Curved Radio Spectra of Weak Cluster Shocks

In order to understand certain observed features of arc-like giant radio relics such as the rareness, uniform surface brightness, and curved integrated spectra, we explore a diffusive shock acceleration (DSA) model for radio relics in which a spherical shock impinges on a magnetized cloud containing fossil relativistic electrons. Toward this end, we perform DSA simulations of spherical shocks with the parameters relevant for the Sausage radio relic in cluster CIZA J2242.8+5301, and calculate the ensuing radio synchrotron emission from re-accelerated electrons. Three types of fossil electron populations are considered: a delta-function like population with the shock injection momentum, a power-law distribution, and a power-law with an exponential cutoff. The surface brightness profile of radio-emitting postshock region and the volume-integrated radio spectrum are calculated and compared with observations. We find that the observed width of the Sausage relic can be explained reasonably well by shocks with speed $u_s \sim 3\times 10^3 \kms$ and sonic Mach number $M_s \sim 3$. These shocks produce curved radio spectra that steepen gradually over $(0.1-10) \nu_{\rm br}$ with break frequency $\nu_{\rm br}\sim 1$ GHz, if the duration of electron acceleration is $\sim 60 - 80$ Myr. However, the abrupt increase of spectral index above $\sim 1.5$ GHz observed in the Sausage relic seems to indicate that additional physical processes, other than radiative losses, operate for electrons with $\gamma_e \gtrsim 10^4$.