Annu. Rev. Astron. Astrophys. 1991. 29:
239-274
Copyright © 1991 by . 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 /
*r*^{2}, 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.