Rayleigh scattering · why the sky is blue

Why is the daytime sky blue and the setting sun red — and can that colour be derived from scratch, without ever assuming the answer?

▶ Run the simulationSee the measured result

Units: dimensionless power-law exponent (⟨P⟩ ∝ λ^−4 ∝ ω^+4)

How the lab tests it

Model an electron bound in an air molecule (natural frequency ω₀ in the deep UV) as a driven damped oscillator ẍ+γẋ+ω₀²x=(e/mₑ)E₀cos(ωt); its steady amplitude x₀(ω)=(eE₀/mₑ)/√((ω₀²−ω²)²+(γω)²) contains NO exponent. The wiggling charge radiates by the Larmor formula ⟨P⟩=e²ω⁴x₀²/(12πε₀c³). Generate the scattered-power curve ⟨P⟩(ω), measure its local log–log slope by finite difference, and extrapolate to ω→0 to read off the exponent p in ⟨P⟩∝λ⁻ᵖ — a recovery in which the number 4 never appears. Repeat across binding stiffness ω₀, damping γ and intensity E₀ (universality), and unbind the electron (ω₀=0, Thomson) as a falsification control. An RK4 lock-in of the actual equation of motion confirms the generator's amplitude. ?world=rayleigh.

What it checks

Rayleigh's law ⟨P⟩∝ω⁴∝λ⁻⁴ — the exponent p=4.000 recovered (to ~1e-4) as the ω→0 limit of the scattered-power slope, and shown to be UNIVERSAL: independent of the binding stiffness ω₀, the damping γ and the intensity E₀, exactly as the law demands. Blue (450 nm) therefore scatters (700/450)⁴≈5.9× more than red — the sky's blue, and the reddened setting sun whose long air path scatters the blue away. The result is decisive because binding is what makes it happen: unbinding the electron (ω₀=0, Thomson scattering) collapses the slope to 0 — a free electron scatters every colour equally (grey), which is why a free-electron plasma does not blue the sky. The same machinery hands back a second textbook constant, the wavelength-independent Thomson cross-section σ_T=(8π/3)r_e²=6.6525e-29 m² (and r_e=2.818e-15 m), to ~1e-11. Distinct from ?world=compton (the inelastic photon–electron energy shift): Rayleigh is elastic dipole re-radiation whose wavelength SCALING makes colour. The scattering branch of the optics arc, after Snell, Brewster and the rainbow.

Rayleigh scattering, λ⁻⁴ exponent & optical-depth calculator

Why the sky is blue, worked out rather than asserted. The exponent is never typed on this page: it is MEASURED, as the ω→0 limit of the log–log slope of the power a driven bound electron radiates — and the formula that generates that power, x₀ = (eE₀/mₑ)/√((ω₀²−ω²)²+(γω)²), has no exponent anywhere in it. What comes back is 3.999886296, and the 1.1e-4 it misses 4 by is not error but the curvature of a ladder that never quite reaches zero. The inverse direction is the one worth keeping: fed nothing but an observed intensity ratio and two wavelengths, ln R / ln(λ_red/λ_blue) hands back the same exponent with no oscillator, no fit and no physical constant in it at all. The cross-section is ASSEMBLED the same way — r_e = e²/(4πε₀mₑc²) rather than 2.818e-15 typed in, so moving any one of those four constants moves everything downstream — and it is where this page gets its cleanest number: σ(450)/σ(700) is a ratio of two polynomials in ω, no logarithm, no power, bit-identical in every browser ever written, while the exponent it encodes is not. That gap is stated rather than hidden: p is printed to ten significant figures against an engine floor of ±1.951e-11 derived on the page from the ulp of ln⟨P⟩, because a world whose answer is a difference of logarithms has no business printing sixteen. The binding dial is the falsification, and it is one number rather than a second formula: b multiplies ω₀ and γ together, b = 1 is the shipped simulation reproduced line for line, and b = 0 is a free electron, whose slope is exactly zero — grey, colour-blind, no sky at all. Binding is what makes the sky blue. The optical-depth direction turns the exponent into the sunset: at one air mass the surviving beam reddens by 1.20×, at the ≈38 of a sun on the horizon by 936×, and nothing about the air changed between them. Two things this page will not do: fix the ABSOLUTE size of the cross-section, which one oscillator cannot (it fixes the scaling, and real air is a sum of them), and derive τ_ref, which is an input measured from a real atmosphere.