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

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4. Estimate of the mass loss rates

To estimate the mass loss rates in white dwarfs, in the discussion of Paper I we used a formula from Blöcker (1995) derived from the calculations of Pauldrach et al. (1988) for hot central stars of planetary nebulae.

[EQUATION]

In Paper I has been shown that with the corresponding mass loss rates the separation of hot hydrogen-rich white dwarfs into DAO's and DA's can be explained. However, the use of this equation for white dwarfs may be criticized for the following reasons. First, it does not take into account the dependance of the mass loss rate on the composition. This dependance is of special importance for our calculations. Secondly, Eq. (33) yields a non-zero mass loss rate for all white dwarfs and thus disregards the existence of a wind limit. Thirdly, the Pauldrach et al. results for hot CSPN have been obtained with their improved CAK (Castor et al. 1975) theory. However, for extremely weak winds expected in white dwarfs the CAK theory is probably not the appropriate one. This is due to the effect of line shadowing discussed by Babel (1996) for the case of main sequence B stars. Whereas dense winds are driven by a large number of weak lines, in thin winds the strong lines contribute significantly to the radiative acceleration. These are, however, affected by their broad photospheric counterparts. As a consequence the radiative acceleration does not only depend on the density and the gradient of the velocity as in the CAK theory, but on the velocity itself in addition. This effect may lead to mass loss rates, which may be lower by more than a factor of ten as predicted by the CAK theory. An additional problem is, that extremely weak winds may be selective, so that only these elements are lost for which the radiative acceleration is large enough. The problem of selective winds has been investigated for A stars by Babel (1995). For similar reasons Abbott's (1982) mass loss formula cannot be expected to be reliable for the case of white dwarfs. Moreover, it is valid for [FORMULA] only.

For purposes of comparison, Eq. (33) will be applied in some cases. In addition we use an alternative estimate of [FORMULA] as described below. It is simple enough to be implemented into our code. The intention is to allow for a dependence of [FORMULA] on the chemical composition, to recognize when the wind limit is reached and to make the calculations formally consistent. The effect of line shadowing would require model atmosphere calculations, however.

The existence of a chemically homogeneous, radiatively driven wind requires that the maximum possible radiative acceleration is at least equal to the gravitational acceleration. If this is the case, we estimate [FORMULA] from the conditions at the sonic point by use of various approximations, which will be described in this section. For an isothermal wind with constant mean particle mass µ the equation for the wind velocity v can be written as (e.g. Mihalas 1978):

[EQUATION]

where g, [FORMULA], [FORMULA] are the gravitational acceleration and the radiative accelerations due to bound-free transitions and lines, respectively. a is the isothermal sound velocity. At the sonic point, where [FORMULA], the right hand side must be zero. In "cool" winds with [FORMULA] the expression [FORMULA] at the sonic point is small, so that the total radiative acceleration must approximately be equal to g. In radiatively driven winds the sonic point is near the photosphere. It is, however, outside of our computation domain. The diffusion velocities in the wind region are assumed to be zero, which is consistent with the assumption of chemically homogeneous winds.

We calculate the radiative acceleration at the sonic point by use of the concept of force multipliers described by CAK and Abbott (1982) and various rough estimates. As no simple and reliable estimate for the occupation numbers in the wind region exists (Pauldrach 1987), the ones at [FORMULA] are assumed to be representative. The emergent fluxes from our own models described in Sect. 2 are used. The line list is the same as used for the diffusion calculations with 280 lines of the elements H, He, C, N and O, which are preferably the strongest lines. The justification for these assumptions is, that for solar composition and wind optical depths [FORMULA] smaller than about -5.5 our force multipliers coincide within a factor of about three with Abbott's ones. For the case of weak winds, which are of special interest for the present investigation, the densities at the sonic point are so low that according to our results [FORMULA] is indeed an upper limit.

The maximum value of the force multiplier and thus the maximum radiative acceleration is reached for [FORMULA]. Abbott (1982) has calculated force multipliers for [FORMULA] and solar composition. At the smallest wind optical depth considered by him ([FORMULA]) the values are of the order [FORMULA]. The dotted line in Fig. 1 would be maximum possible radiative acceleration, if the maximum value of the force multiplier were [FORMULA] for all effective temperatures. Thus only in the region above this line winds can exist. This result may be compared with the short-dashed line, which represents the corresponding wind limit obtained with the force multipliers from our calculations, where the occupation numbers at [FORMULA] for models with [FORMULA] have been used. For [FORMULA] both estimates are of the same order. For [FORMULA], however, according to our results the maximum radiative acceleration even decreases with increasing [FORMULA]. This is because H and He are preferably fully ionized and the CNO elements have noble gas configuration, which makes the radiative acceleration small. The long-dashed line represents the wind limit for helium-rich composition with a solar ratio CNO/He. The long dashed-dotted line is the wind limit for PG 1159 stars with [FORMULA], [FORMULA] and [FORMULA].

[FIGURE] Fig. 1. The predicted wind limits in the [FORMULA] - [FORMULA] diagram for solar compostion (short dashed line), helium-rich composition with solar ratio CNO/He (long dashed), PG 1159 stars (dashed dotted) and for the case with a constant force multiplier of [FORMULA] (dotted line). In addition a [FORMULA] post-EHB track from Dorman et al. (1993) and two post-AGB tracks from Blöcker (1995) are plotted

The results show that for hydrogen-rich white dwarfs with [FORMULA] the existence of chemically homogeneous winds is questionable for the whole cooling sequence. Only for less massive objects they may exist until they have reached a surface gravity of about [FORMULA]. According to our results, for DO's the strong lines of [FORMULA] yield a major contribution to the total radiative acceleration. Mass loss possibly exists until they have cooled down to [FORMULA]. As a consequence of their extremely large metal abundances, for PG 1159 stars the existence of winds is clearly more probable than for any other type of high-gravity stars. They are the only objects for which even at [FORMULA] mass loss may still exist.

We now estimate the mass loss rate from the conditions at the sonic point, where [FORMULA]. Then the right hand side of Eq. (34) must be zero. If the expression [FORMULA] is neglected, we have:

[EQUATION]

In general, the radiative acceleration due to bound-bound transitions, [FORMULA], depends on the density, the gradient of the velocity and the velocity itself. From this equation the density [FORMULA] at the sonic point can be obtained, if an assumption about the velocity gradient is made. We use

[EQUATION]

This assumption is motivated by the fact that for wind models calculated according to the CAK theory a [FORMULA] law is valid: [FORMULA], where [FORMULA] is of the order 0.5. Then between [FORMULA] and [FORMULA] the velocity changes from [FORMULA] to [FORMULA]. To estimate the terminal velocity [FORMULA] we assume a constant ratio [FORMULA]

[EQUATION]

where [FORMULA] is the escape velocity. According to the results for hot central stars of planetary neblue (Pauldrach et al. 1988), hot PG1159 stars (Koesterke et al. 1998) and main sequence B-stars (Babel 1996) this relation between [FORMULA] and [FORMULA] should hold within a factor of two. With [FORMULA] from Eq. (35), the mass loss rate is obtained from the equation of continuity:

[EQUATION]

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Online publication: July 13, 2000
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