Dependence of virial radius (R₂₀₀, left panel), virial velocity (V₂₀₀, center panel), and virial temperature (T₂₀₀, right panel) on halo mass (M₂₀₀) and cosmological redshift (z, see legends). Each panel shows results for four orders of magnitude in halo mass and across 13 Gyr of cosmic time (see legends). These panels illustrate how an estimated value of halo mass can be used to infer the physical and phase-space extent of groups and clusters. Additionally, these plots illustrate how various important halo properties may be inferred directly from halo mass estimates and a redshift measurement. The cosmological parameters used in this figure are as follows: Ω_M = 0.3, Ω_Λ = 0.7, H₀ = 70 [km/s Mpc⁻¹]. For the virial temperature, we additionally assume a mean particle mass of 0.6 m_p, which is correct for a primordial plasma. However, in reality this must be updated by constraints on the individual group or cluster metallically, especially at lower redshifts.