Self‐Similar Gravitational Collapse of Radiatively Cooling Spheres

A new self-similar solution describing spherical implosions of a gaseous sphere under both self-gravity and radiative diffusion is investigated in detail, where the diffusivity is modeled by a power law with respect to density and temperature. The reduced two-dimensional eigenvalue problem is solved to show that there is a unique quantitative relation between the two physical effects for the self-similar dynamics. The resultant spatial and temporal behaviors are also determined uniquely, once the opacity is specified. For a reference case of an inverse bremsstrahlung opacity in a fully ionized hydrogen plasma, the solution predicts that the system evolves within the applicable parameter ranges: 10-11 to 10-8 g cm -3 for the central density, a few to tens of 103 K for the central temperature, a few to tens of years for the collapse period, and a few to a dozen times the solar mass for the core mass. Persistent entropy emission via radiation plays an important role in the core formation with mass accretion, which is contrary to the predictions of implosion models under isothermal or adiabatic assumptions. The mass accretion rate is found to increase with time in a power-law form. The present solution turns out to be convectively stable.