Why is it brighter under a rainbow? part 1

 

You can see from the picture above that the sky is clearly much brighter underneath a rainbow. I started trying to find out why about 5 years ago but found the optical geometry hard and then other stuff got in the way... So I found my way back to my source and have been forcing myself to work through the examples so that I understand what is going on. I started with what is happening inside a water droplet

The black circle is my water droplet. The black radial lines are there because they are normal lines used in the geometrical optics. A ray of light hits the drop at X. Water is optically more dense than air so on entering the drop, the light slows down and bends towards the normal line. Angle alpha > angle beta. There is a reflection at the back of the drop at Y so angle of incidence = angle of reflection. Then exiting the drop on the left at Z is symmetrical to what happened at X.