A critical accretion rate for triggered star formation

We reexamine the similarity solution for a self-gravitating isothermal gas sphere and consider the implications for star formation in turbulent clouds. For adequately chosen parameters, the similarity solution describes an accreting isothermal gas sphere bounded by a spherical shock wave. The mass and radius of the sphere increase in proportion to 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 former has a critical value for a given infall velocity: when the accretion rate lies below this value, there exists a pair of similarity solutions determined by the given combination of accretion rate and infall velocity. One of these solutions is confirmed to be unstable against spherical perturbations. The implication is that the gas sphere will collapse to initiate star formation only when the accretion rate is higher than the critical value. We also examine the stability of the similarity solution against nonspherical perturbations, which are found to be damped.