We know that moons are the worlds that revolve around the worlds that revolve around the Sun.

However, could there possibly be worlds that revolve around the worlds that revolve around the worlds that revolve around the Sun?  Moons orbiting moons (or, MOMs)

As far as we can determine, none of the planets’ moons have moons of their own. However, not only is it possible for a moon to hold onto its own natural satellite -at least in the short term- but some planetary scientists believe that at least two of Saturn’s moons, Rhea and Iapetus, might have had a moon sometime in the past.

RHEA
A Ringed Moon around the Ringed Planet?

In November 2005, data collected by the Cassini orbiter noticed a strange depletion of energetic electrons within Saturn’s magnetosphere around the region of Rhea, Saturn’s second largest moon. Cassini scientist Geraint Jones, co-author of a paper that was published in the March 6, 2008 edition of Science, concluded that a ring system surrounded Rhea. The ring system particles believed to have been located within the space separating Rhea and the Cassini probe must have absorbed these electrons. At least that was the hypothesis. Although subsequent reconnaissance missions detected no evidence to substantiate this conclusion, observations made in 2009 revealed a series of spots visible in the UV band that were distributed along a line around Rhea’s equator. These “spots” might have been formed by the impact of ring material that fell onto the moon’s surface.

Such rings, if they even existed, could have been formed by a moon orbiting around Rhea. If this natural satellite happened to revolve around Rhea at a speed greater than its rotation rate, that moon would have spiraled in toward Rhea. When it reached the Rhea’s Roche limit, the tidal forces would have ripped it apart into fragments to form the rings. These rings would then continue to migrate toward Rhea and would ultimately fall onto its surface.

An artistic depiction of Rhea’s “ring system.” Although the Cassini probe detected a depletion of electrons in the Rhea region consistent with the presence of a ring system, further observations failed to find any evidence of such a system. Image credit: NASA

However, in 2009, “spots” visible in the UV band were observed around Rhea’s equatorial region: evidence, perhaps, of small ring particles that struck Rhea’s surface. Although Rhea has no ring system now, it might have been surrounded by rings sometime in the past. Image credit: NASA/JPL/Space Science Institute/Universities Space Research Association/Lunar & Planetary Institute - NASA / JPL / SSI

Indirect evidence of a moon one having revolved around Rhea? Perhaps.

IAPETUS
A mysterious “bulge”

Iapetus, a strange moon to begin with owing to the stark contrast between its dark and white regions, also sports a mysterious “bulge” and an equatorial mountain range. The bulge might indicate that its rotation rate was faster in the past. (A world’s oblateness -the ratio of its equatorial diameter to its polar diameter- generally increases with rotation rate.) Some models suggest that a moon around this moon could have been responsible for both the rotation rate and the “mountain range.” As the moon drew close to Iapetus, it, too, ventured as close as the Roche limit and then broke apart into rings that eventually scattered along Iapetus’ equator. However, there is no evidence supporting this hypothesis apart from the observations of this mountain and the equatorial bulge.

Could the bulge and the “mountain range” around Iapetus have been caused by a moon that broke apart into rings and then collapsed onto the moon’s surface? Image credit: NASA/JPL

Yet, what is the probability that a moon in the solar system actually has a moon now and where would one most likely find a “moon-orbiting” moon?

From a physical standpoint, it is far easier for a moon to remain around a planet than it is for a moon to remain around a moon of a planet. However, as mentioned at the beginning of this answer, it is possible, provided the conditions are just exactly right.

In order for a moon to have a MOM, it should be:

• Be as massive as possible. We now come to the concept of the Hill Sphere. In order for a body to remain in orbit around another body, the latter must be within the former’s Hill Sphere, a region in which the former’s gravitational influence is dominant. For instance, Earth’s Hill Sphere extends out to a distance of 1.5 million kilometers. At a mean distance of 386 kilometers, the moon is well within Earth’s Hill Sphere. (Despite its gradual recession from Earth at 3.8 cm per year, the moon will never move out of the HS). Were it to venture outside this sphere, it would establish an independent orbit around the Sun. The more massive the moon, the larger the Hill Sphere. ALSO, you want a moon with a higher escape velocity (the velocity a body would need to attain to detach itself from the moon.)
• Far enough away from the parent bodies. You don’t want a MOM to experience too much influence from the moon’s parent body.
• Not too far away from the moon. Modeling shows that a satellite’s orbit will only be stable if it is generally within a region extending out to 1/3 or 1/2 of the object’s radius. A MOM farther away could be easily ejected.
• Not close to a lot of other moons and other perturbing objects. Jovian planets tend to be surrounded by a vast retinue of moons. Jupiter (79) and Saturn (82) have particularly large families. That’s a lot of perturbing bodies. Any MOM would have to be far enough away from both the parent body and other moons so as not to be perturbed out of its orbit,

What are the best candidates? We’ll name three of them.

• GANYMEDE Jupiter’s most massive moon and the most massive moon in the solar system. On the other hand, with the mean distance of 1.07 million kilometers from Jupiter, it might not be far enough away from its parent body to keep a MOM. It also isn’t far enough away from some of the other moons. (Also, it is close to many moons and the orbital resonances it has established with Io and Europa results in a lot of tidal stresses)
• IAPETUS (again) While it is quite a distance from Saturn (3.6 million km), it isn’t particularly massive and so its escape velocity is low and its Hill Sphere small
• TRITON Neptune’s largest moon MIGHT be the best place to look for a MOM. It is both highly massive and revolves at a far distance from Neptune in relation to the planet’s radius. On the downside, it is one of the few geologically active moons and material from its erupting geysers could induce a retarding effect on the orbit of any MOM.

So, even though we haven’t yet found any MOMS, they might be out there.