The full moon that is too bright

The walk that interference can stop followed light scattered many times inside a disordered medium and found the waves’ interference reaching through the disorder — slowing the walk, and in one dimension stopping it. Everything there happened inside a slab of scatterers. This essay stays at the surface of one, and asks about a brightening that is almost entirely geometric: what a surface made of grains looks like from the direction the light is coming from.

The case that made the question famous is the Moon. The Moon’s brightness through its cycle of phases has been measured carefully since the nineteenth century, and it does not do what a smooth, diffusing ball would. It brightens slowly through the crescents, steadily through the quarters, and then, in the last few days before full, it surges. At full it is about ten times as bright as at first or last quarter, although only twice as much lit surface faces the Earth. Nothing on the Moon changes. What changes is where the Earth sits relative to the Sun, and the answer is in the soil.

Standing at the lamp #

A surface made of grains is full of shadows. Every grain lit from one side casts a shadow behind it onto the grains beneath and beside it, and in a loose layer those shadows are long, falling into the gaps between grains. Seen from most directions, a good fraction of the surface an eye can see is in some other grain’s shadow, and the surface looks darker for it.

There is one direction from which none of those shadows can be seen: the direction the light is coming from. An observer standing at the lamp looks along exactly the paths the light took. Any point the observer can see is a point a straight line from the observer reaches unobstructed, and that line is the path the light arrived by; so every point in view is lit. Each shadow lies directly behind the grain that casts it, as seen from the source, and so the grain that casts a shadow is always the grain that hides it.

The drawing makes this concrete. It follows seventy parallel sight lines into a random layer of grains until each meets a surface, and asks, for each point met, whether a straight line from it towards the light is clear. Looking from the light’s direction, all seventy points are lit — not approximately, but exactly, because the second line retraces the first. Move the observer twenty-five degrees round, and twenty-two of the seventy points seen are in shadow. The grains and the light have not moved.

How fast the shadows come into view #

As the observer moves away from the light’s direction — as the phase angle between them grows — the shadows come into view progressively. The drawing measures the share of the visible surface that is lit as the angle opens, for three layers of different packing. All start at one. All fall, steeply at first and then more slowly, towards a floor set by how much of the surface is in shadow when the shadows are fully exposed.

The speed of the fall is the interesting part. In a loose layer the grains are far apart and their shadows are long, so a small change of viewpoint is enough to look past a grain into its shadow: half the fall happens within five degrees. In a dense layer the grains are close, the shadows short and hemmed in, and the observer has to move much further before they can be seen: fourteen degrees for half the fall. The width of the peak measures how loosely the surface is packed. A peak a few degrees wide says the grains occupy a small share of the volume, with long gaps between them; a broad peak says they are packed tightly.

This is the reason the opposition effect is a tool rather than a curiosity. A telescope cannot resolve the grains of an asteroid’s soil or count the pores in the Moon’s, but it can measure how the object’s brightness rises as the Sun comes into line behind the Earth, and the width of that rise says how fluffy the surface is.

Porosity read from a width #

Bruce Hapke gave the effect a closed form in the 1960s and refined it in the 1980s, treating the soil as a random three-dimensional medium and computing the probability that a line of sight and a line of illumination, nearly parallel, pass through the same gaps. His answer makes the brightening a factor that falls from its peak over a width set by a single parameter, $h$, which for grains of one size is $-\tfrac{3}{8}\ln(1 - \phi)$ in terms of the filling factor $\phi$. A soil filling ten per cent of the space gives a peak about four and a half degrees wide; one filling half of it, about thirty.

The drawing sets the computed two-dimensional layers beside Hapke’s curve. They do not agree in their numbers, and should not: a cross-section through a layer of discs is not a volume of spheres, and a line of sight in a plane meets a different distribution of gaps. They agree in the trend, which is what matters for the argument. Denser soil, broader peak. For the Moon the surge’s width implies a surface layer that is mostly empty space — the lunar soil at the very top is a fairy-castle structure of grains touching at a few points, as the astronauts’ bootprints showed by how sharply they held their shape — and measurements of asteroids and the icy moons of the outer planets have been read the same way. The formula needs a correction for grains of mixed sizes, which is where most of the uncertainty in such inferences lies.

Why a smooth, white ball is the wrong model #

The comparison the Moon fails is with a Lambert surface, one that looks equally bright from every direction whatever angle the light arrives at. It is the natural first guess, because it is what an ideal matt white surface does and what the radiation from inside a closed cavity does exactly. A Lambert sphere’s brightness per unit area of disc falls towards the terminator as the cosine of the Sun’s angle of incidence, so at full phase it is brightest in the middle and darkens towards the edge — limb darkening — and at half phase its lit half is dim near the terminator.

A dark, grainy surface does something different, and the reason is how deep the light goes. In a dark soil almost all the light that returns has scattered once, off the first grain it met, and the amount returned depends on how much of the soil the light and the line of sight can both reach. For light arriving and leaving at the same oblique angle, both paths are long and shallow and meet the same thin top layer, so an oblique patch returns about as much light per unit of apparent area as one seen straight on. That is the Lommel–Seeliger law, and it is why the full Moon shows no limb darkening at all. What a telescope measures of each patch is its radiance, the power it sends per unit area per unit solid angle in the telescope’s direction, and a rough dark surface sends far more of it back towards the light than anywhere else.

A whole moon, from full to half #

The drawing turns the local effect into the phases of a whole moon, integrating the brightness over the lit, visible part of a sphere as the phase angle opens from full to half. A perfectly diffusing surface, one that scatters light equally in all directions regardless of how it arrived, makes a sphere 3.1 times brighter at full than at half: at half phase only half the disc is lit, and that half is seen obliquely, where the light strikes it at a grazing angle. A dark, rough soil of the kind the Moon has — whose light comes from single scattering off each grain, described by the Lommel–Seeliger law rather than Lambert’s — does slightly worse, 2.7.

Add the shadow-hiding surge and the ratio doubles, to 5.0. That is still only half of the Moon’s ten. The rest is shadow hiding on larger scales and the scattering of individual grains. The Moon’s surface is rough at every size, from dust to boulders to crater walls, and each scale casts shadows that full phase hides; Hapke’s model adds a term for that large-scale roughness, which for the Moon corresponds to slopes of twenty degrees or so. And the lunar grains themselves, being irregular and full of internal cracks, scatter more light backwards than forwards. Together these make the Moon one of the most strongly backscattering objects in the Solar System, which is why a full moon is bright enough to read by and a quarter moon is not.

The same geometry is at work on the ground. Someone walking across a dry field with the low Sun behind them sees a bright patch round the shadow of their own head, moving as they move: every grain of soil and blade of grass in that direction hides its own shadow from the eye looking along the sunlight. The effect is called the heiligenschein, the “holy glow”, when it appears round a shadow on dewy grass, where a second mechanism — drops acting as lenses that focus light onto the leaf and send it back the way a retroreflector does — adds to it. Photographs taken from aircraft show the same bright spot round the aircraft’s shadow on forests and crops, and satellite images of vegetation carry a “hot spot” in the direction of the Sun that remote-sensing algorithms have to correct for.

A second, narrower peak #

Shadow hiding is geometry and would happen with particles of any size, as long as they are opaque and larger than the wavelength. There is a second mechanism, and it belongs to waves. Light that enters a scattering surface and bounces among several grains before coming out can follow any path. For every such path there is another, the same sequence of grains traversed in reverse. Coming out in an arbitrary direction the two paths have different lengths and their light adds with random phases. Coming out exactly back towards the source they have the same length, whatever the path, because each is the other run backwards — so their light adds in phase, and the brightness in that direction is doubled for multiply scattered light.

This is coherent backscattering, the effect whose three-dimensional generalisation the walk that interference can stop was about, seen from outside the medium. It produces a peak whose width is about the wavelength divided by the distance light travels between scatterings: for fine, bright, weakly absorbing grains, a degree or so, far narrower than the shadow-hiding peak. The drawing adds such a peak to a shadow-hiding curve and shows what an observation near zero phase must separate: a broad geometric rise and a sharp spike on top of it.

The two are separated by properties the geometry lacks. Coherent backscattering depends on wavelength, since its width is proportional to it. It depends on the brightness of the grains, since it needs light to scatter many times before escaping, and dark surfaces absorb it first; shadow hiding, by contrast, needs light to scatter only once. And it carries a polarisation signature — the light coming back near opposition is partly polarised in a characteristic way, negatively with respect to the plane containing the Sun, the reverse of the pattern single scattering writes across the sky — which single scattering from opaque grains does not produce. Measurements of bright icy satellites show a sharp, polarised, wavelength-dependent spike; the dark Moon shows mostly the broad geometric surge.

Other surfaces, other peaks #

Brightening towards the direction of the light is not peculiar to soils, but the mechanisms differ with what the surface is made of, and naming them keeps them apart. A cloud of water droplets, seen from above with the Sun behind the observer, shows a glory: coloured rings round the observer’s shadow, a few degrees across. The droplets are transparent and comparable in size with many wavelengths, so the effect is not shadows but the droplet’s own scattering pattern at the size of the wave, in which light grazing the drop’s edge travels round its surface and emerges nearly backwards. A deep, bright cloud, by contrast, sends light back after a long walk through many scatterings, and that light has forgotten its direction; its surge is the narrow coherent one.

The shadow of each grain is also the other face of a statement about scattering in general. An opaque grain removes from a beam twice the light that falls on its cross-section — half absorbed or scattered away, half diffracted into a narrow forward cone — and the shadow behind it is the part of that removal an observer can see when standing off to one side. Standing at the light, the observer looks straight down the line along which the removal happened and sees none of it. The same geometry that makes a clear sky blue by single scattering from molecules, too small to cast shadows, makes a soil bright at opposition by single scattering from grains large enough to cast them.

Where the geometry is not the whole answer #

Two dimensions. The computed layers are cross-sections through discs, and a real soil is a volume of irregular grains of many sizes. The trend with packing survives; the numbers for any real surface need the three-dimensional theory and a size distribution, and they are uncertain by tens of per cent.

Grains larger than the wavelength. Shadow hiding assumes that grains cast sharp shadows. For grains comparable with the wavelength, light bends round them, shadows blur, and the geometric effect weakens while the wave effect strengthens. Fine dust and bright frost are in that regime.

One scattering. The shadow-hiding calculation follows light that scatters once, off the first grain it meets. In a bright surface most of the light scatters many times, filling the shadows with light that has bounced in from elsewhere, and the surge is weaker for a bright surface than for a dark one with the same structure. That is one reason the dark Moon shows so strong a geometric surge and bright snow so weak a one.

What the phase curves do not show #

The curves report total brightness. They do not show colour, and the opposition surge is slightly bluer or redder than the rest of the light depending on which mechanism dominates, which is one way the two are distinguished. They do not show the detailed shape at the very smallest angles, which from the Earth can never reach zero for the Moon except during a lunar eclipse, when the Earth’s shadow is in the way; the closest approaches were measured from spacecraft. And they do not show how the brightness varies across the disc, which at full phase is nearly uniform from centre to limb — the Moon at full looks like a flat disc rather than a ball, because a rough, dark, backscattering surface returns light towards the source almost regardless of how it is tilted.

Still open: separating the two surges on a body nobody can visit #

For the Moon, returned samples calibrate the soil’s porosity and grain size, and the surge can be checked against them. For asteroids and distant moons it cannot. Whether a measured opposition surge on an asteroid should be read as a porosity, a grain size, a brightness of the grains or a mix of all three depends on how much of the peak is geometric and how much is interference, and fitting a model with enough parameters to allow both often leaves the answer undetermined. Measurements at several wavelengths and in polarised light at very small phase angles narrow it, and the sample-return missions to asteroids since 2020 have given the first ground truth against which telescope-based inferences can be checked. How well those inferences stand up is being worked out now.

The habit worth carrying away is to ask where the observer stands. A rough surface looks brightest from the direction of its light because from there no shadow can be seen, and how quickly the brightness falls as the observer moves away measures how long the shadows are — so a telescope that cannot resolve a single grain can still say how tightly the grains of a distant world are packed, by watching it pass behind the Sun’s line.