Critical Accretion Rate for Triggered Star Formation
  We have reexamined the similarity solution for a self-gravitating isothermal gas sphere and examined implication to star formation in a turbulent cloud. When parameters are adequately chosen, the similarity solution expresses an accreting isothermal gas sphere bounded by a spherical shock wave. The mass and radius of the sphere increases in proportion to the time, while the central density decreases in proportion to the inverse square of time. The similarity solution is specified by the accretion rate and the infall velocity. The accretion rate has an upper limit for a given infall velocity. When the accretion rate is below the upper limit, there exist a pair of similarity solutions for a given set of the accretion rate and infall velocity. One of them is confirmed to be unstable against a spherical perturbation. This means that the gas sphere collapses to initiate star formation only when the accretion rate is larger than the upper limit. We have also examined stability of the similarity solution against non-spherical perturbation. Non-spherical perturbations are found to be damped. 
Tomoyuki Hanawa, Akihito Soeda