Rayleigh Scattering: Why the Sky Is Blue & Sunsets Are Red

1. What Is Scattering?

When electromagnetic radiation encounters a particle smaller than its wavelength, the oscillating electric field of the light drives the electrons in the particle into oscillation — creating an oscillating electric dipole. This oscillating dipole re-radiates energy in all directions: it scatters the light.

The scattered wave has the same frequency as the incident light (elastic scattering) but travels in a new direction — different from the original beam. The intensity and angular distribution of scattered light depend critically on the ratio of particle size to wavelength.

Particle much smaller than λ: Rayleigh scattering (molecules, very fine particles)
Particle comparable to λ: Mie scattering (dust, water droplets, aerosols)
Particle much larger than λ: Geometric optics (reflection, refraction)

2. The Rayleigh Formula

Lord Rayleigh derived the scattering formula in 1871. The intensity I scattered by a single small particle in a direction at angle θ to the incident beam is:

I_s \propto (1 + cos²θ) / λ⁴ The full expression for scattered intensity per unit volume: I_s / I₀ = 8π⁴N α² / (λ⁴ r²) \cdot (1 + cos²θ)/2 N = number density of scatterers (molecules per m³) α = polarisability of the scatterer (m³) λ = wavelength of light (m) r = distance from scatterer to detector θ = scattering angle The crucial factor: λ⁴ in the denominator Blue (450 nm) vs Red (700 nm): I_blue / I_red = (700/450)⁴ = (1.56)⁴ \approx 5.9\times Blue light scatters ~6\times more intensely than red light from a given molecule.

Polarisation: The (1 + cos²θ) factor means scattering is not uniform. At θ = 90° (light scattered perpendicular to the sun), the scattered light is partially polarised. This is why polarised sunglasses reduce sky glare, and why photographers use polarising filters to deepen sky contrast.

3. Why the Sky Is Blue

Sunlight contains all visible wavelengths (roughly 400–700 nm). As it enters the atmosphere, it collides with N₂ and O₂ molecules (diameter ~0.3 nm, far smaller than visible wavelengths of 400–700 nm — perfect Rayleigh regime). Each molecule scatters some light in all directions.

Because the scattering cross-section ∝ 1/λ⁴:

Violet (400 nm) scatters the most — ~9.5× more than red
Blue (450 nm) scatters ~6× more than red
Yellow (580 nm) scatters ~2× more than red
Red (700 nm) scatters the least

Yet the sky appears blue rather than violet — because (1) the Sun emits less violet light than blue, (2) our eyes have lower sensitivity to violet, and (3) some UV/violet is absorbed in the upper atmosphere by ozone. Together, these effects shift the perceived scattered colour from violet to blue.

The sky is brightest and most blue at 90° from the Sun. Near the Sun, forward-scattered light (which contains all wavelengths near equally for small θ) dominates, making the sky near the sun appear whitish-yellow.

4. Sunsets & Red Light

When the Sun is near the horizon, sunlight must travel through a much longer path of atmosphere to reach you. The path length through the atmosphere at sunset is approximately 38× longer than at zenith (directly overhead).

Optical depth τ: τ = σ \cdot N \cdot L σ = Rayleigh scattering cross-section N = molecular number density L = path length through atmosphere Fraction of light remaining: I = I₀ \cdot e⁻τ At zenith (L \approx 8 km effective): τ_blue \approx 0.15, I_blue/I₀ \approx 86% (86% transmitted) τ_red \approx 0.02, I_red/I₀ \approx 98% At sunset (L \approx 300 km effective): τ_blue \approx 5.6, I_blue/I₀ \approx 0.4% (99.6% scattered away!) τ_red \approx 0.75, I_red/I₀ \approx 47% (still 47% transmitted) \to Direct sunlight has almost no blue left at sunset \to appears red/orange

The scattered blue light reaching your eyes comes from dust and aerosols at higher altitudes, which is why the sky away from the sun at sunset can be brilliant purple, pink, or orange — the combination of forward-scattered red/orange direct light and back-scattered blue/violet.

The best sunsets follow volcanic eruptions: volcanic aerosols (fine sulfate particles) in the stratosphere increase Mie scattering, producing deep crimson and purple hues. The 1883 Krakatoa eruption created vivid sunsets globally for 2–3 years.

5. Mie Scattering & White Clouds

When scatterers are comparable in size to the wavelength of light (Mie regime), scattering becomes less wavelength-dependent. Cloud droplets (radius 5–50 μm) and fog droplets are far larger than visible wavelengths — they scatter all wavelengths approximately equally.

Equal scattering of all wavelengths → white appearance. This is why:

Clouds are white (not blue)
Milk is white (fat globules ~0.1–10 μm scatter all wavelengths)
Fog appears white or grey
Glass ground fine is white (powder scatters all colours)

Mie scattering also explains the Tyndall effect: a bright blue appearance of colloidal particles in solution when viewed perpendicular to a white light beam — pure Rayleigh/Mie scattering from colloidal particles. Blue eyes appear blue because of Tyndall scattering from melanin-less iris stroma, not from blue pigment.

6. Sky Colours on Other Planets

The sky colour depends on atmospheric composition, which determines the dominant scattering mechanism:

Mars: Thin CO₂ atmosphere with fine iron oxide dust. The dust (Mie scattering) dominates over Rayleigh. Daytime sky: butterscotch/pinkish-orange. Near sunset: bluish-grey near the sun (forward-scattered blue light from dust).
Venus: Thick CO₂ and H₂SO₄ cloud layers. Sky is a uniform bright yellowish-white. No blue visible from the surface.
Titan (Saturn's moon): Dense N₂ atmosphere with organic aerosol haze (tholin particles). Sky is orange-brown, similar to a heavy smog. Rayleigh scattering from N₂ contributes a blue component, but the aerosol haze dominates.
Uranus/Neptune: Methane (CH₄) absorbs red wavelengths. The remaining scattered light is blue-green (Uranus) and vivid blue (Neptune's slightly different methane bands).

7. Technology Applications

Atmospheric lidar: Pulsed lasers sent into the atmosphere backscatter off molecules and particles. Measuring the returned Rayleigh + Mie signal as a function of time (distance) gives vertical profiles of density, aerosol concentration, and cloud height. Used for weather forecasting, air quality monitoring, and climate research.
Optical fibre communication: Rayleigh scattering in the fibre is the dominant loss mechanism at short wavelengths. Loss ∝ 1/λ⁴, which is why optical fibres operate at 1310 and 1550 nm infrared (where Rayleigh loss is minimised) rather than visible wavelengths.
Stellar atmosphere analysis: Rayleigh scattering in stellar coronas and thin planetary atmospheres is used to diagnose atmospheric composition during exoplanet transits (transmission spectroscopy with JWST).
Nephelometry: Measuring air quality by quantifying the amount of light scattered by particulate matter — used in PM2.5 and PM10 sensors in environmental monitoring networks.