Annu. Rev. Astron. Astrophys. 1991. 29: 239-274 Copyright © 1991 by Annual Reviews. All rights reserved

2.3.2 SELF-CONSISTENT MODELS The separable potentials support no irregular orbits, and therefore Jeans' theorem is strictly valid (40), so that f = f (E, I₂, I₃). Direct calculation of distribution function(s) f (E, I₂, I₃) consistent with a given triaxial density (x, y, z) by solving the fundamental integral equation - which gives as a triple integral of f over the velocities - is a rather intimidating task (49, 74, 75, 110). The individual orbit densities in a separable model are known explicitly (80), however, so that building self-consistent models by means of Schwarzschild's method is straightforward. Statler (320) used this approach, and constructed a large variety of different equilibrium models for a set of 21 triaxial separable models, all with the same density profile and with axial ratios covering all possible shapes. He found that the presence of four major orbit families, each of which can contribute density at any point, provides ample opportunity for exchanging orbits of different shapes while keeping the model density the same. As a result, the distribution functions for self-consistent separable triaxial models are highly non-unique (cf Section 2.2.2), and this is reflected in the variety of kinematic properties displayed by Statler's models: The mean streaming motions range up to values that are comparable to those found for the fastest rotating ellipticals (Section 3.2), showing that ample mean streaming (``rotation'') can occur in triaxial systems without figure rotation. Models with a large fraction of stars on box orbits show differences between the velocity dispersion profiles along the major and minor axes. Detailed kinematic observations may therefore help constrain the distribution functions of elliptical galaxies.