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2013-12-30T21:53:56Z

108.41.20.106: Undid revision 588408268 by 83.24.39.55 (talk)

The '''equatorial coordinate system''' is a widely used [[celestial coordinate system]] used to specify the positions of 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 center of the [[Earth]], a fundamental [[Plane_(geometry)|plane]] consisting of the projection of the Earth's [[equator]] onto the [[celestial sphere]] (forming the [[celestial equator]]), a primary direction towards the vernal [[equinox]], and a [[right-handed]] convention.<ref>
{{cite book
| author = Nautical Almanac Office, U.S. Naval Observatory
| coauthors = H.M. Nautical Almanac Office, Royal Greenwich Observatory
| title = Explanatory Supplement to the Astronomical Ephemeris and the American Ephemeris and Nautical Almanac
| publisher = H.M. Stationery Office, London
| year = 1961
|pages=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
|year=2001
|isbn=1-881883-12-4
|page=157}}</ref>

[[File:Ra and dec demo animation small.gif|right|350px|thumb|The '''equatorial coordinate system''' in [[Spherical coordinate system|spherical coordinates]]. The fundamental plane is formed by projection of the [[Earth]]'s [[equator]] onto the [[celestial sphere]], forming the [[celestial equator]] (blue). The primary direction is established by projecting the [[Earth]]'s [[orbit]] onto the [[celestial sphere]], forming the [[ecliptic]] {red), and setting up the ascending node of the ecliptic on the celestial equator, the vernal [[equinox]]. [[Right ascension]]s are measured eastward along the celestial equator from the equinox, and [[declination]]s are measured positive northward from the celestial equator - two such coordinate pairs are shown here. Projections of the Earth's north and south [[geographic pole]]s form the north and south [[celestial pole]]s, respectively.]]

The origin at the center of the Earth means the coordinates are ''geocentric'', that is, as seen from the center of the Earth as if it were [[Transparency_and_translucency|transparent]] and [[refraction|nonrefracting]].<ref>
{{cite book
| author = U.S. Naval Observatory Nautical Almanac Office
| first =Nautical Almanac Office
| coauthors = U.K. Hydrographic Office, H.M. Nautical Almanac Office
| title = The Astronomical Almanac for the Year 2010
| publisher = U.S. Govt. Printing Office
| year = 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 the Earth's [[equator]] and [[geographic pole|pole]], does not rotate with the Earth, but remains relatively fixed against the background [[star]]s. A right-handed convention means that coordinates are positive toward the north and toward the east in the fundamental plane.

== Primary direction ==
{{also|Axial precession|Nutation}}

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, [[Axial_precession|precession]], causes a slow, continuous turning of the coordinate system westward about the poles of the [[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, [[nutation]].<ref>
''Explanatory Supplement'' (1961), pp. 20, 28</ref>

In order to fix the exact primary direction, these motions necessitate the specification of the [[equinox]] of a particular date, known as an [[Epoch_(astronomy)|epoch]], when giving a position. The three most commonly used are:
*Mean equinox of (a standard epoch, usually [[Epoch_(astronomy)|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 [[Axial_precession|precession]] to its position at "date", but free from the small periodic oscillations of [[nutation]]). Commonly used in planetary [[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
| year = 1991
|page=137
|ISBN=0-943396-35-2 }}</ref>

== Spherical coordinates ==

=== Use in astronomy ===
A [[star]]'s spherical coordinates are often expressed as a pair, [[right ascension]] and [[declination]], without a [[distance]] coordinate. Because of the great distances to most celestial objects, astronomers often have little or no information on their exact distances, and hence use only the direction. 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.

[[Telescope]]s equipped with [[equatorial mount]]s 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.

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

===Declination===
{{Main|Declination}}
Declination (symbol {{math|''δ''}}, 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]].<ref name="calculator28">{{cite book
|title=Practical Astronomy with Your Calculator, third edition
|author=Peter Duffett-Smith
|publisher=Cambridge University Press
|isbn=0-521-35699-7
|pages=28–29
}}</ref><ref name="simple">{{cite book
|title=Astronomy Made Simple
|author=Meir H. Degani
|isbn=0-385-08854-X
|year=1976
|publisher=Doubleday & Company, Inc
|page=216
}}</ref><ref>
''Astronomical Almanac 2010'', p. M4</ref>

===Right ascension===
{{Main|Right ascension}}
Right ascension (symbol {{math|''α''}}, abbreviated RA) measures the angular distance of an object eastward along the [[celestial equator]] from the vernal [[equinox]] to the [[hour circle]] passing through the object. The vernal equinox point is one of the two where the [[ecliptic]] intersects the celestial equator. Analogous to terrestrial [[longitude]], right ascension is usually measured in [[sidereal time|sidereal]] hours, minutes and seconds instead of degrees, a result of the method of measuring right ascensions by [[Meridian circle|timing the passage of objects across the meridian]] as the [[Earth's_rotation|Earth rotates]]. There are (360° / 24h) = 15° in one hour of right ascension, 24h of right ascension around the entire [[celestial equator]].<ref name="calculator28"/><ref>
{{cite web
|url=http://books.google.com/books?id=PJoUAQAAMAAJ&dq=astronomy+moulton&source=gbs_navlinks_s
|title = An Introduction to Astronomy
|last1 = Moulton
|first1 = Forest Ray
|page=127
|year = 1918}}, at [http://books.google.com/books Google books]</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, [[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_(astronomy)|meridian]] to the [[hour circle]] passing through the object. Unlike right ascension, hour angle is always increasing with the rotation of the Earth. Hour angle may be considered a means of measuring the time since an object crossed the meridian.
A star on the observer's [[Meridian (astronomy)|celestial meridian]] is said to have a zero hour angle. One [[Sidereal_time|sidereal hour]] later (approximately 0.9973 [[solar time|solar hours]] later), the [[Earth's rotation]] will carry the star to the west of the meridian, and its hour angle will be +1h. 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
|publisher=Cambridge University Press
|isbn=0-521-35699-7
|pages=34–36
}}</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 center 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 center of the [[Earth]].
*The fundamental [[Plane_(geometry)|plane]] in the plane of the Earth's equator.
*The primary direction (the {{math|''x''}} axis) toward the vernal [[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 {{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, remaining always directed toward the [[equinox]], and drifting over time with the motions of [[Axial_precession|precession]] and [[nutation]].
* In [[astronomy]]:<ref>
''Explanatory Supplement'' (1961), pp. 24-26</ref>
**The [[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''² + ''Y''² + ''Z''²}})}}, 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 {{math|''ξ''}}, {{math|''η''}}, {{math|''ζ''}} and a fourth distance coordinate, {{math|''Δ''}} {{math|({{=}}{{radical|''ξ''² + ''η''² + ''ζ''²}})}}, in units of the [[astronomical unit]].

:These rectangular coordinates are related to the corresponding spherical coordinates by

::<math>{X \over R}</math> or <math>{\xi \over \mathit{\Delta}} = \cos \delta \cos \alpha</math>

::<math>{Y \over R}</math> or <math>{\eta \over \mathit{\Delta}} = \cos \delta \sin \alpha</math>

::<math>{Z \over R}</math> or <math>{\zeta \over \mathit{\Delta}} = \sin \delta</math>.

*In [[astrodynamics]]:<ref>
Vallado (2001), pp. 157, 158</ref>
**The positions of artificial Earth [[satellite]]s are specified in ''geocentric equatorial'' coordinates, also known as ''geocentric equatorial inertial (GEI)'', ''[[Earth-centered 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]]s <math>\hat{I}</math>, <math>\hat{J}</math> and <math>\hat{K}</math>.
**The ''Geocentric Celestial Reference Frame (GCRF)'' is the geocentric equivalent of the [[International Celestial Reference Frame]] (ICRF). Its primary direction is the [[equinox]] of [[Epoch (astronomy)|J2000.0]], and does not move with [[Axial_precession|precession]] and [[nutation]], but it is otherwise equivalent to the above systems.

{| class="wikitable" style="float: right;"
|+ '''Summary of notation for astronomical equatorial coordinates'''<ref>''Explanatory Supplement'' (1961), sec. 1G</ref>
| rowspan="2" bgcolor="#89CFF0" |  
| colspan="3" align="center" bgcolor="#89CFF0" | '''[[Spherical_coordinates|spherical]]'''
| colspan="2" align="center" bgcolor="#89CFF0" | '''[[Cartesian coordinate system|rectangular]]'''
|- bgcolor="#89CFF0" align="center"
| [[right ascension]]
| [[declination]]
| [[distance]]
| general
| special-purpose
|- align="center"
| bgcolor="#89CFF0" | '''geocentric'''
| <math>\alpha</math>
| <math>\delta</math>
| <math>\mathit{\Delta}</math>
| <math>\xi</math>, <math>\eta</math>, <math>\zeta</math>
| <math>X</math>, <math>Y</math>, <math>Z</math> (Sun)
|- align="center"
| bgcolor="#89CFF0" | '''heliocentric'''
|  
|  
|  
| <math>x</math>, <math>y</math>, <math>z</math>
|}

===Heliocentric equatorial coordinates===

In [[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 center of the [[Sun]].
*The fundamental [[Plane_(geometry)|plane]] in the plane of the Earth's equator.
*The primary direction (the {{math|''x''}} axis) toward the vernal [[equinox]].
*A [[right-handed]] convention, specifying a {{math|''y''}} axis 90° to the east in the fundamental plane and a {{math|''z''}} axis along [[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 center of the [[Sun]]. It is commonly used in planetary orbit calculation.
The three astronomical rectangular coordinate systems are related by

:<math>\xi = x + X</math>

:<math>\eta = y + Y</math>

:<math>\zeta = z + Z</math>.<ref>
''Explanatory Supplement'' (1961), pp. 20, 27</ref>

==See also==
* [[Celestial coordinate system]]
*[[Polar distance (astronomy)|Polar distance]]
*[[Spherical astronomy]]

== 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

==References==
{{reflist}}

{{Celestial coordinate systems}}

[[Category:Celestial coordinate system]]

Lévy arcsine law

2012-02-24T02:31:23Z

108.41.173.107: typo

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