Equatorial coordinate system

Spherical coordinates

Use in astronomy

A star's spherical coordinates are often expressed as a pair, right ascension and declination, without a distance coordinate. The direction of sufficiently distant objects is the same for all observers, and it is convenient to specify this direction with the same coordinates for all. In contrast, in the horizontal coordinate system, a star's position differs from observer to observer based on their positions on the Earth's surface, and is continuously changing with the Earth's rotation.

Telescopes equipped with equatorial mounts and setting circles employ the equatorial coordinate system to find objects. Setting circles in conjunction with a star chart or ephemeris allow the telescope to be easily pointed at known objects on the celestial sphere.

Declination

The declination symbol δ, (lower case "delta", abbreviated DEC) measures the angular distance of an object perpendicular to the celestial equator, positive to the north, negative to the south. For example, the north celestial pole has a declination of +90°. The origin for declination is the celestial equator, which is the projection of the Earth's equator onto the celestial sphere. Declination is analogous to terrestrial latitude.

Right ascension

The right ascension symbol α, (lower case "alpha", abbreviated RA) measures the angular distance of an object eastward along the celestial equator from the March equinox to the hour circle passing through the object. The March equinox point is one of the two points where the ecliptic intersects the celestial equator. Right ascension is usually measured in sidereal hours, minutes and seconds instead of degrees, a result of the method of measuring right ascensions by timing the passage of objects across the meridian as the Earth rotates. There are ⁠360°/24^h⁠ = 15° in one hour of right ascension, and 24^h of right ascension around the entire celestial equator.

When used together, right ascension and declination are usually abbreviated RA/Dec.

Hour angle

Alternatively to right ascension, hour angle (abbreviated HA or LHA, local hour angle), a left-handed system, measures the angular distance of an object westward along the celestial equator from the observer's meridian to the hour circle passing through the object. Unlike right ascension, hour angle is always increasing with the rotation of Earth. Hour angle may be considered a means of measuring the time since upper culmination, the moment when an object contacts the meridian overhead.

A culminating star on the observer's meridian is said to have a zero hour angle (0^h). One sidereal hour (approximately 0.9973 solar hours) later, Earth's rotation will carry the star to the west of the meridian, and its hour angle will be 1^h. When calculating topocentric phenomena, right ascension may be converted into hour angle as an intermediate step.

Rectangular coordinates: geocentric equatorial coordinates

There are a number of rectangular variants of equatorial coordinates. All have:

• The origin at the centre of the Earth.
• The fundamental plane in the plane of the Earth's equator.
• The primary direction (the x axis) toward the March equinox, that is, the place where the Sun crosses the celestial equator in a northward direction in its annual apparent circuit around the ecliptic.
• A right-handed convention, specifying a y axis 90° to the east in the fundamental plane and a z axis along the north polar axis.

The reference frames do not rotate with the Earth (in contrast to Earth-centred, Earth-fixed frames), remaining always directed toward the equinox, and drifting over time with the motions of precession and nutation.

• In astronomy:
  • The position of the Sun is often specified in the geocentric equatorial rectangular coordinates X, Y, Z and a fourth distance coordinate, R (= √X² + Y² + Z²), in units of the astronomical unit.
  • The positions of the planets and other Solar System bodies are often specified in the geocentric equatorial rectangular coordinates ξ, η, ζ and a fourth distance coordinate, Δ (equal to √ξ² + η² + ζ²), in units of the astronomical unit.These rectangular coordinates are related to the corresponding spherical coordinates by $${\displaystyle \begin{array}{cc}\frac{X}{R}=\frac{\xi }{\mathit{\Delta }}& =\cos \delta \cos \alpha \\ \frac{Y}{R}=\frac{\eta }{\mathit{\Delta }}& =\cos \delta \sin \alpha \\ \frac{Z}{R}=\frac{\zeta }{\mathit{\Delta }}& =\sin \delta \end{array}}$$
• In astrodynamics:
  • The positions of artificial Earth satellites are specified in geocentric equatorial coordinates, also known as geocentric equatorial inertial (GEI), Earth-centred inertial (ECI), and conventional inertial system (CIS), all of which are equivalent in definition to the astronomical geocentric equatorial rectangular frames, above. In the geocentric equatorial frame, the x, y and z axes are often designated I, J and K, respectively, or the frame's basis is specified by the unit vectors Î, Ĵ and K̂.
  • The Geocentric Celestial Reference Frame (GCRF) is the geocentric equivalent of the International Celestial Reference Frame (ICRF). Its primary direction is the equinox of J2000.0, and does not move with precession and nutation, but it is otherwise equivalent to the above systems.

Generalization: heliocentric equatorial coordinates

In astronomy, there is also a heliocentric rectangular variant of equatorial coordinates, designated x, y, z, which has:

• The origin at the centre of the Sun.
• The fundamental plane in the plane of the Earth's equator.
• The primary direction (the x axis) toward the March equinox.
• A right-handed convention, specifying a y axis 90° to the east in the fundamental plane and a z axis along Earth's north polar axis.

This frame is similar to the ξ, η, ζ frame above, except that the origin is removed to the centre of the Sun. It is commonly used in planetary orbit calculation. The three astronomical rectangular coordinate systems are related by $${\displaystyle \begin{array}{cc}\xi & =x+X\\ \eta & =y+Y\\ \zeta & =z+Z\end{array}}$$