Gravitational lensing by a cluster of galaxies and the central cD galaxy: Measuring the mass profile

We study the constraints on the density profile of the mass in a cluster of galaxies that can be obtained when a radial arc produced by gravitational lensing is observed. We apply this to the cluster MS 2137 - 23, which contains a radial arc, a tangential arc, and three other arclets. As shown by Mellier and coworkers, the positions and elongations of all these five images are well reproduced as arising from two different background galaxies, if the mass distribution in the cluster is elliptical, is exactly centered on the brightest cluster galaxy, and has the same ellipticity and position angle as this galaxy. The galaxy is therefore identified as the center of the cluster potential well. We show that the ratio θᵣ_/θᵢ_ of the distance from the radial arc to the center of the cluster to the distance from the other image of the same source to the center gives a powerful constraint on the form of the dark matter density profile. Models that are favored have well-defined core radii of ~30 h^-1^ kpc, with the slope changing rapidly from almost flat to roughly isothermal around this radius. Density that remain steeper than r^-1^ all the way to the center are probably ruled out, since they predict θᵣ_/θᵢ_ to be smaller than observed. We also show that θᵣ_/θᵢ_ is significantly affected by the mass in stars in the central galaxy. The stellar mass tends to move the radial arc closer to the center, since its profile is very steep. This implies an upper limit to the mass-to-light ratio of the stars, similar to the dynamical measurements of the mass-to- light ratio of elliptical galaxies. The constraints on the profile will be improved once the position of the radial arc is measured more accurately and the arc redshifts are determined. All the models for the density profile that are consistent with the observed position of the radial arc predict a velocity dispersion for the cluster galaxies of ~1200 km s^-1^ and a central velocity dispersion for the stars of ~350 km s^-1^, consistent with the observed velocity dispersions in other cD galaxies. The models also predict that the velocity dispersion of the cD halo should rise rapidly with radius and that the two sources producing the five arcs should be at similar redshifts. This can be tested by future observations.