*Astron. Astrophys. 358, 956-992 (2000)*
## 4. Ionization equilibrium and occupation numbers
The parameterization of the line acceleration at location
requires the knowledge of the local
force-multipliers (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(1991), Lucy (unpublished notes) and Springmann (1997).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 by
with *dilution factor*
accounting for the geometrical dilution of the incident stellar
radiation field. For a spherical star, this expression simplifies to
Eq. (20).
The intensity with radiation
temperature 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
of the Kurucz flux distribution
implies a drastically smaller ionizing flux for wavelengths shortward
of the Lyman edge (see below).
The above approximation for , 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
The asterisk denotes thermodynamic equilibrium values, and
and
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).
© European Southern Observatory (ESO) 2000
Online publication: June 20, 2000
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