<p><strong>TLDR:</strong> The primary ~28 day lunar orbital librations are well modeled by taking into account the changing orbital speed due to the Moon's orbital eccentricity.  The forced physical librations (which include Cassini's 2nd and 3rd laws) can be modeled closed form with assumptions about the Moon's interior layers.  The free librations can be fitted to <a href="https://en.wikipedia.org/wiki/Lunar_Laser_Ranging_experiment" rel="nofollow noreferrer">Lunar Laser Ranging</a> data.</p>
<p><strong>Setup:</strong> Even though the Moon's rotation rate is the same as its orbital period, its orbital speed changes due to its orbital eccentricity.  It moves faster at perigee than apogee in accordance with Kepler's 2nd law.  This miss-match between orbital speed and rotation rate is the primary contributor to lunar libration and (as expected) cycles with the Moon's orbital period.  This libration isn't quite along a single axis due to the 6.8 degree obliquity of the Moon's rotation.</p>
<p>The Moon is tidally deformed by the Earth.  This tidal deformation axis leads or trails the vector from the Moon to the Earth through Apogee and Perigee, respectively, leading to torque on the Moon.</p>
<p>Earth's gravitational field is irregular.  The Earth is well modeled by an oblate spheroid and subject to its own tidal deformations mainly due to the Moon and the Sun. The orientation of the Earth's lunar tidal axis to the Earth/Moon vector is mostly transferring angular momentum from the Earth's spin into the Moon's orbit.  However, it is also imparting torque to the Moon (an irregular shape in an irregular gravitational field).</p>
<p>Gravitation from other bodies in the Solar System, even when treated as point masses will cause tidal torques on the Moon.  I did a little alteration to <a href="https://rads.stackoverflow.com/amzn/click/com/0486600610" rel="nofollow noreferrer" rel="nofollow noreferrer">Bate, Mueller, White's</a> gravitation table to show gravitational acceleration from some of these other bodies on the Moon.</p>
<p><a href="https://i.sstatic.net/jOnM3.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/jOnM3.png" alt="enter image description here" /></a></p>
<p>Note the tidal torques are a function of both gravitational force and gradient, so even though the Sun exerts higher gravitational forces on the Moon than the Earth exerts on the Moon, the Earth exerts more tidal torque since its gravitational gradient is higher. The sum of all these torques affect the rotation of the Moon and thus librations.</p>
<p>Also, we can't separate the long term lunar orbital effects from libration, since eccentricity is a primary cause of libration. Normally in a two body system we would expect circularization of the lunar orbit.  But for our Moon, the eccentricity is actually growing due to gravitational interactions from the Sun!</p>
<p><strong>Answer:</strong>
The librations of the Moon are the result of the changing geometry of the Earth/Moon system due to the sum of forces on the Moon combined with its orbital and angular momentum. The Moon is irregular in shape and composition, and changes shape due to tidal forces.  It also passes through an irregular gravitational field since the Earth is similarly deformed. This causes various torques on the Moon, which result in changes to its rotation.
However, the dominant force of tidal torque prevents the average orbital and rotation periods from ever deviating.  Hence, the librations.</p>
<p><a href="https://arxiv.org/abs/1606.08376" rel="nofollow noreferrer">Pavlov et al. 2018</a> wrote a paper called <em>Determining parameters of Moon's orbital and rotational motion from LLR observations using GRAIL and IERS-recommended models</em>.  Here are the parameters they use:</p>
<p><a href="https://i.sstatic.net/Ej9Fe.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/Ej9Fe.png" alt="enter image description here" /></a></p>
<p>I am surprised they don't explicitly include &quot;solar radiation pressure&quot; explicitly as one of the forces on the Moon, but perhaps it is hidden in one of the other variables.  Instead of using a complete physical model for the free physical librations, they use various models to fit to the LLR (Lunar Laser Ranging) measurements.</p>
<p>For the forced librations (including Cassini's 2nd and 3rd laws), there are some self-consistent formalisms, like in <a href="https://arxiv.org/abs/2201.00803" rel="nofollow noreferrer">https://arxiv.org/abs/2201.00803</a>.   However, they require some assumptions about the mantle and core of the Moon and they are extremely complex.</p>

CC BY-SA 4.0
2022-03-03T22:47:59.197Z
2022-03-07T23:21:10.610Z
2022-03-07T23:21:10.610Z
Connor Garcia
<p>If I understand correctly, in a two-body system with at least one of them more-or-less tidally locked (mean rotational period = mean orbital period) if we draw a line between the centers of mass and look at the point where that line passes through (one of) the locked bodies, that point will roughly periodically migrate around the surface, and that is called, or at least attributed to libration.</p>
<p>The most familiar example is the libration of Earth's Moon, and the primary cause is the Moon's eccentricity and also various axial and orbital tilts (which precess over time as well)</p>
<p>In Space SE:</p>
<ul>
<li><a href="https://space.stackexchange.com/q/22375/12102">What are the &quot;Moon L, B, C&quot; angles shown in this solar eclipse simulation?</a></li>
<li><a href="https://space.stackexchange.com/q/42127/12102">How to get lunar L, B, C parameters from the Moon's 3x3 rotation matrix from the Python package Skyfield?</a> (currently unanswered)</li>
</ul>
<p>In History of Science and Mathematics SE:</p>
<ul>
<li><a href="https://hsm.stackexchange.com/q/8443/3869">How did Cassini measure the &quot;Cassini state&quot; of the Moon? What measurements were made; what did the data look like?</a> (currently unanswered)</li>
</ul>
<p>In addition to the effects of eccentricity and axial and orbital inclinations I suppose that:</p>
<ul>
<li>for a just recently locked body there could still be residual pendulum-like harmonic motion that hasn't yet been damped out. See <a href="https://astronomy.stackexchange.com/q/17956/7982">Is there any residual oscillation left from the Moon rotation?</a></li>
<li>there could be motion excited by third body gravitational effects</li>
<li>there could be &quot;inner sloshing&quot; of magma, liquid core, or subsurface (or surface) oceans</li>
</ul>
<p><strong>Question:</strong> What are all the contributions to libration; is there a self-consistent formalism?</p>
<hr />
<p>Related:</p>
<ul>
<li><a href="https://astronomy.stackexchange.com/q/44361/7982">Moon&#39;s rotation and revolution</a></li>
<li><a href="https://astronomy.stackexchange.com/q/47579/7982">Was lunar libration first observed or first predicted? In either case, who was the responsible party?</a></li>
<li><a href="https://astronomy.stackexchange.com/q/47570/7982">How old is the idea of the far side of the Moon?</a></li>
<li><a href="https://astronomy.stackexchange.com/q/35604/7982">Explanation about the resonance, mean motion resonance and libration</a></li>
</ul>

CC BY-SA 4.0
2022-01-22T01:58:04.433Z
companion question: [For how many solar system bodies do recent JPL Development Ephemerides include libration?](https://astronomy.stackexchange.com/q/48264/7982)
uhoh
CC BY-SA 4.0
2022-01-22T02:02:29.663Z
Also, asked in Space SE: [Just how locked is Titan? Does it exhibit libration due to eccentricity? Have residual oscillations not yet damped out been detected or ruled out?](https://space.stackexchange.com/q/57527/12102) and [Roughly, how much of our Moon's apparent libration amplitude seen from the Geocenter is due to different effects?](https://space.stackexchange.com/q/45366/12102) (currently unanswered) and [How "locked" are Pluto and Charon? How much does each librate as seen from the other?](https://space.stackexchange.com/q/45342/12102)
uhoh
CC BY-SA 4.0
2022-01-22T02:05:06.323Z
[Is there any residual oscillation left from the Moon rotation?](https://astronomy.stackexchange.com/questions/17956/is-there-any-residual-oscillation-left-from-the-moon-rotation/28993#28993)
Keith McClary
CC BY-SA 4.0
2022-01-22T02:05:19.707Z
and here in Astronomy SE [Explanation about the resonance, mean motion resonance and libration](https://astronomy.stackexchange.com/q/35604/7982)
uhoh
CC BY-SA 4.0
2022-03-02T21:24:41.103Z
Is libration really an effect of just two bodies? Or is it one of a special case of a three body system? Honest question, I do not know
Justin T
CC BY-SA 4.0
2022-03-02T21:41:33.407Z
@JustinT Yes it can be. Two bodies in an elliptical orbit with tidal locking results in one kind of libration.
uhoh
CC BY-SA 4.0
2022-03-03T04:11:25.080Z
@uhoh "What are all the contributions to libration; is there a self-consistent formalism" my poor language skills interpret the question in different ways, Can you explain that line a bit ?
Kavin Ishwaran
CC BY-SA 4.0
2022-03-03T05:17:49.723Z
@KavinIshwaran "libration" is a word that's used to describe an observable behavior, but there is more than one thing that can cause or "contribute to" observed libration. The most well known contribution is of course a steady rate of rotation of the Moon about its own axis but variable rate that it revolves around Earth due to its elliptical orbit. But there can be other contributions as well; just for example a moon other than our Moon might not rotate steadily, but instead oscillate slightly in speed due to higher order (tidal) gravitational forces, or it could be in the end stages of...
uhoh
CC BY-SA 4.0
2022-03-03T05:19:51.313Z
@KavinIshwaran ...tidal locking and not yet fully damped down to steady rotation. Because there are multiple possible contributions, some authors may refer to only some subset of these contribution when using the term "libration" and other authors might choose a different subset. By "self-consistent formalism" I'm asking about a formal treatment of all possible contributions to observed libration in a self-consistent (as opposed to self-contradictory) way.
uhoh
<astrophysics><planetary-science><lunar-libration>
What are all the contributions to libration; is there a self-consistent formalism?
uhoh
# What are all the contributions to libration; is there a self-consistent formalism?

(Asked by: username_0 on 2022-01-22)

If I understand correctly, in a two-body system with at least one of them more-or-less tidally locked (mean rotational period = mean orbital period) if we draw a line between the centers of mass and look at the point where that line passes through (one of) the locked bodies, that point will roughly periodically migrate around the surface, and that is called, or at least attributed to libration.

The most familiar example is the libration of Earth's Moon, and the primary cause is the Moon's eccentricity and also various axial and orbital tilts (which precess over time as well)

In Space SE:

* [What are the "Moon L, B, C" angles shown in this solar eclipse simulation?](https://space.stackexchange.com/q/22375/12102)
* [How to get lunar L, B, C parameters from the Moon's 3x3 rotation matrix from the Python package Skyfield?](https://space.stackexchange.com/q/42127/12102) (currently unanswered)

In History of Science and Mathematics SE:

* [How did Cassini measure the "Cassini state" of the Moon? What measurements were made; what did the data look like?](https://hsm.stackexchange.com/q/8443/3869) (currently unanswered)

In addition to the effects of eccentricity and axial and orbital inclinations I suppose that:

* for a just recently locked body there could still be residual pendulum-like harmonic motion that hasn't yet been damped out. See [Is there any residual oscillation left from the Moon rotation?](https://astronomy.stackexchange.com/q/17956/7982)
* there could be motion excited by third body gravitational effects
* there could be "inner sloshing" of magma, liquid core, or subsurface (or surface) oceans

**Question:** What are all the contributions to libration; is there a self-consistent formalism?

---

Related:

* [Moon's rotation and revolution](https://astronomy.stackexchange.com/q/44361/7982)
* [Was lunar libration first observed or first predicted? In either case, who was the responsible party?](https://astronomy.stackexchange.com/q/47579/7982)
* [How old is the idea of the far side of the Moon?](https://astronomy.stackexchange.com/q/47570/7982)
* [Explanation about the resonance, mean motion resonance and libration](https://astronomy.stackexchange.com/q/35604/7982)

**Tags:** astrophysics, planetary-science, lunar-libration
> **Comments:**
> **username_0**: companion question: [For how many solar system bodies do recent JPL Development Ephemerides include libration?](https://astronomy.stackexchange.com/q/48264/7982)  
> **username_0**: Also, asked in Space SE: [Just how locked is Titan? Does it exhibit libration due to eccentricity? Have residual oscillations not yet damped out been detected or ruled out?](https://space.stackexchange.com/q/57527/12102) and [Roughly, how much of our Moon's apparent libration amplitude seen from the Geocenter is due to different effects?](https://space.stackexchange.com/q/45366/12102) (currently unanswered) and [How "locked" are Pluto and Charon? How much does each librate as seen from the other?](https://space.stackexchange.com/q/45342/12102)  
> **username_1**: [Is there any residual oscillation left from the Moon rotation?](https://astronomy.stackexchange.com/questions/17956/is-there-any-residual-oscillation-left-from-the-moon-rotation/28993#28993)  
> **username_0**: and here in Astronomy SE [Explanation about the resonance, mean motion resonance and libration](https://astronomy.stackexchange.com/q/35604/7982)  
> **username_2**: Is libration really an effect of just two bodies? Or is it one of a special case of a three body system? Honest question, I do not know  
> **username_0**: @JustinT Yes it can be. Two bodies in an elliptical orbit with tidal locking results in one kind of libration.  
> **username_3**: @uhoh "What are all the contributions to libration; is there a self-consistent formalism" my poor language skills interpret the question in different ways, Can you explain that line a bit ?  
> **username_0**: @KavinIshwaran "libration" is a word that's used to describe an observable behavior, but there is more than one thing that can cause or "contribute to" observed libration. The most well known contribution is of course a steady rate of rotation of the Moon about its own axis but variable rate that it revolves around Earth due to its elliptical orbit. But there can be other contributions as well; just for example a moon other than our Moon might not rotate steadily, but instead oscillate slightly in speed due to higher order (tidal) gravitational forces, or it could be in the end stages of...  
> **username_0**: @KavinIshwaran ...tidal locking and not yet fully damped down to steady rotation. Because there are multiple possible contributions, some authors may refer to only some subset of these contribution when using the term "libration" and other authors might choose a different subset. By "self-consistent formalism" I'm asking about a formal treatment of all possible contributions to observed libration in a self-consistent (as opposed to self-contradictory) way.  

---

## 1 answers

### Answer by username_4 on 2022-03-03. Score: 2 | Accepted answer

**TLDR:** The primary ~28 day lunar orbital librations are well modeled by taking into account the changing orbital speed due to the Moon's orbital eccentricity. The forced physical librations (which include Cassini's 2nd and 3rd laws) can be modeled closed form with assumptions about the Moon's interior layers. The free librations can be fitted to [Lunar Laser Ranging](https://en.wikipedia.org/wiki/Lunar_Laser_Ranging_experiment) data.

**Setup:** Even though the Moon's rotation rate is the same as its orbital period, its orbital speed changes due to its orbital eccentricity. It moves faster at perigee than apogee in accordance with Kepler's 2nd law. This miss-match between orbital speed and rotation rate is the primary contributor to lunar libration and (as expected) cycles with the Moon's orbital period. This libration isn't quite along a single axis due to the 6.8 degree obliquity of the Moon's rotation.

The Moon is tidally deformed by the Earth. This tidal deformation axis leads or trails the vector from the Moon to the Earth through Apogee and Perigee, respectively, leading to torque on the Moon.

Earth's gravitational field is irregular. The Earth is well modeled by an oblate spheroid and subject to its own tidal deformations mainly due to the Moon and the Sun. The orientation of the Earth's lunar tidal axis to the Earth/Moon vector is mostly transferring angular momentum from the Earth's spin into the Moon's orbit. However, it is also imparting torque to the Moon (an irregular shape in an irregular gravitational field).

Gravitation from other bodies in the Solar System, even when treated as point masses will cause tidal torques on the Moon. I did a little alteration to [Bate, Mueller, White's](https://rads.stackoverflow.com/amzn/click/com/0486600610) gravitation table to show gravitational acceleration from some of these other bodies on the Moon.

[![enter image description here](https://i.sstatic.net/jOnM3.png)](https://i.sstatic.net/jOnM3.png)

Note the tidal torques are a function of both gravitational force and gradient, so even though the Sun exerts higher gravitational forces on the Moon than the Earth exerts on the Moon, the Earth exerts more tidal torque since its gravitational gradient is higher. The sum of all these torques affect the rotation of the Moon and thus librations.

Also, we can't separate the long term lunar orbital effects from libration, since eccentricity is a primary cause of libration. Normally in a two body system we would expect circularization of the lunar orbit. But for our Moon, the eccentricity is actually growing due to gravitational interactions from the Sun!

**Answer:**
The librations of the Moon are the result of the changing geometry of the Earth/Moon system due to the sum of forces on the Moon combined with its orbital and angular momentum. The Moon is irregular in shape and composition, and changes shape due to tidal forces. It also passes through an irregular gravitational field since the Earth is similarly deformed. This causes various torques on the Moon, which result in changes to its rotation.
However, the dominant force of tidal torque prevents the average orbital and rotation periods from ever deviating. Hence, the librations.

[Pavlov et al. 2018](https://arxiv.org/abs/1606.08376) wrote a paper called *Determining parameters of Moon's orbital and rotational motion from LLR observations using GRAIL and IERS-recommended models*. Here are the parameters they use:

[![enter image description here](https://i.sstatic.net/Ej9Fe.png)](https://i.sstatic.net/Ej9Fe.png)

I am surprised they don't explicitly include "solar radiation pressure" explicitly as one of the forces on the Moon, but perhaps it is hidden in one of the other variables. Instead of using a complete physical model for the free physical librations, they use various models to fit to the LLR (Lunar Laser Ranging) measurements.

For the forced librations (including Cassini's 2nd and 3rd laws), there are some self-consistent formalisms, like in <https://arxiv.org/abs/2201.00803>. However, they require some assumptions about the mantle and core of the Moon and they are extremely complex.


---

