Exact numerical methods become costly in terms of computer time when the radiation field is coupled with hydrodynamics. In such situations one needs methods which are fast and give insight into the physics of the problem in an easy and quick manner. Escape probability methods satisfy these requirements to a large extent and therefore became popular. There are ‘first order methods’ due to Biberman, Holstein, Sobolev and Zanstra, which are reviewed by Irons (1979a,b). The methods due to Athay (1972a,b), Rybicki (1972), Frisch and Frisch (1975), Canfield et al. (1981, 1984), Scharmer (1981, 1983, 1984) and others are the so called ‘second order methods’. We shall describe these and others methods in this chapter. These methods have been reviewed by Rybicki (1984).

Nordlund (1984) developed a method for obtaining an iterative solution of radiative transfer in a spherically symmetric atmosphere using a single ray approximation. The convergence is achieved in 2–3 iterations to give an accuracy better than 1% in the source function.

The Monte Carlo technique has been used by several authors (see, for example, Magnan (1970), Panagia and Ranieri (1973); Pozdnyakov et al. (1976)).

Surfaces of constant radial velocity

The geometrical region from which most of the observed emission at a given frequency x comes is likely to be a thin zone centred on a surface of constant radial velocity in such a way that υz = µυr = x, where the term radial velocity means the velocity along the line of sight, which is different from υr the velocity along the radius vector (see Mihalas (1978)) measured from the centre of the star.