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

2.2 Triaxial Models without Figure Rotation

2.2.1 EXISTENCE Triaxial potentials generally admit only one exact isolating integral, the orbital energy E. Although there are three planes of reflection symmetry, there are no symmetry axes, and no component of the angular momentum vector is conserved. In a pioneering study, Schwarzschild (305, 306) showed by numerical orbit calculations that in triaxial potentials relevant for elliptical galaxies most stellar orbits possess two effective integrals, I₂ and I₃, in addition to the energy. The fraction of irregular orbits is small (145). As a result, most orbits belong to one of only a few major families: box orbits, short-axis tube orbits, and long-axis tube orbits (80, 198). Tube orbits around the intermediate axis are unstable (153). The long-axis tubes come in two varieties, bringing the total number of major orbit families to four. Schwarzschild showed for a specific triaxial mass model - first with a stationary figure, and subsequently with a tumbling figure - that it is possible to combine the individual orbital densities in the associated gravitational potential so that they reproduce the original mass model. This is equivalent to finding a distribution function f that is consistent with the mass model (348), and hence Schwarzschild's work demonstrated that self-consistent triaxial galaxy models exist, with and without figure rotation. Similar conclusions were reached on the basis of N-body simulations (2, 248, 250, 251, 371, Section 2.7).

2.2.2 NON-UNIQUENESS The motion of a star on a box orbit in a non-rotating triaxial potential is a combination of oscillations in the three principal directions, so the orbit-averaged angular momentum vanishes. Stars on tube orbits have a definite sense of rotation around either the long axis or the short axis of the model. Clockwise and counterclockwise motion may occur in the same tube orbit. Because the fraction of direct versus retrograde stars may be chosen freely, the total angular momentum vector of a stationary triaxial model may be misaligned from the principal axes of the system: it may point anywhere in the plane containing both the long and the short axis. We shall see in Section 3.4 that there is evidence for such a misalignment in the kinematics of elliptical galaxies.

It is also possible that different combinations of orbits with truly distinct shapes produce the same triaxial density distribution. Thus, there is a large degree of non-uniqueness in the distribution functions consistent with a given three-dimensional mass model. The purpose of the recent work on triaxial models is to explore this freedom in model building and to construct large sets of models which can be compared to observations. The main questions are: what are the permitted intrinsic shapes, figure rotation rates, and streaming velocities, and what constraints on the structure of elliptical galaxies can be deduced from detailed observations? We are still far from answering these questions satisfactorily, but many of the necessary tools have been developed. Specifically, much can be learnt from a study of special models for which sufficient simplification occurs so that whole families of them can be studied. Two useful classes of such models are known. These are the separable or Stäckel models, and the scale-free models. We discuss each of these in turn.