4. The influence of mass loss
The diffusion velocitiy is the mean velocity of the particles of species s in the center of mass system. In the presence of a wind there is a net mass flow, which is added to the diffusion flow:
The wind velocity v at each depth is obtained from the equation of continuity with the assumption of a depth-independent stellar radius :
A stellar mass is usually assumed (with one exception for subdwarfs).
For the calculations with mass loss the numerical method described in Paper II had to be slightly modified. Let j be the total flow (diffusion + wind) of a special species of particles and z the depth variable. We write for the gradient at a grid point i:
For convenience, we use a half-mesh point notation here (in Paper II we denoted the half-mesh points with ). The points i represent the zone centers, the points and denote the inner and outer boundaries, respectively. This equation is equivalent to Eq. (11) of Paper II. This evaluation of the flux at the half-mesh points corresponds to the control volume method described by Hawley et al. (1984) and works sufficiently well for pure diffusion calculations. It is unstable, however, in cases with larger mass loss rates, where in the outer regions the wind velocity v exceeds the diffusion velocity w by several orders of magnitude. This instability is cured by evaluating the wind flow not at the zone boundary, but at the next grid point upstream or upwind. For example, if n is the density of the species of particles under consideration, we write for the flow at the zone boundary :
This means, the diffusion part is treated according to the control volume approach, whereas for the wind part a modified upwind scheme is used. In the usual upwind scheme described by Hawley et al. (1984), the density only is evaluated at the next point upwind. Our method is more appropriate to satisfy the demand of mass conservation in each zone. Compared to the control volume method, the upwind scheme leads to some loss of accuracy, because the derivative of the flow is not centered at the point i. For , and the mass loss rate is still small enough, so that both schemes could be used. The differences are not significant (less than 1%). In addition, for cases with larger mass loss rates a comparison with results obtained with a monotonic transport scheme, which can be considered as an improved upwind scheme with somewhat higher accuracy, did not lead to significant differences (about 3%).
The time steps are such that during one step the composition (e.g. the ratio ) does not change by more than . The drawback of the numerical method is, that during many time steps at the outer grid points the composition oscillates around some mean value, which changes only slowly. One measure to reduce the computational effort is to update such quantities like opacities, radiative forces per particle and resistance coefficients only if the composition at one of the grid points has changed by at least . Only the composition, the ionization equilibrium and the diffusion velocities must be updated after each time step. A second measure is to increase the distance between the grid points, especially in the outer regions. This allows larger time steps. With the gas pressure as independent variable we have used a grid spacing according to:
So from one grid point to another the gas pressure changes by plus an additional constant q. For example, for , and at the outer boundary is used and . So especially the zones in the outer regions are expanded, the inner regions are less affected. This is justified, because in the outer regions is , so that no significant concentration gradients occur. For this example, the regions with are lost within about one year. Within this time, the changes of composition due to diffusion are negligibly small. In spite of all this, the computations are still so time consuming that we can take into account the CNO elements in a few of the mass loss calculations only.
4.1. Cases with log g = 6.0
For lower surface gravities the numerical code works sufficiently well, so that for some of the calculations the CNO elements can be taken into account. We first consider the case , typical for hot central stars of planetary nebulae. At a composition He/H = C/H = N/H = O/H = 0.1 has been assumed. In Table 3 is given the predicted surface composition 10000 y after the onset of mass loss.
Table 3. Results for , at 10000 y after the onset of mass loss. The original number fraction of each element is 0.0714
For a mass loss rate of the number fraction of helium decreases by a factor of 6 instead of a factor of ten in the diffusion calculations. Abundance anomalies of the heavy elements are predicted with a deficiency of carbon in comparison to oxygen and nitrogen. For the number fraction of all elements decreases by about a factor of two only within 10000 y. Stronger winds prevent the elements from sinking. According to the review of Perinotto (1993) and the calculations of Pauldrach et al. (1988) typical mass loss rates for hot CSPN are of the order . Therefore we expect the effect of diffusion to be negligible.
Now we investigate the influence of mass loss for , . These stellar parameters are typical for subdwarfs near the hot end of the extended horizontal branch (EHB). Model atmosphere analysis of subdwarfs of typ sdOB or sdB yields underabundances of helium between a factor of two and 150 (e.g. Heber et al. (1984), Heber (1986), Moehler et al. (1990), Saffer et al. (1994)). In contrast to horizontal branch stars, the contribution of hydrogen burning to the total luminosity is negligible for EHB stars. This is because of the low mass of the hydrogen shell surrounding the helium burning core, which varies between for the sdB stars and to for the sdOB stars near the helium main sequence. The subdwarfs stay near the EHB with almost unchanged stellar parameters for about (Caloi, 1989; Dorman et al., 1993). The helium abundances predicted from the diffusion calculations are clearly too low, in the absence of competing mechanisms helium sinks within 10000 y. Fig. 3 shows the results 40000 y after the onset of mass loss for a weak wind with . At a solar composition has been assumed.
The results differ drastically from those of the diffusion calculations without mass loss. Near helium is now depleted by a factor of ten only, nitrogen has about the solar abundance and carbon is underabundant. A similar result has already been obtained by Michaud et al. (1985) and corresponds to the observed abundance anomalies. The problem, however, is the length of the time scales of stellar evolution. Therefore we have done some additional calculations for various mass loss rates over . The results are summarized in Fig. 4.
To save computation time the heavy elements and the ionization effects have been neglected. The lower boundary has been taken at , which corresponds to a mass depth of about . At a solar helium abundance has been assumed. The results show that a mass loss rate larger than is required to account for the observed abundances. For the surface helium abundance reduces to 0.05 after and 0.02 after . These values are typical for sdOB stars. To investigate the influence of the lower boundary condition on the surface abundance, we have done one calculation with a discontinuity in composition at the lower boundary (dashed line in Fig. 4). Instead of solar composition is assumed at the lower boundary. For the surface number fraction of helium reaches a minimum value of about 0.03 after , just as in the previous case. Then, however, it increases again. After the star is transformed into a helium-rich sdO, for which is a typical value (Dreizler et al., 1990; Thejll et al., 1994).
4.2. DAO white dwarfs
In this section we investigate the time dependence of the helium abundance in the outer regions with a surface layer mass of about for a hydrogen rich white dwarf with and . In view of the numerical effort the CNO elements must be neglected to allow a time integration over .
We now assume that the white dwarf has a thick hydrogen layer with a solar number fraction of helium. In Fig. 5 the results are shown for a weak wind with .
After 10000 y only the outer regions are strongly depleted of helium with a number fraction of about . In this case a weak wind would produce a DA white dwarf. Chayer et al. (1993, 1997) reported on a similar effect for heavy elements in the case of a weak wind. A mass loss rate of is still too low. After 20000 y the helium abundance would reach a value similar as obtained in the diffusion calculations. In Fig. 6 we present the result for .
The ionization effects have been neglected in these calculations. If they were taken into account, helium would sink somewhat more slowly. would then be sufficient to obtain nearly the same result. After the surface abundance is below the solar one by a factor of ten and is reduced by a factor of 50 after . According to Blöcker (1995) and Vassiliadis & Wood (1994) these times are typical time scales of post AGB evolution, in which the star cools to and , respectively. This means that the diffusion time scales in the presence of mass loss with are very similar to those of stellar evolution. The predicted surface abundances are larger than those predicted for the case of zero mass loss by a factor of 10 to 20. The surface abundances after for several mass loss rates are given in Table 4.
Table 4. Surface number fractions of helium for several mass loss rates in for , after
The results show, that for the cooler DAO's with helium abundances below the solar one, should be in the range between and . The calculations have shown, that these results do not differ by more than about a factor of two, if instead of is assumed.
From the results an explanation of the helium abundances in DAO white dwarfs with mass loss and a thick helium layer seems to be possible. However, other possible explanations, e.g. a less massive hydrogen layer and a weaker wind cannot be excluded as long as the mass loss rate is not known independently. The calculations of stellar evolution predict hydrogen layer masses between and , which however, may vary from star to star (Blöcker et al., 1997).
4.3. The transformation from DO's into DA's
In Fig. 2 of Sect. 3 we investigated the transformation of a helium-rich DO white dwarf into a DAO in the absence of mass loss for an initial hydrogen number fraction of . Curve 1 in Fig. 7 shows the H/He abundance profile after 50000 y. A thin hydrogen layer with a mass of about floats ontop of the helium rich regions.
According to Bergeron et al. (1994) this is an upper limit for the mass of the hydrogen layer if the helium abundances of DAO stars were to be explained with stratified atmospheres, in most cases it must even be less massive by a factor of 10 to 100. Now we ask, what would happen with a white dwarf with a stratified atmosphere in the presence of mass loss. Curves 2 and 3 in Fig. 7 show the abundance profiles 1000 and 5000 y, respectively, after the onset of a weak wind with . Within 5000 y the DAO white dwarf would tranform back into a DO. This means that DO's with a hydrogen number fractions of or lower will not transform into DAO's or DA's if at least a weak wind is present. However, the hypothesis, that DAO's have thin hydrogen layers and are the descendants of former DO's cannot be excluded, because there are too many unknowns, the mass loss rate and the hydrogen number fraction in DO's. Under certain conditions a floating up scenario may be possible, as shown in Fig. 8.
We start from a DO with a hydrogen number fraction of and assume a mass loss rate of . After 50000 y it would be tranformed into a hybrid star with in the outer regions. After 150000 y the surface abundance would approximately the solar one. After 250000 y we obtain and the star would appear as a DAO white dwarf. As for the composition is constant within 10%, the atmosphere would appear to be chemically homogenous, although the outer layers are stratified. Finally, after 400000 y the helium abundance sinks below and the star is a DA. These time scales are extended, if the mass loss rate is somewhat larger. For example, for the DO would transform into a hybrid white dwarf with 30 % hydrogen and 70 % helium after 100000 y. A mass loss rate of , however, would prevent the transformation. In this case a white dwarf with a hydrogen layer mass of would be retransformed into a helium-rich one within some thousands of years. A similar result has already been obtained by Michaud (1987).
We conclude that stratified atmospheres with an abundance profile derived from an equilibrium between gravitational settling and ordinary diffusion cannot exist, if at least a weak wind of is present. DO's with hydrogen number fractions of or less do not transform into DA's, if the mass loss rate is larger than about .
© European Southern Observatory (ESO) 1998
Online publication: September 8, 1998