Fun With Science / Globe Deconstruction / Q7 · page 50 / Draft
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.
Q7 is asked on the one night of the month when it cannot be answered. At full phase — and only at full phase — a sphere of lunar dust and a flat disc facing the observer predict the same picture: a uniformly bright circle with no hotspot. Two different geometries, one identical prediction, so the observation cannot count for either. That is the finding of this page, and it holds before any photometry is done. Q7 also never states what a flat Moon should look like; it argues only that the sphere looks wrong, which means even a successful refutation of the sphere would leave the disc unsupported — and where the disc’s predictions can be inferred, at every other phase, they fail badly.
That leaves the question Miller actually asked, which is how. The answer is worth having, and this page spends most of its length on it. He is right that the full Moon shows no centre-brightening, right that a smooth diffusely reflecting sphere lit from behind the observer would show a conspicuous one, and right to reject the usual brush-off that the surface is merely rough. A Lambertian sphere at zero phase falls to 0.60 of its central brightness at 80 per cent of the radius and 0.31 at 95, with a quarter of the disc more than 0.75 magnitudes down and the limb fading to nothing. The Moon does none of that. What the observation falsifies is Lambert’s cosine law, which describes chalk and matte paint, not a dark porous particulate half-space. The right first-order law for that material is Lommel–Seeliger, whose disc function at zero phase is identically 1 out to the limb.
One structural note, since this page answers a shape argument while sending the shape evidence elsewhere. This page does the photometry — the how. Whether the Moon is a ball is settled at Is the Moon a Ball?, and the sixty-second version is in §10 below.
Observation described correctlyLambert expectation correctly derived"It's just rough" rightly rejectedWrong scattering law assumedReproduced in a dish of dustInference to shape does not follow
Where this lands
Miller looks at a full Moon, sees a disc of nearly uniform brightness with a sharp edge, and says that this is not what a lit ball looks like. Against the model he has in mind he is right, and the arithmetic is on his side: a Lambertian sphere lit at zero phase dims to 0.31 of its central brightness by 95% of the radius and fades to nothing at the limb, and the Moon does nothing of the kind. What he has measured, though, is Lambert's cosine law, not the Moon's geometry. Lambert's law describes a bright, dense, multiply scattering diffuser — chalk, matte paint, cloud. Lunar regolith is a dark, porous, particulate half-space, and the first-order solution of the radiative transfer equation for such a medium is the Lommel-Seeliger law, whose disc function at zero phase is identically 1 at every point out to the limb. That law was derived in the nineteenth century for reasons unconnected with the Moon — by Eugen von Lommel in 1887 and Hugo von Seeliger in 1888, two Munich academicians working on the photometry of diffuse reflection, with Seeliger’s longer memoirs of 1887 and 1895 applying the theory to the illumination of Saturn and its rings. It predicts exactly the flat disc Miller finds anomalous. The same flat-at-zero-phase behaviour is measured in a laboratory dish holding a few grams of returned Apollo soil, by a lander standing on the lunar surface, and from an aircraft over Spanish farmland, so it cannot be a fact about the Moon's shape. And the corollary, established in §1: at full phase a Lommel-Seeliger sphere and a flat disc look identical, so this observation does not prove sphericity either, and we will not pretend it does.
Before any photometry, the shape of the argument. At full phase a Lommel–Seeliger sphere and a flat disc facing Earth are photometrically degenerate. On the sphere the angle of incidence equals the angle of emission everywhere, so the disc function is 1. On a flat disc square to the line of sight both angles are zero everywhere, so 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, and that is true no matter how carefully anyone photographs it.
Two consequences, and they cut in both directions. 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, and we are not going to do that here.
Q7 has a second structural feature worth naming, because it is what makes the first one decisive rather than merely awkward. It is 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 argument had succeeded, it would not have established the disc, because the disc has made no prediction to succeed at. And in the places where a flat disc’s predictions can be inferred without his help — every phase that is not full — they fail so badly that §10 disposes of them in a paragraph.
None of which makes the question uninteresting. He asked how the light distributes so evenly, and that has a real answer involving real measurements on real lunar soil. The rest of this page is that answer. But the answer is about the dust, not about the shape, and the reader should know which question is being settled before the arithmetic starts.
Start with what this page grants, because it is a great deal and because most of the replies Miller has received do not deserve to be believed.
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/pi) E cos i, proportional to the cosine of the incidence angle and independent of where you view it from. Light it with a distant point source placed directly behind the observer — zero phase angle, the geometry of a full Moon. Then at every point of the visible disc the incidence angle equals the emission angle, and both equal the angle between the local surface normal and the line of sight. For a sphere that angle satisfies cos i = sqrt(1 - r squared), where r is the fractional distance from the centre of the projected disc.
That is the whole derivation, and it takes one line. The profile it 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 |
The brightness reaches half the central value at r = 0.866, and the area outside that radius is 1 - 0.866 squared = 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 percent 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.
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.
Miller is describing the picture correctly. Anyone who has rendered a matte sphere in a graphics package with the light mounted on the camera has seen precisely the falloff he says is missing, and has seen how badly it fails to resemble a photograph of the Moon.
The second grant is about the standard reply. The usual response to this claim is "the lunar surface is rough," and as normally delivered that is not an argument, it is a noise. Lambert's cosine law is the classical idealisation of a perfectly rough, perfectly diffusing surface; it is derived by assuming the surface scatters isotropically into the hemisphere, which is what perfect diffuse roughness means. Saying "rough" and stopping does not carry you from cos i to flat. It carries you back to cos i. Getting the right answer requires a specific and non-obvious scattering law with a specific physical derivation, and in our experience most people who wave the objection away could not write that law down. Miller is entitled to reject the brush-off, and we are not going to defend it.
The claim as written bundles two things together, and they are worth separating before the physics starts, because one of them is inert.
The first is the absence of a highlight point. This is true. There is no specular glint anywhere on the full Moon, and there never will be one from natural regolith: a mirror-like reflection requires a smooth optical interface, and lunar soil is dark, porous, and microscopically rough at optical scales. But no diffuse sphere of any material or size would show a specular highlight either. The absence of a glint is therefore evidence against a polished ball, which nobody proposes, and evidence for nothing else. It discriminates between no two hypotheses on the table.
The second thing, which is the real content, is the absence of a broad centre-brightening. That is the thing a Lambert sphere would show, that is the thing the table in the previous section quantifies, and that is what the rest of this page addresses. We think Miller's phrasing points at the right absence even though the technical object is not a highlight. A photometrist would not use his words; the reader should not be distracted by that.
We are discussing appearance at optical wavelengths, at the resolution of the naked eye and of an ordinary telescope. Radar, thermal infrared, and polarimetric behaviour follow different laws and are outside the scope of this page.
It is worth pausing on the magnitude of what Miller has noticed, because the effect he says is missing is not a subtlety. It is enormous.
Trim the brightest central patch and the dimmest outer sliver — the 5th to 95th percentiles of brightness by projected area, which is the same statistic the Lommel–Seeliger table below uses. Under Lambert's cos i that band runs from 0.22 to 0.97 of the central value. That is a factor of 4.4, which is 2.5 log10(4.359) = 1.60 magnitudes across the face of the object. If you photographed such a Moon and exposed for the limb, the centre would be four and a half times over. If you exposed for the centre, the outer third of the disc would go dark.
Nothing like this is present in any photograph of the full Moon ever taken. 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. If a cos i gradient were superimposed on the albedo map, that pattern could not survive.
The gradient Miller says is absent is absent. We measured it, and it is not there to a degree that no observer could miss.
The conclusion we draw is not that the Moon is strange. It is that the law is wrong for this material — and the law was never claimed to be right for this material by anyone who thought about it.
Real photographs of the full Moon are additionally compromised by atmospheric scattering, by the instrument's point spread function bleeding light outward from a glaring object, and by the fact that the Moon's albedo map is not uniform. Our claim here is qualitative and comparative: no photograph shows a monotonic radial falloff of the size Lambert requires. We say later, explicitly, what calibrated measurement we were unable to find.
Lambert's law describes a bright, dense, multiply scattering diffuser. Chalk. Matte white paint. A cloud deck. In such a medium a photon bounces many times among the scatterers before it emerges, and the memory of the incoming direction is washed out, which is exactly what "scatters isotropically into the hemisphere" encodes.
Lunar regolith is none of these things. It is dark — the single-scattering albedo 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. In that regime a photon typically scatters once and leaves. Multiple scattering is weak, so the geometry is not washed out, and the appropriate first-order solution of the radiative transfer equation is not Lambert's law but the Lommel-Seeliger law: bidirectional reflectance proportional to mu0/(mu0 + mu), where mu0 = cos i and mu = cos e. Normalised, the disc function is
> D(mu0, mu) = 2 mu0 / (mu0 + mu)
This is quoted verbatim from Schroeder and colleagues' resolved photometry of Vesta, together 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.
Now evaluate it at the geometry in dispute. At zero phase on a sphere, i = e at every single point of the disc — that is forced by the geometry, as we established in the first section. So mu0 = mu everywhere, so
> D = 2 mu0 / (mu0 + mu0) = 1
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. We checked it numerically over a 1200 by 1200 sampled disc and got 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 honest version of this is the small-phase version. 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 is about 5.3 degrees, set by the Moon's orbital inclination to the ecliptic. So at the very worst geometry available, Lommel-Seeliger predicts a spread of 1.081 / 0.901 = 1.1998, which is 2.5 log10(1.1998) = 0.198 magnitudes — about two tenths of a magnitude, or 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. That is not something a naked-eye observer detects. The Lambert spread at the same geometry is eight times larger in magnitude terms.
There is a technicality we would rather raise ourselves. At 5 degrees phase the disc function's extreme values run from near zero toward the analytic limit of 2 — but only in a vanishingly thin sliver along the terminator-side limb, where a few pixels of a real image contain almost no light. The central-90%-of-area figures above are what we are quoting, and we say so rather than let a reader discover the extremes and think we hid them.
This is the section where popular accounts of this subject usually fail, and a planetary scientist reading the page will check it first.
There are two separate observations about lunar brightness, and they have two separate explanations. Conflating them is the commonest error in this literature.
Fact one: the disc is flat at zero phase. That is Lommel-Seeliger, and it is essentially exact, as shown above. 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. We take that figure 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 to the power of (-0.4 x 2.602) = 0.091.A Lambert sphere at 90 degrees phase retains 1/pi = 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.
And here is the point that must not be fudged: Lommel-Seeliger does not explain fact two. Its integrated phase curve is, if anything, slightly shallower than Lambert's, not steeper. Fact two needs additional physics — shadow hiding in a porous unconsolidated surface, a strongly backscattering single-particle phase function, macroscopic roughness at scales above the grain, and, contested, coherent backscatter.
Two notes on the numbers above, one of them now discharged. The closed-form integrated phase law for a Lommel–Seeliger sphere was flagged here as unverified. It is verified now, against Muinonen & Lumme, Astronomy & Astrophysics 584, A23 (2015), whose equation 7 carries the bracket [1 − sin(α/2) tan(α/2) ln(cot(α/4))] — for isotropic single scattering, that bracket is the whole phase dependence. Integrating it gives a phase integral of 1.637, so the 1.64 stands, against exactly 3/2 for a Lambert sphere. We can therefore say more than the qualitative thing we hedged to. A Lommel–Seeliger sphere is very slightly steeper than Lambert below 53° — by at most 2.4 per cent, around 31° — and clearly shallower above it, by 18 per cent at quarter phase and 62 at 120°. Either way it goes the wrong way to explain a phase curve steeper than Lambert’s, which is the point §6 rests on, and it goes the wrong way by a margin now quantified rather than asserted. The second note stands as a caution. The Krisciunas and Schaefer fit is an astronomer’s convenience formula, built from 33 V-band observations at Mauna Kea so that observatories could schedule faint-object work around moonlight; it is symmetric in phase angle by construction, and a term linear in |α| cannot represent an opposition surge at all, so the last few degrees before full are exactly where it should not be trusted. We could not reach the paper itself — it is behind robots exclusions at both IOP and ADS — so its own stated range of validity is unchecked here. Treat 9 per cent and q = 0.63 as indicative. The direction and the rough size of the discrepancy are what the argument needs, and those are safe.
Near zero phase the Moon does something extra, on top of the flat disc: it brightens sharply and non-linearly as the phase angle closes. This is the opposition surge, and it is the second mechanism referred to above.
The measurements exist and they do not perfectly agree, so we will give the range rather than pick a favourite. Clementine produced the first quantitative data at very small phase, with over 90 images taken 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 best-constrained modern fits come from the Lunar Orbiter Laser Altimeter. Barker and colleagues fit Hapke parameters to LOLA's 1064 nm radiometry and obtain, for highlands with FeO below 8%, a shadow-hiding amplitude B_S0 = 1.60 +/- 0.03 with width parameter 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(alpha/2) = h, so the angular half-width is 2 arctan(h): 9.5 degrees for the highlands and 4.8 degrees for the maria. The surge is not a knife-edge spike. It is a broad shoulder several degrees wide, which is why it is present at every full Moon and not just at eclipses.
One detail in Barker's residuals matters more than its size suggests. The Hapke fit reproduces the LOLA data with fractional residuals under 5% for most geometries, and where it drifts, it drifts toward underestimating the data at high emission angle. In other words the real Moon is, if anything, marginally limb-bright relative to the model — the opposite direction from the Lambertian limb darkening under discussion.
Here is the move that settles the matter, and it has nothing to do with astronomy.
If the flat-at-zero-phase behaviour were a property of the Moon's shape, it would not reproduce in situations where shape cannot possibly be relevant. It reproduces everywhere.
In a laboratory dish. Apollo 11 soil sample 10084 and Apollo 16 soil sample 61141 have been measured on the Bloomsburg University Goniometer 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 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 is flat: "lunar BRDFs exhibit strong peaks in reflectance at low phase angles, which is a manifestation of the well-known lunar opposition effect." These measurements were made to support thermal-infrared calibration for 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. Airborne POLDER flown over agricultural fields near Barrax recorded nadir 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 same one stated in the lunar literature: particles at the surface cast shadows on their neighbours, those shadows are visible at large phase angles, and at zero phase every shadow is hidden behind the particle that casts it. 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.Be exact about what that defeats, because the overstatement is tempting and it loses. 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.
Which is what turns §1 from an argument into a measurement. Without the dish a careful critic can still say: uniformity follows from a flat disc under any scattering law, but from a sphere only under the one particular law you chose, so on balance the observation mildly favours flat. The dish closes that off. The law was not chosen to fit the Moon. It was measured on the soil itself, on a bench, with no shape hypothesis anywhere in the apparatus.
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." It fires a pulse at the surface and times the return. Because it is monostatic — the same instrument transmits and receives — it measures reflectance at exactly zero phase angle. The illumination source, the surface element, and the detector are collinear by construction. That is a geometry no Earth-based observer can ever obtain, at any time, except at mid-eclipse.
Lucey and fifteen co-authors used this to build a global normal-albedo map of the Moon at 1064 nm. To do so they had to skip a step that any planetary cartographer would normally take: correcting each measurement for the local slope of the terrain. Their justification is stated in passing, as an assumption too obvious to argue:
> "for the low-albedo Moon, zero-phase reflectance is not sensitive to the local slope [Hapke, 1993], amply demonstrated by the lack of limb darkening exhibited by the full Moon where incidence and emergence angles reach 90 degrees."
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. It comes out with the bulk of the surface near a normal albedo of 0.3, corresponding to typical highlands material; a low-albedo mode near 0.15 between 0 and 30 degrees latitude, which are the major mare basalt deposits; and anomalously bright permanently shadowed regions near 0.36. It resolves crater rays and mare boundaries. It is used as an albedo product.
There is a disc-integrated cross-check available here, and we are going to present it and then demote it ourselves, because it is weaker than it first appears. 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 that is only legitimate if there is no limb darkening. Under Lambert's law the same procedure would overestimate the geometric albedo by a factor of 1/(2/3) = 1.5. Their value lands just above the highest prior Earth-based estimate.
But published Earth-based V-band geometric albedos for the Moon span 0.113 to 0.163, a factor of 1.44 spread that is essentially as large as the effect being tested. A careful reader will notice that 0.169 / 0.113 = 1.50, which is exactly the Lambert offset, and produce it as a rebuttal. So we do not offer this as a falsification. We offer it as consistent with no limb darkening, and we say plainly that it cannot discriminate on its own.
§1 established that the full Moon cannot answer the shape question. The complement is worth stating plainly: every other night of the month answers it easily, and most of the answers cost nothing but a camera and some patience.
A flat disc at quarter phase would be uniformly dim and edge-on, with no terminator anywhere and no relief shadows whatsoever. The first-quarter Moon in a pair of binoculars is nothing of the sort, and the disc-integrated brightness falls to about 9 per cent at quarter phase — 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, taken seriously, would have to apply to other airless bodies. We can point to one we checked directly: Vesta, whose resolved photometry from Dawn was modelled with the Lommel–Seeliger disc function quoted earlier and shows the same near-uniform behaviour at low phase. We had intended to list several; on checking our own notes we could substantiate one, so one is what we claim.
And the standing recommendation, which we mean literally: look at it at first quarter instead. We would rather a reader spent ten minutes at an eyepiece than took our word for any of this.
Five places, in descending order of how much they would cost us.
We were looking under the wrong name, and what we found does not entirely go our way. This item used to say we could not locate a published calibrated centre-to-limb determination for the full Moon. The literature has one; it lives under the vocabulary of photometric disc functions and the Minnaert exponent, and a reader who knows the field would have found it immediately. 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 put it in words we cannot improve on, calling it “the flat-disk effect at full Moon” and noting that the original explanation of it was Galileo’s.
Three qualifications, because the same paper supplies them and they are not all comfortable. First, 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, which is worth knowing before anyone demands one. Second, and this is a real qualification to the flat statement made throughout this page: the Moon does darken measurably toward the photometric poles, which Lommel–Seeliger does not capture — Shkuratov notes the law “ignores the lunar surface darkening at the photometrical poles.” Uniformity is a statement about the equatorial profile, not about the whole disc, and we should have said so. Third, they judge Akimov’s parameter-free function the best fit of the candidates, 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. But the honest summary is that the measurement exists, it is not quite the measurement we asked for, and it names something we had not accounted for.
The falsifier is unchanged. If a calibrated equatorial profile of the full Moon shows a √(1 − r²) fall-off, this page is wrong at its foundation.
The underlying mechanism of the surge is genuinely unresolved, and we will not tidy it away. The split between shadow hiding and coherent backscatter has been argued for decades and is not settled. Hapke's own account shifted over three decades, from attributing the surge to shadow hiding, to proposing coherent backscatter, to settling on 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 — though that figure reaches us through a secondary source and we could not confirm it 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 entirely. This matters less than it sounds for our argument, because the flat disc at zero phase is Lommel-Seeliger and does not depend on which surge mechanism wins. But a page that picked a winner would be wrong in somebody's specialty. Hapke parameters are fitted coefficients, not measured physical quantities. The Hapke model has been the subject of a published critical assessment with a reply and a counter-reply. We can show the problem concretely from our own numbers. LOLA's fitted h_S = 0.083 implies, through Hapke's relation h = -0.375 ln(1 - phi), 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. We do not write "the regolith is 80% porous." We write that the photometric value refers only to the topmost few grain diameters, that the conversion from h to porosity is model-dependent, and that the tension is real and worth reporting rather than hiding. The Moon is not quite uniform even at exact opposition, and the reason is interesting. 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 entirely non-Lambertian origin, in the opposite direction to Lambert's. It is unobservable, because at that instant the Moon is in eclipse. We raise it so that no specialist reads our "uniform" as a claim we have not qualified. Our own numbers where we could not source them. The Lommel-Seeliger integrated phase function has come off this list — it is sourced to Muinonen & Lumme (2015) and the phase integral of 1.637 is integrated from their expression here. What remains is the topocentric correction to the geocentric 5.3-degree maximum is now checked too: the Moon's equatorial horizontal parallax at mean distance is 3,422.6 arcseconds, 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. Direction and size both leave every conclusion here intact. What is left on this list is the exact date and context of the nineteenth-century derivation, and that too is now sourced in section 5 rather than asserted.We have not independently reduced any lunar imagery for this page. Every measurement quoted is somebody else's, read from the source we cite, except the disc-function computations, which are our own numerical evaluations of published closed-form laws and which anyone can reproduce in ten lines of code.
Stated in advance, specifically, and 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 show our disc function is wrong and Miller's expectation right, and we would withdraw the central argument of this page. A dish of dark dust that does not backscatter. A laboratory 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 entirely and force the question back onto geometry. The samples exist, the instruments exist, and the measurement is cheap. It would be decisive. 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 currently report under 5% for most geometries. If the true figure were 30 or 40% we would have to say the law does not describe the Moon. A slope-dependent LOLA albedo map. Our strongest section leans on the claim that zero-phase reflectance is insensitive to local slope. That is directly testable against LOLA's own topography: a correlation between LOLA normal albedo and LOLA-derived local slope at the tens-of-percent level, after controlling for composition, would break it. If that correlation were demonstrated, we would retract the section. 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, from a retroreflector array, or from spacecraft hardware are different objects and would not count. On shape specifically, given §1’s finding that the observation is shape-neutral: a demonstration that a flat disc 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, and the roughly 9% quarter-to-full disc-integrated brightness ratio. We do not expect this to be possible. But that is the test we would accept, and it is the right place for the shape argument to be fought — not at full Moon, where the two hypotheses make the same prediction and neither side learns anything.The full Moon is the one night of the month when this question cannot be settled. Look at it at first quarter instead.