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 has so far been described by four power-law segments divided by three break frequencies nusa<nuc<num. This is valid for a homogeneous electron distribution. However, hot electrons are located right after the shock, while most electrons are farther downstream and have cooled. This spatial distribution changes the optically thick part of the spectrum, introducing a new break frequency, nuac<nusa, and a new spectral slope, Fnu~nu11&solm0;8 for nuac<nu<nusa. The familiar Fnu~nu2 holds only for nu<nuac. This ordering of the break frequencies is relevant for typical gamma-ray burst (GRB) afterglows in an interstellar medium environment. Other possibilities arise for internal shocks or afterglows in dense circumstellar winds. We discuss the possible implications of this spectrum for GRBs and their afterglows in the context of the internal-external shock model. Observations of Fnu~nu11&solm0;8 would enable us to probe scales much smaller than the typical size of the system and to constrain the amount of turbulent mixing behind the shock.