Generalized Airy theory and its region of quantitative validity

Airy theory has long proved to be a remarkably simple analytical model that describes the various features of the atmospheric rainbow. But the stringent assumptions upon which its derivation is based, prevent it from being quantitatively accurate in practical situations. We derive an analytical generalization of Airy theory for both the transverse electric and magnetic polarizations and for an arbitrary number of internal reflections. This generalized analytical model contains both the Airy integral and its first derivative, multiplied by constants of proportionality that are independent of the scattering angle. We find that, for the primary rainbow, it provides a quantitatively accurate approximation to the exact Lorenz-Mie-Debye theory of the rainbow for a much wider range of sizes of spherical water drops than does the original version of Airy theory, but still has stringent limitations for the second-order rainbow and beyond.