Editing Solar azimuth angle (section)

== The formula based on the ''subsolar point'' and the atan2 function==
[[File:Wreath_of_Analemmas.png|thumb|"Wreath of Analemmas". The annual excursion of the position of the Sun defined by the triplet <math>S_{x}</math>, <math>S_{y}</math> and <math>S_{z}</math> at 1-hour step as viewed at the geographic center of the contiguous United States. The gray part indicates it is nighttime.]]

A 2021 publication presents a method that uses a solar azimuth formula based on the [[subsolar point]] and the [[atan2]] function, as defined in [[Fortran 90]], that gives an unambiguous solution without the need for circumstantial treatment.<ref>Zhang, T., Stackhouse, P.W., Macpherson, B., and Mikovitz, J.C., 2021. 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. Renewable Energy, 172, 1333-1340. DOI: https://doi.org/10.1016/j.renene.2021.03.047</ref> The subsolar point is the point on the surface of the Earth where the Sun is overhead.

The method first calculates the [[declination of the Sun]] and [[equation of time]] using equations from The Astronomical Almanac,<ref>The Astronomical Almanac for the Year. The United Naval Observatory, 2019.</ref> then it gives the x-, y- and z-components of the unit vector pointing toward the Sun, through [[vector analysis]] rather than [[spherical trigonometry]], as follows:

:<math>\begin{align}
   \phi_{s} &= \delta, \\
   \lambda_{s} &= -15(T_\mathrm{GMT}-12+E_\mathrm{min}/60), \\
   S_{x} &= \cos \phi_{s} \sin (\lambda_{s}-\lambda_{o}), \\
   S_{y} &= \cos \phi_{o} \sin \phi_{s} - \sin \phi_{o} \cos \phi_{s} \cos (\lambda_{s}-\lambda_{o}), \\
   S_{z} &= \sin \phi_{o} \sin \phi_{s} + \cos \phi_{o} \cos \phi_{s} \cos (\lambda_{s}-\lambda_{o}).
 \end{align}</math>

where

*<math>\delta</math> is the declination of the Sun,
*<math>\phi_{s}</math> is the latitude of the subsolar point,
*<math>\lambda_{s}</math> is the longitude of the subsolar point, 
*<math>T_\mathrm{GMT}</math> is the Greenwich Mean Time or UTC,
*<math>E_\mathrm{min}</math> is the [[equation of time]] in minutes, 
*<math>\phi_{o}</math> is the latitude of the observer, 
*<math>\lambda_{o}</math> is the longitude of the observer, 
*<math>S_{x}, S_{y}, S_{z}</math> are the x-, y- and z-components, respectively, of the unit vector pointing toward the Sun. The x-, y- and z-axises of the coordinate system point to East, North and upward, respectively. 

It can be shown that <math>S_{x}^{2}+S_{y}^{2}+S_{z}^{2}=1</math>. With the above mathematical setup, the solar zenith angle and solar azimuth angle are simply

:<math>Z = \mathrm{acos}(S_{z})</math>, 
:<math>\gamma_{s} = \mathrm{atan2}(-S_{x}, -S_{y})</math>. (South-Clockwise Convention)

where
*<math>Z</math> is the solar zenith angle,
*<math>\gamma_{s}</math> is the solar azimuth angle following the South-Clockwise Convention.

If one prefers North-Clockwise Convention, or East-Counterclockwise Convention, the formulas are

:<math>\gamma_{s} = \mathrm{atan2}(S_{x}, S_{y})</math>, (North-Clockwise Convention)
:<math>\gamma_{s} = \mathrm{atan2}(S_{y}, S_{x})</math>. (East-Counterclockwise Convention)

Finally, the values of <math>S_{x}</math>, <math>S_{y}</math> and <math>S_{z}</math> at 1-hour step for an entire year can be presented in a 3D plot of "wreath of [[analemma]]s" as a graphic depiction of all possible positions of the Sun in terms of solar zenith angle and solar azimuth angle for any given location. Refer to [[sun path]] for similar plots for other locations.