2.9.3. Ellipsoidal clusters
The distribution in equation (2.33) is isotropic in velocity space and spherically symmetric in real space. However, many of the most regular clusters have observed galaxy distributions that are not symmetric on the sky (Section 2.7). In principle, this asymmetry could be due to rotation; however, the velocity fields in clusters show no evidence for dynamically significant rotation (Section 2.6).
Numerical N-body simulations of the formation of clusters show that if the initial distribution of galaxies is aspherical, the final distribution after violent relaxation will be aspherical (Figure 5d; Aarseth and Binney, 1978). The most general final configurations have compact cores and are regular, but the surfaces of constant density are basically triaxial ellipsoids rather than spheres. The triaxial shape is maintained by an ellipsoidal Gaussian velocity distribution p(v) exp[-(2ij)-1 vivj], where is the velocity dispersion tensor. The principal axes of and the spatial figure of the galaxy are parallel and nearly proportional to one another (Binney, 1977), as expected from the tensor virial theorem (Chandrasekhar, 1968). Thus the velocity dispersion is largest parallel to the longest diameter of the cluster.