|MadSci Network: Earth Sciences|
You are suggesting that if you know the distance between the two places where a rainbow touches the ground, you can use trigonometry to calculate the distance to the rainbow. This is correct, but how can you tell what the distance between the points that appear to touch the ground is?
A rainbow is not a physical object with a definite size. It is a pattern of light refracted by raindrops. The colored light is reaching your eyes comes from every raindrop at the correct angle between you and the sun -- some of these raindrops are very near, some are very far away. If it's raining where you're standing, the raindrops right in front of your nose are causing part of the rainbow, and so are raindrops a few kilometers away.
For a typical rainstorm, drops less than a hundred meters away appear too widely spaced to make a noticeable rainbow, and drops more than a few kilometers away obscured by other raindrops and lost in the haze. The light from the rainbow is coming from "somewhere in between". I can't be more precise than that. Also, this could vary a lot with time of day, type of rain and cloud, air pollution, and any number of things.
If you look at the bottom of a rainbow carefully, you'll often see that the "point where the rainbow appears to touch the ground" is awfully hard to pick out. The 'bow is often strongest at the horizon, and gradually fades out as you look below the horizon at nearer and nearer ground-level objects. If you can see a faint bit of rainbow passing through a building a kilometer away and an even fainter bit going through a bush 50 meters away, who's to say whether the rainbow is at the bush or the building?
Now, if the raindrops which are forming the rainbow are in a storm which is not all around you, but a certain distance in a certain direction, say a kilometer or two away, then you can say that the distance to the rainbow is the distance to the raindrops which form it. However, this sort of rainbow is generally too small to touch the ground in two places, which means your technique isn't helpful.
Generally, asking "how far away is a rainbow" is the same sort of thing as asking "how far away is sunshine?"
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