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

2.6 Stability

An accurate delineation of the various instabilities that plague axisymmetric and triaxial equilibrium models, and the way in which these depend on the relative populations of the different orbit families, is of vital importance for our understanding of elliptical galaxies, because the requirement of stability will rule out many distribution functions. Not all viable galaxy models need to be dynamically stable, but if they are not, then the growth rates of the instabilities must be of the order of a Hubble time. This important area of research is still largely unexplored. Our ignorance is caused in part by the lack of available models, and in part by the fact that the required calculations are difficult. Most results for systems which are not spheres or disks have been obtained by means of N-body simulations (132, 239, 242, 278).

Antonov showed that most isotropic spherical galaxy models are stable (5). He also realized that anisotropic spheres which are built mainly with radial orbits are dynamically unstable to the formation of a bar, i.e., become triaxial (6, 279). This fact was rediscovered a decade later by N-body work (19), and stimulated a revival in this area of research (30, 76, 144, 228, 243, 265, 266, 268, 299, 349, 361, 362). The radial orbit instability is important for simulations of dissipationless collapse (Section 2.7), and also rules out certain models that have been fit to the photometric and kinematic observations of M87 (238, Section 3.5.1), illustrating that this type of instability might provide serious constraints on dynamical models for ellipticals. The radial orbit instability persists for nearly spherical triaxial models in which box orbits are heavily occupied (92), but disappears in elongated triaxial models with axial ratios of about 0.7, in agreement with N-body simulations. Schwarzschild's early triaxial models have axial ratios 1:0.625:0.5, contain a large fraction of box orbits, and are stable (314).

Smooth distribution functions which can be used as initial conditions for accurate N-body simulations are available for the thin orbit models (Section 2.3.3). Merritt and his collaborators have undertaken a thorough study of their stability. Oblate thin orbit models with an axial ratio smaller than 0.33 are unstable to formation of rings (93, 246), in agreement with the well-known result that cold disks are similarly unstable (267, 331). All oblate thin orbit models are unstable to lopsided deformations (246). This instability decreases in strength for more nearly spherical models, and can be removed by increasing the radial pressure, i.e., by replacing the thin short-axis tubes by tubes with non-zero radial thickness. Very elongated prolate thin orbit models appear to be unstable against bending perturbations (245); again, the transition to stability occurs at an axial ratio of about 0.3. No results are available yet for triaxial thin orbit models.

A number of instabilities appear to be responsible for limiting the shapes and internal velocity distributions of elliptical galaxies, in particular for the more flattened models (212). This is consistent with the hypothesis that the observed absence of elliptical galaxies flatter than E6 (43, 303) is due to dynamical instabilities (e.g., 132). Strongly flattened triaxial potentials generally support a substantial fraction of stochastic orbits, so that their orbital structure is more complicated than that for the separable systems (79, 135, 339). It is therefore not surprising that equilibrium models of this shape are unstable when perturbed. Hence, extend the stability studies to nonseparable models is important.