A digression on Lissajous orbits


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Aug 13, 2017
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You may have noticed a steady trickle of posts on Lagrange Point orbits has appeared on Lagrange Point orbits. This is, of course, down to me attempting to explain how to use a high-order Lindstedt-Poincaré perturbative solution to the equations of motion in the restricted three-body problem (see Dialing-up arbitrary Lagrange Point orbits.

So far, I've focused on planar and vertical Lyapunov orbits in planar Lyapunov orbits and vertical Lyapunov orbits. These are special cases of general Lissajous orbits.

If you want to see an illustration of a vertical Lyapunov orbit, see the following short vide clip which illustrates vertical Lyapunov orbits and contrasts them with planar Lyapunov orbits:

The planar Lyapunov orbits of the Earth-Moon system have one fundamental frequency and oscillate primarily in the x-y plane of the Moon's orbit around the Earth; and the vertical Lyapunov orbits also have one fundamental (generally a different one) but they oscillate primarily in the out-of-plane 'z' direction. What happens if we combine the two motions>

Well, if we combine the horizontal and vertical motions (with each component oscillating with a different fundamental frequency) we end up with a Lissajous orbit. Here is a short video of a Lissajous orbit generated from the datasets and tools set out in Dialing-up arbitrary Lagrange Point orbits:

(In case you were interested, this was calculated setting 'da1' = 'da2' = 0.02 and 'phi1' = 'phi2' = 0.0 and allowing 'nu' the de facto time variable to vary between 0 and 10*pi - i.e., 5 full revolutions of the Moon around the Earth.)

The Lissajous motion is complicated. At first blush, the calculated Lissajous 'orbit' looks like a tangled heap of spaghetti but if you look closely enough, you can begin to discern some order. That order is choreographed by the two fundamental (but incommensurate) frequencies. Wikipedia has a brief discussion on Lissajous curves and the role of the two frequencies here:https://en.wikipedia.org/wiki/Lissajous_curve.

Although Lissajous orbits are more complicated than the planar and vertical Lyapunov orbits, they are perfectly valid centre manifold solutions to the equations of motion of the elliptical restricted three-body problem. Like the Lyapunov orbits, they are bounded so they are restricted in how far they can move away from the Lagrange Point; and they are unstable requiring some form of active (but minimal) station-keeping to stay on them.They also have stable and unstable manifolds that allow ballistic entry into and departure. But unlike Lyapunov orbits, they form complex three-dimensional structures that are hard to grasp intuitively.

However, mission planners are saved from an ungodly mess because the two fundamental frequencies - one in the x-y plane; and the other in the out-of-plane z-direction are functions of the scale parameters 'da1' and 'da2'. So, if we choose exactly the right combination of 'da1' and 'da2' we can make sure that the two fundamental frequencies are also the same. And if we do that, we end up with a Halo orbit. And Halo orbits are interesting because they allow perpetual direct line-of-sight communication from Earth with a satellite parked on a Halo orbit around L2. An example of Halo orbits is shown in the following short video clip:

Anyway, the next post in this series will return to "Dialing-up a Halo orbit" and will be focusing on using the Lindstedt-Poincaré theory to construct the Orbiter-friendly state vectors of a vessel parked in a Halo orbit.