Gravitational Collapse and Star Formation in Logotropic and Nonisothermal Spheres

We present semianalytical similarity solutions for the inside out, expansion-wave collapse of initially virialized gas clouds with nonisothermal equations of state. Results are given for the family of negative-index polytropes (P ∝ ργ, γ ≤ 1), but we focus especially on the so-called logotrope, P/Pc = 1 + Aln(ρ/ρc). In a separate paper, we have shown this to be the best available phenomenological description of the internal structure and average properties of molecular clouds, as well as dense clumps of both high and low mass. The formalism and interpretation of the present theory are extensions of those in Shu's standard model for accretion in self-gravitating isothermal spheres: a collapse front moves outward 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 the density profiles and velocity fields have the same shape (ρ ∝ r-3/2 and -u ∝ r-1/2) at both small radii in logotropic and isothermal spheres. However, several differences arise from the introduction of a new equation of state. The accretion rate onto a protostar is not constant in a logotrope, but grows as Ṁ∝t3 during the expansion wave. Thus, the formation time for a star of mass M scales as M1/4; 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 initial mass function. In addition, the gas density behind an expansion wave increases with time in our theory, but it would decrease in an isothermal sphere. The infall velocities also grow, but at an initially much slower rate than that found in an isothermal collapse. These results apply to low- and high-mass star formation alike. We briefly discuss how they lead to older inferred collapse ages for Class 0 protostars in general and for the Bok globule B335 in particular.