (3.5.11), because even a completely inelastic collision at D
(in Figure 3.3) will not reduce $v_{\theta}$ to zero. Quantitative
calculation requires a detailed description of the density
distribution of the heterogeneous fluid, which is rarely available.
The reduction of the accretion rate is not likely to be more than a
factor $\sim 2$ unless density contrasts are high, because for $b$
significantly less than that given by (3.5.9) $v_{\theta}$ at D
substantially exceeds $v$, and loss of even half of $v_{\theta}$ would
leave the matter bound to the attracting object. Only in the case of
small isolated particles moving in vacuum without dissipation is
(3.5.3) correct.
\goodbreak
\vskip 2\baselineskip plus2\baselineskip
\goodbreak
\noindent
{\bf 3.6 Accretion Discs}
\def\rightheadline{\vbox{\vskip 0.125truein\line{\tenbf\hfil Accretion
Discs\qquad\folio}}}
\vskip \baselineskip
\noindent
In the preceding section we estimated the rate at which matter may be
gravitationally captured by a star or compact object (degenerate dwarf,
neutron star, or black hole). Its subsequent fate depends on its
angular momentum about the source of gravitational attraction. If it
has no angular momentum then matter will fall radially inward, and will
soon either hit the surface of a star (releasing its gravitational
binding energy as heat), or be swallowed by the event horizon of a
black hole (in which case very little energy may be radiated).
Even a small amount of angular momentum drastically changes the flow.
Matter with angular momentum per unit mass \hbox{$\ell > \sqrt{
2GMR}$} cannot fall directly onto the surface of a star of mass $M$ and
radius $R$. For neutron stars and black holes of comparable mass the
maximum $\ell \approx 10^{16}$ cm$^{2}$/sec, while for degenerate
dwarves it is $\approx 3 \times 10^{17}$ cm$^{2}$/sec. If a neutron
star or a degenerate dwarf has a large magnetic field this may increase
its effective size and maximum $\ell$ for direct accretion; this is
discussed in {\bf 4.2.2}.
In the evolution of a binary system one star may expand beyond its
limiting surface, known as the Roche lobe (or Roche limit), within which
its gravity can confine matter. Outside this surface the gravity of its
companion is sufficient to draw matter away from the expanding star,
and toward the companion. This process, known as Roche lobe overflow,
produces a smoothly flowing stream of matter from one star to the other.
The rate of mass flow depends on the rate of evolution of the expanding
star, the ratio of the stellar masses, and on the rates of angular
momentum transfer and other relaxation processes within the binary;
values over a large range are predicted (or observationally inferred)
in different circumstances. The specific angular momentum $\ell$
of the transferred matter is comparable to that of the orbital
motion, so that \hbox{$\ell \gapp 3 \times 10^{18}$ cm$^{2}$/sec}
(depending on the stellar masses and the size of the orbit).
In accretion from a wind or other extended medium $\ell$ depends on the
medium's heterogeneity. For typical interstellar accretion parameters
$\ell$ may be as high as $\ell \sim bv \delta \rho /\rho \sim 10^{20}
\delta \rho /\rho$ cm$^{2}$/sec, where $\delta\rho /\rho$ is the typical
mean density heterogeneity averaged over the scale of the accretion
impact parameter $b$ (but see Livio 1986).
In accretion from the high velocity wind of a binary companion
star the spherical expansion of the wind guarantees a minimum
$\delta \rho /\rho \sim 2b/D$ where $D$ is the separation between the
centers of the two stars (there may be additional sources of
heterogeneity); for typical parameters the previous estimate gives
$\ell \sim 10^{16}$ cm$^{2}$/sec.
Under some (but not all) plausible circumstances sufficient angular
momentum exists to prevent immediate accretion by the attracting star.
We saw in {\bf 3.2} that the collapse of a protostar from an interstellar
cloud is similarly limited by angular momentum.
Accretion discs were first studied for their application to the early
Solar System; the pre-history of their study goes back to Laplace.
Pringle (1981) presents a review with references to the extensive
literature. The fundamental assumption in the theory of accretion
discs is that matter radiates its internal
energy much faster than it loses angular
momentum. Examination of the rates of the many processes involved
supports this. In fact, the central mystery of accretion disc physics
is the mechanism by which angular momentum is transferred from one
element of mass to another; ordinary viscous torques are insufficient.
The lowest energy state of matter with given angular momentum is a
circular Keplerian orbit; orbits with the same radius but different
orientations collide, the energy of collision is radiated, and the
matter settles into the circular orbit determined by its mean
angular momentum.
Orbits with different radii need not be coplanar unless they are coupled
by some dissipative process (models for some actual accretion
discs in X-ray sources and SS433 require that they not be coplanar).
Frictional torques are small because they are proportional to the
viscosity. Gravitational torques are not dissipative and cannot
directly enforce coplanarity, even if they are large;
self-gravitational torques are small if disc masses are low.
Despite these complications, simple models naturally assume
a planar disc; deviations from flatness are considered as perturbations.
Self-gravity is usually neglected in disc models; this is believed (on
somewhat uncertain grounds) to be justified in accretion discs arising
from mass flows between the components of close binary stars, or from
accretion onto stars and compact objects, but may well be wrong for
protostellar discs. It is also assumed that pressure forces are small
in comparison to the gravitational attraction of the central mass; this
is justified by the assumption that radiation of energy is rapid. In
the almost infinitesimally thin rings of Saturn, composed of solid
particles, this approximation holds extraordinarily well. It is not
empirically known how accurate this approximation is in stellar
accretion discs; it may be incorrect, or at best rough. Finally, we
assume Newtonian gravity; the thorough discussion by Novikov and Thorne
(1972) includes relativistic effects.
The equations of structure of a fluid disc follow from these assumptions.
On circular Keplerian orbits the velocity is entirely in the azimuthal
($\phi$) direction:
$$v_{\phi} = \sqrt{GM/r} . \eqno(3.6.1)$$
This is the fundamental approximation of thin disc theory. Because the
disc matter is not exactly at zero temperature and zero pressure the
disc will have a finite thickness $h$ in the direction ($z$) along the
angular momentum axis. If $h \ll r$ then the vertical and radial
structure may be considered separately.
There is hydrostatic equilibrium between the
$z$-components of the pressure gradient and the gravitational
acceleration (by assumption, entirely that of the central mass).
Expanding the gravitational force to lowest order in $z/r$, we obtain
$${\partial P \over \partial z} = - \rho {GMz \over r^{3}} .
\eqno(3.6.2)$$
This force law is that of a harmonic oscillator. From this we obtain
an estimate of $h$:
$$h \approx \sqrt{{P \over \rho}{r^{3} \over GM}} \approx {c_{s} \over
\Omega}, \eqno(3.6.3)$$
where $c_{s}$ is the sound speed and $\Omega$ is the angular frequency
of Keplerian orbits about the central mass. Because of the square root
in (3.6.3) $h$ varies as the $1/2$ power of the temperature, rather
than being proportional to it, as is
familiar from ordinary atmospheres; $h$ should
not be thought of as a scale height, for the density distribution is
closer to Gaussian than exponential. A disc may, in addition, have a
much thinner atmosphere whose scale height is proportional to its
temperature.
Accretion discs are interesting (and observable) only if accretion
actually takes place; that is, if matter flows through them. In order
for matter to flow inward it must lose angular momentum, which flows
outward. The net binding energy per unit mass of matter in Keplerian
circular orbits is $-G^{2}M^{2}/(2 \ell^{2})$, so that in the state of
lowest energy nearly all the disc mass has $\ell \rightarrow 0$ (it
accumulates at zero radius, or on the surface of the attracting body),
while all the angular momentum resides in an infinitesimal fraction of
the disc mass in orbit at $r \rightarrow \infty$. The problem
is to separate the mass from its angular momentum.
In a steady state in which there is a steady supply of matter with a
finite $\ell > 0$, there is a continual flow of mass inward and of
angular momentum outward. There must be a torque to extract angular
momentum from the inward flowing mass. This torque is described as
the consequence of a viscous stress, whatever its microscopic origin
(which may be magnetic). The central problem of accretion
disc theory is that it is not known, even to order of magnitude, how
to calculate this stress; ordinary viscosity is certainly present
but is much too small to explain the observed accretion rates. This
problem is evaded by describing the stress by a parameter of unknown
value.
The radial structure equations may be written in terms of
quantities which are integrals over the vertical structure. The surface
density is defined $\Sigma \equiv \int \rho \, dz$, and the mean viscous
stress tensor $\langle t_{r\phi} \rangle\equiv\int t_{r\phi}\,dz/(2h)$;
only the $r\phi$ component is important in a thin disc. In a steady
state disc, conservation of mass may be expressed
$$-2 \pi r v_{r} \Sigma = {\dot M} , \eqno(3.6.4)$$
where $\dot M$ is the mass flow rate and $v_{r}$ the (mass-averaged)
mean radial component of velocity. The conservation of angular
momentum takes the form
$${\dot M} \sqrt{GMr} = (2 \pi r)(2 h \langle t_{r\phi} \rangle r) +
{\dot J} . \eqno(3.6.5)$$
The left hand side is the rate at which mass flow carries angular
momentum inward across a cylinder of radius $r$. In steady state the
disc inside this radius does not accumulate angular momentum, so this
flow equals the viscous torque the disc inside $r$ exerts on that
outside $r$ plus the rate $\dot J$ at which angular momentum is taken
up by the central attracting mass.
$\dot J$ depends on the boundary conditions at the
central object, which depend on its mass, angular momentum, and
magnetic field, and on the radius $r_{I}$ of the inner edge of the disc.
Like other uncertainties, it is readily parametrized:
$${\dot J} = \zeta {\dot M} \sqrt{GMr_{I}} . \eqno(3.6.6)$$
The parameter $\zeta$ is expected to be in the range $0 \leq \zeta \leq
1$; \hbox{$\zeta < 0$} would correspond to a rapidly rotating central
object supplying angular momentum to the disc; such a process may occur
(and may be necessary to explain the very long spin periods observed
for some neutron stars), but probably only in a disc in which the
central object does not accrete at all, and all matter flows outwards.
In most cases $\zeta = 1$ is probably a fair assumption, equivalent to
assuming that the central object acquires the angular momentum of the
accreted matter along with its mass.
>From (3.6.5) and (3.6.6) we obtain the mean stress
$$2 h \langle t_{r\phi} \rangle = {{\dot M}\sqrt{GMr} \over 2 \pi r^{2}}
\left( 1- \zeta \sqrt{r_{I} \over r} \right) . \eqno(3.6.7)$$
In order to find the heat released we need the rate-of-strain tensor,
whose $r\phi$ component is given, for Keplerian orbits and $v_{r} \ll
v_{\phi}$, by
$$\hskip 1.06truein\eqalign{
\sigma_{r\phi}& = {1 \over 2} \left( {1 \over r}{\partial
v_{r} \over \partial \phi} + {\partial v_{\phi} \over \partial r} -
{v_{\phi} \over r}\right) \cr &=-{3 \over 4} \sqrt{GM \over r^{3}}.\cr}
\eqno(3.6.8)$$
The rate of viscous heating per unit volume is given by
$$\epsilon = -2 t_{r\phi} \sigma_{r\phi} , \eqno(3.6.9)$$
and the total power released per unit area $2H \equiv \int \epsilon \,
dz$ is
$$2H = {3GM{\dot M} \over 4 \pi r^{3}} \left( 1 - \zeta \sqrt{r_{I}
\over r} \right) . \eqno(3.6.10)$$
Note that for $r_{I} \ll r$ or $\zeta = 0$ (3.6.10) is three times
the rate at which gravitational energy is released by the progression
of matter to more tightly bound orbits; the source of the extra energy
is work done by the viscous stress. Because a disc has two surfaces,
each element of surface radiates a power per unit area $H$.
The total disc luminosity $L$ is found, if the disc extends to very
large radii:
$$\hskip 1.33truein\eqalign{L&=\int_{r_{I}}^{\infty} 2H\ 2\pi r\ dr \cr
&=\left( {3 \over 2} - \zeta \right) { GM{\dot M} \over r_{I}}. \cr}
\eqno(3.6.11)$$
This is the sum of two terms. $GM{\dot M}/(2r_{I})$ is the binding
energy of the Keplerian orbit at radius $r_{I}$, while $(1-\zeta )
GM{\dot M}/r_{I}$ represents work done by the central object on the
disc. The kinetic energy of the Keplerian orbit at $r_{I}$ may be
dissipated and radiated in a narrow boundary layer at the surface of
an accreting star, but because matter in this boundary layer
does not satisfy (3.6.1) disc theory is inapplicable there.
At each point on its surface a disc radiates the flux $H$. Most of the
energy is released at radii of order but not very close to $r_{I}$.
For $\zeta = 1$ half is released between $r_{I}$ and $4r_{I}$, and an
additional 40\% inside $26r_{I}$. As a rough approximation we
assume the disc surface radiates as a black body, in which case
the disc surface effective temperature is given by
$$\eqalign{T_{e}&=\left({H \over \sigma_{SB}} \right) ^{1/4} \cr
&=\biggl\lbrack {3GM{\dot M} \over 8 \pi r^{3} \sigma_{SB}} \left( 1 -
\zeta \sqrt{r_{I} \over r} \right) \biggr\rbrack ^{1/4} \cr &\sim 5
\times 10^{7\ \circ}{\rm K}\ {\dot M}_{17}^{1/4}
\left({M \over M_{\odot}}\right) ^{-1/2} \left({GM \over
rc^{2}} \right) ^{3/4} \left( 1 - \zeta \sqrt{r_{I} \over r} \right)
^{1/4} , } \eqno(3.6.12)$$
where ${\dot M}_{17} \equiv {\dot M} /(10^{17}\;{\rm gm/sec})$.
For typical X-ray source parameters most of the power is radiated
in soft X-rays ($h\nu \sim 3$ KeV). For parameters appropriate to the
supermassive black holes suggested for quasars, the disc radiation
would appear in the ultraviolet.
If $\zeta = 1$ the term proportional to it has the effect of moving the
peak $H$ and $T_{e}$ to $r = {49 \over 36} r_{I}$; the peak $H$ and
$T_{e}$ are respectively .057 and .49 of their values for the case
$\zeta = 0$. If there is a thin boundary layer it will have $H$ and
$T_{e}$ much higher than the disc values given by (3.6.10) and
(3.6.12).
In simple disc models the spectrum is given by an integral over black
body spectra, with $T_{e}$ given by (3.6.12). At low frequencies
($h\nu \ll k_{B}T$, where $T$ is an inner disc temperature), a power
law is predicted. Because of the exponential cutoff of a black body
spectrum, the Planck function (1.7.13) may be very roughly approximated
as $\propto \nu^{2} T$ for $h \nu < 3 k_{B} T$, and zero for $h \nu >
k_{B} T$. Then the integrated spectrum $I_{\nu}$ is estimated, to
order of magnitude, by
$$\hskip 1.17truein\eqalign{
I_{\nu}&=\int_{r_{I}}^{\infty} B_{\nu} \bigl(T(r)\bigr) 2 \pi
r \ dr \cr &\propto \int_{r_{I}}^{r_{max}} \nu^{2} T(r) r \ dr,\cr }
\eqno(3.6.13)$$
where $h \nu = 3 k_{B} T(r_{max})$ defines $r_{max}$. For $r_{max} \gg
r_{I}$ we have $T(r) \propto r^{-3/4}$ over most of the range of
integration, so that
$$\hskip 1.31truein\eqalign{
I_{\nu}&\propto \nu^{2} \int_{r_{I}}^{r_{max}} r^{1/4} \ dr\cr
&\propto \nu^{2} r_{max}^{5/4} .\cr } \eqno(3.6.14)$$
But $r_{max} \propto \nu^{-4/3}$, so that a power law is predicted
$$I_{\nu} \propto \nu^{1/3} . \eqno(3.6.15)$$
Unfortunately, (3.6.12) and (3.6.15) are not supported by any data.
There are few astronomical objects in which the continuum radiation from
an accretion disc can be unambiguously identified. One likely case is
the black hole candidate Cygnus X-1 (a black hole has no stellar surface
as an alternative source of radiation). Its X-ray spectrum requires
temperatures $10-100$ times higher than predicted. The most plausible
explanation is that most of the energy release occurs in very hot
optically thin regions, in disagreement with the assumption of a black
body radiator. There is no good way to predict the properties of
these regions, and Cygnus X-1 shows very different spectra at different
times, so that any single model is at least sometimes wrong.
Stars known as dwarf novae have outbursts in which most of their
radiation is believed to come from an accretion disc. The inferred
temperatures are at least approximately consistent with (3.6.12), but
data are not available over a wide enough range of frequency to test
(3.6.15). Because these are transient outbursts, steady state disc
models should be inapplicable.
We were able to evade the question of disc viscosity by obtaining
results parametrized by the accretion rate. $\dot M$ is directly
related to $L$ and other observable quantities, and may be estimated,
even though the orders of magnitude of $\Sigma$ and $v_{r}$ are unknown.
If assumptions are made about the viscosity these quantities
may be calculated, along with the detailed structure of the disc;
the results are no more certain than the assumptions. It is usually
found that $h \ll r$, typically by one to two orders of magnitude, and
that $h$ is nearly proportional to $r$ in a given disc, except in hot
inner regions; these
conclusions are nearly independent of the numerical values assumed for
the viscosity or $v_{r}$. They are consistent with the assumption of a
thin disc, but are not confirmed by any observational data. Uncertain
arguments and models of SS433 suggest that in the outer regions of its
disc $h/r \sim 0.5$.
It is possible to derive an interesting result for the thickness of a
luminous disc. Begin with the expression
(3.6.3) for $h$, and substitute $P=P_{r}/(1-\beta )$, where $\beta
\equiv P_{g}/P$ is taken to be constant:
$$h \approx \sqrt{{P_{r} \over \rho (1-\beta)}{r^{3} \over GM}} .
\eqno(3.6.16)$$
Now use (1.7.19) and (3.6.10):
$$\hskip 0.9truein\eqalign{
H&={c \over \kappa \rho}{dP_{r} \over dz} \cr
{3GM{\dot M} \over 8 \pi r^{3}}\left( 1-\zeta\sqrt{r_{I}\over r}\right)
&\approx {c P_{r}\over \kappa \rho h},\cr}\eqno(3.6.17)$$
Substituting this result for $P_{r}$ into (3.6.16) leads to
$$h \approx {3 \kappa {\dot M}\over 8\pi c (1-\beta)}
\left( 1 - \zeta \sqrt{r_{I} \over r} \right) . \eqno(3.6.18)$$
Define the energetic efficiency of accretion $\varepsilon$ by $L={\dot M}
c^{2}\varepsilon$, and the characteristic accretion rate ${\dot M}_{E}$
in terms of the Eddington limiting luminosity $L_{E}$ (1.11.6):
$$\eqalign{{\dot M}_{E}&\equiv {L_{E} \over c^{2} \varepsilon} \cr
&={4 \pi GM \over \kappa c \varepsilon} .\cr} \eqno(3.6.19)$$
>From (3.6.11)
$$\varepsilon = \left( {3 \over 2} - \zeta\right){GM \over r_{I} c^{2}}.
\eqno(3.6.20)$$
(3.6.18) may be rewritten, if $\beta \ll 1$ (as is found to be
the case in the inner regions of luminous discs)
$$h \approx r_{I} \left( { 3 \over 3 - 2 \zeta} \right)
\left( {{\dot M} \over {\dot M}_{E}} \right)
\left( 1-\zeta\sqrt{r_{I}\over r} \right).\eqno(3.6.21)$$
Then $h$ varies only slowly with $r$.
The maximum value of $h/r$ is found to be
$$\hskip .10truein \eqalign{
\left({4\over 27 \zeta^{2}-18\zeta^{3}} \right) \left({{\dot M} \over
{\dot M}_{E}} \right) \quad {\rm at} \quad r &= {9 \over 4}\zeta^{2}
r_{I} \qquad {\rm if}\ \zeta \geq {2 \over 3},\cr \left( {3 - 3 \zeta
\over 3 - 2 \zeta} \right) \left( {{\dot M} \over {\dot M}_{E}} \right)
\quad {\rm at} \quad r &=r_{I} \qquad\hskip .29truein {\rm if}\ \zeta
\leq {2 \over 3}.\cr}\eqno(3.6.22)$$
These results imply the existence of a characteristic disc
accretion rate and radiative luminosity, related to the limiting
radiative luminosity of stars. If ${\dot M} \gapp {\dot M}_{E}$ the
inner regions of a disc have $h \gapp r$, and thin disc theory is
inapplicable. Disc accretion at such a high rate may qualitatively
resemble radial accretion, which depends on the nature of the central
object. The energy released by accretion is trapped as radiation
within the accreting matter ({\bf 3.7}). If the attracting object has a
surface, then an extended envelope close to hydrostatic equilibrium
will rapidly accumulate, while in accretion onto a black hole the
radiation is swept into the black hole along with the matter (Eggum,
{\it et al.} 1985). Even if ${\dot M} \gg {\dot M}_{E}$, the emergent
luminosity $L \lapp L_{E}$.
Thin discs and non-rotating stars may be thought of as two opposite
limits of a continuum. In a star the force of gravity is balanced by
a pressure gradient, while in a thin disc gravity is balanced by angular
momentum. Most astronomical objects, other than those in which flows
are chaotic, may be approximately described by one limit or the other.
For this reason the understanding of discs may be as important to
astrophysics as that of stars.
Intermediate configurations may exist; near the stellar end
of the continuum they may be considered rotationally flattened stars,
while near the thin disc end they are discs thickened by their internal
pressure. In a boundary layer between a thin disc and a slowly
rotating star there is a continuous transition between these two limits.
The theory of discs is in a much more primitive state than that of
stars, because one essential constitutive relation is not understood,
their rate $\epsilon$ of viscous heating. This resembles the problem of
stellar structure
prior to the development of nuclear physics in the 1930's. We may be
worse off than this, because so few direct observations of discs are
possible. What little data exist (for example, for discs around likely
black holes like Cygnus X-1) indicates that real discs are not steady
objects radiating from optically thick photospheres (as the theory
assumes), but that they are wildly variable, release much of their
energy in optically thin regions, and may have important nonthermal
processes. It may be appropriate to compare our present understanding
of discs to Galileo's understanding of sunspots and solar activity.
\goodbreak
\vskip 2\baselineskip plus2\baselineskip
\goodbreak
\noindent
{\bf 3.7 Radial Infall}
\def\rightheadline{\vbox{\vskip 0.125truein\line{\tenbf\hfil Radial
Infall \qquad\folio}}}
\vskip \baselineskip
\noindent
There are at least three reasons for considering radial accretion flows,
in which matter with negligible angular momentum falls radially onto a
star, compact object, or black hole. In some circumstances matter with
very little angular momentum may be accreted, and infall is nearly
radial. This need not imply that the flow is spherically symmetric; for
example, flow from the accretion wake shown in Figure 3.3 may be
radially inward, but is concentrated onto a small part of the stellar
surface. The second reason is that if the accreting object has a large
magnetic field this field will force the infall of conducting matter
near it to flow along the field lines, and in particular along
those field lines which extend to great distances. In this case
the flow near the star will be nearly radial, and may therefore
closely approximate radial accretion. This is believed to be a good
description of accretion onto many magnetized neutron stars and
degenerate dwarves. The final reason is that radial accretion is a
relatively simple and calculable problem, which stands at the opposite
extreme from that of an angular momentum-dominated accretion disc.
Even were radial accretion never realized, its study would illuminate
the range of accretion flows possible in more complex circumstances.
In free fall from infinity the infall velocity $v_{r}$ is
$$v_{r} = - \sqrt{2GM \over r} . \eqno(3.7.1)$$
For an accretion rate ${\dot M}$ the density is
$$\rho = {{\dot M} \over \vert v_{r} \vert r^{2} \Omega},\eqno(3.7.2)$$
where the flow has been assumed uniform over a solid angle $\Omega$.
In a spherically symmetric flow $\Omega = 4 \pi$, but in other cases
$\Omega$ may be much less. If the flow is guided by a dipole magnetic
field the solid angle is determined by the size of a bundle of field
lines. Because $B \propto r^{-3}$ the cross-section of such a bundle
is $\propto r^{3}$, and $\Omega \propto r$.
The most important parameter describing radial accretion is ${\dot M}$.
If ${\dot M}$ is small then the accretion luminosity
$$L = {\dot M} c^{2} \varepsilon \eqno(3.7.3)$$
is small, the outward force $F_{rad}$ of radiation pressure on the
infalling matter is negligible, and (3.7.1) and (3.7.2) are
applicable. In spherical geometry
$$F_{rad} = {L \kappa \over 4 \pi r^{2} c}, \eqno(3.7.4)$$
where $\kappa$ is the opacity. Properly, $\kappa$ should be
frequency-averaged, but in most accretion flows the opacity
is almost entirely frequency-independent electron scattering, so
$\kappa$ may be taken to be $\kappa_{es}$.
The ratio of $F_{rad}$ to the inward force of gravity is
$$\eqalign{{F_{rad} \over F_{grav}}&= {L \kappa \over 4 \pi cGM}\cr
&={L \over L_{E}},\cr}\eqno(3.7.5)$$
where the Eddington luminosity $L_{E}=4 \pi cGM/\kappa$ was defined
for stellar interiors in (1.11.6).
If $L \ll L_{E}$ the influence of radiation pressure may be ignored.
If $L > L_{E}$ the net force on matter is directed outward! It is
also possible to define the same Eddington accretion rate ${\dot M}_{E}
\equiv L_{E}/(c^{2} \varepsilon),$ where $c^{2}\varepsilon$ is the
energy release per gram, as we did for disc accretion in (3.6.19).
It is clear that $L_{E}$ is an upper bound on the luminosity emerging
from an accreting object, just as it limits the radiative luminosity
of a star in hydrostatic equilibrium, and the applicability of thin
disc models. It is less obvious what happens if ${\dot M} >
{\dot M}_{E}$; that is, if an accretion flow \lq\lq tries\rq\rq\ to
exceed $L_{E}$. For example, an external source of mass may supply it
at a rate exceeding ${\dot M}_{E}$. In stellar interiors in
hydrostatic equilibrium luminosities in excess of $L_{E}$ can be carried
by convection, but this is inapplicable to accretion flows, which are
very far from hydrostatic.
If the accreted matter falls freely until it hits the stellar surface,
the optical depth $\tau$ along a path radially outward from the surface
to infinity is
$$\eqalign{\tau&= \int_{R}^{\infty} \kappa \rho \ dr \cr &= {\kappa
{\dot M} \over 2 \pi \sqrt{2GMR}}.\cr} \eqno(3.7.6)$$
We have assumed a spherically symmetric inflow. This may be rewritten,
using the definition of ${\dot M}_{E}$ and the Newtonian value of
$\varepsilon = GM /(Rc^{2})$. The result, written in terms of the escape
velocity $v_{esc} = \sqrt{2GM/R} < c$, is
$$\eqalign{\tau&= \left({{\dot M} \over {\dot M}_{E}}\right) \sqrt{2GM
\over Rc^{2}} {1 \over \varepsilon}\cr &= 2 \left({{\dot M} \over
{\dot M}_{E}} \right) {c \over v_{esc}} .\cr} \eqno(3.7.7)$$
As ${\dot M}$ approaches ${\dot M}_{E}$ the flow becomes optically thick.
It is possible to describe crudely the flow of radiative energy through
optically thick matter by a diffusion velocity $v_{diff}$, using
(1.7.15b):
$$\eqalign{v_{diff}&\equiv {H \over {\cal E}_{rad}} \cr &\sim {c \over
3\tau};\cr}\eqno(3.7.8)$$
this result is valid only if $\tau \gapp 1$. Then we may compare the
rate at which free-falling matter flows inward to the rate at which
radiation diffuses outward through the matter:
$${v_{diff} \over v_{esc}} \sim {1 \over 6} \left( {{\dot M} \over
{\dot M}_{E}} \right)^{-1} . \eqno(3.7.9)$$
Before ${\dot M}$ can reach ${\dot M}_{E}$ the matter, optically thick
to the radiation, effectively traps it and sweeps it inward, preventing
its escape. Once ${\dot M} \gapp {\dot M}_{E}$, the approximations used
in deriving (3.7.9) become invalid. The radiation trapped deep within
the flow builds up to a higher energy density, and slows the infall.
Because radiation is trapped within the inner part of the flow, in its
outer part $F_{rad}$ is smaller than (3.7.4) would imply, and there
matter may continue to accrete; it is not blown away by radiation
pressure, as might be imagined.
In the limit ${\dot M} \gg {\dot M}_{E}$ the radiative transport of
energy is a small effect on infall time scales, and the radiation and
matter should be thought of as inseparable components of a single fluid,
with the combined equation of state (1.4.5). When the fluid reaches the
stellar surface a strong shock forms, which reduces its velocity and
increases its density. The compression ratio (3.3.17) is not large,
(an isothermal equation of state is impossible because it would require
efficient radiative losses, and we have seen that radiation is trapped
within the matter). The accreted matter very rapidly builds up an
opaque and voluminous (but low mass) envelope on the star, nearly in
hydrostatic equilibrium; the shock marks the outer boundary of the
envelope. The accreting compact object is very soon lost from view
beneath this envelope, which roughly resembles that of a supergiant
star.
As an approximation to the structure of the envelope we assume that
once matter passes through a strong shock at radius $r$ it remains
at $r$, having converted all its kinetic energy to internal energy;
this ignores its small residual velocity and kinetic energy. Its
internal energy ${\cal E}$ is given by
$${\cal E} = {GM\rho \over r} . \eqno(3.7.10)$$
Then, substituting $P = (\gamma -1) {\cal E}$ ({\bf 1.9}) into the
equation of hydrostatic equilibrium
$${\partial P \over \partial r} =-{GM\rho \over r^{2}} \eqno(3.7.11)$$
yields the solution
$$\rho \propto r^{(\gamma -2)/(\gamma -1)}.
\eqno(3.7.12)$$
Under most conditions of interest radiation pressure is dominant and
$\gamma = 4/3$, so that $\rho \propto r^{-2}$; this is borne out by
radiation-hydrodynamic calculations (Klein, {\it et al.} 1980, Burger
and Katz 1983).
This result should be compared to the structure of a
polytropic envelope. Substituting $P \propto \rho^{\gamma}$ into
(3.7.11) yields
$$\rho \propto r^{-1/( \gamma -1)}.\eqno(3.7.13)$$
This is a steeper variation of $\rho$ with $r$ than (3.7.12),
implying that the envelope structure (3.7.12) is unstable against
convection. If it remains in hydrostatic equilibrium for a long enough
time turbulent convection will make its structure approach that of
(3.7.13).
In accretion onto a black hole there is no surface to support an
envelope, and free fall is likely at any accretion rate.
\goodbreak
\vskip 2\baselineskip plus2\baselineskip
\goodbreak
\noindent
{\bf 3.8 Jets}
\def\rightheadline{\vbox{\vskip 0.125truein\line{\tenbf\hfil Jets\qquad
\folio}}}
\vskip \baselineskip
\noindent
A number of diverse astronomical objects contain two oppositely
directed streams of matter flowing outward from a common source, called
jets (Ferrari and Pacholczyk 1983). This phenomenon is found in
quasars and galaxies with active nuclei (Begelman, {\it et al.} 1984),
in the peculiar binary star SS433 (Margon 1984, Katz 1986), in the
dense dusty gas clouds surrounding protostars called bipolar nebulae
(Lada 1985), and possibly in other kinds of objects. The degree of
collimation of the jets, the nature of the accelerated material, its
speed, and the symmetry between the two opposed jets all vary between
classes of jets, and to a significant extent within a given class.
Extragalactic jets range from tightly collimated ($\sim 1^{\circ}$)
filaments to broad diffuse blobs. They originate in galactic nuclei,
and often terminate in extended lobes of radio emission, the well known
\lq\lq double radio sources.\rq\rq\ Often one jet is much brighter
than the other. They may be broken into bright knots separated by
regions in which the jet is faint or undetectable. The observed
radiation is the synchrotron emission of very energetic electrons in
weak magnetic fields, observed at radio wavelengths (and occasionally
at visible wavelengths). There is little evidence for thermal plasma
in the jets, although it could be present and even dominate the jet
energy flow without being observable.
In SS433 the observed jets are composed of singly ionized thermal
plasma, with a temperature $T \sim 10^{4\ \circ}$K, radiating in the
spectral lines of hydrogen and helium. This matter is moving at a
speed of 78,000 km/sec in jets which are collimated to a few degrees.
Aside from the large Doppler shifts, the spectrum resembles that of
ordinary ionized stellar or interstellar material. A very small
fraction of the jet energy is converted to the energy of relativistic
electrons, which produce observable radio jets by synchrotron emission.
The jets of SS433 do not have fixed orientations, but rather trace out
the surface of a cone of half-angle 20$^{\circ}$ with a period of 164
days. This motion (which has additional complications) provides
important clues to the nature of SS433 and to the jet acceleration
mechanism.
The jets in the bipolar nebulae are usually poorly collimated, generally
resembling diffuse clouds. They consist of molecular gas with
velocities $\sim 100$ km/sec and temperatures $\lapp 100^{\circ}$K.
Nonthermal phenomena (relativistic particles and synchrotron emission)
appear to be absent.
These phenomena are very diverse, but have in common the geometry of
two opposite jets emanating from a common source. The origin of this
geometry must be very general, for it is found for relativistic
particles, thermal plasmas, and neutral gases, in flows ranging over
several orders of magnitude in size, power, energy density, velocity,
sound speed, temperature, optical depth, particle mean free path
(and almost any other parameter one cares to compute). Angular momentum
has the required symmetry, for it determines a preferred axis through a
center of mass. As discussed in {\bf 3.6}, angular momentum is expected
to dominate the dynamics of matter drawn from a large volume by a small
attracting object, or exchanged between two stars in a binary system.
Matter with a significant amount of angular momentum naturally forms a
disc around the center of gravitational attraction. Its axis then
defines two opposite (but equivalent) preferred directions, which may
be identified with the jet directions. Both accretion discs and jets
are consequences of the frequent importance of angular momentum in
astrophysics, just as objects in which it is unimportant are spherical.
The symmetry of jet geometry is easy to explain. It is more difficult
to understand the mechanisms of jet production and collimation, and why
jets have their observed properties. If a bubble of buoyant, high
entropy fluid is produced within a star or an accretion disc it will
rise to the surface, with its buoyancy force directed parallel to the
hydrostatic equilibrium pressure gradient and opposite to the local
effective (including the centripetal potential) acceleration of gravity.
In a spherical star this force is everywhere radially outward; in a
rotationally flattened star it is preferentially directed toward the
poles; in a thin accretion disc or rotationally flattened gas cloud it
is parallel (or anti-parallel) to the rotational axis. The rise of such
buoyant fluid is analogous to the motions of ordinary stellar
convection, but we now consider material whose density is very much less
than, rather than close to, that of its mean surroundings. Possible
examples include a magnetized fluid of relativistic particles produced
by pulsars or other nonthermal processes, or hot thermal fluid heated
by strong shocks. In the course of its rise the buoyant fluid may mix
with or entrain the surrounding denser fluid, a process which should be
described by an (unknown) parameter analogous to the mixing length of
ordinary convection.
Once the buoyant fluid breaches the surface of the disc or gas cloud
it will expand; if its internal energy exceeds the gravitational
binding energy it will escape freely. This flow may be either steady
or intermittent, taking the form of discrete bubbles or persistent
channel flow, depending on whether the supply of fluid is steady or
intermittent, and on values of other hydrodynamic parameters; Norman,
{\it et al.} (1981) present calculations. The expansion is
preferentially normal to the surface (or to surfaces of constant
pressure), and along the angular momentum axis, thus forming jets.
Their degree of collimation depends on the detailed hydrodynamics of
the flow. This is crudely analogous to the behavior
of gas bubbles produced by underwater explosions, whose breaching can
produce prominent vertical spikes of spray.
The expansion of the escaping buoyant fluid converts its internal energy
to the kinetic energy of bulk motion (a process known as adiabatic loss
of internal energy, because the expansion is usually taken to be
adiabatic). Encounter with a surrounding medium or magnetic field may
re-randomize the particle velocities in a collisional or (more likely,
if densities are very low) collisionless shock or instability, possibly
at very great distances from the source.
In an alternative class of models matter is heated or particles are
accelerated at or above the surface of an accretion disc. The hot or
energetic material need not rise through denser fluid. If its internal
energy exceeds the gravitational binding energy it will escape, with its
motion preferentially along the angular momentum axis. The source
of its internal energy may be nonthermal particle acceleration, or
a thermal process; Eggum, {\it et al.} (1985) report calculations
of thermal jets produced by the radiation pressure of accretion discs
with ${\dot M} > {\dot M}_{E}$. The collimation of the jets is
determined by the detailed hydrodynamics of the flow and by the
geometry of the accretion disc; the dense disc provides a boundary
condition on the flow, roughly akin to a rocket nozzle, as well as a
source of radiative acceleration ({\bf 1.15}).
These processes depend on complex, often turbulent, hydrodynamics with
uncertain boundary conditions. In the case of extragalactic jets
consisting of nonthermal particles the poorly understood processes
of nonthermal particle acceleration are central. It is
possible that all the ingredients of complete models of jets are
qualitatively known, but turning them into a quantitative predictive
(or even explanatory) tool remains a difficult problem.
After a jet leaves the vicinity of its source it will interact with
any surrounding medium. This interaction may be complex; in
particular, nonthermal processes may accelerate or scatter energetic
electrons which produce radio (and occasionally visible) synchrotron
radiation. A very rough model based only on momentum balance is
capable of describing the rate at which a jet carves a cavity in the
medium, and therefore relates the apparent length of a jet to the
period over which it has been produced and to the physical parameters
of the jet and of the medium in which it propagates.
The flow is sketched in Figure 3.4. A collimated jet of density
$\rho_{1}$ travelling at speed $u$ enters a semi-infinite medium at
rest. The collision between the jet and the medium slows and deflects
the jet, and pushes the medium aside; the jet creates for itself a
cavity. At the tip of the jet fresh material interacts with the medium.
This interaction region I moves to the right at a speed $U$. It is
possible to calculate $U$ without considering the complex hydrodynamics
of the interaction region.
\topinsert
\vskip 10.5truecm
\ctrline{{\bf Figure 3.4.} Jet propagation.}
\endinsert
In a frame moving with I the flow is stationary, at least when averaged
over any turbulence which may be present. The jet impinges on I from
the left at a speed (assuming all velocities to be nonrelativistic)
$u-U$, and the medium flows towards I from the right at a speed $U$.
Because there are no other sources of momentum and I contains
negligible mass (and is not accelerated), these two momentum fluxes
balance
$$\rho_{1}(u-U)^{2} = \rho_{2}U^{2} . \eqno(3.8.1)$$
Simple algebra gives $U$:
$$U={u \over 1 + \sqrt{\rho_{2} / \rho_{1}}} . \eqno(3.8.2)$$
In a time $t$ the jet penetrates to a depth $D=Ut$. Equation (3.8.2)
has been verified empirically for nearly incompressible fluid flow, if
there is little entrainment of the medium by the jet before it reaches
I. In this case the description of the complex flow pattern
at I by a simple equation of momentum balance appears to be valid.
Real astrophysical jets are much more complex, and the validity of this
simple model of their hydrodynamics is uncertain. For example, because
the flow is supersonic strong shocks form, and will tend both to disrupt
the jet before it reaches region I and to evacuate the medium.
This calculation is readily generalized to relativistic velocities.
The results are not so simple, except in limiting cases.
Define $\rho_{1}$ to be the jet's rest mass density measured in the
laboratory frame. Then
$$U \approx \cases{\displaystyle\sqrt{\rho_{1}/ \rho_{2} \over 1-u^{2}},
&\qquad if $\displaystyle
{\rho_{1} \over \rho_{2}} \ll 1-u^{2}$;\cr \qquad u,&\qquad if
$\displaystyle{\rho_{1} \over \rho_{2}} \gg 1-u^{2}$.\cr}\eqno(3.8.3)$$
Each of these limits may be interpreted as a consequence of momentum
balance.
\goodbreak
\vskip 2\baselineskip plus2\baselineskip
\goodbreak
\noindent
{\bf 3.9 Magnetohydrodynamics}
\def\rightheadline{\vbox{\vskip 0.125truein\line{\tenbf\hfil
Magnetohydrodynamics \qquad\folio}}}
\vskip \baselineskip
\noindent
{\bf 3.9.1} \us{Equations} \quad If a fluid is an electrical conductor,
as is most of the matter dealt with in astronomy, currents may flow in
it and produce magnetic fields. These fields exert forces on the
fluid through the currents it carries. Magnetic fields are observed in
many astronomical objects. Even when they are not observed they may be
present with magnitudes too small to permit direct observation, but large
enough to produce other interesting consequences. For example, magnetic
fields may carry significant torques ({\bf 3.6}), and time-dependent
fields may lead to the acceleration of energetic particles ({\bf 2.5}).
The generalization of the laws of hydrodynamics to a conducting fluid
is called magnetohydrodynamics. In each case the fluid equations are
derived from kinetic equations ({\bf 2.1}) by assuming that the particle
distribution functions are very close to those of thermodynamic
equilibrium. It there are at least two species of particles with
opposite signs of charge, the mean charge density is nearly zero,
displacement currents are neglected, and the equations of
electrodynamics are used to calculate the electromagnetic fields and
the forces on the charged particles, then the equations of
magnetohydrodynamics are obtained. The neglect of the displacement
current and of the net charge density mean that the high frequency
phenomena of electrostatic plasma waves and transverse electromagnetic
waves are excluded.
The equations of magnetohydrodynamics are inapplicable if particle
mean free paths and relaxation times are long, as is generally the case
for energetic particles in low density media, because then the
assumption of thermodynamic equilibrium (the complete description of the
fluid by its pressure, density, velocity, and magnetic field, all of
which are functions of space and time) fails. Even when strictly
inapplicable, the magnetohydrodynamic equations may be a useful
qualitative description of the fluid, although they will not describe
some real phenomena. They may be a much better description of a
magnetized (but nonequilibrium) fluid than the equations of
hydrodynamics are of a similar unmagnetized fluid, for the gyration
of energetic charged particles around the magnetic field lines may in
effect shorten their mean free path to their gyroradius, which is
generally very small.
Following Jackson (1975), the equation of conservation of mass is
(3.1.1):
$${\partial \rho \over \partial t} + \nabla \cdot (\rho {\vec u}) = 0 .
\eqno(3.9.1)$$
The equation of momentum conservation is essentially the same as
(3.1.5), but we write the magnetic (Lorentz) force explicitly:
$${\partial {\vec u} \over \partial t} + ({\vec u} \cdot \nabla){\vec u}
= -{1 \over \rho} \nabla P + {{\vec J} \times {\vec B} \over \rho c} +
{{\vec F} \over \rho} , \eqno(3.9.2)$$
where ${\vec F}$ describes any other body forces, such as gravity and
viscous stress. Neglecting the charge density and the displacement
current, the electromagnetic field equations become
$$\eqalignno{\nabla \times {\vec E}&= -{1 \over c}{\partial {\vec B}
\over \partial t} &(3.9.3)\cr \nabla \times {\vec B}&= {4 \pi \over c}
{\vec J} &(3.9.4)\cr \nabla \cdot {\vec E}&= 0 &(3.9.5)\cr \nabla \cdot
{\vec B}&= 0 . &(3.9.6)\cr}$$
In addition to the relation (3.1.18) which determines the
pressure $P$, another constitutive relation is needed to determine
the current density ${\vec J}$. The simplest possible relation is the
elementary Ohm's law, which ignores the Hall effect and a variety of
other thermomagnetic and galvanomagnetic phenomena. In the rest frame
of an element of fluid this law takes the form
$${\vec J}^{\prime} = \sigma {\vec E}^{\prime} , \eqno(3.9.7)$$
where the primes denote the fluid frame, and $\sigma$ is a scalar
conductivity. Because we are neglecting charge densities, and
assuming that all velocities are nonrelativistic, the current
density in any frame ${\vec J} = {\vec J}^{\prime}$. The electric
field ${\vec E}^{\prime}$ may be expressed in terms of the fields
${\vec E}$ and ${\vec B}$ in an inertial reference frame, so that
$${\vec J} = \sigma ({\vec E} + {{\vec u} \over c} \times {\vec B}) .
\eqno(3.9.8)$$
In real astronomical problems the validity of (3.9.7) and (3.9.8) may
be questionable because they do not correctly describe the response of
collisionless particles to applied fields. They are valid only in the
limit in which particle distribution functions undergo rapid
collisional relaxation. There is no simple remedy for this problem,
and it is likely to be serious whenever the fluid contains energetic
particles.
In good conductors the electric fields are very small, and usually not
directly measurable. They are still important, because without them no
current would flow and there would be no magnetic field. It is usually
of more interest to examine the behavior of the magnetic field, which
is often large and measurable, using equations in which the electric
field does not appear explicitly. The electric field may be
eliminated from (3.9.3) by use of (3.9.8), yielding
$${\partial {\vec B} \over \partial t} = \nabla \times ({\vec u} \times
{\vec B}) - {c \over \sigma} \nabla \times {\vec J} . \eqno(3.9.9)$$
The current density is also hard to measure directly, but may be
eliminated by taking the curl of (3.9.4), using an elementary vector
identity, and substituting into (3.9.9):
$${\partial {\vec B} \over \partial t} = \nabla \times ({\vec u} \times
{\vec B}) + {c^{2} \over 4 \pi \sigma} \nabla^{2}{\vec B}.\eqno(3.9.10)$$
\goodbreak
\vskip 2\baselineskip plus2\baselineskip
\goodbreak
\noindent
{\bf 3.9.2} \us{Resistive Decay} \quad If the fluid is everywhere at
rest (${\vec u} = 0$) then
(3.9.10) becomes a diffusion equation, and the magnetic field
gradually decays to a uniform value. This value is set by the boundary
conditions, and is generally zero for an isolated object. The magnetic
energy appears as resistive heating of the matter. In a body or region
of size $R$ the characteristic decay time $t_{md}$ is approximately
$$t_{md} \sim {4 \pi \sigma R^{2} \over c^{2}} . \eqno(3.9.11)$$
For an ionized nondegenerate hydrogen plasma
$$\sigma \approx {1.4 \times 10^{8}\ T^{3/2} \over \ln \Lambda} \
{\rm sec}^{-1} , \eqno(3.9.12)$$
where $\ln \Lambda$ is the Coulomb logarithm (2.2.10) (Spitzer 1962).
For ordinary (dwarf) stars and for degenerate dwarves $t_{md}$ is
typically in the range $10^{9} - 10^{10}$ years. The value of $t_{md}$
for neutron stars depends on their complex internal structure, but is
probably also very long. Because $\sigma$
is essentially independent of density in nondegenerate ionized matter,
$t_{md}$ is much larger than the age of the universe for very large
objects, such as the interstellar medium. Resistive decay of magnetic
fields is a slow process for objects of astronomical dimensions.
\goodbreak
\vskip 2\baselineskip plus2\baselineskip
\goodbreak
\noindent
{\bf 3.9.3} \us{Dynamos} \quad If the fluid is in motion then the
first term on the right hand side of (3.9.10) becomes important. In a
prescribed steady velocity field this equation is linear in ${\vec B}$,
and its solutions will therefore be eigenmodes which either grow or
decay exponentially as functions of time. The decay found when
${\vec u} = 0$ is a special case of this behavior. A variety of
simple flow patterns may be shown to lead only to decay, but more
complex flows may produce exponential growth. This growth is called
the dynamo mechanism of magnetic field amplification.
In most real astronomical objects, such as stellar convection zones or
the interstellar medium, the flow field is unsteady and turbulent.
Such flows are expected to lead to magnetic field amplification by a
dynamo process, although it is hard to make a confident prediction for
such a complex problem. The characteristic dynamo amplification time
$t_{da}$ may be as short as
$$t_{da} \sim {R \over u} , \eqno(3.9.13)$$
which will generally be very much less than the age of an astronomical
object. If it were proper to speak of eigenmodes of ${\vec B}$ when
${\vec v}$ is a fluctuating quantity, the exponential growth of even a
single eigenmode would soon lead to very large ${\vec B}$.
Once the magnetic stress, of magnitude $B^{2}/8\pi$, becomes comparable
to the hydrodynamic stress $\rho u^{2}$, it is likely that it will
alter the flow pattern so that exponential amplification of the
magnetic field will cease. This is necessary on energetic grounds,
because the fluid motion will then no longer possess enough energy to
amplify the field rapidly. It is often assumed that in a turbulent
conducting fluid the field will be amplified until this condition is
met, so that the field may be estimated
$${B^{2} \over 8 \pi} \sim \rho u^{2} . \eqno(3.9.14)$$
The best observed turbulent conducting flow is probably that of the
Solar convective zone. The magnetic field there reaches the values
implied by (3.9.14) in sunspots, where it also apparently suppresses
the convective motion, in agreement with expectation. Elsewhere on the
visible Solar surface the field is much less, in disagreement with
(3.9.14). The nondegenerate stars with the largest surface magnetic
fields appear to be those whose surfaces are only slightly unstable
against convection, or are weakly stable, while an argument based on
(3.9.14) would suggest that stars with the most vigorous surface
convection should have the strongest fields. In contrast, (3.9.14)
appears to approximately describe the magnitude of interstellar fields.
The only simple summary is that the real world is not simple, and that
this estimate should be used with great skepticism.
\goodbreak
\vskip 2\baselineskip plus2\baselineskip
\goodbreak
\noindent
{\bf 3.9.4} \us{Seed Fields} \quad Exponential growth by a dynamo
process is almost, but not quite, a complete solution to the problem of
the origins ot magnetic fields in astrophysics. There must still be
a seed field to be amplified, although its magnitude may be
extremely small. There are a number of ways in which such a seed field
may be produced. It is a general property of pressure gradients in
ionized gases that they produce small charge densities and electric
fields. To demonstrate this, consider the separate equations of
hydrostatic equilibrium for electrons and ions in a uniform
gravitational field ${\vec g}$, or subjected to a uniform acceleration
$-{\vec g}\,$:
$$\eqalignno{\nabla P_{i}&= n_{i}m_{i}{\vec g} + n_{i}q_{i}{\vec E} +
{\vec F}_{i} &(3.9.15a)\cr \nabla P_{e}&= n_{e}m_{e}{\vec g} + n_{e}
q_{e}{\vec E} + {\vec F}_{e} .&(3.9.15b)\cr}$$
The subscripts $i$ and $e$ refer
to the ions and electrons, $q$, $n$, and $m$ are respectively the
charge, number density, and mass of each species, and ${\vec F}$
is any force other than gravity, inertia, or that of the electrostatic
field. The only such force which needs to be considered is the
frictional force resulting from collisions between the ions and the
electrons. In equilibrium there is no net drift velocity of the
two species with respect to each other, so there is no mean frictional
force between them, and ${\vec F}_{i} = {\vec F}_{e} = 0$.
Addition of (3.9.15a) and (3.9.15b) gives
$$\nabla P = (n_{i}m_{i} + n_{e}m_{e}){\vec g} + (n_{i}q_{i} + n_{e}
q_{e}) {\vec E} , \eqno(3.9.16)$$
where $P \equiv P_{i} + P_{e}$ is the total pressure. If the gas is
nearly electrostatically neutral, as will almost always be the case,
then $n_{i}q_{i} + n_{e}q_{e} = 0$ to an excellent approximation. The
density \hbox{$\rho = n_{i}m_{i} + n_{e}m_{e}$}, so that (3.9.16)
becomes the ordinary equation of hydrostatic equilibrium (1.3.1):
$$\nabla P = \rho {\vec g} . \eqno(3.9.17)$$
Subtraction of (3.9.15b) from (3.9.15a) gives
$$\nabla (P_{i} - P_{e}) = (n_{i}m_{i} - n_{e}m_{e}){\vec g} + (n_{i}
q_{i} - n_{e}q_{e}){\vec E} . \eqno(3.9.18)$$
Assume a nondegenerate perfect gas equation of state for each species,
which in equilibrium are at the same temperature $T$. If we neglect
$m_{e}$ compared to $m_{i}$, the coefficient of ${\vec g}$ is almost
exactly $\rho$. Then the electric field is given by
$${\vec E} = {\nabla \lbrack (n_{i} - n_{e}) k_{B} T \rbrack - \rho
{\vec g} \over n_{i}q_{i} - n_{e}q_{e}} . \eqno(3.9.19)$$
For the special case of a pure hydrogen plasma $n_{i} = n_{e} = n/2$,
$q_{i} = e$, $q_{e} = -e$, and $\rho = n_{i}m_{i}$ (neglecting $m_{e}$),
so that
$${\vec E} = -{m_{i} {\vec g} \over 2 e} . \eqno(3.9.20)$$
The electrostatic potential energy of an ion is then $-{1 \over 2}$
of its gravitational potential energy. At the surface of the Sun $E
\sim 10^{-8}$ Volt/cm, an extremely small field. These fields are
found in all gravitating or accelerated plasmas.
The physical origin of this electric field is clear. Nearly all the
gravitational force acts on the massive ions, but the electrons
contribute substantially to the pressure gradient which opposes it.
In order to keep the electrons from being accelerated to infinity by
the pressure gradient, a small electric field develops, which
confines the electrons, and helps support the ions against gravity.
Temperature gradients also produce small thermoelectric fields.
If the matter is moving (nonrelativistically) with respect to an
observer, that observer will see a magnetic field
$${\vec B} = - {{\vec u} \over c} \times {\vec E} . \eqno(3.9.21)$$
These fields are very small ($\sim 10^{-27}$ gauss for representative
interstellar parameters, and $\sim 10^{-15}$ gauss for the Solar
convective zone), but fewer than 50 $e$-foldings of dynamo amplification
will bring them to the observed values.
\goodbreak
\vskip 2\baselineskip plus2\baselineskip
\goodbreak
\noindent
{\bf 3.9.5} \us{Field Lines} \quad If a fluid is a good conductor
($\sigma \rightarrow \infty$) then (3.9.10) becomes
$${\partial {\vec B} \over \partial t} = \nabla \times ({\vec u} \times
{\vec B}) . \eqno(3.9.22)$$
The rate of change of magnetic flux $\Phi$ through a fixed loop $C$
bounding a surface $S$ may then be found by integration
$$\eqalign{{\partial \Phi \over \partial t}&= {\partial \over \partial t}
\int_{S} {\vec B} \cdot {\hat n}\ da \cr &= \int_{S}\lbrack \nabla \times
({\vec u} \times {\vec B}) \rbrack \cdot {\hat n}\ da \cr
&= \int_{C} ({\vec u} \times {\vec B}) \cdot {\vec {dl}} , \cr}
\eqno(3.9.23)$$
where ${\hat n}$ represents a unit vector normal to $S$, $da$ is an
element of surface $S$, ${\vec {dl}}$ is an element of loop $C$, and
where Stokes' theorem has been used. If the field lines are considered
to move with the fluid as if they were frozen into it, then $({\vec u}
\times {\vec B}) \cdot {\vec {dl}}$ is the rate at which magnetic flux
normal to ${\vec {dl}}$ and to ${\vec u}$ is advected across the element
${\vec {dl}}$ by the flow. The equality (3.9.23) establishes that the
rate of advection of flux across $C$ equals the rate of change of flux
through $S$, so that this picture of magnetic field lines frozen into
the fluid is a valid description of the behavior of the magnetic flux in
a good conductor.
The concept of frozen magnetic flux is a useful tool. Matter can still
bend magnetic field lines (because surface currents flow when it
encounters a field, and these currents add to the pre-existing field),
even though it does not cross them. If the material stress $P + \rho
u^{2}$ is small compared to the magnetic stress $B^{2} / 8 \pi$, even
this bending will be slight, and the field will act nearly as an
immovable obstacle, confining and guiding the flow of matter.
In the opposite case $P + \rho u^{2} \gg B^{2} / 8 \pi$, the field lines
will be swept along passively with the flow of conducting fluid. This
probably describes the behavior of the field in the evolution of stellar
interiors. The conservation of flux within a loop implies that if all
linear scales contract by a factor ${\cal R}$, so that $\rho \propto
{\cal R}^{3}$, then the area of a loop varies $\propto {\cal R}^{-2}$
and the magnetic field $B \propto {\cal R}^{2} \propto \rho^{2/3}$.
The magnetic stress $B^{2} / 8 \pi \propto \rho^{4/3}$, as is expected
for a relativistic fluid ({\bf 1.9}). This scaling of $B$ may account
for the large magnetic fields of neutron stars and degenerate dwarves,
although if it is applied to the contraction of stars from interstellar
material it predicts impossibly large fields. It is possible that
strong fields which violate the condition $P + \rho u^{2} \gg B^{2} / 8
\pi$ channel the contraction parallel to ${\vec B}$, or prevent
contraction until they are reduced by resistive loss or a suitable
(anti-dynamo) flow field.
\goodbreak
\vskip 2\baselineskip plus2\baselineskip
\goodbreak
\noindent
{\bf 3.10 References}
\def\rightheadline{\vbox{\vskip 0.125truein\line{\tenbf\hfil References
\qquad\folio}}}
\parindent=0pt
\vskip \baselineskip
Begelman, M. C., Blandford, R. D., and Rees, M. J. 1984, {\it Rev. Mod.
Phys.} {\bf 56}, 255.
\medskip
Bondi, H. 1952, {\it Mon. Not. Roy. Ast. Soc.} {\bf 112}, 195.
\medskip
Burger, H. L., and Katz, J. I. 1983, {\it Ap. J.} {\bf 265}, 393.
\medskip
Clark, D. H., and Caswell, J. L. 1976, {\it Mon. Not. Roy. Ast. Soc.}
{\bf 174}, 267.
\medskip
Eggum, G. E., Coroniti, F. V., and Katz, J. I. 1985, {\it Ap. J.
(Lett.)} {\bf 298}, L41.
\medskip
Ferrari, A., and Pacholczyk, A. G. eds. 1983, {\it Astrophysical Jets}
(Dordrecht: Reidel).
\medskip
Hoyle, F., and Lyttleton, R. A. 1939, {\it Proc. Camb. Phil. Soc.}
{\bf 35}, 405.
\medskip
Hunt, R. 1971, {\it Mon. Not. Roy. Ast. Soc.} {\bf 154}, 141.
\medskip
Jackson, J. D. 1975, {\it Classical Electrodynamics} (New York: Wiley).
\medskip
Katz, J. I. 1986, {\it Comments Ap.} {\bf 11} in press.
\medskip
Klein, R. I., Stockman, H. S., and Chevalier, R. A. 1980, {\it Ap. J.}
{\bf 237}, 912.
\medskip
Lada, C. J. 1985, {\it Ann. Rev. Astron. Ap.} {\bf 23}, 267.
\medskip
Landau, L. D., and Lifshitz, E. M. 1959, {\it Fluid Mechanics} (Reading,
Mass.: Addison-Wesley).
\medskip
Livio, M. 1986, {\it Comments Ap.} {\bf 11}, 111.
\medskip
Margon, B. 1984, {\it Ann. Rev. Astron. Ap.} {\bf 22}, 507.
\medskip
Norman, M. L., Smarr, L., Wilson, J. R., and Smith, M. D. 1981, {\it Ap.
J.} {\bf 247}, 52.
\medskip
Novikov, I. D., and Thorne, K. S. 1973, in {\it Black Holes} eds. C.
DeWitt and B. S. DeWitt (New York: Gordon and Breach), p. 343.
\medskip
Pringle, J. E. 1981, {\it Ann. Rev. Astron. Ap.} {\bf 19}, 137.
\medskip
Salpeter, E. E. 1964, {\it Ap. J.} {\bf 140}, 796.
\medskip
Spitzer, L. 1962, {\it Physics of Fully Ionized Gases} 2nd ed. (New
York: Interscience).
\medskip
Spitzer, L. 1978, {\it Physical Processes in the Interstellar Medium}
(New York: Wiley).
\medskip
Zeldovich, Ya. B., and Raizer, Yu. P. 1966, {\it Physics of Shock Waves
and High-Temperature Hydrodynamic Phenomena} (New York: Academic Press).
\endpage
\end
\bye