Tertiary Rainbows, etc

Rainbow enclosing bright sky
Click to enlarge
Photo by Binyamin Mellish via Good Free Photos

In my last two posts about rainbows, I discussed the formation of the primary and secondary rainbows, respectively, tracing their origins to specific light paths through falling raindrops.

The primary rainbow ray follows a path like this:
Green rainbow rayFor a raindrop at the apex of the rainbow arc, sunlight enters near the top of the drop, bounces once off the back, and then exits the bottom, descending towards the observer’s eye, making an angle of around 41.5º for the green ray shown.

For the secondary rainbow, sunlight enters near the bottom of the drop, is reflected internally twice, and then exits the front of the drop, descending towards the observer at an angle of around 51º.

Green secondary rainbow rayThese rainbow rays are special, representing the maximum or minimum angles of deflection of the incoming ray. And they are associated with a particular offset of the incoming ray in its interaction with the raindrop. I measured offset like this:Rainbow offset parameterThe rainbow ray for the primary enters the raindrop at an offset of about 0.86; the secondary rainbow ray at about 0.95.

For more about these topics, in particular the importance of maximum and minimum deflections, I direct you back to my previous posts.

I finished my most recent post on this topic with a question: If a single internal reflection produces a primary rainbow, and two internal reflections produce a secondary rainbow, is there such a thing as a tertiary rainbow? And if so, where is it?

The path of the rainbow ray for a tertiary bow looks like this:

Path of rainbow ray for tertiary rainbowThe three reflections carry it almost all the way around the raindrop, so that (in contrast to the primary and secondary bows) it leaves the drop heading away from the sun. This tells us that, to see a tertiary bow, we’d need to look towards the sun, rather than (as for the primary and secondary) towards the antisolar point.

Plotting my usual graph of light deflection and transmission against the full range of ray offsets with three internal reflections, I get this:

Deflection and tranmission in formation of tertiary rainbow
Click to enlarge

In contrast to my graphs for the primary and secondary, this one shows the deflection from the “solar point”—the position of the sun. The maximum occurs at an offset of about 0.97, reaching 42.5º for red light, and 37.7º for violet. The angular distance between red and violet is therefore about two-and-a-half times what we see in the primary bow. The light transmission is scaled to match my previous two graphs, and it shows that, because so much light is lost during multiple internal reflections, only about 1% of the tertiary rainbow ray survives to exit the drop. (But that’s equivalent to about 24% of the transmission for the primary rainbow ray, so not catastrophically dim.)

So it seems straightforward enough. We should see a broad, faint tertiary bow, about the same diameter as the primary but centred on the sun. Have you seen that? No, me neither.

The problem is that there’s a lot of light in the sky around the sun, particularly when rain is falling through the line of sight. Les Cowley at Atmospheric Optics calls this the “zero-order glow”, because it is formed from sunlight that passes through the raindrop without being reflected. So there’s always a directly transmitted, zero-order light ray parallel to the tertiary rainbow ray, like this:Path of rainbow ray for tertiary rainbow, and zero-order glowMuch more light pours through the raindrop in the zero-order glow than survives through three internal reflections. This makes the tertiary rainbow very difficult to see.

But not impossible. There have been sporadic reports over the last few decades, by careful observers who knew what they were looking for. And finally, in 2011, a paper* appeared in the “Light And Color In The Open Air” edition of Applied Optics, entitled “Photographic evidence for the third-order rainbow”. The authors describe taking a photograph under favourable conditions (the sun obscured, a dark cloud in the region that would be occupied by the tertiary bow). The photographer could barely discern a hint of the tertiary rainbow—“only a faint trace of it at the limit of visibility for about 30 seconds”. But after a bit of image processing, a rainbow arc appeared in the resulting photograph. And, after careful analysis, the authors confirmed they had taken the first known photograph of a tertiary rainbow.

You’ll have discerned a pattern. Each successive rainbow (primary, secondary, tertiary) is produced by a rainbow ray which is increasingly offset from the centre of the drop, undergoing one more reflection that its predecessor. The increasing offset means that refraction is greater, with a larger difference between red and violet rays, leading to a broader rainbow. Each additional reflection along the light path means more light lost, and so a fainter rainbow.

Rather than bore you with light paths for higher-order rainbows, I’ll show you them all on one diagram. (The design is based on Jearl Walker’s “Amateur Scientist” column in the July 1977 issue of Scientific American—the calculations and drawing are all my own.)

First twenty rainbows for light entering top of raindrop
Click to enlarge

The rosette shows the first twenty rainbows, produced by the first twenty internal reflections of light that enters the upper half of a water droplet. Light bounces around the drop clockwise, and the difference in deflection for successive rainbows quite quickly converges to settle at a little less than a right angle. So you can see that the primary (labelled “1” in the lower left quadrant) is followed by the secondary in the upper left quadrant, the tertiary in the upper right quadrant, and the quaternary in the lower right. The quinary rainbow (number five) brings the progression almost full circle, and appear just a little anticlockwise of the primary. I won’t bore you with the names for higher orders of reflection, but you should be able to pick out how they go around again, and again, and again, becoming successively wider and fainter.

Because violet is refracted more than red, all the rainbows have the same layout, with violet always lying clockwise of red.

Rainbows in the lower left quadrant are being reflected downwards, and back towards the light source. So an observer would look for them as circles around the antisolar point. Because the red light is descending more steeply than the violet, red will appear on the outer edges of all these rainbows. Rainbows in the lower right quadrant are formed from light that has been reflected downwards and away from the light source—so the observer must look towards the sun to see them. In this position, violet light is descending more steeply than red light, so all these rainbows have violet on their outer edges.

Light in the top half of the diagram is all spraying upwards—these rainbows would not be visible to an observer standing below the drop. But we’ve neglected the light that enters the lower half of the raindrop, and bounces anticlockwise. It produces its own rosette of rainbows, identical to the one above, except mirrored in the horizontal plane, like this:

First twenty rainbows for light entering bottom of raindrop
Click to enlarge

The reversed light path means that violet always lies anticlockwise of red for this family of rainbows. So antisolar rainbows with this light path have violet as their outer colour, and solar rainbows have red outermost.

So the rainbows we see in the sky come from a mixture of the two sets of possible light paths in the two diagrams above. So let’s stack them together, and switch from a view centred on the water drop to a view centred on the observer:

The first twenty rainbows
Click to enlarge

The sky is full of overlapping rainbows! You should think of the upper and lower “copies” of each rainbow as being linked by a vertical circular arc sticking out at right angles to your screen, which the observer sees as a (potential) circular rainbow.

My diagram has the sun directly behind the observer, in which case the rainbows in the lower part of the diagram would be invisible under normal circumstances, superimposed on the ground, and each rainbow would form a semicircular arc against the sky. But the sun is usually some distance above the horizon—as it climbs higher, it pushes the antisolar rainbows lower in the sky, but carries the solar rainbows higher. So antisolar rainbows  generally form less than a semicircular arc, and when the sun is higher above the horizon than the radius of the rainbow they will drop entirely below the horizon. But solar rainbows will be lifted above the horizon by the sun, forming more than a semicircular arc, and at the extreme when the sun is higher above the horizon than the radius of the rainbow they can form complete circles. (Complete circles are also possible for antisolar rainbows, but only when the observer is looking down on water droplets suspended below the horizon, as from a plane flying over clouds.)

Are any of them visible to the naked eye? We know that the tertiary has been spotted very rarely. The quaternary sits right next to it, farther out in the zero-order glow, but additional light losses mean its rainbow ray transmission is just 15% of the primary. I don’t know of anyone who has seen it, but it has been photographed. In fact, the article entitled “Photographic observation of a natural fourth-order rainbow” appeared in the same themed edition of Applied Optics as the tertiary rainbow report. (Early reports of the tertiary photographs had inspired the author to search out the quaternary using image stacking.)

The quinary bow has also been photographed. With rainbow-ray transmission sitting at just 10% of the primary, it is also partially obscured by the brighter secondary bow. But its green, blue and violet portion sit within Alexander’s Dark Band, giving reasonable hope of detecting those colours. And Harald Edens managed to (just) pick out the green stripe of the quinary rainbow in a photograph taken in 2012.

The sixth-order rainbow sits within the bright sky at the inner edge of the primary bow, and seems like a poor candidate for detection. The seventh is better separated from the zero-order glow than the third and fourth, but its rainbow ray transmission is only 6% of the primary. However, it seems likely that some of the higher order bows will yield to the sort of photographic techniques commonly employed in astrophotography these days. Watch this space. Meanwhile, for your delectation, I’ve appended some basic descriptive data for the first twenty rainbows, as featured in my diagrams above.


* Großmann M, Schmidt E, Haußmann A. Applied Optics 50(28): F134-41
Oh, alright, I will. Beyond quinary, five, the sequence goes senary, septenary, octonary, nonary, denary, undenary, duodenary … at which point we need to start making up names until we get to vigenary, twenty.
Theusner M. Applied Optics 50(28): F129-F133

Note: All values, here and in previous posts, are calculated using a refractive index for red light (wavelength 700nm) of 1.33141, and for violet light (wavelength 400nm) of 1.34451. See the page dealing with refractive index on Philip Laven’s excellent website, The Optics Of A Water Drop, for further information and references.


The First Twenty Rainbows
OrderOuter
Radius (º)
Width (º)Outer
Colour
Solar/
Antisolar
Transmission
(% Primary)
142.41.9RedAntisolar100
253.83.4VioletAntisolar42.6
342.54.8RedSolar23.5
448.96.1VioletSolar14.9
552.97.4RedAntisolar10.3
639.68.7VioletAntisolar7.6
765.610.0RedSolar5.8
829.011.2VioletSolar4.5
979.112.5RedAntisolar3.7
1017.813.8VioletAntisolar3.0
1193.1*15.1RedSolar2.6
1210.110.1RedSolar2.2
1390.3*17.6VioletSolar1.8
1424.418.9RedAntisolar1.6
1578.520.1VioletAntisolar1.4
1638.821.4RedSolar1.3
1766.622.7VioletSolar1.2
1853.323.9RedAntisolar1.0
1954.625.2VioletAntisolar0.9
2067.926.5RedSolar0.8

* The order-11 and order-13 rainbows lie predominantly in the solar hemisphere, but extend slightly into the antisolar
The order-12 rainbow spans the solar point, and therefore overlaps itself—a small disc of blue-violet rainbow (radius 6.1º) is superimposed on a larger disc of green-yellow-red rainbow (radius 10.1º)

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