A new calibration of the log D(n)-log sigma relation

The log D_n_-log σ relation combines a distance- dependent parameter (characteristic photometric diameter) with a distance-independent parameter (velocity dispersion of an elliptical galaxy. After calibration with groups of galaxies, this relation can be used for estimating relative distances of ellipticals. In this paper, we develop and apply a new calibration method that considerably reduces the scatter in this relation. For this purpose a new group finding algorithm is implemented and combined with a new optimization procedure with respect to a variable search radius R_s_: For each R_s_, we iteratively calibrate the relation and assign galaxies to groups - using the position estimated by the previous iteration - until the iteration converges. For each R_s_, we estimate the slope of the log D_n_-log σ relation and the uncertainty using a regression model appropriate for measurements with errors in both variables. The search volume is spherical and is an upper bound for the group size at a given search radius. By varying R_s_, we find a radius R_cali_ = 690 +/- 150 km s^-1^ which is optimal in the sense that the error is minimized. The value of the slope x of the relation at R_cali_ is X_cali_ = 0.98 +/- 0.06. We also study the dependence of the estimated average virial mass of groups of R_s_. In the region near the optimal search radius, its value agrees with observations and it attains a local maximum. The behavior of the average virial mass is a good diagnostic for the effectiveness of the optimization procedure and the group finding algorithm: it indicates that, near the optimal search radius, the group finding algorithm finds most of the groups which are held together gravitationally but does not yet contaminate them seriously, as occurs at larger search radii. The results may have important consequences for large-scale structure in cosmology, especially regarding applications to models of large-scale streaming velocities.