2.3.1 ORBITAL STRUCTURE
The HamiltonJacobi equation separates
in ellipsoidal coordinates
(173)
for the general Stäckel potential.
The corresponding mass models have a nonrotating triaxial shape, with an
arbitrary shortaxis density profile, and arbitrary central axial ratios
(81,
90,
197,
198).
On projection,
the ellipticity of the isophotes generally changes with radius, but they show
no twisting
(121).
All relevant models have cores with nonsingular
density profiles. The integrals I_{2} and
I_{3} are related to the angular
momentum integrals of the axisymmetric and spherical limits
(89).
The threedimensional orbital motion is a combination
of three onedimensional motions, each of which is either a libration or an
oscillation in one of the three ellipsoidal coordinates. The orbits can be
divided into four families: boxes, shortaxis tubes, and two families of
longaxis tubes
(80).
These are precisely the four major orbit
families found in Schwarzschild's nonrotating triaxial model. Illustrations
of the four orbital shapes have been presented by Statler
(48,
320).
The orbital structure simplifies in limiting cases with more
symmetry (Figure 1). Prolate Stäckel models
support only inner and outer
longaxis tubes, whereas all orbits are shortaxis
tubes in oblate separable models.

Figure 1. Ellipsoid Land: the plane of all
possible axial ratios b / a and c / a for triaxial
ellipsoids. The limiting cases with more symmetry are indicated. The
dashed lines are curves of constant triaxiality T (see
Section 3.4). Oblate spheroids have
T = 0, prolate spheroids have T = 1.
