Ever since the success of her Clatto Swan photograph, The Boon Companion has been intermittently getting out of her warm bed at some truly God-forsaken hours to photograph sunrises. She recently took some early morning photos on the beach at Saint Andrews. She’ll be a bit annoyed with me for having chosen a glary one to post here—she has some much better views of the same scene. But I have my reasons.

Just right of centre of the frame there’s a little patch of colour in the sky, which could easily be mistaken for lens flare. It appears in the same place in all the views in this direction, but I chose the frame above because it gives the best indication of where the sun is, showing that the little wisp of colour is at the same vertical height above the horizon. And it’s possible to zoom in on it, which is not a feature of lens flare:

A single patch of cloud is alive with light. Here it is in close up:

What The Boon Companion had captured and brought home for me is a sun dog—or a parhelion, to give it its scientific name. They usually come in a pair, equally spaced either side of the sun and moving with it, like hunting dogs flanking a hunter. Unfortunately, the companion to this sun dog was out of sight behind higher ground to the right of the picture.

Sun dogs are formed by light refracting through hexagonal ice crystals, pretty much exactly in the way made familiar (to those of a certain generation) by Hipgnosis’s iconic cover design for Pink Floyd‘s 1973 album The Dark Side Of The Moon.

If you cut the corners off that triangular prism, you have a hexagonal prism, which is the shape of the ice crystals that cause sun dogs.Light goes in at one face, is deflected by refraction, and leaves by another face, being deflected again. There are various ways of describing the light path through prisms of various shapes, but I’m only going to talk about this one, in which the light enters one face of the hexagonal prism and leaves through what’s called an “alternate” face—the next face but one. Let’s call this route through the crystal the parhelion path, for ease of reference. And I’ll call the two relevant faces the entry face and the exit face.

The symmetrical situation I’ve depicted produces the minimum total deflection. For an ice crystal, that happens when the incident ray hits the surface of the prism at an angle of 41º, where an angle of zero degrees would mean that the ray hit the prism at right angles to its surface. (Don’t blame me, I didn’t invent the convention.) It is deflected by about 11º as it enters the crystal, and another 11º when it leaves, making a total deviation of 22º. Since that’s the minimum deviation, that’s the horizontal angle relative to the sun where we’ll start to see light that has been deflected along the parhelion path.

(Actually, that’s only true when the sun is on the horizon, but it’s a good enough approximation for low solar altitudes. The situation at higher altitudes is fearsomely complicated, so I’ve relegated it to a Note and a link at the end of this post.) As Hipgnosis correctly showed, red (long-wavelength) light is deflected less than violet (short-wavelength) light, so we’ll see red light showing up closer to the sun than violet. The difference in minimum deflection amounts to about a degree between the longest and shortest visible wavelengths.

What about the deflection of light when it strikes prisms that are a little rotated relative to the nice symmetrical position in my diagram? We can make a graph showing how deflection varies with angle of incidence (remember, zero degrees incidence means entering the crystal at right angles to its surface; 90º incidence is a light ray that grazes along parallel to the surface). Here’s the graph, for violet, green and red wavelengths of light:

That’s interesting, isn’t it? Between about 25º and 65º incidence, the deflection stays about the same, within a few degrees. So a beam of sunlight hitting an array of randomly rotated hexagonal ice crystals will generate a lot of light coming out in the vicinity of the 22º minimum angle, and then a smear of light out towards a maximum deflection of about 43º. (There’s no deflection at all below an angle of incidence of about 13º—light that comes in at a steeper angle to the entry face ends up being totally internally reflected at the exit face, so it can’t emerge on the parhelion path.)

So that’s the basic explanation for a patch of colour appearing in the sky 22º away from the sun—the cloud in the photograph contains hexagonal ice crystals. But why does the sun dog appear at the same height as the sun?

The crystals that create the sun dog are in the form of flat hexagonal plates, falling horizontally, like falling leaves:

It turns out that means the flat crystals must be between about  0.025mm and 0.25mm across—smaller, and they never get themselves orientated in the turbulent air; larger, and they tend to rotate end-over-end around a diagonal axis, rather than falling flat.

As these horizontal crystals fall level with the sun, they’re neatly orientated to send light towards our eyes. But because they naturally wobble a little, we also see refracted light coming from slightly tilted crystals that are a little higher or a little lower in the sky—the more the falling crystals wobble, the more vertically smeared the sun dog appears.

There’s more, though. The brightness of the sun dog depends on how much light gets through the crystals.  Some light gets reflected away from the parhelion path, at both the entry and exit faces, and that varies according to the angle at which it strikes each face. If we sum these reflection effects, we can plot the amount of light transmission through a triangular prism for various angles of incidence. It turns out that we get good transmission in the middle 25º-to-65º range that we saw producing a concentrated patch of light around 22º from the sun, with less light getting through at the extremes:

That’s what happens in a triangular prism, like the one in the Hipgnosis picture. Things are more complicated in a hexagonal prism, because not every ray that starts off along the parhelion path can find its way across the crystal to the exit face.

For the symmetrical orientation that produces the minimum angle of deflection, things work well—all the light that goes in one face can come out the other, barring the effects of reflection already discussed:

But for other angles of incidence, only some light that enters the crystal along the parhelion path is on a trajectory that connects with the exit face:

In effect, the aperture through which light gets to the sun dog is much diminished at angles that are not close to the minimum deflection angle. (Just make a mental note of that idea of a reduced aperture—it’s going to reappear in a different guise later.) When we put together the effects of reflection with the effects of this “face shuttering”, we find there’s a neat spike of transmission at the angle of incidence that corresponds to minimum deflection:

So it’s clear that the larger angular deviations associated with extreme crystal rotation end up contributing very little light to the sun dog. And so we have a very good explanation of why the sun dog appears as quite a discrete patch of light in the vicinity of 22º from the sun, rather than extending into a long smear out towards the maximum of 43º.

But what about the distribution of colours? Working out the exact colour of the sun dog at different angles from the sun involves plotting the intensity of light at various wavelengths, over various angles. Now that we know deflections close to 22º are the important ones, I’m just going to graph three representative wavelengths (red, green, blue) over the range 21º to 25º:

Now there’s a problem. Although, as I’ve shown, the brightness of the sun dog must fall off rapidly at large angular deviations, in our area of interest, a few degrees across, it doesn’t seem to decline much at all. There’s an initial spike of red at 22º, but when green appears, the red is still present. Those two colours together make yellow. And when blue appears at 22.5º, it has to compete with all the yellow light that’s still hanging around. When the detailed calculations are carried out, using more wavelengths and factoring in the colour perception of human eyes, they confirm first impressions from the graph above. Because the longer (red, yellow) wavelengths are still hanging about out beyond 22.5º where the shorter wavelengths begin to appear, the short-wavelength colours (blue, violet) should never become visible—they should simply cancel down to white. On the basis of refraction alone, the sequence of colours in a sun dog should go (from closest to farthest from the sun): red, orange, pale yellow, very pale greenish-yellow, white.

And sometimes that’s what we see. Here’s one that seems to follow the predicted sequence:

But sometimes not. Here’s the Boon Companion’s sun dog photo again, this time flipped left and right so that it’s orientated the same way as my graph and the image above:

Beyond the hint of greenish-yellow, there’s definitely some pale blue. Some sun dogs show even more extensive colours:

So calculations performed using refraction match the appearance of some, but not all, sun dogs. What’s going on? One of the first people to think about this was S.W. Visser, in a paper (1.3MB pdf) published in the Proceedings of the Royal Netherlands Academy of Arts and Sciences, in 1917. (I used Visser’s data to plot my graph above.)

What Visser realized was that the ice crystals involved were small enough to cause significant diffraction. When light passes through a small aperture, it spreads out on the far side—the smaller the aperture, the greater the spreading. It also develops a characteristic pattern of light and dark streaks, called an interference pattern. Here’s a typical pattern of intensity for two sizes of aperture:

The blue line shows the interference pattern for green light passing through a 100μm aperture; the red line is the same light through a 25μm aperture. These apertures are in the vicinity of the size of ice crystals that produce sun dogs, so each ice crystal is a tiny aperture that causes diffraction and interference in the light that passes through it. And if all the ice crystals producing a sun dog are about the same size, then the diffraction and interference from all the crystals will add together and have an effect on its appearance.

When Visser did the calculations for the appearance of a parhelion with diffraction and interference superimposed on the effects of refraction, the data looked like this:

I’ve plotted on the same scale as before for comparison, although Visser’s data don’t extend beyond 23º. Several things are happening to change the shape of the light intensity curves:

• Diffraction is broadening the main peaks of the curves, so that light starts to appear closer to the sun than is predicted by refraction alone.
• The crystals that contributed to the long rightward tails of the curves are the ones that exhibit “face shuttering”—and the associated small apertures cause fierce diffraction, which spreads their light broadly and thinly over several degrees, so that it gets lost against the sky.
• So instead of the long superimposed tails that combined to form a bright white patch of light, we see the intensity of each colour of light rise to its own peak and then fall off again—not quite separately, but not nearly as intermixed as is predicted by refraction alone.

When Visser did the detailed calculations, it turned out that blue does get a chance to be a dominant colour after all, albeit somewhat diluted by the lingering remnant of red and green wavelengths in the vicinity of 22.5º.

Visser’s calculations are for a uniform population of large crystals, with faces a quarter of a millimetre across, right on the stability limit. A mixed population of sizes would result in a lot of overlapping peaks, washing out the colour separation; a population of smaller crystals would give wider diffraction curves, again intermixing the colours to a greater extent. So Visser is giving us something like the best-case scenario for sun dog formation.

Put it all together and, just by looking at the sun dog picture brought home by the Boon Companion, we can tell that the cloud contains:

• Flat hexagonal plate crystals; which are
• Falling in a horizontal orientation; and
• Oscillating gently. They are
• Of approximately similar sizes (because blue is visible); and
• They’re fairly large (because diffraction hasn’t severely intermixed the colours)

Not a bad series of inferences to be able to draw from a little patch of light in the sky.

---

Note: The 22º parhelion angle (and other parhelic angles discussed in the text) is strictly correct only for the case in which the sun is on the horizon, so that its light travels horizontally through the horizontally orientated crystals. When the sun is higher in the sky, its light necessarily travels through the crystals on a sloping path, which makes their prism angle appear broader than 60º, and pushes the sun dogs farther from the sun. Here’s a plot of where the sun dogs actually appear, against solar altitude:

The sun dogs change shape and become more diffuse as the sun gets higher in the sky. They’re rarely observed above 40º, and optically impossible above 61º. For more on the complicated mathematics of sun dogs, see Roland Stull’s excellent free on-line textbook, Practical Meteorology. You want Chapter 22, Atmospheric Optics (1MB pdf).