The Relation between Gas Density and Velocity Power Spectra in Galaxy Clusters: Qualitative Treatment and Cosmological Simulations

We address the problem of evaluating the power spectrum of the velocity field of the intracluster medium using only information on the plasma density fluctuations, which can be measured today by Chandra and XMM-Newton observatories. We argue that for relaxed clusters there is a linear relation between the rms density and velocity fluctuations across a range of scales, from the largest ones, where motions are dominated by buoyancy, down to small, turbulent scales: (\delta \rho _k/\rho)^2 = \eta _1^2 (V_{1,k}/c_s)^2, where δρ k /ρ is the spectral amplitude of the density perturbations at wavenumber k, V_{1,k}^2=V_k^2/3 is the mean square component of the velocity field, cs is the sound speed, and η1 is a dimensionless constant of the order of unity. Using cosmological simulations of relaxed galaxy clusters, we calibrate this relation and find η1 ≈ 1 ± 0.3. We argue that this value is set at large scales by buoyancy physics, while at small scales the density and velocity power spectra are proportional because the former are a passive scalar advected by the latter. This opens an interesting possibility to use gas density power spectra as a proxy for the velocity power spectra in relaxed clusters across a wide range of scales.