### abstract

- We examine the conditions for the revival of the stalled accretion shock in core-collapse supernovae, in the context of the neutrino heating mechanism. We combine one dimensional simulations of the shock revival process with a derivation of a quasi-stationary approximation, which is both accurate and efficient in predicting the flow. In particular, this approach is used to explore how the evolution of the system depends on the shock radius, $R_S$, and velocity, $V_S$ (in addition to other global properties of the system). We do so through a phase space analysis of the shock acceleration, $a_S$, in the $R_S-V_S$ plane, shown to provide quantitative insights into the initiation of runaway expansion and its nature. In the particular case of an initially stationary ($V_S=0,\;a_S=0$) profile, the prospects for an explosion can be reasonably assessed by the initial signs of the partial derivatives of the shock acceleration, in analogy to a linear damped/anti-damped oscillator. If $\partial a_S/\partial R_S<0$ and $\partial a_S/\partial V_S>0$, runaway expansion will likely occur after several oscillations, while if $\partial a_S/\partial R_S>0$, runaway expansion will commence in a non-oscillatory fashion. These two modes of runaway correspond to low and high mass accretion rates, respectively. We also use the quasi-stationary approximation to assess the advection-to-heating timescale ratio in the gain region, often used as an explosion proxy. Indeed, this ratio does tend to $\sim1$ in conjunction with runaway conditions, but neither this unit value nor the specific choice of the gain region as a point of reference appear to be distinct conditions in this regard.