# How deep is a rainbow?

My cousin shot this photo of a rainbow in front of a mountain.

In the picture it is clear that the mountain is behind the rainbow (the rainbow blocks the view of the mountain), whereas the nearby trees are in front of the rainbow. This got me thinking: how thick (deep) is the rainbow?

To make that more precise, imagine a plane taking off from the top of the mountain behind the rainbow and flying straight towards the camera. You could measure the relative strength of the plane / rainbow in the picture by a simple linear fit (assuming you knew its location). I imagine you'd see something like this:

Graph showing intensities of plain and rainbow http://167.160.167.208/rainbow-depth.png

How thick would this region be?

• The link to the photo doesn't work - can you paste the photo into the question? – DaveInCaz Jun 26 at 9:52

You don't really "look at the rainbow". You look at light that was refracted in rain drops. Where the drops are, the refraction happens. And as long as the density of water drops does not impede your view, the depth is "infinite".

In other words - any drops that present a certain viewing angle (in the triangle sun-drop-eye, the angle subtended from sun to eye is the same) will give the same color, regardless of the distance.

So there is no real answer, because it will depend on the density of the rain drops. Once the probability of a drop being in the way becomes substantial, you reach the limit of your rainbow "depth".

"Heavy rainfall" is considered anything above 0.3" (7.6 mm) per hour. Rain drops fall at rates between 1 m/s and 10 m/s (source). Let's say the average size is 2 mm with a terminal velocity of 6 m/s; This means that in 1 hour, a column of about 20 km (6*3600) "falls", creating a layer of water that is 8 mm thick. When a rain drop (volume $\frac43 \pi r^3$) hits the ground (area $\pi r^2$ it creates a "puddle" of depth $\frac43 r \approx$ 1.3 mm deep. Six of these drops would correspond to 8 mm, so on average mean free path between drops is 20 km / 6 = 3 km.

That means that in "heavy rain" you expect to see a rainbow with a depth up to 3 km. As the rain gets heavier, the depth reduces; if it's lighter, it can be more.

The above is very approximate...

In terms of geometrical optics, the rainbow is a virtual object at infinity, as the rays arriving at the observer's eyes are perfectly parallel. Even if the rainbow is made by nearby droplets from a garden hose, the rainbow is a virtual object at infinity.

Its infinite distance would be more evident if the sun was a point source, and the rainbow was viewed through a monochromatic filter. In those circumstances the rainbow would be a perfectly sharp circle, and photos would be sharp only if the camera was focused at infinity.

It is somewhat similar to looking at the moon through a mirror: the observer is watching a virtual object infinitely far behind the mirror. Similarly the rainbow is infinitely far behind the reflecting raindrops.

The rainbow in your cousin's photo seemed to be in front of the mountain. However, geometrical optics ignores the scenery; you have to extrapolate the rays back to a virtual image behind the mountain. Similarly, the image of the moon in a mirror could be located behind the mountain.

The common idea that the rainbow is located inside the rain curtain, instead of at infinity, is at odds with geometrical optics. That idea results from confusing the reflectors with the image.

• Re, "rainbow...seemed to be in front of the mountain." The rainbow is an illusion, but then, so is everything else that you see. It's all a model, inside your head, built up from rays of light that enter your eyes. The direction of the rainbow rays wants to inform you that the rainbow is infinitely far away, but that's a weak visual cue as compared to our prior experience of not being able to see through mountains. If the source of the rays (raindrops) lies between you and a mountain, then your brain will not let you perceive the rainbow as being behind the mountain. – Solomon Slow Feb 19 at 18:38
• Zero parallax, when watching the rainbow during a train trip, is a strong visual cue for its distance. The apparent origin of the rays is at infinity. It may help to think of the rain curtain as a kind of semi-transparent mirror. The rainbow is superimposed on the landscape. As it is superimposed, the scenery is no longer a visual cue for the distance of the rainbow. People do have experience with superimposed images, as in shop windows (photo). The human brain understands superimposed images. – jkien Feb 19 at 23:23
• As an illustration, a video of rainbow seen though a train window: link. The rainbow appears to be further away than the trees at the horizon, due to its zero parallax. – jkien Feb 20 at 9:10