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Astron. Astrophys. 338, 75-84 (1998)

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5. Discussion

According to the diffusion calculations, for [FORMULA] the surface number fraction of helium reaches a maximum value of about [FORMULA] near [FORMULA]. The momentum transferred on helium by bound-free transitions is essential for these stellar parameters, whereas the ionization effects and the exact shape of the [FORMULA] line profiles turned out to be of minor importance. For [FORMULA] the predicted number fraction is below [FORMULA], which is lower by at least a factor of ten than observed in DAO white dwarfs (Bergeron et al., 1994, Napiwotzki, 1995). According to Chayer et al. (1995) the simplified treatment of the radiative transfer (use of the diffusion approximation as described Papers I and II) gives quite reasonable estimates of the surface abundance of levitating elements. So we expect that a model atmosphere approach would not change the situation significantly.

The failure of the diffusion calculations is not surprising, because the results of Sect. 4 have shown that the helium abundances in hot white dwarfs cannot be understood without the effect of mass loss, which , compared to the diffusion calculations, may lead to over- or underabundances of helium by several orders of magnitude. Recently, Werner et al. (1995) and Dreizler et al. (1995) detected in the spectra of five DO and one DAO white dwarf metal absorption lines of ultrahigh ionization states, which are possibly due to a stellar wind. However, the mass loss rates are not known. To discuss the results we will use a formula from Blöcker (1995), which is based on theoretical mass loss rates for central stars of planetary nebulae (Pauldrach et al., 1988).

[EQUATION]

L is the stellar luminosity in solar units and [FORMULA] is the mass loss rate in [FORMULA]. We use this equation to extrapolate the mass loss rates into the region of white dwarfs and subdwarfs. To account for the observed helium abundaces in sdOB stars, we inferred in Sect. 4.1 for [FORMULA], [FORMULA] a mass loss rate of [FORMULA], which is necessary to prevent helium from sinking too rapidly. For a typical subdwarf mass of [FORMULA] Eq. (10) yields [FORMULA]. The theoretical formula of Abbott (1982) for radiatively driven hot star winds yields [FORMULA] for a solar metallicity. However, it predicts a nearly linear decrease of [FORMULA] with decreasing metallicity. For hot main sequence stars Kudritzki et al. (1987) obtain a mass loss rate reduced by about a factor of three, if the metallicity is reduced by a factor of ten. As the metal abundances in white dwarfs and subdwarfs are different from the solar one, the values for [FORMULA] used in the discussion below are rough estimates only. In the following we will neglect the difference between solar and stellar masses for the mass loss rate.

MacDonald & Arrieta (1994) modelled the evolution of sdB's into sdO's in the presence of mass loss. They have shown that, at least for solar metallicity, the sdB's may loose most of their hydrogen envelope during their evolution into the sdO region in the HRD. Our results indicate, that for a mass loss rate necessary to prevent helium from sinking too rapidly, the transformation of helium-deficient subdwarfs into helium-rich ones is possible. These results favour the hypothesis of an evolutionary link between the sdB's and sdO's, which is expected from the evolutionary tracks of Dorman et al. (1993) and Caloi (1989). For clarification it would be necessary to incorporate mass loss and diffusion into the calculations of stellar evolution. When the star leaves the EHB, the luminosity and thus the mass loss rate tends to increase. This effect will accelerate the transformation into a helium rich one.

Koesterke et al. (1998) investigated the winds of PG 1159 stars with [FORMULA] and [FORMULA]. They derived mass loss rates which are of the order [FORMULA] to [FORMULA]. With the luminosities given by these authors, these order of magnitudes are in accordance with Eq. (10). The results of Sect. 4.1 have shown that for [FORMULA] the surface composition would change by a factor of two only within [FORMULA]. Therefore the effect of diffusion on the surface composition seems to be neglibible small in these stars. This is the probable reason why the diffusion calculations presented in Papers I and II cannot explain the observed abundances. As the influence of diffusion near the stellar surface decreases with increasing mass loss rate, it should be negligible in central stars of planetary nebulae and hot PG 1159 stars.

For [FORMULA] and [FORMULA] between [FORMULA] and [FORMULA] the expected mass loss rates according to Eq. (10) are of the order [FORMULA] to [FORMULA]. For these rates the effect of diffusion is not negligible. Therefore, on the one hand the surface abundance of helium will be lower than the solar one. On the other hand helium sinks so slowly, that the surface abundance after [FORMULA] is still larger than predicted by the diffusion calculations. Therefore the presence of helium in DAO white dwarfs can be explained with the assumptions of a thick hydrogen layer and mass loss rates in this range. If the mass loss rate is below a critical value, we expect a strong depletion of helium. Therefore in the [FORMULA] - [FORMULA] diagram a line should exist, which separates the DAO's from the DA's, as indicated in Fig. 9.

[FIGURE] Fig. 9. Predicted separation line between the DAO's and DA's in the [FORMULA] - [FORMULA] diagram for [FORMULA] according to Eq. (10) and for mass loss rates reduced by a factor of ten (dashed-dotted line). The DAO's analyzed by Napiwotzki (1995) are marked (filled triangles) as well as the DAO's (filled squares) and the DA's (open squares) analyzed by Bergeron et al. (1994). The dashed lines indicate a post-EHB track from Dorman et al. (1993) with [FORMULA] and post AGB-tracks from Schönberner (1983), Koester & Schönberner (1986) and Blöcker (1995), respectively

The line has been obtained from the condition that for all stars below this line the surface helium number fraction decreases below [FORMULA] within [FORMULA]. The mass loss rates given by Eq. (10) have been assumed. In all white dwarfs and subdwarfs above this line helium should be present, if the time scales of stellar evolution are not much larger than [FORMULA]. If the cooling were taken into account, especially for lower values of [FORMULA] we expect the line to be shifted to somewhat lower gravities, because for these stars the cooling ages are significantly larger than [FORMULA]. All single white dwarfs with thick hydrogen layers below the line should be DA's, their helium abundances may be lower than predicted by the diffusion calculations. Bergeron et al. (1994) detected some DAO's, which are clearly in the DA region. These authors suggest that some of them are members of possibly interacting binary systems. The others should be in a DO/DA transition state. Napiwotzki (1995) analyzed some extremely hot objects without detectable helium lines, which are above the separation line. However, as helium is more difficult to detect in the spectra of very hot stars, the upper limits on the helium number fraction are between the solar one and about [FORMULA]. Therefore these results do not rule out the mass loss hypothesis, which hardly could explain the absence of helium in these objects. For almost all hot DA's analyzed by Finley et al. (1997), our results predict the absence of helium, if they have a thick hydrogen layer.

If the mass loss rates are smaller by a factor of ten than obtained from Eq. (10), the separation line would be shifted by about 0.4 dex to lower gravities, as indicated by the dashed-dotted line in Fig. 9. In this case, the presence of helium especially in the hotter and lower gravity DAO's analyzed by Napiwotzki (1995) can still be explained with mass loss, whereas most of the cooler ones analyzed by Bergeron et al. (1994) should be DA's, if they have a thick hydrogen layer. The comparison of the position of the DAO's in the [FORMULA] - [FORMULA] diagram with theoretical evolutionary tracks indicates, that many of the cooler DAO's are post-EHB stars, as suggested by Bergeron et al. (1994). As discussed above, the hydrogen-rich sdB's or sdOB's, respectively, probably evolve into helium rich sdO's. During the cooling, hydrogen will float up and transform them into DAO's. Therefore the outer regions of these objects are possibly stratified. As the H/He abundance profile is smeared out in the presence of mass loss, the atmospheres may appear to be chemically homogenous.

If the DB gap is real, then all DO white dwarfs should be transformed into DA's until they have cooled down to 45000 K. In which stage of evolution the transformation occurs, depends on the mass loss rate and on the number fraction of hydrogen in the DO white dwarf. Dreizler & Werner (1996) note, that the relative number of DA to non-DA stars continuously increases from the hot end of the cooling sequence to the DB gap. The absence of mass loss would require [FORMULA] in hot DO's, because otherwise they would transform into DAO's already before they have cooled down to [FORMULA]. A weak wind of [FORMULA] would be sufficient to prevent the transformation for [FORMULA], [FORMULA] and [FORMULA]. For [FORMULA] and the same stellar parameters [FORMULA] would be required. We have done some additional calculations for [FORMULA]. For [FORMULA], a surface gravity [FORMULA] and according to Eq. (10) [FORMULA] is expected. A DO with [FORMULA] would still be helium rich after some [FORMULA], whereas a DO with [FORMULA] would transform into a DA. So we conclude that the DO's which transform into DA's before they have cooled down to [FORMULA] should have [FORMULA] when they enter the cooling sequence. The others will transform later. An example probably is HD 149499B with [FORMULA], [FORMULA] and [FORMULA] (Napiwotzki et al., 1995, 1996; Dreizler & Werner, 1996). In other DO white dwarfs with similar stellar parameters, however, Dreizler & Werner (1996) did not find any hydrogen. There should be a real chance to detect more hybrid white dwarfs or such with about solar helium abundances, because in the presence of mass loss this transition phase takes about [FORMULA]. The search for hydrogen in helium-rich pre-white dwarfs indicates that a variety of hydrogen abundances may occur. Dreizler et al. (1996) and Napiwotzki & Schönberner (1995) detected some hybrid objects, whereas for two other PG 1159 stars and one DO white dwarfs Werner (1996) gives an upper limit of 5% for the number fraction of hydrogen.

From the results an increase of the number ratio of DA's to non-DA's from the hot end of the cooling sequence to lower effective temperatures seems to be plausible. A continuous distribution of hydrogen abundances in stars at the hot end should lead to a continuous distribution of the time scales, in which hydrogen floats up. To clarify the question if the results can explain quantitatively the observed ratio of DA and non-DA stars and the hydrogen layer masses obtained by asteroseismology of the pulsating ZZ Ceti white dwarfs (see e.g. Bradley & Kleinman, 1997) diffusion and mass loss and the cooling of the white dwarfs must be taken into account simultaneously.

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Online publication: September 8, 1998
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