Has any moon achieved "retrograde equatorial orbit"?

There are many moons which have low (almost negligible) inclination and can be considered to rotate at the planet's equatorial plane. For instance, Galilean moons have almost negligible inclination (< 1$$^\circ$$) and so they are considered to have a "prograde equatorial orbit".

OTOH, "Retrograde equatorial orbit" is that orbit which has an inclination of exactly 180° (or near 180°). Has any moon achieved such kind of orbit? I have found some Saturnian moons (or moonlets) which have inclination around 177° and didn't found anything beyond that. Why is it hard to achieve a "Retrograde equatorial orbit"?

Tangentially related questions: Can a moon rotate perpendicular to planet's plane? [90° inclination] (Theoretically yes, practically hard to achieve; See Kozai Mechanism)

Also related: Are all satellites of all planets in the same plane?

• 177° is pretty darn close to 180°, I'd say it's safe to call those effectively "retrograde". That said, there's likely to be many more moons around Jupiter and Saturn that are just too small to have been detected as of yet, so it's entirely likely that if there are 177° inclinations just in the ones we know about, it's probable there may be higher ones we just haven't discovered yet. Dec 21 '20 at 16:01

2 Answers

This is directly related to another question: Why are asteroids with zero orbital inclination rare?

If captured, irregular moons are randomly oriented in space then there is very little chance of them having either inclination angles near zero or near $$180^\circ$$. This is because, if they are uniformly distributed in space, the fraction of orbits within a given inclination range $$i_1 < i < i_2$$ is given by $$f = \frac{\int^{i_2}_{i_1} \sin i\ di}{\int_{0}^{180^\circ} \sin i\ di}$$ $$f = \frac{1}{2}\left(\cos i_1 - \cos i_2\right)$$

Thus for $$i_1 =177^\circ$$ and $$i_2=180^\circ$$, $$f= 0.00068$$ (i.e. about 0.07%).

Given that there are only $$\sim 100$$ irregular satellites, it is not that surprising that none are found with $$i>177^\circ$$.

Note there are physical reasons why the regular satellites have low inclinations - they are not uniformly distributed and therefore they can have very low inclinations.

As Rob Jeffries says, no moons are in retrograde equatorial orbit in our solar system.

One reason why prograde equatorial orbits are more likely than retrograde equatorial orbits has to do with tidal locking. Our Moon, for example, is tidal locked with the Earth. The Earth is actually also spinning down to tidal lock with the Moon. However, most projections show the Earth will be long gone by that point: Will the Earth ever be tidally locked to the Moon?.

For systems in mutual tidal lock, the moon has to be in a prograde orbit for it to always point to the same face of the main body. The Pluto/Charon system is a great example in our Solar System of mutual tidal locking:

Theoretically, a big moon in retrograde orbit could cause a planet to tidal lock. In this scenario, tidal forces would cause the rotation rate of the planet to go all the way to zero and then speed up in the opposite direction until it matched the orbital period of the moon. However, this eventual state would have the moon in a prograde orbit, since the revolution direction of the planet would have changed.