Angular Emission Function of a City and Skyglow Modeling: A Critical Perspective

Abstract

The radiative transfer equation (RTE) is a common approach to solving the transfer of electromagnetic energy in heterogeneous disperse media, such as atmospheric environment. One-dimensional RTE is a linear boundary value problem that is well suited to plane-parallel atmosphere with no diffuse intensity entering the top of the atmosphere. In nighttime regime, the ground-based light sources illuminate the atmosphere at its bottom interface. However, the light-pollution models conventionally use radiant intensity function rather than radiance. This might potentially result in a number of misconceptions. We focused on similarities and fundamental differences between both functions and clarified distinct consequences for the modeling of skyglow from finite-sized and semi-infinite light-emitting flat surfaces. Minimum requirements to be fulfilled by a City Emission Function (CEF) are formulated to ensure a successful solution of standard and inverse problems. It has been shown that the horizon radiance of a flat surface emitting in accordance with Garstang’s function (GEF) would exceed any limit, meaning that the GEF is not an appropriate tool to model skyglow from distant sources. We developed two alternative CEFs to remedy this problem through correction of direct upward emissions; the most important strengths of the modified CEFs are detailed in this paper. Numerical experiments on sky luminance under well-posed and ill-posed boundary conditions were made for two extreme uplight fractions (F) and for three discrete distances from the city edge. The errors induced by replacing radiance with radiant intensity function in the RTE are generally low (15%–30%) if F is as large as 0.15, but alteration of the luminance may range over 1–3 orders of magnitude if F approaches zero. In the latter case, the error margin can increase by a factor of 10–100 or even 1000, even if the angular structure of luminance patterns suffers only weak changes. This is why such a shift in luminance magnitudes can be mistakenly interpreted as the effect of inaccurate estimate of lumens per head of the population rather than the effect of cosine distortion due to ill-posed inputs to the RTE. For that reason, a thorough revision (and/or remediation) of theoretical and computational models is suggested.