The Luminosity–ep Relation within Gamma–ray Bursts and Implications for Fireball Models

Using a sample of 2408 time–resolved spectra for 91 BATSE GRBs presented by Preece et al., we show that the relation between the isotropic–equivalent luminosity (Liso) and the peak energy (E ′ p) of the νFν spectrum in the cosmological rest frame, Liso ∝ E ′2 p , holds within these bursts, and also holds among these GRBs, assuming that the burst rate as a function of redshift is proportional to the star formation rate. The possible implications of this relation for the fireball models are discussed by defining a parameter ω ≡ (Liso/10 erg s)/(E ′ p/200 keV). It is found that ω is narrowly clustered in 0.1− 1. We constrain some parameters for both the internal shock and external shock models from the requirement of ω ∼ 0.1 − 1, assuming that these model parameters are uncorrelated. The distributions of the parameters suggest that if the prompt gamma–rays are produced from kinetic–energy–dominated internal shocks, they may be radiated from a region around R ∼ 10 − 10 cm (or Lorentz factor ∼ 130 − 410) with a combined internal shock parameter ζi ∼ 0.1 − 1 during the prompt gamma–ray phase, which are consistent with the standard internal shock model; if the prompt gamma–rays of these GRBs are radiated from magnetic–dissipation–dominated external shocks, the narrow cluster of ω requires σ ∼ 1 − 470, Γ ∼ 216 − 511, E ∼ 10 − 10 ergs, n ∼ 0.5 − 470 cm, and ζe ∼ 0.36 − 3.6, where σ is the ratio of the cold-to-hot luminosity components, Γ the bulk Lorentz factor of the fireball, E the total energy release in gamma–ray band, n the medium number density, and ζe a combined external shock parameter, which are also in a good agreement with the fittings to the afterglow data. These results indicate that both the kinetic–energy–dominated internal shock model and the magnetic– dissipation–dominated external shock model can well interpret the Liso ∝ E ′2 p relation and the value of ω. Subject headings: gamma rays: bursts—gamma rays: observations—methods: statistical