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We find small objects (asteroids and dust) in the stable Lagrange points (L4 and L5), but AFAIK no moons (by which I mean a mass that accreted into a body, as opposed to debris that has been captured but lacks sufficient gravity to accrete into a spherical body). Is there something that prevents such a critical accretion-level mass from accumulating at L4 or L5?

(I understand that L4 and L5 are stable only for mass small relative to the two objects they orbit. But I assume that, for example, our gas giants could tow moons at L4/L5 with masses on the order of their conventional moons.)

Edit: From the comments (What is the difference between a trojan asteroid and a "moon?") I see that my underlying question was lost in the definition of "moon." What I meant when I said "moon" was "an object with sufficient mass for its self-gravity to overcome rigid body forces so that it assumes a hydrostatic equilibrium." Maybe this is properly termed a "planetoid?" (I have substituted that term in the title.)

The reason I thought a planetoid at a Lagrange point would be called a "moon" is because its orbit is created by a planet significantly more massive than it. But if the definition of "moon" requires that the object's orbit encircles its planet then I have (again) misused the term.

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    $\begingroup$ What's the largest trojan asteroid and what is it that makes it not a moon? $\endgroup$
    – uhoh
    Nov 16 '21 at 19:20
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    $\begingroup$ @uhoh The largest trojan asteroid is much larger than the smallest moon of Jupiter. What makes it not a moon is that it orbits the Sun, a star, instead of Jupiter, a planet. $\endgroup$ Nov 16 '21 at 19:46
  • $\begingroup$ @Feetweet Trojan objects do not orbit both the Sun and the planet, they only orbit the Sun, though at the same distance as the planet. $\endgroup$ Nov 16 '21 at 19:47
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    $\begingroup$ @uhoh thank you for the prompt. I just edited to clarify the question. Also, your link shows that there are planetoids at the stable Lagrange points of the gas giants, so your comment also provides an answer! $\endgroup$
    – feetwet
    Nov 16 '21 at 20:36
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    $\begingroup$ I think the proper term would be "dwarf planet" rather than "moon" or "planetoid". A gravitationally rounded body at L4 would be a dwarf planet. $\endgroup$
    – James K
    Nov 16 '21 at 20:48
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In astronomy, a trojan is a small celestial body (mostly asteroids) that shares the orbit of a larger one, remaining in a stable orbit approximately 60° ahead of or behind the main body near one of its Lagrangian points L4 and L5. Trojans can share the orbits of planets or of large moons.

https://en.wikipedia.org/wiki/Trojan_(celestial_body)

Whether or not a system of star, planet, and trojan is stable depends on how large the perturbations are to which it is subject. If, for example, the planet is the mass of Earth, and there is also a Jupiter-mass object orbiting that star, the trojan's orbit would be much less stable than if the second planet had the mass of Pluto.

As a rule of thumb, the system is likely to be long-lived if m1 > 100m2 > 10,000m3 (in which m1, m2, and m3 are the masses of the star, planet, and trojan).

https://en.wikipedia.org/wiki/Trojan_(celestial_body)#Stability

If I interpret that rule of thumb correctly, since the Sun has about 330,000 times the mass of Earth, a trojan system would be stable if the mass of the planet was less than 3,300 times the mass of the Earth and if the mass of the trojan object is less than 0.0001 the mass of the planet and lesss than 0.33 times the mass of Earth.

So it seems like the maximum possible mass of a trojan in Earth orbit would be about 0.0001 times the mass of Earth. If such an object had the same overall density as Earth, it would have only 0.0001 times the volume of Earth, and thus diameter of about 0.0465 that of Earth, or 592.506 kilometers, which is larger than any asteroid except for Ceres.

Since Jupiter, the largest planet, has 317.8 times the mass of Earth, the largest possible Jupiter Trojan would have 0.03178 times the mass of Earth. That is a bit larger than Ganymede, the most massive moon in the solar system, with 0.025 times the mass of Earth.

So according to that rule of thumb the planets in our solar system could have objects in their trojan orbits which are many, many times more massive than the most massive objects that are in their trojan orbits.

And the reason for that is probably statistical. For a planet to have an object in a trojan orbit two things have to happen.

  1. The object has to form

  2. The object's orbit has to eventually be modified so that it ends up in a trojan point relative to that planet.

And it is well known that there are only 32 known planetary mass objects in the solar system large enough that their gravity has pulled them into roughly spherical shapes.

They include 1 star, 8 planets, 5 dwarf planets (and a number of other candidates), and 19 moons. Others may be discovered in the future.

https://en.wikipedia.org/wiki/List_of_gravitationally_rounded_objects_of_the_Solar_System

There are at least 198 other known smaller moons, plus at least 1 million known asteroids.

Asteroids vary greatly in size, from almost 1000 km for the largest down to rocks just 1 meter across. The three largest are very much like miniature planets: they are roughly spherical, have at least partly differentiated interiors, and are thought to be surviving protoplanets. The vast majority, however, are much smaller and are irregularly shaped; they are thought to be either battered planetesimals or fragments of larger bodies.

The number of asteroids decreases markedly with size. Although this generally follows a power law, there are 'bumps' at 5 km and 100 km, where more asteroids than expected from a logarithmic distribution are found.

https://en.wikipedia.org/wiki/Asteroid#Size_distribution

So it is much more common for tiny objects a few kilometers across to form than for objects large enough to be the largest possible trojan objects for a planet to form.

In the early solar system there were many more large protoplanets. Gravitational perturbations between them caused most of them to fall into the Sun, or to collide with other protoplanets to form larger protoplanets, or to be ejected from the solar system and become rogue planets in interstellar space.

After the first few hundred million years or so most of the larger protoplanets were gone, one way or the other, leaving only the eight major planets and their moons in stable and widely separated orbits, and there was no longer any chance for large protoplanets to be captured into trojan orbits.

But tiny asteroids such a few kilometers wide were and are many, many, many, times as common as even the smallest spherical protoplanets, and the solar system still has hundreds of thousands of asteroids that size. So statistically it was much more probable for a planet to capture one or more of those tiny asteroids into a trojan orbit than to capture even one large rounded object into a trojan orbit.

I expect that star systems where the largest trojans are merely asteroid sized, as in our solar system, should be much more common than star systems where the largest trojans are planetary mass objects.

Astronomers have discovered so far fewer than 5,000 exoplanets in other star systems, out of the hundreds of billions of planets which should be in the Milky Way Galaxy. They have discovered several hundred star systems with multiple planets.

And they have found that many star systems are very different from ours. So it seems that it is hard to predict what the numbers, masses, and orbital characteristics of the planets in a star system will be. Random chance seems to have played a large role in planetary system formation.

So no doubt there are a small percentage of star systems where one or more planets have trojan objects which are as large as those planets could possibly have.

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  • $\begingroup$ The stability tolerance is not quite that bad because opposing trojans cancel. $\endgroup$
    – Joshua
    Nov 17 '21 at 21:53
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Moons formed in a number of ways.

Many moons are just like the debris that is found at L4 and L5, of the nearly 80 known moons of Jupiter, most are small, irregular and are probably captured asteroids. These small moons are much like the population of Trojans.

There are a few larger moons, the “Galilean moons formed out of the dusty disk left over after Jupiter’s formation” (source), that is they formed in something like a mini-solar system. It was the proximity of Jupiter that formed this dusty disk, and provided the conditions for large moons to form. These conditions didn't exist at L4, so no large bodies formed.

Moons couldn't form at L4/5, because there is no centre of mass there to form an accreting "dusty disk". The region in which Trojans are found is a huge volume of space, not a densely packed disk. The material is at L4 and L5 is simply too widely spread to self gravitate into a single body.

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    $\begingroup$ I think I'm reading the question differently from you. I understand it to mean "Why didn't the trojans self-gravitate into a single large body". Of course that body wouldn't be "a moon" as it would be in orbit around the sun" it would be a dwarf planet. But I'm not lecturing the OP on terminology. Rather looking at the mechanism for the formation of the large moons of Jupiter. $\endgroup$
    – James K
    Nov 16 '21 at 20:23
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    $\begingroup$ There was a specfic mechanism to form large moons around Jupiter, no such mechanism exists to form large bodies at L4/5 so no bodies form. $\endgroup$
    – James K
    Nov 16 '21 at 20:25
  • $\begingroup$ Okay this is helpful, I'll go off and give it some thought, thanks! $\endgroup$
    – uhoh
    Nov 16 '21 at 20:27
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    $\begingroup$ So, to confirm my understanding of your answer: The formation of planetoids requires an accretion disk, and there is nothing about a Lagrange point that provides a gravitational seed for such a disk? (But if, however improbable, sufficient mass accumulated in a sufficiently small area of L4/5, then it would form a planetoid, correct? I.e., there is nothing about L4/5 that disrupts accretion?) $\endgroup$
    – feetwet
    Nov 16 '21 at 20:43
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    $\begingroup$ Yes, there is no "roche limit" type effect at L4 that will tidally (or otherwise) disrupt a body at L4 $\endgroup$
    – James K
    Nov 16 '21 at 20:47
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To add to the other answers, there is a theory (unproven, but fairly widely accepted as plausible) saying that the Earth in fact once, early during the solar system's formation, had a roughly Mars-sized co-orbital companion planet (called Theia) at its L4 or L5 point, and/or possibly in something like a horseshoe orbit.

The theory suggests that eventually, as the numerous protoplanets and smaller lumps of rock in the early solar system merged and grew through collisions and interacted gravitationally with each other, Theia's co-orbital arrangement with the Earth would've been gradually destabilized until it finally, possibly after multiple close passes, ended up colliding with the Earth in a giant impact about 4.5 billion years ago. The iron cores of both planets would've merged (thereby explaining the Earth's anomalously large core), while silicate material from both planets' mantles (but in most simulations primarily from Theia) would've been scattered into orbit around the main planetary body formed in the merger, where much of it would've eventually coalesced into a single body — the Earth's anomalously large moon.

There is still no full agreement on some details of this "giant impact scenario", including the size and original orbit of Theia, and some aspects of it may never be known for certain: as we have no time machine to observe the actual events, the best we can do is reconstruct plausible scenarios, simulate them and see how well the results agree with the Earth–Moon system that we observe today. But the idea of Theia having been co-orbital with Earth does neatly explain various details, such as the relatively low reconstructed impact velocity and the similar chemical and isotopic composition of the Earth's and the Moon's crusts (which would be expected to differ if the Earth and Theia had formed at different distances from the Sun).

So, one facetious answer to your question could be "because we're about 4.5 billion years too late."

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