The theory of synchrotron emission from supernova remnants
  The time-dependent nonlinear kinetic theory for cosmic ray (CR) acceleration in supernova remnants (SNRs) is applied studying the properties of the synchrotron emission from SNRs, in particular, the surface brightness-diameter ($\Sigma-D$) relation. Detailed numerical calculations are performed for the expected range of the relevant physical parameters, namely the ambient density and the supernova explosion energy. The magnetic field in SNRs is assumed to be significantly amplified by the efficiently accelerating nuclear CR component. Due to the growing number of accelerated CRs the expected SNR luminosity increases during the free expansion phase, reaches a peak value at the beginning of the Sedov phase and then decreases again, since in this stage the overall CR number remains nearly constant, whereas the effective magnetic field diminishes with time. The theoretically predicted brightness-diameter relation in the radio range in the Sedov phase is close to $\Sigma_\mathrm{R}\propto D^{-17/4}$. It fits the observational data in a very satisfactory way. The observed spread of $\Sigma_\mathrm{R}$ at a given SNR size $D$ is the result of the spread of supernova explosion energies and interstellar medium densities. 
E.G. Berezhko, H.J. V\"olk