Supernova remnants in clumpy medium: A model of hydrodynamic and radio synchrotron evolution

We present an analytical model of $\Sigma-D$ relation for supernova remnants (SNRs) evolving in a clumpy medium. The model and its approximations were developed using the hydrodynamic simulations of SNRs in environments of low-density bubbles and clumpy media with different densities and volume-filling factors. For calculation of SNR luminosities we developed the synchrotron emission model, implying the test-particle approximation. The goal of this work is to explain the flattened part of $\Sigma-D$ relation for Galactic SNRs at $D\approx14-50$ pc. Our model shows that the shock collision with the clumpy medium initially enhances the brightness of individual SNRs, which is followed by a steeper fall of their $\Sigma-D$ curve. We used the analytical model to generate large SNR samples on $\Sigma-D$ plane, within a span of different densities and distances to clumpy medium, keeping the observed distribution of diameters. After comparison with the Galactic sample, we conclude that the observed $\Sigma-D$ flattening and scatter originates in sporadic emission jumps of individual SNRs while colliding with the dense clumps. Statistically, the significant impact of the clumps starts at diameters of $\approx14$ pc, up to $\sim70$ pc, with the average density jump at clumpy medium of $\sim2-20$ times, roughly depending on the low density of circumstellar region. However, additional analysis considering the selection effects is needed, as well as the improvement of the model, considering radiation losses and thermal conduction.