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

=== 1. I used a Sellmeier-style fit for liquid water (coefficients of the standard fits used on refractiveindex.info), implemented as a function n(λ [μm])n(\lambda\,[\mu\text{m}])n(λ[μm]). ===
# For each wavelength λ\lambdaλ (300–1100 nm grid) I searched numerically for the incidence angle iii that minimizes the geometric deviation D(i)=2i−4r+π,sin⁡r=sin⁡in(λ).D(i) = 2i - 4r + \pi,\qquad\sin r = \frac{\sin i}{n(\lambda)}.D(i)=2i−4r+π,sinr=n(λ)sini. That gives the classical geometric primary-rainbow minimum Dmin⁡(λ)D_{\min}(\lambda)Dmin(λ).
# I computed Planck spectral radiance Bλ(λ,T)B_\lambda(\lambda,T)Bλ(λ,T) (standard Planck formula), used it as a weight, and computed Dmean(T)=∫Bλ(λ,T) Dmin⁡(λ) dλ∫Bλ(λ,T) dλ.D_{\rm mean}(T)=\frac{\int B_\lambda(\lambda,T)\,D_{\min}(\lambda)\,d\lambda}{\int B_\lambda(\lambda,T)\,d\lambda}.Dmean(T)=∫Bλ(λ,T)dλ∫Bλ(λ,T)Dmin(λ)dλ.
# I also built spectra formed by Planck continuum plus narrow Gaussian emission lines (Hα 656.28 nm, Hβ 486.13 nm, NaD ~589.3 nm, [O III] 500.7 nm) and recomputed the weighted mean.

Numeric highlights (degrees):
* Dmean(T=5800 K)≈138.0917∘D_{\rm mean}(T=5800\ \mathrm{K}) \approx 138.0917^\circDmean(T=5800 K)≈138.0917∘  (Δ vs golden ≈ +0.584°)
* Dmean(T=5000 K)≈137.875∘D_{\rm mean}(T=5000\ \mathrm{K}) \approx 137.875^\circDmean(T=5000 K)≈137.875∘
* Dmean(T=4000 K)≈137.579∘D_{\rm_mean}(T=4000\ \mathrm{K}) \approx 137.579^\circDmean(T=4000 K)≈137.579∘
* Dmean(T=3000 K)≈137.283∘D_{\rm_mean}(T=3000\ \mathrm{K}) \approx 137.283^\circDmean(T=3000 K)≈137.283∘

Root-searches:
* Temperature giving golden angle 137.507764∘137.507764^\circ137.507764∘: T≈3765 KT \approx 3765\ \mathrm{K}T≈3765 K.
* Temperature giving inverse fine-structure 137.035999...137.035999...137.035999...: T≈2068 KT \approx 2068\ \mathrm{K}T≈2068 K.

Adding realistic narrow lines (scaled to local Planck continuum) produced only tiny shifts from the 5800K number; to reach the golden angle from 5800 K by lines alone would require line strengths far larger than typical atomic lines (i.e. implausibly intense narrow emission compared to continuum).

I saved the raw computed table (λ(nm),n(λ),Dmin⁡(λ)) (\lambda\text{(nm)}, n(\lambda), D_{\min}(\lambda))(λ(nm),n(λ),Dmin(λ)) to /mnt/data/Dmin_vs_lambda_nm.txt (so you can download it and plot locally). I also displayed a short table of selected wavelengths inside the run.