Solar zenith angle is normally used in combination with the solar azimuth angle to determine the position of the Sun as observed from a given location on the surface of the Earth.
Derivation of the formula using the subsolar point and vector analysis
While the formula can be derived by applying the cosine law to the zenith-pole-Sun spherical triangle, the spherical trigonometry is a relatively esoteric subject.
By introducing the coordinates of the subsolar point and using vector analysis, the formula can be obtained straightforward without incurring the use of spherical trigonometry.[4]
In the Earth-Centered Earth-Fixed (ECEF) geocentric Cartesian coordinate system, let and be the latitudes and longitudes, or coordinates, of the subsolar point and the observer's point, then the upward-pointing unit vectors at the two points, and , are
where , and are the basis vectors in the ECEF coordinate system.
Now the cosine of the solar zenith angle, , is simply the dot product of the above two vectors
Note that is the same as , the declination of the Sun, and is equivalent to , where is the hour angle defined earlier. So the above format is mathematically identical to the one given earlier.
Additionally, Ref. [4] also derived the formula for solar azimuth angle in a similar fashion without using spherical trigonometry.
Minimum and Maximum
At any given location on any given day, the solar zenith angle, , reaches its minimum, , at local solar noon when the hour angle , or , namely, , or . If , it is polar night.
And at any given location on any given day, the solar zenith angle, , reaches its maximum, , at local midnight when the hour angle , or , namely, , or . If , it is polar day.
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
Woolf, Harold M. (1968). "On the computation of solar elevation angles and the determination of sunrise and sunset times". NASA Technical Memorandu, X-1646. Washington, D.C.: 3.
This article uses material from the Wikipedia article Solar_zenith_angle, and is written by contributors.
Text is available under a CC BY-SA 4.0 International License; additional terms may apply. Images, videos and audio are available under their respective licenses.