Aims: We present a study of the statistical properties of three velocity dispersion and mass estimators: biweight, gapper, and standard deviation for a small number of galaxies (Ngal ≤ 75).
Methods: Using a set of 73 numerically simulated galaxy clusters, we first characterised the statistical bias and the variance for each one of the three estimators (biweight, gapper, and standard deviation) in the determination of the velocity dispersion and the dynamical mass of the clusters through the σ-M relation. These results were used to define a new set of unbiased estimators that are able to correct for these statistical biases with a minimum increase in associated variance. We also used the same set of numerical simulations to characterise two other physical biases that affect the estimates: the effect of velocity segregation on the selection of cluster members, and the effect of using cluster members within different physical radii from the cluster centre.
Results: The standard deviation (and its unbiased counterpart) is the estimator with the lowest variance estimator after the biweight and gapper. The effect of velocity segregation in the selection of galaxies within the sub-sample of the most massive galaxies in the cluster introduces a bias of 2% in the velocity dispersion estimate when it is calculated using a quarter of the most massive cluster members. We also find a dependence of the velocity dispersion estimate on the aperture radius as a fraction of R200. This is consistent with previous results in the literature.
Conclusions: The proposed set of unbiased estimators effectively provides a correction of the velocity dispersion and mass estimates from the statistical and physical effects discussed above for small numbers of cluster members. When these new estimators are applied to a subset of simulated observations, they can retrieve bias-corrected values for the mean velocity dispersion and the mean mass; the standard deviation has the lowest variance. Although for a single galaxy cluster the statistical and physical effects discussed here are comparable to or slightly smaller than the bias introduced by interlopers, they are relevant when ensemble properties and scaling relations for large number of clusters are studied.