Rayleigh Scattering
Why the sky is blue and sunsets are red — the physics of how light bounces off tiny atmospheric molecules, and why shorter wavelengths lose that lottery far more often.
- Estimated time
- ~10 min
- Difficulty
- intermediate
- Sources
- 4 sources
The phenomenon: a sky that changes color
Stand outside at noon, then again at sunset. The same sun, the same atmosphere — yet the sky goes from saturated blue to burning orange and red. Something about the path light takes through the air is re-sorting its colors.
Here is the first key observation: the further sunlight travels through the atmosphere, the redder it gets. Overhead at noon, sunlight cuts a short vertical path. At the horizon near sunset, that path stretches to roughly 38 times longer. [Wikipedia: Rayleigh scattering]
The mechanism: oscillating dipoles and the 1/λ⁴ law
When light hits a molecule much smaller than its wavelength, the oscillating electric field drives the electrons into a tiny oscillating dipole. That dipole re-radiates — in all directions, not just forward. This is scattering.
The crucial result: shorter wavelengths (blue, violet) are scattered far more intensely than longer ones (red). How much more?
Plug in blue (λ ≈ 450 nm) and red (λ ≈ 700 nm):
Blue light scatters roughly 9–10 times more than red light. [Classical Electrodynamics, Jackson (3rd ed.)]
Show the classical derivation
A charge q in a harmonic oscillator driven by an electric field radiates with power:
The acceleration a is proportional to ω² (since a = –ω²x for a driven oscillator). Radiated power therefore scales as ω⁴. Because ω = 2πc/λ, this is identical to 1/λ⁴. The angular distribution (1 + cos²θ)/2 comes from the dipole radiation pattern. The full result: [Classical Electrodynamics, Jackson (3rd ed.)]
Where the model breaks: limits and regimes
Rayleigh’s formula is elegant but applies only under specific conditions. Understanding its boundaries is as important as the formula itself.
| Regime | Particle size d vs. λ | Intensity law | Example | |
|---|---|---|---|---|
| Rayleigh | d ≪ λ (d < λ/10) | I ∝ 1/λ⁴ (strongly wavelength-dependent) | N₂, O₂ molecules in air | |
| Mie | d ≈ λ | Complex; weakly wavelength-dependent | Water droplets, aerosols, fog | |
| Geometric | d ≫ λ | Ray optics; wavelength-independent | Raindrops, dust particles |
Why clouds are white: Water droplets in clouds are ~10–100 µm across — comparable to or larger than visible wavelengths. Mie scattering dominates, and it scatters all wavelengths nearly equally, producing the neutral white appearance. [Absorption and Scattering of Light by Small Particles]
Why pollution makes sunsets more vivid: Aerosols and fine particulates are closer to the Mie regime. They scatter more red and orange back toward the viewer compared to clean air, enhancing those hues — a feedback where the model’s failure mode creates a visually striking effect.
A surprising consequence: why Mars has a pink sky
The 1/λ⁴ rule predicts blue skies on any planet with an atmosphere of small molecules. Mars seems to confirm this — but Mars’s sky is actually a dusty salmon-pink, not blue.
Mars has a thin CO₂ atmosphere (also small molecules), but its surface is covered in fine iron-oxide dust, and Martian winds constantly loft that dust into the atmosphere. [Wikipedia: Rayleigh scattering]
The dust particles (~1 µm diameter) are large enough to scatter red wavelengths efficiently — right in the Mie regime for visible light. The result is a reddish haze that overpowers the would-be Rayleigh blue. Earth’s atmosphere is much cleaner; on Mars, the geology wins.
This consequence crystallizes what Rayleigh scattering actually requires: not just small molecules, but a clean atmosphere where those molecules are the dominant scatterers. Change the particle size and you change the sky’s color — the formula is the same, only the regime shifts.