Shock Acceleration Model for the Toothbrush Radio Relic

Abstract

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 approximate to 1.2-1.5) is substantially weaker than that inferred from the radio spectral index (M-rad approximate to 2.8). Toward understanding such a discrepancy, we here consider the following diffusive shock acceleration (DSA) models: (1) weak-shock models with Ms. 2 and a preexisting population of cosmic-ray electrons (CRe) with a flat energy spectrum, and (2) strong-shock models with M-s approximate to 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 the 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 (greater than or similar to 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 (less than or similar to 150 MeV) are preferred for the radio observations of this relic.