Similarity solution for formation of a circumstellar disk through the collapse of a flattened rotating cloud

We present similarity solutions that describe the runaway collapse of a rotating isothermal disk and its subsequent inside-out collapse. The similarity solutions contain the sound speed c(s) and the ratio omega of the specific angular momentum to the mass as model parameters. During the runaway collapse, the surface density of the disk is nearly constant in the central part and inversely proportional to the radius in the tail. As the central surface density increases by collapse, the central high surface density part shrinks its radius. Thus the surface density becomes a 1/r power law at the end of runaway collapse. In the subsequent inside-out collapse phase, the disk has two parts: an inner rotating disk in quasiequilibrium and an outer dynamically infalling envelope. The inner disk and outer envelope are bounded by a shock wave. The mass and outer edge of the inner disk grow at a constant rate. The accretion rate is proportional to the cube of the sound speed, i.e., M over dot proportional to c(s)(3)/G, where G denotes the gravitational constant. The similarity solution of the runaway collapse can reproduce numerical simulations of dynamical collapse of either rotating or magnetized disks. The similarity solution of the inside-out collapse denotes growth of a centrifugally supported circumstellar disk. This solution can apply to a protostar that accretes gas substantially through the disk.