# Why is momentum transferred to the moon?

The moon whizzes around the Earth, dragging water after it and producing tides. In a comment on this question, I asserted this would sap the momentum of the moon causing it to fall to Earth. This was promptly rebutted by people with references, but I don't see why this should be so.

I admit I forgot about planetary rotation and its effect on the water but nevertheless I would have expected tides to dissipate energy from both as friction induced heat.

Now, an assertion that momentum is being transferred to the moon is trivially falsifiable for any astronomical observatory equipped with modern clocks, so I'm going to simply assume it's true. The question then becomes what is the mechanism by which tides transfer momentum to the moon?

Does it have to do with the balance point in a very assymetric two body system transferring energy from a large mass to a small one, akin to a slingshot orbit?

Or does the energy transfer from the body with the water sloshing about to the one without? (And if so, why?)

• This would likely be better suited to our sister site Astronomy (or alternatively Physics) as it pertains to tidal forces (tidal acceleration in this case) between two celestial bodies. But please don't cross-post. If you agree with migration, please note so in the comments and we'll do that for you, if it's not closed as off-topic for Space Exploration by our reviewers here first. Thanks! – TildalWave Jun 12 '14 at 10:31
• I agree that this is a bit off-topic for this site. Besides the two cited by TidalWave, there's a third option, the Earth Science sister site. – David Hammen Jun 12 '14 at 13:54
• He's already got one! On Physics SE and several other instances, too. Just as an aside I would describe the transfer as being of "angular momentum" because the bodies are ballistically coupled. – dmckee Jun 12 '14 at 23:18
• The Earth's tidal bulges are gradually accelerating the Moon in its orbit. The Moon gradually gains orbital momentum and kinetic energy. The Earth gradually loses exactly the same amount of angular momentum from its daily rotation (angular momentum is conserved), but loses more kinetic energy than the Moon gains; the difference being consumed in friction. – Anthony X Jun 13 '14 at 1:15
• The whole point of having different SE sites is that it makes it easier to find experts who know the complete answer to your question. At the moment, you have two contradictory answers, at least one of which written by an amateur (me). Moving the question to Astronomy gives you a better chance to know for sure. Some of the user base here will be interested, but there's a good chance you'll find those people same over on Astronomy and/or Physics as well. – Hobbes Jun 14 '14 at 7:08

The Earth's rotation rate is very gradually slowing down. Over the short term, the Earth's rotation rate changes a bit chaotically, sometimes speeding up, sometimes slowing down. These short term changes are due to transfers of angular momentum amongst various components of the Earth. The long term secular variations in the Earth's rotation rate are due to tidal interactions between the Earth, Sun, and Moon.

The slowing of the Earth's rotation rate means a reduction in angular momentum. Because angular momentum is a conserved quantity, that lost angular momentum doesn't just disappear. It has to be transferred to something else. That something else are the two bodies responsible for the slowing of the Earth's rotation rate, and that is the Moon and the Sun, mostly the Moon.

The textbook explanation for this is the Earth's tidal bulge.

Newton's explained the tides by noting that the tidal force is away from the center of the Earth along the axis connecting the Earth's and Moon's center of mass and is directed inward toward the center of the Earth along the great circle normal to this axis.

Newton reasoned that this force should make Earth's oceans bulge outward along that axial line and draw inward along that circle where the tidal force is inward. This equilibrium theory of the tides gives a nice, simple explanation for transfer of angular momentum from the Earth's rotation to the Moon's orbit. The Moon pulls the tidal bulge slightly forward. This off-axis tidal bulge on the Earth in turn results in a tangential component of the gravitational Earth's gravitational force on the Moon, in the direction of the Moon's velocity.

A similar thing happens when a rocket in circular orbit fires it's thrusters against the velocity vector. The rocket accelerates forward, but in doing so moves to a higher orbit where it slows down. The same thing is happening to the Moon. That continuous forward acceleration very gradually raises the Moon to an every higher orbit.

This is a very slow process. Currently the Moon is receding from the Earth at about 3.8 centimeters/year, and that slow creep is anomalously high. The long term rate based on various fossil records is about 1.7 cm/yr over the last 2 to 3 billion years.

There's one thing wrong with this model: Newton's tidal theory is false. While Newton did develop the correct model for the tidal force, his static model of the tides is just wrong. There is no Newtonian tidal bulge. There can't be, for a number of reasons.

Moreover, Newton's equilibrium theory of the tides is falsified by observations of the tides. Consider the North Sea. Per Newton's equilibrium theory, there would be two high tides per day in the North Sea, one when the Moon is at zenith and the other when it is at nadir. Since the North Sea is rather small, those high tides should occur more or less at the same time all across the North Sea. That isn't what happens. At any time of day, it's always high tide somewhere in the North Sea, low tide somewhere else. The same goes for Patagonia, New Zealand, and a few other spots on the Earth.

Laplace developed the correct theory of the tides. Laplace's dynamic theory of the tides takes into account the very things that say why Newton's tidal bulge cannot exist. Instead of a tidal bulge, the oceans comprise a number of amphidromic systems. There are several amphidromic points in the oceans where tides are nearly non-existent. The tides rotate about these amphidromic points. There are three such points in the North Sea, which explains why tides in the North Sea are so ridiculously complex.

So if Newton's equilibrium theory of the tides and his tidal bulge are false, what explains the transfer of momentum? The interactions at the coasts results in asymmetries in the rotations about these amphidromic points. The net effect of these asymmetries are to induce a tangential component in the Moon's gravitational acceleration.

This is not a "net tidal bulge". There is no tidal bulge. Those amphidromic systems result from the geometry of the oceans and the impossibility of a tidal bulge. The geometry of the oceans has changed drastically over the ages. Currently there are two huge north-south land barriers that impede the tides, the Americas in the western hemisphere and Afro-Eurasia in the eastern hemisphere. This configuration is the reason for the currently anomalously recession rate. The tides have had much freer flow during other times in the Earth's history such as when the Earth's continents were organized as a single supercontinent. The recession rate was anomalously low during those periods.

References

Lambeck, "Tidal Dissipation in the Oceans: Astronomical, Geophysical and Oceanographic Consequences", Phil. Trans. R. Soc. Lond. A 287:1347 (1977)

Proudman and Doodson, "The Principal Constituent of the Tides of the North Sea", Phil. Trans. R. Soc. Lond. A, 224 (1924).

Thompson, "Tide Dynamics: Dynamic Theory of Tides", Lecture notes for Ocean 420 at Univ. of Washington.

Wotal, "Tides and their origin," Lecture notes for Geology 161 at Oberlin College.

• I should add some references, too. I'll do that later today. Part of the problem is that the journal articles that showed Newton's equilibrium theory is wrong are over 200 years old, and most are not in English. – David Hammen Jun 12 '14 at 13:33
• oh man, if I had a nickel for every time somebody used the old I-can't-cite-references-because-they're-hundreds-of-years-old-and-in-another-language excuse... – coburne Jun 12 '14 at 13:51
• I can cite textbooks and college lectures. Journal articles? No. This is very settled science. – David Hammen Jun 12 '14 at 13:52
• Well what do you know. There are some nice journal articles. The Doodson article was a nice find since (a) Doodson is one of the key dudes in the development of the theory of tides, and (b) I mentioned the North Sea in my answer. – David Hammen Jun 12 '14 at 18:02
• @AviCherry - 1.7 cm/year for 2 billion years is 34000 km, not 340000 km ($1.7\,\frac{\text{cm}}{\text{y}} \times \frac{1\,\text{m}}{100\,\text{cm}}\times 2\,\text{billion}\,\text{y} \times\frac{1\,\text{km}}{1000\,\text{m}} = 3.4\times10^4\,\text{km}$, or 34000 km). You added an extra zero somewhere along the way. That said, the Moon is thought to have formed ~4.5 billion years ago at four to six Earth radii from the center of the Earth - 25000 to 38000 km. That means the bulk of the Moon's recession occurred during the first couple of billion years after the Moon formed. – David Hammen Feb 23 at 0:58