opacities

Opacity is a property of a material that describes how it attenuates electromagnetic radiation as it passes through. It is quantified by the opacity or extinction coefficient, often denoted κ or κν for monochromatic radiation. The radiative transfer equation expresses the change in specific intensity Iν along a path s as dIν/ds = -κν ρ Iν + ην, where ρ is density and ην is the emission coefficient. The optical depth is τν = ∫ κν ρ ds, and transmission is Tν = e^{-τν}.

Total opacity κν combines absorption and scattering: κν = κabs,ν + κsca,ν. Absorption arises from bound-bound, bound-free, and free-free transitions, while scattering includes Rayleigh, Thomson, and Mie-type processes. Dust grains can contribute both absorption and scattering, especially at infrared and optical wavelengths.

In many applications, frequency-averaged opacities are used: the Planck mean opacity κP and the Rosseland mean opacity κR. κP weights κν by the Planck function, while κR weights the inverse of κν by the derivative of the Planck function with respect to temperature. These means are useful in modeling energy transport in stellar interiors and atmospheres and in planetary and atmospheric sciences.

Opacity is wavelength dependent and strongly influenced by composition, temperature, and density. It is distinct from simple absorptivity, since scattering can also reduce transmitted radiance. Practical measurements include transmission, absorbance, or attenuation curves across wavelengths, informing material design and interpretation of astronomical spectra.