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^{th} rank galaxy (magnitude m$_n$). We address the question of the universality of the luminosity function for a sample of 28 rich clusters of galaxies ($cz \simeq 20000 km/s$) in order to model the influence on $m_n$ of cluster richness. This luminosity function is found to be universal and the fit of a Schechter profile gives $\alpha = -1.50 \pm 0.11$ and $M_{bj}* = -19.91 \pm 0.21$ in the range [-21,-17]. The uncorrected distance indicator $m_n$ is more efficient for the first ranks n. With n=5, we have a dispersion of 0.61 magnitude for the (m$_n$,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 $\sigma_v$ of cluster radial peculiar motions. At a confidence level of 90%, the dispersion is 0.13 magnitude and $\sigma_v$ is limited to 1200 km/s for our sample of clusters.