Background Information
Orbits and Energy

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The most familiar type of orbit is the Keplerian orbit, which occurs when objects orbit a point mass (i.e., planets orbiting the Sun, or the motion of a binary star system). A Keplerian orbit is a closed ellipse, where the pericenter and apocenter distances are determined by the energy and angular momentum of the orbit.
However, the orbits of stars in a galaxy are not Keplerian, because the mass distribution is not concentrated in a point mass. Rather than form a Keplerian ellipse, stellar orbits are not closed, but slowly fill all allowable space as they move through the galaxy. The exact shape of a stellar orbit depends on a variety of things, including the shape of the galactic potential and the energy and angular momentum of the orbit.
In a spherical galaxy, total energy (E = ½ v ² + phi) and angular momentum (L = r × v) are conserved quantities -- they do not change as the star orbits in the galaxy. Orbits form a rosette, in which a star will eventually pass through every point on an annulus whose inner and outer radii are the pericenter and apocenter distances determined by the energy and angular momentum of the orbit. Larger orbital energies result in large apocenters, while smaller angular momentum gives a smaller pericenter. Orbits like these a referred to as "loop orbits".
loop orbit
Figure 1 - A loop orbit. Over time, the star's path will completely fill in the annulus between rperi and rapo.
If the galaxy is not axisymmetric, angular momentum is not conserved along an orbit; in these triaxial potential the loop orbits are joined by a family of orbits known as "box orbits". In a box orbit, the star oscillates independently along the three different axes as it moves through the galaxy, slowly filling in a (roughly) box-shaped region of space.
box orbit
Figure 2 - A box orbit. Over time, the star's path will completely fill in the rectangular region.

Unlike loop orbits, which loop around the center of the galaxy, stars on box orbits will pass arbitrarily close to the galactic center. Some potentials result in resonant box orbits, in which the orbit is closed. If the oscillation frequencies along two axes are in a 2:1 resonance, a "banana" orbit is formed; a 3:2 orbit forms a "fish"; a 4:3 orbit forms a "pretzel". Just as a closed circular orbit can be thought of as the "parent" orbit of loops, these closed boxlets act as parents of a broader family of boxlets.

banana orbit
Figure 3 - A 2:1 resonant boxlet called a banana. The star will continue to oscillate back and forth along this track.

fish orbit
Figure 4 - A 3:2 resonant boxlet called a fish. The star will continue along this track.

One of the important things to realize here is that the shape of the galaxy is determined by the allowed orbit families (and vice versa). For example, if the potential was perfectly round, so there were no box orbits, the galaxy would appear round since it would be comprised entirely of loop orbits. Alternatively, if you have a triaxial system, you have to have some amount of box or boxlet orbits so that the galaxy will appear to be flattened on the intermediate and minor axes. (We are restricting the discussion here to non-rotating systems. You can also get flattening via rotation, but we won't deal with that here...)
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