On the existence of a luminosity threshold of GRB jets in massive stars

Abstract

Motivated by the many associations of $\gamma$-ray bursts (GRBs) with energetic supernova (SN) explosions, we study the propagation of relativistic jets within the progenitor star in which a SN shock wave may be launched briefly before the jets start to propagate. Based on analytic considerations and verified with an extensive set of 2D axisymmetric relativistic hydrodynamic simulations, we have estimated a threshold intrinsic jet luminosity, $L_{\mathrm{j}}^{\mathrm{thr}}$, for successfully launching a jet. This threshold depends on the structure of the progenitor and, thus, it is sensible to its mass and to its metallicity. For a prototype host of cosmological long GRBs, a low-metallicity star of 35$M_{\odot }$, it is $L_{\mathrm{j}}^{\mathrm{thr}}\simeq 1.35\times 10^{49}$ergs${}^{-1}$. The observed equivalent isotropic $\gamma$-ray luminosity, $L_{\gamma ,\mathrm{iso},\mathrm{BO}}\simeq 4{\epsilon }_{\gamma }{L}_{\mathrm{j}}{\theta }_{\mathrm{BO}}^{-2}$, crucially depends on the jet opening angle after breakout, ${\theta }_{\mathrm{BO}}$, and on the efficiency for converting the intrinsic jet luminosity into $\gamma$-radiation, ${\epsilon }_{\gamma }$. Highly energetic jets can produce low-luminosity events if either their opening angle after the breakout is large, which is found in our models, or if the conversion efficiency of kinetic and internal energy into radiation is low enough. Beyond this theoretical analysis, we show how the presence of a SN shock wave may reduce this luminosity threshold by means of numerical simulations. We foresee that the high-energy transients released by jets produced near the luminosity threshold will be more similar to llGRBs or XRFs than to GRBs.

14 pages, 7 color figures. Submitted to MNRAS

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