Sander,

The likelihood of collision seems intuitive, but there are a bunch of conflicting intuitions going on here. The motions seem random, but space is really big, but time is really long. A collision ends a simulation, but it doesn't end the solar system, but etc.

I don't really know, but it's worth mentioning that things can be random (in the sense of appearing to not be ordered) without being ergodic (in the sense that the system gets near every possible configuration). For there are some of these irregular configurations which go on for a long time, much longer than one would expect to be possible (e.g. just run a few or see the note in Wolfram's book). It is hard to guess what the ratio of safe initial configurations could be without a detailed survey. (Lucas, can you comment?)

One thing that is clear though is that it is easier to avoid collisions in three dimensions than in two dimensions (Most of our solar system is within a few degrees of the ecliptic plane. Of course Jupiter is the main actor.)

Perhaps a quantum mechanics analog could be used to catalog star vibrations (or intensities) with certain probabilities. Observing a planet orbiting a star is going to share many of the same technical difficulties. The planets would represent a quantification of energy in the vibration. Multiple planets and potential sinks would only show up as multiple eigen value energy levels. The added difficulty is that the masses of the planets are not known before hand or required to be integral. Chaos is only going to manifest as a fine structure to the orbital potentials. Unless I miss my guess current methods rely on frequency spikes in the fft of the star motion. Chaotic orbits would randomize an fft, but not the fft/energy envelope. The primary downside to this method of analysis would be a long requisite observation time. I don't know that you could identify individual planets, but you could probably identify mass/energy distributions which would be a good indicator. I think the way to initially approach this would be similar to the way they initially tried to derive quantum mechanics using orbital mechanics with a randomized LRL vector/Energies. Observing the pattern for a length of time would exclude certain LRL vector possibilities and energies. This is roughly analogous to the spherical harmonics solution obtained using traditional quantum methods.
The reason I think this is a similar problem is that most stars are so far away that the data we see is 2D for all intents and purposes, this effectively limits our degrees of freedom to an under-constrained solution (the same thing happens in QM thus we use spin-up/spin-down). I think in order to use this method, one would need to randomly choose a sample mass and then look for 'clumping' in the orbitals. One would then adjust the sample mass till the clumping was resolved into the "simplest solution set possible, but not simpler".

I am an undergrad applied physics student, so I accept any constructive criticism on this idea. My primary region of interest is EM so I may be way off base here.

Yeah Todd, I think you have the basic idea of what I was trying to say. But I don't think you could derive the angular mixed states data from a brightness measurement. A brightness measurement only accounts for a region of space that forms a line between us and the primary. This gives us position, but makes momentum unknowable. It also is dependent on orbital planes passing through that line and completely ignores orbitals that are non-planar. If we had an orbital inferometer, then we could detect red/blue shifting due to sympathetic angular momentum of the parent star with its planets. After accounting for the red/blue shift induced by us, this should give one dimension of vibration on the primary. Additional dimensions of measurement would be dependent on distance between us and the star because it would be dependent on angular resolution. Direct angular measurement of primary vibration is one of the earliest methods of finding planets.

As for the stability of orbits, that is heavily dependent on the ratio of the size of the planets/suns to their distance from each other. I like to think about it the way I think about electrical dipoles. Specifically, n-body electrical interactions with dipoles. From the orbiting planets perspective there are two ways to think about it. The second way is that, there is only one primary acting on it, but its gravity changes periodically (this induces LRL precession in n=3 orbits). When the primary and secondary are aligned with the planet, gravitation is maximized, and when it is tangential, it is minimized. As the distance from the primary increases, the angular distance between the primary and secondary decreases, and the gravitation difference approaches the central limit (in which there is no difference).

In light of this one quickly realizes that chaotic systems would require similarly massive objects in relatively close proximity to a parent star in order to be measurable via non-brightness methods. Even with brightness methods, the ability to measure a chaotic system is literally impossible because of the lack of angular information. A necessary axiom for making the brightness jump to orbital data is that the system is not chaotic. I think the closest once could get using brightness is to make a 3d plot of fourier on one axis and histogram on the other. This might let you see PL-banding if a sufficiently long observation time is available.

Daniel,

Nothing wrong with being a physics undergrad. This is just citizen science, no authorities here.

I don't completely understand what you are saying, but there seem to be a few different ideas that might be fun for you to explore more concretely. Let me try rephrasing what I think you are saying. If one thinks of an atom classically, there are a bunch of electrons flying around. Instead of gravity attracting planets, the electrons repel each other, but it could be that when one goes to quantum atoms (physical ones) the math is the same and one could just as easily treat the planetary systems as quantum mechanical. The brightness signal as a sort of dimensional reduction.

One thing you can do is to run some simulations. For example, here is the 3 body simulation that I posted above run for longer.

It seems to be doing one thing for a while, then one planet goes on a big loop, during which the other two twist around like a well-behaved 2-body system, and then the third body comes back and it starts doing something different. You could interpret this as mixed states, where a pure state might be some identifiable behavior like nearly planar, nearly elliptical orbits.

To pursue the quantum mechanical analog you could quantify things like rotations, or to get fancy, spherical harmonic decompositions.