The position of the Sun in the
sky is a function of both the
time
Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, through the present, into the future. It is a component quantity of various me ...
and the
geographic location of observation 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 sur ...
's surface. As
Earth orbits the
Sun over the course of a
year
A year or annus is the orbital period of a planetary body, for example, the Earth, moving in its orbit around the Sun. Due to the Earth's axial tilt, the course of a year sees the passing of the seasons, marked by change in weather, the h ...
, the Sun appears to move with respect to the
fixed stars
In astronomy, fixed stars ( la, stellae fixae) is a term to name the full set of glowing points, astronomical objects actually and mainly stars, that appear not to move relative to one another against the darkness of the night sky in the backgro ...
on the
celestial sphere
In astronomy and navigation, the celestial sphere is an abstract sphere that has an arbitrarily large radius and is concentric to Earth. All objects in the sky can be conceived as being projected upon the inner surface of the celestial sphe ...
, along a circular path called the
ecliptic.
Earth's rotation
Earth's rotation or Earth's spin is the rotation of planet Earth around its own axis, as well as changes in the orientation of the rotation axis in space. Earth rotates eastward, in prograde motion. As viewed from the northern polar star Po ...
about its axis causes
diurnal motion, so that the Sun appears to move across the sky in a
Sun path that depends on the observer's geographic
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 ...
. The time when the Sun
transits the observer's
meridian depends on the geographic
longitude
Longitude (, ) is a geographic coordinate that specifies the east– west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek let ...
.
To find the Sun's position for a given location at a given time, one may therefore proceed in three steps as follows:
# calculate the Sun's position in the
ecliptic coordinate system,
# convert to the
equatorial coordinate system, and
# convert to the
horizontal coordinate system, for the observer's local time and location. This is the coordinate system normally used to calculate the position of the Sun in terms of
solar zenith 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 the ...
and
solar azimuth angle
The solar azimuth angle is the azimuth (horizontal angle with respect to north) of the Sun's position. This horizontal coordinate defines the Sun's relative direction along the local horizon, whereas the solar zenith angle (or its complementary a ...
, and the two parameters can be used to depict the
Sun path.
This calculation is useful in
astronomy
Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, g ...
,
navigation
Navigation is a field of study that focuses on the process of monitoring and controlling the movement of a craft or vehicle from one place to another.Bowditch, 2003:799. The field of navigation includes four general categories: land navigation ...
,
surveying
Surveying or land surveying is the technique, profession, art, and science of determining the terrestrial two-dimensional or three-dimensional positions of points and the distances and angles between them. A land surveying professional is ...
,
meteorology
Meteorology is a branch of the atmospheric sciences (which include atmospheric chemistry and physics) with a major focus on weather forecasting. The study of meteorology dates back millennia, though significant progress in meteorology did no ...
,
climatology
Climatology (from Greek , ''klima'', "place, zone"; and , '' -logia'') or climate science is the scientific study of Earth's climate, typically defined as weather conditions averaged over a period of at least 30 years. This modern field of stu ...
,
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 essen ...
, and
sundial
A sundial is a horological device that tells the time of day (referred to as civil time in modern usage) when direct sunlight shines by the apparent position of the Sun in the sky. In the narrowest sense of the word, it consists of a f ...
design.
Approximate position
Ecliptic coordinates
These equations, from the ''
Astronomical Almanac'',
can be used to calculate the apparent coordinates of the
Sun,
mean equinox and ecliptic of date, to a precision of about 0°.01 (36″), for dates between 1950 and 2050. These equations are coded into a
Fortran 90 routine in 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] and are used to calculate the
solar zenith 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 the ...
and
solar azimuth angle
The solar azimuth angle is the azimuth (horizontal angle with respect to north) of the Sun's position. This horizontal coordinate defines the Sun's relative direction along the local horizon, whereas the solar zenith angle (or its complementary a ...
as observed from the surface of the Earth.
Start by calculating ''n'', the number of days (positive or negative, including fractional days) since Greenwich noon, Terrestrial Time, on 1 January 2000 (
J2000.0). If the
Julian date for the desired time is known, then
:
The
mean longitude of the Sun, corrected for the
aberration of light, is:
:
The
mean anomaly
In celestial mechanics, the mean anomaly is the fraction of an elliptical orbit's period that has elapsed since the orbiting body passed periapsis, expressed as an angle which can be used in calculating the position of that body in the classica ...
of the Sun (actually, of the Earth in its orbit around the Sun, but it is convenient to pretend the Sun orbits the Earth), is:
:
Put
and
in the range 0° to 360° by adding or subtracting
multiples of 360° as needed.
Finally, the
ecliptic longitude of the Sun is:
:
The
ecliptic latitude of the Sun is nearly:
:
,
as the
ecliptic latitude of the Sun never exceeds 0.00033°,
and the distance of the Sun from the Earth, in
astronomical unit
The astronomical unit (symbol: au, or or AU) is a unit of length, roughly the distance from Earth to the Sun and approximately equal to or 8.3 light-minutes. The actual distance from Earth to the Sun varies by about 3% as Earth orbits ...
s, is:
:
.
Obliquity of the ecliptic
Where the
obliquity of the ecliptic
In astronomy, axial tilt, also known as obliquity, is the angle between an object's rotational axis and its orbital axis, which is the line perpendicular to its orbital plane; equivalently, it is the angle between its equatorial plane and or ...
is not obtained elsewhere, it can be approximated:
:
Equatorial coordinates
,
and
form a complete position of the
Sun in the
ecliptic coordinate system. This can be converted to the
equatorial coordinate system by calculating the
obliquity of the ecliptic
In astronomy, axial tilt, also known as obliquity, is the angle between an object's rotational axis and its orbital axis, which is the line perpendicular to its orbital plane; equivalently, it is the angle between its equatorial plane and or ...
,
, and continuing:
Right ascension
Right ascension (abbreviated RA; symbol ) is the angular distance of a particular point measured eastward along the celestial equator from the Sun at the March equinox to the ( hour circle of the) point in question above the earth.
When pair ...
,
:
, where
is in the same
quadrant as
,
To get RA at the right quadrant on computer programs use double argument Arctan function such as ATAN2(y,x)
:
and
declination
In astronomy, declination (abbreviated dec; symbol ''δ'') is one of the two angles that locate a point on the celestial sphere in the equatorial coordinate system, the other being hour angle. Declination's angle is measured north or south of th ...
,
:
.
Rectangular equatorial coordinates
Right-handed rectangular equatorial coordinates in
astronomical unit
The astronomical unit (symbol: au, or or AU) is a unit of length, roughly the distance from Earth to the Sun and approximately equal to or 8.3 light-minutes. The actual distance from Earth to the Sun varies by about 3% as Earth orbits ...
s are:
:
:
:
:Where
axis is in the direction of the
March equinox
The March equinox or northward equinox is the equinox on the Earth when the subsolar point appears to leave the Southern Hemisphere and cross the celestial equator, heading northward as seen from Earth. The March equinox is known as the ve ...
, the
axis towards
June Solstice
The June solstice is the solstice on Earth that occurs annually between 20 and 22 June according to the Gregorian calendar. In the Northern Hemisphere, the June solstice is the summer solstice (the day with the longest period of daylight), wh ...
, and the
axis towards the
North celestial pole.
Horizontal coordinates
Declination of the Sun as seen from Earth
The Sun appears to move northward during the northern
spring
Spring(s) may refer to:
Common uses
* Spring (season), a season of the year
* Spring (device), a mechanical device that stores energy
* Spring (hydrology), a natural source of water
* Spring (mathematics), a geometric surface in the shape of a h ...
, crossing the
celestial equator on the
March equinox
The March equinox or northward equinox is the equinox on the Earth when the subsolar point appears to leave the Southern Hemisphere and cross the celestial equator, heading northward as seen from Earth. The March equinox is known as the ve ...
. Its
declination
In astronomy, declination (abbreviated dec; symbol ''δ'') is one of the two angles that locate a point on the celestial sphere in the equatorial coordinate system, the other being hour angle. Declination's angle is measured north or south of th ...
reaches a maximum equal to the angle of Earth's
axial tilt
In astronomy, axial tilt, also known as obliquity, is the angle between an object's rotational axis and its orbital axis, which is the line perpendicular to its orbital plane; equivalently, it is the angle between its equatorial plane and orb ...
(23.44°) on the
June solstice
The June solstice is the solstice on Earth that occurs annually between 20 and 22 June according to the Gregorian calendar. In the Northern Hemisphere, the June solstice is the summer solstice (the day with the longest period of daylight), wh ...
, then decreases until reaching its minimum (−23.44°) on the
December solstice
The December solstice, also known as the southern solstice, is the solstice that occurs each December – typically on 21 December, but may vary by one day in either direction according to the Gregorian calendar. In the Northern Hemisphere, the ...
, when its value is the negative of the axial tilt. This variation produces the
season
A season is a division of the year based on changes in weather, ecology, and the number of daylight hours in a given region. On Earth, seasons are the result of the axial parallelism of Earth's tilted orbit around the Sun. In temperate and ...
s.
A
line graph of the Sun's declination during a year resembles a
sine wave
A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the '' sine'' trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often in ...
with an
amplitude
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
of 23.44°, but one lobe of the wave is several days longer than the other, among other differences.
The following phenomena would occur if Earth were a perfect
sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
, in a
circular orbit
A circular orbit is an orbit with a fixed distance around the barycenter; that is, in the shape of a circle.
Listed below is a circular orbit in astrodynamics or celestial mechanics under standard assumptions. Here the centripetal force is ...
around the Sun, and if its axis is tilted 90°, so that the axis itself is on the
orbital plane (similar to
Uranus
Uranus is the seventh planet from the Sun. Its name is a reference to the Greek god of the sky, Uranus ( Caelus), who, according to Greek mythology, was the great-grandfather of Ares (Mars), grandfather of Zeus (Jupiter) and father of ...
). At one date in the year, the Sun would be
directly overhead at the
North Pole
The North Pole, also known as the Geographic North Pole or Terrestrial North Pole, is the point in the Northern Hemisphere where the Earth's axis of rotation meets its surface. It is called the True North Pole to distinguish from the Ma ...
, so its declination would be +90°. For the next few months, the
subsolar point would move toward the
South Pole
The South Pole, also known as the Geographic South Pole, Terrestrial South Pole or 90th Parallel South, is one of the two points where Earth's axis of rotation intersects its surface. It is the southernmost point on Earth and lies antipod ...
at constant speed, crossing the
circles of latitude
A circle of latitude or line of latitude on Earth is an abstract east– west small circle connecting all locations around Earth (ignoring elevation) at a given latitude coordinate line.
Circles of latitude are often called parallels bec ...
at a constant rate, so that the solar declination would decrease ''linearly'' with time. Eventually, the Sun would be directly above the South Pole, with a declination of −90°; then it would start to move northward at a constant speed. Thus, the graph of solar declination, as seen from this highly tilted Earth, would resemble a
triangle wave
A triangular wave or triangle wave is a non-sinusoidal waveform named for its triangular shape. It is a periodic, piecewise linear, continuous real function.
Like a square wave, the triangle wave contains only odd harmonics. However, ...
rather than a sine wave,
zigzag
A zigzag is a pattern made up of small corners at variable angles, though constant within the zigzag, tracing a path between two parallel lines; it can be described as both jagged and fairly regular.
In geometry, this pattern is described as ...
ging between plus and minus 90°, with linear segments between the maxima and minima.
If the 90° axial tilt is decreased, then the absolute maximum and minimum values of the declination would decrease, to equal the axial tilt. Also, the shapes of the maxima and minima on the graph would become less acute, being curved to resemble the maxima and minima of a sine wave. However, even when the axial tilt equals that of the actual Earth, the maxima and minima remain more acute than those of a sine wave.
In reality,
Earth's orbit
Earth orbits the Sun at an average distance of 149.60 million km (92.96 million mi) in a counterclockwise direction as viewed from above the Northern Hemisphere. One complete orbit takes days (1 sidereal year), during which time Earth ...
is
elliptical. Earth moves more rapidly around the Sun near
perihelion, in early January, than near
aphelion, in early July. This makes processes like the variation of the solar declination happen faster in January than in July. On the graph, this makes the minima more acute than the maxima. Also, since perihelion and aphelion do not happen on the exact dates as the solstices, the maxima and minima are slightly asymmetrical. The rates of change before and after are not quite equal.
The graph of apparent solar declination is therefore different in several ways from a sine wave. Calculating it accurately involves some complexity, as shown below.
Calculations
The declination of the
Sun, δ
☉, is the angle between the rays of the Sun and the plane of the Earth's equator. The Earth's
axial tilt
In astronomy, axial tilt, also known as obliquity, is the angle between an object's rotational axis and its orbital axis, which is the line perpendicular to its orbital plane; equivalently, it is the angle between its equatorial plane and orb ...
(called the ''obliquity of the ecliptic'' by astronomers) is the angle between the Earth's axis and a line perpendicular to the Earth's orbit. The Earth's axial tilt changes slowly over thousands of years but its current value of about ε = 23°26' is nearly constant, so the change in solar declination during one year is nearly the same as during the next year.
At the
solstice
A solstice is an event that occurs when the Sun appears to reach its most northerly or southerly excursion relative to the celestial equator on the celestial sphere. Two solstices occur annually, around June 21 and December 21. In many count ...
s, the angle between the rays of the Sun and the plane of the Earth's equator reaches its maximum value of 23°26'. Therefore, δ
☉ = +23°26' at the northern summer solstice and δ
☉ = −23°26' at the southern summer solstice.
At the moment of each
equinox, the center of the Sun appears to pass through the
celestial equator, and δ
☉ is 0°.
The Sun's declination at any given moment is calculated by:
: