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

2. Basic expressions

2.1. Monochromatic radiation flux

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

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

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

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

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

As has been shown in Sect. 5 of Paper I, other radiative quantities of interest, in particular the monochromatic radiative acceleration (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 and the contribution of the individual spectral lines, denoted by the subscript l,

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

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

and

respectively. If the line is sufficiently narrow, i.e. if the relevant distances from the line center are not too large, the profile functions are obtained from the conventional profiles by utilizing (cf. Appendix B of Wehrse et al. 1998); furthermore and where and 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.

© European Southern Observatory (ESO) 2000

Online publication: July 7, 2000
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