<p>The answer to this question is negative, if &quot;almost tidally locked&quot; signifies a small but persistent deviation from an exact resonance. The anwser, however, is positive if &quot;almost tidally locked&quot; implies large-magnitude libration.</p>
<p>Within a tidal two-body problem, a secondary body experiences two torques exerted on it by the primary: a tidal torque <span class="math-container">${\cal{T}}_{tide}$</span> and a torque generated by the permanent dynamical triaxiality of the secondary's figure, <span class="math-container">${\cal{T}}_{tri}$</span> (in practitioners' argo, the &quot;triaxial torque&quot;).</p>
<p>For bodies that are rigid or have a solid crust, the triaxiality-caused elevation above the mean level is typically much larger than the hight of the tidal bulge.  As a result of this, in a spin-orbit resonance (a synchronous resonance, like the Moon; or higher resonance, like Mercury),  <span class="math-container">${\cal{T}}_{tri} \gg {\cal{T}}_{tide}$</span>.</p>
<p>Two important caveats are in order, though:<br />
(a)  In a spin-orbit resonance, the rotator is subject to forced physical libration of a magnitude <span class="math-container">$\simeq e^2$</span>.<br />
(b)  Outside spin-orbit resonances, <span class="math-container">${\cal{T}}_{tri}$</span> averages out due to rotation, and <span class="math-container">$ {\cal{T}}_{tide}$</span> plays a key role.</p>
<p>Usually (the Moon being a notable exception), forced physical libration is predominantly longitudinal. Its evolution is analogous to ordinary pendulum. The torque <span class="math-container">${\cal{T}}_{tri}$</span> splits into two parts, one playing the role of restoring force, another of exterior forcing.  Like a pendulum, the rotator is impelled to deviate from the exact equilibrium position, but is pushed back by the restoring part of the torque.</p>
<p>For gas and liquid bodies lacking dynamical triaxiality, we have <span class="math-container">${\cal{T}}_{tri} =0$</span>.  So dynamics is governed by <span class="math-container">${\cal{T}}_{tide}$</span>.  Calculation demonstrates that in this situation the rotator always ends up in a so-called pseudosynchronous state, a regime wherein the rotator has an excess angular velocity (as compared to the mean motion <span class="math-container">$n$</span>) proportional to <span class="math-container">$6ne^2$</span>.  This calculation is presented, with an error, in <a href="https://rads.stackoverflow.com/amzn/click/com/0521575974" rel="nofollow noreferrer" rel="nofollow noreferrer">this standard text</a>, eqn (5.14).  A correct calculation can be found in <a href="https://link.springer.com/article/10.1007/s10569-012-9446-7" rel="nofollow noreferrer">our paper</a>.
(Be mindful that the value of said excess in angular velocity is rheology-dependent.)</p>
<p>(2)  Now, do pseudosynchronous rotators in the solar system exist?  Not to the best of my knowledge.</p>
<p>(3)   Are there intensely librating objects?  Yes, there are.</p>
<p>Owing to its potatoe shape, Phobos is librating in longitude with a magnitude of <span class="math-container">$1.2^{\,o}$</span>.  Epimetheus has a more regular shape but a larger dynamical triaxiality due to its internal inhomogeneity, so its longitudinal libration magnitude is <span class="math-container">$5.9^{\,o}$</span>.  This is <em>a lot</em>. Libration more than doubles tidal dissipation in Phobos, and boosts tidal dissipation in Epimetheus by a factor of 26 -- more than an order of magnitude.</p>

CC BY-SA 4.0
2023-09-07T03:19:35.220Z
great answer, thank you. I suppose at some point these systems go from very non-uniform rotation to having huge libration with occasional 360 degree turns to having a relatively stable very large libration. As others point out, the period of transition is probably very brief in comparison with other time-scales.  PS. I wasn't aware of Epimetheus/Janus.  That pair sounds quite wild!  Do you think Ttide has played a role in how Venus has ended up or is it all about atmospheric convection?
Roger Wood
CC BY-SA 4.0
2023-09-07T05:31:30.483Z
@RogerWood It goes the opposite way: first rotation, then libration. The Moon was rotating fast after formation. Then $T_{tide}$ slowed it down. At some point the Moon could not perform another full rotation - but changed the direction of spin and performed its first libration, with a magnitude close to 360 deg.  Like a pendulum with friction, rotating about a horizontal pivot, in a gravity field, steadily loosing energy. At some point it fails to pass the upper point - and falls back, performing librations with gradually decreasing magnitude. I can give you references, if you are interested.
Michael_1812
CC BY-SA 4.0
2023-09-07T05:35:13.873Z
@RogerWood The spin state of Venus is an equilibrium between gravitational and thermal atmospheric tidal torques, see https://ui.adsabs.harvard.edu/abs/2003Icar..163....1C/abstract  My colleagues and I are actually working on this topic, developing some dissident idea -- which may work out or may as well fail. Bound by omerta, I cannot say more at this point :)
Michael_1812
CC BY-SA 4.0
2023-09-07T05:43:43.303Z
@RogerWood Speaking of Epimetheus and Janus. Both moons are rotationally synchronised with Saturn (up to libration). What makes them unusual is their *orbital* dance https://www.planetary.org/articles/janus-epimetheus-swap
Michael_1812
CC BY-SA 4.0
2023-09-07T18:59:08.380Z
Yes, I may not have phrased it well, but from rotation to libration.  But I imagine there's likely a period where it becomes chaotic, driven by ellipticity and various other orbital perturbations.  So sometimes it would do a 360 and sometimes it wouldn't. An interesting scenario but one that would likely not last very long.
Roger Wood
CC BY-SA 4.0
2023-09-07T19:02:19.143Z
I always thought Venus' spin was explained by convection rather than tidal effects.  That looks like an interesting paper. Thanks for the feedback.
Roger Wood
CC BY-SA 4.0
2023-09-07T19:21:19.453Z
@RogerWood Yes, chaos also kicks in. If the libration magnitudes about both the 1:1 spin-orbit resonance (SOR) and the neighbouring 3:2 SOR become large, then these resonances may overlap. The rotator will be switching chaotically between the two SORs. When the body is very triaxial, or when forcing is very strong, the spin becomes totally chaotic, with many neighbouring SORs overlapping. To add insult to injury, rotation about the z axis may become unstable -- in which case wobble emerges. You may want to look up Alice Quillen's works on tumble excitation in moons, if interested.
Michael_1812
CC BY-SA 4.0
2023-09-06T00:13:28.073Z
2023-09-06T00:13:28.073Z
Michael_1812
<p>The Moon's rotation is firmly tidally <em>locked</em> to the Earth and the Earth's rotation is firmly tidally <em>unlocked</em> with respect to the Moon. I gather that Mercury's rotation is tidally locked in a 3:2 resonance - which sounds a little precarious. Are there any bodies which are on the verge of being tidally locked (very nonuniform rotation) or are barely tidally locked (very large libration)?  Are there any bodies whose rotation is actually chaotic for such reasons? <br></p>
<p>[Edit] Note, this question is about the physical libration rather than the apparent or optical libration caused by the body being observing from different perspectives.</p>

CC BY-SA 4.0
2020-12-11T08:18:14.687Z
Libration is not an expression of the "quality" of tidal locking. It is an expression of the excentricity and obliquity of the orbit of the tidally-locked body.
planetmaker
CC BY-SA 4.0
2020-12-11T14:52:36.483Z
Great question! This is not the same thing but it's quite an interesting related situation: [Is Venus in some way tidally locked to… Earth?](https://astronomy.stackexchange.com/q/36488/7982)
uhoh
CC BY-SA 4.0
2020-12-11T14:59:30.000Z
@planetmaker that's not correct. You've only described *optical libration*. Now read about [true libration](http://adsabs.harvard.edu/full/1910MmRAS..59..257S) which is what the OP is describing. Our Moon does both, the latter is small now, but it would have been quite wild at some point the past.
uhoh
CC BY-SA 4.0
2020-12-12T04:03:28.587Z
@planetmaker Thanks for the comment.  I'll add a clarification to the question
Roger Wood
CC BY-SA 4.0
2020-12-12T04:14:50.273Z
@uhoh That paper on the Moon's libration is surely a classic.  There's something special about papers like this that were written well before the age of computers.
Roger Wood
CC BY-SA 4.0
2020-12-12T05:43:54.253Z
speaking of using math (or in this case mis-using it) [Why did my vis-viva math solution get so close despite being wrong? Under what conditions would it have been a good approximation?](https://space.stackexchange.com/q/49086/12102)
uhoh
CC BY-SA 4.0
2020-12-12T08:07:50.990Z
The magnitude of tidal forces decreases with a power of 7..8 of the distance of two bodies (as far as I remember). This means that tidal locking either occurs very quickly (on astronomical time scales) or never. The chances to find an object in the middle of the transition from unlocked to locked is thus very small.
Hartmut Braun
CC BY-SA 4.0
2020-12-13T01:59:24.297Z
@uhoh I did put a comment on your other solar sail question, but I see now someone else pointed out the logarithmic spiral solution.  Thrust times velocity divided by mv^2 must give a relevant time constant.  So that's just acceleration from solar-sail divided by orbital velocity.
Roger Wood
CC BY-SA 4.0
2020-12-13T02:04:31.687Z
@ Hartmut Braun That's a very relevant observation.  The tidal force per se will go as the field gradient, so 1/r^3.  But I can well imagine that the rotational losses go as the square or cube of the stress.
Roger Wood
CC BY-SA 4.0
2020-12-19T13:43:15.240Z
Here's one that needs some sorting out and explaining of physics: [What is the YORP effect exactly? Is it just the non-central component of the Yarkovsky effect?](https://astronomy.stackexchange.com/q/40490/7982)
uhoh
<orbital-mechanics><rotation><natural-satellites><tidal-locking>
Are there any bodies in the solar system whose rotation is almost tidally locked or barely tidally locked?
Roger Wood
# Are there any bodies in the solar system whose rotation is almost tidally locked or barely tidally locked?

(Asked by: username_0 on 2020-12-10)

The Moon's rotation is firmly tidally *locked* to the Earth and the Earth's rotation is firmly tidally *unlocked* with respect to the Moon. I gather that Mercury's rotation is tidally locked in a 3:2 resonance - which sounds a little precarious. Are there any bodies which are on the verge of being tidally locked (very nonuniform rotation) or are barely tidally locked (very large libration)? Are there any bodies whose rotation is actually chaotic for such reasons?

[Edit] Note, this question is about the physical libration rather than the apparent or optical libration caused by the body being observing from different perspectives.

**Tags:** orbital-mechanics, rotation, natural-satellites, tidal-locking

> **Comments:**
> **username_1**: Libration is not an expression of the "quality" of tidal locking. It is an expression of the excentricity and obliquity of the orbit of the tidally-locked body.  
> **username_2**: Great question! This is not the same thing but it's quite an interesting related situation: [Is Venus in some way tidally locked to… Earth?](https://astronomy.stackexchange.com/q/36488/7982)  
> **username_2**: @username_1 that's not correct. You've only described *optical libration*. Now read about [true libration](http://adsabs.harvard.edu/full/1910MmRAS..59..257S) which is what the OP is describing. Our Moon does both, the latter is small now, but it would have been quite wild at some point the past.  
> **username_0**: @username_1 Thanks for the comment.  I'll add a clarification to the question  
> **username_0**: @username_2 That paper on the Moon's libration is surely a classic.  There's something special about papers like this that were written well before the age of computers.  
> **username_2**: speaking of using math (or in this case mis-using it) [Why did my vis-viva math solution get so close despite being wrong? Under what conditions would it have been a good approximation?](https://space.stackexchange.com/q/49086/12102)  
> **username_3**: The magnitude of tidal forces decreases with a power of 7..8 of the distance of two bodies (as far as I remember). This means that tidal locking either occurs very quickly (on astronomical time scales) or never. The chances to find an object in the middle of the transition from unlocked to locked is thus very small.  
> **username_0**: @username_2 I did put a comment on your other solar sail question, but I see now someone else pointed out the logarithmic spiral solution.  Thrust times velocity divided by mv^2 must give a relevant time constant.  So that's just acceleration from solar-sail divided by orbital velocity.  
> **username_0**: @ username_3 username_3 That's a very relevant observation.  The tidal force per se will go as the field gradient, so 1/r^3.  But I can well imagine that the rotational losses go as the square or cube of the stress.  
> **username_2**: Here's one that needs some sorting out and explaining of physics: [What is the YORP effect exactly? Is it just the non-central component of the Yarkovsky effect?](https://astronomy.stackexchange.com/q/40490/7982)  

---

## 1 answers

### Answer by username_4 on 2023-09-06. Score: 4 | Accepted answer

The answer to this question is negative, if "almost tidally locked" signifies a small but persistent deviation from an exact resonance. The anwser, however, is positive if "almost tidally locked" implies large-magnitude libration.

Within a tidal two-body problem, a secondary body experiences two torques exerted on it by the primary: a tidal torque ${\cal{T}}\_{tide}$ and a torque generated by the permanent dynamical triaxiality of the secondary's figure, ${\cal{T}}\_{tri}$ (in practitioners' argo, the "triaxial torque").

For bodies that are rigid or have a solid crust, the triaxiality-caused elevation above the mean level is typically much larger than the hight of the tidal bulge. As a result of this, in a spin-orbit resonance (a synchronous resonance, like the Moon; or higher resonance, like Mercury), ${\cal{T}}\_{tri} \gg {\cal{T}}\_{tide}$.

Two important caveats are in order, though:  
(a) In a spin-orbit resonance, the rotator is subject to forced physical libration of a magnitude $\simeq e^2$.  
(b) Outside spin-orbit resonances, ${\cal{T}}\_{tri}$ averages out due to rotation, and $ {\cal{T}}\_{tide}$ plays a key role.

Usually (the Moon being a notable exception), forced physical libration is predominantly longitudinal. Its evolution is analogous to ordinary pendulum. The torque ${\cal{T}}\_{tri}$ splits into two parts, one playing the role of restoring force, another of exterior forcing. Like a pendulum, the rotator is impelled to deviate from the exact equilibrium position, but is pushed back by the restoring part of the torque.

For gas and liquid bodies lacking dynamical triaxiality, we have ${\cal{T}}\_{tri} =0$. So dynamics is governed by ${\cal{T}}\_{tide}$. Calculation demonstrates that in this situation the rotator always ends up in a so-called pseudosynchronous state, a regime wherein the rotator has an excess angular velocity (as compared to the mean motion $n$) proportional to $6ne^2$. This calculation is presented, with an error, in [this standard text](https://rads.stackoverflow.com/amzn/click/com/0521575974), eqn (5.14). A correct calculation can be found in [our paper](https://link.springer.com/article/10.1007/s10569-012-9446-7).
(Be mindful that the value of said excess in angular velocity is rheology-dependent.)

(2) Now, do pseudosynchronous rotators in the solar system exist? Not to the best of my knowledge.

(3) Are there intensely librating objects? Yes, there are.

Owing to its potatoe shape, Phobos is librating in longitude with a magnitude of $1.2^{\,o}$. Epimetheus has a more regular shape but a larger dynamical triaxiality due to its internal inhomogeneity, so its longitudinal libration magnitude is $5.9^{\,o}$. This is *a lot*. Libration more than doubles tidal dissipation in Phobos, and boosts tidal dissipation in Epimetheus by a factor of 26 -- more than an order of magnitude.

> **Comments:**
> **username_0**: great answer, thank you. I suppose at some point these systems go from very non-uniform rotation to having huge libration with occasional 360 degree turns to having a relatively stable very large libration. As others point out, the period of transition is probably very brief in comparison with other time-scales.  PS. I wasn't aware of Epimetheus/Janus.  That pair sounds quite wild!  Do you think Ttide has played a role in how Venus has ended up or is it all about atmospheric convection?  
> **username_4**: @username_0 It goes the opposite way: first rotation, then libration. The Moon was rotating fast after formation. Then $T_{tide}$ slowed it down. At some point the Moon could not perform another full rotation - but changed the direction of spin and performed its first libration, with a magnitude close to 360 deg.  Like a pendulum with friction, rotating about a horizontal pivot, in a gravity field, steadily loosing energy. At some point it fails to pass the upper point - and falls back, performing librations with gradually decreasing magnitude. I can give you references, if you are interested.  
> **username_4**: @username_0 The spin state of Venus is an equilibrium between gravitational and thermal atmospheric tidal torques, see https://ui.adsabs.harvard.edu/abs/2003Icar..163....1C/abstract  My colleagues and I are actually working on this topic, developing some dissident idea -- which may work out or may as well fail. Bound by omerta, I cannot say more at this point :)  
> **username_4**: @username_0 Speaking of Epimetheus and Janus. Both moons are rotationally synchronised with Saturn (up to libration). What makes them unusual is their *orbital* dance https://www.planetary.org/articles/janus-epimetheus-swap  
> **username_0**: Yes, I may not have phrased it well, but from rotation to libration.  But I imagine there's likely a period where it becomes chaotic, driven by ellipticity and various other orbital perturbations.  So sometimes it would do a 360 and sometimes it wouldn't. An interesting scenario but one that would likely not last very long.  
> **username_0**: I always thought Venus' spin was explained by convection rather than tidal effects.  That looks like an interesting paper. Thanks for the feedback.  
> **username_4**: @username_0 Yes, chaos also kicks in. If the libration magnitudes about both the 1:1 spin-orbit resonance (SOR) and the neighbouring 3:2 SOR become large, then these resonances may overlap. The rotator will be switching chaotically between the two SORs. When the body is very triaxial, or when forcing is very strong, the spin becomes totally chaotic, with many neighbouring SORs overlapping. To add insult to injury, rotation about the z axis may become unstable -- in which case wobble emerges. You may want to look up Alice Quillen's works on tumble excitation in moons, if interested.  

---

