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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:

In History of Science and Mathematics SE:

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?
  • 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:

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  • $\begingroup$ companion question: For how many solar system bodies do recent JPL Development Ephemerides include libration? $\endgroup$
    – uhoh
    Commented Jan 22, 2022 at 1:58
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    $\begingroup$ Is there any residual oscillation left from the Moon rotation? $\endgroup$ Commented Jan 22, 2022 at 2:05
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    $\begingroup$ @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 ? $\endgroup$ Commented Mar 3, 2022 at 4:11
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    $\begingroup$ @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... $\endgroup$
    – uhoh
    Commented Mar 3, 2022 at 5:17
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    $\begingroup$ @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. $\endgroup$
    – uhoh
    Commented Mar 3, 2022 at 5:19

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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 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 gravitation table to show gravitational acceleration from some of these other bodies on the Moon.

enter image description here

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 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

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.

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