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

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5. The DAO/DA transition

In this section we investigate the question, for which hydrogen-rich white dwarfs mass loss prevents helium from sinking and when do they transform into DA's with no detectable helium. For the initial abundances of He and the CNO elements the solar ones are assumed. Evolutionary tracks for various masses are considered. A [FORMULA] track from Driebe et al. (1998) represents the evolution of helium white dwarfs. They may be produced in binary systems, if the hydrogen-rich envelope is stripped away by a companion during the red giant branch phase. The remaining mass is too low for the ignition of helium burning. A [FORMULA] track from Dorman et al. (1993) with a core mass [FORMULA], a hydrogen envelope mass on the zero age horizontal branch [FORMULA] and a mass fraction of metals [FORMULA] has been chosen, because it passes through the region of the DAO's investigated by Bergeron et al. (1994). This track represents the evolution from the (extended) horizontal branch directly to the white dwarf region. Most of the DAO's analyzed by Napiwotzki (1999) have masses between 0.5 and [FORMULA]. Therefore we consider a [FORMULA] post-early AGB track from Blöcker (1995) in detail. It represents the evolution of a star which suffers from two late helium flashes. At the cooling sequence, where we expect the onset of gravitational settling, this track is very similar to the [FORMULA] post-AGB track from Schönberner (1983). However, for the latter data are available for [FORMULA] only. In addition we consider various post-AGB tracks from Blöcker (1995) with masses between 0.605 and [FORMULA].

The computations start before the star has reached its maximum effective temperature. This is sufficient, because during the preceding evolution the expected mass loss rates are usually large enough, so that the effect of diffusion is negligible. Exceptions are cases with [FORMULA]. Here the wind limit may be reached even before the "knee" of the tracks, where they begin to cool. In Table 1 for various tracks the start values [FORMULA] and [FORMULA] are given.


[TABLE]

Table 1. Start values [FORMULA] and [FORMULA] for various tracks


For [FORMULA], in Fig. 2 the surface composition is shown as a function of [FORMULA]. The mass loss rates are obtained with the method described in Sect. 4, which allows for a dependance on the composition. They are plotted in Fig. 3 (solid line). The effect of gravitational settling is small until the star approaches the wind limit at [FORMULA], [FORMULA]. Then mass loss cannot prevent or retard any longer the sinking of helium. As decreasing abundances of helium and heavy elements reduce the mass loss rate, we expect a rapid transformation of the DAO into a DA. The small discontinuities in the helium abundance and mass loss rates appear because we update the mass loss rate only if the composition or the stellar parameters have changed by a certain amount (see Sect. 3.7). The CNO elements sink more slowly than helium. Whereas the number fraction of helium decreases by about a factor of ten in the range [FORMULA], the CNO elements are reduced by a factor of two only. The relative abundances of these elements are almost unchanged. At the wind limit the computations have been terminated. Perhaps selective winds still exist for a while, which lead to an outflow of such elements, for which the radiative force is large enough. Finally an equilibrium between gravitational settling and radiative levitation may be reached.

[FIGURE] Fig. 2. Surface number fractions (number of particles of an element l over the total number of heavy particles) as a function of the effective temperature for the track with [FORMULA] and mass loss rates for which the dependence on the composition is taken into account

[FIGURE] Fig. 3. Mass loss rates (in [FORMULA], [FORMULA]) according to this paper (solid line) and Eq. (33) (dashed line)

In Fig. 4 the number fractions of He, C, N and O are plotted as a function of the gas pressure for [FORMULA], [FORMULA]. Whereas the number fraction of helium continuously decreases from the inner to the outer regions, for the CNO elements we recognize the influence of the radiative acceleration, which acts especially on the hydrogen-like ions. For carbon this is near [FORMULA], where the temperature is about [FORMULA]. In regions where the radiative acceleration is very effective, the total velocity (diffusion + wind) has a local maximum. According to the equation of continuity this leads to a low particle density of the corresponding element. In the region with [FORMULA] carbon preferably has helium-like configuration. The radiative acceleration and thus the total velocity decreases. This leads to an accumulation of carbon. So the abundance tends to have a minimum in regions where the radiative force is large and a maximum in the adjoining region where it is small. This situation is somewhat different as expected from diffusion calculations which assume an equilibrium between gravitational and radiative forces.

[FIGURE] Fig. 4. Number fractions as a function of the gas pressure at [FORMULA], [FORMULA]. In the upper part of the figure the Rosseland mean optical depths [FORMULA] and 10 are indicated

In Fig. 5 the surface composition is shown as a function of [FORMULA], but now with the mass loss rates from Eq. (33). [FORMULA] is assumed to depend on the luminosity only. As the corresponding rates (dashed line in Fig. 3) are larger than those from our calculations, helium sinks later. At [FORMULA] it is reduced by a factor of ten. Then the ongoing mass loss leads to a strong depletion of helium. The CNO elements sink more slowly. At [FORMULA] they are reduced by a factor of 10 to 100, whereas no detectable helium is present. The helium abundance is even lower than predicted from the equilibrium diffusion calculations of Vennes et al. (1988). So in this scenario the DAO is transformed into a DA with an extremely low helium abundance, whereas the CNO elements are clearly less reduced. However, it requires that mass loss goes on independently of the low abundances.

[FIGURE] Fig. 5. Surface number fractions as a function of the effective temperature for the track with [FORMULA] and mass loss rates according to Eq. (33)

In the [FORMULA] - [FORMULA] diagram of Fig. 6 the results obtained from the calculations along tracks with various masses are summarized. At the upper solid line the helium abundance would be reduced by a factor of two according to our mass loss calculations. So just before the tracks cross this line we expect the onset of gravitational settling. The dashed-dotted line is the wind limit. For all objects above this line, our results predict the existence of winds. After the onset of gravitational settling, the wind limit is reached rapidly. Therefore both lines are close together. If the mass loss rates according to Eq. (33) are used, at the dotted line the helium abundance would be reduced by a factor of two. At the lower solid line would be He/H = [FORMULA]. In addition the DAO's analyzed by Napiwotzki (1999) and Bergeron et al. (1994) are shown (filled symbols), as well as the DA's (open symbols) from the compilation in Table A2 from Napiwotzki (1999). These objects, for which it is [FORMULA] within the limits of uncertainty given by the authors, are represented by filled squares. In accordance with our results, they are preferably near or above the upper solid line, where the helium abundance should be reduced by not more than a factor of two. The objects below this line, for which the onset of gravitational settling is predicted, all are clearly helium deficient. In the region below the lower solid line both estimates predict the absence of helium, at least the abundance should be lower than [FORMULA]. Indeed, in the great majority of all hydrogen-rich white dwarfs in this region no helium has been detected. The few DAO's may either be influenced by accretion of matter (e.g. from a companion) or are possibly in a DO/DA transition state. For the two hottest known DA's (EGB 1 and WeDe 1) with [FORMULA] and [FORMULA] Napiwotzki (1999) gives an upper limit [FORMULA]. From the estimate of the radiative acceleration in the wind region, we clearly expect the absence of mass loss. Therefore they should indeed be DA's. Eq. (33) yields a mass loss rate of the order [FORMULA]. Then, however, helium should be reduced by less than a factor of two and thus be detectable. This shows that [FORMULA] is an upper limit for these two objects.

[FIGURE] Fig. 6. Summary of the results in the [FORMULA] - [FORMULA] diagram. If the dependance of the mass loss rates on the composition is taken into account it is [FORMULA] at the upper solid line and the wind limit is reached at the dashed-dotted line. With the rates from Eq. (33) it is [FORMULA] at the dotted line and [FORMULA] at the lower solid line. The filled squares represent DAO's with [FORMULA], the filled circles helium deficient DAO's and the open circles DA's, respectively. In addition several tracks are introduced (labelled with [FORMULA])

If all progenitors of the white dwarfs had the same composition, the DAO's and DA's should be clearly separated in the [FORMULA]-[FORMULA] diagram. Otherwise the mass loss rates will differ even for objects with the same mass. Therefore, in dependance of the initial composition, in some objects helium sinks earlier, in others later. This leads to the existence of a transition region, where DAO's as well as DA's with similar stellar parameters exist. Thus this coexistence of both types is not in contradiction to our results, but probably is a consequence of different evolutionary histories.

Napiwotzki (1999) found a relation between the helium abundance and the luminosity of hot DAO's. In Fig. 7 our predictions are shown. The solid lines represent the results obtained with the mass loss rates, for which the dependance on the composition is taken into account. The two left solid lines are the predictions for the tracks with 0.546 and [FORMULA], respectively. As the tracks are similar, the results are similar, too. Therefore the two curves are close together. 16 out of 23 DAO's analyzed by Napiwotzki (1999) have masses in the range [FORMULA] and are represented by circles in Fig. 7. The results show that they are close to our predictions. For objects with luminosities larger than [FORMULA] the helium abundance is near the start value [FORMULA]. Here gravitational settling is negligible. To lower luminosities the helium abundance decreases slowly first, then there is a sharp cutoff at [FORMULA]. The two right solid lines are the predictions for the 0.467 and 0.414 tracks, respectively. In these cases, the luminosities, for which helium begins to sink, are lower. Most of the DAO's analyzed by Bergeron et al. (1994; represented by squares) have helium abundances in the range [FORMULA] and masses below [FORMULA]. According to our predictions they are beyond the wind limit. If the mass loss rates from Eq. (33) are used, however, for some of these DAO's an explanation with diffusion and mass loss seems to be still possible. The corresponding predictions for the tracks with 0.605, 0.467 and [FORMULA] are represented by dashed lines. All these result show that at least in the luminosity range between about 10 and [FORMULA] some relation between the helium abundance and the luminosity is indeed expected. A certain scatter will appear, if the various objects have very different masses and possibly initial compositions.

[FIGURE] Fig. 7. The helium abundance as a function of the luminosity. The solid lines represent the predictions with mass loss rates from this paper for four tracks with (from left to the right) [FORMULA] and [FORMULA], respectively. The dashed lines are the predictions with mass loss rates according to Eq. (33) for (from left to the right) [FORMULA] and [FORMULA]. The DAO's from Napiwotzki (1999) with masses in the range [FORMULA] are represented by circles, those with other masses by triangles. In addition the DAO's from Bergeron et al. (1994) are shown (squares)

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© European Southern Observatory (ESO) 2000

Online publication: July 13, 2000
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