Why does a rainbow only form with a 42° angle? [duplicate]

Question

Why do rainbows only form with a 42° angle? Is there something special about this angle?

Answer 1

Yes, indeed. But first we need to look into Rainbows. According to Wikipedia,

A rainbow is not located at a specific distance from the observer, but comes from an optical illusion caused by any water droplets viewed from a certain angle relative to a light source. Thus, a rainbow is not an object and cannot be physically approached. Indeed, it is impossible for an observer to see a rainbow from water droplets at any angle other than the customary one of 42 degrees from the direction opposite the light source.

Light rays enter a raindrop from one direction (typically a straight line from the sun), reflect off the back of the raindrop, and fan out as they leave the raindrop. The light leaving the rainbow is spread over a wide angle, with a maximum intensity at the angles 40.89–42°. The most intense effect can be observed at only 42°.

Rainbow single reflection (View Image)

The reason the returning light is most intense at about 42° is that this is a turning point – light hitting the outermost ring of the drop gets returned at less than 42°, as does the light hitting the drop nearer to its centre. There is a circular band of light that all gets returned right around 42°. If the sun were a laser emitting parallel, monochromatic rays, then the luminance (brightness) of the bow would tend toward infinity at this angle (ignoring interference effects).

Image source: Wikipedia

EDIT: I also remember studying the fact that the radius of a Titan rainbow would be 49° instead of the usual 42°, because the fluid in that cold atmosphere is methane instead of water.

Also, interestingly, the red light exits the raindrop subtending an angle of 42° with the incoming sunlight ray. This is because the red light has more photon flux than other colours. Therefore it has more intensity. Since we want that angle for the most intense light, we get 42°.

Angles formed by light in raindrops

P.S: Red light has more photon flux in this case because here all colours are falling with the same intensity (that of white light). So, the number of photons of lower frequency will be higher than the number of photons of higher frequency. Way back in the 1630s, Rene Descartes used the laws of optics and calculated that the three stage refraction-reflection-refraction pattern that light undergoes when passing through a raindrop produces a concentration of outgoing rays along a line that is 42 degrees above the head of an observer's shadow. This concentration of light rays is the rainbow that we see.