Editing Astronomical coordinate systems (section)

== 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.