The Synchrotron Spectrum of Fast Cooling Electrons Revisited

We discuss the spectrum arising from synchrotron emission by fast cooling (FC) electrons, when fresh electrons are continually accelerated by a strong blast wave, into a power law distribution of energies. The FC 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 farther down stream and have cooled. This spatial distribution changes the optically thick part of the spectrum, introducing a new break frequency, ${\nu }_{ac}<{\nu }_{sa}$, and a new spectral slope, $F_{\nu }\propto \nu^{11/8}$for$\nu_{ac} < \nu < \nu_{sa}$\mathrm{.}Thefamiliar$F_{\nu }\propto \nu ^{2}$holdsonlyfor$\nu < \nu_{ac}$. This ordering of the break frequencies is relevant for typical gamma-ray burst (GRB) afterglows 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 their afterglows, in the context of the internal-external shock model. Observations of $F_{\nu }\propto {\nu }^{11\mathrm{/}8}$ 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.