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From what I know, the Moon is accelerating away from the Earth. Do we know when it will reach escape velocity? How do we calculate this?

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It will never reach such a high velocity. The moon is drifting further from the earth due to tidal acceleration. This process is, at the same time, slowing the rotation of the earth. Once the earth's rotational period matches the moon's orbital period, the earth-moon system will be tidally locked to each other (note: the moon is already tidally locked to the earth), and the acceleration will cease.

To briefly explain the mechanism, the gravitational pull between the earth and the moon causes tidal "bulges" to extend out on both bodies (just like the ocean tides, except that the entire surface moves slightly, not just the water). Since the earth rotates faster than the moon completes one orbit, the bulge on the earth lies slightly ahead of the earth-moon line, because the earth is rotating so quickly. This bulge gravitationally pulls on the moon, speeding it up, while at the same time the moon pulls on the bulge, creating a torque on the earth and slowing down its rotation.

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Fred Hoyle has suggested that the Earth's rotation isn't in fact slowing down, due to resonance effects in the atmosphere, which prompted another question of mine on Astro.SE. –  Brian Hooper Jun 4 '11 at 3:45
    
Thanks for the answer. May I ask though, if the Earth's rotation is slowing, and the acceleration of the Moon is to eventually cease, is this to say they will reach equilibrium? –  Grant Thomas Jun 4 '11 at 20:58
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@Mr. Disappointment: Reach equilibrium in what sense? If the moon's acceleration is to cease (per the standard theory), then the earth's rotation rate will also stop slowing down. That will happen when both bodies become locked to each other, showing the same side at all times. The Pluto-Charon binary system is a good example of this. If that's what you meant, then yes they will reach equilibrium. –  voithos Jun 4 '11 at 23:17
    
Do you have a source for the Earth becoming tidally locked to the moon before the moon reaches escape velocity? I don't see it in either of your WP articles. Using the rate given in the first of 2.3ms/century (6.4hours/billion years) lengthening of the day is too slow for the Earth to become tidally locked before the sun becomes a red giant and swallows both the Earth and Moon. –  Dan Neely Feb 15 '12 at 15:35
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@voithos: really, all you need to do is show that the rotational KE of the earth is less than the gravitational binding energy of the earth-moon system. –  Jerry Schirmer Dec 31 '12 at 19:16
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