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# declination

[dek-luh-ney-shuhn]
declination, in astronomy, one of the coordinates in the equatorial coordinate system. The declination of a celestial body is its angular distance north or south of the celestial equator measured along its hour circle.

In astronomy, declination (abbrev. dec or δ) is one of the two coordinates of the equatorial coordinate system, the other being either right ascension or hour angle. Dec is comparable to latitude, projected onto the celestial sphere, and is measured in degrees north and south of the celestial equator. Therefore, points north of the celestial equator have positive declinations, while those to the south have negative declinations.

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.

A celestial object that passes over zenith, has a declination equal to the observer's latitude, with northern latitudes yielding positive declinations. A pole star therefore has the declination +90° or -90°. Conversely, at northern latitudes φ > 0, celestial objects with a declination greater than 90° - φ, are always visible. Such stars are called circumpolar stars, while the phenomenon of a sun not setting is called midnight sun.

If instead of measuring from and along the equator the angles are measured from and along the horizon, the angles are called azimuth and altitude (elevation).

### Stars

Because a star lies in a nearly constant direction as viewed from earth, its declination is approximately constant from year to year. However, both the right ascension and declination do change gradually due to the effects of precession of the equinoxes and proper motion.

## Varying declination

The declinations of all solar system objects change much more quickly than those of stars.

### Sun

The declination of the Sun (δ) is the angle between the rays of the sun and the plane of the earth's equator. Since the angle between the earth axis and the plane of the earth orbit is nearly constant, δ varies with the seasons and its period is one year, that is the time needed by the earth to complete its revolution around the sun.

When the projection of the earth axis on the plane of the earth orbit is on the same line linking the earth and the sun, the angle between the rays of the sun and the plane of the earth equator is maximum and its value is 23°27'. This happens at the solstices. Therefore δ = +23°27' at the northern hemisphere summer solstice and δ = -23°27' at the northern hemisphere winter solstice. Due to the changes in the tilt of the Earth's axis, the angle between the rays of the sun and the plane of the earth equator is slightly decreasing.

When the projection of the earth axis on the plane of the earth orbit is perpendicular to the line linking the earth and the sun, the angle between the rays of the sun and the plane of the earth equator is null. This happens at the equinoxes. Therefore δ is 0° at the equinoxes.

Sun's declination is equal to inverse sine of the product of sine of Sun's maximum declination and sine of Sun's tropical longitude at any given moment. Instead of computing sun's tropical longitude, if we need sun's declination in terms of days, following procedure is used.

Since the eccentricity of the earth orbit is quite low, it can be approximated to a circle, and δ is approximately given by the following expression:

$delta = -23.45^circ cdot cos left \left[frac\left\{360^circ\right\}\left\{365\right\} cdot left \left(N + 10 right \right) right \right]$

where cos operates on degrees; if cos operates on radians, 360° in the equation needs to be replaced with 2π and will still output δ in degree; $N$ is Day of the Year, that is the number of days spent since January 1.

An alternative form is given as:

$delta = 23.45^circ cdot sin left \left[frac\left\{360^circ\right\}\left\{365\right\} cdot left \left(N + 284 right \right) right \right]$

A more precise formula is given by:

$delta = frac\left\{180^circ\right\}\left\{pi\right\} cdot \left(0.006918 - 0.399912 cos gamma + 0.070257 sin gamma - 0.006758 cos 2gamma + 0.000907 sin 2gamma - 0.002697 cos 3gamma + 0.00148 sin 3gamma\right)$

where

$gamma = frac\left\{2pi\right\}\left\{365\right\} \left(N - 1 \right)$

is the fractional year in radians.

More accurate daily values from averaging the four years of a leap-year cycle are given in the Table of the Declination of the Sun

### Moon

Declination of the Moon is computed by adding Sun's declination (which is called Declination of Place while computing declination of other planets and Moon) to Moon's latitude. Sun's declination (± 23.44°) is much larger in magnitude than Moon's latitude (± 5.14°). Therefore Moon's declination can be said to have an annual cycle synchronous with that of the Sun starting with the vernal equinox.

Moon's latitude is a function of the difference between True Moon and its ascending node. Since lunar nodes make one revolution in nearly 19 years, lunar latitude has an approximately 19 year long cycle. Lunar latitude is equal to inverse sine of the product of sine of maximum lunar latitude and sine of difference between Moon and its node.

For greater accuracy, Reduced Latitude is used instead of Moon's true latitude, which is obtained by multiplying lunar latitude with a multiplier having a maximum value of 1 for tropical Moon at 180° and 0.91745 for tropical Moon at 0°. This is caused by a third cycle in lunar declination which has a period of one lunar month and a maximum range of ± 0.425°. Summing all three componets gives a range of maximum declination from +28°35' to +18°18' and the minimum from -18°18' to -28°35' for lunar declination.

The third component of lunar declination is computed from following formula : Multiplier = Cos D / [Cos {Sin¯(Sin M * Sin D)}] where D is Sun's maximum declination (± 23.44°) and M is Moon's tropical longitude. This multiplier is multiplied into Moon's latitude to get Reduced Latitude. The minimum value of Multiplier is for tropical Moon at zero longitude, which is equal to cosine of Sun's maximum declination, being equal to 0.91745.

This multiplier is used to determine the reduced latitude of other planets as well.