Declination

# Declination

Overview
In astronomy
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...

, declination (abbrev. dec or δ) is one of the two coordinates of the equatorial coordinate system
Equatorial coordinate system
The equatorial coordinate system is a widely-used method of mapping celestial objects. It functions by projecting the Earth's geographic poles and equator onto the celestial sphere. The projection of the Earth's equator onto the celestial sphere is called the celestial equator...

, the other being either right ascension
Right ascension
Right ascension is the astronomical term for one of the two coordinates of a point on the celestial sphere when using the equatorial coordinate system. The other coordinate is the declination.-Explanation:...

or hour angle
Hour angle
In astronomy and celestial navigation, the hour angle is one of the coordinates used in the equatorial coordinate system to give the position of a point on the celestial sphere....

. Declination in astronomy is comparable to geographic latitude
Latitude
In geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...

, but projected onto the celestial sphere
Celestial sphere
In astronomy and navigation, the celestial sphere is an imaginary sphere of arbitrarily large radius, concentric with the Earth and rotating upon the same axis. All objects in the sky can be thought of as projected upon the celestial sphere. Projected upward from Earth's equator and poles are the...

. Declination is measured in degrees north and south of the celestial equator
Celestial equator
The celestial equator is a great circle on the imaginary celestial sphere, in the same plane as the Earth's equator. In other words, it is a projection of the terrestrial equator out into space...

. Points north of the celestial equator have positive declinations, while those to the south have negative declinations.
• An object on the celestial equator
Celestial equator
The celestial equator is a great circle on the imaginary celestial sphere, in the same plane as the Earth's equator. In other words, it is a projection of the terrestrial equator out into space...

has a declination of 0°.
• An object at the celestial north pole
North Pole
The North Pole, also known as the Geographic North Pole or Terrestrial North Pole, is, subject to the caveats explained below, defined as the point in the northern hemisphere where the Earth's axis of rotation meets its surface...

has a declination of +90°.
• An object at the celestial south pole
South Pole
The South Pole, also known as the Geographic South Pole or Terrestrial South Pole, is one of the two points where the Earth's axis of rotation intersects its surface. It is the southernmost point on the surface of the Earth and lies on the opposite side of the Earth from the North Pole...

has a declination of −90°.

The sign is customarily included even if it is positive.
Discussion

Encyclopedia
In astronomy
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...

, declination (abbrev. dec or δ) is one of the two coordinates of the equatorial coordinate system
Equatorial coordinate system
The equatorial coordinate system is a widely-used method of mapping celestial objects. It functions by projecting the Earth's geographic poles and equator onto the celestial sphere. The projection of the Earth's equator onto the celestial sphere is called the celestial equator...

, the other being either right ascension
Right ascension
Right ascension is the astronomical term for one of the two coordinates of a point on the celestial sphere when using the equatorial coordinate system. The other coordinate is the declination.-Explanation:...

or hour angle
Hour angle
In astronomy and celestial navigation, the hour angle is one of the coordinates used in the equatorial coordinate system to give the position of a point on the celestial sphere....

. Declination in astronomy is comparable to geographic latitude
Latitude
In geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...

, but projected onto the celestial sphere
Celestial sphere
In astronomy and navigation, the celestial sphere is an imaginary sphere of arbitrarily large radius, concentric with the Earth and rotating upon the same axis. All objects in the sky can be thought of as projected upon the celestial sphere. Projected upward from Earth's equator and poles are the...

. Declination is measured in degrees north and south of the celestial equator
Celestial equator
The celestial equator is a great circle on the imaginary celestial sphere, in the same plane as the Earth's equator. In other words, it is a projection of the terrestrial equator out into space...

. Points north of the celestial equator have positive declinations, while those to the south have negative declinations.
• An object on the celestial equator
Celestial equator
The celestial equator is a great circle on the imaginary celestial sphere, in the same plane as the Earth's equator. In other words, it is a projection of the terrestrial equator out into space...

has a declination of 0°.
• An object at the celestial north pole
North Pole
The North Pole, also known as the Geographic North Pole or Terrestrial North Pole, is, subject to the caveats explained below, defined as the point in the northern hemisphere where the Earth's axis of rotation meets its surface...

has a declination of +90°.
• An object at the celestial south pole
South Pole
The South Pole, also known as the Geographic South Pole or Terrestrial South Pole, is one of the two points where the Earth's axis of rotation intersects its surface. It is the southernmost point on the surface of the Earth and lies on the opposite side of the Earth from the North Pole...

has a declination of −90°.

The sign is customarily included even if it is positive. Any unit of angle can be used for declination, but it is often expressed in degrees, minutes, and seconds of arc
Minute of arc
A minute of arc, arcminute, or minute of angle , is a unit of angular measurement equal to one sixtieth of one degree. In turn, a second of arc or arcsecond is one sixtieth of one minute of arc....

.

As seen from locations in the Earth's Northern Hemisphere, celestial objects with declinations greater than 90° − φ (φ = observer's latitude) are always above the horizon. This similarly occurs in the Southern Hemisphere for objects with declinations less than, i.e. more negative than, -90° − φ. Such stars appear to circle daily around the celestial pole without dipping below the horizon, and are therefore called circumpolar star
Circumpolar star
A circumpolar star is a star that, as viewed from a given latitude on Earth, never sets , due to its proximity to one of the celestial poles...

s. An extreme example is the pole star
Pole star
The term "Pole Star" usually refers to Polaris, which is the current northern pole star, also known as the North Star.In general, however, a pole star is a visible star, especially a prominent one, that is approximately aligned with the Earth's axis of rotation; that is, a star whose apparent...

which has a declination near to +90°, so it is circumpolar as seen from anywhere in the Northern Hemisphere except very close to the equator. The Sun's declination varies with the seasons (see below). As seen from arctic or antarctic latitudes, the Sun is circumpolar near the local summer solstice, leading to the phenomenon of it being above the horizon at midnight, which is called midnight sun
Midnight sun
The midnight sun is a natural phenomenon occurring in summer months at latitudes north and nearby to the south of the Arctic Circle, and south and nearby to the north of the Antarctic Circle where the sun remains visible at the local midnight. Given fair weather, the sun is visible for a continuous...

.

When an object is directly overhead, at the zenith
Zenith
The zenith is an imaginary point directly "above" a particular location, on the imaginary celestial sphere. "Above" means in the vertical direction opposite to the apparent gravitational force at that location. The opposite direction, i.e...

, its declination is almost always within 0.01 degree of the observer's latitude. It would be exactly equal except for three complications. The first complication applies to all celestial objects. The object's declination equals its astronomic latitude, i.e. the latitude of its position on the celestial sphere. The term "latitude" ordinarily means geodetic latitude, which is the latitude shown on maps and GPS devices, and takes into account the non-spherical shape of the Earth. A line from the center of the Earth to the object is not quite perpendicular to the Earth's surface. This effect usually doesn't exceed a few thousandths of a degree. The second complication occurs mainly in a few places such as the big island of Hawaii, where it can exceed 0.01 degree. Dense underground rocks affect the local direction of the force of gravity, changing the direction that is perceived as "overhead". For practical purposes the third complication only applies to nearby objects. Declination is ordinarily defined relative to the center of the earth. If, because of the preceding two effects, an observer at the surface is not exactly on the line between the centre of the Earth and the celestial object, he will see the object in a slightly different direction because of parallax. The importance of this complication is inversely proportional to the object's distance from the Earth. For most practical purposes it's only a concern with the moon, with asteroids that pass nearby, and with artificial spacecraft close to the Earth. In the case of the moon, this effect never exceeds 0.005 degrees.

## Stars

A star
Star
A star is a massive, luminous sphere of plasma held together by gravity. At the end of its lifetime, a star can also contain a proportion of degenerate matter. The nearest star to Earth is the Sun, which is the source of most of the energy on Earth...

lies in a nearly constant direction as viewed from Earth
Earth
Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...

, with its declination roughly constant from year to year, but right ascension
Right ascension
Right ascension is the astronomical term for one of the two coordinates of a point on the celestial sphere when using the equatorial coordinate system. The other coordinate is the declination.-Explanation:...

and declination do both change gradually due to precession of the equinoxes
Precession of the equinoxes
In astronomy, axial precession is a gravity-induced, slow and continuous change in the orientation of an astronomical body's rotational axis. In particular, it refers to the gradual shift in the orientation of Earth's axis of rotation, which, like a wobbling top, traces out a pair of cones joined...

, proper motion
Proper motion
The proper motion of a star is its angular change in position over time as seen from the center of mass of the solar system. It is measured in seconds of arc per year, arcsec/yr, where 3600 arcseconds equal one degree. This contrasts with radial velocity, which is the time rate of change in...

, and annual parallax
Parallax
Parallax is a displacement or difference in the apparent position of an object viewed along two different lines of sight, and is measured by the angle or semi-angle of inclination between those two lines. The term is derived from the Greek παράλλαξις , meaning "alteration"...

.

## Varying declination

The declinations of all solar system
Solar System
The Solar System consists of the Sun and the astronomical objects gravitationally bound in orbit around it, all of which formed from the collapse of a giant molecular cloud approximately 4.6 billion years ago. The vast majority of the system's mass is in the Sun...

objects change much more quickly than those of stars.

## Declination of the Sun as seen from Earth

The declination of the Sun
Sun
The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...

, δ, is the angle between the rays of the Sun and the plane of the Earth's equator. The Earth's axial tilt
Axial tilt
In astronomy, axial tilt is the angle between an object's rotational axis, and a line perpendicular to its orbital plane...

(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 gradually over thousands of years, but its current value is about ε = 23°26'. Because this axial tilt is nearly constant, solar declination (δ) varies with the season
Season
A season is a division of the year, marked by changes in weather, ecology, and hours of daylight.Seasons result from the yearly revolution of the Earth around the Sun and the tilt of the Earth's axis relative to the plane of revolution...

s and its period is one year
Year
A year is the orbital period of the Earth moving around the Sun. For an observer on Earth, this corresponds to the period it takes the Sun to complete one course throughout the zodiac along the ecliptic....

.

At the solstice
Solstice
A solstice is an astronomical event that happens twice each year when the Sun's apparent position in the sky, as viewed from Earth, reaches its northernmost or southernmost extremes...

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
Equinox
An equinox occurs twice a year, when the tilt of the Earth's axis is inclined neither away from nor towards the Sun, the center of the Sun being in the same plane as the Earth's equator...

, the center of the Sun appears to pass through the celestial equator
Celestial equator
The celestial equator is a great circle on the imaginary celestial sphere, in the same plane as the Earth's equator. In other words, it is a projection of the terrestrial equator out into space...

, and δ is 0°.

The Sun's declination at any given moment is calculated by:

Where EL is the ecliptic longitude. (On some calculator keyboards, and elsewhere, arcsin is written as sin−1.) Since the Earth's orbital eccentricity
Orbital eccentricity
The orbital eccentricity of an astronomical body is the amount by which its orbit deviates from a perfect circle, where 0 is perfectly circular, and 1.0 is a parabola, and no longer a closed orbit...

is small, its orbit can be approximated as a circle which causes up to 1 degree of error. The circle approximation means the EL would be 90 degrees ahead of the solstices in Earth's orbit (at the equinoxes), so that sin(EL) can be written as sin(90+NDS)=cos(NDS) where NDS is the number of days after the December solstice. By also using the approximation that arcsin[sin(d)*cos(NDS)] is close to d*cos(NDS), the following frequently used formula is obtained:

where N is the day of the year beginning with N=0 at midnight Coordinated Universal Time
Coordinated Universal Time
Coordinated Universal Time is the primary time standard by which the world regulates clocks and time. It is one of several closely related successors to Greenwich Mean Time. Computer servers, online services and other entities that rely on having a universally accepted time use UTC for that purpose...

as January 1st begins. The number 10, in (N+10), is the approximate number of days after the December solstice to January 1st. This equation overestimates the declination near the September equinox by up to +1.5 degrees. The sine function approximation by itself leads to an error of up to 0.26 degrees and has been discouraged for use in solar energy applications. The 1971 Spencer formula (based on a fourier series) is also discouraged for having an error of up to 0.28 degrees. An additional error of up to 0.5 degrees can occur in all equations around the equinoxes if not using a decimal place when selecting N to adjust for the time after Coordinated Universal Time midnight for the beginning of that day. So the above equation can have up to 2.0 degrees of error, about 4 times the Sun's angular width, depending on how it's used.

The declination can be more accurately calculated by not making the two approximations, using the parameters of the Earth's orbit to more accurately estimate EL:

where N = 0 at midnight Coordinated Universal Time as January 1st begins and can include decimals to adjust for local times later or earlier in the day. The number 2, in (N-2), is the approximate number of days after January 1 to the Earth's perihelion. The number 0.0167 is the current value of the eccentricity
Orbital eccentricity
The orbital eccentricity of an astronomical body is the amount by which its orbit deviates from a perfect circle, where 0 is perfectly circular, and 1.0 is a parabola, and no longer a closed orbit...

of the Earth's orbit. The eccentricity varies very slowly over time, but for dates fairly close to the present, it can be considered to be constant. The largest errors in this equation are less than +/- 0.2 degrees, but are less than +/- 0.03 degrees for a given year if the number 10 is adjusted up or down in fractional days as determined by how far the previous year's December solstice occurred before or after noon on December 22nd. These accuracies are compared to NOAA's advanced calculations which are based on the 1999 Jean Meeus algorithm that is accurate to within 0.01 degree.

(The above calculation is related to a reasonably simple and accurate calculation of the Equation of Time, which is described here.)

More complicated algorithms correct for changes to the ecliptic longitude by using terms in addition to the 1st-order eccentricity correction above. They also correct the 23.44-degree obliquity which changes very slightly with time. Corrections may also include the effects of the moon in offsetting the Earth's position from the center of the pair's orbit around the Sun. After obtaining the declination relative to the center of the Earth, a further correction for parallax
Parallax
Parallax is a displacement or difference in the apparent position of an object viewed along two different lines of sight, and is measured by the angle or semi-angle of inclination between those two lines. The term is derived from the Greek παράλλαξις , meaning "alteration"...

is applied, which depends on the observer's distance away from the center of the Earth. This correction is less than 0.0025 degrees. The error in calculating the position of the center of the Sun can be less than 0.00015 degrees. For comparison, the Sun's width is about 0.5 degrees. The declination calculations do not include the effects of the refraction of light in the atmosphere, which causes the apparent angle of elevation of the Sun as seen by an observer to be higher than the actual angle of elevation, especially at low Sun elevations. For example, when the Sun is at an elevation of 10 degrees, it appears to be at 10.1 degrees. The declination can be used in calculating the Sun's azimuth and its true elevation, which can then be corrected for refraction to give the Sun's apparent position.

• Celestial coordinate system
Celestial coordinate system
In astronomy, a celestial coordinate system is a coordinate system for mapping positions on the celestial sphere.There are different celestial coordinate systems each using a system of spherical coordinates projected on the celestial sphere, in analogy to the geographic coordinate system used on...

• Ecliptic
Ecliptic
The ecliptic is the plane of the earth's orbit around the sun. In more accurate terms, it is the intersection of the celestial sphere with the ecliptic plane, which is the geometric plane containing the mean orbit of the Earth around the Sun...

• Geographic coordinate system
Geographic coordinate system
A geographic coordinate system is a coordinate system that enables every location on the Earth to be specified by a set of numbers. The coordinates are often chosen such that one of the numbers represent vertical position, and two or three of the numbers represent horizontal position...

• Inclination
Inclination
Inclination in general is the angle between a reference plane and another plane or axis of direction.-Orbits:The inclination is one of the six orbital parameters describing the shape and orientation of a celestial orbit...

• Lunar standstill
Lunar standstill
At a major lunar standstill, which takes place every 18.6 years, the range of the declination of the Moon reaches a maximum. As a result, at high latitudes, the Moon appears to move in just two weeks from high in the sky to low on the horizon...

• Setting circles
Setting circles
Setting circles are used on telescopes equipped with an equatorial mount to find astronomical objects in the sky by their equatorial coordinates often used in star charts or ephemeris.-Description:...

• Euler angles
Euler angles
The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body. To describe such an orientation in 3-dimensional Euclidean space three parameters are required...

• Equation of time
Equation of time
The equation of time is the difference between apparent solar time and mean solar time. At any given instant, this difference will be the same for every observer...