# Solar azimuth angle

The solar azimuth angle is the azimuth angle of the Sun's position.[1][2][3] This horizontal coordinate defines the Sun's relative direction along the local horizon, whereas the solar zenith angle (or its complementary angle solar elevation) defines the Sun's apparent altitude.

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 line 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 counterclockwise angles as negative.[5]

## 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.

${\displaystyle \sin \phi _{\mathrm {s} }={\frac {-\sin h\cos \delta }{\sin \theta _{\mathrm {s} }}}.}$

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]

{\displaystyle {\begin{aligned}\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{aligned}}}

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

${\displaystyle {\text{compass }}\phi _{\mathrm {s} }=360-\phi _{\mathrm {s} }.}$

The formulas use the following terminology:

• ${\displaystyle \phi _{\mathrm {s} }}$ is the solar azimuth angle
• ${\displaystyle \theta _{\mathrm {s} }}$ is the solar zenith angle
• ${\displaystyle h}$ is the hour angle, in the local solar time
• ${\displaystyle \delta }$ is the current sun declination
• ${\displaystyle \Phi }$ is the local latitude

## References

1. Sukhatme, S. P. (2008). Solar Energy: Principles of Thermal Collection and Storage (3rd ed.). Tata McGraw-Hill Education. p. 84. ISBN 978-0070260641.
2. Seinfeld, John H.; Pandis, Spyros N. (2006). Atmospheric Chemistry and Physics, from Air Pollution to Climate Change (2nd ed.). Wiley. p. 130. ISBN 978-0-471-72018-8.
3. Duffie, John A.; Beckman, William A. (2013). Solar Engineering of Thermal Processes (4th ed.). Wiley. pp. 13, 15, 20. ISBN 978-0-470-87366-3.
4. Reda, I., Andreas, A. (2004). "Solar Position Algorithm for Solar Radiation Applications". Solar Energy. 76 (5): 577–89. Bibcode:2004SoEn...76..577R. doi:10.1016/j.solener.2003.12.003. ISSN 0038-092X.
5. "Sun Azimuth". Landsat Data Dictionary. USGS.
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