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. With reference to the 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--> 0(+) to a final stage t-->+infinity. 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 feature concurrent interior core collapse and exterior envelope expansion. The interior collapse towards the central core approaches a free-fall state as the radius r--> 0, while the exterior envelope expansion gradually approaches a constant radial flow speed as r-->+infinity. 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. Without crossing the sonic critical line where the travel speed of non-linear disturbances relative to the radial flow is equal to the sound speed, there exists a continuous band of infinitely many EECC solutions with only one supersonic stagnation point as well as a continuous band of infinitely many similarity solutions for 'envelope contraction with core collapse' (ECCC) without stagnation point. Crossing the sonic critical line twice analytically, there are infinitely many discrete EECC solutions with one or more subsonic stagnation points. Such discrete EECC similarity solutions generally allow radial oscillations in the subsonic region between the central core collapse and the outer envelope expansion. In addition, we obtained complementary discrete ECCC similarity solutions that cross the sonic critical line twice with subsonic oscillations. In all these discrete solutions, subsonic spherical stagnation surfaces resulting from similarity oscillations travel outward at constant yet different speeds in a self-similar manner. With specified initial boundary or shock conditions, it is possible to construct an infinite number of such EECC similarity solutions, which are conceptually applicable to various astrophysical problems involving gravitational collapses and outflows. We mention potential applications of EECC similarity solutions to the formation process of protoplanetary nebulae connecting the asymptotic giant branch phase and the planetary nebula phase to H II clouds surrounding star formation regions, and to a certain evolution phase of galaxy clusters.