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Does our moon have any captured objects orbiting it that have been discovered, such as asteroids or other debris?
If it doesn't, is it possible that it could capture them in the future, given the right circumstances? If so, what would be the maximum size of such an object?
It does not have any known objects. It seems unlikely that it would happen, although it could theoretically happen for a short period of time. There are few stable orbits around the Moon, and even fewer that are likely to remain stable for any length of time. Even Lunar orbiting missions don't remain stable, and they have a better chance than a random object would of orbiting the Moon, as they are deliberately placed away from the moon.
Size would not be an issue, the maximum size limit would be rather large. There are no known Earth crossing objects that are that large.
An object would have to have a roughly circular, but high, polar retrograde orbit to remain in Lunar orbit. That seems very unlikely to happen by pure chance.
The main problem is that the Moon has many mascons, not a uniform sphere. Close orbits around it tend to get perturbed. They keep the same semi-major axis but become more and more elliptical until they hit the Moon. Early satellites sometimes hit the moon within weeks, while others lasted for months or longer, and it took them a while to figure out why. Contrary to what some articles say on the web, its Hill sphere is easily large enough to permit moonlets to orbit the moon, the problem is, the mascons.
from the NASA page: Bizarre Lunar Orbits:
"There are actually a number of 'frozen orbits' where a spacecraft can stay in a low lunar orbit indefinitely. They occur at four inclinations: 27º, 50º, 76º, and 86º"
That's especially so for surface skimming orbits, or ones that get quite clos. For more distant orbits - the most stable orbits are retrograde. A retrograde orbit around the moon, looked at another way is an elliptical prograde orbit around the Earth, with the same orbital period as the Moon. When closer to the Earth then it orbits it more quickly than the moon and when further away, then it orbits more slowly than the moon, so combined effect is that when seen from the moon it orbits it in a retrograde direction.
You can get retrograde orbits both inside and outside of the lunar L1 and L2 positions - i.e. either inside both of them, or outside both of them, relative to the moon.
These orbits are stable over time periods of about a century or so. But not necessarily over longer time periods. With the NASA asteroid capture mission, one of the proposals is to return it to a lunar retrograde orbit.
So far though there are no known natural objects in any lunar orbits.
However we wouldn't be able to see really small objects of just a few meters in diameter if they were orbiting the moon. So that leads to a natural question - could it have temporary satellites that we don't know about, which eventually either hit it, or escape to interplanetary space via the Sun Earth L2 (by coincidence the Sun Earth L2 is accessible to either of the lunar L1 or L2 points by a low energy transfer with almost no delta v).
You do get temporary extra moons of the Earth, which orbit it for a few months and then leave it again. Such as 2006 RH120, 2-3 meters across which was a satellite of the Earth for nearly a year: September 2006 to June 2007. It's estimated that at any particular time there is probably at least one temporary moon of the Earth one meter in diameter, and perhaps 700 or so tiny moons at least 10 cm in diameter.
Also there may be dust in the L4 and L5 positions in the Kordylewski cloud if it exists - temporary because those are unstable in the Earth Moon system.
So given that population of tiny moonlets of the Earth, and these L4 and L5 transient continually replenished and dissipating clouds (if they exist), could there be some moonlets at least 10 cm in diameter in 28 day elliptical orbits around Earth that make them temporary retrograde moonlets of the moon?
But I don't know of any similar result for the Moon.
Indeed there are, so far, no known moons anywhere in the solar system that themselves have moons or rings. At one point Rhea, moon of Saturn was thought to possibly have rings. It still remains a possibility, though the evidence is indirect and inconclusive and a search for a ring hasn't turned up anything. A ring is basically a large number of very small moonlets.
We don't yet know of any moon in our solar system with moonlets or rings orbiting it, or indeed, don't know of any "double moon" although double asteroids are quite common.The nearest to this are Saturn's two co-orbital moons Epimetheus and Janus.
You might be interested in my article on this topic, which I wrote as New Horizons was approaching Pluto hence the title - to make it topical - but of course New Horizons didn't find any new moons.
Can Moons Have Moonlets? Or Rings? Moonlets Of Pluto's Moons?
We might count spacecraft sent to be deliberately captured into relatively stable lunar orbits. One such craft currently in such an orbit is Chandrayaan-2.
Aside from such cases, the Earth and Moon combined might capture objects into librations around the L$_4$ and L$_5$ Lagrange points, which may be regarded as simultaneously orbiting the Earth and Moon. The Kordylewski clouds have been controversial, but polarimetric measurements have indicated their existence at both of these Lagrange points[1,2,3].
The Saturnian moons Tethys and Dione are known to have co-orbiting moons at their stable Lagrange points with Saturn.
References
Slíz-Balogh, Judit; Barta, András; Horváth, Gábor (11 November 2018). "Celestial mechanics and polarization optics of the Kordylewski dust cloud in the Earth–Moon Lagrange point L5 – I. Three-dimensional celestial mechanical modelling of dust cloud formation". Monthly Notices of the Royal Astronomical Society. 480 (4): 5550–5559. arXiv:1910.07466. Bibcode:2018MNRAS.480.5550S. doi:10.1093/mnras/sty2049. S2CID 125609141.
Slíz-Balogh, Judit; Barta, András; Horváth, Gábor (1 January 2019). "Celestial mechanics and polarization optics of the Kordylewski dust cloud in the Earth–Moon Lagrange point L5 – Part II. Imaging polarimetric observation: new evidence for the existence of Kordylewski dust cloud". Monthly Notices of the Royal Astronomical Society. 482 (1): 762–770. arXiv:1910.07471. Bibcode:2019MNRAS.482..762S. doi:10.1093/mnras/sty2630.
Judit Slíz-Balogh, Attila Mádai, Pál Sári, András Barta, Gábor Horváth (February 2023). "First polarimetric evidence of the existence of the Kordylewski Dust Cloud at the L4 Lagrange point of the Earth–Moon system". Monthly Notices of the Royal Astronomical Society, 518(4), 5236–5241. https://doi.org/10.1093/mnras/stac3429
At the moment the Earth's moon currently does not have any natural satellites. Although it is possible for the moon to capture an object, the proximity of the moon to Earth makes it unlikely that an orbit would be stable. Perhaps this is the reason no natural satellites currently orbiting the moon.
