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Astron. Astrophys. 358, 956-992 (2000)

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4. Ionization equilibrium and occupation numbers

The parameterization of the line acceleration at location [FORMULA] requires the knowledge of the local force-multipliers [FORMULA] (Eq. (24) and Sect. 5). To calculate these quantities via the frequency and line strength distribution of the contributing lines, we determine the ionization structure and occupation numbers in the wind by employing an approximate method developed by Abbott & Lucy (1985), Schmutz (1991), Lucy (unpublished notes) and Springmann (1997). In contrast to the "exact" self-consistent non-LTE solution, this simplified Ansatz allows a comparatively fast local solution of the rate equations (within reasonable physical approximations). This is essential with respect to the substantial computational effort required by our multidimensional problem.

Ionizing radiation field. We calculate the ionizing radiation field [FORMULA] by

[EQUATION]

with dilution factor

[EQUATION]

accounting for the geometrical dilution of the incident stellar radiation field. For a spherical star, this expression simplifies to Eq. (20).

The intensity [FORMULA] with radiation temperature [FORMULA] describes the frequency-dependent photon distribution emerging from the photosphere and is taken either as Planckian or from Kurucz flux distributions (Kurucz 1992). Both flux distributions give rise to different ionization structures, since the frequency dependence [FORMULA] of the Kurucz flux distribution implies a drastically smaller ionizing flux for wavelengths shortward of the Lyman edge (see below).

The above approximation for [FORMULA], of course, is only valid in the case of an optically thin continuum in the wind, as discussed here.

Ionization equilibrium. With the electron temperature taken as a constant fraction of the effective temperature (typically 0.8) and the radiation temperature as either the effective one (Planck case) or significantly lower shortward of the flux maximum (Kurucz fluxes, see Sect. 4), the ionization equilibrium reads

[EQUATION]

The asterisk denotes thermodynamic equilibrium values, and [FORMULA] and [FORMULA] are the fraction of recombination processes leading directly to the ground and meta-stable levels, respectively. For details, see Springmann (1997).

Occupation numbers. Having determined the ionization structure, we can calculate the occupation numbers (e.g., Puls et al. 2000). For the wind dynamics, we consider (in agreement with Abbott & Lucy 1985) only the crucial lines (giving rise to more than 90% of the line acceleration), which are those with the lower level as a ground, meta-stable or so-called "1st order" subordinate state. The latter are defined as those subordinate levels which have a direct transition either to the ground or to a meta-stable state. Since we neglect stimulated emission, we only need the occupation numbers for the lower levels defined in this way.

Atomic data. In order to perform the above calculations, we employ the Munich atomic data bank provided by A. Pauldrach and collaborators. For a description, see Pauldrach et al. (1998).

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© European Southern Observatory (ESO) 2000

Online publication: June 20, 2000
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