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.[1][2] 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.[3]
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
NREL in their solar position calculators[4] and is also the convention used in the
formulas presented here. However,
Landsat photos and other
USGS products, while also defining azimuthal angles relative to due north, take counter
clockwise angles as negative.[5]
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. x = sin−1 y or x = arcsin y, has multiple solutions, only one of which will be correct.
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, h, is negative (morning) and the angle between 180 and 360 degrees when the hour angle, h, is positive (afternoon). (These two formulas are equivalent if one assumes the "
solar elevation angle" approximation formula).[2][3][4]
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
In addition, dividing the above sine formula by the first cosine formula gives one the tangent formula as is used in The Nautical Almanac.[6]
The formula based on the subsolar point and the atan2 function
"Wreath of Analemmas". The annual excursion of the position of the Sun defined by the triplet , and 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.[7] The subsolar point is the point on the surface of the Earth where the Sun is overhead.
are the x-, y- and z-components, respectively, of the unit vector pointing toward the Sun.
It can be shown that . With the above mathematical setup, the solar zenith angle and solar azimuth angle are simply
,
. (South-Clockwise Convention)
where
is the solar zenith angle,
is the solar azimuth angle following the South-Clockwise Convention.
If one prefers North-Clockwise Convention, or East-Counterclockwise Convention, the formulas are
, (North-Clockwise Convention)
. (East-Counterclockwise Convention)
Finally, the values of , and at 1-hour step for an entire year can be presented in a 3D plot of "wreath of
analemmas" 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.
^
abSukhatme, S. P. (2008). Solar Energy: Principles of Thermal Collection and Storage (3rd ed.). Tata McGraw-Hill Education. p. 84.
ISBN978-0070260641.
^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
^The Astronomical Almanac for the Year. The United Naval Observatory, 2019.