Thermalization in relativistic outflows and the correlation between spectral hardness and apparent luminosity in gamma-ray bursts

We present an interpretation of the phenomenological relations between the spectral peak, isotropic luminosity, and duration of long gamma-ray bursts that have been discovered by Amati and coworkers, Ghirlanda and coworkers, Firmani and coworkers, and Liang & Zhang. In our proposed model, a jet undergoes internal dissipation which prevents its bulk Lorentz factor from exceeding 1/θ (θ being the jet opening angle) until it escapes from the core of its progenitor star, at a radius of order 10¹⁰ cm; dissipation may continue at larger radii. The dissipated radiation will be partially thermalized, and we identify its thermal peak (Doppler boosted by the outflow) with E_pk. The radiation comes, in effect, from within the jet photosphere. The nonthermal, high-energy part of the GRB emission arises from Comptonization of this radiation by relativistic electrons and positrons outside the effective photosphere. This model can account naturally not only for the surprisingly small scatter in the various claimed correlations, but also for the normalization, as well as the slopes. It then has further implications for the jet energy, the limiting jet Lorentz factor, and the relation of the energy, opening angle and burst duration to the mass and radius of the stellar stellar progenitor. The observed relation between pulse width and photon frequency can be reproduced by inverse Compton emission at ∼10¹⁴ cm from the engine, but there are significant constraints on the energy distribution and isotropy of the radiating particles.