Self-Similar Shocks in Polytropic Gas Flows around Star-Forming Regions
  Self-similar shock solutions in spherically symmetric polytropic gas flows are constructed and analyzed in contexts of proto-star formation processes. Among other possible solutions, we model a similarity shock across the sonic critical curve with an inner free-fall core collapse and a simultaneous outer expansion of the extended envelope; the separation or stagnation surface between these two flow zones travels outwards in a self-similar manner at a variable speed. One readily obtains accretion shock solutions. Semi-complete self-similar solutions across the sonic critical curve either once or twice without shocks can also be constructed. Features of star formation clouds of our polytropic model include the mass density scaling in the outer flow zone $\rho\propto r^{-2/(2-\gamma)}$, the temperature scalings of the inner flow zone $T\propto r^{-3(\gamma-1)/2}$ and of the outer flow zone $T\propto r^{-2(\gamma-1)/(2-\gamma)}$, and the variable central mass accretion rate $\dot{M}=k^{3/2}t^{(3-3\gamma)}m_0/G$ where $\gamma$ is the polytropic index, $k$ is a constant, $m_0$ is the core mass, and $G$ is the gravitational constant. Spectral line profiles characteristic of the `envelope expansion with core collapse' (EECC) shock solutions are expected. Random magnetic field permeated in a partially ionized cloud can be incorporated into this theoretical polytropic model framework. We discuss briefly our results in context of the oft-observed starless B335 cloud system as an example. 
Yu-Qing Lou and Yang Gao