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

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2. Improvements on the diffusion calculations

For the elements hydrogen and helium ionization effects are taken into account. This momentum transfer by collisions originates from the different diffusion velocities of the hydrogen and helium particles in the various ionization states. Our method is similar to the one described by Michaud et al. (1979) and used by Vennes et al. (1988). However, it allows to account for non-zero diffusion velocities not only for helium, but for the neutral and ionized particles of the hydrogen background plasma in addition. Babel & Michaud (1991) have shown that ambipolar diffusion of hydrogen could be one of the major processes affecting the chemical abundances supported in the atmospheres of Ap stars. If thermal diffusion is neglected, from the equations of Geiss & Bürgi (1986) and Burgers (1969)


can be derived, where [FORMULA] and [FORMULA] are the z components of the diffusion velocities of the particle species t and s.

The resistance coefficients [FORMULA] for the interaction between charged particles are obtained from Paquette et al. (1986). The most important mechanism of interaction between H and [FORMULA] for the plasma conditions considered here is resonant charge exchange (RCE), the interaction due to polarization will be neglected. For the RCE momentum transfer cross section (see e.g. Eq. 15.4 of Suchy, 1984) of the ground state of hydrogen a constant value of [FORMULA] has been adopted, which neglects the approximately logarithmic dependence on the relative velocity of the colliding particles. According to the results of Newman et al. (1982) and Hunter & Kuriyan (1977) this is a typical value for the plasma conditions important in our work. For the excited states a scaling factor [FORMULA] proposed by Peterson & Theys (1981) has been used. The other charged and neutral particles mainly interact via polarization. With polarizabilities from Radzig & Smirnov (1985) and a polarization potential proportional to [FORMULA] (Michaud et al., 1978), the collision integrals have been evaluated according to Eq. 10.31,7 of Chapman & Cowling (1970). For the interaction between the neutral atoms H and He a hard sphere approximation has been used.

[FORMULA] is the transformation rate of particle species t into s by ionization or recombination processes. To obtain the rates for particles in a certain ionization state, mean values over all excitation states are used, e.g. the ionization rate of hydrogen is:


[FORMULA] and [FORMULA] are the collisional and radiative ionization rate (see Eqs. 5-79 and Eqs. 5-40 of Mihalas (1978) or Appendix C of Gonzales et al. (1995)) The number [FORMULA] of bound states is obtained from the Debye screening formalism and a cutoff procedure as described in Paper I. The recombination rates are obtained from detailed balancing arguments.

For each of the elements C, N and O one momentum equation, averaged over all ionization states , is used. As one of them is redundant, the one for protons is replaced by the condition of zero mass flow. These equations and the equation of radiative transfer (diffusion approximation) are solved just as described in Paper II.

An important improvement concerns the momentum transferred from photons to matter by photoionization processes. In Papers I and II has been assumed that all the photon momentum is transferred to the heavy particle. However, as the photoelectrons are preferentially ejected in forward direction, part of the momentum is given to the photoelectron. For hydrogen-like particles, this effect has been evaluated in detail by Massacrier (1996). The fraction [FORMULA] is given for the various excitation states. So we obtain for the momentum [FORMULA] per unit volume and unit time transferred by photoionization processes to the particles in ionization state k:


The factor [FORMULA] represents the monochromatic absorption cross section per particle in ionization state [FORMULA] corrected by the factor [FORMULA]. It is a weighted mean value over the various excitation states. The first expression on the right hand side of the equation above represents the momentum transferred on the particles in ionization state k by photoionizations of particles in ionization state k-1. The residual photon momentum is given to the free electrons. The second term accounts for the recoil on the particles in ionization state k which is due to photorecombination processes of particles in ionization state [FORMULA]. It has been derived from an emission rate obtained from detailed balancing arguments and the boson factor, which predicts, that photons are preferably emitted in direction of the radiation flux (for details see e.g. Oxenius, 1986). For neutral helium the values of [FORMULA] for the hydrogenic case have been used. According to Massacrier & El-Murr (1996) they do not differ very much from the exact values.

In the corresponding Eq (25) of Paper I for [FORMULA] is a misprint, the right hand side of which must be multiplied with a factor [FORMULA]. (The results of Paper I are not affected, however, because the corresponding equation in the computational code has been correct).

In contrast to Paper I, the line profiles for the lines of [FORMULA] and [FORMULA] have been evaluated in detail with the data from Underhill & Waddell (1959) and a probability distribution function for the electric microfield at a charged point from Hooper (1967). In the computation of the radiative force on [FORMULA] due to bound-bound transitions the Lyman, Balmer and Paschen series are taken into account, for hydrogen the Lyman series only. The total radiative force due to each series is computed by numerical integration from the absorption edge to some maximal wavelength (e.g. [FORMULA] for the Lyman series of [FORMULA]) with about 500 frequency points. The effect of line blends between the various lines of a series, which tends to reduce the radiative force, has been taken into account. For neutral helium the computation of the radiative force is similar as in Paper I with line broadening parameters of Dimitrijevic & Sahal-Bréchot (1984).

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© European Southern Observatory (ESO) 1998

Online publication: September 8, 1998