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Astron. Astrophys. 359, 780-787 (2000)

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3. Radiative transfer equation for slowly moving 3D media

In order to obtain the transfer equation for a moving medium, we start from the static equation for a 3D medium (1) and apply to it the "simplified" Lorentz transformation

[EQUATION]

which, in fact, is a Galilei transformation in combination with the linear Doppler formula. Quantities referring to the comoving frame are denoted by the subscript 0 here (in this section only).

By applying (24) and (25) we restrict ourselves to sufficiently small velocities [FORMULA] so that we may assume for the Lorentz factor [FORMULA], and furthermore neglect the aberration and advection, i.e. keep [FORMULA] unchanged. These assumptions are justified as long as the velocity change over the mean free path length [FORMULA] of the photons in the continuum (denoted by c) or, equivalently, over a distance corresponding to unit optical depth, [FORMULA], in the continuum is sufficiently small, i.e. if

[EQUATION]

In the diffusion limit of radiation fields this condition is usually fulfilled.

Proceeding with the derivation of the comoving-frame transfer equation, we now consider any vector [FORMULA] to depend on the variables [FORMULA] and [FORMULA] rather than directly on the length variable s, i.e.

[EQUATION]

Then the nabla operator in Eq. (1) has to be replaced by

[EQUATION]

with

[EQUATION]

according to Eq. (24). Introducing this expression into the transfer equation then yields

[EQUATION]

where

[EQUATION]

Since in the following we are dealing exclusively with comoving-frame quantities, we for simplicity drop the subscript 0 in our notation so that our basic 3D radiative transfer equation now reads, in coordinate-free form,

[EQUATION]

with

[EQUATION]

Thus the motions of the medium enter the transfer equation only in the form of the "velocity gradient" w, and consequently all their effects upon the radiative quantities can be expressed in terms of w. We further emphasize that in a moving medium the comoving frame is the relevant "natural" description for the radiative transfer equation since in particular all thermodynamic quantities are defined in this frame.

Expressed in terms of w, the condition (26) for the diffusion limit now reads

[EQUATION]

since [FORMULA] is at most of the order of [FORMULA].

Eq. (32) has the same mathematical structure as the 1D comoving-frame transfer equation used by Baschek et al. (1997b) for plane-parallel and spherical media so that its analytical solution (Baschek et al. 1997a,b) can also be applied here. That equation was derived as limiting case for small [FORMULA] and for the directions [FORMULA] from the full (spherically symmetric) relativistic transfer equation given by Mihalas & Weibel-Mihalas (1984). We note that the meaning of the coefficients in the two papers, however, is different: our [FORMULA] in Eq. (32) reads [FORMULA] in Baschek et al. (1997b), and their w is simply equal [FORMULA].

In the diffusion limit, however, at which we aim in this paper, we can neglect the term [FORMULA] in the transfer equation since [FORMULA] and hence [FORMULA] from the condition (34). (Incidentally, the "transport-type" term [FORMULA], acting as an additional extinction, would not appear in the transfer equation if the relativistically invariant intensity [FORMULA] had been used instead of I, cf. also Wehrse & Baschek 1999.)

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© European Southern Observatory (ESO) 2000

Online publication: July 7, 2000
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