# Declination

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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 unto the celestial sphere, and is measured in degrees north and south of the celestial equator. Therefore, points north of the celestial equator have positive declination, while those to the south have negative declination.

Note that the sign must be included even if positive.

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, celestial objects with a declination higher than 90° - φ, where φ is the latitude, are visible the whole sidereal day. Such stars are called circumpolar stars, while the phenomenon of a sun not setting is called midnight sun.

## Varying declination

The declination of all celestial objects vary over time, in different periods.

### Sun

The declination of the sun (δ) is the angle between the rays of the sun and the plane of the earth 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.

When the projection of the earth axis on the plane of the earth orbit is perpendicular to 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.

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:

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

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.

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.

The errors caused by this approximation are then contemplated by the Equation of Time.

### Moon

The Moon also has an annual cycle, with maximum declination at northern hemisphere midwinter and minimum at midsummer. There is also an approximately 19 year long cycle, varying the maximum declination from +28°35' to +18°18' and the minimum from -18°18' to -28°35'.

### Stars

The stars have approximately the same declination from year to year, but they do have proper motion that can be measured in whole degrees after the passing of centuries.