## 3. Radiative transfer equation for slowly moving 3D media
In order to obtain the transfer equation for a 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 so that we may
assume for the Lorentz factor , and
furthermore neglect the aberration and advection, i.e. keep
unchanged. These assumptions are
justified as long as the velocity change over the mean free path
length of the photons in the
continuum (denoted by 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
to depend on the variables
Then the nabla operator in Eq. (1) has to be replaced by with according to Eq. (24). Introducing this expression into the transfer equation then yields where Since in the following we are dealing exclusively with
with Thus the motions of the medium enter the transfer equation only in
the form of the "velocity gradient" Expressed in terms of since is at most of the order of . Eq. (32) has the same In the diffusion limit, however, at which we aim in this paper, we
can neglect the term in the transfer
equation since and hence
from the condition (34).
(Incidentally, the "transport-type" term
, acting as an additional extinction,
would not appear in the transfer equation if the relativistically
invariant intensity had been used
instead of © European Southern Observatory (ESO) 2000 Online publication: July 7, 2000 |