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Unpredictable Solar Systems

Posted 4 years ago
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Are there solar systems with chaotic orbits? When astronomers look for exoplanets they look for periodic signals in the brightness of the central star. The analysis relies on predictable behaviors. But I have always wondered whether there are unpredictable solar systems out there.

The reason for posting this now is the news of a star whose signal appears to be unpredictable (KIC 8462852). Some articles have suggested this is due to aliens. On the other hand, Stephen Wolfram has said many times (e.g. in his New Kind of Science book) that it is pretty easy for nature to produce unpredictable sequences, and regular signals would be a better sign of civilization.

You can simulate hypothetical solar systems relatively easily in Wolfram language (search the demonstrations for three body problem). At the Wolfram Science Summer School in 2013, Nicholas Lucas did a systematic survey. He produced a nice phase-space type of diagram and in the process found a class of behaviors which were not regular in any sense except that all of the planets did not fly away. The possibility of planets zooming off to infinity is a possible explanation for the prevalence of regularity and order (at least when stars are far apart). This is an example of an irregular solution:

irregular paths of 3 bodies

This code is a simple two body version:

s = NDSolve[{x''[t] == 8 (y[t] - x[t])/Norm[y[t] - x[t]]^3, y''[t] == 8 (x[t] - y[t])/Norm[x[t] - y[t]]^3,
     x[0] == {-2, 0, 0}, y[0] == {2, 0, 0}, x'[0] == {0, 1, 0}, y'[0] == {0, -1, 0}}, {x, y}, {t, 0, 4}][[1]];
ParametricPlot3D[{Evaluate[x[t] /. s], Evaluate[y[t] /. s]}, {t, 0, 4}]

Theoretically, from the study of simple rules (see Wolfram's book), one expects the possibility of long transients, but also that most transients are short. From Wolfram's principle of computational equivalence, one expects that solar system dynamics can be computationally universal even from simple initial conditions (for more see Wolfram's note).

Maybe someone on Community knows more about this star. Is there data out there for KIC 8462852? In principle one should be able to take the orbital paths from a simulation and derive the brightness signal that someone would see from Earth, and do it systematically.

9 Replies

Hi everyone,

perhaps it would be good to look at the available data, too. For KIC 8462852 we have a lot of data available, for example on this website. We can plot some example light curves like so:

GraphicsRow[{ListLinePlot[
   Import["http://archive.stsci.edu/missions/kepler/lightcurves/0084/\
008462852/kplr008462852-2013098041711_llc.fits", "Data"][[1, All, 4]],
    PlotRange -> All, ImageSize -> Medium], 
  ListLinePlot[
   Import["http://archive.stsci.edu/missions/kepler/lightcurves/0084/\
008462852/kplr008462852-2013131215648_llc.fits", "Data"][[1, All, 4]],
    PlotRange -> All, ImageSize -> Medium]}]

enter image description here. A long time ago we organised a summer school on super computational physics on the stability of extrasolar planetary systems. We simulated the orbits of test-particles in observed extrasolar planetary systems; the idea being that at that point they would only observe Jupiter-like planets and the question is whether stable orbits of earth-like planets could exist (in the habitable zone).

We used simple complexity measures to determine stability of orbits. You can then study the stability of orbits in different parameter spaces. Some time ago I had a student on a summer project who implemented that in the Wolfram Language, but I have not been able to find the code.

There is also an interesting book, that contains Mathematica code, that might be relevant.

Cheers,

Marco

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.

I would guess that any chaotic solar system shows collisions after sufficient time, such that they only have finite lifetime as being chaotic...

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.)

Thanks Sam. Yes, I think that story is relevant (the original article is in Nature) I think in some way this must be similar to what happens with the Galilean moons, where each one speeds up or down depending on its neighboring moons. The end result is chaos in the technical sense of sensitivity to initial conditions.

I am wondering about chaos in the sense of irregular. Does anyone know how to go from the planetary positions to the brightness of the planetary system as seen from Earth? Does a dip of brightness only occur when the planet transits the disk of the star?

It should be straightforward to go from the n-body simulation to a simulation of the brightness signal.

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.

enter image description here

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.

I guess another thing that makes solar systems 'stable' is that (for our case) the sun is much heavier than all the others (Jupiter is ~1000 less heavy then the sun, and the other planets are even much lighter). If they would be roughly equal, the trajectories get more complex as the movement of planets (significantly) move the sun, and that way the sun couples back to all the others, giving chaotic motion.

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.

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