Editing Solar zenith angle (section)

== 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.<ref name="Zhangetal">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>

In the Earth-Centered Earth-Fixed ([[ECEF]]) geocentric Cartesian coordinate system, let <math>(\phi_{s}, \lambda_{s})</math> and <math>(\phi_{o}, \lambda_{o})</math> 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, <math>\mathbf{S}</math> and <math>\mathbf{V}_{oz}</math>, are

<math display="block">\mathbf{S}=\cos\phi_{s}\cos\lambda_{s}{\mathbf i}+\cos\phi_{s}\sin\lambda_{s}{\mathbf j}+\sin\phi_{s}{\mathbf k},</math>
<math display="block">\mathbf{V}_{oz}=\cos\phi_{o}\cos\lambda_{o}{\mathbf i}+\cos\phi_{o}\sin\lambda_{o}{\mathbf j}+\sin\phi_{o}{\mathbf k}.</math>

where <math>{\mathbf i}</math>, <math>{\mathbf j}</math> and <math>{\mathbf k}</math> are the basis vectors in the ECEF coordinate system.

Now the cosine of the solar zenith angle, <math>\theta_{s}</math>, is simply the [[dot product]] of the above two vectors

<math display="block">\cos\theta_{s} = \mathbf{S}\cdot\mathbf{V}_{oz} = \sin\phi_{o}\sin\phi_{s} + \cos\phi_{o}\cos\phi_{s}\cos(\lambda_{s}-\lambda_{o}).</math>

Note that <math>\phi_{s}</math> is the same as <math>\delta</math>, the declination of the Sun, and <math>\lambda_{s}-\lambda_{o}</math> is equivalent to <math>-h</math>, where <math>h</math> is the hour angle defined earlier. So the above format is mathematically identical to the one given earlier. 

Additionally, Ref. <ref name="Zhangetal" /> also derived the formula for [[solar azimuth angle]] in a similar fashion without using spherical trigonometry.

=== Minimum and Maximum ===
[[File:Solar Zenith Angle min.png|thumb|The daily minimum of the solar zenith angle as a function of latitude and day of year for the year 2020.]]

[[File:Solar Zenith Angle max.png|thumb|The daily maximum of the solar zenith angle as a function of latitude and day of year for the year 2020.]]

At any given location on any given day, the solar zenith angle, <math>\theta_{s}</math>, reaches its minimum, <math>\theta_\text{min}</math>, at local solar noon when the hour angle <math>h = 0</math>, or <math>\lambda_{s}-\lambda_{o}=0</math>, namely, <math>\cos\theta_\text{min} = \cos(|\phi_{o}-\phi_{s}|)</math>, or <math>\theta_\text{min} = |\phi_{o}-\phi_{s}|</math>. If <math>\theta_\text{min} > 90^{\circ}</math>, it is polar night.

And at any given location on any given day, the solar zenith angle, <math>\theta_{s}</math>, reaches its maximum, <math>\theta_\text{max}</math>, at local midnight when the hour angle <math>h = -180^{\circ}</math>, or <math>\lambda_{s}-\lambda_{o}=-180^{\circ}</math>, namely, <math>\cos\theta_\text{max} = \cos(180^{\circ}-|\phi_{o}+\phi_{s}|)</math>, or <math>\theta_\text{max} = 180^{\circ}-|\phi_{o}+\phi_{s}|</math>. If <math>\theta_\text{max} < 90^{\circ}</math>, it is polar day.

===Caveats===
The calculated values are approximations due to the distinction between [[Latitude#Geodetic and geocentric latitudes|common/geodetic latitude]] and [[Latitude#Geocentric latitude|geocentric latitude]]. However, the two values [[Latitude#Comparison of selected types|differ]] by less than 12 [[minutes of arc]], which is less than the apparent angular radius of the sun.

The formula also neglects the effect of [[atmospheric refraction]].<ref>{{cite journal | title = On the computation of solar elevation angles and the determination of sunrise and sunset times | page = 3 | first = Harold M. | last = Woolf | journal = NASA Technical Memorandu, X-1646 | date = 1968 | location = Washington, D.C.}}</ref>