The synchrotron spectrum of fast cooling electrons revisited

Abstract

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: Fν∝ν2, ν1/3, ν−1/2 and ν−p/2, divided by three break frequencies: νsa<νc<νm. 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. We show that this spatial distribution introduces a new break frequency, νac, in the optically thick regime: νac<νsa<νc<νm, and a new spectral slope: Fν∝ν11/8 for νac<ν<νsa, while the familiar Fν∝ν2 is obtained only for ν<νac. While the above ordering of the break frequencies is relevant for afterglows with typical parameters in an ISM environment, other possibilities may also be relevant for internal shocks or afterglows in dense circumstellar winds. We discuss possible applications for gamma-ray bursts (GRBs) and their afterglows, in the context of the internal-external shock model. This may explain spectral slopes steeper than 1/3 seen in the 1–10 keV range in some bursts, if νsa reaches the X-ray.