Gravitational Collapse and Star Formation in Logotropic and Non-Isothermal Spheres

We present semi-analytical similarity solutions for the inside-out, expansion-wave collapse of initially virialized gas clouds with non-isothermal equations of state. Results are given for the family of negative-index polytropes, but we focus especially on the so-called logotrope, P/P_c=1+A ln(rho/rho_c). The formalism and interpretation of the present theory are extensions of those in Shu's (1977) standard model for accretion in self-gravitating isothermal spheres: a collapse front moves outwards into a cloud at rest, and the gas behind it falls back to a collapsed core, or protostar. The infalling material eventually enters free-fall, so that, at small radii, the density profiles and velocity fields have the same power-law forms in logotropic and isothermal spheres both. However, the accretion rate onto a protostar is not constant in a logotrope, but grows in proportion to t^3 during the expansion wave. Thus, low-mass stars are accreted over longer times, and high-mass stars over shorter times, than expected in isothermal clouds. This result has implications for the form and origin of the stellar IMF. We also find that infall velocities grow more slowly with time in a collapsing logotrope. These results lead to older inferred collapse ages for Class 0 protostars in general, and for the Bok globule B335 in particular.