The Synchrotron Spectrum of Fast Cooling Electrons Revisited

Abstract

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 $u_{sa} < u_{c} < u_{m}$. 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, $u_{ac} < u < u_{sa}$. The familiar $F_{u }\propto u ^{2}$ holds only for $u