Analytic insight into the physics of SASI I. Shock instability in a non-rotating stellar core

During the core collapse of a massive star just before its supernova explosion, the amplification of asymmetric motions by the standing accretion shock instability (SASI) imprints on the neutrino flux and the gravitational waves a frequency signature carrying direct information on the explosion process. The physical interpretation of this multi-messenger signature requires a detailed understanding of the instability mechanism. A perturbative analysis is used to characterize the properties of SASI, and assess the effect of the region of neutronization above the surface of the proto-neutron star. The eigenfrequencies of the most unstable modes are compared to those obtained in an adiabatic approximation where neutrino interactions are neglected above the neutrinosphere. The differential system is solved analytically using a Wronskian method and approximated asymptotically for a large shock radius. The oscillation period of SASI is well fitted with a simple analytic function of the shock radius, the radius of maximum deceleration and the mass of the proto-neutron star. The oscillation period is weakly dependent on the parametrized cooling function which however affects the SASI growth rate. The general properties of SASI eigenmodes are described using an adiabatic model. In this approximation the eigenvalue problem is formulated as a self-forced oscillator. The forcing agent is the radial advection of baroclinic vorticity perturbations and entropy perturbations produced by the shock oscillation. The differential system defining the eigenfrequencies is reduced to a single integral equation. Its analytical approximation sheds light on the radially extended character of the region of advective-acoustic coupling. The simplicity of this adiabatic formalism opens new perspectives to investigate the effect of stellar rotation and non-adiabatic processes on SASI.