Low Frequency Spectra of Gamma-Ray Bursts

Particles with energies below the mean energy $E_0$ in relativistic shocked plasmas should assume an equilibrium energy distribution. This leads to a synchrotron spectrum $F_\nu \propto \nu^{1/3}$ up to approximately the critical frequency $\nu_0$ of an electron with the energy $E_0$. Application to GRBs implies that a burst with $10^{-5}$ erg/cm$^2$s of soft gamma-rays and $h \nu_0 = 300$ KeV should be about 18th magnitude in visible light and a few $\mu$Jy at 1 GHz (less if self-absorbed).