The shape of spectral breaks in gamma-ray burst afterglows

Gamma-ray burst afterglows are well described by synchrotron emission from relativistic blast waves expanding into an external medium. The blast wave is believed to amplify the magnetic field and accelerate the electrons into a power-law distribution of energies promptly behind the shock. These electrons then cool both adiabatically and by emitting synchrotron and inverse Compton radiation. The resulting spectra are known to consist of several power-law segments, which smoothly join at certain break frequencies. Here, we give a complete description of all possible spectra under those assumptions and find that there are five possible regimes, depending on the ordering of the break frequencies. The flux density is calculated by integrating over all of the contributions to a given photon arrival time from all of the shocked region using the Blandford & McKee solution. This allows us to calculate more accurate expressions for the value of these break frequencies and describe the shape of the spectral breaks around them. This also provides the shape of breaks in the light curves caused by the passage of a break frequency through the observed band. These new, more exact, estimates are different from more simple calculations by typically a factor of a few, and they describe some new regimes that were previously ignored.