Indirect Measurements of Gas Velocities in Galaxy Clusters: Effects of Ellipticity and Cluster Dynamic State

While awaiting direct velocity measurement of gas motions in the hot intracluster medium, we rely on indirect probes, including gas perturbations in galaxy clusters. Using a sample of $\sim 80$ clusters in different dynamic states from Omega500 cosmological simulations, we examine scaling relations between the fluctuation amplitudes of gas density, $\delta\rho/\rho$, pressure, $\delta P/P$, X-ray surface brightness, Sunyaev-Zeldovich (SZ) y-parameter, and the characteristic Mach number of gas motions, $M_{\rm 1d}$. In relaxed clusters, accounting for halo ellipticities reduces $\delta\rho/\rho$ or $\delta P/P$ by a factor of up to 2 within $r_{500c}$. We confirm a strong linear correlation between $\delta\rho/\rho$ (or $\delta P/P$) and $M_{\rm 1d}$ in relaxed clusters, with the proportionality coefficient $\eta \approx 1$. For unrelaxed clusters, the correlation is less strong and has a larger $\eta\approx 1.3\pm 0.5$ ($1.5\pm0.5$) for $\delta\rho/\rho$ ($\delta P/P$). Examination of the power-law scaling of $M_{\rm 1d}$ with $\delta\rho/\rho$ shows that it is almost linear for relaxed clusters, while for the unrelaxed ones, it is closer to $\delta\rho/\rho\propto M_{\rm 1d}^2$, supporting an increasing role of non-linear terms and compressive modes. In agreement with previous studies, we observe a strong correlation of $M_{\rm 1d}$ with radius. Correcting for these correlations leaves a residual scatter in $M_{\rm 1d}$ of $\sim 4 (7)$ per cent for relaxed (perturbed) clusters. Hydrostatic mass bias correlates with $M_{\rm 1d}$ as strongly as with $\delta\rho/\rho$ in relaxed clusters. The residual scatters after correcting for derived trends is $\sim 6-7$ per cent. These predictions can be verified with existing X-ray and SZ observations of galaxy clusters combined with forthcoming velocity measurements with X-ray microcalorimeters.