Phase-space shapes of clusters and rich groups of galaxies
  Clusters and groups of galaxies are highly aspherical, with shapes
approximated by nearly prolate ellipsoids of revolution. An equally fundamental
property is the shape of these objects in velocity space which is the
anisotropy of the global velocity dispersion tensor. Here we make use of
kinematical data comprising around 600 nearby clusters and rich groups of
galaxies from the SDSS to place constraints on the phase-space shapes of these
objects, i.e. their shapes in both position and velocity space. We show that
the line of sight velocity dispersion normalised by a mass dependent velocity
scale correlates with the apparent elongation, with circular (elongated)
clusters exhibiting an excessive (decremental) normalised velocity dispersion.
This correlation holds for dynamically young or old clusters and, therefore, it
originates from projecting their intrinsic phase-space shapes rather than from
dynamical evolution. It signifies that clusters are preferentially prolate not
only in position space, but also in velocity space. The distribution of the
axial ratios in position space is found to be well approximated by a Gaussian
with a mean 0.66+/-0.01 and a dispersion 0.07+/-0.008. The velocity ellipsoids
representing the shapes in velocity space are more spherical, with a mean axial
ratio of 0.78+/-0.03. This finding has important implications for mass
measurements based on the line of sight velocity dispersion profiles in
individual clusters. For typical axial ratios of the velocity ellipsoids in the
analysed cluster sample, systematic errors on the mass estimates inferred from
the line of sight velocity dispersions become comparable to statistical
uncertainties for galaxy clusters with as few as 40 spectroscopic redshifts.
