Self-similar collapse of isothermal spheres and star formation

Similarity solutions are obtained which describe the gravitational collapse of isothermal spheres that originated as gas clouds not far removed from the condition of marginal stability. It is argued that the similarity solution of Larson and Penston (1969) for the stages before core formation is physically artificial, but the gasdynamic flow subsequent to core formation exhibits self-similar properties. Similarity solutions are determined for the collapse of singular isothermal spheres, minus solutions without critical points are obtained by imposing the condition that the fluid velocities are negligible at the 'initial instant', and an expansion-wave collapse solution is evaluated. The results are illustrated with a numerical example roughly corresponding to conditions appropriate for Bok globules or the central regions of a nonmagnetic molecular cloud. Two possible applications of the solutions are discussed: analyzing the stability to gravitational fragmentation of collapsing pressure-free gas spheres and determining the amount of energy radiated away during protostar formation.