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

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6. The PG 1159 / DO transition

In this section the transformation of PG 1159 stars into DO's will be investigated. An initial composition [FORMULA], [FORMULA] and [FORMULA] (number ratios) and [FORMULA] is assumed. Cases with hydrogen will be considered in the next section. The results for the [FORMULA] born-again track from Blöcker (1995) are discussed in detail. This track is used, because the data for the corresponding 0.625 track are available for [FORMULA] only and all known "cool" PG 1159 stars have masses smaller than about [FORMULA].

In Fig. 8 the surface number fractions of the elements He, C, N and O are plotted as a function of [FORMULA]. The corresponding mass loss rates have been obtained with the method described in Sect. 4 and are shown in Fig. 9 (solid line). We see that mass loss with [FORMULA] effectively prevents the CNO elements from sinking. Until the star has cooled to [FORMULA] the number fractions are reduced by a factor of about two only. Then, as a consequence of the increasing gravity and the decreasing mass loss rates, the effect of gravitational settling becomes more and more apparent. Because of the ongoing depletion of heavy elements, finally the mass loss rate decreases drastically. At [FORMULA], the PG 1159 star is transformed into a DO. Here the CNO elements are reduced by about a factor of 20 and mass loss cannot exist any longer. During the cooling, the relative abundances of the CNO elements remain nearly unchanged. If an initial value of [FORMULA] instead of 0.01 is used, the results would be similar with the only difference that C and O sink somewhat earlier, because the mass loss rates are slightly lower. In this case the wind limit is reached at (at [FORMULA]). The relative abundances of the CNO elements remain nearly unchanged. Therefore the signifcant differences of the nitrogen abundances in the various PG 1159 stars found by Dreizler & Heber (1998) cannot be due to diffusion processes.

[FIGURE] Fig. 8. Surface number fractions as a function of the effective temperature for the track with [FORMULA] and mass loss rates for which the dependance on the composition is taken into account

[FIGURE] Fig. 9. Comparison of the mass loss rates (in [FORMULA], [FORMULA]) according to this paper (solid line) to the rates from Eq. (33)

For comparison, in Fig. 10 the corresponding results obtained with the mass loss rates according to Eq. (33) are shown. Again the PG 1159 star is transformed into a DO at [FORMULA] with a somewhat smoother transition, however. This is because in this case [FORMULA] does not depend on the composition and decreases continiously (see Fig. 10). The ongoing mass loss beyond our predicted wind limit finally leads to a strong depletion of the CNO elements at [FORMULA].

[FIGURE] Fig. 10. Surface number fractions as a function of the effective temperature for the track with [FORMULA] and mass loss rates according to Eq. (33)

It is a typical result that mass loss rates of the order [FORMULA] or somewhat lower lead to strong underabundances of the CNO elements. In depths where the temperature is between about 200000 and [FORMULA] the CNO elements have noble gas configuration. The radiative acceleration is extremely small, which leads to negative diffusion velocities directed to the stellar interior. When the mass loss rate falls below a critical value, even the total velocity (diffusion+wind) may become negative. Thus the particles of these elements lost at the surface cannot be replaced any longer, because the supply from the stellar interior is interrupted. This is true for helium-rich as well as for hydrogen-rich white dwarfs.

In Fig. 11 and Fig. 12 the number fractions of the various elements are plotted as a function of the gas pressure for two examples. At [FORMULA], [FORMULA] (Fig. 11) the composition still is approximately homogeneous. The surface number fractions of the CNO elements are reduced by not more than a factor of two in comparison to the initial composition. The mass loss rates are of the order [FORMULA]. This is enough to prevent the formation of composition gradients. This is quite different from the situation at [FORMULA], [FORMULA] shown in Fig. 12. According to Eq. (33) it is [FORMULA] here. As explained above, this leads to strongly reduced abundances in the outer regions. At the stellar surface the number fraction of C is of the order [FORMULA], these for N and O are even lower. Dreizler & Werner (1996) have analyzed several DO's with [FORMULA]. Indeed, their upper limits for the abundances of the CNO elements are of the order [FORMULA]. This is lower than the predictions from the equilibrium diffusion theory, which is expected to be valid in the absence of mass loss or other competing processes. So the strong depletion of the CNO elements seems to be compatible with our predictions, if the mass loss rates are assumed to depend on the luminosity only. However, in the absence of heavy elements, chemically homogeneous winds could exist only if they were driven by the lines of helium alone. According to our estimate the radiative acceleration at the sonic point is too low (this would even be true for ratios [FORMULA]). In addition we have estimated the radiative acceleration on helium at the sonic point for [FORMULA] with the fluxes from a (static) NLTE model atmosphere with pure helium from Napiwotzki (priv. comm.). It is too low by a factor of the order 100. To summarize, an explanation of the low abundances of these DO's with diffusion and mass loss is possible. However, the driving mechanism of these winds is not clear.

[FIGURE] Fig. 11. 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

[FIGURE] Fig. 12. The same as Fig. 11 at [FORMULA], [FORMULA]

In Fig. 13 all results of this section are summarized. Apart from the 0.529 track discussed above, we have done calculations for a 0.836 born-again track from Blöcker (1995). To cover the intermediate mass range two tracks for hydrogen burners have been used with [FORMULA] and [FORMULA], respectively. The tracks for hydrogen and helium burners are very different during the pre-white dwarf evolution. On the cooling sequence, however, when we expect the onset of gravitational settling, they are similar. The surface gravities differ by a factor of about 1.2 only and the cooling ages are approximately the same.

[FIGURE] Fig. 13. Summary of the results in the [FORMULA] - [FORMULA] diagram. The solid line is the predicted wind limit for PG 1159 stars, the dashed line the wind limit obtained with mass loss rates reduced by a factor of 10. With the mass loss rates from Eq. (33) carbon would be reduced by a factor of two at the upper dotted line and by a factor of ten at the lower dotted line. The squares symbols represent PG 1159 stars, the triangles DO's. The thin lines represent the evolutionary tracks used for the computations (labelled with [FORMULA])

Let us consider the two dotted lines in Fig. 13. With the mass loss rates from Eq. (33) carbon would be reduced by a factor of two at the upper dotted line and by a factor of ten at the lower one. Thus even if the dependance of the mass loss rate on the composition is neglected, we expect a narrow transition zone between PG 1159 stars and DO's. If we allow for a dependance of [FORMULA] on the composition, [FORMULA] decreases drastically when the heavy elements sink. This in turn favours gravitational settling so that we expect an even sharper transition. Therefore for these cases we have plotted the wind limit only, this is the line where we expect the transition. The solid line represents the wind limit if the mass loss rates from the method described in Sect. 4 are used. The dashed line is the wind limit from the same estimate, however with mass loss rates reduced by a factor of ten. The squares in Fig. 13 represent the PG 1159 stars from the list of Dreizler et al. (1995b), where the results of the new analyses of some objects by Dreizler & Heber (1998) have been taken into account. The triangles represent the DO's analyzed by Dreizler & Werner (1996). Clearly no PG 1159 star exists below the predicted transition region. Near a horizontal line at [FORMULA] several PG 1159 stars and DO's exist with similar stellar parameters. Some of these PG 1159 tendencially have somewhat lower abundances of C and O as it is typical for their hotter counterparts. This may be considered as an indication for the onset of gravitational settling. Two of the DO's (RE0503-289 with [FORMULA], [FORMULA] and PG0108+101 with [FORMULA], [FORMULA]) have number ratios [FORMULA]. These abundances are clearly lower than in PG 1159 stars, but more than a factor of ten larger than the predictions from diffusion calculations without mass loss (Chayer et al. 1995a; Dreizler 1999). A natural explanation of these results is that they are transition objects. So from the observational side comes some evidence that the transition region is near a line with [FORMULA]. If this line represents the transition region, this would require that our estimated mass loss rates are too large by more than a factor of ten. In Sect. 4 we have discussed the effect of line shadowing, which is of great importance in the case of thin winds. This may be one reason. Secondly, the densities in the wind region are much lower than at [FORMULA], where we have evaluated the occupation numbers. The lower densities favour higher ionization states such as [FORMULA], [FORMULA], [FORMULA] and [FORMULA]. For the CNO elements with helium-like configuration the radiative acceleration is low, however. For [FORMULA] with increasing [FORMULA] more and more particles tend to be in these ionization states. So it is well possible that the mass loss rates are approximately the same for objects with [FORMULA] and similar gravities, although the luminosities are very different. The estimates in Fig. 1 have shown that the maximum possible radiative acceleration in the wind region even tends to decrease for [FORMULA]. This could explain why in the [FORMULA]-[FORMULA] diagram the transition region is near a horizontal line with constant gravity.

According to our results the existence of PG 1159 and DO's with various composition with similar stellar parameters is not surprising, because we expect a sharp transition. Small differences in the initial composition may be the reason that the mass loss rates are somewhat different. Because of the resulting variations of the mass loss rates some objects may transform into DO's somewhat earlier, others later. So a certain overlap of the PG 1159 and DO region in the [FORMULA] - [FORMULA] diagram is expected and can even be considered as an indication for the existence of an evolutionary link between both types.

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© European Southern Observatory (ESO) 2000

Online publication: July 13, 2000
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