We have calculated synchrotron spectra of relativistic blast waves , and find predicted characteristic frequencies that are more than an order of magnitude different from previous calculations .
For the case of an adiabatically expanding blast wave , which is applicable to observed gamma-ray burst ( GRB ) afterglows at late times , we give expressions to infer the physical properties of the afterglow from the measured spectral features .

We show that enough data exist for GRB 970508 to compute unambiguously the ambient density , n = 0.03 \mbox { \thinspace cm$ { } ^ { -3 } $ } , and the blast wave energy per unit solid angle , { \cal E } = 3 \times 10 ^ { 52 } \mbox { \thinspace erg$$ } / 4 \pi \mbox { \thinspace sr$$ } .
We also compute the energy density in electrons and magnetic field .
We find that they are 12 % and 9 % , respectively , of the nucleon energy density and thus confirm for the first time that both are close to but below equipartition .

For GRB 971214 , we discuss the break found in its spectrum by Ramaprakash et al .
( 1998 ) .
It can be interpreted either as the peak frequency or as the cooling frequency ; both interpretations have some problems , but on balance the break is more likely to be the cooling frequency .
Even when we assume this , our ignorance of the self-absorption frequency and presence or absence of beaming make it impossible to constrain the physical parameters of GRB 971214 very well .