Self-similar collapse of magnetized molecular cloud cores with ambipolar diffusion and the "magnetic flux problem" in star formation

We investigate the dynamic collapse of magnetized singular isothermal spheres in the presence of ambipolar diffusion, using a self-similarity technique. The spherical geometry, a crucial idealization first introduced by Safier, McKee, & Stahler to study the evolution of magnetized clouds, is made possible by ignoring magnetic tension forces. We find that, in the limit of a complete coupling between magnetic fields and neutral matter, magnetized spheres collapse via an expansion-wave solution, as in the (nonmagnetized) singular isothermal case. The presence of ambipolar diffusion modifies the collapse in two interesting ways. First, it smooths out the kinky wave head into an expansion wave of continuous (or "C") type. Second, ambipolar diffusion allows magnetic fields to decouple from rapidly collapsing neutral matter at small radii near the central point mass, thereby reducing the amount of magnetic flux dragged into the origin. We obtain viable collapse solutions with various amounts of magnetic flux at the center, including special ones that have no central flux at all. In such special cases, the long-standing "magnetic flux problem" in star formation is completely resolved. Furthermore, we show that the decoupled magnetic flux drives a hydromagnetic accretion shock against the dynamically collapsing envelope, as suggested previously by Li & McKee. Depending on the degree of coupling between magnetic fields and neutral matter and the amount of magnetic flux released from the central compact object, the accretion shock (which modifies the flow dynamics significantly) could either be of purely C type or have an embedded J subshock.