Self-similar evolution of supercritical cores

We use a semi-analytic 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 (e.g., 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, and we identify the key role of ambipolar diffusion in accelerating the collapse. We find that the collapse is dynamic and supersonic velocities are achieved in the innermost region of the core by the time of protostar formation. The mass accretion rate is significantly greater than the canonical C-3/G 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.