Closed-Form Equation for the Duration of Daily Insolation on Uniformly Sloping Terrain

Aims: Derivation of a closed-form expression for the duration of the daily insolation on surfaces of arbitrary uniform slope and aspect located anywhere on Earth, anytime of the year.
Study Method: It is shown that the sunrise- and sunset-hour angles and the duration of daily insolation depend on the roots of the equation A cosΦ + B sinΦ + C = 0, in which Φ is the hour angle and A, B, and C are coefficients that involve the slope of the surface, the aspect of the sloping surface, the solar declination, and the latitude of a point of interest on the sloping surface.
Results: The method to calculate the duration of daily insolation developed in this article is applicable to any sequence or combinations of days to obtain the total number of daylight hours over arbitrary periods. Solutions for double sunrise and double sunset situations are also derived in this paper.
Conclusion: The closed-form equations developed in this paper can be used in conjunction with measurements of atmospheric transmissivity to calculate the direct solar insolation on a surface of arbitrary slope and aspect, yielding a powerful tool for agricultural, meteorologic, hydrologic, ecologic, and climatic studies and modeling.