Behavior of the reflection function of a plane-parallel medium for directions of incidence and reflection tending to horizontal directions

TL;DR

This paper derives general relations for the reflection function of plane-parallel media with scattering particles, especially when incidence and reflection directions are nearly horizontal, highlighting potential inaccuracies in common approximations.

Contribution

It presents comprehensive formulas for the reflection function in various media configurations, emphasizing the limitations of existing approximations near horizontal directions.

Findings

Relations are valid for semi-infinite and finite media.

Common approximations like Lambert's law can be inaccurate for horizontal directions.

Results are applicable to planetary atmospheres and reflection nebulae.

Abstract

The atmospheres of (exo) planets and moons, as well as reflection nebulae, contain in general independently scattering particles in random orientation and are often supposed to be plane-parallel. Relations are presented for the (bidirectional) reflection function and several related functions of such a medium in case the directions of incidence and reflection both tend to horizontal directions. The results are quite general. The medium may be semi-infinite or finite, with or without a reflecting surface underneath, and vertically homogeneous or inhomogeneous. Some approximative formulae for the reflection function of a plane-parallel medium with independently scattering particles in random orientation, including Lambert's law, may be very inaccurate if the directions of incidence and reflection are both nearly horizontal.