opticaldepth

Optical depth, τ, is a dimensionless measure of a medium’s transparency along a line of sight. It is defined by τ = ∫ α(s) ds, where α is the extinction (attenuation) coefficient along the path. If α = κν ρ, with κν the mass extinction coefficient and ρ the density, then τν = ∫ κν ρ ds. The transmitted intensity is related by I = I0 e^{-τ}, and the corresponding extinction in magnitudes is Aν = 1.086 τν. Because κν depends on wavelength, τ is generally wavelength dependent.

In practical terms, optical depth distinguishes optically thin (τ ≪ 1) and optically thick (τ ≫ 1) regimes. In radiative transfer, the basic equation dI/ds = -α I + j governs the propagation, where j is the emission coefficient. The concept applies to stellar and planetary atmospheres, interstellar dust clouds, galaxies, and other media where radiation interacts with matter.

In astronomy and cosmology, τ is used to quantify extinction and scattering, infer column densities, and describe effects such as the Thomson scattering optical depth τe relevant to the cosmic microwave background and reionization. τ depends on the material’s composition, density distribution, and geometry, as well as on wavelength, making it a key parameter in interpreting observations across the electromagnetic spectrum.