Spherical Isothermal Self-Similar Shock Flows
  We explore self-similar dynamical processes in a spherical isothermal self-gravitational fluid with an emphasis on shocks and outline astrophysical applications of such shock solutions. The previous similarity shock solutions of Tsai & Hsu and of Shu et al. may be classified into two types: Class I solutions with downstream being free-fall collapses and Class II solutions with downstream being Larson-Penston (LP) type solutions. By the analyses of Lou & Shen and Shen & Lou, we further construct similarity shock solutions in the `semi-complete space'. These general shock solutions can accommodate and model dynamical processes of radial outflows (wind), inflows (accretion or contraction), subsonic oscillations, and free-fall core collapses all with shocks in various settings such as star-forming molecular clouds, `champagne flows' in H{\sevenrm II} regions around luminous massive OB stars or surrounding quasars, dynamical connection between the asymptotic giant branch phase to the proto-planetary nebula phase with a central hot white dwarf as well as accretion shocks around compact objects such as white dwarfs, neutron stars, and supermassive black holes. By a systematic exploration, we are able to construct families of infinitely many discrete Class I and Class II solutions matching asymptotically with a static outer envelope of singular isothermal sphere; the shock solutions of Tsai & Hsu form special subsets. These similarity shocks travel at either subsonic or supersonic constant speeds. We also construct twin shocks as well as an `isothermal shock' separating two fluid regions of two different yet constant temperatures. 
Fu-Yan Bian, Yu-Qing Lou