Synchrotron Emission Modeling of Radio Relics in the Cluster Outskirts

Radio relics are diffuse radio sources found in the outskirts of galaxy clusters and they are thought to trace synchrotron-emitting relativistic electrons accelerated at shocks. We explore a diffusive shock acceleration (DSA) model for radio relics in which a spherical shock with the parameters relevant for the Sausage radio relic in cluster CIZA J2242.8+5301 impinges on a magnetized cloud containing fossil relativistic electrons. This model is expected to explain some observed characteristics of giant radio relics such as the relative rareness, uniform surface brightness along the length of thin arc-like radio structure, and spectral curvature in the integrated radio spectrum. We find that the observed surface brightness profile of the Sausage relic can be explained reasonably well by shocks with speed $u_s{\sim}3{\times}10^3km/s$ and sonic Mach number $M_s{\sim}3$. These shocks also produce curved radio spectra that steepen gradually over $(0.1-10){\nu}_{br}$ with a break frequency ${\nu}_{br}{\sim}1GHz$, if the duration of electron acceleration is ~60-80 Myr. However, the abrupt increase in the spectral index above ~1.5 GHz observed in the Sausage relic seems to indicate that additional physical processes, other than radiative losses, operate for electrons with the Lorentz factor, ${\gamma}_e$ > $10^4$.