The 42° Bow

Plate 24 speaks for itself: “Every rainbow is a cone of 42 degrees around your shadow.”

The Physics

Sunlight refracts into a raindrop, reflects once inside and refracts out. The deviation has a minimum, and light piles up there, so the primary bow sits about 42° from the antisolar point; two internal reflections give the secondary bow near 51°.

D = 180° + 2θ_i − 4θ_r

• D: deviation for one internal reflection, °
• θ_i: angle of incidence, °
• θ_r: angle of refraction, sin θ_i = n sin θ_r, °
• n: refractive index of water, dimensionless

n = 1.333: primary 42°, secondary 51°. Descartes traces the rays, 1637.

Honest caveat: The exact angle shifts slightly with colour, since n depends on wavelength.

Sources

• R. Descartes, Les Météores (1637), Discourse 8.

On the poster

Plate 024, “The 42° Bow”: the pixel-art plate with its slogan “Every rainbow is a cone of 42 degrees around your shadow.”, the key equation and the facts panel shown above. Pixelated Physics. Physics. Personified.