Opacity distribution in static and moving media

¹ Institut für Theoretische Astrophysik, Tiergartenstrasse 15, 69121 Heidelberg, Germany
² Interdisziplinäres Zentrum für Wissenschaftliches Rechnen, Im Neuenheimer Feld 368, 69120 Heidelberg, Germany
³ Institut für Angewandte Mathematik, Im Neuenheimer Feld 294, 69120 Heidelberg, Germany

Corresponding author: R. Wehrse, wehrse@ita.uni-heidelberg.de

Received: 11 January 2001
Accepted: 21 March 2001

Abstract

The use of the conventional opacity distribution function (ODF) to deal with very many spectral lines is restricted to static media. In this paper, its generalization to differentially moving media is derived from the analytical solution of the comoving-frame radiative transfer equation. This generalized ODF depends on only two parameters, on the wavelength position (as in the static case) and in addition on a wavelength interval Δ over which the line extinction is averaged. We present two methods for the calculation of the generalized ODF: (i) in analogy to the static case, it is derived from the mean values of the extinction coefficients over wavelength intervals Δ, (ii) it is calculated under the assumption that the lines follow a Poisson point process. Both approaches comprise the conventional static case as the limit of vanishing velocities, i.e. of . The averages of the extinction for all values of Δ contain the necessary information about the Doppler shifts and about the correlations between the extinction at different wavelengths. The flexible statistical approximation of the lines by a Poisson point process as an alternative to calculating the averages over all Δ from a deterministic "real"spectral line list, has the advantage that the number of parameters is reduced, that analytical expressions allow a better insight into the effects of the lines on the radiative transfer, and that the ODFs and their corresponding characteristic functions can be calculated efficiently by (fast) Fourier transforms. Numerical examples demonstrate the effects of the relevant parameters on the opacity distribution functions, in particular that with increasing line density and increasing Δ the ODF becomes narrower and its maximum is shifted to larger extinction values.