The Non-Relativistic Evolution of GRBs 980703 and 970508: Beaming-Independent Calorimetry

Abstract

We use the Sedov-Taylor self-similar solution to model the radio emission from the gamma-ray bursts (GRBs) 980703 and 970508, when the blastwave has decelerated to non-relativistic velocities. This approach allows us to infer the energy independent of jet collimation. We find that for GRB 980703 the kinetic energy at the time of the transition to non-relativistic evolution, t_NR ~ 40 d, is E_ST ~ (1-6)e51 erg. For GRB 970508 we find E_ST ~ 3e51 erg at t_NR ~ 100 d, nearly an order of magnitude higher than the energy derived in Frail, Waxman and Kulkarni (2000). This is due primarily to revised cosmological parameters and partly to the maximum likelihood fit we use here. Taking into account radiative losses prior to t_NR, the inferred energies agree well with those derived from the early, relativistic evolution of the afterglow. Thus, the analysis presented here provides a robust, geometry-independent confirmation that the energy scale of cosmological GRBs is about 5e51 erg, and additionally shows that the central engine in these two bursts did not produce a significant amount of energy in mildly relativistic ejecta at late time. Furthermore, a comparison to the prompt energy release reveals a wide dispersion in the gamma-ray efficiency, strengthening our growing understanding that E_gamma is a not a reliable proxy for the total energy.