### Bright zones

The following figure shows the 3 graphs of the function that associates the direction of the ray to each impact parameter, for \(C_{2}\), \(C_{3}\) and \(C_{4}\) in red.

The first, relative to \(C_{2}\), represents an increasing function, with no angle (the y-coordinate in the graph) with a much higher concentration of emerging ray directions. The greater slope, close to the two extremes of the function domain, corresponds to the greater rarefaction of rays in the corresponding directions. But in each of the other cases (\(C_{3}\) and \(C_{4}\)), there is a maximum and a minimum (absolute) of the respective function, at points interior to the domain, and this means that in an entire neighborhood of each of the respective points of impact, all emerging rays come out with directions very close to each other (the maximum and minimum respectively), and this causes an exceptionally bright (red) zone, precisely at the extremes of the arcs representing the directions (to \(C_{3}\) and \(C_{4}\)). Something similar happens for the other colors, with the values of the angles (maximum and minimum) being different for the different colors, as can be seen in the following figure.

This detail means that, when we apply what we have just mentioned to the specific situation that interests us, of a solar ray made up of a mixture of rays of different colors, we have areas of brightness (of different colors), which do not overlap^{5}. As is clear from the analysis of the previous figure, the angles of the "bright" directions for the other colors are between red and violet (between \(137.5^{\circ}\) and \(139,1 ^{\circ}\) to \(C_{3}\) and between \(127^{\circ}\) and \(129,9^{\circ}\) to \(C_{4}\) ). In the horizontal range between the red of \(C_{3}\) \((137.5^{\circ})\) and the red of \(C_{4}\) \((129,9^{\circ}\)) there is no ray (of class \(C_{3}\) or \(C_{4}\)) of any color emerging in that direction. As for the other directions, what generally happens is that, for each ray emerging of a given color, there is another ray emerging of another color, with the same direction, coming from another point of impact. The overall balance, for the rays emitted by the drop in these other directions, is, therefore, that of a white light without particularly bright areas.^{6}

The following interactive application allows you to verify this fact.

^{5}These are the ones that will be at the origin of the rainbow (\(C_{3}\)) and the secondary rainbow (\(C_{4}\)), as we will see.

^{6}For reasons of simplicity, we do not take into account other light phenomena linked to the rainbow, whose explanation requires considerations (of a wave nature) more sophisticated than the refraction phenomena we have presented here. The interested reader can consult [2], which is also a good reference for this Atractor text.