The relation between velocity dispersion and temperature in clusters: Limiting the velocity bias

The observed galaxy velocity dispersion in clusters of galaxies is compared with the X-ray temperature of the intracluster gas for a large sample of clusters. A strong correlation is found between the observables which is best fitted by the relation σ = 332 ± 52(kT)^0.6±0.1 km s^-1; here σ is the rms radial velocity dispersion in the cluster, and kT is the gas temperature in keV. The relation is consistent with that expected from an isothermal model where both the gas and the galaxies are in hydrostatic equilibrium with the binding cluster potential, i.e., σ ∝ T^0.5. The observed relation is used to determine the best-fit average β parameter for clusters, β = σ²/(kT/μm_p), which describes the ratio of energy per unit mass in the galaxies to that in the gas (where μm_p is the mean particle mass); we find β= 0.94 + 0.08 [and β(median) = 0.98]. This implies that the galaxies and the gas trace the same cluster potential with approximately equal energy per unit mass (i.e., β = 1). The best-fit observed β also suggests that no significant velocity bias exists in clusters of galaxies, i.e., the galaxy velocity dispersion in clusters is a fair tracer of the dark matter velocity dispersion. The best-fit velocity bias is b_υ = σ_gal/σ_DM ≃ β^0.5 = 0.97 ± 0.04 (with b_υ = 1 corresponding to no bias).