One more moon thread - the separating aspect

[hiebram]

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:

The Moon is thought to have formed when a rogue Mars—sized planet collided with the Earth 4.6 billion years ago.

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

[Podkayne]

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.

[rwjefferson]

@Podkayne's other thread

Ignoring the Sun and its evolution, the Moon would move away for a few billion years, and evenutally it will be far enough away that it will drift out of the Earth’s influence and go on its own orbit around the Sun.

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

[Exapno_Mapcase]

@hiebram

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.

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

[Podkayne]

@rwjefferson

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.

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.

[John_W.Kennedy]

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.

[scotandrsn]

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

Gravitational forces in the Earth-Moon-Sun system cause tides, or bulges in the shape of Earth and the Moon. The closer two objects are, the stronger the gravitational force each exerts on the other. Thus, the side of the Moon facing the Earth has a stronger force on it than the far side, because the facing side is closer. …

…As a result, the Moon stretches slightly along this line, limited by the small elasticity of its rock interior. This stretching is called a body tide. Similar forces acting on Earth produce not only a body tide but also an ocean tide in the fluid layer of water on Earth’s surface. These tides take the form of bulges on the front and back side of both the Earth and the Moon, since each body comes into equilibrium with the gravitational forces by stretching along the Earth-Moon line…

…The effect of tides explains why the Moon keeps one side facing Earth … <snip>… any elongation in the Moon’s shape would tend to make one side always face Earth. Tides guarantee such an elongation, since they create bulges. …

…The second effect is a low tidal recession of the Moon away from the Earth because of gravitational forces on tidal bulges. <snip> Earth’s rapid rotation drags its tidal bulge slightly ahead of the Earth-Moon line by some angle <snip> The gravitational effect of the bulge [on the side nearest the Moon- s] exceeds that of [the bulge on the opposite side of the Earth], because [the near side] is closer to the Moon. Because [the near side] is in front of the Moon, the net force from [the near side] tends to pull the moon ahead in its orbit. This accelerates the Moon forward and makes it spiral very slowly out from the Earth <snip> The Moon was much closer to the Earth several billion years ago. Analyses do not indicate the exact date, but most researchers believe the Moon was closest to Earth about 4.6 billion years ago, during the formation of the two bodies…

…The third effect of tides is that the Moon slows Earth’s rotation, just as Earth has slowed the Moon to synchronous rotation. <snip> Studies indicate that billins of years ago, when the Moon was closer, the Earth’s day was only about 5 or 6 hours long…

…As the Earth slows and the Moon recedes in the future, it might eventually recede so far that Earth’s gravitational pull would be very weak. Before it can escape into and orbit around the sun, however, the whole process will be stopped by small tidal bulges that are constantly being raised on Earth by the Sun. These tides will slow Earth’s rotation so that the length of the day and the month will become equal, stoppin the tidal recession. Earth and the moon will then both be in synchronous rotation. Eventually, because of this small solar effect, Earth’s rotation will slow so much that the day will exceed the month and the tidal process will reverse: The Moon will begin to approach Earth again.

Similarly, from Nine Planets:

The gravitational forces between the Earth and the Moon cause some interesting effects. The most obvious is the tides. The Moon’s gravitational attraction is stronger on the side of the Earth nearest to the Moon and weaker on the opposite side. Since the Earth, and particularly the oceans, is not perfectly rigid it is stretched out along the line toward the Moon. From our perspective on the Earth’s surface we see two small bulges, one in the direction of the Moon and one directly opposite. The effect is much stronger in the ocean water than in the solid crust so the water bulges are higher. And because the Earth rotates much faster than the Moon moves in its orbit, the bulges move around the Earth about once a day giving two high tides per day. (This is a greatly simplified model; actual tides, especially near the coasts, are much more complicated.)

But the Earth is not completely fluid, either. The Earth’s rotation carries the Earth’s bulges slightly ahead of the point directly beneath the Moon. This means that the force between the Earth and the Moon is not exactly along the line between their centers producing a torque on the Earth and an accelerating force on the Moon. This causes a net transfer of rotational energy from the Earth to the Moon, slowing down the Earth’s rotation by about 1.5 milliseconds/century and raising the Moon into a higher orbit by about 3.8 centimeters per year. (The opposite effect happens to satellites with unusual orbits such as Phobos and Triton).

The asymmetric nature of this gravitational interaction is also responsible for the fact that the Moon rotates synchronously, i.e. it is locked in phase with its orbit so that the same side is always facing toward the Earth. Just as the Earth’s rotation is now being slowed by the Moon’s influence so in the distant past the Moon’s rotation was slowed by the action of the Earth, but in that case the effect was much stronger. When the Moon’s rotation rate was slowed to match its orbital period (such that the bulge always faced toward the Earth) there was no longer an off-center torque on the Moon and a stable situation was achieved. The same thing has happened to most of the other satellites in the solar system. Eventually, the Earth’s rotation will be slowed to match the Moon’s period, too, as is the case with Pluto and Charon.

[Chronos]

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.

[John_W.Kennedy]

@Chronos

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

[Lumpy]

@Podkayne

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

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.

[rwjefferson]

@scotandrsn

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

Eventually, because of this small solar effect, Earth’s rotation will slow so much that the day will exceed the month and the tidal process will reverse: The Moon will begin to approach Earth again.

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