Although many of the observed properties of giant radio relics detected in the outskirts of galaxy clusters can be explained by relativistic electrons accelerated at merger-driven shocks , significant puzzles remain .
In the case of the so-called Toothbrush relic , the shock Mach number estimated from X-ray observations ( M _ { X } \approx 1.2 - 1.5 ) is substantially weaker than that inferred from the radio spectral index ( M _ { rad } \approx 2.8 ) .
Toward understanding such a discrepancy , we here consider the following diffusive shock acceleration ( DSA ) models : ( 1 ) weak-shock models with M _ { s } \la 2 and a preexisting population of cosmic-ray electrons ( CRe ) with a flat energy spectrum , and ( 2 ) strong-shock models with M _ { s } \approx 3 and either shock-generated suprathermal electrons or preexisting fossil CRe .
We calculate the synchrotron emission from the accelerated CRe , following the time evolution of the electron DSA , and subsequent radiative cooling and postshock turbulent acceleration ( TA ) .
We find that both models could reproduce reasonably well the observed integrated radio spectrum of the Toothbrush relic , but the observed broad transverse profile requires the stochastic acceleration by downstream turbulence , which we label “ turbulent acceleration ” or TA to distinguish it from DSA .
Moreover , to account for the almost uniform radio spectral index profile along the length of the relic , the weak-shock models require a preshock region over 400 kpc with a uniform population of preexisting CRe with a high cutoff energy ( \ga 40 GeV ) .
Due to the short cooling time , it is challenging to explain the origin of such energetic electrons .
Therefore , we suggest the strong-shock models with low-energy seed CRe ( \la 150 MeV ) are preferred for the radio observations of this relic .