The Synchrotron Spectrum of Fast Cooling Electrons Revisited

We discuss the spectrum arising from synchrotron emission by fast cooling electrons, when fresh electrons are continually accelerated by a strong blast wave, into a power law distribution of energies. The fast cooling spectrum was so far described by four power law segments divided by three break frequencies: $\nu_{sa} < \nu_c < \nu_m$. This is valid for a homogeneous electron distribution. However, hot electrons are located right after the shock, while most electrons are located farther down stream and have cooled down. This spatial distribution alters the observed spectrum in the optically thick regime, leading to a new break frequency, $\nu_{ac}$: $\nu_{ac} < \nu_{sa} < \nu_c < \nu_m$, and a new spectral slope: $F_{\nu} \propto \nu^{11/8}$ for $\nu_{ac} < \nu <\nu_{sa}$. The familiar: $F_{\nu} \propto \nu^2$ holds only for $\nu < \nu_{ac}$. This ordering of the break frequencies is relevant for GRB afterglows with typical parameters in an ISM environment. Other possibilities arise for internal shocks or afterglows in dense circumstellar winds. We discuss possible implications of this spectrum for GRBs and for their afterglow, in the context of the internal-external shock model. These self absorbed spectra might explain the appearance of spectral slopes larger than 1/3, in the 1-10 keV. More generally, observations of the new 11/8 spectral slope would enable us to probe scales much smaller than the typical size of the system, and constrain the amount of turbulent mixing behind the shock.