The radio to X-ray spectrum of GRB 970508 on May 21.0 UT

We have reconstructed the spectrum of the afterglow of GRB 970508, on May 21.0 UT (12.1 days after the GRB), on the basis of observations spanning the X-ray to radio range. The low-frequency power law index of the spectrum, $\alpha=0.44\pm0.07$ ($F_\nu\propto\nu^\alpha$), is in agreement with the expected value $\alpha=1/3$ for optically thin synchrotron radiation. The 1.4 GHz emission is self-absorbed. We infer constraints on the break frequecies $\nu_{c}$ and $\nu_{m}$ on May 21.0 UT from a spectral transition from $F_{\nu} \sim \nu^{-0.6}$ to $F_{\nu} \sim \nu^{-1.1}$ in the optical passband around 1.4 days. A model of an adiabatically expanding blast wave, emitting synchrotron radiation, in which a significant fraction of the electrons cool rapidly provides a successful and consistent description of the afterglow observations over nine decades in frequency, ranging in time from trigger until several months later.