A Revised Analysis of Gamma Ray Bursts' prompt efficiencies

Abstract: The prompt Gamma-Ray Bursts' (GRBs) efficiency is an important clue on the emission mechanism producing the $\gamma$-rays. Previous estimates of the kinetic energy of the blast waves, based on the X-ray afterglow luminosity $L_X$, suggested that this efficiency is large, with values above 90\% in some cases. This poses a problem to emission mechanisms and in particular to the internal shocks model. These estimates are based, however, on the assumption that the X-ray emitting electrons are fast cooling and that their Inverse Compton (IC) losses are negligible. The observed correlations between $L_X$ (and hence the blast wave energy) and $E_{\gamma\rm ,iso}$, the isotropic equivalent energy in the prompt emission, has been considered as observational evidence supporting this analysis. It is reasonable that the prompt gamma-ray energy and the blast wave kinetic energy are correlated and the observed correlation corroborates, therefore, the notion $L_X$ is indeed a valid proxy for the latter. Recent findings suggest that the magnetic field in the afterglow shocks is significantly weaker than was earlier thought and its equipartition fraction, $\epsilon_B$, could be as low as $10^{-4}$ or even lower. Motivated by these findings we reconsider the problem, taking now IC cooling into account. We find that the observed $L_X-E_{\gamma\rm ,iso}$ correlation is recovered also when IC losses are significant. For small $\epsilon_B$ values the blast wave must be more energetic and we find that the corresponding prompt efficiency is significantly smaller than previously thought. For example, for $\epsilon_B\sim10^{-4}$ we infer a typical prompt efficiency of $\sim15\%$.