ABUNDANCES FROM INTEGRATED LIGHT
If a galaxy is too distant for individual stars to be spectroscopically analyzed one might still hope to learn something from the properties of the integrated starlight summed over all the stars along the line of sight. Broadband colors or low-resolution spectra can be readily obtained for galaxies of sufficient surface brightness. (The surface brightness constraint disqualifies dwarf spheroidals and other low surface brightness galaxies.) The question then becomes how to interpret a radial color profile or a radial absorption line strength profile in terms of abundance. This is done by examining the behavior of "simple" stellar populations with age and abundance. A "simple" stellar population is an abstraction of a star cluster, characterized by a single age and a single abundance. One could use real cluster spectra as templates, if one knew the clusters' ages and abundances (see, e.g., Bica 1988) and if the cluster library brackets the age-metallicity space of interest. The more usual approach is to use a theoretical stellar evolutionary isochrone as the template population. The isochrone gives direct (but theoretical) information on age and abundance.
The first point to realize is that a young stellar population will contain massive stars. O-type stars for the first few tens of millions years, B-type stars for the first few hundred million years, A-type stars until roughly 1 billion years of age. These hot stars make the integrated colors blue, and absorption line strengths weak (except for hydrogen lines, which are strongest in A-type stars). Furthermore, young populations are very bright compared with old populations, so they dominate the integrated light if they are present. In the case of spiral galaxies, with their ongoing star formation, getting abundance information from integrated starlight is difficult or impossible because the depth of the absorption features is very strongly modulated by age effects.
Integrated light is useful, therefore, only in "dead" stellar populations where star formation has not occurred for some time, so that the OBA-type stars are gone. Most of the light (at optical wavelengths) then comes from FG-type stars at the main-sequence turnoff, and KM-type stars on the red giant branch, in roughly equal proportions. Practically speaking, this means that the bulges of spirals and E and S0 galaxies are the targets for integrated-light abundance work.
Even in old populations, age effects are the major complication in
getting abundance information. One can never be sure that traces of
young stars are completely absent, and if they are present they skew the
abundance result toward lower values because the metal lines will be
weaker. Most isochrone synthesis models
(Aaronson et al. 1978;
Worthey 1994;
Bressan et al. 1994;
Vazdekis et al. 1996)
show that age effects are muted compared with abundance effects in their
impact on the resultant spectrum shape, colors, and line strengths. The
null-spectral-change line is
log(Age) /
log Z
- 3/2, so that a
factor of 3 age change produces the same spectral change as a factor of
2 change in metallicity. This -3/2 slope is approximate and can be
different (but not wildly so) for different colors or spectral
indices. In global terms, this is not so bad. For example, if you can be
reasonably certain that an object formed in the first half of the
universe's history, then your age will be, at most, a factor of 100%
uncertain. This translates to a factor of 2/3 × 100 = 66%
uncertainty in abundance, if your isochrone model is calibrated
correctly. This is 0.2 dex–an impressive accuracy given the large
age range allowed. The qualification "if your isochrone model is
calibrated correctly" is important, because scatter among different
models amounts to ± 35% in age (or roughly ± 25% in Z
via the 3/2 rule)
(Charlot, Worthey, & Bressan 1996).
Models for integrated light are conceptually simple addition problems,
but the ingredients rest on complicated input physics. Stellar
evolutionary isochrones are computed from evolutionary tracks of
different masses in order to construct a snapshot of the population at a
single age. The tracks, in turn, depend on opacities, equations of
state, theories of convection and mass loss, and the numerical methods
of making a model star. With the addition of an IMF the isochrone
specifies star number, luminosity, mass, and temperature in stellar bins
along the curve. The integrated luminosity at a single wavelength is
L
=
isochrone
bins
l,bindN, where the number of stars dN is
obtained from an IMF 
(M) via dN =
(M)dM, and
where
l,bin is the monochromatic luminosity of one star in
one bin on the isochrone. Perhaps most commonly these days,
l,bin is obtained from a grid of theoretical model
atmosphere fluxes, but empirical libraries are also used.
In dealing with integrated light models, the critical parameter is temperature: for a good model one must make sure that the temperatures along the isochrone are correct at all points and that the conversion from temperature to color (or flux or line strength) is solid. Relatively small errors in stellar color can propagate almost unattenuated to the integrated colors (Worthey 1994).