Fun With Science / Globe Deconstruction Review

Is the Moon a Ball?

A foundation page. The shape of the Moon, settled from the ground, before any argument about how it is lit.

Several claims in this book turn on the Moon: that its full face is too evenly lit for a sphere, that it tilts the wrong way, that its silhouette is missing before an eclipse. Each of those is answered on its own page. What none of those pages should have to do is re-establish that the Moon is a ball in the first place, so this page does that once. It is deliberately built out of things a person can check from their own garden, and it is built out of ratios rather than distances wherever it can be, because the ratios are what decide the question and they need no scale at all. Where a figure in kilometres does appear it rests on the Earth–Moon distance, which is measured to millimetres by professional radar and laser ranging — but not only by them, as §4 shows: an amateur bouncing a signal off the Moon times the round trip at about two and a half seconds and has the distance directly.

Foundation, not a verdictGround-basedAmateur-reproducible
What this page establishes, and what it does not

Establishes: that the Moon is a solid body of near-spherical figure, from five independent lines of evidence, four of which need no space agency and three of which need nothing but a camera. The strongest single result is that the Moon is as deep as it is wide — measured by radar, from the ground, in 1965 — to within four parts in ten thousand.

Does not establish: anything about what lights it. Miller’s Q7, that a full Moon is too uniformly bright for a sphere, is a real and well-posed observation, and it is answered on its own page. This page shows only that the shape it is an argument against is not in doubt, so the answer has to come from the surface rather than from the figure.

Conceded up front: a smooth Lambertian sphere really would show limb darkening, and the Moon really does not. That is his observation and it is correct. §6 says what it does and does not imply, and the short version is that it is a prediction of the sphere model, not a problem for it.

1 · The terminator, which a facing disc cannot have at all

Start with the thing everyone has seen. The lit fraction of the Moon is not a free quantity: it tracks the Sun–Moon–Earth angle α exactly, and the US Naval Observatory has published it for every date from 1700 to 2100 on the strength of that.

For a sphere, the boundary between day and night is a great circle. Seen from Earth it projects to a semi-ellipse with semi-major axis R and semi-minor axis R |cos α|. The lit area is half the disc plus half that ellipse, so

lit fraction = (1 + cos α) / 2

which is exactly the relation the ephemerides use. Now the flat model. A disc facing Earth, lit by a distant Sun, presents the same angle of incidence at every point of its surface. It has no terminator anywhere, at any time. It is uniformly lit, its brightness falls as cos α, and it goes dark altogether once α reaches 90°.

The two models at the same phase angle. The sphere is three-quarters lit with a curved terminator whose bow is exactly half the disc radius. The facing disc is fully lit and merely dimmer — it has no boundary to draw. Measure the bow on your own photograph as a fraction of the radius and you are measuring cos α.

Phase angle α	Sphere: lit fraction	Sphere: terminator bow / R	Facing disc: lit fraction	Facing disc: brightness
0° — full	1.000	1.000	1	1.000
45°	0.854	0.707	1	0.707
90° — quarter	0.500	0 (straight)	1	0.000
120°	0.250	0.500	0	0
150° — thin crescent	0.067	0.866	0	0

A crescent Moon is a refutation of the facing disc, and it hangs in the afternoon sky about ten days a month. At α = 150° the sphere predicts a disc 6.7 per cent lit. The facing disc predicts no Moon at all. There is no adjustment available, because the flat model’s brightness is cos α and cos α is negative there.

The obvious rescue is to move the Sun close, so that its rays fan across the disc and one edge is lit more steeply than the other. We worked out what that costs. Across a disc of the Moon’s radius, a Sun at 20,000 km gives an edge-to-centre difference in incidence angle of 4.96°; at 5,000 km, 19.2°. To drive the solar altitude to zero at one edge — which is what a terminator is — the Sun has to sit within roughly one lunar radius of the surface, which makes it far smaller than the Moon and produces a bright patch rather than a lit hemisphere. And nothing in that family gives a straight terminator through the exact centre with exactly half the disc lit, twice a month, on a published schedule.

Check it yourself. A phone on a tripod, eight nights spread across a month. Measure the terminator’s maximum bow as a fraction of the disc radius and compare it with |cos α|, where α comes from any almanac. No telescope, no distance, no trust required.

2 · Shadows that run away at the terminator, and peaks lit beyond it

Galileo’s 1610 observation, and still the best hands-on demonstration on this page. Crater rims near the terminator throw shadows that lengthen enormously as the terminator approaches; near the sub-solar point they shrink to nothing. And past the terminator, in the dark, isolated bright points appear detached from the lit region, then grow over the following hours until they merge with it.

First-quarter Moon photographed through an amateur telescope: crater relief is deeply shadowed and three-dimensional along the terminator down the left, and fades to almost no shadow toward the sunlit limb on the right. — **The whole argument, in one amateur photograph.** Watch how the shadowing changes across the face. Along the terminator, down the left, every rim throws a shadow long enough to see at this scale and the ground reads as violently three-dimensional. Move right toward the sunlit limb and the shadows shorten to nothing; the same kind of terrain goes smooth and washed out. That gradient is the measurement. Shadow length depends on where you are on the surface, so the Sun stands at a different altitude at each place, so the surface is turning away from it. **On a plane there is nothing for the gradient to be made of:** one distant source over a flat face means one solar altitude at every point of it, so equal rims would cast equal shadows everywhere on the disc at every moment — uniformly stark or uniformly flat, never both at once in one frame. Notice as well that the terminator is a curve and not a straight chord, which is §1 visible in the same picture. First-quarter Moon by Aisy Maffaz, own work, CC BY 4.0. A ground-based amateur frame, which is the only kind this page uses.

On a sphere the solar altitude at a point is 90° minus its angular distance from the sub-solar point, so shadow length L = h/tan(altitude) diverges as the terminator is reached — and a summit of height h stays lit while its base is dark, because its horizon is depressed by √(2h/R).

Solar altitude at the point	Shadow length / feature height
45°	1.0×
30°	1.7×
10°	5.7×
5°	11.4×
2°	28.6×
1°	57.3×

Which is the arithmetic behind what the photograph shows. The one thing a flat Moon cannot supply is the variation: with the solar altitude equal at every point of a plane, there is nowhere for shadow length to blow up, and no way for a summit to be lit while the ground beside it is dark, because there is no curvature to put that ground in shadow.

The measurement runs backwards too, and this is what undergraduate labs actually do with Earth-based images: crater height H = L × D / R at first quarter, where D is the feature’s distance from the terminator. Look at what sits in the denominator. The lunar radius is the measuring stick, and the method is self-consistent only because sin(solar altitude) = D/R on a sphere. Photograph the same crater on three consecutive nights: the shadow shortens each night, and the height you derive from it does not move.

Galileo ran the same geometry forwards. From a lit peak roughly a twentieth of a lunar diameter beyond the terminator he obtained h = R(sec 0.1 − 1), which with the modern radius is 8.7 km. The largest relief since measured by laser altimetry is 10.75 km. We have not seen a facsimile of his exact wording, so we paraphrase rather than quote him.

The objection that does not survive. “Those are surface features lit at a grazing angle.” Grazing light across the whole face cannot coexist with a high sub-solar point on that same face for one light source on a plane — and nothing on a plane produces detached lit points inside the dark region that later join up with the lit part. That needs a summit seeing the Sun over a curved horizon while its surroundings do not.

3 · The Moon rocks, and its outline does not deform

Over a month the Moon nods and rolls: features near the limb turn into and out of view, and the point facing Earth wanders by several degrees. The amplitudes are 7.9° in longitude on a 27.55-day period and 6.68° in latitude on a 27.21-day one, so Earth traces a rectangle roughly 13.4° by 15.8° in the Moon’s sky over a six-year cycle.

This is the sharpest ground-based discriminator available, because the two models differ in where the motion appears and in whether the silhouette changes.

Under a 7° tilt	Sphere	Flat disc
Feature at the disc centre	114″	0″
Feature at 30° from centre	95″	−3.5″
Feature at 45°	76″	−4.9″
Feature at 60°	51″	−6.0″
The outline itself	unchanged circle	ellipse, axis ratio 0.9925

A sphere moves a feature at longitude λ from R sin λ to R sin(λ+L) — a first-order displacement, largest at the disc centre — while the silhouette is untouched, because a sphere looks like a circle from every direction. A flat disc tilted by L merely multiplies every position by cos L: a second-order squeeze that is zero at the centre, and which necessarily compresses the outline into an ellipse, about three-quarters of a per cent on the diameter, with the compression axis rotating monthly.

The two models have opposite signatures, and it is the outline that gives it away. Tilt a sphere and every surface feature shifts — most of all the ones near the middle of the face — while the silhouette stays exactly circular, because a ball looks like a circle from every direction. Tilt a flat disc and the reverse happens: a feature at the centre does not move at all, and the outline itself is compressed into an ellipse. Observation gives us the first. Drawn at 25° so both effects are visible. The real optical libration is 7°, which moves a central feature by 114 arcseconds and would squash a disc’s outline by 0.75 per cent of its diameter, once a month, with the compression axis rotating.

Those two signatures point in opposite directions, and the observed one is the sphere’s. Features near the middle of the face swing by a hundred arcseconds a month; the outline stays a circle. The disc model predicts the exact reverse: a still centre and a breathing rim.

The rocking is also why more of the Moon is visible over time than at any one moment. We recomputed the swept fraction numerically, integrating the union of visible caps over the libration rectangle on an equal-area grid: 58.1 per cent from optical libration alone, rising to 59.1 per cent once diurnal libration is included — the extra tilt you get from standing on the Earth’s surface rather than at its centre, worth 0.951°. The familiar “59 per cent” is right, but about a percentage point of it is contributed by the observer’s own position, which is a pleasing thing to be able to say and is usually left out.

Check it yourself. Photograph the Moon at the same phase each month for a year, register the frames on the disc outline, and blink them. Mare Crisium slides visibly toward and away from the limb. The outline does not deform. A single pair of nights at the two extremes is enough to see it.

4 · The Moon is as deep as it is wide

The strongest result on this page, and the only one that measures the dimension a facing disc does not have: extent along the line of sight.

Aim a radar pulse at the Moon and the echo does not come back as a single return. It comes back smeared over 11.6 milliseconds — a bright leading edge from the point nearest us, then a decaying tail as echoes from successively wider annuli arrive, ending at the limb. For a sphere that duration is exactly 2R/c. We checked: 2 × 1738.1 km / c = 11.595 ms.

Turn it round and use the echo to measure the Moon rather than the other way about. The observed 11.6 ms gives a radial half-depth of 1738.8 km. The transverse radius, from the angular diameter, is 1738.1 km.

depth / width = 1.0004 (a facing disc predicts 0)

The obvious objection is that a flat disc is not infinitely thin either — rays to its edge travel further than rays to its centre. That is true and it is nowhere near enough. For a disc of the Moon’s radius at the Moon’s distance, the extra two-way path to the rim is R²/d = 7.86 km, which is 26 microseconds. The observed echo is 11.6 milliseconds. The two are a factor of 440 apart, and surface roughness of a few kilometres adds tens of microseconds, not milliseconds.

The one measurement that reaches along the line of sight. A radar pulse reflects first from the nearest point and last from the limb, which is one lunar radius further away, so a sphere returns an echo smeared over 2R/c — 11.6 milliseconds, which is what is observed. A disc facing us has almost no depth to spread the echo across: the extra path to its rim is 7.86 km, worth 26 microseconds, a factor of 440 short. Echo shape indicative; the decay is set by the surface scattering law, the width by the geometry.

Half of this is amateur, and it is the half people find hardest to believe. Hundreds of radio amateurs bounce signals off the Moon — “EME”, earth-moon-earth — and the libration of a sphere spreads the returned frequency by an amount that should grow linearly with the transmitted frequency. Measured medians from the amateur community: 2.2 Hz at 144 MHz, 7 at 432, 19 at 1296, 35 at 2304, 156 at 10368. Scaling the 144 MHz figure by frequency alone predicts 6.6, 19.8, 35.2 and 158. That is agreement to a few per cent across a 72-fold range of frequency, from equipment in people’s gardens.

Two things an EME operator hears are worth separating. The round trip takes about 2.56 seconds, and timing it gives the distance straight away — 10 ms of timing error is 1,500 km, so even a rough measurement lands inside a per cent or so. It borrows the speed of light and nothing else. The frequency spread measures something different and more useful here: the rotation of a ball, as a ratio, with no scale involved. The delay measurement itself needs short pulses and real power, so it is professional radar — but it is entirely ground-based, it was done at several unaffiliated observatories in different countries, and it predates the space programme. No agency’s testimony is involved.

5 · A hard limb with relief, mapped from the ground before spacecraft

When the Moon passes in front of a star, the star does not fade. It vanishes in about twenty milliseconds — consistent with the Fresnel scale at that distance, √(λd) = 14.5 m at 550 nm, swept past the observer at a few hundred metres a second. Along a narrow path on Earth the star instead blinks out and back repeatedly as lunar mountains and valleys cross it, and a line of observers strung across that path reconstructs the limb in profile. C. B. Watts turned a 1950s photographic survey into limb charts that are still in use, and the relief they record is about ±3 arcseconds, roughly a third of a per cent of the radius.

A flat disc could have a ragged rim too, so a single occultation does not decide anything. What decides it is that the same limb feature comes back at the same selenographic position when libration brings it round again, and that the profile you must use for a given event depends on the libration at that moment. A fixed rim on a fixed disc has one profile for ever. The silhouette and the visible face are the same object seen from different angles, and libration is what proves it.

This is a live amateur discipline. Total occultations need binoculars and a stopwatch, good to about ±0.05 s; video with GPS time insertion gets residuals inside a tenth of an arcsecond, and those observations feed the professional catalogues. Grazes need a team spread along a predicted line, which is a weekend of organising rather than a laboratory.

6 · The one thing that looks like a problem, and where it is answered

At full Moon the disc is very nearly uniformly bright right out to the edge. There is no darkening toward the rim, what variation there is tracks albedo — dark maria against bright highlands — rather than geometry, and a photograph of it does look remarkably like a flat coin. This is the strongest observation on the other side of the argument, it is entirely real, and a reader who has heard it will not get past this page until it is dealt with.

It refutes exactly one model, and that model is not the sphere. It refutes a Lambertian sphere — a body that scatters like matte paper. The Moon is a bed of dark porous dust, and for that material the appropriate first-order law is Lommel–Seeliger, under which a sphere at zero phase is predicted to be exactly uniform to the limb: the cosine that would darken a Lambertian surface cancels against a cosine in the denominator. The uniform full Moon is a prediction of the sphere model rather than an embarrassment to it.

The derivation is not on this page, and deliberately so. That is a photometry question, not a shape question, and it has its own answer at Why the Full Moon Has No Hotspot — with the working, the Lambertian comparison, the opposition surge, and the parts that are still uncomfortable, including a porosity tension with the Apollo cores and the fact that we could not obtain a calibrated centre-to-limb scan.

Two results from it belong here, because without them this section is just an assertion. First: the scattering law was not chosen to fit the Moon. It is measured on the actual soil, in a dish on a bench, in a geometry where the Moon’s shape cannot be the cause of anything. Second, and specific to this page: §4’s radar independently finds the same medium. At wavelengths from a millimetre to ten metres it returns a dielectric constant implying that solid material occupies only about thirty per cent of the volume — a porous bed — and a limb scattering law going as cos φ, which is Lommel–Seeliger again, arrived at from radio rather than light.

And then the part that matters most for this page: the full Moon cannot decide the shape either way. At full phase a Lommel–Seeliger sphere and a flat disc facing Earth predict the same picture, so its appearance is evidence for neither. That is exactly why nothing above rests on it. This page is built on the terminator, the shadows, the libration and the radar echo — and on §1’s crescent, which the facing disc forbids outright, and which is therefore worth more than the full Moon ever could be.

7 · What we are deliberately not resting on

Two further lines of evidence exist and are strong, and we are demoting both on purpose, because the audience for this page has reason to weigh them differently from the rest.

Lunar laser ranging. Five retroreflectors on the Moon, ranged from Earth to about 1.4 mm. On a sphere a reflector at angular distance θ from the sub-Earth point lies R(1 − cos θ) further away, which for the five sites predicts extra round-trip delays from 556 microseconds to 4.1 milliseconds; on a coplanar disc they would be nearly equal. Three of the five targets are Apollo hardware and the reflector coordinates come from the same programme, so we present this as consistency rather than proof. Two things about it are worth having anyway: two of the targets are Soviet, the ranging stations are in several countries, and the technique requires observations of at least three reflectors to solve the Moon’s orientation — a plane would need no such thing. One number in the literature is quietly decisive on its own: the papers quote one nanoradian of orientation error as 1.7 mm of motion at the lunar surface, which is a lever arm of 1,700 km. That is a lunar radius, falling out of a figure quoted for an entirely unrelated purpose.

Laser altimetry. Japan’s Kaguya gives a mean radius of 1737.15 ± 0.01 km with a maximum relief of 0.62 per cent of that, and NASA’s LRO agrees. Two national programmes, different hardware, same figure — and the same figure the Earth-based Watts limb charts of the 1950s already gave to a fraction of a per cent, which is the part that matters, because charts compiled from ground photography could not have been reverse-engineered from data that did not exist yet. A corroborating footnote, and we use it as no more than that.

8 · Where this page could be wrong

The kilometre figures rest on the Earth–Moon distance, and the precise value of it is professional work. Millimetre ranging needs a 3.5 m telescope and a pulsed laser. We had this listed as a flat limitation — that the distance is not available to an amateur at all — and that was too strong, as §4 makes obvious once you look at it: an EME operator times a round trip of about 2.56 seconds and has the distance from c, with each 10 ms of timing error worth 1,500 km, or 0.4 per cent. It borrows the speed of light, which is not a quantity anyone derived from a globe. The honest statement is narrower: the precision is institutional, the distance itself is not, and the shape arguments on this page need neither, because they are ratios.
We could not find a published measurement of the lunar outline’s circularity as a function of libration angle. This is the weakest link in §3. The geometry is not in doubt: a tilted disc must show a three-quarter-per-cent monthly ellipticity. We are confident nobody has ever observed that, because someone would have said so, and stacked amateur images would show it. But confident is not cited, and we would rather have the paper. If you know of one, we want it.
§6 asserts the limb-darkening answer rather than deriving it. That is on purpose — the derivation lives on the full-moon-lighting page — but it means a reader has to follow a link to check the one part of this page that concedes something. The known weaknesses travel with it: Lommel–Seeliger is a first-order law where real lunar photometry needs Hapke’s, with roughness, porosity, coherent backscattering and shadow-hiding; and no published centre-to-limb radiance profile with error bars could be obtained, so neither page states a tolerance on “uniform”.
The Earth-based heliometry we are not using would have been the cleanest ground-only result — Kazan measured 32 craters in 429 observations and solved for depth below the tangent plane, which a flat Moon has none of — but the reduction assumed a spherical model from the outset, so it demonstrates that a sphere fits rather than deriving one. We mention it and lean on nothing.
We paraphrase Galileo rather than quoting him, because we have not seen a facsimile of Sidereus Nuncius and secondary summaries disagree on the details.

9 · What would change our mind

— A crescent Moon that is uniformly lit across its full circular outline rather than being a crescent — i.e. a Moon that dims without ever showing a terminator. That is the facing disc’s actual prediction, and one photograph would settle it.
— A measured monthly ellipticity of the lunar outline of around three-quarters of a per cent in diameter, with the compression axis rotating in step with libration. That is the tilted disc’s signature and we say it is absent. Stacked images across a year would show it if it were there.
— Features that hold still at the disc centre while the rim moves. That is the disc’s libration signature and it is the exact inverse of the observed one. Blink two frames a fortnight apart.
— A radar echo from the Moon of a few tens of microseconds rather than 11.6 milliseconds. That is what a facing disc returns, and the measurement has been repeated at many observatories since 1946.
— Crater shadows of the same length regardless of distance from the terminator. Three nights, one telescope, one crater.

10 · The bottom line

Five independent lines, four of them ground-based and three doable with a camera and patience. The terminator exists at all, which a facing disc forbids. Shadows diverge where the light grazes, and summits catch it beyond the terminator. Features swing across the middle of the face by a hundred arcseconds a month while the outline stays a perfect circle — the exact opposite of what a tilting disc must do. The radar echo says the Moon is as deep as it is wide to four parts in ten thousand. And the limb has three-dimensional relief that returns to the same place when libration brings it round.

None of that is affected by how the Moon is lit, which is the useful part. The uniform full Moon is a real observation and a good question, and it has a real answer about the surface. But it is not an argument about the figure, because the figure was settled from the ground, by five methods that do not talk to each other, before the question was asked.

Sources & further reading

US Naval Observatory, Astronomical Applications Department, ‘Fraction of the Moon Illuminated’ — https://aa.usno.navy.mil/data/MoonFraction, and the phases FAQ at https://aa.usno.navy.mil/faq/moon_phases. Establishes that the lit fraction is a computed function of phase angle, published for every date 1700–2100, and defined as illuminated area over total disc area. The relation (1 + cos α)/2 and the terminator’s semi-minor axis R|cos α| are derived from scratch on this page and reproduce it.
Department of Physics, University of Hong Kong, ‘Measuring the Height of Lunar Craters’ — https://www.physics.hku.hk/upload/page/86/self/65eed497690b7.pdf. Establishes the Earth-based shadow-height method used in §2, H = LD/R at first quarter and H = L tan(sin⁻¹(D/R)) away from it, together with the instrument used (16-inch f/8.4, Apogee U9 CCD, 9 µm pixels). The point taken from it here is that the lunar radius is the method’s own measuring stick.
M. Wagner et al., Atmospheric Measurement Techniques 16, 1527 (2023) — https://amt.copernicus.org/articles/16/1527/2023/amt-16-1527-2023.pdf. Establishes the libration amplitudes used in §3: 7.9° in longitude on the 27.55-day anomalistic month, 6.68° in latitude on the 27.21-day draconic month, and the 13.4° × 15.8° rectangle traced over six years. The two models’ predicted displacements, the 0.9925 axis ratio, and the swept-fraction integral are computed here.
D. B. Taylor et al., ‘Computation of the Quantities Describing the Lunar Librations’, US Naval Observatory — https://aa.usno.navy.mil/downloads/reports/Tayloretal2011.pdf. Corroborates the amplitudes with a worked ephemeris instance and documents the machinery.
‘Radar Studies of the Moon’, Journal of Research of the National Bureau of Standards 69D — https://nvlpubs.nist.gov/nistpubs/jres/69D/jresv69Dn12p1637_A1b.pdf. Establishes the 11.6 ms echo duration, the sphere mapping φ = cos⁻¹(1 − ct/2a), the mean distance 384,400.2 ± 1.1 km, the cos φ limb scattering law, and the dielectric constants of 2.13–2.79 implying roughly 30 per cent solid packing — the porosity result that §6 relies on. The depth-over-width ratio and the 52 µs flat-disc figure are computed here.
Radio Society of Great Britain, ‘Moonbounce (EME)’ — https://rsgb.org/main/technical/space-satellites/moonbounce/. The amateur entry point for the measurement in §4, and cited so a reader can see that this is a hobby rather than a facility: it gives the round trip as “approximately 2.5 seconds”, names Doppler and libration among the propagation effects an operator has to work with, and notes that digital modes have “permitted many more radio amateurs to become involved in EME”. The American Radio Relay League hosts a companion beginner’s guide at https://www.arrl.org/weak-signal-vhf-dx-meteor-scatter-eme-moonbounce.
J. Taylor (K1JT), ‘Frequency-Dependent Characteristics of the EME Path’, EME 2010 — https://wsjt.sourceforge.io/EME2010_K1JT.pdf. Establishes the amateur Doppler-spread medians quoted in §4 (2.2 Hz at 144 MHz through 156 Hz at 10368 MHz) and states the echo duration 2R/c = 11.6 ms independently. The linearity check across the 72-fold frequency range is computed here.
NASA Goddard, ‘The Lunar Limb Profile and Eclipse Predictions’ — https://eclipse.gsfc.nasa.gov/SEhelp/limb.html, with NASA/TP-1999-209484 at https://eclipse.gsfc.nasa.gov/SEpubs/20010621/text/lunar-limb-profile.html. Establishes the ±3-arcsecond limb relief, the Watts charts and their systematic errors to 0.4″ from a slightly elliptical datum, and the resulting eclipse-contact accuracies.
British Astronomical Association, Lunar Occultations — https://britastro.org/section_information_/lunar-section-overview/lunar-occultations/lunar-occultations. Establishes the amateur timing performance in §5: ±0.05 s visually, residuals within a tenth of an arcsecond with video and GPS insertion.
J. B. Tatum & D. Fairbairn, Planetary Photometry, ch. 3, University of Victoria — https://www.astro.uvic.ca/~tatum/plphot/plphot03.pdf. Establishes the Lommel–Seeliger reflectance law behind the statement in §6 that a sphere of this material is uniform to the limb at zero phase. The derivation and the comparison against a Lambertian sphere are not on this page — they are on the full-moon-lighting page, which is where that argument belongs.
M. Vilenius et al., Astronomy & Astrophysics (2011) — https://www.aanda.org/articles/aa/full_html/2011/07/aa16115-10/aa16115-10.html. Establishes that lunar photometry is strongly non-Lambertian and Lommel–Seeliger-based, with a double Henyey–Greenstein backscattering phase function, and attributes most of the opposition surge to coherent backscattering within roughly 100 µm volume elements.
T. W. Murphy et al., ‘APOLLO: millimeter lunar laser ranging’, Classical and Quantum Gravity 29, 184005 (2012) — http://tmurphy.physics.ucsd.edu/papers/CQG-29-184005.pdf. Establishes the 1.4 mm median nightly uncertainty, the requirement to observe at least three reflectors to solve lunar orientation, and the 1.7 mm-per-nanoradian conversion from which the 1,700 km lever arm in §7 is derived here.
H. Araki et al., ‘The Lunar Global Topography by the Laser Altimeter (LALT) Onboard Kaguya’, LPSC XL, abstract 1432 — https://www.lpi.usra.edu/meetings/lpsc2009/pdf/1432.pdf. Establishes the Kaguya figure quoted in §7: mean radius 1737.15 ± 0.01 km, relief from −9.06 to +10.75 km, radial accuracy ±4.1 m.
EclipseWise, ‘Solar Eclipse Predictions and the Mean Lunar Radius’ — https://www.eclipsewise.com/solar/SEhelp/SEradius.html. Source of the IAU value k = 0.2725076 used throughout, and of the note that it derives from Earth-based limb observations averaging peaks and valleys.
Companion pages on this site: docs/full-moon-lighting/index.html (Q7, the uniform full Moon, where the photometry is done properly); docs/luminaries/index.html (Q9, the missing silhouette before an eclipse); docs/sky-rate-and-shadows/index.html (Q6, Jupiter’s moon shadows). Working notes and the unverified items are in MOON-SPHERE-SPINE.md in the repository.