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 the 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 $u_s\sim 3\times {10}^{3}km{s}^{-1}$ and sonic Mach number $M_s\sim 3$. These shocks produce curved radio spectra that steepen gradually over (0.1–10)${\nu }_{br}$ with a break frequency ${\nu }_{br}\sim 1$ GHz 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 ${\gamma }_e\gtrsim {10}^4$.