Axisymmetric two-integral models of many elliptical galaxies don't fit the observed kinematics, implying that these systems have a third, non-classical integral. Schwarzschild's method permits the construction of both axisymmetric and triaxial models which implicitly depend on a third integral. But minor orbit families and chaotic orbits both limit the range of axial ratios permitted for `cuspy' elliptical galaxies; systems with steep power-law cusps probably evolve toward more axisymmetric shapes over tens of orbital times.

Numerical calculations show that most orbits in `plausible'
axisymmetric potentials have a *third integral*, since their
shapes are not fully specified by the classical integrals, energy and
the z-component of angular momentum. No general expressions for this
*non-classical* integral are known, although for nearly spherical
systems it is approximated by `|J|`, the magnitude of the total
angular momentum, while in very flattened systems the energy invested
in vertical motion may be used.

Despite the existence of third integrals in most axisymmetric potentials, it is reasonable to ask if models based on just two integrals can possibly describe real galaxies. In such models the distribution function has the form

(1) f = f(E,J_z) .One immediate result is that the distribution function depends on the

_______ _______ (2) v_R v_R = v_z v_z ,This equality does

**From f(E,J_z) to rho(R,z)**: Much as in the
spherically symmetric case described before, one may adopt a plausible
guess for

**From rho(R,z) to f(E,J_z)**: Conversely, one
may try to find a distribution function which generates a given

An `*unbelievably simple*' and analytic distribution function
exists for the mass distribution which generates the axisymmetric
logarithmic potential (Evans 1993). This potential, introduced to
describe the halos of galaxies (Binney 1981, BT87, Chapter 2.2.2), has
the form

1 2 2 2 2 2 (3) Phi = - v_0 ln(R_c + R + z / q ) , 2where

2 4 E 4 E 2 E (4) f(E,J_z) = A J_z exp(-----) + B exp(-----) + C exp(-----) , v_0^2 v_0^2 v_0^2where

But even if kinematic data is available, this approach is not very
practical for modeling observed galaxies. The reason is that the
transformation from density (and streaming velocity) to distribution
function is *unstable*; small errors in the input data can
produce huge variations in the results (*e.g.* Dejonghe 1986,
BT87). A few two-integral distribution functions are known for
analytic density distributions, and recent developments have removed
some mathematical obstacles to the construction of more models (Hunter
& Qian 1993).

Since we can't construct distribution functions for real galaxies,
consider the simpler problem of modeling observed systems using the
Jeans equations. If we *assume* that the underlying distribution
function depends only on `E` and `J_z` we can simplify
the Jeans equations, since the radial and vertical dispersions must be
everywhere equal; thus

d _______ nu _______ ___________ dPhi (5) -- nu v_R v_R + -- (v_R v_R - v_phi v_phi) = - nu ---- , dR R dR d _______ dPhi (6) -- nu v_R v_R = - nu ---- . dz dzAt each

The Jeans equations don't tell how to divide up the azimuthal velocities into streaming and random components. One popular choice is

_____ 2 ___________ _______ (7) v_phi = k (v_phi v_phi - v_R v_R) ,where

2 ___________ _____ 2 (8) sigma_phi = v_phi v_phi - v_phi .Note that if

Jeans equations models have been constructed of a number of elliptical galaxies (Binney, Davies, & Illingworth 1990, van der Marel, Binney, & Davies 1990, van der Marel 1991). The procedure is to:

- Observe the surface brightness
`Sigma(x',y')`; - Deproject to get the stellar density
`nu(R,z)`, assuming an inclination angle; - Compute the potential
`Phi(R,z)`, assuming a constant mass-to-light ratio; - Solve the Jeans equations for the mean squared velocities;
- Divide the azimuthal motion into streaming and random parts;
- Project the velocities back on to the plane of the
sky to get the line-of-sight velocity and dispersion
`v_los(x',y')`and`sigma_los(x',y')`; - Compare the predicted and observed kinematics.

Some conclusions following from this exercise are that:

- Isotropic oblate rotators (
`k = 1`) generally*don't fit*, even though some galaxies lie close to the expected relation between`v_0/sigma_0`and`epsilon`; - Some galaxies (
*e.g.*NGC 1052)*are*well-fit by two-integral Jeans equation models; - The models predict major-axis velocity dispersions in excess of those observed in most galaxies;
- Consequently, most of the galaxies must have a third integral,
or may even be
*triaxial*.

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's
not clear if such models can be in true equilibrium.

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 \ (9) M(c) = ) Q_i P_i(c) / -- i = 1simultaneously for all cells, where

In general Eq. 9 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.*).

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 thus lie anywhere in the plane containing the short and long axes (Levison 1987).

Using Schwarzschild's method, it is possible to numerically
determine orbit populations 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, de Zeeuw, & Hunter 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 (Franx, Illingworth, & Heckman 1989b).

In scale-free models, box orbits tend to be replaced by minor
orbital families known as *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.

The appearance of boxlets instead of boxes poses a problem for
model building because boxlets are `centrophobic' (meaning that they
avoid the center) and 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).

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 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`. But
these models are not true equilibria; over long times they will become
rounder and less triaxial.

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 (Merritt & Valluri 1996).

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 only `10^6` years (de Zeeuw 1995).

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

Last modified: March 6, 1997