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Astron. Astrophys. 358, 1058-1068 (2000)

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5. Nebulae and their central stars

5.1. HR diagram

Fig. 9 shows the positions of the observed PNe in the HR diagram, for the four populations. The luminosities are not well determined: they will be underestimated if the PN is optically thin in some directions, i.e. if it leaks. Overplotted is an evolutionary track of Blöcker (1995), for a core mass of 0.6 [FORMULA] (Gorny et al. 1997). Only one object is located on the cooling track: all other objects are consistent with the horizontal, hydrogen-burning part of the post-AGB track, allowing for uncertainty in the luminosities. The size of the symbols is proportional to [FORMULA]. Among the Bulge PNe there are a few young objects (with cool central stars) with low expansion velocities, but otherwise little correlation is seen.

[FIGURE] Fig. 9. HR diagram for the PNe. Symbols as in Fig. 7. Size of the symbol is proportional to the expansion velocity.

The luminosities are insufficiently accurate to draw quantitative conclusions for the central stars of our sample.

5.2. Dynamical ages

The dynamical age of a PN is normally defined as the radius divided by the expansion velocity. This ignores the effects of the velocity gradient, as well as the acceleration over time. Dopita et al. (1996) calculate correction factors, based on semi-empirical relations, to convert dynamical ages to actual ages: they are typically around 1.5.

We calculate the dynamical age based on the average of [FORMULA] and [FORMULA], as a first-order estimate of the effect of acceleration. [FORMULA] is calculated as before, but where the metallicity is not known we took [FORMULA]. To account for the velocity gradient, we used a radius of 0.8 times the outer radius (since our velocity is based on a mass-weighted average). The resulting dynamical ages are plotted against [FORMULA] of the star (which is well determined from our models) in Fig. 10.

[FIGURE] Fig. 10. The corrected dynamical age of the PNe, versus effective temperature of the star. The tracks correspond to theoretical models with core masses of 0.565, 0.6 and 0.625 [FORMULA]. Symbols as in Fig. 7.

The lines show predicted ages versus temperature for three post-AGB models with core masses of 0.565, 0.6 and 0.625 [FORMULA] (adopted from Schönberner 1983; Gorny et al. 1997 and Blöcker 1995 respectively). Even over this small range of stellar masses, the evolutionary time scales for the increase in stellar temperature varies by a factor of 10. The models include the effect of post-AGB mass loss. For stars which leave the AGB as helium burners, stellar evolution slows down by about a factor of 3. A few objects could be explained as helium burners, but most Bulge PNe are well fitted by the models.

The uncertainty in the dynamical ages come from two factors. First, the use of [OIII ] lines only which do not allow us to determine the expected faster acceleration in the outer nebular regions. To first order this is corrected for because we use a radius which is smaller than the outer radius, and corresponds to the way the mass-weighted expansion velocity is defined. Hydrodynamical calculations which would allow a more accurate treatment are not presently available. Second, the direct averaging between the PN expansion velocity and the AGB velocity is ad-hoc: the associated uncertainty can reach 50% which would result in an uncertainty of 0.2 in [FORMULA].

5.3. Masses of nebular cores

The derived dynamical ages plotted in Fig. 10 show that, in this sample, average core masses are in the range 0.60-0.625 [FORMULA]. PNe associated with the disk appear to have slightly higher core masses than Bulge PNe. Of the two halo PNe, one (M2-29) has a low core mass of around 0.58 [FORMULA]. The two PNe in Sgr have somewhat higher central-star masses of approximately 0.62 [FORMULA]. The uncertainties discussed introduce an systematic error in the derived masses of about 0.02 [FORMULA]. The relative uncertainties are much smaller since the uncertainties are derived from the way the ages are calculated.

The core masses agree well with Gorny et al. (1997) and Stasinska et al. (1997) for Galactic disk PNe: For an assumed nebular mass of 0.1 [FORMULA], their median core mass is 0.60 [FORMULA]. Higher assumed nebular masses would shift the mass down to 0.58 [FORMULA] for 0.4-[FORMULA] nebulae. As noted by Stasinska et al., the most likely range for nebular masses is 0.1-0.15 [FORMULA]. Best agreement is indeed obtained for this range.

Sofar we have only used the relative location of the PNe with respect to the theoretical tracks. To obtain more precise values for the stellar mass, we use the fact that the three tracks are almost parallel for [FORMULA]. In this range, the location of the tracks can be fitted with the following relation: [FORMULA] where [FORMULA]. The resulting masses are shown in Fig. 11, as a histogram. The Bulge and disk objects are shown by the drawn and dashed lines respectively. Fig. 11 indicates a tendency towards higher core masses for disk PNe. This can qualitatively be understood, since disk PNe will on average come from younger stars: higher initial mass would be expected to give higher final mass.

[FIGURE] Fig. 11. Histogram of estimated core masses for Bulge and disk planetary nebulae. Systematic uncertainty on the masses is 0.02 [FORMULA].

Fig. 11 shows that four Bulge PNe in our sample have core masses around [FORMULA]. The sample includes the most metal-rich Bulge PN. The radial velocities confirm that these objects really are located in the Bulge. These stars may have belonged to the inner disk population, having recently diffused into the Bulge through interaction with a Bar (Norman et al. 1996). The results of Ortolani et al. (1995) are based on clusters in the Bulge, and a younger diffused stellar population in the Bulge would not necessarily be contradicted by the old globular clusters.

We finally note that the two PNe in Sgr also have somewhat higher core masses of approximately 0.62 [FORMULA]. This is consistent with the star formation history of Sgr: the progenitor stars are expected to be around 5 Gyr old (Dudziak et al. 2000). The halo PN with very low metallicity, on the other hand, has a lower core mass than the Bulge PNe. This could indicate that its progenitor star formed earlier than those of the Bulge PNe in our sample.

The core masses are strictly related to theoretical, hydrogen-burning evolutionary tracks. There is some uncertainty in these models regarding post-AGB mass loss. This would mainly affect the early part of the post-AGB evolution: at the temperature range in our sample, stellar mass loss is observed to be much less than the nuclear burning rate. The effect of model uncertainties are beyond the scope of the present paper. However, we note that Dudziak et al. (2000) determined accurate luminosities for the Sgr PNe, which are consistent with core masses around [FORMULA]. Thus, the combined errors from models and our determination may be within [FORMULA].

5.4. Comparison with white dwarf masses

The core mass of the central star of a PNe is (within [FORMULA]) the same as the mass of the resulting white dwarf. The Bulge PNe therefore measure the initial-final mass relation for low-mass stars. The Bulge population is among the oldest in the Galaxy (Ortolani et al. 1995) with a well-determined turn-off mass: [FORMULA] and [FORMULA]. The metallicity is also known to be not far from solar. For these numbers, the initial-final mass relation proposed by Weidemann (1987) predicts final masses around [FORMULA]. Our results indicate that this is too low by about 10 per cent. The good agreement with Stasinska et al. confirms our higher value.

We can also compare the core mass distribution to the mass distribution of white dwarfs in the solar neighbourhood. Bragaglia et al. (1995) find that the local white dwarf mass distribution shows a peak at [FORMULA], with tails towards both higher and lower masses. Napiwotzki et al. (1999) find a peak at higher masses, at [FORMULA], which is within the uncertainties of the present determination for the Bulge. It should be noted that the determinations should not necessarily agree: Stars in the Bulge could evolve to somewhat higher core masses than stars in the solar neighbourhood, if the AGB superwind is less efficient at lower metallicity. Also, PNe trace the present main-sequence turn-off stars while the white dwarf population has accumulated over time. The effect is complicated by cooling which renders the white dwarf eventually undetectable, and is dependent on the mass (Hansen 1999). The masses of local white dwarfs may also have been underestimated. This is discussed in Bragaglia et al. (1995) who find that if the hydrogen layer in the white dwarfs is much thicker than assumed, the spectroscopic masses should be increased by [FORMULA] [FORMULA].

In view of these uncertainties, the agreement especially with the result of Napiwotzki is satisfactory. The discrepancy with the initial-final mass relations is however much larger than the present uncertainties, and must be considered significant.

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Online publication: June 20, 2000