On the energy of gamma-ray bursts

We show that gamma -ray burst (GRB) afterglow observations strongly suggest, within the fireball model framework, that radiating electrons are shock accelerated to a power-law energy distribution, dn(e)/d gamma (e) proportional to gamma (-p)(e), with universal index p approximate to 2.2, and that the fraction of shock energy carried by electrons, xi (e), is universal and close to equipartition, xi (e) similar to 1/3. For universal p and xi (e), 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, epsilon, averaged over a conical section of the fireball of opening angle theta similar to 0.1. The energy estimate epsilon varies by similar to 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 4 pi epsilon = 10(51.5)-10(53.5) ergs; (2) the ratio of observed gamma -ray to total fireball energy per unit solid angle, epsilon (gamma)/epsilon, is of order unity, satisfying \log(10) (epsilon (gamma)/epsilon)\ less than or similar to 0.5; and (3) if fireballs are jet-like, their opening angle should satisfy theta greater than or similar to 0.1. Our results imply that if typical opening angles are theta similar to 0.1, a value consistent with our analysis, the total energy associated with a GRB event is in the range of 10(50)-10(51.5) ergs.