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Here's the basic gist about our moon--it's one-quarter the width and one-eighth the gravity of Earth, it's rocky, it has no atmosphere and it orbits Earth from a distance of approximately 240,000 miles.

It's something of a trope to ask the question, "What if the Earth has two moons?" However, this question concerns a second moon the exact same size as the first moon orbiting Earth from the exact same distance as the first moon. In short, a Trojan. In this question, our moon is placed between the Lagrangian points L1 and L2 and the second moon in L3. Regarding stability, could this orbital arrangement be stable in the long term?

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    $\begingroup$ No, it is unstable. Even when there are some stable solutions for 3 body problem, if u recall potencial way moon was formed, all the trash after collision, epic collision, is long gone., no traces left. $\endgroup$
    – MolbOrg
    May 23, 2021 at 16:59
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    $\begingroup$ Also, the cancelling of the gravitaional forces of the 2 main bodies in the L-points only work for a very small 3rd mass, neglectible in comparison. A third mass the size of a proper moon and things get chaotic again. $\endgroup$
    – user78828
    May 23, 2021 at 17:03
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    $\begingroup$ In L3 it won't orbit Earth, it would coorbit the Sun with Earth $\endgroup$
    – L.Dutch ♦
    May 23, 2021 at 17:06
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    $\begingroup$ "Lagrangian points" of what system? Earth-Moon, Earth-Sun? The Lagrange point L3 of the Earth-Sun system is on the other side of the Sun... (If you mean a sort-of anti-Moon on the other side of Earth, then consider what exactly is cancelling the gravitational attraction between Moon One and Moon Two.) $\endgroup$
    – AlexP
    May 23, 2021 at 17:38
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    $\begingroup$ L1, L2, and L3 are not even stable in the short term, let alone the long term. NASA does have some spacecraft at L1 and L2 - L1 is good for viewing the sun, L2 for deep space astronomy - but they have to use active maneuvering to stay there. Per NASA, they're unstable at ~23 days: solarsystem.nasa.gov/resources/754/what-is-a-lagrange-point $\endgroup$
    – jamesqf
    May 23, 2021 at 18:54

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There are many wrong points in your reasoning.

First of all, if a body is in the L3 point of the Earth-Sun system, it is orbiting the Sun along the same orbit of Earth, so it cannot be a second Moon.

Moreover, L3 is way more distant than the Moon from Earth, which is also one of your requirements.

Last but not least, to answer your question about stability

The Sun–Earth L3 is unstable and could not contain a natural object, large or small, for very long. This is because the gravitational forces of the other planets are stronger than that of Earth (Venus, for example, comes within 0.3 AU of this L3 every 20 months).

If instead by any chance your mean the L3 of the Earth-Moon system, same consideration will hold: the forces induced by the first Moon (which you place somewhere between L1 and L2) and the Sun would quickly perturb any body happening to be in L3.

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  • $\begingroup$ Idk, maybe a last nail to the bucket, moon orbit by itself isn't particularly stable it moves 6cm a year away, if I recall correctly. So another source of perturbances. Which may look strange in idealistic model, in which OP operates, and in which it may look reasonable to have another opposite body to orbit, but in reality moon orbits sun, aaand its orbit by about 20% influenced by earth. $\endgroup$
    – MolbOrg
    May 23, 2021 at 19:54
  • $\begingroup$ @MolbOrg: It moves away because Earth is neither tidally locked nor a rigid body; otherwise, it is pretty stable -- it has been orbiting the Earth for billions of years. (Earth is not tidally locked to the Moon; the tidal forces of the Moon flex Earth, producing friction, which dissipates rotational energy, slowing Earth's rotation; the angular momentum has to go somewhere, and it goes into the Earth-Moon system, widening the orbit.) $\endgroup$
    – AlexP
    May 23, 2021 at 22:18
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A small asteroid could orbit Earth at the distance of the Moon in the L4 point, 60 degrees ahead of the Moon, or the L5 point, 60 degrees behind the Moon. Orbits in L4 or L5 points are called Trojan orbits.

https://en.wikipedia.org/wiki/Lagrange_point[1]

The lunar L4 and L5 points are also considered to be stable points for artificial space habitats, so explaining the name of the L5 Society.

https://en.wikipedia.org/wiki/L5_Society[2]

But I have the impression that in Trojan obits the main object, in this case the Earth, should have many times the mass of the secondary object, in this case the Moon, which in turn has to have many times the mass of the tertiary object in the L4 or l5 point.

As far as I know the only objects in the L4 and L5 points of the Moon are concentrations of interplanetary dust called the Kordylewski clouds, so faint that though they were first detected in 1956 they were not confirmed until 2018. Their mass must be minute compared to that of the Moon.

Perhaps the Earth could have two moons of equal mass in the same orbit, 60 degrees apart, one moon being in the L4 point 60 degrees ahead of the other, and the other moon being in the L5 position 6 degrees behind the other. But I don't know if such a situation would be stable, and I know of no examples of such a situation.

It has been claimed that two planets with similar mass could be stable in the same orbit if they were 60 degrees apart.

Two planets with similar masses can also share the same orbit if they orbit 60 degrees apart. This means that each is in the other’s L4/L5 Lagrange point. This kind of configuration comes out of our computer simulations, and we expect to find one of these setups among exoplanet systems.

https://planetplanet.net/2020/11/19/cohorts/[3]

If that is correct, two identical mass moons ought to also be able to share the same orbit, spaced 60 degrees apart.

Here is some additional information.

The PlanetPlanet blog has a set of posts called The Ultimate Solar System, designing solar systems which have as many habitable planets as possible.

The post The Ultimate Engineered Solar System designs a solar system which doesn't have single planets in each orbit, but rings of planets in each orbit.

https://planetplanet.net/2017/05/03/the-ultimate-engineered-solar-system/[4]

It seems that a solar system could have a number of planets sharing the same orbit, as long as the planets have equal mass and are equally spaced in the orbit. The source is this paper:

https://ui.adsabs.harvard.edu/abs/2010CeMDA.107..487S/abstract[5]

Apparently such a ring of planets could be stable with seven to forty two planets in a single orbit.

And what is stable for a ring of planets around a star would also be stable for a ring of moons around a planet. Except that the gravity of the star would be a perturbing factor.

The Moon has a mass of 0.012300 of the Earth's mass, and the Moon's orbit with a semi-major axis of 384,399 kilometers would have a circumference of approximately 2,415,248.1 kilometers if it was circular.

So if there were 7 moons at the distance of the Moon and with the same mass as the Moon, they would have a total mass of 0.0861 Earth mass, and they would be spaced 51.4285 degrees, or about 345,035.44 kilometers, apart in their shared orbit.

So if there were 42 moons at the distance of the Moon and with the same mass as the Moon, they would have a total mass of 0.5166 Earth mass, and they would be spaced 8.5714 degrees, or about 57,505.904 kilometers, apart in their shared orbit.

In another post, Cohorts of Co-Orbital Planets, it was proposed that arcs of planets could share stable orbits and they didn't need to be complete rings.

https://planetplanet.net/2020/11/19/cohorts/[3]

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