This work indicates that bright planetary nebulae are excellent tools for probing the chemical evolution of galaxies. This conclusion is based upon previous studies of star-forming galaxies (Richer 1993), where a direct comparison with the interstellar medium is possible, and the predicted properties of planetary nebula populations in galaxies where star formation has ceased.
Although the comparisons of the abundances in planetary nebulae and the interstellar medium for star-forming galaxies are encouraging, it is in galaxies where star formation has stopped that planetary nebulae have their greatest value as abundance probes. To investigate whether, in galaxies lacking star formation, planetary nebulae remain faithful probes of the interstellar medium abundances that persisted when star formation stopped, a code was developed to model the planetary nebula populations in galaxies. This code generates planetary nebula populations based upon the host galaxy's history of star formation, which itself is based upon a self consistent galaxy modelling code (Arimoto & Yoshii 1986). The models were evaluated by comparing their predicted planetary nebula luminosity functions with those observed in galaxies both with and without star formation, and by comparing the mean densities and oxygen abundances they predicted for bright planetary nebulae in the LMC with those observed. To simultaneously satisfy these requirements, it was necessary to couple the evolution of the nebular shell and the central star, and to impose a relation between the central star mass and the nebular covering factor. Our final models reproduce the first 4 mag of the PNLFs observed in galaxies of different types, as well as the mean densities and abundances observed for planetary nebulae in the LMC (for the LMC model).
Given these successes, the models were used to
investigate how well the oxygen abundances in bright planetary
nebulae probe the last epoch interstellar medium abundances in
galaxies where star formation has long since ceased. The models
predict that a gap develops between the abundances observed in
bright planetary nebulae and those that existed in the interstellar
medium when star formation stopped. The magnitude of this gap
increases as the oxygen abundance in the interstellar medium
increases. Extending the duration of the star-forming epoch beyond
the 1 Gyr that we adopted for our simplest models also increases
the size of the abundance gap. The abundance gap is unaffected by
changes to the yield of oxygen in the model, so these results are
not affected by the slope for the initial mass function. The maximum
size of the abundance gap is modest, attaining only 0.35dex at an
interstellar medium oxygen abundance of 
(6.6 times the solar value). Thus, the models predict that,
even in galaxies where star formation stopped long ago, bright
planetary nebulae should be reliable probes of the oxygen abundance
that persisted in the interstellar medium when star formation
stopped.
The predicted abundance gaps were compared with those observed in the Magellanic Clouds and the disk of the Milky Way. In the Magellanic Clouds, the observed and predicted abundance gaps are statistically compatible. For the Milky Way disk, models predict an abundance gap of 0.14dex, which is identical to the value observed.
In light of these results, we re-analyzed the oxygen abundances for the planetary nebulae in NGC 185, NGC 205, and Fornax from RM95 to investigate whether diffuse ellipticals could be related to dwarf irregulars by evolution. Accounting for the abundance gap only served to exaggerate the differences found by RM95: when star formation stops, diffuse ellipticals have systematically higher oxygen abundances than equally luminous dwarf irregulars, and higher [O/Fe] ratios than dwarf irregulars with similar oxygen abundances. The simplest explanation for both of these trends is that diffuse ellipticals consume their gas more rapidly than dwarf irregulars before expelling or losing what remains of it. Consequently, diffuse ellipticals cannot be the faded remnants of dwarf irregulars, for the star formation histories in these two types of galaxies are fundamentally different.
Acknowledgements
MGR would like to thank Dr. Emanuel Vassiliadis for making his evolutionary tracks for planetary nebula central stars available well before publication, and Dr. Robin Ciardullo for communicating important results prior to publication. MGR would also like to acknowledge the Ontario Ministry of Colleges and Universities, Marshall McCall, and the Natural Sciences and Engineering Research Council of Canada for their financial support. MLM thanks the Natural Sciences and Engineering Research Council of Canada for its continuing support. This work was financially supported in part by Grant-in-Aid (Nos. 06640349, 072222206) for the Scientific Research of the Japanese Ministry of Education, Culture, Sport, and Science.