<p>user177107 should be commended for asking this clever question.</p>
<p>In short, the described configuration is possible, at least in principle.</p>
<p>First of all, let us recall that the solar tides in a planet whose spin is synchronised with the star are always working to ensure descent, see eqn (150) in <a href="https://arxiv.org/abs/1904.02253" rel="nofollow noreferrer">this work</a>. On the other hand, the solar tides in a nonsynchronised planet are working to repel the planet from the star.</p>
<p>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 solar 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 <span class="math-container">$n$</span> about the star is smaller or larger than the planet's spin rate. As demonstrated <a href="https://www.aanda.org/component/article?access=doi&amp;doi=10.1051/0004-6361/202245533" rel="nofollow noreferrer">here</a> the spin rate of a planet synchronised by a moon will, typically, be several times higher than the mean motion <span class="math-container">$n$</span> of the planet about the star. So the solar tides in the planet will be repulsive -- and this is one possible mechanism helping a close-in planet to avoid engulfment.</p>
<p>Two technical warnings to those who decide to integrate this problem.</p>
<p>(1) If a prograde moon starts 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 <a href="https://www.nature.com/articles/s41550-021-01306-2" rel="nofollow noreferrer">to fall down through the synchronous orbit</a>.) 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 may be lost to the star. Naively, this should happen when its semimajor axis crosses the Hill radius <span class="math-container">$r_H$</span>. In reality, the orbit becomes unstable  already at <span class="math-container">$0.49 r_H$</span>, for a prograde-orbiting moon, and <span class="math-container">$0.93r_H$</span>, for a retrograde-orbiting one. Keep this detail in mind if you try to integrate.</p>
<p>(2) If a prograde moon starts below the synchronous radius, it will be falling down and, at the same time, accelerating the planet's rotation. It may result in synchronisation of the planet with the moon -- provided this end state is attained before the descending moon crosses the Roche radius.</p>

CC BY-SA 4.0
2022-07-02T16:07:51.490Z
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).
PM 2Ring
CC BY-SA 4.0
2023-07-21T20:06:40.450Z
Surely the effects also depend on the rotation rate of the Sun and whether that is faster or slower than the Earth/Moon orbital period. The Sun's rotation rate is basically unaffected by its tidal interaction with the Earth.
ProfRob
CC BY-SA 4.0
2023-07-21T21:22:46.820Z
@ProfRob The rotation rate of the Sun will determine in which direction the solar tides will be pushing the Earth. Given that the Sun's rotation is faster than the Earth's mean motion about it, the Earth-generated tides in the Sunn will be working to push the Earth away (a feeble effect actually, given the large distance). I do not see, however, how the rotation rate of Sun can influence the working of the solar tides in the Earth. That can depend only on the Earth's rotation rate. Please correct me if I am missing something.
Michael_1812
CC BY-SA 4.0
2022-06-25T04:32:10.513Z
2023-07-21T21:25:06.387Z
2023-07-21T21:25:06.387Z
Michael_1812
<p>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.</p>

<p>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>

<p><strong>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.</strong></p>

CC BY-SA 4.0
2020-05-20T13:04:31.247Z
Regarding the PS, you are asking us to scientifically explain something that cannot scientifically happen. That doesn't make sense. Voting to close.
David Hammen
CC BY-SA 4.0
2020-05-20T13:05:09.523Z
I’m voting to close this question because the question asks us to use science to explain something that science says will not happen.
David Hammen
CC BY-SA 4.0
2020-05-20T13:34:56.387Z
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.
usernumber
CC BY-SA 4.0
2020-05-20T13:36:51.603Z
All problems in physics make some simplifying assumptions. This one doesn't seem exuberant, so I vote to leave open.
usernumber
CC BY-SA 4.0
2020-05-20T13:48:14.493Z
@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.
David Hammen
CC BY-SA 4.0
2020-05-25T18:31:11.790Z
I wonder if the question could be rephrased to a hypothetical where the Earth-Moon system is orbiting and early K dwarf that is not evolving as quickly as the sun. If the "slow down" continued at the same rate as today the "locking" would take 4.84 billion years, but it will slow down as the moon gets farther away. So, estimates are tens of billions of years. Maybe even a K dwarf might not give enough time.
Jack R. Woods
CC BY-SA 4.0
2020-10-05T05:06:09.037Z
See: https://en.wikipedia.org/wiki/Tidal_acceleration#Effect_of_the_Sun
Nilay Ghosh
CC BY-SA 4.0
2022-06-25T04:55:21.710Z
@NilayGhosh In this paragraph of the Wiki article, both formulae are in error. In the first eqn, the numerical factor must be not 9/2 but 3, see eqn (148) in https://arxiv.org/abs/1904.02253      
In the second eqn, the coefficient must be not 27/4, but 15/4. See eqn (120) in https://arxiv.org/abs/1105.6086 , and also keep in mind that the moment of inertia of a homogeneous sphere is $(2/5) M R^2$. There are other inaccuracies in that article.
Michael_1812
<the-moon><earth><tidal-locking><tidal-forces>
How will the Solar tides affect the Earth's rotation once it is tidally locked to the Moon?
user177107
# How will the Solar tides affect the Earth's rotation once it is tidally locked to the Moon?

(Asked by: username_0 on 2020-05-20)

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.**

**Tags:** the-moon, earth, tidal-locking, tidal-forces

> **Comments:**
> **username_1**: Regarding the PS, you are asking us to scientifically explain something that cannot scientifically happen. That doesn't make sense. Voting to close.  
> **username_1**: I’m voting to close this question because the question asks us to use science to explain something that science says will not happen.  
> **username_2**: 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.  
> **username_2**: All problems in physics make some simplifying assumptions. This one doesn't seem exuberant, so I vote to leave open.  
> **username_1**: @username_2 - 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.  
> **username_3**: I wonder if the question could be rephrased to a hypothetical where the Earth-Moon system is orbiting and early K dwarf that is not evolving as quickly as the sun. If the "slow down" continued at the same rate as today the "locking" would take 4.84 billion years, but it will slow down as the moon gets farther away. So, estimates are tens of billions of years. Maybe even a K dwarf might not give enough time.  
> **username_4**: See: https://en.wikipedia.org/wiki/Tidal_acceleration#Effect_of_the_Sun  
> **username_5**: @username_4 In this paragraph of the Wiki article, both formulae are in error. In the first eqn, the numerical factor must be not 9/2 but 3, see eqn (148) in https://arxiv.org/abs/1904.02253       In the second eqn, the coefficient must be not 27/4, but 15/4. See eqn (120) in https://arxiv.org/abs/1105.6086 , and also keep in mind that the moment of inertia of a homogeneous sphere is $(2/5) M R^2$. There are other inaccuracies in that article.  

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## 1 answers

### Answer by username_5 on 2022-06-25. Score: 3

username_0 should be commended for asking this clever question.

In short, the described configuration is possible, at least in principle.

First of all, let us recall that the solar tides in a planet whose spin is synchronised with the star are always working to ensure descent, see eqn (150) in [this work](https://arxiv.org/abs/1904.02253). On the other hand, the solar tides in a nonsynchronised planet are working to repel the planet from the star.

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 solar 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 $n$ about the star is smaller or larger than the planet's spin rate. As demonstrated [here](https://www.aanda.org/component/article?access=doi&doi=10.1051/0004-6361/202245533) the spin rate of a planet synchronised by a moon will, typically, be several times higher than the mean motion $n$ of the planet about the star. So the solar tides in the planet will be repulsive -- and this is one possible mechanism helping a close-in planet to avoid engulfment.

Two technical warnings to those who decide to integrate this problem.

(1) If a prograde moon starts 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](https://www.nature.com/articles/s41550-021-01306-2).) 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 may 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 try to integrate.

(2) If a prograde moon starts below the synchronous radius, it will be falling down and, at the same time, accelerating the planet's rotation. It may result in synchronisation of the planet with the moon -- provided this end state is attained before the descending moon crosses the Roche radius.

> **Comments:**
> **username_6**: Ok, but as username_1 username_1 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).  
> **username_7**: Surely the effects also depend on the rotation rate of the Sun and whether that is faster or slower than the Earth/Moon orbital period. The Sun's rotation rate is basically unaffected by its tidal interaction with the Earth.  
> **username_5**: @username_7 The rotation rate of the Sun will determine in which direction the solar tides will be pushing the Earth. Given that the Sun's rotation is faster than the Earth's mean motion about it, the Earth-generated tides in the Sunn will be working to push the Earth away (a feeble effect actually, given the large distance). I do not see, however, how the rotation rate of Sun can influence the working of the solar tides in the Earth. That can depend only on the Earth's rotation rate. Please correct me if I am missing something.  

---

