View source for Solar zenith angle

{{Short description|Angle between the zenith and the centre of the Sun's disc}}

The '''solar zenith angle''' is the [[zenith angle]] of the [[sun]], i.e., the angle between the sun’s rays and the [[vertical direction]]. It is the [[complement angle|complement]] to the '''solar altitude''' or '''solar elevation''', which is the [[altitude angle]] or [[elevation angle]] between the sun’s rays and a [[horizontal plane]].<ref>{{cite book | page = [https://archive.org/details/fundamentalsatmo00jaco/page/n332 317] | title = Fundamentals of Atmospheric Modeling | url = https://archive.org/details/fundamentalsatmo00jaco | url-access = limited | first = Mark Z. | last = Jacobson | publisher = [[Cambridge University Press]] | date = 2005 | isbn = 0521548659 | edition = 2nd}}</ref><ref name="hartmann">{{cite book | title = Global Physical Climatology | url = https://archive.org/details/globalphysicalcl00hart | url-access = limited | first = Dennis L. | last = Hartmann | publisher = [[Academic Press]] | page = [https://archive.org/details/globalphysicalcl00hart/page/n41 30] | date = 1994 | isbn = 0080571638}}</ref> At [[solar noon]], the altitude angle (complement of the solar angle) is at a minimum and is equal to latitude minus [[solar declination]] angle. This is the basis by which ancient mariners navigated the oceans.<ref>{{cite book |last1=Bonan |first1=Gordon |title=Ecological climatology: concepts and applications |date=2005 |publisher=Cambridge University Press |isbn=9781316425190 |page=62 |url=https://www.cambridge.org/core/books/ecological-climatology/D146443B007985BC366B2512345692C0 |accessdate=13 November 2019}}</ref>{{TOC right}} 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.

==Formula==
<math display="block"> \cos \theta_s = \sin \alpha_s = \sin \Phi \sin \delta + \cos \Phi \cos \delta \cos h</math>

where
* <math>\theta_s</math> is the ''solar zenith angle''
* <math>\alpha_s</math> is the ''solar altitude angle'', <math>\alpha_s = 90^\circ - \theta_s</math>
* <math>h</math> is the [[hour angle]], in the local [[solar time]].
* <math>\delta</math> is the current [[declination of the Sun]]
* <math>\Phi</math> is the local [[latitude]].

== 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>

==Applications==

===Sunrise/Sunset===
{{Main articles|Sunrise equation}}
Sunset and sunrise occur (approximately) when the zenith angle is 90°, where the hour angle ''h''<sub>0</sub> satisfies<ref name="hartmann" />
<math display="block">\cos h_0 = -\tan \Phi \tan \delta.</math>

Precise times of sunset and [[Sunrise#Angle|sunrise]] occur when the upper limb of the Sun appears, as refracted by the atmosphere, to be on the horizon.

===Albedo===
A weighted daily average zenith angle, used in computing the local [[albedo of the Earth]], is given by
<math display="block">\overline{\cos \theta_s} = \frac{\displaystyle \int_{-h_0}^{h_0} Q \cos \theta_s \, \text{d}h}{\displaystyle \int_{-h_0}^{h_0} Q \, \text{d}h}</math>
where ''Q'' is the instantaneous [[irradiance]].<ref name="hartmann" />

===Summary of special angles===
{{Subsolar point date graph|300px|float=right}}
For example, the solar elevation angle is:
* 90° at the [[subsolar point]], which occurs, for example, at the equator on a day of equinox at solar noon
* near 0° at the sunset or at the sunrise
* between −90° and 0° during the night (midnight)

An exact calculation is given in [[position of the Sun]]. Other approximations exist elsewhere.<ref>{{cite web|last=livioflores-ga|url=http://answers.google.com/answers/threadview/id/782886.html|title=Equation to know where the Sun is at a given place at a given date-time|accessdate=9 March 2013}}</ref>

==See also==
* [[Azimuth]]
* [[Solar azimuth angle]]
* [[Horizontal coordinate system]]
* [[List of orbits]]
* {{slink|Photovoltaic mounting system#Orientation and inclination}}
* [[Position of the Sun]]
* [[Sun path]]
* [[Sunrise]]
* [[Sunset]]
* [[Sun transit time]]

==References==

{{Reflist|30em}}
{{Portal bar|Astronomy|Stars|Spaceflight|Outer space|Solar System}}
{{DEFAULTSORT:Solar Elevation Angle}}
[[Category:Horizontal coordinate system]]
[[Category:Sun]]
[[Category:Solar energy]]