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