On the Energy of Gamma‐Ray Bursts

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Abstract
We show that γ-ray burst (GRB) afterglow observations strongly suggest, within the fireball model framework, that radiating electrons are shock accelerated to a power-law energy distribution, dne/dγe ∝ γ, with universal index p ≈ 2.2, and that the fraction of shock energy carried by electrons, ξe, is universal and close to equipartition, ξe ~ . For universal p and ξe, a single measurement of the X-ray afterglow flux on the timescale of a day provides a robust estimate of the fireball energy per unit solid angle, ε, averaged over a conical section of the fireball of opening angle θ ~ 0.1. The energy estimate ε varies by ~50% as p is modified over the range of values p = 2.2 ± 0.2 typically inferred from observations. Applying our analysis to BeppoSAX afterglow data, we find that (1) fireball energies are in the range of 4πε = 1051.5-1053.5 ergs; (2) the ratio of observed γ-ray to total fireball energy per unit solid angle, εγ/ε, is of order unity, satisfying |(εγ/ε)| 0.5; and (3) if fireballs are jet-like, their opening angle should satisfy θ 0.1. Our results imply that if typical opening angles are θ ~ 0.1, a value consistent with our analysis, the total energy associated with a GRB event is in the range of 1050-1051.5 ergs.
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