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

=== 1. I modelled the primary rainbow deviation D for a single raindrop (geometrical optics, one internal reflection). - For a given refractive index n(λ) and an incidence angle i, the primary-deviation formula used was the standard geometric optics one: D(i,λ)=180∘+2i−4r,r=arcsin⁡(sin⁡i/n(λ))D(i,\lambda) = 180^\circ + 2i - 4 r,\qquad r=\arcsin\big(\sin i / n(\lambda)\big)D(i,λ)=180∘+2i−4r,r=arcsin(sini/n(λ)) - For each wavelength λ I found the stationary (minimum) deviation D_min(λ) by sampling the incidence angle i and locating the minimum. That gives the wavelength-dependent rainbow angle θ(λ). ===
# I used a simple analytic dispersion model for water (rough, illustrative): n(λ)≈1.3330+0.0040λμm2n(\lambda)\approx 1.3330 + \frac{0.0040}{\lambda_{\mu m}^2}n(λ)≈1.3330+λμm20.0040 (λ in µm). This is approximate — good enough to show the qualitative trend of θ(λ) with λ.
# I computed a set of spectra S(λ) and evaluated spectrum-weighted average rainbow angles θˉ=∫θ(λ) S(λ) dλ∫S(λ) dλ.\bar\theta = \frac{\int \theta(\lambda)\,S(\lambda)\,d\lambda}{\int S(\lambda)\,d\lambda}.θˉ=∫S(λ)dλ∫θ(λ)S(λ)dλ. Spectra used: - Planck (blackbody) at 5800 K (Sun-like) and 3000 K. - Narrow emission lines (Gaussian kernels) for Hα (656.28 nm), Hβ (486.13 nm), Na D doublet (~589 nm). - A small sample plasma set (O III 500.7 nm and He II 468.6 nm). - Mixtures: Planck (5800 K) + Hα with increasing line amplitude.
# Plots were produced (θ(λ) vs λ, normalized spectra) and the weighted mean angles printed.