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\hfil}}}
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\noindent
{\twelveb Chapter 1}
\vskip .25truein
\noindent
{\twelveb Stars}
\vskip 1.44truein plus.06truein
\noindent
{\bf 1.1 Generalities}
\vskip \baselineskip
\noindent
This book was not meant to be about stars. But stars are the most
familiar, best studied, and arguably most important objects in
the astrophysicist's universe. They are therefore the building
blocks of many theories of more exotic objects. More fundamentally,
the study of stars is the study of the competition between
gravity and pressure. Astrophysics is distinguished from nearly
all of the rest of physics by the importance of gravity, so that
an understanding of the principles of stellar structure is
necessary in order to understand most other astronomical objects.
The study of stellar structure and evolution is an elaborate and
mature subject. The underlying physical principles are mostly
well-known, and have been developed in great detail. Powerful
numerical methods produce quantitative results for the properties
and evolution of stars. Numerous texts and a very extensive
research literature document this field. I refer the reader to
three standard texts; although not new they have aged very well, and
it would be both pointless and presumptuous to attempt to improve
on them. Chandrasekhar (1939) reviews the classical
mathematical theory of stellar structure, whose beginnings are
now more than a century old. Schwarzschild (1958) presents a less
mathematical description of the physical principles of stellar
structure and evolution, with more attention to the observed
phenomenology. This is probably the best book for a general
introduction to the properties of stars and their governing physics. I
recommend it (supplemented by any of the numerous recent
descriptive astronomy books) as a reference for the physicist
without astronomical background. Clayton (1968)
is particularly concerned with processes of
nucleosynthesis and thermonuclear energy generation.
There are still a number of outstanding problems in the theory of
ordinary stars. Many of these arise from a single area of
theoretical difficulty: the problem of quantitatively describing
turbulent flows. This problem arises in the formation of stars from
diffuse gas clouds, in stellar atmospheres, for rotating stars and
accretion discs (which may be thought of as the limiting case of
rapidly rotating stars), in interacting binary stars, in stars with
surface abundance anomalies, and in stellar collapse and
explosion. If turbulent flows have a material effect on the
properties of a star, quantitative theory must usually be
supplemented by rough approximations, and confident calculation
becomes uncertain and approximate phenomenology. This is even
more true of the more exotic objects which are the subject of
this book.
\footline={\hfil}
The problems of turbulent flow appear in two distinct forms. In
the first form, a turbulent flow arises in an otherwise
well-understood configuration, and may even resemble the turbulent
flows known to hydrodynamicists; the problem is the calculation
of some property, usually an effective transport coefficient, of
the flow. The most familiar example of this is turbulent
convection in the solar surface layers. In the second form, the
initial or boundary conditions of a flow are not known; it may
not be turbulent in the hydrodynamicist's sense of eddies or
nonlinear wave motion on a broad range of length scales, but
quantitative calculation is still impossible. The formation of
stars is an example of this kind of flow. A variety of assumptions,
approximations, and models, generally of uncertain validity and unknown
accuracy, are used to study turbulent flows in astrophysics.
This chapter on stars has two purposes. One is to illustrate
some of those physical principles of stellar structure which are
useful in understanding stars and other astrophysical objects.
The other is to develop the kind of rough (often order-of-magnitude)
estimates and dimensional analysis which are widely used
in modelling novel astrophysical phenomena. Some of this
material follows Schwarzschild (1958).
\goodbreak
\vskip 2\baselineskip plus2\baselineskip
\goodbreak
\noindent
{\bf 1.2 Phenomenology}
\def\rightheadline{\vbox{\vskip .125truein\line{\tenbf\hfil Phenomenology
\qquad\folio}}}
\vskip \baselineskip
\noindent
Hundreds of years of observations of stars have produced an
enormous body of data and revealed a wide variety of phenomena
which are discussed in numerous texts and monographs and a
voluminous research literature. Here we will summarize only the
tiny fraction of those data essential to the astrophysicist who
wishes to use stars in models of high energy astrophysical
phenomena.
The luminosities and surface temperatures of stars are often
described by their place on a Hertzsprung-Russell diagram, such
as that shown in Figure 1.1. In this theoretician's version the
abscissa is the stellar effective surface temperature
$T_{e}$, defined as the temperature of a black body which
radiates the same power per unit area as the actual stellar
surface; the ordinate is the stellar photon luminosity in units
of the Solar luminosity
$L_{\odot} = 3.9 \times 10^{33}$ erg/sec. There are also observers'
versions in which the abscissa is a \lq\lq color index,\rq\rq\
a directly observable measure of the spectrum of the emitted
radiation, and the ordinate may be the absolute or apparent
stellar magnitude in some observable part of the spectrum.
Accurate conversion between these two versions requires a
quantitative knowledge of the spectrum of emitted radiation,
which is approximately (but not exactly) that of a black body.
\topinsert
\vskip 13.5truecm
\ctrline{{\bf Figure 1.1.} Hertzsprung-Russell diagram.}
\endinsert
Most stars are found to lie on a narrow strip called the main
sequence. These stars (occasionally referred to as dwarves)
produce energy by the thermonuclear transmutation of hydrogen
into helium near their centers. Their positions along the main
sequence are determined by their masses, which vary monotonically
from about
$30 M_{\odot}$ (where the solar mass
$M_{\odot} = 2\times 10^{33}$ gm) at the upper left to
$0.1 M_{\odot}$
in the lower right. The Sun lies on the main sequence near its
middle.
Stars found above and to the right of the main sequence are
called giants and supergiants; their higher luminosities (and
their names) are accounted for by large radii, ranging in extreme
cases up to $10^{14}$ cm, about 1000 times that of the Sun.
These stars have exhausted the hydrogen at their centers and produce
energy by thermonuclear reactions in shells close to, but outside,
their centers. Stars of nearly equal ages (such as the members of a
single cluster of stars, formed nearly simultaneously) will be
distributed along a narrow track in the giant and supergiant
region, a track whose form reflects their complex evolutionary path.
Stars of a broad range of ages, such as the totality of stars in
the solar neighborhood, will mostly be found on the main
sequence; those in the giant and supergiant regions will be
broadly distributed rather than lying on a narrow track. There
are no sharp distinctions among main sequence (dwarf) stars,
giants, and supergiants, and intermediate cases are found.
Degenerate (traditionally called white) dwarves are faint, dense stars
in whose interiors the electrons are Fermi-degenerate, resembling
the state of an ideal metal or metallic liquid. They generally produce
negligible thermonuclear energy, having converted essentially all their
hydrogen (and probably also their helium) to heavier elements. Their
meager luminosity is supplied by their thermal energy content,
possibly augmented by the latent heat of crystallization, the
gravitational energy released by the sedimentation of their heavier
elements, and other minor sources. They cool steadily as these energy
sources are exhausted. Degenerate dwarves move to the lower right
along a track parallel to lines of constant radius as they
cool. Their radii depend on their masses (roughly as their
reciprocals), but because their masses are believed to span a
moderate range (perhaps
$0.4 M_{\odot}$ to $1.2 M_{\odot}$) they all lie in a strip of
moderate width. These masses are less than those which these stars
had when young, but the amount of mass lost is controversial and may
range from a few percent of to nearly all the initial mass. It is not
known whether the mass in the degenerate dwarf stage is a monotonic
function of or even determined by the mass at birth; it may be random
and unpredictable. Very few stars other than degenerate dwarves
are found much below and to the left of the main
sequence; most of these few are probably evolving rapidly into
degenerate dwarves.
An extrapolation of the main sequence to the lower right leads to
stars of mass too low to produce thermonuclear energy, generally called
brown dwarves. These objects slowly evolve into degenerate dwarves of
very low mass and lie near (but above, because of their low masses) an
extrapolation of the degenerate dwarf strip. Jupiter may be
regarded as an extreme case. These objects are nearly unobservable
because of their low luminosities, and only a few, if any, can
be identified with confidence. Their properties are uncertain because
the properties of matter under brown dwarf conditions are not well
known; few data are available to test the uncertain calculations.
Objects at the upper left end of the main sequence are very rare,
with their rarity increasing with increasing mass and luminosity. As a
consequence, extrapolation beyond masses of $50 M_{\odot}$ is largely
limited to theory.
\goodbreak
\vskip 2\baselineskip plus2\baselineskip
\goodbreak
\noindent
{\bf 1.3 Equations}
\def\rightheadline{\vbox{\vskip .125truein\line{\tenbf\hfil Equations
\qquad\folio}}}
\vskip \baselineskip
\noindent
A star may be defined as a luminous self-gravitating gas cloud.
If it is also spherical, in hydrostatic equilibrium, and in
thermal steady state it is described by the classical equations
of stellar structure:
$$\eqalignno{
{dP(r) \over dr}&= - {\rho (r) GM(r) \over r^{2}} &(1.3.1)\cr
{dM(r) \over dr}&= \ \ 4\pi r^{2} \rho (r) &(1.3.2)\cr
{dL(r) \over dr}&= \ \ 4\pi r^{2} \rho (r) \epsilon (r) &(1.3.3)\cr
{dT(r) \over dr}&= - {3 \kappa (r) \rho (r) L(r) \over 16\pi ac T^{3}
(r)r^{2}} . &(1.3.4)\cr}$$
Here
$P(r)$ is the pressure,
$M(r)$ is the mass enclosed by a sphere of radius $r$,
$\rho (r)$ is the density,
$L(r)$ is the luminosity produced within a sphere of radius $r$,
$\epsilon (r)$ is the rate of nuclear energy release per gram,
$T(r)$ is the temperature,
$\kappa (r)$ is the Rosseland mean opacity (defined in {\bf 1.7.2})
in cm$^{2}$/gm, and $a$ is the radiation constant. The first three
of these equations are elementary; (1.3.4) is derived in {\bf 1.7}.
Numerous assumptions and approximations have been made: spherical
symmetry, Newtonian gravity, a star in a stationary (unchanging)
state, and a flow of energy by the diffusion of radiation only.
Various of these
assumptions may be relaxed if the equations are appropriately
modified. It is frequently necessary to allow for the transport
of energy by turbulent convection (most familiarly, in the outer
layers of the Sun) or by conduction (in electron-degenerate matter).
These equations must be supplemented by three constitutive
relations, derived from the microscopic physics of the stellar
material. For any given chemical composition they take the form:
$$\eqalignno{
P&= P(\rho, T) &(1.3.5)\cr
\epsilon&= \epsilon (\rho ,T) &(1.3.6)\cr
\kappa&= \kappa(\rho ,T) . &(1.3.7)\cr}$$
These equations of stellar structure may be solved numerically,
which is necessary to obtain quantitative results. It is
illuminating, however, to make order-of-magnitude estimates. If
we did not have computers available (and were unwilling to
integrate these equations numerically by hand), or did not know
the quantitative form of the constitutive relations, these rough
estimates would be the best that we could do. Until the development
of quantitative theories of thermonuclear reactions and opacity, no
detailed calculation was possible. Even today, rough estimates are the
basis of most qualitative understanding. In novel circumstances they
are the first step toward building a quantitative model.
\goodbreak
\vskip 2\baselineskip plus2\baselineskip
\goodbreak
\noindent
{\bf 1.4 Estimates}
\def\rightheadline{\vbox{\vskip .125truein\line{\tenbf\hfil Estimates
\qquad\folio}}}
\vskip \baselineskip
\noindent
{\bf 1.4.1} \us{Order of Magnitude Equations} \quad
In order to make rough approximations to the differential equations
(1.3.1--4) we replace them by algebraic equations in
which the variables
$P$, $M$, $L$, and $T$ represent their mean or characteristic
values in the star, the continuous variable
$r$ is replaced by the stellar radius
$R$, and the derivative
$d/dr$ is replaced by the multiplicative factor
$1/R$. In most cases this level of approximation produces useful
rough results, although it is occasionally disastrous; with
intelligent choice of the numerical constants it can be
remarkably accurate, though usually only when a quantitative
solution is available as a guide.
The equations become:
$$\eqalignno{
P&= \rho {GM \over R} &(1.4.1)\cr
M&= {4\over 3} \pi R^{3}\rho &(1.4.2)\cr
L&= {4\over 3} \pi R^{3}\rho \epsilon &(1.4.3)\cr
T^{4}&= {3\kappa \rho L \over 16\pi acR} . &(1.4.4)\cr}$$
We now assume the perfect nondegenerate gas constitutive relation for
pressure
$$\eqalign{P&=P_{g} + P_{r}\cr
&= {\rho N_{A}k_{B}T \over \mu} + {aT^{4} \over 3} ,\cr}\eqno(1.4.5)$$
where $P_{g}$ and $P_{r}$ are the gas and radiation pressures
respectively,
$\mu$ is the mean molecular weight (the number of atomic mass
units per free particle),
$N_{A}$ is Avogadro's number per gram,
$k_{B}$ is Boltzmann's constant, and
$a$ is the radiation constant. Combination of (1.4.1), (1.4.2), and
(1.4.5) (ignoring the radiation pressure term in 1.4.5,
an excellent approximation for stars like the Sun)
yields results for the characteristic values of
$\rho$, $P$, and $T$:
$$\eqalignno{
\rho&= {3M \over 4\pi R^{3}} &(1.4.6)\cr
P&= {3GM^{2} \over 4\pi R^{4}} &(1.4.7)\cr
T&= {GM \over R} {\mu \over N_{A}k_{B}} . &(1.4.8)\cr}$$
\goodbreak
\vskip 2\baselineskip plus2\baselineskip
\goodbreak
\noindent
{\bf 1.4.2} \us{Application to the Sun} \quad
In Table 1.1 we compare the numerical estimates for
$\rho$, $P$, and $T$ obtained by substituting the solar mass, radius,
and molecular weight, to the quantitative values found for the center of
the Sun in a numerical integration (Schwarzschild 1958) of the equations
(1.3.1)--(1.3.7). More recent calculations
(Bahcall, {\it et al.} 1982) produce slightly different numbers, but
the difference is of no importance when we are examining the validity
of order-of-magnitude estimates. We use $R = 6.95 \times 10^{10}$ cm,
$M = 2 \times 10^{33}$ gm, and $\mu = 0.6$.
The estimated value of
$T$ is remarkably accurate (probably fortuitously so), while the
estimates of
$\rho $ and
$P$ are low by two orders of magnitude (note that the estimated
$\rho $ is nothing more than the mean stellar density). This large
discrepancy reflects the concentration of mass towards the center
of a star, and is a consequence of the compressibility of gases
and the inverse-square law of Newtonian gravity. The discrepancy also
reflects a deliberate obtuseness on our part in comparing the estimated
values of $\rho $ and $P$ to the calculated central values. Had we been
more cunning we could have chosen to compare to a suitable chosen
\lq\lq mean\rq\rq\ point in the numerical integration, and would have
obtained truly impressive (but deceptive) agreement.
\topinsert
\vskip 0.5truein
\ctrline {\bf{Table 1.1}}
\bigskip
\hrule
\bigskip
\settabs 4 \columns
\+ &&Estimate &Solar Center \cr
\bigskip
\hrule
\bigskip
\+\hskip.6in $\rho$ (gm/cm$^{3}$) &&$1.42$ &$134$ \cr
\medskip
\+\hskip.6in P (dyne/cm$^2$)&&$2.73\times 10^{15}$&$2.24\times 10^{17}$
\cr\medskip
\+\hskip.6in T ($^{\circ}$K)&&$1.39\times 10^{7}$ &$1.46\times 10^{7}$
\cr\medskip
\+\hskip.6in $\kappa$ (cm$^{2}$/gm)&&$2.18\times 10^{3}$ &$1.07$ \cr
\medskip
\+\hskip.6in $\epsilon$ (erg/gm/sec)&&$1.95$ &$14$ \cr
\medskip
\hrule
\endinsert
In all stars the
central density greatly exceeds the mean density. In stars of
similar structure this ratio is nearly constant, and the greatest
use of eqs. (1.4.6--8) is as scaling relations among stars
of differing mass and radius. Rough estimates and qualitative
understanding may be obtained readily; numerical integrations are
always possible when quantitative results are needed.
For giant and supergiant stars the ratio of central to mean
density may be as much as
$10^{16}$. Such enormous ratios indicate a complete breakdown
of the approximations (1.4.1--4); the interior structure of
such stars is very different from that of stars like the Sun; it
can be roughly described by simple relations, but requires an
understanding of their peculiar structure. In fact, their condensed
central cores and very dilute outer layers may each be separately
described by equations (1.4.6--8) with reasonable accuracy; disaster
strikes only when one attempts to describe both these regions together.
Equations (1.4.3) and (1.4.4) may also be used to estimate
$\kappa$ and $\epsilon$ given the estimates for
$\rho$, $P$, and $T$. For the Sun we use\break
$L = 3.9 \times 10^{33}$ erg/sec. These numerical values
are also compared in Table 1.1 to quantitative values at the
Solar center (Schwarzschild, 1958). The estimated value of
$\epsilon$ is just the Solar (mass-weighted) mean; the actual
central value is several times higher because thermonuclear
reaction rates are steeply increasing functions of temperature,
which peaks at the center. The estimated value of
$\kappa$ is far wrong; this is in part because of the
hundredfold concentration of density at the center, and in part
because of the concentration into a small central core of
thermonuclear energy generation. Equation (1.3.4) shows that using
an erroneously low estimated
$\rho$ and high
$R$ produces an erroneously large estimate for
$\kappa$.
Except for temperature, our rough estimates have been very inaccurate.
Approximations like those of equations (1.3.6) and (1.3.7)
are still useful, particularly when only scaling laws are needed
for a qualitative understanding. They can also produce
semiquantitative results when some additional understanding is
inserted into the equations in the form of intelligently chosen
numerical coefficients. We have deliberately refrained from
doing so in order to show the pitfalls as well as the utility of
rough estimates; when aided by intuition and guided by experience
they can do much better.
\goodbreak
\vskip 2\baselineskip plus2\baselineskip
\goodbreak
\noindent
{\bf 1.4.3} \us{Minimum and Maximum Stellar Surface Temperatures} \quad
The observed range of stellar surface temperatures is
approximately $2500^{\circ}$K to $50,000^{\circ}$K. These limits
each have simple explanations.
The continuum opacity of stellar atmospheres is largely
attributable to bound-free (photoionization) and free-free
(inverse bremsstrahlung) processes. For the visible and
near-infrared photons carrying most of the black-body flux at low
stellar temperatures the most important bound-free transition is
that of the
H$^{-}$ ion, which has a threshold of 0.75 eV. At temperatures
of a few thousand degrees matter consists largely of neutral atoms and
molecules, and the small equilibrium (Saha equation)
free-electron density is very sensitive to
temperature, dropping precipitously with further decreases in
temperature. The
H$^{-}$ abundance, in equilibrium with the free electrons,
drops nearly as steeply. The atmosphere approaches the very
transparent molecular gas familiar from the Earth's atmosphere.
As a consequence of this steep drop in opacity, the photosphere
(the layer in which the emitted radiation is produced) of a very
cool star forms at a temperature around
$2500^{\circ}$K, below which there is hardly enough opacity and
emissivity to absorb or emit radiation. This temperature bound is
insensitive to other stellar parameters, and amounts to an outer
boundary condition on integrations of the stellar structure equations
for cool stars.
The maximum stellar surface temperature has a different
explanation. In luminous stars the radiation pressure far exceeds
the gas pressure, and the luminosity is nearly the Eddington
limiting luminosity
$L_{E}$ ({\bf 1.11}), at which the outward force of
radiation pressure equals the attraction of gravity:
$$ L \approx L_{E} \equiv {4\pi cGM \over \kappa} , \eqno(1.4.9)$$
where
$\kappa$ is the opacity. Under these conditions the opacity is
predominantly electron scattering, and
$\kappa = 0.34$ cm$^{2}$/gm, essentially independent of other
parameters. The effective (surface) temperature
$T_{e}$ is then approximately given by
$$T_{e}^{4} = {cGM \over \kappa \sigma_{SB} R^{2}} , \eqno(1.4.10)$$
where $\sigma_{SB}$ is the Stefan-Boltzmann constant.
In order to estimate
$R$ we approximate the pressure by the radiation pressure
$$P \approx {a \over 3} T^{4} , \eqno(1.4.11)$$
where $T$ is an estimate of the central temperature. Note that here we
neglect the gas pressure; in obtaining equation (1.4.8) we neglected the
radiation pressure. Eliminating
$P$ and $\rho$ from (1.4.1), (1.4.6), and (1.4.11) produces an
estimate for $R$:
$$R^{4} = {9GM^{2} \over 4\pi aT^{4}}. \eqno(1.4.12)$$
Substituting this result in (1.4.10) gives
$$T_{e}^{4} = {T^{2}c\over \kappa \sigma_{SB}} \sqrt{{4\over 9}\pi acG} .
\eqno(1.4.13)$$
Because thermonuclear reaction rates are usually very steeply
increasing functions of temperature, the condition that
thermonuclear energy production balances radiative losses acts as a
thermostat; detailed calculation shows that
$T \approx 4 \times 10^{7 \circ}$K, nearly independent of
other parameters for these very massive and luminous stars.
Numerical evaluation of (1.4.13) then gives
$$ T_{e} \approx 90,000^{\circ}K. \eqno(1.4.14)$$
This numerical value is about twice as large as the results of
detailed calculations, but they confirm the qualitative result of a
mass-independent upper bound to $T_{e}$ for hydrogen burning stars.
\goodbreak
\vskip 2\baselineskip plus2\baselineskip
\goodbreak
\noindent
{\bf 1.5 Virial Theorem}
\vskip \baselineskip
\noindent
For stars (defined as self-gravitating spheres in hydrostatic
equilibrium) it is easy to prove a virial theorem, so named because it
is closely related to the
virial theorem of point-mass mechanics. Begin with the equation
(1.3.1) of hydrostatic equilibrium and assume it is always valid:
$$-\rho (r) {GM(r) \over r^{2}} = {dP(r) \over dr}. \eqno(1.5.1)$$
Multiply each side by
$4\pi r^{3}$, and integrate over
$r$, integrating by parts:
$$\hskip .25truein\eqalign{
-\int_{0}^{R} \rho (r) {GM(r) \over r} 4\pi r^{2}dr&= \int_{0}^{R}
{dP(r) \over dr} 4\pi r^{3}dr \cr
&= -\int_{0}^{R}12\pi r^{2}P(r)dr +
4\pi r^{3}P(r) \biggr\vert _{0}^{R} .\cr} \eqno(1.5.2)$$
The definition of the stellar radius
$R$ is that
$P(R) = 0$. Hence
$$- \int_{0}^{R} \rho (r) {GM(r) \over r} 4 \pi r^{2}dr = -3\int_{0}^{R}
P(r) 4 \pi r^{2}dr. \eqno(1.5.3)$$
The left hand side is the integrated gravitational binding energy of the
star $E_{grav}$. For a gas which satisfies a relation $P \propto
\rho^{\gamma}$ for adiabatic processes we can use the thermodynamic
relation (see {\bf 1.9.1})
$$P = (\gamma -1){\cal E} , \eqno(1.5.4)$$
where ${\cal E}$ is the internal energy per unit volume. If we
denote the integrated internal energy content of the star
by $E_{in}$ we obtain
$$E_{grav} = -3(\gamma -1)E_{in} . \eqno(1.5.5)$$
Denoting the total energy
$E = E_{in} + E_{grav}$ we have
$$E = E_{in}(4-3\gamma) = E_{grav} \left( {3\gamma -4 \over 3\gamma -3}
\right) \leq 0 . \eqno(1.5.6)$$
The inequality comes from the requirement that a star be energetically
bound. This simple relation is very useful in qualitatively
understanding stellar stability and energetics.
\def\rightheadline{\vbox{\vskip .125truein\line{\tenbf\hfil
Virial Theorem\qquad\folio}}}
For perfect monotonic nonrelativistic gases (including the fully
ionized material which constitutes most stellar interiors)
$\gamma = 5/3$; this applies even if the electrons are
Fermi-degenerate. For a perfect gas of relativistic particles or
photons
$\gamma = 4/3$; this is a good description of gases whose
pressure is largely that of radiation. Gases in which new
degrees of freedom appear as the temperature is raised (for
example, those undergoing dissociation, ionization, or pair
production) may have still lower values of
$\gamma$, approaching 1. Interatomic forces reduce
$\gamma$ if attractive, or increase it if repulsive (as for the
nucleon-nucleon repulsion of neutron star matter).
If $\gamma = 5/3$, as is accurately the case for stars like the Sun,
and more roughly so for most
degenerate (white) dwarves and for neutron stars, then
$E = {1 \over 2} E_{grav} = -E_{in} < 0$. Such a star is
gravitationally bound with a large net binding energy, and
resists disruption. It is also stable and resists dynamical collapse,
because in a smaller and denser state $\vert E_{grav} \vert $ and
$\vert E \vert $ would be larger. In order to reach
such a state it would have to reduce its total energy
$E$, but on dynamical time scales energy is conserved. Energy can only
be lost by slow radiative processes (including emission of neutrinos);
in most cases it is stably replenished from thermonuclear sources.
A star with
$\gamma > 4/3$ may be thought of as having negative specific
heat, because an injection of energy increases
$E$, which reduces $\vert E \vert $, $\vert E_{grav} \vert $
and $E_{in}$ (see 1.5.6). Because
temperature is a monotonically increasing function of
$E_{in}$ (and depends only on $E_{in}$ for perfect nondegenerate matter)
this injection of energy
leads to a reduction in temperature; similarly, the radiative
loss of energy from the stellar surface, if not replenished
internally, leads to increasing internal temperature. The
reason for this somewhat surprising behavior, described as a negative
effective specific heat, is the fixed relation (1.5.5) between
$E_{in}$ and $E_{grav}$, which holds so long as the
assumption of hydrostatic equilibrium is strictly maintained.
The negative effective specific heat is also the reason
thermonuclear energy release, which increases rapidly with
temperature, is usually stably self-regulating.
In a degenerate star the relation between
$E_{in}$ and temperature is complicated by the presence of a
Fermi energy and the effective specific heat is positive
when thermonuclear or radiative processes are considered;
thermonuclear energy release is either insignificant or
unstable, and radiation produces steady cooling. On dynamical time
scales processes are adiabatic and the star is stable, just as is a
nondegenerate star. $E_{in}$ is related to the Fermi energy which
varies in proportion to the temperature for adiabatic processes, and the
effective specific heat is again negative.
A star with
$\gamma = 4/3$ has
$E = 0$; the addition of 1 erg is sufficient to disrupt it
entirely, and the removal of 1 erg to produce collapse. Of
course, stars with
$\gamma$ exactly equal to 4/3 do not exist (and cannot exist, for this
reason), but as $\gamma$ approaches 4/3 a star becomes more and more
prone to various kinds of instability. Stars with
$\gamma$ very close to 4/3 include very massive stars whose
pressure is almost entirely derived from radiation, and
degenerate dwarves near their upper mass (Chandrasekhar) limit.
A star with
$\gamma < 4/3$ would have positive energy and would be exploding or
collapsing. Such stars do not exist, but localized regions with
$\gamma < 4/3$ do. They are found in cool stellar atmospheres
(especially those of giants and supergiants) in which matter is partly
ionized, and possibly in the cores of evolved stars which are hot enough
for thermal pair production or dense enough for nuclei to undergo
inverse $\beta$-decay. Such regions tend to destabilize a star, though
the response of the entire star must be calculated to determine if it is
unstable; instability is a property of an entire star in hydrostatic
equilibrium, not of a subregion of it.
\goodbreak
\vskip 2\baselineskip plus2\baselineskip
\goodbreak
\noindent
{\bf 1.6 Time Scales}
\def\rightheadline{\vbox{\vskip .125truein\line{\tenbf\hfil Time Scales
\qquad\folio}}}
\vskip \baselineskip
\noindent
A star is characterized by a number of time scales. The shortest
is the hydrodynamic time scale
$t_{h}$, which is defined
$$t_{h} \equiv \sqrt{ {R^{3}\over GM} } . \eqno(1.6.1)$$
This is approximately equal to the time required for
the star to collapse if its internal pressure were suddenly set
to zero. The fundamental mode of vibration has a period
comparable to
$t_{h}$, as does a circular Keplerian orbit skimming the
stellar surface. For phenomena with time scale much longer than
$t_{h}$ the star may be considered to be in hydrostatic
equilibrium, and eq. (1.3.1) applies. On shorter time scales the
application of (1.3.1) is in general not justified. For the Sun
$t_{h} \approx 26$ minutes.
The thermal time scale $t_{th}$ is defined
$$t_{th} \equiv {E \over L} , \eqno(1.6.2)$$
where $E$ is the total energy (gravitational plus internal) of
the star, as defined in {\bf 1.5}, and $L$ is its luminosity.
This is the time which would be required for a star to
substantially change its internal structure if its thermonuclear
energy supply were suddenly set to zero. For phenomena with time
scales longer than
$t_{th}$ the star may be considered to be in thermal
equilibrium, and eq. (1.3.3) applies. The application of (1.3.3)
on shorter time scales is in general not justified. For the Sun
$t_{th} \approx 2 \times 10^{7}$ years.
The longest time scale is the thermonuclear time
$t_{n}$, defined by
$$t_{n} \equiv {M \varepsilon c^{2} \over L} , \eqno(1.6.3)$$
where
$\varepsilon c^{2}$ is the energy per gram available from
thermonuclear reactions of stellar material. This measures the
life expectancy of a star in a state of thermal equilibrium.
After a time of order
$t_{n}$ its fuel will be exhausted and its production of radiant
energy will end; a wide variety of ultimate fates are
conceivable, including cooling to invisibility, explosion, and
gravitational collapse. For ordinary stellar composition
$\varepsilon \approx 0.007$; about 3/4 of this is accounted for by the
conversion of hydrogen to helium and about 1/4 by the conversion of
helium to heavier elements. For the Sun
$t_{n} \approx 10^{11}$ years; its actual life will be
about ten times shorter because after the exhaustion of the
hydrogen in a small region at the center,
$L$ will begin to increase rapidly and its remaining life will
be brief. The Sun is presently near the midpoint of its life.
There is an additional time scale
$t_{E}$ which characterizes stars in general. In {\bf 1.4.3} we
saw that there is a characteristic luminosity
$L_{E}$ (Eq. 1.4.9) which serves as an upper bound on stellar
luminosities. Define the Eddington time
$t_{E}$ as the thermonuclear time
$t_{n}$ for a hypothetical star of luminosity
$L_{E}$. Then
$$t_{E} \equiv {\varepsilon c \kappa \over 4\pi G} =
{2 \varepsilon e^{4} \over 3Gc^{3}m_{e}^{2}m_{p}\mu _{e}} ,\eqno(1.6.4)$$
where we have written the electron scattering opacity
$\kappa$ in terms of fundamental constants and
$\mu _{e}$ is the mean number of nucleons per electron. For
ordinary stellar composition
$t_{E} \approx 3 \times 10^{6}$ years. This is an approximate lower
bound on the lifespan of a star. Because it nearly four orders
of magnitude shorter than the age of the universe, luminous stars
have passed through many generations, manufacturing nearly all
the elements heavier than helium. The luminosities of stars range
over at least nine orders of magnitude, so lower luminosity stars
have lifetimes very much longer than $t_{E}$, and even much longer than
the present age of the universe.
A quantity analogous to the Eddington time is also an important
parameter in the study of rapidly accreting masses (for example, in
models of X-ray sources and quasars; Salpeter 1964). The luminosity is
given by $L = {\dot M}c^{2}\varepsilon$. The Salpeter time is defined
as the $e$-folding time of the mass $M$, if $L=L_{E}$:
$$t_{S} \equiv {M \over {\dot M}} = {\varepsilon c \kappa \over 4 \pi G}
. \eqno(1.6.5)$$
It is usually estimated that $\varepsilon \sim .1$, so that $t_{S} \sim
4 \times 10^{7}$ years. This is the characteristic lifetime of such a
luminous accreting object.
Finally, there is a simple ``light travel'' time scale $t_{lt}$ which
may be defined for any object of size R:
$$t_{lt} \equiv {R \over c} . \eqno(1.6.6)$$ It is generally
not possible for an object of size $R$ to change substantially
(by a factor of $\sim 2$) its emission on a time scale shorter
than $t_{lt}$, because that is the shortest time in which signals from
a single triggering event can propagate throughout the object, and hence
the shortest time on which its emission can vary coherently. A small
change, by a factor $1 + \delta$ with $\delta \ll 1$, can occur in a
time $\sim \delta t_{lt}$. If the velocity of propagation were the
sound speed (or, equivalently, a free-fall speed) rather than $c$, then
$t_{lt}$ would be the hydrodynamic time $t_{h}$ given by (1.6.1).
The time scale $t_{lt}$ is chiefly used in models of transient or
rapidly variable objects in high energy astrophysics, such as variable
quasars and active galactic nuclei, $\gamma$-ray bursts, and rapidly
fluctuating X-ray sources. The observation of a substantial variation
in the radiation of an object in a time $t_{var}$ is evidence that
its size $R$ satisfies
$$R \lapp c t_{var} . \eqno(1.6.7)$$
Such an upper bound on $R$ may then be combined with the luminosity
to place a lower bound on the radiation flux and energy density
within the object, and therefore to constrain models of it.
These arguments contain loopholes. It is possible to synchronize
clocks connected to energy release mechanisms and distributed over a
large volume so that they all simultaneously trigger a sudden release
of energy (because the clocks are at rest with respect to each other
there is no difficulty in defining simultaneity). A distant observer
would not see the energy release to be simultaneous, but rather
spread over a time $t_{lt}$, where $R$ is the difference in the path
lengths between him and the various clocks. However, if the clocks
have appropriately chosen delays which cancel the differences in path
lengths, he will see the signals of all the clocks simultaneously,
violating (1.6.7). This would require a conspiracy among the clocks
which is unlikely to occur except by intelligent design, and would
produce a signal violating (1.6.7) only for observers in a narrow cone.
Other loopholes are more likely to occur in nature. A strong brief
pulse of laser light propagating through a medium with a population
inversion depopulates the excited state at the moment of its passage.
Nearly all of the medium's stored energy may appear in a thin sheet of
electromagnetic energy, whose thickness may be much less than $R$, and
whose duration measured by an observer at rest may violate (1.6.7).
This is a familiar phenomenon in the laser laboratory, in which
nanosecond (or shorter) pulses of light may be produced by arrays of
lasing medium more than a meter long.
Analogous to a thin sheet of laser light is a spherical shell of
relativistic particles streaming outward from a central source (Rees
1966). If they produce radiation collimated outward (radiation produced
by relativistic particles is usually directed nearly parallel to the
particle velocity) the shell of particles will be accompanied by a shell
of radiation. This radiation shell will propagate freely, and will
eventually sweep over a distant observer, who may see a rapidly varying
source of radiation whose duration violates (1.6.7). The factor by
which it is violated depends on the detailed kinematics of the
radiating particles. In general, (1.6.7) is inapplicable when there
is bulk relativistic motion, even if only of energetic particles;
conversely, its violation implies bulk relativistic motion.
\goodbreak
\vskip 2\baselineskip plus2\baselineskip
\goodbreak
\noindent
{\bf 1.7 Radiative Transport}
\vskip \baselineskip
\noindent
{\bf 1.7.1} \us{Fundamental Equations} \quad
The most important means by which energy is transported in
astrophysics is by the flow of radiation from regions of high radiant
energy density to those of lesser; radiation carries energy from
stellar interiors to their surfaces, and from their surfaces to dark
space. The complete theory of this process is unmanageably and
incalculably complex and cumbersome, but a variety of approximations
make it tractable and useful. Fortunately, these approximations are
well justified in most (but not all) circumstances of interest, so
that the theory is not only tractable but also powerful and successful.
Here we will be concerned principally with the simplest limit,
applicable to stellar interiors, in which matter is dense and opaque,
and radiation diffuses slowly. There is another, even simpler limit,
that of vacuum, through which radiation streams freely at the speed $c$.
Between these limits there are the more complex problems
of radiative transport in stellar atmospheres (by definition, the
regions in which the observed photons are produced). This is a large
field of research blessed with an abundance of observational data;
several texts exist (for example, Mihalas 1978).
Consider in spherical coordinates the propagation of a beam of
radiation, so that $r$ measures the distance from the center of the
coordinate system and $\vartheta$ is the angle between the beam and
the local radius vector. In general, the radiation intensity $I$
will depend on the point of measurement $(r,\theta ,\phi )$ (note that
$\vartheta$ must be distinguished from the polar angle $\theta$),
on the polarization, and the the photon frequency $\nu$. In most
cases it is possible either to assume spherical symmetry (so that
there is no dependence on $\theta$ and $\phi$), or to treat the
problem at different $\theta$ and $\phi$ locally, so that these
angles enter only as parameters of the solution, like the chemical
composition of the star being studied. In either case it is not
necessary to consider $\theta$ and $\phi$ explicitly, and they will
be ignored, along with any dependence of the intensity on the
azimuthal angle $\varphi$ of its propagation direction. Problems
in which these approximations are not permissible are difficult, and
generally their solution requires Monte Carlo methods (in which the
paths of large numbers of test photons are followed on a computer
in order to determine the mean flow of radiation). I also neglect
polarization because it does not significantly affect the flow of
radiative energy; it is worth calculating in some stellar atmospheres
because it is sometimes observable for nonspherical stars or during
eclipses (symmetry implies that the radiative flux integrated over the
surface of a spherical star is unpolarized). The frequency dependence
of the radiation field is important, although it will not always be
written explicitly.
\def\rightheadline{\vbox{\vskip .125truein\line{\tenbf\hfil
Radiative Transport \qquad\folio}}}
In travelling a small distance $dl$ a beam loses a fraction $\kappa
\rho dl$ of its intensity, where $\kappa$ is the mass extinction
coefficient (with dimensions of cm$^{2}$/gm), and $\rho$ is the matter
density. We consider a beam with intensity $I(r,\vartheta )$ (with
dimensions erg/cm$^{2}$/sec/steradian, where the element of solid
angle refers to the direction of propagation, not to the geometry
of the spherical star); the power crossing an element of area $ds$
normal to the direction of propagation, and propagating in an element
$d\Omega$ of solid angle, is $I(r,\vartheta )dsd\Omega$. In the short
path $dl$ a power $I(r,\vartheta )\kappa \rho dldsd\Omega$ is removed
from the beam by matter in the right cylinder defined by $ds$ and $dl$,
where we have taken $d\Omega \ll ds/dl^{2}$. Matter also emits
radiation, and the volume emissivity $j$ is defined so that the power
emitted by the volume $dlds$ into the beam solid angle $d\Omega$ is
$j\rho dlds{d\Omega \over 4\pi}$. The units of $j$ are erg/gm/sec
and the emission is assumed isotropic, as is the case unless there
is a very large magnetic field.
After travelling the distance $dl$ the radiation field transports
energy out of the cylinder with a power $I(r+dr,\vartheta + d\vartheta)
dsd\Omega$, where it has been essential to note that a straight ray
(we neglect refraction) changes its angle to the local radius vector
as it propagates. In a steady state the energy contained in the
cylinder does not change with time, so that the sum of sources and
sinks is zero:
$$I(r,\vartheta )dsd\Omega - I(r,\vartheta )\kappa \rho dldsd\Omega
+ j\rho dlds{d\Omega \over 4\pi} - I(r+dr,\vartheta + d\vartheta)
dsd\Omega = 0. \eqno(1.7.1)$$
>From elementary geometry
$$\eqalignno{
dr&= dl \cos \vartheta &(1.7.2a)\cr
d\vartheta&= -dl \sin \vartheta /r. &(1.7.2b)\cr}$$
These equations are a complete description of the trivial problem
of the propagation of a ray in vacuum, and may be combined and
integrated to yield the solution
$$r = r_{\circ} \csc \vartheta , \eqno(1.7.3)$$
where $r_{\circ}$ is the distance of closest approach of the ray to
the center of the sphere. If the polar axis of the spherical
coordinates is chosen to pass through the point at which the ray is
tangent to the sphere of radius $r_{\circ}$ then the path of the ray
in spherical coordinates is given by
$$\theta = \pi /2 -\vartheta = \pi /2 -\sin^{-1}(r_{\circ}/r) .
\eqno(1.7.4)$$
If we expand $I(r,\vartheta )$ in a Taylor series:
$$I(r+dr,\vartheta +d\vartheta ) = I(r,\vartheta ) + {\partial I(r,
\vartheta ) \over \partial r}dr + {\partial I(r,\vartheta ) \over
\partial \vartheta} d\vartheta + \cdots , \eqno(1.7.5)$$
keep only first order terms in small quantities, and substitute this
and the expressions 1.7.2 into 1.7.1, we obtain the basic equation of
radiative transport:
$${\partial I_{\nu}(r,\vartheta ) \over \partial r} \cos \vartheta -
{\partial I_{\nu}(r,\vartheta ) \over \partial \vartheta}{\sin \vartheta
\over r}+\kappa_{\nu}\rho I_{\nu}(r,\vartheta )-{j_{\nu}\rho \over 4\pi}
= 0 . \eqno(1.7.6)$$
The subscript $\nu$ denotes the dependence of $I$, $\kappa$, and $j$ on
photon frequency; properly $I_{\nu}$ and $j_{\nu}$ are defined per
unit frequency interval.
Henceforth we do not make this subscript or the arguments $(r,\vartheta )
$ explicit unless they are being discussed.
We are usually more interested in quantities like the energy density
of the radiation field and the rate at which it transports energy
than in the full dependence of $I$ on angle. Fortunately, these
quantities may be represented as angular integrals over $I$, and are
intrinsically much simpler quantities which satisfy much simpler
equations than (1.7.6). Only in the very detailed study of stellar
atmospheres is the full angular dependence of $I$ significant. The
following quantities are important:
$$\eqalignno{
{4\pi \over c}J \equiv {\cal E}_{rad}&\equiv {1 \over c}\int I\ d\Omega
&(1.7.7a)\cr
H&\equiv \int I \cos \vartheta\ d\Omega &(1.7.7b)\cr
{4\pi \over c}K \equiv P_{rad}&\equiv {1 \over c}\int I \cos^{2}
\vartheta\ d\Omega . &(1.7.7c)\cr}$$
In (1.7.7a) and (1.7.7c) two symbols have been defined because both are
in common use. Sometimes $H$ is defined as ${1 \over 4 \pi}$ times the
definition in (1.7.7b). The integrals in (1.7.7) are called the angular
moments of $I$; clearly an infinite number of such moments may be
defined, but these three are usually the only important ones. It is
evident that ${\cal E}_{rad}$ is the energy density of the radiation
field, $H$ is the radiation flux (the rate at which radiation carries
energy across a unit surface normal to the $\vartheta = 0$ direction),
and $P_{rad}$ is the radiation pressure. As defined these quantities
are functions of frequency, but formally identical relations apply
to their integrals over frequency.
In general the $n$-th
moment (where $n$ is the power of $\cos\vartheta$ appearing in the
integrand) is a tensor of rank $n$; the scalar expressions of
(1.7.7b) and (1.7.7c) refer to the $z$ component of the flux vector and
the $zz$ component of the radiation stress tensor, where $\hat z$
is the unit vector along the $\vartheta = 0$ axis. In practice,
the $z$ component of $H$ is usually the only nonzero one and the
stress tensor is usually nearly isotropic so that it may be described
by a scalar $P_{rad}$.
It is now easy to obtain differential equations for the simpler
quantities ${\cal E}_{rad}$, $H$, $P_{rad}$ by taking angular moments of
equation (1.7.6); that is, by applying $\int \cos^{n} \vartheta \
d\Omega$ to the entire equation and carrying out the integrals.
The zeroth and first moments are
$$\eqalignno{
{dH \over dr} + {2 \over r}H + c\kappa \rho {\cal E}_{rad} - j\rho&= 0
&(1.7.8a)\cr
{dP_{rad} \over dr} + {1 \over r} (3P_{rad} - {\cal E}_{rad}) +
{\kappa \rho \over c} H&= 0 . &(1.7.8b)\cr}$$
There is an evident problem with this procedure: we have two equations
for the three quantities ${\cal E}_{rad}$, $H$, and $P_{rad}$. If we
obtain a third equation by taking the second moment of (1.7.6) we must
evaluate integrals like $\int I \cos^{3} \vartheta \ d\Omega$, which
introduce a fourth quantity, the third moment of $I$. It is evident
that this problem will not be solved exactly by taking any finite
number of moments; it arises very generally in moment expansions in
physics.
In practice moment expansions are truncated; only a small finite number
of moments are taken, and some other information, usually approximate,
is used to supply the missing equation. In order to do this expand $I$
in a power series in $\cos \vartheta$:
$$I = I_{0} + I_{1}\cos\vartheta + I_{2}\cos^{2}\vartheta + \cdots .
\eqno(1.7.9)$$
We could also expand in Legendre polynomials, which would have the
advantage of being orthogonal functions, but for the argument to be
made here this is unnecessary. Substitute this power series into
(1.7.6), and equate the coefficients of each power of $\vartheta$ in
the resulting expression to zero. There results an infinite series
of algebraic equations whose first three members are:
$$\eqalignno{
{I_{1} \over r} + \kappa \rho I_{0} &= {j\rho \over 4\pi}
&(1.7.10a)\cr
{\partial I_{0} \over \partial r} + {2I_{2} \over r} + \kappa \rho
I_{1}&= 0 &(1.7.10b)\cr
{\partial I_{1} \over \partial r} - {I_{1} \over r} +{3I_{3} \over r}
+ \kappa \rho I_{2}&= 0 . &(1.7.10c)\cr}$$
We now need only to estimate the order of magnitude of the $I_{n}$, so
we may replace ${\partial \over \partial r}$ by $1/l$ and $r$ by $l$
where $l$ is a characteristic length (noting that ${\partial \over
\partial r}$ and $-1/r$ do not cancel because this is only an
order-of-magnitude replacement---instead, their sum is still of order
$1/l$). Again, we have one more variable
than equations. However, these equations have an approximate solution
for which terms involving the extra variable become insignificant.
This solution is
$$\eqalignno{
I_{0}&\approx {j \over 4\pi \kappa} &(1.7.11a)\cr
I_{n}&\sim I_{0}(\kappa \rho l)^{-n}\qquad n \geq 1.&(1.7.11b)\cr}$$
The factor $(\kappa \rho l)$ is generally very large ($\sim 10^{10}$
in the Solar interior) so the higher terms in (1.7.9) become small
exceedingly rapidly. As a result (1.7.11a) holds very accurately, while
(1.7.11b) is only an order of magnitude expression. It is evident that
the terms in (1.7.10) which bring in more variables than equations (those
of the form $nI_{n}/r$) are smaller than the other terms by a factor of
order $(\kappa \rho l)^{-2}$ and are completely insignificant.
(1.7.11b) is a rough approximation only because of the replacement
of ${\partial \over \partial r}$ by $1/l$, not because of the
neglect of the terms of the form $nI_{n}/r$.
Because of (1.7.11b), (1.7.9) may be truncated after the $n=1$ term, and
${\cal E}_{rad}$, $H$, and $P_{rad}$ expressed to high accuracy in
terms of $I_{0}$ and $I_{1}$ alone, reducing the three variables to
two. The important result is that
$$P_{rad} = {4 \pi \over 3c} I_{0} = {1 \over 3} {\cal E}_{rad} .
\eqno(1.7.12)$$
This relation between $P_{rad}$ and ${\cal E}_{rad}$ is known as the
Eddington approximation. By relating two of the moments of the
radiation field it \lq\lq closes\rq\rq\ the moment expansion
(1.7.8). It holds to high accuracy everywhere except in stellar
atmospheres (in which $\kappa \rho l \sim 1$).
It might be thought that more accurate results could be obtained by
taking more terms in the moment expansions. In stellar interiors this
is unnecessary. Where (1.7.12) is not accurate, taking higher terms
does not lead to rapid improvement. Expansions which do not converge
rapidly often do not converge at all. A numerical description of
the full $\vartheta$ dependence of $I$ is a better approach.
The form of (1.7.12) is no surprise; it expresses the relation between
radiation pressure and energy density in thermodynamic equilibrium, which
should hold deep in a stellar interior. Similarly, if the matter at any
point is locally in thermal equilibrium and there are no photon
scattering processes the right hand side of (1.7.11a) equals
(by the condition of detailed-balance) the black-body radiation
spectrum (also called the Planck function) $B_{\nu}$:
$${j_{\nu} \over 4 \pi \kappa_{\nu}} = B_{\nu} = {2h\nu^{3} \over c^{2}}
{1 \over \exp(h\nu / k_{B}T) -1} . \eqno(1.7.13)$$
The condition that the matter is in local thermal equilibrium
(abbreviated LTE) holds to high accuracy in stellar interiors.
It may fail in stellar atmospheres where the radiation field is
strongly anisotropic, being mostly directed upward; such a radiation
field is not in equilibrium (the Planck function is isotropic), and
may drive populations of atomic levels away from equilibrium. This
often produces observable effects in stellar spectra, but does not
have significant effects on the gross energetics of radiative energy
flow.
Scattering presents a different problem. It is simple enough to
include scattering out of the beam in the opacity $\kappa$, but the
source term $j$ is more difficult, because radiation is scattered
{\it into} the beam from all other directions (and, in some cases,
from other frequencies). In general, a term of the form
$$\int d\Omega^{\prime}d\nu^{\prime}{d\sigma (\Omega ,\Omega^{\prime} ,
\nu ,\nu^{\prime}) \over d\Omega^{\prime}} I(\Omega^{\prime},
\nu^{\prime}) \eqno(1.7.14)$$
must be added to $j_{\nu}$ in (1.7.6), where $\sigma$ is the scattering
cross-section, and the solid angles $\Omega$ and $\Omega^{\prime}$
describe the pairs of angles $(\vartheta ,\varphi )$ and $(\vartheta^
{\prime},\varphi^{\prime})$. The azimuthal angles must be included to
completely describe the geometry of scattering. This term is
complicated; worse, it turns the relatively simple differential
equation (1.7.6) into an integral equation which is much harder to solve.
If the radiation field equals the Planck function, as is accurately the
case in stellar interiors, then the relation (1.7.13) holds even in the
presence of scattering, and it is not necessary to consider the messy
integral (1.7.14).
In stellar interiors we may use the Eddington approximation (1.7.12)
to reduce equations (1.7.8) to the form
$$\eqalignno{
{d(Hr^{2}) \over dr} + c \kappa \rho {\cal E}_{rad} - j \rho&= 0
&(1.7.15a)\cr
H + {c \over 3\kappa \rho}
{d{\cal E}_{rad} \over dr}&=0 . &(1.7.15b)\cr}$$
\goodbreak
\vskip 2\baselineskip plus2\baselineskip
\goodbreak
\noindent
{\bf 1.7.2} \us{Spectral Averaging and Energy Flow} \quad
In stellar interiors we are concerned with the flow of energy, and not
with its detailed frequency dependence. We therefore wish to consider
frequency integrals of our previous results. Define the luminosity $L
\equiv \int 4 \pi r^{2} H_{\nu}\ d\nu$, and note that in steady state
there is no net exchange of energy between the radiation and the matter,
so that $\int j_{\nu} d\nu = \int c \kappa_{\nu} {\cal E}_{rad\nu} d\nu$.
Then (1.7.15a) states that $L$
is independent of $r$. For a star in steady state (as we have assumed)
this is just the conservation of energy. In discussing radiative
transport we have neglected nuclear energy generation; if it were
included we would obtain (1.3.3).
It is more interesting to integrate (1.7.15b) over frequency. Define
$H_{av} \equiv \int H_{\nu}d\nu$ and ${\cal E}_{av} \equiv \int {\cal E}_
{rad\nu}d\nu$ so that
$$\hskip 1.02truein\eqalign{ H_{av}
&= -{c \over 3\rho}\int{1\over \kappa_{\nu}}{d{\cal E}_{rad\nu} \over dr}
d\nu\cr &= -{c \over 3\rho}{d{\cal E}_{av} \over dr}{\displaystyle
\int{1\over \kappa_{\nu}}{d{\cal E}_{rad\nu} \over dr}d\nu \over
\displaystyle\int{d{\cal E}_{rad\nu} \over dr}d\nu}.\cr}\eqno(1.7.16)$$
Because the radiation field $I_{\nu}$ is very close to that of a black
body $B_{\nu}$ we may write ${\cal E}_{rad\nu} = {4 \pi \over c}B_{\nu}$.
Then (1.7.16) may be written in the simple form
$$H_{av} = -{c \over 3 \kappa_{R} \rho}{d{\cal E}_{av} \over dr} ,
\eqno(1.7.17)$$
where we have defined the Rosseland mean opacity
$$
\kappa_{R} \equiv {\displaystyle \int{dB_{\nu} \over dr}d\nu \over
\displaystyle\int{1 \over \kappa_{\nu}}{dB_{\nu} \over dr}d\nu}
= {\displaystyle \int{dB_{\nu} \over dT}d\nu \over \displaystyle
\int{1 \over \kappa_{\nu}}{dB_{\nu} \over dT}d\nu}. \eqno(1.7.18)$$
These integrals may be computed from the atomic properties of the
matter and the Planck function.
The Rosseland mean $\kappa_{R}$ is a harmonic mean, and
therefore is sensitive to any \lq\lq windows\rq\rq\ (frequencies at
which $\kappa_{\nu}$ is small), but is insensitive to spectral lines
at which $\kappa_{\nu}$ is large. This behavior is very different
from that of the frequency-integrated microscopic emissivity of matter
(which gives the power radiated by low density matter for which
absorption in unimportant); this emissivity is proportional to the
arithmetic mean of $\kappa_{\nu}$ so that lines are important but
windows are not. The spectrum of matter usually contains many
absorption lines, but not windows, because there generally are
processes which provide some absorption across very broad ranges
of frequency. The Rosseland mean is therefore not very sensitive to
uncertainties in $\kappa_{\nu}$, which is fortunate, because
$\kappa_{\nu}$ is hard to calculate accurately. Because of the
frequency dependences of ${dB_{\nu} \over dT}$ and of typical $\kappa_
{\nu}$, $\kappa_{R}$ is most sensitive to the values of $\kappa_{\nu}$
at frequencies for which ${h\nu \over k_{B}T} \sim $ 3--10.
>From (1.7.17) we obtain
$$H_{av} = -{c \over \kappa_{R} \rho}{dP_{r} \over dr} ,\eqno(1.7.19)$$
where $P_{r}$ is the frequency-integrated radiation pressure. This
relates the rate at which radiation carries energy to the gradient
of radiation pressure. If the black body relation $P_{r} = {a \over 3}
T^{4}$ is substituted in (1.7.19) and the definition of $L$ is used then
(1.3.4) is obtained.
In general $0>{dP_{r} \over dr} \geq {dP \over dr}$ (unless the gas
pressure were to {\it increase} outward, an unlikely event which would
require that the density also increase outward, an unstable situation;
see {\bf 1.8.1}). The equation
of hydrostatic equilibrium (1.3.1) gives ${dP \over dr}$, so
that (1.7.19) implies an upper bound on $H_{av}$ and on $L$ for a star
in hydrostatic equilibrium. This is the origin of the Eddington limit
on stellar luminosities $L_{E}$ used in {\bf 1.4.3}.
\goodbreak
\vskip 2\baselineskip plus2\baselineskip
\goodbreak
\noindent
{\bf 1.7.3} \us{Scattering Atmospheres} \quad
An interesting application of these equations is to the problem of
an atmosphere in which the opacity is predominantly frequency-conserving
scattering, rather than absorption. This is a good approximation for
hot luminous stars, X-ray sources, and the hotter parts of accretion
discs, but also for visible radiation in very cool stellar and
planetary atmospheres. Define the single-scattering albedo $\varpi$
of the material as the fraction of the opacity attributable to
scattering; then $1-\varpi \ll 1$ is the fraction attributable to
absorption.
Begin with equations (1.7.8), assume
a nearly isotropic radiation field and the Eddington approximation
(1.7.12), and consider the case of a plane-parallel atmosphere of uniform
temperature, so that ${1 \over r} \ll {d \over dr}$ and $B$ is
independent of space. Equations (1.7.8) become
$$\eqalignno{
{dH \over dr} + c\kappa \rho {\cal E}_{rad} - j\rho&= 0 &(1.7.20a)\cr
{dP_{rad} \over dr} + {\kappa \rho \over c}H&= 0 . &(1.7.20b)\cr}$$
The source term $j$ is now given by
$$j = 4 \pi \kappa B(1-\varpi ) + \kappa{\cal E}_{rad}c\varpi ;
\eqno(1.7.21)$$
substitution leads to
$${1 \over \kappa \rho}{dH \over dr} + {\cal E}_{rad}c(1-\varpi )
- 4 \pi B (1-\varpi ) = 0 . \eqno(1.7.22)$$
Define the optical depth $\tau$ by
$$d\tau \equiv -\kappa \rho dr, \eqno(1.7.23)$$
with $\tau = 0$ outside the atmosphere (above essentially all its
material); this definition is used in all radiative transfer problems.
Equations (1.7.22) and (1.7.20b) become
$$\eqalignno{
{dH \over d\tau}&= ({\cal E}_{rad}c -4 \pi B)(1-\varpi )
&(1.7.24a)\cr
{1 \over 3}{d{\cal E} \over d\tau}&= {H \over c} . &(1.7.24b)\cr}$$
Differentiation of (1.7.24b) and substitution into (1.7.24a) leads to
$${d^{2}({\cal E}_{rad}-4\pi B/c) \over d\tau^{2}} = 3(1-\varpi )
({\cal E}_{rad} - 4\pi B/c) . \eqno(1.7.25)$$
Applying the boundary condition that ${\cal E}_{rad} \rightarrow
4\pi B/c$ as $\tau \rightarrow \infty$ leads to the solution
$${\cal E}_{rad} = {4\pi B \over c}\Bigl\lbrack 1-\exp\bigl(
-\sqrt{3(1-\varpi )} \tau \bigr) \Bigr\rbrack . \eqno(1.7.26)$$
One consequence of this result is that the radiation field does not
approach the black body radiation field until $\tau \gapp \lbrack
3(1-\varpi ) \rbrack^{-1/2} \gg 1$; in an atmosphere with largely
absorptive opacity the corresponding condition is $\tau \gapp 1$.
Another consequence is found when we compute the emergent radiant
power $H(\tau = 0)$ from (1.7.24b):
$$H = {4\pi B \over 3} \sqrt{3(1-\varpi )} . \eqno(1.7.27)$$
This should be compared to the result for a black body radiator
$H=\pi B$, which is obtained from (1.7.7b) if $I=B$ for $\vartheta \leq
\pi /2$, and $I=0$ for $\vartheta > \pi /2$. The scattering atmosphere
radiates a factor of ${4 \over 3} \sqrt{3(1-\varpi)} \ll 1$ as much
power as a black body at the same temperature. This may be described
as an emissivity $\varsigma = {4 \over 3} \sqrt{3(1-\varpi)} \ll 1$ of
the scattering atmosphere; by the condition of detailed balance such an
atmosphere has an angle-averaged albedo (the fraction of incident
flux returned to space after one or more scatterings) of $1-\varsigma$.
If it has an effective temperature $T_{e}$, its actual
temperature $T \approx \varsigma^{-1/4}T_{e} \approx 0.81(1-\varpi )^
{-1/8}T_{e}$, where we have assumed that $\varpi$ and $\varsigma$ are
not strongly frequency dependent.
The high albedo of a medium whose opacity is mostly scattering is
observed in everyday life when one adds cream to coffee. The extract of
coffee we drink is a nearly homogeneous substance whose opacity is
almost entirely absorptive; its albedo is very low. The mixture of
coffee and cream is visibly lighter in appearance because of the high
scattering cross-sections of globules of milk fat. The reduced
emissivity of the mixture is unobservable, because the Planck function
is infinitesimal at visible wavelengths and room temperature.
Equation
(1.7.27) appears to imply $\varsigma > 1$ if $\varpi \rightarrow 0$, but
this thermodynamically impossible result is incorrect because the
assumption of the Eddington approximation is invalid for $\tau \lapp 1$,
which is the important region in determining the emergent flux from an
absorbing atmosphere. In a scattering atmosphere, optical
depths up to $\lbrack 3(1-\varpi )\rbrack ^{-1/2} \gg 1$ are
important; the Eddington approximation is valid over most of this range.
\goodbreak
\vskip 2\baselineskip plus2\baselineskip
\goodbreak
\noindent
{\bf 1.8 Turbulent Convection}
\vskip \baselineskip
\noindent
If we heat the bottom and cool the top of a reservoir of fluid at rest,
heat will flow upward. The central regions of stars are heated by
thermonuclear reactions and their surfaces are cooled by radiation.
If the rate of heat flow is low, it will flow by a combination of
radiation and conduction. Conduction is usually dominant in everyday
liquids and in degenerate stellar material, and radiation is usually
dominant in gases, at high temperatures, and in nondegenerate stellar
interiors. At high heat fluxes a new process appears, in which
macroscopic fluid motions transport warmer material upward and cooler
material downward. This process is called convection. For limited
parameter ranges convection may take the form of a laminar flow, but
in astronomy it is almost always turbulent, if it occurs at all. We
must ask when it occurs and what are its consequences.
\goodbreak
\vskip 2\baselineskip plus2\baselineskip
\goodbreak
\noindent
{\bf 1.8.1} \us{Criteria} \quad
Two criteria must be satisfied in order to have convection.
The first is that viscosity not be large enough to prevent it. This is
an important effect in small laboratory systems, and successful
quantitative theories exist, but in stellar heat transport the
influence of viscosity is negligible; if convection takes place at all
Reynolds numbers usually exceed $10^{10}$.
\def\rightheadline{\vbox{\vskip .125truein\line{\tenbf\hfil
Turbulent Convection\qquad\folio}}}
The more important criterion is that the thermodynamic state of the
stellar interior be such that convective motions release energy, rather
than requiring energy to drive them. In other words, convection will
occur if it carries heat from hotter regions to cooler ones (given the
well-justified assumption that viscosity is a negligible retarding
force), but not if it were to carry heat from cooler regions to hotter
ones.
To make this criterion more quantitative we compare the thermodyamic
state of the star at two radii separated by a small radius increment
$dr$; at $r_{l}$ the pressure is $P_{l}$ and the density is $\rho_{l}$,
while at $r_{u}$ the pressure is $P_{u}$ and the density is $\rho_{u}$.
We assume that the chemical composition is uniform and that densities
and opacities are high enough that radiative transport of energy is
negligible on the time-scales of convective motions; these assumptions
are usually (but not always) justified in stellar interiors, but fail
in stellar atmospheres. We also relate adiabatic variations in the
pressure and density of the fluid by an equation of state of the form
$$P \propto \rho^{\gamma} . \eqno(1.8.1)$$
Such a fluid is known as a \lq\lq$\gamma$-law\rq\rq\ gas; $\gamma$ is
discussed in {\bf 1.9.1} and is usually between 4/3 and 5/3. It is here
only necessary to assume that the form (1.8.1) holds for adiabatic
processes over small ranges of $P$ and $\rho$; this will be the case
for any fluid except near a phase transition.
Now consider raising an element of fluid from the lower level to the
upper one, with all fluid velocities slow (much slower than the sound
speed) so that the fluid element remains in hydrostatic equilibrium
with its mean surroundings. When it reaches the upper level it has a
density $\rho_{u}^{\prime}$ given by
$$\rho_{u}^{\prime} = \rho_{l} \left( {P_{u} \over P_{l}} \right)^{1
\over \gamma} \approx \rho_{l} \left( 1 + {1 \over \gamma P_{l}}
{dP \over dr} dr \right) . \eqno(1.8.2)$$
If $\rho_{u}^{\prime} > \rho_{u}$ then the raised fluid element is
denser than its surroundings and will tend to fall back to its
initial position. In this case the fluid is stable against convective
displacement. A more quantitative analysis would calculate the
frequency of sinusoidal perturbations of the horizontal fluid layers
(analogous to water surface waves, but allowing for the continuous
variation of $P$ and $\rho$), and would find their frequency to be
real.
If $\rho_{u}^{\prime} < \rho_{u}$ the raised fluid is less dense than
its surroundings, and experiences a further buoyancy force which
accelerates its rise. A similar calculation of the density of a
fluid element descending from the upper layer shows that for it
$\rho_{l}^{\prime} > \rho_{l}$, so negative buoyancy accelerates its
descent. In this case the fluid is unstable, and convective motions
begin. In the more quantitative analysis the perturbations of the
horizontally layered structure have imaginary frequencies of both signs,
and grow exponentially.
For small $dr$ we may write $\rho_{u} \approx \rho_{l} + {d\rho \over dr}
dr$ so that the stability condition becomes
$$-{1 \over \gamma P}{dP \over dr} < - {1 \over \rho}{d\rho \over dr} .
\eqno(1.8.3)$$
This awkward-appearing form with minus signs on each side has been
chosen because the derivatives are both negative.
The definition of an incompressible fluid is that $\gamma \rightarrow
\infty$; then the stability criterion
(1.8.3) becomes ${d\rho \over dr}< 0$, a familiar result.
It is apparent that for compressible fluids as well ${d\rho \over dr} >0$
would make stability impossible (because the equation 1.3.1 of
hydrostatic equilibrium requires ${dP \over dr} < 0$). For an adiabatic
equation of state of the form (1.8.1) the entropy $S \propto \ln (P/\rho
^{\gamma} )$, and the stability condition takes the form
$$0 < {dS \over dr} . \eqno(1.8.4)$$
These stability conditions are local; it is clear that if an unstable
interchange is possible between two widely separated layers (1.8.3)
and (1.8.4) will be violated for at least a portion of the region
between the layers.
The bound
(1.8.3) may be transformed into a bound on ${dT \over dr}$ by use of
(1.4.5); the result is messy unless one of the terms in (1.4.5) is
negligible. More generally, if $P \propto \rho^{\alpha} T^{\beta}$ (in
contrast to 1.8.1, this refers to the functional form of $P(\rho ,T)$,
and {\it not} to its variation under adiabatic processes) we can
readily obtain
$$- \left( 1 - {\alpha \over \gamma} \right) {1 \over P} { dP \over dr}
> -{\beta \over T}{dT \over dr} . \eqno(1.8.5)$$
This is known as the Schwarzschild criterion for stability.
In this derivation we have assumed uniform chemical composition and
have ignored angular momentum. Either of these may make the problem
much more difficult. For example, if the matter in layer $l$ has higher
molecular weight than that in layer $u$ this will tend to stabilize the
fluid against convection. A more subtle process called
semi-convection may still occur even when ordinary convection does not;
it depends on the ability of energy to flow radiatively out of the
denser fluid, and thus to separate itself from the stabilizing influence
of the higher molecular weight. Semi-convection is one of a large class
of \lq\lq double-diffusive\rq\rq\ and \lq\lq multi-diffusive\rq\rq\
processes known to astrophysicists and geophysicists.
The criterion (1.8.5) shows that there is
instability when $\bigl\vert {dT \over dr} \bigr\vert$
is large, and (1.3.4) shows that this tends to occur when $\kappa$ or
$L/r^{2}$ are large. Detailed calculations show that (1.8.3--5) are
violated in the outer layers of stars with cool surfaces (including the
Sun) because at low temperatures $\kappa$ is large, and near the
energy-producing regions of luminous stars, where $L/r^{2}$ is large.
\goodbreak
\vskip 2\baselineskip plus2\baselineskip
\goodbreak
\noindent
{\bf 1.8.2} \us{Consequences} \quad
Suppose (1.8.3--5) are violated; what then? It is clear that the
interchange of elements of matter which are unstable against
interchange will tend to reduce $\rho_{u}$ and to increase $\rho_{l}$,
and to increase $S_{u}$ and to decrease $S_{l}$. The limiting state
of this process is to turn the violated inequalities (1.8.3--5) into
equalities whch then describe the variation of $P$, $\rho$, $T$, and $S$
in the star. Any one of these equalities (they are all equivalent) then
replaces (1.3.4) in describing the thermal structure of the star. In
other words, the effect of convective instability is to eliminate the
conditions which gave rise to it. This is a natural and plausible
hypothesis which is widely assumed in turbulent flow problems. It
cannot be exactly true; some small excess $\bigl\vert {dT \over dr}
\bigr\vert $ must remain to drive the convective flow.
A crude argument exists to estimate the accuracy of this approximation;
the estimate is based on an adaptation of Prandtl's mixing length theory
of turbulent flows. Although reality is surely more complex, imagine
that the turbulent flow is composed of discrete fluid elements which
rise or fall without drag forces (but remain in pressure equilibrium
with their surroundings) for a distance $\ell$ from their origins.
After travelling this distance they mix with their new surroundings
and lose their identity. Denote the excess of the temperature gradient
over the value given by (1.8.5) (taken as an equality) by $\Delta \nabla
T$; it is this quantity (called the superadiabatic temperature gradient)
we must estimate. After a rising fluid element has travelled a distance
$dr$ its temperature exceeds that of its mean surroundings by an amount
$\Delta \nabla Tdr$; its own thermodynamic state has varied exactly
adiabatically and it remains in pressure equilibrium with its mean
surroundings (both by assumption). A falling fluid element is
similarly cooler than its mean surroundings by $\Delta \nabla Tdr$.
The combination of rising warmer fluid and falling cooler fluid
produces a mean convective heat flux
$$H_{conv} \sim \Delta \nabla T dr c_{P} \rho v , \eqno(1.8.6)$$
where $v$ is a typical flow velocity and $c_{P}$ is the specific heat
at constant $P$.
In order to estimate $v$ we use the assumption that the only forces
acting on fluid elements are those of buoyancy. We have
$${\Delta \nabla \rho \over \rho} = \left({\beta \over \gamma - \alpha}
\right) {\Delta \nabla T \over T} \sim {\rho \over T} \Delta \nabla T ,
\eqno(1.8.7)$$
and the buoyancy force (which is proportional to $dr$) leads to a
velocity
$$v^{2} = {GM(r) \over r^{2}}{\Delta \nabla \rho \over \rho} (dr)^{2}
\sim {GM(r) \over r^{2}} {\Delta \nabla T \over T} (dr)^{2}.
\eqno(1.8.8)$$
Now evaluate these expressions after fluid elements have travelled half
of the mixing length, so that $dr = \ell /2$:
$$H_{conv} \sim {c_{P} \rho \ell^{2} \over 4} \sqrt{GM(r) \over r^{2}T}
(\Delta \nabla T)^{3/2}. \eqno(1.8.9)$$
A sensible choice of $\ell$ is a matter of guesswork; it is usually taken
to be comparable to the pressure scale height $\bigl\vert{d\ln P \over
dr}\bigr\vert^{-1}$. Observations of the Solar surface show that the
convective motions are very complex. The visible surface is divided
into a network of small polygonal cells, called granules, which are
columns of rising fluid bounded by regions of descending fluid. There
is also a larger scale pattern of supergranulation. These observations
do not provide direct evidence concerning the vertical mixing
length, and flows in the observable Solar atmosphere (where the scale
height is small) may not resemble those in deeper layers.
If $\ell$ is the pressure scale height and $H_{conv} = L/(4\pi r^{2})
- H_{av}$\break (where $H_{av}$ is the radiative flux calculated in
{\bf 1.7}) then we can evaluate $\Delta \nabla T$ and $v$ at various
places in a star. Our results may be manipulated to yield
$$\eqalignno{
\Delta \nabla T&\sim \biggl\vert {dT \over dr} \biggr\vert \left(
{\ell \over r} \right)^{-4/3} \left( {t_{h} \over t_{th}} \right)^{2/3}
&(1.8.10a)\cr
v^{2}&\sim c_{s}^{2}\left( {T_{c} \over T}\right) \left({\ell \over r}
\right)^{2/3} \left( {t_{h} \over t_{th}} \right)^{2/3}
&(1.8.10b)\cr}$$
where the thermal time $t_{th}$ has been redefined (from 1.6.2) to
include only the thermal energy content of the convective region, $T_{c}$
is the central temperature, and $c_{s}$ is the sound speed. For the
convective regions of the Sun (but not its surface layers)
$\Delta \nabla T \sim 10^{-6} \bigl\vert {dT \over dr} \bigr\vert$ and
$v \sim 10^{-4}c_{s} \sim 30\ $m/sec. Thus the adiabatic approximation
to the structure of a convective zone---the adoption of (1.8.3--5) as
equalities---is usually justified to high accuracy, even though the
estimates (1.8.6--9) are very crude. Similarly, characteristic
hydrodynamic stresses are $\sim \rho v^{2} \sim 10^{-8}P$, which
establishes that the assumption that fluid elements remain in
hydrostatic equilibrium also holds to high accuracy. The time for
fluid to circulate through the Solar convective region is $\sim \ell /v
\sim 1\ $month, which is short enough to guarantee complete mixing.
These approximations break down in the surface layers of stars, as
shown by equations (1.8.10). In these layers the scale height and
$\ell$ become small, as do $\rho$, $T$, and $t_{th}$ ($t_{th} \approx
c_{P} \rho T \ell /H$). It is not possible to calculate quantitatively
the structure of these layers. This problem is most severe for cool
giants and supergiants, where $T$ and especially $\rho$ become very
small. Their surfaces may not be spherical or in hydrostatic
equilibrium, but may rather consist of geysers or fountains
of gas which erupts, radiatively cools, and then falls back.
It is important to realize that $H_{conv}$ (1.8.9) is not directly
related to or limited by the pressure gradient, unlike the radiative
$H_{av}$ (1.7.17). This means that in stellar interiors convection may
carry a nearly arbitrarily large luminosity, and the Eddington limit
$L_{E}$ does not apply.
Near stellar surfaces this problem is more complicated because there $
\Delta \nabla T$ becomes large for large $H_{conv}$. In the low densities
of stellar atmospheres convection is incapable of carrying a large heat
flux because the thermal energy content of the matter is low, and energy
must flow by radiation. For hot stars the opacity is essentially
constant and radiative transport in the upper atmosphere imposes the
upper bound $L_{E}$ on the stellar luminosity. For cool giants and
supergiants the opacity in the upper atmosphere may be extremely small,
and no simple bound on the luminosity exists. The actual luminosity of
fully convective stars is determined by these surface layers in which
the approximation of nearly adiabatic convection breaks down, and no
satisfactory theory exists.
\goodbreak
\vskip 2\baselineskip plus2\baselineskip
\goodbreak
\noindent
{\bf 1.9 Constitutive Relations}
\def\rightheadline{\vbox{\vskip .125truein\line{\tenbf\hfil
Constitutive Relations\qquad\folio}}}
\vskip \baselineskip
\noindent
Each of the constitutive relations (1.3.5--7) is an extensive field
of research which extends far beyond the scope of this book. This
section presents only the sketchiest overview of a few qualitative
conclusions which should be familiar to every astrophysicist.
\goodbreak
\vskip 2\baselineskip plus2\baselineskip
\goodbreak
\noindent
{\bf 1.9.1} \us{Adiabatic Exponent} \quad
Here we derive a few useful results. Because stars are large and
opaque, and $t_{th}$ is usually long, we are often concerned with
the properties of matter undergoing adiabatic processes.
Consider a perfect gas which satisfies the equation of state (1.4.5)
$$P = {\rho N_{A} k_{B} T \over \mu} \eqno(1.9.1)$$
where we now neglect radiation pressure. For a gram of gas
undergoing a reversible process
$$dQ = d {\cal U} + PdV \eqno(1.9.2)$$
where $dQ$ is an infinitesimal increment of heat, ${\cal U}(V,T)$ is
the internal energy per gram, and $V \equiv 1/\rho $ is the volume per
gram. We define a perfect gas by the condition that ${\cal U}$ depend
only on $T$: \hbox{${\cal U}(V,T) = {\cal U}(T)$.}
The specific heats at constant pressure and at constant volume, $c_{P}$
and $c_{V}$ respectively, are defined:
$$c_{P} \equiv {dQ \over dT} \biggr\vert_{P} \eqno(1.9.3a)$$
$$c_{V} \equiv {dQ \over dT} \biggr\vert_{V} , \eqno(1.9.3b)$$
where the subscript denotes the thermodynamic variable to be held
constant. From (1.9.2), using (1.9.1) to eliminate $P$
$$\eqalignno{
c_{V}&= {d{\cal U} \over dT} &(1.9.4a)\cr
c_{P}&= {d{\cal U} \over dT} + {N_{A}k_{B} \over \mu}. &(1.9.4b)\cr}$$
The definition of an adiabatic process is that $dQ=0$. From the
preceding equations and definitions we find for such a process
$$0 = c_{V}dT + (c_{P} - c_{V}) {T \over V} dV . \eqno(1.9.5)$$
Defining $\gamma \equiv c_{P} / c_{V}$ yields
$$0 = d \ln T + (\gamma -1) d \ln V . \eqno(1.9.6)$$
Integrating this equation, using the definition of $V$ and (1.9.1),
yields
$$P \propto \rho^{\gamma} . \eqno(1.9.7)$$
The ratio of specific heats depends on the atoms or molecules making
up the gas. By explicit calculation of ${\cal U}$ for a perfect gas
it is easy to see that
$$\gamma = {q+2 \over q} \eqno(1.9.8)$$
where $q$ is the number of degrees of freedom excited per atom or
molecule. For a monatomic gas $q=3$, for a diatomic gas in which the
vibrational degrees of freedom are not excited (such as air under
ordinary conditions) $q=5$, while for a gas of large molecules or one
undergoing temperature-sensitive dissociation or ionization $q
\rightarrow \infty$. In stellar interiors we may usually take $q=3$
and $\gamma = 5/3$, except in regions of partial ionization or where
radiation pressure or relativistic degeneracy are important.
In this simple derivation it was necessary to assume a perfect gas and
to exclude radiation pressure. These may be included, but lead to
much more complex results. For a gas consisting only of radiation this
derivation is invalid because $c_{P} \rightarrow \infty$; $T$ is a
unique function of $P$ so that at fixed $P$ no amount of added energy
can raise the temperature.
>From the relation (1.9.7) describing adiabatic processes we can derive
a relation between $P$ and the internal energy per volume ${\cal E}$.
Taking logarithmic derivatives of (1.9.7) and using the definition of
$V$ we obtain
$$VdP = -\gamma PdV . \eqno(1.9.9)$$
Adding $PdV$ to each side gives
$$\eqalignno{
VdP + PdV&= -(\gamma -1)PdV &(1.9.10a)\cr
d \left( {PV \over \gamma -1} \right)&= -PdV . &(1.9.10b)\cr}$$
In an adiabatic process the work done by the fluid on the outside world
is $-PdV$, so that (1.9.10b) has the form of a condition of conservation
of energy for the fluid, with the left hand side being the increment in
internal energy. Then the internal energy per unit volume ${\cal E}$
is given by
$${\cal E} = {P \over \gamma -1} . \eqno(1.9.11)$$
The order of the manipulations between (1.9.7) and (1.9.11) may be
reversed, so that these two relations are equivalent.
It is important to note that the equivalence between (1.9.7) and (1.9.11)
does not require the assumption of a perfect gas or the definition of
the specific heats, so that it applies even where it is not possible to
derive $\gamma$ as a ratio of specific heats. The most important
application of this is to radiation. From (1.7.12) (or 1.7.7), for a
black body radiation field ${\cal E}_{rad} = 3P_{rad}$, so that
$\gamma = 4/3$ and (1.9.7) describes adiabatic processes in a gas of
equilibrium radiation.
\goodbreak
\vskip 2\baselineskip plus2\baselineskip
\goodbreak
\noindent
{\bf 1.9.2} \us{Degeneracy} \quad
The matter in degenerate dwarves, the cores of some giant and
supergiant stars, and in neutron stars is Fermi-degenerate. By this
we mean that the thermal energy $k_{B}T$ is much less than the Fermi
energy $\epsilon _{F}$ (or, more properly, the chemical potential of
the degenerate species), so that states with energies up to $\epsilon
_{F}$ are nearly all occupied, and those with higher energies are
nearly all empty. This resembles the familiar metallic state of matter.
The degenerate species is usually the electron; in neutron stars free
neutrons are also degenerate, hence their name.
The density $n_{d}$ of the degenerate fermion species is given by
$$n_{d} = 2 \left( {4 \over 3} \pi p_{F}^{3} \right) {1 \over h^{3}} ,
\eqno(1.9.12)$$
where $p_{F}$ is the momentum corresponding to the Fermi energy
$\epsilon_{F}$. This is a standard result of elementary statistical
mechanics, obtained by counting volumes in phase space, or by
calculating the eigenstates of free particles in a box. The factor of
$2$ comes from the statistical weight of spin $1/2$ particles.
For noninteracting nonrelativistic particles of mass $m_{d}$ we have
$$\epsilon_{F}= {p_{F}^{2} \over 2 m_{d}} \propto n_{d}^{2/3} ,
\eqno(1.9.13)$$ while
characteristic Coulomb energies vary with density as $\epsilon_{C}
\propto e^{2}n_{d}^{1/3}$. Thus at high densities $\epsilon_{F} \gg
\epsilon_{C}$ and degenerate electrons may be accurately treated as
non-interacting particles. This makes the calculation of their
equation of state easy and accurate, because the complex band structure
of ordinary metals (for which $\epsilon_{F} \sim \epsilon_{C}$) may be
neglected. The cohesion of ordinary metals (the fact that they have
$P=0$ at finite $n_{d}$) requires that $\epsilon_{C}$ be comparable to
$\epsilon_{F}$.
The pressure and internal energy of noninteracting degenerate
nonrelativistic particles are found by integrating over their
distribution function:
$$\eqalign{
P&= \int_{0}^{p_{F}} p_{x} v_{x} {2 \over h^{3}} d^{3}p\cr &= {1 \over 3}
\int_{0}^{p_{F}} m_{d} v^{2} {2\over h^{3}} d^{3}p\cr &= {8 \pi p_{F}^{5}
\over 15 m_{d} h^{3}}\cr &\propto \rho^{5/3}\cr} \eqno(1.9.14a)$$
$$\eqalign{
{\cal E}&= \int_{0}^{p_{F}} {m_{d} v^{2} \over 2} {2 \over h^{3}}
d^{3}p\cr &= {3 \over 2} P,\cr} \eqno(1.9.14b)$$
where we have used the fact that $\langle p_{x} v_{x} \rangle = {1 \over
3} \langle p_{x} v_{x} + p_{y} v_{y} + p_{z} v_{z} \rangle = {1 \over 3}
\langle pv \rangle $ for a distribution function which is isotropic in
3-dimensional momentum space; here unsubscripted $p$ and $v$ denote
their magnitudes. The relation between ${\cal E}$ and $P$, which
corresponds to $\gamma = 5/3$, depends only on the fact that the
particle energy $\epsilon_{p} = {1 \over 2} p v$, and not on the form
of the distribution function; hence it applies to all noninteracting
gases of nonrelativistic particles, whether degenerate, nondegenerate,
or partially degenerate ($\epsilon_{F} \approx k_{B}T$).
If the density is very high most of the particles are relativistic,
$\epsilon_{p} \approx pc$ and $v_{x} \approx c p_{x} / p$.
If we assume this relation holds exactly over the entire
distribution function then
$$\eqalign{
P&= \int_{0}^{p_{F}} {p_{x}^{2}c \over p} {2 \over h^{3}} d^{3}p\cr &=
{1 \over 3} \int_{0}^{p_{F}} pc {2 \over h^{3}} d^{3}p\cr &= {2\pi c
p_{F}^{4} \over 3 h^{3}}\cr &\propto \rho^{4/3}\cr} \eqno(1.9.15a)$$
$$\eqalign{
{\cal E}&= \int_{0}^{p_{F}} pc {2 \over h^{3}} d^{3}p\cr &= 3 P.\cr}
\eqno(1.9.15b)$$
The relation between ${\cal E}$ and $P$, which corresponds to $\gamma
= 4/3$, depends only on the relativistic relation $\epsilon_{p} = pc$,
and not on the form of the distribution function; hence it applies to
all noninteracting relativistic gases whether degenerate or not; it
even applies to bosons, which is why we recover the relation (1.7.12)
for photons.
Between the nonrelativistic and relativistic limits is a regime in
which neither (1.9.14) nor (1.9.15) is accurate, and $4/3< \gamma <5/3$.
This transition occurs for $p_{F} \approx m_{d}c$, which by (1.9.12)
occurs at a density
$$n_{d} \approx {8\pi m_{d}^{3} c^{3} \over 3 h^{3}} . \eqno(1.9.16)$$
For degenerate electrons this corresponds to $\rho \approx 2 \times
10^{6}$ gm/cm$^{3}$, while for neutrons $\rho \approx 10^{16}$ gm/cm$^{3}
$. These are, to order of magnitude, the characteristic densities of
degenerate dwarves and neutron stars respectively.
The regions in the $\rho$ - $T$ plane in which various approximations
to the equation of state hold are shown in Figure 1.2. Quantitative
calculations exist for the intermediate cases. The regions occupied
by the centers and deep interiors of ordinary stars and of degenerate
dwarves are shown.
\topinsert
\vskip 10.5truecm
\ctrline{{\bf Figure 1.2.} Equation of State Regimes.}
\endinsert
The results (1.9.14) and (1.9.15) are only rough approximations for
degenerate neutrons, because neutrons interact by strong nuclear forces,
which are attractive at relatively large distances (several $\times
10^{-13}$ cm) but which are strongly repulsive at shorter distances.
\goodbreak
\vskip 2\baselineskip plus2\baselineskip
\goodbreak
\noindent
{\bf 1.9.3} \us{Opacity} \quad
A quantitative calculation of the opacity of stellar material requires
elaborate calculations involving the absorption cross-sections of the
ground and many excited states of many ionic species. Such calculations
have been performed, and their results are available for quantitative
work. It is still important to be aware of a few qualitative principles.
In all ionized matter free electrons scatter radiation, a process called
Thomson or Compton scattering. For nondegenerate electrons, in the
limits $h\nu \ll m_{e}c^{2}$ and $k_{B}T \ll m_{e}c^{2}$ the scattered
radiation has the same frequency as the incident radiation, and carries
no net momentum. The scattering is not isotropic, but for all $0 \leq
\psi \leq \pi /2$ scattering by angles $\psi$ and by $\pi - \psi$ is
equally likely; for most purposes it may be treated as if it were
isotropic. The total scattering cross-section ({\bf 2.6.3})
is ${8\pi e^{4} \over 3 m_{e}^{2} c^{4}} = 6.65 \times 10^{-25}$
cm$^{2}$. For matter of the usual stellar composition (70\% hydrogen
by mass) this produces an electron scattering opacity
$$\kappa_{es} = 0.34 \ {\rm cm^{2}/gm} . \eqno(1.9.17)$$
Because this opacity is essentially independent of frequency and
temperature in fully ionized matter, (1.9.17) is usually a lower bound
on the Rosseland mean opacity. The only circumstances in which the
opacity of stellar matter may be significantly less than this value are
when it is degenerate (electron scattering is suppressed because most
outgoing electron states are occupied), or when it is cool enough
that most of the electrons are bound to atoms. The total opacity drops
below the value given by (1.9.17) for $T \lapp 6000^{\circ}$K.
A free electron moving in the Coulomb field of an ion may absorb
radiation; this process is called free-free absorption or inverse
bremsstrahlung. Its quantitative calculation is rather lengthy, but
a simple semiclassical result is informative. This may be obtained
by using the classical expression (2.6.12) or (2.6.15)
for the power radiated by an
accelerated charge (an electron in the Coulomb field of the ion) to
calculate the emissivity, and using the condition of detailed-balance
(1.7.13) to obtain from this the opacity. The resulting
cross-section per electron is proportional to $n_{i}v^{-1}\nu^{-3}$,
where $n_{i}$ is the ion density, $v$ is the electron velocity, and
$\nu$ is the photon frequency. For a typical electron $v$ will be
comparable to the thermal velocity, so $v \propto T^{1/2}$, and for
a representative photon $h\nu \propto T$. Rough numerical evaluation
of the Rosseland mean leads to
$$\kappa_{R} \sim 10^{23} {\rho \over T^{7/2}}\ {\rm cm^{2}/gm} ;
\eqno(1.9.18)$$
this expression is only approximate. The functional form of (1.9.18)
is known as Kramers' law.
The photoionization of bound electrons (from both ground and excited
states) produces bound-free absorption. Its frequency dependence above
its energy threshold is usually similar to the $\nu^{-3}$ of free-free
absorption, but the abundances of the various ions, ionization states,
and excitation levels must be considered too. The resulting mean
opacity roughly follows Kramers' law, and is of the same order of
magnitude as that attributable to free-free absorption.
Any Kramers' law opacity is large at low temperature and high density.
At high temperature or low density electron scattering is the principal
opacity. The dividing line is approximately given by $T \sim 5 \times 10
^6\rho^{2/7}\ ^{\circ}$K. At low temperatures ($T\lapp 10000^{\circ}$K)
the number of free electrons becomes small and most photons have
insufficient energy to ionize atoms; consequently, the opacity drops
precipitously and falls below $\kappa_{es}$.
The serious user of quantitative opacity information will use the
tables which have been computed, but a few further qualitative points
should be made:
Because the Rosseland mean is a harmonic mean, the various
contributions to the mean opacity are not additive unless they have
the same frequency dependence.
Absorption opacities contain a factor $\bigl\lbrack 1 - \exp (-h
\nu /k_{B}T) \bigr\rbrack $ whose physical origin is the effect
of stimulated emission. This must be included when the Rosseland mean
is computed; it is implied by the factor of this form contained in
$B_{\nu}$ in (1.7.13); LTE of the atomic and ionic levels has been
assumed.
Scattering opacities do not contain a stimulated emission factor
if the scattering conserves frequency. The total rate of scattering
from state $i$ to state $f$ is proportional to $n_{i}(1+n_{f})$, where
$n_{i}$ and $n_{f}$ are the occupation numbers of the corresponding
photon states; $n_{i}n_{f}$ is the rate of stimulated scattering.
>From this must be subtracted the rate $n_{f}(1+n_{i})$ of scatterings
from $f$ to $i$. The net rate is proportional to $n_{i}-n_{f}$, where
$n_{i}$ gives the scattering rate implied by the scattering cross-section
without any stimulated scattering term, and $n_{f}$ gives the the
scattering contribution to the source term $j$. The absence of an
explicit stimulated scattering factor is of little importance in stellar
interiors, but may be significant in laser experiments in which $n_{i}$
and $n_{f}$ may be very large.
Degenerate matter, like ordinary metals, is a good conductor of heat,
and in it the radiative transport of energy is usually insignificant.
Because the conductive heat flux is proportional to the temperature
gradient, a relation like (1.3.4) may be defined in which $\kappa$
includes also the effects of conduction.
\goodbreak
\vskip 2\baselineskip plus2\baselineskip
\goodbreak
\noindent
{\bf 1.9.4} \us{Thermonuclear Energy Generation} \quad
Many nuclear reactions are involved in the thermonuclear production
of energy and the transmutation of lighter elements into heavier ones.
Each presents special problems. Here I briefly discuss a few general
principles. Quantitative calculation of reaction rates in stellar
interiors requires more careful attention to many details; see, for
example, Clayton (1968) and Harris {\it et al.} (1983).
The radius of a nucleus containing $A$ nucleons is approximately given
by
$$R \approx 1.4 \times 10^{-13} A^{1/3} {\rm cm} . \eqno(1.9.19)$$
The electrostatic energy required to bring two rigid and unpolarizable
spherical nuclei of radii $R_{1}$ and $R_{2}$ and atomic numbers $Z_{1}$
and $Z_{2}$ into contact, if their charges are concentrated at
their centers, is
$$E_{C} = {Z_{1}Z_{2}e^{2} \over R_{1}+R_{2}} \approx {Z_{1}Z_{2} \over
A_{1}^{1/3}+A_{2}^{1/3}}\ {\rm MeV} . \eqno(1.9.20)$$
Once the nuclei touch strong attractive nuclear forces take over. In
the centers of main sequence stars $k_{B}T$ is in the range ${1 \over 2}$
-- 4 KeV so that it is evident that conquering the Coulomb barrier
is the chief obstacle to thermonuclear reactions.
The Coulomb barrier is overcome by tunnelling, in a manner first
calculated by Gamow; nuclei with energies much less than $E_{C}$ may
(infrequently) react. We work in the center-of-mass frame of the two
nuclei, so that $m={M_{1}M_{2} \over M_{1} + M_{2}}$ is their reduced
mass, $r$ their separation, and $k=\sqrt{2mE_{\circ}}/\hbar$ and $E_{
\circ}$ are the wave-vector and kinetic energy at infinite separation.
The barrier tunnelling probability $P_{0}$ is calculated in the W. K. B.
approximation as
$$P_{0} \sim \exp \left( -2 \int_{R}^{r_{\circ}} \sqrt{{2me^{2}Z_{1}
Z_{2} \over \hbar^{2} r} - k^{2}}\ dr \right) \equiv \exp(-{\cal I}) ,
\eqno(1.9.21)$$
where we write only the very sensitive exponential term, neglecting
more slowly varying factors. Here $R = R_{1}+R_{2}$ is the
separation at contact (within which the nuclear interactions make
the potential attractive), $r_{\circ}={2me^{2}Z_{1}Z_{2} \over
\hbar^{2} k^{2}}$ is the classical turning point (at which the integrand
is zero), and the subscript $0$ indicates that we consider only the
$l=0$ partial wave. Higher angular momentum states produce much
smaller $P_{l}$.
The exponent in (1.9.21) may be calculated:
$$\eqalign{
{\cal I}&= 2k \int_{R}^{r_{\circ}} \sqrt{{r_{\circ} \over r} -1}\ dr\cr
&= 4kr_{\circ} \int_{\sqrt{R/r_{\circ}}}^{1} \sqrt{1-\zeta^{2}}\ d\zeta
, \cr} \eqno(1.9.22)$$
where $\zeta \equiv \sqrt{r/r_{\circ}}$. Now $\sqrt{R/r_{\circ}}\ll 1$
so that we may expand the integral in a power series in $\sqrt{R/r_{\circ
}}$ with the result:
$$\hskip .4truein\eqalign{
{\cal I}&= 4kr_{\circ} \left( \int_{0}^{1} \sqrt{1-\zeta^{2}}\ d\zeta -
\int_{0}^{\sqrt{R/r_{\circ}}} 1 \ d\zeta + \cdots \right)\cr&=4kr_{\circ}
\left( {\pi \over 4} - \sqrt{{R \over r_{\circ}}} + \cdots \right).\cr}
\eqno(1.9.23)$$
The leading term in (1.9.23) does not depend on $R$ at all; this is
fortunate because it implies that to a good approximation the result
is independent of the nuclear sizes or to the form of the potential
near nuclear contact, where it is poorly known. We now have
$${\cal I} = {\pi Z_{1}Z_{2}e^{2} \over \hbar} \sqrt{2m \over E_{\circ}}
- 4 {e \over \hbar} \sqrt{2mZ_{1}Z_{2}R} + \cdots . \eqno(1.9.24)$$
The second term is independent of energy; it affects the reaction rate
but we do not consider it further. The third and higher terms are
small. The first term is large and after exponentiation makes the
reaction rate a sensitive function of $E_{\circ}$.
We now must average the reaction rate over the thermal equilibrium
distribution of nuclear kinetic energies. When we transform variables
from the velocities of the reacting nuclei to the center-of-mass and
relative velocities $v_{cm}$ and $v_{rel}$, we find that the kinetic
energy ${1 \over 2}M_{1}v_{1}^{2} + {1 \over 2}M_{2}v_{2}^{2} =
{1 \over 2}(M_{1}+M_{2})v_{cm}^{2} + {1 \over 2}mv_{rel}^{2}$, so that
the distribution function of the relative motion of the reduced mass
$m$ is Maxwellian at the particle temperature $T$. Then the total
reaction rate is given by the average over the distribution function
$\langle \sigma v_{rel} \rangle$, where $\sigma$ is the reaction
cross-section and contains the critical factor $\exp(-{\cal I})$.
Aside from slowly varying factors this leads to
$$\langle \sigma v_{rel} \rangle \sim \int_{0}^{\infty} \exp \left(
-{E \over k_{B}T} - {B \over \sqrt{E}} \right) dE , \eqno(1.9.25)$$
where $B \equiv \pi Z_{1}Z_{2}e^{2}\sqrt{2m}/\hbar$.
The first term in the exponent in (1.9.25) declines rapidly with
increasing $E$, while the second increases rapidly. For $B^{2} \gg
k_{B}T$ (almost always the case) their sum has a fairly narrow maximum,
and when exponentiated the peak is very narrow. We therefore find the
maximum and expand around it. By elementary calculus
$$-{E \over k_{B}T} - {B \over \sqrt{E}} = -{3E_{G} \over k_{B}T} -
{3 \over 8}{B \over E_{G}^{5/2}} (E-E_{G})^{2} + \cdots , \eqno(1.9.26)$$
where the Gamow energy $E_{G}$ has been defined
$$E_{G} \equiv \left( {Bk_{B}T \over 2} \right)^{2/3} .\eqno(1.9.27)$$
Now the integral in (1.9.25) may be carried out by taking only the
first two terms of (1.9.26) and extending the lower limit of integration
to $-\infty$, with the result
$$\hskip .79truein\eqalign{
\langle \sigma v_{rel} \rangle&\sim \sqrt{{8\pi E_{G}^{5/2} \over 3B}}
\exp \left( -{3E_{G} \over k_{B}T} \right)\cr &\sim \exp \biggl\lbrack
-3 \left( {\pi^{2}Z_{1}^{2}Z_{2}^{2}e^{4}m \over 2\hbar^{2}k_{B}T}
\right)^{1/3} \biggr\rbrack ,\cr} \eqno(1.9.28)$$
where in the last expression the slowly varying factor has been dropped,
as similar factors were before, leaving only the dominant exponential
dependence. This result gives the dominant temperature dependence of
nonresonant thermonuclear reactions.
Under typical conditions of interest the argument of the cube root in
(1.9.28) is $\sim 10^{4}$. It is therefore apparent that $P_{0}$ and
$\langle \sigma v_{rel} \rangle$ are very small, as must be the case,
in order that the nuclei in a dense stellar interior survive for
$10^{6}$--$10^{10}$ years before reacting. It is then evident that
the reaction rate is a steeply increasing function of $T$, and a
steeply decreasing function of $Z_{1}Z_{2}$. The sensitivity to $T$
implies that thermonuclear energy generation acts nearly as a
thermostat when in a star whose effective specific heat is negative (see
{\bf 1.5}), and tends to produce rapid instability when the effective
specific heat is positive (as is the case in degenerate matter or for
thin shells). It also means that when energy is produced by a given
nuclear reaction $T$ is a weak function of the other parameters. The
sensitivity to $Z_{1}Z_{2}$ implies that in most circumstances the
reactions which proceed most rapidly are those with the smallest product
$Z_{1}Z_{2}$.
Real nuclear physics makes the problem more complex. If the reaction
of interest is resonant at near-thermal energies (as some important
ones are) this may increase the reaction rates by a large factor. The
peculiar properties of nuclei with $A=2$, $5$, and $8$ are also worthy
of note:
The only stable nucleus with $A=2$ is the deuteron. To produce it from
protons requires the reaction
$$p + p \rightarrow D + e^{+} + \nu_{e} . \eqno(1.9.29)$$
Because this reaction depends on the weak interaction (it amounts to a
$\beta$-decay from an unbound diproton state), its rate is many orders
of magnitude lower than would otherwise be the case. Yet there is no
other direct way of combining two protons; the diproton is not a bound
nucleus at all, but is better described as a pole of the $p$-$p$
scattering matrix. Were the diproton bound, stars (and the universe)
would be very different. Because (1.9.29) is so slow, a catalytic
process known as the CNO cycle proceeds more rapidly in stars
more massive than the Sun, even though it requires reactions with
$Z_{1}Z_{2} = 7$.
There are no stable nuclei with $A=5$ or $8$, so that helium nuclei
cannot react with each other or with protons. More exotic reactions
(such as $^{3}$He + $^{4}$He, or He + Li) also do not cross the
$A=8$ barrier. The only way to build nuclei heavier than $A=8$ is
by the process
$$\alpha + \alpha + \alpha \rightleftharpoons ^{12}{\rm \! C}^{*}
\rightarrow ^{12}{\rm \! C} + \gamma + \gamma^{\prime} , \eqno(1.9.30)$$
where the asterisk denotes the 7.654 MeV excited state and the right
hand side indicates two successive radiative decays. This process
is resonant because the energy of $^{12}$C$^{*}$ is only $E_{*} = 379$
KeV above that of three $\alpha$-particles. In (1.9.30) the decay
rate $\Gamma_{\alpha}$ of $^{12}$C$^{*}$ to the left is much faster than
that $\Gamma_{\gamma}$ to the right; the excited state is
in thermal equilibrium with the $\alpha$-particles, and its density
$n_{*}$ may be calculated from the Saha equation, with the result:
$$n_{*} = n_{\alpha}^{3} \left( {h^{2} \over 2\pi k_{B}T} \right)^{3}
\left( {3m_{\alpha} \over m_{\alpha}^{3}} \right)^{3/2} \exp (-E_{*}/
k_{B}T) , \eqno(1.9.31)$$
where $n_{\alpha}$ and $m_{\alpha}$ are the $\alpha$-particle density
mass.
The exponential in (1.9.31) contains the critical temperature dependence,
which is characteristic of resonant reaction rates and is even steeper
than that of (1.9.28). The factor $P_{0}$ need not be calculated
explicitly because it enters in both directions on the left hand side of
(1.9.30). A steady state abundance of $^{12}$C$^{*}$ is achieved in a
time $\sim \Gamma_{\alpha}^{-1} \sim 10^{-15}$ sec. In practice,
(1.9.30) proceeds through the
unbound $^{8}$Be nucleus (a scattering resonance only 92 KeV above the
energy of 2 $\alpha$-particles), rather than through a triple collision,
but this does not affect the thermodynamic argument or the result. The
reaction rate is $n_{*}\Gamma_{\gamma}$. The presence of an excited
state of $^{12}$C at the right energy to facilitate (1.9.30) is the
reason carbon is a relatively abundant element in the universe; this is
apparently fortuitous unless one attributes it to divine intervention,
or argues that if it were not there we would not be present to observe
its absence.
\goodbreak
\vskip 2\baselineskip plus2\baselineskip
\goodbreak
\noindent
{\bf 1.10 Polytropes}
\def\rightheadline{\vbox{\vskip .125truein\line{\tenbf\hfil Polytropes
\qquad\folio}}}
\vskip \baselineskip
\noindent
The solution of the equations (1.3.1--4) of stellar structure is
complicated, because the equation of hydrostatic equilibrium (1.3.1)
is coupled to the equation of energy flow (1.3.4) through (1.3.3)
and the constitutive relation among $P$, $\rho$, and $T$. This
problem is now readily handled numerically, even if some of the
assumptions (most importantly, that of a thermal steady state) made
in deriving (1.3.1--4) are relaxed. In the early (pre-computer)
decades of stellar structure research this was not possible, and
calculations of models simplified still further were performed. These
methods are of more than historical interest, because the very
simplified models which they produced are still powerful qualitative
tools in understanding stars. They cannot replace modern computational
methods of obtaining quantitative results, but they are much
more transparent than a table of numbers, and therefore are very
helpful to the astrophysicist who needs a qualitative understanding
of the properties of self-gravitating configurations of matter.
A {\it polytrope} is a solution of the equation of hydrostatic
equilibrium (1.3.1) under the assumption that the pressure $P$ and the
density $\rho$ are everywhere related by the condition
$$P = K \rho^{n+1 \over n} . \eqno(1.10.1)$$
The quantity $n$ is called the polytropic index.
This relation is formally identical to the adiabatic relation (1.9.7) if
$\gamma = {n+1 \over n}$, but their meanings are quite different.
Equation (1.9.7) describes the variation of the properties of a fluid
element undergoing an adiabatic process. Equation (1.10.1) constrains
the variations of $P$ and $\rho$ with radius in a star, because
if $r$ is introduced as a
parameter it relates $P(r)$ and $\rho (r)$. A star may be described
by (1.10.1) even if the thermodynamic properties of its constituent
matter are described by an adiabatic exponent $\gamma$ different from
${n+1 \over n}$.
Equations
(1.10.1) and (1.9.7) are equivalent if a star is neutrally stable
(equivalently, marginally unstable) against convection, so that the
actual dependence of $P$ on $\rho$ in the star is the same as the
adiabatic one. This will be the case in a star which is completely
convectively mixed, as is believed to be the case for very low mass
main-sequence stars ($M \lapp 0.2 M_{\odot}$). The envelopes of
red giants and supergiants are mixed, and also resemble polytropes
if the gravitational influence of their dense cores may be neglected
(a fair approximation if the envelope is very massive). In each of
these cases $n \approx 3/2$; the deep convective envelope is a
consequence of the high radiative opacity in the surface layers.
Very luminous and massive stars also possess extensive mixed inner
regions, and their envelopes are not far from convective instability.
For these stars $n \approx 3$; convection is a consequence of their
large luminosity.
The assumption of (1.10.1) in place of (1.3.4) permits the stellar
structure equations to be reduced to a single nonlinear ordinary
differential equation characterized by the parameter $n$. This
equation is readily integrated numerically (even without computers!).
Eliminating $M$ from (1.3.1) and (1.3.2), we obtain
$${1 \over r^{2}}{d \over dr}\left( {r^{2} \over \rho}{dP \over dr}
\right) = -4 \pi G \rho . \eqno(1.10.2)$$
Dimensionless variables are defined: $\phi^{n} \equiv \rho / \rho_{c}$
and $\xi \equiv r / \alpha$, where $\rho_{c}$ is the central density,
and the characteristic length (not the radius) $\alpha \equiv
\bigl\lbrack {(n+1)K\rho_{c}^{(1-n)/n} \over 4 \pi G} \bigr\rbrack ^{1/2}
$. Substitution of these variables and (1.10.1) into (1.10.2) yields
the Lane-Emden equation:
$${1 \over \xi^{2}}{d \over d\xi} \left( \xi^{2} {d\phi \over d\xi}
\right) = - \phi^{n} . \eqno(1.10.3)$$
The boundary conditions at $\xi =0$ are $\phi =1$ and ${d\phi
\over d\xi} = 0 $. The surface is defined as the smallest value of
$\xi$ for which $\phi = 0$ (the solution for larger $\xi$ is of no
physical significance). Once a numerical integration in the
dimensionless variables has been tabulated, it is readily applied to a
star of specified $\rho_{c}$ and $K$ by using the definitions of $\phi$
and $\xi$.
Polytropes with certain
values of $n$ are of special interest. The ratios of the central
density $\rho_{c}$ to the mean density $\langle \rho \rangle$
indicate the degree to which mass is concentrated in their centers,
and are a convenient one-parameter description of their structure.
If $n=0$ then (1.10.1) corresponds to an incompressible fluid (only
one value of $\rho$ is permitted) and $\rho_{c} / {\langle \rho \rangle}
=1$. The definitions of $\phi$, $\alpha$, and $K$ become indeterminate;
with a little care they could be redefined, but there are easier ways
of calculating the radius and pressure distribution of a sphere of
incompressible fluid.
If $n=1$ (1.10.3) is linear and may be integrated analytically,
with the result $\phi = \sin \xi / \xi$. Here $\rho_{c} / \langle \rho
\rangle = 3.29$.
If $n=3/2$ (1.10.1) corresponds to an adiabatic star with $\gamma = 5/3$,
and is therefore a good description of fully convective stars with this
equation of state. The calculated $\rho_{c} / \langle \rho \rangle =
5.99$ is the lowest such value which may be obtained for stars composed
of perfect gases.
If $n=3$ (1.10.1) corresponds to an adiabatic star with $\gamma = 4/3$,
and is therefore a good description of fully convective (or nearly
convective) stars with this equation of state. It also turns out that
an $n=3$ polytrope is a fair description of the density structure
$\rho (r)$ of stars in the middle and upper main sequence. Their
deep interiors have steeper density gradients than they would if
they were convective, but the adiabatic $\gamma$ is larger than that
of a fully convective $n=3$ polytrope (for which $\gamma$ must be 4/3);
these two effects roughly cancel. For an $n=3$ polytrope $\rho_{c} /
\langle \rho \rangle = 54.2$. In the present-day Sun this ratio is
calculated to be close to 100, while when the Sun was young it was
about 60 (the difference results from the depletion of hydrogen and
the increase in the molecular weight in the core). The structure and
properties of an $n=3$ polytrope are widely used when a rough but
convenient model of a star is needed for more complex calculations.
If $n=5$ (1.10.3) may also be solved analytically, with the result
$\phi = (1+\xi^{2}/3)^{-1/2}$. For $n \geq 5$ the radius is infinite
because $\xi$ never drops to zero.
If $n \rightarrow \infty$ (1.10.1) approaches an isothermal equation
of state. The definition of $\phi$ becomes improper, but (1.10.2) is
readily integrated without using (1.10.3). At large $r$, $\rho \propto
r^{-2}$ and $M(r) \propto r$, so that both the radius and the total
mass diverge. Such configurations do not describe stars. The upper
atmospheres of stars
may be isothermal but their structure does not approach
an $n=\infty$ polytrope except at very large radii and extremely small
density. Long before this the assumption of hydrostatic equilibrium
will have failed because of the forces applied by the interstellar
medium. These $n=\infty$ polytropes may describe the structure of
gravitating clusters of collisionless objects (clusters of stars or
of galaxies, for example).
\goodbreak
\vskip 2\baselineskip plus2\baselineskip
\goodbreak
\noindent
{\bf 1.11 Mass-Luminosity Relations}
\vskip \baselineskip
\noindent
In {\bf 1.4} we derived scaling relations and made order-of-magnitude
estimates for the characteristic $\rho$, $P$, and $T$ of a star of
given mass $M$ and radius $R$. We now make similar approximations
to estimate the relation between $L$ and $M$ of a main sequence star.
As in {\bf 1.4}, our results are not meant to be numerically accurate,
but rather to be an illuminating guide to the governing physics of stars
of various masses.
We begin by defining $\beta$, the ratio of the gas
pressure to the total pressure:
$$\eqalignno{P_{g}&=\beta P&(1.11.1a)\cr
P_{r}&=(1-\beta)P.&(1.11.1b)\cr}$$
The parameter $\beta$ is a function of $T$ and $\rho$ and, in general,
varies from place to place within a star. Here we assume that it is
a constant throughout a given star. This is true for an $n=3$ perfect
gas polytrope, because in such a polytrope the variations in $\rho$ and
$T$ are related by $\rho \propto T^{n}$, so that the two terms in
(1.4.5) vary in proportion. Stars on the middle and upper main
sequence are approximately described as $n=3$ polytropes, so that for
them our results, derived assuming a constant $\beta$, are fair
approximations to reality.
\def\rightheadline{\vbox{\vskip .125truein\line{\tenbf\hfil
Mass-Luminosity Relations \qquad\folio}}}
Now rewrite (1.3.4) or (1.7.19) in the form
$${dP_{r}(r) \over dr} = {\kappa (r) \rho (r) L(r) \over 4\pi c r^{2}},
\eqno(1.11.2)$$
and divide this equation by (1.3.1). The result is
$${dP_{r} \over dP} = {\kappa (r) L(r) \over 4\pi cGM(r)}. \eqno(1.11.3)
$$
Drop the explicit dependence on $r$, and use (1.11.1) to rewrite this
in terms of a constant $\beta$:
$$L={4\pi cGM \over \kappa} (1-\beta ) . \eqno(1.11.4)$$
This equation is a fundamental relation among $L$, $M$, $\kappa$, and
$\beta$. Because $\beta > 0$ it implies an upper limit on the radiative
luminosity of a star.
In hot, luminous stars $\kappa \approx \kappa_{es}$ (1.9.17), so that
$$L=L_{E}(1-\beta ), \eqno(1.11.5)$$
where the Eddington limiting luminosity $L_{E}$ is defined
$$\hskip 1.0truein\eqalign{
L_{E} \equiv {4\pi cGM \over \kappa_{es}}&=1.47\times 10^{38} {{\rm erg}
\over {\rm sec}\ M_{\odot}}\cr&=3.77\times 10^{4}\left(
{M \over M_{\odot}} \right) L_{\odot}.\cr} \eqno(1.11.6)$$
Therefore, $L_{E}$ is the upper limit to the radiative luminosity of hot
stars. As discussed in {\bf 1.8}, it does not properly apply to the
convective luminosity; it probably does still limit the luminosity of
hot convective stars because their luminosity must flow radiatively
through their atmospheres, where convection is ineffective.
Cool supergiants may perhaps evade the limit (1.11.6) because $\kappa$
may be very small in their cool atmospheres, but there is no evidence
that they actually do so.
We can also express $\beta$ in terms of $\rho$ and $T$, and by so doing
obtain a unique (though very approximate) relation between $L$ and $M$.
>From the definitions of $P_{g}$, $P_{r}$, and $P$ (1.4.5) we obtain,
after eliminating $T$,
$$P = \biggl\lbrack {3 \over a} \left( {N_{A} k_{B} \over \mu} \right)
^{4} {1-\beta \over \beta^{4}} \biggr\rbrack^{1/3} \rho^{4/3} .
\eqno(1.11.7)$$
Now use the relations (1.4.6,7) to express the dependence of $P$ and
$\rho$ on $M$ and $R$. In order to obtain a more useful numerical
result we take the actual values of the coefficients which have been
calculated for an $n=3$ polytrope. The result is
$${1-\beta \over \beta^{4}} = 2.979 \times 10^{-3} \mu^{4} \left( {M
\over M_{\odot}} \right)^{2} . \eqno(1.11.8)$$
This is known as Eddington's quartic equation. From it we may obtain
$\beta (M)$ and $L(M)$. Note that $\beta$ and $L$ do not depend
explicitly on $R$.
At low masses ($M\mu^{2} \ll 20 M_{\odot}$, which includes nearly all
stars) $\beta \rightarrow 1$ and $1-\beta \propto \mu^{4}M^{2}$.
>From (1.11.4), dropping the $\mu$ dependence,
we obtain the mass-luminosity relation for constant $\kappa$:
$$L\propto M^{3}; \eqno(1.11.9)$$
this describes main sequence stars with $\kappa \approx \kappa_{es}$
and holds for \hbox{$M_{\odot} \ll M \ll 50 M_{\odot}$}.
For stars of yet lower mass,
$\kappa$ is roughly described by Kramers' law (1.9.18). If we use
(1.4.6,8) to determine $T$ and $\rho$ in Kramers' law, then
$$L \propto M^{11/2}R^{-1/2} \propto M^{5} , \eqno(1.11.10)$$
where the last relation assumed $M \propto R$, which is implied by the
approximation {\bf (1.9.4)} that thermonuclear energy generation makes
the central temperature nearly independent of $M$.
The Sun is very near the transition between
(1.11.9) and (1.11.10), and has $\beta \approx 0.9996$.
Very low mass stars ($M \lapp 0.2 M_{\odot}$) are fully convective
and their luminosity is determined by their surface boundary condition;
the relations of this section do not apply.
Although these results are only approximate, it is evident that $L$ is
a steeply increasing function of $M$; massive stars are
disproportionately luminous and short-lived, and low mass stars are
disproportionately faint. Very massive stars are also much rarer in
the Galaxy than low mass stars, so that they do not overwhelmingly
dominate the total luminosity produced by stars; stars of moderate
(Solar) mass are not insignificant. If one picks a photon of visible
starlight in the Galaxy (or, similarly, chooses a star randomly on the
sky), there is a significant chance that it will have come from a star
of moderate mass. Very low mass stars, however, are so faint (1.11.10)
that they contribute little to the starlight of the night sky.
For very large masses ($M \gapp 50M_{\odot}$) $\beta \propto M^{-1/2}
\rightarrow 0$ and \hbox{$L \rightarrow L_{E}$,} so that
$$L \propto M . \eqno(1.11.11)$$
Stars this massive are very rare or nonexistent, but (1.11.11)
represents a limiting relation which is approached by the most massive
and luminous stars.
The relations in this section are inapplicable to stars far from the
main sequence. In degenerate dwarfs the pressure is almost entirely
that of electron degeneracy, which was not included in (1.4.5). As a
result $T$ is much lower than (1.4.8) would suggest for these dense
stars, and $L$ is lower by several orders of magnitude. This was a
puzzle until electron degeneracy pressure was understood. White
dwarfs slowly cool to a state in which $T=0$, $\beta =1$, and $L=0$,
in complete contradiction to (1.11.8).
The internal structures of giants and supergiants differ drastically
from those of $n=3$ polytropes, with $\rho_{c} / \langle \rho \rangle$
larger by many orders of magnitude. As a result, the approximate
relations (1.4.6,7) fail completely. The structures of these stars
are discussed in {\bf 1.13}. An analogue of (1.11.8) may be obtained
if, instead of (1.4.7), we write
$$P \sim {GM \over R_{c}}{M \over R^{3}}, \eqno(1.11.12)$$
where $R_{c} = \zeta R$ is the core radius. Then we obtain
$${1-\beta \over \beta^{4}} \sim {\mu^{4}M^{2} \over \zeta^{3}} .
\eqno(1.11.13)$$
Because $\zeta \ll 1$, the limit $\beta \rightarrow 0$ is approached for
much smaller $M$ than would otherwise be the case; this crudely
describes the high luminosity of giant and supergiant stars.
\goodbreak
\vskip 2\baselineskip plus2\baselineskip
\goodbreak
\noindent
{\bf 1.12 Degenerate Stars}
\def\rightheadline{\vbox{\vskip .125truein\line{\tenbf\hfil
Degenerate Stars\qquad\folio}}}
\vskip \baselineskip
\noindent
The basic theory of cold degenerate stars was developed by
Chandrasekhar, shortly after the development of quantum mechanics and
the Pauli exclusion principle made possible the calculation of
degenerate equations of state. His work was concerned with stars in
which the electrons are degenerate, known to astronomers as white
dwarves, and the discussion of this section generally refers to them.
The results and conclusions are also qualitatively (but not
quantitatively) applicable to neutron stars, in which degenerate
neutrons contribute most of the pressure.
The theory of degenerate stars quantitatively predicts a relation
between their masses and radii. It is possible to consider also a
number of small effects not included in the basic theory, such as the
effect of nonzero temperature, the structure of the nondegenerate
atmosphere, the thermodynamics of the ion liquid and its
crystallization, gravitational sedimentation in the atmosphere and
in the deep interior, $\ldots\ $, and to make detailed predictions about
luminosities, spectra, cooling histories, and other properties.
Unfortunately, the quality of the extant data is inadequate to test
either the basic mass-radius relation or these more sophisticated
theories. Reliable masses are known for only a very few degenerate
dwarves, and accurate radii for fewer (if any). Therefore, we are
here concerned chiefly with their most basic properties, for which
the theory, based only on quantum mechanics and Newtonian gravity,
may be assumed with confidence.
In order to calculate the relation between the masses and radii of
degenerate stars, we should calculate the zero-temperature equation of
state $P(\rho)$ for arbitrary density, including the important regime
of $\rho \sim 10^{6} $ gm/cm$^{3}$ lying between the relativistic
(1.9.15) and nonrelativistic (1.9.14) limits. These calculations
exist (see Chandrasekhar 1939), but a qualitative approach using
the virial theorem may be more illuminating.
The total energy $E$ of a star is
$$E = E_{grav} + E_{in} . \eqno(1.12.1)$$
The quantitative value of each of these terms depends on the detailed
forms of $\rho(r)$, $M(r)$, and ${\cal E}(r)$. Their scaling with $M$
and $R$ may be simply written, using relations like (1.4.6,7)
$$\eqalignno{E_{grav}&= - \int_{0}^{R} \rho (r) {GM \over r}
4 \pi r^{2}\ dr \equiv -{\cal A} {GM^{2} \over R}&(1.12.2a)\cr
E_{in}&= \int_{0}^{R} {\cal E} 4 \pi r^{2}\ dr \equiv {\cal B}K \left(
{M \over R^{3}} \right)^{\gamma} R^{3},&(1.12.2b)\cr}$$
where ${\cal A}$ and ${\cal B}$ are dimensionless numbers of order
unity, and we have written $P = K \rho^{\gamma} \propto (M/R^{3})^
{\gamma}$, as is appropriate for adiabatic changes.
For our qualitative considerations, we will assume that
${\cal A}$ and ${\cal B}$ are independent of changes in $R$, although
this is not accurate except in the extreme nonrelativistic
and extreme relativistic limits.
To compute the dynamical equilibrium radius of the star we find the
minimum of the function $E(R)$. If $\gamma = 5/3$ there is a stable
minimum $E$ at
$$R = {2{\cal B} K \over {\cal A} G M^{1/3}} . \eqno(1.12.3)$$
This result is strictly applicable only in the limit $\rho \rightarrow
0$ (in order that $\gamma = 5/3$ hold exactly), $R \rightarrow \infty$,
and $M \rightarrow 0$.
(1.12.3) describes the mass-radius relation of low mass degenerate
dwarves, for which $\gamma = 5/3$ is a good approximation.
(1.12.3) applies also to any series of $n=3/2$ polytropes
with a given value of $K$ (equivalently, with a given specific
entropy); if one adds to the outside of such a star matter with the
same $K$ as that inside, it will shrink. If mass is removed it expands.
This is true both of
degenerate dwarves (for which $S=0$) and of low mass nondegenerate
stars. The appearance of $M$ in the denominator of (1.12.3) may be
surprising; it is a consequence of the compressibility of matter and
the increase of the gravitational force with increasing mass.
For small bodies, like those of everyday life, the density is set by
their atomic properties, (1.9.14) is inapplicable, and $R \propto M^{1/3}
$ (this may be taken as the definition of a planet). Jupiter is near the
dividing line between these two regimes, and thus has approximately the
largest radius possible for {\it any} cold body.
If $\gamma = 4/3$ the condition of minimum $E$ is an equation for $M$,
in which $R$ does not appear:
$$M =\left({{\cal B} K \over {\cal A} G} \right)^{3/2} . \eqno(1.12.4)$$
Such a configuration is an $n=3$ polytrope, and ${\cal A}$ and ${\cal B}$
may be calculated from the known properties of polytropes. We know (see
{\bf 1.5}) that if $\gamma = 4/3$ then $E = 0$, independently of $R$,
so the absence of $R$ from (1.12.4) is no surprise. Because the binding
energy is zero and independent of $R$ the radius is indeterminate.
More remarkable is the fact that a solution exists for only one
allowable mass! This mass is called the Chandrasekhar mass $M_{Ch}$.
Numerical evaluation for the relativistic degenerate equation of state
(1.9.15) gives
$$\eqalign{M_{Ch}&= 5.75 M_{\odot} / \mu_{e}^{2}\cr
&\sim \left({\hbar c \over G m_{P}^{2}}\right)^{3/2} m_{P}.\cr}
\eqno(1.12.5)$$
Calculations of stellar evolution and nucleosynthesis indicate that
real degenerate dwarves will be composed principally of carbon and
oxygen; in the special case in which they are built up by the gradual
accretion of matter supplied from the outside they may be principally
helium. For all of these elements the molecular weight per electron
$\mu_{e} = 2$. $M_{Ch}$ is reduced slightly below the value given in
(1.12.5) by some small effects; the final numerical result is $M_{Ch}
= 1.40 M_{\odot}$ (Hamada and Salpeter 1961).
The unique mass (1.12.4,5) and indeterminate radius apply only in the
limit $R \rightarrow 0$ and $\rho \rightarrow \infty$, because only
in this limit is $\gamma = 4/3$ exactly. Between this singular
solution and the low density limit (1.12.3) there are solutions in
which $4/3 < \gamma < 5/3$, and the equation of state is only partly
relativistic. These solutions are not polytropes (because $\gamma$
is not constant within them), but are readily calculated. Observed
degenerate dwarves are believed to lie in the range $0.4 M_{\odot} \lapp
M \lapp 1.2 M_{\odot}$, and to be in this semirelativistic regime.
Calculations show that for these masses $R \approx
6000 (M_{\odot} / M)$ km is a fair approximation; their characteristic
density is $\rho \sim 2 \times 10^{6}$ gm/cm$^{3}$ (1.9.16). By using
the virial theorem ({\bf 1.5}) we can also estimate the surface
gravitational potential $GM/R \sim m_{e}c^{2}$ (actual calculated values
are $\sim 100$ KeV/amu).
If $M > M_{Ch}$ no zero-temperature hydrostatic solutions exist. This
is probably the most important result in astrophysics, because it means
that stars more massive than $M_{Ch}$ must either reduce their masses
below $M_{Ch}$, end their lives in an explosion, or ultimately collapse.
Equations
(1.12.3,4) apply to nondegenerate stars as well. For example, (1.12.4)
describes the dependence of $K$ on $M$ for very massive stars, which
approximate $n=3$ polytropes because of the importance of radiation
pressure. The factor $K$ has larger values for nondegenerate matter
than for degenerate matter, which has the lowest possible $P$ at a
given $\rho$.
The discussion of this section also applies qualitatively to neutron
stars. Their characteristic density is determined by (1.9.16), and is
$\sim (m_{n}/m_{e})^{3}$ times larger than that of degenerate dwarves,
and their radii are $\sim m_{e}/m_{n}$ times as large. Because $K$ is
independent of $m_{d}$ in the relativistic regime (1.9.15),
(1.12.4) predicts essentially
the same limiting mass for neutron stars as for degenerate
dwarves. Their surface gravitational potential $GM/R \sim m_{n}c^{2}$
(actual numerical values are believed to be $\sim 100$ MeV/amu). The
strong interactions between neutrons make (1.9.14,15) and (1.12.4) rough
approximations at best; the equation of state of neutron matter is
controversial. However, the conclusion that as $\rho \rightarrow \infty$
the Fermi momentum $p_{F} \rightarrow \infty$ and $\gamma \rightarrow
4/3$, which implies an upper mass limit $M_{Ch}^{ns}$, is inescapable.
The effects of general relativity are also significant, and tend to
increase the strength of gravity and to reduce $M_{Ch}^{ns}$, though
they are not as large as the uncertainties in the equation of state.
Most calculations agree that for neutron stars $R \approx 10$ km,
approximately independent of mass for $0.5M_{\odot} \lapp M \lapp
M_{Ch}^{ns}$. The value of $M_{Ch}^{ns}$ is also controversial,
but it is probably in the range $1.40 M_{\odot} < M_{Ch}^{ns} \lapp
2.5 M_{\odot}$. The lower bound on $M_{Ch}^{ns}$ is firm, and is
obtained from the observation of neutron stars of this mass in the
binary pulsar PSR 1913+16, for which relativistic orbital effects
permit accurate determination of the the pulsar mass (this is the
only accurately determined neutron star mass). Because it is hard
to imagine the production of neutron stars except as a consequence of
the collapse of degenerate dwarves (or the degenerate dwarf cores
of larger stars), it is likely that most neutron stars have $M \geq
M_{Ch}$, which also implies $M_{Ch}^{ns} \geq M_{Ch}$. The upper
bound on $M_{Ch}^{ns}$ is less certain, but uncontroversial properties
of the equation of state imply that it cannot much exceed $2.5M_{\odot}$.
It is frequently pointed out in nontechnical astronomy books that a
teaspoon (5 cm$^{3}$) of typical white dwarf matter has a mass of about
10 tons. It is not usually added that the internal energy of this
teaspoonful is equivalent to that released by about 20 megatons of
high explosive.
\goodbreak
\vskip 2\baselineskip plus2\baselineskip
\goodbreak
\noindent
{\bf 1.13 Giants and Supergiants}
\def\rightheadline{\vbox{\vskip .125truein\line{\tenbf\hfil
Giants and Supergiants\qquad\folio}}}
\vskip \baselineskip
\noindent
Main sequence and degenerate stars may be approximately described as
polytropes. For giants and supergiants polytropic models
and the rough approximations of {\bf 1.4} fail
completely. These stars contain dense cores, resembling degenerate
dwarves, and very dilute extended envelopes. The ratio $\rho_{c} /
\langle \rho \rangle$, which is 54.2 for an $n=3$ polytrope, may be
$\sim 10^{12}$ (or more, in extreme cases).
The development of giant structure in a star is the outcome of complex
couplings among the equations (1.3.1--7). Their solutions, obtained
numerically, are the only proper explanation of giant structure, but
it is useful to consider rough arguments. If the core and envelope are
considered separately, the approximations of {\bf 1.4}, and simple
models, may still be qualitatively informative.
A main sequence star will eventually exhaust the hydrogen at its center,
leaving a core of nearly pure helium. For stars of masses approximately
equal to or exceeding that of the Sun, this happens in less than the age
of the Galaxy. Stars have presumably been born throughout that time
(there are few quantitative data), so that there now exist stars of a
variety of masses which have helium cores. Because the star continues
to radiate energy, in a thermal steady state hydrogen must continue to
be transformed to helium. This will happen in the hottest part of the
star which contains hydrogen, a thin shell just outside the helium core.
The helium core will be essentially inert. In steady state it is
isothermal at the temperature of the hydrogen burning shell at its outer
surface. Because of the thermostatic properties ({\bf 1.9.4}) of
thermonuclear energy release, we may roughly regard this shell as
having a fixed $T_{\circ} \approx 4 \times 10^{7 \circ}$K.
Once the core has accumulated a significant fraction (typically 8\%)
of the stellar mass, its temperature $T_{\circ}$ is insufficient to
satisfy the equation (1.3.1) of hydrostatic equilibrium. Equation
(1.4.8) explains why; $T$ is set by the shell temperature, and hence
by the structure of the outer star, but the core has a larger value
of $\mu$ (4/3 for helium) and its higher density leads to a large $M/R$.
It then contracts, producing a higher $T$ (this process is stable, by
the arguments of {\bf 1.5}). Now heat flows outward, which leads to yet
higher $T$ (the negative effective specific heat discussed in {\bf 1.5}).
The heat flow reduces the entropy of the core, until its equation
of state approaches that of a degenerate electron gas; the core
comes to resemble a degenerate dwarf inside the larger star.
Core contraction will be interrupted when the temperature becomes high
enough ($T \gapp 10^{8 \circ}$K) for reaction (1.9.30) to take place,
and exothermically to convert helium to carbon (auxiliary reactions
also produce oxygen and rarer elements). This leaves an inert
carbon-oxygen core surrounded by a double shell, the outer shell
burning hydrogen and the inner shell burning helium. Such double
shells have a complex and unstable evolution, but this is irrelevant
to our rough description of the structure of a giant star.
The combination of a degenerate dwarf core with a thermostatic
boundary condition produces the extended low density envelope of a
giant star. A simple argument uses the scale height of the matter
overlying the core. If $L$ is not close to $L_{E}$ radiation pressure
is unimportant (see 1.11.5). An isothermal gas, supported in
hydrostatic equilibrium by gas pressure against a uniform acceleration
of gravity $g = GM_{c}/R_{c}^{2}$, has a density which varies as
$$\rho \propto \exp (-r/h) , \eqno(1.13.1)$$
where the scale height $h$ is
$$\eqalign{h&={R_{c}^{2} N_{A} k_{B} T \over GM_{c}\mu}\cr
&={R_{c} k_{B} T \over E_{b} \mu},\cr} \eqno(1.13.2)$$
and $E_{b}$ is the gravitational binding energy per nucleon. The matter
is not accurately isothermal and $g$ is not strictly constant, but for
$h \ll R_{c}$ these are good approximations. The approximations made in
{\bf 1.4} were equivalent to assuming $h \sim R$ everywhere in the
stellar interior, and fail at the core-envelope boundary where $h \ll
R_{c} \ll R$.
For a degenerate core with $M_{c} = 0.7 M_{\odot}$, at $r=R_{c}$ we find
$h/R_{c} \approx 0.055 \ll 1$. As a result, the density drops by a
large factor in the region just outside the core boundary, where $g$ is
large. If the envelope contains a significant amount of mass, as it
will in most giants, then this low density requires it to have a large
volume and a large radius. Very crudely, we might expect the
radius to be larger than that of a main sequence star (which the
envelope would otherwise resemble) by a factor $\sim \exp \bigl(R_{c}
/(3h) \bigr)
\sim 10^{2}$ -- $10^{3}$, which is consistent with the radii of large
red giants. If the core is more massive the density will be yet lower
and the radius yet larger. The actual radius and $T_{e}$ of a red
giant are determined by the surface boundary conditions on its outer
convective zone.
This argument is not applicable when $L \approx L_{E}$,
because then the scale height is larger by a factor
$\beta^{-1} \gg 1$. Instead, we equate the pressure of radiation to
the pressure produced by the weight of the overlying matter, so that
$${a \over 3} T_{\circ}^{4} \sim {GM_{c}\rho \over R_{c}}.\eqno(1.13.3)$$
For $M_{c} = 1.2 M_{\odot}$ ($\beta \ll 1$ only as $M_{c}
\rightarrow M_{Ch}$) and $T_{\circ} = 4 \times 10^{7 \circ}$K we
estimate $\rho \sim 0.02$ gm/cm$^{3}$. If the envelope roughly
resembles an $n=3$ polytrope, as is likely, then its radius will be
$\sim 20 R_{\odot}$. Such a star is not nearly as large as a red giant
or supergiant, but possesses a less extreme form of their structure of
dense core and extended envelope. Because of its high luminosity
and moderate radius its surface temperature is high. These
stars are found in a region of the Hertzsprung-Russell diagram between
the red supergiants and the upper main sequence, called the horizontal
branch (most horizontal branch stars are produced differently, when
rapid helium burning increases $R_{c}$ and $h$).
\goodbreak
\vskip 2\baselineskip plus2\baselineskip
\goodbreak
\noindent
{\bf 1.14 Spectra}
\vskip \baselineskip
\noindent
The study of astronomical spectra is a large field of research. Here
we only draw a few qualitative conclusions useful in modelling novel
objects and phenomena.
The radiation we observe from stars is produced in their atmospheres,
and its spectrum reflects the physical conditions there. These
atmospheres may usually be approximated as plane-parallel layers, so
that in the equation (1.7.6) of radiative transfer we may neglect
the term containing $1/r$. Then
$${\partial I(\tau ,\vartheta )\over \partial \tau} \cos \vartheta
- I(\tau ,\vartheta ) + S(\tau ) = 0 , \eqno(1.14.1)$$
where the source function $S(\tau ) \equiv j(\tau ) / 4 \pi \kappa
(\tau )$, and the optical depth $\tau$ is defined by $d\tau \equiv
\kappa \rho dr$, and $\tau \rightarrow 0$ as $r \rightarrow \infty$.
$I$, $j$, $\kappa$, and $\tau$ all implicitly depend on $\nu$. For
$\cos \vartheta > 0$ this equation has the formal solution
$$I(\tau ,\vartheta ) = \int_{\tau}^{\infty} S(\tau^{\prime}) \exp
\bigl\lbrack - (\tau^{\prime} - \tau )\sec \vartheta \bigr\rbrack
\sec \vartheta \ d\tau^{\prime} . \eqno(1.14.2)$$
The emergent flux is that at $\tau = 0$:
$$I(0,\vartheta ) = \int_{0}^{\infty} S(\tau^{\prime}) \exp (-\tau^
{\prime} \sec \vartheta ) \sec \vartheta \ d\tau^{\prime}.\eqno(1.14.3)$$
The emergent flux is a weighted average of $S$ over the atmosphere,
with most of the contribution coming from the range $0\leq \tau^{\prime}
\lapp \cos \vartheta$.
\def\rightheadline{\vbox{\vskip .125truein\line{\tenbf\hfil Spectra
\qquad\folio}}}
The opacity $\kappa_{\nu}$ of matter typically has the form shown in
Figure 1.3, with sharp atomic lines superposed on a slowly varying
continuum. The lines are those of the species abundant in the
atmosphere, which depend on its chemical composition, density, and
(most sensitively) temperature. In hot stars the strong lines are those
of species like He II and C III, in somewhat cooler stars those of
He I or H I, in yet cooler stars Ca I and Fe I, and in the coolest
stars those of molecules like TiO.
\topinsert
\vskip 13.5truecm
\ctrline{{\bf Figure 1.3.} Varieties of Spectra.}
\endinsert
In the simplest stellar atmospheres matter is in thermodynamic
equilibrium, there is no scattering, $S=B$ (the Planck function), and
the temperature increases monotonically inward. Then $I$ reflects the
value of $B$ in the region $\tau \sim 1$, and we may approximate
$I_{\nu}(\tau =0) \approx B_{\nu}\bigl(T(\tau_{\nu}=2/3)\bigr)$.
At a line frequency $\nu_{l}$ the opacity $\kappa_{\nu_{l}}$ is large
and $\tau_{\nu_{l}}=2/3$ high in the atmosphere, where $T$ and $B$
are low, while outside the line $\kappa_{\nu}$ is small and $\tau_{\nu}
=2/3$ much deeper in the atmosphere. The result is an absorption line
spectrum, as shown in the figure.
In many stars the upper atmosphere is much hotter than the rest of
the atmosphere. In the Sun the upper atmosphere and corona are heated
by acoustic (or magneto-acoustic) waves generated within the convective
zone. In a few stars a strong radiation flux from a luminous binary
companion heats the upper atmosphere; this is found in some
companions to strong X-ray sources. When the temperature profile
is inverted in this manner there results an emission line spectrum,
as shown in the figure. Often a weak emission line spectrum from the
highest levels of the atmosphere is superposed on a stronger absorption
line spectrum.
If line scattering opacity is important it may also produce an
absorption line, regardless of the temperature gradient in the
atmosphere. The mechanism is outlined in {\bf 1.7.3}; the presence of
scattering reduces the emissivity. At such frequencies the diffuse
reflectivity of the atmosphere is significant, so that a fraction of
the flux is the (zero) reflected flux of the dark sky. If there is
significant scattering opacity in the continuum, but the line opacity
is absorptive, then the sky is reflected in the continuum and the line
will appear in emission. These processes are known as the Schuster
mechanism.
In a dilute gas cloud the upper limit in the integral (1.14.3) is
$\tau_{max}$, the total optical depth integrated through the cloud.
Often the cloud is so rarefied and transparent that $\tau_{max} \ll 1$
at all frequencies. Then (1.14.3) may be approximated
$$I(0,\vartheta ) \approx {j_{\nu} \over 4\pi} \sec \vartheta
\int \rho \ dr . \eqno(1.14.4)$$
The frequency dependence of the emergent spectrum is that of the
emissivity $j_{\nu}$. Under these conditions LTE is usually inaccurate;
the emergent spectrum qualitatively resembles that of the opacity
$\kappa_{\nu}$, although quantitative results require a calculation of
the various atomic and ionic processes. There is an emission
line spectrum in which the lines are extremely strong, carrying a
significant fraction of the total flux. Such spectra are observed
from interstellar clouds, winds flowing outward from stars, the
debris of stellar explosions, stellar coronae, laboratory gas discharge
lamps, and in other circumstances in which $\int \rho \ dr$ is very
small. Because the emitting volume may be large, the total mass and
radiated power need not be small, despite the low density.
These classes of spectra are very different, and may often be
identified at a glance, even though they are not usually found in their
pure states. This is useful in attempting to construct a rough model of
a novel astronomical object, because the densities, dimensions, and
directions of energy flow are readily constrained. Images are
not available for many interesting astronomical objects, because of
their small angular sizes, so that the first step in understanding them
is the identification of their components and the construction of a
rough model of their geometry, their physical parameters, and of the
important physical processes.
\goodbreak
\vskip 2\baselineskip plus2\baselineskip
\goodbreak
\noindent
{\bf 1.15 Mass Loss}
\vskip \baselineskip
\noindent
Spectroscopic observations show that many stars lose mass. Typically,
the observations show emission lines whose Doppler widths indicate
the flow velocity. In most cases the line shape does not directly
establish that the mass is flowing outward, only that the star is
surrounded by a dilute cloud of gas with the appropriate distribution
of velocities; it is usually not possible to determine from the data
which velocities are found at which points in space, but outflow is
often the only plausible interpretation. In some cases the outflowing
gas absorbs an observable amount of the stellar line radiation, and
the resulting complex (P Cygni) line profiles may be interpreted
unambiguously as mass outflow.
Some stars are observed in ordinary photographs (or infrared images)
to be surrounded by luminous gas clouds they have expelled; in some
cases these clouds have visibly expanded since the first photographs
were taken. Many different kinds of stars lose mass by a variety of
mechanisms and at widely varying rates. Even the Sun loses mass at
the very small rate of $\sim 10^{-15} M_{\odot}$/year in the Solar
wind, produced by the thermal expansion of its hot corona. All
stars with convective surface layers are expected to have coronae,
whose mass loss rates should be much greater in larger stars with lower
surface gravity.
It is known that some stars born with $M$ substantially larger than
$M_{Ch}$ have evolved into degenerate dwarves; this establishes that,
in some cases, a star may lose the greater part of its mass. In this
section I briefly and qualitatively discuss mass loss mechanisms which
may occur in luminous stars, where the mass loss rate is often high.
Most of these processes are not understood quantitatively.
\def\rightheadline{\vbox{\vskip .125truein\line{\tenbf\hfil Mass Loss
\qquad\folio}}}
In a very luminous star the radiation pressure approaches the total
pressure, and $\beta \rightarrow 0$ (1.11.1). How closely a star
approaches the neutrally stable limit $\beta =0$ depends on the detailed
calculation of its structure; we know (see 1.11.8) that very massive
stars and giant stars with dense degenerate cores have small $\beta$.
>From the equation of hydrostatic equilibrium we have
$$-{\beta GM\rho \over r^{2}} = {dP_{g} \over dr} , \eqno(1.15.1)$$
so that in this limit the gradient of the gas pressure becomes zero.
Essentially the entire weight of the matter is supported by the
gradient of radiation pressure; in other words, the force of gravity
and the force of radiation pressure cancel. If $\beta =0$ exactly,
nothing is left to resist the gradient of $P_{g}$, and the stellar
material will float off into space. This argument suggests that
very luminous stars are likely to lose mass.
This conclusion is at least qualitatively correct, and may be reached
on simple energetic grounds by noting that as $\beta \rightarrow 0$
we have $\gamma \rightarrow 4/3$, and that if $\gamma = 4/3$ the binding
energy $E=0$ (see {\bf 1.5}). It is possible to show, by manipulation
of the stellar structure equations, that $n=3$ polytropes (which stars
approach as $\beta \rightarrow 0$) with a constant $\beta$ are
neutrally stable against convection if $\gamma = 4/3$ (also approached
as $\beta \rightarrow 0$); it is unsurprising that a star with zero
binding energy should be neutrally stable against the interchange of
its parts.
Should $L$ exceed $4\pi cGM/\kappa$, the star becomes
unstable against convection, and if convection is efficient it carries
the excess flux. The radiative luminosity does not exceed $4\pi cGM/
\kappa$ and the gradient of radiation pressure does not exceed the
force of gravity. In fact, $L > 4\pi cGM/\kappa$ in the envelopes
of many cool giants and supergiants, where $\kappa$ is large; these
stars generally do not lose mass rapidly. Only if convection is
incapable of carrying the heat flux does excess radiation pressure
drive a mass efflux.
It is comparatively easy to disrupt a star with $\beta\ll 1$ if it can be
disturbed, but reliable calculation is difficult. Possible disturbances
include fluctuations and instability in the nuclear energy generation
rate (known to occur in supergiants with degenerate cores and double
burning shells), and the inefficient convection present in the outer
layers of cool giants and supergiants. Such stars may lose their
entire envelopes in response to modest disturbances (most notably in
the formation of planetary nebulae by supergiant stars), but it is
also necessary to consider less dramatic mass loss processes. These
are easier to observe (because they last longer) and to calculate.
The most important factor leading to steady mass loss is probably an
increase in $\kappa$ in optically thin regions above the photosphere.
Because the density and optical depth are low, convection cannot
transport heat effectively, and probably does not take place. Instead,
matter can actually be subject to a force of radiation pressure
exceeding that of gravity (a situation which would not occur in a
stellar interior in hydrostatic equilibrium). At least
two kinds of physical processes, changing ionization balance and grain
formation, may produce such an abrupt jump in $\kappa$.
The temperature of a grey body (one whose opacity is independent of
frequency) just outside a photosphere will be lower
than that of one just inside by a factor of about $2^{-1/4} = 0.84$;
outside, the black body radiation field only fills the $2\pi$ steradians
of outward-directed rays, while the $2\pi$ steradians of inward-directed
rays have little intensity. The opacity of stellar atmospheres is not
accurately grey, but this is still a reasonable estimate of the
temperature drop. Such a drop may be sufficient to shift substantially
the ionization balance, and therefore the opacity. In addition, the
Rosseland mean opacity, derived for stellar interiors (in which $\tau_
{\nu} \gapp 1$ at all frequencies) is inapplicable in optically thin
regions. In the opposite limit, $\tau_{\nu} \lapp 1$ at all
frequencies, the radiation force is proportional to $\int I_{\nu}
\kappa_{\nu}\ d\nu$; the arithmetic mean opacity exceeds the Rosseland
mean. Strong atomic or ionic lines may now make a large
contribution to the force of radiation pressure, and calculations show
that in the upper atmospheres of hot luminous stars the net
acceleration may be upward.
A simple argument makes it possible to estimate the mass efflux.
Suppose the matter is accelerated by radiation pressure in a spectral
line of rest frequency $\nu_{\circ}$. Radiation between $\nu_{\circ}$
and $\nu_{\circ}(1-v/c)$ may be absorbed or scattered
by the outflowing wind; the total pressure
the radiation field can exert on the matter may be $\sim
H_{\nu_{\circ}}\nu_{\circ}v/c^{2}$. Equate this to the momentum
efflux rate per unit area $\dot m v$ (where $\dot m$ is the rate of
mass loss per unit area) to obtain the total mass loss rate $\dot M$:
$$\eqalign{{\dot M} &=4\pi R^{2} {\dot m}\cr
&\sim 4\pi R^{2} H_{\nu_{\circ}} \nu_{\circ}/c^{2}\cr
&\sim L/c^{2},\cr} \eqno(1.15.2)$$
where we have approximated $L \equiv \int H_{\nu}\ d\nu \approx
H_{\nu_{\circ}}{\nu_{\circ}}$. This result is an upper bound, because
not all of the radiation at the frequencies of the Doppler-shifted line
will be absorbed or scattered, and because gravity has been neglected.
${\dot M}$ is independent of $v$; calculations
usually show that $v$ is a few times the stellar surface escape
velocity. If $N$ strong lines contribute to the
absorption of radiation, then ${\dot M}$ may be larger by a factor
$N$, which may be $\gg 1$, but not by orders of magnitude.
The mass efflux rate (1.15.2) is small, although readily observable
spectroscopically. It is roughly the same as the equivalent mass
carried off by the radiation field itself; we know that during a star's
life thermonuclear reactions convert less than 1\% of its mass to
energy. If $L \approx L_{E}$ then ${\dot M} \lapp 10^{-9} M$/year.
A luminous star with a very cool surface (a red supergiant) may lose
mass in a related, but more effective way. Above its photosphere the
temperature may be cool enough for carbon (and other elements or
molecules) to condense into grains; this is probably the origin of
interstellar grains. These grains (in particular, those of carbon)
are very effective absorbers of visible and near-infrared radiation
across the entire spectrum ($\kappa \sim 10^{5}$ cm$^{2}$/gm), so that
the pressure of the radiation on the matter may be $\sim \int H_{\nu}
\ d\nu/c$; the Doppler shift factor $v/c \ll 1$ does not enter. Then
we obtain
$${\dot M} \sim {L \over vc} . \eqno(1.15.3)$$
For a red supergiant $v \sim \sqrt{GM/R} \sim 30$ km/sec, so this
result is $\sim 10^{4}$ times as large as (1.15.2). The time required
to halve $M$ may be as short as $\sim$ 30,000 years. Such a large mass
loss rate may change the evolutionary history of the star; for
example, it may reduce $M$ below $M_{Ch}$. Unfortunately, it has not
been possible to quantitatively calculate mass loss by this process,
although observations indicate it does take place.
The highest estimate of mass loss comes if the energy of the star's
radiation may be efficiently used to overcome the gravitational
binding energy and to provide kinetic energy, so that
$${\dot M} \sim {L \over v^{2}} . \eqno(1.15.4)$$
In order for this to occur the radiation must be trapped between an
expanding optically thick outflow and the luminous stellar core, and
be the working fluid in a heat engine. The required optical depth
at all frequencies is $\tau \gapp c/v \gg 1$; the acceleration occurs
in the stellar interior rather than in the atmosphere. However, such
a radiatively accelerated optically thick shell will probably be
unstable to convection if it is in hydrostatic equilibrium. Mass loss
rates as high as (1.15.4) may be obtained when hydrostatic equilibrium
does not apply; for example if $L$ rises significantly above $L_{E}$
in a time $