Editing Astronomical coordinate systems

{{short description|System for specifying positions of celestial objects}}
{{Infobox
 | title   = Orientation of astronomical coordinates
 | image   = [[File:Ecliptic equator galactic anim.gif|250px]]
 | caption = A [[star]]'s {{colorbox|yellow}}{{nbsp}}[[galactic coordinate system|galactic]], {{colorbox|red}}{{nbsp}}[[ecliptic coordinate system|ecliptic]], and {{colorbox|blue}}{{nbsp}}[[equatorial coordinate system|equatorial]] coordinates, as projected on the [[celestial sphere]]. Ecliptic and equatorial coordinates share the {{colorbox|magenta}}{{nbsp}}[[equinox (celestial coordinates)|March equinox]] as the [[primary direction]], and galactic coordinates are referred to the {{colorbox|yellow}}{{nbsp}}galactic center. The origin of coordinates (the "center of the sphere") is ambiguous; see [[celestial sphere]] for more information.
}}

In [[astronomy]], [[coordinate systems]] are used for specifying [[position (geometry)|positions]] of [[astronomical object|celestial objects]] ([[natural satellite|satellites]], [[planet]]s, [[star]]s, [[galaxy|galaxies]], etc.) relative to a given [[reference frame]], based on physical reference points available to a situated observer (e.g. the true [[horizon]] and [[true north|north]] to an observer on Earth's surface).<ref>{{cite journal |last=Kanas|first=Nick |title=Star and Solar System Maps: A History of Celestial Cartography |journal=Research Notes of the AAS |publisher=[[American Astronomical Society]] |year=2021 |volume=5 |issue=4 |page=69 |doi=10.3847/2515-5172/abf35c |bibcode=2021RNAAS...5...69K |s2cid=233522547 |doi-access=free}}</ref> Coordinate systems in astronomy can specify an object's relative position in [[three-dimensional space]] or [[plot (graphics)|plot]] merely by its [[direction (geometry)|direction]] on a [[celestial sphere]], if the object's distance is unknown or trivial.

[[spherical coordinate system|Spherical coordinates]], projected on the celestial sphere, are analogous to the [[geographic coordinate system]] used on the surface of [[Earth]]. These differ in their choice of [[fundamental plane (spherical coordinates)|fundamental plane]], which divides the celestial sphere into two equal [[sphere|hemisphere]]s along a [[great circle]]. [[Cartesian coordinate system|Rectangular coordinates]], in appropriate [[units of measurement|units]], have the same fundamental ({{math|''x, y''}}) plane and [[primary direction|primary ({{math|''x''}}-axis) direction]], such as an [[rotation around a fixed axis|axis of rotation]]. Each coordinate system is named after its choice of fundamental plane.

== Coordinate systems ==
The following table lists the common coordinate systems in use by the astronomical community. The [[Fundamental plane (spherical coordinates)|fundamental plane]] divides the [[celestial sphere]] into two equal [[Celestial sphere|hemispheres]] and defines the baseline for the latitudinal coordinates, similar to the [[equator]] in the [[geographic coordinate system]]. The poles are located at ±90° from the fundamental plane. The primary direction is the starting point of the longitudinal coordinates. The origin is the zero distance point, the "center of the celestial sphere", although the definition of [[celestial sphere]] is ambiguous about the definition of its center point.

{| class="wikitable" style="text-align:center;"
|-
! rowspan="2" | Coordinate system<ref>{{cite web 
 |url=http://www.faculty.virginia.edu/ASTR5610/lectures/COORDS/coords.html 
 |title=Coordinate Systems 
 |last=Majewski 
 |first=Steve 
 |publisher=UVa Department of Astronomy 
 |access-date=19 March 2011 
 |archive-date=12 March 2016 
 |archive-url=https://web.archive.org/web/20160312195329/http://faculty.virginia.edu/ASTR5610/lectures/COORDS/coords.html 
 |url-status=dead 
 }}</ref>
! rowspan="2" | Center point<br/>(origin)
! rowspan="2" | Fundamental plane<br/>(0° latitude)
! rowspan="2" | Poles
! colspan="2" | Coordinates
! rowspan="2" | Primary direction<br/>(0° longitude)
|-
! Latitude
! Longitude
|-
| [[Horizontal coordinate system|Horizontal]] (also called {{abbr|alt|altitude}}-{{abbr|az|azimuth}} or {{abbr|el|elevation}}-{{abbr|az|azimuth}})
| Observer
| [[Horizon]]
| [[Zenith]], [[nadir (astronomy)|nadir]]
| Altitude ({{math|''a''}}) or elevation
| [[Azimuth]] ({{math|''A''}})
| [[North]] or [[south]] point of horizon
|-
| [[Equatorial coordinate system|Equatorial]]
| rowspan="2" | Center of the [[Earth]]{{nbsp}}(geocentric), or [[Sun]]{{nbsp}}(heliocentric)
| [[Celestial equator]]
| [[Celestial pole]]s
| [[Declination]] ({{math|''δ''}})
| [[Right ascension]] ({{math|''α''}})<br/>or [[hour angle]] ({{math|''h''}})
| rowspan="2" | [[Equinox (celestial coordinates)|March equinox]]
|-
| [[Ecliptic coordinate system|Ecliptic]]
| [[Ecliptic]]
| [[Ecliptic pole]]s
| [[Ecliptic latitude]] ({{math|''β''}})
| [[Ecliptic longitude]] ({{math|''λ''}})
|-
| [[Galactic coordinate system|Galactic]]
| Center of the [[Sun]]
| [[Galactic plane]]
| [[Galactic pole]]s
| Galactic latitude ({{math|''b''}})
| Galactic longitude ({{math|''l''}})
| [[Galactic Center]]
|-
| [[Supergalactic coordinate system|Supergalactic]]
|
| [[Supergalactic plane]]
| Supergalactic poles
| Supergalactic latitude ({{math|''SGB''}})
| Supergalactic longitude ({{math|''SGL''}})
| Intersection of supergalactic plane and galactic plane
|}

===Horizontal system===
{{Main|Horizontal coordinate system}}
[[File:Equatorial_and_horizontal_celestial_coordinates_E.svg|350px|thumb|Equatorial (red) and horizontal (blue) celestial coordinates]]
The ''horizontal'', or [[Horizontal coordinate system|altitude-azimuth]], system is based on the position of the observer on Earth, which revolves around its own axis once per [[sidereal day]] (23 hours, 56 minutes and 4.091 seconds) in relation to the star background. The positioning of a celestial object by the horizontal system varies with time, but is a useful coordinate system for locating and tracking objects for observers on Earth.  It is based on the position of stars relative to an observer's ideal horizon.

===Equatorial system===
{{Main|Equatorial coordinate system}}
The ''equatorial'' coordinate system is centered at Earth's center, but fixed relative to the celestial poles and the [[Equinox (celestial coordinates)|March equinox]]. The coordinates are based on the location of stars relative to Earth's equator if it were projected out to an infinite distance.  The equatorial describes the sky as seen from the [[Solar System]], and modern star maps almost exclusively use equatorial coordinates.

The ''equatorial'' system is the normal coordinate system for most professional and many amateur astronomers having an equatorial mount that follows the movement of the sky during the night. Celestial objects are found by adjusting the telescope's or other instrument's scales so that they match the equatorial coordinates of the selected object to observe.

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 when 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 [[astronomical nutation|nutation]], and "true of date," which include nutation.

===Ecliptic system===
{{Main|Ecliptic coordinate system}}
The fundamental plane is the plane of the Earth's orbit, called the ecliptic plane. There are two principal variants of the ecliptic coordinate system: geocentric ecliptic coordinates centered on the Earth and heliocentric ecliptic coordinates centered on the center of mass of the Solar System.

The geocentric ecliptic system was the principal coordinate system for ancient astronomy and is still useful for computing the apparent motions of the Sun, Moon, and planets.<ref>[[Asger Aaboe|Aaboe, Asger]]. 2001 ''Episodes from the Early History of Astronomy.'' New York: Springer-Verlag., pp. 17–19.</ref> It was used to define the twelve [[astrological sign]]s of the [[zodiac]], for instance.

The heliocentric ecliptic system describes the planets' orbital movement around the Sun, and centers on the [[Center of mass#Barycenter in astrophysics and astronomy|barycenter]] of the Solar System (i.e. very close to the center of the Sun). The system is primarily used for computing the positions of planets and other Solar System bodies, as well as defining their [[orbital elements]].

===Galactic system===
{{Main|Galactic coordinate system}}
The galactic coordinate system uses the approximate plane of the Milky Way Galaxy as its fundamental plane.  The Solar System is still the center of the coordinate system, and the zero point is defined as the direction towards the [[Galactic Center]]. Galactic latitude resembles the elevation above the galactic plane and galactic longitude determines direction relative to the center of the galaxy.

===Supergalactic system===
{{Main|Supergalactic coordinate system}}
The supergalactic coordinate system corresponds to a fundamental plane that contains a higher than average number of local galaxies in the sky as seen from Earth.

== Converting coordinates ==
{{see also|Euler angles|Rotation matrix}}
Conversions between the various coordinate systems are given.<ref name=Meeus>
{{cite book 
 | last = Meeus
 | first = Jean
 | title = Astronomical Algorithms
 | publisher = Willmann-Bell, Inc., Richmond, VA
 | year = 1991
 |isbn=0-943396-35-2 }}, chap. 12
</ref> See the [[#Notes on conversion|notes]] before using these equations.

=== Notation ===
{{div col}}
*Horizontal coordinates
** {{mvar|A}}, [[azimuth]]
** {{mvar|a}}, [[Horizontal coordinate system|altitude]]
*Equatorial coordinates
** {{mvar|α}}, [[right ascension]]
** {{mvar|δ}}, [[declination]]
** {{mvar|h}}, [[hour angle]]
*Ecliptic coordinates
** {{mvar|λ}}, [[ecliptic longitude]]
** {{mvar|β}}, [[ecliptic latitude]]
*Galactic coordinates
** {{mvar|l}}, [[galactic longitude]]
** {{mvar|b}}, [[galactic latitude]]
*Miscellaneous
** {{math|''λ''<sub>o</sub>}}, [[longitude|observer's longitude]]
** {{math|''ϕ''<sub>o</sub>}}, [[latitude|observer's latitude]]
** {{mvar|ε}}, [[Axial tilt#Earth|obliquity of the ecliptic]] (about 23.4°)
** {{math|''θ''<sub>L</sub>}}, [[sidereal time|local sidereal time]]
** {{math|''θ''<sub>G</sub>}}, [[sidereal time|Greenwich sidereal time]]
{{div col end}}

=== Hour angle ↔ right ascension ===

:<math>\begin{align}
       h &= \theta_\text{L} - \alpha & &\mbox{or} &      h &= \theta_\text{G} + \lambda_\text{o} - \alpha \\
  \alpha &= \theta_\text{L} - h      & &\mbox{or} & \alpha &= \theta_\text{G} + \lambda_\text{o} - h
\end{align}</math>

=== Equatorial ↔ ecliptic ===
The classical equations, derived from [[spherical trigonometry]], for the longitudinal coordinate are presented to the right of a bracket; dividing the first equation by the second gives the convenient tangent equation seen on the left.<ref name=ExplSupp>
{{cite book 
 | last1 = U.S. Naval Observatory
 | first1=Nautical Almanac Office
 | last2 = H.M. Nautical Almanac Office
 | title = Explanatory Supplement to the Astronomical Ephemeris and the American Ephemeris and Nautical Almanac
 | publisher = H.M. Stationery Office, London
 | year = 1961
}}, sec. 2A</ref>  The rotation matrix equivalent is given beneath each case.<ref>
{{cite book 
 | last1 = U.S. Naval Observatory
 | first1=Nautical Almanac Office
 | editor = P. Kenneth Seidelmann
 | title = Explanatory Supplement to the Astronomical Almanac
 | publisher = University Science Books, Mill Valley, CA
 | year = 1992
 | isbn = 0-935702-68-7
}}, section 11.43</ref> This division is ambiguous because tan has a period of 180° ({{pi}}) whereas cos and sin have periods of 360° (2{{pi}}).

:<math>\begin{align}
   \tan\left(\lambda\right) &= {\sin\left(\alpha\right) \cos\left(\varepsilon\right) + \tan\left(\delta\right) \sin\left(\varepsilon\right) \over \cos\left(\alpha\right)}; \qquad\begin{cases}
    \cos\left(\beta\right) \sin\left(\lambda\right) = \cos\left(\delta\right) \sin\left(\alpha\right) \cos\left(\varepsilon\right) + \sin\left(\delta\right) \sin\left(\varepsilon\right); \\
    \cos\left(\beta\right) \cos\left(\lambda\right) = \cos\left(\delta\right) \cos\left(\alpha\right).
  \end{cases} \\
     \sin\left(\beta\right) &= \sin\left(\delta\right) \cos\left(\varepsilon\right) - \cos\left(\delta\right) \sin\left(\varepsilon\right) \sin\left(\alpha\right) \\[3pt]
  \begin{bmatrix}
    \cos\left(\beta\right)\cos\left(\lambda\right) \\
    \cos\left(\beta\right)\sin\left(\lambda\right) \\
    \sin\left(\beta\right)
              \end{bmatrix} &= \begin{bmatrix}
    1 &  0                            & 0                            \\
    0 &  \cos\left(\varepsilon\right) & \sin\left(\varepsilon\right) \\
    0 & -\sin\left(\varepsilon\right) & \cos\left(\varepsilon\right)
  \end{bmatrix}\begin{bmatrix}
    \cos\left(\delta\right)\cos\left(\alpha\right) \\
    \cos\left(\delta\right)\sin\left(\alpha\right) \\
    \sin\left(\delta\right)
  \end{bmatrix} \\[6pt]
    \tan\left(\alpha\right) &= {\sin\left(\lambda\right) \cos\left(\varepsilon\right) - \tan\left(\beta\right) \sin\left(\varepsilon\right) \over \cos\left(\lambda\right)} ; \qquad \begin{cases}
    \cos\left(\delta\right) \sin\left(\alpha\right) = \cos\left(\beta\right) \sin\left(\lambda\right) \cos\left(\varepsilon\right) - \sin\left(\beta\right) \sin\left(\varepsilon\right); \\
    \cos\left(\delta\right) \cos\left(\alpha\right) = \cos\left(\beta\right) \cos\left(\lambda\right).
  \end{cases} \\[3pt]
    \sin\left(\delta\right) &= \sin\left(\beta\right) \cos\left(\varepsilon\right) + \cos\left(\beta\right) \sin\left(\varepsilon\right) \sin\left(\lambda\right). \\[6pt]
  \begin{bmatrix}
    \cos\left(\delta\right)\cos\left(\alpha\right) \\
    \cos\left(\delta\right)\sin\left(\alpha\right) \\
    \sin\left(\delta\right)
  \end{bmatrix} &= \begin{bmatrix}
    1 & 0                            &  0                            \\
    0 & \cos\left(\varepsilon\right) & -\sin\left(\varepsilon\right) \\
    0 & \sin\left(\varepsilon\right) &  \cos\left(\varepsilon\right)
  \end{bmatrix}\begin{bmatrix}
    \cos\left(\beta\right)\cos\left(\lambda\right) \\
    \cos\left(\beta\right)\sin\left(\lambda\right) \\
    \sin\left(\beta\right)
  \end{bmatrix}.
\end{align}</math>

=== Equatorial ↔ horizontal ===
Azimuth ({{mvar|A}}) is measured from the south point, turning positive to the west.<ref>
{{cite book
 | last1 = Montenbruck
 | first1 = Oliver
 | last2 = Pfleger
 | first2 = Thomas
 | title = Astronomy on the Personal Computer
 | publisher = Springer-Verlag Berlin Heidelberg
 | year = 2000
 | isbn = 978-3-540-67221-0}}, pp 35-37</ref>
Zenith distance, the angular distance along the [[great circle]] from the [[zenith]] to a celestial object, is simply the [[Complementary angles|complementary angle]] of the altitude: {{math|90° − ''a''}}.<ref>
{{cite book 
 | last1 = U.S. Naval Observatory
 | first1=Nautical Almanac Office
 | first2 = H.M. Nautical Almanac Office
 | last2 = U.K. Hydrographic Office
 | title = The Astronomical Almanac for the Year 2010
 | publisher = U.S. Govt. Printing Office
 | year = 2008
 |isbn = 978-0160820083
 |page=M18}}
</ref>

:<math>\begin{align}
  \tan\left(A\right) &= {\sin\left(h\right) \over \cos\left(h\right) \sin\left(\phi_\text{o}\right) - \tan\left(\delta\right) \cos\left(\phi_\text{o}\right)}; \qquad \begin{cases}
    \cos\left(a\right) \sin\left(A\right) = \cos\left(\delta\right) \sin\left(h\right) ;\\
    \cos\left(a\right) \cos\left(A\right) = \cos\left(\delta\right) \cos\left(h\right) \sin\left(\phi_\text{o}\right) - \sin\left(\delta\right) \cos\left(\phi_\text{o}\right)
  \end{cases} \\[3pt]
  \sin\left(a\right) &= \sin\left(\phi_\text{o}\right) \sin\left(\delta\right) + \cos\left(\phi_\text{o}\right) \cos\left(\delta\right) \cos\left(h\right);
\end{align}</math>

In solving the {{math|tan(''A'')}} equation for {{math|''A''}}, in order to avoid the ambiguity of the [[arctangent]], use of the [[atan2|two-argument arctangent]], denoted {{math|atan2(''x'',''y'')}}, is recommended. The two-argument arctangent computes the arctangent of {{math|{{sfrac|''y''|''x''}}}}, and accounts for the quadrant in which it is being computed. Thus, consistent with the convention of azimuth being measured from the south and opening positive to the west,
:<math>A = -\operatorname{atan2}(y,x)</math>,

where
:<math>\begin{align}
  x &= -\sin\left(\phi_\text{o}\right) \cos\left(\delta\right) \cos\left(h\right) + \cos\left(\phi_\text{o}\right) \sin\left(\delta\right) \\
  y &=  \cos\left(\delta\right) \sin\left(h\right)
\end{align}</math>.

If the above formula produces a negative value for {{math|''A''}}, it can be rendered positive by simply adding 360°.

:<math>\begin{align}
  \begin{bmatrix}
    \cos\left(a\right) \cos\left(A\right) \\
    \cos\left(a\right) \sin\left(A\right) \\
    \sin\left(a\right)
            \end{bmatrix} &= \begin{bmatrix}
    \sin\left(\phi_\text{o}\right) & 0 & -\cos\left(\phi_\text{o}\right) \\
    0                              & 1 &  0                              \\
    \cos\left(\phi_\text{o}\right) & 0 &  \sin\left(\phi_\text{o}\right)
  \end{bmatrix}\begin{bmatrix}
    \cos\left(\delta\right)\cos\left(h\right) \\
    \cos\left(\delta\right)\sin\left(h\right) \\
    \sin\left(\delta\right)
  \end{bmatrix} \\
                          &= \begin{bmatrix}
    \sin\left(\phi_\text{o}\right) & 0 & -\cos\left(\phi_\text{o}\right) \\
    0                              & 1 &  0                              \\
    \cos\left(\phi_\text{o}\right) & 0 &  \sin\left(\phi_\text{o}\right)
  \end{bmatrix}\begin{bmatrix}
    \cos\left(\theta_L\right) &  \sin\left(\theta_L\right) & 0 \\
    \sin\left(\theta_L\right) & -\cos\left(\theta_L\right) & 0 \\
    0            &  0            & 1
  \end{bmatrix}\begin{bmatrix}
    \cos\left(\delta\right)\cos\left(\alpha\right) \\
    \cos\left(\delta\right)\sin\left(\alpha\right) \\
    \sin\left(\delta\right)
  \end{bmatrix}; \\[6pt]
       \tan\left(h\right) &= {\sin\left(A\right) \over \cos\left(A\right) \sin\left(\phi_\text{o}\right) + \tan\left(a\right) \cos\left(\phi_\text{o}\right)}; \qquad \begin{cases}
    \cos\left(\delta\right) \sin\left(h\right) = \cos\left(a\right) \sin\left(A\right); \\
    \cos\left(\delta\right) \cos\left(h\right) = \sin\left(a\right) \cos\left(\phi_\text{o}\right) + \cos\left(a\right) \cos\left(A\right) \sin\left(\phi_\text{o}\right)
  \end{cases} \\[3pt]
  \sin\left(\delta\right) &= \sin\left(\phi_\text{o}\right) \sin\left(a\right) - \cos\left(\phi_\text{o}\right) \cos\left(a\right) \cos\left(A\right);
\end{align}</math>{{efn|Depending on the azimuth convention in use, the signs of {{math|cos ''A''}} and {{math|sin ''A''}} appear in all four different combinations. Karttunen et al.,<ref name=Karttunen/> Taff,<ref name=Taff/> and Roth<ref name=Roth/> define {{math|''A''}} clockwise from the south. Lang<ref name=Lang/> defines it north through east, Smart<ref name=Smart/> north through west. Meeus (1991),<ref name=Meeus/> p.&nbsp;89: {{math|sin ''δ'' {{=}} sin ''φ'' sin ''a'' − cos ''φ'' cos ''a'' cos ''A''}}; ''Explanatory Supplement'' (1961),<ref name=ExplSupp/> p.&nbsp;26: {{math|sin ''δ'' {{=}} sin ''a'' sin ''φ'' + cos ''a'' cos ''A'' cos ''φ''}}.}}

Again, in solving the {{math|tan(''h'')}} equation for {{math|''h''}}, use of the two-argument arctangent that accounts for the quadrant is recommended. Thus, again consistent with the convention of azimuth being measured from the south and opening positive to the west,
: <math>h = \operatorname{atan2}(y, x)</math>,

where
:<math>\begin{align}
              x &= \sin\left(\phi_\text{o}\right)\cos\left(a\right) \cos\left(A\right) + \cos\left(\phi_\text{o}\right)\sin\left(a\right) \\
              y &= \cos\left(a\right)\sin\left(A\right) \\[3pt]
  \begin{bmatrix}
    \cos\left(\delta\right)\cos\left(h\right) \\
    \cos\left(\delta\right)\sin\left(h\right) \\
    \sin\left(\delta\right)
  \end{bmatrix} &= \begin{bmatrix}
     \sin\left(\phi_\text{o}\right) & 0 & \cos\left(\phi_\text{o}\right) \\
     0                              & 1 & 0                              \\
    -\cos\left(\phi_\text{o}\right) & 0 & \sin\left(\phi_\text{o}\right)
  \end{bmatrix}\begin{bmatrix}
    \cos\left(a\right) \cos\left(A\right) \\
    \cos\left(a\right) \sin\left(A\right) \\
    \sin\left(a\right)
  \end{bmatrix} \\
  \begin{bmatrix}
    \cos\left(\delta\right) \cos\left(\alpha\right) \\
    \cos\left(\delta\right) \sin\left(\alpha\right) \\
    \sin\left(\delta\right)
  \end{bmatrix} &= \begin{bmatrix}
    \cos\left(\theta_L\right) &  \sin\left(\theta_L\right) & 0 \\
    \sin\left(\theta_L\right) & -\cos\left(\theta_L\right) & 0 \\
    0                         &  0                         & 1
  \end{bmatrix}\begin{bmatrix}
     \sin\left(\phi_\text{o}\right) & 0 & \cos\left(\phi_\text{o}\right) \\
     0                              & 1 & 0                              \\
    -\cos\left(\phi_\text{o}\right) & 0 & \sin\left(\phi_\text{o}\right)
  \end{bmatrix}\begin{bmatrix}
    \cos\left(a\right) \cos\left(A\right) \\
    \cos\left(a\right) \sin\left(A\right) \\
    \sin\left(a\right)
  \end{bmatrix}.
\end{align}</math>

=== Equatorial ↔ galactic ===
These equations<ref>{{Cite arXiv |title=Transformation of the equatorial proper motion to the Galactic system|last=Poleski|first=Radosław|year=2013 |eprint=1306.2945 |class=astro-ph.IM}}</ref> are for converting equatorial coordinates to Galactic coordinates.

:<math>\begin{align}
  \cos\left(l_\text{NCP} - l\right)\cos(b) &= \sin\left(\delta\right) \cos\left(\delta_\text{G}\right) - \cos\left(\delta\right)\sin\left(\delta_\text{G}\right)\cos\left(\alpha - \alpha_\text{G}\right) \\
  \sin\left(l_\text{NCP} - l\right)\cos(b) &= \cos(\delta)\sin\left(\alpha - \alpha_\text{G}\right) \\
                        \sin\left(b\right) &= \sin\left(\delta\right) \sin\left(\delta_\text{G}\right) + \cos\left(\delta\right) \cos\left(\delta_\text{G}\right) \cos\left(\alpha - \alpha_\text{G}\right)
\end{align}</math>

<math>\alpha_\text{G}, \delta_\text{G}</math> are the equatorial coordinates of the North Galactic Pole and <math>l_\text{NCP}</math> is the Galactic longitude of the North Celestial Pole. Referred to [[Epoch (astronomy)|J2000.0]] the values of these quantities are:
: <math>\alpha_G = 192.85948^\circ \qquad \delta_G = 27.12825^\circ \qquad l_\text{NCP}=122.93192^\circ</math>

If the equatorial coordinates are referred to another [[Equinox (celestial coordinates)|equinox]], they must be [[Axial precession|precessed]] to their place at J2000.0 before applying these formulae.

These equations convert to equatorial coordinates referred to [[Epoch (astronomy)|B2000.0]].

:<math>\begin{align}
  \sin\left(\alpha - \alpha_\text{G}\right)\cos\left(\delta\right) &= \cos\left(b\right) \sin\left(l_\text{NCP} - l\right) \\
  \cos\left(\alpha - \alpha_\text{G}\right)\cos\left(\delta\right) &= \sin\left(b\right) \cos\left(\delta_\text{G}\right) - \cos\left(b\right) \sin\left(\delta_\text{G}\right)\cos\left(l_\text{NCP} - l\right) \\
                                      \sin\left(\delta\right) &= \sin\left(b\right) \sin\left(\delta_\text{G}\right) + \cos\left(b\right) \cos\left(\delta_\text{G}\right) \cos\left(l_\text{NCP} - l\right)
\end{align}</math>

=== Notes on conversion ===

* Angles in the degrees ( ° ), minutes ( ′ ), and seconds ( ″ ) of [[Minute of arc|sexagesimal measure]] must be converted to decimal before calculations are performed. Whether they are converted to decimal [[Degree (angle)|degrees]] or [[radian]]s depends upon the particular calculating machine or program. Negative angles must be carefully handled; {{nowrap|−10° 20′ 30″}} must be converted as {{nowrap|−10° −20′ −30″}}.
* Angles in the hours ( <sup>h</sup> ), minutes ( <sup>m</sup> ), and seconds ( <sup>s</sup> ) of time measure must be converted to decimal [[Degree (angle)|degrees]] or [[radian]]s before calculations are performed. 1<sup>h</sup>&nbsp;=&nbsp;15°; 1<sup>m</sup>&nbsp;=&nbsp;15′; 1<sup>s</sup>&nbsp;=&nbsp;15″
* Angles greater than 360° (2{{pi}}) or less than 0° may need to be reduced to the range 0°–360° (0–2{{pi}}) depending upon the particular calculating machine or program.
* The cosine of a latitude (declination, ecliptic and Galactic latitude, and altitude) are never negative by definition, since the latitude varies between −90° and +90°.
* [[Inverse trigonometric functions]] arcsine, arccosine and arctangent are [[quadrant (plane geometry)|quadrant]]-ambiguous, and results should be carefully evaluated. Use of  the [[Atan2|second arctangent function]] (denoted in computing as {{mono|atn2(''y'',''x'')}} or {{mono|atan2(''y'',''x'')}}, which calculates the arctangent of {{math|{{sfrac|''y''|''x''}}}} using the sign of both arguments to determine the right quadrant) is recommended when calculating longitude/right ascension/azimuth. An equation which finds the [[Trigonometric functions|sine]], followed by the [[Inverse trigonometric functions|arcsin function]], is recommended when calculating latitude/declination/altitude.
* Azimuth ({{math|''A''}}) is referred here to the south point of the [[horizon]], the common astronomical reckoning. An object on the [[Meridian (astronomy)|meridian]] to the south of the observer has {{math|''A''}} = {{math|''h''}} = 0° with this usage. However, n [[Astropy]]'s AltAz, in the [[Large Binocular Telescope]] FITS file convention, in [[XEphem]], in the [[International Astronomical Union|IAU]] library [[SOFA (astronomy)|Standards of Fundamental Astronomy]] and Section B of the [[Astronomical Almanac]] for example, the azimuth is East of North.  In [[navigation]] and some other disciplines, azimuth is figured from the north.
* The equations for altitude ({{math|''a''}}) do not account for [[atmospheric refraction]].
* The equations for horizontal coordinates do not account for [[diurnal parallax]], that is, the small offset in the position of a celestial object caused by the position of the observer on the [[Earth]]'s surface. This effect is significant for the [[Moon]], less so for the [[planet]]s, minute for [[star]]s or more distant objects.
* Observer's longitude ({{math|''λ''<sub>o</sub>}}) here is measured positively eastward from the [[prime meridian]], accordingly to current [[International Astronomical Union|IAU]] standards.

==See also==
*[[Apparent longitude]]
*{{annotated link|Azimuth}}
*{{annotated link|Barycentric and geocentric celestial reference systems}}
*{{annotated link|Celestial sphere}}
*{{annotated link|International Celestial Reference System and its realizations}}
*{{annotated link|Orbital elements}}
*{{annotated link|Planetary coordinate system}}
*{{annotated link|Terrestrial reference frame}}

==Notes==
{{notelist}}

==References==
{{Reflist|refs=
<ref name=Smart>{{cite book|first1=William Marshall |last1=Smart
|title=Text-book on spherical astronomy
|publisher=[[Cambridge University Press]]
|year=1949
|bibcode=1965tbsa.book.....S
}}</ref>
<ref name=Lang>{{cite book| first1= Kenneth R. |last1=Lang
|title=Astrophysical Formulae
|year=1978
|publisher=Springer
|isbn=3-540-09064-9
|bibcode=1978afcp.book.....L
}}</ref>
<ref name=Taff>{{cite book|first1=L. G. |last1=Taff
|title =Computational spherical astronomy
|year=1981
|publisher=Wiley
|bibcode=1981csa..book.....T
|isbn=0-471-06257-X
}}</ref>
<ref name=Karttunen>{{cite book| first1=H. |last1=Karttunen
|first2=P. |last2=Kröger
|first3=H. |last3=Oja
|first4=M. |last4=Poutanen
|first5=H. J. |last5=Donner
|title=Fundamental Astronomy
|year=2006
|publisher=Springer
|edition=5
|isbn=978-3-540-34143-7
|bibcode=2003fuas.book.....K
}}</ref>
<ref name=Roth>{{cite book|first1=G. D. |last1=Roth
|title=Handbuch für Sternenfreunde
|date=23 October 1989
|isbn=3-540-19436-3
|publisher=Springer
}}</ref>
}}

==External links==
{{Commons category|Celestial coordinate systems|Astronomical coordinate systems}}
* [https://aa.usno.navy.mil/software/novas_info NOVAS], the [[United States Naval Observatory]]'s Vector Astrometry Software, an integrated package of subroutines and functions for computing various commonly needed quantities in positional astronomy.
* [https://smithsonian.github.io/SuperNOVAS SuperNOVAS] a maintained fork of NOVAS C 3.1 with bug fixes, improvements, new features, and online documentation.
* [http://www.iausofa.org/ SOFA], the [[International Astronomical Union|IAU]]'s Standards of Fundamental Astronomy, an accessible and authoritative set of algorithms and procedures that implement standard models used in fundamental astronomy.
* This article was originally based on Jason Harris' ''Astroinfo'', which is accompanied by [[KStars]], a [http://edu.kde.org/kstars/ KDE Desktop Planetarium] for [[Linux]]/[[KDE]].

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