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The Moon’s gravity produces tidal deformations or “bulges” in the Earth. Because of the Earth’s rotation, the line that goes through the bulges is not aligned with the line between the Earth and the Moon. This misalignment produces a torque that transfers angular momentum . I am not clear how the transfer of angular momentum occurs

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  • $\begingroup$ It's the same thing as transferring regular momentum, which is what happens whenever you push on something. Here, the Moon just pulls on the tidal bulges to make the torque. $\endgroup$ – knzhou May 5 '16 at 4:49
  • $\begingroup$ In the same way that the gravitational force between two objects transfers linear momentum between them. $\endgroup$ – Emilio Pisanty May 5 '16 at 8:21
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The angular momentum is conserved in central force motion (like what we have in the case of Earth-Moon system). In such a case, the force $\vec{F}$ and the radius vector $\vec{r}$ are parallel so that the resultant torque is zero

$$\vec{\tau}=\vec{r}\times\vec{F}=o$$

This means the angular momentum ($\vec{L}$) of the of the Moon about the center is a constant (since $\displaystyle{\vec{\tau}=\frac{d\vec{L}}{dt}}$). Hence the moon while orbiting earth conserves it's angular momentum. The angular momentum is given by

$$\vec{L}=\vec{r}\times\vec{p}$$

This means, to conserve the angular momentum the moon exchange the distance and velocity as they move about Earth. The Earth's angular momentum is distributed between it's spin (rotation) and it's motion along the orbit (revolution). Hence to keep the angular momentum conserved, the spin and orbital angular momentum of earth is exchanged by several mechanisms. The moon could exert tidal forces on Earth. This acceleration causes a gradual recession of Moon in a prograde orbit away from Earth (direction of motion same as rotation of Earth). How the tidal forces of moon are exerted on earth, you can see here. To conserve the angular momentum, the Earth have to manage to counteract the tidal torque on it.

The extra torque acts in the direction of the angular momentum of earth due to the spin of Earth. So, to nullify this additional torque to zero the Earth's rotation slow down and cause a gradual increase in the Moon's orbit. This is how angular momentum is exchanged in between the Earth-Moon system by exchanging the radius and velocities.

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  • $\begingroup$ How does slowing down of earth's rotation increase moon's orbit ? $\endgroup$ – user115495 May 5 '16 at 5:10
  • $\begingroup$ When earth slows down, it creates an additional torque in the opposite direction so that it cancels the additional torque. i.e., in effect the earth add some torque on moon in the reverse direction of what the moon applied to earth. So the effective torque exerted by moon now decrease. To avoid that and to conserve momentum, the moon increase it's orbit so as to retain the torque. $\endgroup$ – UKH May 5 '16 at 5:17
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It's worth thinking about why the tidal bulges1 are not lined up with the line between the bodies (which is where you would naively expect them to) and then thinking about how that affects the gravitational interaction between them.

Because the moon takes about one month to orbit and the Earth takes one day to turn, the naive location of the tidal bulges should move east-to-west across the surface of the Earth in order to stay on the line between the bodies. However, that would represent an actual flow of water and the water in motion interacts with the ocean beds and shores. The net results is a frictive force between ground and water that (on average2) retards the motion of the water and causes the bulges to be east of the line between the bodies.

Finally, the bulges (which on average leading the Moon around the Earth) represent an aspherical component of the Earth's mass distribution and their gravitational effect on the moon can't be reduced to a point-mass approximation. The near bulge leads the moon pulling it slightly forward. The far bulge trails and pulls slightly backward, but it is farther away so it's effect is smaller. The net result is that the moon is pulled forward in its orbit and the Earth is slowed.

It is the usual contrariness of orbital mechanics that turns a steady forward force on the moon into a combination of outward motion and reduced speed.3


1 It's possible also worth mentioning that the real tidal bulges are not the same as the idealized one often presented in a first treatment of this subject because the oceans are not deep enough to treat the bulges as free moving surface waves.

2 In addition to friction the bulk motion can excite resonant modes in the ocean basis, and cause the phase angle between the actual location of the excess water and the line between the bodies to either lead of trail. With the right arrangement of continents the process can actually run backward.

3 This has been treated several times in accessible language in science fiction. Heinlein did a nice job in The Cat Who Walks Through Walls, but I think the pithest version comes from The Integral Trees:

"East takes you out, out takes you west, west takes you in, and in takes you east." --Larry Niven

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