  In Kang (2015) we calculated the acceleration of cosmic-ray electrons and the
ensuing radio synchrotron emission at weak spherical shocks that are expected
to form in the outskirts of galaxy clusters.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
injected electrons decrease in time. In this study, we consider spherical blast
waves propagating into a constant density core surrounded by an isothermal halo
with a decreasing density profile in order to explore how the deceleration rate
of the shock speed affects the radio emission from accelerated electrons. The
surface brightness profile and the volume-integrated 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 7 microgauss, at the shock
age of ~50 Myr, the volume-integrated radio spectrum steepens gradually with
the spectral index from alpha_{inj} to alpha_{inj}+0.5 over 0.1-10 GHz, where
alpha_{inj} is the injection index at the shock positionexpected 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. (2014), if the relic shock
is young enough so that the break frequency falls in around 1 GHz.
