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Astron. Astrophys. 327, 736-742 (1997)

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2. Derivation of the PNN masses

2.1. Summary of the method

The method is extensively described in GST97, and we give here only the essentials. The basic idea - shared by all the methods for deriving PNN masses - is that central stars of different masses [FORMULA] differ in their basic properties (effective temperature [FORMULA] and total luminosity [FORMULA]) at any time t. The PNN masses can then be estimated by comparing the observed nuclei with a grid of evolutionary models of post-AGB stars of different [FORMULA]. Since, generally, the available observations do not allow a direct determination of [FORMULA] and [FORMULA] and of the time, t, one uses the surrounding nebulae to approach the physical properties of the central star. However, the parameters one recovers are generally different from [FORMULA], [FORMULA] and t, mainly because the distance to the PN is unknown and because the nebula may be ionization- or density-bounded.

Our approach is to compare the properties of the observed objects with those of simple models which receive exactly the same treatment as the observations in the estimation of [FORMULA], [FORMULA] and t. Our models consist of a spherical nebula of given total mass [FORMULA] expanding at a constant velocity [FORMULA] around an evolving star of mass [FORMULA]. The covering factor is taken equal to 1, the thickness of the gaseous shell is [FORMULA] and the density is uniform. The central star evolution is interpolated from the theoretical grid of Blöcker (1995) and Schönberner (1983), relevant for H-burning central stars, which are believed to constitute the majority of PNN. A blackbody radiation field is assumed. We can then compute at each time, the radius and mass of the ionized part of the nebula, and its luminosity in H [FORMULA].

In the traditional H-R diagram, if the stellar temperature and luminosity are obtained by the Zanstra method using the Shklovsky distance, this amounts to comparing the positions of the objects to "apparent" theoretical tracks for stars of different masses. These apparent tracks differ significantly from the pure stellar tracks (see Fig. 1a below), and are strongly dependent on [FORMULA] and [FORMULA] (see GST97). This is why the conclusions on PNN masses derived by authors using the H-R diagram (e.g. Kaler et al. 1990, Stanghellini et al. 1993, Cazetta & Maciel 1994) need to be reexamined. A similar comment applies to the ([FORMULA], t) diagram (used by Schönberner 1981, Heap & Augensen 1987 and Weidemann 1989) where [FORMULA] is the stellar absolute visual magnitude.

Many PN have their expansion velocity measured. Such objects can be directly compared with a grid of apparent tracks built with the same [FORMULA] as observed. The only unknown parameter in our approach is [FORMULA], and the estimated value of [FORMULA] depends on the assumed [FORMULA]. The advantage of our method is that it is self-consistent and that all the assumptions can be clearly stated.

2.2. The sample of PN

We have considered all the PN for which the relevant data (stellar apparent visual magnitude [FORMULA], total nebular flux F(H [FORMULA]), nebular angular radius [FORMULA] and expansion velocity [FORMULA]) are available in the literature and of good quality, and for which the central stars are not known to be close binaries or to have H-poor atmospheres. The observational data, together with the references, are listed in Tylenda & Stasi[FORMULA]ska (1994), and we use the same extinction correction procedure as in that paper. This results in a sample of 125 objects.

Fig. 1a shows all these objects in the ([FORMULA], [FORMULA]) plane, where [FORMULA] is the Zanstra temperature of the star derived from the H [FORMULA] line, and [FORMULA] is its Zanstra luminosity calculated with the Shklovsky distance. Superimposed are the apparent tracks for central stars of masses 0.565, 0.605 and 0.645M [FORMULA], surrounded by a nebula of total mass 0.2M [FORMULA] whose outer rim is expanding at 20km/sec (continuous curves). The Zanstra luminosity and temperature of the central star, as well as the Shklovsky distance are derived in exactly the same way as in the observations (i.e. assuming a shining mass of 0.2M [FORMULA] and a filling factor of 0.5). For illustration, we also show (dotted curves) the original theoretical tracks for the same stars, that is the variations of [FORMULA] as a function of [FORMULA].

Fig. 1b is similar to Fig. 1a but in the ([FORMULA], [FORMULA]) plane, where [FORMULA] is the absolute stellar visual magnitude, and [FORMULA] the radius of the ionized part of the nebula, both calculated with the Shklovsky distance. Fig. 1c shows the same, but in the ([FORMULA], [FORMULA]) plane introduced by GST97, where [FORMULA] is the nebular surface brightness in H [FORMULA], and [FORMULA] is defined as [FORMULA], where [FORMULA] is the stellar flux in the V band.

Note that not only Fig. 1c is distance independant. Figs. 1a and 1b are also independent of the true distances, since they use the "Shklovsky distance", which is a mere combination of observed parameters.

[FIGURE] Fig. 1. Distribution of PN in three analyzed diagrams. a  Zanstra luminosity, [FORMULA], (in solar units) calculated with the Shklovsky distance, versus Zanstra temperature, [FORMULA], (in Kelvins). Observed PN are represented by circles. Continuous curves: apparent tracks (see text) for central stars of masses 0.565, 0.605 and 0.645M [FORMULA], surrounded by a nebula of 0.2M [FORMULA] expanding at 20km/sec. Dotted curves: the original theoretical tracks for the same stars, i.e. the variations of [FORMULA] as a function of [FORMULA]. b  Total visual magnitude of the central stars, [FORMULA], calculated at the Shklovsky distance, versus radius of the ionized part of the nebula, [FORMULA] (in pc). c  Nebular surface brightness in H [FORMULA], [FORMULA], versus [FORMULA] (see text).

2.3. The PNN masses of our sample

As stated above, the only parameter on which we do not have any observational constraint in our representation is the total nebular mass. We derive the central star masses, [FORMULA], assuming a total nebular mass of 0.2M [FORMULA]. Table 1 (available also in electronic form) gives the values of [FORMULA] for all the PN of our sample that do not appear in Table 1 of GST97. If more than one value of [FORMULA] is possible for a given object, the lowest value is adopted, and the upper limit compatible with the data is indicated in parenthesis. As an information, Table 1 also lists, in the same format as Table 1 of GST97, the values of the following parameters: the morphological type as defined in GST97 1, log  [FORMULA], log  [FORMULA] (in solar units), [FORMULA], log  [FORMULA] (in pc), log  [FORMULA] (in erg cm-2 sec-1 sr-1), log  [FORMULA] (in erg cm-2 sec-1 sr-1), the Shklovsky distance [FORMULA] (in kpc), as well as the evolutionary time [FORMULA] (in units of [FORMULA] yrs) and the distance d (in kpc) (the latter two parameters being obtained once [FORMULA] has been determined, as explained in GST97).


Table 1. Planetary nebulae properties and derived central stars masses, evolutionary ages and distances

The uncertainty in PNN mass resulting from a change in [FORMULA] by a factor two downwards or upwards is represented in Fig. 2a. We see that, for most objects, it becomes important only if [FORMULA] is larger than about 0.65M [FORMULA]. At high central masses, the uncertainty may become very large if the PN is in an evolutionary stage corresponding to a region of crowding of apparent tracks.

The effects of the observational uncertainties in F(H [FORMULA]), [FORMULA] and [FORMULA] on the determination of [FORMULA] are of [FORMULA] 0.01M [FORMULA] at most for [FORMULA] [FORMULA] 0.65M [FORMULA]. They are much larger at higher central star masses. One may consider that for [FORMULA] [FORMULA] 0.65M [FORMULA], the problem of deriving central star masses with our method becomes degenerate, and that masses above 0.65M [FORMULA] are very uncertain.

Additional causes of uncertainty are of course deviations from our adopted model. For instance, masses derived under the assumption of a covering factor smaller than one are larger than if the covering factor is assumed equal to one. Fig. 2b shows the effect of adopting a covering factor of 0.3 for all the PN. While this leaves practically unchanged the derived central star masses at [FORMULA] [FORMULA]  0.56M [FORMULA] and some of those at about 0.60M [FORMULA], it may increase significantly the derived masses in other cases, especially those with [FORMULA] [FORMULA] 0.65M [FORMULA]. Actually, the true covering factor of many PN is probably close to one, so that the errors are not as large as Fig. 2b might express. Covering factors drastically smaller than one are expected for some bipolar PN (those of subclass B following the nomenclature of GST97), whose proportion among PN with [FORMULA] [FORMULA] 0.65M [FORMULA] is rather small (see GST97), and for irregular PN, which constitute only a small fraction of the PN.

[FIGURE] Fig. 2. a The uncertainty in PNN mass resulting from a change of [FORMULA] by a factor two downwards or upwards. Taking a higher [FORMULA] results in a lower derived PNN mass (in all but two cases). The objects are ordered by increasing [FORMULA] (value obtained assuming [FORMULA] =0.2M [FORMULA]). b The uncertainty in PNN mass resulting from using a different covering factor. Circles: PNN masses obtained with [FORMULA] =0.2 and a covering factor of 1. Dashes (or arrows): PNN masses (or limits) obtained with [FORMULA] =0.2 and a covering factor of 0.3.

In Fig. 3, we plot the values of the evolutionary time [FORMULA], as a function of [FORMULA] (assuming [FORMULA] =0.2M [FORMULA]), for the low [FORMULA] objects of our sample. We see a trend similar to the one found in GST97. PN with small [FORMULA] (around 0.56M [FORMULA]) are seen at ages between from 5000 years to over 20000 years. PN with [FORMULA] about 0.62M [FORMULA] are seen between 2000 and 5000 years. Two curves are drawn in Fig. 3. The lower one represents the time when central stars have a temperature of 20000K. The other one, which is a sort of upper envelope of the PN ages in our sample, was obtained by dividing the sample into 5 bins of approximately 20 objects each, and by drawing a smooth curve below which are found 90% of the objects in each bin. One can see that both curves decrease with [FORMULA]. The behaviour of the lower one simply reflects that low mass PNN evolve more slowly and become able to ionize the surrounding nebula only at larger ages. The behaviour of the upper curve is better understood when considering Fig. 4, which represents [FORMULA] as a function of [FORMULA]. This plot confirms the trend shown in GST97 that, for objects with [FORMULA] [FORMULA] 0.62M [FORMULA], [FORMULA] tends to be larger for PN with nuclei of larger masses. Thus, at the same age, PN with central stars of higher masses tend to be more diluted, which discriminates against them in a sample where the H [FORMULA] fluxes have to be measured.

[FIGURE] Fig. 3. Estimated age [FORMULA], as a function of PNN mass [FORMULA] (assuming [FORMULA] =0.2M [FORMULA]) for objects with [FORMULA] [FORMULA] 0.64M [FORMULA].

[FIGURE] Fig. 4. Observed PN expansion velocity, [FORMULA] as a function of PNN mass [FORMULA] (assuming [FORMULA] =0.2M [FORMULA]) for objects with [FORMULA] [FORMULA] 0.64M [FORMULA].

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© European Southern Observatory (ESO) 1997

Online publication: April 6, 1998