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

Motivated by recent observations that show starless molecular cloud coresexhibit subsonic inward velocities, we revisit the collapse problem forpolytropic gaseous spheres. In particular, we provide a generalized treatmentof protostellar collapse in the spherical limit and find semi-analytic(self-similar) solutions, corresponding numerical solutions, and purelyanalytic calculations of the mass infall rates (the three approaches are ingood agreement). This study focuses on collapse solutions that exhibit nonzeroinward velocities at large radii, as observed in molecular cloud cores, andextends previous work in four ways: (1) The initial conditions allow nonzeroinitial inward velocity. (2) The starting states can exceed the density ofhydrostatic equilibrium so that the collapse itself can provide the observedinward motions. (3) We consider different equations of state, especially thosethat are softer than isothermal. (4) We consider dynamic equations of statethat are different from the effective equation of state that produces theinitial density distribution. This work determines the infall rates over a widerange of parameter space, as characterized by four variables: the initialinward velocity $\vin$, the overdensity $\overdense$ of the initial state, theindex $\Gamma$ of the static equation of state, and the index $\gamma$ of thedynamic equation of state. For the range of parameter space applicable toobserved cores, the resulting infall rate is about a factor of two larger thanfound in previous theoretical studies (those with hydrostatic initialconditions and $\vin = 0$).Comment: 48 pages, 13 figures, accepted for publication to The Astrophysical Journa