Magic speed under radiation drag for the optically thick outflows and SS 433 jets

One of the promising mechanisms to explain the stable jet speed of SS 433 is the magic speed, which is established by the balance between radiation pressure and radiation drag (and gravity). We examine the magic speed in the optically thick sub-relativistic flows for several typical situations, such as plane-parallel and spherical cases, and show that it is difficult to establish the magic speed in the simple optically thick flows in contrast to the optically thin ones. Instead, we propose the funnel jets as an improved model. That is, we consider the optically thick flows in the funnel, which is formed by the supercritical accretion disk, and mass and radiation energy are injected from the funnel wall. In this model the mass-accretion rate determines the configuration of the funnel, the increasing mass-loss rate of jets, the radiative environments, and then the sub-relativistic magic speed is naturally established.