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Would it be possible for a planet to naturally have a moon or large natural satellite at one of its Lagrange points?

If so would it then also be able to have a second moon that would be orbiting this hypothetical planet?

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    $\begingroup$ "hypocritical" - you probably mean "hypothetical" $\endgroup$ – StephenG Jul 15 at 20:41
  • $\begingroup$ By "lagrange points" you mean specifically L1 or L2, rather than L3, L4 or L5? $\endgroup$ – Alexander Jul 15 at 20:47
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    $\begingroup$ Jupiter has a large number of trojans (= small celestial bodies that share the orbit of a larger one, remaining in a stable orbit approximately 60° ahead or behind the main body near one of its Lagrangian points L4 and L5) and an impressive set of satellites. What does the question ask which this immediately available real-world example doesn't provide? $\endgroup$ – AlexP Jul 15 at 20:52
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    $\begingroup$ Do bear in mind that the Earth's L4 and L5 points are as far from the Earth as the sun is. If our moon was at one of those points, it would be a tiny and dim point of light in the sky, though it would be visible with the naked eye — my back-of-the envelope calculation gives it a magnitude of about 3 or 4, since we’d also never see it at full phase. $\endgroup$ – Mike Scott Jul 16 at 6:32
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Yes it could

The Wikipedia article on the subject is quite detailed and offers the exact answer to your question.

The L4 and L5 points are stable provided that the mass of the primary body (e.g. the Earth) is at least 25 times the mass of the secondary body (e.g. the Moon). The Earth is over 81 times the mass of the Moon (the Moon is 1.23% of the mass of the Earth.)

Thus having quite a large L4 or/and L5 satellite is perfectly reasonable. The biggest possible Lagrange satellite would orbit a 13.8 Jupiter-mass superjovian (this is the upper mass limit for planets since above it Deuterium fusion would happen, turning the planet into a Brown Dwarf) and would thus have a mass of 0.552 jupiter-masses. For an earth-like world this would be an 0.04 earth-mass object, 3.24 times more massive than Luna.

As for L1, L2 and L3 satellites it is gonna be a no.

Although the L1, L2, and L3 points are nominally unstable, there are (unstable) periodic orbits called "halo" orbits around these points in a three-body system. A full n-body dynamical system such as the Solar System does not contain these periodic orbits.

Concerning what you'll usually find at a planets L4 and L5.

It is common to find objects at or orbiting the L4 and L5 points of natural orbital systems. These are commonly called "The Trojans".

The giant planets of the outer solar system do have extensive Trojan asteroid clouds, although they are poorly studied. Even Mars, Earth and Venus do have some garbage flying around these points. I suggest you read yourself through the List of objects at Lagrangian points.

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TheLuckless's comment

A comment asked what the odds of having a large object instead of a swarm of rubble are. While I wasn't able to find any papers on that subject, the ones I read about the stability issue seemed to assume one big object of that mass. Furthermore, running a back-of-the-envelope calculation based on solar system data could give one a rough value for that chance. Note that I ignore stability effects due to the objects distance to the sun and many other, most likely important factors.

The list of gravitationally rounded objects in the solar system tells me that there are roughly 37 eligible objects in the solar system. Of these, Mars and the giant planets have significant Trojan debris clouds, telling me that 10 out of 74 possible positions are occupied by debris clouds; thus there is a 13.5% chance for it. Lagrange moons near the mass limit can only be found orbiting the Saturian moons Tethis and Dione, namely Telesto, Calypso and Helene (not counting Polydeuces, since it is extremely small). Thus 3 out of 74 possible positions are occupied by Lagrangian moons, a 4% chance. It therefore follows that there is a 24% chance that Lagrangian objects are a massive moon.

Orbital behavior

I once asked a professor of astrophysics how an L4/L5 satellite would behave, and he said that big Lagrangian objects will sit still at the exact L4/L5 point, unlike many of the asteroids which orbit the points and may diverge as far as 5° in the orbit from them.

Moons of L4 and L5 satellites

I somehow missed this during my original answer. Moons of moons are always a tricky subject. My answer here goes into detail on how many moons a planet could have. The Tldr version is that, in a normal setting, the inner half of the Hill-sphere offers stable orbits for moons. I suspect that this zone will be much smaller due to the already volatile setup of the system. While I would consider moons which are in hydrostatic equilibrium highly unlikely, captured asteroid moons as they can be found around Mars, Pluto, the giant planets and most of the Kuiper belt objects are quite likely. Spacerocks naturally converge at L4 and L5 as the big Trojan asteroid swarms show. That some of them get captured by a big moon sitting there is very likely. In fact I would bet a lot of money on us finding a moonlet around one of the bigger Trojan asteroids if we were to send a probe there.

More math

While researching this topic I came across this and this. They discuss the stability of the Lagrange Points and the mass limit of a satellite there in great detail and with lots of math. Should you really wanna get into the details of the subject they could be very interesting.

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    $\begingroup$ A useful expansion might be if someone has math to show any probability one way or the other over whether L4 and L5 could reasonably gather a 1/25th bit of rock as a stable orbit, or if 'trash fields of small stone' are the only thing likely to be able to form at those points. $\endgroup$ – TheLuckless Jul 15 at 22:56
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    $\begingroup$ @TheLuckless I planned on doing that until I learned that my bus has no option to charge my phone... As soon as I have reliable electricity I'll continue my answer. $\endgroup$ – TheDyingOfLight Jul 16 at 5:54
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    $\begingroup$ @TheLuckless I'll look into it and post more details, but I've read theories that Theia formed at L4 or L5 and an Astrophysicsprofessor I asked the questions onc3 said that the interaction of a significant will likely lead to planet formation if the massive object wasn't captured to begin with. $\endgroup$ – TheDyingOfLight Jul 16 at 6:01
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    $\begingroup$ Worth noting that this 'moon' wouldn't orbit the planet in the normal sense; it would lead or trail the planet by 60 degrees in its orbit, If two large planets orbited each other, they could feasibly have a moon in their common L4 or L5 points. $\endgroup$ – Klaus Æ. Mogensen Jul 16 at 7:37
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    $\begingroup$ Also check out 3753 Cruithne, sometimes dubbed "Earth's second moon". en.wikipedia.org/wiki/3753_Cruithne $\endgroup$ – Klaus Æ. Mogensen Jul 16 at 7:39
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If so would it then also be able to have a second moon that would be orbiting this hypothetical planet?

It would have to.

The Lagrange points are born between two orbiting bodies; a single body has no Lagrange points. If you have the sun and the planet, then the Lagrange points are the solar Lagrange points and a celestial body there would not be a moon (it would orbit the sun, so it would be a sort of a sister planet. On the other hand, the Aten asteroid 3753 Cruithne is often incorrectly referred to as "Earth's second moon", so it would depends on how strict you want to be).

To have a small moon in a Lagrangian position, you need a larger moon orbiting the planet, and this would give you a set of Lagrange points related to the planet and the large moon. The L4 (Greek) and L5 (Trojan) points would be stable if the large moon's mass is below 4% of the planet's, and the smaller Greek or Trojan moon's mass is below 0.16% of the planet's.

The two moons would orbit the planet at the same distance and with the same speed (on average. The smaller moon would actually describe a small kidney-shaped orbit around the L4 or L5 point). One would simply be much smaller than the other.

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    $\begingroup$ The question appears to be talking about Lagarnge points of the Sun-Planet system of orbits, not of a Planet-Moon system. ["a moon or large natural satellite at one of its Lagrange points"] Earth's L-Points orbit the sun, the Moon's orbit Earth. Status of something at a planet's L-Point as 'A Moon' or not is a less than overly useful academic debate about excessively arbitrary concepts. $\endgroup$ – TheLuckless Jul 15 at 22:52
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    $\begingroup$ I disagree about 3753 being an Earth's-moon type object. It shares Earth's orbit more than it orbits Earth: en.wikipedia.org/wiki/3753_Cruithne#/media/… $\endgroup$ – Draco18s Jul 16 at 15:06
  • $\begingroup$ @Draco18s for what is worth, I share your opinion. But as TheLuckless noted, someone could argue that such an object is a "sort of moon". Indeed, Cruithne is only one of Earth's "claimed" moons ( see en.wikipedia.org/wiki/Claimed_moons_of_Earth ) $\endgroup$ – LSerni Jul 16 at 16:06
  • $\begingroup$ @LSerni even that page refers to Cruithne as a "quasi-satellite" as it is in 1:1 resonance with Earth, but orbits the sun. Trojans are a special class of quasi-satellite that lives near (Earth-Sun) L4 or L5. The key difference between a satellite and a quasi-satellite is that the orbit of a satellite of Earth fundamentally depends on the gravity of the Earth–Moon system, whereas the orbit of a quasi-satellite would negligibly change if Earth and the Moon were suddenly removed because a quasi-satellite is orbiting the Sun on an Earth-like orbit in the vicinity of Earth. $\endgroup$ – Draco18s Jul 16 at 16:11

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