Generalized Collapse Solutions with Nonzero Initial Velocities for Star Formation in Molecular Cloud Cores

Motivated by recent observations that show starless molecular cloud cores exhibit subsonic inward velocities, we revisit the collapse problem for polytropic gaseous spheres. In particular, we provide a generalized treatment of protostellar collapse in the spherical limit and find semi-analytic (self-similar) solutions, corresponding numerical solutions, and purely analytic calculations of the mass infall rates (the three approaches are in good agreement). This study focuses on collapse solutions that exhibit nonzero inward velocities at large radii, as observed in molecular cloud cores, and extends previous work in four ways: (1) The initial conditions allow nonzero initial inward velocity. (2) The starting states can exceed the density of hydrostatic equilibrium so that the collapse itself can provide the observed inward motions. (3) We consider different equations of state, especially those that are softer than isothermal. (4) We consider dynamic equations of state that are different from the effective equation of state that produces the initial density distribution. This work determines the infall rates over a wide range of parameter space, as characterized by four variables: the initial inward velocity $\vin$, the overdensity $\overdense$ of the initial state, the index $\Gamma$ of the static equation of state, and the index $\gamma$ of the dynamic equation of state. For the range of parameter space applicable to observed cores, the resulting infall rate is about a factor of two larger than found in previous theoretical studies (those with hydrostatic initial conditions and $\vin = 0$).