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Sun
The antisolar point is as far below the horizon as the sun is above it.
Raindrop
Real raindrops are 0.1 to 3 mm; fog droplets about 0.02 mm.
Ray through the drop
Pick one ray; the other colours follow the same geometry with their own refractive index.
Rain shower
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The science
One raindrop
Sunlight refracts into a spherical drop (Snell: sin i = n sin r), reflects once at the back (k = 1, primary bow) or twice (k = 2, secondary bow) and refracts out. The total turning angle is D = 2(i - r) + k(180 deg - 2r). Rays with impact parameter s = sin i pile up where D is stationary (Descartes' ray): cos²i = (n² - 1) / (k(k + 2)). For n = 1.3336 this gives Dmin = 138.0 deg, so the primary bow sits 42.0 deg from the antisolar point; the secondary has Dmin = 231.0 deg, 51.0 deg away on the other side.
Dispersion and colour order
The refractive index of water falls from about 1.343 (violet) to 1.331 (red): the sim uses the Cauchy fit n = 1.3251 + 2946 / λ² (nm). Red therefore leaves at the larger angle (42.4 deg) and violet at the smaller (40.6 deg): red on the outside of the primary. The secondary is reversed, and the dark Alexander's band lies between the bows because no single-reflection or double-reflection rays reach those angles.
Brightness: Airy theory
Near the stationary ray the phase is cubic in the impact parameter, giving I(ε) ∝ E · Ai²(-(D - Dmin) K) with K = (2πa/λ)2/3 (D''/2)-1/3 and Fresnel factor E = (1 - R)²Rk. Big drops give narrow, bright bows with supernumerary fringes; tiny drops (below 0.1 mm) smear the colours, and 0.02 mm fog droplets give a nearly white fogbow. The disc of the sun (0.53 deg) is convolved in. Because Rs > Rp near Brewster's angle the primary bow is about 96 % polarised along the bow: the polarising-filter slider shows it vanishing at 90 deg.
Geometry in the sky
A bow is a circle of radius 42 deg (51 deg) centred on the antisolar point, which lies at the same angle below the horizon as the sun is above it. Its highest point is therefore 42 deg minus the sun elevation; with the sun higher than 42 deg no primary bow is visible from the ground. The visible fraction of the circle follows from cos φ = tan(sun elevation) / tan(42 deg).
Honest simplifications: spherical drops all of one size (real rain flattens, which blurs the bow and lowers the red); the Airy cubic approximation near the caustic only, no full Mie solution (so no third-order effects, no Alexander-band diffuse light beyond a stylised haze); the sky, the rain curtain and the brightness scale are artistic; the reflected-light bow, twinned bows and the tertiary bow are not shown.
What to try
- Drag the sun down: the bow climbs until it is a semicircle at 0 deg elevation; above 42 deg it disappears.
- Shrink the drop from 1 mm to 0.02 mm and watch the bow broaden and fade to a white fogbow.
- Move the impact-parameter slider: all rays crowd near s = 0.86 where D is smallest. Press "Sweep the ray".
- Switch on the polarising filter and turn it to 90 deg.