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It is my understanding that the tidal forces of the Moon acting on Earth cause it to slow down its rotation and, because angular momentum is conserved, the Moon's orbit subsequently expands. This continues until the Earth's rotation is synchronous with the Moon's orbit leaving both bodies tidally locked to each other.

However once that happens, how will the tidal forces exerted by the Sun on Earth (which are weaker than the Moon's) affect the Earth's rotation? Would they cause the Earth to start rotating again and thus cause the Moon's orbit to shrink?

P.S. I do realize the Earth is not destined to be tidally locked to the Moon until after the Sun evolves into a red giant, so for the sake of simplicity assume the Earth-Moon system survives unscathed without getting disturbed.

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  • $\begingroup$ Regarding the PS, you are asking us to scientifically explain something that cannot scientifically happen. That doesn't make sense. Voting to close. $\endgroup$ May 20, 2020 at 13:04
  • $\begingroup$ I’m voting to close this question because the question asks us to use science to explain something that science says will not happen. $\endgroup$ May 20, 2020 at 13:05
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    $\begingroup$ Although hypothetical scenarios are typically off-topic for this site, this seems like an answer that could get interesting answers. The evolution of the Sun into a red giant has no impact on the tidal locking of the Earth and the Moon. So tidal locking can be considered independently from the Sun's evolution. $\endgroup$
    – usernumber
    May 20, 2020 at 13:34
  • $\begingroup$ All problems in physics make some simplifying assumptions. This one doesn't seem exuberant, so I vote to leave open. $\endgroup$
    – usernumber
    May 20, 2020 at 13:36
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    $\begingroup$ @usernumber - One key reason that the Earth will never become tidally locked to the Moon is that the Earth's oceans are responsible for almost all of the Earth's rotational deceleration. Thank's to the ever increasing so-called solar constant, the Earth will lose its oceans in about a billion years, well before the Sun turns into a red giant, and well before the Earth could ever become tidally locked to the Moon. $\endgroup$ May 20, 2020 at 13:48

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user177107 should be commended for asking this clever question.

In short, the described configuration is possible, at least in principle. It will bring two elements into the dynamics.

First, a large moon orbiting a not too large planet may synchronise this planet's rotation. Planet's rotation synchronised with the moon will not be synchronised with the star. As a result of this, the tides in the planet may be contributing either to the tidal ascent or to the tidal descent to the star, dependent on whether the planet's (to be exact, the planet-moon barycentre's) mean motion $n$ is smaller or larger than the planet's spin rate.

[Recall that the tides in a planet whose spin is synchronised with the star are always working to ensure descent, see eqn (150) in this work.]

Second, if the moon is orbiting above the synchronous radius (like Deimos or the Moon), it will usually be ascending. I am saying `usually', because different is a situation where the eccentricity is noticeable. (And it is likely that Phobos was at some point in this exceptional situation -- which helped it to fall down through the synchronous orbit.) Assuming that nothing prevents the orbit from circularisation, and that the moon is ascending, we have to take into account that at some point the moon will be lost to the star. Naively, this should happen when its semimajor axis crosses the Hill radius $r_H$. In reality, the orbit becomes unstable already at $0.49 r_H$, for a prograde-orbiting moon, and $0.93r_H$, for a retrograde-orbiting one. Keep this detail in mind if you decide to integrate this problem.

Foir further reading, this work may be of use.

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  • $\begingroup$ Ok, but as David Hammen mentioned in the question comments, the tidal motion on Earth will be extremely reduced in a billion years, when the oceans are gone. (Also, tectonic plate motion will virtually cease). And in a few more billion years, the mass loss of the Sun will surely have an effect on planetary orbits, especially in the inner Solar system. Perhaps the Earth & Moon can survive the early stages of the Sun's red giant phase, I doubt they'll survive the later stages (but of course that's just a guess). $\endgroup$
    – PM 2Ring
    Jul 2 at 16:07

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