Envelope Expansion with Core Collapse II. Quasi-Spherical Self-Similar Solutions for an Isothermal Magnetofluid

We investigate self-similar magnetohydrodynamic (MHD) processes in an isothermal self-gravitating fluid with a quasi-spherical symmetry and extend the envelope expansion with core collapse (EECC) solutions of Lou & Shen by incorporating a random magnetic field. Magnetized expansion-wave collapse solutions (MEWCS) can be readily constructed as a special case. This inside-out MHD collapse occurs at the magnetosonic speed and magnetized EECC solutions are obtained systematically. Stagnation surfaces of EECC solutions that seperate core collapse and envelope expansion propagate at constant speeds either sub-magnetosonically or super-magnetosonically. These similarity MHD solutions show various behaviours, such as radial inflow or contraction, core collapse, oscillations and outflow or wind as well as shocks. For solutions to go across the magnetosonic line smoothly, we carefully analyze topological properties of magnetosonic critical points. Without crossing the magnetosonic critical line, continuous spectra of magnetized EECC and envelope contraction with core collapse (ECCC) solutions are readily obtained. Crossing the magnetosonic line twice analytically, there exists an infinite number of discrete magnetized EECC and ECCC solutions. One or more sub-magnetosonic stagnation surfaces associated with these discrete solutions travel outward at constant yet different speeds in a self-similar manner. In addition to the EECC shock solution which could change the central accretion rate, the magnetic field can also affect the core accretion rate. As the magnetic parameter λ increases, the core accretion rate appropriate for the MHD EWCS becomes larger. Under the frozen-in approximation, magnetic fields in the envelope expansion portion would scale as B ∝ r, while in the core collapse portion they would scale as B ∝ r. We discuss several astrophysical applications of EECC similarity solutions to the formation process of proto-planetary nebulae connecting the AGB phase and the planetary nebula, to supernova remnants, collapse of magnetized molecular cloud, to H II clouds surrounding massive OB stars and to a certain evolution phase of galaxy clusters.