The general problem of modeling triaxial galaxies is illustrated
with a couple of special cases. In *separable* models the
allowed orbits are fairly simple, and the job of populating orbits so
as to produce the density distribution is well-understood. In
*scale-free* models the allowed orbits are more complex, and it
is not clear if such models can be in true equilibrium. If not,
*secular evolution* over many tens of dynamical times may play a
role in shaping elliptical galaxies.

Schwarzschild (1979) invented a powerful method for constructing
equilibrium models of galaxies *without explicit knowledge of the
integrals of motion*. To use this method,

- Specify the mass model
`rho(x)`and find the corresponding potential; - Construct a grid of
`K`cells in position space; - Chose initial conditions for a set of
`N`orbits, and for each one,- integrate the equations of motion for many orbital periods, and
- keep track of how much time the orbit spends in each cell, which is a measure of how much mass the orbit contributes to that cell;

- Determine non-negative weights for each orbit such that the
summed mass in each cell is equal to the mass implied by the
original
`rho(x)`.

The last step is the most difficult. Formally, let `P_i(c)`
be the mass contributed to cell `c` by orbit `i`; the
task is then to find `N` non-negative quantities `Q_i`
such that

-- N \ (1) M(c) = | Q_i P_i(c) / -- i = 1simultaneously for all cells, where

In general Eq. 1 has *many* solutions, reflecting the fact
that many *different* distribution functions are consistent with
a given mass model. Some methods allow one to specify additional
constraints so as to select solutions with special properties (maximum
rotation, radial anisotropy, *etc.*).

Very roughly speaking, a *separable* potential is one in which
the orbit of a star may be decomposed into three separate motions (or
two separate motions, in two dimensions). Motion in separable
potentials is tractable analytically, and exact expressions for the
integrals of motion are known. Separable potentials are rather
special, mathematically speaking; it is highly unlikely that real
galaxies have such potentials. However, numerical experiments
indicate that most triaxial density distributions *with cores*
generate potentials which share certain key features with separable
potentials.

The orbits in a separable potential may be classified into distinct
*families*. Each family is associated with a set of closed and
stable orbits. In two dimensions, for example, there are two types of
closed, stable orbits; one type (i) oscillates back and forth along
the major axis, and the other type (ii) loops around the center.
Because these orbits are stable, other orbits which start nearby will
remain nearby at later times. The families associated with types (i)
and (ii) are known as *box* and *loop* orbits, respectively
(see BT87, Chapter 3.3.1).

In three dimensions, a separable potential permits four distinct orbit families:

*box*orbits,*short-axis tube*orbits,*inner long-axis tube*orbits, and*outer long-axis tube*orbits.

The time-averaged angular momentum of a star on a box orbit vanishes; such orbits therefore do not contribute to the net rotation of the system. Short-axis and long-axis tube orbits, on the other hand, preserve a definite sense of rotation about their respective axes; consequently, their time-averaged angular momenta do not vanish. The total angular momentum vector of a non-rotating triaxial galaxy may lie anywhere in the plane containing the short and long axes (Levison 1987).

Using Schwarzschild's method, it is possible to numerically
determine distribution functions `f(E,I_2,I_3)` corresponding
to separable triaxial models (Statler 1987). A somewhat more
restricted set of models can be constructed *exactly*; these
models make use of all available box orbits, but only those tube
orbits with zero radial thickness (Hunter & de Zeeuw 1992). Apart
from the choice of streaming motion, thin tube models are
*unique*. One use of such models is to illustrate the effects of
streaming motion by giving all tube orbits the same sense of rotation;
the predicted velocity fields display a wide range of possibilities
(Arnold *et al.* 1994). Non-zero streaming on the
projected minor axis is a generic feature of such models; a number of
real galaxies exhibit such motions and thus *must* be triaxial.

In scale-free models all properties have either power-law or
logarithmic dependence on radius. In particular, scale-free models
with density profiles scaling as `r^-2` have logarithmic
potentials and flat rotation curves. While real galaxies are not
entirely scale-free, power-law density distributions are reasonable
approximations to the central regions of some elliptical galaxies and
to the halos of galaxies in general.

If the density falls as `r^-2` or faster, then all box
orbits are replaced by minor orbital families called *boxlets*
(Gerhard & Binney 1985, Miralda-Escude & Schwarzschild 1989).
Each boxlet family is associated with a closed and stable orbit
arising from a resonance between the motions in the `x` and
`y` directions.

Moreover, some scale-free potentials have *irregular* orbits;
these have no integrals of motion apart from the energy `E`.
In principle, such an orbit can wander everywhere on the phase-space
hypersurface of constant `E`, but in actuality such orbits show
a complicated and often fractal-like structure.

The appearance of boxlets instead of boxes poses a problem for
model builders because boxlets are `centrophobic' (meaning that they
avoid the center) and consequently do not provide the elongated
density distributions of the box orbits they replace. As a result,
the very *existence* of scale-free triaxial systems is open to
doubt (de Zeeuw 1995).

The scale-free elliptic disk is a relatively simple two-dimensional
analog of a scale-free triaxial system. Because the model is
scale-free, the radial dimension can be folded out when applying
Schwarzschild's method; thus the calculations are fast (Kuijken 1993).
The result is that self-consistent models *can* be built using
the available boxlets, loops, and irregular orbits, but the minimum
possible axial ratio `b/a` increases as the numerical
resolution of the calculation is improved.

Similar results hold for scale-free models in three dimensions.
Models have been constructed for triaxial logarithmic potentials with
a range of axial ratios `b/a` and `c/a` (Schwarzschild
1993). Tubes and regular boxlets provide sufficient variety to
produce models with `c/a > 0.5`, but not flatter. However,
over intervals of `~50` dynamical times, irregular orbits
behave like `fuzzy regular orbits', and by including them it becomes
possible to build near-equilibrium models as flat as `c/a =
0.3`.

Figure rotation adds a new level of complexity to the orbit structure of triaxial systems. A few models have been constructed using Schwarzschild's method, but little is known about the existence and stability of such systems. N-body experiments indicate that at least some such systems are viable models of elliptical galaxies. Rotation tends to steer orbits away from the center and so may lessen the effects of central density cusps.

Realistic potentials are likely to have some irregular or chaotic
orbits, and there is no reason to think that such orbits are
systematically avoided by processes of galaxy formation. Over
`~10^2` or more dynamical times, such orbits tend to produce
nearly round density distributions.

Consequently, it is likely that secular evolution over timescales
of `~10^2` dynamical times may be changing the structures of
elliptical galaxies (Binney 1982, Gerhard 1986). The outer regions
are not likely to be affected since dynamical times are long at large
radii, but significant changes may occur in the central cusps where
dynamical times are `~10^6` years (de Zeeuw 1995).

- Arnold, R.A., de Zeeuw, P.T., & Hunter, C. 1994,
*M.N.R.A.S.*, submitted. - Binney, J.J. 1982,
*M.N.R.A.S.***201**, 15. - de Zeeuw, P.T. 1995, in
*The Formation and Evolution of Galaxies*, ed. C. Munoz-Tunon & F. Sanchez, p. 231. - Gerhard, O.E. 1986,
*M.N.R.A.S.***219**, 373. - Gerhard, O.E. & Binney, J.J. 1985,
*M.N.R.A.S.***216**, 467. - Hunter, C. & de Zeeuw, P.T. 1992,
*Ap.J***398**, 79. - Kuijken, K. 1993,
*Ap.J.***409**, 68. - Levison, H.F. 1987,
*Ap.J.***320**, L93. - Miralda-Escude, J. & Schwarzschild, M. 1989,
*Ap.J.***339**, 752. - Newton, A. & Binney, J.J. 1984,
*M.N.R.A.S.***210**, 711. - Pfenniger, D. 1984,
*Astr.Ap.***141**, 171. - Richstone, D.O. & Tremaine, S.D. 1988,
*Ap.J.***327**, 82. - Schwarzschild, M. 1979,
*Ap.J.***232**, 236. - Schwarzschild, M. 1993,
*Ap.J.***409**, 563. - Statler, T.S. 1987,
*Ap.J.***321**, 113.

Joshua E. Barnes (barnes@zeno.ifa.hawaii.edu)

Last modified: February 14, 1995