Stability of accretion flows with stalled shocks in core-collapse supernovae

With an eye toward application to the theory of core-collapse supernovae, we perform a global linear analysis of the stability of spherically symmetric accretion flows through a standing shock wave onto a proto-neutron star. For the unperturbed flows, we adopt spherically symmetric, steady solutions obtained with a realistic equation of state and realistic neutrino reaction rates. These solutions are characterized by the mass accretion rate and neutrino luminosity. Then we solve the equations for linear perturbations numerically, obtaining the eigenfrequencies and eigenfunctions. We find the following: (1) The flows are stable for all modes if the neutrino luminosity is lower than a certain value, e.g., ∼1 × 10⁵² ergs s^-1 for Ṁ = 1.0 M _⊙ s^-1. (2) For higher luminosities, nonradial instabilities are induced, probably through advective-acoustic cycles. Interestingly, modes with l=2 and l = 3 first become unstable for relatively low neutrino luminosities, e.g., ∼(2-3) × 10⁵² ergs s ^-1 for the same accretion rate, whereas the l = 1 mode is the most unstable for higher luminosities, ∼(3-7) × 10⁵² ergs s ^-1. These are all oscillatory modes. (3) For still larger luminosities, ≳7 × 10⁵² ergs s^-1 for Ṁ = 1.0 M_⊙ s^-1, nonoscillatory modes, both radial and nonradial, become unstable. These nonradial modes are identified as convective, and their growth rates have a peak at l = 5-11, depending on the luminosity. We confirm the result from numerical simulations that the instabilities induced by advective-acoustic cycles are more important than convection for lower neutrino luminosities. Furthermore, we investigate changing the inner boundary conditions and find that while the effects are nonnegligible, the existence of the instabilities does not qualitatively change for a variety of conditions.