Rainbow · Descartes' 42° caustic

Why does the rainbow always sit at the same angle from the sun (~42°), whatever the size of the raindrops — and why is it split into colours?

How the lab tests it

Vector-trace a real ray (refract in · reflect once off the back wall · refract out) through a spherical water drop for a fan of impact parameters b = R·sin i, and record the total deviation D(i) = π + 2i − 4r (sin i = n sin r). Find the minimum of D(i) both by a fine numeric scan and by the closed form cos i = √((n²−1)/3), and repeat for seven wavelengths using Cauchy n(λ) = A + B/λ².

What it checks

DESCARTES' RAINBOW — D(i) has a MINIMUM, so rays pile up at the smallest deviation: a caustic. That bright arc sits at the rainbow angle φ = 180° − D_min = 42.0° for water (numeric and closed-form i_min agree), independent of drop size. Dispersion gives each colour its own caustic — red 42.4° (outer), violet 40.6° (inner), Δ≈1.8° — and a second reflection makes the secondary bow at ~51° (colours reversed) with Alexander's dark band between