View source for Astronomy:Equatorial coordinate system

{{short description|celestial coordinate system used to specify the positions of celestial objects}}
[[File:Ra and dec demo animation small.gif|right|350px|thumb|The '''equatorial coordinate system''' using [[Spherical coordinate system|spherical coordinates]]. The [[Fundamental plane (spherical coordinates)|fundamental plane]] is formed by [[Projective geometry|projection]] of Earth's equator onto the [[Celestial sphere|celestial sphere]], forming the {{colorbox|lightblue}}{{nbsp}}[[Celestial equator|celestial equator]]. The primary direction is established by projecting Earth's orbit onto the [[Celestial sphere|celestial sphere]], forming the {{colorbox|red}}{{nbsp}}ecliptic, and setting up the ascending node of the ecliptic on the celestial equator, forming the vernal equinox. [[Right ascension|Right ascension]] is measured eastward along the celestial equator from the equinox, and [[Declination|declination]] is measured positive northward from the celestial equator. (Two such coordinate pairs are shown here.) Projections of the Earth's north and south geographic poles form the north and south celestial poles, respectively.]]

The '''equatorial coordinate system''' is a [[Astronomy:Celestial coordinate system|celestial coordinate system]] widely used to specify the positions of [[Astronomy:Astronomical object|celestial objects]]. It may be implemented in [[Spherical coordinate system|spherical]] or [[Cartesian coordinate system|rectangular]] coordinates, both defined by an [[Origin (mathematics)|origin]] at the centre of [[Astronomy:Earth|Earth]], a [[Fundamental plane (spherical coordinates)|fundamental plane]] consisting of the [[Projective geometry|projection]] of Earth's [[Astronomy:Equator|equator]] onto the [[Celestial sphere|celestial sphere]] (forming the [[Celestial equator|celestial equator]]), a primary direction towards the vernal [[Astronomy:Equinox (celestial coordinates)|equinox]], and a [[Physics:Right-hand rule|right-handed]] convention.<ref>
{{cite book
 | url = https://archive.org/details/astronomicalalmanac1961
 | author = Nautical Almanac Office, U.S. Naval Observatory
 | author2 = H.M. Nautical Almanac Office
 | author3 = Royal Greenwich Observatory
 | title = Explanatory Supplement to the Astronomical Ephemeris and the American Ephemeris and Nautical Almanac
 | publisher = H.M. Stationery Office, London (reprint 1974)
 | date = 1961
 | pages = [https://archive.org/details/astronomicalalmanac1961/page/n34 24], 26
}}</ref><ref name="Vallado">
{{cite book
 | title=Fundamentals of Astrodynamics and Applications
 | first=David A.
 | last=Vallado
 | publisher=Microcosm Press, El Segundo, CA
 | date=2001
 | isbn=1-881883-12-4
 | page=157
}}</ref>

The origin at the centre of Earth means the coordinates are ''[[Geocentric model|geocentric]]'', that is, as seen from the centre of Earth as if it were [[Transparency and translucency|transparent]].<ref>
{{cite book
 | author = U.S. Naval Observatory Nautical Almanac Office
 | author2 = U.K. Hydrographic Office
 | author3 = H.M. Nautical Almanac Office
 | title = The Astronomical Almanac for the Year 2010
 | publisher = U.S. Govt. Printing Office
 | date = 2008
 | page = M2, "apparent place"
 | isbn = 978-0-7077-4082-9
}}</ref> The fundamental plane and the primary direction mean that the coordinate system, while aligned with Earth's [[Astronomy:Equator|equator]] and pole, does not rotate with the Earth, but remains relatively fixed against the background [[Astronomy:Fixed stars|stars]]. A right-handed convention means that coordinates increase northward from and eastward around the fundamental plane.

== Primary direction ==

This description of the [[Orientation (geometry)|orientation]] of the reference frame is somewhat simplified; the orientation is not quite fixed. A slow motion of Earth's axis, [[Astronomy:Axial precession|precession]], causes a slow, continuous turning of the coordinate system westward about the poles of the [[Astronomy:Ecliptic|ecliptic]], completing one circuit in about 26,000 years. Superimposed on this is a smaller motion of the ecliptic, and a small oscillation of the Earth's axis, [[Astronomy:Astronomical nutation|nutation]].<ref>''Explanatory Supplement'' (1961), pp. 20, 28</ref>

In order to fix the exact primary direction, these motions necessitate the specification of the [[Astronomy:Equinox|equinox]] of a particular date, known as an [[Astronomy:Epoch|epoch]], when giving a position. The three most commonly used are:
; Mean equinox of a standard epoch (usually [[Astronomy:Epoch|J2000.0]], but may include B1950.0, B1900.0, etc.): is a fixed standard direction, allowing positions established at various dates to be compared directly.
; Mean equinox of date: is the intersection of the ecliptic of "date" (that is, the ecliptic in its position at "date") with the ''mean'' equator (that is, the equator rotated by precession to its position at "date", but free from the small periodic oscillations of nutation). Commonly used in planetary [[Physics:Orbit|orbit]] calculation.
; True equinox of date: is the intersection of the ecliptic of "date" with the ''true'' equator (that is, the mean equator plus nutation). This is the actual intersection of the two planes at any particular moment, with all motions accounted for.

A position in the equatorial coordinate system is thus typically specified ''true equinox and equator of date'', ''mean equinox and equator of J2000.0'', or similar. Note that there is no "mean ecliptic", as the ecliptic is not subject to small periodic oscillations.<ref>
{{cite book
 | last = Meeus
 | first = Jean
 | title = Astronomical Algorithms
 | publisher = Willmann-Bell, Inc., Richmond, VA
 | date = 1991
 |page=137
 |isbn=0-943396-35-2
}}</ref>

== Spherical coordinates ==

=== Use in astronomy ===
A [[Astronomy:Star|star]]'s spherical coordinates are often expressed as a pair, [[Right ascension|right ascension]] and [[Declination|declination]], without a [[Distance|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|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.

[[Astronomy:Telescope|Telescope]]s equipped with [[Astronomy:Equatorial mount|equatorial mount]]s and [[Astronomy:Setting circles|setting circles]] employ the equatorial coordinate system to find objects. Setting circles in conjunction with a [[Astronomy:Star chart|star chart]] or [[Astronomy:Ephemeris|ephemeris]] allow the telescope to be easily pointed at known objects on the celestial sphere.

===Declination===
{{Main|Declination}}
The declination symbol {{math|''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 [[Earth:Latitude|latitude]].<ref name="calculator28">
{{cite book
 | title=Practical Astronomy with Your Calculator, third edition
 | author=Peter Duffett-Smith
 | year=1988
 | publisher={{wipe|Cambridge University Press}}
 | isbn=0-521-35699-7
 | pages=[https://archive.org/details/practicalastrono0000duff/page/28 28–29]
 | url=https://archive.org/details/practicalastrono0000duff/page/28
 }}</ref><ref name="simple">
{{cite book
 | title=Astronomy Made Simple
 | url=https://archive.org/details/astronomymadesim00hamb
 | url-access=registration
 | author=Meir H. Degani
 | isbn=0-385-08854-X
 | date=1976
 | publisher=Doubleday & Company, Inc
 | page=[https://archive.org/details/astronomymadesim00hamb/page/216 216]
}}</ref><ref>
''Astronomical Almanac 2010'', p. M4
</ref>

===Right ascension===
[[File:Hour angle still1.png|thumb|right|300px|As seen from above the Earth's north pole, a star's {{colorbox|cyan}}{{nbsp}}[[Hour angle|local hour angle]] (LHA) for an {{colorbox|red}}{{nbsp}}observer near New York. Also depicted are the star's {{colorbox|green}}{{nbsp}}[[Right ascension|right ascension]] and {{colorbox|blue}}{{nbsp}}Greenwich hour angle (GHA), the {{colorbox|magenta}}{{nbsp}}local mean sidereal time (LMST) and {{colorbox|purple}}{{nbsp}}Greenwich mean sidereal time (GMST). The symbol ʏ identifies the vernal equinox direction.]]

{{Main|Right ascension}}
The right ascension symbol {{math|''α''}}, (lower case "alpha", abbreviated RA) measures the angular distance of an object eastward along the [[Celestial equator|celestial equator]] from the vernal [[Astronomy:Equinox|equinox]] to the [[Hour circle|hour circle]] passing through the object. The vernal equinox point is one of the two points where the [[Astronomy:Ecliptic|ecliptic]] intersects the celestial equator. Right ascension is usually measured in [[Astronomy:Sidereal time|sidereal]] hours, minutes and seconds instead of degrees, a result of the method of measuring right ascensions by [[Astronomy:Meridian circle|timing the passage of objects across the meridian]] as the [[Earth's rotation|Earth rotates]]. There are {{sfrac|360°|24<sup>h</sup>}} = 15° in one hour of right ascension, and 24<sup>h</sup> of right ascension around the entire [[Celestial equator|celestial equator]].<ref name="calculator28"/><ref>
{{cite book
 | url=https://books.google.com/books?id=PJoUAQAAMAAJ
 | title = An Introduction to Astronomy
 | last1 = Moulton
 | first1 = Forest Ray
 | page=127
 | date = 1918
}}</ref><ref>
''Astronomical Almanac 2010'', p. M14
</ref>

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

===Hour angle===
{{Main|Hour angle}}

Alternatively to [[Right ascension|right ascension]], [[Hour angle|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|celestial equator]] from the observer's [[Astronomy:Meridian|meridian]] to the [[Hour circle|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 [[Astronomy:Culmination|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<sup>h</sup>). One [[Astronomy:Sidereal time|sidereal hour]] (approximately 0.9973 [[Astronomy:Solar time|solar hours]]) later, Earth's rotation will carry the star to the west of the meridian, and its hour angle will be 1<sup>h</sup>. When calculating [[Horizontal coordinate system|topocentric]] phenomena, right ascension may be converted into hour angle as an intermediate step.<ref>
{{cite book
 | title=Practical Astronomy with Your Calculator, third edition
 | author=Peter Duffett-Smith
 | year=1988
 | publisher=Cambridge University Press
 | isbn=0-521-35699-7
 | pages=[https://archive.org/details/practicalastrono0000duff/page/34 34–36]
 | url=https://archive.org/details/practicalastrono0000duff/page/34
 }}</ref><ref>''Astronomical Almanac 2010'', p. M8</ref><ref>Vallado (2001), p. 154</ref>

== Rectangular coordinates ==

===Geocentric equatorial coordinates===

[[File:Ra and dec rectangular.png|thumb|300px|Geocentric equatorial coordinates. The [[Origin (mathematics)|origin]] is the centre of the Earth. The fundamental [[Plane (geometry)|plane]] is the plane of the Earth's equator. The primary direction (the {{math|''x''}} axis) is the vernal equinox. A right-handed convention specifies a {{math|''y''}} axis 90° to the east in the fundamental plane; the {{math|''z''}} axis is the north polar axis. The reference frame does not rotate with the Earth, rather, the Earth rotates around the {{math|''z''}} axis.]]

There are a number of [[Cartesian coordinate system|rectangular]] variants of equatorial coordinates. All have:
* The [[Origin (mathematics)|origin]] at the centre of the [[Astronomy:Earth|Earth]].
* The fundamental [[Plane (geometry)|plane]] in the plane of the Earth's equator.
* The primary direction (the {{math|''x''}} axis) toward the vernal [[Astronomy:Equinox|equinox]], that is, the place where the [[Astronomy:Sun|Sun]] crosses the [[Celestial equator|celestial equator]] in a northward direction in its annual apparent circuit around the [[Astronomy:Ecliptic|ecliptic]].
* A right-handed convention, specifying a {{math|''y''}} axis 90° to the east in the fundamental plane and a {{math|''z''}} axis along the north polar axis.

The reference frames do not rotate with the Earth (in contrast to [[Engineering:ECEF|Earth-centred, Earth-fixed]] frames), remaining always directed toward the [[Astronomy:Equinox|equinox]], and drifting over time with the motions of [[Astronomy:Axial precession|precession]] and [[Astronomy:Astronomical nutation|nutation]].
* In [[Astronomy:Astronomy|astronomy]]:<ref>
''Explanatory Supplement'' (1961), pp. 24–26
</ref>
** The [[Astronomy:Position of the Sun|position of the Sun]] is often specified in the geocentric equatorial rectangular coordinates {{math|''X''}}, {{math|''Y''}}, {{math|''Z''}} and a fourth distance coordinate, {{math|''R''}} {{math|({{=}} {{radical|''X''{{isup|2}} + ''Y''{{isup|2}} + ''Z''{{isup|2}}}})}}, in units of the [[Astronomy:Astronomical unit|astronomical unit]].
** The positions of the planets and other [[Astronomy:Solar System|Solar System]] bodies are often specified in the geocentric equatorial rectangular coordinates {{math|''ξ''}}, {{math|''η''}}, {{math|''ζ''}} and a fourth distance coordinate, {{math|''Δ''}} (equal to {{math|{{radical|''ξ''{{isup|2}} + ''η''{{isup|2}} + ''ζ''{{isup|2}}}}}}), in units of the [[Astronomy:Astronomical unit|astronomical unit]].{{paragraph}}These rectangular coordinates are related to the corresponding spherical coordinates by
**: <math>\begin{align}
\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{align}</math>
* In astrodynamics:<ref>
Vallado (2001), pp. 157, 158
</ref>
** The positions of artificial Earth [[Astronomy:Satellite|satellite]]s are specified in ''geocentric equatorial'' coordinates, also known as ''geocentric equatorial inertial (GEI)'', ''[[Astronomy:Earth-centered inertial|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 {{math|''x''}}, {{math|''y''}} and {{math|''z''}} axes are often designated {{math|''I''}}, {{math|''J''}} and {{math|''K''}}, respectively, or the frame's [[Basis (linear algebra)|basis]] is specified by the [[Unit vector|unit vector]]s {{math|''Î''}}, {{math|''Ĵ''}} and {{math|''K̂''}}.
** The ''Geocentric Celestial Reference Frame (GCRF)'' is the geocentric equivalent of the [[Astronomy:International Celestial Reference Frame|International Celestial Reference Frame]] (ICRF). Its primary direction is the [[Astronomy:Equinox|equinox]] of [[Astronomy:Epoch|J2000.0]], and does not move with [[Astronomy:Axial precession|precession]] and [[Astronomy:Astronomical nutation|nutation]], but it is otherwise equivalent to the above systems.

{| class="wikitable floatright" style="text-align:center;"
|+ Summary of notation for astronomical equatorial coordinates<ref>''Explanatory Supplement'' (1961), sec. 1G</ref>
! rowspan="2" style="background:#89cff0;"                    | &nbsp;
! colspan="3" style="text-align:center; background:#89cff0;" | Spherical
! colspan="2" style="text-align:center; background:#89cff0;" | [[Cartesian coordinate system|Rectangular]]
|-
! style="background:#89cff0;" | [[Right ascension]]
! style="background:#89cff0;" | [[Declination]]
! style="background:#89cff0;" | [[Distance]]
! style="background:#89cff0;" | General
! style="background:#89cff0;" | Special-purpose
|-
! style="background:#89cff0;" | Geocentric
| {{mvar|α}}
| {{mvar|δ}}
| {{mvar|Δ}}
| {{mvar|ξ}}, {{mvar|η}}, {{mvar|ζ}}
| {{mvar|X}}, {{mvar|Y}}, {{mvar|Z}} (Sun)
|- style="text-align:center;"
! style="background:#89cff0;" | Heliocentric
| &nbsp;
| &nbsp;
| &nbsp;
| colspan=2 | {{mvar|x}}, {{mvar|y}}, {{mvar|z}}
|}

===Heliocentric equatorial coordinates===

In [[Astronomy:Astronomy|astronomy]], there is also a heliocentric [[Cartesian coordinate system|rectangular]] variant of equatorial coordinates, designated {{math|''x''}}, {{math|''y''}}, {{math|''z''}}, which has:

*The [[Origin (mathematics)|origin]] at the centre of the [[Astronomy:Sun|Sun]].
*The fundamental [[Plane (geometry)|plane]] in the plane of the Earth's equator.
*The primary direction (the {{math|''x''}} axis) toward the vernal [[Astronomy:Equinox|equinox]].
*A right-handed convention, specifying a {{math|''y''}} axis 90° to the east in the fundamental plane and a {{math|''z''}} axis along [[Astronomy:Earth|Earth]]'s north polar axis.

This frame is in every way equivalent to the {{math|''ξ''}}, {{math|''η''}}, {{math|''ζ''}} frame, above, except that the origin is removed to the centre of the [[Astronomy:Sun|Sun]]. It is commonly used in planetary orbit calculation. The three astronomical rectangular coordinate systems are related by<ref>''Explanatory Supplement'' (1961), pp.&nbsp;20, 27</ref>
:<math>\begin{align}
    \xi &= x + X \\
   \eta &= y + Y \\
  \zeta &= z + Z
\end{align}</math>

==See also==
*[[Astronomy:Celestial coordinate system|Celestial coordinate system]]
*[[Astronomy:Planetary coordinate system|Planetary coordinate system]]
*[[Astronomy:Polar distance|Polar distance]]
*[[Astronomy:Spherical astronomy|Spherical astronomy]]
*[[Astronomy:Star position|Star position]]

==References==
{{reflist}}

== External links ==
* [http://stars.astro.illinois.edu/celsph.html MEASURING THE SKY A Quick Guide to the Celestial Sphere] James B. Kaler, University of Illinois
* [http://astro.unl.edu/naap/motion1/cec_units.html Celestial Equatorial Coordinate System] University of Nebraska-Lincoln
* [http://astro.unl.edu/naap/motion1/cec_both.html Celestial Equatorial Coordinate Explorers] University of Nebraska-Lincoln

{{Celestial coordinate systems}}
[[Category:Celestial coordinate system]]

{{Sourceattribution|Equatorial coordinate system|1}}