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update: Searching for "choreographies" I found this Physics SE question which is related but different because it asks only if periodic solutions can be proven to be periodic numerically and my current question is broader.


@MarkAdler's answer to Was the Apollo spacecraft always gravitationally bound to the Earth-Moon system? is very specific and does not directly apply to this more general question.

To truly answer your question, you would need to propagate the trajectory from every state between maneuvers, potentially for a very long time, to determine its ultimate fate. There will often not be enough accuracy in the known state, as well as uncertainty in solar pressure perturbations, for that to even be deterministic.

If we have a two-body orbit we can look at the specific energy or the $C_3$ parameter using only separation and speed and know immediately if it is bound or unbound. From this answer:

$C_3$ the characteristic energy is twice the total energy (kinetic plus potential) $E$ of a body with respect to a larger gravitational body

$$E = \frac{1}{2}v^2 - \frac{GM}{r}$$

$$C_3 = v^2 - 2\frac{GM}{r}$$

Question: But for a three body orbit, even in the CR3BP limit, are there some three-body orbits that definitely can or can't be escaped? Are there any instantaneous configurations (system state vector) where we can say "yep, that one's definitely gonna separate some day" or "nope, that one will definitely live forever", or no matter what we must always propagate to separation to know anything definitive, because propagation a long time ultimately only yields a new state vector to be explored.

Assume three point masses, Newtonian gravity and no losses.


https://en.wikipedia.org/wiki/Three-body_problem#Restricted_three-body_problem

note: The paragraph above applies to chaotic orbits or any orbit that is not closed and periodic. For example there are some stable halo orbits in the CR3BP that we can show will remain forever mathematically; see Are some Halo Orbits actually Stable? These closed and periodic trajectory solutions should be mentioned in the answer as trivial cases; I'm really interested in all the others.


Here's an example of one that got away, from “Pythagorean Three Body Problem” - need some points from an accurate solution for comparison

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    $\begingroup$ How do you classify 3-body configurations where simulation for a small number of orbital periods yields a collision? $\endgroup$ – Russell Borogove May 1 at 2:45
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    $\begingroup$ @RussellBorogove "Assume three point masses..." mostly (but not rigorously) addressed that; some initial states can certainly lead to exact intersections in space and time. In terms of how I would classify them, I'd call them "unlucky orbits" ;-) I'm adding a close call to the question for fun. It has several near-misses that are a challenge for numerical integrators (but not for Mark Adler and Mathematica). $\endgroup$ – uhoh May 1 at 4:02
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    $\begingroup$ Boo, I overlooked "point masses". $\endgroup$ – Russell Borogove May 1 at 4:17
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There are really two questions here:

  1. Do there exist $n$-body systems with long-term stability?
  2. Can a third body (massive or not) be shown, a-priori, to be bounded or to escape—without resorting to numerical simulation?

1. Stability of $n$-body systems

It is widely known that $n$-body systems are "chaotic" when $n>2$. However, this must be unpacked mathematically to be useful.

There are several basic cases:

  • In a stable system, the behavior returns to the same equilibrium, even in the presence of small perturbing forces.
    • In a metastable system, the behavior is stable, but is not at the lowest-possible energy level (for a problem-specific definition of "energy").
  • In a neutrally stable system, the long-term behavior is altered by small perturbing forces, but that difference in behavior remains small.
  • In an unstable system, the system's long-term behavior is strongly affected by perturbations. Small changes cause large differences in long-term behavior.

A real discussion of nonlinear dynamics is out of scope (if you want one, I highly recommend Nonlinear Dynamics and Chaos by Strogatz), but roughly speaking, a "chaotic" system is an unstable system. However, unstable systems also tend to have at least some neutrally stable regions. Many also have stable regions. For example, in-general, the Mandelbrot set (perhaps the canonical chaotic system) iterate $z_n:=z_{n-1}^2+c, ~~z_0=c$ is chaotic. However, there are many stable and neutrally stable cases (e.g. trivially, $c:=0$).


So now that we know what we're talking about, what of the stability of $n$-body systems? We know, from centuries of study, that they are chaotic, but are they always unstable?

A moment's thought will reveal that $n$-body systems have some elements of neutral stability: perturbations in planetary orbits are abundant, but the planets are neither thrown out to infinity, nor crash into the Sun constantly! However, these perturbations have lasting effects over the long term: indeed, astrophysicists make inferences about planetary science by reading clues of ancient perturbations written in alterations of orbits that persist today.

What this tells us is that $n$-body systems are unstable, with regions of neutral stability. Orbits are fundamentally chaotic, but in some situations the behavior can be more predictable (albeit still subject to small long-term changes by small perturbations).


2. Will [some initial condition] remain bounded?

The mathematical techniques to definitively answer this question flatly do not exist. Indeed, characterizing the long-term behavior of chaotic systems beyond coarse stability is believed to be actually impossible. For example, there is no possible finite-time algorithm to correctly predict the long-term behavior of an arbitrary $c$ in the above example of the Mandelbrot set in all cases—a sequence of $z_k$s which are, fittingly, called "orbits".

However, this is defeatist, and it would be misleading to say that nothing can be done.


The easy case is to look at the total orbital energy. This is invariant, neglecting collisions and outside forces. For the two-body case, it can be expressed in the famous vis-viva equation, but energy conservation is true with any number of bodies as well.

A sufficient condition for the system to remain bounded is to imagine what would happen if one body somehow got all of the orbital energy (imagine: all points at the origin; all bodies motionless except for one). If this body's orbital energy is less than the gravitational binding energy tying it to the rest of the system, and this is true for any body so chosen, then no bodies can escape, and the system will remain bounded. However, the converse is not true: even if a body could escape, does not mean the system is necessarily unbounded.


To analyze that case, we turn to various some special cases which have been concocted over the years.

Most configurations, such as the central configuration, are unstable: the slightest perturbation, and the system devolves into unstable behavior. While the long-term behavior of an individual element in a chaotic system in a region of instability cannot be predicted in general—that's what "chaotic" means, after all—it is difficult to say what will happen. It seems to be, in practice however, that bodies will eventually get ejected until only two (or one) remain.

Heuristically, this makes some amount of sense. Gravity assists transfer energy from one body to another; since the orbits are essentially unpredictable (read: "random"-ish) in an $n$-body system, the bodies divide up the available energy, effectively randomly, until one of them just happens to get enough energy to escape the binding energy of the others. The system will sometimes walk through regions of neutral stability (such as our solar system right now) that can be relatively long-lived but will eventually become unstable. This isn't a proof, but it describes the qualitative behavior of most $n$-body systems (and is the motivation for the sufficient condition for boundedness stated above).

There are a few other configurations, however, such as the figure-8 configuration, which are actually (meta)stable within a small region:

                             figure-8-frame

                                                                 (still frame from animation at link above)

Theoretically, this orbit is stable forever, and would never exhibit chaotic behavior unless it was disturbed—and disturbed significantly—from the outside.


Conclusion

So, TL;DR: there exist a few examples of three-body systems that are (meta)stable (i.e. resistant to small perturbations over long periods of time). However, in most cases, three-body (or $n$-body) systems are only neutrally stable at best (i.e., small perturbations have lasting, but not destabilizing effects), and for most initial conditions are unstable (i.e. small perturbations have long-lasting dramatic effects), with the former eventually becoming the latter.

For boundedness questions, beyond simple analysis of the orbital energy, the knowledge that a system is unstable is not sufficient to say that it is unbounded (although it probably is).

Characterizing the behavior of such systems beyond this in general is somewhere between "beyond our knowledge" and "actually impossible".

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  • $\begingroup$ Thank you for the thorough and well-written answer! The application of conservation laws is indeed useful; should angular momentum be considered also? To the question asked, conservation laws certainly provide a possibility to declare "can't escape" for some initial states, but in cases where they allow it and the orbits are not closed and periodic, then will the answer sometimes be "impossible to know either way"? $\endgroup$ – uhoh May 1 at 9:05
  • $\begingroup$ Just to prank the cosmologists & string theorists, imagine $c$ or the fine-structure constant varying over time, and then try to predict bounded orbits :-) $\endgroup$ – Carl Witthoft May 1 at 11:49
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    $\begingroup$ @uhoh There may be something to be said about (angular) momentum, but I think energy is the easier/more-useful quantity to analyze for this problem. And yes, to summarize: if a conservation law allows escape, and the orbits are not one of the very special cases known to be (meta)stable, then the system is probably-unstable / maybe-neutrally-stable, and will decay, but this is not guaranteed. $\endgroup$ – imallett May 2 at 5:38
  • $\begingroup$ @imallett thanks, so the next question should probably ask if it is possible for a three-body orbit to be energetically unbound but bounded by angular momentum conservation, and if there exist any proofs either way. I'm not sure if something like that is better asked in Math SE, Physics SE or Math Overflow, because I still don't know if the answer is trivial or deep. $\endgroup$ – uhoh May 2 at 6:02
  • $\begingroup$ @uhoh It's more like, angular momentum is always conserved, even for collision, and I don't think it says anything at all about boundedness, except as it relates to orbital energy, which is what we're discussing already. So the question doesn't seem well-posed. $\endgroup$ – imallett May 2 at 20:36
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Assume three point masses, Newtonian gravity and no losses.

If we can also assume no perturbations of any other kind, and perfect placement of the bodies in the initial conditions, then a 3-body Klemperer rosette -- three bodies of equal mass, in an equilateral triangle, with any rotationally symmetrical initial velocities comfortably below barycentric escape velocity -- should remain stable.

I believe that given point masses, infinitesimal perturbations, and infinite time, all configurations must lead to escape, but a pair of partners orbiting closely plus a third orbiting far away can behave like two independent 2-body problems for extremely long periods of time.

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  • $\begingroup$ It looks like the link says the smallest number of bodies for a proper Klemperer rosette with alternating masses is 4. I think there are several three-body choreographies, all closed and periodic and therefore trivial but important bound solutions. Here are some more fun 3-body closed, periodic choreographies: analyticphysics.com/Three-Body%20Problem/… $\endgroup$ – uhoh May 1 at 4:46

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