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This is the link of what I have in mind, https://www.youtube.com/watch?v=jKvnn1r-9Iw , can a planet rotate chaotically through this system and still be habitable? If it's going to collapse after a period of time, how much time does it take for the planet to fall into something or drift away?

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    $\begingroup$ See "The Three-Body Problem"... won the Hugo Award in 2015. $\endgroup$ – SRM - Reinstate Monica Jan 10 '17 at 17:52
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    $\begingroup$ Are we talking "stable long enough to develop intelligent life from basic particles" or "stable long enough for a spacefaring species to land (just in time for dramatic disasters!)"? $\endgroup$ – Tin Man Jan 10 '17 at 23:33
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First, the figure-of-8 orbit, as pictured, is a solution to the three body problem. However it is not a "stable" solution, in that if you disturb the orbit by a tiny amount, the orbit won't remain in a figure of 8, but rapidly become chaotic, and end with either one sun colliding with another or being ejected. This is an example of the "butterfly effect". Since it is impossible to avoid disturbing the orbit, there will never be a figure-of-8 orbit in the universe.

Next, if a planet is moving "chaotically" then pretty soon it will either be ejected, or collide with one of the suns. This would happen pretty quickly. (the actual number can't be calculated, but it would be a matter of years, not millennia)

The only three body systems that are stable are

  • One large star, with two smaller stars orbiting it (like a star with two planets.)
  • Two distant stars, with a third star orbiting very close to one of them stars (like a star-planet-moon system)
  • Exceptional co-orbital configurations such as stars at Lagrange L4 or L5 points.

Any other orbital configuration is not dynamically stable.

If you go ahead and have a system like this, you will have to deal with extreme variation in the amount of sunlight the planet receives, and consequently, there will be massive and unpredictable variation in climate. This is a hard planet for life.

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  • $\begingroup$ Regarding the variation in sunlight for a 'figure-of-8' orbit, I provide graphs of that in this answer. Pretty variable, but not unpredictable. Of course, once the planet inevitably falls out of its orbit, then things will get pretty unpredictable. $\endgroup$ – kingledion Jan 10 '17 at 20:43
  • $\begingroup$ As per the OP, the planet is in addition to three stars in a figure-of-8, and "rotating chaotically through the system" hence pretty unpredictable. BTW, nice answer on that question. $\endgroup$ – James K Jan 10 '17 at 20:58
  • $\begingroup$ Ah, three stars, yes very chaotic indeed then. $\endgroup$ – kingledion Jan 10 '17 at 20:59
  • $\begingroup$ We demand horseshoe co-orbits! $\endgroup$ – Steve Jessop Jan 11 '17 at 2:32
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The whole point of the three body problem is that there is no general solution that proves the stability of a certain orbit. The number of terms it takes for the power series analytical solution to converge is beyond our capacity to compute, at least for now. Given that we cannot prove the orbital stability of the planet in your setup, it is hard to tell what the conditions on that planet would be.

As @Aron pointed out in comments, there exist numerical solutions to the n-body problem (as well as many families of general solutions). But a numerical simulation (using a program like Rebound, though presumably more advanced) alone only proves the stability of a single scenario with all the inputs provided to the simulation. In 'real life,' there are nearly infinite small gravitational perturbations, caused by motion in other planets, uneven density distributions in the main star, passing comets, etc. Without modeling these inputs exactly as they happen over billions of years, the numerical solution cannot prove stability. Thus, without a general analytical solution, we cannot prove stability of a real-world n-body orbital setup.

As far as we know, it takes billions of years to develop intelligent life. While the probability of a planetary life-cleansing event (like, say, plunging into a star) is very small on a per-year basis, I conjecture that on the timescale of billions of years, the likelihood of a life-cleansing event happening to a planet in a chaotic orbit are high.

So the answer is: we don't know the answer to your question, but I, at least, think the answer is no.

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  • $\begingroup$ I completely utterly disagree with your entire answer. n-body problems are a solved problem space. We can efficiently calculate them using a range of approximation that are as accurate as you want to make them. The only problem even close to what you pointed out is that a subset of the n-body setups will result in chaotic behavior (which is still solvable and deterministic). $\endgroup$ – Aron Jan 11 '17 at 3:04
  • $\begingroup$ @Aron First, hopefully we can agree that while there are families of solutions to the n-body problem, there is no general solution. Second, I agree that there are numerical solutions: I can plug the numbers into Rebound and get a result. However, I argue that a numerical solution is insufficient to model a stable star system. There are always gravitational perturbations, and a numerical solution cannot prove stability in the face of perturbations. An analytical solution can. So, a numerical solution to the n-body problem is insufficient. $\endgroup$ – kingledion Jan 11 '17 at 13:12
  • $\begingroup$ I agree with your comment. I guess the main problem is the inaccurate language you used. The problem isn't so much that we cannot model a chaotic system, since they are deterministic. The problem is that chaotic systems are very sensitive to initial conditions, which is the opposite of stability. $\endgroup$ – Aron Jan 12 '17 at 6:12
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Yes it can, but the inhabitants will have to come from somewhere else. They will have to bring everything with them to set up their environment, right down to the microbes. Finally, they will need to stay ready to leave quickly in case their unpredictable planet starts heading towards something lethal.

On the positive note, the inhabitants will probably be able to hide from the rest of the universe, since no one with enough science to understand evolution would expect to find intelligent life under such conditions.

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