Editing Ecliptic coordinate system

{{Short description|Celestial coordinate system used to describe Solar System objects}}
{{distinguish|Elliptic coordinate system}}

In [[astronomy]], the '''ecliptic coordinate system''' is a [[celestial coordinate system]] commonly used for representing the [[apparent place|apparent positions]], [[orbit]]s, and [[Poles of astronomical bodies|pole]] orientations<ref name=Cunningham1985/> of [[Solar System]] objects. Because most [[planet]]s (except [[Mercury (planet)|Mercury]]) and many [[Small Solar System body|small Solar System bodies]] have orbits with only slight [[inclination]]s to the [[ecliptic]], using it as the [[fundamental plane (spherical coordinates)|fundamental plane]] is convenient. The system's [[origin (mathematics)|origin]] can be the center of either the [[Sun]] or [[Earth]], its primary direction is towards the [[March equinox|March]] [[equinox (celestial coordinates)|equinox]], and it has a [[right-hand rule|right-hand convention]]. It may be implemented in [[spherical coordinate system|spherical]] or [[Cartesian coordinate system|rectangular coordinates]].<ref>
{{cite book
 | url = https://archive.org/details/astronomicalalmanac1961
 | author1 = Nautical Almanac Office, U.S. Naval Observatory
 | author2 = 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 (reprint 1974)
 | year = 1961
 |pages=[https://archive.org/details/astronomicalalmanac1961/page/n34 24]–27}}</ref>

[[File:Ecliptic grid globe.png|thumb|[[geocentric model|Earth-centered]] '''ecliptic coordinates''' as seen from outside the [[celestial sphere]]. 
{{legend|red|Ecliptic longitude; measured along the [[ecliptic]] from the March [[equinox (celestial coordinates)|equinox]]}}
{{legend|yellow|Ecliptic latitude; measured [[perpendicular]] to the ecliptic}}
{{legend-line|solid blue|[[Celestial equator]]}}
A full globe is shown here, although [[polar regions of Earth|high-latitude]] coordinates are seldom seen except for certain [[comet]]s and [[asteroid]]s.]]

== Primary direction ==

[[File:Ecliptic vs equator small.gif|thumb|The apparent motion of the [[Sun]] along the ecliptic (red) as seen on the inside of the [[celestial sphere]]. Ecliptic coordinates appear in (red). The [[celestial equator]] (blue) and the [[Equatorial coordinate system|equatorial coordinates]] (blue), being inclined to the ecliptic, appear to wobble as the Sun advances.]]

{{see also|Axial precession|Astronomical nutation}}

The [[celestial equator]] and the [[ecliptic]] are slowly moving due to [[Perturbation (astronomy)|perturbing forces]] on the [[Earth]], therefore the [[Orientation (geometry)|orientation]] of the primary direction, their intersection at the [[March equinox]], 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, [[astronomical nutation|nutation]].<ref>
''Explanatory Supplement'' (1961), pp. 20, 28</ref><ref>
{{cite book
 | url = https://books.google.com/books?id=uJ4JhGJANb4C&pg=PA11
 | 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 (reprint 2005)
 | year = 1992 
 | isbn = 1-891389-45-9
 |pages=11–13}}
</ref>

In order to reference a coordinate system which can be considered as fixed in space, these motions require specification of the [[equinox]] of a particular date, known as an [[Epoch (astronomy)|epoch]], when giving a position in ecliptic coordinates. The three most commonly used are:
;Mean equinox of a standard epoch: (usually the [[Epoch (astronomy)|J2000.0 epoch]], 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 [[astronomical nutation|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 [[astronomical nutation|nutation]]). This is the actual intersection of the two planes at any particular moment, with all motions accounted for.
A position in the ecliptic coordinate system is thus typically specified ''true equinox and ecliptic of date'', ''mean equinox and ecliptic 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 ==

{| class="wikitable" style="float:right; text-align:center;"
|+ Summary of notation for ecliptic coordinates<ref>''Explanatory Supplement'' (1961), sec. 1G</ref>
! rowspan="2" style="background-color:#89CFF0;" |
! colspan="3" style="background-color:#89CFF0;" | Spherical
! rowspan="2" style="background-color:#89CFF0;" | Rectangular
|-
! style="background-color:#89CFF0;" | Longitude
! style="background-color:#89CFF0;" | Latitude
! style="background-color:#89CFF0;" | Distance
|-
! style="background-color:#89CFF0;" | Geocentric
| {{math|''λ''}}
| {{math|''β''}}
| {{math|''Δ''}}
|
|-
! style="background-color:#89CFF0;" | Heliocentric
| {{math|''l''}}
| {{math|''b''}}
| {{math|''r''}}
| {{math|''x''}}, {{math|''y''}}, {{math|''z''}}<ref group="note">Occasional use; {{math|''x''}}, {{math|''y''}}, {{math|''z''}} are usually reserved for [[Equatorial coordinate system|equatorial coordinates]].</ref>
|-
| colspan="5" | {{Reflist|group="note"}}
|}

;Ecliptic longitude
:''Ecliptic longitude'' or ''celestial longitude'' (symbols: heliocentric {{mvar|l}}, geocentric {{mvar|λ}}) measures the angular distance of an object along the [[ecliptic]] from the primary direction. Like [[right ascension]] in the [[equatorial coordinate system]], the primary direction (0° ecliptic longitude) points from the Earth towards the Sun at the [[March equinox]]. Because it is a right-handed system, ecliptic longitude is measured positive eastwards in the fundamental plane (the ecliptic) from 0° to 360°. Because of [[axial precession]], the ecliptic longitude of most "fixed stars" (referred to the equinox of date) increases by about 50.3 [[arcsecond]]s per year, or 83.8 [[arcminute]]s per century, the speed of general precession.<ref>{{cite journal|author1=N. Capitaine|author2=P.T. Wallace|author3=J. Chapront|title=Expressions for IAU 2000 precession quantities|journal=Astronomy & Astrophysics|date=2003|volume=412|issue=2|page=581|doi=10.1051/0004-6361:20031539|url=http://syrte.obspm.fr/iau2006/aa03_412_P03.pdf |archive-url=https://web.archive.org/web/20120325210757/http://syrte.obspm.fr/iau2006/aa03_412_P03.pdf |archive-date=2012-03-25 |url-status=live|bibcode=2003A&A...412..567C|doi-access=free}}</ref><ref>J.H. Lieske ''et al.'' (1977), "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1977A%26A....58....1L&amp;db_key=AST&amp;data_type=HTML&amp;format=&amp;high=46303c7cf308007 Expressions for the Precession Quantities Based upon the IAU (1976) System of Astronomical Constants]".  ''Astronomy & Astrophysics'' '''58''', pp. 1-16</ref> However, for stars near the ecliptic poles, the rate of change of ecliptic longitude is dominated by the slight movement of the ecliptic (that is, of the plane of the Earth's orbit), so the rate of change may be anything from minus infinity to plus infinity depending on the exact position of the star.
;Ecliptic latitude
:''Ecliptic latitude'' or ''celestial latitude'' (symbols: heliocentric {{mvar|b}}, geocentric {{mvar|β}}), measures the angular distance of an object from the [[ecliptic]] towards the north (positive) or south (negative) [[ecliptic pole]]. For example, the [[Ecliptic pole|north ecliptic pole]] has a celestial latitude of +90°. Ecliptic latitude for "fixed stars" is not affected by precession.
;Distance
:''Distance'' is also necessary for a complete spherical position (symbols: heliocentric {{mvar|r}}, geocentric {{mvar|Δ}}). Different distance units are used for different objects. Within the [[Solar System]], [[astronomical unit]]s are used, and for objects near the [[Earth]], [[Earth radius|Earth radii]] or [[kilometer]]s are used.

=== Historical use ===
From antiquity through the 18th&nbsp;century, ecliptic longitude was commonly measured using twelve [[Astrological sign|zodiacal signs]], each of 30°&nbsp;longitude, a practice that continues in modern [[astrology]]. The signs approximately corresponded to the [[constellation]]s crossed by the ecliptic. Longitudes were specified in signs, degrees, minutes, and seconds. For example, a longitude of {{nowrap|♌ 19°&thinsp;55′&thinsp;58″}} is 19.933°&nbsp;east of the start of the sign [[Leo (astrology)|Leo]]. Since Leo begins 120°&nbsp;from the March equinox, the longitude in modern form is {{nowrap|139°&thinsp;55′&thinsp;58″}}.<ref>{{cite book |url=https://archive.org/details/acompleatsystem01leadgoog |title=A Compleat System of Astronomy |first=Charles |last=Leadbetter |year=1742 |publisher=J. Wilcox, London |page=[https://archive.org/details/acompleatsystem01leadgoog/page/n102 94]}}; numerous examples of this notation appear throughout the book.</ref>

{{See also|solar term}}

In China, ecliptic longitude is measured using 24&nbsp;[[Solar term]]s, each of 15°&nbsp;longitude, and are used by [[Chinese calendar|Chinese lunisolar calendars]] to stay synchronized with the seasons, which is crucial for agrarian societies.

==Rectangular coordinates==
[[File:Heliocentric rectangular ecliptic.png|thumb|[[heliocentrism|Heliocentric]] ecliptic coordinates. The [[origin (mathematics)|origin]] is the [[Sun]]'s center, the [[plane of reference]] is the [[ecliptic]] plane, and the primary direction (the {{mvar|x}}-axis) is the March [[equinox (celestial coordinates)|equinox]]. A [[right-handed rule]] specifies a {{mvar|y}}-axis 90° to the east on the fundamental plane. The {{mvar|z}}-axis points toward the north [[ecliptic pole]]. The reference frame is relatively stationary, aligned with the March equinox.]]

A [[Cartesian coordinate system|rectangular variant]] of ecliptic coordinates is often used in [[orbit]]al calculations and simulations. It has its [[origin (mathematics)|origin]] at the center of the [[Sun]] (or at the [[barycenter]] of the [[Solar System]]), its [[fundamental plane (spherical coordinates)|fundamental plane]] on the [[ecliptic]] plane, and the {{mvar|x}}-axis toward the March [[equinox (celestial coordinates)|equinox]]. The coordinates have a [[right-handed rule|right-handed convention]], that is, if one extends their right thumb upward, it simulates the {{mvar|z}}-axis, their extended index finger the {{mvar|x}}-axis, and the curl of the other fingers points generally in the direction of the {{mvar|y}}-axis.<ref>
''Explanatory Supplement'' (1961), pp. 20, 27</ref>

These rectangular coordinates are related to the corresponding spherical coordinates by

:<math>\begin{align}
x &= r \cos b \cos l \\
y &= r \cos b \sin l \\
z &= r \sin b \end{align}</math>

== Conversion between celestial coordinate systems ==
{{main|Celestial coordinate system}}

=== Converting Cartesian vectors ===

==== Conversion from ecliptic coordinates to equatorial coordinates ====
<math display="block">
 \begin{bmatrix}
  x_\text{equatorial} \\
  y_\text{equatorial} \\
  z_\text{equatorial} \\
 \end{bmatrix}
 =
 \begin{bmatrix}
  1 & 0 & 0 \\
  0 & \cos \varepsilon & -\sin \varepsilon \\
  0 & \sin \varepsilon &  \cos \varepsilon \\
 \end{bmatrix}
 \begin{bmatrix}
  x_\text{ecliptic} \\
  y_\text{ecliptic} \\
  z_\text{ecliptic} \\
 \end{bmatrix}
</math><ref>
''Explanatory Supplement'' (1992), pp. 555-558</ref>

==== Conversion from equatorial coordinates to ecliptic coordinates ====
<math display="block">
 \begin{bmatrix}
  x_\text{ecliptic} \\
  y_\text{ecliptic} \\
  z_\text{ecliptic} \\
 \end{bmatrix}
 =
 \begin{bmatrix}
  1 & 0 & 0 \\
  0 & \cos \varepsilon & \sin \varepsilon \\
  0 & -\sin \varepsilon & \cos \varepsilon \\
 \end{bmatrix}
 \begin{bmatrix}
  x_\text{equatorial} \\
  y_\text{equatorial} \\
  z_\text{equatorial} \\
 \end{bmatrix}
</math>
where {{mvar|ε}} is the [[obliquity of the ecliptic]].

==See also==
* [[Celestial coordinate system]]
* [[Ecliptic]]
* [[Orbital pole#Ecliptic pole|Ecliptic pole]], where the ecliptic latitude is ±90°
* [[Equinox]]
** [[Equinox (celestial coordinates)]]
** [[March equinox]]

== Notes and references ==
{{reflist|refs=

<ref name=Cunningham1985>{{cite journal
 | title=Asteroid Pole Positions: A Survey
 | last=Cunningham | first=Clifford J.
 | journal=The Minor Planet Bulletin
 | volume=12 | pages=13–16 | date=June 1985
 | bibcode=1985MPBu...12...13C }}</ref>

}}

== External links ==
* [https://johnlucey.webspace.durham.ac.uk/users/solar-year/ The Ecliptic: the Sun's Annual Path on the Celestial Sphere] Durham University Department of Physics
* [https://frostydrew.org/utilities.dc/convert/tool-eq_coordinates/ Equatorial ↔ Ecliptic coordinate converter]
* [http://stars.astro.illinois.edu/celsph.html MEASURING THE SKY A Quick Guide to the Celestial Sphere] James B. Kaler, University of Illinois
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[[Category:Astronomical coordinate systems]]