# How stable are orbits of light objects around twin large bodies?

I know the two-body problem has a stable solution and the three-body problem does not.

In the case that there are two comparable large bodies (twin planets) in a stable mutual orbit, what happens to a small, light particle's trajectory in that system?

If there were a number of tiny moons around either of these planets, would they remain in stable orbits, or would small petruburances cause all the tiny moons to fall into one or the other large bodies in "short" time?

EDIT: To clarify "stable", I mean that although the trajectory of any tiny moon is likely to be chaotic, it will "probably" avoid colliding with either planet for a "long" time. I realise this is delegating to other imprecise terms but consider that if we gave our own moon a sudden kick which changed its velocity by a few percent we would not expect it to collide with the earth within a decade but rather find a new, different, long-term-stable, approximately elliptical orbit. Could the same be said for a tiny satellite in a twin planet system?

• Depends what you mean by "stable." Some people might think that our solar system has been "stable" for quite a long time, and it has a lot more than just two bodies. But, the motion of our solar system actually is chaotic, which means, there's no closed-form that predicts the exact configuration for all future time, and even numerical techniques can only take you so far. Commented Mar 15 at 14:34
• @AgniusVasiliauskas I suspect the OP did not meant "photon" by "small, light particle" - I think they meant a particle with small mass. Commented Mar 15 at 16:00
• It depends very much on the ratios of mass to distance to "how long is short", so it's hard to answer. I'd suggest to find a simulator and experiment. I haven't tried any of them, but searching "gravity simulator program" got me a whole page of free results. I've played a little with Universe Sandbox, which is not free and won't run on a phone, but is cool if you have a PC and a little spare cash.
– g s
Commented Mar 15 at 16:44

As you can see in the description, for two equal mass planets in a circular orbit with distance $$D$$ between them, this is $$R_H = \frac{\small 1}{\small 2} \sqrt[{\Large 3}]{\frac{\small 1}{\small 6}} \ D \approx 0.28 \ D$$ and the article also shows that this is related to the Lagrange points of the twin planets, the orbit of the moons should never come into the neighborhood of those points.