How far away is a rainbow?

As a biologist, I am not an expert on rainbows, but I thought I knew how they work.

A rainbow occurs when the sun is shining into rain, and each water droplet acts as a little spherical prism, splitting the sun’s white light into the colours of the rainbow in a process called refraction.

The sunlight is hitting each rain droplet at the same angle, regardless of where it is in the sky. But the refracted light comes out of the droplet at different angles, depending on its colour (wavelength).

So when we look at one particular droplet, we only see one colour – the one that happens to be refracted in our direction. And if we look at another droplet in a different position, our viewing angle is different, so we see a different colour.

This means that when we look at a rainbow, all the bits of the same colour are droplets for which the angle between the sun and us, the observer, is the same. (Of course, it’s really a succession of droplets falling through that location…)

Now, because it’s the angle to the sun that counts, a rainbow doesn’t move when we move; Just like the sun, it stays put in the same direction. Our brain has learnt that this is the case for objects that are very far away – when we move, stationary things at the horizon appear to move much less than things nearby, something we call motion parallax.

So I was always under the impression, as I’m sure most people are, that rainbows are way up there in the sky. Imagine my surprise when I saw this, from the third-floor balcony of my mother’s apartment. The trees accross the street are less than twenty metres away, and the rainbow must be nearer than that!


Of course, now that I thought about it, the location of the rainbow, if there is such a thing, is the location of the rain droplets that happen to be at the appropriate angle to the sun. And in this case, that was right in front of me, in the space between the balcony and those trees.

2 Comments Add yours

  1. Wulf says:

    Hi Andy,

    I would argue that rainbows are very far away, indeed. What you see is a reflection of the sun in a “mirror” of rain drops.

    In optics, we call this a “virtual image”, and because it is virtual, it can lie well behind the row of trees that would block the view to any far-away “real image” or object. What is in front of the trees is not the rainbow, but the reflecting raindrop curtain that creates it.

    This is similar to looking into the bathroom mirror and finding your image way beyond where the bathroom wall is.

    1. Andy Giger says:

      Yes, Wulf, if we understand the rainbow as a virtual image of the sun, we could say that it is about 149,597,870,700 metres away, give or take a few kilometres… However, unlike a reflection in a mirror, that “image” would not be in any one location but scattered in many (indeed infinite) directions, which would make it hard to locate. But as an abstract concept, that image is indeed behind the trees, and not in front. (Or is it actually inside my eye? That would be even closer!)

      In any case, this does not diminish my surprise I describe above, though. Just replace “rainbow” with “volume of atmosphere containing rain droplets refracting the sunlight and creating a virtual image of the sun” in the last two paragraphs, and you’re set. ; )

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