Why the Sky Is Blue (Rayleigh)

Equivalent forms

\sigma = \frac{8\pi}{3}\left(\frac{2\pi}{\lambda}\right)^4 \alpha^2

\frac{I_{blue}}{I_{red}} = \left(\frac{\lambda_{red}}{\lambda_{blue}}\right)^4

A single exponent of four explains the color of the sky, the red of sunsets, and the polarization of skylight — all at once.

Symbol	Name	SI unit	Dimension	Range
$lambda$	Wavelength Wavelength of the scattered light	nm	L	380 – 750
$lambda_ref$	Reference wavelength Comparison wavelength (e.g. red)	nm	L	380 – 750
$ratio$	Scattering ratiooutput Relative scattered intensity (lambda_ref/lambda)^4	dimensionless	1	0 – 20

Unit systems

SI:: lambda in m; intensity ratio dimensionless
CGS:: lambda in cm
Imperial:: ratio dimensionless

Where it holds

Valid when scatterers are much smaller than the wavelength (gas molecules,

\sim 0.3\,\mathrm{nm}

<< 500 nm). Fails for aerosols/droplets comparable to lambda, where Mie scattering takes over (haze, white clouds).

Discovery

Lord Rayleigh · 1871

Rayleigh derived the 1/lambda^4 dependence from dimensional and electromagnetic arguments, explaining both the blue of the sky and the polarization of skylight. He later used it to estimate Avogadro's number from the sky's brightness.

Try this

Why is the daytime sky blue but the setting sun red?

Air molecules scatter short wavelengths far more strongly than long ones — Rayleigh's 1/lambda^4 law. Blue light is flung across the whole sky; at sunset the long path strips it away, leaving red.

Research status: stable

Common misconceptions

The sky is not blue because it reflects the ocean, and not violet despite violet scattering even more — the Sun emits less violet and our eyes are less sensitive to it, so the mix reads blue.

Derivation

An induced dipole

p = \alpha E re

-radiates power proportional to

(d^2\,\mathrm{p/dt}^2)^2 \sim \omega ^4

.

Since

\omega = 2\,\mathrm{pi} c/\lambda

, scattered intensity

\sim 1/\lambda ^4

.

Integrating the dipole pattern gives the Rayleigh cross-section

\sigma = (8\,\mathrm{pi}/3)(2\,\mathrm{pi}/\lambda )^4 \alpha ^2

.

Limiting cases

Particle size >> lambda (clouds)⟶ Mie scattering — wavelength-independent, so clouds look white

Long slant path (sunset)⟶ Blue removed by repeated scattering; transmitted light reddens

How Rainbows Form→Intensity of an Electromagnetic Wave→Malus's Law→Photon Energy (Planck-Einstein Relation)→