THE SOLUTION TOPOLOGY OF RADIATION-DRIVEN WINDS .1. THE X-TYPE NATURE OF THE CAK CRITICAL-POINT

The radiatively driven wind theory of Castor, Abbott, & Klein (CAK) is now widely accepted as the mechanism producing the mass loss from early-type (OB) stars. Unfortunately, finding solutions that satisfy the CAK critical point conditions can be a somewhat uncertain and difficult task, especially when additional forces are included. In contrast, the fluid equation describing the solar wind, which contains a so-called X-type critical point, was easily solved by Parker using an analysis of the solution topology of the differential equation. Unfortunately, the CAK wind equation is a nonlinear equation for the velocity gradient. As a consequence of the multiple roots of this equation, the origin and nature of the topology of the CAK critical point have been unclear. Employing a commonly used change of variables, we obtain a quasi-linear differential equation whose solution topology is easily found. This analysis indicates that there are a total of five critical points (three X-type, a spiral, and a cusp-shaped node), but two of these are unphysical. We find that the CAK critical point is indeed an X-type singularity like the Parker critical point. In addition to the transcritical solution found by CAK, which has a monotonically increasing velocity, there are nonmonotonic solutions, analogous to the Chamberlain breeze solutions for the solar wind, as well as a trans-critical monotonically decreasing solution. Both the monotonically increasing and monotonically decreasing solutions may be used for either outflow or infall, but the CAK breeze solutions can only be employed for outflow. However, the only outflow solution that satisfies the boundary condition of zero pressure at infinite radius is the original CAK solution. Thus, the additional trans-critical solutions are relevant only for outflows with a finite back-pressure or for accretion flows fed by an external source such as a mass transfer binary.