RADIATIVE-TRANSFER IN SPHERICALLY SYMMETRIC SYSTEMS .3. FUNDAMENTALS OF LINE FORMATION

A generalization of the variable Eddington factor method is presented that makes possible the solution of line formation problems in extended spherical atmospheres whose constitutive properties depend on radius in an arbitrary way. Extensive numerical results for Doppler broadening in models with power law opacities (n = 0, 2, 3) are presented and interpreted. Very substantial deviations are found from the solutions of analogous plane-parallel models. The single-flight escape probability is derived for a general opacity law and is shown to exceed that for an analogous plane-parallel slab by no more than a factor of approximately two for Doppler broadening, or three-halves for Lorentz broadening. However, it is shown that each time a photon is scattered, it has a probability greater than one-half of ending its flight at a radius larger than that at which it was emitted. This effect is peculiar to spherical geometries and may be important in aiding the escape of photons from optically thick systems. Finally the effects of dilution are considered and some properties of the infinite radius, finite optical depth models are inferred. An appendix contains the solution of the line transfer problem for a homogeneous sphere by the kernel-approximation method. © 1974 Royal Astronomical Society.