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

**2.1 Spherical and Axisymmetric Models**

A spherical potential admits four isolating integrals of motion: the orbital
energy *E* and the three components of the angular momentum vector
**L** = (*L*_{x}, *L*_{y},
*L*_{z}). One
usually considers models without any preferred axis, so
that *f* can depend only on |**L**|, i.e., *f = f (E,
L*^{2}). Such models have
anisotropic velocity distributions. It is possible to find the unique
*f = f (E)*
that generates a given spherical density (*r*) by means of Eddington's
inversion formula
(103).
If (*r*) falls off with radius
sufficiently rapidly, this *f (E)* is nowhere negative, and represents the
unique isotropic model. Anisotropic models can sometimes be found by similar
inversion methods
(70),
but they are usually constructed by
assumption of a special functional form for *f (E, L*^{2})
(e.g., 139,
236,
247,
323)
or by numerical techniques
(280,
290,
294).
Although few elliptical galaxies are spherical
(43,
303)
there is ample room for further work on the properties of anisotropic models.
In particular, much information is contained in the line profiles, i.e., the
distribution of stars as a function of line-of-sight radial velocity at
different projected distances from the center. These profiles can now be
measured (25,
122),
but so far few theoretical profiles have been calculated
(71).

Axisymmetric potentials admit two exact isolating integrals, the energy
*E* and one angular momentum component, *L*_{z}
say. Lynden-Bell
(219)
generalized Eddington's formula, and showed that for a given axisymmetric
density (*R,
z*) there is a unique distribution function *f (E,
L*_{z}^{2}). In
such two-integral models the velocity dispersion <
*v _{R}*

Orbits with *L*_{z} 0
have a definite sense of rotation around the symmetry
axis, but both clockwise and counterclockwise motion may occur. It follows
that non-zero mean streaming is possible around the symmetry axis of any
axisymmetric model, and that its magnitude is not fixed by the density, since
in each orbit the fraction of direct versus retrograde stars may be chosen
freely. In two-integral models < *v*_{} > is often
chosen such that the azimuthal dispersion < *v*_{}^{2} >
- < *v*_{}
>^{2} = < *v _{R}*

Most orbits in realistic axisymmetric potentials are tubes around the symmetry
axis, and possess an extra isolating integral of motion,
*I*_{3} (e.g.,
64,
226,
288).
The remaining phase-space is
generally made up of a host of minor orbit families and irregular orbits. The
latter do not have a third integral, and Binney has argued that for this
reason Jeans' theorem is not valid for such systems
(40,
275),
implying that no true equilibrium solutions may exist. In many
cases of interest the fraction of irregular orbits turns out to be small, and
they are nearly indistinguishable from regular orbits on time scales of the
order of a Hubble time. For practical purposes one may therefore probably
still use Jeans' theorem, and construct approximate equilibrium models. The
potentials relevant for elliptical galaxies are nearly spherical, and
*I*_{3} is
related closely to *L*^{2}
(172,
298).
This fact has been used to
construct approximate three-integral models with density profiles appropriate
for elliptical galaxies
(271,
272).
Models based on special
potentials which support an *exact* third integral are discussed in
Section 2.3.

Kinematic properties of dynamical models are often calculated by direct
solution of the Jeans equations which relate the velocity dispersions to the
density and the potential of the model
(48,
180).
Various
solutions for spherical systems have been used to model the kinematics of
elliptical galaxies, e.g., in connection with the search for nuclear black
holes
(45,
329,
Section 3.5.1). An
explicit solution of the Jeans
equations is available also for two-integral axisymmetric models
(163),
but anisotropic axisymmetric solutions cannot be derived unless certain
assumptions are made
(7,
9,
116,
Section 3.3.2).
The resulting dispersion profiles are not always guaranteed to
correspond to a distribution function *f* 0, however