2. Static case and nomenclature
For this purpose we start with the general form of the transfer equation for a static 3D medium for the monochromatic intensity at a wavelength along a ray described by or the unit vector , respectively,
Here S is the source function and the monochromatic extinction coefficient comprising absorption as well as scattering. In cartesian coordinates Eq. (1) reads
with (cf. Oxenius 1986).
We introduce logarithmic wavelengths
which are mostly used throughout this paper in place of the usual wavelengths in order to obtain Doppler shifts that depend only on spatial coordinates and angles. Then , .
In the diffusion limit, the mean intensities are very close to the Planck function of the local temperature . This implies that LTE conditions prevail and the source function is identical to the Planck function,
In the following we need also the wavelength-integrated Planck function denoted by
This weighting function, which enters the Rosseland mean opacity (Eq. (19)), is normalized according to and is independent of n . It decreases exponentially with for very large as well as for very small (Fig. 1).
are sufficiently slowly varying functions of over a few mean free photon paths (in the continuum).
2.1. Radiative flux
It is evident from Eq. (12) that the essential physics is contained in the antisymmetric average of the specific intensities which has a flux-like character (cf. Mihalas 1978). In fact, it is the (monochromatic) flux in the two-stream approximation. In the following we therefore will concentrate on this average and - in order to simplify the notation - subsequently write
2.2. Radiative acceleration
2.3. Static radiative quantities in the diffusion limit
the classical result of Rosseland (1924).
and according to Eqs. (15) and (16)
© European Southern Observatory (ESO) 2000
Online publication: July 7, 2000