Electron acceleration 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 relatve 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 $\epsilon_e$ exists so far. Here we demonstrate that it inevitably follows from the Weibel shock theory that $\epsilon_e\simeq\sqrt{\epsilon_B}$. The GRB afteglow data fully agree with this theoretical prediction. Our result explains why the electrons are close to equipartition in GRBs. The $\epsilon_e-\epsilon_B$ relation can be used to reduce the number of free parameters in afterglow models.