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\noindent
{\twelveb Chapter 2}
\vskip .25truein
\noindent
{\twelveb Non-Equilibrium Thermodynamics}
\vskip 1.44truein plus.06truein
\noindent
{\bf 2.1 Kinetic Equations}
\vskip \baselineskip
\noindent
A great many astrophysical problems involve distributions of
particles which are not in thermodynamic equilibrium. In one
sense this statement is trivial and tautological---the entire
universe would be in equilibrium only if ``heat death'' had
occurred. That clearly is not the case, and the universe would not
be very interesting if it were, nor would we be around to care.
More seriously, what is meant is that the local distributions of
particles are frequently not close to equilibrium, and that the
ways in which they approach equilibrium are interesting. This is
not often the case in everyday life, where matter densities and
collision rates are high, so that relaxation to equilibrium is
rapid and deviations from it are small (a striking everyday
exception is radiation; because the atmosphere is transparent, the sky
is dark, and the Sun is bright, the visible radiation field around
us is very far from any equilibrium distribution). In contrast,
many astrophysical problems involve very low densities and low
collisional relaxation rates, and deviations from equilibrium are
large, important, and interesting (stellar interiors are an exception).
Matter which is not in thermodynamic equilibrium is described by kinetic
equations; a standard textbook is that by Liboff (1969).
In order to describe completely the state of a classical gas of
identical particles which are not in thermodynamic equilibrium it is
necessary to specify the coordinates and momenta of all its particles.
The probability that one particle will be found in a cell of volume
$d^{3}x_{1}$ at position
${\vec x}_{1}$, and in a momentum cell of volume
$d^{3}p_{1}$ at momentum
${\vec p}_{1}$, that a second particle will be in a volume
$d^{3}x_{2}$ at
${\vec x}_{2}$, and
$d^{3}p_{2}$ at ${\vec p}_{2}$, $\ldots\,$, and so on, up to the
$N$-th, is given by an
$N$-particle distribution function
$f_{N}$ multiplied by the volume in phase space:
$$f_{N}({\vec x}_{1},\ldots,{\vec x}_{N},{\vec p}_{1},\ldots,{\vec p}_N,
t)\; d^{3}x_{1}\ldots d^{3}x_{N}d^{3}p_{1}\ldots d^{3}p_{N} .
\eqno(2.1.1)$$
Clearly $f_{N}$ is a horribly complex and unwieldy object, for
$N$ is typically of the order of Avogadro's number.
If the motion of any one particle is completely independent of
any and all other particles, then instead of specifying
$f_{N}$ it is sufficient to specify a single particle
distribution function
$f$, where
$$f({\vec x}, {\vec p}, t)\, d^{3}xd^{3}p \eqno(2.1.2)$$
is the probability of finding a single particle in the phase
space volume
$d^{3}xd^{3}p$ around the point in phase space $({\vec x},
{\vec p}\,)$; equivalently, $f$ is the mean or expected density
of particles there. The radical approximation of using (2.1.2) instead
of (2.1.1) is exact for noninteracting particles and good for
weakly interacting particles. It is usually justified for
dilute gases, but is quite wrong for liquids and solids.
\headline={\ifodd\pageno\rightheadline \else\leftheadline\fi}
\def\leftheadline{\vbox{\vskip 0.125truein\line{\tenbf\folio\qquad
Non-Equilibrium Thermodynamics \hfil}}}
\def\rightheadline{\vbox{\vskip 0.125truein\line{\tenbf\hfil
Kinetic Equations \qquad\folio}}}
\footline={\hfil}
The single particle distribution function (2.1.2) is simple enough to
be tractable. If there are no interactions between the particles
and they are nonrelativistic it obeys the Liouville equation
$${df \over dt} = {\partial f \over \partial t} + {{\vec p} \over m}
\cdot {\partial f \over \partial {\vec x}} + {\vec F}({\vec x},{\vec p},
t)\cdot {\partial f \over \partial {\vec p}} = 0 , \eqno(2.1.3)$$
where ${\vec F}$ is any force field which may be present,
and the arguments $({\vec x},{\vec p},t)$ of
$f$ have not been written for the sake of brevity. This equation is
only the statement that because particles are conserved so is the
particle density in phase space.
There are two ways in which (2.1.3) must be modified to include
interactions between particles, because these interactions may be
divided into two kinds. The first kind is the interaction of a
single particle with the mean distribution of the other
particles. If we assume that the mean distribution is completely
specified by (2.1.2) then its effect on
$f$ may be included by adding a force field which depends on
$f$, so that (2.1.3) now takes the form
$${df \over dt} = {\partial f \over \partial t} + {{\vec p} \over m}
\cdot {\partial f \over \partial {\vec x}} + {\vec F}({\vec x},{\vec p},
t,f)\cdot {\partial f \over \partial {\vec p}} = 0 . \eqno(2.1.4)$$
This is known as the Vlasov equation, and is widely used in both
plasma physics and stellar dynamics.
$F$ now includes the electric, magnetic and gravitational fields
produced by the mean distribution of particles
$f$, and is calculated from additional equations (Newton's or Maxwell's).
The second kind of interparticle interaction has a random and
statistical character, and so cannot be described by a mean force
field
${\vec F}$. Instead, it is assumed to take the form of
instantaneous collisions between two particles, independent of
any other particles in the fluid. This assumption is again exact
in the trivial case of noninteracting particles, and is a good
approximation for dilute gases with short range forces. It is
generally a reasonable approximation in plasmas where the
interaction is long range but screened at long distances; this
screening, properly a three-particle interaction, is reasonably
accounted for by modifying the interparticle force law.
The assumption of instantaneous binary collisions is also believed
to be a good approximation for systems of gravitating particles if a
large number of particles are present.
Collisions remove particles from a cell in phase space (or supply
them to it), so in (2.1.3) and (2.1.4) they take the form of a source or
sink term in place of the zero on the right hand side. We first
calculate the rate at which collisions scatter particles out of the
phase space volume
$d^{3}xd^{3}p$ about the point
$({\vec x},{\vec p}\, )$.
Consider a collision between particles with momenta
${\vec p}$ and ${\vec p}_{1}$, producing particles with momenta
${\vec p}^{\,\prime}$ and ${\vec p}^{\,\prime}_{1}$. Ignore quantum
statistics, so that the particles are distinguishable, though of
the same kind. In order to have a collision with impact
parameter between
$s$ and $ds$ in a time
$dt$ the colliding particle must be found in a cylindrical volume
element
$2 \pi s ds dt \vert {\vec p} - {\vec p}_{1}\vert /m$; before collision
the first particle is on the axis of this cylinder, which is
oriented along the relative velocity vector
$({\vec p} - {\vec p}_{1})/m$. The rate at which collisions remove
particles of momentum
${\vec p}$ is then
$$\int f_{2}({\vec x},{\vec x},{\vec p},{\vec p}_{1})\; d^{3}p_{1}\;
2\pi s\ ds \left\vert{{\vec p}-{\vec p}_{1} \over m}\right\vert ,
\eqno(2.1.5)$$
where $f_{2}({\vec x},{\vec x},{\vec p},{\vec p}_{1})$
is the two particle distribution function, giving the
probability that there are two particles simultaneously present
in a unit volume of phase space at coordinate
${\vec x}$ and at momenta
${\vec p}$ and ${\vec p}_{1}$.
In order to make any progress with (2.1.5) it is necessary to have
(or at least assume) some information about
$f_{2}$. It is possible to write an equation like (2.1.3) for
$f_{2}$, and then to add a collision term to its right hand
side. This would be useless, because the collision term would
involve the rate at which collisions with a third particle change
$f_{2}$; the rate of such triple collisions depends on $f_{3}$.
An infinite hierarchy of equations would result, unless drastic
action were taken. This resembles the closure problem of the moment
equations of radiative transport theory ({\bf 1.7}), but is much worse,
because of the increasing complexity of successive $f_{n}$.
Boltzmann took drastic action. He approximated
$$f_{2}({\vec x},{\vec x},{\vec p},{\vec p}_{1}) = f({\vec x},{\vec p}\,)
f({\vec x},{\vec p}_{1}) . \eqno(2.1.6)$$
This is known as the ``assumption of molecular chaos,'' and is
justified in the dilute gas approximation we have been using all
along. It says that there is no correlation between the momentum
distributions of two colliding particles. It is not possible to
justify this rigorously, but it is believed to hold in a dilute fluid
because between scatterings a particle travels many
times the mean nearest neighbor distance. The scattering
particle (incident with momentum
${\vec p}_{1}$) almost certainly has scattered from many other
particles and has thoroughly randomized its velocity since the
previous time (if ever) it scattered from the particle of momentum ${\vec
p}$ whose distribution we are calculating. The scattering particle may
therefore be considered to have been drawn randomly from the particle
distribution function $f$, so that its momentum is uncorrelated with
that of the test particle from which it scatters.
In contrast, in a liquid where two particles may find themselves
encaged by their neighbors and repeatedly scattering from each
other, and the assumption of molecular chaos would not be justified.
The total rate of removal of particles from the phase space cell
$d^{3}xd^{3}p$ is
$$\int f({\vec x},{\vec p}\,)\; d^{3}p_{1}\ 2 \pi s \; ds \left\vert
{{\vec p} - {\vec p}_{1} \over m} \right\vert . \eqno(2.1.7)$$
>From this we must subtract the rate of the inverse process of scattering
from momenta
${\vec p}^{\,\prime}$, ${\vec p}^{\,\prime}_{1}$ to ${\vec p}$,
${\vec p}_{1}$, which is clearly
$$ \int f({\vec x},{\vec p}^{\,\prime})f({\vec x},{\vec p}^{\,\prime}
_{1})\; 2 \pi s^{\prime} \; ds^{\prime} \left\vert{{\vec p}^{\,\prime} -
{\vec p}^{\,\prime}_{1} \over m} \right\vert . \eqno(2.1.8)$$
(2.1.8) is the rate of removal from the phase space cell
$d^{3}x^{\prime}d^{3}p^{\prime}$. For a nonrelativistic elastic
collision
$\vert {\vec p} - {\vec p}_{1} \vert = \vert {\vec p}^{\,\prime} -
{\vec p}^{\,\prime}_{1} \vert $, $s\, ds = s^{\prime}\, ds^{\prime}$,
\break $d^{3}p_{1} = d^{3}p^{\prime}_{1}$,
$d^{3}p = d^{3}p^{\prime}$, and
$d^{3}x = d^{3}x^{\prime}$. Then (2.1.7) and (2.1.8) may be
combined and added to (2.1.3), dropping the explicit
${\vec x}$ dependence, to give the Boltzmann equation
$$\hskip .21truein
\eqalign{{df \over dt}&= {\partial f \over \partial t} + {{\vec p}
\over m} \cdot {\partial f \over \partial {\vec x}} + {\vec F} \cdot
{\partial f \over \partial {\vec p}}\cr &= \int d^{3}p_{1} \left\vert
{{\vec p} - {\vec p}_{1} \over m}\right\vert {d\sigma \over d\Omega} \;
d\Omega \; \left\lbrack f({\vec p}^{\,\prime}_{1}) f({\vec p}^{\,\prime})
- f({\vec p}_{1}) f({\vec p}\,) \right\rbrack , \cr} \eqno(2.1.9)$$
where ${\vec p}^{\,\prime}_{1}$
and ${\vec p}^{\,\prime}$ are implicitly functions of
${\vec p}_{1}$, ${\vec p}$, and the scattering angle $\Omega$ (determined
by the conservation of momentum and energy), and $d\sigma \equiv 2 \pi s
\, ds$. As before, in most terms the arguments of $f$ are not written
explicitly unless they differ from (${\vec x},{\vec p},t$).
There is an implicit relation between
$s$ and $\Omega$, which may be calculated from the force law
between the particles; the differential cross-section
$d\sigma / d\Omega$ may be obtained from this relation. Because we have
ignored quantum mechanics, scattering is deterministic; (2.1.9) is also
correct for quantum mechanical $d\sigma / d\Omega$, but our elementary
derivation involving impact parameters is inapplicable.
The Boltzmann equation (2.1.9) is a nonlinear integrodifferential
equation, and is therefore rather difficult to solve. It is harder to
solve than the equation of radiation transport for scattering
opacity, because the scattering of radiation by matter is linear in
the radiation field (see 1.7.14), while the right hand side of (2.1.9)
is nonlinear in $f$. The difficulty lies partly in the nonlinearity
and partly in the fact that $df({\vec p}\,) / dt$ is related
to an integral of $f({\vec p}_{1})$ over all values of
${\vec p}_{1}$; equivalently, one may say that the scattering is
nonlocal in momentum space. If
$f$ is known a brute-force numerical evaluation of the right hand
side is not difficult, but this is usually not very illuminating.
Certain special cases may be solved explicitly, at least to a
useful approximation. In one such case
$f$ is everywhere nearly Maxwellian, but the mean fluid
parameters of density, velocity, and temperature are slowly
varying functions of space. Then an expansion (by Chapman and Enskog;
see Chapman and Cowling 1960) of
$f$ about a Maxwellian leads to a linear integral equation which
is easier to solve, and eventually to the calculation of the
macroscopic transport coefficients. For a gas containing only one
species of uncharged particle these are the thermal conductivity and the
viscosity. If more than one species is present there are also diffusion
coefficients and additional coefficients (thermal diffusion, diffusion
thermo-effect, and cross-diffusion) coupling the fluxes of the several
scalar quantities (heat and the concentrations of the species). If the
particles are charged these are also coupled to the flow of current
and the electrostatic potential by the thermoelectric effect.
This procedure is a remarkable accomplishment of formal
nonequilibrium statistical mechanics, but it is often inadequate for
the astrophysicist. In many astrophysical problems
the dimensions are so large and gradients of temperature,
velocity, and composition are so small that the flow of heat,
momentum, and material by these processes is insignificant. Notable
exceptions include heat conduction in white dwarfs and neutron stars and
diffusion in some stellar photospheres. In many other problems the
transport of heat, momentum, and matter are important, but the density
is usually so low and the mean free paths to collisions so long
that the distribution functions are very far from Maxwellian, and
the Chapman-Enskog procedure (and the textbook values of
transport coefficients) are inapplicable. One example of this is
heat conduction in the Solar wind, in which electron mean free
paths may be longer than the size of the solar system. Processes
other than two body collisions, such as the excitation of plasma
waves, then become dominant in determining the actual flow of
heat. Another example is fluid turbulence. Locally the
particle distribution functions are nearly Maxwellian, but on
larger spatial scales the fluid velocity, temperature, (and
sometimes composition) vary irregularly in both space and time.
The result is a turbulent flow which transports heat, momentum,
(and sometimes composition) many orders of magnitude faster than
the microscopic transport processes. Unfortunately, no
quantitative understanding of these turbulent transport processes
exists. Semi-empirical models (such as ``mixing-length
theory'' {\bf 1.8}) work in the laboratory and are very useful in
engineering, where they can be tested and calibrated by experiment.
Their extrapolation to flows on astrophysical
scales, though widely assumed, is a matter of guesswork.
A second special case of the Boltzmann equation is very useful in
astrophysics. This is the limit in which
${\vec p}^{\,\prime}$ is close to
${\vec p}$, and ${\vec p}^{\,\prime}_{1}$ is close to
${\vec p}_{1}$; collisions individually produce small momentum
transfers. If the potential between particles is $\propto 1/r$, as is
the case for electrostatic and gravitational interactions, then the
integral in (2.1.9) is dominated by distant collisions which
individually produce small momentum
transfers. It is then possible to expand the right hand
side of (2.1.9) in powers of
$\Delta {\vec p} = {\vec p}^{\,\prime}-{\vec p}$, and to carry out the
integral. The result involves derivatives of
$f$ which enter from the Taylor series expansion of
$f$ about
$f({\vec p}\,)$, multiplied by algebraic functions of
${\vec p}$ and $f$. The resulting differential equation is
much easier to deal with. The calculation of charged particle
equilibration in {\bf 2.2} and the derivation of the Kompaneets
equation in {\bf 2.3} are examples of this kind of procedure. The
result of an expansion of (2.1.9) in powers of small momentum
transfers, including only first and second derivatives, is of the form:
$$\hskip .67truein
\eqalign{{df \over dt}&={\partial f \over \partial t} + {{\vec p}
\over m} \cdot {\partial f \over \partial {\vec x}} + {\vec F} \cdot
{\partial f \over \partial {\vec p}} \cr &= - {\partial \over \partial
{\vec p}}\cdot \Bigl({\vec a}({\vec p}\,)f\Bigr)+{1\over 2} {\partial^{2}
\over\partial{\vec p}\partial{\vec p}}:\Bigl(\hbox{\bf b}({\vec p}\,)
f\Bigr),\cr}\eqno(2.1.10)$$
where the double dot product is between a tensor and a tensor operator.
It is necessary to include second derivatives in the expansion of $f$,
but generally not higher derivatives.
Equation (2.1.10) is purely formal; all the physics is contained in the
coefficients ${\vec a}({\vec p}\,)$ and $\hbox{\bf b}({\vec p}\,)$,
and their calculation is usually quite
difficult (the derivation of the Kompaneets equation is one of the
easier examples). In general, they will depend on
$f$. Equations of the form of (2.1.10) are known as Fokker-Planck
equations. They are powerful tools in the study of
nonequilibrium distribution functions, and are generally valid
whenever the evolution of a particle's momentum (or other
conserved dynamical parameters) is the result of many small
independent random events.
The significance of the coefficients
${\vec a}$ and $\hbox{\bf b}$ is apparent. Consider only the
${\vec a}$ term for a spatially homogeneous
$f$ with no external force. Then
$${\partial f \over \partial t} + {\partial \over \partial {\vec p}}
\cdot ({\vec a} f) = 0 . \eqno(2.1.11)$$
This is the equation of conservation of particles, if
${\vec a}$ is their ``velocity'' in momentum space (the rate at
which their momentum is changing, or the mean force on them
resulting from collisions). Therefore,
${\vec a}$ is generally called the coefficient of dynamical
friction. It is usually directed opposite to
${\vec p}$, as a particle loses its initial momentum in randomizing
collisions.
If we consider only the
$\hbox{\bf b}$ term in (2.1.10), again for a spatially homogeneous
$f$ with no external force, and also take $\hbox{\bf b}$
to be a scalar constant $b$ times the unit tensor, we obtain
$${\partial f \over \partial t} = {1 \over 2} b \nabla^{2}_{p} f ,
\eqno(2.1.12)$$
where
$\nabla^{2}_{p}$ is the Laplacian with respect to the momentum
coordinates. We see that
$b$ represents a diffusion coefficient in momentum space, as a
particle suffers a random series of impulses. If
$f({\vec p}\,)$ is very narrowly peaked, then its second
derivatives are much larger than its first derivatives, and the
peak spreads out in a Gaussian shape according to (2.1.12). As the
peak broadens the
${\vec a}$ term becomes significant, and begins to
counteract the broadening.
If $f$ is evolving by collisions with a background equilibrium
distribution of particles at temperature $T$, then a stationary solution
to (2.1.10) must be proportional to a Maxwellian distribution $f_{0}
\equiv \exp (-p^{2}/2mk_{B}T)$ at that temperature. The coefficients
${\vec a}$ and $\hbox{\bf b}$ must satisfy
$$0 = - {\partial \over \partial {\vec p}} \cdot ({\vec a} f_{0}) +
{1 \over 2} {\partial^{2} \over \partial {\vec p} \partial {\vec p}} :
(\hbox{\bf b} f_{0}) . \eqno(2.1.13)$$
If the background distribution function is not Maxwellian, then no
general constraint on
${\vec a}$ and $\hbox{\bf b}$ is possible, because its deviation from
thermodynamic equilibrium may maintain a stationary but nonequilibrium
(and {\it a priori} unknown) form for $f$.
\goodbreak
\vskip 2\baselineskip plus2\baselineskip
\goodbreak
\noindent
{\bf 2.2 Charged Particle Equilibration}
\vskip \baselineskip
\noindent
In this section I present explicit results for some processes of
astrophysical interest by which nonequilibrium distribution functions
relax to their equilibrium form, or by which individual test particles
lose their initial momentum and energy, and come to be described by an
equilibrium probabilistic distribution function. The complete
calculations are generally rather lengthy, and numerous cases must be
considered, so that I present abbreviated accounts of the derivations
and summaries of the salient results. Rosenbluth, {\it et al.} (1957)
present a particularly clear derivation of the theory of the
Fokker-Planck coefficients for Coulomb (or gravitating) gases,
Trubnikov (1965) contains a detailed discussion of this problem, and
Spitzer (1962) provides a convenient summary of the explicit results
required in many practical applications.
\goodbreak
\vskip 2\baselineskip plus2\baselineskip
\goodbreak
\def\rightheadline{\vbox{\vskip 0.125truein\line{\tenbf\hfil Charged
Particle Equilibration \qquad\folio}}}
\noindent
{\bf 2.2.1} \us{Dynamical Friction} \quad
The simplest problem to consider is that of a nonrelativistic fast
charged test particle slowing down as it moves through a background
equilibrium plasma. The results calculated for this problem may be
applied to gravitating neutral particles if the electrostatic potential
$Z_{1}Z_{2}e^{2}/r$ is replaced by the gravitational potential
$Gm_{1}m_{2}/r$. This process is called Coulomb drag in a plasma, and
dynamical friction in gravitating systems.
Consider a test particle with mass $m_{1}$, charge $Z_{1}e$, and
velocity ${\vec v} = v{\hat x}$ moving through a plasma consisting of
randomly distributed ``field'' particles of mass $m_{2}$, charge
$Z_{2}e$, and mean number density $n_{2}$. All plasmas must contain
more than one species of particle in order to maintain
charge-neutrality, but the effects of the several species are almost
exactly additive, so it is only necessary to calculate
the effects of one species.
The calculations are much simplified if the thermal velocities
${\vec v}^{\,\prime}$ of the plasma particles may be neglected. This is
a good approximation if their thermal energies are much less than the
test particle kinetic energy.
The differential cross-section for classical Coulomb scattering
(Eisberg 1961) is
$${d\sigma \over d\Omega} = {Z_{1}^{2} Z_{2}^{2} e^{4}\over 4 m_{12}^{2}
u^{4}} \left\lbrack \sin (\theta /2) \right\rbrack ^{-4} , \eqno(2.2.1)$$
where $m_{12} \equiv m_{1}m_{2}/(m_{1}+m_{2})$ is the reduced mass,
$\theta$ is the scattering angle in the center of mass frame, and
$u = \vert {\vec u} \vert$, where
${\vec u} \equiv {\vec v} - {\vec v}^{\,\prime}$ is the relative velocity
of the test particle with respect to the field particle from which it
scatters. If ${\vec u}$ suffers a change $\Delta {\vec u}$ in a
collision then the change in ${\vec v}$ is
$$\Delta {\vec v} = {m_{2} \over m_{1} + m_{2}} \Delta {\vec u} ;
\eqno(2.2.2)$$
the remaining fraction $m_{1} / (m_{1} + m_{2})$ of $\Delta {\vec u}$
appears as recoil velocity of the field particle.
Because the scattering is elastic $u$ does not change as the result of
a collision, but ${\vec u}$ is rotated by the scattering angle $\theta$.
If the scattering plane makes an angle $\varphi$ to the $x$-$y$ plane,
then the components of $\Delta {\vec u}$ are readily found from
trigonometry to be
$$\hskip .54truein
\eqalign{\Delta u_{x}&=-u(1-\cos \theta) = -2u\sin^{2}(\theta /2)\cr
\Delta u_{y}&= u\sin\theta\cos\varphi= 2u\sin (\theta /2) \cos
(\theta /2)\cos\varphi \cr \Delta u_{z}&= u\sin\theta\sin\varphi= 2u\sin
(\theta /2)\cos (\theta /2) \sin\varphi \cr} \eqno(2.2.3)$$
To compute the mean rate of change of velocity of the test particle,
integrate over all scattering angles
$${d\langle{\vec v}\,\rangle\over dt}={m_{2} \over m_{1}+m_{2}}
\int d\Omega\ {d\sigma \over d\Omega}\ n_{2}v\ \Delta{\vec u}(\Omega).
\eqno(2.2.4)$$
The factor $n_{2}v$ is the rate at which the test particle encounters
field particles, and is required by the normalization of ${d\sigma /
d\Omega}$ to a single scatterer. It is evident that $\Delta v_{y}$ and
$\Delta v_{z}$ average to zero when the integral over $\varphi$ is
performed, because the impulse perpendicular to the path of the incident
test particle is equally likely to be in the $+{\hat y}$ direction as
$-{\hat y}$, and in the $+{\hat z}$ direction as $-{\hat z}$. Then
$${d\langle{\vec v}\rangle\over dt} = - {\hat x} {\pi
Z_{1}^{2}Z_{2}^{2}e^{4}n_{2} \over m_{1} m_{12} v^{2}} \int_{0}^{\pi}
{\sin \theta\ d\theta \over \sin^{2} (\theta /2)} . \eqno(2.2.5)$$
The indefinite integral of the integrand in (2.2.5) is $4\ln\sin
(\theta /2)$, so that the definite integral diverges logarithmically
at its lower limit. It is necessary to introduce a cutoff on the range
of integration. The potential surrounding a static test charge $q$ in
a thermal equilibrium plasma is not the bare potential $q/r$, but is
found (by combining Poisson's equation and the thermal
equilibrium distribution of charges in a potential) to be
$$\phi (r) = {q \over r} \exp (-r/\lambda_{D}) , \eqno(2.2.6)$$
where the Debye length
$$\lambda_{D} \equiv \sqrt{k_{B}T \over 4 \pi \sum_{i} n_{i} q_{i}^{2}}.
\eqno(2.2.7)$$
The sum runs over all the species making up the plasma (this sum is
the reason the effects of the various plasma species in slowing the
test particle are not strictly additive). It is therefore sensible
to cut off the divergent integration in (2.2.5) by making the lower
bound of the integral the angle $\theta_{min}$, at which the impact
parameter of the collision equals $\lambda_{D}$. From an elementary
impulse-approximation analysis this angle is found to be
$$\theta_{min} = {2Z_{1}Z_{2}e^{2} \over m_{12}u^{2}\lambda_{D}} .
\eqno(2.2.8)$$
Equation (2.2.5) then becomes
$${d\langle{\vec v}\,\rangle \over dt} = -{\hat x} {4 \pi Z_{1}^{2}
Z_{2}^{2} e^{4} n_{2}\ln\Lambda \over m_{1} m_{12} v^{2}},\eqno(2.2.9)$$
where $\Lambda$ is the argument of the ``Coulomb logarithm'':
$$\Lambda = {m_{12}v^{2}\lambda_{D}\over Z_{1}Z_{2}e^{2}}.\eqno(2.2.10)$$
The potential (2.2.6) is not strictly applicable to a moving charge, and
a proper treatment requires consideration of the dynamic, rather than
static, shielding properties of the plasma (Montgomery and Tidman 1964).
The use of an abrupt cutoff is also not strictly correct; in principle
${d\sigma / d\Omega}$ could be calculated for the dynamically
shielded potential. If either test or field particles are electrons
then the cross-section should be calculated quantum mechanically
(Spitzer 1962). Fortunately, a logarithm is very forgiving of such
corrections, so long as its argument $\Lambda \gg 1$. In practice it
is rarely necessary to be scrupulously careful in evaluating $\Lambda$.
Values of $\ln \Lambda$ typically are in the range $20 - 30$ in
interstellar plasmas, are $\sim 10$ in hot accretion flows and stellar
winds, but may be $\lapp 1$ in stellar interiors. Our derivation has
implicitly assumed $\Lambda \gg 1$, and becomes inapplicable when this
condition is not met. When it is applicable, neglected effects
introduce fractional inaccuracies ${\cal O}(1/\ln \Lambda)$; these are
referred to as non-dominant terms.
The logarithmic divergence of the integral in (2.2.5) means that each
decade (or octave) of scattering angle $\theta$ between $\theta_{min}$
and $\pi$ contributes equally to the slowing down of the test particle.
Over nearly all of this range $\theta \ll 1$, so that most (all but a
fraction $\sim 1/\ln\Lambda$) of the slowing down is a consequence of
small angle collisions. Much of it is also a consequence of collisions
with impact parameters $b \gapp n_{2}^{-1/3}$. Because in such a
collision the acceleration of the test particle extends over a time
$\sim b/u$ and distance $\sim b$, many such wide collisions take place
simultaneously. As long as they cumulatively produce little change in
${\vec u}$, they may still be regarded as independent additive events,
each described by the differential cross-section (2.2.1).
If we ignore the dependence of $\ln \Lambda$ on $u$ (which is weak
because it is logarithmic), then (2.2.9) is readily integrated to give
the time $t_{s}$ and length $\ell_{s}$ required for the test particle
to stop:
$$\eqalignno{t_{s}&={m_{1}m_{12}v^{3} \over 12 \pi Z_{1}^{2}Z_{2}^{2}
e^{4}n_{2}\ln\Lambda}&(2.2.11)\cr \ell_{s}&={m_{1}m_{12}v^{4} \over 16
\pi Z_{1}^{2}Z_{2}^{2}e^{4}n_{2}\ln\Lambda}.&(2.2.12)\cr}$$
Numerical evaluation for a fast electron of energy $E$ interacting with
a hydrogen plasma, taking $\Lambda = 10$, yields
$$n_{2}\ell_{s} \approx 2.0 \times 10^{17} \left({E \over 1\ {\rm KeV}}
\right)^{2} {\rm cm}^{-2} . \eqno(2.2.13)$$
For a fast proton
$$n_{2}\ell_{s} \approx 2.0 \times 10^{20} \left({E \over 1\ {\rm MeV}}
\right)^{2} {\rm cm}^{-2} . \eqno(2.2.14)$$
The rapid increase of stopping length with $v$ and $E$ is apparent.
The stopping length is greater for an electron than for a proton of the
same energy, but greater for a proton than for an electron of the same
velocity, in each case by a factor $\sim m_{p}/m_{e} \sim 10^{3}$.
These results are inapplicable if the test particle is relativistic.
Then $\ell_{s}$ is roughly proportional to energy for electrons.
Energetic nucleons and nuclei lose their energy in violent nuclear
collisions, whose cross-sections are roughly independent of energy,
and which for protons exceed Coulomb slowing in importance for
$E\gapp 100$ MeV.
Because the test particle is deflected as well as slowed in collisions,
the stopping length $\ell_{s}$ is measured along its actual path. It
may be seen by considering its deflections ({\bf 2.2.2}) that in most
cases the path is approximately straight, at least until nearly all its
energy has been lost, so that $\ell_{s}$ is a good approximation to the
depth of penetration into the stopping material.
The most important exceptions to this are
fast electrons in a medium with $Z_{2} \gg 1$, which
randomize their directions before they lose much of their energy.
The slowing and energy loss of fast particles is predominantly the
effect of field electrons. In a collision of given impact parameter
a field ion (of $Z_{2}=1$) and electron will receive essentially the
same impulse $\Delta p$, but the acquired kinetic energy $\Delta p^{2}
/(2m_{2})$ is much greater for the field electron. The passage of fast
particles through cold matter predominantly heats the electrons. This
effect is much reduced if the electron thermal velocity becomes
comparable to the test particle speed (in which case it is no longer
``fast''). The deflection of the test particles results, in comparable
amounts (if $Z_{2}=1$), from the influence of field ions and field
electrons; unless $Z_{2} \gg 1$ deflection is usually unimportant.
A result very similar to (2.2.9) applies to the passage of fast
particles through neutral matter. The lower cutoff on the integral in
(2.2.5) must now be taken at collisions which impart enough energy to a
bound electron to ionize it. For impacts significantly more violent
than these the electronic binding is negligible and (2.2.1) is
approximately correct. More distant encounters are ineffective, so that
the appropriate value of $\ln \Lambda$ is less than it would be in an
ionized medium.
The rare collisions with $\theta \sim 1$ are not completely
insignificant; they contribute a fraction ${\cal O}(1/\ln \Lambda)$ of
the total slowing. Because a fast particle undergoes only a few such
collisions in the course of its slowing, their contribution
is not accurately described by an integral over the differential
cross-section, but varies from particle to particle.
As a result the actual range of a particle may differ slightly
from its mean value (2.2.12); this phenomenon is called straggling.
In cold dense matter evaluation of (2.2.10) for thermal particles
(${1 \over 2}m_{1}v^{2} \approx {3 \over 2} k_{B}T$) implies
$\ln\Lambda \lapp 1$. This occurs in the interiors of low mass stars;
at the Solar center $\ln \Lambda \approx 2$. When $\ln \Lambda$ is
this small the theory is inapplicable. The plasma is ``strongly
coupled,'' meaning that the Coulomb energies of nearest neighbors are
comparable to (or exceed) $k_{B}T$. The ions and electrons do not move
freely, and the plasma is better described as a liquid than a gas.
Stopping lengths of thermal particles are smaller than the mean
interparticle separation\break $\sim n_{2}^{-1/3}$, and the theory of
this section is inapplicable.
If this theory is to be applied to gravitating systems $\lambda_{D}$
must be redefined. The phenomenon of Debye shielding does not exist
because the particles cannot be in thermodynamic equilibrium (in
equilibrium the most probably state would have particles with zero
separation and infinite binding energy). If the substitution \hbox{
$q^{2}\rightarrow Gm^{2}$} is made in (2.2.7), and the thermal velocity
of the particles $\sqrt{k_{B}T/m}$ is taken to be the sound speed,
then $\lambda_{D}$ becomes the Jeans length $\lambda_{J}$ (see
{\bf 3.2}). We know that no stable self-gravitating configuration can
exceed $\lambda_{J}$ in size. A gas of point masses cannot be confined
by external pressure, and will necessarily have a size close
to $\lambda_{J}$. It is therefore customary and plausible to take the
lower cutoff on the angular integration (2.2.5) in a gravitating
system to be at angles for which the impact parameter is the size of
the system; there are few field particles at greater distances.
A particularly simple result is obtained for gravitating systems in
the limit $m_{2} \ll m_{1}$. Substitution into (2.2.9) gives
$${d\langle{\vec v}\,\rangle \over dt} = - {\hat x} {4 \pi
G^{2} m_{1} \rho \ln\Lambda \over v^{2}} , \eqno(2.2.15)$$
where the mean background density $\rho = n_{2}m_{2}$. This should
be compared to the result (3.5.10) for the rate at which a mass moving
through a fluid medium accretes additional mass, a process which also
slows it. The slowing by dynamical friction, calculated in (2.2.15),
exceeds the accretional slowing by a factor $\ln\Lambda$; this
calculation of dynamical friction is applicable to supersonic motion
through collisional fluids as well as to free particles.
In a collisional fluid accretion
occurs only from orbits whose deflection exceeds $2 \sin^{-1}(1/\sqrt{5})
\approx 53^{\circ}$, which represent only $\sim 1/\ln\Lambda$ of the
integral in (2.2.5). The rate of slowing by dynamical friction is
proportional to the test particle mass (the drag force and the accretion
rate are
proportional to its square), so that dynamical friction may be important
for massive bodies, such as large molecular clouds, star clusters, and
galaxies, even when it is insignificant for individual stars.
\goodbreak
\vskip 2\baselineskip plus2\baselineskip
\goodbreak
\noindent
{\bf 2.2.2} \us{Equipartition Times} \quad
There is another limit, complementary to that of a fast test particle
moving through a cold plasma, in which it is easy to analyse the effects
of Coulomb collisions on the velocity distribution function of test
particles. Consider test particles of mass $m_{1}$ and charge $Z_{1}e$
which are initially at rest (${\vec v} = 0$) in a plasma of particles
of mass $m_{2}$, charge $Z_{2}e$, and number density $n_{2}$, in thermal
equilibrium at temperature $T_{2}$. As in {\bf 2.2.1}, the effects of
several species are simply additive (except that they all contribute to
the Debye length), so we consider only one.
Because initially ${\vec v} = 0$ and an equilibrium plasma is isotropic,
there is no preferred direction, and we must have
$${d\langle{\vec v}\,\rangle \over dt} = 0 . \eqno(2.2.16)$$
However, Coulomb collisions do impart velocity to the test particle, so
we can calculate the increase in its velocity dispersion $\langle v^{2}
\rangle$. As long as the test particle may be considered to be nearly
at rest the rate of increase of $\langle v^{2} \rangle$ is independent
of the direction of the colliding field particle (again, because there
is no preferred direction). For algebraic simplicity take the field
particles to be travelling in the $-{\hat x}$ direction, so the results
(2.2.3), together with (2.2.2), may be used to calculate the change of
$\langle v^{2} \rangle$. Note that (2.2.3) would give erroneous results
for each of $\langle v_{x}^{2} \rangle$, $\langle v_{y}^{2} \rangle$,
and $\langle v_{z}^{2} \rangle$, but it gives the correct average (and
sum $\langle v^{2} \rangle$), because taking this average is equivalent
to averaging over all possible directions of impact of the field
particles.
The rate of change of the velocity dispersion is then given by
$${d\langle v^{2} \rangle \over dt} = \left({m_{2} \over m_{1} + m_{2}}
\right)^{2} \int d\Omega\ d^{3}v_{2}\ {d\sigma \over d\Omega} f(v_{2})
\lbrack \Delta u^{2}(\Omega,v_{2}) \rbrack , \eqno(2.2.17)$$
where the equilibrium distribution function is
$$f(v_{2}) = n_{2} \left({m_{2} \over 2 \pi k_{B} T_{2}} \right)^{3/2}
\exp \left( - {m_{2} v_{2}^{2} \over 2 k_{B} T_{2}} \right) ,
\eqno(2.2.18)$$
and from (2.2.3) we find $\Delta u^{2} = 4 u^{2} \sin^{2}(\theta / 2)$.
Substituting (2.2.1), $\Delta u^{2}$, and $f(v_{2})$ into (2.2.17) gives
$$\eqalign{
{d\langle v^{2} \rangle \over dt}&= \left({m_{2} \over m_{1} + m_{2}}
\right)^{2} \left({m_{2} \over 2 \pi k_{B} T_{2}}\right)^{3/2}
{8\pi^{2}n_{2}Z_{1}^{2}Z_{2}^{2}e^{4}\over m_{12}^{2}} \cr &
\qquad\times\int {\sin \theta\ d\theta\over\sin^{2}(\theta /2)} \int
dv_{2}\ v_{2}\exp\left(-{m_{2} v_{2}^{2} \over 2 k_{B} T_{2}} \right)
. \cr}\eqno(2.2.19)$$
The integral over $\theta$ is exactly the same
integral evaluated in (2.2.5), and gives $4 \ln \Lambda$. Now $\Lambda$
is defined for an encounter with the mean thermal energy ${3 \over 2}
k_{B} T$ (where $T$ is an average of the test particle temperature, here
taken to be zero, and $T_{2}$, weighted according to the reciprocals of
$m_{1}$ and $m_{2}$ in order to give the mean kinetic energy
of relative motion):
$$\Lambda = {3 k_{B} T \lambda_{D} \over Z_{1} Z_{2} e^{2}} .
\eqno(2.2.20)$$
Properly $\Lambda$ varies with $v_{2}$, but this gives an insignificant
correction (smaller than other neglected effects mentioned in
{\bf 2.2.1}) to the final result. We now have
$$\eqalign{
{d \langle v^{2} \rangle \over dt}&= \left({m_{2} \over m_{1} + m_{2}}
\right)^{2} \left({m_{2} \over 2 \pi k_{B} T_{2}} \right)^{3/2} {32\pi^2
n_{2} Z_{1}^{2} Z_{2}^{2} e^{4} \ln \Lambda \over m_{12}^{2}}\cr&\qquad
\times\int_{0}^{\infty} dv_{2}\ v_{2} \exp \left( - {m_{2} v_{2}^{2}
\over 2 k_{B}T_{2}} \right) . \cr}\eqno(2.2.21)$$
This integral over velocity is elementary. Performing it, and
collecting factors gives
$${d\langle v^{2} \rangle \over dt} = {16 \pi n_{2} Z_{1}^{2} Z_{2}^{2}
e^{4} \ln \Lambda \over m_{1}^{2}} \sqrt{\mathstrut m_{2} \over 2
\pi k_{B} T_{2}} . \eqno(2.2.22)$$
We now define an equipartition time $t_{eq}$ as the characteristic time
for the mean kinetic energy to relax to the value ${3 \over 2} k_{B} T$
it has if the test particles are in equilibrium with the field particles:
$$\eqalign{t_{eq}&\equiv{{3 \over 2} k_{B} T_{2}\over{1\over 2}m_{1}
{d\over dt}\langle v^{2}\rangle}\cr&={3 m_{1} (k_{B} T_{2})^{3/2} \over
8\sqrt{2 \pi m_{2}} n_{2} Z_{1}^{2} Z_{2}^{2} e^{4} \ln \Lambda} . \cr}
\eqno(2.2.23)$$
The $3/2$ power of $T$ appearing in this equation resembles the $v^{3}$
dependence in (2.2.11), and has a similar origin in the velocity
dependence of the cross-sections.
These results are applicable only in the limit in which the test
particles move slowly compared to the field particles; unless
$m_{1} \gg m_{2}$ this approximation will break down before
equipartition is achieved. In the more general case in which the test
particles are neither very fast ({\bf 2.2.1}) nor very slow ({\bf 2.2.2})
compared to the field particles, the kinematics of the collisions
becomes more complex. Spitzer (1962) gives the result
$$t_{eq} = {3 m_{1} m_{2} \over 8 \sqrt{2 \pi} n_{2} Z_{1}^{2} Z_{2}^{2}
e^{4} \ln \Lambda} \left( {k_{B} T_{1} \over m_{1}} + {k_{B} T_{2} \over
m_{2}} \right)^{3/2} \eqno(2.2.24)$$
for the equipartition time between a distribution of test particles
which have an equilibrium distribution with temperature $T_{1}$ and
field particles at temperature $T_{2}$. Equation (2.2.23) describes the
case $T_{1} = 0$. The result (2.2.24) may also be applied to the
relaxation towards equilibrium of a distribution of particles with
itself.
The parentheses in (2.2.24) correspond to the $v^{3}$ or $T^{3/2}$
dependence of (2.2.11) or (2.2.23), but indicate that the larger of
the two thermal velocities is to be used. A very fast ion passing
through a cool plasma principally gives up its energy to the plasma
electrons, until it slows below the electron thermal velocity. If
\hbox{$v \lapp (k_{B}T/m_{e})^{1/2}(m_{e}/m_{p})^{1/3}
=(k_{B}T/m_{p})^{1/2}(m_{p}/m_{e})^{1/6}$}
(which may still correspond to kinetic energies far exceeding $k_{B}T_{2}
$), the test particle predominantly heats the plasma ions, because of
the effect of a small $m_{2} = m_{e}$ in the denominator of (2.2.24)
when applied to its interaction with the plasma electrons.
\goodbreak
\vskip 2\baselineskip plus2\baselineskip
\goodbreak
\noindent
{\bf 2.3 Comptonization}
\vskip \baselineskip
\noindent
Comptonization is the name given to the process by which
electron scattering brings a photon gas to equilibrium. Because
photons have a negligible cross-section for scattering by other
photons (although for photons with frequencies $\nu_{1}$ and $\nu_{2}$
satisfying $h^{2}\nu_{1}\nu_{2} \geq m_{e}^{2}c^{4}$ photon-photon pair
production and subsequent annihilation have the effect of photon-photon
scattering), they can only come to equilibrium by interaction with
matter. In the laboratory the walls of an enclosure are available to
absorb and re-emit the radiation, but in most astrophysical problems
there are no walls and absorptive processes in the matter
are often very slow. In hot fully ionized matter,
the most frequent and important process is
Compton scattering. In the limit that the electrons are
infinitely massive (compared to
$h\nu / c^{2}$) and slow-moving, Compton scattering would make
the photon angular distribution isotropic, but would not affect
the frequency spectrum. Because electrons are of finite mass and
do have random thermal velocities, the photon spectrum may
change as a result of scattering, and net energy transfer between
photons and electrons may occur. The finite electron mass leads
to an electron recoil as a result of Compton scattering, and
tends, on average, to transfer energy from photons to electrons.
The random electron velocities produce random Doppler shifts of
the scatterers in the laboratory frame, which, on average, tend
to increase the photon energy at the expense of the electrons.
The former effect increases with photon frequency, and the latter
with electron energy; if the photons have a thermal equilibrium
distribution at the same temperature as that of the electrons, the
two effects will balance.
\def\rightheadline{\vbox{\vskip 0.125truein\line{\tenbf\hfil
Comptonization \qquad\folio}}}
The term Comptonization is used if the electrons are in thermal
equilibrium at temperature $T$, and if both
$k_{B}T$ and $h \nu$ are much less than
$m_{e}c^{2}$, where
$\nu$ is the frequency of a typical photon. In this
nonrelativistic limit a number of powerful approximations are
possible, and a differential equation of Fokker-Planck form for
the time evolution of the photon occupation number
$n(\nu)$, assumed isotropic, is obtained. This
equation was first published by Kompaneets (1957), and is generally
referred to as the Kompaneets Equation. The fully relativistic
case is computationally much more complex, and cannot be reduced
to a Fokker-Planck equation because the photon frequency shift
$\Delta\nu$ upon scattering does not satisfy $\Delta\nu \ll \nu$.
Comptonization is likely to be important when the temperature is
high and the density is low, so that matter is fully ionized, and
absorption processes (generally proportional to the square of the
particle density) are less important than scattering (proportional
to the density). Comptonization occurs whenever photons and electrons
in the same volume are described by different temperatures. It is of
particular interest when the temperature of a low density electron gas
much exceeds the temperature of a Planck function with the same energy
density, and the absorption optical depth is low enough that the photon
spectrum falls far below a Planck function at the electron temperature;
typically some low energy ($h\nu \lapp k_{B}T$) photons are present.
Energy flows from the electrons to the photons.
Three likely astrophysical sites have been suggested:
the early universe, accretion flows onto compact objects (such as occur
in compact X-ray sources), and quasars. Comptonization has been used
in theoretical modelling of each of these (though synchrotron radiation
by relativistic electrons is probably more important for quasars).
Comptonization will also occur when cool electrons are found in a
radiation field of high energy ($h\nu \gg k_{B}T$), in which case the
net energy transfer is from the photons to the matter.
Despite the wide use of Comptonization by theorists, no clear
observational evidence for it exists. There can be no doubt that
the process occurs, for it is derived from elementary relativistic
kinetics, but its actual importance in astronomical objects is
unproven. It produces no unique spectral signature, so it may be
important without that fact being apparent. In some circumstances it
may bound the parameters of observable objects; if Compton scattering
and energy transfer are effective, the
object may not be practicably observable because they may drain energy
from an observable form, such as electrons which may emit synchrotron
radiation, to a less observable form.
Kompaneets' original paper and the book by Rybicki and
Lightman (1979) are rather terse, so I will try to explain the
derivation in more detail. The elementary scattering
process begins with an electron of momentum
${\vec p}$ and energy $E$, and a photon of frequency
$\nu$ directed along the unit vector
${\hat n}$, and scatters these into
${\vec p}^{\prime}$, $E^{\prime}$,
$\nu^{\prime}$, and
${\hat n}^{\prime}$. Define the frequency shift
$\Delta \equiv \nu^{\prime}-\nu$. Conservation of energy and
momentum state
$$\eqalignno{h\nu + {{\vec p}^{\, 2} \over 2 m_{e}} &= h\nu^{\prime} +
{{\vec p}^{\,\prime 2}\over 2 m_{e}} & (2.3.1)\cr {h\nu \over c} {\hat n}
+ {\vec p} &= {h\nu^{\prime} \over c} {\hat n}^{\prime} + {\vec p}
^{\prime} . & (2.3.2)\cr}$$
Eliminate ${\vec p}^{\,\prime}$ by regrouping and squaring (2.3.2) and
substituting it into (2.3.1). Collecting terms and ignoring the term in
$\Delta^{2}$ (valid in the nonrelativistic limit) leads to a linear
equation for $\Delta$ with the solution
$$ h\Delta = - {h\nu c {\vec p} \cdot ({\hat n}-{\hat n}^{\prime}) +
h^{2}\nu^{2} (1-{\hat n}\cdot{\hat n}^{\prime}) \over m_{e}c^{2} +
h\nu(1-{\hat n}\cdot {\hat n}^{\prime}) - c {\vec p}\cdot{\hat n}
^{\prime}} . \eqno(2.3.3)$$
Kompaneets (1957) contains typographical errors in the
corresponding expression and elsewhere.
Now take $h\nu \sim k_{B}T \sim {\cal O}(m_{e}v^{2})$, where
$v$ is a characteristic electron thermal velocity. The second
term in the numerator is an
${\cal O}(v/c)$ correction to its first term, as is the third term in
the denominator to its first term. The second term in the
denominator is an
${\cal O}(v^{2}/c^{2})$ correction to its first term. We also see that
$h\Delta/m_{e}c^{2} \sim {\cal O}(v^{3}/c^{3})$. Had we included the
$\Delta^{2}$ term in deriving (2.3.3), it would have led to an
${\cal O}(v^{3}/c^{3})$ fractional correction, by adding an
additional
$h\Delta /2$ to the denominator; equivalently, the ratio
$\Delta/\nu$ which is
${\cal O}(v/c)$ would have its value changed by an amount ${\cal O}
(v^{4}/c^{4})$. This is clearly negligible. As will be explained
later, we only need consider the leading term in (2.3.3), which is
$$h\Delta \approx - {h\nu {\vec p} \cdot ({\hat n}-{\hat n}^{\prime})
\over m_{e}c} . \eqno(2.3.4)$$
Now we must write down the Boltzmann (kinetic) equation governing
the evolution of the photon occupation number $n(\nu)$. For an infinite
homogeneous volume $(\partial /\partial{\vec x}\,)=0$ and ${\vec F}=0$,
so that
$${\partial n(\nu)\over\partial t}=\!\int\! d^{3}p\ cd\sigma\bigl\lbrack
n(\nu)\bigl(1+n(\nu^{\prime})\bigr)N(E) - n(\nu^{\prime})\bigl(1+n(\nu)
\bigr)N(E^{\prime})\bigr\rbrack . \eqno(2.3.5)$$
$N(E)$ is the electron distribution function per unit momentum space
and unit real space. Because the electrons are in thermal equilibrium
$N(E)$ depends only on energy and not on direction. \hbox{$d\sigma =
(d\sigma / d\Omega)d\Omega$} is the infinitesimal element of
cross-section for scattering a photon into the element of solid angle $d
\Omega$; in this scattering $(\nu, {\vec p}\,)\rightarrow (\nu^{\prime}
, {\vec p}^{\,\prime})$. ${d\sigma / d\Omega}$ is the differential
scattering cross-section. The first term in the brackets represents the
scattering of photons from frequency $\nu$ to frequency
$\nu^{\prime}$ by electrons of energy
$E$. Because photons are bosons the factor
$\bigl(1+n(\nu^{\prime})\bigr)$ represents the effect of stimulated
scattering. The electrons are assumed
nondegenerate ($N \ll 1$), so the corresponding fermion
factor $\bigl(1-N(E^{\prime})\bigr)$ is ignored. The second term
similarly represents scattering from
$\nu^{\prime}$ to $\nu$. The argument of
$N$ in this term is
$E^{\prime} = E + h\nu - h\nu^{\prime} = E -h\Delta$
because in this scattering the electron energy changes from
$E^{\prime}$ to $E$. This term represents the process which is the
exact inverse of that of the first term; it
proceeds at a different rate only because the occupation numbers
$n(\nu)$ and $N(E)$ are different from
$n(\nu^{\prime})$ and $N(E^{\prime})$. The differential cross-section
$d\sigma$ is the same, because if it were not an equilibrium gas of
photons and electrons could drive itself away from equilibrium,
violating the second law of thermodynamics. For a given
$\nu$ and ${\hat n}$ the electron momentum
${\vec p}$ and the scattering angle
$\Omega$ together determine
$\nu^{\prime}$ and ${\hat n}^{\prime}$, so there is no additional
integration over these variables. Such an integration could have been
included, but then
the kinematic constraints (2.3.1) and (2.3.2) would have introduced a
Dirac-$\delta$ function, which would have made the integrals trivial.
It is easy to see that if a general Bose distribution
$$ n(\nu) = {1 \over \exp (a+h\nu/k_{B}T) - 1} \eqno(2.3.6)$$
is substituted into (2.3.5) and
$N$ is taken to be Maxwellian at the same temperature
$T$ then the right hand side of (2.3.5) vanishes, as it must by the
second law of thermodynamics. It is necessary to introduce an arbitrary
chemical potential
$-ak_{B}T$ into $n(\nu)$ because Comptonization conserves photons;
equilibrium is achieved only subject to this constraint. The photon
distribution cannot relax to the Planck function ($a=0$) by Compton
scattering alone. The assumption that the electrons are fully relaxed
and nondegenerate is generally well justified when Comptonization is
of interest.
Equation (2.3.5) is an integrodifferential equation for
$n(\nu)$, which is hard to solve, except by ``brute force''
numerical techniques. Such a solution gives little physical
insight. The reason for this difficulty is that as it is written
(2.3.5) relates
$\partial n(\nu)/\partial t$ to the values of
$n$ at all other frequencies
$\nu^{\prime}$; it is nonlocal in frequency. In the nonrelativistic
limit the only
$\nu^{\prime}$ which contribute are close to
$\nu$, so the right hand side of (2.3.5) may be expanded in powers of
$\Delta /\nu \ll 1$. The integrals may then be
performed explicitly, leaving a differential equation which is
easy to solve, and which is readily interpreted. It must be remembered
that the Kompaneets equation is based on an expansion in powers of
$\Delta /\nu$, which is
${\cal O}(v/c)$. This is not a very small ratio under conditions in
which Comptonization is of interest; for
$T = 10^{8\ \circ}$K it is about 0.2. Although the
Kompaneets equation is strictly correct in the nonrelativistic
limit, it is not very accurate under the conditions in which it
is usually used.
It will be necessary to expand (2.3.5) to terms of the second power
in $\Delta$. Use the Taylor expansions
$$\eqalignno{n(\nu^{\prime})&= n(\nu )+ {h\Delta \over k_{B}T}{\partial n
\over \partial x} + {1 \over 2} \left({h\Delta \over k_{B}T}\right)^{2}
{\partial^{2} n \over \partial x^{2}} + \cdots\ \ \ \ \ \ \ &(2.3.7)\cr
N(E-h\Delta )&= N(E)\left\lbrack 1 + {h\Delta \over k_{B}T} +
{1 \over 2} \left({h\Delta \over k_{B}T}\right)^{2} + \cdots\right\rbrack
, &(2.3.8)\cr}$$
where a nonrelativistic Maxwellian $N(E)$ has been assumed and \hbox{
$x \equiv h\nu /k_{B}T$} is a convenient scaled frequency variable.
Substitute these expressions into (2.3.5) and collect terms to obtain
$$\eqalign{{\partial n(\nu )\over\partial t}
&= {h \over k_{B}T}\left({\partial n \over \partial x} + n + n^{2}\right)
{\cal I}_{1}\cr &\qquad + {1\over 2} \left({h \over k_{B}T}\right)^{2}
\left({\partial^{2} n \over \partial x^{2}} + 2(1+n){\partial n \over
\partial x} + n + n^{2}\right){\cal I}_{2} + \cdots\ ,\cr}\eqno(2.3.9a)$$
where
$$\eqalignno{
{\cal I}_{1}&\equiv \int d^{3}p\ d\sigma\ cN(E)\Delta&(2.3.9b)\cr
{\cal I}_{2}&\equiv \int d^{3}p\ d\sigma\ cN(E)\Delta^{2}.&(2.3.9c)\cr}$$
This has been written as a power series in $\Delta$. The term in
$\Delta^{0}$ is zero, as it must be, for if
$\Delta = 0$ no photon can change its frequency. The crucial step in
reducing the integrodifferential equation has been the replacement of
$n(\nu^{\prime})$ and $N(E^{\prime})$ by functions of
$n(\nu)$, $N(E)$, and their derivatives, which have been brought out of
the integrals. The only function of
$\nu^{\prime}$ left in the integrals is
$\Delta$, which has been simply expressed in terms of other
variables by (2.3.4).
Examination of (2.3.4) shows why it was necessary to include the
$\Delta^{2}$ term in (2.3.9). For any
$({\hat n} - {\hat n}^{\prime})$, and to lowest order in $v/c$,
${\cal I}_{1}$ is proportional to
$\int d^{3}p\ {\vec p}\cdot ({\hat n} - {\hat n}^{\prime}) N(E)$, which
is zero because
$N(E)$ is independent of direction (it is sufficient that the mean
electron momentum be zero). Thus, to lowest order in $v/c$,
${\cal I}_{1}$ is zero. This must
be so because this term would correspond to a systematic Doppler shift of
${\cal O}(v/c)$ in each scattering, which cannot be present for the
assumed isotropic distribution of electrons. The nonzero contribution
from ${\cal I}_{1}$ depends on a better estimate of
$\Delta$ than (2.3.4), and is smaller by a factor
${\cal O}(v/c)$. Equation (2.3.3) could be used to estimate
$\Delta$ to higher order in
$v/c$, but the integration of the resulting expression would be
difficult. Fortunately (but not miraculously) the integration of the
$\Delta^{2}$ term in (2.3.9) is tractable, and makes the explicit
integration of the
$\Delta$ term unnecessary. In the $\Delta^{2}$ term the first
approximation (2.3.4) to $\Delta$ is sufficient.
Substitute (2.3.4) into the integral ${\cal I}_{2}$ (2.3.9c) to obtain
$$\hskip .51truein
\eqalign{{\cal I}_{2}&=\int d^{3}p\ cd\sigma\ N(E)\Delta^{2}\cr
&=\left({\nu \over m_{e}c}\right)^{2}\int cd\sigma\ d^{3}p\ N(E) \bigl(
{\vec p}\cdot({\hat n} - {\hat n}^{\prime})\bigr)^{2}.\cr}\eqno(2.3.10)$$
In the last integral write
${\vec p} \cdot ({\hat n} - {\hat n}^{\prime}) = p\,\vert {\hat n} -
{\hat n}^{\prime} \vert \cos \psi$, where $\psi$ is the included angle
between these two vectors. The quantity $\vert {\hat n} -
{\hat n}^{\prime} \vert^{2}$ depends on the scattering angle
but not on ${\vec p}$, and may be removed from the integral over
electron momentum space. Because the remaining factors in the integral
are isotropic ($N(E)$ by assumption), the angular part of the momentum
space integral is just the angular integral of $\cos^{2}\psi$, which is
$4\pi /3$ for any polar axis (the direction of
${\hat n} - {\hat n}^{\prime}$). Performing the integral over the
direction of the electron momentum ${\vec p}$ gives
$${\cal I}_{2} = {1\over 3} \left({\nu \over m_{e} c}\right)^{2} \int c
d\sigma\vert {\hat n} - {\hat n}^{\prime} \vert^{2} \int 4 \pi p^{2}\,dp
\,N(E)p^{2}. \eqno(2.3.11)$$
The last integral in (2.3.11) is just $n_{e}$ times $\langle p^{2}
\rangle$, or $2m_{e}$ times the electron kinetic energy density.
Because $N(E)$ is Maxwellian this is $3k_{B}Tm_{e}n_{e}$. Then
$${\cal I}_{2} = \left({\nu \over m_{e}c}\right)^{2} k_{B}Tm_{e}n_{e}c
\int d\Omega {d\sigma \over d\Omega} \vert {\hat n} - {\hat n}^{\prime}
\vert^{2} . \eqno(2.3.12)$$
The exact differential cross-section for Compton scattering is given by
the Klein-Nishina formula. Because we are taking the
nonrelativistic limit we approximate it by the Thomson differential
cross-section $d\sigma / d\Omega = {1\over 2} r_{e}^{2} \bigl(
1+({\hat n} \cdot {\hat n}^{\prime})^{2}\bigr)$, where
$r_{e} \equiv e^{2}/(m_{e}c^{2})$
is the classical electron radius. This introduces a fractional error
${\cal O}(h\nu / m_{e}c^{2})\sim {\cal O}(v^{2}/c^{2})$, which may be
ignored because in general we are only considering
the leading terms in expansions whose successive terms are in the
ratio ${\cal O}(v/c)$. Similarly, we ignore the Lorentz transformations
between the laboratory and scattering electron's frames in evaluating
${d\sigma / d\Omega}$ and $d\Omega$, because in the nonrelativistic
limit these frames become identical. A fully relativistic treatment
would include these transformations, and is cumbersome. (2.3.12)
becomes
$${\cal I}_{2} = \left({\nu \over m_{e}c}\right)^{2} k_{B}T m_{e} n_{e} c
\int d\Omega {1 \over 2} r_{e}^{2} \bigl(1 + ({\hat n}\cdot{\hat n}^
{\prime})^{2}\bigr) \vert {\hat n} - {\hat n}^{\prime} \vert ^{2} .
\eqno(2.3.13)$$
Substitute $\vert{\hat n}-{\hat n}^{\prime}\vert^{2} = {\hat n}^{2} +
{\hat n}^{\prime\, 2} - 2({\hat n}\cdot{\hat n}^{\prime}) = 2 - 2
({\hat n}\cdot{\hat n}^{\prime})$, and integrate over the photon
scattering angle $d\Omega$. Integrals over a sphere
of odd powers of ${\hat n}\cdot {\hat n}^{\prime} = \cos\theta$
are zero, while the integral
of $\cos^{2}\theta$ is ${4 \pi / 3}$. Finally, rewrite the result
in terms of the angle-integrated Thomson cross-section $\sigma_{es} =
{8 \over 3}\pi r_{e}^{2}$ to obtain
$${\cal I}_{2} = 2 \left({\nu \over m_{e}c}\right)^{2}k_{B}T
m_{e}n_{e}\sigma_{es}c. \eqno(2.3.14)$$
Substituting (2.3.14) into (2.3.9) shows that the integral of $\Delta
^{2}$ contributes to $\partial n / \partial t$ a term
proportional to $x^{2} (\partial^{2} n / \partial x^{2})$, while
the integral of $\Delta$ does not.
This result will permit the derivation of the
Kompaneets equation without any further computation of integrals.
The procedure is analogous to using (2.1.13) to determine the
Fokker-Planck coefficient ${\vec a}$ from $\hbox{\bf b}$.
Compton scattering conserves photons. Therefore the occupation number
$n(\nu)$ must satisfy a conservation law in three dimensional photon
momentum space. For an isotropic photon distribution $n$ depends only
on the magnitude of the photon momentum, and such a law takes the form
$${\partial n \over \partial t} = - {1 \over x^{2}}{\partial (x^{2} j)
\over \partial x} , \eqno(2.3.15)$$
where $j$ is the ``current density'' of photons. Equation
(2.3.9a) contains a term equal to ${\partial^{2} n / \partial
x^{2}}$ times a function of $x$ (but not of $n$). Because (2.3.15)
describes the same function it must have the same form, so that any term
in $j$ proportional to ${\partial n / \partial x}$ cannot contain
any other dependence on $n$. Hence $j$ must be of the form
$$j(n,x) = g(x) \left( {\partial n \over \partial x} + h(n,x) \right),
\eqno(2.3.16)$$
where $g$ and $h$ are functions yet to be determined.
The Bose equilibrium distribution function (2.3.6) must give zero when
substituted into (2.3.9) (as it is a stationary solution of 2.3.5).
Because there are no photon sources or sinks,
$j=0$ for a stationary solution.
For $n$ of the form (2.3.6)
$${\partial n \over \partial x} = - n - n^{2} \eqno(2.3.17)$$
holds identically. Therefore, the condition that $j=0$ in equilibrium
is satisfied if for all
$n$ and $x$
$$h(n,x) = n + n^{2} . \eqno(2.3.18)$$
To determine $g$ compare the coefficient of ${\partial^{2} n / \partial
x^{2}}$ in (2.3.15) and (2.3.16) to that in (2.3.9). In (2.3.9) it
is $x^{2}$ times constants (the $x^{2}$ comes from the $\nu^{2}$ in
2.3.14), so that $g(x) \propto x^{2}$. The constant factors are found
from (2.3.9) and (2.3.14), so that
$$g(x) =-{k_{B}T\over m_{e}c^{2}}n_{e}\sigma_{es}cx^{2}.\eqno(2.3.19)$$
Combining (2.3.15), (2.3.16), (2.3.18), and (2.3.19), and defining a
dimensionless scaled variable $y$ in place of $t$
$$y \equiv t {k_{B}T \over m_{e}c^{2}}n_{e}\sigma_{es}c \eqno(2.3.20)$$
gives the Kompaneets equation:
$${\partial n \over \partial y} = {1 \over x^{2}} {\partial \over
\partial x} \left\lbrack x^{4} \left({\partial n \over \partial x}
+ n + n^{2} \right) \right\rbrack . \eqno(2.3.21)$$
It is possible to determine ${\cal I}_{1}$ explicitly, now that
(2.3.21) and (2.3.14) are known, by comparing the coefficients of
${\partial n / \partial x}$. The result is
$${\cal I}_{1} = {k_{B}T \over m_{e}c^{2}} \sigma_{es} n_{e} x (4-x) .
\eqno(2.3.22)$$
This result may also be obtained by comparing the coefficients of
$n$ and $n^{2}$; the answers are necessarily the same.
In most astrophysically interesting Comptonization problems the
photon density is far below its equilibrium value, so
$n \ll 1$ and the nonlinear term in (2.3.21) may be neglected. Then
it is possible to derive a simple result for the variation of the
total photon energy density with time, if nearly all the energy is
at low frequencies $x \ll 1$. This energy density is given by
$${\cal E}_{r} = {2 (k_{B}T)^{4} \over h^{3}c^{3}} \int_{0}^{\infty}
n(x) x\ 4 \pi x^{2}dx . \eqno(2.3.23)$$
Then $$\hskip .54truein
\eqalign{{h^{3}c^{3} \over 8 \pi (k_{B}T)^{4}} {d{\cal E}_{r} \over
dy}&={\partial \over \partial y} \int_{0}^{\infty}nx^{3}\ dx \cr &=\int
_{0}^{\infty}dx\ x^{3}{\partial n \over \partial y} \cr &=\int_{0}
^{\infty} dx\ x {\partial \over \partial x}\left\lbrack x^{4}\left({
\partial n \over \partial x} + n \right)\right\rbrack\cr &= \int_{x=0}
^{\infty} x\ d\left\lbrack x^{4} \left({\partial n \over \partial x}
+ n \right) \right\rbrack ,\cr}\eqno(2.3.24)$$
where the $n^{2}$ term in (2.3.21) has been neglected.
Integrate by parts, and assume that
$n(x)$ drops off sufficiently rapidly (it will usually decline
exponentially) as $x \rightarrow \infty$, but does not rise too rapidly
as $x \rightarrow 0$, so that $x^{5}\left({\partial n \over \partial x}
+ n \right) \rightarrow 0$ in both these limits. Then
$$\hskip .44truein
\eqalign{{h^{3}c^{3} \over 8 \pi (k_{B}T)^{4}}{d{\cal E}_{r} \over
dy} &= - \int_{0}^{\infty} x^{4}\left({\partial n \over \partial x} + n
\right)\ dx\cr &=-\int_{0}^{\infty}nx^{4}\ dx - \int_{0}^{\infty}x^{4}
{\partial n \over \partial x}\ dx . \cr}\eqno(2.3.25)$$
Again integrating by parts,
$${h^{3}c^{3} \over 8 \pi (k_{B}T)^{4}}{d{\cal E}_{r} \over dy} =
-\int_{0}^{\infty} nx^{4}\ dx + 4 \int_{0}^{\infty} nx^{3}\ dx .
\eqno(2.3.26)$$
If the photon energy density is concentrated at low frequencies
$x \ll 1$, the first integral in (2.3.26) is much less than the
second. Neglect the first integral to obtain the approximate result
$$\eqalignno{{d{\cal E}_{r} \over dy} &= 4{\cal E}_{r} &(2.3.27)\cr
\noalign{\hbox{or}}
{\cal E}_{r}&={\cal E}_{0} \exp (4y) . &(2.3.28)\cr}$$
The energy density grows exponentially (even though photon number is
conserved). The $e$-folding time is
$$t_{C\gamma}={t \over 4y}={m_{e}c^{2} \over 4 k_{B}T}{1 \over n_{e}
\sigma_{es} c} . \eqno(2.3.29)$$
The characteristic time scale (2.3.20 and 2.3.29) is proportional to
$(n_{e}\sigma_{es}c)^{-1}$, a photon's mean time between
Compton scatterings, and inversely proportional to
$k_{B}T/m_{e}c^{2}$, because the mean photon energy gain per scattering
is ${\cal O}(k_{B}T/m_{e}c^{2})\sim{\cal O}(v^{2}/c^{2})$.
The exponential growth indicated by (2.3.28) does not continue
indefinitely. Eventually a significant fraction of the photon
energy comes to be in photons for which
$x \gapp 1$. Then the negative term in (2.3.26) is no longer
insignificant, and the growth slows.
A realistic astronomical problem is unlikely to start at some
initial time with a soft photon spectrum whose energy
density then grows exponentially according to (2.3.28). It is more
plausible to think of an object of finite size and finite optical depth
with a steady source of low frequency photons. As the photons
increase their mean energy according to (2.3.28) they also diffuse
outward and escape. A steady state is obtained if there is a
steady heat source for the matter to replenish the energy it gives up
to the photons. It is straightforward to
compute numerically the emergent spectrum. Qualitatively, it
depends on the ratio of the size of the Compton scattering cloud
to a critical size approximately equal to
$\sqrt{m_{e}c^{2}/k_{B}T}(n_{e}\sigma_{es})^{-1}$.
For significantly larger clouds the energy growth
saturates, and the emergent spectrum resembles a Wien law
$n(\nu) \propto \exp (-h\nu / k_{B}T)$ (the low $n$ or large $a$
limit of the equilibrium Bose distribution 2.3.6).
Smaller clouds are inefficient energy multipliers, and produce
nearly power law spectra, steeply decreasing with increasing
frequency. Because the critical size depends on
$T$, the emergent radiative power is a steeply increasing function of
$T$, and wide variations in the photon source or the power
supplied to the matter are accommodated by modest changes in the
steady state temperature.
In some circumstances the effect of Comptonization on the
electrons is of more interest than its effect on the photons.
This is a much simpler problem; because we have assumed that the
electron distribution function quickly relaxes to a Maxwellian of
temperature
$T$, we only need to know the mean rate of energy
transfer, and not the effect of Compton scattering as a function
of electron momentum. The complete answer in the limit
$n(\nu) \ll 1$ is supplied by (2.3.26), noting that each erg supplied to
the photons is drawn from the electron thermal energy.
The energy transfer may be calculated explicitly for special cases of
$n(\nu)$. For a general Bose distribution the integrals must be
expressed as infinite series, and if $a \lapp 1$ the approximation
$n(\nu) \ll 1$ is not valid throughout the spectrum. In astrophysics
one often deals with diluted black body radiation fields, in which the
spectral shape resembles that of a black body but the energy density
is much lower. The dilution is generally the result of spherical
divergence, and is found in regions illuminated by a small distant
source. Dilution reduces the energy density, but does not change
$n(\nu)$ for photon states whose momenta are directed from the source;
other states have
$n=0$, giving an anisotropic
$n$. It is easier and usually a good approximation to
consider an isotropic Wien spectrum of arbitrary intensity
characterized by a temperature
$\alpha T$; then
$n(\nu ) = n_{0} \exp (-x/\alpha )$. Insertion of this
into (2.3.26) (which implicitly assumes $n \ll 1$) leads to the result
$${d{\cal E}_{e} \over dt} = {8 \over 3} {\cal E}_{e} {{\cal E}_{r}
\sigma_{es}\over m_{e}c}(\alpha -1) , \eqno(2.3.30)$$
where
${\cal E}_{e}$ is the electron thermal energy density. For very soft
photon spectra
($\alpha \ll 1$) the electrons cool at the rate
$${d{\cal E}_{e} \over dt} = - {8 \over 3} {\cal E}_{e} {{\cal E}_{r}
\sigma_{es}\over m_{e} c} . \eqno(2.3.31)$$
The result
(2.3.31) may be obtained regardless of the shape of the photon spectrum
by considering the Thomson scattering drag force on the thermal motion
of the electrons, allowing for Doppler shifts and aberration to first
order in $v/c$; for $h\nu \ll k_{B}T$ the increase in electron velocity
dispersion resulting from scattering recoil is negligible. These
results hold for any angular distribution of radiation,
provided the mean radiation pressure has been subtracted out and
the electron distribution remains isotropic.
An example of the application of (2.3.30) and (2.3.31) is to the
accretion of matter onto a white dwarf or a neutron star. The
stellar photosphere is a source of radiation with temperature
$T_{e} \approx (L/4\pi r^{2} \sigma_{SB})^{1/4}$, where
$L$ is the luminosity and
$\sigma_{SB}$ is the Stefan-Boltzmann constant; this may typically be
$\sim 3 \times 10^{5\ \circ}$K for a white dwarf and
$\sim 10^{7\ \circ}$K for a neutron star. If matter falling freely from
infinity is stopped in a shock at the stellar surface, its temperature
is of order $T \sim (GM\mu /k_{B}R)$, where
$\mu$ is the molecular weight; this is
$\sim 10^{9\ \circ}$K for a white dwarf and could range as high as
$\sim 10^{12\ \circ}$K for a neutron star (though a simple accretion
shock is not expected). Thus very hot matter is immersed in a
radiation field of much lower color temperature, and (2.2.31) may be
used to calculate the Compton cooling of the matter (provided the
electron temperature satisfies $k_{B}T \ll m_{e}c^{2}$).
>From (2.3.31) we see that if no heat sources are present the electron
thermal energy decays exponentially on a characteristic Compton cooling
time
$$t_{Ce} = {3 m_{e}c \over 8 {\cal E}_{r} \sigma_{es}} . \eqno(2.3.32)$$
If ${\cal E}_{r}=L/(4 \pi r^{2} c)$, corresponding to a soft photon
luminosity $L$ flowing radially outward, then
$$t_{Ce} = {L_{E} \over L}{3 m_{e} \over 8 m_{H}}{r^{2}c \over GM} ,
\eqno(2.3.33)$$
where $L$ has been expressed in terms of the Eddington luminosity $L_{E}$
(1.11.6) for pure hydrogen composition and $m_{H}$ is the mass of a
hydrogen atom. This cooling time should be compared to the hydrodynamic
time $t_{h}$ (1.6.1), which characterizes free fall or periods of
vibration. Their ratio is
$${t_{Ce} \over t_{h}} = {L_{E} \over L}{3 m_{e} \over 8 m_{H}} \sqrt{
r c^{2} \over GM} . \eqno(2.3.34)$$
It is evident that if $L$ is large $t_{Ce} \ll t_{h}$, so that Compton
cooling is very rapid. It may dominate the energy balance in accretion
flows onto compact objects, and therefore may determine material
temperatures and the spectrum of observable radiation.
\goodbreak
\vskip 2\baselineskip plus2\baselineskip
\goodbreak
\noindent
{\bf 2.4 Evolution and Collapse of Star Clusters}
\vskip \baselineskip
\noindent
There are two kinds of astronomical objects which have led to the
study of the evolution and collapse of clusters of stars. The
first of these is the globular clusters. These spectacularly
beautiful objects typically contain
$10^{5}$ stars in a region perhaps 10 parsecs across; the central
density of stars may be as much as
$10^{6}$ times higher than that in the Solar neighborhood. Only a small
fraction of these stars appear in a photograph of a
globular cluster, for they span a wide range of brightness and most
are too faint to detect. Frequently the inner parts of a cluster
resemble a single over-exposed blob of overlapping stellar images.
Despite this, even the densest inner regions are quite empty; an
observer at the center would see only
$\sim 10^{-11}$ of the sky covered with stars. The escape velocity
and random stellar velocity of globular clusters are quite small
and difficult to measure, even though their central densities are
high, because the total mass is modest. Typical velocities are
probably around 10 km/sec, which should be contrasted to
velocities of 250 km/sec in galaxies. Globular clusters are
dynamically fragile objects.
\def\rightheadline{\vbox{\vskip 0.125truein\line{\tenbf\hfil Evolution
and Collapse of Star Clusters \qquad\folio}}}
The stars in globular clusters in our Galaxy are among the oldest known.
(The Magellanic Clouds contain globular clusters, apparently recently
formed, in which the stars are young.) Astronomers have been
interested in globular clusters for nearly a century because
they have been very useful in understanding stellar evolution and the
extragalactic distance scale, and because some have hoped that
they might illuminate the history of the early universe and the
formation of our Galaxy. Interest in globular clusters surged in
1975, when it was discovered that they are about 100
times richer in compact X-ray sources than our galaxy as a whole. It was
then widely speculated that globular clusters might contain 1\% of their
mass in a massive central black hole. This hypothesis is
now largely discounted because the X-ray emission appears to be
more characteristic of accreting neutron stars in binary systems
({\bf 4.2}; most globular cluster X-ray sources show characteristic
bursts; Lewin and Joss 1983), and because the X-ray sources are not
found in the exact cluster centers, implying that they are not very
massive (Grindlay, {\it et al.} 1984). The overabundance of X-ray
sources, and even their presence at all, are still unexplained.
This problem is part of the more general problem of binary stars in
globular clusters, about which there are few data, but which may
be important for the dynamical evolution of the clusters.
The second astronomical application of the theory of star
clusters is to elliptical galaxies, and possibly to
quasars ({\bf 4.7}). Elliptical galaxies typically contain
$10^{11}$ stars in a region
$10^{4}$ parsecs across. The stars themselves resemble those
in globular clusters but they are distributed at a much lower
density, and with much higher velocities (typically 250 km/sec).
The evolutionary and collapse phenomena which are believed to be
important for globular clusters are negligibly slow for
elliptical galaxies. Interest lies in the possibility that some
galaxies (either elliptical or spiral) contain much denser cores,
intermediate in parameters between globular clusters and the
galaxies as a whole, in which these phenomena may be important.
There is evidence that some nearby galaxies (for example, Andromeda)
have such cores, for photographs underexposed in order to study their
inner regions show very compact nuclei with nearly star-like images (see
also Light, {\it et al.} 1974). I will call these ``pseudo-stellar
nuclei;'' the word ``quasi-stellar'' generally refers to quasars, which
are very different (though possibly related) objects.
The fundamental problem in the theory of star clusters (reviewed by
Lightman and Shapiro 1978) is the inability of the individual stars,
considered as particles constituting a gas,
to come to full thermodynamic equilibrium. Because the
depth of the potential well is finite, in any Maxwellian
distribution of stellar velocities there will always be a small
fraction of the stars moving faster than the escape velocity. It
is possible to estimate this fraction if one uses the virial
theorem to relate the random thermal velocity to the mean
gravitational potential of the cluster. The quantitative result
depends on how the structure is modeled (for example, how the
mean radius is defined in estimating the depth of the potential);
typically, about 1\% of the stars in a Maxwellian would have
enough energy to escape. In practice, the stellar
distribution function is cut off at the escape energy. As the
distribution function collisionally relaxes towards a complete
Maxwellian, there is a steady efflux of stars with positive
energy. The remainder of the cluster contracts as it loses mass
and its total binding energy grows in magnitude. The loss of
mass is more important, and would imply a deepening of
the gravitational potential well even were no energy lost
to evaporation.
This fundamental problem of cluster thermodynamics also appears if one
considers the equilibrium distribution of the orbital parameters
of a binary star in thermal contact with a heat bath (the cluster)
at temperature
$T$. In equilibrium, the probability of being in a state of
energy
$E$ is proportional to
$\exp (-E/k_{B}T)$. The energy
$E = -Gm_{1}m_{2}/(2a)$, where
the stars have masses $m_{1}$ and $m_{2}$, and
$a$ is the semi-major axis of the orbit of their separation vector.
As $a \rightarrow 0$,
$E \rightarrow -\infty$, and the probability diverges. It is not
possible to sum over these probabilities to obtain a partition
function. Thermodynamic equilibrium is not possible, for
essentially all the equilibrium probability density is in states of
infinite binding energy. In practice, a binary star would not rapidly
contract to an orbit of infinitesimal size, because as its orbit
becomes smaller, its rate of relaxation by encounters with the stars
making up the thermal bath rapidly decreases. The distribution of
orbital parameters is always determined by the rates of relaxation
processes, and never by a thermal equilibrium distribution
function. The atomic version of this equilibrium catastrophe
(aggravated by the rapid radiation of an orbiting classical electron)
was a fundamental problem of physics resolved by the quantization
of atomic states. Discs of matter orbiting a central object ({\bf 3.6})
present an analogous problem; a dissipative process (viscosity) draws
matter into the central gravitational potential well, as the
distribution of matter relaxes towards an (unattainable) equilibrium.
The inner parts of a globular cluster may be compared to the
hypothetical binary star, with the outer parts representing the
heat bath. Essentially all the equilibrium probability is in
states which are ``down the black hole'' of infinitely tight
binding. The cluster never actually comes to equilibrium, and
the calculation of its properties requires the calculation of its
nonequilibrium relaxation processes.
The collisional relaxation time for gravitating masses, such as
stars in a cluster, may be calculated in essentially the same way
as for charged particles in a plasma ({\bf 2.2}). The stars themselves
only rarely collide, but their paths are changed by their mutual
gravitational interaction. In carrying out the integration over
impact parameters the upper cutoff is not the Debye length, for there is
no Debye shielding in gravitational interactions (in the unattainable
state of complete thermodynamic equilibrium there would be
anti-shielding). Instead, the integration is cut off at the
geometrical size of the cluster, or that of the dense core region
in which collisional relaxation is important. The result (Lightman and
Shapiro 1978) for clusters all of whose stars have the same mass is
a characteristic relaxation time
$$\hskip .41truein
\eqalign{t_{r}&= {v_{m}^{3} \over 15.4 G^{2} m^{2} n \ln (0.4N)}\cr
&\approx 7 \times 10^{8}\ {\rm yr} \left({N \over 10^{5}}\right)^{1/2}
\left({m \over M_{\odot}}\right)^{-1/2} \left({R \over 5\ {\rm pc}}
\right)^{3/2} , \cr}\eqno(2.4.1)$$
which has been evaluated at the radius $R$ which includes half of the
cluster mass; in the logarithm $N$ was taken equal to $10^{5}$.
In (2.4.1) $v_{m}$ is the dispersion of the stellar velocities,
$m$ the stellar mass,
$n$ the mean density of stars inside of
$R$, and $N$ the total number of stars in the cluster. The
coefficient differs somewhat from that obtained in {\bf 2.2}
because an attempt has been made to
allow for the nonuniformity of the cluster. The argument of the
logarithm may be obtained from (2.2.10), replacing $Z_{1}Z_{2}e^{2}$
by $Gm^{2}$, $\lambda_{D}$ by $R$, and using the virial theorem to
estimate the velocity dispersion; the (uncertain) coefficient of $N$
depends on the quantitative structure of the cluster.
The most important single implication of (2.4.1) is its order of
magnitude. For a globular cluster the factors in parentheses are
of order unity, so that the relaxation time is much shorter than
the age of the cluster (globular cluster ages are estimated from
stellar evolutionary arguments to be about
$10^{10}$ years, close to the age of the Galaxy). Dynamical
evolution is important for globular clusters. In contrast, for
elliptical galaxies
$N \sim 10^{11}$, $R \sim 10{,}000$ pc, and
$t_{r}$ is more than
$10^{16}$ years; dynamical evolution is completely
insignificant, unless there is an inner region with much
higher density. Such a region must have $n$ much greater than that of
a globular cluster because of the factor
$v_{m}^{3}$ in (2.4.1), which is much larger in elliptical galaxies.
There would be no reason to suspect the existence of such a region,
were it not for the unexpected existence of quasars, other non-stellar
activity in galactic nuclei, and a few observed pseudo-stellar nuclei.
There is a second time scale of interest for a globular cluster,
the dynamical time
$t_{d}$ required for a star to cross the cluster:
$$t_{d} \equiv R/v_{m} . \eqno(2.4.2)$$
There is another significance to
$t_{d}$. On time scales much shorter than
$t_{r}$ collisions may be ignored, and the Boltzmann equation (2.1.9)
for the stellar distribution function $f$ becomes the Vlasov equation
(2.1.4) (closely related to the Vlasov equation used in the theory of
collisionless plasmas):
$${df \over dt} = {\partial f \over \partial t} + {\vec v} \cdot
{\partial f \over \partial {\vec r}} - \nabla \Phi \cdot {\partial
f \over \partial {\vec v}} = 0 , \eqno(2.4.3)$$
where
$\Phi$ is the cluster gravitational potential. Some distribution
functions imply
$\partial f / \partial t = 0$, but some do not. If
not, then the characteristic time scale on which
$f$ changes is of order
$R/v$ or $v/\nabla \Phi$, each of which is of order
$t_{d}$. This rapid change in
$f$ is called violent relaxation.
There is no complete criterion, other than (2.4.3) itself, for the
occurrence of violent relaxation. Thermodynamic equilibrium
guarantees its absence, but it will not occur for many nonequilibrium
$f$. Because $f$ is a function of 6 variables (aside from time) it may
be very complex. Several well-known examples of nonequilibrium
distribution functions which do not undergo violent relaxation exist.
For example, begin with a spherically
symmetric cluster in equilibrium, with the distribution function
isotropic and Maxwellian everywhere (except for the missing
positive-energy tail). Define an arbitrary axis through the cluster
center. Then, for each particle with positive azimuthal velocity
about that axis, reverse the sign of the azimuthal component of
its velocity. This bizarre distribution function will not undergo
violent relaxation.
$\nabla \Phi$ has no azimuthal component, because the density is
spherically symmetric.
$\partial f/\partial {\vec v}$ is not changed by the velocity
reversal, except for its azimuthal component, so $\nabla \Phi \cdot
(\partial f / \partial {\vec v}\,)$ is unaffected by the velocity
reversals. Similarly,
$\partial f / \partial {\vec r}$ has no azimuthal component, while it
is only the azimuthal component of
${\vec v}$ which changes, so ${\vec v} \cdot (\partial f /\partial
{\vec r}\,)$ is likewise unaffected. Hence, if
$\partial f /\partial t = 0$ before the reversal,
it is still so afterwards. It is even more remarkable that the cluster
remains spherically symmetric despite having a large angular
momentum. Only on the longer collisional time scale
$t_{r}$ will
$f$ change, and the cluster shape will flatten. Recall that for
elliptical galaxies
$t_{r}$ is extremely long. This has led to the suggestion that
the shapes of elliptical galaxies, in contrast to those of stars,
planets, and globular clusters, may not be simply related to their
angular momenta. Another example is a disc-like (but non-rotating)
distribution of stars with a large isotropic velocity dispersion in
the disc plane, but negligible velocities perpendicular to its plane.
There may also be bar-like structures with one-dimensional velocity
dispersions, and superpositions of two or three orthogonal bars or discs.
When violent relaxation occurs its consequences are not simple. It is
natural to assume that the distribution function acquires a
fine-grained structure in phase space as a result of rapid
``scrambling,'' and that if one averages over this structure the
resulting coarse-grained distribution function will then satisfy the
Vlasov equation with
$\partial f /\partial t = 0$. This averaging is reminiscent of the
turbulent mixing of two diffusionless fluids, or of Gibbs's
explanation of the entropy of mixing. Because
no thermodynamic principle determines the end point of violent
relaxation, it is not possible to specify it in advance. Violent
relaxation is therefore very different from collisional relaxation,
which we believe rapidly leads to a Maxwellian distribution. The latter
belief is founded on the knowledge that a Maxwellian is a stationary
solution to the Boltzmann equation, and is the solution of
highest entropy. Except in pathological cases (for example, when
one degree of freedom is completely uncoupled) it is probably the
only stationary solution. No such governing principle is
applicable to violent relaxation. Fortunately, it is rapid, and
therefore is feasible to calculate numerically; some results
are reviewed by Lightman and Shapiro. One consequence of the
absence of a thermodynamic principle is that the endpoint may
preserve a memory of the initial conditions in complex and
subtle ways. Observation of a system which has undergone violent,
but not collisional, relaxation, such as an elliptical galaxy,
may give interesting information about its formation.
The ratio of the dynamical to the collisional time scale may be
calculated from (2.4.1) and (2.4.2):
$${t_{d} \over t_{r}} \approx {26 \log_{10}(0.4N) \over N}.\eqno(2.4.4)$$
The number of stars
$N$ is a measure of the smoothness of the gravitational potential,
and of the validity of the separation of violent
(collective) and collisional (microscopic) relaxation time
scales. In order that this separation be valid it is necessary
that
$N \gg 100$. Clusters of fewer members are best regarded as
complex multiple stars. This includes many open clusters and
clusters of galaxies (where the fact that galaxies are not point
masses adds a further complication). Clusters of many members
have a clear separation of time scales, which permits the
simplification of calculating each process separately. Once
violent relaxation is over, it is possible to calculate the
effects of collisional relaxation by assuming that the particles
move in a static potential, in orbits of nearly constant energy
(and nearly constant angular momentum, if the potential is azimuthally
symmetric).
Because direct measurements of the velocities of individual stars
in globular clusters are difficult, most of our
understanding of the structure and evolution of clusters is based
on theory. Although the theory may be, and has been, tested
against counts of stars, measurements of stellar velocities, or the
distribution of light in clusters, they do not test it completely.
Like all theoretical ``understanding'' of physical phenomena, it
is subject to later empirical revision. It may be more secure
than our similar ``understanding'' of stellar structure and
evolution, because cluster dynamics is founded only upon
Newtonian mechanics. Take this last sentence skeptically; I
might not have written it had I not known of the unexpected
result of the solar neutrino experiment.
The innermost core region of a globular cluster is isothermal,
because the stars within it undergo rapid relaxation of their
distribution function and are spatially mixed. Because the density
is highest here, relaxation is even more rapid than indicated by
(2.4.1), which is an average. Poisson's equation for the
gravitational potential
$\Phi$ in such a spherically symmetric region is
$${1 \over r^{2}}{d \over dr} \left( r^{2} {d\Phi \over dr} \right)
= 4 \pi G \rho . \eqno(2.4.5)$$
Using a thermal velocity
$v_{th}$ to describe the stellar velocity distribution, the
equilibrium density
$\rho(r)$ may be written
$$\rho(r) = \rho_{0} \exp \bigl(-\Phi(r)/v_{th}^{2}\bigr).\eqno(2.4.6)$$
Then define the dimensionless potential $\varphi$ and radius $\xi$
$$\eqalignno{\varphi&\equiv\Phi /v_{th}^{2}&(2.4.7)\cr
\xi&\equiv r \sqrt{4 \pi \rho_{0}G/v_{th}^{2}},&(2.4.8)\cr}$$
and use (2.4.6) to rewrite (2.4.5):
$${1 \over \xi^{2}}{d \over d\xi}\left(\xi^{2}{d\varphi \over d\xi}
\right) = \exp (-\varphi) . \eqno(2.4.9)$$
Equation (2.4.9) describes the isothermal core region of all relaxed
star clusters, and need only be numerically integrated once, just like
the Lane-Emden equation for polytropes of a given polytropic index
({\bf 1.10}). Its derivation would take exactly the same form for an
isothermal collisional gas in hydrostatic equilibrium, so that
it is also the Lane-Emden equation for an isothermal star.
>From Earth we cannot directly measure the density
$\rho(r)$. We can measure the density
$\varrho (r)$ projected along a line of sight which passes a distance
$r$ from the cluster center. By elementary geometry
$$\varrho(r) = 2 \int_{r}^{\infty} \rho(r^{\prime}) { r^{\prime}\ d
r^{\prime} \over \sqrt{r^{\prime\, 2} - r^{2}}} . \eqno(2.4.10)$$
Given a tabulated
$\rho(r)$ the calculation of
$\varrho (r)$ is simple. Less easy is the inverse problem,
of inverting an observed
$\varrho (r)$ to obtain
$\rho (r)$. This requires the solution of (2.4.10) as an integral
equation for the unknown
$\rho (r)$. If one replaces the integral by a finite sum then
$\rho (r)$ (defined only at a finite number of points, as
$\varrho (r)$ is observed) may be determined from a set of linear
equations, solved by matrix inversion. Unfortunately, inverting
a matrix of data, containing observational error, is much more
treacherous than inverting a matrix of exactly known numbers.
Observations of the inner parts of globular clusters
are at least consistent with the isothermal core model.
Outside the isothermal core is a region known as the halo. This
is roughly defined by the condition that collisional relaxation
is very slow, because the density is lower than in the core. Two
kinds of stellar orbits should be distinguished. Those of high
angular momentum never enter the core, and play no part in its
dynamical evolution. Stars may have entered such orbits at an
earlier epoch in the evolution of the cluster, or been born in them.
Orbits of low angular momentum enter the core, and undergo dynamical
relaxation there, though most of life of a star in such an orbit is
spent in the halo. These orbits may be considered to be the
highest energy part of the core's distribution function, having
nearly enough energy to escape the cluster entirely. The mass
contained in the halo is not large, so that the potential may be
considered to vary as $1/r$.
Stars in halo orbits have energies negative in sign, but of small
absolute magnitude. As the cluster core contracts, stars diffuse
through this region of near-zero energy, with a net flux towards
$E = 0$. At
$E = 0$ there is a sink in energy space, as stars freely escape the
cluster. The density of stars in nearly-zero
energy orbits may be estimated roughly but simply (following
Lightman and Shapiro). For an isotropic distribution function the
Fokker-Planck equation (2.1.10), in which the variables are $t$,
${\vec x}$, and ${\vec p}$, may be transformed by a change of variable
into a simpler Fokker-Planck equation in which the variables are $t$,
$r$, and $E$. The diffusion coefficient in energy varies
smoothly through $E = 0$, and may be regarded as nearly constant in a
small interval around $E = 0$. There is nothing ``special'' about this
energy when a star is in the cluster core, where nearly all its
dynamical relaxation takes place; relaxation depends on the kinetic
energy $E - m\Phi$. The ``specialness'' is only present
in the behavior of the potential as $r \rightarrow \infty$.
The diffusion term in the Fokker-Planck equation
dominates the dynamical friction term because of the abrupt cutoff
of the distribution function at
$E = 0$; near such a cutoff second derivatives are much larger
than first derivatives. Therefore, an energy-independent flux of stars
toward zero energy and escape requires that in the cluster core $\partial
f / \partial E$ be constant for small $\vert E \vert$, or
$$f(E) \propto - E . \eqno(2.4.11)$$
A star with energy
$E$ has an orbit which extends to a radius $r \propto \vert E \vert
^{-1}$, and which has a period proportional to $r^{3/2}$ or
$\vert E \vert^{-3/2}$, for the orbit outside of the cluster core is
nearly Keplerian. All low angular momentum halo stars move through
the cluster core at nearly the same speed, so the fraction of
their lifetime spent in the core is proportional to
$\vert E \vert ^{3/2}$. In order to calculate the total number of stars
per unit energy $N(E)$ the cluster contains, the density in the core
$f(E)$ must be divided by the fraction of its life a star spends in the
core, so that (2.4.11) implies
$$N(E) \propto \vert E \vert^{-1/2} . \eqno(2.4.12)$$
We want $N(r)$, the number of stars whose orbits extend to radius
$r$, per unit radius. Most of the stars found at
$r$ have orbits whose major axes are comparable to
$r$. Then
$$N(r)\, dr = N(E) {dE \over dr} dr \propto \vert E \vert^{3/2}\, dr
\propto r^{-3/2}\, dr . \eqno(2.4.13)$$
These stars are spread over a volume
$\sim 4 \pi r^{2}\, dr$, so their volume density
$n(r)$ is
$$n(r) \equiv {N(r)\, dr \over 4 \pi r^{2}\, dr} \propto r^{-7/2} .
\eqno(2.4.14)$$
Observation agrees with this theoretical estimate. This confirms the
assumption that most of the stars in the halo have low angular momentum
orbits, and enter these orbits as a result of the diffusion of core
stars to zero energy and escape. Higher angular momentum stars
would not undergo significant dynamical evolution, and their density
would depend on their orbital parameters when formed; they need not
follow (2.4.14).
The halo is cut off at a finite radius by the tidal effects of
the Galactic gravitational field. Beyond that radius, the only
stars are those which are freely escaping, or background
(``field'') stars accidentally encountering the cluster.
As stars escape the cluster the total binding energy of those left
behind must increase. Because the mass left behind is
decreasing, it must contract into an ever-smaller volume, at
ever-increasing density. A proper calculation of this process
requires integrating a Fokker-Planck equation, but a much simpler
calculation reveals its qualitative nature. Assume that the
escaping stars carry away zero energy; this approximation is
rather good, and leads only to a slight underestimate of the rate
of cluster evolution. Use (2.4.1) to estimate the cluster
relaxation time, ignoring the variation of the logarithm, and use
the virial theorem to estimate the velocity dispersion.
Conservation of total cluster energy implies
$$R \propto N^{2} , \eqno(2.4.15)$$
while the virial theorem implies
$$v^{2} \propto {N \over R} \propto N^{-1} . \eqno(2.4.16)$$
Then the variation of
$N$ is given by
$${dN \over dt} \sim {N \over t_{r}} \propto N^{-5/2} . \eqno(2.4.17)$$
This may be integrated to give
$${N \over N_{0}}=\left(1 - {t \over t_{0}}\right)^{2/7}.\eqno(2.4.18)$$
The cluster evaporates entirely in a finite time $t_{0}$, except for an
infinitesimal fraction of its mass which contains the full initial
binding energy. More detailed calculations show that
$t_{0}$ is typically 10 to 30 times the initial relaxation time $t_{r}$.
In the course of this evaporation the density and velocity
dispersion diverge as the radius contracts to zero.
This result is startling and intriguing, particularly because
estimates of core relaxation times in observed globular clusters
are as low, in some cases, as
$10^{7}$ years. These clusters should collapse soon, and,
unless we are at a ``special'' and preferred cosmological moment,
others have collapsed in the past. What actually happens, and
what does the remnant look like afterward?
One obvious oversimplification we have made has been to treat the
integer
$N$ as a continuous variable. If a cluster is reduced to a
single very tightly bound binary star
$(N = 2)$, evolution stops. To absorb all the binding energy
of a cluster of
$N$ stars, such a binary would have to have an orbit whose size is
$\sim N^{-2}$ times the initial core radius. This is not possible
for ordinary stars, which are too large, but a binary made of
degenerate stars or black holes could be sufficiently compact.
Even ordinary binaries supply energy to a cluster as their binding
energy grows. The presence of a small number of binaries may inject
enough energy to slow or reverse core collapse.
Another oversimplification has been to ignore the finite sizes of the
stars. When a cluster contracts sufficiently a significant number of
collisions may occur. Their effect is complex, for many
processes are possible: partial or complete disruption of the stars,
re-accretion of disrupted material by stars, or stellar coalescence.
The qualitative effect of these collisions is probably to accelerate
the cluster evolution and to produce a small number of more massive
stars. These rapidly evolve to supernova explosion or collapse. If
these processes are important, a cluster core may ultimately either
destroy itself as its stars explode, or leave behind a single black hole.
Such processes taking place in the nuclei of galaxies
may be the genesis of quasars.
The endpoint of globular cluster evolution remains controversial.
There is no clear evidence for any relics of core collapse,
suggesting either that the relics are not recognizable as
globular clusters, or that they are recognized as globular clusters and
are not obviously distinguishable from clusters which have not undergone
core collapse. The former
possibility seems unlikely, for there should be some halo stars
in orbits of high angular momentum. These stars never
enter the core, and suffer no dynamical evolution, regardless of
what happens in the core. They would remain behind as a
dilute globular cluster, with little central condensation. It may be
that many of the globular clusters we observe are such relics, and that
their masses were once much larger than they are today.
\goodbreak
\vskip 2\baselineskip plus2\baselineskip
\goodbreak
\noindent
{\bf 2.5 Nonthermal Particle Acceleration}
\vskip \baselineskip
\noindent
{\bf 2.5.1} \us{Spectral Shapes} \quad If a population of classical
particles (or photons) has relaxed to thermodynamic equilibrium at
temperature $T$, then the probability $n(E)$ that a state of energy
$E$ will be occupied is proportional to the Boltzmann factor
$\exp (-E/k_{B}T)$. At high particle density the effects of quantum
statistics become significant, and this exponential is replaced by
$\lbrack \exp (a+E/k_{B}T) \pm 1 \rbrack^{-1}$, where $-ak_{B}T$ is
the chemical potential, and $+1$ applies to fermions and $-1$ to bosons;
in this discussion we need consider only the classical Boltzmann factor.
The rate of emission $F(\nu)$ of photons of frequency $\nu$ from such
an equilibrium gas will be of the form
$$F(\nu) = \int d^{3}p\ n(E) {\cal F}(\nu , E), \eqno(2.5.1)$$
where ${\cal F}$ is the emission rate per unit frequency interval of
photons of frequency $\nu$ by a particle of energy $E$. For emission
by interactions between two particles $n$ is the density of particle
pairs (analogous to $f_{2}$ defined in {\bf 2.1}) whose relative
motion has kinetic energy $E$. For continuum emission ${\cal F}(\nu,E)$
will be a smoothly varying function of $\nu$, dropping to zero at
$h\nu = E$. A very rough approximation to $F(\nu)$ may be obtained by
writing
$${\cal F}(\nu ,E) \sim \cases{{\cal F}_{0},&for $h\nu\leq E$;\cr
0,&for $h\nu > E$.\cr}\eqno(2.5.2)$$
Then, using the nonrelativistic relation between $p$ and $E$
(equilibrium is rarely achieved at relativistic temperatures),
$$\eqalign{F(\nu)&\sim {\cal F}_{0} \int_{E=h\nu}^{\infty}d^{3}p\ \exp
(-E/k_{B}T)\cr&\sim {\cal F}_{0} \int_{h\nu}^{\infty}\sqrt{E}\ dE\ \exp
(-E/k_{B}T)\cr&\sim {\cal F}_{0}\exp (-h\nu / k_{B}T);\cr}
\eqno(2.5.3)$$
in the same spirit as (2.5.2) slowly varying factors have been ignored.
If ${\cal F}(\nu ,E)$ consists of narrow spectral lines (2.5.3) is the
envelope of a series of narrow spikes corresponding to the spectral
lines.
\def\rightheadline{\vbox{\vskip 0.125truein \line{\tenbf\hfil Nonthermal
Particle Acceleration \qquad\folio}}}
The exponential dependence of $n(E)$ on $E$ and $T$ produces a spectrum
$F(\nu) \sim \exp (-h\nu /k_{B}T)$. If $h\nu \gapp k_{B}T$ this
exponential usually gives the dominant frequency dependence of
$F(\nu)$ (other than the shapes of individual spectral lines, if
present), because it is usually a much more sensitive function of $\nu$
than any dependence of ${\cal F}(\nu ,E)$ on $\nu$ which we neglected in
(2.5.2). Radiation produced by matter in thermal equilibrium almost
invariably has the exponential frequency dependence of (2.5.3) at
frequencies $\nu \gapp k_{B}T/h$. At lower frequencies $F(\nu)$
depends more on the form of ${\cal F}(\nu ,E)$ and on the optical
depth of the source, approaching the Planck function (1.7.13) if the
optical depth is high.
When plotted on a log-log plot, (2.5.3) has a characteristic curved
shape, and defines the characteristic energy $k_{B}T$ (Figure 2.1).
Many astronomical objects have spectra like this, and it is often
possible to recognize the shape (2.5.3) and to estimate $k_{B}T$ at a
glance, even from poor quality data.
\topinsert
\vskip 10.5truecm
\ctrline{{\bf Figure 2.1.} Spectral shapes.}
\endinsert
Other astronomical objects have spectra which are nearly straight lines
on a log-log plot, corresponding to power law spectra
$$F(\nu) \propto \nu^{-s} , \eqno(2.5.4)$$
where the spectral index $s$ is nearly constant. Such a spectrum
(and only such a spectrum) defines {\it no} characteristic frequency,
and hence no characteristic energy of the radiating particles. An
example is shown in Figure 2.1.
A power law spectrum of the form (2.5.4) cannot be extrapolated
indefinitely to both high and low frequency, because if it were the
integrated power $\int F(\nu)\, d\nu$ would diverge. There must be
at least one change in $s$, and the frequency $\nu_{0}$ at which
it occurs defines a characteristic energy $h\nu_{0}$. However, a
power law spectrum is often found over several decades of frequency,
which indicates that over a large range in energy of the radiating
particles their distribution function $n(E)$ also has no characteristic
energy. Such a $n(E)$ is also a power law, and may have this form over
a very wide range in energy, but similarly must have at least one break
in its slope in order that the integrated number density and energy
density be finite.
Bodies in thermal equilibrium may produce power law spectra at
frequencies for which $h\nu \ll k_{B}T$. At low energies $n(E)$ is
nearly constant (or proportional to $E$ for bosons with zero chemical
potential), which are (uninteresting) examples of distributions
with no characteristic energy. The simplest and most familiar example
of a thermal equilibrium power law spectrum is that of a black body at
$h\nu \ll k_{B}T$, called a Jeans spectrum, for which $s = -2$;
the power law slope is broken at $h\nu \approx k_{B}T$. Optically
thin thermal emitters produce power law spectra for $h\nu \ll k_{B}T$
if their emissivity integrated over the distribution function
is a power law function of $\nu$; bremsstrahlung
({\bf 2.6.1}) is a familiar example. Thermal emitters with power law
distributions of temperature also produce power law spectra, as
discussed in {\bf 3.6} for accretion discs. In this case the power
law slope is broken at $h\nu \approx k_{B}T_{c}$, where $T_{c}$ is a
characteristic temperature (typically a break in the distribution of
temperature).
The most interesting sources of power law spectra are those in which
there is no characteristic energy because the particles are {\it not}
in thermal equilibrium, but instead $n(E)$ is a power law over a wide
range of energies. Usually these are very relativistic energies \hbox
{$E \gg mc^{2}$}; the rest mass is another characteristic energy and will
generally interrupt both a power law $n(E)$ and the power law spectrum
it radiates. It is equally possible to have a power law $n(E)$ for \hbox
{$E \ll mc^{2}$}, but such lower energy nonthermal particles are usually
less interesting as sources of astronomical radiation; they are
significant in laboratory and astrophysical plasma physics. The
presence of power law $n(E)$ are inferred from observations of power
law $F(\nu)$ in quasars, active galactic nuclei, extragalactic radio
sources, many Galactic radio sources (particularly those associated
with supernova remnants), and a variety of compact objects. A power law
$n(E)$ is directly observed for cosmic rays, with energies extending
to\break $\gapp 10^{19}$ eV.
\goodbreak
\vskip 2\baselineskip plus2\baselineskip
\goodbreak
\noindent
{\bf 2.5.2} \us{Particle Acceleration} \quad In order to understand
the origin of a nonequilibrium particle distribution function $n(E)$
it is necessary to understand the processes by which particles gain
and lose energy. In {\bf 2.2} and {\bf 2.3} we considered the
relaxation towards thermal equilibrium of distributions of charged
particles and photons in contact with a heat bath. These processes
cannot produce power law distributions of very energetic particles,
because their energies exceed the thermal energy per particle of any
conceivable heat bath. We are here concerned with processes which can
accelerate energetic particles as the consequence of many small
increments to their energy.
Fermi (1949) suggested a mechanism for the acceleration of energetic
cosmic rays (Longair 1981); it may be applied to particles scattering
within any
dilute moving medium, such as discs ({\bf 3.6}) or accretion flows
({\bf 3.7}). This mechanism is called stochastic acceleration because it
involves numerous uncorrelated random events. Particles are accelerated
as a result of their scattering by moving massive objects. In the
interstellar medium these objects could be interstellar clouds or
magnetohydrodynamic waves; charged particles are elastically scattered by
clouds by being deflected by their magnetic fields. This process may
be thought of as the relaxation of the particles and the clouds
(considered as single very massive particles) towards thermal
equilibrium. Because the kinetic energies of the clouds are enormous
($\sim 10^{45}$ erg for plausible interstellar clouds, corresponding
to $T \sim 10^{61\ \circ}$K), equilibrium can never be reached, but
there will be a net energy flow from the kinetic energy of the clouds
to that of relativistic particles. There is, in principle, no
characteristic energy scale between the particle rest mass energy
$mc^{2}$ and the kinetic energy of an entire cloud, so that a power law
distribution of particle energy may be achieved over a very large
range in energy (in practice, various energy loss mechanisms have
thresholds which may serve to define characteristic energies).
A relativistic particle of energy $E$ elastically scattered by an
object of velocity $v \ll c$ suffers an energy change
$$\Delta E \sim {\cal O}\left(E{v \over c}\right). \eqno(2.5.5)$$
This change in energy is as likely to be positive as negative, for
isotropically distributed cloud velocities. There will, however, be a
mean positive energy shift
$$\langle \Delta E \rangle \sim {\cal O}\left( E {v^{2} \over c^{2}}
\right) . \eqno(2.5.6)$$
The numerical coefficient in (2.5.6) is of order unity but depends on
the detailed geometry and kinematics of the scattering, which depend on
the (unknown) details of cloud magnetic field geometry. This energy
transfer process is analogous to Comptonization ({\bf 2.3}) in the
soft photon limit, with clouds
instead of electrons and the relativistic particles instead of low
energy photons; the rate of energy transfer from the electrons (clouds)
to the photons (particles) is proportional to the product of the photon
(particle) energy density and the electron (cloud) kinetic energy
density (2.3.31).
After ${\cal N} \gg 1$ scatterings a relativistic particle whose
initial energy was $E_{0}$ has an energy of approximately
$$E \approx E_{0} \exp(\beta_{s}^{2}{\cal N}) , \eqno(2.5.7)$$
where the parameter $\beta_{s} \approx v/c$. If $E \gg E_{0}$ the
dispersion in $\ln (E/E_{0})$ will be less than its mean value, so it is
reasonable to take the mean value and ignore the dispersion, as in
(2.5.7).
Particles will not gain energy forever. Eventually losses will become
significant. Consider sudden random catastrophic losses, such as
escape from the region in which acceleration takes place, and (for
strongly interacting particles) nuclear collisions in which most of
their energy is lost. It the characteristic loss time is $T$,
independent of energy, then the fraction of particles surviving for
times between $t$ and $t+dt$ is
$$N(t)\ dt = {\exp (-t/T) \over T} dt . \eqno(2.5.8)$$
If the mean rate of scattering of particles by clouds is $1/\tau$, then
${\cal N} \approx t/\tau$ (which holds accurately for $t \gg \tau$),
and after a time $t$ a particle has the energy
$$E(t) \approx E_{0} \exp (\beta_{s}^{2} t /\tau) . \eqno(2.5.9)$$
The energy distribution is found
$$\eqalign{N(E)\ dE &=N(t){dt \over dE}dE\cr &\approx {\exp (-t/T) \over
T}{\tau \over \beta_{s}^{2}E}dE . \cr}\eqno(2.5.10)$$
Using (2.5.9) to express $t$ in terms of $E$ yields
$$N(E) \approx {\tau \over \beta_{s}^{2}TE_{0}}\left({E \over E_{0}}
\right)^{-(1+\tau /\beta_{s}^{2}T)} . \eqno(2.5.11)$$
This is a power law distribution, and has the spectral index \hbox{$p=1+
\tau / \beta_{s}^{2}T$}. $N(E)$ is the total number of particles per
unit energy interval; for relativistic particles or photons $N(E)
\propto E^{2}n(E)$.
The absence of a characteristic energy, as expressed in the
exponential functions of (2.5.8) and (2.5.9), makes the power law form
inevitable. A very similar derivation applies to soft photon
Comptonization ({\bf 2.3}) in a finite volume, in which escape from the
volume is the loss process, and power laws are similarly predicted.
In problems of interest the spectral index $p$ is of order unity;
for cosmic rays it is observed to be close to 2.6, and for relativistic
electrons in sources of synchrotron radiation it is typically between
2 and 3. There is probably no deep significance to the fact that $p$
is generally found in such a narrow range. For $p \leq 2$ the total
energy density diverges as $E \rightarrow \infty$, so values smaller
than 2 are unlikely to describe $N(E)$ over a very wide range of energy.
For $p \gapp 3$ the energy contained in very energetic particles is
very small; the high energy parts of such steep distribution functions
therefore produce little radiation, which is less likely to be
detected than that produced by regions or objects in which $p$ is
smaller.
There is no obvious {\it a priori} reason why the parameters $\tau$,
$\beta_{s}$, and $T$ should be as closely constrained as is $p$. For
cosmic rays it is known from studies of their nuclear abundances that $T
\sim 10^{7}$ yr, while from observations of the interstellar medium
$\beta_{s}\approx 3\times 10^{-5}$; $\tau$ depends on the microstructure
of interstellar magnetic fields and can only be guessed at. For regions
in which energetic electrons are accelerated all these parameters are
very uncertain. The likely explanation is that the energetics of the
accelerating process is self-regulating. The mean energy per particle
is $E_{0}(p-1)/(p-2)$. As $p \rightarrow 2$ the power required to
accelerate the energetic particles diverges, so particle acceleration
becomes a strong damper on the cloud motions or waves which drive it.
As $p \rightarrow 3$ the power required to accelerate the particles
becomes a small multiple of the power required to inject them at energy
$E_{0}$. $\beta_{s}$ and $\tau$ (which depend on the detailed
distribution of fluid velocity, wave amplitude, and magnetic field) may
adjust themselves so that the power supplied to the energetic particles
equals that supplied to the accelerating clouds or waves (for cosmic
rays in our Galaxy probably the kinetic energy of supernovae). This
naturally leads to $2 < p \lapp 3$ for a wide range of parameters.
We have so far assumed an energy change quadratic in $\beta_{s} \ll 1$.
A particle may occasionally be trapped between two converging reflectors.
Then the energy change on each scattering is positive, and the mean
energy increment is linear in $\beta_s$. In a chaotic flow periods when
a particle is trapped between converging reflectors are balanced, in
part, by those when it is trapped between diverging reflectors (but
not completely balanced, because as reflectors converge the mean time
between scatterings becomes progressively smaller, while as they
diverge it becomes larger). In an ordered converging flow the particles
will always be in a region of convergence, in which case $\beta_{s}^{2}$
should be replaced by $\beta_{s}$ in the previous results. This leads
to much more rapid acceleration. An example of such a converging flow
is that around a shock ({\bf 3.3}); if the fluid on each side contains
magnetic disturbances capable of scattering the particles they will
gain energy rapidly. Shocks are therefore promising locations for
particle acceleration. It may even be that much of the kinetic energy
released in some shocks appears in accelerated high energy particles
rather than the internal energy of the shocked fluid.
Fermi acceleration is capable of accelerating high energy particles
to yet higher energy, but does not answer the question of their
initial acceleration. The theory is applicable to nonrelativistic
particles, if their speed $v_{p}$ is used in place of $c$, but at
low energy their rate of slowing by Coulomb drag ({\bf 2.2}) exceeds
their rate of acceleration. Approximately, the loss rates by Coulomb
drag, hard nuclear collisions, and escape may be added to give $T^{-1}$;
if $T \sim \tau / \beta_{s}^{2}$ at relativistic energies, as must be
the case, then for $E \ll mc^{2}$ Coulomb drag will probably overwhelm
any acceleration. However, it is conceivable that if there are no
relativistic particles $\tau /\beta_s^2$ will decrease until it equals
the Coulomb slowing time for a few of the fastest thermal particles; once
these particles are accelerated to relativistic energy it will increase
again as they rapidly gain energy and
the energy balance regulates itself. If this does not occur
(and there is no evidence for it), there is a large gap in energy
between thermal particles and those moving fast enough for Fermi
acceleration to be effective.
There is another process, analogous to Fermi acceleration but
involving plasma oscillations instead of magnetized gas clouds or
hydromagnetic waves, which may be capable of bridging the energy gap.
This process draws energy from the plasma waves and is called Landau
damping, but because it accelerates particles it may also be called
Landau acceleration. Its essential property is that it is resonant,
meaning that a given plasma wave can accelerate only particles with a
very narrow range of velocity. As a result, it may be
effective in accelerating low energy particles, where rapid
acceleration is necessary to overcome Coulomb drag, without an
excessive amount of energy being drained from the plasma waves by the
further acceleration
of particles which are already relativistic. The theory of Landau
damping is subtle, but is explained in innumerable books on plasma
physics; see, for example, Nicholson (1983). Here we present only
a simple and elementary argument.
Consider a particle of charge $e$ and velocity $v$ suddenly placed in
an electric field
$$E(x,t) = E_{0}\cos (kx-\omega t) . \eqno(2.5.12)$$
We ignore the subtlety of the problem by replacing a careful
investigation of the initial conditions with a ``suddenly.'' The
particle is accelerated
$$m{dv \over dt} = eE_{0} \cos (kx-\omega t) . \eqno(2.5.13)$$
If $E$ is in the $x$-direction (a longitudinal wave) $x \sim \int v\,
dt$, and it is necessary to
expand $v$ in a series in powers of $eE_{0}/m\omega$, keeping terms of
the second order. A similar result is obtained much more simply for
transverse waves. In this case we can write $x=v_{x}t$ and let (2.5.13)
describe the particle motion in the direction of the electric field,
taken to be $z$. Then
$$v_{z} = {eE_{0} \over m (kv_{x}-\omega)}\sin \lbrack (kv_{x} - \omega )
t\rbrack , \eqno(2.5.14)$$
and the particle's transverse motion has a kinetic energy
$${1 \over 2} m v_{z}^{2} = {e^{2}E_{0}^{2} \over 2 m (kv_{x}-\omega )^
{2}}\sin^{2}\lbrack (kv_{x} -\omega )t\rbrack . \eqno(2.5.15)$$
The resonant nature of the acceleration is evident; particles whose
velocity $v_{x}$ is close to the wave phase velocity $\omega /k$ receive
a great deal of energy. We assume nonrelativistic motion, and integrate
(2.5.15) over the velocity distribution function $f(v_{x})$, taken to be
slowly varying near $v_{x} = \omega /k$.
Then the total kinetic energy imparted to the particles by the wave is
$$\hskip .73truein
\eqalign{K&= \int {1 \over 2}mv_{z}^{2}f(v_{x})\ dv_{x} \cr
&\approx {e^{2}E_{0}^{2}f(\omega /k) \over 2m} \int {\sin^{2} \lbrack
(kv_{x} - \omega )t\rbrack \over (kv_{x} - \omega)^{2}}dv_{x}\cr
&\approx {e^{2}E_{0}^{2}f(\omega /k)t\over 2mk} \int_{-\infty}^{\infty}
{\sin^{2}u \over u^{2}}du\cr &\approx {\pi e^{2}E_{0}^{2}f(\omega /k)
t \over 2mk},\cr}\eqno(2.5.16)$$
and the power is
$$P \approx {\pi e^{2}E_{0}^{2}f(\omega /k) \over 2mk} . \eqno(2.5.17)$$
The approximations in these equations approach equality as $t \rightarrow
\infty$ and the integrand becomes sharply peaked.
The calculation for longitudinal waves (Nicholson 1983) shows that
particles with $v_{x} < \omega /k$ gain energy from the wave, while
those with $v_{x} > \omega /k$ give up energy to it. The total power
supplied to the particles resembles (2.5.17), but in place of
$f(\omega /k)$ there is $(\omega /k)f^{\prime}(\omega /k)$ (which is
generally of the same order of magnitude).
Particles moving with the phase velocity of plasma waves will be
accelerated by the waves. Because this acceleration is random in
direction and sign (depending on phase and polarization
for transverse waves, and phase and the sign of $\omega /k - v_{x}$
for longitudinal waves), particles diffuse
in momentum space. This resembles the ${\hbox{\bf b}}$ term in the
Fokker-Planck equation (2.1.10), the $\partial^{2} n / \partial x^{2}$
term in the Kompaneets equation (2.3.21), and the scattering of
relativistic particles by moving magnetized clouds or hydromagnetic
waves (2.5.7). If the plasma waves are only excited to a thermal level
(an average of ${1 \over 2}k_{B}T$ per degree of freedom) this
diffusion is only a small increment to the diffusion (2.2.17) produced
by encounters between the particles, and is exactly cancelled by an
additional drag term resulting from the emission of plasma waves by
particles. However, if plasma waves are excited to a high intensity
by a plasma instability, the rate of momentum diffusion may be large and
the rate of acceleration rapid. A variety of plasma waves exist with
a broad range of phase velocities and may be excited to high intensity,
so it may be possible by this mechanism to accelerate particles from
the high velocity tail of a Maxwellian distribution to much higher
(possibly even relativistic)
velocities. This phenomenon is observed in many laboratory plasma
experiments, and is a frequent consequence of plasma instability.
It is also possible to accelerate energetic particles with a single
large potential drop, without the necessity of their diffusing in
energy. This is believed to occur in pulsars ({\bf 4.4}). It may
not occur elsewhere, because astrophysical plasmas are usually good
conductors, and it is not easy to produce large potential drops,
although they may occur transiently as a consequence of complex
magnetohydrodynamic flows called ``field reconnection,'' in which
current densities are high and scattering by plasma waves impedes the
flow of current and leads to a temporarily large resistivity.
Acceleration by a single potential drop does not lead to the observed
power law particle distribution functions, but the effects of many
potential drops, with a power law distribution of accelerating
potentials, may.
Colgate and Johnson (1960) suggested that purely hydrodynamic processes
could accelerate power law distributions of energetic particles. As a
shock travels into a medium of progressively decreasing density (the
atmosphere of an exploding star {\bf 4.3}, or an interstellar density
gradient) the velocity
and energy (per particle) of the shocked fluid progressively increase.
Although locally it may be in thermal equilibrium, the overall
distribution of kinetic energy per particle approximates a power law.
This mechanism has difficulty explaining some of the observed
properties of cosmic rays, for which it was proposed, but it may be
capable of providing the initial acceleration across the velocity
range in which Coulomb drag is important.
A great wealth of detailed mechanisms for particle acceleration have
been discussed (Arons, {\it et al.} 1979); it is likely that different
ones are important in different objects. All of them depend on the
detailed distribution of fluid velocity, magnetic field, or plasma
waves. For this reason acceleration is difficult to calculate
even in laboratory and Solar System plasmas, where direct probes and
abundant data exist. In a remote astronomical object the problem is
much harder.
Phenomena involving nonthermal particles pose the most difficult
problems in astrophysics. They are nearly ubiquitous, and are observed
in the solar photosphere as a consequence of turbulent convection, in
planetary magnetospheres as a consequence of interaction with the Solar
wind, as well as in more exotic and distant places.
Much of the power of some pulsars, quasars, and $\gamma$-ray sources
appears as energetic particles, but there is no
quantitative understanding or theory. Although it is not possible to
predict the properties of nonthermal phenomena, the astrophysicist
should not be surprised by their appearance. They are likely to be
found wherever fluid motions are available as a source of free energy,
whether turbulent or ordered, as in accretion, shocks, or
the differential rotation of a disc.
\goodbreak
\vskip 2\baselineskip plus2\baselineskip
\goodbreak
\noindent
{\bf 2.6 Radiation Processes}
\vskip \baselineskip
\noindent
Matter absorbs, emits, and scatters radiation by a variety of processes.
These are usually the chief mechanisms by which radiation and matter
relax towards thermal equilibrium. Nearly all our knowledge of
astronomical objects is obtained from the study of their radiation,
and their properties and structure usually are determined by the
emission and absorption of radiation within them. Fortunately, most
radiation processes are well understood. Their calculation is often
tedious or difficult, but quantitative results are available. In this
section I present a few simple results for important processes. I have
drawn upon the book of Rybicki and Lightman (1979), which contains
more complete discussions and derivations of most of these results,
and references to the literature.
\def\rightheadline{\vbox{\vskip 0.125truein \line{\tenbf\hfil Radiation
Processes \qquad\folio}}}
\goodbreak
\vskip 2\baselineskip plus2\baselineskip
\goodbreak
\noindent
{\bf 2.6.1} \us{Bremsstrahlung} \quad When a charged particle is
accelerated it radiates. An important example is the acceleration
of an electron by a Coulomb field, in which case the
radiation is called bremsstrahlung. In the nonrelativistic
limit the instantaneous power radiated by a system of charges is
$$P = {2 {\ddot d}^{2} \over 3 c^{3}},\eqno(2.6.1)$$
where $d$ is any one component of the electric dipole moment ${\vec d}$
of the system; contributions from each component are added. In this
limit the acceleration of an electron by another electron produces no
radiation, because the opposite accelerations of the two electrons
produce no net change in ${\vec d}$ (At relativistic energies
electron-electron bremsstrahlung is comparable to electron-ion
bremsstrahlung). The spectral density radiated in a single encounter is
$${dW \over d\omega} = {8 \pi \omega^{4} \over 3 c^{3}} \bigl\vert {\hat
d}(\omega )\bigr\vert^{2} , \eqno(2.6.2)$$
where ${\hat d}$ is the Fourier transform of $d$:
$${\hat d}(\omega ) = {1 \over 2 \pi} \int_{-\infty}^{\infty}
e^{i\omega t}d(t)\ dt . \eqno(2.6.3)$$
A quantitative calculation of bremsstrahlung is lengthy (and must be
performed quantum mechanically). Instead, we use these classical
expressions and make rough approximations in order to provide an
illustriative guide. The dipole moment ${\vec d}$ represented by
an accelerated electron of charge $-e$ at position ${\vec r}$ is
$${\vec d} = -e{\vec r} . \eqno(2.6.4)$$
The fixed origin of coordinates is irrelevant, because we are only
interested in time derivatives of $d$, but may be conveniently
taken at the ion of charge $Ze$ from which the electron scatters, or
the center of mass of the two particles. Taking two time derivatives
of (2.6.4) and considering only one component gives
$${\ddot d} = -e {\dot v}. \eqno(2.6.5)$$
The Fourier transform of (2.6.5) is
$$\int e^{i \omega t}{\ddot d}\ dt = -e\int e^{i \omega t}{\dot v}\
dt . \eqno(2.6.6)$$
Integrating the left hand side by parts twice, and using (2.6.3),
yields
$$\omega^{2} {\hat d}(\omega ) = {e \over 2\pi}\int e^{i\omega
t}{\dot v}\ dt . \eqno(2.6.7)$$
Now consider an electron approaching an ion at speed $v$ and impact
parameter $b$. If its deflection is not large we may calculate the
force on and acceleration of the electron as if its path were a straight
line. Most (55\%) of its radiation occurs during a time interval $\tau
\equiv b/v$ when the electron is within a distance $b\sqrt{5/4}$ of
the ion; outside this interval the power is less than 0.64 of its peak
value. The cumulative impulse and change in velocity $\Delta v$ is
found by integrating along the undeflected trajectory the component
of acceleration normal to it:
$$\eqalign{\Delta v&={-Ze^{2}\over m_e} \int_{-\infty}^{\infty} {b\ dt
\over (b^{2} + v^{2}t^{2})^{3/2}}\cr &=-{2Ze^{2} \over m_{e}bv};\cr}
\eqno(2.6.8)$$
the impulse parallel to the trajectory integrates to zero in this
approximation.
For very high frequencies $\omega\tau \gg 1$ and the exponential in
(2.6.7) has many cycles of oscillation over the range over which
${\dot v}$ varies significantly, so the positive and negative phases
of the integrand nearly cancel. At low frequencies $\omega\tau \ll 1$
and the exponential may be taken to be unity where ${\dot v}$ is
significantly different from zero. Considering only these two limits,
and ignoring the more difficult intermediate regime, gives
$${\hat d}(\omega ) \approx \cases{ \displaystyle
{e\Delta v \over 2\pi\omega^{2}}&
for $\omega\tau\ll 1$;\cr 0& for $\omega\tau\gg 1$.\cr}\eqno(2.6.9)$$
Substitution into (2.6.2) gives
$${dW \over d\omega} \approx\cases{\displaystyle {8Z^{2}e^{6} \over
3 \pi c^{3}m_{e}^{2}b^{2}v^{2}}& for $b \ll v/\omega $;\cr 0& for
$b \gg v/\omega$.\cr}\eqno(2.6.10)$$
It is now necessary to integrate over the distribution of inpact
parameters to obtain the spectral density $dP / d\omega$
radiated by an electron with speed $v$ passing through a gas of ions
of density $n_{i}$:
$$\eqalign{{dP \over d\omega}&= n_{i}v\int_{0}^{\infty}{dW\over d\omega}
2\pi b\ db\cr &\approx {16n_{i}Z^{2}e^{6} \over 3 c^{3}m_{e}^{2}v}
\int_{b_{min}}^{v/\omega}{db \over b} . \cr}\eqno(2.6.11)$$
A lower cutoff $b_{min}$ has been introduced into the integration to
avoid the logarithmic divergence at $b \rightarrow 0$. Physically,
the origin of $b_{min}$ is the breakdown of either the small deflection
approximation for $b \lapp Ze^{2}/m_{e}v^{2}$ or of the classical
approximation for $b \lapp h/m_{e}v$. A more quantitative calculation
must include both these effects, and is rather lengthy. The ratio
between the accurate result and a rough approximation like ours is
called the Gaunt factor $g$. This factor has been calculated in detail,
and is usually of order unity. The spectral density is
$${dP \over d\omega} = {16 \pi n_{i} Z^{2} e^{6} \over 3 \sqrt{3}
m_{e}^{2} v}g(v,\omega) , \eqno(2.6.12)$$
where the extra factor of $\pi /\sqrt{3}$ is required by the standard
definition of $g$.
To compute the total power radiated integrate (2.6.12) over frequency,
cutting off the integration at $\hbar \omega = {1 \over 2}mv^{2}$
because photon energy is quantized (without this cutoff, which is
implicit in $g$, the power would diverge). The result is
$$P_{brems} = {8 \pi n_{i} Z^{2} e^{6} v \langle g \rangle \over 3
\sqrt{3} c^{3} m_{e} \hbar} . \eqno(2.6.13)$$
This power may be compared to the rate at which the electron loses
energy to other electrons by Coulomb drag (using 2.2.9; $n_{e}=Zn_{i}$
and $m_{12}=m_{e}/2$). Their ratio is
$${P_{brems} \over P_{drag}} = {Z \langle g \rangle \over 3 \sqrt{3}
\ln \Lambda}{v^{2} \over c^{2}}{e^{2} \over \hbar c} . \eqno(2.6.14)$$
The last factor is the fine structure constant $\alpha$, and is nearly
equal to 1/137. It is evident that for nonrelativistic electrons (the
only case to which these results are applicable), bremsstrahlung
energy loss is very small compared to Coulomb drag.
Now integrate (2.6.12) over a Maxwellian distribution of electron
velocities. To do this quantitatively requires knowledge of $g(v,
\omega )$; in the result this is absorbed into a new integrated Gaunt
factor $g(T,\omega )$. The integration begins at $v=\sqrt{2\hbar\omega
/m}$, because lower speed electrons cannot produce a photon of frequency
$\omega$. The resulting emissivity is
$$j(T,\omega ) = 7 \times 10^{-38}\ {n_{e}n_{i}Z^{2} \over
T^{1/2}} \exp(-\hbar\omega /k_{B}T)g(T,\omega ){{\rm erg}\over{\rm
cm}^{3}\ {\rm sec}\ {\rm Hz}} . \eqno(2.6.15)$$
The physical origin of the $T^{-1/2}$ factor is the $v^{-1}$ in
(2.6.12), which in turn comes from the square of the $v^{-1}$ in
(2.6.8) and the $v$ in (2.6.11). The characteristic exponential
comes from the Maxwellian distribution function, as discussed in
{\bf 2.5.1}. Because this contains the only strong dependence on
$\omega$ in (2.6.15), the spectrum of thermal bremsstrahlung resembles
the exponential shown in figure 2.1. Such a spectrum is actually
observed from dilute clouds of ionized interstellar gas. Bremsstrahlung
is believed to be important in many more compact objects, but in these
objects the optical depth (1.7.23) is large, re-absorption of the
radiation is important, and the emergent spectrum ({\bf 1.14}) does not
resemble the source function (2.6.15); in the limit of large optical
depth (for example, a star) the emergent spectrum is close to a Planck
function (1.7.13).
\goodbreak
\vskip 2\baselineskip plus2\baselineskip
\goodbreak
\noindent
{\bf 2.6.2} \us{Magnetic Radiation} \quad An electron moving across a
magnetic field ${\vec B}$ is accelerated by the field, and therefore
radiates. This is known as cyclotron emission. The electron follows
a helical path, gyrating about ${\vec B}$ with an angular frequency
$$\omega_{B} = {eB \over \gamma m_{e} c},\eqno(2.6.16)$$
where $\gamma\equiv (1-v^{2}/c^{2})^{-1/2}$, and a radius of gyration
$$r_{g} = {\gamma m_{e} c v_{\perp} \over eB} , \eqno(2.6.17)$$
where $v_{\perp}$ is the magnitude of the component of velocity
perpendicular to ${\vec B}$.
For nonrelativistic motion ($\gamma \rightarrow 1$) the radiated power is
found from (2.6.1). There are two oscillating components of ${\vec d}$,
each varying sinusoidally with amplitude $er_{g}$ and frequency $\omega
_{B}$, but $90^{\circ}$ out of phase. The total radiated power is then
$$\eqalign{P_{cyc}&= {2 \omega_{B}^{4} e^{2}r_{g}^{2} \over 3 c^{3}}\cr
&={2e^{4}B^{2}v_{\perp}^{2} \over 3m_{e}^{2}c^{5}}.\cr}\eqno(2.6.18)$$
Cyclotron emission by ions is
generally negligible because of the two powers of $m$ in the denominator.
Because the motion of a gyrating electron is sinusoidal in time, in the
nonrelativistic limit the radiation is monochromatic at the frequency
$\nu_{B} = \omega_{B}/2\pi$. The component $v_{\parallel}$ of electron
velocity parallel to ${\vec B}$ leads to significant Doppler broadening
of the radiation received by all observers except those in directions
exactly perpendicular to ${\vec B}$. In a real astronomical object the
magnetic fields are curved, and all observers will receive
Doppler-broadened radiation; $B$ and $\nu_{B}$ will also vary from place
to place within the emission region.
The kinetic energy ${1 \over 2}m_{e}v_{\perp}^{2}$ of perpendicular
motion is reduced by cyclotron radiation, and is easily seen to decay
exponentially with an $e$-folding time
$$\hskip .82truein
\eqalign{t_{cyc}&={3m_{e}^{3}c^{5} \over 4e^{4}B^{2}}\cr &= 2.6 \times
10^{-4} \left({B \over 10^{6}\ {\rm gauss}}\right)^{-2}\ {\rm sec}.\cr}
\eqno(2.6.19)$$
This time is short for large fields, such as those found in magnetic
white dwarves ($B \sim 10^{7}$ gauss) and magnetic neutron stars ($B \sim
10^{12}$ gauss), and implies that electrons rapidly radiate the kinetic
energy of their perpendicular motion. In such large fields cyclotron
radiation is usually the most rapid radiation process. If $t_{cyc}$ is
small compared to $t_{eq}$ (2.2.24), and the cyclotron radiation freely
escapes, the electron distribution function will become strongly
anisotropic.
Cyclotron radiation is, in general, elliptically polarized. This is a
general property of radiation processes in strong magnetic fields, and
is unusual in astrophysics. The elliptical polarization of a few
white dwarves led to the recognition that they have large magnetic
fields, although cyclotron emission is not usually the dominant
source of radiation.
Because electrons in large magnetic fields are such efficient radiators,
they are also efficient absorbers of radiation at their cyclotron
frequency. It is generally incorrect to apply (2.6.18) or (2.6.19)
directly to the radiation of a gas of electrons, because the radiation
emitted by one will be efficiently absorbed by its neighbors. In
order to estimate the opacity we need to assume a finite line width
$\Delta \nu$. The Doppler width will typically range from $\sim .001
\nu$ for cool white dwarves to $\sim .1\nu$ for hot accreting neutron
stars; the cyclotron line will also be broadened by a variety of
collisional and plasma processes. From (2.6.18) and (1.7.13) we
obtain the cyclotron absorption opacity
$$\hskip .32truein
\eqalign{\kappa_{cyc}&={8\pi^{3}\over 3}\left({\nu \over \Delta \nu}
\right){k_{B}Tm_{e}c \over B^{2} hm_{p}\mu_{e}}\Bigl(1-\exp (-h\nu_{B}
/k_{B}T)\Bigr) \cr &\approx 3 \times 10^{11}
\left({.01 \nu \over \Delta \nu}\right)
\left({B \over 10^{6}\ {\rm gauss}}\right)^{-1} {\rm cm}^{2}/{\rm gm} ,
\cr}\eqno(2.6.20)$$
where $\mu_{e}$ is the molecular weight per electron. In the numerical
evaluation $\mu_{e}=1.2$ (ordinary stellar matter) and $h\nu_{B} \ll
k_{B}T$ were assumed. The corresponding cross-section is $\sim e^{2} /
(\Delta \nu m_{e} c)$, which is the natural cross-section for
absorption by an electron considered as a classical oscillator.
This is usually a very large
opacity, much exceeding that produced by other processes, but applicable
only within the spectral width $\Delta \nu$ of the line. Cyclotron
line radiation therefore flows very slowly through a magnetized plasma,
and the rate at which it radiates is usually much less than (2.6.18)
or (2.6.19) would imply. For a thermal electron distribution function
the spectral power density produced by cyclotron emission cannot
exceed the Planck function (1.7.13). At frequencies far from the line
center there is no emission at all. When the optical depth at the
line center is high, the width of the line becomes important, and
broadening processes must be considered.
Because of special relativistic kinematics
the motion of an electron seen by an unaccelerated observer is not
exactly harmonic. Consequently, the radiation is not strictly harmonic
at the frequency $\nu_{B}$. Because the motion is still periodic
(if radiation damping and collisions are ignored) the spectrum is a
series of harmonics of $\nu_{B}$, with the strength of the $n$-th
harmonic varying $\sim (v_{\perp}^{2}/c^{2})^{n}$. The dependence on
electron energy and frequency $n\nu_{B}$ is steep, and cyclotron
harmonic radiation therefore does not follow (2.5.3).
The importance of
the harmonics is that the spectral density may approach the Planck
function $B_{\nu}$ in each of them. If $\Delta \nu / \nu$ is
independent of $n$ and $nh\nu_{B} \ll k_{B}T$, then the power in the
$n$-th harmonic is $\sim \Delta\nu B_{\nu} \sim \nu^{3} \sim
n^{3}$, while the total power in harmonics $1-n$ is $\sim n^{4}$.
Because radiation at the fundamental frequency is so strong, rather
high harmonics may be produced, especially at the high temperatures
characteristic of accreting degenerate dwarves, neutron stars, and
laboratory plasma machines; harmonics up to $n \sim 10-100$ may
be optically thick. Under these conditions the harmonic lines usually
overlap because of Doppler broadening to form a smooth continuum,
$\Delta\nu$ is effectively constant, and the total radiated power
is $\sim n^{3}$. Quantitative calculations are intricate; Petrosian
(1981) gives some results and references to the earlier literature.
The radiation produced by a relativistic electron in a magnetic field,
called synchrotron radiation, is also of interest. Most inferences
of the presence and acceleration of energetic electrons are based on
the observation of their synchrotron radiation, usually at radio
frequencies (in supernova remnants and extragalactic radio sources)
or in visible light (in many quasars and active galactic nuclei, and
the Crab nebula), but occasionally in X-rays (in the Crab nebula, and
probably other objects). This radiation is readily identified because
it is strongly linearly polarized, and has a featureless power law
spectrum (when produced by electrons with a power law distribution of
energies); there is usually no plausible alternative way of producing
radiation with these properties.
The relativistic generalization of (2.6.18) is (Rybicki and Lightman
1979, Jackson 1975)
$$P_{synch} = {2 \gamma^{2}e^{4}B^{2}v_{\perp}^{2} \over 3 m_{e}^{2}
c^{5}}.\eqno(2.6.21)$$
Most of the emitted radiation is at frequencies a few times lower than
a characteristic frequency (defined differently by different authors)
$$\eqalign{\nu_{c}&\equiv 3\gamma^{3}\nu_{B} \sin \alpha \cr &= {3
\gamma^{2}eB\sin \alpha \over 2\pi m_{e}c},\cr}\eqno(2.6.22)$$
where $\alpha$ is the pitch angle of the electron's helical motion
($\alpha = 0$ for motion parallel to ${\vec B}$, and $\alpha = \pi /2$
for circular motion in a plane normal to ${\vec B}$). This
corresponds to harmonic numbers $n \sim 3 \gamma^{3} \sin \alpha$; for
$\gamma \gapp 1$ the harmonics overlap and the spectrum is a smoothly
varying continuum.
The integrated spectrum produced by electrons with a power law
distribution of energies
$$N(E) = N_{0}E^{-p} \eqno(2.6.23)$$
is given by (2.5.1). Using $E = \gamma m_{e}c^{2}$ and $n(E)\, d^{3}p
= N(E)\, dE$, and taking $\gamma \gg 1$, we obtain
$$F(\nu ) \sim \int N_{0} \gamma^{-p} {\cal F}(\nu ,\gamma )\
d\gamma . \eqno(2.6.24)$$
Now
$${\cal F}(\nu ,\gamma ) \sim {P_{synch} \over \nu_{c}}
{\cal S} (\nu / \nu_{c}) , \eqno(2.6.25)$$
where the single function ${\cal S}$ describes the shape of the
emission spectrum as a function of $\nu / \nu_{c}$ at all
relativistic energies. Then, using (2.6.21) and (2.6.22), and
noting that $\nu$ is a variable independent of $\gamma$ or $\nu_{c}$,
$$\hskip .30truein
\eqalign{F(\nu )&\sim \int \gamma^{-p} {\cal S}(\nu / \nu_{c})
\ d\gamma \cr &\sim \int \nu_{c}^{-p/2} {\cal S}(\nu / \nu_{c})
\nu_{c}^{3/2}\ d(\nu_{c}^{-1}) \cr &\sim \nu^{-(p-1)/2} \int
\left({\nu / \nu_{c}}\right)^{(p-3)/2} {\cal S}(\nu /
\nu_{c})\ d\left({\nu / \nu_{c}}\right).\cr}\eqno(2.6.26)$$
The last integral depends only on the function ${\cal S}$ and the limits
of integration; if the form (2.6.23) extends over a wide range in $E$
then over a wide range of $\nu$ these limits may be taken to be 0 and
$\infty$, and the integral is a number independent of $\nu$. The
integrated spectrum is then a power law (2.5.4) with spectral index
$$s = {p-1 \over 2} . \eqno(2.6.27)$$
This power law and linear polarization are the characteristic
signatures of a synchrotron source; most frequently $0.5 < s < 1.0$,
corresponding to $2 < p < 3$. In visible and ultraviolet light such a
spectrum is readily distinguishable from stellar spectra, even by a
crude comparison of the colors measured through broad filters. Stellar
spectra have a pronounced thermal curvature (Figure 2.1), and therefore
stars are almost always fainter in the ultraviolet than power law
sources with similar colors in visible light. This permits the quick
identification of candidate nonthermal sources (such as quasars) from
crude photographic measures of their colors.
The synchrotron power radiated per unit volume is $\sim N_{0}\gamma^{2}
B^{2}$. For radiation of a given frequency $\nu$, $\gamma$ and $B$ are
related by (2.6.22); eliminating $\gamma$, the power is $\sim {\cal E}_
{e}{\cal E}_{B}^{3/4}$, where ${\cal E}_{e} \sim N_{0} \gamma $
and ${\cal E}_{B} \sim B^{2}$ are
respectively the relativistic electron and magnetic energy densities.
For a given total energy density ${\cal E} = {\cal E}_{e}+{\cal E}_{B}$
the maximum power is obtained if ${\cal E}_{e}={4 \over 3}{\cal E}_{B}$,
close to the ``equipartition'' condition ${\cal E}_{e} = {\cal E}_{B}$.
Sources in which equipartition holds, at least roughly, are more
efficient radiators and more likely to be observed than those in which
it does not hold. Astronomers frequently assume equipartition in order
to estimate source parameters; these parameters describe a typical
source if (and only if) {\it some} sources are close to equipartition.
\goodbreak
\vskip 2\baselineskip plus2\baselineskip
\goodbreak
\noindent
{\bf 2.6.3} \us{Compton Scattering} \quad An electron in an
electromagnetic wave will be accelerated by the wave's electric field
(Unless the fields are extremely strong, the motion is nonrelativistic
and
the effect of the magnetic field may be ignored). Such an accelerated
electron will radiate. If $\hbar \omega \ll m_{e}c^{2}$ a classical
description is adequate. The electron's motion is given by (2.5.14).
Integrate this expression with $v_{x} = 0$, and use (2.6.4) to obtain
$$d(t)= {e^{2}E_{0}\over m_{e}\omega^{2}}\cos (\omega t).\eqno(2.6.28)$$
Using (2.6.1) and integrating over time leads to a mean radiated power
$$\langle P \rangle = {1 \over 3} {e^{4}E_{0}^{2} \over m_{e}^{2}c^{3}}.
\eqno(2.6.29)$$
This power is drawn from the power of the incident electromagnetic wave,
is at the same frequency, and may be described as its scattering by the
electron, called Thomson scattering. The mean power density of the
electromagnetic wave (including both ${\vec E}$ and ${\vec B}$, and
averaging over phase) is $E_{0}^{2}c/8\pi$. The ratio of $\langle P
\rangle$ to this power density is the electron scattering cross-section
$$\sigma_{es}={8\pi\over 3}{e^{4}\over m_{e}^{2}c^{4}}.\eqno(2.6.30)$$
In order to calculate the angular dependence of electron scattering it
is necessary to consider the angular dependence of the electron's
radiated field. The result (Rybicki and Lightman 1979) for the
differential scattering cross-section is
$${d\sigma_{es} \over d\Omega} = {1 \over 2}{e^{4} \over m_{e}^{2}c^{4}}
(1 + \cos^{2} \theta ) , \eqno(2.6.31)$$
where $\theta$ is the angle between the incident and scattered photon
directions. Because (2.6.31) is reflection-symmetric about $\theta =
\pi /2$ the scattered radiation carries no momentum in the
nonrelativistic limit; for most purposes (including Comptonization
{\bf 2.3}) correct results would be obtained even if electron scattering
were taken to be isotropic.
>From (2.6.30) the electron scattering opacity is
$$\eqalign{\kappa_{es}&= {\sigma_{es} \over \mu_{e}m_{p}}\cr &= .20
(1 + X)\ {\rm cm}^{2}/{\rm gm} , \cr} \eqno(2.6.32)$$
where $\mu_{e}$ is the molecular weight (in atomic mass units) per
electron and $X$ is the mass fraction of hydrogen in the matter. For
ordinary stellar composition $X=0.7$ and $\kappa_{es} = .34$ cm$^{2}$/gm.
It is also necessary to consider scattering by electrons moving at
relativistic speeds. The photon frequency $\nu^{\prime}$ in the
electron's frame is related to its frequency $\nu$ in the laboratory
frame by the Lorentz transformation
$$\nu^{\prime} = \nu \gamma (1 - \beta\cos\vartheta ) , \eqno(2.6.33)$$
where $v$ is the electron velocity, $\beta \equiv v/c$, $\gamma \equiv
(1 - \beta^{2})^{-1/2}$, and $\vartheta$ is the angle (in the
laboratory frame) between the
unscattered photon and electron directions. If $h\nu^{\prime} \ll m_{e}
c^{2}$ the scattering may be described as Thomson scattering in the
electron's frame. In that frame the frequency shift on scattering is
small and may be neglected, as may be the recoil velocity of the
electron. The frequency of the scattered photon in the laboratory frame
is then
$$\nu^{\prime\prime} = \nu^{\prime}\gamma (1 + \beta \cos \vartheta
^{\prime\prime}) , \eqno(2.6.34)$$
where $\vartheta^{\prime\prime}$ is the angle (in the electron's
frame) between its velocity and the
the scattered photon's direction. The angle $\vartheta$ is typically
$\sim \pi /2$, if the photon and electron distributions are initially
isotropic, and $\vartheta^{\prime\prime}$ is also typically $\sim \pi
/2$ because in the electron's frame the scattering follows the
Thomson law (2.6.31); neither of these angles is affected by
relativistic beaming. Therefore the factors in parentheses in (2.6.33)
and (2.6.34) are generally of order unity and
$$\nu^{\prime\prime} \sim \gamma^{2}\nu . \eqno(2.6.35)$$
For relativistic electrons the photon frequency in the laboratory frame
is multiplied by a very large factor. Scattering is roughly equivalent
to reflection by a mirror moving at the relativistic electron's speed.
Typically, the scattered photon will be an X-ray or $\gamma$-ray,
even if the unscattered photon was visible light or from the \hbox{3
$^{\circ}$K} background radiation. Because of
relativistic beaming nearly all the scattered photons travel in nearly
the same direction as the electrons from which they scattered; photons
scattered by an isotropic electron distribution are also isotropic.
The result of a more quantitative calculation (Rybicki and Lightman
1979) is that an electron loses energy at the rate
$$P_{Compt} = {4 \over 3} \sigma_{es} c \gamma^{2} \beta^{2} {\cal E}_{r}
, \eqno(2.6.36)$$
where ${\cal E}_{r}$ is the radiation energy density. This may be
compared to the energy loss by the synchrotron process (2.6.21),
assuming an isotropic electron distribution function so that $\langle
v_{\perp}^{2} \rangle = 2 \beta^{2} c^{2} /3$, and using (2.6.30):
$${P_{synch} \over P_{Compt}} = {B^{2} / 8 \pi \over {\cal E}_{r}}.
\eqno(2.6.37)$$
The powers are in the same ratio as the magnetic to the photon energy
density, and are often comparable. Synchrotron radiation is observed
much more often because it is usually emitted at radio frequencies,
where detectors are very sensitive. $P_{synch}$ equals $P_{Compt}$
produced by scattering the 3 $^{\circ}$K background radiation if
\hbox{$B\approx 3\times 10^{-6}$ gauss,} a typical interstellar field.
The result (2.6.37) should not be a surprise, because both processes
involve the scattering of electromagnetic energy by relativistic
electrons. Each process multiplies photon frequencies by a factor
$\sim \gamma^{2}$; in the case of synchrotron radiation the photons
are not real, but are effectively present in the acceleration of the
electron gyrating around the magnetic field.
Relativistic Compton scattering also produces power law photon spectra
from power law electron energy distributions. There is usually little
danger of confusing these with synchrotron power law spectra because
Compton scattered radiation is generally at much higher frequency.
These results for relativistic Compton scattering only apply if
$h\nu^{\prime\prime} \ll \gamma m_{e}c^{2}$ (or, equivalently,
$h\nu^{\prime} \ll m_{e}c^{2}$). If these conditions are not met, then
the relativistic Klein-Nishina formula must be used for the differential
scattering cross-section. Even more important, simple conservation of
energy limits the scattered photon energy to $(\gamma -1)m_{e}c^{2} +
h\nu$, and (2.6.35) no longer holds.
\goodbreak
\vskip 2\baselineskip plus2\baselineskip
\goodbreak
\noindent
{\bf 2.7 References}
\def\rightheadline{\vbox{\vskip 0.125truein\line{\tenbf\hfil References
\qquad\folio}}}
\vskip \baselineskip
\parindent=0pt
Arons, J., McKee, C., and Max, C., eds. 1979, {\it Particle Acceleration
Mechanisms in Astrophysics} (New York: American Institute of Physics).
\medskip
Chapman, S., and Cowling, T. G. 1960, {\it The Mathematical Theory of
Non-Uniform Gases} 2nd ed. (Cambridge: Cambridge University Press).
\medskip
Colgate, S. A., and Johnson M. H. 1960, {\it Phys. Rev. Lett.} {\bf 5},
235.
\medskip
Eisberg, R. M. 1961, {\it Fundamentals of Modern Physics} (New York:
Wiley).
\medskip
Fermi, E. 1949, {\it Phys. Rev.} {\bf 75}, 1169.
\medskip
Grindlay, J. E., Hertz, P., Steiner, J. E., Murray, S. S., and Lightman,
A. P. 1984, {\it Ap. J. (Lett.)} {\bf 282}, L13.
\medskip
Jackson, J. D. 1975, {\it Classical Electrodynamics} 2nd ed. (New York:
Wiley).
\medskip
Kompaneets, A. S. 1957, {\it Sov. Phys.---JETP} {\bf 4}, 730.
\medskip
Lewin, W. H. G., and Joss, P. C. 1983, in {\it Accretion Driven Stellar
X-Ray Sources}, eds. W. H. G. Lewin and E. P. J. van den Heuvel
(Cambridge: Cambridge University Press), p. 41.
\medskip
Liboff, R. L. 1969, {\it Introduction to the Theory of Kinetic Equations}
(New York: Wiley).
\medskip
Light, E. S., Danielson, R. E., and Schwarzschild, M. 1974, {\it Ap. J.}
{\bf 194}, 257.
\medskip
Lightman, A. P., and Shapiro, S. L. 1978, {\it Rev. Mod. Phys.} {\bf 50},
437.
\medskip
Longair, M. S. 1981, {\it High Energy Astrophysics} (Cambridge:
Cambridge University Press).
\medskip
Montgomery, D. C., and Tidman, D. A. 1964, {\it Plasma Kinetic Theory}
(New York: McGraw-Hill).
\medskip
Nicholson, D. R. 1983, {\it Introduction to Plasma Theory} (New York:
Wiley).
\medskip
Petrosian, V. 1981, {\it Ap. J.} {\bf 251}, 727.
\medskip
Rosenbluth, M. N., MacDonald, W. M., and Judd, D. L. 1957, {Phys. Rev.}
{\bf 107}, 1.
\medskip
Rybicki, G. B., and Lightman, A. P. 1979, {\it Radiative Processes in
Astrophysics} (New York: Wiley).
\medskip
Spitzer, L. 1962 {\it Physics of Fully Ionized Gases} 2nd ed. (New York:
Interscience).
\medskip
Trubnikov, B. A. 1965, {\it Reviews of Plasma Physics} {\bf 1}, 105.
\medskip
\endpage
\end
\bye