Conditions for jet breakout in neutron stars’ mergers

Abstract

ABSTRACT We consider conditions for jet breakout through ejecta following mergers of neutron stars and provide simple relations for the breakout conditions. We demonstrate that: (i) break-out requires that the isotropic-equivalent jet energy Ej exceeds the ejecta energy Eej by Ej ≥ Eej/βej, where βej = Vej/c, Vej is the maximum velocity of the ejecta. If the central engine terminates before the breakout, the shock approaches the edge of the ejecta slowly ∝ 1/t; late breakout 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 ≈ 5/4 (increasing with time delay td); (iii) the delay time td between the ejecta and the jet’s launching is important for $t_\mathrm{ d} \gt t_0= ({3 }/{16}) {c M_{\mathrm{ ej}} V_{\mathrm{ ej}}}/{L_\mathrm{ j}} = 1.01 {\rm \mathrm{ s}} M_{\mathrm{ ej}, -2} L_{\mathrm{ j}, 51} ^{-1} \left({\beta _{\mathrm{ ej}}} /{0.3} \right)$, where Mej is ejecta mass, Lj is the jet luminosity (isotropic equivalent). For small delays, t0 is also an estimate of the break-out time.