A semianalytic model for supercritical core collapse: Self-similar evolution and the approach to protostar formation

We use a semianalytic model to examine the collapse of supercritical cores (i.e., cores with a mass-to-flux ratio exceeding a critical value). Recent numerical simulations of the formation and contraction of supercritical cores show that the inner solution tends toward self-similar evolution. We use this feature to develop analytic expressions for quantities such as the density, angular velocity, and magnetic field. All forces involved in the problem (gravitational, magnetic, thermal, and centrifugal) can be calculated analytically in the thin-disk geometry of the problem. The role of each force during the contraction is analyzed. We identify the key role of ambipolar diffusion in producing a departure from an exact similarity solution. The slow leakage of magnetic flux during the supercritical phase is enough to significantly accelerate an otherwise near-quasi-static contraction. This leads to dynamic collapse with supersonic infall speeds in the innermost region of the core by the time of protostar formation. We find a time-dependent semianalytic solution for the late supercritical phase, and asymptotic forms are obtained for important profiles at the moment that a central protostar is formed. We obtain estimates for the rotational velocity, infall velocity, and mass accretion rate at this moment. The mass accretion rate is significantly greater than the canonical C-3/G (where C is the isothermal sound speed and G is the universal gravitational constant) at the moment of protostar formation, although we argue that it is time-dependent and will eventually decrease. Comparisons are made with the predictions of existing spherical similarity solutions.