Exact, Algebraic Solutions of the Thin-Shell Two-Wind Interaction Problem

Abstract

We have developed a formalism based on considerations of linear and angular momentum conservation for solving axisymmetric, steady, "thin-shell" problems, which is applicable to problems of interactions of nonaccelerated flows. This formalism yields a system of algebraic equations that can be solved to obtain the shape of the thin shell, its mass surface density, and the velocity along the shell. We first use this approach to obtain the solution (obtained with a somewhat different approach by Wilkin 1996) to the problem of an isotropic stellar wind interacting with a plane-parallel stream. Second, we find an exact (implicit) and approximate (explicit) analytic solution to the problem of the interaction of two isotropic stellar winds. Our solution of the two-wind problem is a step forward from previous numerical solutions based on a ram-pressure balance argument since it is analytic and, furthermore, includes centrifugal effects. This solution has clear applications to problems of interacting winds in binary stars as well as in young stellar objects.