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*Astron. Astrophys. 359, 780-787 (2000)*
## 1. Introduction
There are many differentially moving, optically thick astronomical
objects in which the radiation field is important for the energy
and/or momentum balance and therefore the total fluxes and/or
radiative accelerations have to be calculated accurately in a
modeling. Typical celestial systems of this kind are novae,
supernovae, collapsing molecular clouds, and accretion discs. Ideally,
one would solve the full appropriate radiative transfer equation which
is possible only numerically and is unfortunately extremely CPU time
and memory consuming if many spectral lines contribute to the opacity;
in fact, in non-stationary models this solution is virtually
impossible. However, in many cases one can exploit the large optical
thickness and derive a simpler form of the solution of the transfer
equation that is valid only in this limit: the radiation diffusion
equation.
For static 1D media the problem has been solved already many years
ago (Rosseland 1924). Karp et al. (1977) were the first to discuss
differentially moving media. They used infinitely narrow lines and
showed that the *expansion opacity* , i.e. the modification of
the opacity due to the motions that has to be employed to Rosseland's
formula for the flux, may be quite large. Subsequently, the problem
has been addressed - mainly in the context of supernova explosions,
spectra and light-curves - e.g. by Eastman & Kirshner (1989),
Höflich (1990, 1995),
Höflich et al. (1993), Eastman &
Pinto (1993), Blinnikov & Bartunov (1993), Jeffery (1995),
Blinnikov (1996a,b), Baron et al. (1996), and recently Pinto &
Eastman (2000). These authors make use of the particular conditions in
these objects as e.g. the small intrinsic width of the lines and the
coincidence of the directions of the velocity and of the temperature
gradients.
Here we present the first paper of a series which is devoted to the
discussion of the effects of differential motions on the radiative
quantities in optically thick absorbing and scattering media of
general shape with arbitrary non-relativistic velocities. In
particular the directions of the flow and of the temperature gradient
may be different which introduces additional components to the flux
vector. The expressions are derived in a rigorous way from the
comoving-frame transfer equation. Our approach leads to the correct
limit of static media, and can handle spectral lines of arbitrary
shape and strength as well as edges and continua. In addition, it
allows easy physical interpretations of the effects of the motion on
the radiative quantities.
In Sect. 2 of *this* paper we first recall the results of the
long known "conventional", static diffusion limit and give the
definitions of the radiative quantities, the abbreviations etc. used
in this series. In Sect. 3 we derive the transfer equation for
*slowly* differentially moving 3D media. In the diffusion limit
it is sufficient in most cases to consider small velocities *v*
since only velocity differences over one free mean path length in the
continuum are relevant. This simplifies the equations significantly as
only first order terms in (*c*
velocity of light) have to be considered. In addition, the
aberration/advection terms are so small that they can be neglected,
and the characteristics continue to be straight lines. Subsequently,
in Sect. 4 we obtain the solution of the transfer equation for the
limit of large optical depths, and in Sect. 5 give expressions for the
flux and the radiative force in the diffusion limit, and present a
generalized version of the Rosseland opacity valid for differentially
moving media. The essential effects of the velocity field upon the
flux are then demonstrated in Sect. 6 on the basis of a numerical
evaluation of the simple case that the extinction is due to a
continuum and a single line only. Finally, Sect. 7 contains the
conclusions and an outlook.
In Paper II of this series we discuss the radiative quantities
in the diffusion approximation for moving media for the
*limiting* cases of *large* as well as of *small*
velocity gradients for deterministic spectral lines of *finite*
width. Paper III then will deal with *stochastic*
distributions of lines (of finite width) described by a Poisson point
process which has been shown by Wehrse et al. (1998) to be flexible
and adequate for the treatment of very many spectral lines. In
Paper IV *infinitely sharp* lines - which had e.g. been used
by Karp et al. (1977) in their discussion of the expansion opacity -
will be considered.
© European Southern Observatory (ESO) 2000
Online publication: July 7, 2000
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