*Astron. Astrophys. 359, 788-798 (2000)*
## 1. Introduction
The comoving-frame radiative transfer equation has been derived and
solved by Wehrse et al. (2000) (Paper I of this series) for
differential motions (with velocities much smaller than the speed of
light) in a medium which is optically thick so that the diffusion
limit of the radiation applies, and expressions for radiative
quantities such as the wavelength-integrated flux and radiative
acceleration have been given. These expressions become particularly
simple if the absolute value of the velocity gradient
is either very large or very small,
and if the extinction due to the spectral lines can be described in a
determinisitic way. In the present paper (II) a detailed derivation
and discussion of the radiative quantities for these two limiting
cases is given; preliminary results have already been reported in a
review-like paper by Wehrse & Baschek (1999).
We first formulate, in this Introduction, the conditions for the
validity of the limiting cases of large and small *w*,
respectively. In Sect. 2, we then recall some basic results from
Paper I and describe the deterministic extinction coefficient
used here. For a complete list of the assumptions made and for the
notation, we refer to Paper I; an equation (*n*) of that
paper is cited here by (I:*n*). Subsequently we treat the
limiting case of large *w* in Sect. 3, and - in much more detail
- that for small *w* in Sect. 4. For the latter, more important
case the effects of the motions are demonstrated by selected examples
of extinction coefficients: a power-law continuum, a spectral edge, a
narrow single line, and many, isolated as well as overlapping lines.
While in the previous sections the flux is considered only in a given
direction, Sect. 5 is devoted to the intricate behavior of the flux
*vector* if the directions of the temperature gradient and of the
velocity gradient do not coincide. The essential featurers of these
anisotropic mean free photon paths are demonstrated for a
plane-parallel stratification in the limit of small *w*. Finally,
the conclusions and an outlook are given in Sect. 6.
The *limiting cases* of a large or a small velocity gradient
are realized if the Doppler shift
occurring over a mean free path length in the continuum is much larger
or much smaller than the intrinsic width
of the narrowest spectral line, i.e.
if
respectively. Here we have expressed shifts and line widths in the
logarithmic wavelength scale (I:3)
which we use throughout this paper.
We express the Doppler shift in
terms of the velocity gradient (I:33)
with *s* being the path element in the ray direction. Since
the mean free path of a photon in the continuum is
with
being the continuous extinction coefficient, the change of the
velocity (*c* velocity of light)
over a mean free path is , and hence
. On the other hand, the intrinsic
line width is of the order either of
the damping constant (in the
scale) for a Lorentz profile
(Eq. (10)) or of the Doppler width *W* for a Gauss and for a
Voigt profile (Eq. (11)).
In this paper we adopt mostly *Lorentz* profiles for the
discussion of our examples so that the limiting cases (1) imply for
the velocity gradient
respectively. In the *diffusion* limit the condition
(I:34), of course, is fulfilled in
addition.
© European Southern Observatory (ESO) 2000
Online publication: July 7, 2000
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