Sonic surfaces for adiabatic flow in a stationary axisymmetric spacetime

Abstract

The equations governing the steady flow of a rotating adiabatic fluid in an axisymmetric, stationary background spacetime are cast in the form of a quasilinear second-order partial differential equation. When the flow is sonic (as measured by a corotating observer) the equation is parabolic. If the stream-lines are assumed to be orthogonal to the sonic surface (in poloidal projection) then (i) the total mass flux through the surface is locally extremal, and (ii) the surface itself can be constructed by solving a fourth-order ordinary differential equation. This is done analytically for small deviations from the spherical in the Schwarzschild and weak Kerr metrics, and numerically for more rapidly rotating flows in the Schwarzschild metric and non-rotating flows outside more extreme Kerr holes. Although the distribution of angular momentum on the sonic surface must be specified before the surface is uniquely determined, a simple theorem is introduced which enables this distribution to be calculated from a knowledge of conditions in the flow at infinity. Finally, it is shown that there can be at most one transonic solution with a given stream pattern, specific energy, angular momentum, entropy and equation of state.