On thermalization in gamma-ray burst jets and the peak energies of photospheric spectra

The low-energy spectral slopes of the prompt emission of most gamma-ray bursts (GRBs) are difficult to reconcile with radiatively efficient optically thin emission models irrespective of the radiation mechanism. An alternative is to ascribe the radiation around the spectral peak to a thermalization process occurring well inside the Thomson photosphere. This quasi-thermal spectrum can evolve into the observed non-thermal shape by additional energy release at moderate to small Thomson optical depths, which can readily give rise to the hard spectral tail. The position of the spectral peak is determined by the temperature and Lorentz factor of the flow in the thermalization zone, where the total number of photons carried by the jet is established. To reach thermalization, dissipation alone is not sufficient and photon generation requires an efficient emission/absorption process in addition to scattering. We perform a systematic study of all relevant photon production mechanisms searching for possible conditions in which thermalization can take place. We find that a significant fraction of the available energy should be dissipated at intermediate radii, ≳10¹⁰ to a few ×10¹¹ cm, and the flow there should be relatively slow: the bulk Lorentz factor could not exceed a few tens for all but the most luminous bursts with the highest E _pk values. The least restrictive constraint for successful thermalization, Γ ≲ 20, is obtained if synchrotron emission acts as the photon source. This requires, however, a non-thermal acceleration deep below the Thomson photosphere transferring a significant fraction of the flow energy to relativistic electrons with Lorentz factors between 10 and 100. Other processes require bulk flow Lorentz factors of order of a few for typical bursts. We examine the implications of these results to different GRB photospheric emission models.