EXERCISE 5: CRUDE ESTIMATE OF THE IRRADIANCE VERSUS DEFLECTION ANGLE FOR A SINGLE WAVELENGTH: PRIMARY RAINBOW

Assume that the each outgoing ray produces an irradiance that is equal to $$ I(θ)=I_{0} \exp{−(θ−δ)^{2}} $$ where here the angles are assumed to be in degrees and the implied width of this distribution is 1°. (For simplicity, we neglect the loss of intensity during refraction and internal reflection. This is a huge oversimplification, but it allows us to focus on adding up contributions from each ray.)
Sum up the contributions from rays uniformly distributed over the scaled impact parameter $$ \tilde{b}=\frac{b}{R} $$to compute the overall irradiance as a function of deflection angle for light of wavelength 400 nm.
Plot the resulting irradiance distribution versus deflection angle for a single wavelength.
How does this plot help explain why the rainbow is bright?