Envelope Expansion with Core Collapse. I. Spherical Isothermal Similarity Solutions

We investigate self-similar dynamical processes in an isothermal self-gravitational fluid with spherical symmetry. In reference to earlier complementary solution results of Larson, Penston, Shu, Hunter and Whitworth & Summers, we further explore the semi-complete solution space' from an initial instant $t\to 0^{+}$ to a final stage $t\to +\infty$. These similarity solutions can describe and accommodate physical processes of radial inflow, core collapse, oscillations and envelope expansion (namely, outflow or wind) or contraction as well as shocks. In particular, we present new classes of self-similar solutions, referred to as envelope expansion with core collapse’ (EECC) solutions, that are featured by an interior core collapse and an exterior envelope expansion concurrently. The interior collapse towards the central core approaches a free-fall state as the radius $r\to 0$, while the exterior envelope expansion gradually approaches a constant radial flow speed as $r\to +\mathrm{\infty }$. There exists at least one spherical stagnation surface of zero flow speed that separates the core collapse and the envelope outflow and that travels outward at constant speed, either subsonically or supersonically, in a self-similar manner.