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

2.4 Scale-free Models

The second set of special models that has received considerable attention in the last decade are those with density profiles that are single power-laws, and have scale-free dynamical properties. This simplifies the construction of self-consistent equilibria, and is the main reason for the popularity of these models. Most studies have considered oblate and (non-rotating) triaxial models with 1 / r², which have a flat rotation curve, and can be thought of as deformed singular isothermal spheres (37, 91, 249, 253, 287, 333).

Motion in the scale-free potentials is not separable. Numerical orbit calculations reveal that generally nearly all orbits are regular (288). The orbital structure of non-rotating triaxial models of this kind differs fundamentally from that of models with homogeneous cores, which include all the separable systems. The three families of tube orbits still exist, but the box orbits are replaced by a multitude of minor orbit families. These are associated with higher-order resonances between the oscillation frequencies along and perpendicular to the principal axes (140), and have been christened boxlets (252, 276). Although boxlets have a variety of shapes, it appears that in substantially flattened triaxial models they cannot reproduce the characteristics of very elongated box orbits that remain close to the long axis. These are needed in any self-consistent model, owing to the fact that all tube orbits are elongated opposite to the figure of the model. The (preliminary) conclusion, based on inspection of orbits rather than on the actual construction of models, is that whereas exact oblate self-consistent scale-free models do exist (213, 214, 287, 289), flattened triaxial equilibria of this kind may not (210). The ``scale-free'' triaxial models that were built with Schwarzschild's method used orbits calculated in a potential that was strongly softened in the center, which reinstated most box orbits. As a result, these models are approximate, and not truly scale-free (215).

The appearance of boxlets is not confined to scale-free models. Other triaxial models that have been studied usually have density profiles that fall off proportional to a power of 1 / r at large distances. Models which contain a central point mass have also received attention. Boxlets occur at large distances in many of these models, typically at 50-100 core radii (136, 140, 210, 276). On the basis of the various numerical experiments done to date, it appears that non-separable potentials which support boxlets need to be avoided when building self-consistent models (137, 307). One way of doing this is to let the potential become at least axisymmetric at large radii, if not spherical as in the case of the separable models (which do not have boxlets). It is important to supplement the investigations of the orbital structure in various potentials that support boxlets with attempts to construct equilibria. This will provide valuable information on the possible shapes of the outer parts of elliptical galaxies, and the halos of spirals.