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Radio relics are the manifestation of electrons presumably being shock(re-)accelerated to high energies in the outskirts of galaxy clusters. However,estimates of the shocks' strength yield different results when measured withradio or X-ray observations. In general, Mach numbers obtained from radioobservations are larger than the corresponding X-ray measurements. In thiswork, we investigate this Mach number discrepancy. For this purpose, we usedthe cosmological code ENZO to simulate a sample of galaxy clusters that hostbright radio relics. For each relic, we computed the radio Mach number from theintegrated radio spectrum and the X-ray Mach number from the X-ray surfacebrightness and temperature jumps. Our analysis suggests that the differences inthe Mach number estimates follow from the way in which different observablesare related to different parts of the underlying Mach number distribution:radio observations are more sensistive to the high Mach numbers present only ina small fraction of a shock's surface, while X-ray measurements reflect theaverage of the Mach number distribution. Moreover, X-ray measurements are verysensitive to the relic's orientation. If the same relic is observed fromdifferent sides, the measured X-ray Mach number varies significantly. On theother hand, the radio measurements are more robust, as they are unaffected bythe relic's orientation.

Exploring the spectral properties of radio relics I: Integrated spectral index and Mach number