energy loss by nuclear dissociation $\varepsilon$ is allowed at the shock, increasing the compression ratio [@thompson00]. The solution is uniquely determined by specifying the ratio $r_*/r_{\rm s0}$, the nuclear dissociation parameter $\varepsilon$, the upstream Mach number $\mathcal{M}_1$, and the heating rate $B$ (see @FT09b for a sample of initial density profiles). In all models, we set $r_*/r_{\rm s0} = 0.4$.

Throughout this paper, we adopt the initial shock radius without heating, $r_{\rm s0}$, the free fall speed at this radius, $v_{\rm ff0}^2 = 2GM/r_{\rm s0}$, and the upstream density $\rho_1$ as the basic system of units. Full-scale simulations yield characteristic values $r_{\rm s0}\simeq 150$ km, $M\simeq 1.3M_\odot$, and $\dot{M}\simeq 0.3M_\sun$ s$^{-1}$, with a resulting free-fall speed $v_{\rm ff0} \simeq 5\times 10^9$ cm s$^{-1}$, dynamical time $t_{\rm ff0} = r_{\rm s0}/v_{\rm ff0} \simeq 3$ ms, and upstream density $\rho_1 \simeq 10^8$ g cm$^{-3}$. Setting the heating term in equation (\[eq:neutrino\_source\_term\]) equal to the approximation from @janka01 commonly used in ‘lightbulb’ heating studies (e.g., @murphy08 [@couch2013]), one obtains a relation between $B$ and the electron neutrino luminosity, $$\label{eq:B_dimensional}
B \simeq 0.009\, L_{\nu_e,52}\, T_{\nu,4}^2\, 
\left(\frac{r_{\rm s0}}{150\textrm{ km}}\right)^{1/2}\left(\frac{1.3M_\odot}{M} \right)^{3/2},$$ where $L_{\nu_e,52}$ is the electron neutrino luminosity in units of $10^{52}$ erg s$^{-1}$, and $T_{\nu,4}$ 