**ABUNDANCES FROM STELLAR SPECTRA**

We do not wish to include a complete description of stellar atmospheres
in this review as most readers have completed a stellar atmospheres
course or have textbooks like
Mihalas (1978)
on their shelves or have read shorter introductions like chapter 12 of
Cowley (1995).
However, in the interest of keeping up to date, we provide a brief
description with emphasis on oft-heard buzzwords. Converting measured
absorption feature strengths into abundances requires a model stellar
atmosphere that gives the run of physical variables like temperature and
pressure with optical depth. With today's fast computers, it is no
longer much of a computational burden to compute fairly realistic LTE
model atmospheres, most of which have the following features. (1)
Plane-parallel geometry. This is a good approximation for most stars,
but a spherical geometry is needed for M giants and other stars that
have extended envelopes. (2) A modern model atmosphere (see, e.g.,
Gustafsson 1989;
Kurucz 1993)
will be line-blanketed. That is, individual transitions from (usually
millions) of atomic and molecular lines are explicitly included in the
frequency-dependent opacity calculations along with the various sources
of continuous opacity like electron scattering or H^{-}. (3)
Convection is often treated in the mixing length approximation, but
alternatives are always being tested. (4) For many stars, local
thermodynamic equilibrium (LTE) is assumed. In this case the local
radiation field and the local thermodynamic state (which can be
described by temperature and pressure alone) of the matter are
equilibrated. The alternative is non-LTE (NLTE), in which the radiation
field decouples from the matter. The complicating part of this is that
atomic level occupation numbers then depend mostly on the radiation
field rather than the local matter thermodynamics, and the radiation
field *is not local* and therefore a global self-consistent
solution must be sought. Extreme NLTE prevails in the case of nebulae
and hot stars, while most stars can be treated with the LTE assumption
for most lines. However, a temperature inversion above the photosphere
(i.e., a chromosphere) will introduce certain NLTE effects in some
lines. If these lines are going to be used for abundance analysis, they
need to be treated appropriately. See
Mihalas (1978) or
Kudritzki & Hummer (1990)
for a description of NLTE methods in hot stars.

Except for those stars where spherical geometry effects become
important, effective temperature *T*_{eff}, surface gravity
log *g* (where *g* is expressed in cm s^{-2}), and
abundance are sufficient parameters to begin the calculation of a model
star. Temperatures can be gotten from calibrations of broadband or
narrowband colors, by Balmer line strength, or by consideration of two
or more ionization states of the same species (by requiring consistent
abundance results from all ionization states). Abundance analysis is
often not very sensitive to surface gravity.

There is also the matter of Doppler broadening of lines. Thermal motion
of gas particles causes a broadening of the line profile that is almost
always larger than the width of the line as broadened by pressure and by
radiation damping. In the LTE approximation inclusion of this effect is
trivially accomplished by convolving the intrinsic Lorentzian line
profile with the Gaussian thermal Doppler profile. But there is also the
matter of bulk motions in the atmosphere, in the Sun seen as
prominences, spicules, granules, acoustical waves, and other
phenomena. These add Doppler width to lines, but their velocity
distribution is not known. In the face of the total unknown, we follow
historical precedent and assume Gaussian random motion, and an
empirically adjusted term is added to the width of the Gaussian Doppler
profile. That is, inside the Gaussian broadening function
*e*^{-( / D)},

(B1) |

where is the central
wavelength of the line,
_{D} is the
Doppler width, *R* is the gas constant, *T* is the
temperature, *µ* is the mean atomic weight, *c* is the
light speed, and
_{t} is
called the *microturbulent velocity*, meant to account for
Gaussian-random, optically thin turbulent motion in the atmosphere. The
final Doppler motion to be considered is *macroturbulence*, very
large moving elements that can be considered independent
atmospheres. Such motions will not change the equivalent widths of
absorption lines, but they will change the line profile.

Atomic (or molecular) parameters are often crucial to a reliable
abundance. Parameters may include damping constants for radiation and
van der Waals forces. The damped line profile is usually approximated by
a Lorentzian function. The atomic parameters that are mentioned most
often are the "*gf*-values," where *g* is the statistical
weight of the level and *f* is the oscillator strength; they are
usually combined as "log *gf*." The *gf*-values can be
calculated with varying degrees of accuracy, and many can be measured in
the laboratory, but the usual method of getting accurate values is to
compute a model atmosphere for the Sun. The equivalent widths of the
line transitions one is interested in studying are measured from the
solar atlas, and also computed from the model atmosphere using standard
meteoritic+photospheric abundances, for example those of
Grevesse et al. (1996).
The *gf*-values are then adjusted until the model equivalent widths
match those of the Sun. This method begins to fail for stars too
dissimilar from the Sun in temperature because the line of interest will
be invisible in one of the two stars or on a different part of the curve
of growth.

The curve of growth is the increase in equivalent width of a line as a function of abundance. When lines are weak, they grow linearly with the abundance. When the line center nears its maximum depth, little growth in equivalent width is seen until the line is so saturated that the weak wings of the line profile contribute. Thus the textbook curve of growth has a linear-with-abundance portion, an almost flat portion, and then a proportional-to-the-square-root-of-the-abundance portion. For accurate abundance work, it is therefore advantageous to choose weak lines in the linear-growth part of the curve.

Observational material for spectroscopic abundance work will vary
according to circumstance. In the ideal case, one would have high-S/N,
high-resolution (*R* =
/
= 50,000-100,000) spectra
compared with near-perfect models. In reality, the models have defects
and many/most of the interesting stars are too far away for both
high-S/N and high-resolution spectra. The usual solution is to go to
lower resolution. The ultimate low-resolution spectrum is broadband
photometry (*R* ~ 10), which can be coaxed to yield good
information about *Z* with suitable assumptions and calibrations,
but little information about individual elements. A resolution at which
some elemental abundance information becomes available is about *R*
~ 500, but only for the strongest, cleanest absorption features. This
is the approximate resolution needed for integrated starlight studies.