A solar azimuth formula that renders circumstantial treatment unnecessary without compromising mathematical rigor: Mathematical setup, application and extension of a formula based on the subsolar point and atan2 function. (July 2021) - Zhang, Taiping; Stackhouse, Paul W.; Macpherson, Bradley; Mikovitz, J. Colleen
Abstract: A conceptually and mathematically concise formula for computing the solar azimuth angle has been used by a subgroup of scientists, but for lack of documentation and publication, it has not been well circulated. This paper introduces this formula which is based on the idea of a unit vector, S, originating from the observer's location and pointing toward the center of the Sun. The vector is completely determined by the coordinates of the subsolar point and of the observer. The x- and y-components of the vector determine the solar azimuth angle, and their use along with the function atan2, which is available in a number of programming/scripting languages, including Fortran and Python, renders any circumstantial treatment absolutely unnecessary. The z-component of the vector, at the same time, determines the solar zenith angle. The use of the unit vector also facilitates a figure that can be used as a full 3D graphic depiction of the daily and annual cycles of the Sun's position for any given location, and this figure can be called "wreath of analemmas". Highlights: This paper introduces a solar azimuth formula that has not been published before. This formula is based on the subsolar point and the atan2 function. This formula, without losing mathematical rigor, renders any circumstantial treatment of results unnecessary. The declination, the Earth-Sun distance and the equation of time are from the Astronomical Almanac for the Year 2019. The "wreath of analemmas" is a full 3D depiction of the daily and annual cycles of the Sun's position for any location.