The position of the Sun in the
sky
The sky is an unobstructed view upward from the surface of the Earth. It includes the atmosphere and outer space. It may also be considered a place between the ground and outer space, thus distinct from outer space.
In the field of astronomy, ...
is a function of both the
time
Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, to ...
and the
geographic location
In geography, location or place are used to denote a region (point, line, or area) on Earth's surface or elsewhere. The term ''location'' generally implies a higher degree of certainty than ''place'', the latter often indicating an entity with an ...
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 surfa ...
's surface. As
Earth orbits
A geocentric orbit or Earth orbit involves any object orbiting Earth, such as the Moon or artificial satellites. In 1997, NASA estimated there were approximately 2,465 artificial satellite payloads orbiting Earth and 6,216 pieces of space debris ...
the
Sun
The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared radi ...
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 hou ...
, 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 sphere, ...
, along a circular path called the
ecliptic
The ecliptic or ecliptic plane is the orbital plane of the Earth around the Sun. From the perspective of an observer on Earth, the Sun's movement around the celestial sphere over the course of a year traces out a path along the ecliptic again ...
.
Earth's rotation
Earth's rotation or Earth's spin is the rotation of planet Earth around its own Rotation around a fixed axis, axis, as well as changes in the orientation (geometry), orientation of the rotation axis in space. Earth rotates eastward, in retrograd ...
about its axis causes
diurnal motion
Diurnal motion (, ) is an astronomical term referring to the apparent motion of celestial objects (e.g. the Sun and stars) around Earth, or more precisely around the two celestial poles, over the course of one day. It is caused by Earth's rotatio ...
, 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 pol ...
. The time when the Sun
transits the observer's
meridian
Meridian or a meridian line (from Latin ''meridies'' via Old French ''meridiane'', meaning “midday”) may refer to
Science
* Meridian (astronomy), imaginary circle in a plane perpendicular to the planes of the celestial equator and horizon
* ...
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 letter l ...
.
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
The ecliptic coordinate system is a celestial coordinate system commonly used for representing the apparent positions, orbits, and pole orientations of Solar System objects. Because most planets (except Mercury) and many small Solar System bodi ...
,
# convert to the
equatorial coordinate system
The equatorial coordinate system is a celestial coordinate system widely used to specify the positions of celestial objects. It may be implemented in spherical or rectangular coordinates, both defined by an origin at the centre of Earth, a fund ...
, and
# convert to the
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 th ...
, 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 s ...
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 ...
, 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 astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...
,
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 ca ...
,
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 not ...
,
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 stud ...
,
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 essenti ...
, 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 flat ...
design.
Approximate position
Ecliptic coordinates
These equations, from the ''
Astronomical Almanac
''The Astronomical Almanac''The ''Astronomical Almanac'' for the Year 2015, (United States Naval Observatory/Nautical Almanac Office, 2014) . is an almanac published by the United States Naval Observatory (USNO) and His Majesty's Nautical Almanac ...
'',
can be used to calculate the apparent coordinates of the
Sun
The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared radi ...
,
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 s ...
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 ...
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
In astronomy, an epoch or reference epoch is a moment in time used as a reference point for some time-varying astronomical quantity. It is useful for the celestial coordinates or orbital elements of a celestial body, as they are subject to pertu ...
). If the
Julian date
The Julian day is the continuous count of days since the beginning of the Julian period, and is used primarily by astronomers, and in software for easily calculating elapsed days between two events (e.g. food production date and sell by date).
...
for the desired time is known, then
:
The
mean longitude Mean longitude is the ecliptic longitude at which an orbiting body could be found if its orbit were circular and free of perturbations. While nominally a simple longitude, in practice the mean longitude does not correspond to any one physical angle ...
of the Sun, corrected for the
aberration of light
In astronomy, aberration (also referred to as astronomical aberration, stellar aberration, or velocity aberration) is a phenomenon which produces an apparent motion of celestial objects about their true positions, dependent on the velocity of t ...
, 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 classical ...
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
The ecliptic coordinate system is a celestial coordinate system commonly used for representing the apparent positions, orbits, and pole orientations of Solar System objects. Because most planets (except Mercury) and many small Solar System bodi ...
of the Sun is:
:
The
ecliptic latitude
The ecliptic coordinate system is a celestial coordinate system commonly used for representing the apparent positions, orbits, and pole orientations of Solar System objects. Because most planets (except Mercury) and many small Solar System bod ...
of the Sun is nearly:
:
,
as the
ecliptic latitude
The ecliptic coordinate system is a celestial coordinate system commonly used for representing the apparent positions, orbits, and pole orientations of Solar System objects. Because most planets (except Mercury) and many small Solar System bod ...
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 t ...
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 o ...
is not obtained elsewhere, it can be approximated:
:
Equatorial coordinates
,
and
form a complete position of the
Sun
The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared radi ...
in the
ecliptic coordinate system
The ecliptic coordinate system is a celestial coordinate system commonly used for representing the apparent positions, orbits, and pole orientations of Solar System objects. Because most planets (except Mercury) and many small Solar System bodi ...
. This can be converted to the
equatorial coordinate system
The equatorial coordinate system is a celestial coordinate system widely used to specify the positions of celestial objects. It may be implemented in spherical or rectangular coordinates, both defined by an origin at the centre of Earth, a fund ...
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 o ...
,
, 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 paired w ...
,
:
, 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 the ...
,
:
.
Rectangular equatorial coordinates
Right-handed
In human biology, handedness is an individual's preferential use of one hand, known as the dominant hand, due to it being stronger, faster or more dextrous. The other hand, comparatively often the weaker, less dextrous or simply less subjecti ...
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 t ...
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 verna ...
, 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), whil ...
, and the
axis towards the
North celestial pole
The north and south celestial poles are the two points in the sky where Earth's axis of rotation, indefinitely extended, intersects the celestial sphere. The north and south celestial poles appear permanently directly overhead to observers at ...
.
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 ...
, crossing the
celestial equator
The celestial equator is the great circle of the imaginary celestial sphere on the same plane as the equator of Earth. This plane of reference bases the equatorial coordinate system. In other words, the celestial equator is an abstract projectio ...
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 verna ...
. 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 the ...
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 orbi ...
(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), whil ...
, 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 pol ...
s.
A
line graph
In the mathematical discipline of graph theory, the line graph of an undirected graph is another graph that represents the adjacencies between edges of . is constructed in the following way: for each edge in , make a vertex in ; for every ...
of the Sun's declination during a year resembles a
sine wave
A sine wave, sinusoidal wave, or just sinusoid is a curve, mathematical curve defined in terms of the ''sine'' trigonometric function, of which it is the graph of a function, graph. It is a type of continuous wave and also a Smoothness, smooth p ...
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 amplit ...
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 Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
, 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
The orbital plane of a revolving body is the geometric plane in which its orbit lies. Three non-collinear points in space suffice to determine an orbital plane. A common example would be the positions of the centers of a massive body (host) an ...
(similar to
Uranus
Uranus is the seventh planet from the Sun. Its name is a reference to the Greek god of the sky, Uranus (mythology), Uranus (Caelus), who, according to Greek mythology, was the great-grandfather of Ares (Mars (mythology), Mars), grandfather ...
). 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 Mag ...
, so its declination would be +90°. For the next few months, 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 the sun ...
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 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, the ...
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 a ...
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
Elliptical may mean:
* having the shape of an ellipse, or more broadly, any oval shape
** in botany, having an elliptic leaf shape
** of aircraft wings, having an elliptical planform
* characterised by ellipsis (the omission of words), or by conc ...
. Earth moves more rapidly around the Sun near
perihelion
An apsis (; ) is the farthest or nearest point in the orbit of a planetary body about its primary body. For example, the apsides of the Earth are called the aphelion and perihelion.
General description
There are two apsides in any ellip ...
, in early January, than near
aphelion
An apsis (; ) is the farthest or nearest point in the orbit of a planetary body about its primary body. For example, the apsides of the Earth are called the aphelion and perihelion.
General description
There are two apsides in any ellip ...
, 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
The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared radi ...
, 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 orbi ...
(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 countr ...
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
A solar equinox is a moment in time when the Sun crosses the Earth's equator, which is to say, appears directly above the equator, rather than north or south of the equator. On the day of the equinox, the Sun appears to rise "due east" and se ...
, the center of the Sun appears to pass through the
celestial equator
The celestial equator is the great circle of the imaginary celestial sphere on the same plane as the equator of Earth. This plane of reference bases the equatorial coordinate system. In other words, the celestial equator is an abstract projectio ...
, and δ
☉ is 0°.
The Sun's declination at any given moment is calculated by:
: