Steady-state conduction-driven temperature profile in clusters of galaxies

The temperature profile (TP) of the intracluster medium (ICM) is of primeval importance for deriving the dynamical parameters of the largest equilibrium systems known in the universe, in particular their total mass profile. Analytical models of the ICM often assume that the ICM is isothermal or parametrize the TP with a polytropic index γp. This parameter is ajusted to observations, but has in fact poor physical meaning for values other than 1 or 5/3, when considering monoatomic gases. In this article, I present a theoretical model of a relaxed cluster where the TP is instead structured by electronic thermal conduction. Neglecting cooling and heating terms, the stationnary energy conservation equation reduces to a second order differential equation, whose resolution requires two boundary conditions, taken here as the inner radius and the ratio between inner and outer temperature. Once these two constants are chosen, the TP has a fixed analytical expression, which reproduces nicely the observed “universal” TP obtained by Markevitch et al. (1998) from ASCA data. Using observed X-ray surface brightnesses for two hot clusters with spatially resolved TP, the local polytropic index and the hot gas fraction profile are predicted and compare very well with ASCA observations (Markevitch et al 1999). Moreover, the total density profile derived from observed X-ray surface brightness, hydrostatic equilibrium and the conduction-driven TP is very well fit by three analytical profiles found to describe the structure of galactic or cluster halos in numerical simulations of collisionless matter (Hernquist, 1991; Navarro et al. 1995, 1997; Burkert 1995). The suppression of the heat conduction several orders of magnitude below the Spitzer rate is an important assumption of the cooling-flow models, in order to ensure the thermal instability ability to trigger further condensation and cooling of density perturbations, although no definitive theoretical picture of this reduction has yet been put forward. In consequence, electronic heat conduction has seldom been considered for the structure of the main volume of the cluster, outside the cooling flow radius. However, the physical situation outside the cooling flow differs widely from the one inside, the temperature gradient being much shallower, the magnetic field intensity much smaller (as shown by the Faraday rotation measures and predicted by Soker & Sarazin, 1990) and the cooling time higher than the mean age of the structure. Thus, it is not obvious that the mechanism reducing the heat flux in the cooling flow is as highly effective in the main body of a cluster. If the TP decline in clusters is confirmed by the new generation of X-ray telescopes (Chandra and XMM-Newton), this simple conduction-driven model of the cluster ICM equilibrium could give useful insights on the physical situation in this region and the predicted shape of the TP (related to the temperature dependance of the heat flux for a collisionally-ionised plasma) will be tested directly against observations.