Background Information
Surfaces of Section

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We would like to envision the phase space motion of these orbits in different potentials. Unfortunately envisioning 6-d phase space coordinates (x, y, z, vx, vy, vz) is difficult. Instead, we make a number of cuts in phase space to make the problem tractable:
  • First, we will concentrate on 2D motion in a plane, which gives us 4D space: (x, y, vx, vy)
  • Second, since energy is conserved, for a given energy the 4 coordinates are not independant. We can specify orbits of a given energy, then just plot 3 independent variables, say x, y, vy. But even drawing 3-d is not straightforward (at least w/out fancy graphics). So...
  • Third, we only plot the phase space coordinate of the star when it crosses the y axis (i.e., when x = 0).
So here's the recipe: for a given orbital energy, examine orbits in the x,y plane, and plot vy vs v every time the star crosses the y axis. Plots such as these are known as surfaces of section, and can be used to study the orbit content of different gravitational potentials. Let's look at how this works.

Take a circular orbit, with radius r = 1 and velocity v = 0.5. Every time it crosses the y axis going to negative x, its phase space coordinate is (x, y, vx, vy)=(0, 1, -0.5, 0). So we would plot a point on the surface of section at (y, vy)=(1, 0). When it crosses the y axis going to positive x, its phase space coordinate is (x, y, vx, vy)=(0, -1, 0.5, 0), and we we would plot a point on the surface of section at (y, vy)=(-1, 0). This would continue ad infinitum, drawing these same two points. Not terribly exciting.

But now let's draw a loop orbit. Since it's not a perfect circle, it won't always cross the y axis with the same phase space coordinate, so we won't be drawing that same point. Instead the points trace out a closed surface around the point defined by the circular orbit, because the circular orbit is the parent of the loop orbit. Resonant parent orbits lie in the center of the closed regions describing the general orbits.
Loop Orbit with Surface of Section Plot
Figure 5 - left: a loop orbit.   right: the surface of section for this orbit. the (circular) parent orbit would generate a point on the surface of section at the center of the area enclosed by the loop orbit's track.
Now let's look at a box orbit. Because of the fact that it fills a region of space, it can cross the y axis at any value of y or with any value of vy. But because it is subject to conservation of energy, the (y, vy) phase space pair at y crossing is not arbitrary. Instead, it traces out a closed surface on the surface of section around the origin.
Box Orbit with Surface of Section Plot
Figure 6 - left: a box orbit. right: the surface of section for this box.
Boxlets are even more bizarre. Because they are resonant orbits, they have only certain allowable (y,vy) phase space coordinates when they cross the y axis, and they trace out different shapes on the surface of section. As with loop orbits, the parents of the box and boxlet orbits can be found at the center of the surfaces drawn on the surface of section.
Boxlet Banana Orbit with Surface of Section Plot
Figure 7 - left: a banana orbit. right: the surface of section for this orbit. the pure 2:1 resonant parent banana would generate a point on the surface of section at the center of the area enclosed by this orbit's track.

Boxlet Fish Orbit with Surface of Section Plot
Figure 8 - left: a fish orbit. right: the surface of section for this orbit. the pure 3:2 resonant parent fish would generate a point on the surface of section at the center of the area enclosed by this orbit's track.
A surface of section is a visual way, therefore, of looking at the orbit content of a given gravitational potential. If the potential is not "scale-free" -- that is, if it changes shape or density profile with radius -- the surface of section will change as a function of energy. This is because orbits with different amounts of energy are restricted to different parts of the galaxy. Orbits with very negative energy are tightly bound, and move through the inner portions of the galaxy. Orbits which are loosely bound (so have less negative energy) can move to larger radius. So if the potential changes shape with radius, the orbit families will change as a function of energy.
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