Entropy of gas and dark matter in galaxy clusters

On the basis of a large-scale 'adiabatic', namely non-radiative and non-dissipative, cosmological smooth particle hydrodynamic simulation we compare the entropy profiles of the gas and the dark matter (DM) in galaxy clusters. We employ the quantity K-g = 3k(B)T(g)rho(-2/3)(g)/(mu m(p)) = sigma(2)(g)rho(-2/3)(g) as measure for the entropy of the intracluster gas. By analogy the DM entropy is defined as K-DM = sigma(2)(DM)rho(-2/3)(DM)(sigma(2)(DM) is the 3D velocity dispersion of the DM). The DM entropy is related to the DM phase-space density by K-DM proportional to Q(DM)(-2/3). In accord with other studies, the radial DM phase-space density profile follows a power-law behaviour, Q(DM) proportional to r(-1.82), which corresponds to K-DM proportional to r(1.21). The simulated intracluster gas has a flat entropy core within (0.8 +/- 0.4)R-s, where R-s is the Navarro-Frenk-White scale radius. The outer profile follows the DM behaviour, K-g proportional to r(1.21), in close agreement with X-ray observations. Upon scaling the DM and gas densities by their mean cosmological values, we find that outside the entropy core a constant ratio of K-g/K-DM = 0.71 +/- 0.18 prevails. By extending the definition of the gas temperature to include also the bulk kinetic energy the ratio of the DM and gas extended entropy is found to be unity for r greater than or similar to 0.8R(s). The constant ratio of the gas thermal entropy to that of the DM implies that observations of the intracluster gas can provide an almost direct probe of the DM.