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

¹ Astronomical Institute “Anton Pannekoek”, University of Amsterdam, Kruislaan 403, 1098 SJ Amsterdam, The Netherlands e-mail: J.W.Hovenier@uva.nl
² DEOS, Department of Aerospace Engineering, TU Delft, Kluyverweg 1, 2629 HS, Delft, The Netherlands
³ SRON Netherlands Institute for Space Research, Sorbonnelaan 2, 3584 CA Utrecht, The Netherlands

Received: 12 August 2008
Accepted: 7 September 2008

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.