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 separate 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 analyse 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 outwards 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 lambda 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 proportional to r(-1), while in the core collapse portion they would scale as B proportional to r(-1/2). We discuss several astrophysical applications of EECC similarity solutions to the formation process of protoplanetary nebulae connecting the asymptotic giant branch phase and the planetary nebula, to supernova remnants, to collapse of magnetized molecular clouds, to H II clouds surrounding massive OB stars and to a certain evolution phase of galaxy clusters.