Gravitational Collapse and Star Formation in Logotropic and Nonisothermal Spheres

We present semi-analytical similarity solutions for the inside-out,expansion-wave collapse of initially virialized gas clouds with non-isothermalequations of state. Results are given for the family of negative-indexpolytropes, but we focus especially on the so-called logotrope, P/P_c=1+Aln(rho/rho_c). The formalism and interpretation of the present theory areextensions of those in Shu's (1977) standard model for accretion inself-gravitating isothermal spheres: a collapse front moves outwards into acloud at rest, and the gas behind it falls back to a collapsed core, orprotostar. The infalling material eventually enters free-fall, so that, atsmall radii, the density profiles and velocity fields have the same power-lawforms in logotropic and isothermal spheres both. However, the accretion rateonto a protostar is not constant in a logotrope, but grows in proportion to t^3during 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. Wealso find that infall velocities grow more slowly with time in a collapsinglogotrope. These results lead to older inferred collapse ages for Class 0protostars in general, and for the Bok globule B335 in particular.Comment: 39 pages, LaTeX with 5 ps figures, uses aaspp4.sty. ApJ, in pres