Self‐similar Collapse of an Isopedic Isothermal Disk

Abstract

We study the gravitational collapse of an isothermal disk which is isopedically magnetized (i.e., with a mass-to-flux ratio that is spatially constant). The two theorems concerning magnetic forces in such a disk proven in a companion paper (Shu & Li 1997), plus the self-similar nature of the overall problem, allow a semianalytical treatment. The inflow occurs in an inside-out manner similar to that which applies in the collapse of the unmagnetized singular isothermal sphere (Shu 1977). These two cases (singular sphere and disk) bracket the range of possible collapse behaviors expected for the family of isopedic singular isothermal toroids described by Li & Shu (1996b). Although the strong magnetic fields dilute the effects of self-gravity in the isopedic isothermal disk, they do not prevent its outer parts (envelope) from falling onto the central condensed object (protostar) at a fixed infall rate M dot , even when the field is perfectly frozen to the matter. Indeed, the higher densities supported by the fields in the equilibrium state increase M dot during collapse in comparison with the unmagnetized case. The larger effective speed of sound due to magnetization produces a smaller effect. The flattened geometry enforced by the strong magnetic fields introduces a complication: the appearance of an outwardly propagating shock wave that runs ahead of the region of infall (also studied by Tsai & Hsu 1995 in a different context). We discuss the implications of our results for the magnetic-flux problem and for the formation of centrifugally supported disks in the presence of rotation.