proportional to the neutrino luminosity. The cooling function, first introduced by @BM06 and subsequently used by @F07, models electron and positron capture in an optically thin environment $(\propto \rho T^6)$ assuming a radiation-dominated gas $(p\propto T^4)$. The exponential suppression at a low entropy $s_{\rm min}$ is introduced to prevent runaway cooling at the base of the flow, and the cutoff at high Mach number $\mathcal{M}_0=2$ is used to suppress heating and cooling in the upstream flow [@FT09b].

The initial condition consists of a steady-state spherical accretion shock at a radius $r_s$, below which the fluid settles subsonically onto a protoneutron star of radius $r_*$. Given a boundary condition at the shock, the normalization of the cooling function $A$ is determined by demanding that the radial velocity vanishes at $r=r_*$. The upstream flow is supersonic and adiabatic, with zero Bernoulli parameter. The Mach number upstream of the shock is set to $\mathcal{M}_1 = 5$ at a radius $r_{\rm s0}$ equal to the shock radius obtained with zero heating ($B=0$). To connect with previous studies (e.g., @FT09b), the adiabatic index is set to $\gamma=4/3$, even though a more realistic flow would have this index varying within the range $1.4-1.6$. A constant specific 