Trojans have been found at the L4 and L5 points of most planets. However Tethys and Dione, both moons of Saturn, are the only known moons in the Solar System to have trojans of their own.
Is this just a coincidence? Or is there something unique separating Tethys and Dione from the rest of the moons in the Solar System?
This is a case where size matters.
Tethys and Dione are rather large as moons go. They are large enough to be roughly spherical planetary mass objects.
Tethys has dimensions of 1076.8 by 1057.4 by 1052.6 kilometers.
Dione has dimensions of 1128.8 by 1122.6 by 1119.2 kilometers.
Sorting solar system objects by radius, from the Sun to asteroids, Dione is number 29 or 30 and Tethys is 31 or 32.
https://en.wikipedia.org/wiki/List_of_Solar_System_objects_by_size
There are hundreds of thousands of known bodies which are smaller than Tethys and Dione. And millions of small undiscovered objects.
At the present time there are 219 confirmed moons in the solar system orbiting around 6 planets and 11 dwarf planets and dwarf planet candidate.
https://en.wikipedia.org/wiki/List_of_Solar_System_objects_by_size
There are also 309 moons of asteroids and 119 moons of trans Neptunian objects Including 16 orbiting around dwarf planets and dwarf planet candidates.
Dione is the 15th largest moon and Tethys is the 16th largest. There are only 19 moons presently known to be large enough to be gravitationally rounded like Dione and Tethys.
The smallest moon that is more or less rounded is Mimas, with a radius of about 198.2 kilometers and diameter of about 396.4 kilometers. The only other planetary moons with radii over 100 kilometers are Nereid, Hyperion, and Phobe.
The majority of moons have radii below 10 kilometers.
And there is a rule of thumb for estimating the long term stability of a trojan orbital arrangement.
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).
In the case being considered, the mass of the planet should be at least 100 time the mass of the large moon, which should be at least 100,000 times the mass of the smaller moon in the trojan orbit.
Since Tethys has a mean radius of about 531.1 kilometers, an object which the same average density and less than 100,000th the mass of Tethys, would have to have i00,000th of the volume of Tethys, which would require a radius of less than about 1 46th that of Tethys or less than about 11.545652 kilometers. Telesto is about 32 by 23 by 20 kilometers, while Calypso is about 30 by 23 by 14 kilometers. So Telesto and Calypso are close to the estimated maximum size of Tethys trojans, and might be less dense than Tethys.
Since Dione has a mean radius of about 561.4 kilometers, an object which the same average density and less than 100,000th the mass of Dione, would have to have i00,000th of the volume of Tethys, which would require a radius of less than about 1 46th that of Dione, or less than about 12.204 kilometers. Helene is about 43 by 38 by 26 kilometers, while Polydeuces is about 3 by 2.5 by 2 kilometers. Helene might slightly ec xceed the maximum mass of a trojan of Dione according to the rule of thumb, though it might be less dense than Dione, and Polydueces is much smaller than that.
The majority of moons have radii of less than 10 kilometers, and thus diameters less than 20 kilometers. So the maximum diameter of a trojan with 100,000th the mass of such a small moon would be about 0.43 kilometers, or 430 meters, or 1,410 feet. And such small objects would be hard to detect at the distances of the outer planets. So if there are any such tiny trojans of moons, they probably wouldn't have been detected yet.
But there still about 15 moons which could have trojans about the size of the trojans of Tethys and Dione, but don't have any detected trojans.
The early solar system included countless planetesimals that formed out of the protoplanetary disc and which gradually clumped together to form planets, dwarf, planets, moons and asteroids.
A good reason why there are no known trojans of moons of Neptune is that Neptune has only a few moons, mostly small one. And the reason that is that Neptune captured its largest moon, Triton, and in a retrograde orbit, a process which may have perturbed the orbits of most of Neptune's moons, and any trojans they might have had, so they fell into Neptune, collides with Triton, or were ejected from orbit.
Any object in a trojan orbit with Triton would have had to have been captured by Neptune like Triton was, and after Triton was captured, instead of being an original moonlet of Neptune.
So that explains why Neptune, one of the four giant planets has no known trojan moons.
The capture of Triton by Neptune was not the largest such event in the history of the solar system. The orbits of planets and other objects where not as stable soon after they formed, and so during the early solar system there were many planetary mass objects whose orbits were perturbed by other objects and which fell into the Sun, collided with other planetary mass objects, or were ejected from the solar system.
The process of planets, moons, and other objects adjusting their orbits by mutual gravitational attraction caused the loss of many solar system objects through collision or ejection, until the solar system settled do to having a lot fewer objects, most of them with orbits that are stable over geological time, and thus reducing the chances that an object with move into a trojan orbit with a moon of a planet.
So there is a strong probability that random factors in the early solar system caused Tethys and Dione, and only Tethys and Dione, out of all the larger moons, to acquire objects in their trojan L4 and L5 orbital positions.
It is probably just a coincidence that the only two moons with known trojan objects orbit the same planet in adjacent orbits.
It is possible that there is special about the position of the orbits of Dione and Tethys in the ring system and satellite system of Saturn which makes them more probable to have trojan orbits than any other satellites. But if that is the case I don't know what it is.
