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# Celestial coordinate system

In astronomy, a celestial coordinate system is a coordinate system for mapping positions in the sky. 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 the surface of the Earth. The coordinate systems differ only in their choice of the fundamental plane, which divides the sky into two equal hemispheres along a great circle. For example, the fundamental plane of the geographic system is the Earth's equator. Each coordinate system is named for its choice of fundamental plane.

## Coordinate systems

Coordinate system Fundamental plane Poles Coordinates Epoch
Horizontal horizon zenith/nadir elevation - azimuth - meridian
Equatorial celestial equator celestial poles declination - right ascension or hour angle B1950, J2000
Ecliptic ecliptic ecliptic poles ecliptic latitude - ecliptic longitude
Galactic galactic plane galactic poles
Supergalactic supergalactic plane

### Equatorial coordinate system

Popular choices of pole and equator are the older B1950 and the modern J2000 systems, but a pole and equator "of date" can also be used, meaning one appropriate to the date under consideration, such as that at which a measurement of the position of a planet or spacecraft is made. There are also subdivisions into "mean of date" coordinates, which average out or ignore nutation, and "true of date," which include nutation.

## Elevation angle

Elevation angle, also referred to as altitude, refers to the vertical angle measured from the geometric horizon (0°) towards the zenith (+90°). It can also take negative values for objects below the horizon, down to the nadir (-90°). Although some will use the term height instead of elevation, this is not recommended as height is usually understood to be a linear distance unit, to be expressed in meters (or any other length unit), and not an angular distance.

The term zenith distance is more often used in astronomy and is the complement of the elevation. That is: 0° in the zenith, 90° on the horizon, up to 180° at the nadir.

## Converting coordinates

### Equatorial to horizontal coordinates

Let δ be the declination and $H$ the hour angle.

Let φ be the observer's latitude.

Let El be the elevation angle and Az the azimuth angle.

Let θ be the zenith (or zenith distance, i.e. the 90° complement of Alt).

Then the equations of the transformation are:

$sin mathrm\left\{El\right\} = cos theta = sin phi cdot sin delta + cos phi cdot cos delta cdot cos H$

$cos mathrm\left\{Az\right\} = frac\left\{cos phi cdot sin delta - sin phi cdot cos delta cdot cos H\right\}\left\{cos mathrm\left\{El\right\}\right\}.$

Use the inverse trigonometric functions to get the values of the coordinates.

NOTE: Inverse cosine is dual valued, i.e. 160° and 200° both have the same cosine. The above needs to be corrected. If H < 180 (or Pi radians) then Az = 360 - Az as derived from the above equation.