Rainbow Physics: Why Every Rainbow Forms at 42°

A rainbow is a cone, not an object

You always see a rainbow as an arc centred on the antisolar point — the point exactly opposite the Sun from your eye. A rainbow is really a cone of light: every raindrop sitting at 42° from that antisolar point simultaneously sends red light toward you, and the arc you see is where that cone intersects the sky in front of you. From an aeroplane, high enough that the ground doesn't get in the way, that cone becomes a full circle. Because the geometry is defined relative to your own eye, no two observers — even standing side by side — ever see exactly the same rainbow.

Descartes' ray theory and the minimum-deviation angle

In 1637 René Descartes traced a single ray through a spherical raindrop: refraction on entry (Snell's law, n·sin r = sin i), one internal reflection off the back of the drop, then refraction again on exit. The total deviation of the ray depends on its angle of incidence i, and that function has a minimum — the angle where neighbouring rays bunch together instead of spreading out, concentrating the light into a bright edge rather than a smooth gradient.

D(i) = π + 2i − 4·arcsin(sin(i)/n)      n ≈ 1.333 (water, yellow light)

Minimum deviation:  i_min = arccos(√((n²−1)/3)) ≈ 59.5°
                     D_min ≈ 137.5°  →  rainbow angle = 180° − D_min ≈ 42.5°

That pile-up at ~42° is the entire reason the primary bow appears at a fixed angle no matter the size of the raindrops involved — only the brightness and fringe pattern change with drop size, never the basic angle.

Dispersion: why red sits on the outside

Water is dispersive — its refractive index depends slightly on wavelength. Violet light (n ≈ 1.342) bends more than red (n ≈ 1.331), so each colour has its own minimum-deviation angle: about 40.5° for violet up to 42.5° for red. That shifts the exit angle of each wavelength by a fraction of a degree, spreading white sunlight into a roughly 2°-wide spectral band. Because red exits at the largest angle from the antisolar point, it lands on the outer edge of the arc, with violet innermost.

The secondary bow and Alexander's dark band

Some rays undergo a second internal reflection before exiting, producing a fainter secondary bow at roughly 51° with the colour order reversed — red on the inside this time — and about 40% dimmer, since every extra reflection loses light to transmission. Between the primary (42°) and secondary (51°) bows lies a visibly darker strip of sky known as Alexander's dark band: the primary sends light only inside 42°, the secondary only outside 51°, so the region between receives no rainbow-deviated light at all and looks darker by contrast with the sky around it.

Airy's wave theory and supernumerary arcs

Descartes' geometry predicts one sharp bow, but real rainbows — especially in small, uniform droplets like fog or drizzle — often show faint pink-and-green fringes just inside the primary arc, called supernumerary arcs. George Airy explained these in 1838 as wave interference: two slightly different ray paths through the droplet exit in nearly the same direction and interfere constructively or destructively depending on their path-length difference. Larger, more variable raindrops smear the fringes out, which is why supernumeraries are easiest to spot in fine, uniform mist rather than a heavy shower.

Frequently asked questions

Try it live

Everything above runs in your browser — open Rainbow & Optics and drag the sun position to watch the arc rise and fall, switch liquids to see the angle shift, and toggle the secondary bow and its dark band. Nothing is installed, nothing is uploaded.