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In Kang (2015) we calculated the acceleration of cosmic-ray electrons and theensuing radio synchrotron emission at weak spherical shocks that are expectedto form in the outskirts of galaxy clusters.There we demonstrated that, atdecelerating spherical shocks, the volume integrated spectra of both electronsand radiation deviate significantly from the test-particle power-laws predictedfor constant planar shocks, because the shock compression ratio and the flux ofinjected electrons decrease in time. In this study, we consider spherical blastwaves propagating into a constant density core surrounded by an isothermal halowith a decreasing density profile in order to explore how the deceleration rateof the shock speed affects the radio emission from accelerated electrons. Thesurface brightness profile and the volume-integrated radio spectrum of themodel shocks are calculated by assuming a ribbon-like shock surface on aspherical shell and the associated downstream region of relativistic electrons.If the postshock magnetic field strength is about 7 microgauss, at the shockage of ~50 Myr, the volume-integrated radio spectrum steepens gradually withthe spectral index from alpha_{inj} to alpha_{inj}+0.5 over 0.1-10 GHz, wherealpha_{inj} is the injection index at the shock positionexpected from thediffusive shock acceleration theory. Such gradual steepening could explain thecurved radio spectrum of the radio relic in cluster A2266, which wasinterpreted as a broken power-law by Trasatti et al. (2014), if the relic shockis young enough so that the break frequency falls in around 1 GHz.

Radio emission from weak spherical shocks in the outskirts of galaxy clusters