Generalization of the Sobolev method for flows with nonlocal radiative coupling

The Sobolev, or escape-probability, method for solving radiative transfer problems in moving atmospheres is generalized to treat flows in which the line-of sight component of the flow velocity is not monotonic; for these cases the purely local nature of the approximation is lost, and radiative coupling between distant parts of the atmosphere must be taken into account. The method is formulated for a general three-dimensional flow. For spherically symmetric cases in which the relative projected flow velocity on a line of sight goes through zero 2,3,..., N times, an integral equation for the source function is obtained. In the simplest nontrivial case when N=2, it is shown that the normalization of the kernel is such that an iterative solution of the integral equation always converges rapidly. For spherically symmetric flows with N=2, the kernel of the integral equation is expressed in closed form. Extensive numerical results for inverse power-law velocity fields are presented to illustrate the magnitude of the coupling between different parts of the atmosphere. Errors in the magnitude of the flux peak of 50% or larger are readily made if this coupling is neglected