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

=== 1. I used the standard primary-rainbow geometry: the deviation as a function of entry angle iii and refraction rrr (the usual textbook form, the extremum gives the rainbow direction). — (the extremal condition used: cos⁡icos⁡r=n2\dfrac{\cos i}{\cos r} = \dfrac{n}{2}cosrcosi=2n, snell: sin⁡r=sin⁡i/n\sin r = \sin i / nsinr=sini/n, and the deviation D(i)=π+2i−4rD(i) = \pi + 2i - 4rD(i)=π+2i−4r; the observed rainbow angle is θ=π−Dmin⁡\theta=\pi-D_{\min}θ=π−Dmin). ===
# I built a simple (approximate) dispersion model for liquid water by fitting a three-term Cauchy/Sellmeier-like function to typical literature points (the values taken are typical of Hale & Querry / refractiveindex.info data in visible band).
# I computed the primary-rainbow viewing angle θ(λ)\theta(\lambda)θ(λ) across the visible band (≈380–780 nm). The rainbow angle depends on λ\lambdaλ because n(λ)n(\lambda)n(λ) disperses.
# I tested several spectral weightings S(λ)S(\lambda)S(λ) and computed the weighted mean angle

θˉ  =  ∫S(λ) θ(λ) dλ∫S(λ) dλ\bar\theta \;=\; \frac{\int S(\lambda)\,\theta(\lambda)\,d\lambda}{\int S(\lambda)\,d\lambda}θˉ=∫S(λ)dλ∫S(λ)θ(λ)dλ
— Baseline: Planck (blackbody) at T ⁣≈ ⁣5800T\!\approx\!5800T≈5800 K (solar-like).
— Additions: narrow Gaussian emission lines (examples chosen: H-alpha 656.28 nm, H-beta 486.13 nm, Na D doublet ≈589 nm, a few others).
— Plasma example: a strongly H-alpha–dominated line set (a plausible plasma spectrum is often dominated by H-α).
# I measured how θˉ\bar\thetaθˉ shifts between cases.

(If you want the raw code or the plots I produced I can attach them or rerun with any precise refractive-index table you prefer.)