Marginally fast cooling synchrotron models for prompt GRBs

Previous studies have considered synchrotron as the emission mechanism for prompt Gamma-Ray Bursts (GRBs). These works have shown that the electrons must cool on a timescale comparable to the dynamic time at the source in order to satisfy spectral constraints while maintaining high radiative efficiency. We focus on conditions where synchrotron cooling is balanced by a continuous source of heating, and in which these constraints are naturally satisfied. Assuming that a majority of the electrons in the emitting region are contributing to the observed peak, we find that the energy per electron has to be $E\gtrsim 20$ GeV and that the Lorentz factor of the emitting material has to be very large $10^3\lesssim \Gamma_{\rm em} \lesssim 10^4$, well in excess of the bulk Lorentz factor of the jet inferred from GRB afterglows. A number of independent constraints then indicate that the emitters must be moving relativistically, with $\Gamma'\approx 10$, relative to the bulk frame of the jet and that the jet must be highly magnetized upstream of the emission region, $\sigma_{\rm up}\gtrsim 30$. The emission radius is also strongly constrained in this model to $R\gtrsim 10^{16}$cm. These values are consistent with magnetic jet models where the dissipation is driven by magnetic reconnection that takes place far away from the base of the jet.