The Nonrelativistic Evolution of GRBs 980703 and 970508: Beaming‐independent Calorimetry

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Abstract
We use the Sedov-Taylor self-similar solution to model the radio emission from the γ-ray bursts GRB 980703 and GRB 970508 when the blast wave has decelerated to nonrelativistic velocities. This approach allows us to infer the energy of the GRBs independent of jet collimation. We find that for GRB 980703 the kinetic energy at the time of the transition to nonrelativistic evolution, tNR ≈ 40 days, is EST ≈ (1-6) × 1051 ergs. For GRB 970508 we find EST ≈ 3 × 1051 ergs at tNR ≈ 100 days, nearly an order of magnitude higher than the energy formerly derived by Frail, Waxman, and Kulkarni. 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 tNR, 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 5 × 1051 ergs and also shows that the central engine in these two bursts did not produce a significant amount of energy in mildly relativistic ejecta at late times. Furthermore, a comparison to the prompt energy release reveals a wide dispersion in the γ-ray efficiency, strengthening our growing understanding that Eγ is a not a reliable proxy for the total energy.
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