The ring round the Moon that sizes a droplet

On a night of thin, high cloud the Moon can sit in a small disc of milky light, a few degrees across, its edge a dull brown or reddish ring, and outside that, if the cloud is right, a ring of blue and green and another of red. It is easy to confuse with the much larger halo, the 22° ring that the ring at twenty-two degrees traced to refraction through hexagonal ice crystals. The two have nothing to do with each other. The halo is geometry: its size is fixed by the 60° angle between faces of an ice prism and does not change from night to night. The corona is diffraction, and its size changes with the cloud, because it is set by the size of the particles the light passes.

That makes the corona something rare in everyday optics: a phenomenon whose dimensions are a measurement. Its rings are a ruler laid against the drops of the cloud, read with nothing but a protractor held up to the sky.

A drop as an opaque disc #

A cloud droplet is a sphere of water ten or so micrometres across. Light passing near it is affected in many ways — refracted, reflected, scattered — and when the particle is the size of the wave followed the full theory of a sphere that does all of them at once. But for a drop many wavelengths across and for angles of a few degrees from the forward direction, one effect dominates: the drop blocks a disc of the incoming wave, and the wave diffracts around it.

The pattern from an opaque disc follows from Babinet’s principle, which says that an obstacle and a hole of the same shape diffract light into the same pattern away from the forward direction, because the two diffracted waves added together must give the undisturbed wave. So a drop of radius $a$ diffracts like a circular aperture of radius $a$, and its pattern is the one how far apart two things have to be drew for a telescope’s lens: Airy’s,

$$I(\theta) \propto \left[\frac{2J_1(x)}{x}\right]^2, \qquad x = \frac{2\pi a}{\lambda}\sin\theta,$$

with a bright central disc, the first dark ring where $x = 3.83$, and a sequence of fainter rings outside. A cloud contains enormous numbers of drops at random positions, so their patterns add in intensity rather than amplitude, and the pattern of the whole cloud is the pattern of one drop, brightened.

That picture explains the colours. Every wavelength makes the same rings, but at angles proportional to the wavelength, so the patterns slide over one another. The central disc, the aureole, is white near the Moon where all colours overlap; its edge is reddish brown, because the blue light has already reached its first dark ring while the red has not; the next bright ring is blue on its inner side and red on its outer side, and so on, the colours growing purer outward as the patterns separate. The halo’s colours run the other way, red inside, because refraction bends red light least. A coloured ring round the Moon that is red outside is a corona; one that is red inside is a halo. That single observation settles which mechanism is at work.

A ruler for drops #

The size of the aureole is the most useful number in the corona. The first dark ring lies at

$$\sin\theta_1 = 1.22\,\frac{\lambda}{d},$$

with $d$ the drop’s diameter. For yellow-green light near 570 nm, a cloud of 10 µm drops gives an aureole with its edge 4° from the Moon, eight Moon diameters; a cloud of 40 µm drops gives one only 1° across.

So a photograph of a corona, with the Moon’s half-degree disc for scale, measures the droplet size of a cloud kilometres away. Meteorologists have used exactly this: coronae have been measured to estimate droplet sizes in thin clouds since the nineteenth century, and the method was refined in the twentieth for clouds that instruments on aircraft could not easily reach. Small drops make large coronae, and the largest and most brilliant are seen in young, thin clouds such as altocumulus lenticularis, whose drops have just condensed and are only a few micrometres across. Thick clouds of large old drops give small coronae, if any, close round the Moon where the glare hides them.

The classroom version is a glass plate dusted with lycopodium powder, the spores of a club moss, which are unusually uniform at about 30 µm across. Held in front of a small bright lamp, the plate surrounds it with a corona of several clear rings, a degree or so in radius, and measuring the rings gives the size of the spores to a few per cent. A misted window in front of a streetlight does the same thing less tidily, its condensed droplets giving a soft aureole whose size changes as they grow or dry. The same method, in instruments, is called laser diffraction particle sizing. A laser beam passes through a cloud of particles — a spray, a powder, a suspension — and a ring of detectors records the forward diffraction pattern; inverting it gives the size distribution. It is the standard way to size particles from a micrometre to a millimetre in industry, and it is the corona with a laser for a Moon.

When the drops are not all alike #

A real cloud does not contain drops of a single size, and that is why most coronae show only the aureole.

The $n$th ring of a drop of diameter $d$ sits at an angle roughly proportional to $n/d$. A spread of drop sizes of, say, ten per cent spreads the first ring over ten per cent of its own angle, which is less than the spacing between rings, so the ring survives. The third ring is three times further out and is spread three times as widely, enough to fill the dark gap beside it, and it disappears. The aureole itself survives almost any spread, because it is a single bright lobe whose edge merely softens.

So the number of rings is a second measurement. A corona with three clear rings outside the aureole says the cloud’s drops differ in size by only a few per cent, which is a statement about the cloud’s history. Drops that condensed together, in a single pulse of rising air, grow at the same rate from the same vapour and stay alike; drops in an old, mixed, turbulent cloud have grown and evaporated along different paths and differ widely. Lenticular clouds over mountains, where air rises and condenses in a steady wave, give the most perfect coronae and also the iridescent colours on their edges, which are the same diffraction seen at larger angles from the Sun.

The narrowness of the spread that rings require is the reason a corona is a sign of something unusual. Why the sky is blue and the sunset is not dealt with scatterers far smaller than the wavelength, where every particle does the same thing and nothing about their sizes is visible. A corona is the opposite regime: the particles are larger than the wavelength, their size is written into the angle of the light, and the cloud has to be unusually orderly for the writing to stay legible.

Pollen and the shape of the obstacle #

Coronae also appear around the Sun and Moon in spring, without cloud, when the air is full of pollen. Pollen grains are larger than cloud droplets, twenty to sixty micrometres, so pollen coronae are small, a degree or two. And they are often visibly oval, or show bright spots on their rings.

The explanation is in the diffraction of a non-circular obstacle. The pattern of an elliptical disc is Airy’s pattern stretched: it is widest in the direction in which the obstacle is narrowest, by the inverse of its width in that direction. A cloud of round drops, in any orientation, gives a round pattern. A cloud of elongated grains gives a round pattern too if their orientations are random. But if they settle through the air with a preferred orientation — as pine and birch pollen do, with the grain’s air sacs or its shape making it fall in a particular attitude — their patterns line up and the corona becomes an oval. The spikes a bright star wears met the same principle in a telescope, where the struts holding the secondary mirror diffract starlight into spikes perpendicular to the struts. An obstacle’s shape is printed on its diffraction pattern, rotated and inverted in scale.

So an oval corona, or one with bright points on its ring, is diagnostic: it cannot be made by liquid drops, which are spheres, and it points to solid, shaped particles falling in step. Observers of pollen coronae have identified the trees responsible from the size of the rings, since each species’ pollen has a characteristic size, and the oval shape distinguishes the bisaccate pines, with two air sacs either side of the grain, from the more nearly spherical grains of other species.

Three rings from one drop #

A single water drop sends coloured rings in three directions, and each is a different mechanism with a different relation to the drop’s size. The corona is forward, within a few degrees of the source, and is diffraction round the drop’s outline. The rainbow is at 138° from the source, 42° from the antisolar point, and is refraction and one internal reflection, its angle fixed by the refractive index of water and independent of size, as the angle the rainbow has to be showed. The glory is almost exactly backward, a small set of rings round the shadow of an aircraft on a cloud below it, and the rings no ray can reach traced it to waves travelling round the drop’s surface, beyond anything a ray picture can produce.

Two of the three are rulers. The corona’s rings scale as $1/d$, as here. The glory’s rings scale the same way, since they too are a diffraction pattern of something the size of the drop, and pilots who photograph glories on clouds below have provided droplet sizes that agree with those from coronae in similar clouds. The rainbow by itself is not a ruler, but its supernumerary bows are: the fringes below the rainbow found that the spacing of the faint bands inside the primary bow is set by interference between two rays through the drop, whose path difference grows with its size, so that small drops give widely spaced supernumeraries and large drops crowd them against the bow until they vanish. Rainbows with clear supernumeraries, like coronae with clear rings, require drops of nearly one size.

So a cloud of uniform drops writes its size three times over: forward in the corona, sideways in the supernumeraries, backward in the glory, each by a different physical mechanism, and the three agree. It is a satisfying thing to see in the sky, and it is also a check that the theory of each is right, since an error in any one of them would show up as a disagreement about the same drops. The halo stands apart from all three. Made by ice crystals much larger than the wavelength, through prisms with fixed angles, it carries no information about size at all, which is why its radius is the same everywhere on Earth and has been since the first person noticed it.

The same rings in a telescope, an eye and a laser #

Airy’s pattern turns up in three places in this subject, and the corona ties them together. In a telescope, it is the image of a star through a circular lens: its size sets the resolving power. In a corona, it is the pattern of a drop: its size measures the drop. And in the rings that belong to the edge, it was traced to the boundary of the obstacle, the wave diffracted from the rim, which is why a drop and a hole of the same size give the same rings away from the forward direction. The mathematics is identical; only the question changes. A telescope maker wants the pattern small, to separate two stars. A cloud physicist wants it measured, to read off the drops. Light diffracting round a drop does not know which question it is being asked.

There is a curious consequence in how much light the drops remove. Everything a scatterer removes found that a large particle takes twice its geometric cross-section out of a beam: once by blocking, and once again by diffraction into the forward lobe. The corona is that second cross-section made visible. The light in the aureole and rings is the light the drops diffracted, an amount equal to the light they physically intercepted, and it is all sent into a cone a few degrees across round the source. A cloud dim enough to see the Moon through puts a large fraction of the Moon’s light into the corona.

What the picture cannot show #

The figures treat each drop as an opaque disc and use Fraunhofer diffraction. That is a good approximation for the forward lobe and the first few rings for drops many wavelengths across, but a water drop is transparent, not opaque. Light passing through it is refracted and emerges at larger angles, contributing the rainbow and the glory at other angles entirely, and interfering slightly with the diffracted light even in the forward direction. For drops below about five micrometres the disc approximation becomes poor, and the full Mie theory, which treats the sphere exactly, is needed. Coronae from very small drops therefore have ring positions and colours that the simple formula gets slightly wrong.

The size distribution in the figures is a gamma distribution, a reasonable shape for cloud drops but not the only one, and the contrast at which a ring counts as lost — a tenth — is a choice. The thresholds of 8, 10 and 16 per cent depend on both. Real coronae also include multiple scattering in thicker clouds, which washes the pattern out independently of the size spread, and the Moon’s disc has a finite size of half a degree, which blurs each ring by about that much and alone can hide the outer rings of a small corona.

The colours have not been computed, only the patterns in three wavelengths. The colour actually seen depends on the spectrum of moonlight, on how the eye’s colour vision works at night, which is weak, and on contrast with the sky. Coronae round the Sun are more colourful than round the Moon for that reason, and are best seen in reflection in a dark pane of glass or behind a building that hides the Sun itself.

Still open: what the iridescence at a cloud’s edge records #

Iridescent clouds — patches of pastel colour at the edges of thin clouds, often well away from the Sun — are diffraction from the same drops at larger angles, where the outer rings of the corona would be. Because the colour at each point depends on the local drop size, an iridescent cloud is in principle a map of droplet sizes across the cloud, showing where drops are just forming and where they have grown. Reading that map quantitatively from photographs has been attempted, but separating the effect of drop size from the effects of the spread of sizes, the cloud’s thickness and the varying angle to the Sun is difficult, and no routine method has emerged. The physics needed is known; whether a camera can turn a cloud’s colours into a reliable measurement of how it grew is a question that has only begun to be explored.

A corona is Airy’s pattern from cloud droplets acting as opaque discs, so its aureole ends at sin θ = 1.22 λ/d — 4° for 10 µm drops, 1° for 40 µm — with red outside, the reverse of a refraction halo; its outer rings survive only if the drops differ by less than about ten per cent, and aligned elongated pollen stretches it into an oval across the grains’ short axis. The habit worth carrying is to ask what sets the size of a pattern, because when the answer is the object making it, the pattern is a measurement of that object.