### Is any 3 body system known?

• https://en.m.wikipedia.org/wiki/Three-body_problem

Have three or more celestial bodies rotating each other in a stable manner ever been observed, or it is only a theoretical problem?

I'd like to clarify: The three body problem is so called because for 3 or more bodies there are no exact solutions to Newtons gravitational equations. However there are known systems with more than three bodies, that are stable on very long timescales. Would you accept an answer that had more than three bodies? Does the system need to be mathematically proved stable, or would empirical answer be acceptable?

Our solar system comprises one star, four giant planets, four terrestrial planets, plus moons, dwarf planets, and small bodies. Jupiter by itself has many moons (most likely more than JPL's SPICE software can handle) and an even larger number of asteroids trapped in its stable Lagrange points.

@jamesk yes I would accept more than three bodies. I thought to find only three was enough difficult or never happened.

Well, there's those damn Trisolarians about to conquer Earth :-)

@carl that's exactly why I'm asking, Trisolarians. :) Nice book.

5 years ago

Answering in the spirit of the question, I think he's asking if there is ever a chaotic 3 body system that's long term stable, or, to put it another way, a 3 body system without a standard hierarchy where it's stable.

The answer is no. 3 body systems without Heirarchy are never stable for very long. They can certainly exist for a while, but they aren't long term stable.

What do we mean by Hierarchy

The Sun-Earth-Moon is an example of a long-term-stable 3 body system in heirarchy. It will remain stable for billions of years. The Sun is pretty much in the center. The Earth-Moon system orbits the sun and the Moon also orbits the Earth. In a sense, the Moon orbits both the Sun and the Earth. The Moon can orbit the Earth because it's inside the stable part of Earth's Hill Sphere.

Hierarchies beyond 3 might exist, such as Sun, Planet, Moon, Moon's satellite but there are no natural satellites of Moons in our solar system.

Multiple planets

Obviously, multiple planets around a comparatively massive Central star can be long term stable, and just for clarity, lets call a system stable if it lasts a billion years. Mercury, Venus, Earth, Mars, Juptier, Saturn, Uranus, Neptune are obviously stable and mathematical models say that they should remain stable for at least the next 3.5 billion years, after which, there's a chance Mercury could become chaotic and fly out of it's orbit, flying into the sun, crashing into another planet or perhaps leaving the solar system.

Any of these orbital systems, for any kind of long term prediction of where the planets will be in a billion years, requires n-body calculations which are very complex and inexact, but they can be calculated within a range of error, which makes it possible to say that a system is stable, for a couple billion years anyway. You can't, for example, ignore the other planets and calculate where the Earth will be in 50 million years. The planets affect Earth's orbit but they don't destabilize it. This is called perturbation theory.

And the final example, binary star system with a planet orbiting the two stars.

This XKCD discusses the binary star system, and from same:

A planet orbiting two stars can't get too close to them or its orbit
becomes unstable. If it gets too close, the irregular tugging from the
gravity of the two stars as they orbit will eventually cause the
planet to crash into one of them or get flung out of the system.

For a system with two similar-sized stars, this "critical radius" is
around six times the distance between the two stars.

What this means is that any 3 body system has to follow the rules if it's going to be stable. You can have several planets orbiting a central star, and those planets will perturb each other, sometimes casting a planet out but you can also have stability.

You can have Sun, Planet, Moon, but the Moon must be inside the stable region of the Hill Sphere.

And you can have one object orbiting two central objects, like Tatooine, or, like the 4 satellites that orbit Pluto-Charon.

But anything outside those rules is an unstable 3 body or n-body system and it may take some time, but it's ultimately unstable and unlikely to last anywhere close to the billion(s) of years or so that stable orbits tend to last.

So a star with two planets who's orbits cross each other (note, Pluto and Neptune don't actually cross, they just appear to cross when drawn in 2 dimensions). Their orbits don't come close to crossing.

Horeshoe orbits are 3 body orbits that might last a fairly long time, but I don't believe they are truly long-term stable (if anyone can verify that, please do), but intuitively, I don't see how horeshoe orbits could last very long.

When you have a 3 body system where there's no structured orbit and all 3 bodies kind of dance around each other. That system isn't stable.

If properly balanced, a 3 body system outside the rules of stability might last for hundreds or thousands or orbits, but in celestial mechanical time that's not stable.

Mathematically some 3 body orbits have been worked out, but they are stable in the same way that a pencil is stable if you balance it perfectly on it's point in a room with no wind. While this article says that they found 13 solutions, none of these solutions would actually work because space isn't perfectly smooth. It's full of objects with gravity. So, like the balanced pencil that gets knocked over by a slight gust of wind, a passing rogue planet would throw any of these "solutions" out of whack and they would progressively destabilize. Where as a stable system (sun/earth/moon) can hande some perterbations and remain stable.

This diagram below is an example of a perfectly stable 3 body orbit in a computer program, but like the pencil, a small push on any one of the 3 objects would send it towards instability.

Hope that's not too long. I can clean up if needed.

Thanks! That's what exactly my question, non hierarchical systems. Perfect answer.