Self-similar stellar collapse

Self-similar solutions are found for the collapse of a stellar core with a polytropic equation of state p = κργ, and 6/5 ≤ γ ≤ 4/3. These solutions show that the core splits into a homologously collapsing inner core and an outer core, which is infalling supersonically, at about half the free-fall velocity, and with a Mach number ≡2-3. The self-similar solutions yield two important qualitative conclusions regarding supernovae in massive stars: (1) the mass of the inner core is about 10% higher than the Chandrasekhar mass, and this determines the energy transferred by the inner core to the shocked outer core during rebound at nuclear density; (2) the closer γ is to 4/3, the smaller the density of the infalling outer core, and hence the easier it is for the shock to move against its ram pressure.