In Kang (2015) we calculated the acceleration of cosmic-ray electrons at weak spherical shocks that are expected to form in the cluster outskirts, and estimated the diffuse synchrotron radiation emitted by those electrons. There we demonstrated that, at decelerating spherical shocks, the volume integrated spectra of both electrons and radiation deviate significantly from the test-particle power-laws predicted for constant planar shocks, because the shock compression ratio and the flux of inject electrons decrease in time. In this study, we consider spherical blast waves propagating through a constant density core surrounded by an isothermal halo with  ∝ r−n in order to explore how the deceleration of the shock affects the radio emission from accelerated electrons. The surface brightness profile and the volumeintegrated radio spectrum of the model shocks are calculated by assuming a ribbon-like shock surface on a spherical shell and the associated downstream region of relativistic electrons. If the postshock magnetic field strength is about 0.7 or 7 μG, at the shock age of ∼ 50 Myr, the volume-integrated radio spectrum steepens gradually with the spectral index from inj to inj + 0.5 over 0.1–10 GHz, where inj is the injection index at the shock position expected from the diffusive shock acceleration theory. Such gradual steepening could explain the curved radio spectrum of the radio relic in cluster A2266, which was interpreted as a broken power-law by Trasatti et al. (2015), if the relic shock is young enough so that the break frequency is around 1 GHz.

Abstract
1. INTRODUCTION
2. NUMERICAL CALCULATIONS
    2.1. Basic Equations
    2.2. Simulation Set-up
    2.3. Synchrotron Radiation
3. DSA SIMULATION RESULTS
    3.1. Shocks with DifferentMagnetic Field Profiles
    3.2. Shocks with Different Background Density Profile
4. SUMMARY
REFERENCES

• Hyesung Kang(Department of Earth Sciences, Pusan National University) Corresponding author