The driving force behind the interest in
PNLFs comes from the near constancy of the peak luminosity, which makes the PNLF
useful as a standard candle. So far, the only property that is observed to have
any effect upon the PNLF peak luminosity is the nebular oxygen abundance (Richer
1993). Both the maximum 
luminosity and the
maximum 
ratio attained by planetary nebulae are
a function of their oxygen abundance (RM95). The effect of metallicity seems
to be restricted to low oxygen abundances, however, since all bright galaxies,
which are presumably oxygen-rich, appear to have similar PNLF peak
luminosities.
There is no evidence that the PNLF peak luminosity is sensitive to the age of the underlying stellar population. A comparison of the PNLF peak luminosities in the LMC and the bulge of M 31 is the most stringent evidence that young stars do not produce brighter planetary nebulae. Undoubtedly, the planetary nebula progenitors in the LMC are younger and more massive than those in the bulge of M 31, but M 31 has the higher PNLF peak luminosity (RM95). In fact, the difference in peak luminosities between M 31 and the LMC is compatible with the expected difference in oxygen abundance (RM95). Further evidence that the PNLF peak luminosity is insensitive to its progenitor population comes from the similar PNLF peak luminosities observed in the bulge of M 31 and the disk of the Milky Way (Méndez et al. 1993), and the very similar PNLF distances that are derived for galaxies of different morphological types in the same group or cluster (e.g., Ciardullo et al. 1989a; McMillan et al. 1993).
That the maximum 
luminosity is independent of the presence of ongoing star formation is
critically important, for it implies that, excluding abundance effects, the
maximum 
luminosity is constant throughout the life of all
galaxies. Fundamentally, this observation constrains the way in which planetary
nebulae evolve. Some mechanism must exist that prevents the young, massive
planetary nebula progenitors in star-forming galaxies from producing more luminous
planetary nebulae than the old, low mass planetary nebula progenitors in galaxies
where star formation ended long ago.
The luminosity function for the planetary nebulae in the Magellanic Clouds has been studied to fainter levels than that for any other planetary nebula population. Jacoby (1980) constructed the deepest PNLF for the Magellanic Clouds, attaining luminosities some six magnitudes below the peak. However, subsequent work has shown that a substantial number of Jacoby's (1980) planetary nebulae were, in fact, not planetary nebulae (Boroson & Liebert 1989). Because an accurate understanding of the PNLF is crucial to constraining models of planetary nebula populations, we first re-analyze the PNLF in the Magellanic Clouds to determine what effect interlopers had upon the luminosity function Jacoby (1980) obtained.
In addition to excluding the interlopers from Jacoby's (1980)
sample, we re-calibrated his original photometry to the Jacoby
(1989) system. This is necessary to define the shape of the PNLF as
precisely as possible. To effect this re-calibration, we used the 
fluxes for the Jacoby (1980) objects found in
the compilation in Richer (1993), but enlarged to include the
photometric fluxes from Vassiliadis et al. (1992) and
Jacoby & Kaler (1993). Henceforth, this enlarged compilation
will be referred to as the complete sample (Richer 1994;
see also RM95 for further details).
To generate 
magnitudes for the Jacoby (1980) objects
lacking photometry on the Jacoby (1989) system, we adopted the
following procedure. First, the fluxes for Jacoby's (1980)
planetary nebulae appearing in the complete sample were transformed to 
magnitudes using
(Allen 1973; Jacoby
1989). Second, we fit these 
magnitudes as a
function of the Jacoby (1980) magnitudes. We fit the data for
planetary nebulae in the LMC and the SMC separately, since the original
magnitude scales were slightly different. There were 8 and 13 planetary
nebulae in the SMC and LMC, respectively, to use in this transformation. The
relations between the original and new magnitude systems are shown in
Fig. 1 (click here). The dispersions about the fits are 0.42 mag for the SMC and
0.34 mag for the LMC. Finally, we used the relations shown in Fig. 1 (click here)
to calculate 
magnitudes for the Jacoby
(1980) planetary nebulae that are missing from the complete sample.
Figure 1: The re-calibration of Jacoby's (1980) magnitude system to that defined by Eq.
(1). The different symbols are defined in the legend. Although the LMC fit is
effectively a zero-point shift, the slope of the SMC fit departs significantly from unity
regardless of whether the faintest planetary nebula is included
Since there
are only 38 planetary nebulae in the Jacoby (1980) sample, we
pooled the LMC and SMC data to create a composite PNLF that was representative
of the planetary nebula population in the Magellanic Clouds. Thus, for each
planetary nebula in the Jacoby (1980) sample, we calculated the
difference between its 
magnitude and the PNLF peak
luminosity, adopting apparent PNLF peak magnitudes of 
and 
in the LMC and the SMC, respectively
(Jacoby et al. 1990). Using these magnitude differences, we then
computed a composite PNLF for the Magellanic Clouds that we shall refer to as
the Jacoby PNLF. The Jacoby PNLF is tabulated in Col. 2 of
Table 1 (click here) and shown by the heavy line in Fig. 2 (click here). (The
luminosity functions in Table 1 (click here), like all those that follow, are
cumulative luminosity functions, i.e., the fraction of all planetary nebulae
exceeding a particular luminosity threshold). The Jacoby PNLF is more linear
than the one Jacoby (1980) derived, especially for the first
5 mag, because many of Jacoby's (1980) fainter planetary nebula
candidates were interlopers. Jacoby (1980) found that the
theoretical luminosity function of Henize & Westerlund (1963),
, the dashed line in
Fig. 2 (click here), fit his data well. Over the 6 mag interval shown in
Fig. 2 (click here), however, a chi- squared test shows that the Henize &
Westerlund (1963) luminosity function is statistically distinguishable
from the Jacoby PNLF, i.e., 
.
Table 1: PNLFs for the Magellanic Clouds
Figure 2: The cumulative PNLF for Magellanic Cloud planetary nebulae.
The Jacoby
(1980) PNLF is based upon re-calibrated magnitudes for the objects not included in the
complete sample (see text). The PNLFs for the LMC and SMC planetary nebulae are
based upon objects in the complete sample only. Compared to the Jacoby (1980)
sample, the LMC PNLF becomes seriously incomplete beyond about 4 mag below the
luminosity function peak, while that for the SMC becomes incomplete beyond about
3 mag below the luminosity function peak. Both the Jaboby and LMC PNLFs are
statistically different from the Henize & Westerlund (1963) luminosity function
Table 1 (click here) also tabulates the global luminosity functions for the LMC and SMC based upon the complete sample. The global luminosity functions complement the Jacoby PNLF in that the bright end is much better defined on account of the complete sample's much larger areal coverage. The global luminosity functions have been normalized relative to the Jacoby PNLF by forcing the fractions of planetary nebulae within 2 mag of the luminosity function peak to be equal. Assuming that the shape of the Jacoby PNLF is representative of the shape of the true PNLF in the Magellanic Clouds, Fig. 2 (click here) shows that the global luminosity function for the LMC is approximately complete for planetary nebulae within 4 mag of the luminosity function peak. By the same reasoning, the SMC's global luminosity function is approximately complete for planetary nebulae within 3 mag of the luminosity function peak. As might be suspected from Fig. 2 (click here), a chi-squared test shows that, over the first 4 mag, the LMC's global luminosity function is also statistically different from the Henize & Westerlund (1963) luminosity function. However, the number of SMC objects within the first 3 mag of its global luminosity function is too small to rule out compatibility with the Henize & Westerlund (1963) luminosity function.
Recently, Ciardullo (personal communication; see also Ciardullo 1995) has obtained a deep luminosity function for the bulge of M 31 that extends to 4 mag below the peak luminosity. Their preliminary analysis indicates that the PNLF in the bulge of M 31 is statistically compatible with the Henize & Westerlund (1963) luminosity function. Thus, the shapes of the PNLFs in the LMC and the bulge of M 31 are statistically different. Indeed, using sample A from Ciardullo et al. (1989b) to define the first 2.5 mag of PNLF in the bulge of M 31, both chi-squared and Kolmogorov-Smirnov tests indicate that M 31's PNLF differs statistically from the LMC's global PNLF. Presumably, the variation in the shape of the PNLF is a result of the different evolution of planetary nebulae derived from progenitors of different masses.
Observations now clearly demonstrate that the specific density of bright planetary nebulae (the number per unit luminosity) depends upon the underlying stellar population. Peimbert (1990) was the first to notice that galaxies with higher luminosities or redder colours have a lower specific density of bright planetary nebulae. Further work has only reinforced these trends (Hui et al. 1993). Both trends invite suspicion that they are the result of abundance variations since larger, more luminous galaxies are also more metal-rich.
Peimbert (1990) argued that a change of the central star mass
with metallicity is not responsible for the specific density trends, because
galaxies in groups are observed to have different specific densities but
similar maximum 
luminosities. Another suggestion is
that bright planetary nebulae arise from a subset of the planetary nebula
progenitors that produce planetary nebulae with more massive central stars, and
that the fraction of progenitors that produce more massive central stars
decreases as the galaxy's luminosity or metallicity increases (Peimbert
1990; Ciardullo et al. 1991).
The oxygen abundances observed in bright planetary nebulae in
the Magellanic Clouds constrain planetary nebula evolution through the number of
planetary nebulae arising from oxygen-rich progenitors, i.e., recent star
formation. The number of such planetary nebulae depends upon the history of star
formation, the initial-to-final mass relation for planetary nebula progenitors,
and the evolutionary time scales for planetary nebula central stars. These three
factors affect the number of planetary nebulae expected from recently formed
progenitors in different ways. First, the death rate from younger stellar
populations will be higher than that from older populations (e.g., Renzini &
Buzzoni 1986). Thus, per unit mass, young, metal-rich stellar
populations will produce more planetary nebulae than older, more metal-poor
populations, which will bias the mean oxygen abundance for bright planetary
nebulae upward. Second, if the central star mass decreases significantly as
the initial metallicity increases, as many stellar evolution models predict,
the 
luminosities of planetary nebulae with lower
oxygen abundances can be similar to those with higher abundances because the
low metallicity central stars will be brighter and able to compensate for
their nebula's lower 
emission efficiency. This would
cause a great deal of population mixing among bright planetary nebulae,
lowering their mean abundance. Third, all evolutionary models for planetary
nebula central stars indicate that, while they still derive energy from
nuclear fusion, massive central stars evolve much more rapidly than lower mass
central stars. This effect will also tend to decrease the mean abundance
observed for bright planetary nebulae by reducing the time during which the
planetary nebulae produced by young progenitors are bright. Thus, the similar
oxygen abundances in the interstellar medium and in bright planetary nebulae
in the Magellanic Clouds (e.g., Richer 1993) reflects the
balance between these three effects that the models must strive to attain.
Quantitatively, the
abundance parameters that the models must match are set by the planetary nebula
population in the LMC. For later comparison with models, the mean oxygen
abundances and their dispersions in the Magellanic Clouds, both for the
interstellar medium and for planetary nebulae within 2 mag of the PNLF peak, are
given in Table 2 (click here). Given the number of planetary nebulae within 1 mag
of the PNLF peak in the LMC, the models are constrained to predict abundances
for the planetary nebulae within 1 mag of the PNLF peak that are within 
of the interstellar medium value (99% confidence limit). For
the SMC, the equivalent constraint is 
.
Table 2: Oxygen abundances in the Magellanic Clouds