A scaling relation between the SZ decrement and the Thomson depth in clusters of galaxies

In a recent study of dark mater N-body simulations, a scaling relation between the SZ decrement and the Thomson depth of a cluster of galaxies of the form ΔT _r ∝τ _T ² has been found (Diaferio et al. 2000). In this paper, it will be shown that such a scaling relation arises if the intracluster gas is distributed similar to the dark matter density described by the NFW-profile and the finite spatial resolution of the numerical simulation is taken into account. It is furthermore investigated whether the ΔT _r ∝τ _T ² relation holds for analytical models of an isothermal gas sphere in the gravitational potential of a dark matter halo distributed according to the NFW-profile, the available experimental data of SZE observations, and recent results from cosmological gas-dynamical simulations of clusters of galaxies. Combining such a relation with temperature estimates from X-ray observations would provide information about a dependence of T _e on τ_T. The Thomson depth might therefore emerge as another important scaling parameter in studies of clusters of galaxies.