Cluster luminosity function and n th ranked magnitude as a distance indicator ⋆ ⋆ ⋆

We define here a standard candle to determine the distance of clusters of galaxies and to investigate their peculiar velocities by using the n rank galaxy (magnitude mn). We address the question of the universality of the luminosity function for a sample of 28 rich clusters of galaxies (cz≃20000 km.s) in order to model the influence on mn of cluster richness. This luminosity function is found to be universal and the fit of a Schechter profile gives α=-1.50±0.11 and Mbj∗=-19.91 ±0.21 in the range [-21,-17]. The uncorrected distance indicator mn is more efficient for the first ranks n. With n=5, we have a dispersion of 0.61 magnitude for the (mn,5log(cz)) relation. When we correct for the richness effect and subtract the background galaxies we reduce the uncertainty to 0.21 magnitude with n=15. Simulations show that a large part of this dispersion originates from the intrinsic scatter of the standard candle itself. These provide upper bounds on the amplitude σv of cluster radial peculiar motions. At a confidence level of 90 %, the dispersion is 0.13 magnitude and σv is limited to 1200 km.s −1 for our sample of clusters.