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href="#Torque-induced"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Torque-induced</span> </div> </a> <button aria-controls="toc-Torque-induced-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 Torque-induced subsection</span> </button> <ul id="toc-Torque-induced-sublist" class="vector-toc-list"> <li id="toc-Classical_(Newtonian)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Classical_(Newtonian)"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Classical (Newtonian)</span> </div> </a> <ul id="toc-Classical_(Newtonian)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relativistic_(Einsteinian)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relativistic_(Einsteinian)"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Relativistic (Einsteinian)</span> </div> </a> <ul id="toc-Relativistic_(Einsteinian)-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Astronomy" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Astronomy"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Astronomy</span> </div> </a> <button aria-controls="toc-Astronomy-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 Astronomy subsection</span> </button> <ul id="toc-Astronomy-sublist" class="vector-toc-list"> <li id="toc-Axial_precession_(precession_of_the_equinoxes)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Axial_precession_(precession_of_the_equinoxes)"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Axial precession (precession of the equinoxes)</span> </div> </a> <ul id="toc-Axial_precession_(precession_of_the_equinoxes)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Apsidal_precession" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Apsidal_precession"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Apsidal precession</span> </div> </a> <ul id="toc-Apsidal_precession-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Nodal_precession" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Nodal_precession"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Nodal precession</span> </div> </a> <ul id="toc-Nodal_precession-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">4</span> <span>See also</span> </div> </a> <ul id="toc-See_also-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" > <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">Precession</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 57 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-57" 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">57 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%A8%D8%A7%D8%AF%D8%B1%D8%A9_(%D8%AD%D8%B1%D9%83%D8%A9)" title="مبادرة (حركة) – Arabic" lang="ar" hreflang="ar" data-title="مبادرة (حركة)" 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/Precesi%C3%B3n" title="Precesión – Asturian" lang="ast" hreflang="ast" data-title="Precesión" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9F%D1%80%D1%8D%D1%86%D1%8D%D1%81%D1%96%D1%8F" title="Прэцэсія – Belarusian" lang="be" hreflang="be" data-title="Прэцэсія" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9F%D1%80%D0%B5%D1%86%D0%B5%D1%81%D0%B8%D1%8F" title="Прецесия – Bulgarian" lang="bg" hreflang="bg" data-title="Прецесия" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Precessi%C3%B3" title="Precessió – Catalan" lang="ca" hreflang="ca" data-title="Precessió" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Precese" title="Precese – Czech" lang="cs" hreflang="cs" data-title="Precese" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Pr%C3%A6cession" title="Præcession – Danish" lang="da" hreflang="da" data-title="Præcession" 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/Pr%C3%A4zession" title="Präzession – German" lang="de" hreflang="de" data-title="Präzession" 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/Pretsessioon" title="Pretsessioon – Estonian" lang="et" hreflang="et" data-title="Pretsessioon" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Precesi%C3%B3n" title="Precesión – Spanish" lang="es" hreflang="es" data-title="Precesión" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Precesio" title="Precesio – Esperanto" lang="eo" hreflang="eo" data-title="Precesio" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Prezesio" title="Prezesio – Basque" lang="eu" hreflang="eu" data-title="Prezesio" 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%AD%D8%B1%DA%A9%D8%AA_%D8%AA%D9%82%D8%AF%DB%8C%D9%85%DB%8C" title="حرکت تقدیمی – Persian" lang="fa" hreflang="fa" data-title="حرکت تقدیمی" 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/Pr%C3%A9cession" title="Précession – French" lang="fr" hreflang="fr" data-title="Précession" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Luain%C3%ADocht" title="Luainíocht – Irish" lang="ga" hreflang="ga" data-title="Luainíocht" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Precesi%C3%B3n" title="Precesión – Galician" lang="gl" hreflang="gl" data-title="Precesión" 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%84%B8%EC%B0%A8_%EC%9A%B4%EB%8F%99" title="세차 운동 – Korean" lang="ko" hreflang="ko" data-title="세차 운동" 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/%D5%8A%D6%80%D5%A5%D6%81%D5%A5%D5%BD%D5%AB%D5%A1" title="Պրեցեսիա – Armenian" lang="hy" hreflang="hy" data-title="Պրեցեսիա" 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%85%E0%A4%AF%E0%A4%A8_(%E0%A4%96%E0%A4%97%E0%A5%8B%E0%A4%B2%E0%A4%B5%E0%A4%BF%E0%A4%9C%E0%A5%8D%E0%A4%9E%E0%A4%BE%E0%A4%A8)" title="अयन (खगोलविज्ञान) – Hindi" lang="hi" hreflang="hi" data-title="अयन (खगोलविज्ञान)" 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/Precesija" title="Precesija – Croatian" lang="hr" hreflang="hr" data-title="Precesija" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Precesiono" title="Precesiono – Ido" lang="io" hreflang="io" data-title="Precesiono" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Presesi" title="Presesi – Indonesian" lang="id" hreflang="id" data-title="Presesi" 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-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Precessione" title="Precessione – Italian" lang="it" hreflang="it" data-title="Precessione" 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%A0%D7%A7%D7%99%D7%A4%D7%94" title="נקיפה – Hebrew" lang="he" hreflang="he" data-title="נקיפה" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9E%E1%83%A0%E1%83%94%E1%83%AA%E1%83%94%E1%83%A1%E1%83%98%E1%83%90" title="პრეცესია – Georgian" lang="ka" hreflang="ka" data-title="პრეცესია" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9F%D1%80%D0%B5%D1%86%D0%B5%D1%81%D1%81%D0%B8%D1%8F_%D0%B6%D3%99%D0%BD%D0%B5_%D0%BD%D1%83%D1%82%D0%B0%D1%86%D0%B8%D1%8F" title="Прецессия және нутация – Kazakh" lang="kk" hreflang="kk" data-title="Прецессия және нутация" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Presesyon" title="Presesyon – Haitian Creole" lang="ht" hreflang="ht" data-title="Presesyon" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%9F%D1%80%D0%B5%D1%86%D0%B5%D1%81%D1%81%D0%B8%D1%8F" title="Прецессия – Kyrgyz" lang="ky" hreflang="ky" data-title="Прецессия" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Praecessio" title="Praecessio – Latin" lang="la" hreflang="la" data-title="Praecessio" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Precesszi%C3%B3" title="Precesszió – Hungarian" lang="hu" hreflang="hu" data-title="Precesszió" 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%9F%D1%80%D0%B5%D1%86%D0%B5%D1%81%D0%B8%D1%98%D0%B0" title="Прецесија – Macedonian" lang="mk" hreflang="mk" data-title="Прецесија" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%AA%E0%B5%81%E0%B4%B0%E0%B4%B8%E0%B5%8D%E0%B4%B8%E0%B4%B0%E0%B4%A3%E0%B4%82" title="പുരസ്സരണം – Malayalam" lang="ml" hreflang="ml" data-title="പുരസ്സരണം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-xmf mw-list-item"><a href="https://xmf.wikipedia.org/wiki/%E1%83%9E%E1%83%A0%E1%83%94%E1%83%AA%E1%83%94%E1%83%A1%E1%83%98%E1%83%90" title="პრეცესია – Mingrelian" lang="xmf" hreflang="xmf" data-title="პრეცესია" data-language-autonym="მარგალური" data-language-local-name="Mingrelian" class="interlanguage-link-target"><span>მარგალური</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Liukan" title="Liukan – Malay" lang="ms" hreflang="ms" data-title="Liukan" 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-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Precessie" title="Precessie – Dutch" lang="nl" hreflang="nl" data-title="Precessie" 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/%E6%AD%B3%E5%B7%AE" title="歳差 – Japanese" lang="ja" hreflang="ja" data-title="歳差" 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/Presesjon" title="Presesjon – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Presesjon" 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/Presesjon" title="Presesjon – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Presesjon" 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/Precession" title="Precession – Occitan" lang="oc" hreflang="oc" data-title="Precession" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Pr%C3%A4zession" title="Präzession – Low German" lang="nds" hreflang="nds" data-title="Präzession" data-language-autonym="Plattdüütsch" data-language-local-name="Low German" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Precesja" title="Precesja – Polish" lang="pl" hreflang="pl" data-title="Precesja" 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/Precess%C3%A3o" title="Precessão – Portuguese" lang="pt" hreflang="pt" data-title="Precessão" 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/Precesie" title="Precesie – Romanian" lang="ro" hreflang="ro" data-title="Precesie" 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%9F%D1%80%D0%B5%D1%86%D0%B5%D1%81%D1%81%D0%B8%D1%8F" title="Прецессия – Russian" lang="ru" hreflang="ru" data-title="Прецессия" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Precesija" title="Precesija – Slovenian" lang="sl" hreflang="sl" data-title="Precesija" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Precesija" title="Precesija – Serbian" lang="sr" hreflang="sr" data-title="Precesija" 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/Precesija" title="Precesija – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Precesija" 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/Prekessio" title="Prekessio – Finnish" lang="fi" hreflang="fi" data-title="Prekessio" 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/Precession" title="Precession – Swedish" lang="sv" hreflang="sv" data-title="Precession" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%9F%D1%80%D0%B5%D1%86%D0%B5%D1%81%D1%81%D0%B8%D1%8F" title="Прецессия – Tatar" lang="tt" hreflang="tt" data-title="Прецессия" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B8%AB%E0%B8%A1%E0%B8%B8%E0%B8%99%E0%B8%84%E0%B8%A7%E0%B8%87" title="การหมุนควง – Thai" 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class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Periodic change in the direction of a rotation axis</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">For other uses, see <a href="/wiki/Precession_(disambiguation)" class="mw-disambig" title="Precession (disambiguation)">Precession (disambiguation)</a>.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Gyroscope_precession.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/Gyroscope_precession.gif/220px-Gyroscope_precession.gif" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/8/82/Gyroscope_precession.gif 1.5x" data-file-width="300" data-file-height="300" /></a><figcaption>Precession of a <a href="/wiki/Gyroscope" title="Gyroscope">gyroscope</a><sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="This image, while it moves, does not clarify which aspect of the gyroscope is demonstrating precession! This could be improved with a caption but would be better improved by a label in the image. (November 2022)">clarification needed</span></a></i>]</sup></figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Praezession.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bb/Praezession.svg/220px-Praezession.svg.png" decoding="async" width="220" height="264" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bb/Praezession.svg/330px-Praezession.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bb/Praezession.svg/440px-Praezession.svg.png 2x" data-file-width="500" data-file-height="600" /></a><figcaption><style data-mw-deduplicate="TemplateStyles:r981673959">.mw-parser-output .legend{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .legend-color{display:inline-block;min-width:1.25em;height:1.25em;line-height:1.25;margin:1px 0;text-align:center;border:1px solid black;background-color:transparent;color:black}.mw-parser-output .legend-text{}</style><div class="legend"><span class="legend-line mw-no-invert" style="display: inline-block; vertical-align: middle; width: 1.67em; height: 0; border-style: none; border-top: 2px dotted black;border-top:green solid 2px;"> </span> <a href="/wiki/Rotation" title="Rotation">Rotation</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r981673959"><div class="legend"><span class="legend-line mw-no-invert" style="display: inline-block; vertical-align: middle; width: 1.67em; height: 0; border-style: none; border-top: 2px dotted black;border-top:blue solid 2px;"> </span> Precession</div><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r981673959"><div class="legend"><span class="legend-line mw-no-invert" style="display: inline-block; vertical-align: middle; width: 1.67em; height: 0; border-style: none; border-top: 2px dotted black;border-top:red solid 2px;"> </span> <a href="/wiki/Nutation" title="Nutation">Nutation</a></div> in <a href="/wiki/Obliquity" class="mw-redirect" title="Obliquity">obliquity</a> of a planet</figcaption></figure> <p><b>Precession</b> is a change in the <a href="/wiki/Orientation_(geometry)" title="Orientation (geometry)">orientation</a> of the rotational axis of a <a href="/wiki/Rotation" title="Rotation">rotating</a> body. In an appropriate <a href="/wiki/Frame_of_reference" title="Frame of reference">reference frame</a> it can be defined as a change in the first <a href="/wiki/Euler_angles" title="Euler angles">Euler angle</a>, whereas the third Euler angle defines the <a href="/wiki/Rotation_around_a_fixed_axis" title="Rotation around a fixed axis">rotation itself</a>. In other words, if the axis of rotation of a body is itself rotating about a second axis, that body is said to be precessing about the second axis. A motion in which the second Euler angle changes is called <i><a href="/wiki/Nutation" title="Nutation">nutation</a></i>. In <a href="/wiki/Physics" title="Physics">physics</a>, there are two types of precession: <a href="/wiki/Torque" title="Torque">torque</a>-free and torque-induced. </p><p>In astronomy, <i>precession</i> refers to any of several slow changes in an astronomical body's rotational or orbital parameters. An important example is the steady change in the orientation of the axis of rotation of the <a href="/wiki/Earth" title="Earth">Earth</a>, known as the <a href="/wiki/Axial_precession" title="Axial precession">precession of the equinoxes</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Torque-free_or_torque_neglected">Torque-free or torque neglected</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Precession&action=edit&section=1" title="Edit section: Torque-free or torque neglected"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Torque-free precession implies that no external moment (torque) is applied to the body. In torque-free precession, the <a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a> is a constant, but the <a href="/wiki/Angular_velocity" title="Angular velocity">angular velocity</a> vector changes orientation with time. What makes this possible is a time-varying <a href="/wiki/Moment_of_inertia" title="Moment of inertia">moment of inertia</a>, or more precisely, a time-varying <a href="/wiki/Moment_of_inertia#The_inertia_tensor" title="Moment of inertia">inertia matrix</a>. The inertia matrix is composed of the moments of inertia of a body calculated with respect to separate <a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">coordinate axes</a> (e.g. <span class="texhtml"><i>x</i></span>, <span class="texhtml"><i>y</i></span>, <span class="texhtml"><i>z</i></span>). If an object is asymmetric about its principal axis of rotation, the moment of inertia with respect to each coordinate direction will change with time, while preserving angular momentum. The result is that the <a href="/wiki/Vector_component#Decomposition" class="mw-redirect" title="Vector component">component</a> of the angular velocities of the body about each axis will vary inversely with each axis' moment of inertia. </p><p>The torque-free precession rate of an object with an axis of symmetry, such as a disk, spinning about an axis not aligned with that axis of symmetry can be calculated as follows:<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> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\omega }}_{\mathrm {p} }={\frac {{\boldsymbol {I}}_{\mathrm {s} }{\boldsymbol {\omega }}_{\mathrm {s} }}{{\boldsymbol {I}}_{\mathrm {p} }\cos({\boldsymbol {\alpha }})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">I</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> </mrow> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">I</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">α</mi> </mrow> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\omega }}_{\mathrm {p} }={\frac {{\boldsymbol {I}}_{\mathrm {s} }{\boldsymbol {\omega }}_{\mathrm {s} }}{{\boldsymbol {I}}_{\mathrm {p} }\cos({\boldsymbol {\alpha }})}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23d3bebebf8a38234f9bc019a1a594f9997ca158" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:16.16ex; height:6.176ex;" alt="{\displaystyle {\boldsymbol {\omega }}_{\mathrm {p} }={\frac {{\boldsymbol {I}}_{\mathrm {s} }{\boldsymbol {\omega }}_{\mathrm {s} }}{{\boldsymbol {I}}_{\mathrm {p} }\cos({\boldsymbol {\alpha }})}}}"></span> where <span class="texhtml"><i><b>ω</b></i><sub>p</sub></span> is the precession rate, <span class="texhtml"><i><b>ω</b></i><sub>s</sub></span> is the spin rate about the axis of symmetry, <span class="texhtml"><i><b>I</b></i><sub>s</sub></span> is the moment of inertia about the axis of symmetry, <span class="texhtml"><i><b>I</b></i><sub>p</sub></span> is moment of inertia about either of the other two equal perpendicular principal axes, and <span class="texhtml mvar" style="font-style:italic;"><b>α</b></span> is the angle between the moment of inertia direction and the symmetry axis.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>When an object is not perfectly <a href="/wiki/Rigid_body_dynamics" title="Rigid body dynamics">rigid</a>, inelastic dissipation will tend to damp torque-free precession,<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> and the rotation axis will align itself with one of the inertia axes of the body. </p><p>For a generic solid object without any axis of symmetry, the evolution of the object's orientation, represented (for example) by a rotation matrix <span class="texhtml mvar" style="font-style:italic;"><b>R</b></span> that transforms internal to external coordinates, may be numerically simulated. Given the object's fixed internal <a href="/wiki/Moment_of_inertia_tensor" class="mw-redirect" title="Moment of inertia tensor">moment of inertia tensor</a> <span class="texhtml"><i><b>I</b></i><sub>0</sub></span> and fixed external angular momentum <span class="texhtml mvar" style="font-style:italic;"><b>L</b></span>, the instantaneous angular velocity is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\omega }}\left({\boldsymbol {R}}\right)={\boldsymbol {R}}{\boldsymbol {I}}_{0}^{-1}{\boldsymbol {R}}^{T}{\boldsymbol {L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">R</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">R</mi> </mrow> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">I</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">L</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\omega }}\left({\boldsymbol {R}}\right)={\boldsymbol {R}}{\boldsymbol {I}}_{0}^{-1}{\boldsymbol {R}}^{T}{\boldsymbol {L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70e23f214a812564f581144c30eedbf7429889a4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.967ex; height:3.343ex;" alt="{\displaystyle {\boldsymbol {\omega }}\left({\boldsymbol {R}}\right)={\boldsymbol {R}}{\boldsymbol {I}}_{0}^{-1}{\boldsymbol {R}}^{T}{\boldsymbol {L}}}"></span> Precession occurs by repeatedly recalculating <span class="texhtml mvar" style="font-style:italic;"><b>ω</b></span> and applying a small <a href="/wiki/Rotation_representation_(mathematics)#Euler_axis_and_angle_(rotation_vector)" class="mw-redirect" title="Rotation representation (mathematics)">rotation vector</a> <span class="texhtml"><i><b>ω</b> dt</i></span> for the short time <span class="texhtml"><i>dt</i></span>; e.g.: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {R}}_{\text{new}}=\exp \left(\left[{\boldsymbol {\omega }}\left({\boldsymbol {R}}_{\text{old}}\right)\right]_{\times }dt\right){\boldsymbol {R}}_{\text{old}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>new</mtext> </mrow> </msub> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msub> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mrow> <mo>(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>old</mtext> </mrow> </msub> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>×</mo> </mrow> </msub> <mi>d</mi> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>old</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {R}}_{\text{new}}=\exp \left(\left[{\boldsymbol {\omega }}\left({\boldsymbol {R}}_{\text{old}}\right)\right]_{\times }dt\right){\boldsymbol {R}}_{\text{old}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c45443183dd325f055c351ab8a0453e303d1ccca" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.652ex; margin-bottom: -0.186ex; width:31.179ex; height:2.843ex;" alt="{\displaystyle {\boldsymbol {R}}_{\text{new}}=\exp \left(\left[{\boldsymbol {\omega }}\left({\boldsymbol {R}}_{\text{old}}\right)\right]_{\times }dt\right){\boldsymbol {R}}_{\text{old}}}"></span> for the <a href="/wiki/Cross_product#Conversion_to_matrix_multiplication" title="Cross product">skew-symmetric matrix</a> <span class="texhtml">[<i><b>ω</b></i>]<sub>×</sub></span>. The errors induced by finite time steps tend to increase the rotational kinetic energy: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\left({\boldsymbol {R}}\right)={\boldsymbol {\omega }}\left({\boldsymbol {R}}\right)\cdot {\frac {\boldsymbol {L}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">R</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">R</mi> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="bold-italic">L</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\left({\boldsymbol {R}}\right)={\boldsymbol {\omega }}\left({\boldsymbol {R}}\right)\cdot {\frac {\boldsymbol {L}}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9b681d22c66b10e601f7d8272578da562081d9c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.303ex; height:5.343ex;" alt="{\displaystyle E\left({\boldsymbol {R}}\right)={\boldsymbol {\omega }}\left({\boldsymbol {R}}\right)\cdot {\frac {\boldsymbol {L}}{2}}}"></span> this unphysical tendency can be counteracted by repeatedly applying a small rotation vector <span class="texhtml mvar" style="font-style:italic;"><b>v</b></span> perpendicular to both <span class="texhtml mvar" style="font-style:italic;"><b>ω</b></span> and <span class="texhtml mvar" style="font-style:italic;"><b>L</b></span>, noting that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\left(\exp \left(\left[{\boldsymbol {v}}\right]_{\times }\right){\boldsymbol {R}}\right)\approx E\left({\boldsymbol {R}}\right)+\left({\boldsymbol {\omega }}\left({\boldsymbol {R}}\right)\times {\boldsymbol {L}}\right)\cdot {\boldsymbol {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mrow> <mo>(</mo> <mrow> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>×</mo> </mrow> </msub> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">R</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mo>≈</mo> <mi>E</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">R</mi> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">R</mi> </mrow> <mo>)</mo> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">L</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\left(\exp \left(\left[{\boldsymbol {v}}\right]_{\times }\right){\boldsymbol {R}}\right)\approx E\left({\boldsymbol {R}}\right)+\left({\boldsymbol {\omega }}\left({\boldsymbol {R}}\right)\times {\boldsymbol {L}}\right)\cdot {\boldsymbol {v}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/375c3aed2b4653aa66b2e2aae1b06f2f98adeb48" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.652ex; margin-bottom: -0.186ex; width:42.778ex; height:2.843ex;" alt="{\displaystyle E\left(\exp \left(\left[{\boldsymbol {v}}\right]_{\times }\right){\boldsymbol {R}}\right)\approx E\left({\boldsymbol {R}}\right)+\left({\boldsymbol {\omega }}\left({\boldsymbol {R}}\right)\times {\boldsymbol {L}}\right)\cdot {\boldsymbol {v}}}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Torque-induced">Torque-induced</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Precession&action=edit&section=2" title="Edit section: Torque-induced"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Torque-induced precession (<b>gyroscopic precession</b>) is the phenomenon in which the <a href="/wiki/Axis_of_rotation" class="mw-redirect" title="Axis of rotation">axis</a> of a spinning object (e.g., a <a href="/wiki/Gyroscope" title="Gyroscope">gyroscope</a>) describes a <a href="/wiki/Cone_(geometry)" class="mw-redirect" title="Cone (geometry)">cone</a> in space when an external <a href="/wiki/Torque" title="Torque">torque</a> is applied to it. The phenomenon is commonly seen in a <a href="/wiki/Spinning_top" title="Spinning top">spinning toy top</a>, but all rotating objects can undergo precession. If the <a href="/wiki/Speed" title="Speed">speed</a> of the rotation and the <a href="/wiki/Magnitude_(mathematics)" title="Magnitude (mathematics)">magnitude</a> of the external torque are constant, the spin axis will move at <a href="/wiki/Right_angle" title="Right angle">right angles</a> to the <a href="/wiki/Direction_(geometry,_geography)" class="mw-redirect" title="Direction (geometry, geography)">direction</a> that would intuitively result from the external torque. In the case of a toy top, its weight is acting downwards from its <a href="/wiki/Center_of_mass" title="Center of mass">center of mass</a> and the <a href="/wiki/Normal_force" title="Normal force">normal force</a> (reaction) of the ground is pushing up on it at the point of contact with the support. These two opposite forces produce a torque which causes the top to precess. </p> <figure class="mw-halign-right" typeof="mw:File/Frame"><a href="/wiki/File:Gyroscopic_precession_256x256.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/68/Gyroscopic_precession_256x256.png" decoding="async" width="256" height="256" class="mw-file-element" data-file-width="256" data-file-height="256" /></a><figcaption>The response of a rotating system to an applied torque. When the device swivels, and some roll is added, the wheel tends to pitch.</figcaption></figure> <p>The device depicted on the right is <a href="/wiki/Gimbal" title="Gimbal">gimbal</a> mounted. From inside to outside there are three axes of rotation: the hub of the wheel, the gimbal axis, and the vertical pivot. </p><p>To distinguish between the two horizontal axes, rotation around the wheel hub will be called <i>spinning</i>, and rotation around the gimbal axis will be called <i>pitching</i>. Rotation around the vertical pivot axis is called <i>rotation</i>. </p><p>First, imagine that the entire device is rotating around the (vertical) pivot axis. Then, spinning of the wheel (around the wheelhub) is added. Imagine the gimbal axis to be locked, so that the wheel cannot pitch. The gimbal axis has sensors, that measure whether there is a <a href="/wiki/Torque" title="Torque">torque</a> around the gimbal axis. </p><p>In the picture, a section of the wheel has been named <span class="texhtml"><i>dm</i><sub>1</sub></span>. At the depicted moment in time, section <span class="texhtml"><i>dm</i><sub>1</sub></span> is at the <a href="/wiki/Perimeter" title="Perimeter">perimeter</a> of the rotating motion around the (vertical) pivot axis. Section <span class="texhtml"><i>dm</i><sub>1</sub></span>, therefore, has a lot of angular rotating <a href="/wiki/Velocity" title="Velocity">velocity</a> with respect to the rotation around the pivot axis, and as <span class="texhtml"><i>dm</i><sub>1</sub></span> is forced closer to the pivot axis of the rotation (by the wheel spinning further), because of the <a href="/wiki/Coriolis_effect" class="mw-redirect" title="Coriolis effect">Coriolis effect</a>, with respect to the vertical pivot axis, <span class="texhtml"><i>dm</i><sub>1</sub></span> tends to move in the direction of the top-left arrow in the diagram (shown at 45°) in the direction of rotation around the pivot axis.<sup id="cite_ref-Teodorescu_4-0" class="reference"><a href="#cite_note-Teodorescu-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Section <span class="texhtml"><i>dm</i><sub>2</sub></span> of the wheel is moving away from the pivot axis, and so a force (again, a Coriolis force) acts in the same direction as in the case of <span class="texhtml"><i>dm</i><sub>1</sub></span>. Note that both arrows point in the same direction. </p><p>The same reasoning applies for the bottom half of the wheel, but there the arrows point in the opposite direction to that of the top arrows. Combined over the entire wheel, there is a torque around the gimbal axis when some spinning is added to rotation around a vertical axis. </p><p>It is important to note that the torque around the gimbal axis arises without any delay; the response is instantaneous. </p><p>In the discussion above, the setup was kept unchanging by preventing pitching around the gimbal axis. In the case of a spinning toy top, when the spinning top starts tilting, gravity exerts a torque. However, instead of rolling over, the spinning top just pitches a little. This pitching motion reorients the spinning top with respect to the torque that is being exerted. The result is that the torque exerted by gravity – via the pitching motion – elicits gyroscopic precession (which in turn yields a counter torque against the gravity torque) rather than causing the spinning top to fall to its side. </p><p>Precession or gyroscopic considerations have an effect on <a href="/wiki/Bicycle" title="Bicycle">bicycle</a> performance at high speed. Precession is also the mechanism behind <a href="/wiki/Gyrocompass" title="Gyrocompass">gyrocompasses</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Classical_(Newtonian)"><span id="Classical_.28Newtonian.29"></span>Classical (Newtonian)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Precession&action=edit&section=3" title="Edit section: Classical (Newtonian)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:PrecessionOfATop.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2b/PrecessionOfATop.svg/256px-PrecessionOfATop.svg.png" decoding="async" width="256" height="299" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2b/PrecessionOfATop.svg/384px-PrecessionOfATop.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2b/PrecessionOfATop.svg/512px-PrecessionOfATop.svg.png 2x" data-file-width="300" data-file-height="350" /></a><figcaption>The <a href="/wiki/Torque" title="Torque">torque</a> caused by the normal force – <span class="texhtml"><b>F</b><sub>g</sub></span> and the weight of the top causes a change in the <a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a> <span class="texhtml"><b>L</b></span> in the direction of that torque. This causes the top to precess.</figcaption></figure> <p>Precession is the change of <a href="/wiki/Angular_velocity" title="Angular velocity">angular velocity</a> and <a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a> produced by a torque. The general equation that relates the torque to the rate of change of angular momentum is: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\tau }}={\frac {\mathrm {d} \mathbf {L} }{\mathrm {d} t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">τ</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\tau }}={\frac {\mathrm {d} \mathbf {L} }{\mathrm {d} t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f6a00f564197e2555b50691d0a91a79fb2afcc7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.253ex; height:5.509ex;" alt="{\displaystyle {\boldsymbol {\tau }}={\frac {\mathrm {d} \mathbf {L} }{\mathrm {d} t}}}"></span> where <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 {\boldsymbol {\tau }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">τ</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\tau }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/beb1359149db88a28c86f0a3030894b71610a224" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.418ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {\tau }}}"></span> 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 \mathbf {L} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {L} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f5f750865376a1a4ae2b15a00b4ff9c75a66630" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.608ex; height:2.176ex;" alt="{\displaystyle \mathbf {L} }"></span> are the torque and angular momentum vectors respectively. </p><p>Due to the way the torque vectors are defined, it is a vector that is perpendicular to the plane of the forces that create it. Thus it may be seen that the angular momentum vector will change perpendicular to those forces. Depending on how the forces are created, they will often rotate with the angular momentum vector, and then circular precession is created. </p><p>Under these circumstances the angular velocity of precession is given by: <sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <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 {\boldsymbol {\omega }}_{\mathrm {p} }={\frac {\ mgr}{I_{\mathrm {s} }{\boldsymbol {\omega }}_{\mathrm {s} }}}={\frac {\tau }{I_{\mathrm {s} }{\boldsymbol {\omega }}_{\mathrm {s} }\sin(\theta )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mi>m</mi> <mi>g</mi> <mi>r</mi> </mrow> <mrow> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>τ</mi> <mrow> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\omega }}_{\mathrm {p} }={\frac {\ mgr}{I_{\mathrm {s} }{\boldsymbol {\omega }}_{\mathrm {s} }}}={\frac {\tau }{I_{\mathrm {s} }{\boldsymbol {\omega }}_{\mathrm {s} }\sin(\theta )}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a547f68ba4e78bdd18e4b28bf2f9a5545ba7359f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:26.065ex; height:5.676ex;" alt="{\displaystyle {\boldsymbol {\omega }}_{\mathrm {p} }={\frac {\ mgr}{I_{\mathrm {s} }{\boldsymbol {\omega }}_{\mathrm {s} }}}={\frac {\tau }{I_{\mathrm {s} }{\boldsymbol {\omega }}_{\mathrm {s} }\sin(\theta )}}}"></span></dd></dl> <p>where <span class="texhtml"><i>I</i><sub>s</sub></span> is the <a href="/wiki/Moment_of_inertia" title="Moment of inertia">moment of inertia</a>, <span class="texhtml"><i><b>ω</b></i><sub>s</sub></span> is the angular velocity of spin about the spin axis, <span class="texhtml mvar" style="font-style:italic;">m</span> is the mass, <span class="texhtml"><i>g</i></span> is the acceleration due to gravity, <span class="texhtml mvar" style="font-style:italic;">θ</span> is the angle between the spin axis and the axis of precession and <span class="texhtml"><i>r</i></span> is the distance between the center of mass and the pivot. The torque vector originates at the center of mass. Using <span class="texhtml"><i><b>ω</b></i> = <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">2π</span><span class="sr-only">/</span><span class="den"><i>T</i></span></span>⁠</span></span>, we find that the <a href="/wiki/Frequency" title="Frequency">period</a> of precession is given by:<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> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{\mathrm {p} }={\frac {4\pi ^{2}I_{\mathrm {s} }}{\ mgrT_{\mathrm {s} }}}={\frac {4\pi ^{2}I_{\mathrm {s} }\sin(\theta )}{\ \tau T_{\mathrm {s} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> </mrow> <mrow> <mtext> </mtext> <mi>m</mi> <mi>g</mi> <mi>r</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mtext> </mtext> <mi>τ</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\mathrm {p} }={\frac {4\pi ^{2}I_{\mathrm {s} }}{\ mgrT_{\mathrm {s} }}}={\frac {4\pi ^{2}I_{\mathrm {s} }\sin(\theta )}{\ \tau T_{\mathrm {s} }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1e43a94d2c35c0ddec297422b4318564c97ac1c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:28.993ex; height:6.343ex;" alt="{\displaystyle T_{\mathrm {p} }={\frac {4\pi ^{2}I_{\mathrm {s} }}{\ mgrT_{\mathrm {s} }}}={\frac {4\pi ^{2}I_{\mathrm {s} }\sin(\theta )}{\ \tau T_{\mathrm {s} }}}}"></span> </p><p>Where <span class="texhtml"><i>I</i><sub>s</sub></span> is the <a href="/wiki/Moment_of_inertia" title="Moment of inertia">moment of inertia</a>, <span class="texhtml"><i>T</i><sub>s</sub></span> is the period of spin about the spin axis, and <span class="texhtml mvar" style="font-style:italic;"><b>τ</b></span> is the <a href="/wiki/Torque" title="Torque">torque</a>. In general, the problem is more complicated than this, however. </p> <div class="mw-heading mw-heading3"><h3 id="Relativistic_(Einsteinian)"><span id="Relativistic_.28Einsteinian.29"></span>Relativistic (Einsteinian)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Precession&action=edit&section=4" title="Edit section: Relativistic (Einsteinian)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The special and general theories of <a href="/wiki/Theory_of_relativity" title="Theory of relativity">relativity</a> give three types of corrections to the Newtonian precession, of a gyroscope near a large mass such as Earth, described above. They are: </p> <ul><li><a href="/wiki/Thomas_precession" title="Thomas precession">Thomas precession</a>, a special-relativistic correction accounting for an object (such as a gyroscope) being accelerated along a curved path.</li> <li><a href="/wiki/Geodetic_effect" title="Geodetic effect">de Sitter precession</a>, a general-relativistic correction accounting for the Schwarzschild metric of curved space near a large non-rotating mass.</li> <li><a href="/wiki/Lense%E2%80%93Thirring_precession" title="Lense–Thirring precession">Lense–Thirring precession</a>, a general-relativistic correction accounting for the frame dragging by the Kerr metric of curved space near a large rotating mass.</li></ul> <p>The <a href="/wiki/Schwarzschild_geodesics" title="Schwarzschild geodesics">Schwarzschild geodesics</a> (sometimes Schwarzschild precession) is used in the prediction of the <a href="/wiki/Anomalous_perihelion_precession" class="mw-redirect" title="Anomalous perihelion precession">anomalous perihelion precession</a> of the planets, most notably for the accurate prediction of the <a class="mw-selflink-fragment" href="#Apsidal_precession">apsidal precession</a> of Mercury </p> <div class="mw-heading mw-heading2"><h2 id="Astronomy">Astronomy</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Precession&action=edit&section=5" title="Edit section: Astronomy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In astronomy, precession refers to any of several gravity-induced, slow and continuous changes in an astronomical body's rotational axis or orbital path. Precession of the equinoxes, perihelion precession, changes in the <a href="/wiki/Axial_tilt#Earth" title="Axial tilt">tilt of Earth's axis</a> to its orbit, and the <a href="/wiki/Orbital_eccentricity" title="Orbital eccentricity">eccentricity</a> of its orbit over tens of thousands of years are all important parts of the astronomical theory of <a href="/wiki/Ice_age" title="Ice age">ice ages</a>. <i>(See <a href="/wiki/Milankovitch_cycles" title="Milankovitch cycles">Milankovitch cycles</a>.)</i> </p> <div class="mw-heading mw-heading3"><h3 id="Axial_precession_(precession_of_the_equinoxes)"><span id="Axial_precession_.28precession_of_the_equinoxes.29"></span>Axial precession (precession of the equinoxes)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Precession&action=edit&section=6" title="Edit section: Axial precession (precession of the equinoxes)"><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/Axial_precession" title="Axial precession">Axial precession</a></div> <p>Axial precession is the movement of the rotational axis of an astronomical body, whereby the axis slowly traces out a cone. In the case of Earth, this type of precession is also known as the <i>precession of the equinoxes</i>, <i>lunisolar precession</i>, or <i>precession of the equator</i>. Earth goes through one such complete precessional cycle in a period of approximately 26,000 years or 1° every 72 years, during which the positions of stars will slowly change in both <a href="/wiki/Equatorial_coordinates" class="mw-redirect" title="Equatorial coordinates">equatorial coordinates</a> and <a href="/wiki/Ecliptic_longitude" class="mw-redirect" title="Ecliptic longitude">ecliptic longitude</a>. Over this cycle, Earth's north axial pole moves from where it is now, within 1° of <a href="/wiki/Polaris" title="Polaris">Polaris</a>, in a circle around the <a href="/wiki/Ecliptic_pole" class="mw-redirect" title="Ecliptic pole">ecliptic pole</a>, with an angular radius of about 23.5°. </p><p>The <a href="/wiki/Greek_astronomy" class="mw-redirect" title="Greek astronomy">ancient Greek astronomer</a> <a href="/wiki/Hipparchus" title="Hipparchus">Hipparchus</a> (c. 190–120 BC) is generally accepted to be the earliest known astronomer to recognize and assess the precession of the equinoxes at about 1° per century (which is not far from the actual value for antiquity, 1.38°),<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> although there is some minor dispute about whether he was.<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> In <a href="/wiki/Ancient_China" class="mw-redirect" title="Ancient China">ancient China</a>, the <a href="/wiki/Jin_dynasty_(265%E2%80%93420)" class="mw-redirect" title="Jin dynasty (265–420)">Jin-dynasty</a> scholar-official <a href="/wiki/Yu_Xi" title="Yu Xi">Yu Xi</a> (<abbr title="floruit ('flourished' – known to have been active at a particular time or during a particular period)">fl.</abbr> 307–345 AD) made a similar discovery centuries later, noting that the position of the Sun during the <a href="/wiki/Winter_solstice" title="Winter solstice">winter solstice</a> had drifted roughly one degree over the course of fifty years relative to the position of the stars.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> The precession of Earth's axis was later explained by <a href="/wiki/Classical_mechanics" title="Classical mechanics">Newtonian physics</a>. Being an <a href="/wiki/Oblate_spheroid" class="mw-redirect" title="Oblate spheroid">oblate spheroid</a>, Earth has a non-spherical shape, bulging outward at the equator. The gravitational <a href="/wiki/Tidal_force" title="Tidal force">tidal forces</a> of the <a href="/wiki/Moon" title="Moon">Moon</a> and <a href="/wiki/Sun" title="Sun">Sun</a> apply torque to the equator, attempting to pull the <a href="/wiki/Equatorial_bulge" title="Equatorial bulge">equatorial bulge</a> into the plane of the <a href="/wiki/Ecliptic" title="Ecliptic">ecliptic</a>, but instead causing it to precess. The torque exerted by the planets, particularly <a href="/wiki/Jupiter" title="Jupiter">Jupiter</a>, also plays a role.<sup id="cite_ref-Bradt_10-0" class="reference"><a href="#cite_note-Bradt-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <style data-mw-deduplicate="TemplateStyles:r1237032888/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner img{background-color:white}}</style><div class="thumb tmulti tnone center"><div class="thumbinner multiimageinner" style="width:658px;max-width:658px"><div class="trow"><div class="tsingle" style="width:160px;max-width:160px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Earth_precession.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/43/Earth_precession.svg/158px-Earth_precession.svg.png" decoding="async" width="158" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/43/Earth_precession.svg/237px-Earth_precession.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/43/Earth_precession.svg/316px-Earth_precession.svg.png 2x" data-file-width="670" data-file-height="764" /></a></span></div></div><div class="tsingle" style="width:310px;max-width:310px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Equinox_path.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Equinox_path.png/308px-Equinox_path.png" decoding="async" width="308" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Equinox_path.png/462px-Equinox_path.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Equinox_path.png/616px-Equinox_path.png 2x" data-file-width="638" data-file-height="373" /></a></span></div></div><div class="tsingle" style="width:182px;max-width:182px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Precession_N.gif" class="mw-file-description"><img alt="Small white disks representing the northern stars on a black background, overlaid by a circle showing the position of the north pole over time" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/16/Precession_N.gif/180px-Precession_N.gif" decoding="async" width="180" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/16/Precession_N.gif/270px-Precession_N.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/16/Precession_N.gif/360px-Precession_N.gif 2x" data-file-width="1134" data-file-height="1134" /></a></span></div></div></div><div class="trow" style="display:flex"><div class="thumbcaption">Precessional movement of the axis (left), precession of the equinox in relation to the distant stars (middle), and the path of the north celestial pole among the stars due to the precession. Vega is the bright star near the bottom (right).</div></div></div></div> <div class="mw-heading mw-heading3"><h3 id="Apsidal_precession">Apsidal precession</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Precession&action=edit&section=7" title="Edit section: Apsidal precession"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Precessing_Kepler_orbit_280frames_e0.6_smaller.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/89/Precessing_Kepler_orbit_280frames_e0.6_smaller.gif/280px-Precessing_Kepler_orbit_280frames_e0.6_smaller.gif" decoding="async" width="280" height="243" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/8/89/Precessing_Kepler_orbit_280frames_e0.6_smaller.gif 1.5x" data-file-width="345" data-file-height="300" /></a><figcaption><a href="/wiki/Apsidal_precession" title="Apsidal precession">Apsidal precession</a>—the orbit rotates gradually over time.</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Apsidal_precession" title="Apsidal precession">Apsidal precession</a></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/Anomalous_perihelion_precession" class="mw-redirect" title="Anomalous perihelion precession">Anomalous perihelion precession</a></div> <p>The <a href="/wiki/Orbit" title="Orbit">orbits</a> of planets around the <a href="/wiki/Sun" title="Sun">Sun</a> do not really follow an identical ellipse each time, but actually trace out a flower-petal shape because the major axis of each planet's elliptical orbit also precesses within its orbital plane, partly in response to perturbations in the form of the changing gravitational forces exerted by other planets. This is called perihelion precession or <a href="/wiki/Apsidal_precession" title="Apsidal precession">apsidal precession</a>. </p><p>In the adjunct image, Earth's apsidal precession is illustrated. As the Earth travels around the Sun, its elliptical orbit rotates gradually over time. The eccentricity of its ellipse and the precession rate of its orbit are exaggerated for visualization. Most orbits in the Solar System have a much smaller eccentricity and precess at a much slower rate, making them nearly circular and nearly stationary. </p><p>Discrepancies between the observed perihelion precession rate of the planet <a href="/wiki/Mercury_(planet)" title="Mercury (planet)">Mercury</a> and that predicted by <a href="/wiki/Classical_mechanics" title="Classical mechanics">classical mechanics</a> were prominent among the forms of experimental evidence leading to the acceptance of <a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a>'s <a href="/wiki/Theory_of_Relativity" class="mw-redirect" title="Theory of Relativity">Theory of Relativity</a> (in particular, his <a href="/wiki/General_relativity" title="General relativity">General Theory of Relativity</a>), which accurately predicted the anomalies.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> Deviating from Newton's law, Einstein's theory of gravitation predicts an extra term 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>A</i></span><span class="sr-only">/</span><span class="den"><i>r</i><sup>4</sup></span></span>⁠</span></span>, which accurately gives the observed excess turning rate of 43 <a href="/wiki/Arcsecond" class="mw-redirect" title="Arcsecond">arcseconds</a> every 100 years. </p> <div class="mw-heading mw-heading3"><h3 id="Nodal_precession">Nodal precession</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Precession&action=edit&section=8" title="Edit section: Nodal precession"><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/Nodal_precession" title="Nodal precession">Nodal precession</a></div> <p><a href="/wiki/Orbital_node" title="Orbital node">Orbital nodes</a> also <a href="/wiki/Nodal_precession" title="Nodal precession">precess</a> over time. </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">For the precession of the Moon's orbit, see <a href="/wiki/Lunar_precession" title="Lunar precession">lunar precession</a>.</div> <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=Precession&action=edit&section=9" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Larmor_precession" title="Larmor precession">Larmor precession</a></li> <li><a href="/wiki/Nutation" title="Nutation">Nutation</a></li> <li><a href="/wiki/Polar_motion" title="Polar motion">Polar motion</a></li> <li><a href="/wiki/Precession_(mechanical)" title="Precession (mechanical)">Precession (mechanical)</a></li> <li><a href="/wiki/Foucault_pendulum#Precession_as_a_form_of_parallel_transport" title="Foucault pendulum">Precession as a form of parallel transport</a></li></ul> <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=Precession&action=edit&section=10" title="Edit section: References"><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"> <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="CITEREFSchaub2003" class="citation cs2"><a href="/wiki/Hanspeter_Schaub" title="Hanspeter Schaub">Schaub, Hanspeter</a> (2003), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=qXvESNWrfpUC"><i>Analytical Mechanics of Space Systems</i></a>, AIAA, pp. 149–150, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781600860270" title="Special:BookSources/9781600860270"><bdi>9781600860270</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Analytical+Mechanics+of+Space+Systems&rft.pages=149-150&rft.pub=AIAA&rft.date=2003&rft.isbn=9781600860270&rft.aulast=Schaub&rft.aufirst=Hanspeter&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DqXvESNWrfpUC&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrecession" 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="CITEREFBoal2001" class="citation web cs1">Boal, David (2001). <a rel="nofollow" class="external text" href="https://www.sfu.ca/~boal/211lecs/211lec26.pdf">"Lecture 26 – Torque-free rotation – body-fixed axes"</a> <span class="cs1-format">(PDF)</span><span class="reference-accessdate">. Retrieved <span class="nowrap">2008-09-17</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Lecture+26+%E2%80%93+Torque-free+rotation+%E2%80%93+body-fixed+axes&rft.date=2001&rft.aulast=Boal&rft.aufirst=David&rft_id=https%3A%2F%2Fwww.sfu.ca%2F~boal%2F211lecs%2F211lec26.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrecession" 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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSharmaBurnsHui2005" class="citation journal cs1">Sharma, Ishan; Burns, Joseph A.; Hui, C.-H. (2005). <a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1365-2966.2005.08864.x">"Nutational damping times in solids of revolution"</a>. <i>Monthly Notices of the Royal Astronomical Society</i>. <b>359</b> (1): 79. <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/2005MNRAS.359...79S">2005MNRAS.359...79S</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/10.1111%2Fj.1365-2966.2005.08864.x">10.1111/j.1365-2966.2005.08864.x</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Monthly+Notices+of+the+Royal+Astronomical+Society&rft.atitle=Nutational+damping+times+in+solids+of+revolution&rft.volume=359&rft.issue=1&rft.pages=79&rft.date=2005&rft_id=info%3Adoi%2F10.1111%2Fj.1365-2966.2005.08864.x&rft_id=info%3Abibcode%2F2005MNRAS.359...79S&rft.aulast=Sharma&rft.aufirst=Ishan&rft.au=Burns%2C+Joseph+A.&rft.au=Hui%2C+C.-H.&rft_id=https%3A%2F%2Fdoi.org%2F10.1111%252Fj.1365-2966.2005.08864.x&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrecession" class="Z3988"></span></span> </li> <li id="cite_note-Teodorescu-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-Teodorescu_4-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTeodorescu2002" class="citation book cs1">Teodorescu, Petre P (2002). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=aXCBlHOtO3kC&pg=PA396"><i>Mechanical Systems, Classical Models: Volume II: Mechanics of Discrete and Continuous Systems</i></a>. Springer Science & Business Media. p. 420. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4020-8988-6" title="Special:BookSources/978-1-4020-8988-6"><bdi>978-1-4020-8988-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mechanical+Systems%2C+Classical+Models%3A+Volume+II%3A+Mechanics+of+Discrete+and+Continuous+Systems&rft.pages=420&rft.pub=Springer+Science+%26+Business+Media&rft.date=2002&rft.isbn=978-1-4020-8988-6&rft.aulast=Teodorescu&rft.aufirst=Petre+P&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DaXCBlHOtO3kC%26pg%3DPA396&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrecession" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMoebsLingSanny2016" class="citation book cs1">Moebs, William; Ling, Samuel J.; Sanny, Jeff (Sep 19, 2016). <a rel="nofollow" class="external text" href="https://openstax.org/books/university-physics-volume-1/pages/11-4-precession-of-a-gyroscope"><i>11.4 Precession of a Gyroscope - University Physics Volume 1 | OpenStax</i></a>. 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Houston, Texas<span class="reference-accessdate">. Retrieved <span class="nowrap">23 October</span> 2020</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=11.4+Precession+of+a+Gyroscope+-+University+Physics+Volume+1+%7C+OpenStax&rft.place=Houston%2C+Texas&rft.date=2016-09-19&rft.aulast=Moebs&rft.aufirst=William&rft.au=Ling%2C+Samuel+J.&rft.au=Sanny%2C+Jeff&rft_id=https%3A%2F%2Fopenstax.org%2Fbooks%2Funiversity-physics-volume-1%2Fpages%2F11-4-precession-of-a-gyroscope&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrecession" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: location missing publisher (<a href="/wiki/Category:CS1_maint:_location_missing_publisher" title="Category:CS1 maint: location missing publisher">link</a>)</span></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="CITEREFBarbieri2007" class="citation book cs1">Barbieri, Cesare (2007). <i>Fundamentals of Astronomy</i>. New York: Taylor and Francis Group. p. 71. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7503-0886-1" title="Special:BookSources/978-0-7503-0886-1"><bdi>978-0-7503-0886-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+of+Astronomy&rft.place=New+York&rft.pages=71&rft.pub=Taylor+and+Francis+Group&rft.date=2007&rft.isbn=978-0-7503-0886-1&rft.aulast=Barbieri&rft.aufirst=Cesare&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrecession" class="Z3988"></span></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="CITEREFSwerdlow1991" class="citation book cs1">Swerdlow, Noel (1991). <i>On the cosmical mysteries of Mithras</i>. Classical Philology, 86, (1991), 48–63. p. 59.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=On+the+cosmical+mysteries+of+Mithras&rft.pages=59&rft.pub=Classical+Philology%2C+86%2C+%281991%29%2C+48%E2%80%9363&rft.date=1991&rft.aulast=Swerdlow&rft.aufirst=Noel&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrecession" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">Sun, Kwok. (2017). <i>Our Place in the Universe: Understanding Fundamental Astronomy from Ancient Discoveries</i>, second edition. Cham, Switzerland: Springer. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-54171-6" title="Special:BookSources/978-3-319-54171-6">978-3-319-54171-6</a>, p. 120; see also Needham, Joseph; Wang, Ling. (1995) [1959]. <i>Science and Civilization in China: Mathematics and the Sciences of the Heavens and the Earth</i>, vol. 3, reprint edition. Cambridge: Cambridge University Press. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-05801-5" title="Special:BookSources/0-521-05801-5">0-521-05801-5</a>, p. 220.</span> </li> <li id="cite_note-Bradt-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-Bradt_10-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBradt2007" class="citation book cs1">Bradt, Hale (2007). <i>Astronomy Methods</i>. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. p. 66. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-53551-9" title="Special:BookSources/978-0-521-53551-9"><bdi>978-0-521-53551-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Astronomy+Methods&rft.pages=66&rft.pub=Cambridge+University+Press&rft.date=2007&rft.isbn=978-0-521-53551-9&rft.aulast=Bradt&rft.aufirst=Hale&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrecession" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><a href="/wiki/Max_Born" title="Max Born">Max Born</a> (1924), <i>Einstein's Theory of Relativity</i> (The 1962 Dover edition, page 348 lists a table documenting the observed and calculated values for the precession of the perihelion of Mercury, Venus, and Earth.)</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20180807131603/http://www.dailygalaxy.com/my_weblog/2008/03/18-billion-suns.html">"An even larger value for a precession has been found, for a black hole in orbit around a much more massive black hole, amounting to 39 degrees each orbit"</a>. 18 March 2008. Archived from the original on 2018-08-07<span class="reference-accessdate">. Retrieved <span class="nowrap">2023-11-15</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=An+even+larger+value+for+a+precession+has+been+found%2C+for+a+black+hole+in+orbit+around+a+much+more+massive+black+hole%2C+amounting+to+39+degrees+each+orbit.&rft.date=2008-03-18&rft_id=http%3A%2F%2Fwww.dailygalaxy.com%2Fmy_weblog%2F2008%2F03%2F18-billion-suns.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrecession" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_web" title="Template:Cite web">cite web</a>}}</code>: CS1 maint: bot: original URL status unknown (<a href="/wiki/Category:CS1_maint:_bot:_original_URL_status_unknown" title="Category:CS1 maint: bot: original URL status unknown">link</a>)</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=Precession&action=edit&section=11" 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"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/40px-Wikibooks-logo-en-noslogan.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/60px-Wikibooks-logo-en-noslogan.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/80px-Wikibooks-logo-en-noslogan.svg.png 2x" data-file-width="400" data-file-height="400" /></span></span></div> <div class="side-box-text plainlist">Wikibooks has a book on the topic of: <i><b><a href="https://en.wikibooks.org/wiki/Rotational_Motion" class="extiw" title="wikibooks:Rotational Motion">Rotational Motion</a></b></i></div></div> </div> <ul><li><span class="noviewer" 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/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span> Media related to <a href="https://commons.wikimedia.org/wiki/Category:Precession" class="extiw" title="commons:Category:Precession">Precession</a> at Wikimedia Commons</li> <li><a rel="nofollow" class="external text" href="http://hyperphysics.phy-astr.gsu.edu/hbase/top.html">Explanation and derivation of formula for precession of a top</a></li> <li><a rel="nofollow" class="external text" href="http://www.phy6.org/stargaze/Sprecess.htm">Precession and the Milankovich theory</a> <a rel="nofollow" class="external text" href="http://www.phy6.org/stargaze/Sintro.htm">From Stargazers to Starships</a></li></ul> <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 .portal-bar-bordered{padding:0 2em;background-color:#fdfdfd;border:1px solid #a2a9b1;clear:both;margin:1em auto 0}.mw-parser-output .portal-bar-related{font-size:100%;justify-content:flex-start}.mw-parser-output .portal-bar-unbordered{padding:0 1.7em;margin-left:0}.mw-parser-output .portal-bar-header{margin:0 1em 0 0.5em;flex:0 0 auto;min-height:24px}.mw-parser-output .portal-bar-content{display:flex;flex-flow:row wrap;flex:0 1 auto;padding:0.15em 0;column-gap:1em;align-items:baseline;margin:0;list-style:none}.mw-parser-output .portal-bar-content-related{margin:0;list-style:none}.mw-parser-output .portal-bar-item{display:inline-block;margin:0.15em 0.2em;min-height:24px;line-height:24px}@media screen and (max-width:768px){.mw-parser-output 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portal-bar-bordered" role="navigation" aria-label="Portals"><span class="portal-bar-header"><a href="/wiki/Wikipedia:Contents/Portals" title="Wikipedia:Contents/Portals">Portals</a>:</span><ul class="portal-bar-content"><li class="portal-bar-item"><span class="nowrap"><span typeof="mw:File"><a href="/wiki/File:Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/17px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png" decoding="async" width="17" height="19" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/26px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png 1.5x, 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class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/He1523a.jpg/16px-He1523a.jpg" decoding="async" width="16" height="19" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/He1523a.jpg/25px-He1523a.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5f/He1523a.jpg/33px-He1523a.jpg 2x" data-file-width="180" data-file-height="207" /></a></span> </span><a href="/wiki/Portal:Stars" title="Portal:Stars">Stars</a></li><li class="portal-bar-item"><span class="nowrap"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/RocketSunIcon.svg/19px-RocketSunIcon.svg.png" decoding="async" width="19" height="19" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/RocketSunIcon.svg/29px-RocketSunIcon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d6/RocketSunIcon.svg/38px-RocketSunIcon.svg.png 2x" data-file-width="128" data-file-height="128" /></span></span> </span><a href="/wiki/Portal:Spaceflight" title="Portal:Spaceflight">Spaceflight</a></li><li class="portal-bar-item"><span class="nowrap"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Earth-moon.jpg/21px-Earth-moon.jpg" decoding="async" width="21" height="17" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Earth-moon.jpg/32px-Earth-moon.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Earth-moon.jpg/42px-Earth-moon.jpg 2x" data-file-width="3000" data-file-height="2400" /></span></span> </span><a href="/wiki/Portal:Outer_space" title="Portal:Outer space">Outer space</a></li><li class="portal-bar-item"><span class="nowrap"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Solar_system.jpg/15px-Solar_system.jpg" decoding="async" width="15" height="19" class="mw-file-element" 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