Opacity distribution in static and moving media

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 Delta 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 Delta, (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 Delta --> 0. The averages of the extinction for all values of Delta 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 Delta 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 Delta the ODF becomes narrower and its maximum is shifted to larger extinction values.