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class="vector-toc-numb">3</span> <span>Rectangular coordinates</span> </div> </a> <ul id="toc-Rectangular_coordinates-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Conversion_between_celestial_coordinate_systems" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Conversion_between_celestial_coordinate_systems"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Conversion between celestial coordinate systems</span> </div> </a> <button aria-controls="toc-Conversion_between_celestial_coordinate_systems-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Conversion between celestial coordinate systems subsection</span> </button> <ul id="toc-Conversion_between_celestial_coordinate_systems-sublist" class="vector-toc-list"> <li id="toc-Converting_Cartesian_vectors" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Converting_Cartesian_vectors"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Converting Cartesian vectors</span> </div> </a> <ul id="toc-Converting_Cartesian_vectors-sublist" class="vector-toc-list"> <li id="toc-Conversion_from_ecliptic_coordinates_to_equatorial_coordinates" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Conversion_from_ecliptic_coordinates_to_equatorial_coordinates"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1.1</span> <span>Conversion from ecliptic coordinates to equatorial coordinates</span> </div> </a> <ul id="toc-Conversion_from_ecliptic_coordinates_to_equatorial_coordinates-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Conversion_from_equatorial_coordinates_to_ecliptic_coordinates" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Conversion_from_equatorial_coordinates_to_ecliptic_coordinates"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1.2</span> <span>Conversion from equatorial coordinates to ecliptic coordinates</span> </div> </a> <ul id="toc-Conversion_from_equatorial_coordinates_to_ecliptic_coordinates-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes_and_references" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes_and_references"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Notes and references</span> </div> </a> <ul id="toc-Notes_and_references-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Ecliptic coordinate system</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 42 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-42" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">42 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%86%D8%B8%D8%A7%D9%85_%D8%A7%D9%84%D8%A5%D8%AD%D8%AF%D8%A7%D8%AB%D9%8A%D8%A7%D8%AA_%D8%A7%D9%84%D9%83%D8%B3%D9%88%D9%81%D9%8A%D8%A9" title="نظام الإحداثيات الكسوفية – Arabic" lang="ar" hreflang="ar" data-title="نظام الإحداثيات الكسوفية" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%AD%D0%BA%D0%BB%D1%96%D0%BF%D1%82%D1%8B%D1%87%D0%B0%D1%8F_%D1%81%D1%96%D1%81%D1%82%D1%8D%D0%BC%D0%B0_%D0%BA%D0%B0%D0%B0%D1%80%D0%B4%D1%8B%D0%BD%D0%B0%D1%82" title="Экліптычая сістэма каардынат – Belarusian" lang="be" hreflang="be" data-title="Экліптычая сістэма каардынат" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Coordenades_ecl%C3%ADptiques" title="Coordenades eclíptiques – Catalan" lang="ca" hreflang="ca" data-title="Coordenades eclíptiques" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Ekliptik%C3%A1ln%C3%AD_sou%C5%99adnice" title="Ekliptikální souřadnice – Czech" lang="cs" hreflang="cs" data-title="Ekliptikální souřadnice" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Ekliptiske_koordinatsystem" title="Ekliptiske koordinatsystem – Danish" lang="da" hreflang="da" data-title="Ekliptiske koordinatsystem" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Ekliptikales_Koordinatensystem" title="Ekliptikales Koordinatensystem – German" lang="de" hreflang="de" data-title="Ekliptikales Koordinatensystem" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Ekliptilised_koordinaadid" title="Ekliptilised koordinaadid – Estonian" lang="et" hreflang="et" data-title="Ekliptilised koordinaadid" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Coordenadas_ecl%C3%ADpticas" title="Coordenadas eclípticas – Spanish" lang="es" hreflang="es" data-title="Coordenadas eclípticas" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Ekliptika_koordinata_sistemo" title="Ekliptika koordinata sistemo – Esperanto" lang="eo" hreflang="eo" data-title="Ekliptika koordinata sistemo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Koordenatu_ekliptiko" title="Koordenatu ekliptiko – Basque" lang="eu" hreflang="eu" data-title="Koordenatu ekliptiko" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AF%D8%B3%D8%AA%DA%AF%D8%A7%D9%87_%D9%85%D8%AE%D8%AA%D8%B5%D8%A7%D8%AA_%D8%AF%D8%A7%D8%A6%D8%B1%D8%A9%D8%A7%D9%84%D8%A8%D8%B1%D9%88%D8%AC%DB%8C" title="دستگاه مختصات دائرةالبروجی – Persian" lang="fa" hreflang="fa" data-title="دستگاه مختصات دائرةالبروجی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Syst%C3%A8me_de_coordonn%C3%A9es_%C3%A9cliptiques" title="Système de coordonnées écliptiques – French" lang="fr" hreflang="fr" data-title="Système de coordonnées écliptiques" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%99%A9%EB%8F%84%EC%A2%8C%ED%91%9C%EA%B3%84" title="황도좌표계 – Korean" lang="ko" hreflang="ko" data-title="황도좌표계" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Eklipti%C4%8Dki_koordinatni_sustav" title="Ekliptički koordinatni sustav – Croatian" lang="hr" hreflang="hr" data-title="Ekliptički koordinatni sustav" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-ig mw-list-item"><a href="https://ig.wikipedia.org/wiki/Usoro_nhazi_nke_Ecliptic" title="Usoro nhazi nke Ecliptic – Igbo" lang="ig" hreflang="ig" data-title="Usoro nhazi nke Ecliptic" data-language-autonym="Igbo" data-language-local-name="Igbo" class="interlanguage-link-target"><span>Igbo</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Sistem_koordinat_ekliptika" title="Sistem koordinat ekliptika – Indonesian" lang="id" hreflang="id" data-title="Sistem koordinat ekliptika" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/S%C3%B3lbaugshnit" title="Sólbaugshnit – Icelandic" lang="is" hreflang="is" data-title="Sólbaugshnit" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Sistema_di_coordinate_eclittiche" title="Sistema di coordinate eclittiche – Italian" lang="it" hreflang="it" data-title="Sistema di coordinate eclittiche" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A2%D7%A8%D7%9B%D7%AA_%D7%94%D7%A7%D7%95%D7%90%D7%95%D7%A8%D7%93%D7%99%D7%A0%D7%98%D7%95%D7%AA_%D7%94%D7%90%D7%A7%D7%9C%D7%99%D7%A4%D7%98%D7%99%D7%AA" title="מערכת הקואורדינטות האקליפטית – Hebrew" lang="he" hreflang="he" data-title="מערכת הקואורדינטות האקליפטית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%93%D0%B5%D0%BB%D0%B8%D0%BE%D0%B1%D0%BE%D1%80%D0%B4%D1%83%D0%BA_%D0%BA%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82%D0%B0%D0%BB%D0%B0%D1%80" title="Гелиобордук координаталар – Kyrgyz" lang="ky" hreflang="ky" data-title="Гелиобордук координаталар" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Ekliptin%C4%97_koordina%C4%8Di%C5%B3_sistema" title="Ekliptinė koordinačių sistema – Lithuanian" lang="lt" hreflang="lt" data-title="Ekliptinė koordinačių sistema" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Sistem_koordinat_gerhana" title="Sistem koordinat gerhana – Malay" lang="ms" hreflang="ms" data-title="Sistem koordinat gerhana" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E9%BB%84%E9%81%93%E5%BA%A7%E6%A8%99" title="黄道座標 – Japanese" lang="ja" hreflang="ja" data-title="黄道座標" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Ekliptiske_koordinater" title="Ekliptiske koordinater – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Ekliptiske koordinater" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Ekliptikksystemet" title="Ekliptikksystemet – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Ekliptikksystemet" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Geliotsentrik_koordinatalar" title="Geliotsentrik koordinatalar – Uzbek" lang="uz" hreflang="uz" data-title="Geliotsentrik koordinatalar" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Uk%C5%82ad_ekliptyczny" title="Układ ekliptyczny – Polish" lang="pl" hreflang="pl" data-title="Układ ekliptyczny" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Sistema_ecl%C3%ADptico_de_coordenadas" title="Sistema eclíptico de coordenadas – Portuguese" lang="pt" hreflang="pt" data-title="Sistema eclíptico de coordenadas" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Coordonate_astronomice_ecliptice" title="Coordonate astronomice ecliptice – Romanian" lang="ro" hreflang="ro" data-title="Coordonate astronomice ecliptice" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%AD%D0%BA%D0%BB%D0%B8%D0%BF%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0_%D0%BA%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82" title="Эклиптическая система координат – Russian" lang="ru" hreflang="ru" data-title="Эклиптическая система координат" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Ekliptik%C3%A1lna_s%C3%BAstava_s%C3%BAradn%C3%ADc" title="Ekliptikálna sústava súradníc – Slovak" lang="sk" hreflang="sk" data-title="Ekliptikálna sústava súradníc" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Eklipti%C4%8Dni_koordinatni_sistem" title="Ekliptični koordinatni sistem – Slovenian" lang="sl" hreflang="sl" data-title="Ekliptični koordinatni sistem" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%95%D0%BA%D0%BB%D0%B8%D0%BF%D1%82%D0%B8%D1%87%D0%BA%D0%B8_%D0%BA%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82%D0%BD%D0%B8_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC" title="Еклиптички координатни систем – Serbian" lang="sr" hreflang="sr" data-title="Еклиптички координатни систем" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Eklipti%C4%8Dki_koordinatni_sistem" title="Ekliptički koordinatni sistem – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Ekliptički koordinatni sistem" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Ekliptiska_koordinater" title="Ekliptiska koordinater – Swedish" lang="sv" hreflang="sv" data-title="Ekliptiska koordinater" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Sistemang_ekliptiko_ng_mga_koordinado" title="Sistemang ekliptiko ng mga koordinado – Tagalog" lang="tl" hreflang="tl" data-title="Sistemang ekliptiko ng mga koordinado" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li 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href="/w/index.php?title=Ecliptic_longitude&redirect=no" class="mw-redirect" title="Ecliptic longitude">Ecliptic longitude</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Celestial coordinate system used to describe Solar System objects</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Elliptic_coordinate_system" title="Elliptic coordinate system">Elliptic coordinate system</a>.</div> <p>In <a href="/wiki/Astronomy" title="Astronomy">astronomy</a>, the <b>ecliptic coordinate system</b> is a <a href="/wiki/Celestial_coordinate_system" class="mw-redirect" title="Celestial coordinate system">celestial coordinate system</a> commonly used for representing the <a href="/wiki/Apparent_place" title="Apparent place">apparent positions</a>, <a href="/wiki/Orbit" title="Orbit">orbits</a>, and <a href="/wiki/Poles_of_astronomical_bodies" title="Poles of astronomical bodies">pole</a> orientations<sup id="cite_ref-Cunningham1985_1-0" class="reference"><a href="#cite_note-Cunningham1985-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> of <a href="/wiki/Solar_System" title="Solar System">Solar System</a> objects. Because most <a href="/wiki/Planet" title="Planet">planets</a> (except <a href="/wiki/Mercury_(planet)" title="Mercury (planet)">Mercury</a>) and many <a href="/wiki/Small_Solar_System_body" title="Small Solar System body">small Solar System bodies</a> have orbits with only slight <a href="/wiki/Inclination" class="mw-redirect" title="Inclination">inclinations</a> to the <a href="/wiki/Ecliptic" title="Ecliptic">ecliptic</a>, using it as the <a href="/wiki/Fundamental_plane_(spherical_coordinates)" title="Fundamental plane (spherical coordinates)">fundamental plane</a> is convenient. The system's <a href="/wiki/Origin_(mathematics)" title="Origin (mathematics)">origin</a> can be the center of either the <a href="/wiki/Sun" title="Sun">Sun</a> or <a href="/wiki/Earth" title="Earth">Earth</a>, its primary direction is towards the <a href="/wiki/March_equinox" title="March equinox">March</a> <a href="/wiki/Equinox_(celestial_coordinates)" title="Equinox (celestial coordinates)">equinox</a>, and it has a <a href="/wiki/Right-hand_rule" title="Right-hand rule">right-hand convention</a>. It may be implemented in <a href="/wiki/Spherical_coordinate_system" title="Spherical coordinate system">spherical</a> or <a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">rectangular coordinates</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Ecliptic_grid_globe.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Ecliptic_grid_globe.png/220px-Ecliptic_grid_globe.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Ecliptic_grid_globe.png/330px-Ecliptic_grid_globe.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/50/Ecliptic_grid_globe.png/440px-Ecliptic_grid_globe.png 2x" data-file-width="1100" data-file-height="1100" /></a><figcaption><a href="/wiki/Geocentric_model" title="Geocentric model">Earth-centered</a> <b>ecliptic coordinates</b> as seen from outside the <a href="/wiki/Celestial_sphere" title="Celestial sphere">celestial sphere</a>. <style data-mw-deduplicate="TemplateStyles:r981673959">.mw-parser-output .legend{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .legend-color{display:inline-block;min-width:1.25em;height:1.25em;line-height:1.25;margin:1px 0;text-align:center;border:1px solid black;background-color:transparent;color:black}.mw-parser-output .legend-text{}</style><div class="legend"><span class="legend-color mw-no-invert" style="background-color:red; color:black;"> </span> Ecliptic longitude; measured along the <a href="/wiki/Ecliptic" title="Ecliptic">ecliptic</a> from the March <a href="/wiki/Equinox_(celestial_coordinates)" title="Equinox (celestial coordinates)">equinox</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r981673959"><div class="legend"><span class="legend-color mw-no-invert" style="background-color:yellow; color:black;"> </span> Ecliptic latitude; measured <a href="/wiki/Perpendicular" title="Perpendicular">perpendicular</a> to the ecliptic</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r981673959"><div class="legend"><span class="legend-line mw-no-invert" style="display: inline-block; vertical-align: middle; width: 1.67em; height: 0; border-style: none; border-top: 2px dotted black;border-top:solid blue;"> </span> <a href="/wiki/Celestial_equator" title="Celestial equator">Celestial equator</a></div> A full globe is shown here, although <a href="/wiki/Polar_regions_of_Earth" title="Polar regions of Earth">high-latitude</a> coordinates are seldom seen except for certain <a href="/wiki/Comet" title="Comet">comets</a> and <a href="/wiki/Asteroid" title="Asteroid">asteroids</a>.</figcaption></figure> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Primary_direction">Primary direction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ecliptic_coordinate_system&action=edit&section=1" title="Edit section: Primary direction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Ecliptic_vs_equator_small.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Ecliptic_vs_equator_small.gif/220px-Ecliptic_vs_equator_small.gif" decoding="async" width="220" height="263" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/a/a3/Ecliptic_vs_equator_small.gif 1.5x" data-file-width="300" data-file-height="359" /></a><figcaption>The apparent motion of the <a href="/wiki/Sun" title="Sun">Sun</a> along the ecliptic (red) as seen on the inside of the <a href="/wiki/Celestial_sphere" title="Celestial sphere">celestial sphere</a>. Ecliptic coordinates appear in (red). The <a href="/wiki/Celestial_equator" title="Celestial equator">celestial equator</a> (blue) and the <a href="/wiki/Equatorial_coordinate_system" title="Equatorial coordinate system">equatorial coordinates</a> (blue), being inclined to the ecliptic, appear to wobble as the Sun advances.</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Axial_precession" title="Axial precession">Axial precession</a> and <a href="/wiki/Astronomical_nutation" title="Astronomical nutation">Astronomical nutation</a></div> <p>The <a href="/wiki/Celestial_equator" title="Celestial equator">celestial equator</a> and the <a href="/wiki/Ecliptic" title="Ecliptic">ecliptic</a> are slowly moving due to <a href="/wiki/Perturbation_(astronomy)" title="Perturbation (astronomy)">perturbing forces</a> on the <a href="/wiki/Earth" title="Earth">Earth</a>, therefore the <a href="/wiki/Orientation_(geometry)" title="Orientation (geometry)">orientation</a> of the primary direction, their intersection at the <a href="/wiki/March_equinox" title="March equinox">March equinox</a>, is not quite fixed. A slow motion of Earth's axis, <a href="/wiki/Axial_precession" title="Axial precession">precession</a>, causes a slow, continuous turning of the coordinate system westward about the poles of the <a href="/wiki/Ecliptic" title="Ecliptic">ecliptic</a>, completing one circuit in about 26,000 years. Superimposed on this is a smaller motion of the <a href="/wiki/Ecliptic" title="Ecliptic">ecliptic</a>, and a small oscillation of the Earth's axis, <a href="/wiki/Astronomical_nutation" title="Astronomical nutation">nutation</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>In order to reference a coordinate system which can be considered as fixed in space, these motions require specification of the <a href="/wiki/Equinox" title="Equinox">equinox</a> of a particular date, known as an <a href="/wiki/Epoch_(astronomy)" title="Epoch (astronomy)">epoch</a>, when giving a position in ecliptic coordinates. The three most commonly used are: </p> <dl><dt>Mean equinox of a standard epoch</dt> <dd>(usually the <a href="/wiki/Epoch_(astronomy)" title="Epoch (astronomy)">J2000.0 epoch</a>, but may include B1950.0, B1900.0, etc.) is a fixed standard direction, allowing positions established at various dates to be compared directly.</dd> <dt>Mean equinox of date</dt> <dd>is the intersection of the <a href="/wiki/Ecliptic" title="Ecliptic">ecliptic</a> of "date" (that is, the ecliptic in its position at "date") with the <i>mean</i> equator (that is, the equator rotated by <a href="/wiki/Axial_precession" title="Axial precession">precession</a> to its position at "date", but free from the small periodic oscillations of <a href="/wiki/Astronomical_nutation" title="Astronomical nutation">nutation</a>). Commonly used in planetary <a href="/wiki/Orbit" title="Orbit">orbit</a> calculation.</dd> <dt>True equinox of date</dt> <dd>is the intersection of the <a href="/wiki/Ecliptic" title="Ecliptic">ecliptic</a> of "date" with the <i>true</i> equator (that is, the mean equator plus <a href="/wiki/Astronomical_nutation" title="Astronomical nutation">nutation</a>). This is the actual intersection of the two planes at any particular moment, with all motions accounted for.</dd></dl> <p>A position in the ecliptic coordinate system is thus typically specified <i>true equinox and ecliptic of date</i>, <i>mean equinox and ecliptic of J2000.0</i>, or similar. Note that there is no "mean ecliptic", as the ecliptic is not subject to small periodic oscillations.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Spherical_coordinates">Spherical coordinates</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ecliptic_coordinate_system&action=edit&section=2" title="Edit section: Spherical coordinates"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable" style="float:right; text-align:center;"> <caption>Summary of notation for ecliptic coordinates<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </caption> <tbody><tr> <th rowspan="2" style="background-color:#89CFF0;"> </th> <th colspan="3" style="background-color:#89CFF0;">Spherical </th> <th rowspan="2" style="background-color:#89CFF0;">Rectangular </th></tr> <tr> <th style="background-color:#89CFF0;">Longitude </th> <th style="background-color:#89CFF0;">Latitude </th> <th style="background-color:#89CFF0;">Distance </th></tr> <tr> <th style="background-color:#89CFF0;">Geocentric </th> <td><span class="texhtml"><i>λ</i></span> </td> <td><span class="texhtml"><i>β</i></span> </td> <td><span class="texhtml"><i>Δ</i></span> </td> <td> </td></tr> <tr> <th style="background-color:#89CFF0;">Heliocentric </th> <td><span class="texhtml"><i>l</i></span> </td> <td><span class="texhtml"><i>b</i></span> </td> <td><span class="texhtml"><i>r</i></span> </td> <td><span class="texhtml"><i>x</i></span>, <span class="texhtml"><i>y</i></span>, <span class="texhtml"><i>z</i></span><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>note 1<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td colspan="5"><style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">Occasional use; <span class="texhtml"><i>x</i></span>, <span class="texhtml"><i>y</i></span>, <span class="texhtml"><i>z</i></span> are usually reserved for <a href="/wiki/Equatorial_coordinate_system" title="Equatorial coordinate system">equatorial coordinates</a>.</span> </li> </ol></div></div> </td></tr></tbody></table> <dl><dt>Ecliptic longitude</dt> <dd><i>Ecliptic longitude</i> or <i>celestial longitude</i> (symbols: heliocentric <span class="texhtml mvar" style="font-style:italic;">l</span>, geocentric <span class="texhtml mvar" style="font-style:italic;">λ</span>) measures the angular distance of an object along the <a href="/wiki/Ecliptic" title="Ecliptic">ecliptic</a> from the primary direction. Like <a href="/wiki/Right_ascension" title="Right ascension">right ascension</a> in the <a href="/wiki/Equatorial_coordinate_system" title="Equatorial coordinate system">equatorial coordinate system</a>, the primary direction (0° ecliptic longitude) points from the Earth towards the Sun at the <a href="/wiki/March_equinox" title="March equinox">March equinox</a>. 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 <a href="/wiki/Axial_precession" title="Axial precession">axial precession</a>, the ecliptic longitude of most "fixed stars" (referred to the equinox of date) increases by about 50.3 <a href="/wiki/Arcsecond" class="mw-redirect" title="Arcsecond">arcseconds</a> per year, or 83.8 <a href="/wiki/Arcminute" class="mw-redirect" title="Arcminute">arcminutes</a> per century, the speed of general precession.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> 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.</dd> <dt>Ecliptic latitude</dt> <dd><i>Ecliptic latitude</i> or <i>celestial latitude</i> (symbols: heliocentric <span class="texhtml mvar" style="font-style:italic;">b</span>, geocentric <span class="texhtml mvar" style="font-style:italic;">β</span>), measures the angular distance of an object from the <a href="/wiki/Ecliptic" title="Ecliptic">ecliptic</a> towards the north (positive) or south (negative) <a href="/wiki/Ecliptic_pole" class="mw-redirect" title="Ecliptic pole">ecliptic pole</a>. For example, the <a href="/wiki/Ecliptic_pole" class="mw-redirect" title="Ecliptic pole">north ecliptic pole</a> has a celestial latitude of +90°. Ecliptic latitude for "fixed stars" is not affected by precession.</dd> <dt>Distance</dt> <dd><i>Distance</i> is also necessary for a complete spherical position (symbols: heliocentric <span class="texhtml mvar" style="font-style:italic;">r</span>, geocentric <span class="texhtml mvar" style="font-style:italic;">Δ</span>). Different distance units are used for different objects. Within the <a href="/wiki/Solar_System" title="Solar System">Solar System</a>, <a href="/wiki/Astronomical_unit" title="Astronomical unit">astronomical units</a> are used, and for objects near the <a href="/wiki/Earth" title="Earth">Earth</a>, <a href="/wiki/Earth_radius" title="Earth radius">Earth radii</a> or <a href="/wiki/Kilometer" class="mw-redirect" title="Kilometer">kilometers</a> are used.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Historical_use">Historical use</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ecliptic_coordinate_system&action=edit&section=3" title="Edit section: Historical use"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>From antiquity through the 18th century, ecliptic longitude was commonly measured using twelve <a href="/wiki/Astrological_sign" title="Astrological sign">zodiacal signs</a>, each of 30° longitude, a practice that continues in modern <a href="/wiki/Astrology" title="Astrology">astrology</a>. The signs approximately corresponded to the <a href="/wiki/Constellation" title="Constellation">constellations</a> crossed by the ecliptic. Longitudes were specified in signs, degrees, minutes, and seconds. For example, a longitude of <span class="nowrap">♌ 19° 55′ 58″</span> is 19.933° east of the start of the sign <a href="/wiki/Leo_(astrology)" title="Leo (astrology)">Leo</a>. Since Leo begins 120° from the March equinox, the longitude in modern form is <span class="nowrap">139° 55′ 58″</span>.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Solar_term" title="Solar term">solar term</a></div> <p>In China, ecliptic longitude is measured using 24 <a href="/wiki/Solar_term" title="Solar term">Solar terms</a>, each of 15° longitude, and are used by <a href="/wiki/Chinese_calendar" title="Chinese calendar">Chinese lunisolar calendars</a> to stay synchronized with the seasons, which is crucial for agrarian societies. </p> <div class="mw-heading mw-heading2"><h2 id="Rectangular_coordinates">Rectangular coordinates</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ecliptic_coordinate_system&action=edit&section=4" title="Edit section: Rectangular coordinates"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Heliocentric_rectangular_ecliptic.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Heliocentric_rectangular_ecliptic.png/220px-Heliocentric_rectangular_ecliptic.png" decoding="async" width="220" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Heliocentric_rectangular_ecliptic.png/330px-Heliocentric_rectangular_ecliptic.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Heliocentric_rectangular_ecliptic.png/440px-Heliocentric_rectangular_ecliptic.png 2x" data-file-width="800" data-file-height="435" /></a><figcaption><a href="/wiki/Heliocentrism" title="Heliocentrism">Heliocentric</a> ecliptic coordinates. The <a href="/wiki/Origin_(mathematics)" title="Origin (mathematics)">origin</a> is the <a href="/wiki/Sun" title="Sun">Sun</a>'s center, the <a href="/wiki/Plane_of_reference" class="mw-redirect" title="Plane of reference">plane of reference</a> is the <a href="/wiki/Ecliptic" title="Ecliptic">ecliptic</a> plane, and the primary direction (the <span class="texhtml mvar" style="font-style:italic;">x</span>-axis) is the March <a href="/wiki/Equinox_(celestial_coordinates)" title="Equinox (celestial coordinates)">equinox</a>. A <a href="/wiki/Right-handed_rule" class="mw-redirect" title="Right-handed rule">right-handed rule</a> specifies a <span class="texhtml mvar" style="font-style:italic;">y</span>-axis 90° to the east on the fundamental plane. The <span class="texhtml mvar" style="font-style:italic;">z</span>-axis points toward the north <a href="/wiki/Ecliptic_pole" class="mw-redirect" title="Ecliptic pole">ecliptic pole</a>. The reference frame is relatively stationary, aligned with the March equinox.</figcaption></figure> <p>A <a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">rectangular variant</a> of ecliptic coordinates is often used in <a href="/wiki/Orbit" title="Orbit">orbital</a> calculations and simulations. It has its <a href="/wiki/Origin_(mathematics)" title="Origin (mathematics)">origin</a> at the center of the <a href="/wiki/Sun" title="Sun">Sun</a> (or at the <a href="/wiki/Barycenter" class="mw-redirect" title="Barycenter">barycenter</a> of the <a href="/wiki/Solar_System" title="Solar System">Solar System</a>), its <a href="/wiki/Fundamental_plane_(spherical_coordinates)" title="Fundamental plane (spherical coordinates)">fundamental plane</a> on the <a href="/wiki/Ecliptic" title="Ecliptic">ecliptic</a> plane, and the <span class="texhtml mvar" style="font-style:italic;">x</span>-axis toward the March <a href="/wiki/Equinox_(celestial_coordinates)" title="Equinox (celestial coordinates)">equinox</a>. The coordinates have a <a href="/wiki/Right-handed_rule" class="mw-redirect" title="Right-handed rule">right-handed convention</a>, that is, if one extends their right thumb upward, it simulates the <span class="texhtml mvar" style="font-style:italic;">z</span>-axis, their extended index finger the <span class="texhtml mvar" style="font-style:italic;">x</span>-axis, and the curl of the other fingers points generally in the direction of the <span class="texhtml mvar" style="font-style:italic;">y</span>-axis.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p>These rectangular coordinates are related to the corresponding spherical coordinates by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}x&=r\cos b\cos l\\y&=r\cos b\sin l\\z&=r\sin b\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>r</mi> <mi>cos</mi> <mo>⁡</mo> <mi>b</mi> <mi>cos</mi> <mo>⁡</mo> <mi>l</mi> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>r</mi> <mi>cos</mi> <mo>⁡</mo> <mi>b</mi> <mi>sin</mi> <mo>⁡</mo> <mi>l</mi> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>r</mi> <mi>sin</mi> <mo>⁡</mo> <mi>b</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x&=r\cos b\cos l\\y&=r\cos b\sin l\\z&=r\sin b\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5695ceb1543dc5a294296dd302b2614eefdeb31b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:15.69ex; height:8.843ex;" alt="{\displaystyle {\begin{aligned}x&=r\cos b\cos l\\y&=r\cos b\sin l\\z&=r\sin b\end{aligned}}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Conversion_between_celestial_coordinate_systems">Conversion between celestial coordinate systems</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ecliptic_coordinate_system&action=edit&section=5" title="Edit section: Conversion between celestial coordinate systems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Celestial_coordinate_system" class="mw-redirect" title="Celestial coordinate system">Celestial coordinate system</a></div> <div class="mw-heading mw-heading3"><h3 id="Converting_Cartesian_vectors">Converting Cartesian vectors</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ecliptic_coordinate_system&action=edit&section=6" title="Edit section: Converting Cartesian vectors"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Conversion_from_ecliptic_coordinates_to_equatorial_coordinates">Conversion from ecliptic coordinates to equatorial coordinates</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ecliptic_coordinate_system&action=edit&section=7" title="Edit section: Conversion from ecliptic coordinates to equatorial coordinates"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\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}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>equatorial</mtext> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>equatorial</mtext> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>equatorial</mtext> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>ε</mi> </mtd> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>ε</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>ε</mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>ε</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>ecliptic</mtext> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>ecliptic</mtext> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>ecliptic</mtext> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\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}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3298294eda69fa0aefd6c6acea96376f71f93eff" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:46.982ex; height:9.843ex;" alt="{\displaystyle {\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}}}"></span><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Conversion_from_equatorial_coordinates_to_ecliptic_coordinates">Conversion from equatorial coordinates to ecliptic coordinates</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ecliptic_coordinate_system&action=edit&section=8" title="Edit section: Conversion from equatorial coordinates to ecliptic coordinates"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\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}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>ecliptic</mtext> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>ecliptic</mtext> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>ecliptic</mtext> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>ε</mi> </mtd> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>ε</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>ε</mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>ε</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>equatorial</mtext> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>equatorial</mtext> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>equatorial</mtext> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\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}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15a3bb4fd53a8be7811f737044a80796584a2991" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:46.982ex; height:9.843ex;" alt="{\displaystyle {\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}}}"></span> where <span class="texhtml mvar" style="font-style:italic;">ε</span> is the <a href="/wiki/Obliquity_of_the_ecliptic" class="mw-redirect" title="Obliquity of the ecliptic">obliquity of the ecliptic</a>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ecliptic_coordinate_system&action=edit&section=9" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Celestial_coordinate_system" class="mw-redirect" title="Celestial coordinate system">Celestial coordinate system</a></li> <li><a href="/wiki/Ecliptic" title="Ecliptic">Ecliptic</a></li> <li><a href="/wiki/Orbital_pole#Ecliptic_pole" title="Orbital pole">Ecliptic pole</a>, where the ecliptic latitude is ±90°</li> <li><a href="/wiki/Equinox" title="Equinox">Equinox</a> <ul><li><a href="/wiki/Equinox_(celestial_coordinates)" title="Equinox (celestial coordinates)">Equinox (celestial coordinates)</a></li> <li><a href="/wiki/March_equinox" title="March equinox">March equinox</a></li></ul></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes_and_references">Notes and references</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ecliptic_coordinate_system&action=edit&section=10" title="Edit section: Notes and references"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-Cunningham1985-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Cunningham1985_1-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFCunningham1985" class="citation journal cs1">Cunningham, Clifford J. (June 1985). "Asteroid Pole Positions: A Survey". <i>The Minor Planet Bulletin</i>. <b>12</b>: 13–16. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1985MPBu...12...13C">1985MPBu...12...13C</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Minor+Planet+Bulletin&rft.atitle=Asteroid+Pole+Positions%3A+A+Survey&rft.volume=12&rft.pages=13-16&rft.date=1985-06&rft_id=info%3Abibcode%2F1985MPBu...12...13C&rft.aulast=Cunningham&rft.aufirst=Clifford+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEcliptic+coordinate+system" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNautical_Almanac_Office,_U.S._Naval_ObservatoryH.M._Nautical_Almanac_Office,_Royal_Greenwich_Observatory1961" class="citation book cs1">Nautical Almanac Office, U.S. Naval Observatory; H.M. Nautical Almanac Office, Royal Greenwich Observatory (1961). <a rel="nofollow" class="external text" href="https://archive.org/details/astronomicalalmanac1961"><i>Explanatory Supplement to the Astronomical Ephemeris and the American Ephemeris and Nautical Almanac</i></a>. H.M. Stationery Office, London (reprint 1974). pp. <a rel="nofollow" class="external text" href="https://archive.org/details/astronomicalalmanac1961/page/n34">24</a>–27.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Explanatory+Supplement+to+the+Astronomical+Ephemeris+and+the+American+Ephemeris+and+Nautical+Almanac&rft.pages=24-27&rft.pub=H.M.+Stationery+Office%2C+London+%28reprint+1974%29&rft.date=1961&rft.au=Nautical+Almanac+Office%2C+U.S.+Naval+Observatory&rft.au=H.M.+Nautical+Almanac+Office%2C+Royal+Greenwich+Observatory&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fastronomicalalmanac1961&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEcliptic+coordinate+system" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"> <i>Explanatory Supplement</i> (1961), pp. 20, 28</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFU.S._Naval_Observatory1992" class="citation book cs1">U.S. Naval Observatory, Nautical Almanac Office (1992). P. Kenneth Seidelmann (ed.). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=uJ4JhGJANb4C&pg=PA11"><i>Explanatory Supplement to the Astronomical Almanac</i></a>. University Science Books, Mill Valley, CA (reprint 2005). pp. 11–13. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-891389-45-9" title="Special:BookSources/1-891389-45-9"><bdi>1-891389-45-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Explanatory+Supplement+to+the+Astronomical+Almanac&rft.pages=11-13&rft.pub=University+Science+Books%2C+Mill+Valley%2C+CA+%28reprint+2005%29&rft.date=1992&rft.isbn=1-891389-45-9&rft.aulast=U.S.+Naval+Observatory&rft.aufirst=Nautical+Almanac+Office&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DuJ4JhGJANb4C%26pg%3DPA11&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEcliptic+coordinate+system" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMeeus1991" class="citation book cs1">Meeus, Jean (1991). <i>Astronomical Algorithms</i>. Willmann-Bell, Inc., Richmond, VA. p. 137. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-943396-35-2" title="Special:BookSources/0-943396-35-2"><bdi>0-943396-35-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Astronomical+Algorithms&rft.pages=137&rft.pub=Willmann-Bell%2C+Inc.%2C+Richmond%2C+VA&rft.date=1991&rft.isbn=0-943396-35-2&rft.aulast=Meeus&rft.aufirst=Jean&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEcliptic+coordinate+system" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><i>Explanatory Supplement</i> (1961), sec. 1G</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFN._CapitaineP.T._WallaceJ._Chapront2003" class="citation journal cs1">N. Capitaine; P.T. Wallace; J. Chapront (2003). <a rel="nofollow" class="external text" href="http://syrte.obspm.fr/iau2006/aa03_412_P03.pdf">"Expressions for IAU 2000 precession quantities"</a> <span class="cs1-format">(PDF)</span>. <i>Astronomy & Astrophysics</i>. <b>412</b> (2): 581. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2003A&A...412..567C">2003A&A...412..567C</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1051%2F0004-6361%3A20031539">10.1051/0004-6361:20031539</a></span>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120325210757/http://syrte.obspm.fr/iau2006/aa03_412_P03.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2012-03-25.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Astronomy+%26+Astrophysics&rft.atitle=Expressions+for+IAU+2000+precession+quantities&rft.volume=412&rft.issue=2&rft.pages=581&rft.date=2003&rft_id=info%3Adoi%2F10.1051%2F0004-6361%3A20031539&rft_id=info%3Abibcode%2F2003A%26A...412..567C&rft.au=N.+Capitaine&rft.au=P.T.+Wallace&rft.au=J.+Chapront&rft_id=http%3A%2F%2Fsyrte.obspm.fr%2Fiau2006%2Faa03_412_P03.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEcliptic+coordinate+system" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">J.H. Lieske <i>et al.</i> (1977), "<a rel="nofollow" class="external text" href="http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1977A%26A....58....1L&db_key=AST&data_type=HTML&format=&high=46303c7cf308007">Expressions for the Precession Quantities Based upon the IAU (1976) System of Astronomical Constants</a>". <i>Astronomy & Astrophysics</i> <b>58</b>, pp. 1-16</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLeadbetter1742" class="citation book cs1">Leadbetter, Charles (1742). <a rel="nofollow" class="external text" href="https://archive.org/details/acompleatsystem01leadgoog"><i>A Compleat System of Astronomy</i></a>. J. Wilcox, London. p. <a rel="nofollow" class="external text" href="https://archive.org/details/acompleatsystem01leadgoog/page/n102">94</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Compleat+System+of+Astronomy&rft.pages=94&rft.pub=J.+Wilcox%2C+London&rft.date=1742&rft.aulast=Leadbetter&rft.aufirst=Charles&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Facompleatsystem01leadgoog&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEcliptic+coordinate+system" class="Z3988"></span>; numerous examples of this notation appear throughout the book.</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"> <i>Explanatory Supplement</i> (1961), pp. 20, 27</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"> <i>Explanatory Supplement</i> (1992), pp. 555-558</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ecliptic_coordinate_system&action=edit&section=11" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://johnlucey.webspace.durham.ac.uk/users/solar-year/">The Ecliptic: the Sun's Annual Path on the Celestial Sphere</a> Durham University Department of Physics</li> <li><a rel="nofollow" class="external text" href="https://frostydrew.org/utilities.dc/convert/tool-eq_coordinates/">Equatorial ↔ Ecliptic coordinate converter</a></li> <li><a rel="nofollow" class="external text" href="http://stars.astro.illinois.edu/celsph.html">MEASURING THE SKY A Quick Guide to the Celestial Sphere</a> James B. 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title="Aquarius (astrology)">Aquarius</a></li> <li><a href="/wiki/Pisces_(astrology)" title="Pisces (astrology)">Pisces</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Concepts in <br /><a href="/wiki/Western_astrology" title="Western astrology">Western astrology</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Astrology_and_the_classical_elements" title="Astrology and the classical elements">Element</a></li> <li><a href="/wiki/House_(astrology)" title="House (astrology)">House</a></li> <li><a href="/wiki/Astrological_sign#The_three_modalities" title="Astrological sign">Modality</a></li> <li><a href="/wiki/Astrological_sign#Polarity" title="Astrological sign">Polarity</a></li> <li><a href="/wiki/Triplicity" title="Triplicity">Triplicity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Astronomy" title="Astronomy">Astronomy</a></li> <li><a href="/wiki/Axial_precession" title="Axial precession">Axial precession</a></li> <li><a href="/wiki/Constellation" title="Constellation">Constellations</a> of the <a href="/wiki/Ecliptic" title="Ecliptic">Ecliptic</a></li> <li><a class="mw-selflink selflink">Ecliptic coordinate system</a></li> <li><a href="/wiki/Ophiuchus_(astrology)" title="Ophiuchus (astrology)">Ophiuchus</a></li> <li><a href="/wiki/Planets_in_astrology" title="Planets in astrology">Planets in astrology</a> <ul><li><a href="/wiki/Planetary_symbols" title="Planetary symbols">symbols</a></li></ul></li> <li><a href="/wiki/Zodiacal_light" title="Zodiacal light">Zodiacal light</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" 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