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Astron. Astrophys. 321, 485-491 (1997)

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4. Results

Fig. 2 shows the results for [FORMULA], [FORMULA]. At a time [FORMULA] we assume number fractions C/He=N/He/=O/He=Ne/He= [FORMULA] for all heavy elements. The lower boundary condition is such that the compositition remains constant there, so that for long times the abundance distribution should converge to the stationary case, for which we predicted an atmosphere dominated by heavy elements with traces of helium only (see Fig. 4a in Paper I). After 10000 y the atmosphere is still far from a stationarity as can be seen from Fig. 2a. This time is comparable to the time scales of stellar evolution of post-AGB stars in the corresponding region of the HRD (according the evolutionary track in Fig. 1d of Wood & Faulkner, 1986, for [FORMULA]). Thus it becomes clear immediately that the transformation of a helium-rich into a metal-rich atmosphere is not possible in times which are short in comparison to the time scales of stellar evolution. The number fraction of carbon is still below [FORMULA] everywhere in the atmosphere. However, the number fraction of oxygen in the outer regions near [FORMULA] is 0.05, which is only slightly lower than the typical oxygen abundance of PG 1159 stars. Although the influence of stellar evolution cannot be neglected for longer times, for this example we proceed with the calculations until the atmosphere is transformed into a metal-rich one. After 50000 y maximal number fractions of oxygen and neon of 0.2 and 0.3, respectively, are obtained in a region with [FORMULA] (Fig. 2a). After 200000 y (Fig. 2c) neon becomes more abundant than helium near [FORMULA]. The carbon number fraction has a maximum value of 0.23 at [FORMULA] (the temperature there is 380000 K). Only in this part of the atmosphere where the state of ionization changes from [FORMULA] to [FORMULA] a carbon enrichment by radiative forces is possible. It takes longer than 200000 y until a carbon abundance typical for PG 1159 stars is possible. After this time, however, the star should already have cooled down to effective temperatures below 100000 K. Therefore it seems to be impossible to explain the observed number ratios of carbon by diffusion, at least if we start from approximately solar number fractions. After 800000 y (Fig. 2c) we expect a thin metal-rich region of about [FORMULA] floating ontop of the helium-rich mantle. A comparison with Fig. 4a of Paper I shows that stationary conditions are still not possible. Only in the outer regions with [FORMULA] the number fractions of carbon, oxygen and neon are close to their stationary value. The nitrogen abundance has a sharp minimum near [FORMULA]. The rapidly increasing radiative forces lead to an increasing diffusion flow of nitrogen in outward direction, which in turn leads to a depletion of the region between the two abundance maxima. At [FORMULA] nitrogen diffuses still outwards with a high diffusion velocity of [FORMULA] m/s. The diffusion velocities of the elements C, O and Ne are lower by about four orders of magnitude. The number fraction of helium is still much larger than predicted for the stationary cases. It still diffuses inwards with a diffusion velocity of about [FORMULA] m/s.

[FIGURE] Fig. 2a-d. Number fractions of the elements He (full drawn line), C (- - - -), N (- [FORMULA] - [FORMULA] -), O (- - - -), Ne (- [FORMULA] - [FORMULA] -) as a function of the gas pressure for [FORMULA], [FORMULA] after various times. The number fractions are on a logarithmic scale and are defined as the ratio of all particles of an element to all heavy particles. The gas pressure is in SI-units! The three tick marks in the upper part of each figure show where the Rosseland mean optical depth is [FORMULA], respectively.

Fig. 3 shows the results for [FORMULA], [FORMULA], again with C/He=N/He=O/He=Ne/He= [FORMULA] at time [FORMULA]. After 100 y (Fig. 3a) the atmosphere is still helium-rich. After 500 y (Fig. 3b) carbon and neon are more abundant than helium near [FORMULA] and smaller optical depths. According to the evolutionary track of a 0.89 [FORMULA] post-AGB remnant of Wood & Faulkner (1986) this time is in the same order of magnitude similar as the time scales of stellar evolution. So we see that a transformation from a helium-rich into a metal-rich atmosphere is indeed possible in this case, at least if there is no mass-loss. After 1000 y (Fig. 3b) the helium number fraction in the photosphere has dropped below 0.1 and carbon and neon are the most abundant elements. A comparison with Fig. 7a of Paper I shows that the atmosphere is still far away from a diffusive equilibrium state. Here again time-independent calculations are inadequate.

[FIGURE] Fig. 3a-c. Results for [FORMULA], [FORMULA] (for explanations see Fig. 2)

In Figs. 4 and 5 we take number ratios typical for PG 1159 stars at time [FORMULA]: C/He=0.5 and O/He=0.1. Furthermore is assumed N/He= [FORMULA] and Ne/He= [FORMULA]. Two cases are considered, both with gravities [FORMULA]: [FORMULA] (Fig. 4) and [FORMULA] (Fig. 5). In Paper I surface abundances of carbon and oxygen were predicted which are clearly lower than the observed ones. Therefore it is an interesting question if these discrepancies are simply due to long diffusion time scales or not. A look at Figs. 4 and 5 reveals that in both cases carbon as well as oxygen should sink in time scales which are clearly short compared to these of stellar evolution.

[FIGURE] Fig. 4a-c. Results for [FORMULA], [FORMULA] (for explanations see Fig. 2)
[FIGURE] Fig. 5a-c. Results for [FORMULA], [FORMULA] (for explanations see Fig. 2)

For [FORMULA], carbon is significantly depleted already after 100 y in the outermost regions near [FORMULA]. Its number fraction has decreased by about a factor of three. After 1000 y (Fig. 4b) carbon is a trace element in regions with [FORMULA]. At [FORMULA] all number fractions are already very close to the values which have been predicted in Paper I for the stationary case. The calculations have been continued until [FORMULA] (Fig. 4c). It is remakable that the depletion of carbon goes on only slowly. This is due to the radiative forces acting in regions where carbon is mainly [FORMULA]. In depths with [FORMULA] almost the original abundance is maintained. The carbon-depleted region comprises a surface layer mass of not more than [FORMULA]. The number fraction of oxygen has a maximum value of 0.10 near [FORMULA] (the temperature there is [FORMULA]). Therefore in this region it has slightly increased in time (original number fraction at t =0: 0.062), which is also due to the effective radiative forces in regions where oxygen has hydrogen-like configuration.

For [FORMULA] the depletion of carbon and oxygen proceeds still more quickly. After 100 y (Fig. 5a), all number fractions in the photosphere near [FORMULA] are below [FORMULA], after 1000 y (Fig. 5b) they have reached their value which has been predicted for the stationary case. The dramatic decrease of the neon abundance in the outermost regions is not realistic. This is because we have entirely neglected the radiative forces for [FORMULA] and states of lower ionization. In Paper I the contribution of the bound-bound transitions of [FORMULA] has been taken into account, therefore the predictions there are more accurate for this case. After [FORMULA] a strong depletion of heavy elements is predicted in the outer region, which comprises a surface layer mass of about [FORMULA] (at [FORMULA]). For comparison, Dreizler et al. (1994) obtain from the model atmosphere analysis of HS 0704+6153 number ratios C/He = 0.2, O/He [FORMULA] 0.05. Also in the inner regions carbon and oxygen tend to sink. Only near [FORMULA] the oxygen abundance is still the original one (more exactly: it has increased by [FORMULA]). The temperature in this depth is [FORMULA].

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998