Conditions for jet break-out in NS mergers

Abstract: We consider conditions for jet break-out through ejecta following mergers of neutron stars and provide simple relations for the break out conditions. We demonstrate that: (i) break-out requires that the isotropic-equivalent jet energy $E_j$ exceeds the ejecta energy $E_{ej}$ by $E_j \geq E_{ej}/ \beta_0$, where $ \beta_0 = V_{ej}/c$, $V_{ej}$ is the maximum velocity of the ejecta. If the central engine terminates before the break out, the shock approaches the edge of the ejecta slowly $\propto 1/t$; late break out occurs only if at the termination moment the head of the jet was relatively close to the edge. (ii) If there is a substantial delay between the ejecta's and the jet's launching, the requirement on the jet power increases. (iii) The forward shock driven by the jet is mildly strong, with Mach number $M\approx 5/4$ (increasing with time delay $t_d$); (iii) the delay time $t_d$ between the ejecta and the jet's launching is important for $t_d > t_0= ({3 }/{16} ) {c M_{ej} V_{ej}}/{L_j} = 1.01 {\rm sec} M_{ej, -2} L_{j, 51} ^{-1} \left ( {\beta_{ej}} /{0.3} \right)$, where $M_{ej}$ is ejecta mass, $L_j$ is the jet luminosity (isotropic equivalent). For small delays, $t_0 $ is also an estimate of the break-out time.