Thomson cross section

σ_T = (8π/3)r_e² is the effective target area a free electron presents to a photon whose energy is far below 511 keV, derived classically by J. J. Thomson from an electron shaken by an oscillating field and re-radiating. It is independent of frequency, which is exactly why it fails for hard photons: above roughly 100 keV, Compton recoil takes over and the Klein-Nishina formula drives the cross section down.

This constant runs astrophysics. It fixes the Eddington luminosity, the point at which radiation pressure on electrons balances gravity and a star blows off its own envelope; it sets the optical depth of the ionised intergalactic medium, whose Thomson scattering of the cosmic microwave background is measured by Planck as τ ≈ 0.054 and used to date reionisation; and it governs how photons random-walked out of the primordial plasma before recombination. In the barn units nuclear physicists use, σ_T = 0.665 b.