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

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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 [FORMULA] 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 [FORMULA] occurring over a mean free path length in the continuum is much larger or much smaller than the intrinsic width [FORMULA] of the narrowest spectral line, i.e. if

[EQUATION]

respectively. Here we have expressed shifts and line widths in the logarithmic wavelength scale [FORMULA] (I:3) which we use throughout this paper.

We express the Doppler shift [FORMULA] in terms of the velocity gradient (I:33)

[EQUATION]

with s being the path element in the ray direction. Since the mean free path of a photon in the continuum is [FORMULA] with [FORMULA] being the continuous extinction coefficient, the change of the velocity [FORMULA] (c velocity of light) over a mean free path is [FORMULA], and hence [FORMULA]. On the other hand, the intrinsic line width [FORMULA] is of the order either of the damping constant [FORMULA] (in the [FORMULA] 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

[EQUATION]

respectively. In the diffusion limit the condition [FORMULA] (I:34), of course, is fulfilled in addition.

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© European Southern Observatory (ESO) 2000

Online publication: July 7, 2000
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