Ytdyklly

I am wondering whether rainbows are three dimensional, and if so, what determine their depth? How to calculate the depth of a rainbow, given its radius?

From what I understand, all rainbows are circular and their radius are determined by the distance between the observer and the water drops. How can one determine the depth based on this information, assuming the depth is non zero?

Edit: As stated in a comment by Solomon Slow, a rainbow is not really an object. My question still applies, the rays of light that make us see a rainbow are coming from droplets. What is the depth extension of these (real) droplets?

A rainbow doesn't strictly have a radius.

First: the obvious. A rainbow isn't a 'circle' but an annulus. The thickness comes from the range of frequencies (colors). However, if we restrict ourselves to one frequency, it's a circle.

No matter where you stand, the rainbow will appear as a circle centered around the antisolar point with an angle of $theta_{rm{crit}}approx 42^o$ (note, the image above shows 2 different angles - this is a double rainbow. You can ignore the outer rainbow.). If the rainbow was a definite distance $d$ from you, it would have a definite radius $r=d tan(theta_{rm{crit}})$. But it isn't!

The light which reaches your eye comes from any droplet of water located on a cone that has yourself as the vertex, the direction of sunlight as an axis and an angle of $theta_{rm{crit}}$. If the rain was confined to a thin sheet perpendicular to the sunlight (an obviously artificial configuration), only then would it's intersection with the cone be a definite circle with a definite radius.

As for determining thickness: roughly speaking a fainter rainbow should correspond to less thickness (fewer droplets contributing to the cone of light).

In theory, if conditions are right, a rainbow can have a depth of several kilometres. But for this you would need the rainclouds to have just the right structure so that enough sunlight could get to the rain that they were producing, and then back to your eye. They would need to form a kind of enormous funnel. So that's not very likely. Also, the back of such a rainbow would contribute little to the spectacle, because the intervening rain would block out most of the reflected light.

In general, I would expect a decent rainbow to be of the order of hundreds of metres deep. This is not based on any scientific measurements, just on an understanding of the geometry involved.

No.

Rainbows aren't objects, so they don't have dimensions in the naïve sense.

Yes.

Rainbows are visible because droplets of water reflect and refract the sunlight into your eyes. The colors are refracted at slightly different angles, approximately at 42°. All droplets that are on a ray originating in your eye will refract the same color into your eye and depending on the current state of the atmosphere, there may be more or less droplets and they will probably be located in a certain volume that could be considered "the rainbow".