Editing Openai/6897769e-4ee4-800f-aba5-69cca34f701c (section)

=== Goal: compute the primary-rainbow deflection (deviation) angle D(λ)D(\lambda)D(λ) for a spherical water droplet using geometric optics and the wavelength-dependent refractive index n(λ)n(\lambda)n(λ). For each wavelength we find the angle of minimum deviation Dmin⁡(λ)D_{\min}(\lambda)Dmin(λ). For ordinary water Dmin⁡(λ)D_{\min}(\lambda)Dmin(λ) ≈ 138° (primary rainbow), corresponding to the ~42° apparent rainbow angle measured from the anti-solar point. ===

Then weight Dmin⁡(λ)D_{\min}(\lambda)Dmin(λ) by the irradiance spectrum I(λ)I(\lambda)I(λ) to obtain a spectrum-weighted mean deviation angle:

Dˉ=∫Dmin⁡(λ) I(λ) dλ∫I(λ) dλ.\bar{D} = \frac{\int D_{\min}(\lambda)\,I(\lambda)\,d\lambda}{\int I(\lambda)\,d\lambda}.Dˉ=∫I(λ)dλ∫Dmin(λ)I(λ)dλ.
Compare Dˉ\bar{D}Dˉ with the “golden-angle region” 137.5078° and with 1/α ≈ 137.035999…. Next insert narrow emission lines (e.g. Hα, Hβ, Na D) into the spectrum and recompute Dˉ\bar{D}Dˉ to see how lines pull the mean.

We will use high-precision refractive index n(λ)n(\lambda)n(λ) (Hale & Querry or refractiveindex.info data) because the rainbow angle is sensitive to dispersion.