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

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2. Basic assumptions

The most important improvement in comparison to Paper I is, that the stellar parameters are allowed to vary. [FORMULA] and [FORMULA] as a function of time are taken from evolutionary tracks, [FORMULA] is estimated as described in Sect. 4. Thus the effect of cooling on the stellar structure can be taken into account. We neglect, however, the possible effect of the changing composition on the stellar parameters and the cooling ages. This would require the implementation of the method into a stellar evolution code. As nuclear burning is not taken into account, the calculations are restricted to the outer regions. The inner boundary of our computation domain, characterized by a fixed gas pressure scale, is at [FORMULA]. As during the cooling the gravity increases, the mass within the computation domain decreases. For [FORMULA] it is about [FORMULA]. The composition at the inner boundary is held fixed. As we know from our previous results, usually the diffusion time scales at this depth are large in comparison to the cooling ages of about [FORMULA]. An exception may be the case with traces of hydrogen in a helium-rich background plasma. In addition, for typical mass loss rates of the order [FORMULA] and cooling ages of [FORMULA] the total mass loss is small compared to the mass within the computation domain. Thus we expect that the inner boundary condition has negligible influence on the predicted surface composition. At the beginning of the calculations, that is just before the pre-white dwarfs enter the cooling sequence, the composition within the computation domain is assumed to be homogeneous.

The temperature structure is obtained from

[EQUATION]

with [FORMULA] at the outer boundary. In contrast to Paper I radial effects are taken into account now. To account for the effect of the composition on the temperature structure we calculate monochromatic continuum opacities as described in Unglaub & Bues (1996), from which the Rosseland mean opacity [FORMULA] is evaluated. Eq. (1) is a sufficient approximation for our purposes. For the mass loss rates of interest the matter of the stellar atmosphere is removed within a few years or even less. Within this time diffusion cannot change the composition significantly. Then the atmosphere is replaced by matter from underlying regions. Therefore in diffusion calculations with mass loss especially the conditions in deeper regions are important. This is in contrast to calculations without mass loss, where the surface abundance can be predicted only if the radiative accelerations in the stellar atmosphere are known.

The radiative accelerations for heavy elements are obtained as described in Unglaub & Bues (1996) with the method and line list similar as in Vauclair et al. (1979). For [FORMULA] and [FORMULA] the evaluation of the line profiles has been improved (see Paper I). For hydrogen-like ions the factor [FORMULA] from Massacrier (1996) is taken into account. With this factor the momentum transferred to the heavy particles in photoionization processes is calculated as described in Paper I. For the mass loss rates and stellar parameters of interest only the radiative accelerations for helium- and hydrogenlike ions of the CNO elements are of major importance. Lower ionization states are preferably present in the outermost regions near the stellar atmosphere. As explained above, the conditions in these regions have little influence on our results, however.

The diffusion velocities [FORMULA] are calculated as in Unglaub & Bues (1997) from the system of linear equations, which consists of the various momentum equations and the condition of zero net mass flow. For an element l the momentum equation reads

[EQUATION]

The expressions on the left represent the momentum per unit volume and unit time transferred to the element by the gradient of the partial pressure, gravity, the electric field E and the radiative force [FORMULA]. Ionization effects and thermal diffusion are neglected. The former may be of importance only in regions of partial ioniziation and thus are negligible at least for H and He. Paquette et al. (1986a,b) have shown that for plasma conditions typical for hot white dwarfs thermal diffusion is less effective than gravitational settling. For our calculations the relative magnitude of the diffusion velocity and the drift velocity due to mass loss (see Eq. (3)) is of special importance. If, for example, the inward diffusion velocity of an element would be larger by a factor of two, this could be compensated by an increase of the mass loss rate by a similar factor. As the mass loss rates are obtained from rough estimates, these simplifications in the diffusion calculations seem to be justified. The dependence of the gravity g on the distance r from the stellar center is approximated by [FORMULA]. The resistance coefficients [FORMULA] are obtained according to Paquette et al. (1986a). The charge [FORMULA] of an element is the mean value over all ionization states.

Although we take into account mass loss, the total stellar mass is assumed to be constant. This is justified, because for the considered mass loss rates and time scales the total mass loss is negligible in comparison to [FORMULA]. With this assumption the mean velocity due to mass loss can be obtained from the equation of continuity:

[EQUATION]

where [FORMULA] is the density and [FORMULA] the wind velocity.

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© European Southern Observatory (ESO) 2000

Online publication: July 13, 2000
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