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Astron. Astrophys. 359, 1042-1058 (2000)

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7. The floating up of hydrogen

In this section we again assume initial number ratios [FORMULA], [FORMULA], [FORMULA], but now with various admixtures of hydrogen. For the track with [FORMULA] the results for initial ratios hydrogen [FORMULA] and 0.2 will be discussed in detail. Then the results for all tracks are summarized.

In Fig. 14 the surface number fractions of the various elements are shown as a function of [FORMULA] for an initial number ratio [FORMULA]. The corresponding mass loss rates are obtained with the method described in Sect. 4 and are shown in Fig. 16. For [FORMULA] the results are similar as for the case without hydrogen discussed in the previous section. Mass loss with [FORMULA] keeps the stellar surface helium-rich and prevents the gravitational settling of the CNO elements. Then, at [FORMULA] and [FORMULA], hydrogen floats up. The number fractions of the CNO elements are still reduced by a factor of two only, so that we now have a hybrid PG 1159 star. The enrichment of hydrogen in the outer regions favours the gravitational settling of the heavy elements. This is because the resistance coefficients in Eq. (2) are proportional to the square of the charge of the particles. This leads to larger diffusion velocities in hydrogen-rich surroundings in comparison to the helium-rich case. This has the consequence that in hydrogen-rich PG 1159 stars the heavy elements sink earlier than in helium-rich ones. For the example shown in Fig. 14 we expect the wind limit at [FORMULA], [FORMULA]. The corresponding internal structure is shown in Fig. 15. We now have a DAO with [FORMULA]. Accidentally, N and O have nearly the solar abundance, only C is still somewhat overabundant. It seems to be impossible to find out spectroscopically, if such a DAO is in a transition state or a descendant from a hydrogen-rich progenitor. The tick marks in the upper left part in the figure indicate the mean optical depths [FORMULA] and 10, respectively. It can be seen that the stellar atmosphere is in very good approximation chemically homogeneous.

[FIGURE] Fig. 14. Surface number fractions of H, He, C, N and O as a function of the effective temperature for the track with [FORMULA] and an initial number fraction [FORMULA]

[FIGURE] Fig. 15. Number fractions as a function of the gas pressure at [FORMULA], [FORMULA] for the case with [FORMULA]. In the upper part of the figure the Rosseland mean optical depths [FORMULA] and 10 are indicated

[FIGURE] Fig. 16. Mass loss rates (in [FORMULA]; [FORMULA]) as a function of the effective temperature. The two solid lines represent the rates obtained in this paper for the cases [FORMULA] and 0.01, respectively. For comparison, the results from Eq. (33) are shown in addition (dashed line)

In Fig. 17 the surface number fractions are shown as a function of [FORMULA] for an initial ratio [FORMULA], the corresponding mass loss rates are plotted in Fig. 16. Hydrogen floats up at [FORMULA], [FORMULA]. Then the CNO elements sink rapidly and the wind limit is reached at [FORMULA], [FORMULA]. So hydrogen floats up almost at the same time, when the CNO elements sink. The PG 1159 star directly evolves into a DAO, whereas the DO state is left out. In Fig. 18 the internal structure at the wind limit is shown. The outer regions are stratified with a hydrogen layer mass of about [FORMULA] floating on top of the helium-rich regions. Again the stellar atmosphere is approximately chemically homogeneous, because during the transformation phase a weak wind is still present.

[FIGURE] Fig. 17. Surface number fractions as a function of the effective temperature for the track with [FORMULA] and [FORMULA]

[FIGURE] Fig. 18. Number fractions as a function of the gas pressure at [FORMULA], [FORMULA] and the case [FORMULA]. In the upper part of the figure the Rosseland mean optical depths [FORMULA] and 10 are indicated

In Fig. 19 all results of this section are summarized. In addition we have done calculations for the tracks from Blöcker (1995) for [FORMULA] and [FORMULA] (hydrogen burners) and [FORMULA] (helium burners). The two dashed lines indicate where a number ratio [FORMULA] is expected for initial ratios [FORMULA] and 0.01, respectively. These two lines have been derived with the mass loss rates from Eq. (33). (The corresponding results obtained with the mass loss rates from this paper are similar, they are shown in Fig. 20.) With the rates according to this paper we expect the wind limit for the two cases at the solid lines in Fig. 19. So PG 1159 stars with an initial number ratio [FORMULA] will transform into DAO's somewhere near the region between the upper dashed line and the lower solid line. For [FORMULA] the results would be very similar to the case without hydrogen discussed in the previous section. The heavy elements sink and the wind limit is reached before hydrogen floats up. So only these PG 1159 stars will be transformed into DO's.

[FIGURE] Fig. 19. Summary of the results in the [FORMULA] - [FORMULA] diagram. The dashed lines indicate, where (with [FORMULA] from Eq. (33)) it is [FORMULA] for initial ratios [FORMULA] (upper line) and 0.01 (lower line). The solid lines indicate the wind limits for both cases, derived with [FORMULA] from this paper (the upper line is for [FORMULA], the lower line for 0.01, respectively). In addition are introduced the same DAO's (filled symbols) and DA's (open symbols) as in Fig. 6 and two evolutionary tracks labelled with the mass in [FORMULA]

[FIGURE] Fig. 20. Summary of the results in the [FORMULA]-[FORMULA] diagram. The dashed line represents the wind limit for hydrogen-rich white dwarfs. Near this line we expect the transition from DAO's into DA's. The solid line is the wind limit for PG 1159 stars, which can exist above this line only. For PG 1159 stars with an initial ratios [FORMULA] and 0.01, we expect the floating up of hydrogen ([FORMULA]) at the upper and lower dotted line, respectively. The filled circles represent the DAO's from Napiwotzki (1999) and Bergeron et al. (1994), the open symbols the DA's from the compilation of Napiwotzki (1999). The filled squares represent the PG 1159 stars from Dreizler et al. (1995b) and Dreizler & Heber (1998), the triangles the DO's from Dreizler & Werner (1996)

The most important conclusion from the results shown in Fig. 19 is, that the majority of all DAO's cannot be descendants of hydrogen-poor PG 1159 stars with [FORMULA]. This is true especially for objects with [FORMULA] or even lower surface gravities. For the expected mass loss rates hydrogen would not yet float up. The DAO's with [FORMULA] probably have evolved directly from the extented horizontal branch (EHB) to the white dwarf region. Thus they are descendants from sdB, sdOB or sdO stars (for a review see Heber 1992). The H/He ratio of these objects varies from helium-poor with [FORMULA] to helium-rich with no detectable hydrogen. Although in this section we have not investigated the post-EHB evolution, it is plausible that these variations have consequences for the composition of the corresponding white dwarfs.

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© European Southern Observatory (ESO) 2000

Online publication: July 13, 2000
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