Morphology and Mach Number Distribution of Merger Shock Surfaces in Merging Galaxy Clusters

In a binary merger of two subclusters with comparable masses, a pair of merger shocks are typically generated, often manifesting as double radio relics. Using cosmological hydrodynamic simulations, we identify major merger events with mass ratio $\mathcal{M}_1/\mathcal{M}_2\lesssim4$ and impact parameter $b/r_{\rm vir,1}\lesssim1$, where $r_{\rm vir,1}$ is the virial radius of the larger subcluster. We analyze merger shock surfaces approximately 1 Gyr after the pericenter passage, focusing on their morphology and the distribution of the Mach number, $M_s$, of their constituent shock zones. The shock surfaces exhibit an elongated shape with a minor-to-major axis ratio of $\sim0.6-0.9$ and cover the area of $\sim5-20\%$ of the enclosed sphere. The area ratio of the two shock surfaces roughly scales with $\mathcal{M}_1/\mathcal{M}_2$, typically positioning the larger shock ahead of the smaller subcluster. The axis connecting the two subclusters generally does not pass through the centers of the shock surfaces, due to the nonzero impact parameter and the turbulent flows around them. The distribution of $M_s$ of shock zones on each surface can be approximated by a log-normal function, peaking at $M_{s,\rm{peak}}\approx2-4.5$ and extending up to $\sim10$. The surface-area-weighted and X-ray-emissivity-weighted average Mach numbers are comparable, with ${\langle{M_s}\rangle}_{\rm{area}}\approx2.3-4.4$ and ${\langle{M_s}\rangle}_{X}\approx2-4$. In contrast, the cosmic-ray-energy-flux-weighted average Mach numbers are higher with ${\langle{M_s}\rangle}_{\rm{CR}}\approx3-5$. This discrepancy aligns with the differences between Mach numbers derived from X-ray and radio observations of radio relic shocks. On the other hand, we find that mostly ${\langle{M_s}\rangle}_{X}\gtrsim2$ for simulated merger shocks, although shocks with $M_{\rm X-ray}\lesssim2$ are often reported in observations.