A careful inspection of Table 4 (click here) reveals
that the difference between the oxygen abundance in the interstellar
medium and the mean abundance for bright planetary nebulae
(henceforth, the abundance gap) is a function of the
interstellar medium abundance. (Here, bright planetary
nebulae are those within 1 mag of the PNLF peak). The abundance
gap is displayed as a function of the interstellar medium oxygen
abundance in Fig. 5 (click here). The abundance gap is plotted for ages
of 15 Gyr for all the models from Table 4 (click here). The different
model series are described in the figure legend. The dashed line
in Fig. 5 (click here) is a straight line fit to the rapid and flat IMF
elliptical series (the RX and FX series from
Table 4 (click here)), and is given by
where
and 
.
Two general trends are apparent in Fig. 5 (click here). First, within a model series, the abundance gap increases as the oxygen abundance in the interstellar medium increases. Second, the abundance gap increases along the sequence of model series from fast elliptical, slow elliptical, constant SFR elliptical, to star forming galaxies, which is a sequence in which the mean time scale for star formation increases. Thus, the abundance gap also increases as the star formation time scale increases.
The first trend has a simple
origin. As the oxygen abundance increases, the abundance
sensitivity of the 
ratio
decreases. Consequently, there is less luminosity differentiation
among the planetary nebulae from different stellar generations,
allowing a larger fraction of all stellar generations to contribute
to the bright planetary nebula population. As a result, the spread
in the oxygen abundances among bright planetary nebulae increases as
the oxygen abundance in the interstellar medium increases, causing a
departure of the mean oxygen abundance for the bright planetary
nebulae from the interstellar medium abundance.
The reason that the
abundance gap increases as the time scale for star formation
increases is due to the nebular covering factor. At any epoch, if we
ignore the effect of metallicity upon stellar lifetimes, planetary
nebula central stars with masses of 
arise from stellar
generations that were born at some specific time, 
. Thus, the
longer star formation continues beyond 
, the larger is
the fraction of planetary nebulae with central stars whose masses
exceed 
and whose luminosities are effectively
identical on account of the covering factor. For planetary nebulae
whose central star masses exceed 
, the dependence of
the 
ratio upon oxygen
abundance and the expansion velocity are the only properties
differentiating their 
luminosities. The
effect of the oxygen abundance is not strong, however. At a given
optical depth, all planetary nebulae with oxygen abundances
will attain maximum 
luminosities within 1 mag of the PNLF peak (see
Eq. (3) of RM95), if they contain a central star with a mass of at
least 
. Consequently, the longer star formation
persists, the wider the abundance range for the brightest planetary
nebulae becomes, and the more the mean abundance departs from that
in the interstellar medium. This effect is muted in the elliptical
models because of the cutoff in the expansion velocities. Except for
the two-burst ellipticals, the planetary nebula progenitors in these
models all have masses below 
, so the youngest, most
metal-rich progenitors are also the most optically thick, which
biases their contribution to the population of bright planetary
nebulae.
The abundance gap is not sensitive to the yield of oxygen from dying stars. The value for the yield of oxygen affects how far chemical evolution must proceed if a given interstellar medium abundance is to be reached. With a higher value for the yield, chemical evolution must stop at a larger gas fraction if the interstellar medium is to have the same final oxygen abundance. The yield is controlled by the slope of the initial mass function, which fixes the fraction of stars that return newly synthesized oxygen to the interstellar medium. In Fig. 5 (click here), the RX and FX elliptical model series have similar abundance gaps in the abundance range where they overlap, even though their final gas fractions are very different.
The magnitude of the abundance gap is almost independent of the intrinsic abundance dispersion adopted for each stellar generation (Sect. 3.2.3). Varying the intrinsic dispersion from 0.066dex to 0.20dex decreases the abundance gap by at most 0.03dex. Thus, the abundance gap is not an artifact of our particular choice for the intrinsic abundance dispersion among stars from each stellar generation.
Figure 6 (click here) shows the temporal behaviour of the abundance
gaps in the star- forming models. The solid line is a fit to the
abundance gaps for the 15 Gyr points, and is given by
where
and 
have the same meanings as in Eq.
(15). For comparison, the fit to the 15 Gyr points for the RX and
FX elliptical models (Eq. 15) is also shown in Fig. 6 (click here).
Figure 6: Here, we plot the abundance gaps for the star-forming models, showing how they
evolve with age. For these models, the abundance gap depends upon both the duration
and the rate of star formation. The solid line is a straight line fit to the 15 Gyr points
(Eq. 16). The dashed line is the fit to the rapid and flat IMF elliptical galaxy models
(Eq. 15)
Figure 6 (click here) demonstrates that the abundance gap depends upon both the age of the model and the star formation rate. For a constant star formation rate, the rate at which the oxygen abundance changes in the interstellar medium decreases with time. (Adding a fixed amount of newly synthesized oxygen makes a greater change to the interstellar medium abundance if the abundance is initially low). At all times in the star-forming models, the age difference between the youngest and oldest stellar generations that contribute to the population of bright planetary nebulae remains approximately constant. Consequently, the abundance gap decreases with time because the range of ages for the brightest planetary nebulae maps to a smaller range of abundances at older ages. The star formation rate simply exaggerates this effect: stellar generations occupying a fixed range of ages will span a wider range of oxygen abundances as the star formation rate increases.
Given the effect of age in Fig. 6 (click here), it is not surprising that the two-burst ellipticals have larger abundance gaps than models that formed their stars in single star-forming episode (see Fig. 5 (click here)). Not only are later star formation episodes equivalent to stretching out the period of star formation, but they are also equivalent to observing the galaxy at a younger age. As a result, Eq. (15) could perhaps be regarded as a minimum difference between the last epoch oxygen abundance and the mean abundance measured in the brightest planetary nebulae.
Equation (15) may also underestimate the true abundance gap if there exist metallicity-dependent processes that prevent stars from becoming planetary nebulae, e.g., the AGB-manqué evolution. If this were the case, the metal-rich planetary nebulae in our models would be removed from the population of bright planetary nebulae, which would increase the abundance gaps.
Presently, the planetary nebulae in the
Magellanic Clouds and the Milky Way are the only possible
observational connection between real galaxies and Figs. 5 (click here)
and 6 (click here). For the Magellanic Clouds, the relevant data for
the HII regions and the planetary nebulae within 1 mag of the PNLF
peak are given in Table 2 (click here). From these data, the observed
abundance gap is 
in the LMC and 
in the SMC, whereas Table 4 (click here) predicts
abundance gaps of 
and 
in the LMC and SMC, respectively. Statistically, these
observed and predicted abundance gaps are indistinguishable (see
Sect. 2.4). For the Milky Way, the mean abundance for the six
planetary nebulae within 1 mag of the luminosity function peak in
the Méndez et al. (1993) sample is
. Using the distances from
Méndez et al. (1993) and the observed positions of
these objects (e.g., Kaler 1976), their mean
galactocentric radius is 7.06 kpc (adopting 7.8 kpc for the solar
radius; Feast 1987). Taking the difference in the
galactocentric distance scales into account, the observations of
19 H II regions by Shaver et al. (1983) indicate a
mean H II region abundance of 
at a galactocentric distance of 7.06 kpc. A t-test shows that
this differs from the planetary nebula mean at the 98.5%
confidence level. For 
, Eq. (16) predicts
an abundance gap of 0.14 dex, identical to that observed.
Since the abundance gap
(Eq. 15) is more sensitive to the final interstellar medium oxygen
abundance than to the details of how that final oxygen abundance was
achieved, it is straightforward to use Eq. (15) to estimate the
interstellar medium oxygen abundance when ellipticals stopped
forming stars. From observations, one can determine the mean oxygen
abundance for planetary nebulae within 1 mag of the PNLF peak. If
(see Fig. 5 (click here)), Eq. (15) can be
solved for the oxygen abundance in the interstellar medium. If 
, the abundance gap will depend upon the star formation history,
but will be less than 
, based upon the
star-forming series, so it is probably safer to make no correction in
this case.