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Abstract
Radio relics are diffuse arc-like synchrotron radio sources found in the outskirts of galaxy clusters. It is suggested that merger shocks power such radio sources by accelerating charged particles in the intracluster medium (ICM) through diffusive shock acceleration (DSA). In a number of studies, the strength of the associated merger shocks (Mach number) is estimated through (i) the radio injection spectral indices with the assumption of simple DSA theory, or (ii) the density discontinuities at the merger shock front revealed in X-ray emissions. However, the estimated Mach number through radio observations tends to be larger than that through X-rays. It is discussed in several studies that DSA is not sufficient to accelerate particles with weak shock to required energies that correspond to the brightness of observed radio sources. Substitutional models consider the reviving and evolution of pre-existing relativistic particles during the compression caused by merger shocks. Such particles potentially reside in AGN-driven bubbles or radio lobes of radio galaxies. In this work, we use hydrodynamic simulations to explore how bubbles, when interacting with a low-Mach merger shock (M = 2 ∼ 3), can result in a non-uniform Mach number distributing along the shock front. We find that the Mach number near the centerline of the bubble can be larger than that at the edge by a factor of 2 when the bubble is spherical and has a density contrast of 100. We find such non-uniform Mach number distribution is subject to the density contrast and shape of the bubble. We show that the shock non-uniform distribution lasts for a considerably long time so that observations may capture such features. This effect biases the Mach number measurements in X-rays and makes the comparison of shock Mach numbers inferred from the X-ray and radio observations non-trivial.