1 The physical model

We consider emission from the whole volume behind an adiabatic highly relativistic spherical blast wave expanding into a cold and uniform medium. The hydrodynamics is described by the Blandford-McKee (1976 denoted BM hereafter) self similar solution. For typical parameters, the evolution becomes adiabatic fairly early, about an hour after the initial burst (Sari et al. 1998; Granot et al. 1999 and 1998, hereafter GPSa and GPSb, respectively). The BM solution is valid from this time, and as long as $\gamma\ \hbox{\rlap{$^\gt$}$_\sim$}\ 2$ (Kobayashi et al. 1998), typically a few months after the burst.

We assume that $\nu_{\rm a} \ll \nu_{\rm m}$, where $\nu_{\rm m}$ is the peak frequency and $\nu_{\rm a}$ is the self absorption frequency, which is reasonable for the first few months. The dominant radiation emission mechanism is assumed to be synchrotron radiation, while Compton scattering and electron cooling are ignored. We denote quantities measured in the local rest frame of the matter with a prime, while quantities without a prime are measured in the observer frame.

We assume that the energy of the electrons is everywhere a constant fraction of the internal energy: $e'_{\rm el}=\epsilon_{\rm e} e'$, and consider a power law electron distribution: $N(\gamma_{\rm e})\propto\gamma_{\rm e}^{-p}$for $\gamma_{\rm e} \ge \gamma_{\rm min}$. The magnetic field is also assumed to hold a constant fraction of the internal energy: $e'_{\rm B}=\epsilon_{\rm B} e'$, where $e_{\rm B}=B^2/8\pi$ is the energy density of the magnetic field. Alternative magnetic field models were considered in GPSa and GPSb, and we obtained that our results are not sensitive to the assumptions on the magnetic field.