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Astron. Astrophys. 359, 788-798 (2000)

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2. Basic expressions

2.1. Monochromatic radiation flux

The monochromatic flux vector [FORMULA] at depth [FORMULA] is obtained by integration over all directions (I:12) from the flux [FORMULA] in the direction n (I:13). In the diffusion limit we write the latter for a moving medium in the form (I:42)

[EQUATION]

with [FORMULA] being the corresponding static value, so that the correction (I:44) for the inclusion of a velocity field is

[EQUATION]

Here [FORMULA] is the monochromatic extinction coefficient, [FORMULA] the corresponding spectral thickness (I:37) with [FORMULA],

[EQUATION]

the spatial derivative (I:6) of the the Planck function [FORMULA] for the temperature [FORMULA] at depth [FORMULA], and

[EQUATION]

the weighting function (I:8) entering the Rosseland mean opacity. [FORMULA] is the wavelength integrated Planck function, and [FORMULA] ([FORMULA] Stefan-Boltzmann constant).

As has been shown in Sect. 5 of Paper I, other radiative quantities of interest, in particular the monochromatic radiative acceleration [FORMULA] (I:16), as well as the wavelength-integrated expressions (I:11,15) can be derived from the monochromatic flux.

2.2. Extinction coefficient

The extinction coefficient copmprising absorption and scattering - can be written in the deterministic description - as the sum of a continuous part [FORMULA] and the contribution of the individual spectral lines, denoted by the subscript l,

[EQUATION]

with [FORMULA] being the strength of the line l at the position [FORMULA], and [FORMULA] its profile function, normalized to [FORMULA], and the assumption that [FORMULA] does not vary significantly over the range of any given spectral line. The corresponding spectral thickness is

[EQUATION]

For the profile function [FORMULA] we may choose either Lorentz L or Doppler D profiles, which - in the dimensionless logarithmic wavelength scale - read

[EQUATION]

and

[EQUATION]

respectively. If the line is sufficiently narrow, i.e. if the relevant distances [FORMULA] from the line center are not too large, the profile functions are obtained from the conventional profiles [FORMULA] by utilizing [FORMULA] (cf. Appendix B of Wehrse et al. 1998); furthermore [FORMULA] and [FORMULA] where [FORMULA] and [FORMULA] are the usual damping constant and Doppler width, respectively, in the ordinary wavelength scale.

In astrophysical applications one frequently has to treat a very large number of spectral lines. In this case one can, on the one hand, choose to apply large lists of "real" lines which may result in very time-consuming calculations, if this is possible at all. On the other hand, the deterministic treatment of very many lines may be replaced by a statistic approach. As Wehrse et al. (1998) have shown, the distribution of spectral lines can well be described by a Poisson point process. The effects in the diffusion limit will be discussed in a subsequent paper of this series.

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© European Southern Observatory (ESO) 2000

Online publication: July 7, 2000
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