MadSci Network: Physics
Query:

Re: The size of a rainbow

Date: Mon Aug 3 16:44:22 2009
Posted By: Chris Peterson, Faculty, Denver Museum of Nature and Science
Area of science: Physics
ID: 1247787608.Ph
Message:

I was just looking at a rainbow and got the thought "what factors go into the actual size and magnitude of a rainbow?" I already know how a rainbow is formed, but I now need to know what makes rainbows the size they are. Is it the angle of the sun to the earth at a particular time of day (ie after sunrise or before sunset). I know a bit about angles and rotation, but have never learned about light being cast in such a manor as to create the specific size of rainbows. My math ed goes just below calc, so I should be able to understand any angle jargon, but please just stay away from derivatives in your answer (if possible). In any case, just wondering....and thank you.

All rainbows are the same size, although they may appear differently- either because we can only see part of them, or because of an optical illusion.

You say you understand how a rainbow forms, but let me describe the optics involved. A rainbow is actually a full circle, and it appears in the sky around the antisolar point. The antisolar point is the location on the sky that is directly opposite the Sun. This is pretty obviously below the horizon any time the Sun is above it. When you are facing away from the Sun, raindrops internally reflect sunlight back towards you over a fairly narrow range of angles. This produces the circle of the rainbow, which has an angular diameter of about 84°. Every rainbow has this same dimension. For comparison, the Sun and Moon themselves have an angular diameter of about 0.5°. So you could fit 168 Moons across the full diameter of a rainbow.

You seldom can see much more than half of a full rainbow, and you'll only see that when the Sun is very close to the horizon. Perceptually, you might consider such a rainbow "big". As the Sun gets higher in the sky, the antisolar point gets farther below the horizon and you see less and less of the full rainbow. When the Sun is 42° above the horizon, you'll only be able to see a tiny segment of the rainbow. Perceptually, that might be considered a "small" rainbow. These variations in apparent size only reflect the fact that you are seeing more or less of the entire circle.

Separately, the size of a rainbow may seem large or small simply because of the references you see around it- hills, trees, buildings, etc. This is similar to the Moon illusion, which makes a rising Moon seem larger than when it gets higher in the sky, even though the angular diameter of the Moon is nearly constant regardless of where in the sky you see it.

This site, www.atoptics.co.uk/rainbows/primcone.htm, has a nice image visualizing a rainbow as the rays traveling on a cone (with a 42° half angle) where the eye is at the apex.


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