15 Breakdown of Local Thermodynamic Equilibrium

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Stellar Atmospheres



Breakdown of Local

Thermodynamic

Equilibrium

. . .

Thus far we have made considerable use of the concepts of equilibrium. In

the stellar interior, the departures from a steady equilibrium distribution for the
photons and gas particles were so small that it was safe to assume that all the
constituents of the gas behaved as if they were in STE. However, near the surface of
the star, photons escape in such a manner that their energy distribution departs from
that expected for thermodynamic equilibrium, producing all the complexities that are
seen in stellar spectra. However, the mean free path for collisions between the
particles that make up the gas remained short compared to that of the photons, and so
the collisions could be regarded as random. More importantly, the majority of the
collisions between photons and the gas particles could be viewed as occurring
between particles in thermodynamic equilibrium. Therefore, while the radiation field
departs from that of a black body, the interactions determining the state of the gas
continue to lead to the establishment of an energy distribution for the gas particles
characteristic of thermodynamic equilibrium. This happy state allowed the complex
properties of the gas to be determined by the local temperature alone and is known as
LTE.

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Breakdown of Local Thermodynamic Equilibrium

However, in the upper reaches of the atmosphere, the density declines to such

a point that collisions between gas particles and the remaining "equilibrium" photons
will be insufficient for the establishment of LTE. When this occurs, the energy level
populations of the excited atoms are no longer governed by the Saha-Boltzmann
ionization-excitation formula, but are specified by the specific properties of the
atoms and their interactions.

Although the state of the gas is still given by a time-independent distribution

function and can be said to be in steady or statistical equilibrium, that equilibrium
distribution is no longer the maximal one determined by random collisions. We have
seen that the duration of an atom in any given state of excitation is determined by the
properties of that atomic state. Thus, any collection of similar atoms will attempt to
rearrange their states of excitation in accordance with the atomic properties of their
species. Only when the interactions with randomly moving particles are sufficient to
overwhelm this tendency will the conditions of LTE prevail. When these interactions
fail to dominate, a new equilibrium condition will be established that is different
from LTE. Unfortunately, to find this distribution, we have to calculate the rates at
which excitation and de-excitation occur for each atomic level in each species and to
determine the population levels that are stationary in time. We must include
collisions that take place with other constituents of the gas as well as with the
radiation field while including the propensity of atoms to spontaneously change their
state of excitation. To do this completely and correctly for all atoms is a task of
monumental proportions and currently is beyond the capability of even the fastest
computers. Thus we will have to make some approximations. In order for the
approximations to be appropriate, we first consider the state of the gas that prevails
when LTE first begins to fail.

A vast volume of literature exists relating to the failure of LTE and it would

be impossible to cover it all. Although the absorption of some photon produced by
bound-bound transitions occurs in that part of the spectrum through which the
majority of the stellar flux flows, only occasionally is the absorption by specific lines
large enough to actually influence the structure of the atmosphere itself. However, in
these instances, departures from LTE can affect changes in the atmosphere's structure
as well as in the line itself. In the case of hydrogen, departures in the population of
the excited levels will also change the "continuous" opacity coefficient and produce
further changes in the upper atmosphere structure. To a lesser extent, this may also
be true of helium. Therefore, any careful modeling of a stellar atmosphere must
include these effects at a very basic level. However, the understanding of the physics
of non-LTE is most easily obtained through its effects on specific atomic transitions.
In addition, since departures from LTE primarily occur in the upper layers of the
atmosphere and therefore affect the formation of the stellar spectra, we concentrate
on this aspect of the subject.

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Stellar Atmospheres

15.1 Phenomena Which Produce Departures from Local

Thermodynamic

Equilibrium

Principle of Detailed Balancing

Under the assumption of LTE, the material particles of the gas are

assumed to be in a state that can be characterized by a single parameter known as the
temperature. Under these conditions, the populations of the various energy levels of
the atoms of the gas will be given by Maxwell-Boltzmann statistics regardless of the
atomic parameters that dictate the likelihood that an electron will make a specific
transition. Clearly the level populations are constant in time. Thus the flow into any
energy level must be balanced by the flow out of that level. This condition must hold
in any time-independent state. However, in thermodynamic equilibrium, not only
must the net flow be zero, so must the net flows that arise from individual levels.
That is, every absorption must be balanced by an emission. Every process must be
matched by its inverse. This concept is known as the

principle of detailed balancing

Consider what would transpire if this were not so. Assume that the values of

the atomic parameters governing a specific set of transitions are such that absorptions
from level 1 to level 3 of a hypothetical atom having only three levels are vastly
more likely than absorptions to level 2 (see figure 15.1). Then a time-independent
equilibrium could only be established by transitions from level 1 to level 3 followed
by transitions from level 3 to level 2 and then to level 1. There would basically be a
cyclical flow of electrons from levels 1

→

1. The energy to supply the

absorptions would come from either the radiation field or collisions with other
particles. To understand the relation of this example to LTE, consider a radiation-less
gas where all excitations and de-excitations result from collisions. Then such a
cyclical flow would result in energy corresponding to the 1

→

3 transition being

systematically transferred to the energy ranges corresponding to the transitions 3

→

2 and 2

→

1. This would lead to a departure of the energy momentum distribution

from that required by Maxwell-Boltzmann statistics and hence a departure from
LTE. But since we have assumed LTE, this process cannot happen and the upward
transitions must balance the downward transitions. Any process that tends to drive
the populations away from the values they would have under the principle of detailed
balancing will generate a departure from LTE. In the example, we considered the
case of a radiationless gas so that the departures had to arise in the velocity
distributions of the colliding particles. In the upper reaches of the atmosphere, a
larger and larger fraction of the atomic collisions are occurring with photons that are
departing further and further from the Planck function representing their
thermodynamic equilibrium distribution. These interactions will force the level
populations to depart from the values they would have under LTE.

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Breakdown of Local Thermodynamic Equilibrium

Figure 15.1

shows the conditions that must prevail in the case of detailed

balancing (panel a) and for interlocking (panel b).The transition from
level(1) to level (3) might be a resonance line and hence quite strong.
The conditions that prevail in the atmosphere can then affect the line
strengths of the other lines that otherwise might be accurately described
by LTE.

Interlocking

Consider a set of lines that have the same upper level (see Figure

15.1). Any set of lines that arise from the same upper level is said to be interlocked
(see R.Woolley and D.Stibbs

). Lines that are interlocked are subject to the cyclical

processes such as we used in the discussion of detailed balancing and are therefore
candidates to generate departures from LTE. Consider a set of lines formed from
transitions such as those shown in Figure 15.1. If we assume that the transition from
1

3 is a resonance line, then it is likely to be formed quite high up in the atmosphere

where the departures from LTE are the largest. However, since this line is
interlocked with the lines resulting from transitions 3

→

2 and 3

→

1, we can expect

the departures affecting the resonance line to be reflected in the line strengths of the
other lines. In general, the effect of a strong line formed high in the atmosphere
under conditions of non-LTE that is interlocked with weaker lines formed deeper in
the atmosphere is to fill in those lines, so that they appear even weaker than would
otherwise be expected. A specific example involves the red lines of Ca II(

λλ

8498,

λλ

8662,

λλ

8542), which are interlocked with the strong Fraunhofer H and K

resonance lines. The red lines tend to appear abnormally weak because of the
photons fed into them in the upper atmosphere from the interlocked Fraunhofer H &
K lines.

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Stellar Atmospheres

Collisional versus Photoionization

We have suggested that it is the relative dominance of the interaction

of photons over particles that leads to departures from LTE that are manifest in the
lines. Consider how this notion can be quantified. The number of photoionizations
from a particular state of excitation that takes place in a given volume per second
will depend on the number of available atoms and the number of ionizing photons.
We can express this condition as

(15.1.1)

The frequency

corresponds to the energy required to ionize the atomic state under

consideration. The integral on the far right-hand side is essentially the number of
ionizing photons (modulo 4

), so that this expression really serves as a definition of

as the rate coefficient for photoionizations from the ith state to the continuum. In

a similar manner, we may describe the number of collisional ionizations by

(15.1.2)

Here,

is the rate at which atoms in the ith state are ionized by collisions with

particles in the gas. The quantity

(v) is the collision cross section of the particular

atomic state, and it must be determined either empirically or by means of a lengthy
quantum mechanical calculation; and

(v) is the velocity distribution function of the

particles.

In the upper reaches of the atmosphere, the energy distribution functions of

the constituents of the gas depart from their thermodynamic equilibrium values. The
electrons are among the last particles to undergo this departure because their mean
free path is always less than that for photons and because the electrons suffer many
more collisions per unit time than the ions. Under conditions of thermodynamic
equilibrium, the speeds of the electrons will be higher than those of the ions by (m

)

as a result of the equipartition of energy. Thus we may generally ignore

collisions of ions of atomic weight

with anything other than electrons. Since the

electrons are among the last particles to depart from thermodynamic equilibrium, we
can assume that the velocity distribution

(v) is given by Maxwell-Boltzmann

statistics. Under this assumption

Ω

will depend on atomic properties and the

temperature alone. If we replace J

with B

(T), then we can estimate the ratio of

photoionizations to collisional ionizations R

under conditions that prevail in the

atmospheres of normal stars. Karl Heintz Böhm

has used this procedure along with

the semi-classical Thomson cross section for the ion to estimate this ratio. Böhm
finds that only for the upper-lying energy levels and at high temperatures and
densities will collisional ionizations dominate over photoionizations. Thus, for most

402

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Breakdown of Local Thermodynamic Equilibrium

lines in most stars we cannot expect electronic collisions to maintain the atomic-level
populations that would be expected from LTE. So we are left with little choice but to
develop expressions for the energy-level populations based on the notion that the
sum of

all

transitions into and out of a level must be zero. This is the weakest

condition that will yield an atmosphere that is time-independent.

15.2 Rate Equations for Statistical Equilibrium

The condition that the sum of all transitions into and out of any specific level must be
zero implies that there is no net change of any level populations. This means that we
can write an expression that describes the flow into and out of each level,
incorporating the detailed physics that governs the flow from one level to another.
These expressions are known as the

rate equations for statistical equilibrium

. The

unknowns are the level populations for each energy level which will appear in every
expression for which a transition between the respective states is allowed. Thus we
have a system of n simultaneous equations for the level populations of n states.
Unfortunately, as we saw in estimating the rates of collisional ionization and
photoionization, it is necessary to know the radiation field to determine the
coefficients in the rate equations. Thus any solution will require self-consistency
between the radiative transfer solution and the statistical equilibrium solution.
Fortunately, a method for the solution of the radiative transfer and statistical
equilibrium equations can be integrated easily in the iterative algorithm used to
model the atmosphere (see Chapter 12). All that is required is to determine the source
function in the line appropriate for the non-LTE state.

Since an atom has an infinite number of allowed states as well as an infinite

number of continuum states that must be considered, some practical limit will have
to be found. For the purpose of showing how the rate equations can be developed, we
consider two simple cases.

Two-Level Atom

It is possible to describe the transitions between two bound states we

did for photo- and collisional ionization. Indeed, for the radiative processes, basically
we have already done so in (Section 11.3) through the use of the Einstein
coefficients. However, since we are dealing with only two levels, we must be careful
to describe exactly what happens to a photon that is absorbed by the transition from
level 1 to level 2. Since the level is not arbitrarily sharp, there may be some
redistribution of energy within the level. Now since the effects of non-LTE will
affect the level populations at various depths within the atmosphere, we expect these
effects will affect the line profile as well as the line strength. Thus, we must be clear
as to what other effects might change the line profile. For that reason, we assume
complete redistribution of the line radiation. This is not an essential assumption, but
rather a convenient one.

403

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Stellar Atmospheres

If we define the probability of the absorption of a photon at frequency

' by

(15.2.1)

and the probability of reemission of a photon at frequency

(15.2.2)

then the concept of the redistribution function describes to what extent these photons
are correlated in frequency. In Chapter 9, we introduced a fairly general notion of
complete redistribution by stating that

' and

would not be correlated. Thus,

(15.2.3)

Under the assumption of complete redistribution, we need only count radiative
transitions by assuming that specific emissions are unrelated to particular
absorptions. However, since the upward radiative transitions in the atom will depend
on the availability of photons, we will have to develop an equation of radiative
transfer for the two-level atom.

Equation of Radiative Transfer for the Two-Level Atom

In Chapter 11

[equations (11.3.6) and (11.3.7)] we described the emission and absorption
coefficients, j

and

, respectively, in terms of the Einstein coefficients. Using these

expressions, or alternatively just balancing the radiative absorptions and emissions,
we can write an equation of radiative transfer as

(15.2.4)

This process of balancing the transitions into and out of levels is common to any
order of approximation in dealing with statistical equilibrium. As long as all the
processes are taken into account, we will obtain an expression like equation (15.2.4)
for the transfer equation for multilevel atoms [see equation (15.2.25)]. Equation
(15.2.4) can take on a somewhat more familiar form if we define

(15.2.5)

Then the equation of transfer becomes

404

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Breakdown of Local Thermodynamic Equilibrium

(15.2.6)

where

(15.2.7)

Making use of the relationships between the Einstein coefficients determined in
Chapter 11 [equation (11.3.5)], we can further write

(15.2.8)

Under conditions of LTE

(15.2.9)

so that we recover the expected result for the source function, namely

(15.2.10)

Two-Level-Atom Statistical Equilibrium Equations

The solution to

equation (15.2.6) will provide us with a value of the radiation field required to
determine the number of radiative transitions. Thus the total number of upward
transitions in the two-level atom is

(15.2.11)

Similarly, the number of downward transitions is

(15.2.12)

The requirement that the level populations be stationary means that

(15.2.13)

so that the ratio of level populations is

(15.2.14)

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Stellar Atmospheres

Now consider a situation where there is no radiation field and the collisions

are driven by particles characterized by a maxwellian energy distribution. Under
these conditions, the principle of detailed balancing requires that

(15.2.15)

(15.2.16)

This argument is similar to that used to obtain the relationships between the Einstein
coefficients and since the collision coefficients depend basically on atomic constants,
equation (15.2.16) must hold under fairly arbitrary conditions. Specifically, the result
will be unaffected by the presence of a radiation field. Thus we may use it and the
relations between the Einstein coefficients [equations (11.3.5)] to write the line
source function as

15.2.17)

If we let

(15.2.18)

then the source function takes on the more familiar form

(15.2.19)

The quantity

is, in some sense, a measure of the departure from LTE and is

sometimes called the

departure coefficient

. A similar method for describing the

departures from LTE suffered by an atom is to define

(15.2.20)

where N

is the level population expected in LTE so that

is just the ratio of the

actual population to that given by the Saha-Boltzmann formula. From that definition,
equation (15.2.14), and the relations among the Einstein coefficients we get

(15.2.21)

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Breakdown of Local Thermodynamic Equilibrium

Two-Level Atom plus Continuum

The addition of a continuum increases the algebraic difficulties of the

above analysis. However, the concepts of generating the statistical equilibrium
equations are virtually the same. Now three levels must be considered. We must keep
track of transitions to the continuum as well as the two discrete energy levels. Again,
we assume complete redistribution within the line so that the line source function is
given by equation (15.2.8), and the problem is to find the ratio of the populations of
the two levels.

We begin by writing the rate equations for each level which balance all

transitions into the level with those to the other level and the continuum. For level 1,

(15.2.22)

The parameter R

is the photoionization rate defined in equation (15.1.1), while R

is the analogous rate of photorecombination. When the parameter

Ω

contains the

subscript k, it refers to collisional transitions to or from the continuum. The term on
the left-hand side describes all the types of transitions from level 1 which are photo-
and collisional excitations followed by the two terms representing photo- and
collisional, ionizations respectively. The two large terms on the right-hand side
contain all the transitions into level 1. The first involves spontaneous and stimulated
radiative emissions followed by collisionly stimulated emissions. The second term
describes the recombinations from the continuum. The parameter N

will in general

represent those ions that have been ionized from the ith state.

We may write a similar equation

(15.2.23)

for level 2 by following the same prescription for the meaning of the various terms.
Again letting the terms on the left-hand side represent transitions out of the two
levels while terms on the right-hand side denote inbound transitions, we find the rate
equation for the continuum is

(15.2.24)

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Stellar Atmospheres

However, this equation is not linearly independent from the other two and can be
generated simply by adding equations (15.2.22) and (15.2.23). This is an expression
of continuity and will always be the case regardless of how many levels are
considered. There will always be one less independent rate equation than there are
levels. An electron that leaves one state must enter another, so its departure is not
independent from its arrival. If all allowed levels are counted, as they must be if the
equations are to be complete, this interdependence of arrivals and departures of
specific transitions will make the rate equation for one level redundant. Noting that
the same kind of symmetry described by equation (15.2.15) also holds for the
collisional ionization and recombination coefficients, we may solve equations
(15.2.22) and (15.2.23) for the population ratio required for the source function given
by equation (15.2.8). The algebra is considerably more involved than for the two
levels alone and yields

a source function of the form

If the terms involving

dominate the source function, the lin is said to be

collisionly dominated, while if the terms involving

are the largest, the line is

said to be dominated by photoionization. If

)

(

but

(or vice

versa), the line is said to be mixed. Some examples of lines in the solar spectrum
that fall into these categories are given in Table 15.1.

│

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Breakdown of Local Thermodynamic Equilibrium

Multilevel Atom

A great deal of effort has gone into approximating the actual case

of many levels of excitation by setting up and solving the rate equations for three
and four levels or approximating any particular transition of interest by an
"equivalent two level atom" (see D.Mihalas

, pp. 391-394). However, the advent

of modern, swift computers has made most of these approximations obsolete.
Instead, one considers an n-level atom (with continuum) and solves the rate
equations directly. We have already indicated that this procedure can be
integrated into the standard algorithm for generating a model atmosphere quite
easily. Consider the generalization of equations (15.2.22) through (15.2.24).
Simply writing equations for each level, by balancing the transitions into the level
with those out of the level, will yield a set of equations which are linear in the
level populations. However, as we have already indicated, these equations are
redundant by one. So far we have only needed population ratios for the source
function, but if we are to find the population levels themselves, we will need an
additional constraint. The most obvious constraint is that the total number of
atoms and ions must add up to the abundance specified for the atmosphere.
Mihalas

suggests using charge conservation, which is a logically equivalent

constraint. Whatever additional constraint is chosen, it should be linear in the
level populations so that the linear nature of the equations is not lost.

It is clear that the equations are irrevocably coupled to the radiation field

through the photoexcitation and ionization terms. It is this coupling that led to the
rather messy expressions for the source functions of the two-level atom. However,
if one takes the radiation field and electron density as known, then the rate
equations have the form

(15.2.26)

where

is a matrix whose elements are the coefficients multiplying the population

levels and

is a vector whose elements are the populations of the energy levels for

all species considered in the calculation. The only nonzero element of the constant
vector

arises from the additional continuity constraint that replaced the redundant

level equation. These equations are fairly sparse and can be solved quickly and
accurately by well-known techniques.

409

Since the standard procedure for the construction of a model atmosphere is

an iterative one wherein an initial guess for the temperature distribution gives rise to
the atmospheric structure, which in turn allows for the solution of the equation of
radiative transfer, the solution of the rate equations can readily be included in this
process. The usual procedure is to construct a model atmosphere in LTE that nearly

⋅

Stellar Atmospheres

satisfies radiative equilibrium. At some predetermined level of accuracy, the rate
equations are substituted for the Saha-Boltzmann excitation and ionization equations
by using the existing structure (electron density and temperature distribution) and
radiation field. The resulting population levels are then used to calculate opacities
and the atmospheric structure for the next iteration. One may even chose to use an
iterative algorithm for the solution of the linear equations, for an initial guess of the
LTE populations will probably be quite close to the correct populations for many of
the levels that are included. The number of levels of excitation that should be
included is somewhat dictated by the problem of interest. Depending on the state of
ionization, four levels are usually enough to provide sufficient accuracy. However,
some codes routinely employ as many as eight. One criterion of use is to include as
many levels as is necessary to reach those whose level populations are adequately
given by the Saha-Boltzmann ionization-excitation formula.

Many authors consider the model to be a non-LTE model if hydrogen alone

has been treated by means of rate equations while everything else is obtained from
the Saha-Boltzmann formula. For the structure of normal stellar atmospheres, this is
usually sufficient. However, should specific spectral lines be of interest, one should
consider whether the level populations of the element in question should also be
determined from a non-LTE calculation. This decision will largely be determined by
the conditions under which the line is formed. As a rule of thumb, if the line occurs
in the red or infra-red spectral region, consideration should be given to a non-LTE
calculation. The hotter the star, the more this consideration becomes imperative.

Thermalization Length

Before we turn to the solution of the equation of radiative transfer for

lines affected by non-LTE effects, we should an additional concept which helps
characterize the physical processes that lead to departures from LTE. It is known as
the

thermalization length

. In LTE all the properties of the gas are determined by the

local values of the state variables. However, as soon as radiative processes become
important in establishing the populations of the energy levels of the gas, the problem
becomes global. Let

be the mean free path of a photon between absorptions or

scatterings and

ℒ

be the mean free path between collisional destructions. If

scatterings dominate over collisions, then

ℒ

will not be a "straight line" distance

through the atmosphere. Indeed,

ℒ

if A

>> C

. That is, if the probability of

radiative de-excitation is very much greater than the probability of collisional de-
excitation, then an average photon will have to travel much farther to be destroyed by
a collision than by a radiative interaction. However,

ℒ

if C

>> A

. In this

instance, all photons that interact radiatively will be destroyed by collisions.

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Breakdown of Local Thermodynamic Equilibrium

If the flow of photons is dominated by scatterings, then the character of the

radiation field will be determined by photons that originate within a sphere of radius

ℒ

rather than

. However,

ℒ

should be regarded as an upper limit because many

radiative interactions are pure absorptions that result in the thermalization of the
photon as surely as any collisional interaction. In the case when

ℒ

, some

photons will travel a straight-line distance equal to

ℒ

, but not many. A better

estimate for an average length traveled before the photon is thermalized would
include other interactions through the notion of a "random walk". If n is the ratio of
radiative to collisional interactions, then a better estimate of the thermalization length
would be

(15.2.27)

If the range of temperature is large over a distance corresponding to the

thermalization length

, then the local radiation field will be characterized by a

temperature quite different from the local kinetic gas temperature. These departures
of the radiation field from the local equilibrium temperature will ultimately force the
gas out of thermodynamic equilibrium. Clearly, the greatest variation in temperature
within the thermalization sphere will occur as one approaches the boundary of the
atmosphere. Thus it is no surprise that these departures increase near the boundary.
Let us now turn to the effects of non-LTE on the transfer of radiation.

15.3 Non-LTE Transfer of Radiation and the Redistribution

Function

While we did indicate how departures of the populations of the energy levels from
their LTE values could be included in the construction of a model atmosphere so that
any structural effects are included, the major emphasis of the effects of non-LTE has
been on the strengths and shapes of spectral lines. During the discussion of the two
level atom, we saw that the form of the source function was somewhat different from
what we discussed in Chapter 10. Indeed, the equation of transfer [equation (15.2.6)]
for complete redistribution appears in a form somewhat different from the customary
plane-parallel equation of transfer. Therefore, it should not be surprising to find that
the effects of non-LTE can modify the profile of a spectral line. The extent and
nature of this modification will depend on the nature of the redistribution function as
well as on the magnitude of the departures from LTE. Since we already introduced
the case of complete redistribution [equations (15.2.1) and (15.2.2), we begin by
looking for a radiative transfer solution for the case where the emitted and absorbed
photons within a spectral line are completely uncorrelated.

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Stellar Atmospheres

a Complete

Redistribution

In Chapter 14, we devoted a great deal of effort to developing

expressions for the atomic absorption coefficient for spectral lines that were
broadened by a number of phenomena. However, we dealt tacitly with absorption
and emission processes as if no energy were exchanged with the gas between the
absorption and reemission of the photon. Actually this connection was not necessary
for the calculation of the atomic line absorption coefficient, but this connection is
required for calculating the radiative transfer of the line radiation. Again, for the case
of pure absorption there is no relationship between absorbed and emitted photons.
However, in the case of scattering, as with the Schuster-Schwarzschild atmosphere,
the relationship between the absorbed and reemitted photons was assumed to be
perfect. That is, the scattering was assumed to be completely coherent. In a stellar
atmosphere, this is rarely the case because micro-perturbations occurring between
the atoms and surrounding particles will result in small exchanges of energy, so that
the electron can be viewed as undergoing transitions

within

the broadened energy

level. If those transitions are numerous during the lifetime of the excited state, then
the energy of the photon that is emitted will be uncorrelated with that of the absorbed
photon. In some sense the electron will "lose all memory" of the details of the
transition that brought it to the excited state. The absorbed radiation will then be
completely redistributed throughout the line. This is the situation that was described
by equations (15.2.1) through (15.2.3), and led to the equation of transfer (15.2.6) for
complete redistribution of line radiation.

Although this equation has a slightly different form from what we are used

to, it can be put into a familiar form by letting

(15.3.1)

It now takes on the form of equation (10.1.1), and by using the classical solution
discussed in Chapter 10, we can obtain an integral equation for the mean intensity in
the line in terms of the source function.

(15.3.2)

This can then be substituted into equation (15.2.19) to obtain an integral equation for
the source function in the line.

(15.3.3)

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Breakdown of Local Thermodynamic Equilibrium

The integral over

results from the integral of the mean intensity over all

frequencies in the line [see equation (15.2.19)]. Note the similarity between this
result and the integral equation for the source function in the case of coherent
scattering [equation (9.1.14)]. Only the kernel of the integral has been modified by
what is essentially a moment in frequency space weighted by the line profile function

(t). This is clearly seen if we write the kernel as

(15.3.4)

so that the source function equation becomes a Schwarzschild-Milne equation of the
form

(15.3.5)

Since

(15.3.6)

the kernel is symmetric in

and t, so that

(

,t) =

(t,

). This is the same

symmetry property as the exponential integral E

│

-t

│

in equation 10.1.14.

Unfortunately, for an arbitrary depth dependence of

(t), equations (15.3.4) and

(15.3.5) must be solved numerically. Fortunately, all the methods for the solution of
Schwarzschild-Milne equations discussed in Chapter 10 are applicable to the
solution of this integral equation.

While it is possible to obtain some insight into the behavior of the solution

for the case where

(t)

≠

(t) (see Mihalas

, pp.366-369), the insight is of dubious

value because it is the solution for a special case of a special case. However, a
property of such solutions, and of noncoherent scattering in general, is that the core
of the line profile is somewhat filled in at the expense of the wings. As we saw for
the two-level atom with continuum, the source function takes on a more complicated
form. Thus we turn to the more general situation involving partial redistribution.

Hummer Redistribution Functions

The advent of swift computers has made it practical to model the

more complete description of the redistribution of photons in spectral lines.
However, the attempts to describe this phenomenon quantitatively go back to
L.Henyey

who carried out detailed balancing

within an energy level

to describe the

way in which photons are actually redistributed across a spectral line. Unfortunately,
the computing power of the time was not up to the task, and this approach to the
problem has gone virtually unnoticed. More recently, D.Hummer

has classified the

problem of redistribution into four main categories which are widely used today. For

413

⋅

Stellar Atmospheres

these cases, the energy levels are characterized by Lorentz profiles which are
appropriate for a wide range of lines. Regrettably, for the strong lines of hydrogen,
many helium lines as well as most strong resonance lines, this characterization is
inappropriate (see Chapter 14) and an entirely different analysis must be undertaken.
This remains one of the current nagging problems of stellar astrophysics. However,
the Hummer classification and analysis provides considerable insight into the
problems of partial redistribution and enables rather complete analyses of many lines
with Lorentz profiles produced by the impact phase-shift theory of collisional
broadening.

Let us begin the discussion of the Hummer redistribution functions with a

few definitions. Let

(

be the probability that an absorbed photon having

frequency

' is scattered into the frequency interval

ξ→

+ d

. Furthermore, let the

probability density function

(

) be normalized so that

∫

) d

' = 1. That is,

the absorbed photon must go somewhere. If this is not an appropriate result for the
description of some lines, the probability of scattering can be absorbed in the
scattering coefficient (see Section 9.2). In addition, let

(n^',^n ) be the probability

density function describing scattering from a direction ^n ' into n^ , also normalized so
that the integral over all solid angles, [

∫

(n^', ^n) d

Ω

]/(4

) = 1. For isotropic scattering,

(n^',n^ ) = 1, while in the case of Rayleigh Scattering

(n^', n^ ) = 3[1 + (n^'

⋅

n^)

]/4. We

further define

(

') d

' as the relative [that is,

∫

(

') d

' = 1 ] probability that a photon

with frequency

' is absorbed. These probability density functions can be used to

describe the redistribution function introduced in Chapter 9.

In choosing to represent the redistribution function in this manner, it is tacitly

assumed that the redistribution of photons in frequency is independent of the
direction of scattering. This is clearly not the case for atoms in motion, but for an
observer located in the rest frame of the atom it is

usually

a reasonable assumption.

The problem of Doppler shifts is largely geometry and can be handled separately.
Thus, the probability that a photon will be absorbed at frequency

' and reemitted at a

frequency

(15.3.7)

David Hummer

has considered a number of cases where

, and

, take on special

values which characterize the energy levels and represent common conditions that
are satisfied by many atomic lines.

Emission and Absorption Probability Density Functions for the Four

Cases Considered by Hummer

Consider first the case of coherent scattering

where both energy levels are infinitely sharp. Then the absorption and reemission
probability density functions will be given by

414

⋅

Breakdown of Local Thermodynamic Equilibrium

(15.3.8)

If the lower level is broadened by collisional radiation damping but the upper level
remains sharp, then the absorption probability density function is a Lorentz profile
while the reemission probability density function remains a delta function, so that

15.3.9)

If the lower level is sharp but the upper level is broadened by collisional radiation
damping, then both probability density functions are given Lorentz profiles since the
transitions into and out of the upper level are from a broadened state. Thus,

(15.3.10)

Since

' and

are uncorrelated, this case represents a case of complete

redistribution of noncoherent scattering. Hummer gives the joint probability of
transitions from a broadened lower level to a broadened upper level and back again
as

(15.3.11)

This probability must be calculated as a unit since

' is the same for both

and

Unfortunately a careful analysis of this function shows that the lower level is
considered to be sharp for the reemitted photon and therefore is inconsistent with the
assumption made about the absorption. Therefore, it will not satisfy detailed
balancing in an environment that presupposes LTE. A correct quantum mechanical
analysis

gives

415

⋅

Stellar Atmospheres

(15.3.12)

A little inspection of this rather messy result shows that it is symmetric in

and

, which it must be if it is to obey detailed balancing. In addition, the function

has two relative maxima at

' =

and

' =

. Since the center of the two energy

levels represent a very likely transition, transitions from the middle of the lower level
and back again will be quite common. Under these conditions

' =

and the

scattering is fully coherent. On the other hand, transitions from the exact center of the
lower level (

' =

) will also be very common. However, the return transition can be

to any place in the lower level with frequency

. Since the function is symmetric in

and

, the reverse process can also happen. Both these processes are fully

noncoherent so that the relative maxima occur for the cases of fully coherent and
noncoherent scattering with the partially coherent photons being represented by the
remainder of the joint probability distribution function.

Effects of Doppler Motion on the Redistribution Functions

Consider an

atom in motion relative to some fixed reference frame with a velocity

. If a photon

has a frequency

' as seen by the atom, the corresponding frequency in the rest frame

(15.3.13)

Similarly, the photon emitted by the atom will be seen in the rest frame, Doppler-
shifted from its atomic value by

(15.3.14)

Thus the redistribution function that is seen by an observer in the rest frame is

(15.3.15)

416

⋅

Breakdown of Local Thermodynamic Equilibrium

We need now to relate the scattering angle determined from

to the

angle between the atomic velocity and the directions of the incoming and outgoing
scattered photon. Consider a coordinate frame chosen so that the x-y plane is the
scattering plane and the x axis lies in the scattering plane midway between the
incoming and outgoing photon (see Figure 15.2). In this coordinate frame, the
directional unit vectors n^ and n^ ' have Cartesian components given by

nˆ

•

′

(15.3.16)

Figure 15.2

displays a Cartesian coordinate frame where the x-axis

bisects the angle between the incoming and outgoing photon and the
x-y plane is the scattering plane.

417

⋅

Stellar Atmospheres

Now if we assume that the atoms have a maxwellian velocity distribution

(15.3.17)

we can obtain the behavior of an ensemble of atoms by averaging equation (15.3.15)
over all velocity. First it is convenient to make the variable transformations

(15.3.18)

so that the components of the particle's velocity projected along the directions of the
photon's path become

(15.3.19)

and the velocity distribution is

(15.3.20)

The symbol

means du

We define the ensemble average over the velocity of the redistribution

function as

(15.3.21)

(15.3.22)

A coordinate rotation by y/2 about the y axis so that n^ ' is aligned with x^ (see Figure
15.2) leads to the equivalent, but useful, form

418

⋅

Breakdown of Local Thermodynamic Equilibrium

(15.3.23)

We are now in a position to evaluate the effects of thermal Doppler motion

on the four cases given by Hummer

, represented by equations (15.3.8) through

(15.3.12). The substitution of these forms of f(

') and p(

) into equation (15.3.22)

or equation (15.3.23) will yield the desired result. The frequencies

' and

must be

Doppler shifted according to equations (15.3.13) and (15.3.14) and some difficulty
may be encountered for the case of direct forward or back scattering (that is,

= 0)

and when one of the distribution functions is a delta function (i.e., for a sharp energy
level). The fact that

= 0 for these cases should be invoked before any variable

transformations are made for the purposes of evaluating the integrals.

Making a final transformation to a set of dimensionless frequencies

(15.3.24)

we can obtain the following result for Hummer's case I:

(15.3.25)

Consider what the emitted radiation would look like for an ensemble of

atoms illuminated by an isotropic uniform radiation field I

. Substitution of such a

radiation field into equation (9.2.29) would yield

(15.3.26)

which after some algebra gives

(15.3.27)

This implies that the emission of the radiation would have exactly the same form as
the absorption profile. But this was our definition of complete redistribution [see
equation (15.2.3)]. Thus, although a single atom behaves coherently, an ensemble of
thermally moving atoms will produce a line profile that is equivalent to one suffering
complete redistribution of the radiation over the Doppler core. Perhaps this is not too
surprising since the motion of the atoms is totally uncorrelated so that the Doppler
shifts produced by the various motions will mimic complete redistribution.

419

⋅

Stellar Atmospheres

As one proceeds with the progressively more complicated cases, the results

become correspondingly more complicated to derive and express. Hummer's cases II
and III yield

(15.3.28)

and

(15.3.29)

respectively. There is little point in giving the result for case IV as given by equation
(15.3.11). But the result for the correct case IV (sometimes called case V) that is
obtained from equation (15.3.12) is of some interest and is given by McKenna

(15.3.30)

where

(15.3.31)

and the function

K(a,x),

which is known as the shifted Voigt function is defined by

(15.3.32)

420

⋅

Breakdown of Local Thermodynamic Equilibrium

Unfortunately, all these redistribution functions contain the scattering angle

explicitly and so by themselves are difficult to use for the calculation of line

profiles. Not only does the scattering angle appear in the part of the redistribution
function resulting from the effects of the Doppler motion, but also the scattering
angle is contained in the phase function

(n^', ). Thus, the Doppler motion can be

viewed as merely complicating the phase function. While there are methods for
dealing with the angle dependence of the redistribution function (see McKenna

nˆ

they are difficult and beyond the present scope of this discussion. They are, however,
of considerable importance to those interested in the state of polarization of the line
radiation. For most cases, the phase function is assumed to be isotropic, and we may
remove the angle dependence introduced by the Doppler motion by averaging the
redistribution function over all angles, as we did with velocity. These averaged forms
for the redistribution functions can then be inserted directly into the equation of
radiative transfer. As long as the radiation field is nearly isotropic and the angular
scattering dependence (phase function) is also isotropic, this approximation is quite
accurate. However, always remember that it is indeed an approximation.

Angle-Averaged Redistribution Functions

We should remember from

Chapter 13 [equation (13.2.14)], and the meaning of the redistribution function [see
equation (9.2.29)], that the equation of transfer for line radiation can be written as

(15.3.33)

Here the parameter

ℒ

is not to be considered constant with depth as it was for the

Milne-Eddington atmosphere. If we assume that the radiation field is nearly
isotropic, then we can integrate the equation of radiative transfer over

and write

(15.3.34)

If we define the angle-averaged redistribution function as

(15.3.35)

then in terms of the absorption and reemission probabilities f(

') and p(

) it

becomes

(15.3.36)

421

⋅

Stellar Atmospheres

The phase function

(n^', n^ ) must be expressed in the coordinate frame of the

observer, that is, in terms of the incoming and outgoing angles that the photon makes
with the line of sight (see Figure 15.3).

Figure 15.3

describes the scattering event as seen in the coordinate

frame of the observer. The

k - axis points along the normal to the

atmosphere or the observer's line-of-sight. The angle

is the angle

between the scattered photon and the observer's line-of-sight, while
the angle

' is the corresponding angle of the incoming photon. The

quantities

and

' are just the cosines of these respective angles.

422

⋅

Breakdown of Local Thermodynamic Equilibrium

We may write the phase function

(n^', n^ ) as

(15.3.37)

so that the angle-averaged redistribution function becomes

(15.3.38)

The two most common types of phase functions are isotropic scattering and

Rayleigh scattering. Although the latter occurs more frequently in nature, the former
is used more often because of its simplicity. Evaluating these phase functions in
terms of the observer's coordinate frame yields

(15.3.39)

In general, the appropriate procedure for calculating the angle-averaged

redistribution functions involves carrying out the integrals in equation (15.3.38) and
then applying the effects of Doppler broadening so as to obtain a redistribution
function for the four cases described by Hummer. For the first two cases, the delta
function representing the upper and lower levels requires that some care be used in
the evaluation of the integrals (see Mihalas

, pp. 422-433). In terms of the

normalized frequency x, the results of all that algebra are, for case I

(15.3.40)

For case II the result is somewhat more complicated where

(15.3.41)

while for case III it is more complex still:

(15.3.42)

Note that for all these cases the redistribution function is symmetric in x and

x'. From equations (15.2.1) through (15.2.3), it is clear that the angle-averaged

423

⋅

Stellar Atmospheres

redistribution functions will yield a complete redistribution profile in spite of the fact
that case I is completely coherent.

To demonstrate the effect introduced by an anisotropic phase function, we

give the results for redistribution by electrons. Although we have always considered
electron scattering to be fully coherent in the atom's coordinate frame, the effect of
Doppler motion can introduce frequency shifts that will broaden a spectral line. This
is a negligible effect when we are calculating the flow of radiation in the continuum,
but it can introduce significant broadening of spectral lines. If we assume that the
scattering function for electrons is isotropic, then the appropriate angle-averaged
redistribution function has the form

(15.3.43)

However, the correct phase function for electron scattering is the Rayleigh phase
function given in the observer's coordinate frame by the second of equations
(15.3.39). The angle-averaged redistribution function for this case has been
computed by Hummer and Mihalas

and is

Clearly the use of the correct phase function causes a significant increase in the
complexity of the angle-averaged redistribution function. Since the angle-averaged
redistribution function itself represents an approximation requiring an isotropic
radiation field, one cannot help but wonder if the effort is justified.

We must also remember that the entire discussion of the four Hummer cases

relied on the absorption and reemission profiles being given by Lorentz profiles in
the more complicated cases. While considerable effort has been put into calculating
the Voigt functions and functions related to them that arise in the generation of the
redistribution functions

, some of the most interesting lines in stellar astrophysics

are poorly described by Lorentz profiles. Perhaps the most notable example is the
lines of hydrogen. At present, there is no quantitative representation of the
redistribution function for any of the hydrogen lines. While noncoherent scattering is
probably appropriate for the cores of these lines, it most certainly is not for the
wings. Since a great deal of astrophysical information rests on matching theoretical
line profiles of the Balmer lines to those of stars, greater effort should be made on the
correct modeling of these lines, including the appropriate redistribution functions.

The situation is even worse when one tries to estimate the polarization to be

expected within a spectral line. It is a common myth in astrophysics that the radiation

424

⋅

Breakdown of Local Thermodynamic Equilibrium

in a spectral line should be locally unpolarized. Hence, the global observation of
spectral lines should show no net polarization. While this is true for simple lines that
result only from pure absorption, it is not true for lines that result from resonant
scattering. The phase function for a line undergoing resonant scattering is essentially
the same as that for electron scattering - the Rayleigh phase function. While
noncoherent scattering processes will tend to destroy the polarization information,
those parts of the line not subject to complete redistribution will produce strong local
polarization. If the source of the radiation does not exhibit symmetry about the line
of sight, then the sum of the local net polarization will not average to zero as seen by
the observer. Thus there should be a very strong wavelength polarization through
such a line which, while difficult to model, has the potential of placing very tight
constraints on the nature of the source. Recently McKenna

has shown that this

polarization, known to exist in the specific intensity profiles of the sun, can be
successfully modeled by proper treatment of the redistribution function and a careful
analysis of the transfer of polarized radiation. So it is clear that the opportunity is
there remaining to be exploited. The existence of modern computers now makes this
feasible.

15.4 Line Blanketing and Its Inclusion in the Construction of

Model Stellar Atmospheres and Its Inclusion in the

Construction of Model Stellar Atmospheres

In Chapter 10, we indicated that the presence of myriads of weak spectral lines could
add significantly to the total opacity in certain parts of the spectrum and virtually
blanket the emerging flux forcing it to appear in other less opaque regions of the
spectrum. This is particularly true for the early-type stars for which the major
contribution from these lines occurs in the ultraviolet part of the spectrum, where
most of the radiative flux flows from the atmosphere. Although it is not strictly a
non-LTE effect, the existence of these lines generally formed high in the atmosphere
can result in structural changes to the atmosphere not unlike those of non-LTE. The
addition of opacity high up in the atmosphere tends to heat the layers immediately
below and is sometimes called

backwarming

425

Because of their sheer number, the inclusion of these lines in the calculation

of the opacity coefficient poses some significant problems. The simple approach of
including sufficient frequency points to represent the presence of all these lines
would simply make the computational problem unmanageable with even the largest
of computing machines that exist or can be imagined. Since the early attempts of
Chandrasekhar

, many efforts have been made to include these effects in the

modeling of stellar atmospheres. These early efforts incorporated approximating the
lines by a series of frequency "pickets". That is, the frequency dependence would be
represented by a discontinuous series of opaque regions that alternate with
transparent regions. One could then average over larger sections of the spectrum to
obtain a mean line opacity for the entire region. However, this did not represent the

⋅

Stellar Atmospheres

effect on the photon flow through the alternatingly opaque and relatively transparent
regions with any great accuracy. Others tried using harmonic mean line opacities to
reduce this problem. Of these attempts, two have survived and are worthy of
consideration.

Opacity Sampling

This conceptually simple method of including line blanketing takes

advantage of the extremely large number of spectral lines. The basic approach is to
represent the frequency-dependent opacity of all the lines as completely as possible.
This requires tabulating a list of all the likely lines and their relative strengths. For an
element like iron, this could mean the systematic listing of several million lines. In
addition, the line shape for each line must be known. This is usually taken to be a
Voigt function for it represents an excellent approximation for the vast majority of
weak lines. However, its use requires that some estimate of the appropriate damping
constant be obtained for each line. In many cases, the Voigt function has been
approximated by the Doppler broadening function on the assumption that the
damping wings of the line are relatively unimportant. At any frequency the total line
absorption coefficient is simply the sum of the significant contributions of lines that
contribute to the opacity at that frequency, weighted by the relative abundance of the
absorbing species. These abundances are usually obtained by assuming that LTE
prevails and so the Saha-Boltzmann ionization-excitation equation can be used.

If one were to pick a very large number of frequencies, this procedure would

yield an accurate representation of the effects of metallic line blanketing. However, it
would also require prodigious quantities of computing time for modeling the
atmosphere. Sneden et al.

have shown that sufficient accuracy can be obtained by

choosing far fewer frequency points than would be required to represent each line
accurately. Although the choice of randomly distributed frequency points which
represent large chunks of the frequency domain means that the opacity will be
seriously overestimated in some regions and underestimated in others, it is possible
to obtain accurate structural results for the atmosphere if a large enough sample of
frequency points is chosen. This sample need not be anywhere near as large as that
required to represent the individual lines, for what is important for the structure is
only the net flow of photons. Thus, if the frequency sampling is sufficiently large to
describe the photon flow over reasonably large parts of the spectrum, the resulting
structure and the contribution of millions of lines will be accurately represented. This
procedure will begin to fail in the higher regions of the atmosphere where the lines
become very sharp and non-LTE effects become increasingly important. In practice,
this procedure may require the use of several thousand frequency points whereas the
correct representation of several million spectral lines would require tens of millions
of frequency points. For this reason (and others), this approach has been extremely
successful as applied to the structure of late-type model atmospheres where the

426

⋅

Breakdown of Local Thermodynamic Equilibrium

opacity is dominated by the literally millions of bound-bound transitions occurring in
molecules. The larger the number of weak lines and the more uniform their
distribution, the more accurate this procedure becomes. However, the longer the lists
of spectral lines, the more computer time will be required to carry out the calculation.
This entire procedure is generally known as

opacity sampling

and it possesses a great

degree of flexibility in that all aspects of the stellar model that may affect the line
broadening can be included

ab initio

for each model. This is not the case with the

competing approach to line blanketing.

Opacity Distribution Functions

This approach to describing the absorption by large numbers of lines

also involves a form of statistical sampling. However, here the statistical
representation is carried out over even larger regions of the spectra than was the case
for the opacity sampling scheme. This approach has its origins in the mean opacity
concept alluded to earlier. However, instead of replacing the complicated variation of
the line opacity over some region of the spectrum with its mean, consider the fraction
of the spectral range that has a line opacity less than or equal to some given value.
For small intervals of the range, this may be a fairly large number since small
intervals correspond to the presence of line cores. If one considers larger fractions of
the interval, the total opacity per unit frequency interval of this larger region will
decrease, because the spaces between the lines will be included. Thus, an opacity
distribution function represents the probability that a randomly chosen point in the
interval will have an opacity less than or equal to the given value (see Figure 15.4).
The proper name for this function should be the inverse cumulative opacity
probability distribution function, but in astronomy it is usually referred to as just the

opacity distribution function

or (ODF). Carpenter

gives a very complete description

of the details of computing these functions while a somewhat less complete picture is
given by Kurucz and Pettymann

and by Mihalas

(pp. 167-169).

The ODF gives the probability that the opacity is a particular fraction of a

known value for any range of the frequency interval, and the ODF may be obtained
from a graph that is fairly simple to characterize by simple functions. This approach
allows the contribution to the total opacity due to spectral lines appropriate for that
range of the interval to be calculated. Unfortunately, the magnitude of that given
value will depend on the chemical composition and the details of the individual line-
broadening mechanisms. Thus, any change in the chemical composition, turbulent
broadening, etc., will require a recalculation of the ODF.

427

⋅

Stellar Atmospheres

Figure 15.4

schematically shows the opacity of a region of the spectrum

represented in terms of the actual line opacity (panel a) and the opacity
distribution function (panel b).

In addition, ODFs must be calculated as a function of temperature and pressure (or
alternatively, electron density), and so their tabular representation can be extremely
large. Their calculation also represents a significant computational effort. However,
once ODF's exist, their inclusion in a stellar atmosphere code is fairly simple and the
additional computational load for the construction of a model atmosphere is not
great, particularly compared to the opacity sampling technique. This constitutes the
primary advantage of this approach for the generation of model stellar atmospheres.
For stars where the abundances and kinematics of the atmospheres are well known,
ODF's provide by far the most efficient means of including the effects of line
blanketing. This will become increasingly true as the number of spectral lines for
which atomic parameters are known grows; although the task of calculating the
opacity distribution functions will also increase.

Considerations such as these will enable the investigator to include the

effects of line blanketing and thereby to create reasonably accurate models of the
stellar atmosphere which will represent the structure correctly through the line
forming region of a normal star. These, when combined with the model interiors
discussed in the first six chapters of the book, will allow for the description of normal
stars from the center to the surface. While this was the goal of the book, We cannot
resist the temptation to demonstrate to the conservative student that the concepts
developed so far will allow the models to be extended into the region above stars and
to determine some properties of the stellar radiation field that go beyond what is
usually considered to be part of the normal stellar model. So in the last chapter we
will consider a few extensions of the ideas that have already been developed

428

⋅

Breakdown of Local Thermodynamic Equilibrium

Problems

Estimate the ratio of collisional ionization to photoionization for hydrogen
from the ground state, and compare it to the ratio from the second level.
Assume the pressure is 300 bars. Obtain the physical constants you may need
from the literature, but give the appropriate references.

Calculate the Doppler-broadened angle-averaged redistribution function for

Hummer's case I, but assuming a Rayleigh phase function [i.e., find

<R(x,x')>

I,B

] and compare it to <R(x,x')>

I,A

and the result for electron

scattering.

Show that

is indeed a solution to

and obtain an integral equation for

Describe the mechanisms which determine the Ly

profile in the sun. Be

specific about the relative importance of these mechanisms and the parts of
the profile that they affect.

Given a line profile of the form

find

. Assume complete redistribution of the line radiation. State what

further assumptions you may need; indicate your method of solution and
your reasons for choosing it.

Show explicitly how equation (15.2.21) is obtained.

Show how equation (15.3.25) is implied by equation (15.3.15).

How does equation (15.3.27) follow from equation (15.3.26).

Derive equations (15.3.28) and (15.3.29).

10.

Use equation (15.3.30) to obtain the angle-averaged form of <R

IV,A

(x',x)>.

11.

Show explicitly how equations (15.3.40) and (15.3.41) are obtained.

429

⋅

Stellar Atmospheres

References and Supplemental Reading

Woolley, R.v.d.R., and Stibbs, D.W.N.

The Outer Layers of a Star

, Oxford

University Press, London, 1953, p. 152.

2.

Böhm, K.-H. "Basic Theory of Line Formation",

Stellar Atmospheres

, (Ed.:

J. Greenstein), St

ars and Stellar Systems: Compendium of Astronomy and

Astrophysics,

Vol.6, University of Chicago Press, 1960, pp. 88 - 155.

3. Mihalas,

Stellar Atmospheres

, W.H. Freeman, San Francisco, 1970,

pp. 337 - 378.

Mihalas, D.

Stellar Atmospheres

, 2d ed., W.H. Freeman, San Francisco,

1978, pp. 138.

Henyey, L.

Near Thermodynamic Radiative Equilibrium

, Ap.J. 103, 1946,

pp. 332 - 350.

Hummer, D.G.

Non-Coherent Scattering

, Mon. Not. R. astr. Soc. 125, 1962,

pp. 21 - 37.

Omont, A., Smith, E.R., and Cooper, J.

Redistribution of Resonance

Radiation I. The Effect of Collisions

, Ap.J. 175, 1972, pp. 185 - 199.

McKenna,S.

A Reinvestigation of Redistribution Functions R

III

and R

, Ap.

J. 175, 1980, pp. 283 - 293.

McKenna, S.

The Transfer of Polarized Radiation in Spectral

Lines:Formalism and Solutions in Simple Cases

, Astrophy. & Sp. Sci., 108,

1985, pp. 31 - 66.

10. Hummer, D.G., and Mihalas, D.

Line Formation with Non-Coherent

Electron Scattering in O and B Stars

, Ap.J. Lett. 150, 1967, pp. 57 - 59.

11.

McKenna, S.

A Method of Computing the Complex Probability Functionand

Other Related Functions over the Whole Complex Plane

, Astrophy. & Sp.

Sci. 107, 1984, pp. 71 - 83.

12.

McKenna, S.

The Transfer of Polarized Radiation in Spectral Lines: Solar-

Type Stellar Atmospheres

, Astrophy. & Sp. Sci., 106, 1984, pp. 283 - 297.

430

⋅

Breakdown of Local Thermodynamic Equilibrium

431

13.

Chandrasekhar,S.

The Radiative Equilibrium of the Outer Layers of a Star

with Special Reference to the Blanketing Effect of the Reversing Layer

, Mon.

Not. R. astr. Soc. 96, 1936, pp. 21 - 42.

14. Sneden, C., Johnson, H.R., and Krupp, B.M.

A Statistical Method for

Treating Molecular Line Opacities

, Ap. J. 204, 1976, pp. 281 - 289.

15.

Carpenter,K.G.

A Study of Magnetic, Line-Blanketed Model Atmospheres

doctoral dissertation: The Ohio State University, Columbus, 1983.

16.

Kurucz, R., and Peytremann, E.

A Table of Semiemperical gf Values Part 3

SAO Special Report #362, 1975.

Although they have been cited frequently, the serious student of departures

from LTE should read both these:

Mihalas,

D.:

Stellar Atmospheres

, W.H.Freeman, San Francisco, 1970,

chaps. 7-10, 12, 13.

and

Mihalas,

D.:

Stellar Atmospheres

, 2d ed., W.H.Freeman, San Francisco,

1978, chaps. 11-13.

A somewhat different perspective on the two-level and multilevel atom can be found
in:

Jefferies, J.T.:

Spectral Line Formation

, Blaisdell, New York, 1968,

chaps. 7, 8.

Although the reference is somewhat old, the physical content is such that I would
still recommend reading the entire chapter:

Böhm, K.-H.: "Basic Theory of Line Formation",

Stellar Atmospheres

, (Ed.:

J.Greenstein),

Stars and Stellar Systems: Compendium of Astronomy and

Astrophysics,

Vol.6, University of Chicago Press, Chicago, 1960, chap. 3.