Abstract
That the Moon shows one face is familiar, and that more than half is nonetheless visible is explained by libration. What this paper shows is that the width of that libration is fixed by a single condition: which side of the ecliptic normal the two normals lie on. No new theorem or law is claimed.
First, the choice of side gives two inclinations——from $5.145$ degrees for the orbit and $1.543$ for the equator, opposite sides give $\mathbf{6.688}$ degrees and the same side gives $\mathbf{3.602}$ (Section 2).
Second, observation selects opposite sides——the libration in latitude is about $6.7$ degrees, within $0.2$ of $\mathbf{6.688}$ and more than $3$ degrees from $\mathbf{3.602}$ (Section 2).
Third, the libration in longitude follows from eccentricity——solving Kepler’s equation at the mean eccentricity $0.0549$ gives $\mathbf{6.2933}$ degrees, close to the first-order $2e$ value of $\mathbf{6.2911}$ (Section 3).
Fourth, which falls short of the observed $7.9$ degrees——reaching $7.9$ requires an eccentricity of $\mathbf{0.068903}$, inside the range from $0.026$ to $0.077$ that the Moon actually takes (Section 3).
Fifth, and this is the core. The visible fraction changes with the choice of side——$\mathbf{0.580675}$ for opposite sides and $\mathbf{0.563750}$ for the same side, a difference of $\mathbf{0.016925}$ (Section 4).
Sixth, the separator is the side of the normal——not the size of the inclinations but which side they lie on fixes the area (Section 5).
Keywords: Cassini’s laws, lunar libration, lunar spin axis, visible fraction, celestial mechanics.
Scope (what is not claimed)
No new theorem or law is claimed——Cassini’s three laws and the classification of librations are standard. The dynamics of the Cassini state are not treated——why the Moon settled there, through tidal damping and capture, is not entered. Nodal precession is not treated——the $18.6$-year motion of the nodes is averaged over so that only the mean geometry is used. Diurnal libration is not counted separately——the observer is placed at the centre of the Earth, so the extra degree from parallax is excluded. Physical libration is not treated——the Moon’s own rotational wobble, an order of magnitude smaller, is excluded. No observing guidance is given——when and from where a given region becomes visible is not the subject. The customary figure of $59$ per cent is not adopted——the value here is $\mathbf{0.580675}$ without diurnal libration, which the customary figure includes. Relation to earlier papers: Paper 406 showed that seeing one face is the evidence that the Moon rotates——that paper argued about whether it rotates; this one shows that where its axis points fixes how much is seen. Paper 328 showed that an ellipse three months out was evidence of a different cause——the same shape of argument, with orientation separating causes. Paper 95 separated convention from fact——which side the normals lie on is fact and not a convention one may choose, and so it can be refuted. Paper 300 showed that sharing a root is decidable——the two librations are separate roots under the criterion of what a quantity carries. What is added is putting the consequence of the third law as a choice between $\mathbf{6.688}$ and $\mathbf{3.602}$ degrees, obtaining the longitudinal libration from Kepler’s equation as $\mathbf{6.2933}$ degrees, inverting the shortfall to an eccentricity of $\mathbf{0.068903}$, integrating the visible area both ways for a difference of $\mathbf{0.016925}$, and placing the separator on the side of the normal.
1. Notation and the Question
Setting: in 1693 Cassini stated three laws for the Moon’s rotation: that its rotation period equals its orbital period, that its equator keeps a constant inclination to the ecliptic, and that its rotation axis, the normal to its orbit and the normal to the ecliptic are coplanar.
Observation 1.1 only the third law carries an orientation. Not merely coplanar: the two normals lie on opposite sides of the ecliptic normal——and same side or opposite changes the inclination between equator and orbit.
Observation 1.2 that inclination is the amplitude of the libration in latitude. The visible area follows from it——so the third law contains a refutable claim about area.
Question: which orientation matches observation?
2. The Third Law Fixes the Orientation
Computation 2.1: taking $5.145$ degrees for orbit to ecliptic and $1.543$ degrees for equator to ecliptic, and computing the equator-to-orbit inclination both ways.
| side of the normals | equator to orbit | gap from the observed $6.7$ degrees |
|---|---|---|
| opposite (the third law) | $\mathbf{6.688}$ degrees | $\mathbf{0.012}$ degrees |
| same | $\mathbf{3.602}$ degrees | $\mathbf{3.098}$ degrees |
Observation 2.2 it is a matter of adding or subtracting. $5.145+1.543$ against $5.145-1.543$——one sign in a formula, more than three degrees in an observable.
Observation 2.3 observation excludes one. The libration in latitude is about $6.7$ degrees——the same-side assumption is refuted, so the third law is a refutable claim rather than a classification.
Observation 2.4 by the division of Paper 95 this lies on the side of fact. Which side the normals lie on cannot be chosen by convention——and because it cannot be chosen, it can be wrong.
3. Longitudinal Libration Follows from Eccentricity
Computation 3.1: with uniform rotation and an elliptical orbit, the apparent orientation swings by the equation of the centre. Solving Kepler’s equation and taking the largest difference between true and mean anomaly.
| quantity | value |
|---|---|
| amplitude at eccentricity $0.0549$ | $\mathbf{6.2933}$ degrees |
| first-order value $2e$ | $\mathbf{6.2911}$ degrees |
| observed libration in longitude | about $7.9$ degrees |
| eccentricity required for $7.9$ degrees | $\mathbf{0.068903}$ |
| range the Moon’s eccentricity actually takes | $0.026$ to $0.077$ |
Observation 3.2 the mean eccentricity is not enough. $\mathbf{6.2933}$ degrees against about $7.9$——short by $1.6$ degrees, too much for higher-order terms to supply.
Observation 3.3 the shortfall was inverted rather than guessed at. Reaching $7.9$ degrees requires an eccentricity of $\mathbf{0.068903}$——and that value lies inside the range from $0.026$ to $0.077$ that solar perturbation actually gives the Moon.
Observation 3.4 so the eccentricity was not constant. Neither “higher-order terms are missing” nor “the observation is poor”——the largest libration occurs when the eccentricity is large, so a mean value must always fall short.
Observation 3.5 agreement with the first-order value is a separate check. $\mathbf{6.2933}$ against $\mathbf{6.2911}$——confirming that Kepler’s equation was solved correctly. What is missing is the input, not the method.
4. The Visible Area
Computation 4.1: taking the sub-Earth point to move within a rectangle of $\pm7.9$ degrees in longitude and $\pm b$ in latitude, and integrating over the sphere the fraction ever visible.
| case | visible fraction |
|---|---|
| no libration (one hemisphere) | $\mathbf{0.500000}$ |
| as the third law has it ($b=\mathbf{6.688}$ degrees) | $\mathbf{0.580675}$ |
| were the normals on the same side ($b=\mathbf{3.602}$ degrees) | $\mathbf{0.563750}$ |
| difference | $\mathbf{0.016925}$ |
Observation 4.2 this is the core. Swapping the side of the normals moves the visible area by $\mathbf{0.016925}$, about $1.7$ per cent of the Moon’s surface——the third law predicts a measurable area.
Observation 4.3 no libration gives exactly one half. $\mathbf{0.500000}$——a check on the integration, not a claim.
Observation 4.4 this differs from the customary $59$ per cent. The $\mathbf{0.580675}$ here excludes diurnal libration——the extra degree from the Earth’s radius should be counted separately.
Observation 4.5 longitude and latitude contribute differently. The longitudinal amplitude is larger, and yet the latitudinal contribution to area is substantial——on a sphere, the same angle covers less area near the poles.
5. The Separator
Observation 5.1 the separator is the side of the normal. Give both inclinations and the area is still undetermined——$\mathbf{0.580675}$ or $\mathbf{0.563750}$ remain.
Observation 5.2 this continues Paper 406. There, seeing one face was the evidence of rotation——here, where that rotation axis points fixes how much is seen.
Observation 5.3 separate roots under Paper 300. Longitudinal libration carries eccentricity; latitudinal libration carries an axial tilt——both are called libration and answer to different things.
Observation 5.4 the same shape as Paper 328. There a three-month phase offset separated causes——here an orientation separates areas.
Observation 5.5 what is not said. Why the Moon settled into this state is not treated——tidal damping and capture are not entered, and neither physical nor diurnal libration is included.
6. Roll Call and Closing
- opposite sides give $\mathbf{6.688}$ degrees (Computation 2.1): same side gives $\mathbf{3.602}$.
- observation is about $6.7$ (Observation 2.3): same side is refuted.
- one sign, three degrees (Observation 2.2): add or subtract.
- longitudinal libration $\mathbf{6.2933}$ (Computation 3.1): first order $\mathbf{6.2911}$.
- short of the observed $7.9$ (Observation 3.2): by $1.6$ degrees.
- eccentricity required $\mathbf{0.068903}$ (Computation 3.1): inside the real range.
- the eccentricity was not constant (Observation 3.4): not higher-order terms.
- visible fraction $\mathbf{0.580675}$ (Computation 4.1): the core.
- same side gives $\mathbf{0.563750}$ (Computation 4.1): difference $\mathbf{0.016925}$.
- no libration gives $\mathbf{0.500000}$ (Observation 4.3): a check.
- the separator is the side (Observation 5.1): not the size.
Closing: That the Moon shows one face is familiar, and more than half is nonetheless visible because of libration. The third law fixes the orientation——with the orbit at $5.145$ degrees to the ecliptic and the equator at $1.543$, normals on opposite sides give an equator-to-orbit inclination of $\mathbf{6.688}$ degrees and the same side gives $\mathbf{3.602}$. As a formula it is only a matter of adding or subtracting, and as an observable it is a gap of more than three degrees. The observed libration in latitude of about $6.7$ degrees refutes the same-side assumption, so the third law is a refutable claim and not a classification. By the division of Paper 95 this lies on the side of fact, since which side the normals lie on cannot be chosen, and what cannot be chosen can be wrong. Longitudinal libration follows from eccentricity, and the mean is not enough——solving Kepler’s equation at the mean eccentricity $0.0549$ gives $\mathbf{6.2933}$ degrees, close to the first-order $\mathbf{6.2911}$. The observation is about $7.9$ degrees, short by $1.6$, too much for higher-order terms. Inverting rather than guessing, reaching $7.9$ degrees requires an eccentricity of $\mathbf{0.068903}$, which lies inside the range from $0.026$ to $0.077$ that solar perturbation actually gives. So what was missing was the input rather than the method: the eccentricity is not constant, and the largest libration happens when it is large. The orientation fixes the visible area——integrating over the sphere with the sub-Earth point in a rectangle of $\pm7.9$ degrees by $\pm b$ gives $\mathbf{0.580675}$ as the third law has it and $\mathbf{0.563750}$ on the same side, a difference of $\mathbf{0.016925}$. That no libration returns exactly $\mathbf{0.500000}$ checks the integration. The figure differs from the customary $59$ per cent because diurnal libration, about one degree, is excluded here. The separator is the side of the normal——give both inclinations and $\mathbf{0.580675}$ or $\mathbf{0.563750}$ still remain. Paper 406 took one face as evidence of rotation; here the direction of that axis fixes how much is seen. Under Paper 300, longitudinal libration carries eccentricity and latitudinal libration an axial tilt, so two things called libration answer to different quantities. Paper 328 let a three-month offset separate causes; here an orientation separates areas. To be honest——why the Moon settled into this state was not treated, and tidal damping and capture were not entered. The $18.6$-year motion of the nodes was averaged over, and neither physical nor diurnal libration is included, the observer being placed at the centre of the Earth, so the figure differs from the customary one. No observing guidance was given. What can be said is that one orientation, the side of a normal, moves the visible area by $\mathbf{0.016925}$ through the width of the libration in latitude, and no further.
References
- Cassini, J.-D. (1693). De l’origine et du progrès de l’astronomie et de son usage dans la géographie et dans la navigation. In Recueil d’observations faites en plusieurs voyages. Paris.
- Colombo, G. (1966). Cassini’s Second and Third Laws. Astronomical Journal.
- Meeus, J. (1998). Astronomical Algorithms (2nd ed.). Willmann-Bell.
- Peale, S. J. (1969). Generalized Cassini’s Laws. Astronomical Journal.
- Yamagishi, Y. (2026). Convention and Fact [Paper 95]. Zenodo. 10.5281/zenodo.21758621
- Yamagishi, Y. (2026). The Three Hundredth Census: Whether Two Things Share a Root Is Decidable [Paper 300]. Zenodo. 10.5281/zenodo.22171914
- Yamagishi, Y. (2026). That the Same Ellipse Was Three Months Out Was the Evidence of a Different Cause [Paper 328]. Zenodo. 10.5281/zenodo.22210340
- Yamagishi, Y. (2026). That the Same Face Is Seen Is the Evidence That the Moon Rotates [Paper 406]. Zenodo. 10.5281/zenodo.22699455
Reference
Yamagishi, Y. (2026). What Fixes How Much of the Moon Is Visible Is Which Side Two Normals Lie On [Paper 535]. YOI: https://www.yamagishiyuuki.com/yoi/535
DOI: 10.5281/zenodo.23014119
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 535. This paper is a preprint and has not been peer reviewed.