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Bará, S., Ribas, S., & Kocifaj, M. (2015). Modal evaluation of the anthropogenic night sky brightness at arbitrary distances from a light source. J. of Optics, 17, 105607.
Abstract: The artificial emissions of light contribute to a high extent to the observed brightness of the night sky in many places of the world. Determining the all-sky radiance of anthropogenic origin requires solving the radiative transfer equation for ground-level light sources, generally resorting to a double-scattering approximation in order to account for the observed radiance patterns with a reasonable degree of accuracy. Since the all-sky radiance distribution produced by an elementary light source depends on the distance to the observer in a way that is not immediately obvious, the contributions of sources located at different distances have to be computed on an individual basis, solving for each one the corresponding scattering integrals. In this paper we show that these calculations may be significantly alleviated by using a modal approach, whereby the hemispheric night-sky radiance is expanded in terms of a convenient basis of two-dimensional (2D) orthogonal functions. Since the modal coefficients of this expansion do vary smoothly with the distance to the observer, the all-sky brightness distributions produced by light sources located at arbitrary intermediate distances can be efficiently estimated by interpolation, provided that the coefficients at a discrete set of distances are accurately determined beforehand.