A Nozzle Analysis of Slow-Acceleration Solutions in One-dimensional Models of Rotating Hot-Star Winds

One-dimensional (1D) stellar wind models for hot stars rotating at ≳75% of the critical rate show a sudden shift to a slow-acceleration mode, implying a slower, denser equatorial outflow that might be associated with the dense equatorial regions inferred for B[e] supergiants. Here we analyze the steady 1D flow equations for a rotating stellar wind based on a "nozzle" analogy for terms that constrain the local mass flux. For low rotation, we find the nozzle minimum occurs near the stellar surface, allowing a transition to a standard, CAK-type steep-acceleration solution; but for rotations ≳75% of the critical rate, this inner nozzle minimum exceeds the global minimum, implying near-surface supercritical solutions would have an overloaded mass-loss rate. In steady, analytic models in which the acceleration is assumed to be monotonically positive, this leads the solution to switch to a slow-acceleration mode. However, time-dependent simulations using a numerical hydrodynamics code show that, for rotation rates 75%-85% of critical, the flow can develop abrupt "kink" transitions from a steep acceleration to a decelerating solution. For rotations above 85% of critical, the hydrodynamic simulations confirm the slow acceleration, with the lower flow speed implying densities 5-30 times higher than the polar (or a nonrotating) wind. Still, when gravity darkening and 2D flow effects are accounted for, it seems unlikely that rotationally modified equatorial wind outflows could account for the very large densities inferred for the equatorial regions around B[e] supergiants.