Astronomical coordinate systems

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<ul id="toc-Coordinate_systems-sublist" class="vector-toc-list"> <li id="toc-Horizontal_system" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Horizontal_system"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Horizontal system</span> </div> </a> <ul id="toc-Horizontal_system-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Equatorial_system" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Equatorial_system"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Equatorial system</span> </div> </a> <ul id="toc-Equatorial_system-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ecliptic_system" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ecliptic_system"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Ecliptic system</span> </div> </a> <ul id="toc-Ecliptic_system-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Galactic_system" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Galactic_system"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Galactic system</span> </div> </a> <ul id="toc-Galactic_system-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Supergalactic_system" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Supergalactic_system"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.5</span> <span>Supergalactic system</span> </div> </a> <ul id="toc-Supergalactic_system-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Converting_coordinates" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Converting_coordinates"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Converting coordinates</span> </div> </a> <button aria-controls="toc-Converting_coordinates-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 Converting coordinates subsection</span> </button> <ul id="toc-Converting_coordinates-sublist" class="vector-toc-list"> <li id="toc-Notation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Notation"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Notation</span> </div> </a> <ul id="toc-Notation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hour_angle_↔_right_ascension" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hour_angle_↔_right_ascension"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Hour angle ↔ right ascension</span> </div> </a> <ul id="toc-Hour_angle_↔_right_ascension-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Equatorial_↔_ecliptic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Equatorial_↔_ecliptic"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Equatorial ↔ ecliptic</span> </div> </a> <ul id="toc-Equatorial_↔_ecliptic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Equatorial_↔_horizontal" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Equatorial_↔_horizontal"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Equatorial ↔ horizontal</span> </div> </a> <ul id="toc-Equatorial_↔_horizontal-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Equatorial_↔_galactic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Equatorial_↔_galactic"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Equatorial ↔ galactic</span> </div> </a> <ul id="toc-Equatorial_↔_galactic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes_on_conversion" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Notes_on_conversion"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.6</span> <span>Notes on conversion</span> </div> </a> <ul id="toc-Notes_on_conversion-sublist" class="vector-toc-list"> </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">3</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>References</span> </div> </a> <ul id="toc-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">6</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" title="Table of Contents" > <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">Astronomical coordinate systems</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 66 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-66" 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">66 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Lugko%C3%B6rdinaatstelsel" title="Lugkoördinaatstelsel – Afrikaans" lang="af" hreflang="af" data-title="Lugkoördinaatstelsel" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><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%A5%D8%AD%D8%AF%D8%A7%D8%AB%D9%8A%D8%A7%D8%AA_%D9%81%D9%84%D9%83%D9%8A" title="نظام إحداثيات فلكي – Arabic" lang="ar" hreflang="ar" data-title="نظام إحداثيات فلكي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Coordenaes_celestes" title="Coordenaes celestes – Asturian" lang="ast" hreflang="ast" data-title="Coordenaes celestes" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/F%C9%99za_koordinat_sistemi" title="Fəza koordinat sistemi – Azerbaijani" lang="az" hreflang="az" data-title="Fəza koordinat sistemi" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A1%D1%96%D1%81%D1%82%D1%8D%D0%BC%D0%B0_%D0%BD%D1%8F%D0%B1%D0%B5%D1%81%D0%BD%D1%8B%D1%85_%D0%BA%D0%B0%D0%B0%D1%80%D0%B4%D1%8B%D0%BD%D0%B0%D1%82" title="Сістэма нябесных каардынат – Belarusian" lang="be" hreflang="be" data-title="Сістэма нябесных каардынат" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%A1%D1%8B%D1%81%D1%82%D1%8D%D0%BC%D0%B0_%D0%BD%D1%8F%D0%B1%D0%B5%D1%81%D0%BD%D1%8B%D1%85_%D0%BA%D0%B0%D0%B0%D1%80%D0%B4%D1%8B%D0%BD%D0%B0%D1%82" title="Сыстэма нябесных каардынат – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Сыстэма нябесных каардынат" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9D%D0%B5%D0%B1%D0%B5%D1%81%D0%BD%D0%B0_%D0%BA%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82%D0%BD%D0%B0_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0" title="Небесна координатна система – Bulgarian" lang="bg" hreflang="bg" data-title="Небесна координатна система" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Nebeski_koordinatni_sistem" title="Nebeski koordinatni sistem – Bosnian" lang="bs" hreflang="bs" data-title="Nebeski koordinatni sistem" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Coordenades_astron%C3%B2miques" title="Coordenades astronòmiques – Catalan" lang="ca" hreflang="ca" data-title="Coordenades astronòmiques" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Koordinatsystem_for_himmelrummet" title="Koordinatsystem for himmelrummet – Danish" lang="da" hreflang="da" data-title="Koordinatsystem for himmelrummet" 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/Astronomische_Koordinatensysteme" title="Astronomische Koordinatensysteme – German" lang="de" hreflang="de" data-title="Astronomische Koordinatensysteme" 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/Taevakoordinaatide_s%C3%BCsteem" title="Taevakoordinaatide süsteem – Estonian" lang="et" hreflang="et" data-title="Taevakoordinaatide süsteem" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9F%CF%85%CF%81%CE%AC%CE%BD%CE%B9%CE%B5%CF%82_%CF%83%CF%85%CE%BD%CF%84%CE%B5%CF%84%CE%B1%CE%B3%CE%BC%CE%AD%CE%BD%CE%B5%CF%82" title="Ουράνιες συντεταγμένες – Greek" lang="el" hreflang="el" data-title="Ουράνιες συντεταγμένες" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Coordenadas_celestes" title="Coordenadas celestes – Spanish" lang="es" hreflang="es" data-title="Coordenadas celestes" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Koordenatu_sistema_astronomiko" title="Koordenatu sistema astronomiko – Basque" lang="eu" hreflang="eu" data-title="Koordenatu sistema astronomiko" 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%B3%D9%85%D8%A7%D9%88%DB%8C" title="دستگاه مختصات سماوی – Persian" lang="fa" hreflang="fa" data-title="دستگاه مختصات سماوی" 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_c%C3%A9lestes" title="Système de coordonnées célestes – French" lang="fr" hreflang="fr" data-title="Système de coordonnées célestes" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Coordenadas_celestes" title="Coordenadas celestes – Galician" lang="gl" hreflang="gl" data-title="Coordenadas celestes" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%B2%9C%EA%B5%AC%EC%A2%8C%ED%91%9C%EA%B3%84" title="천구좌표계 – Korean" lang="ko" hreflang="ko" data-title="천구좌표계" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B5%D6%80%D5%AF%D5%B6%D5%A1%D5%B5%D5%AB%D5%B6_%D5%AF%D5%B8%D5%B8%D6%80%D5%A4%D5%AB%D5%B6%D5%A1%D5%BF%D5%B6%D5%A5%D6%80%D5%AB_%D5%B0%D5%A1%D5%B4%D5%A1%D5%AF%D5%A1%D6%80%D5%A3%D5%A5%D6%80" title="Երկնային կոորդինատների համակարգեր – Armenian" lang="hy" hreflang="hy" data-title="Երկնային կոորդինատների համակարգեր" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%96%E0%A4%97%E0%A5%8B%E0%A4%B2%E0%A5%80%E0%A4%AF_%E0%A4%A8%E0%A4%BF%E0%A4%B0%E0%A5%8D%E0%A4%A6%E0%A5%87%E0%A4%B6%E0%A4%BE%E0%A4%82%E0%A4%95_%E0%A4%AA%E0%A4%A6%E0%A5%8D%E0%A4%A7%E0%A4%A4%E0%A4%BF" title="खगोलीय निर्देशांक पद्धति – Hindi" lang="hi" hreflang="hi" data-title="खगोलीय निर्देशांक पद्धति" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Nebeski_koordinatni_sustavi" title="Nebeski koordinatni sustavi – Croatian" lang="hr" hreflang="hr" data-title="Nebeski koordinatni sustavi" 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_mbara_igwe" title="Usoro nhazi mbara igwe – Igbo" lang="ig" hreflang="ig" data-title="Usoro nhazi mbara igwe" data-language-autonym="Igbo" data-language-local-name="Igbo" class="interlanguage-link-target"><span>Igbo</span></a></li><li class="interlanguage-link interwiki-ilo mw-list-item"><a href="https://ilo.wikipedia.org/wiki/Sistema_ti_nainlangitan_a_nagsasabtan" title="Sistema ti nainlangitan a nagsasabtan – Iloko" lang="ilo" hreflang="ilo" data-title="Sistema ti nainlangitan a nagsasabtan" data-language-autonym="Ilokano" data-language-local-name="Iloko" class="interlanguage-link-target"><span>Ilokano</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Tata_koordinat_langit" title="Tata koordinat langit – Indonesian" lang="id" hreflang="id" data-title="Tata koordinat langit" 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/Himinhvolfshnitakerfi" title="Himinhvolfshnitakerfi – Icelandic" lang="is" hreflang="is" data-title="Himinhvolfshnitakerfi" 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/Coordinate_celesti" title="Coordinate celesti – Italian" lang="it" hreflang="it" data-title="Coordinate celesti" 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%A7%D7%95%D7%90%D7%95%D7%A8%D7%93%D7%99%D7%A0%D7%98%D7%95%D7%AA_%D7%A9%D7%9E%D7%99%D7%9E%D7%99%D7%AA" title="מערכת קואורדינטות שמימית – Hebrew" lang="he" hreflang="he" data-title="מערכת קואורדינטות שמימית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%90%D1%81%D0%BF%D0%B0%D0%BD_%D0%BA%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82%D1%82%D0%B0%D1%80_%D0%B6%D2%AF%D0%B9%D0%B5%D1%81%D1%96" title="Аспан координаттар жүйесі – Kazakh" lang="kk" hreflang="kk" data-title="Аспан координаттар жүйесі" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Majiranukta_ya_anga" title="Majiranukta ya anga – Swahili" lang="sw" hreflang="sw" data-title="Majiranukta ya anga" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Systema_coordinatarum_caelestium" title="Systema coordinatarum caelestium – Latin" lang="la" hreflang="la" data-title="Systema coordinatarum caelestium" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Himmelskoordinaten" title="Himmelskoordinaten – Luxembourgish" lang="lb" hreflang="lb" data-title="Himmelskoordinaten" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxembourgish" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Csillag%C3%A1szati_koordin%C3%A1ta-rendszer" title="Csillagászati koordináta-rendszer – Hungarian" lang="hu" hreflang="hu" data-title="Csillagászati koordináta-rendszer" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9D%D0%B5%D0%B1%D0%B5%D1%81%D0%B5%D0%BD_%D0%BA%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82%D0%B5%D0%BD_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC" title="Небесен координатен систем – Macedonian" lang="mk" hreflang="mk" data-title="Небесен координатен систем" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Sistem_koordinat_cakerawala" title="Sistem koordinat cakerawala – Malay" lang="ms" hreflang="ms" data-title="Sistem koordinat cakerawala" 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-min mw-list-item"><a href="https://min.wikipedia.org/wiki/Sistem_koordinat_astronomi" title="Sistem koordinat astronomi – Minangkabau" lang="min" hreflang="min" data-title="Sistem koordinat astronomi" data-language-autonym="Minangkabau" data-language-local-name="Minangkabau" class="interlanguage-link-target"><span>Minangkabau</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%9E%D0%B4%D0%BE%D0%BD_%D0%BE%D1%80%D0%BD%D1%8B_%D0%BA%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82%D1%8B%D0%BD_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC" title="Одон орны координатын систем – Mongolian" lang="mn" hreflang="mn" data-title="Одон орны координатын систем" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%94%E1%80%80%E1%80%B9%E1%80%81%E1%80%90%E1%80%B9%E1%80%90%E1%80%97%E1%80%B1%E1%80%92%E1%80%86%E1%80%AD%E1%80%AF%E1%80%84%E1%80%BA%E1%80%9B%E1%80%AC_%E1%80%80%E1%80%AD%E1%80%AF%E1%80%A9%E1%80%92%E1%80%AD%E1%80%94%E1%80%AD%E1%80%90%E1%80%BA%E1%80%85%E1%80%94%E1%80%85%E1%80%BA" title="နက္ခတ္တဗေဒဆိုင်ရာ ကိုဩဒိနိတ်စနစ် – Burmese" lang="my" hreflang="my" data-title="နက္ခတ္တဗေဒဆိုင်ရာ ကိုဩဒိနိတ်စနစ်" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Astronomisch_co%C3%B6rdinatenstelsel" title="Astronomisch coördinatenstelsel – Dutch" lang="nl" hreflang="nl" data-title="Astronomisch coördinatenstelsel" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%A4%A9%E7%90%83%E5%BA%A7%E6%A8%99%E7%B3%BB" title="天球座標系 – Japanese" lang="ja" hreflang="ja" data-title="天球座標系" 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/Himmelkoordinat" title="Himmelkoordinat – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Himmelkoordinat" 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/Himmelkoordinatar" title="Himmelkoordinatar – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Himmelkoordinatar" 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-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Sist%C3%A8ma_de_coordenadas_cel%C3%A8stas" title="Sistèma de coordenadas celèstas – Occitan" lang="oc" hreflang="oc" data-title="Sistèma de coordenadas celèstas" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Astronomik_koordinatalar" title="Astronomik koordinatalar – Uzbek" lang="uz" hreflang="uz" data-title="Astronomik 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_wsp%C3%B3%C5%82rz%C4%99dnych_astronomicznych" title="Układ współrzędnych astronomicznych – Polish" lang="pl" hreflang="pl" data-title="Układ współrzędnych astronomicznych" 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_de_coordenadas_celestes" title="Sistema de coordenadas celestes – Portuguese" lang="pt" hreflang="pt" data-title="Sistema de coordenadas celestes" 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" title="Coordonate astronomice – Romanian" lang="ro" hreflang="ro" data-title="Coordonate astronomice" 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%A1%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0_%D0%BD%D0%B5%D0%B1%D0%B5%D1%81%D0%BD%D1%8B%D1%85_%D0%BA%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82" title="Система небесных координат – Russian" lang="ru" hreflang="ru" data-title="Система небесных координат" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Celestial_coordinate_system" title="Celestial coordinate system – Simple English" lang="en-simple" hreflang="en-simple" data-title="Celestial coordinate system" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Astronomick%C3%A9_s%C3%BAradnicov%C3%A9_syst%C3%A9my" title="Astronomické súradnicové systémy – Slovak" lang="sk" hreflang="sk" data-title="Astronomické súradnicové systémy" 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/Nebesni_koordinatni_sistem" title="Nebesni koordinatni sistem – Slovenian" lang="sl" hreflang="sl" data-title="Nebesni 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-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%B3%DB%8C%D8%B3%D8%AA%DB%95%D9%85%DB%8C_%D9%BE%DB%86%D8%AA%D8%A7%D9%86%DB%8C_%D8%A6%D8%A7%D8%B3%D9%85%D8%A7%D9%86%DB%8C" title="سیستەمی پۆتانی ئاسمانی – Central Kurdish" lang="ckb" hreflang="ckb" data-title="سیستەمی پۆتانی ئاسمانی" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://sr.wikipedia.org/wiki/%D0%9D%D0%B5%D0%B1%D0%B5%D1%81%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%D0%B8" title="Небески координатни системи – Serbian" lang="sr" hreflang="sr" data-title="Небески координатни системи" 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/Nebeski_koordinatni_sistemi" title="Nebeski koordinatni sistemi – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Nebeski koordinatni sistemi" 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-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/T%C3%A4htitieteellinen_vertausj%C3%A4rjestelm%C3%A4" title="Tähtitieteellinen vertausjärjestelmä – Finnish" lang="fi" hreflang="fi" data-title="Tähtitieteellinen vertausjärjestelmä" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Astronomiska_koordinatsystem" title="Astronomiska koordinatsystem – Swedish" lang="sv" hreflang="sv" data-title="Astronomiska koordinatsystem" 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_panlangit_ng_mga_koordinado" title="Sistemang panlangit ng mga koordinado – Tagalog" lang="tl" hreflang="tl" data-title="Sistemang panlangit ng mga koordinado" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%A3%E0%B8%B0%E0%B8%9A%E0%B8%9A%E0%B8%9E%E0%B8%B4%E0%B8%81%E0%B8%B1%E0%B8%94%E0%B8%97%E0%B8%A3%E0%B8%87%E0%B8%81%E0%B8%A5%E0%B8%A1%E0%B8%97%E0%B9%89%E0%B8%AD%E0%B8%87%E0%B8%9F%E0%B9%89%E0%B8%B2" title="ระบบพิกัดทรงกลมท้องฟ้า – Thai" lang="th" hreflang="th" data-title="ระบบพิกัดทรงกลมท้องฟ้า" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/G%C3%B6ky%C3%BCz%C3%BC_koordinat_sistemi" title="Gökyüzü koordinat sistemi – Turkish" lang="tr" hreflang="tr" data-title="Gökyüzü koordinat sistemi" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A1%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8_%D0%BD%D0%B5%D0%B1%D0%B5%D1%81%D0%BD%D0%B8%D1%85_%D0%BA%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82" title="Системи небесних координат – Ukrainian" lang="uk" hreflang="uk" data-title="Системи небесних координат" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%B3%D9%85%D8%A7%D9%88%DB%8C_%D9%85%D8%AA%D9%86%D8%A7%D8%B3%D9%82_%D9%86%D8%B8%D8%A7%D9%85" title="سماوی متناسق نظام – Urdu" lang="ur" hreflang="ur" data-title="سماوی متناسق نظام" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/H%E1%BB%87_t%E1%BB%8Da_%C4%91%E1%BB%99_thi%C3%AAn_v%C4%83n" title="Hệ tọa độ thiên văn – Vietnamese" lang="vi" hreflang="vi" data-title="Hệ tọa độ thiên văn" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Sistema_pan-langit_hin_mga_koordinato" title="Sistema pan-langit hin mga koordinato – Waray" lang="war" hreflang="war" data-title="Sistema pan-langit hin mga koordinato" data-language-autonym="Winaray" data-language-local-name="Waray" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%A4%A9%E7%90%83%E5%9D%90%E6%A0%87%E7%B3%BB%E7%BB%9F" title="天球坐标系统 – Wu" lang="wuu" hreflang="wuu" data-title="天球坐标系统" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%A4%A9%E7%90%83%E5%9D%90%E6%A8%99%E7%B3%BB%E7%B5%B1" title="天球坐標系統 – Cantonese" lang="yue" hreflang="yue" data-title="天球坐標系統" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%A4%A9%E7%90%83%E5%9D%90%E6%A0%87%E7%B3%BB%E7%BB%9F" title="天球坐标系统 – Chinese" lang="zh" hreflang="zh" data-title="天球坐标系统" data-language-autonym="中文" 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.infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox"><caption class="infobox-title">Orientation of astronomical coordinates</caption><tbody><tr><td colspan="2" class="infobox-image"><span typeof="mw:File"><a href="/wiki/File:Ecliptic_equator_galactic_anim.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Ecliptic_equator_galactic_anim.gif/250px-Ecliptic_equator_galactic_anim.gif" decoding="async" width="250" height="254" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Ecliptic_equator_galactic_anim.gif/375px-Ecliptic_equator_galactic_anim.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Ecliptic_equator_galactic_anim.gif/500px-Ecliptic_equator_galactic_anim.gif 2x" data-file-width="1181" data-file-height="1200" /></a></span><div class="infobox-caption">A <a href="/wiki/Star" title="Star">star</a>'s <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><span class="legend-color mw-no-invert" style="background-color:yellow; color:black;"> </span><span class="nowrap"> </span><a href="/wiki/Galactic_coordinate_system" title="Galactic coordinate system">galactic</a>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r981673959" /><span class="legend-color mw-no-invert" style="background-color:red; color:black;"> </span><span class="nowrap"> </span><a href="/wiki/Ecliptic_coordinate_system" title="Ecliptic coordinate system">ecliptic</a>, and <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r981673959" /><span class="legend-color mw-no-invert" style="background-color:blue; color:white;"> </span><span class="nowrap"> </span><a href="/wiki/Equatorial_coordinate_system" title="Equatorial coordinate system">equatorial</a> coordinates, as projected on the <a href="/wiki/Celestial_sphere" title="Celestial sphere">celestial sphere</a>. Ecliptic and equatorial coordinates share the <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r981673959" /><span class="legend-color mw-no-invert" style="background-color:magenta; color:black;"> </span><span class="nowrap"> </span><a href="/wiki/Equinox_(celestial_coordinates)" title="Equinox (celestial coordinates)">March equinox</a> as the <a href="/wiki/Primary_direction" class="mw-redirect" title="Primary direction">primary direction</a>, and galactic coordinates are referred to the <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r981673959" /><span class="legend-color mw-no-invert" style="background-color:yellow; color:black;"> </span><span class="nowrap"> </span>galactic center. The origin of coordinates (the "center of the sphere") is ambiguous; see <a href="/wiki/Celestial_sphere" title="Celestial sphere">celestial sphere</a> for more information.</div></td></tr></tbody></table> <p>In <a href="/wiki/Astronomy" title="Astronomy">astronomy</a>, <a href="/wiki/Coordinate_systems" class="mw-redirect" title="Coordinate systems">coordinate systems</a> are used for specifying <a href="/wiki/Position_(geometry)" title="Position (geometry)">positions</a> of <a href="/wiki/Astronomical_object" title="Astronomical object">celestial objects</a> (<a href="/wiki/Natural_satellite" title="Natural satellite">satellites</a>, <a href="/wiki/Planet" title="Planet">planets</a>, <a href="/wiki/Star" title="Star">stars</a>, <a href="/wiki/Galaxy" title="Galaxy">galaxies</a>, etc.) relative to a given <a href="/wiki/Reference_frame" class="mw-redirect" title="Reference frame">reference frame</a>, based on physical reference points available to a situated observer (e.g. the true <a href="/wiki/Horizon" title="Horizon">horizon</a> and <a href="/wiki/True_north" title="True north">north</a> to an observer on Earth's surface).<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Coordinate systems in astronomy can specify an object's relative position in <a href="/wiki/Three-dimensional_space" title="Three-dimensional space">three-dimensional space</a> or <a href="/wiki/Plot_(graphics)" title="Plot (graphics)">plot</a> merely by its <a href="/wiki/Direction_(geometry)" title="Direction (geometry)">direction</a> on a <a href="/wiki/Celestial_sphere" title="Celestial sphere">celestial sphere</a>, if the object's distance is unknown or trivial. </p><p><a href="/wiki/Spherical_coordinate_system" title="Spherical coordinate system">Spherical coordinates</a>, projected on the celestial sphere, are analogous to the <a href="/wiki/Geographic_coordinate_system" title="Geographic coordinate system">geographic coordinate system</a> used on the surface of <a href="/wiki/Earth" title="Earth">Earth</a>. These differ in their choice of <a href="/wiki/Fundamental_plane_(spherical_coordinates)" title="Fundamental plane (spherical coordinates)">fundamental plane</a>, which divides the celestial sphere into two equal <a href="/wiki/Sphere" title="Sphere">hemispheres</a> along a <a href="/wiki/Great_circle" title="Great circle">great circle</a>. <a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">Rectangular coordinates</a>, in appropriate <a href="/wiki/Units_of_measurement" class="mw-redirect" title="Units of measurement">units</a>, have the same fundamental (<span class="texhtml"><i>x, y</i></span>) plane and <a href="/wiki/Primary_direction" class="mw-redirect" title="Primary direction">primary (<span class="texhtml"><i>x</i></span>-axis) direction</a>, such as an <a href="/wiki/Rotation_around_a_fixed_axis" title="Rotation around a fixed axis">axis of rotation</a>. Each coordinate system is named after its choice of fundamental plane. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Coordinate_systems">Coordinate systems</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Astronomical_coordinate_systems&action=edit&section=1" title="Edit section: Coordinate systems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The following table lists the common coordinate systems in use by the astronomical community. The <a href="/wiki/Fundamental_plane_(spherical_coordinates)" title="Fundamental plane (spherical coordinates)">fundamental plane</a> divides the <a href="/wiki/Celestial_sphere" title="Celestial sphere">celestial sphere</a> into two equal <a href="/wiki/Celestial_sphere" title="Celestial sphere">hemispheres</a> and defines the baseline for the latitudinal coordinates, similar to the <a href="/wiki/Equator" title="Equator">equator</a> in the <a href="/wiki/Geographic_coordinate_system" title="Geographic coordinate system">geographic coordinate system</a>. The poles are located at ±90° from the fundamental plane. The primary direction is the starting point of the longitudinal coordinates. The origin is the zero distance point, the "center of the celestial sphere", although the definition of <a href="/wiki/Celestial_sphere" title="Celestial sphere">celestial sphere</a> is ambiguous about the definition of its center point. </p> <table class="wikitable" style="text-align:center;"> <tbody><tr> <th rowspan="2">Coordinate system<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> </th> <th rowspan="2">Center point<br />(origin) </th> <th rowspan="2">Fundamental plane<br />(0° latitude) </th> <th rowspan="2">Poles </th> <th colspan="2">Coordinates </th> <th rowspan="2">Primary direction<br />(0° longitude) </th></tr> <tr> <th>Latitude </th> <th>Longitude </th></tr> <tr> <td><a href="/wiki/Horizontal_coordinate_system" title="Horizontal coordinate system">Horizontal</a> (also called <abbr title="altitude">alt</abbr>-<abbr title="azimuth">az</abbr> or <abbr title="elevation">el</abbr>-<abbr title="azimuth">az</abbr>) </td> <td>Observer </td> <td><a href="/wiki/Horizon" title="Horizon">Horizon</a> </td> <td><a href="/wiki/Zenith" title="Zenith">Zenith</a>, <a href="/wiki/Nadir_(astronomy)" class="mw-redirect" title="Nadir (astronomy)">nadir</a> </td> <td>Altitude (<span class="texhtml"><i>a</i></span>) or elevation </td> <td><a href="/wiki/Azimuth" title="Azimuth">Azimuth</a> (<span class="texhtml"><i>A</i></span>) </td> <td><a href="/wiki/North" title="North">North</a> or <a href="/wiki/South" title="South">south</a> point of horizon </td></tr> <tr> <td><a href="/wiki/Equatorial_coordinate_system" title="Equatorial coordinate system">Equatorial</a> </td> <td rowspan="2">Center of the <a href="/wiki/Earth" title="Earth">Earth</a><span class="nowrap"> </span>(geocentric), or <a href="/wiki/Sun" title="Sun">Sun</a><span class="nowrap"> </span>(heliocentric) </td> <td><a href="/wiki/Celestial_equator" title="Celestial equator">Celestial equator</a> </td> <td><a href="/wiki/Celestial_pole" title="Celestial pole">Celestial poles</a> </td> <td><a href="/wiki/Declination" title="Declination">Declination</a> (<span class="texhtml"><i>δ</i></span>) </td> <td><a href="/wiki/Right_ascension" title="Right ascension">Right ascension</a> (<span class="texhtml"><i>α</i></span>)<br />or <a href="/wiki/Hour_angle" title="Hour angle">hour angle</a> (<span class="texhtml"><i>h</i></span>) </td> <td rowspan="2"><a href="/wiki/Equinox_(celestial_coordinates)" title="Equinox (celestial coordinates)">March equinox</a> </td></tr> <tr> <td><a href="/wiki/Ecliptic_coordinate_system" title="Ecliptic coordinate system">Ecliptic</a> </td> <td><a href="/wiki/Ecliptic" title="Ecliptic">Ecliptic</a> </td> <td><a href="/wiki/Ecliptic_pole" class="mw-redirect" title="Ecliptic pole">Ecliptic poles</a> </td> <td><a href="/wiki/Ecliptic_latitude" class="mw-redirect" title="Ecliptic latitude">Ecliptic latitude</a> (<span class="texhtml"><i>β</i></span>) </td> <td><a href="/wiki/Ecliptic_longitude" class="mw-redirect" title="Ecliptic longitude">Ecliptic longitude</a> (<span class="texhtml"><i>λ</i></span>) </td></tr> <tr> <td><a href="/wiki/Galactic_coordinate_system" title="Galactic coordinate system">Galactic</a> </td> <td>Center of the <a href="/wiki/Sun" title="Sun">Sun</a> </td> <td><a href="/wiki/Galactic_plane" title="Galactic plane">Galactic plane</a> </td> <td><a href="/wiki/Galactic_pole" class="mw-redirect" title="Galactic pole">Galactic poles</a> </td> <td>Galactic latitude (<span class="texhtml"><i>b</i></span>) </td> <td>Galactic longitude (<span class="texhtml"><i>l</i></span>) </td> <td><a href="/wiki/Galactic_Center" title="Galactic Center">Galactic Center</a> </td></tr> <tr> <td><a href="/wiki/Supergalactic_coordinate_system" title="Supergalactic coordinate system">Supergalactic</a> </td> <td> </td> <td><a href="/wiki/Supergalactic_plane" class="mw-redirect" title="Supergalactic plane">Supergalactic plane</a> </td> <td>Supergalactic poles </td> <td>Supergalactic latitude (<span class="texhtml"><i>SGB</i></span>) </td> <td>Supergalactic longitude (<span class="texhtml"><i>SGL</i></span>) </td> <td>Intersection of supergalactic plane and galactic plane </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Horizontal_system">Horizontal system</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Astronomical_coordinate_systems&action=edit&section=2" title="Edit section: Horizontal system"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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">Main article: <a href="/wiki/Horizontal_coordinate_system" title="Horizontal coordinate system">Horizontal coordinate system</a></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Equatorial_and_horizontal_celestial_coordinates_E.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/db/Equatorial_and_horizontal_celestial_coordinates_E.svg/350px-Equatorial_and_horizontal_celestial_coordinates_E.svg.png" decoding="async" width="350" height="438" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/db/Equatorial_and_horizontal_celestial_coordinates_E.svg/525px-Equatorial_and_horizontal_celestial_coordinates_E.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/db/Equatorial_and_horizontal_celestial_coordinates_E.svg/700px-Equatorial_and_horizontal_celestial_coordinates_E.svg.png 2x" data-file-width="720" data-file-height="900" /></a><figcaption>Equatorial (red) and horizontal (blue) celestial coordinates.</figcaption></figure> <p>The <i>horizontal</i>, or <a href="/wiki/Horizontal_coordinate_system" title="Horizontal coordinate system">altitude-azimuth</a>, system is based on the position of the observer on Earth, which revolves around its own axis once per <a href="/wiki/Sidereal_day" class="mw-redirect" title="Sidereal day">sidereal day</a> (23 hours, 56 minutes and 4.091 seconds) in relation to the star background. The positioning of a celestial object by the horizontal system varies with time, but is a useful coordinate system for locating and tracking objects for observers on Earth. It is based on the position of stars relative to an observer's ideal horizon. </p> <div class="mw-heading mw-heading3"><h3 id="Equatorial_system">Equatorial system</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Astronomical_coordinate_systems&action=edit&section=3" title="Edit section: Equatorial system"><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/Equatorial_coordinate_system" title="Equatorial coordinate system">Equatorial coordinate system</a></div> <p>The <i>equatorial</i> coordinate system is centered at Earth's center, but fixed relative to the celestial poles and the <a href="/wiki/Equinox_(celestial_coordinates)" title="Equinox (celestial coordinates)">March equinox</a>. The coordinates are based on the location of stars relative to Earth's equator if it were projected out to an infinite distance. The equatorial describes the sky as seen from the <a href="/wiki/Solar_System" title="Solar System">Solar System</a>, and modern star maps almost exclusively use equatorial coordinates. </p><p>The <i>equatorial</i> system is the normal coordinate system for most professional and many amateur astronomers having an equatorial mount that follows the movement of the sky during the night. Celestial objects are found by adjusting the telescope's or other instrument's scales so that they match the equatorial coordinates of the selected object to observe. </p><p>Popular choices of pole and equator are the older <a href="/wiki/B1950" class="mw-redirect" title="B1950">B1950</a> and the modern <a href="/wiki/J2000" class="mw-redirect" title="J2000">J2000</a> systems, but a pole and equator "of date" can also be used, meaning one appropriate to the date under consideration, such as when a measurement of the position of a planet or spacecraft is made. There are also subdivisions into "mean of date" coordinates, which average out or ignore <a href="/wiki/Astronomical_nutation" title="Astronomical nutation">nutation</a>, and "true of date," which include nutation. </p> <div class="mw-heading mw-heading3"><h3 id="Ecliptic_system">Ecliptic system</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Astronomical_coordinate_systems&action=edit&section=4" title="Edit section: Ecliptic system"><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/Ecliptic_coordinate_system" title="Ecliptic coordinate system">Ecliptic coordinate system</a></div> <p>The fundamental plane is the plane of the Earth's orbit, called the ecliptic plane. There are two principal variants of the ecliptic coordinate system: geocentric ecliptic coordinates centered on the Earth and heliocentric ecliptic coordinates centered on the center of mass of the Solar System. </p><p>The geocentric ecliptic system was the principal coordinate system for ancient astronomy and is still useful for computing the apparent motions of the Sun, Moon, and planets.<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> It was used to define the twelve <a href="/wiki/Astrological_sign" title="Astrological sign">astrological signs</a> of the <a href="/wiki/Zodiac" title="Zodiac">zodiac</a>, for instance. </p><p>The heliocentric ecliptic system describes the planets' orbital movement around the Sun, and centers on the <a href="/wiki/Center_of_mass#Barycenter_in_astrophysics_and_astronomy" title="Center of mass">barycenter</a> of the Solar System (i.e. very close to the center of the Sun). The system is primarily used for computing the positions of planets and other Solar System bodies, as well as defining their <a href="/wiki/Orbital_elements" title="Orbital elements">orbital elements</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Galactic_system">Galactic system</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Astronomical_coordinate_systems&action=edit&section=5" title="Edit section: Galactic system"><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/Galactic_coordinate_system" title="Galactic coordinate system">Galactic coordinate system</a></div> <p>The galactic coordinate system uses the approximate plane of the Milky Way Galaxy as its fundamental plane. The Solar System is still the center of the coordinate system, and the zero point is defined as the direction towards the <a href="/wiki/Galactic_Center" title="Galactic Center">Galactic Center</a>. Galactic latitude resembles the elevation above the galactic plane and galactic longitude determines direction relative to the center of the galaxy. </p> <div class="mw-heading mw-heading3"><h3 id="Supergalactic_system">Supergalactic system</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Astronomical_coordinate_systems&action=edit&section=6" title="Edit section: Supergalactic system"><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/Supergalactic_coordinate_system" title="Supergalactic coordinate system">Supergalactic coordinate system</a></div> <p>The supergalactic coordinate system corresponds to a fundamental plane that contains a higher than average number of local galaxies in the sky as seen from Earth. </p> <div class="mw-heading mw-heading2"><h2 id="Converting_coordinates">Converting coordinates</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Astronomical_coordinate_systems&action=edit&section=7" title="Edit section: Converting coordinates"><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">See also: <a href="/wiki/Euler_angles" title="Euler angles">Euler angles</a> and <a href="/wiki/Rotation_matrix" title="Rotation matrix">Rotation matrix</a></div> <p>Conversions between the various coordinate systems are given.<sup id="cite_ref-Meeus_4-0" class="reference"><a href="#cite_note-Meeus-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> See the <a href="#Notes_on_conversion">notes</a> before using these equations. </p> <div class="mw-heading mw-heading3"><h3 id="Notation">Notation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Astronomical_coordinate_systems&action=edit&section=8" title="Edit section: Notation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col"> <ul><li>Horizontal coordinates <ul><li><span class="texhtml mvar" style="font-style:italic;">A</span>, <a href="/wiki/Azimuth" title="Azimuth">azimuth</a></li> <li><span class="texhtml mvar" style="font-style:italic;">a</span>, <a href="/wiki/Horizontal_coordinate_system" title="Horizontal coordinate system">altitude</a></li></ul></li> <li>Equatorial coordinates <ul><li><span class="texhtml mvar" style="font-style:italic;">α</span>, <a href="/wiki/Right_ascension" title="Right ascension">right ascension</a></li> <li><span class="texhtml mvar" style="font-style:italic;">δ</span>, <a href="/wiki/Declination" title="Declination">declination</a></li> <li><span class="texhtml mvar" style="font-style:italic;">h</span>, <a href="/wiki/Hour_angle" title="Hour angle">hour angle</a></li></ul></li> <li>Ecliptic coordinates <ul><li><span class="texhtml mvar" style="font-style:italic;">λ</span>, <a href="/wiki/Ecliptic_longitude" class="mw-redirect" title="Ecliptic longitude">ecliptic longitude</a></li> <li><span class="texhtml mvar" style="font-style:italic;">β</span>, <a href="/wiki/Ecliptic_latitude" class="mw-redirect" title="Ecliptic latitude">ecliptic latitude</a></li></ul></li> <li>Galactic coordinates <ul><li><span class="texhtml mvar" style="font-style:italic;">l</span>, <a href="/wiki/Galactic_longitude" class="mw-redirect" title="Galactic longitude">galactic longitude</a></li> <li><span class="texhtml mvar" style="font-style:italic;">b</span>, <a href="/wiki/Galactic_latitude" class="mw-redirect" title="Galactic latitude">galactic latitude</a></li></ul></li> <li>Miscellaneous <ul><li><span class="texhtml"><i>λ</i><sub>o</sub></span>, <a href="/wiki/Longitude" title="Longitude">observer's longitude</a></li> <li><span class="texhtml"><i>ϕ</i><sub>o</sub></span>, <a href="/wiki/Latitude" title="Latitude">observer's latitude</a></li> <li><span class="texhtml mvar" style="font-style:italic;">ε</span>, <a href="/wiki/Axial_tilt#Earth" title="Axial tilt">obliquity of the ecliptic</a> (about 23.4°)</li> <li><span class="texhtml"><i>θ</i><sub>L</sub></span>, <a href="/wiki/Sidereal_time" title="Sidereal time">local sidereal time</a></li> <li><span class="texhtml"><i>θ</i><sub>G</sub></span>, <a href="/wiki/Sidereal_time" title="Sidereal time">Greenwich sidereal time</a></li></ul></li></ul> </div> <div class="mw-heading mw-heading3"><h3 id="Hour_angle_↔_right_ascension"><span id="Hour_angle_.E2.86.94_right_ascension"></span>Hour angle ↔ right ascension</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Astronomical_coordinate_systems&action=edit&section=9" title="Edit section: Hour angle ↔ right ascension"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <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}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{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>h</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> </msub> <mo>−</mo> <mi>α</mi> </mtd> <mtd></mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>or</mtext> </mstyle> </mrow> </mtd> <mtd> <mi>h</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>G</mtext> </mrow> </msub> <mo>+</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>o</mtext> </mrow> </msub> <mo>−</mo> <mi>α</mi> </mtd> </mtr> <mtr> <mtd> <mi>α</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> </msub> <mo>−</mo> <mi>h</mi> </mtd> <mtd></mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>or</mtext> </mstyle> </mrow> </mtd> <mtd> <mi>α</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>G</mtext> </mrow> </msub> <mo>+</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>o</mtext> </mrow> </msub> <mo>−</mo> <mi>h</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}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{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3dabf3cb4acc8de2573ce32fb10ceaa423719cef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:40.157ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}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{aligned}}}" /></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Equatorial_↔_ecliptic"><span id="Equatorial_.E2.86.94_ecliptic"></span>Equatorial ↔ ecliptic</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Astronomical_coordinate_systems&action=edit&section=10" title="Edit section: Equatorial ↔ ecliptic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The classical equations, derived from <a href="/wiki/Spherical_trigonometry" title="Spherical trigonometry">spherical trigonometry</a>, 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.<sup id="cite_ref-ExplSupp_5-0" class="reference"><a href="#cite_note-ExplSupp-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> The rotation matrix equivalent is given beneath each case.<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> This division is ambiguous because tan has a period of 180° (<span class="texhtml mvar" style="font-style:italic;">π</span>) whereas cos and sin have periods of 360° (2<span class="texhtml mvar" style="font-style:italic;">π</span>). </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}\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{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 0.6em 0.9em 0.6em 0.9em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>tan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>λ</mi> <mo>)</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>α</mi> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>ε</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>tan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>δ</mi> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>ε</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>α</mi> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <mo>;</mo> <mspace width="2em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>β</mi> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>λ</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>δ</mi> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>α</mi> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>ε</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>δ</mi> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>ε</mi> <mo>)</mo> </mrow> <mo>;</mo> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>β</mi> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>λ</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>δ</mi> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>α</mi> <mo>)</mo> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>β</mi> <mo>)</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>δ</mi> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>ε</mi> <mo>)</mo> </mrow> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>δ</mi> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>ε</mi> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>α</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>β</mi> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>λ</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>β</mi> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>λ</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>β</mi> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mtd> <mtd> <mi></mi> <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> <mrow> <mo>(</mo> <mi>ε</mi> <mo>)</mo> </mrow> </mtd> <mtd> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>ε</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>ε</mi> <mo>)</mo> </mrow> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>ε</mi> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>δ</mi> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>α</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>δ</mi> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>α</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>δ</mi> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>tan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>α</mi> <mo>)</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>λ</mi> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>ε</mi> <mo>)</mo> </mrow> <mo>−</mo> <mi>tan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>β</mi> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>ε</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>λ</mi> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <mo>;</mo> <mspace width="2em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>δ</mi> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>α</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>β</mi> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>λ</mi> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>ε</mi> <mo>)</mo> </mrow> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>β</mi> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>ε</mi> <mo>)</mo> </mrow> <mo>;</mo> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>δ</mi> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>α</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>β</mi> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>λ</mi> <mo>)</mo> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>δ</mi> <mo>)</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>β</mi> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>ε</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>β</mi> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>ε</mi> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>λ</mi> <mo>)</mo> </mrow> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>δ</mi> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>α</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>δ</mi> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>α</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>δ</mi> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mtd> <mtd> <mi></mi> <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> <mrow> <mo>(</mo> <mi>ε</mi> <mo>)</mo> </mrow> </mtd> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>ε</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>ε</mi> <mo>)</mo> </mrow> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>ε</mi> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>β</mi> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>λ</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>β</mi> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>λ</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>β</mi> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\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{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32e64f7f6ef1f8a0eda9cf775fa41f0023c54d4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -21.005ex; width:107.792ex; height:43.176ex;" alt="{\displaystyle {\begin{aligned}\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{aligned}}}" /></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Equatorial_↔_horizontal"><span id="Equatorial_.E2.86.94_horizontal"></span>Equatorial ↔ horizontal</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Astronomical_coordinate_systems&action=edit&section=11" title="Edit section: Equatorial ↔ horizontal"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Azimuth (<span class="texhtml mvar" style="font-style:italic;">A</span>) is measured from the south point, turning positive to the west.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Zenith distance, the angular distance along the <a href="/wiki/Great_circle" title="Great circle">great circle</a> from the <a href="/wiki/Zenith" title="Zenith">zenith</a> to a celestial object, is simply the <a href="/wiki/Complementary_angles" class="mw-redirect" title="Complementary angles">complementary angle</a> of the altitude: <span class="texhtml">90° − <i>a</i></span>.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </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}\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{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="0.6em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>tan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>A</mi> <mo>)</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>h</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>h</mi> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>o</mtext> </mrow> </msub> <mo>)</mo> </mrow> <mo>−</mo> <mi>tan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>δ</mi> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>o</mtext> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <mo>;</mo> <mspace width="2em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>A</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>δ</mi> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>h</mi> <mo>)</mo> </mrow> <mo>;</mo> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>A</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>δ</mi> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>h</mi> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>o</mtext> </mrow> </msub> <mo>)</mo> </mrow> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>δ</mi> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>o</mtext> </mrow> </msub> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>o</mtext> </mrow> </msub> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>δ</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>o</mtext> </mrow> </msub> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>δ</mi> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>h</mi> <mo>)</mo> </mrow> <mo>;</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\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{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24591754e1a4d05bbcb52fd2b752ad2fa281ad42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:103.036ex; height:10.176ex;" alt="{\displaystyle {\begin{aligned}\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{aligned}}}" /></span></dd></dl> <p>In solving the <span class="texhtml">tan(<i>A</i>)</span> equation for <span class="texhtml"><i>A</i></span>, in order to avoid the ambiguity of the <a href="/wiki/Arctangent" class="mw-redirect" title="Arctangent">arctangent</a>, use of the <a href="/wiki/Atan2" title="Atan2">two-argument arctangent</a>, denoted <span class="texhtml">arctan(<i>x</i>,<i>y</i>)</span>, is recommended. The two-argument arctangent computes the arctangent of <span class="texhtml"><style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num"><i>y</i></span><span class="sr-only">/</span><span class="den"><i>x</i></span></span>⁠</span></span>, 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, </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 A=-\arctan(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mo>−</mo> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=-\arctan(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d320d9f2cf97eb81179dceedded00d18b41db8e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.831ex; height:2.843ex;" alt="{\displaystyle A=-\arctan(x,y)}" /></span>,</dd></dl> <p>where </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&=-\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{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> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>o</mtext> </mrow> </msub> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>δ</mi> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>h</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>o</mtext> </mrow> </msub> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>δ</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>δ</mi> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>h</mi> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}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{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5acc5f844c748270660fdb347d9bb1d38fcdcf2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:43.783ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}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{aligned}}}" /></span>.</dd></dl> <p>If the above formula produces a negative value for <span class="texhtml"><i>A</i></span>, it can be rendered positive by simply adding 360°. </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}{\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{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 0.9em 0.6em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>A</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>A</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>o</mtext> </mrow> </msub> <mo>)</mo> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>o</mtext> </mrow> </msub> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>o</mtext> </mrow> </msub> <mo>)</mo> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>o</mtext> </mrow> </msub> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>δ</mi> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>h</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>δ</mi> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>h</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>δ</mi> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>o</mtext> </mrow> </msub> <mo>)</mo> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>o</mtext> </mrow> </msub> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>o</mtext> </mrow> </msub> <mo>)</mo> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>o</mtext> </mrow> </msub> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo>)</mo> </mrow> </mtd> <mtd> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo>)</mo> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo>)</mo> </mrow> </mtd> <mtd> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo>)</mo> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>δ</mi> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>α</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>δ</mi> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>α</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>δ</mi> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>;</mo> </mtd> </mtr> <mtr> <mtd> <mi>tan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>h</mi> <mo>)</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>A</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>A</mi> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>o</mtext> </mrow> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>tan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>o</mtext> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <mo>;</mo> <mspace width="2em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>δ</mi> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>h</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>A</mi> <mo>)</mo> </mrow> <mo>;</mo> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>δ</mi> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>h</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>o</mtext> </mrow> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>A</mi> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>o</mtext> </mrow> </msub> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>δ</mi> <mo>)</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>o</mtext> </mrow> </msub> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>o</mtext> </mrow> </msub> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>A</mi> <mo>)</mo> </mrow> <mo>;</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\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{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0b38b9e388153fa974a286f72be3bf6906a5fe3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -14.864ex; margin-bottom: -0.307ex; width:113.943ex; height:31.509ex;" alt="{\displaystyle {\begin{aligned}{\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{aligned}}}" /></span><sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup></dd></dl> <p>Again, in solving the <span class="texhtml">tan(<i>h</i>)</span> equation for <span class="texhtml"><i>h</i></span>, 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, </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 h=\arctan(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h=\arctan(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c681c53478905b51d2d10ccd5c6b1c267fdcc23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.232ex; height:2.843ex;" alt="{\displaystyle h=\arctan(x,y)}" /></span>,</dd></dl> <p>where </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&=\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{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 0.6em 0.3em 0.3em" 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>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>o</mtext> </mrow> </msub> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>A</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>o</mtext> </mrow> </msub> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>A</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>δ</mi> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>h</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>δ</mi> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>h</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>δ</mi> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>o</mtext> </mrow> </msub> <mo>)</mo> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>o</mtext> </mrow> </msub> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>o</mtext> </mrow> </msub> <mo>)</mo> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>o</mtext> </mrow> </msub> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>A</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>A</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>δ</mi> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>α</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>δ</mi> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>α</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>δ</mi> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo>)</mo> </mrow> </mtd> <mtd> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo>)</mo> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo>)</mo> </mrow> </mtd> <mtd> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo>)</mo> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>o</mtext> </mrow> </msub> <mo>)</mo> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>o</mtext> </mrow> </msub> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>o</mtext> </mrow> </msub> <mo>)</mo> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>o</mtext> </mrow> </msub> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>A</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>A</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}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{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e80cd665d08444dadd652d53b0f1bfd9bb35e9c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.671ex; width:91.321ex; height:26.509ex;" alt="{\displaystyle {\begin{aligned}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{aligned}}}" /></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Equatorial_↔_galactic"><span id="Equatorial_.E2.86.94_galactic"></span>Equatorial ↔ galactic</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Astronomical_coordinate_systems&action=edit&section=12" title="Edit section: Equatorial ↔ galactic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>These equations<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> are for converting equatorial coordinates to Galactic coordinates. </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}\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{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>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>NCP</mtext> </mrow> </msub> <mo>−</mo> <mi>l</mi> </mrow> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>δ</mi> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>G</mtext> </mrow> </msub> <mo>)</mo> </mrow> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>δ</mi> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>G</mtext> </mrow> </msub> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>α</mi> <mo>−</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>G</mtext> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>NCP</mtext> </mrow> </msub> <mo>−</mo> <mi>l</mi> </mrow> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>δ</mi> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>α</mi> <mo>−</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>G</mtext> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>b</mi> <mo>)</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>δ</mi> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>G</mtext> </mrow> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>δ</mi> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>G</mtext> </mrow> </msub> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>α</mi> <mo>−</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>G</mtext> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\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{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78994b1e8705517659dd8b13f262157d178e8bd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:65.71ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}\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{aligned}}}" /></span>run_going</dd></dl> <p><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 \alpha _{\text{G}},\delta _{\text{G}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>G</mtext> </mrow> </msub> <mo>,</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>G</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{\text{G}},\delta _{\text{G}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7737134f568aa3fa4819f07bc987cc7a38d1f0a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.599ex; height:2.676ex;" alt="{\displaystyle \alpha _{\text{G}},\delta _{\text{G}}}" /></span> are the equatorial coordinates of the North Galactic Pole and <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 l_{\text{NCP}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>NCP</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l_{\text{NCP}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/209d2070bf725ee618f562e417c381ed52164b65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.464ex; height:2.509ex;" alt="{\displaystyle l_{\text{NCP}}}" /></span> is the Galactic longitude of the North Celestial Pole. Referred to <a href="/wiki/Epoch_(astronomy)" title="Epoch (astronomy)">J2000.0</a> the values of these quantities are: </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 \alpha _{G}=192.85948^{\circ }\qquad \delta _{G}=27.12825^{\circ }\qquad l_{\text{NCP}}=122.93192^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> </mrow> </msub> <mo>=</mo> <msup> <mn>192.85948</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mspace width="2em"></mspace> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> </mrow> </msub> <mo>=</mo> <msup> <mn>27.12825</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mspace width="2em"></mspace> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>NCP</mtext> </mrow> </msub> <mo>=</mo> <msup> <mn>122.93192</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{G}=192.85948^{\circ }\qquad \delta _{G}=27.12825^{\circ }\qquad l_{\text{NCP}}=122.93192^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22238e19106e46827927d43047ec4a02431da4c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:60.457ex; height:2.676ex;" alt="{\displaystyle \alpha _{G}=192.85948^{\circ }\qquad \delta _{G}=27.12825^{\circ }\qquad l_{\text{NCP}}=122.93192^{\circ }}" /></span></dd></dl> <p>If the equatorial coordinates are referred to another <a href="/wiki/Equinox_(celestial_coordinates)" title="Equinox (celestial coordinates)">equinox</a>, they must be <a href="/wiki/Axial_precession" title="Axial precession">precessed</a> to their place at J2000.0 before applying these formulae. </p><p>These equations convert to equatorial coordinates referred to <a href="/wiki/Epoch_(astronomy)" title="Epoch (astronomy)">B2000.0</a>. </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}\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{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>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>α</mi> <mo>−</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>G</mtext> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>δ</mi> <mo>)</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>b</mi> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>NCP</mtext> </mrow> </msub> <mo>−</mo> <mi>l</mi> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>α</mi> <mo>−</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>G</mtext> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>δ</mi> <mo>)</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>b</mi> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>G</mtext> </mrow> </msub> <mo>)</mo> </mrow> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>b</mi> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>G</mtext> </mrow> </msub> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>NCP</mtext> </mrow> </msub> <mo>−</mo> <mi>l</mi> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>δ</mi> <mo>)</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>b</mi> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>G</mtext> </mrow> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>b</mi> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>G</mtext> </mrow> </msub> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>NCP</mtext> </mrow> </msub> <mo>−</mo> <mi>l</mi> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\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{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fa4c034f04f6b86c445367c1e21650fbd8d33c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:65.659ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}\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{aligned}}}" /></span>>laft_spasse>11.3</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Notes_on_conversion">Notes on conversion</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Astronomical_coordinate_systems&action=edit&section=13" title="Edit section: Notes on conversion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Angles in the degrees ( ° ), minutes ( ′ ), and seconds ( ″ ) of <a href="/wiki/Minute_of_arc" class="mw-redirect" title="Minute of arc">sexagesimal measure</a> must be converted to decimal before calculations are performed. Whether they are converted to decimal <a href="/wiki/Degree_(angle)" title="Degree (angle)">degrees</a> or <a href="/wiki/Radian" title="Radian">radians</a> depends upon the particular calculating machine or program. Negative angles must be carefully handled; <span class="nowrap">–10° 20′ 30″</span> must be converted as <span class="nowrap">−10° −20′ −30″</span>.</li> <li>Angles in the hours ( <sup>h</sup> ), minutes ( <sup>m</sup> ), and seconds ( <sup>s</sup> ) of time measure must be converted to decimal <a href="/wiki/Degree_(angle)" title="Degree (angle)">degrees</a> or <a href="/wiki/Radian" title="Radian">radians</a> before calculations are performed. 1<sup>h</sup> = 15°; 1<sup>m</sup> = 15′; 1<sup>s</sup> = 15″</li> <li>Angles greater than 360° (2<span class="texhtml mvar" style="font-style:italic;">π</span>) or less than 0° may need to be reduced to the range 0°−360° (0–2<span class="texhtml mvar" style="font-style:italic;">π</span>) depending upon the particular calculating machine or program.</li> <li>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°.</li> <li><a href="/wiki/Inverse_trigonometric_functions" title="Inverse trigonometric functions">Inverse trigonometric functions</a> arcsine, arccosine and arctangent are <a href="/wiki/Quadrant_(plane_geometry)" title="Quadrant (plane geometry)">quadrant</a>-ambiguous, and results should be carefully evaluated. Use of the <a href="/wiki/Atan2" title="Atan2">second arctangent function</a> (denoted in computing as <style data-mw-deduplicate="TemplateStyles:r886049734">.mw-parser-output .monospaced{font-family:monospace,monospace}</style><span class="monospaced">atn2(<i>y</i>,<i>x</i>)</span> or <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">atan2(<i>y</i>,<i>x</i>)</span>, which calculates the arctangent of <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" /><span class="sfrac">⁠<span class="tion"><span class="num"><i>y</i></span><span class="sr-only">/</span><span class="den"><i>x</i></span></span>⁠</span></span> using the sign of both arguments to determine the right quadrant) is recommended when calculating longitude/right ascension/azimuth. An equation which finds the <a href="/wiki/Trigonometric_functions" title="Trigonometric functions">sine</a>, followed by the <a href="/wiki/Inverse_trigonometric_functions" title="Inverse trigonometric functions">arcsin function</a>, is recommended when calculating latitude/declination/altitude.</li> <li>Azimuth (<span class="texhtml"><i>A</i></span>) is referred here to the south point of the <a href="/wiki/Horizon" title="Horizon">horizon</a>, the common astronomical reckoning. An object on the <a href="/wiki/Meridian_(astronomy)" title="Meridian (astronomy)">meridian</a> to the south of the observer has <span class="texhtml"><i>A</i></span> = <span class="texhtml"><i>h</i></span> = 0° with this usage. However, n <a href="/wiki/Astropy" title="Astropy">Astropy</a>'s AltAz, in the <a href="/wiki/Large_Binocular_Telescope" title="Large Binocular Telescope">Large Binocular Telescope</a> FITS file convention, in <a href="/wiki/XEphem" title="XEphem">XEphem</a>, in the <a href="/wiki/International_Astronomical_Union" title="International Astronomical Union">IAU</a> library <a href="/wiki/SOFA_(astronomy)" class="mw-redirect" title="SOFA (astronomy)">Standards of Fundamental Astronomy</a> and Section B of the <a href="/wiki/Astronomical_Almanac" title="Astronomical Almanac">Astronomical Almanac</a> for example, the azimuth is East of North. In <a href="/wiki/Navigation" title="Navigation">navigation</a> and some other disciplines, azimuth is figured from the north.</li> <li>The equations for altitude (<span class="texhtml"><i>a</i></span>) do not account for <a href="/wiki/Atmospheric_refraction" title="Atmospheric refraction">atmospheric refraction</a>.</li> <li>The equations for horizontal coordinates do not account for <a href="/wiki/Diurnal_parallax" class="mw-redirect" title="Diurnal parallax">diurnal parallax</a>, that is, the small offset in the position of a celestial object caused by the position of the observer on the <a href="/wiki/Earth" title="Earth">Earth</a>'s surface. This effect is significant for the <a href="/wiki/Moon" title="Moon">Moon</a>, less so for the <a href="/wiki/Planet" title="Planet">planets</a>, minute for <a href="/wiki/Star" title="Star">stars</a> or more distant objects.</li> <li>Observer's longitude (<span class="texhtml"><i>λ</i><sub>o</sub></span>) here is measured positively westward from the <a href="/wiki/Prime_meridian" title="Prime meridian">prime meridian</a>; this is contrary to current <a href="/wiki/International_Astronomical_Union" title="International Astronomical Union">IAU</a> standards.</li></ul> <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=Astronomical_coordinate_systems&action=edit&section=14" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Apparent_longitude" title="Apparent longitude">Apparent longitude</a></li> <li><a href="/wiki/Azimuth" title="Azimuth">Azimuth</a> – Horizontal angle from north or other reference cardinal direction</li> <li><a href="/wiki/Barycentric_and_geocentric_celestial_reference_systems" title="Barycentric and geocentric celestial reference systems">Barycentric and geocentric celestial reference systems</a> – Celestial coordinate system</li> <li><a href="/wiki/Celestial_sphere" title="Celestial sphere">Celestial sphere</a> – Imaginary sphere of arbitrarily large radius, concentric with the observer</li> <li><a href="/wiki/International_Celestial_Reference_System_and_its_realizations" title="International Celestial Reference System and its realizations">International Celestial Reference System and its realizations</a> – Current standard celestial reference system and frame</li> <li><a href="/wiki/Orbital_elements" title="Orbital elements">Orbital elements</a> – Parameters that define a specific orbit</li> <li><a href="/wiki/Planetary_coordinate_system" title="Planetary coordinate system">Planetary coordinate system</a> – Coordinate system for planets</li> <li><a href="/wiki/Terrestrial_reference_frame" class="mw-redirect" title="Terrestrial reference frame">Terrestrial reference frame</a> – Reference frame for measuring location<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Astronomical_coordinate_systems&action=edit&section=15" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <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 reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">Depending on the azimuth convention in use, the signs of <span class="texhtml">cos <i>A</i></span> and <span class="texhtml">sin <i>A</i></span> appear in all four different combinations. Karttunen et al.,<sup id="cite_ref-Karttunen_9-0" class="reference"><a href="#cite_note-Karttunen-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Taff,<sup id="cite_ref-Taff_10-0" class="reference"><a href="#cite_note-Taff-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> and Roth<sup id="cite_ref-Roth_11-0" class="reference"><a href="#cite_note-Roth-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> define <span class="texhtml"><i>A</i></span> clockwise from the south. Lang<sup id="cite_ref-Lang_12-0" class="reference"><a href="#cite_note-Lang-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> defines it north through east, Smart<sup id="cite_ref-Smart_13-0" class="reference"><a href="#cite_note-Smart-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> north through west. Meeus (1991),<sup id="cite_ref-Meeus_4-1" class="reference"><a href="#cite_note-Meeus-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> p. 89: <span class="texhtml">sin <i>δ</i> = sin <i>φ</i> sin <i>a</i> − cos <i>φ</i> cos <i>a</i> cos <i>A</i></span>; <i>Explanatory Supplement</i> (1961),<sup id="cite_ref-ExplSupp_5-1" class="reference"><a href="#cite_note-ExplSupp-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> p. 26: <span class="texhtml">sin <i>δ</i> = sin <i>a</i> sin <i>φ</i> + cos <i>a</i> cos <i>A</i> cos <i>φ</i></span>.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Astronomical_coordinate_systems&action=edit&section=16" title="Edit section: 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-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</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="CITEREFKanas2021" class="citation journal cs1">Kanas, Nick (2021). <a rel="nofollow" class="external text" href="https://doi.org/10.3847%2F2515-5172%2Fabf35c">"Star and Solar System Maps: A History of Celestial Cartography"</a>. <i>Research Notes of the AAS</i>. <b>5</b> (4). <a href="/wiki/American_Astronomical_Society" title="American Astronomical Society">American Astronomical Society</a>: 69. <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/2021RNAAS...5...69K">2021RNAAS...5...69K</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/10.3847%2F2515-5172%2Fabf35c">10.3847/2515-5172/abf35c</a></span>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:233522547">233522547</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Research+Notes+of+the+AAS&rft.atitle=Star+and+Solar+System+Maps%3A+A+History+of+Celestial+Cartography&rft.volume=5&rft.issue=4&rft.pages=69&rft.date=2021&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A233522547%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.3847%2F2515-5172%2Fabf35c&rft_id=info%3Abibcode%2F2021RNAAS...5...69K&rft.aulast=Kanas&rft.aufirst=Nick&rft_id=https%3A%2F%2Fdoi.org%2F10.3847%252F2515-5172%252Fabf35c&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAstronomical+coordinate+systems" 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="CITEREFMajewski" class="citation web cs1">Majewski, Steve. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160312195329/http://faculty.virginia.edu/ASTR5610/lectures/COORDS/coords.html">"Coordinate Systems"</a>. UVa Department of Astronomy. Archived from <a rel="nofollow" class="external text" href="http://www.faculty.virginia.edu/ASTR5610/lectures/COORDS/coords.html">the original</a> on 12 March 2016<span class="reference-accessdate">. Retrieved <span class="nowrap">19 March</span> 2011</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Coordinate+Systems&rft.pub=UVa+Department+of+Astronomy&rft.aulast=Majewski&rft.aufirst=Steve&rft_id=http%3A%2F%2Fwww.faculty.virginia.edu%2FASTR5610%2Flectures%2FCOORDS%2Fcoords.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAstronomical+coordinate+systems" 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"><a href="/wiki/Asger_Aaboe" title="Asger Aaboe">Aaboe, Asger</a>. 2001 <i>Episodes from the Early History of Astronomy.</i> New York: Springer-Verlag., pp. 17–19.</span> </li> <li id="cite_note-Meeus-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-Meeus_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Meeus_4-1"><sup><i><b>b</b></i></sup></a></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. <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.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%3AAstronomical+coordinate+systems" class="Z3988"></span>, chap. 12</span> </li> <li id="cite_note-ExplSupp-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-ExplSupp_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-ExplSupp_5-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFU.S._Naval_ObservatoryH.M._Nautical_Almanac_Office1961" class="citation book cs1">U.S. Naval Observatory, Nautical Almanac Office; H.M. Nautical Almanac Office (1961). <i>Explanatory Supplement to the Astronomical Ephemeris and the American Ephemeris and Nautical Almanac</i>. H.M. Stationery Office, London.</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.pub=H.M.+Stationery+Office%2C+London&rft.date=1961&rft.aulast=U.S.+Naval+Observatory&rft.aufirst=Nautical+Almanac+Office&rft.au=H.M.+Nautical+Almanac+Office&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAstronomical+coordinate+systems" class="Z3988"></span>, sec. 2A</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"> <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.). <i>Explanatory Supplement to the Astronomical Almanac</i>. University Science Books, Mill Valley, CA. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-935702-68-7" title="Special:BookSources/0-935702-68-7"><bdi>0-935702-68-7</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.pub=University+Science+Books%2C+Mill+Valley%2C+CA&rft.date=1992&rft.isbn=0-935702-68-7&rft.aulast=U.S.+Naval+Observatory&rft.aufirst=Nautical+Almanac+Office&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAstronomical+coordinate+systems" class="Z3988"></span>, section 11.43</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMontenbruckPfleger2000" class="citation book cs1">Montenbruck, Oliver; Pfleger, Thomas (2000). <i>Astronomy on the Personal Computer</i>. Springer-Verlag Berlin Heidelberg. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-67221-0" title="Special:BookSources/978-3-540-67221-0"><bdi>978-3-540-67221-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Astronomy+on+the+Personal+Computer&rft.pub=Springer-Verlag+Berlin+Heidelberg&rft.date=2000&rft.isbn=978-3-540-67221-0&rft.aulast=Montenbruck&rft.aufirst=Oliver&rft.au=Pfleger%2C+Thomas&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAstronomical+coordinate+systems" class="Z3988"></span>, pp 35-37</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="CITEREFU.S._Naval_ObservatoryU.K._Hydrographic_Office2008" class="citation book cs1">U.S. Naval Observatory, Nautical Almanac Office; U.K. Hydrographic Office, H.M. Nautical Almanac Office (2008). <i>The Astronomical Almanac for the Year 2010</i>. U.S. Govt. Printing Office. p. M18. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0160820083" title="Special:BookSources/978-0160820083"><bdi>978-0160820083</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Astronomical+Almanac+for+the+Year+2010&rft.pages=M18&rft.pub=U.S.+Govt.+Printing+Office&rft.date=2008&rft.isbn=978-0160820083&rft.aulast=U.S.+Naval+Observatory&rft.aufirst=Nautical+Almanac+Office&rft.au=U.K.+Hydrographic+Office%2C+H.M.+Nautical+Almanac+Office&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAstronomical+coordinate+systems" class="Z3988"></span></span> </li> <li id="cite_note-Karttunen-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-Karttunen_9-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKarttunenKrögerOjaPoutanen2006" class="citation book cs1">Karttunen, H.; Kröger, P.; Oja, H.; Poutanen, M.; Donner, H. J. (2006). <i>Fundamental Astronomy</i> (5 ed.). Springer. <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/2003fuas.book.....K">2003fuas.book.....K</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-34143-7" title="Special:BookSources/978-3-540-34143-7"><bdi>978-3-540-34143-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamental+Astronomy&rft.edition=5&rft.pub=Springer&rft.date=2006&rft_id=info%3Abibcode%2F2003fuas.book.....K&rft.isbn=978-3-540-34143-7&rft.aulast=Karttunen&rft.aufirst=H.&rft.au=Kr%C3%B6ger%2C+P.&rft.au=Oja%2C+H.&rft.au=Poutanen%2C+M.&rft.au=Donner%2C+H.+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAstronomical+coordinate+systems" class="Z3988"></span></span> </li> <li id="cite_note-Taff-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-Taff_10-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFTaff1981" class="citation book cs1">Taff, L. G. (1981). <i>Computational spherical astronomy</i>. Wiley. <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/1981csa..book.....T">1981csa..book.....T</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-06257-X" title="Special:BookSources/0-471-06257-X"><bdi>0-471-06257-X</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computational+spherical+astronomy&rft.pub=Wiley&rft.date=1981&rft_id=info%3Abibcode%2F1981csa..book.....T&rft.isbn=0-471-06257-X&rft.aulast=Taff&rft.aufirst=L.+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAstronomical+coordinate+systems" class="Z3988"></span></span> </li> <li id="cite_note-Roth-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-Roth_11-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRoth1989" class="citation book cs1">Roth, G. D. (23 October 1989). <i>Handbuch für Sternenfreunde</i>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-19436-3" title="Special:BookSources/3-540-19436-3"><bdi>3-540-19436-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbuch+f%C3%BCr+Sternenfreunde&rft.pub=Springer&rft.date=1989-10-23&rft.isbn=3-540-19436-3&rft.aulast=Roth&rft.aufirst=G.+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAstronomical+coordinate+systems" class="Z3988"></span></span> </li> <li id="cite_note-Lang-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-Lang_12-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLang1978" class="citation book cs1">Lang, Kenneth R. (1978). <i>Astrophysical Formulae</i>. Springer. <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/1978afcp.book.....L">1978afcp.book.....L</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-09064-9" title="Special:BookSources/3-540-09064-9"><bdi>3-540-09064-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Astrophysical+Formulae&rft.pub=Springer&rft.date=1978&rft_id=info%3Abibcode%2F1978afcp.book.....L&rft.isbn=3-540-09064-9&rft.aulast=Lang&rft.aufirst=Kenneth+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAstronomical+coordinate+systems" class="Z3988"></span></span> </li> <li id="cite_note-Smart-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-Smart_13-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSmart1949" class="citation book cs1">Smart, William Marshall (1949). <i>Text-book on spherical astronomy</i>. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <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/1965tbsa.book.....S">1965tbsa.book.....S</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Text-book+on+spherical+astronomy&rft.pub=Cambridge+University+Press&rft.date=1949&rft_id=info%3Abibcode%2F1965tbsa.book.....S&rft.aulast=Smart&rft.aufirst=William+Marshall&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAstronomical+coordinate+systems" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPoleski2013" class="citation arxiv cs1">Poleski, Radosław (2013). "Transformation of the equatorial proper motion to the Galactic system". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1306.2945">1306.2945</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/astro-ph.IM">astro-ph.IM</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Transformation+of+the+equatorial+proper+motion+to+the+Galactic+system&rft.date=2013&rft_id=info%3Aarxiv%2F1306.2945&rft.aulast=Poleski&rft.aufirst=Rados%C5%82aw&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAstronomical+coordinate+systems" class="Z3988"></span></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=Astronomical_coordinate_systems&action=edit&section=17" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output 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typeof="mw:File"><a href="/wiki/File:Commons-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Celestial_coordinate_systems" class="extiw" title="commons:Category:Celestial coordinate systems">Astronomical coordinate systems</a></span>.</div></div> </div> <ul><li><a rel="nofollow" class="external text" href="https://aa.usno.navy.mil/software/novas_info">NOVAS</a>, the <a href="/wiki/United_States_Naval_Observatory" title="United States Naval Observatory">United States Naval Observatory</a>'s Vector Astrometry Software, an integrated package of subroutines and functions for computing various commonly needed quantities in positional astronomy.</li> <li><a rel="nofollow" class="external text" href="https://smithsonian.github.io/SuperNOVAS">SuperNOVAS</a> a maintained fork of NOVAS C 3.1 with bug fixes, improvements, new features, and online documentation.</li> <li><a rel="nofollow" class="external text" href="http://www.iausofa.org/">SOFA</a>, the <a href="/wiki/International_Astronomical_Union" title="International Astronomical Union">IAU</a>'s Standards of Fundamental Astronomy, an accessible and authoritative set of algorithms and procedures that implement standard models used in fundamental astronomy.</li> <li>This article was originally based on Jason Harris' <i>Astroinfo</i>, which is accompanied by <a href="/wiki/KStars" title="KStars">KStars</a>, a <a rel="nofollow" class="external text" href="http://edu.kde.org/kstars/">KDE Desktop Planetarium</a> for <a href="/wiki/Linux" title="Linux">Linux</a>/<a href="/wiki/KDE" title="KDE">KDE</a>.</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output 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.navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Gravitational_orbits256" style="padding:3px"><table class="nowraplinks hlist mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Orbits" title="Template:Orbits"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Orbits" title="Template talk:Orbits"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Orbits" title="Special:EditPage/Template:Orbits"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Gravitational_orbits256" style="font-size:114%;margin:0 4em">Gravitational <a href="/wiki/Orbit" title="Orbit">orbits</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/List_of_orbits" title="List of orbits">Types</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:6em">General</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Box_orbit" title="Box orbit">Box</a></li> <li><a href="/wiki/Parabolic_trajectory" title="Parabolic trajectory">Capture</a></li> <li><a href="/wiki/Circular_orbit" title="Circular orbit">Circular</a></li> <li><a href="/wiki/Elliptic_orbit" title="Elliptic orbit">Elliptical</a> / <a href="/wiki/Highly_elliptical_orbit" title="Highly elliptical orbit">Highly elliptical</a></li> <li><a href="/wiki/Parabolic_trajectory" title="Parabolic trajectory">Escape</a></li> <li><a href="/wiki/Horseshoe_orbit" title="Horseshoe orbit">Horseshoe</a></li> <li><a href="/wiki/Hyperbolic_trajectory" title="Hyperbolic trajectory">Hyperbolic trajectory</a></li> <li><a href="/wiki/Inclined_orbit" title="Inclined orbit">Inclined</a> / <a href="/wiki/Non-inclined_orbit" class="mw-redirect" title="Non-inclined orbit">Non-inclined</a></li> <li><a href="/wiki/Kepler_orbit" title="Kepler orbit">Kepler</a></li> <li><a href="/wiki/Lagrange_point" title="Lagrange point">Lagrange point</a></li> <li><a href="/wiki/Osculating_orbit" title="Osculating orbit">Osculating</a></li> <li><a href="/wiki/Parabolic_trajectory" title="Parabolic trajectory">Parabolic trajectory</a></li> <li><a href="/wiki/Parking_orbit" title="Parking orbit">Parking</a></li> <li><a href="/wiki/Retrograde_and_prograde_motion" title="Retrograde and prograde motion">Prograde / Retrograde</a></li> <li><a href="/wiki/Synchronous_orbit" title="Synchronous orbit">Synchronous</a> <ul><li><a href="/wiki/Semi-synchronous_orbit" title="Semi-synchronous orbit">semi</a></li> <li><a href="/wiki/Subsynchronous_orbit" title="Subsynchronous orbit">sub</a></li></ul></li> <li><a href="/wiki/Hohmann_transfer_orbit" title="Hohmann transfer orbit">Transfer orbit</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em"><a href="/wiki/Geocentric_orbit" title="Geocentric orbit">Geocentric</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Geosynchronous_orbit" title="Geosynchronous orbit">Geosynchronous</a> <ul><li><a href="/wiki/Geostationary_orbit" title="Geostationary orbit">Geostationary</a></li> <li><a href="/wiki/Geostationary_transfer_orbit" title="Geostationary transfer orbit">Geostationary transfer</a></li></ul></li> <li><a href="/wiki/Graveyard_orbit" title="Graveyard orbit">Graveyard</a></li> <li><a href="/wiki/High_Earth_orbit" title="High Earth orbit">High Earth</a></li> <li><a href="/wiki/Low_Earth_orbit" title="Low Earth orbit">Low Earth</a></li> <li><a href="/wiki/Medium_Earth_orbit" title="Medium Earth orbit">Medium Earth</a></li> <li><a href="/wiki/Molniya_orbit" title="Molniya orbit">Molniya</a></li> <li><a href="/wiki/Near-equatorial_orbit" title="Near-equatorial orbit">Near-equatorial</a></li> <li><a href="/wiki/Orbit_of_the_Moon" title="Orbit of the Moon">Orbit of the Moon</a></li> <li><a href="/wiki/Polar_orbit" title="Polar orbit">Polar</a></li> <li><a href="/wiki/Sun-synchronous_orbit" title="Sun-synchronous orbit">Sun-synchronous</a></li> <li><a href="/wiki/Transatmospheric_orbit" title="Transatmospheric orbit">Transatmospheric</a></li> <li><a href="/wiki/Tundra_orbit" title="Tundra orbit">Tundra</a></li> <li><a href="/wiki/Very_low_Earth_orbit" title="Very low Earth orbit">Very low Earth</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em">About<br />other points</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li>Mars <ul><li><a href="/wiki/Areocentric_orbit" title="Areocentric orbit">Areocentric</a></li> <li><a href="/wiki/Areosynchronous_orbit" title="Areosynchronous orbit">Areosynchronous</a></li> <li><a href="/wiki/Areostationary_orbit" title="Areostationary orbit">Areostationary</a></li></ul></li> <li>Lagrange points <ul><li><a href="/wiki/Distant_retrograde_orbit" title="Distant retrograde orbit">Distant retrograde</a></li> <li><a href="/wiki/Halo_orbit" title="Halo orbit">Halo</a></li> <li><a href="/wiki/Lissajous_orbit" title="Lissajous orbit">Lissajous</a></li> <li><a href="/wiki/Libration_point_orbit" title="Libration point orbit">Libration</a></li></ul></li> <li><a href="/wiki/Lunar_orbit" title="Lunar orbit">Lunar</a></li> <li>Sun <ul><li><a href="/wiki/Heliocentric_orbit" title="Heliocentric orbit">Heliocentric</a> <ul><li><a href="/wiki/Earth%27s_orbit" title="Earth's orbit">Earth's orbit</a></li></ul></li> <li><a href="/wiki/Mars_cycler" title="Mars cycler">Mars cycler</a></li> <li><a href="/wiki/Sun-synchronous_orbit" title="Sun-synchronous orbit">Heliosynchronous</a></li></ul></li> <li>Other <ul><li><a href="/wiki/Lunar_cycler" title="Lunar cycler">Lunar cycler</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Orbital_elements" title="Orbital elements">Parameters</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:6em"><div class="hlist"><ul><li>Shape</li><li>Size</li></ul></div></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><span class="texhtml mvar" style="font-style:italic;">e</span>  <a href="/wiki/Orbital_eccentricity" title="Orbital eccentricity">Eccentricity</a></li> <li><span class="texhtml mvar" style="font-style:italic;">a</span>  <a href="/wiki/Semi-major_and_semi-minor_axes" title="Semi-major and semi-minor axes">Semi-major axis</a></li> <li><span class="texhtml mvar" style="font-style:italic;">b</span>  <a href="/wiki/Semi-major_and_semi-minor_axes" title="Semi-major and semi-minor axes">Semi-minor axis</a></li> <li><span class="texhtml mvar" style="font-style:italic;">Q</span>, <span class="texhtml mvar" style="font-style:italic;">q</span>  <a href="/wiki/Apsis" title="Apsis">Apsides</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em">Orientation</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><span class="texhtml mvar" style="font-style:italic;">i</span>  <a href="/wiki/Orbital_inclination" title="Orbital inclination">Inclination</a></li> <li><span class="texhtml mvar" style="font-style:italic;">Ω</span>  <a href="/wiki/Longitude_of_the_ascending_node" title="Longitude of the ascending node">Longitude of the ascending node</a></li> <li><span class="texhtml mvar" style="font-style:italic;">ω</span>  <a href="/wiki/Argument_of_periapsis" title="Argument of periapsis">Argument of periapsis</a></li> <li><span class="texhtml mvar" style="font-style:italic;">ϖ</span>  <a href="/wiki/Longitude_of_the_periapsis" class="mw-redirect" title="Longitude of the periapsis">Longitude of the periapsis</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em">Position</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><span class="texhtml mvar" style="font-style:italic;">M</span>  <a href="/wiki/Mean_anomaly" title="Mean anomaly">Mean anomaly</a></li> <li><span class="texhtml mvar" style="font-style:italic;">ν</span>, <span class="texhtml mvar" style="font-style:italic;">θ</span>, <span class="texhtml mvar" style="font-style:italic;">f</span>  <a href="/wiki/True_anomaly" title="True anomaly">True anomaly</a></li> <li><span class="texhtml mvar" style="font-style:italic;">E</span>  <a href="/wiki/Eccentric_anomaly" title="Eccentric anomaly">Eccentric anomaly</a></li> <li><span class="texhtml mvar" style="font-style:italic;">L</span>  <a href="/wiki/Mean_longitude" title="Mean longitude">Mean longitude</a></li> <li><span class="texhtml mvar" style="font-style:italic;">l</span>  <a href="/wiki/True_longitude" title="True longitude">True longitude</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em">Variation</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><span class="texhtml mvar" style="font-style:italic;">T</span>  <a href="/wiki/Orbital_period" title="Orbital period">Orbital period</a></li> <li><span class="texhtml mvar" style="font-style:italic;">n</span>  <a href="/wiki/Mean_motion" title="Mean motion">Mean motion</a></li> <li><span class="texhtml mvar" style="font-style:italic;">v</span>  <a href="/wiki/Orbital_speed" title="Orbital speed">Orbital speed</a></li> <li><span class="texhtml"><i>t</i><sub>0</sub></span>  <a href="/wiki/Epoch_(astronomy)" title="Epoch (astronomy)">Epoch</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Orbital_maneuver" title="Orbital maneuver">Maneuvers</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/Bi-elliptic_transfer" title="Bi-elliptic transfer">Bi-elliptic transfer</a></li> <li><a href="/wiki/Collision_avoidance_(spacecraft)" title="Collision avoidance (spacecraft)">Collision avoidance (spacecraft)</a></li> <li><a href="/wiki/Delta-v" title="Delta-v">Delta-v</a></li> <li><a href="/wiki/Delta-v_budget" title="Delta-v budget">Delta-v budget</a></li> <li><a href="/wiki/Gravity_assist" title="Gravity assist">Gravity assist</a></li> <li><a href="/wiki/Gravity_turn" title="Gravity turn">Gravity turn</a></li> <li><a href="/wiki/Hohmann_transfer_orbit" title="Hohmann transfer orbit">Hohmann transfer</a></li> <li><a href="/wiki/Orbital_inclination_change" title="Orbital inclination change">Inclination change</a></li> <li><a href="/wiki/Low-energy_transfer" title="Low-energy transfer">Low-energy transfer</a></li> <li><a href="/wiki/Oberth_effect" title="Oberth effect">Oberth effect</a></li> <li><a href="/wiki/Orbit_phasing" title="Orbit phasing">Phasing</a></li> <li><a href="/wiki/Tsiolkovsky_rocket_equation" title="Tsiolkovsky rocket equation">Rocket equation</a></li> <li><a href="/wiki/Space_rendezvous" title="Space rendezvous">Rendezvous</a></li> <li><a href="/wiki/Trans-lunar_injection" title="Trans-lunar injection">Trans-lunar injection</a></li> <li><a href="/wiki/Transposition,_docking,_and_extraction" title="Transposition, docking, and extraction">Transposition, docking, and extraction</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Orbital_mechanics" title="Orbital mechanics">Orbital<br />mechanics</a></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 class="mw-selflink selflink">Astronomical coordinate systems</a></li> <li><a href="/wiki/Characteristic_energy" title="Characteristic energy">Characteristic energy</a></li> <li><a href="/wiki/Escape_velocity" title="Escape velocity">Escape velocity</a></li> <li><a href="/wiki/Ephemeris" title="Ephemeris">Ephemeris</a></li> <li><a href="/wiki/Equatorial_coordinate_system" title="Equatorial coordinate system">Equatorial coordinate system</a></li> <li><a href="/wiki/Ground_track" class="mw-redirect" title="Ground track">Ground track</a></li> <li><a href="/wiki/Hill_sphere" title="Hill sphere">Hill sphere</a></li> <li><a href="/wiki/Interplanetary_Transport_Network" title="Interplanetary Transport Network">Interplanetary Transport Network</a></li> <li><a href="/wiki/Kepler%27s_laws_of_planetary_motion" title="Kepler's laws of planetary motion">Kepler's laws of planetary motion</a></li> <li><a href="/wiki/Kozai_mechanism" title="Kozai mechanism">Kozai mechanism</a></li> <li><a href="/wiki/Lagrange_point" title="Lagrange point">Lagrangian point</a></li> <li><a href="/wiki/N-body_problem" title="N-body problem"><i>n</i>-body problem</a></li> <li><a href="/wiki/Orbit_equation" title="Orbit equation">Orbit equation</a></li> <li><a href="/wiki/Orbital_state_vectors" title="Orbital state vectors">Orbital state vectors</a></li> <li><a href="/wiki/Perturbation_(astronomy)" title="Perturbation (astronomy)">Perturbation</a></li> <li><a href="/wiki/Retrograde_and_prograde_motion" title="Retrograde and prograde motion">Retrograde and prograde motion</a></li> <li><a href="/wiki/Specific_orbital_energy" title="Specific orbital energy">Specific orbital energy</a></li> <li><a href="/wiki/Specific_angular_momentum" title="Specific angular momentum">Specific angular momentum</a></li> <li><a href="/wiki/Two-line_element_set" title="Two-line element set">Two-line elements</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="List-Class article"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/16px-Symbol_list_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/23px-Symbol_list_class.svg.png 1.5x, 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href="/wiki/Equatorial_coordinate_system" title="Equatorial coordinate system">Equatorial</a></li> <li><a href="/wiki/Ecliptic_coordinate_system" title="Ecliptic coordinate system">Ecliptic</a></li> <li><a href="/wiki/Invariable_plane" title="Invariable plane">Invariable</a></li> <li><a href="/wiki/Galactic_coordinate_system" title="Galactic coordinate system">Galactic</a></li> <li><a href="/wiki/Supergalactic_coordinate_system" title="Supergalactic coordinate system">Supergalactic</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a class="mw-selflink-fragment" href="#Converting_coordinates">Converting coordinates</a></li> <li><a href="/wiki/Coordinate_system" title="Coordinate system">Coordinate system</a></li></ul> </div></td></tr></tbody></table></div> <style data-mw-deduplicate="TemplateStyles:r1130092004">.mw-parser-output .portal-bar{font-size:88%;font-weight:bold;display:flex;justify-content:center;align-items:baseline}.mw-parser-output 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Astronomical coordinate systems - Wikipedia