# Escaping moons conflict with what I understand of gravity

I have known that the Earth's Moon, let's call it Luna (tip of the hat to The Expanse), has been slowly but surely increasing it's distance from the Earth by a small measure each year.

This week I was surprised to learn that NASA has determined that at least one of the moons of Saturn is also creeping away from it's home planet. In this case it's Saturn's largest moon Titan.

While this news actually supports an idea/question I had around 8 years ago which I was never able to reconcile. I had always wondered why the moon spinning around Earth isn't being tugged away from the Earth at a constant rate, done so on account of the centrical force realized by Luna's circular orbit path.

I suppose this is one question for both celestial bodies, Luna and Titan. I had always been taught that when all the calculations for any object's stable orbital mechanics, one will nearly always find that the orbit is in a perpetual state of orbital decay. As is the case with the ISS and the majority of satellites, this necessitates a periodic and deliberate intervention in the orbit usually in the form of a few precisely calculated and timed manoeuvring thrusters.

What is it which causes these two moons to drift outward into space and against their respective host planet's gravity wells, as opposed to the activity observed more commonly observed with the orbital patterns of most other satellites, characterized by the slow tightening of the orbital pattern as they would drift inwards towards their respective host planet's gravity wells with their orbital tracks slowly decay?

• NASA's JPL calls it Luna too! btw searching Astronomy SE for "Moon moving" yields a rich selection of answers to this
– uhoh
Jun 11 '20 at 8:03
• Will do, Thank you!! Jun 11 '20 at 8:27
• Note that the orbital decay of the ISS and other near-Earth satellites isn't a consequence of orbital mechanics & tides, it's due to friction from the small traces of atmosphere that exist at those levels. Jun 11 '20 at 21:21

A very good question!

The reason is essentially to do with tides. And a slightly over-simplified summary is: If the moon orbits more slowly than the rotation of the parent body (as our Moon does, 12 degrees per day while the Earth rotates about 360 degrees per day) then the moon will gradually orbit further and further away. If the moon orbits faster than the rotation of the parent body, then the moon will gradually orbit closer and closer and eventually crash.

If there were no tides, none of this would happen. If both bodies are perfectly rigid and perfectly spherical, they will orbit each other for ever, with no change.

Tides

The Earth is soft, and stretches in response to gravitational forces. By "the Earth" here, I mean mostly the oceans; but the rock also stretches (much less) in response to gravity.

Let us assume for a moment that the Earth is perfectly fluid, and that that fluid is also perfectly frictionless and has no inertia. In that case, there will be a "bulge" just beneath the Moon, caused by the fact that this position is closer to the Moon than the centre of the Earth is, and therefore is more strongly attracted by the Moon's gravity. Similarly there is a "bulge" on the side furthest away from the Moon, caused by the fact that this position is further from the Moon than the centre of the Earth is, and therefore is less strongly attracted to the Moon by the Moon's gravity. (From memory, the height of these bulges is about 50cm).

Thus as the Moon orbits a fluid, frictionless, inertia-free Earth, the Earth becomes slightly elliptical, and the "bulge" follows the sub-lunar point exactly. There is therefore no effect on the Moon's motion.

The Earth is not perfectly fluid. The motion of its component materials (especially water) is not frictionless. Real materials do have inertia. So the description I have just given is completely false.

In the real world, tides are higher than 50cm. This is because the water sloshes around - for a simple example, take a shallow tray, fill it with water, and try carrying it: the small irregularities in the way you walk become huge irregularities in the way the water moves, and you end up spilling most of the water.

In the real world, since the Earth is rotating more rapidly (360°/day) than the Moon is orbiting (12°/day), the bulge is being carried too far forward by the Earth's rotation. Omitting a lot of accurate detail, this means that the Moon "sees" beneath itself a slightly elliptical Earth with its bulge slightly ahead of the sub-lunar point. Thus the Moon is always being pulled slightly forwards in its orbit.

Pulling a satellite forwards in its orbit makes it orbit higher, and also makes its orbital period longer. Since action and reaction are equal and opposite, this is also pulling the Earth backwards in its rotation, which is why the days are gradually getting longer. Two interesting consequences: since the Moon moves further away, it gets smaller in the sky, and one day it will get small enough that there will be no more total eclipses of the Sun. Since the days are getting longer, one day the days will be $$\frac 1 {365}$$ of a year in length, and there will be no more 29 February.

Fast moons

When a moon orbits faster than its parent planet's day, exactly the opposite happens. The moon will be "seeing" beneath itself a slightly elliptical planet whose bulge is "too far behind" and pulls it backwards in its orbit. This makes the orbit lower, and makes its orbital period shorter. There is no end to this process and eventually the moon will fall low enough to be caught by the planet's atmosphere, and crash.

Summary

Because of tides, the orbits of moons and satellites tend to decay away from the "one orbit = one day" position. A moon that is outside that position will move further and further outside it. A moon that is inside that position will move further and further inside it.

How fast this process happens depends on the nature of the planet. If the planet were made of a perfectly rigid substance then the effect would not happen at all because it would not be changing shape because of the moon's gravity. If it were perfectly fluid and inertialess then the effect would not happen at all because the bulge could be kept exactly under the moon with no effort at all. The Earth is a good candidate for orbital decay because of its oceans. Saturn is a good candidate for orbital decay because it is mostly gas. Mars is not a good candidate because it is mostly rock which, although flexible, is not as flexible as water or gas.

Postscript: The Earth also orbits the Moon, and the overall effect has been that the Moon's day has got lengthened until it equals the time the Earth takes to orbit the Moon. The Sun also raises tides on the Earth (and the Earth on the Sun) and so, since the Sun rotates faster than once per year, the Earth is being pulled forward in its orbit, it is moving steadily away from the Sun, and the year is getting steadily longer.

• Interesting answer! Do you have any information on the tidal forces of solid land masses? I'm not sure what to search for, but I'd be interested in some real world examples of this if there are any Jun 11 '20 at 18:23
• It's also worth noting that in an idealized world, the Earth-Moon system will eventually become tidally locked, where the Earth will rotate with the same period as the Moon's orbit. At this point, the Moon will stop moving away from the Earth. Jun 11 '20 at 19:32
• (The "rough estimate" in the linked article, adjusted for a initial period of 24 hours rather than 12, leads to an order-of-magnitude estimate of about 2 billion years for this to happen. So don't lose any sleep over it.) Jun 11 '20 at 19:40
• @MichaelSeifert But at what point does it become tidally locked? The moon escapes if it goes out of Earth's Hill Sphere before locking occurs. Jun 12 '20 at 5:09
• Interesting to think that leap days will become less and less frequent as the earth's orbital period approaches exactly 365 days. As it goes just under 365 days, we'd then have an "anti-leap day", where we drop Feb 28 from the calendar periodically (every 10 years for a period of 364.9 days, for example). As it goes below 364.5 days, Feb 28 then gets dropped from the regular calendar entirely, and becomes the new leap day that only appears in certain years. Jun 12 '20 at 14:19

Orbits beneath synchronous orbits have a higher angular velocity than their planets rotation, orbits above have a slower angular velocity. Drag (atmospheric or tidal) would try to match the angular velocity to the planets rotation. So below a synchronous orbit objects get slower, above it they would speed up (and slow down the rotation of the body they are interacting with). Phobos for example is below the synchronous orbit of Mars and actually gets closer to Mars over time.

The Wikipedia article on Tidal acceleration/deceleration has further examples.

• Makes complete sense, thank you Jun 11 '20 at 8:26