How Rainbows Form

Equivalent forms

\cos^2\theta_i = \frac{n^2 - 1}{3}

\theta_{rainbow} \approx 42^\circ

A single internal reflection plus dispersion turns every raindrop into a tiny prism, and a billion of them into a 42-degree arc.

Symbol	Name	SI unit	Dimension	Range
$theta_i$	Incidence angle Angle of the sun ray hitting the drop	deg	1	0 – 90
$n$	Water index Refractive index of the droplet	dimensionless	1	1.3 – 1.36
$theta$	Deflection angleoutput Total scattering angle of the exiting ray	deg	1	0 – 90

Unit systems

SI:: angles in radians (shown in degrees)
CGS:: angles dimensionless
Imperial:: degrees

Where it holds

Geometric-optics single-reflection model. The supernumerary fringes inside the primary bow require wave (Airy) theory; very small drops (fog) wash the bow into a white fogbow.

Discovery

Rene Descartes · 1637

Descartes traced thousands of rays through a model drop and found the concentration of deviated rays near 42 degrees. Newton then added dispersion, showing why the band is colored, completing the geometric theory of the rainbow.

Try this

Why is a rainbow always a 42-degree arc with red on the outside?

Sunlight refracts into a raindrop, reflects off the back, and refracts out — bunching up at a maximum deflection. That caustic sits near 42 degrees, and dispersion splits the colors across it.

Research status: stable

Common misconceptions

A rainbow is not at a fixed place — it is an angle, so each observer sees their own bow centered on their antisolar point. The sky is brighter inside the bow (Alexander's dark band lies between the primary and secondary).

Derivation

Total deviation for one internal reflection is

D = 2(\theta _i - \theta _r) + (pi - 2 \theta _r)

, with

\sin \theta _i = n \sin \theta _r

.

Setting

dD/d \theta _i = 0

gives

\cos ^2 \theta _i = (n^2 - 1)/3

.

For

n = 1.333

this yields

\theta _i \sim 59.6\,\mathrm{deg}

and a minimum deviation -> rainbow angle

\sim 42\,\mathrm{deg}

.

Limiting cases

Two internal reflections⟶ Secondary

bow at \sim 51

degrees with reversed colors (red inside)

n red vs n blue⟶

Red bow at \sim 42.4\,\mathrm{deg}

, violet

at \sim 40.6\,\mathrm{deg}

—

the \sim 2

degree color spread

Snell's Law→Optical Dispersion→Why the Sky Is Blue (Rayleigh)→Critical Angle (Total Internal Reflection)→