Bidirectional reflectance spectroscopy

An analytical model is developed for the opposition effects (heiligenshein) in the case of light scattering from a semi-infinite, particulate medium with particles that are large relative to the wavelength. The effect is common for natural materials, and comprises a bright surge in light diffusively reflected from a surface at near zero phase. A generalized expression is devised for the extinction coefficient of a particulate medium. Models are developed for step function and hyperbolic tangent distributions of light scattered from a stratified medium and exhibiting the opposition effect. A maximum brightness amplitude increase of 0.753 is projected for the effect. Greater values must have other causes. To illustrate the theory it is fitted to observations of the Moon, an asteroid, and a satellite of Uranus; Europa is also discussed.