Self-similar Collapse of Rotating Magnetic Molecular Cloud Cores

Abstract

We present self-similar solutions that describe the gravitational collapse of rotating, isothermal, magnetic molecular-cloud cores, relevant to the formation of rotationally supported protostellar disks. This work focuses on the evolution after a point mass first forms at the center and generalizes previous results by Contopoulos, Ciolek, & Konigl that did not include rotation. Our model includes ambipolar diffusion and magnetic braking and allows us to examine the full range of expected behaviors and their dependence on the physical parameters. For typical parameter values, the inflow first passes through an ambipolar-diffusion shock (at a radius r_a, where the magnetic flux decouples from the matter), and later through a centrifugal shock at r=r_c, the outer edge of a rotationally supported disk of mass M_d. By the time (~10^5 yr) that the central mass M_c grows to ~1 M_\sun, r_a may be larger than 1000 AU, r_c larger than 100 AU, and M_d/M_c smaller than 10%. Disk properties are consistent with data on T Tauri systems, and our results imply that protostellar disks may well be Keplerian also during earlier phases. We show that the disk is likely to drive centrifugal outflows transporting angular momentum and mass, and we incorporate these effects into the model. We verify that gravitational torques and magnetorotational instability-induced turbulence typically do not play an important role in the angular momentum transport. We also present solutions for the limiting cases of fast rotation (where collapse results in a massive disk with such a large outer radius that it traps the ambipolar diffusion front) and strong braking (where no disk is formed and the collapse resembles that of a nonrotating core at small radii), as well as solutions for the rotational collapse of ideal-MHD and nonmagnetic model cores.