Dark matter distribution in the Coma cluster from galaxy kinematics: breaking the mass-anisotropy degeneracy

We study velocity moments of elliptical galaxies in the Coma cluster using Jeans equations. The dark matter distribution in the cluster is modelled by a generalised formula based upon the results of cosmological N-body simulations. Its inner slope (cuspy or flat), concentration, and mass within the virial radius are kept as free parameters, as well as the velocity anisotropy, assumed independent of position. We show that the study of line-of-sight velocity dispersion alone does not allow to constrain the parameters. By a joint analysis of the observed profiles of velocity dispersion and kurtosis we are able to break the degeneracy between the mass distribution and velocity anisotropy. We determine the dark matter distribution at radial distances larger than 3% of the virial radius and we find that the galaxy orbits are close to isotropic. Due to limited resolution, different inner slopes are found to be consistent with the data and we observe a strong degeneracy between the inner slope $\alpha$ and concentration $c$: the best-fitting profiles have the two parameters related with $c=19 - 9.6 \alpha$. Our best-fitting NFW profile has concentration $c=9$, which is 50% higher than standard values found in cosmological simulations for objects of similar mass. The total mass within the virial radius of $2.9 h_{70}^{-1}$ Mpc is $1.4 \times 10^{15} h_{70}^{-1} M_{\sun}$ (with 30% accuracy), 85% of which is dark. At this distance from the cluster centre, the mass-to-light ratio in the blue band is $351 h_{70}$ solar units. The total mass within the virial radius leads to estimates of the density parameter of the Universe, assuming that clusters trace the mass-to-light ratio and baryonic fraction of the Universe, with $\Omega_0=0.29 \pm 0.1$.