More About Converging Rainbows

Reflected-light rainbows over Morecambe Bay, by Mick Shaw
Click to enlarge
Photo credit: Mick Shaw

A couple of months ago I received this lovely picture from Mick Shaw, which I use with his permission. The sun is reflecting off a thin layer of sea-water covering the sand-flats of Morecambe Bay, and producing a pair of reflected-light rainbows in tandem with the usual primary and secondary arcs.

Reflected-light rainbows were the subject of one of my first posts on this blog, back in 2015, when I posted my own photograph of the phenomenon:

Double rainbow from reflected sun
Click to enlarge

In that photograph, you can appreciate how still the waters of the Tay estuary were—an essential condition for this sort of rainbow to form, because the water surface has to be flat enough to reflect a near-perfect image of the sun.

Once the water surface reflects a clear image, falling drops of rain are illuminated by two suns—one above the horizon, and one below. So two rainbows can form, each centred on an antisolar point—the point directly opposite the sun. Rainbows formed by direct sunlight are centred on a point as far below the horizon as the sun is above it—this angular distance between sun and horizon is called the solar altitude, in astronomical jargon, and that’s what I’ll call it from now on. But the reflected-light rainbow arcs are centred on an antisolar point that is at the same height in the sky as the sun. Like this:

How double rainbows form

Rainbows form a circle around the antisolar point, but the sections of their arc that fall below the horizon are generally invisible, unless the observer is in a high place looking down on clouds. The reflected-light rainbow forms an arc that is identical in shape and size to the portion of the normal rainbow that lies below the horizon, except it is flipped vertically to sit above the horizon. So the two arcs, from direct and reflected sunlight, converge exactly at the horizon, forming a prominent lopsided “V” shape, evident in the photographs above.

A little bit of spherical trigonometry lets me plot the divergence between our two intersecting rainbows—that is, the angle at the base of the “V”. Here it is for the primary rainbows:

Divergence of directed and reflected-light rainbows at horizon, by solar altitude

With a solar altitude of zero, there is no divergence. The sun is sitting on the horizon, and the direct and reflected antisolar points have merged on the opposite horizon, so that the direct and reflected-light rainbows are precisely superimposed, forming a single semicircular arc. With the sun farther above the horizon, the “V” between the two rainbows opens up steadily, until suddenly shooting up towards 180 degrees as the sun approaches an altitude of 42 degrees. This critical angle of 42 degrees corresponds to the radius of the primary rainbow. With the antisolar points 42 degrees above and below the horizon, the two rainbows are circles that touch each other at the horizon, with the primary rainbow invisible below the horizon, and the reflected-light rainbow forming a complete circle in the sky, sitting on the horizon. Like this:

Direct and reflected light primary rainbows at solar altitude of 42 degrees

Have you ever seen such a thing? Me neither. So I began to wonder why that doesn’t seem to happen.

There are a couple of practical considerations that strongly limit our opportunities to see a circular reflected-light rainbow entirely above the horizon. One is that rainbows are huge. A circular primary rainbow would stretch from the horizon almost all the way to the zenith. We’re used to the upper part of even a normal rainbow fading out, as it extends from a region of sky near the horizon where we have long sight-lines through falling rain, producing bright rainbow “legs”, into a region where our sight-lines are blocked by the rain-clouds themselves, so that the upper part of the arc is faint or invisible, like this:

Rainbow at sunset, Kylerhea
Click to enlarge
Rainbow at sunset, Kylerhea, © 2021, The Oikofuge

Compared to the sunset bow above, a full circular rainbow would extend twice as far upwards into the sky, taking it into regions that normally provide few raindrops along our line of sight.

There’s also the issue that a reflected-light rainbow needs a sizeable reflective surface. Areas simultaneously large enough and calm enough to produce a complete circular rainbow are probably fairly rare.

But there’s a more fundamental reason that precludes our seeing a perfect circular reflected-light rainbow, hanging in the sky. It has to do with the reflective properties of water. The amount of light a flat water surface reflects depends on the angle at which the incoming light hits the water surface. Here’s a graph of the amount of reflection of (unpolarized) sunlight according to solar altitude:

Proportion of sunlight reflected by water surface, by solar altitude

You can see that, by the time the sun is thirty degrees above the horizon, its reflected image is only a fraction of the brightness of the real sun—about 6%, if we do the sums. So the reflected-light rainbow will be comparably reduced in brightness.

Another, more minor, influence on the brightness of the reflected-light rainbow is that the light reflected from a water surface is usually quite strongly polarized:

Degree of polarization of sunlight reflected from water

Now, as I described in my post about rainbow rays, the repeated process of reflection and refraction inside a raindrop mean that the rainbow itself is polarized, in a fashion that follows the arc of the rainbow. The horizontal top of the rainbow arc is polarized in the same sense as the reflected light from the water surface; the vertical sides of the rainbow are polarized transversely relative to the reflected light. So the upper part of a reflected-light rainbow should be a little brighter than its “legs”, if all other things are equal (which they’re generally not).

Here’s what happens when I feed the reflected light from a flat water surface into the light-path of the primary “rainbow rays”:

Brightness of reflected-light rainbow arcs, by solar altitude

The centre of the graph corresponds to the sunlight that reaches us after reflecting off the water surface and then passing through the horizontal top of the rainbow arc. The sides of the graph show the same information for the vertical “legs” on either side of the rainbow.

At the top of the graph is the line corresponding to a solar altitude of zero. With the sun on the horizon, the water surface reflects all the light that falls on it, so there’s no polarization—illumination from the reflected sun is the same as that from the real sun, and our reflected rainbow is as bright as a normal rainbow (though that means only about 4.5% of light rays striking the reflecting surface find their way back to our eyes in the rainbow).

But the rapid fall in reflection from the water surface with increasing solar altitude means that our reflected-light rainbow fades out quickly. The onset of polarization also means that the “legs” of the rainbow grow fainter faster than does the top of its curve. With the sun just 10 degrees above the horizon, the reflected-light rainbow is already fainter than a normal secondary rainbow, which we know is frequently faint or invisible.

So our best chance of seeing a reflected-light rainbow occurs when the sun is close to the horizon, because these rainbows fade into invisibility as the sun gets higher. We generally see them as they appear in the two photographs at the head of this post—as a fainter rainbow that echoes the “leg” of a normal rainbow, while converging with it at the horizon. The supposedly brighter top of the reflected-light arc is often invisible, for the same reason that top of a conventional rainbow is often invisible—because it extends higher than the region in which long sight-lines extend under the rain-clouds. So I’ve never seen the upper arc of a reflected bow, but I live in hope.

2 thoughts on “More About Converging Rainbows”

  1. Swoooo! I so admire your grasp of physics, maths, and the language to explicate them. Or do I just mean explain?
    Thank you for this which I shall enjoy sharing with friends who can appreciate and follow your logical process more thoroughly than I…

    1. I’m not sure if there’s a specific threshold at which explanation turns into explication; but I’m glad you enjoyed the post.

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