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,
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 M(c) is the integral of rho(x) over cell c. This may be accomplished by taking N > K, so as to obtain a reasonably rich set of `basis functions', and using any one of a number of numerical techniques, including linear programming (Schwarzschild 1979), non-negative least squares (Pfenniger 1984), Lucy's method (Newton & Binney 1984), or maximum entropy (Richstone & Tremaine 1988).
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:
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).
Last modified: February 14, 1995