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Is there a good way to measure the orbital stability of two moons around a planet?

Here I mean something like a mathematical constant or an equation that can approximate the stability of the orbits of the two moons since the two moons are relatively close to each other and I need to calculate how stable their orbits are.

My question does not involve a specific planetary system, It is a general scientific question.

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    $\begingroup$ What sort of stability are you looking for. That's a three body problem, so they are by definition unstable by astronomy's definitions. But they may be "stable enough" by your terms $\endgroup$
    – Cort Ammon
    Nov 7, 2021 at 23:30
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    $\begingroup$ You need to clarify what you mean by "the two moons are relatively close to each other." Moons that orbit at the same distance from the planet will also have the same orbital period. See Kepler's 3rd Law. The moment your two moons have a different distance from the planet, they will have different orbital periods. That means they'll spend some of the time on the 'same side' of the planet, and other times across from each other on opposite sides for planet. $\endgroup$ Nov 8, 2021 at 3:57
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    $\begingroup$ @fasterthanlight Questions that can be asked on another stack are not automatically off-topic on the WB.SE. Please see here for more details. $\endgroup$
    – Otkin
    Nov 8, 2021 at 16:54
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    $\begingroup$ @fasterthanlight I'm quite certain we've a lot of quite technical answers about astronomy and astrophysics right here on WB SE, including those using the hard-science tag. $\endgroup$ Nov 8, 2021 at 17:10
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    $\begingroup$ FYI: Orbital resonance $\endgroup$
    – Alexander
    Nov 8, 2021 at 17:10

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Numerical Modelling Is The Way

As stated in comments, the three body problem is an unsolved problem in physics, which includes all two-moons-and-planet system. No "nice" equation exists, although the numerical model method give you an answer with some "nice" concepts and a thing to measure.

The basic approach is to figure out the six gravitational forces in the system, apply those forces to each body (moon/planet) over a small time step to get new positions, and repeat.

And repeat.

And repeat.

And repeat.

...

And repeat.

Until the heat death of the universe, the power goes out, the computer breaks, things crash into each other (in your model or otherwise), or you get tired of it.

Where is the payoff? Well, you count how many steps until one of those above conditions happen in/to your model, and that is how you measure stability.

There are a lot of little questions when doing this that get the pendants really going. Like... is your time step short enough? Did you check for collisions? Did you check for bodies passing through each other? Did you go long enough? Did you account for any liquid rock which may be going on? Did you model them anywhere along the sliding scale from three solid bodies to every atom in each body? (They will make uncomfortable noises no matter what you say: either uncomfortable for you or uncomfortable for them.) What if each body has a magnetic field? And it goes on...

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