Semi-dynamical approach to the shock revival in core-collapse supernovae

We develop a new semi-dynamical method to study shock revival by neutrino heating in core- collapse supernovae. Our new approach is an extension of the previous studies that employ spherically symmetric, steady, shocked accretion flows together with the light bulb approximation. The latter has been widely used in the supernova community for the phenomenological investigation of the criteria for successful supernova explosions. In the present approach, on the other hand, we get rid of the steady-state condition and take into account shock wave motions instead. We have in mind the scenario that not the critical luminosity but the critical fluctuation generated by hydrodynamical instabilities such as SASI and neutrino-driven convection in the post-shock region determines the onset of shock revival. After confirming that the new approach indeed captures the dynamics of revived shock wave qualitatively, we then apply the method to various initial conditions and find that there is a critical fluctuation for shock revival, which can be well fit by the following formula: f_crit ~ 0.8 * (M_in/1.4M_sun) * {1- (rsh/10^8cm)}, in which fcrit denotes the critical pressure fluctuation normalized by the unperturbed post-shock value. Min and rsh stand for the mass of the central compact object and the shock radius, respectively. The critical fluctuation decreases with the shock radius, whereas it increases with the mass of the central object. We discuss the possible implications of our results for 3D effects on shock revival, which is currently controversial in the supernova community.