Converging Rainbows

Double rainbow from reflected sun
Click to enlarge

A familiar pair of primary and secondary rainbows is always concentric, and the outer rainbow has its colours in the reverse order from the primary. But these two have their colours in the same order, and are converging to meet on the horizon. What’s going on there?

I was walking home from work a couple of months ago when I saw this pair of rainbows sticking up above the roofs of the houses like some sort of cosmic V-sign. I lurched to a halt, stared for a few seconds, and then broke into a jog—the sun was going to be setting soon, and I wanted to get a proper view of the pair. Once chez Oikofuge, I body-checked my way past the Boon Companion’s customary greeting (sorry, my love), grabbed a camera, and ran up the stairs to take this photograph looking out over our local river estuary. The photo provides a hint as to what’s causing the unusual rainbow.

Here’s a reminder of how a standard primary rainbow forms:

How rainbows form

It’s actually a bit of a Just-So story. Quite why a spherical raindrop chooses to turn an incoming ray of red light back on itself at an angle of  42º (and a blue ray at 40º) is rather complicated, and I’m working on a little programming project to try to explain it clearly—watch this space. But for now, we just accept that if you look directly away from the sun, towards what is called the antisolar point (handily marked by the shadow of your head), then every raindrop that happens to be at 42º from your line of sight will be directing red light towards your eyes (and every raindrop at 40º will be directing blue light towards you). In principle, then, a primary rainbow should form a complete circle in the sky, centred on the shadow of your head, and 42º in angular radius. In practice, the parts below the horizon become progressively more difficult to see, because there are fewer and fewer raindrops along your line of sight as you shift your gaze downwards.

Notice that the antisolar point is always below the horizon, because the sun can only illuminate raindrops when it’s above the horizon. (D’oh!)

Now, look at how still the water is in my photo. (That’s unusual, hereabouts.) The rainbow-forming raindrops on the far side of the estuary are not just being  exposed to direct sunlight, they’re also being illuminated by light coming from the image of the sun reflected in the still water. Although I can’t see that extra “sun”, it’s nevertheless providing me with another antisolar point and an associated rainbow. This reflected antisolar point is precisely as far above the horizon as the real antisolar point is below it. So the two rainbows have to meet exactly at the horizon, as in my diagram:

How double rainbows form

What you’re seeing in my photo is the little V formed by those two rainbows coming together just above the horizon. (The V is noticeably narrower in the photograph than in the diagram, because the sun is lower and the antisolar points are closer together—but when I tried to reproduce the real situation in the diagram, it got less clear and harder to label properly.)

I stood and watch the display for a while. In theory, the angle of the V should get progressively narrower as the sun gets lower in the sky and the two antisolar points approach each other. And as the light of the setting sun gets redder, the associated rainbow should lose its bluer shades.

What actually happened was that the sun dropped below a bank of cloud, and the two rainbows winked out of existence.

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