# One more moon thread - the separating aspect

http://www.straightdope.com/classics/a2_112.html

Cecil says that the moon is pulling away at about four inches per month. Nasa and the Melbourne Planetarium say the moon is, on average, about 384,400 miles away. Here, I begin to wonder about the moon in the past. Four inches per month turns into four feet per year, so I finally noodled out the following:

4 ft/yr * x yrs / 5280 ft/mi = 384,400 miles

Melbourne also says:

But the numbers I have found don’t even come close to matching up with that scenario. The moon (by its present speed) should have left earth about 500 million years ago. Add to that, I would assume that if some Mars-sized planet rammed a big hunk of earth into orbit, it would originally be traveling much faster, quickly falling into something that closely approximates its present orbit. So my questions are:

(1) Am I just an idiot who’s using completely bunk numbers, or is there a more sophisticated algorithm that puts this problem in the proper perspective?
(2) How solid is the “Mars-sized planet collision” theory?
(3) Does the moon appear to be leaving earth any more quickly/slowly than in the past, or can’t we tell?
(4) Does it look like the moon will ever leave us completely, or fall back to earth?
(5) Just how delicate is a satellite’s orbit? As I understand it, there is a very specific distance-to-speed ratio required in order to maintain orbit. Is it even possible for the moon to have been in a stable orbit at half its present distance (while gradually receding to its present orbit)?
Hiebram

It is incorrect to assume that the rate of recession is constant. It depends on several things (some obvious, some not) that vary over time. One is the distance between the Earth and the Moon. The farther the Moon gets away from the Earth, the slower the recession rate. Also, the physical properties of the Earth and Moon are important (how thick their crusts, mantles and cores are.

It turns out that, surprisingly, the shape of the Earth’s ocean basins are also very important.

The Pacific Basin right now is about the right size that a wave can slosh across the basin in about the same time it takes for the Earth to turn underneath the Moon. This sets up a resonant effect, and makes the process of moving the Earth’s rotational energy into the Moon’s orbit very efficient. Since plate tectonics causes the shape of ocean basins, this changes the rate of the Moon’s recession.

So, to address your questions,

1. Cecil provided you with some bad numbers. See my thread. However, even using the correct figures (4 cm/year) you still run into this problem, because, yes, you can’t just assume the recession rate is constant.

2. Somewhat solid. Some astronomers don’t like it because it requires a fairly specific, somewhat uncommon event. That doesn’t mean it didn’t happen, but astronomers prefer explaining what we see by invoking ordinary, common events rather than singular, surprising ones. Dr. Robin Canup (now at the Southwest Research Institute, I believe) has done some pretty convincing computer simulations which have shored up support for this hypothesis. Also, there’s no strong competing hypothesis that explains the Moon’s composition.

3. The rate of recession right now is unusually large. Tidal rhythmites show that the rate was slower millions of years ago. It’s hard to predict the rate in the future due to the uncertainty in predicting plate tectonics.

4. The Moon’s orbit will keep increasing until the Sun becomes a red giant. Assuming that will have no effect (unlikely!) the Moon would leave the Earth and orbit on its own around the Sun a few billion years later.

5. As long as the changes to the Moon’s orbit are gradual, it can change quite a lot without catastrophic results.

Do you have a cite that there is that much angular momentum in the Earth/Moon system? The Earth’s rotation is slowing. Eventually, the moon will remain fixed over the supercontinent AmerEurAsia and tides will cease. At that time, the moon’s orbit will begin to decay.

Just asking

rwj

One big problem might be that your cite says that the moon is, on average, about 384,400 kilometers away.

My apologies, I cannot find a cite. I’m remembering this from a grad-level course in celestial mechanics. I thought my notes were here at my office; it appears they are at my other office, where I can’t lay hands on them easily at the moment, and I can’t find a corresponding citation (either supporting or refuting that result) in the literature.

The basic idea is that yes, the Earth-Moon system will evolve toward the Earth being synchronized to the Moon. The question is, where will the Moon be when this happens? The result of the calculation I remember doing (quite possibly as a homework problem) was that the Moon would be outside the Earth’s Hill sphere at that time, and thus would end up escaping from the Earth.

However: 1) I could have gotten the homework problem wrong though I think I’d have remebered that. 2) The calculation was definitely an over-
simplified one, as it was a pencil-and-paper calculation, not a full-blown numerical simulation. 3) Even if it was full-blown n-body numerical simulation, there would still have been a lot of assumptions going into it.

The figure I’m seeing thrown around a lot on the web for the Moon’s orbital period (and the Earth’s rotational period) when the system is fully synched up is about 44 days, which appears to comes from a 2-body conservation of momentum argument, which is what you’re invoking, too. That would make the Moon’s orbital radius being about 50% bigger than its current radius—still within the Hill sphere.

If you prefer to take that over my half-remembered solution to a homework problem, I wouldn’t blame you. As I can’t support it, I withdraw the assertion that the Moon will escape from the Earth. If I ever find my friggin’ notes, I may take up the banner again.

For those who are unaware of these things, the awkward fact is that, although Newton’s Law of Gravity seems simple, it has never been completely solved as a calculus problem except for a simplified universe with only two bodies in it. Therefore, solving orbital problems requires either a thundering lot of simplifying assumptions, a thundering lot of raw arithmetic, or both.

From “Astronomy, the Cosmic Journey” (standard undergrad text):

Similarly, from Nine Planets:

Wow, I never realized that the possibility of being wrong would make Podkayne so happy :).

And there are a few special cases of orbits which have been solved exactly with more than two bodies, but they’re freakishly simple and specific, and most of them are unstable, so what John W. Kennedy is saying is still in essense correct. In fact, for the general three body problem, it’s my understanding that not only do we not have an exact solution, but an exact explicit solution does not even exist. However, for many cases, it is possible to construct numerical solutions to arbitrary accuracy, so exact solutions aren’t generally necessary.

In fact, what I said was exactly correct; I put the word “completely” in there for a reason.

Haven’t heard the term “Hill sphere” before- definition please? From context, I would presume it’s the limit of how far a satellite can be from Earth before it would enter a solar orbit.

Thanks, that Clarifies my Understanding:
The Earth, but also the Sun, is slowing the Moon’s Revolution and the Earth’s Rotation.
Sun Tides will again Predominate as Earth-Moon Tides diminish.

rwj