Do moons of rocky oblate planets migrate to low inclination over time? If so, does the time it takes depend on the Moon's size?

Question

Discussions elsewhere suggest that this happens and that smaller moons would tend to low inclination with respect to the planet's equator faster than larger ones.

For example, Mars' small moons are inclined only by 1° or so while Earth's moon's inclination varies between 23-5° and 23+5° with an 18.6 year period.

But this is not a good example because the origin and history of these moons is very different.

Question: So if we had two Earths around two Suns at 1 AU each, and each had a rocky moon like our own inclined at say 10° with respect to the Earth's oblate equator at the same distances except that one moon was big and one was small:

1. Would they migrate to low inclination over time?
2. If so, does the time it takes depend on the Moon's size?
3. If so, which would tend to migrate to low inclination faster?

Answer 1

To answer this question, we have to recall the definition of the Laplace surface. Conventionally called Laplace plane, this surface is not actually a plane. In the vicinity of a planet, it coincides with this planet's equatorial plane, while further from the planet it coincides with the planet's orbit about the Sun. As Wikipedia says,

• The Laplace plane arises because the equatorial oblateness of the parent planet tends to cause the orbit of the satellite to precess around the polar axis of the parent planet's equatorial plane, while the solar perturbations tend to cause the orbit of the satellite to precess around the polar axis of the parent planet's orbital plane around the Sun. The two effects acting together result in an intermediate position for the reference axis for the satellite orbit's precession.

Locally, the Laplace plane (or, better to say, Laplace surface) is perpendicular to the axis about which the orbital plane of the satellite is precessing.

Being remote from Earth, the Moon "feels" the pull of the Sun stronger than it "feels" the Earth's oblateness. So its (local) Laplace surface is close to the ecliptic. The normal to the lunar orbit precesses about the normal to ecliptic. (Actually, so does the lunar spin axis, because the Moon is in a Cassini state.)

Now, it we place the Moon on a slightly more inclined orbit (say, $i_0=$10$^{\circ}$), the tidal forces will start working to reduce $i$. However, due to the presence of the Sun, they will reduce $i$ not to zero but to the present configuration, where the orbit will be precessing about the Laplace surface.

As our collocutor eshaya rightly pointed out, a larger moon raises a larger tidal bulge on the Earth. Since the tidal interactions are caused by the tidal bulge, the timescale of the tidal evolution would be shorter for a more massive moon.

All said about the Moon pertains to the outher satellites of the giant planets.

The case of the inner satellites or Phobos and Deimos is different. They "feel" the oblateness of their host planet stronger than the pull of the Sun. So, in their case, the tidal interaction is working to nullify the inclination.

Now, how about Triton? Aside from being retrograde, its orbit is strongly inclined to Neptune's equator. How can it remain stable? To solve this puzzle, we may have to challenge the common belief that the tidal forces are always working to reduce the inclination. To see that this is not always so, please have a look at eqn (165) in this paper, and read the subsequent paragraph. If you manage to prove, using this knowledge, that Triton's orbit is tidally stable, your result will be publishable in Nature, certainly in Nature Astronomy. For accurate analysis, you will have to use the full expression (162), and will have to develop a physically reasonable tidal model for Neptune. (In the first approximation, I would probably neglect the tides in Triton, though I am not 100% sure if this is permissible.) Also, you will have to demonstrate that the entire set of initial conditions contains a finite -- and not negligibly small! -- subset leading to the present configuration. I would estimate this work as 1/3 - 1/2 of a good PhD thesis. Could be a whole boina fide thesis, if contains a detailed tidal model for Neptune.