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 v _ { \infty } , the overdensity { \Lambda } 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 v _ { \infty } = 0 ) .