Phase-space distribution of accelerated electrons in relativistic GRB shocks*

The shock model of gamma-ray bursts (GRBs) contains two equipartition parameters: the magnetic energy density and the kinetic energy density of the electrons relative to the total energy density of the shock, $\epsilon_B$ and $\epsilon_e$, respectively. These are free parameters within the model. Whereas the Weibel shock theory and numerical simulations fix $\epsilon_B$ at the level of $\sim$few$\times(10^{-3}...10^{-4})$, no understanding of $\epsilon_e$ existed until recently. Medvedev (2006) has demonstrated that it inevitably follows from the Weibel shock theory that $\epsilon_e\simeq\sqrt{\epsilon_B}$, thus explaining why the electrons are close to equipartition in GRBs. The GRB afterglow data fully agree with this theoretical prediction. It has been suggested that the $\epsilon_e- \epsilon_B$ relation can be used to reduce the number of free parameters in afterglow models. Here we further develop the model of non-Fermi acceleration of electrons in prompt GRBs. We developed a numerical code, which computes full phase space distribution of electrons in Weibel electromagnetic fields. This distribution is further used to compute the electron energy distribution, the distribution over pitch-angle, the angular pattern of jitter emissivity, and so on. Relevance of the results to modeling of GRBs is discussed.