Self-similar collapse of nonrotating magnetic molecular cloud cores

We obtain self-similar solutions that describe the gravitational collapse of nonrotating, isothermal, magnetic molecular cloud cores. We use simplifying assumptions but explicitly include the induction equation, and the semianalytic solutions we derive are the first to account for the effects of ambipolar diffusion and its critical dependence on the magnetic tension force following the formation of a central point mass. Our results demonstrate that, after the protostar first forms, ambipolar diffusion causes the magnetic flux to decouple in a growing region around the center. The decoupled held lines remain approximately stationary and drive a hydromagnetic C-shock that moves outward at a fraction of the speed of sound (typically a few tenths of a kilometer per second), reaching a distance of a few thousand AU at the end of the main accretion phase for a solar-mass star. We also show that, in the absence of held diffusivity, a contracting core will not give rise to a shock if, as is Likely to be the case, the inflow speed near the origin is nonzero at the time of point-mass formation. Although the evolution of realistic molecular cloud cores will not be exactly self similar, our results reproduce the main qualitative features found in detailed core-collapse simulations.