Rayleigh Scattering — Why the Sky is Blue

A molecule as a tiny radio antenna

Sunlight is an oscillating electromagnetic field. When it passes a particle much smaller than its own wavelength — a nitrogen or oxygen molecule, roughly 0.3 nm across, against visible light's 400-700 nm — the field drives the particle's electrons into oscillation, and that oscillating charge re-radiates energy in essentially every direction. That re-radiation is scattering: the light keeps its original frequency (it is elastic) but leaves in a new direction. This is the regime Lord Rayleigh analysed in 1871, and it is only one of three regimes light-matter interaction can fall into, sorted purely by how the particle size compares to the wavelength.

particle size << λ   → Rayleigh scattering   (air molecules)
particle size ≈ λ    → Mie scattering       (dust, water droplets, aerosols)
particle size >> λ   → geometric optics      (reflection, refraction, raindrops)

The λ⁴ law

Rayleigh's result for the intensity a single small particle scatters, at angle θ from the incoming beam and distance r from the detector, is dominated by one term: the scattered intensity is inversely proportional to the fourth power of the wavelength.

I_s / I₀  =  8π⁴Nα² / (λ⁴r²) · (1 + cos²θ)/2

N = number density of scatterers (molecules/m³)   α = molecular polarisability
λ = wavelength                                     θ = scattering angle

blue (450 nm) vs red (700 nm):  I_blue/I_red = (700/450)⁴ ≈ 5.9×

A molecule scatters blue light roughly six times more strongly than red light, and violet (400 nm) close to ten times more strongly. The (1 + cos²θ)/2 factor means the scattering is not the same in every direction — it is symmetric front-to-back but weakest at 90° to the beam, where the scattered light is also strongly polarised. That is the physical reason polarising sunglasses and camera filters cut sky glare so effectively: they are rejecting the specific polarisation state that 90°-scattered blue light carries.

Why blue, not violet

Sunlight enters the atmosphere carrying the full visible spectrum, and every wavelength scatters off every N₂ and O₂ molecule it meets, with violet scattering hardest and red scattering least. If the eye simply reported the wavelength that scatters most, the sky would look violet. It looks blue instead for three compounding reasons: the Sun's own spectrum puts out less power in violet than in blue to begin with; the human eye's cone cells are markedly less sensitive to violet wavelengths; and ozone high in the atmosphere absorbs a slice of the violet and ultraviolet light before it ever reaches the scattering layer we see. Net effect: blue wins the perceptual competition even though it is not the physically strongest scatterer.

The sky is also brightest and most saturated at 90° from the Sun, for the same (1 + cos²θ) reason above. Close to the Sun's disc, forward scattering (small θ) mixes all wavelengths together nearly equally, which is why the sky right around the Sun looks pale and whitish rather than deep blue.

Sunsets: the same law over a much longer path

When the Sun sits near the horizon, its light has to cross far more atmosphere to reach an observer than it does at noon — roughly 38 times more air, by effective path length. The fraction of light surviving that path follows Beer's law, I = I₀e^(−τ), where the optical depth τ = σNL grows with the scattering cross-section σ, the molecular density N, and the path length L.

              zenith (L ≈ 8 km)        sunset (L ≈ 300 km)
blue light    τ ≈ 0.15 → 86% left      τ ≈ 5.6 → 0.4% left
red light     τ ≈ 0.02 → 98% left      τ ≈ 0.75 → 47% left

At zenith the difference between blue and red survival is modest — the sky above you is blue precisely because a small but real fraction is being scattered out of the direct beam and into your eye from every direction. At sunset, almost all the blue in the direct beam has been scattered away long before it reaches the observer, while roughly half the red light still makes it straight through — so the direct disc of the Sun and the clouds it lights up turn deep orange and red. The blue you still see away from the Sun at sunset is light that was scattered sideways off molecules and high-altitude aerosols along that long path, arriving from a different direction than the direct beam. After major volcanic eruptions such as Krakatoa in 1883, extra sulfate aerosol in the stratosphere adds strong Mie scattering on top of the usual Rayleigh scattering, and produces the vivid purple and crimson sunsets that were reported worldwide for two to three years afterward.

When particles are bigger: Mie scattering and white clouds

The λ⁴ dependence only holds while the scatterer is much smaller than the wavelength. Cloud droplets and fog droplets are 5-50 micrometres across — tens to hundreds of times larger than a visible-light wavelength — which puts them squarely in the Mie regime, where the strong wavelength dependence washes out and all colours scatter at close to the same strength. Equal scattering across the spectrum reads as white to the eye, which is why clouds, fog and milk (fat globules scattering light the same way) all look white rather than blue. The same Mie physics, viewed from colloidal particles in solution, produces the Tyndall effect — the blue glow you get shining a light sideways through a suspension, and part of the reason blue eyes appear blue: light scattering off the melanin-poor iris stroma rather than any blue pigment being present.

A useful diagnostic, on Earth and off it

Because Rayleigh scattering depends so predictably on wavelength, particle size, and density, it doubles as a measurement tool. Atmospheric lidar fires laser pulses skyward and reads the timing and strength of the backscattered Rayleigh and Mie signal to profile air density, aerosol load and cloud height for weather and air-quality monitoring. Optical fibre communication runs into the same 1/λ⁴ law as a loss mechanism, which is exactly why fibre-optic networks operate at 1310 nm and 1550 nm infrared rather than visible wavelengths — scattering loss there is far lower. And beyond Earth, sky colour itself becomes a diagnostic: Mars's iron-oxide dust sits in the Mie regime and gives it a butterscotch daytime sky with a bluish glow near a setting Sun — the reverse of Earth's pattern — while Uranus and Neptune owe their blue-green and vivid blue skies to methane absorbing red light out of whatever gets scattered back.

Frequently asked questions

Try it live

Everything above runs in your browser — open Rayleigh Scattering — Why the Sky is Blue and drag the Sun's elevation from noon down to the horizon to watch the sky colour shift live, ray by ray. Nothing is installed, nothing is uploaded, the whole model lives in one tab.