The solar azimuth angle is the
azimuth
An azimuth (; from ar, اَلسُّمُوت, as-sumūt, the directions) is an angular measurement in a spherical coordinate system. More specifically, it is the horizontal angle from a cardinal direction, most commonly north.
Mathematicall ...
(horizontal angle with respect to north) of the
Sun's position.
This
horizontal coordinate defines the
Sun's
relative direction
In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents the position of a Point (geometry), point ''P'' in space#Classical mechanics, space in relation to an arbitrary refer ...
along the local
horizon
The horizon is the apparent line that separates the surface of a celestial body from its sky when viewed from the perspective of an observer on or near the surface of the relevant body. This line divides all viewing directions based on whether ...
, whereas the
solar zenith angle (or its
complementary angle solar
elevation
The elevation of a geographic location is its height above or below a fixed reference point, most commonly a reference geoid, a mathematical model of the Earth's sea level as an equipotential gravitational surface (see Geodetic datum § ...
) defines the Sun's apparent
altitude
Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context ...
.
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
South is one of the cardinal directions or compass points. The direction is the opposite of north and is perpendicular to both east and west.
Etymology
The word ''south'' comes from Old English ''sūþ'', from earlier Proto-Germanic ''*sunþa ...
and the shadow cast by a vertical rod on
Earth
Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surf ...
. This convention states the angle is positive if the shadow is
east
East or Orient is one of the four cardinal directions or points of the compass. It is the opposite direction from west and is the direction from which the Sun rises on the Earth.
Etymology
As in other languages, the word is formed from the fa ...
of south and negative if it is
west
West or Occident is one of the four cardinal directions or points of the compass. It is the opposite direction from east and is the direction in which the Sun sets on the Earth.
Etymology
The word "west" is a Germanic word passed into some ...
of south.
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
Solar irradiance is the power per unit area (surface power density) received from the Sun in the form of electromagnetic radiation in the wavelength range of the measuring instrument.
Solar irradiance is measured in watts per square metre (W/m ...
, e.g. for
solar energy
Solar energy is radiant light and heat from the Sun that is harnessed using a range of technologies such as solar power to generate electricity, solar thermal energy (including solar water heating), and solar architecture. It is an ...
applications, is clockwise from due
north
North is one of the four compass points or cardinal directions. It is the opposite of south and is perpendicular to east and west. ''North'' is a noun, adjective, or adverb indicating direction or geography.
Etymology
The word ''no ...
, 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
In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwe ...
presented here. However,
Landsat
The Landsat program is the longest-running enterprise for acquisition of satellite imagery of Earth. It is a joint NASA / USGS program. On 23 July 1972, the Earth Resources Technology Satellite was launched. This was eventually renamed to L ...
photos and other
USGS
The United States Geological Survey (USGS), formerly simply known as the Geological Survey, is a scientific agency of the United States government. The scientists of the USGS study the landscape of the United States, its natural resources, ...
products, while also defining azimuthal angles relative to due north, take counter
clockwise
Two-dimensional rotation can occur in two possible directions. Clockwise motion (abbreviated CW) proceeds in the same direction as a clock's hands: from the top to the right, then down and then to the left, and back up to the top. The opposite ...
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
In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). Spec ...
, i.e. or , 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, , 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
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 to the solar altitude or solar elevation, which is the altitude angle or elevation angle between ...
" approximation formula).
:
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
:
The formulas use the following terminology:
*
is the solar azimuth angle
*
is the
solar zenith angle
*
is the
hour angle, in the local
solar time
Solar time is a calculation of the passage of time based on the position of the Sun in the sky. The fundamental unit of solar time is the day, based on the synodic rotation period. Two types of solar time are apparent solar time (sundial t ...
*
is the current
sun declination
*
is the local
latitude
In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north po ...
In addition, dividing the above sine formula by the first cosine formula gives one the tangent formula as is used in ''The Nautical Almanac''.
The formula based on the ''subsolar point'' and the atan2 function
A 2021 publication presents a method that uses a solar azimuth formula based on the
subsolar point
The subsolar point on a planet is the point at which its sun is perceived to be directly overhead (at the zenith); that is, where the sun's rays strike the planet exactly perpendicular to its surface. It can also mean the point closest to th ...
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
The position of the Sun in the sky is a function of both the time and the geographic location of observation on Earth's surface. As Earth orbits the Sun over the course of a year, the Sun appears to move with respect to the fixed stars on the cel ...
and
equation of time using equations from The Astronomical Almanac,
[The Astronomical Almanac for the Year. The United Naval Observatory, 2019.] then it gives the x-, y- and z-components of the unit vector pointing toward the Sun, through
vector analysis
Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subjec ...
rather than
spherical trigonometry
Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are grea ...
, as follows:
:
where
*
is the declination of the Sun,
*
is the latitude of the subsolar point,
*
is the longitude of the subsolar point,
*
is the Greenwich Mean Time or UTC,
*
is the
equation of time in minutes,
*
is the latitude of the observer,
*
is the longitude of the observer,
*
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
analemma
In astronomy, an analemma (; ) is a diagram showing the position of the Sun in the sky as seen from a fixed location on Earth at the same mean solar time, as that position varies over the course of a year. The diagram will resemble a figu ...
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
The horizontal coordinate system is a celestial coordinate system that uses the observer's local horizon as the fundamental plane to define two angles: altitude and azimuth.
Therefore, the horizontal coordinate system is sometimes called as t ...
*
Hour angle
*
Position of the Sun
The position of the Sun in the sky is a function of both the time and the geographic location of observation on Earth's surface. As Earth orbits the Sun over the course of a year, the Sun appears to move with respect to the fixed stars on th ...
*
Solar time
Solar time is a calculation of the passage of time based on the position of the Sun in the sky. The fundamental unit of solar time is the day, based on the synodic rotation period. Two types of solar time are apparent solar time (sundial t ...
*
Solar tracker
A solar tracker is a device that orients a payload toward the Sun. Payloads are usually solar panels, parabolic troughs, fresnel reflectors, lenses or the mirrors of a heliostat.
For flat-panel photovoltaic systems, trackers are used to m ...
*
Sun path
*
Sunrise
Sunrise (or sunup) is the moment when the upper rim of the Sun appears on the horizon in the morning. The term can also refer to the entire process of the solar disk crossing the horizon and its accompanying atmospheric effects.
Terminology ...
*
Sunset
Sunset, also known as sundown, is the daily disappearance of the Sun below the horizon due to Earth's rotation. As viewed from everywhere on Earth (except the North and South poles), the equinox Sun sets due west at the moment of both the spr ...
*
Zenith
The zenith (, ) is an imaginary point directly "above" a particular location, on the celestial sphere. "Above" means in the vertical direction ( plumb line) opposite to the gravity direction at that location ( nadir). The zenith is the "high ...
References
External links
Solar Position Calculators by National Renewable Energy Laboratory (NREL)Solar Position Algorithm for Solar Radiation Applications (NREL)An Excel workbookwith VBA functions for solar azimuth, solar elevation, dawn, sunrise, solar noon, sunset, and dusk, b
translated from NOAA's online calculators fo
an
An Excel workbookwith a solar position and solar radiation time-series calculator, b
Free on-line tool to estimate the position of the sun with three different algorithms.
PVCDROMAzimuth Angle - online material regarding Photovoltaics by UNSW, ASU, NSF et al.
{{DEFAULTSORT:Solar Azimuth Angle
Horizontal coordinate system
Sun
Solar energy