Editing Solar azimuth angle

{{Short description|Azimuth angle of the Sun's position}}
The '''solar azimuth angle''' is the [[azimuth]] (horizontal angle with respect to north) of the [[position of the Sun|Sun's position]].<ref name="sukhatme"/><ref name="Seinfeld and Pandis"/><ref name="Duffie"/> This [[horizontal coordinate system|horizontal coordinate]] defines the [[Sun]]'s [[relative direction]] along the local [[horizon]], whereas the [[solar zenith angle]] (or its [[Angle#Combining angle pairs|complementary angle]] solar [[elevation]]) defines the Sun's apparent [[altitude]].

==Conventional sign and origin==
There are several conventions for the solar azimuth; however, it is traditionally defined as the angle between a line due [[south]] and the shadow cast by a vertical rod on [[Earth]]. This convention states the angle is positive if the shadow is [[east]] of south and negative if it is [[west]] of south.<ref name="sukhatme">{{cite book |title=Solar Energy: Principles of Thermal Collection and Storage |first=S. P. |last=Sukhatme |date=2008 |edition=3rd |publisher=Tata McGraw-Hill Education |isbn=978-0070260641 |page=84}}</ref><ref name="Seinfeld and Pandis">{{cite book |title=Atmospheric Chemistry and Physics, from Air Pollution to Climate Change |first1=John H. |last1=Seinfeld |first2=Spyros N. |last2=Pandis |date=2006 |edition=2nd |publisher=Wiley |isbn=978-0-471-72018-8 |page=130 |url=http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471720186.html |access-date=2013-05-01 |archive-date=2013-09-06 |archive-url=https://web.archive.org/web/20130906094320/http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471720186.html |url-status=dead }}</ref> For example, due east would be 90° and due west would be -90°. Another convention is the reverse; it also has the origin at due south, but measures angles clockwise, so that due east is now negative and west now positive.<ref name="Duffie">{{cite book |title=Solar Engineering of Thermal Processes |first1=John A. |last1=Duffie |first2=William A. |last2=Beckman |date=2013 |edition=4th |publisher=Wiley |isbn=978-0-470-87366-3 |pages=13, 15, 20 |url=http://www.wiley.com/WileyCDA/WileyTitle/productCd-0470873663.html}}</ref>

However, despite tradition, the most commonly accepted convention for analyzing [[solar irradiation]], e.g. for [[solar energy]] applications, is clockwise from due [[north]], so east is 90°, south is 180°, and west is 270°. This is the definition used by [[National Renewable Energy Laboratory|NREL]] in their solar position calculators<ref name="SPA">{{cite journal |last=Reda, I. |first=Andreas, A. |title=Solar Position Algorithm for Solar Radiation Applications |journal=Solar Energy |volume=76 |issue=5 |date=2004 |pages=577–89 |doi=10.1016/j.solener.2003.12.003 |issn=0038-092X |bibcode=2004SoEn...76..577R}}</ref> and is also the convention used in the [[#Formulas|formulas]] presented here. However, [[Landsat]] photos and other [[United States Geological Survey|USGS]] products, while also defining azimuthal angles relative to due north, take counter[[clockwise]] angles as negative.<ref>{{cite web |url=https://lta.cr.usgs.gov/landsat_dictionary.html#sun_azimuth |title=Sun Azimuth |work=Landsat Data Dictionary |publisher=[[United States Geological Survey|USGS]]}}</ref>

==Conventional Trigonometric Formulas==
The following formulas assume the north-clockwise convention. The solar azimuth angle can be calculated to a good approximation with the following formula, however angles should be interpreted with care because the [[inverse sine]], i.e. {{math|1=''x'' = sin<sup>−1</sup> y}} or {{math|1=''x'' = arcsin ''y''}}, has multiple solutions, only one of which will be correct.
:<math>\sin \phi_\mathrm{s} = \frac{-\sin h \cos \delta}{\sin \theta_\mathrm{s}}.</math>

The following formulas can also be used to approximate the solar azimuth angle, but these formulas use cosine, so the azimuth angle as shown by a calculator will always be positive, and should be interpreted as the angle between zero and 180 degrees when the hour angle, {{mvar|h}}, is negative (morning) and the angle between 180 and 360 degrees when the hour angle, {{mvar|h}}, is positive (afternoon). (These two formulas are equivalent if one assumes the "[[solar elevation angle]]" approximation formula).<ref name="Seinfeld and Pandis" /><ref name="Duffie" /><ref name="SPA" />

:<math>\begin{align}
   \cos \phi_\mathrm{s} &= \frac{\sin \delta \cos \Phi - \cos h \cos \delta \sin \Phi}{\sin \theta_\mathrm{s}} \\[5pt]
   \cos \phi_\mathrm{s} &= \frac{\sin \delta - \cos \theta_\mathrm{s}\sin \Phi}{\sin \theta_\mathrm{s}\cos \Phi}.
 \end{align}</math>

So practically speaking, the compass azimuth which is the practical value used everywhere (in example in airlines as the so called course) on a compass (where North is 0 degrees, East is 90 degrees, South is 180 degrees and West is 270 degrees) can be calculated as 

:<math>\text{compass } \phi_\mathrm{s} = 360 - \phi_\mathrm{s}.</math>

The formulas use the following terminology:
*<math>\phi_\mathrm{s}</math> is the solar azimuth angle
*<math>\theta_\mathrm{s}</math> is the [[solar zenith angle]]
*<math>h</math> is the [[hour angle]], in the local [[solar time]]
*<math>\delta</math> is the current [[Position of the Sun|sun declination]]
*<math>\Phi</math> is the local [[latitude]]

In addition, dividing the above sine formula by the first cosine formula gives one the tangent formula as is used in ''The Nautical Almanac''.<ref>The Nautical Almanac https://thenauticalalmanac.com/Formulas.html</ref>

== 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.

== See also ==
* [[Equation of time]]
* [[Horizontal coordinate system]]
* [[Hour angle]]
* [[Position of the Sun]]
* [[Solar time]]
* [[Solar tracker]]
* [[Sun path]]
* [[Sunrise]]
* [[Sunset]]
* [[Zenith]]

== References ==

{{reflist}}

==External links==
* [http://www.nrel.gov/midc/solpos/  Solar Position Calculators by National Renewable Energy Laboratory (NREL)]
* [http://www.nrel.gov/docs/fy08osti/34302.pdf Solar Position Algorithm for Solar Radiation Applications (NREL)]
* [https://web.archive.org/web/20061208171818/http://www.ecy.wa.gov/programs/eap/models/twilight.zip An Excel workbook] with VBA functions for solar azimuth, solar elevation, dawn, sunrise, solar noon, sunset, and dusk, by [https://web.archive.org/web/20070525051436/http://www.ecy.wa.gov/programs/eap/models.html Greg Pelletier], translated from NOAA's online calculators for [http://www.srrb.noaa.gov/highlights/sunrise/azel.html solar position] and [http://www.srrb.noaa.gov/highlights/sunrise/sunrise.html sunrise/sunset]
* [https://web.archive.org/web/20070305050844/http://www.ecy.wa.gov/programs/eap/models/solrad.zip An Excel workbook] with a solar position and solar radiation time-series calculator, by [https://web.archive.org/web/20070525051436/http://www.ecy.wa.gov/programs/eap/models.html Greg Pelletier]
* [http://www.volker-quaschning.de/datserv/sunpos/index_e.html Sun Position Calculator] Free on-line tool to estimate the position of the sun with three different algorithms.
* [https://web.archive.org/web/20130501041741/http://pveducation.org/pvcdrom/properties-of-sunlight/azimuth-angle PVCDROM] Azimuth Angle - online material regarding Photovoltaics by UNSW, ASU, NSF et al.

{{DEFAULTSORT:Solar Azimuth Angle}}
[[Category:Horizontal coordinate system]]
[[Category:Sun]]
[[Category:Solar energy]]