Let's say I know how to compute the apparent radius of a rainbow from the viewpoint of the observer: take a photo of the scene, measure the distance to a known reference object, and its dimensions. Using triangle similarity, I can extrapolate the radius of the rainbow.
But my question is: which physical phenomenon determines the radius?
It depends on where the sun is. If it is near the horizon (behind you) and in front of you there are water droplets, then you will see a rainbow with a radius (in angular measure) of about 42 degrees, because each water droplet returns a cone of light, whose axis is parallel to the direction to the sun and whose aperture is roughly $2 \cdot 42 = 84$ degrees.
I've never seen better explanations of dozens of phenomena concerning rainbows than in Walter Lewin's lectures.
It depends on the position of the sun in the sky; the higher, the smaller the rainbow radius. Likewise it depends on the temperature of the water droplets and atmosphere; the lower the temperature the smaller the rainbow radius. Also, depends on the impurities in the water droplets; the denser the greater the radius. Also depends on the viewers proximity to the water droplets; the closer, the greater the radius. Also depends on the atmospheric pressure between water droplets (similar to temperature effect). Also depends on the seasonal position of the earth on the ecliptic orbit. Once you have this data you can formulate an approximate radius at a point in time.
Occurs when a light beam is intercepted by a drop of water suspended in the atmosphere. The drop breaks it down into all its colors at the same time it deviates (refracts it to drop into and out). Because of these refractions Ray turns to the sky when the sun is.
The straw acts just like a prism, the first refraction separates colors containing the beam and the second refraction increases further this separation.
Part of the light refracted into the drop, reaching the other interior wall, is refracted again and exits to the atmosphere and the other part is reflected inside.
In turn, a portion of the reflected light on the inside of the drop forward and back out into the atmosphere refracts. The rays from this portion of incident light are those that form the rainbow (as seen by the eye). The red form with the sunbeam incident angle of 42 degrees, blue 40 ° and in this range 2 ° out all the radiation we know as the rainbow.
Make sure that when your eyes intercept the separated colors that depart from the raindrops, the red rays are the rays from incidents that slightly higher angles (42 º) than the blue rays are (40 º). The beam coming from the sun and the ideal line goes from observer to the center of the rainbow are parallel and therefore we see that the red beam is $42 °$ to the horizontal. Then the ratio of the radius of the rainbow is directly linked to inlcinacion angle of the sun.
What you see in the rainbow is a rainbow of colors with red located on the outside, then the other orange, yellow, green and finally blue is on the inside of the arch.
As the angle to see the rainbow is always 42 degrees, the lower the Sun is highest is the rainbow, becoming the arc in a circle visible when the sun is above the horizon.
The reason that the rainbow in the sky draws an arc is the angle between the incident sunlight and the refracted light of any color is not necessarily the same for each drop being the greatest for primary rainbow light red (opened wider), 42 º and less for the violet, 40. For these angles remain visible to our eyes the drops that must be sent in a circle. That circle is the base of a cone with vertex in our eyes and axis parallel to the rays of the sun affect the drops.
