Fun With Science / Globe Deconstruction / Q7 · page 50
The observation is correct and the expectation behind it is correct. The law being measured against is the wrong law for dark dust — and the night it is measured on is the one night of the month that cannot settle the question it is asked about.
Observation correctWrong scattering lawSays nothing about shape
Where this lands
The claim. A sunlit ball should show a bright centre fading toward the edge; the full Moon shows a flat disc with a sharp rim; so the Moon is not a sunlit ball.
The verdict. Miller is right about the picture, right that a smooth diffusely reflecting sphere lit from behind the observer would show a conspicuous centre-brightening, and right to reject the brush-off that the surface is merely rough. A Lambertian sphere at full phase dims to 0.60 of its central brightness at 80 per cent of the radius, 0.31 at 95, and nothing at the limb; the Moon does nothing of the kind. What that falsifies is Lambert’s cosine law, which describes chalk. Lunar regolith is dark, porous, particulate dust, and the first-order scattering law for that material — Lommel–Seeliger, published in 1887 and 1888 for reasons unconnected with the Moon — gives a disc function — the brightness at each point of the disc, relative to its centre — of exactly 1 at every point out to the limb. The uniform full Moon is that law’s closed-form prediction. And at full phase a Lommel–Seeliger sphere and a flat disc facing us predict the same picture, so the observation is shape-neutral: it cannot count for a flat Moon, and it cannot count for a round one either.
A few grams of lunar soil in a dish on a bench in Pennsylvania reproduce the effect that is said to prove the Moon is flat.
This page answers the how. It assumes the Moon is a sphere and does not argue for it; that is done, from five independent ground-based lines, at Is the Moon a Ball?, with the sixty-second version in §7 below.
Before any photometry, the shape of the argument. At full phase the Sun is behind the observer, so at every point of the surface light arrives and leaves along the same direction. On a Lommel–Seeliger sphere the angle of incidence then equals the angle of emission everywhere, and the disc function is 1. On a flat disc square to the line of sight both angles are zero everywhere, and the disc function is also 1. Two geometries, one prediction.
An observation that both hypotheses predict is evidence for neither. The likelihood ratio is exactly one. Whatever the full Moon looks like, it was never going to tell you the Moon’s shape, however carefully anyone photographs it. Anyone offering the uniform full Moon as evidence for a flat Moon is reading a scattering law as a geometry; so is anyone offering it as evidence for a round one.
Q7 is also a purely negative argument. Miller never derives what a flat Moon should look like — not its brightness profile, not its phase curve, not its behaviour at any other geometry; he argues only that the sphere looks wrong. Even if that succeeded it would not establish the disc, which has made no prediction to succeed at; and where a flat disc’s predictions can be inferred — every phase that is not full — they fail so badly that §7 disposes of them in a paragraph. He asked how the light distributes so evenly, and the answer is about the dust, not the shape.
Take a sphere whose surface is a perfect Lambertian diffuser: it scatters incident light isotropically into the hemisphere above it, so its radiance is L = (A/π) E cos i, proportional to the cosine of the incidence angle i and independent of viewing direction. Light it from directly behind the observer — zero phase angle, the Sun–Moon–observer angle, which is the geometry of a full Moon. Then at every point of the disc the incidence angle equals the angle between the local surface normal and the line of sight, and for a sphere cos i = √(1 − r²), where r is the fractional distance from the centre of the projected disc. The profile that produces is severe.
| r/R | 0 | 0.25 | 0.5 | 0.7 | 0.8 | 0.9 | 0.95 | 0.99 | 1.0 |
|---|---|---|---|---|---|---|---|---|---|
| relative brightness | 1.000 | 0.968 | 0.866 | 0.714 | 0.600 | 0.436 | 0.312 | 0.141 | 0.000 |
Brightness reaches half the central value at r = 0.866, and the area outside that radius is 1 − 0.866² = 0.250 of the disc. A flux ratio of 0.5 is 2.5 log10(2) = 0.753 magnitudes. So a full quarter of the visible face would be more than three quarters of a magnitude fainter than the middle, and the outer few per cent of the radius would fade toward nothing, giving the object a soft, dissolved edge rather than a rim. Averaged over the projected disc, cos i comes to exactly 2/3. Trim the brightest central patch and the dimmest outer sliver — the 5th to 95th percentiles of brightness by projected area, the same statistic the table in §3 uses — and the band still runs from 0.22 to 0.97 of the central value: a factor of 4.4, which is 2.5 log10(4.359) = 1.60 magnitudes across the face of the object. Expose for the limb and the centre is four and a half times over; expose for the centre and the outer third of the disc goes dark.
The full Moon plainly does not look like that. Its edge is sharp, its limb is bright, and Grimaldi and the mare boundaries near the edge hold their contrast against the highlands beside them.
No photograph of the full Moon shows a gradient of that size, and none could hide one. What variation there is tracks composition, not position: the maria are dark because they are basalt, the highlands bright because they are anorthositic, and those boundaries run across the disc without regard to distance from the centre — Mare Crisium sits near the limb and is dark, Aristarchus sits near the limb and is bright. The claim here is comparative, not a measurement of our own: no published photograph shows a monotonic radial falloff of the size Lambert requires, and §8 names the calibrated profile that does exist. The conclusion is not that the Moon is strange. It is that the law is wrong for this material.
Two clarifications. Miller’s word is “highlight”, and a specular, mirror-like glint is indeed absent — but no diffuse sphere of any material shows one, so that absence discriminates between nothing on the table; the real content of Q7 is the absence of the broad centre-brightening the table quantifies. And the usual reply, “the lunar surface is rough,” is not an answer: Lambert’s cosine law is the classical idealisation of a perfectly rough, perfectly diffusing surface, so saying “rough” and stopping carries you back to cos i, not away from it. The right answer needs a specific scattering law with a specific physical derivation, and one already exists.
Scope: appearance at optical wavelengths, at the resolution of the naked eye and an ordinary telescope; radar, thermal infrared and polarimetric behaviour follow different laws, and real photographs are further affected by atmospheric scattering, by the instrument’s point spread function, and by the non-uniform albedo map. One distinction to hold onto: a Lambertian emitter shows no limb darkening at all, because radiance is defined per unit projected area — a different problem from a Lambertian scatterer lit by a distant source, where cos i is an illumination factor. §6 returns to the emitter.
Lambert’s law describes a bright, dense, multiply scattering diffuser: chalk, matte white paint, a cloud deck. A photon bounces many times among the scatterers before it emerges, and the memory of the incoming direction is washed out. Lunar regolith is none of those things. It is dark — the single-scattering albedo, the fraction of light a grain reflects, fitted to lunar data at 1064 nm runs around 0.49 for highlands and 0.36 for maria — and it is a highly porous, unconsolidated particulate half-space. A photon typically scatters once and leaves. Multiple scattering is weak, the geometry is not washed out, and the first-order solution of the radiative transfer equation is the Lommel–Seeliger law: bidirectional reflectance proportional to μ0/(μ0 + μ), where μ0 = cos i and μ = cos e, the cosines of the incidence and emission angles. Normalised, the disc function is
quoted verbatim from Schroeder and colleagues’ resolved photometry of Vesta, with their statement that it “naturally arises from the radiative transfer theory of a particulate medium when considering only single scattering.” It is not a curve fitted to the Moon; it is a solution to a scattering problem. At zero phase on a sphere i = e at every point of the disc, so μ0 = μ everywhere, so
identically — at the centre, at half radius, at 95% of the radius, and at the limb where both i and e approach 90 degrees. Not approximately uniform: uniform by construction, with no free parameters and nothing fitted. A numerical check over a 1200 by 1200 sampled disc returns minimum equal to maximum equal to 1.0000, as it must.
The Moon is never at exactly zero phase except during a lunar eclipse, so the working version is the small-phase one. Sampling the illuminated disc numerically and taking the 5th to 95th percentiles of the disc function by projected area:
| phase angle | central 90% of disc area spans | equivalent range |
|---|---|---|
| 1 degree | 0.982 to 1.018 | about 0.04 mag |
| 3 degrees | 0.943 to 1.051 | about 0.12 mag |
| 5 degrees | 0.901 to 1.081 | about 0.20 mag |
| Lambert, any of these | 0.22 to 0.97 | about 1.6 mag |
The largest geocentric phase angle a full Moon reaches — geocentric meaning as seen from the Earth’s centre; a ground observer’s topocentric view differs by up to the Moon’s parallax, bounded in §8 — is about 5.3 degrees, set by the Moon’s orbital inclination to the ecliptic. At that worst geometry Lommel–Seeliger predicts a spread of 1.081 / 0.901 = 1.1998, which is 2.5 log10(1.1998) = 0.198 magnitudes — within roughly 0.1 mag of the disc mean either way, across the whole face of a glaring object with a strongly varying albedo map. No naked-eye observer detects that. The Lambert spread at the same geometry is eight times larger in magnitude terms. (At 5 degrees the disc function’s extremes run from near zero toward the analytic limit of 2, but only in a vanishingly thin sliver along the terminator-side limb that holds almost no light; the central-90%-of-area figures are the ones quoted.)
A careful critic can still say: uniformity follows from a flat disc under any scattering law, but from a sphere only under the one law you chose, so on balance the observation mildly favours flat. The reply has nothing to do with astronomy. The law was not chosen to fit the Moon. It has been measured on the soil itself, in settings where the Moon’s shape cannot possibly be the cause, and the soil does the same thing everywhere: reflectance rises steeply as the phase angle closes toward zero, because surface particles shadow their neighbours, those shadows show at large phase angles, and at zero phase every shadow hides behind the particle that casts it.
In a laboratory dish. Apollo 11 soil sample 10084 and Apollo 16 soil sample 61141 have been measured on the Bloomsburg University Goniometer — a bench instrument that records reflectance at set angles — over 3 to 140 degrees phase — extending to 155 degrees only at high incidence — and both show strong reflectance peaks at low phase. The Hapke fits — fits of the standard multi-parameter model of regolith reflectance — return an opposition amplitude B0 = 2.5 for both samples, with single-scattering albedos of 0.324 and 0.588 respectively. The paper’s own summary: “lunar BRDFs exhibit strong peaks in reflectance at low phase angles, which is a manifestation of the well-known lunar opposition effect.” The measurements were made for thermal-infrared calibration of the Diviner instrument; nobody involved was arguing about the Moon’s shape.
In a returned sample from a different mission. Chang’e-5 regolith, measured on a bench, roughly doubles in reflectance between about 70 degrees and 3 degrees phase at normal incidence, and nearly triples at 60 degrees incidence.
Standing on the surface. Chang’E-4 measured phase curves in situ from the floor of Von Karman crater over 1 to 144 degrees phase. The in-situ reduced reflectance from the Chang’E-3 and Chang’E-4 rovers rises by roughly a factor of eight into the opposition region. A camera on the ground, looking at dirt a few metres away, sees the same physics.
Over a wheat field in Spain. In terrestrial remote sensing the identical phenomenon is a well-known nuisance called the BRDF hot spot (BRDF: bidirectional reflectance, how a surface’s brightness depends on the lighting and viewing angles together). Airborne POLDER flown over agricultural fields near Barrax recorded nadir (straight-down) reflectance rising by up to 120% in the backscattering direction at 45 degrees view zenith, with forward-scattering reflectance falling by 20 to 40%. From space, hot-spot angular half-widths run 0.8 to 2.0 degrees for most cover types and 1.0 to 4.0 degrees for some forest and desert surfaces. The mechanism stated in that literature is the one stated in the lunar literature.
With your own eyes. It is the heiligenschein — the bright halo around the shadow of your own head on dry dusty ground or dewy grass. Your head is at zero phase angle. Nothing else in the scene is.
And on the Moon itself. Near zero phase the Moon brightens sharply as the phase angle closes: the opposition surge, opposition being the full-Moon geometry with the Moon directly opposite the Sun. It is a broad shoulder several degrees wide, present at every full Moon and not just at eclipses, and it was an engineering nuisance for the Ranger camera designers in 1964 before anyone understood it. The measured amplitudes, fitted widths and disputed mechanism are in the method notes; the argument rests on the flat disc, not the surge.
Be exact about what that defeats. It does not defeat a flat Moon. A flat dusty Moon accounts for the dish perfectly well — dust is dust on any shape. What the dish defeats is the inference: uniformity, therefore flat. The uniformity is a property of the material, measured on the material, in a geometry where the Moon’s shape cannot possibly be the cause. It carries no shape information for anybody. (The Hapke fits quoted above are coefficients within a model; what is measured, and model-independent, is the raw rise every one of these instruments records.)
The strongest single item on this page was produced by people who were not thinking about this question at all. The Lunar Orbiter Laser Altimeter aboard Lunar Reconnaissance Orbiter is, in its own description, “principally a laser altimeter used for quantitative topography and related cartographic and geodetic applications.” Because the same instrument transmits and receives, it measures reflectance at exactly zero phase angle — source, surface and detector collinear by construction, a geometry no Earth-based observer can obtain except at mid-eclipse. Lucey and fifteen co-authors used it to build a global normal-albedo map of the Moon at 1064 nm, and to do so they skipped a step any planetary cartographer would normally take — correcting each measurement for the local slope of the terrain — on a justification stated in passing:
And, in the same vein: “no photometric normalization is required to compare measurements of different portions of the lunar surface because of the constant phase angle.” The map works. The bulk of the surface comes out near a normal albedo of 0.3, typical highlands material; a low-albedo mode near 0.15 between 0 and 30 degrees latitude marks the major mare basalt deposits; permanently shadowed regions are anomalously bright near 0.36. It resolves crater rays and mare boundaries, and it is used as an albedo product.
Barker and colleagues’ Hapke fit to LOLA’s phase data reproduces it with fractional residuals under 5% for most geometries, and where it drifts, it drifts toward underestimating the data at high emission angle: the real Moon is, if anything, marginally limb-bright relative to the model, the opposite direction from Lambert. A disc-integrated cross-check from the same map — consistent with no limb darkening, but unable to discriminate on its own — and two more instruments built for unrelated purposes are in the method notes.
Claim #3 of the book, of which Q7 is the first of four exhibits, does not conclude that the Moon is flat. It concludes that the four observations “make a strong case that the Sun is not the source of illumination” (p. 2). So the alternative the uniform disc is being asked to support is a self-luminous Moon — and the full Moon cannot support that either, for the same reason it cannot support a shape. A uniformly emitting surface shows no limb darkening whatever its shape, because radiance is defined per unit projected area: a glowing ball and a glowing disc both present a flat disc, exactly as a Lommel–Seeliger ball and a sunlit disc do. Four hypotheses, one picture.
Every other night separates them, and three observations settle it. The half Moon comes at 90°. The lit fraction of a ball lit by a distant source is f = (1 − cos E)/2, where E is the elongation, the Sun–Moon angle in your own sky: half lit at 90° from the Sun, full at 180°, dark at 0° — and that is what the Moon does. A light that “powers up and down” has no reason to be half dark with a sharp great-circle boundary on the side facing away from the Sun, nor for that boundary to swing round the sky to follow the Sun; the moon-tilt page measures the boundary’s orientation and works through the “self-luminous but solar-powered” repair. The dark side is not dark. A thin crescent shows the whole rest of the disc, maria and all, faintly lit by earthshine — sunlight bounced off the Earth’s cloud tops, roughly one ten-thousandth of direct sunlight, about 10–15 lux against 130,000 — on the very hemisphere a self-luminous Moon has supposedly switched off, and strongest near new Moon when the Earth is fullest in the lunar sky. The new Moon occults the Sun. At a solar eclipse the Moon is a black disc against the Sun, and beside the Sun it is no brighter than the sky around it: whatever the Moon emits on its own is below what the eye can register, on the one occasion its face is turned fully away from the Sun and fully toward us. The luminaries page has the earthshine photograph and the eclipse silhouette, with the numbers.
None of that is a photometric subtlety. It is the plain behaviour of a ball of dust lit from outside, and Q7 is asked on the one night when the ball, the disc, the mirror and the lamp all look alike.
The full Moon cannot answer the shape question; every other night of the month answers it easily, for the price of a camera and some patience. A flat disc at quarter phase would be uniformly dim, with no terminator and no relief shadows. The first-quarter Moon in binoculars is nothing of the sort, and the disc-integrated brightness falls to about 9 per cent at quarter phase (arithmetic in the method notes) — the signature of a lit hemisphere turning away, not of a flat disc being dimmed. Five more, each reproducible without anyone’s permission:
The full Moon is the one night of the month when the shape question cannot be settled. The other twenty-seven settle it for free. The deep version of all of the above — with the numbers, the sources and the objections — is at Is the Moon a Ball?
The same argument would have to apply to other airless bodies. Vesta is the one we checked: its resolved photometry from Dawn was modelled with the Lommel–Seeliger disc function quoted in §3 and shows the same near-uniform behaviour at low phase. The standing recommendation, meant literally: look at the Moon at first quarter instead.
Five places, in descending order of cost.
The calibrated measurement exists, and it does not entirely go this page’s way. The literature carries a centre-to-limb determination for the full Moon under the vocabulary of photometric disc functions and the Minnaert exponent k — a fitted number describing how brightness depends on viewing angle, unrelated to the refraction coefficient called k elsewhere on this site. The Minnaert disc function is D = μ₀k μk−1, which at zero phase collapses to μ2k−1: k = 1 is Lambert and k = 0.5 is a uniform disc, so the whole question of this page is a question about one number, and the lunar value at opposition is the one that means uniform. Shkuratov and colleagues call it “the flat-disk effect at full Moon” and credit the original explanation to Galileo. Three qualifications from the same paper. Their calibrated equatorial profile is a ratio of images taken at ±16.2°, not at zero phase — a true zero-phase measurement from Earth is impossible outside a lunar eclipse. The Moon does darken measurably toward the photometric poles, which Lommel–Seeliger “ignores”, so uniformity is a statement about the equatorial profile, not the whole disc. And they judge Akimov’s parameter-free function the best fit, better than Lommel–Seeliger and better than Minnaert, so the law this page uses is the right kind of law and not the best available one. None of that rescues Lambert, which is not in contention anywhere in that literature, and none of it touches §1. The falsifier: a calibrated equatorial profile of the full Moon showing a √(1 − r²) fall-off would make this page wrong at its foundation.
The mechanism of the surge is unresolved. The split between shadow hiding and coherent backscatter has been argued for decades; Hapke’s own account shifted over three decades, from shadow hiding, to coherent backscatter, to roughly equal contributions. Clementine’s colour analysis, which finds the surge 3 to 4% larger at 0.41 micron than at 1.00 micron, is read as favouring shadow hiding — a figure taken from a secondary source, unconfirmed against the primary. The Chang’E-4 in-situ fit returns a negative coherent-backscatter amplitude, B_C0 = −0.05, and the authors drop the term. The flat disc at zero phase does not depend on which mechanism wins, so this costs the argument little; but a page that picked a winner would be wrong in somebody’s specialty.
Hapke parameters are fitted coefficients, not measured physical quantities. The model has been the subject of a published critical assessment with a reply and a counter-reply, and the problem shows in this page’s own numbers. LOLA’s fitted h_S = 0.083 (the Hapke shadow-hiding width) implies, through Hapke’s relation h = −0.375 ln(1 − φ), a filling factor of 0.199 and hence about 80% porosity; the maria value h_S = 0.042 implies a filling factor of 0.106, or 89% porosity. Apollo core tubes, driven into the actual soil for soil-mechanics reasons, give roughly 50% porosity in the top 15 cm falling to about 43% below. This page does not write “the regolith is 80% porous”: the photometric value refers only to the topmost few grain diameters, and the conversion from h to porosity is model-dependent.
The Moon is not quite uniform even at exact opposition. The Moon subtends about 0.52 degrees, so at the instant of geocentric opposition the disc centre is at 0 degrees phase while the limb is at arctan(1737.4 / 384400) = 0.259 degrees. Given a surge of around 20% below 0.25 degrees, the Moon should show a real centre-to-limb brightening of order 10 to 20% at that instant — a genuine radial gradient, of non-Lambertian origin, in the opposite direction to Lambert’s. It is unobservable, because at that instant the Moon is in eclipse.
What is checked, and what is not. The topocentric correction to the geocentric 5.3-degree maximum: the Moon’s equatorial horizontal parallax at mean distance is 3,422.6 arcseconds (an arcsecond is 1/3600 of a degree), which is 0.95 degrees, so a ground observer’s worst case is about 6.25 degrees, where the Lommel–Seeliger central band spans 0.25 magnitudes; every conclusion survives. The Lommel–Seeliger integrated phase function is sourced to Muinonen & Lumme (2015), and the phase integral — a body’s brightness summed over all phase angles, relative to full phase — of 1.637 is integrated here from their expression. The law’s form and derivation come from Schroeder et al. and from Tatum and Fairbairn’s open textbook; the 1887 and 1888 originals, and Seeliger’s Saturn memoirs, are cited from the bibliographic record and have not been read here. And this page has not independently reduced any lunar imagery: every measurement quoted is somebody else’s, read from the source cited, except the disc-function computations, which are numerical evaluations of published closed-form laws that anyone can reproduce in ten lines of code.
In the order that would hurt most.
A calibrated centre-to-limb scan showing the Lambert profile. A published radiance profile of the full Moon which, after division by an independently derived albedo map from Clementine, Kaguya, or LROC WAC, shows brightness at r/R = 0.95 falling to roughly 31% of the disc-centre value rather than staying within about 10% of it. That would make the disc function wrong and Miller’s expectation right.
A dish of dark dust that does not backscatter. A goniometer measurement of a well-characterised dark particulate sample — returned Apollo or Chang’e soil, or a standard lunar simulant — showing no backscatter peak and instead following cos i at small phase. That would remove the material explanation and force the question back onto geometry. The samples exist, the instruments exist, and the measurement is cheap.
Large position-correlated residuals. Systematic residuals of tens of percent, correlated with position on the disc, after fitting the Lommel–Seeliger or Hapke disc function to disc-resolved lunar imagery at small phase. Published fits report under 5% for most geometries; at 30 or 40% the law would not describe the Moon.
A slope-dependent LOLA albedo map. §5 leans on the claim that zero-phase reflectance is insensitive to local slope. A correlation between LOLA normal albedo and LOLA-derived local slope at the tens-of-percent level, after controlling for composition, would break it, and the section with it.
A specular glint on natural regolith. A mirror-like highlight that tracks the sub-solar point across the disc as the phase changes. We predict none exists; glints from fresh impact glass, a retroreflector array or spacecraft hardware would not count.
On shape, and on self-luminosity: a demonstration that a flat disc, or a self-lit body, reproduces the entire phase sequence — the curvature of the terminator and its reversal between crescent and gibbous, the lengthening of crater shadows toward the terminator, libration rotating limb features into and out of view, earthshine on the unlit hemisphere, and the roughly 9% quarter-to-full disc-integrated brightness ratio. We do not expect this to be possible, but it is the test we would accept, and the right place for the argument to be fought — not at full Moon, where every hypothesis on the table makes the same prediction.
The full Moon is the one night of the month when this question cannot be settled. Look at it at first quarter instead.
Two facts, two explanations. Fact one, the disc is flat at zero phase: Lommel–Seeliger, essentially exact (§3). Fact two, the Moon’s disc-integrated brightness falls off very steeply with phase: at quarter phase the Moon is only about 9% as bright as at full, from the Krisciunas and Schaefer empirical lunar phase term as reproduced by Jones and colleagues: the magnitude penalty is 0.026 times the phase angle in degrees plus 4.0e-9 times the fourth power, which at 90 degrees gives 2.340 + 0.262 = 2.602 magnitudes, and 10−0.4 × 2.602 = 0.091. A Lambert sphere at 90 degrees phase retains 1/π = 0.318 of its full-phase brightness, with a phase integral q = 3/2; so the real Moon is about 3.5 times steeper at quarter phase than a Lambert sphere — 0.3183 / 0.0910 = 3.50 — and integrating the Krisciunas and Schaefer fit gives an implied phase integral near 0.63, roughly 2.4 times smaller than Lambert’s 1.50. Lommel–Seeliger does not explain fact two; it explains the flat disc at small phase and nothing else. Its integrated phase function, from Muinonen & Lumme, Astronomy & Astrophysics 584, A23 (2015), equation 7, carries the bracket [1 − sin(α/2) tan(α/2) ln(cot(α/4))], integrates to a phase integral of 1.637, against exactly 3/2 for a Lambert sphere: very slightly steeper than Lambert below 53° (by at most 2.4 per cent, around 31°) and clearly shallower above it (18 per cent at quarter phase, 62 at 120°) — the wrong direction to explain a phase curve steeper than Lambert’s. Fact two needs shadow hiding in a porous unconsolidated surface, a strongly backscattering single-particle phase function, macroscopic roughness above the grain scale, and, contested, coherent backscatter. The Krisciunas and Schaefer fit is a convenience formula built from 33 V-band observations at Mauna Kea so that observatories could schedule around moonlight; it is symmetric in phase angle by construction, a term linear in |α| cannot represent an opposition surge, and the paper was not reachable (robots exclusions at IOP and ADS), so its stated range of validity is unchecked. Treat 9 per cent and q = 0.63 as indicative; §7 needs only the direction and rough size.
The opposition surge, measured and disputed. Clementine produced the first quantitative data at very small phase, over 90 images below 0.5 degrees. Buratti and colleagues report roughly a 40% brightness increase between 4 degrees and 0 degrees phase, and about 20% below 0.25 degrees alone; Yokota and colleagues report 20 to 30% over the same 4-to-0 interval. The Icarus primaries for the 4-to-0 figures were not reachable, so “over 40%” and “20 to 30%” come through secondary sources that agree with each other; “about 20% below 0.25 degrees” and the count of over 90 sub-0.5-degree images are read directly from the NASA technical report server abstract. The best-constrained modern fits are Barker and colleagues’ Hapke parameters for LOLA’s 1064 nm radiometry: for highlands with FeO below 8%, shadow-hiding amplitude B_S0 = 1.60 ± 0.03 with width h_S = 0.083 ± 0.002; for maria with FeO above 14%, B_S0 = 1.50 ± 0.02 and h_S = 0.042 ± 0.001. Hapke’s shadow-hiding function drops to half its peak where tan(α/2) = h, so the angular half-width is 2 arctan(h): 9.5 degrees for the highlands and 4.8 degrees for the maria. Willingham’s 1964 JPL report for the Ranger impactor programme records that “nearly every area reach[es] maximum brightness at 0 degrees phase” and notes the “extremely large slope between 0 and 5-degree phase angles,” while refusing to extrapolate to zero because doing so was “to say the least, a questionable procedure.”
The disc-integrated cross-check, and why it is demoted. Lucey and colleagues obtain a V-band geometric albedo of 0.169 by taking the orthographic disc mean of their normal-albedo map, a step only legitimate if there is no limb darkening; under Lambert’s law the same procedure would overestimate the geometric albedo by 1/(2/3) = 1.5, and their value lands just above the highest prior Earth-based estimate. But published Earth-based V-band geometric albedos span 0.113 to 0.163, a factor of 1.44 spread essentially as large as the effect being tested, and 0.169 / 0.113 = 1.50 is exactly the Lambert offset. So it is consistent with no limb darkening, not a falsification. Two more instruments in the same category: the USGS ISIS cartographic pipeline, built to make planetary mosaics match at the seams, encodes a “lunar-Lambert” photometric function B(phase) × [(1 − L) μ0 + 2 L μ0/(μ0 + μ)], a blend of Lambert and Lommel–Seeliger that collapses to pure Lommel–Seeliger at L = 1 (the tabulation of L against phase angle was not read from the primary source, so no claim is made about its value at small phase); and the NIST lunar spectral irradiance programme, which uses the Moon as an on-orbit radiometric standard at a target uncertainty of 0.5% over 320 to 2500 nm, states that its calibration model must include “non-Lambertian reflectance properties such as the backscatter increase at small phase angles known as the opposition effect.”