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Astron. Astrophys. 338, 75-84 (1998)

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3. Results of the diffusion calculations

Fig. 1 illustrates the diffusion calculations for [FORMULA] and [FORMULA]. At time [FORMULA] in all depths He/H=0.005 and number fractions of the CNO elements of 0.0005 have been assumed. At the lower boundary at [FORMULA] the composition remains the original one during the calculations. The gas pressure is related to the surface mass fraction q approximately by

[EQUATION]

So the mass of the layer considered is about [FORMULA]. [FORMULA] after the onset of diffusion, the number fraction of helium near [FORMULA] is about [FORMULA] and does not decrease significantly during the following time, due to an equilibrium state between gavitational settling and radiative levitation is reached in the outer regions. In deeper layers, however, the separation of elements continues.

[FIGURE] Fig. 1. Results of the diffusion calculations for a hydrogen rich white dwarf with [FORMULA], [FORMULA] and a solar helium abundance at the start. The number fraction of helium is plotted as a function of the gas pressure (in SI units) after (from top to the bottom) 10, 100, 1000, 5000 and 10000 y. In the upper part of the figure the Rosseland mean optical depths [FORMULA], respectively, are indicated for the case after 10000 y

In Table 1 the results of the diffusion calculations at [FORMULA] are given for several effective temperatures and gravities.


[TABLE]

Table 1. From the diffusion calculations predicted logarithmic number fractions of elements at a Rosseland mean optical depth [FORMULA]


For cases with [FORMULA] a time of [FORMULA] has usually been necessary until the outer regions are near a diffusive equilibrium state. For cases with [FORMULA] time integrations over [FORMULA] and [FORMULA] for [FORMULA] have been sufficient. To avoid numerical problems the initial abundances of helium and the heavy elements at [FORMULA] have been chosen somewhat larger than the expected ones, so that the elements sink during the diffusion calculations. For high effective temperatures and lower gravities the numerical code works sufficiently well, whereas the calculations are extremely time consuming in cases with lower [FORMULA] and high gravities. The helium abundance for [FORMULA], [FORMULA] is an upper limit, because in this case we integrated over a time scale of four years only. Nevertheless the helium number fraction decreases by two orders of magnitude from an initial abundance of [FORMULA]. The elements C and N tend to be more abundant than helium. This is in favour of the result that heavy elements are responsible for the flux deficiency in the soft X-ray and EUV wavelength range of many DA white dwarfs with [FORMULA] (Barstow et al. (1997) and references therein). In white dwarfs with [FORMULA] or in those with [FORMULA] and surface gravities [FORMULA], however, detectable amounts of helium should be present according to the diffusion calculations.

The results do not differ significantly from those of Vennes et al. (1988). Compared to observational results, the predictions for helium are too low by at least a factor of ten in the case of DAO stars (typically [FORMULA], [FORMULA] or somewhat larger) and central stars of planetary nebulae ([FORMULA], [FORMULA]). For sdOB type stars ([FORMULA], [FORMULA]) they are too low by more than a factor of 100 and cannot reproduce the typical abundance anomalies observed with deficiencies of carbon and solar abundances of nitrogen (Lamontagne et al., 1987).

With the data given in Table 2 the results for [FORMULA], [FORMULA] at [FORMULA] will be considered in detail. Helium is preferably doubly ionized. All ionization states have been taken into account in the calculations, the effect of neutral helium on the momentum balance is, however, negligibly small.


[TABLE]

Table 2. Data at [FORMULA] for [FORMULA], [FORMULA]. The gas pressure at this depth is [FORMULA], the temperature [FORMULA], the chemical composition as given in Table 1. In the left part of the table is given the momentum per unit volume and unit time transferred on helium by gravitation, radiation (bound-bound and bound-free transitions), the electric field, partial pressure gradients, [FORMULA] and [FORMULA] collisions, respectively. The right part of the table shows the density of particles (in [FORMULA]) in the ionization states of hydrogen and helium


The momentum per unit volume and unit time transferred to helium by gravity must be compensated by the momentum transferred by photons due to bound-bound and bound-free transitions, by the electric field, the gradient of the partial pressure and the momentum transfer due to collisions with all species of particles taken into account.

For this example it can be seen that the radiative force [FORMULA] due to bound-free transitions exceeds the radiative force due to lines [FORMULA] by about a factor of 2.6. For larger gravities and lower effective temperatures helium sinks until the lines become desaturated and thus the bound-bound transitions become more important. For [FORMULA] and [FORMULA] both contributions are approximately equal, for [FORMULA] and [FORMULA] [FORMULA] exceeds [FORMULA] by a factor of 15 (the contribution of neutral helium is still small even for [FORMULA]). If the contribution of the electrons to the line broadening would be neglected, this would lead to a momentum transfer by bound-bound transitions of 0.0083 instead of the value 0.0092 given in Table 2, which was obtained with the assumption of a quasistatic line broadening due to ions and electrons. Therefore uncertainties in line broadening theory are not of crucial importance, at least in this case. The momentum exchange due to collisions between helium and protons is small compared to the other contributions to the momentum equation. This indicates that for the given plasma conditions the ionization effects are negligible.

Fig. 2 shows the transformation of a helium-rich white dwarf with [FORMULA], [FORMULA] into a hydrogen-rich one by diffusion. An original number ratio [FORMULA] has been assumed in an outer layer of about [FORMULA], the heavy elements have been neglected. After 1000 y the white dwarf is still helium rich, after 10000 y there is [FORMULA] at [FORMULA] or [FORMULA], respectively. According to Eq. (4) this corresponds to a hydrogen layer mass of about [FORMULA]. After 100000 y the mass of the hydrogen layer would be about [FORMULA], after 200000 y the mass would be [FORMULA]. For the latter case, the helium number fraction in the outer regions with [FORMULA] is below [FORMULA]. So we conclude that, if the hydrogen number ratio in a DO white dwarf is [FORMULA] or larger, in the absence of mass loss it would transform into a DAO with an ultrathin hydrogen layer in not more than 10000 y. After 100000 y the surface composition would be very similar to the one predicted by the diffusion calculations for a thick hydrogen layer (see Table 1).

[FIGURE] Fig. 2. Transformation of a helium-rich white dwarf with [FORMULA], [FORMULA] into a hydrogen-rich one by diffusion. The number fraction H/He is plotted as a function of the gas pressure (in SI units) after (from top to the bottom) 200000, 100000, 10000 and 1000 y. In the upper part of the figure the Rosseland mean optical depths [FORMULA], respectively, are indicated for the case after 100000 y

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© European Southern Observatory (ESO) 1998

Online publication: September 8, 1998
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