The group of sunshine modes on any inclined surface

Summary

It is easy to compute the diurnal arc or day length. We show that the same simplicity exists for the “solar arc” on an inclined surface, i.e. the theoretical interval during which it sees the sun. By transposing the daily extraterrestrial irradiance onto the celestial sphere, we demonstrate the concept of image, sunshine modes, transitions from one mode to another and annual sequence of modes which are, in a sense, the generalization of astronomical seasons for inclined surfaces. The modes are the elements of a group, or the focus of the theory. We give the foundation of a simple and exhaustive theory on the subject for any surface the following explicit formulae: image, mode, declination of transition between two modes, sunrise and sunset, solar arc or extraterrestrial sunshine, and daily irradiance. This subject has been previously broached in some infrequently referenced papers as well as in more recent papers by Revfeim (1976, 1978, 1982), Swift (1976), and Sato (1984). However, due to their lack of structure, those contributions to a geometrical theory have been largely neglected and the empirical point of view still prevails.