Over the years, extragalactic planetary nebulae have been used sporadically as probes of the chemical evolution of their parent galaxies. The most extensively studied extragalactic planetary nebulae are those in the Magellanic Clouds. In both the LMC and the SMC, planetary nebulae and HII regions have similar oxygen abundances (e.g., Dopita & Meatheringham 1991a,b). Planetary nebulae have also seen limited duty as probes of the chemical evolution of the bulge and disk of M 31. Jacoby & Ford (1986) obtained oxygen abundances for three planetary nebulae, one of which had a radial velocity that led them to conclude that it was a disk object. Since this disk planetary nebula had an oxygen abundance similar to that in HII regions at the same radius, Jacoby & Ford (1986) concluded that the high oxygen abundance in M 31's outer disk was attained early in its evolution. The two halo planetary nebulae in M 31 had very different oxygen abundances, which Jacoby & Ford (1986) interpreted as a manifestation of the heterogeneous line strengths previously found among M 31's globular clusters (van den Bergh 1969; Huchra et al. 1982).
Richer (1993) provided the basis for using bright planetary nebulae as probes of chemical evolution in star-forming galaxies. However, it is in galaxies where star formation ceased long ago that planetary nebulae present their most exciting uses as chemical evolution probes. Here, we investigate whether, in galaxies without star formation, bright planetary nebulae can be used to determine the oxygen abundance in the interstellar medium at the time when star formation stopped.
From the viewpoint of chemical evolution, the oxygen abundance in the interstellar medium when star formation ceased (henceforth, the last epoch abundance) is particularly interesting. If bright planetary nebulae were reliable probes of the last epoch abundance, many questions concerning the formation and evolution of galaxies lacking recent star formation could be investigated. Among these: What fraction of their gas did ellipticals turn into stars? How does this fraction vary with the luminosity or the total mass? Was the stellar initial mass function during the star-forming epoch similar to that observed in star-forming galaxies today? What was the time scale over which these galaxies formed? Are diffuse elliptical galaxies (spheroidals) the faded remnants of dwarf irregular galaxies? Excluding the last one, these questions could equally well be asked concerning the bulges of spiral galaxies.
Although
bright planetary nebulae are excellent probes of the current oxygen abundance in
the interstellar medium in the Magellanic Clouds (Richer 1993),
it is not clear that they will be such effective indicators of the last epoch
abundance in galaxies where star formation stopped long ago. In the
Magellanic Clouds and the Milky Way, the maximum attainable 
luminosities and 
ratios increase as the
oxygen abundance increases, but they become insensitive to changes in the
oxygen abundance at high abundances (Richer & McCall 1995,
henceforth RM95). Depending upon the history of star formation, in galaxies
where the oxygen abundance is higher, the bright planetary nebulae may
originate from stars occupying a broader range of oxygen abundances. Were
this to happen, their mean abundance would become a progressively less
reliable measure of the oxygen abundance in the interstellar medium when star
formation ceased. Unfortunately, it is impossible to determine directly
whether planetary nebulae measure the last epoch abundance in galaxies where
star formation stopped long ago. In these galaxies, the only way to
investigate how the present abundances in bright planetary nebulae are related
to the last epoch abundances is to build models of the planetary nebula
population. Fortunately, observations cover a sufficient range of conditions
in galaxies, both with and without star formation, that they can place robust
constraints upon the free parameters.
Previous studies of planetary nebula luminosity functions (PNLFs; e.g., Jacoby 1989; Méndez et al. 1993) have successfully accounted for the near constancy of the peak luminosity by incorporating a narrow central star mass distribution. Such a mass distribution is reasonable given the narrow mass distribution for white dwarfs (e.g., Weidemann 1990), which are the direct descendants of planetary nebula central stars. Since these investigations sought to explain the existence and behaviour of PNLFs, primarily at the bright end, the lack of direct coupling between the central star mass distribution and the history of star formation was not a shortcoming. In order to relate the abundances in bright planetary nebulae to the last epoch abundances, however, it is critical to maintain the explicit connection between the star formation history of the host galaxy and its planetary nebula central star mass distribution. It is in this respect that the present work differs from past efforts. In our models, the planetary nebula population is generated from a self-consistent treatment of the history of star formation and chemical evolution of its parent galaxy.
Before considering models of planetary nebula populations, we review the observed properties of planetary nebula populations in galaxies to establish the characteristics the models must display (Sect. 2). Following this, we develop the models (Sect. 3). Once a suitable model is chosen, we consider its implications regarding the utility of bright planetary nebulae as probes of the last epoch oxygen abundance (Sect. 4). As an application of these results, we re-investigate whether diffuse ellipticals and dwarf irregulars may be related by evolution (Sect. 5). Finally, we present our conclusions (Sect. 6).