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Gamma-ray burst spectra from continuously accelerated electrons

We discuss here constraints on the particle acceleration models from the observed gamma-ray bursts spectra. The standard synchrotron shock model assumes that some fraction of available energy is given instantaneously to the electrons which are injected at high Lorentz factor. The emitted spectrum in that case corresponds to the spectrum of cooling electrons, F_ν~ ν^{-1/2}, is much too soft to account for the majority of the observed spectral slopes. We show that continuous heating of electrons over the life-time of a source is needed to produce hard observed spectra. In this model, a prominent peak develops in the electron distribution at energy which is a strong function of Thomson optical depth τ_T of heated electrons (pairs). At τ_T>1, a typical electron Lorentz factor γ~ 1-2 and quasi-thermal Comptonization operates. It produces spectrum peaking at a too high energy. Optical depths below 10^{-4} would be difficult to imagine in any physical scenario. At τ_T =10^{-4}-10^{-2}, γ~ 30-100 and synchrotron self-Compton radiation is the main emission mechanism. The synchrotron peak should be observed at 10--100 eV, while the self-absorbed low-energy tail with F_ν~ ν^2 can produce the prompt optical emission (like in the case of GRB 990123). The first Compton scattering radiation by nearly monoenergetic electrons peaks in the BATSE energy band and can be as hard as F_ν~ ν^1 reproducing the hardness of most of the observed GRB spectra. The second Compton peak should be observed in the high-energy gamma-ray band, possibly being responsible for the 10-100 MeV emission detected in GRB 941017. A significant electron-positron pair production reduces the available energy per particle, moving spectral peaks to lower energies as the burst progresses.