The solar azimuth angle is the azimuth (horizontal angle with respect to north) of the Sun's position. 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.

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 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.

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. or , 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, , is negative (morning) and the angle between 180 and 360 degrees when the hour angle, , is positive (afternoon). (These two formulas are equivalent if one assumes the "solar elevation angle" approximation formula).

The formula based on the subsolar point and the atan2 function

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. 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, 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 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.

thumb|The solar azimuth angle can be used to find the directions of sunrise and sunset at Stonehenge (𝜙 is its latitude and 𝜀 is Earth's axial tilt)

See also

  • Equation of time
  • Horizontal coordinate system
  • Hour angle
  • Position of the Sun
  • Solar time
  • Solar tracker
  • Sun path
  • Sunrise
  • Sunset
  • Zenith

References

  • Solar Position Calculators by National Renewable Energy Laboratory (NREL)
  • Solar Position Algorithm for Solar Radiation Applications (NREL)
  • An Excel workbook with VBA functions for solar azimuth, solar elevation, dawn, sunrise, solar noon, sunset, and dusk, by Greg Pelletier, translated from NOAA's online calculators for solar position and sunrise/sunset
  • An Excel workbook with a solar position and solar radiation time-series calculator, by Greg Pelletier
  • Sun Position Calculator Free on-line tool to estimate the position of the sun with three different algorithms.
  • PVCDROM Azimuth Angle - online material regarding Photovoltaics by UNSW, ASU, NSF et al.