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vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Mean-motion_resonances_in_the_Solar_System"> <div class="vector-toc-text"> 3 Mean-motion resonances in the Solar System </div> </a> <button aria-controls="toc-Mean-motion_resonances_in_the_Solar_System-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Mean-motion resonances in the Solar System subsection </button> <ul id="toc-Mean-motion_resonances_in_the_Solar_System-sublist" class="vector-toc-list"> <li id="toc-Laplace_resonance" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Laplace_resonance"> <div class="vector-toc-text"> 3.1 Laplace resonance </div> </a> <ul id="toc-Laplace_resonance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Plutino_resonances" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Plutino_resonances"> <div class="vector-toc-text"> 3.2 Plutino resonances </div> </a> <ul id="toc-Plutino_resonances-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Naiad:Thalassa_73:69_resonance" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Naiad:Thalassa_73:69_resonance"> <div class="vector-toc-text"> 3.3 Naiad:Thalassa 73:69 resonance </div> </a> <ul id="toc-Naiad:Thalassa_73:69_resonance-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Mean-motion_resonances_among_extrasolar_planets" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Mean-motion_resonances_among_extrasolar_planets"> <div class="vector-toc-text"> 4 Mean-motion resonances among extrasolar planets </div> </a> <ul id="toc-Mean-motion_resonances_among_extrasolar_planets-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Coincidental_'near'_ratios_of_mean_motion" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Coincidental_'near'_ratios_of_mean_motion"> <div class="vector-toc-text"> 5 Coincidental 'near' ratios of mean motion </div> </a> <ul id="toc-Coincidental_'near'_ratios_of_mean_motion-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Possible_past_mean-motion_resonances" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Possible_past_mean-motion_resonances"> <div class="vector-toc-text"> 6 Possible past mean-motion resonances </div> </a> <ul id="toc-Possible_past_mean-motion_resonances-sublist" class="vector-toc-list"> </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"> 7 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 8 Notes </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 9 References </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"> 10 External links </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </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">Orbital resonance</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 48 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-48" aria-hidden="true" > 48 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Baanresonansie" title="Baanresonansie – Afrikaans" lang="af" hreflang="af" data-title="Baanresonansie" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target">Afrikaans</a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Bahnresonanz" title="Bahnresonanz – Alemannic" lang="gsw" hreflang="gsw" data-title="Bahnresonanz" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target">Alemannisch</a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B1%D9%86%D9%8A%D9%86_%D9%85%D8%AF%D8%A7%D8%B1%D9%8A" title="رنين مداري – Arabic" lang="ar" hreflang="ar" data-title="رنين مداري" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%90%D1%80%D0%B1%D1%96%D1%82%D0%B0%D0%BB%D1%8C%D0%BD%D1%8B_%D1%80%D1%8D%D0%B7%D0%B0%D0%BD%D0%B0%D0%BD%D1%81" title="Арбітальны рэзананс – Belarusian" lang="be" hreflang="be" data-title="Арбітальны рэзананс" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9E%D1%80%D0%B1%D0%B8%D1%82%D0%B0%D0%BB%D0%B5%D0%BD_%D1%80%D0%B5%D0%B7%D0%BE%D0%BD%D0%B0%D0%BD%D1%81" title="Орбитален резонанс – Bulgarian" lang="bg" hreflang="bg" data-title="Орбитален резонанс" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Resson%C3%A0ncia_orbital" title="Ressonància orbital – Catalan" lang="ca" hreflang="ca" data-title="Ressonància orbital" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Dr%C3%A1hov%C3%A1_rezonance" title="Dráhová rezonance – Czech" lang="cs" hreflang="cs" data-title="Dráhová rezonance" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Baneresonans" title="Baneresonans – Danish" lang="da" hreflang="da" data-title="Baneresonans" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Bahnresonanz" title="Bahnresonanz – German" lang="de" hreflang="de" data-title="Bahnresonanz" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Resonancia_orbital" title="Resonancia orbital – Spanish" lang="es" hreflang="es" data-title="Resonancia orbital" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Orbita_resonanco" title="Orbita resonanco – Esperanto" lang="eo" hreflang="eo" data-title="Orbita resonanco" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B1%D8%B2%D9%86%D8%A7%D9%86%D8%B3_%D9%85%D8%AF%D8%A7%D8%B1%DB%8C" title="رزنانس مداری – Persian" lang="fa" hreflang="fa" data-title="رزنانس مداری" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/R%C3%A9sonance_orbitale" title="Résonance orbitale – French" lang="fr" hreflang="fr" data-title="Résonance orbitale" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Resonancia_orbital" title="Resonancia orbital – Galician" lang="gl" hreflang="gl" data-title="Resonancia orbital" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B6%A4%EB%8F%84_%EA%B3%B5%EB%AA%85" title="궤도 공명 – Korean" lang="ko" hreflang="ko" data-title="궤도 공명" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%88%D6%82%D5%B2%D5%A5%D5%AE%D6%80%D5%A1%D5%B5%D5%AB%D5%B6_%D5%BC%D5%A5%D5%A6%D5%B8%D5%B6%D5%A1%D5%B6%D5%BD" title="Ուղեծրային ռեզոնանս – Armenian" lang="hy" hreflang="hy" data-title="Ուղեծրային ռեզոնանս" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%95%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A5%80%E0%A4%AF_%E0%A4%85%E0%A4%A8%E0%A5%81%E0%A4%A8%E0%A4%BE%E0%A4%A6" title="कक्षीय अनुनाद – Hindi" lang="hi" hreflang="hi" data-title="कक्षीय अनुनाद" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Orbitalna_rezonancija" title="Orbitalna rezonancija – Croatian" lang="hr" hreflang="hr" data-title="Orbitalna rezonancija" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Resonansi_orbit" title="Resonansi orbit – Indonesian" lang="id" hreflang="id" data-title="Resonansi orbit" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Risonanza_orbitale" title="Risonanza orbitale – Italian" lang="it" hreflang="it" data-title="Risonanza orbitale" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%AA%D7%94%D7%95%D7%93%D7%94_%D7%9E%D7%A1%D7%9C%D7%95%D7%9C%D7%99%D7%AA" title="תהודה מסלולית – Hebrew" lang="he" hreflang="he" data-title="תהודה מסלולית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Orbit%C4%81l%C4%81_rezonanse" title="Orbitālā rezonanse – Latvian" lang="lv" hreflang="lv" data-title="Orbitālā rezonanse" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target">Latviešu</a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Bunnresonanz" title="Bunnresonanz – Luxembourgish" lang="lb" hreflang="lb" data-title="Bunnresonanz" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxembourgish" class="interlanguage-link-target">Lëtzebuergesch</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Orbitinis_rezonansas" title="Orbitinis rezonansas – Lithuanian" lang="lt" hreflang="lt" data-title="Orbitinis rezonansas" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9E%D1%80%D0%B1%D0%B8%D1%82%D0%B0%D0%BB%D0%BD%D0%B0_%D1%80%D0%B5%D0%B7%D0%BE%D0%BD%D0%B0%D0%BD%D1%86%D0%B0" title="Орбитална резонанца – Macedonian" lang="mk" hreflang="mk" data-title="Орбитална резонанца" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target">Македонски</a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%A2%D0%BE%D0%B9%D1%80%D0%B3_%D0%B7%D0%B0%D0%BC%D1%8B%D0%BD_%D1%82%D0%B0%D1%82%D0%B0%D1%85_%D1%85%D2%AF%D1%87_%D0%B3%D0%B0%D0%B4%D0%BD%D1%8B_%D0%B1%D0%B8%D0%B5%D1%82%D1%8D%D1%8D%D1%81_%D0%B1%D0%BE%D0%BB%D0%B6_%D3%A9%D3%A9%D1%80%D1%87%D0%BB%D3%A9%D0%B3%D0%B4%D3%A9%D1%85_%D3%A9%D3%A9%D1%80%D1%87%D0%BB%D3%A9%D0%BB%D1%82" title="Тойрг замын татах хүч гадны биетээс болж өөрчлөгдөх өөрчлөлт – Mongolian" lang="mn" hreflang="mn" data-title="Тойрг замын татах хүч гадны биетээс болж өөрчлөгдөх өөрчлөлт" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target">Монгол</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Baanresonantie" title="Baanresonantie – Dutch" lang="nl" hreflang="nl" data-title="Baanresonantie" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E8%BB%8C%E9%81%93%E5%85%B1%E9%B3%B4" title="軌道共鳴 – Japanese" lang="ja" hreflang="ja" data-title="軌道共鳴" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Baneresonans" title="Baneresonans – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Baneresonans" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Resson%C3%A0ncia_orbitala" title="Ressonància orbitala – Occitan" lang="oc" hreflang="oc" data-title="Ressonància orbitala" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target">Occitan</a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Bahnresonanz" title="Bahnresonanz – Low German" lang="nds" hreflang="nds" data-title="Bahnresonanz" data-language-autonym="Plattdüütsch" data-language-local-name="Low German" class="interlanguage-link-target">Plattdüütsch</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Rezonans_orbitalny" title="Rezonans orbitalny – Polish" lang="pl" hreflang="pl" data-title="Rezonans orbitalny" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Resson%C3%A2ncia_orbital" title="Ressonância orbital – Portuguese" lang="pt" hreflang="pt" data-title="Ressonância orbital" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Rezonan%C8%9B%C4%83_orbital%C4%83" title="Rezonanță orbitală – Romanian" lang="ro" hreflang="ro" data-title="Rezonanță orbitală" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9E%D1%80%D0%B1%D0%B8%D1%82%D0%B0%D0%BB%D1%8C%D0%BD%D1%8B%D0%B9_%D1%80%D0%B5%D0%B7%D0%BE%D0%BD%D0%B0%D0%BD%D1%81" title="Орбитальный резонанс – Russian" lang="ru" hreflang="ru" data-title="Орбитальный резонанс" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Orbital_resonance" title="Orbital resonance – Simple English" lang="en-simple" hreflang="en-simple" data-title="Orbital resonance" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Dr%C3%A1hov%C3%A1_rezonancia" title="Dráhová rezonancia – Slovak" lang="sk" hreflang="sk" data-title="Dráhová rezonancia" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Orbitalna_resonanca" title="Orbitalna resonanca – Slovenian" lang="sl" hreflang="sl" data-title="Orbitalna resonanca" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9E%D1%80%D0%B1%D0%B8%D1%82%D0%B0%D0%BB%D0%BD%D0%B5_%D1%80%D0%B5%D0%B7%D0%BE%D0%BD%D0%B0%D0%BD%D1%86%D0%B5" title="Орбиталне резонанце – Serbian" lang="sr" hreflang="sr" data-title="Орбиталне резонанце" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Keskiliikeresonanssi" title="Keskiliikeresonanssi – Finnish" lang="fi" hreflang="fi" data-title="Keskiliikeresonanssi" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Banresonans" title="Banresonans – Swedish" lang="sv" hreflang="sv" data-title="Banresonans" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%9E%D1%80%D0%B1%D0%B8%D1%82%D0%B0%D0%BB%D1%8C_%D1%80%D0%B5%D0%B7%D0%BE%D0%BD%D0%B0%D0%BD%D1%81" title="Орбиталь резонанс – Tatar" lang="tt" hreflang="tt" data-title="Орбиталь резонанс" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target">Татарча / tatarça</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%AA%E0%B8%B1%E0%B9%88%E0%B8%99%E0%B8%9E%E0%B9%89%E0%B8%AD%E0%B8%87%E0%B8%82%E0%B8%AD%E0%B8%87%E0%B8%A7%E0%B8%87%E0%B9%82%E0%B8%84%E0%B8%88%E0%B8%A3" title="การสั่นพ้องของวงโคจร – Thai" lang="th" hreflang="th" data-title="การสั่นพ้องของวงโคจร" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Y%C3%B6r%C3%BCngesel_rezonans" title="Yörüngesel rezonans – Turkish" lang="tr" hreflang="tr" data-title="Yörüngesel rezonans" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9E%D1%80%D0%B1%D1%96%D1%82%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D1%80%D0%B5%D0%B7%D0%BE%D0%BD%D0%B0%D0%BD%D1%81" title="Орбітальний резонанс – 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dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Regular and periodic mutual gravitational influence of orbiting bodies</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 the science fiction novel, see <a href="/wiki/Orbital_Resonance_(novel)" title="Orbital Resonance (novel)">Orbital Resonance (novel)</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Update plainlinks metadata ambox ambox-content ambox-Update" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Ambox_current_red_Americas.svg/42px-Ambox_current_red_Americas.svg.png" decoding="async" width="42" height="34" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Ambox_current_red_Americas.svg/63px-Ambox_current_red_Americas.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/53/Ambox_current_red_Americas.svg/84px-Ambox_current_red_Americas.svg.png 2x" data-file-width="360" data-file-height="290" /></div></td><td class="mbox-text"><div class="mbox-text-span">This article needs to be updated. The reason given is: missing exoplanet resonances, several of which are known. Please help update this article to reflect recent events or newly available information. (January 2021)</div></td></tr></tbody></table> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Galilean_moon_Laplace_resonance_animation_2.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Galilean_moon_Laplace_resonance_animation_2.gif/275px-Galilean_moon_Laplace_resonance_animation_2.gif" decoding="async" width="275" height="185" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/e/e5/Galilean_moon_Laplace_resonance_animation_2.gif 1.5x" data-file-width="365" data-file-height="245" /></a><figcaption>The three-body Laplace resonance exhibited by three of Jupiter's <a href="/wiki/Galilean_moons" title="Galilean moons">Galilean moons</a>. <a href="/wiki/Conjunction_(astronomy)" title="Conjunction (astronomy)">Conjunctions</a> are highlighted by brief color changes. There are two Io-Europa conjunctions (green) and three Io-Ganymede conjunctions (grey) for each Europa-Ganymede conjunction (magenta). This diagram is not to scale.</figcaption></figure> In <a href="/wiki/Celestial_mechanics" title="Celestial mechanics">celestial mechanics</a>, orbital resonance occurs when <a href="/wiki/Orbit" title="Orbit">orbiting</a> bodies exert regular, periodic <a href="/wiki/Gravitational" class="mw-redirect" title="Gravitational">gravitational</a> influence on each other, usually because their <a href="/wiki/Orbital_period" title="Orbital period">orbital periods</a> are related by a ratio of small <a href="/wiki/Integer" title="Integer">integers</a>. Most commonly, this relationship is found between a pair of objects (binary resonance). The physical principle behind orbital resonance is similar in concept to pushing a child on a <a href="/wiki/Swing_(seat)" title="Swing (seat)">swing</a>, whereby the orbit and the swing both have a <a href="/wiki/Natural_frequency" title="Natural frequency">natural frequency</a>, and the body doing the "pushing" will act in periodic repetition to have a cumulative effect on the motion. Orbital resonances greatly enhance the mutual gravitational influence of the bodies (i.e., their ability to alter or constrain each other's orbits). In most cases, this results in an unstable interaction, in which the bodies exchange <a href="/wiki/Momentum" title="Momentum">momentum</a> and shift orbits until the resonance no longer exists. Under some circumstances, a resonant system can be self-correcting and thus stable. Examples are the 1:2:4 resonance of <a href="/wiki/Jupiter" title="Jupiter">Jupiter</a>'s moons <a href="/wiki/Ganymede_(moon)" title="Ganymede (moon)">Ganymede</a>, <a href="/wiki/Europa_(moon)" title="Europa (moon)">Europa</a> and <a href="/wiki/Io_(moon)" title="Io (moon)">Io</a>, and the 2:3 resonance between <a href="/wiki/Neptune" title="Neptune">Neptune</a> and <a href="/wiki/Pluto" title="Pluto">Pluto</a>. Unstable resonances with <a href="/wiki/Saturn" title="Saturn">Saturn</a>'s inner moons give rise to gaps in the <a href="/wiki/Rings_of_Saturn" title="Rings of Saturn">rings of Saturn</a>. The special case of 1:1 resonance between bodies with similar orbital radii causes large <a href="/wiki/Planetary_system" title="Planetary system">planetary system</a> bodies to eject most other bodies sharing their orbits; this is part of the much more extensive process of <a href="/wiki/Clearing_the_neighbourhood" title="Clearing the neighbourhood">clearing the neighbourhood</a>, an effect that is used in the current <a href="/wiki/Definition_of_planet" title="Definition of planet">definition of a planet</a>.<a href="#cite_note-1">[1]</a> A binary resonance ratio in this article should be interpreted as the ratio of number of orbits completed in the same time interval, rather than as the ratio of orbital periods, which would be the inverse ratio. Thus, the 2:3 ratio above means that Pluto completes two orbits in the time it takes Neptune to complete three. In the case of resonance relationships among three or more bodies, either type of ratio may be used (whereby the smallest whole-integer ratio sequences are not necessarily reversals of each other), and the type of ratio will be specified. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2>[<a href="/w/index.php?title=Orbital_resonance&action=edit&section=1" title="Edit section: History">edit</a>]</div> Since the discovery of <a href="/wiki/Newton%27s_law_of_universal_gravitation" title="Newton's law of universal gravitation">Newton's law of universal gravitation</a> in the 17th century, the <a href="/wiki/Stability_of_the_Solar_System" title="Stability of the Solar System">stability of the Solar System</a> has preoccupied many mathematicians, starting with <a href="/wiki/Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Pierre-Simon Laplace</a>. The stable orbits that arise in a <a href="/wiki/N-body_problem" title="N-body problem">two-body approximation</a> ignore the influence of other bodies. The effect of these added interactions on the stability of the <a href="/wiki/Solar_System" title="Solar System">Solar System</a> is very small, but at first it was not known whether they might add up over longer periods to significantly change the orbital parameters and lead to a completely different configuration, or whether some other stabilising effects might maintain the configuration of the orbits of the planets. It was Laplace who found the first answers explaining the linked orbits of the <a href="/wiki/Galilean_moon" class="mw-redirect" title="Galilean moon">Galilean moons</a> (see below). Before Newton, there was also consideration of ratios and proportions in orbital motions, in what was called "the music of the spheres", or <a href="/wiki/Musica_universalis" title="Musica universalis">musica universalis</a>. The article on <a href="/wiki/Resonant_interaction" title="Resonant interaction">resonant interactions</a> describes resonance in the general modern setting. A primary result from the study of <a href="/wiki/Dynamical_system" title="Dynamical system">dynamical systems</a> is the discovery and description of a highly simplified model of mode-locking; this is an oscillator that receives periodic kicks via a weak coupling to some driving motor. The analog here would be that a more massive body provides a periodic gravitational kick to a smaller body, as it passes by. The mode-locking regions are named <a href="/wiki/Arnold_tongue" title="Arnold tongue">Arnold tongues</a>. <div class="mw-heading mw-heading2"><h2 id="Types_of_resonance">Types of resonance</h2>[<a href="/w/index.php?title=Orbital_resonance&action=edit&section=2" title="Edit section: Types of resonance">edit</a>]</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:TheKuiperBelt_75AU_All.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/TheKuiperBelt_75AU_All.svg/300px-TheKuiperBelt_75AU_All.svg.png" decoding="async" width="300" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/TheKuiperBelt_75AU_All.svg/450px-TheKuiperBelt_75AU_All.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/86/TheKuiperBelt_75AU_All.svg/600px-TheKuiperBelt_75AU_All.svg.png 2x" data-file-width="800" data-file-height="400" /></a><figcaption>The <a href="/wiki/Semimajor_axis" class="mw-redirect" title="Semimajor axis">semimajor axes</a> of <a href="/wiki/Resonant_trans-Neptunian_object" title="Resonant trans-Neptunian object">resonant trans-Neptunian objects</a> (red) are clumped at locations of low-integer resonances with <a href="/wiki/Neptune" title="Neptune">Neptune</a> (vertical red bars near top), in contrast to those of <a href="/wiki/Cubewano" class="mw-redirect" title="Cubewano">cubewanos</a> (blue) and nonresonant (or not known to be resonant) <a href="/wiki/Scattered_disk" class="mw-redirect" title="Scattered disk">scattered objects</a> (grey).</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Kirkwood_Gaps.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d3/Kirkwood_Gaps.svg/300px-Kirkwood_Gaps.svg.png" decoding="async" width="300" height="232" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d3/Kirkwood_Gaps.svg/450px-Kirkwood_Gaps.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d3/Kirkwood_Gaps.svg/600px-Kirkwood_Gaps.svg.png 2x" data-file-width="792" data-file-height="612" /></a><figcaption>A chart of the distribution of <a href="/wiki/Asteroid" title="Asteroid">asteroid</a> semimajor axes, showing the <a href="/wiki/Kirkwood_gap" title="Kirkwood gap">Kirkwood gaps</a> where orbits are destabilized by resonances with <a href="/wiki/Jupiter" title="Jupiter">Jupiter</a></figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:PIA10452_-_Saturn_A_ring_spiral_density_waves.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/PIA10452_-_Saturn_A_ring_spiral_density_waves.jpg/300px-PIA10452_-_Saturn_A_ring_spiral_density_waves.jpg" decoding="async" width="300" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/PIA10452_-_Saturn_A_ring_spiral_density_waves.jpg/450px-PIA10452_-_Saturn_A_ring_spiral_density_waves.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5c/PIA10452_-_Saturn_A_ring_spiral_density_waves.jpg/600px-PIA10452_-_Saturn_A_ring_spiral_density_waves.jpg 2x" data-file-width="1014" data-file-height="1014" /></a><figcaption><a href="/wiki/Spiral_density_wave" class="mw-redirect" title="Spiral density wave">Spiral density waves</a> in <a href="/wiki/Rings_of_Saturn#A_Ring" title="Rings of Saturn">Saturn's A Ring</a> excited by resonances with <a href="/wiki/Moons_of_Saturn#Ring_shepherds" title="Moons of Saturn">inner moons</a>. Such waves propagate away from the planet (towards upper left). The large set of waves just below center is due to the 6:5 resonance with <a href="/wiki/Janus_(moon)" title="Janus (moon)">Janus</a>.</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:PIA17173_Titan_resonances_in_Saturn%27s_C_ring.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/PIA17173_Titan_resonances_in_Saturn%27s_C_ring.jpg/200px-PIA17173_Titan_resonances_in_Saturn%27s_C_ring.jpg" decoding="async" width="200" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/PIA17173_Titan_resonances_in_Saturn%27s_C_ring.jpg/300px-PIA17173_Titan_resonances_in_Saturn%27s_C_ring.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6f/PIA17173_Titan_resonances_in_Saturn%27s_C_ring.jpg/400px-PIA17173_Titan_resonances_in_Saturn%27s_C_ring.jpg 2x" data-file-width="980" data-file-height="980" /></a><figcaption>The eccentric <a href="/wiki/Rings_of_Saturn#Colombo_Gap_and_Titan_Ringlet" title="Rings of Saturn">Titan Ringlet</a><a href="#cite_note-Porco1984-2">[2]</a> in the Columbo Gap of Saturn's <a href="/wiki/Rings_of_Saturn#C_Ring" title="Rings of Saturn">C Ring</a> (center) and the inclined orbits of resonant particles in the bending wave<a href="#cite_note-Rosen1988-3">[3]</a><a href="#cite_note-Chakrabarti2001-4">[4]</a> just inside it have <a href="/wiki/Apsidal_precession" title="Apsidal precession">apsidal</a> and <a href="/wiki/Nodal_precession" title="Nodal precession">nodal</a> precessions, respectively, commensurate with <a href="/wiki/Titan_(moon)" title="Titan (moon)">Titan</a>'s mean motion.</figcaption></figure> In general, an orbital resonance may <ul><li>involve one or any combination of the orbit parameters (e.g. <a href="/wiki/Orbital_eccentricity" title="Orbital eccentricity">eccentricity</a> versus <a href="/wiki/Semimajor_axis" class="mw-redirect" title="Semimajor axis">semimajor axis</a>, or eccentricity versus <a href="/wiki/Orbital_inclination" title="Orbital inclination">inclination</a>).</li> <li>act on any time scale from short term, commensurable with the orbit periods, to <a href="/wiki/Secular_phenomena" class="mw-redirect" title="Secular phenomena">secular</a>, measured in 104 to 106 years.</li> <li>lead to either long-term stabilization of the orbits or be the cause of their destabilization.</li></ul> A mean-motion orbital resonance occurs when two bodies have periods of <a href="/wiki/Orbit" title="Orbit">revolution</a> that are a simple integer ratio of each other. It does not depend only on the existence of such a ratio, and more precisely the ratio of periods is not exactly a rational number, even averaged over a long period. For example, in the case of <a href="/wiki/Pluto" title="Pluto">Pluto</a> and <a href="/wiki/Neptune" title="Neptune">Neptune</a> (see below), the true equation says that the average rate of change of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3\alpha _{P}-2\alpha _{N}-\varpi _{P}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>−</mo> <mn>2</mn> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>ϖ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3\alpha _{P}-2\alpha _{N}-\varpi _{P}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/302db2830516c41717c34acd7ef0a74b7d7bfdf8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.53ex; height:2.509ex;" alt="{\displaystyle 3\alpha _{P}-2\alpha _{N}-\varpi _{P}}"> is exactly zero, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{P}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{P}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d572a5f43d4f585a2295a9e39ec565df5bc5120a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.954ex; height:2.009ex;" alt="{\displaystyle \alpha _{P}}"> is the longitude of Pluto, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{N}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{N}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78ce2991e751be93196abd6d0304ca5374ec0636" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.179ex; height:2.009ex;" alt="{\displaystyle \alpha _{N}}"> is the longitude of Neptune, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varpi _{P}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϖ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varpi _{P}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c158e0220f032ba07585b6cf73f7560ab0dc5d59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.391ex; height:2.009ex;" alt="{\displaystyle \varpi _{P}}"> is the longitude of Pluto's <a href="/wiki/Perihelion" class="mw-redirect" title="Perihelion">perihelion</a>. Since the rate of motion of the latter is about 0.97×10−4 degrees per year, the ratio of periods is actually 1.503 in the long term.<a href="#cite_note-williams71-5">[5]</a> Depending on the details, mean-motion orbital resonance can either stabilize or destabilize the orbit. Stabilization may occur when the two bodies move in such a synchronised fashion that they never closely approach. For instance: <ul><li>The orbits of <a href="/wiki/Pluto" title="Pluto">Pluto</a> and the <a href="/wiki/Plutino" title="Plutino">plutinos</a> are stable, despite crossing that of the much larger <a href="/wiki/Neptune" title="Neptune">Neptune</a>, because they are in a 2:3 resonance with it. The resonance ensures that, when they approach perihelion and Neptune's orbit, Neptune is consistently distant (averaging a quarter of its orbit away). Other (much more numerous) Neptune-crossing bodies that were not in resonance were ejected from that region by strong <a href="/wiki/Perturbation_(astronomy)" title="Perturbation (astronomy)">perturbations</a> due to Neptune. There are also smaller but significant groups of <a href="/wiki/Resonant_trans-Neptunian_object" title="Resonant trans-Neptunian object">resonant trans-Neptunian objects</a> occupying the 1:1 (<a href="/wiki/Neptune_trojan" title="Neptune trojan">Neptune trojans</a>), <a href="/wiki/Resonant_Kuiper_belt_object#3:5_resonance_(period_~275_years)" class="mw-redirect" title="Resonant Kuiper belt object">3:5</a>, <a href="/wiki/Resonant_Kuiper_belt_object#4:7_resonance_(period_~290_years)" class="mw-redirect" title="Resonant Kuiper belt object">4:7</a>, 1:2 (<a href="/wiki/Resonant_Kuiper_belt_object#1:2_resonance_("twotinos",_period_~330_years)" class="mw-redirect" title="Resonant Kuiper belt object">twotinos</a>) and <a href="/wiki/Resonant_Kuiper_belt_object#2:5_resonance_(period_~410_years)" class="mw-redirect" title="Resonant Kuiper belt object">2:5</a> resonances, among others, with respect to Neptune.</li> <li>In the <a href="/wiki/Asteroid_belt" title="Asteroid belt">asteroid belt</a> beyond 3.5 AU from the Sun, the 3:2, 4:3 and 1:1 resonances with <a href="/wiki/Jupiter" title="Jupiter">Jupiter</a> are populated by clumps of asteroids (the <a href="/wiki/Hilda_family" title="Hilda family">Hilda family</a>, the few <a href="/wiki/Thule_asteroid" class="mw-redirect" title="Thule asteroid">Thule asteroids</a>, and the numerous <a href="/wiki/Jupiter_trojan" title="Jupiter trojan">Trojan asteroids</a>, respectively).</li></ul> Orbital resonances can also destabilize one of the orbits. This process can be exploited to find energy-efficient ways of <a href="/wiki/Deorbit" class="mw-redirect" title="Deorbit">deorbiting</a> spacecraft.<a href="#cite_note-Witze2018-6">[6]</a><a href="#cite_note-Daquin2016-7">[7]</a> For small bodies, destabilization is actually far more likely. For instance: <ul><li>In the <a href="/wiki/Asteroid_belt" title="Asteroid belt">asteroid belt</a> within 3.5 AU from the Sun, the major mean-motion resonances with <a href="/wiki/Jupiter" title="Jupiter">Jupiter</a> are locations of gaps in the asteroid distribution, the <a href="/wiki/Kirkwood_gap" title="Kirkwood gap">Kirkwood gaps</a> (most notably at the 4:1, 3:1, 5:2, 7:3 and 2:1 resonances). <a href="/wiki/Asteroid" title="Asteroid">Asteroids</a> have been ejected from these almost empty lanes by repeated perturbations. However, there are still populations of asteroids temporarily present in or near these resonances. For example, asteroids of the <a href="/wiki/Alinda_family" class="mw-redirect" title="Alinda family">Alinda family</a> are in or close to the 3:1 resonance, with their orbital eccentricity steadily increased by interactions with Jupiter until they eventually have a close encounter with an inner planet that ejects them from the resonance.</li> <li>In the <a href="/wiki/Rings_of_Saturn" title="Rings of Saturn">rings of Saturn</a>, the <a href="/wiki/Rings_of_Saturn#Cassini_Division" title="Rings of Saturn">Cassini Division</a> is a gap between the inner <a href="/wiki/Rings_of_Saturn#B_Ring" title="Rings of Saturn">B Ring</a> and the outer <a href="/wiki/Rings_of_Saturn#A_Ring" title="Rings of Saturn">A Ring</a> that has been cleared by a 2:1 resonance with the moon <a href="/wiki/Mimas_(moon)" class="mw-redirect" title="Mimas (moon)">Mimas</a>. (More specifically, the site of the resonance is the <a href="/wiki/Rings_of_Saturn#Huygens_Gap" title="Rings of Saturn">Huygens Gap</a>, which bounds the outer edge of the <a href="/wiki/Rings_of_Saturn#B_Ring" title="Rings of Saturn">B Ring</a>.)</li> <li>In the rings of Saturn, the <a href="/wiki/Rings_of_Saturn#Encke_Gap" title="Rings of Saturn">Encke</a> and <a href="/wiki/Rings_of_Saturn#Keeler_Gap" title="Rings of Saturn">Keeler</a> gaps within the A Ring are cleared by 1:1 resonances with the embedded moonlets <a href="/wiki/Pan_(moon)" title="Pan (moon)">Pan</a> and <a href="/wiki/Daphnis_(moon)" title="Daphnis (moon)">Daphnis</a>, respectively. The A Ring's outer edge is maintained by a destabilizing 7:6 resonance with the moon <a href="/wiki/Janus_(moon)" title="Janus (moon)">Janus</a>.</li></ul> Most bodies that are in resonance orbit in the same direction; however, the <a href="/wiki/Retrograde_motion" class="mw-redirect" title="Retrograde motion">retrograde</a> asteroid <a href="/wiki/514107_Ka%CA%BBepaoka%CA%BBawela" title="514107 Kaʻepaokaʻawela">514107 Kaʻepaokaʻawela</a> appears to be in a stable (for a period of at least a million years) 1:−1 resonance with Jupiter.<a href="#cite_note-Wieger2017-8">[8]</a> In addition, a few retrograde <a href="/wiki/Damocloid_asteroid" class="mw-redirect" title="Damocloid asteroid">damocloids</a> have been found that are temporarily captured in mean-motion resonance with <a href="/wiki/Jupiter" title="Jupiter">Jupiter</a> or <a href="/wiki/Saturn" title="Saturn">Saturn</a>.<a href="#cite_note-Morais_2013-9">[9]</a> Such orbital interactions are weaker than the corresponding interactions between bodies orbiting in the same direction.<a href="#cite_note-Morais_2013-9">[9]</a><a href="#cite_note-Morais2013cmda-10">[10]</a> The <a href="/wiki/Trans-Neptunian_object" title="Trans-Neptunian object">trans-Neptunian object</a> <a href="/wiki/471325_Taowu" title="471325 Taowu">471325 Taowu</a> has an orbital inclination of 110<a href="/wiki/Degree_(angle)" title="Degree (angle)">°</a> with respect to the planets' <a href="/wiki/Orbital_plane" title="Orbital plane">orbital plane</a> and is currently in a 7:9 polar resonance with Neptune.<a href="#cite_note-Morais_Namouni_2017-11">[11]</a> A Laplace resonance is a three-body resonance with a 1:2:4 orbital period ratio (equivalent to a 4:2:1 ratio of orbits). The term arose because <a href="/wiki/Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Pierre-Simon Laplace</a> discovered that such a resonance governed the motions of Jupiter's moons <a href="/wiki/Io_(moon)" title="Io (moon)">Io</a>, <a href="/wiki/Europa_(moon)" title="Europa (moon)">Europa</a>, and <a href="/wiki/Ganymede_(moon)" title="Ganymede (moon)">Ganymede</a>. It is now also often applied to other 3-body resonances with the same ratios,<a href="#cite_note-Gargaud2011-12">[12]</a> such as that between the <a href="/wiki/Extrasolar_planet" class="mw-redirect" title="Extrasolar planet">extrasolar planets</a> <a href="/wiki/Gliese_876" title="Gliese 876">Gliese 876</a> c, b, and e.<a href="#cite_note-rivera2010-13">[13]</a><a href="#cite_note-14">[14]</a><a href="#cite_note-MartiGiuppone2013-15">[15]</a> Three-body resonances involving other simple integer ratios have been termed "Laplace-like"<a href="#cite_note-ShowalterHamilton2015-16">[16]</a> or "Laplace-type".<a href="#cite_note-MurrayDermott1999-17">[17]</a> A <a href="/wiki/Lindblad_resonance" title="Lindblad resonance">Lindblad resonance</a> drives <a href="/wiki/Density_wave_theory" title="Density wave theory">spiral density waves</a> both in <a href="/wiki/Galaxies" class="mw-redirect" title="Galaxies">galaxies</a> (where stars are subject to <a href="/wiki/Harmonic_oscillator" title="Harmonic oscillator">forcing</a> by the spiral arms themselves) and in <a href="/wiki/Rings_of_Saturn" title="Rings of Saturn">Saturn's rings</a> (where ring particles are subject to forcing by <a href="/wiki/Moons_of_Saturn" title="Moons of Saturn">Saturn's moons</a>). A <a href="/wiki/Secular_resonance" title="Secular resonance">secular resonance</a> occurs when the <a href="/wiki/Precession#Astronomy" title="Precession">precession</a> of two orbits is synchronised (usually a precession of the <a href="/wiki/Perihelion" class="mw-redirect" title="Perihelion">perihelion</a> or <a href="/wiki/Ascending_node" class="mw-redirect" title="Ascending node">ascending node</a>). A small body in secular resonance with a much larger one (e.g. a <a href="/wiki/Planet" title="Planet">planet</a>) will precess at the same rate as the large body. Over long times (a million years, or so) a secular resonance will change the <a href="/wiki/Eccentricity_(orbit)" class="mw-redirect" title="Eccentricity (orbit)">eccentricity</a> and <a href="/wiki/Inclination" class="mw-redirect" title="Inclination">inclination</a> of the small body. Several prominent examples of secular resonance involve Saturn. There is a near-resonance between the precession of Saturn's rotational axis and that of Neptune's orbital axis (both of which have periods of about 1.87 million years), which has been identified as the likely source of Saturn's large <a href="/wiki/Axial_tilt" title="Axial tilt">axial tilt</a> (26.7°).<a href="#cite_note-18">[18]</a><a href="#cite_note-19">[19]</a><a href="#cite_note-20">[20]</a> Initially, Saturn probably had a tilt closer to that of Jupiter (3.1°). The gradual depletion of the Kuiper belt would have decreased the precession rate of Neptune's orbit; eventually, the frequencies matched, and Saturn's axial precession was captured into a spin-orbit resonance, leading to an increase in Saturn's obliquity. (The angular momentum of Neptune's orbit is 104 times that of Saturn's rotation rate, and thus dominates the interaction.) However, it seems that the resonance no longer exists. Detailed analysis of data from the <a href="/wiki/Cassini_spacecraft" class="mw-redirect" title="Cassini spacecraft">Cassini spacecraft</a> gives a value of the moment of inertia of Saturn that is just outside the range for the resonance to exist, meaning that the spin axis does not stay in phase with Neptune's orbital inclination in the long term, as it apparently did in the past. One theory for why the resonance came to an end is that there was another moon around Saturn whose orbit destabilized about 100 million years ago, perturbing Saturn.<a href="#cite_note-21">[21]</a><a href="#cite_note-22">[22]</a> The <a href="/wiki/Secular_resonance#?6_resonance" title="Secular resonance">perihelion secular resonance</a> between <a href="/wiki/Asteroid" title="Asteroid">asteroids</a> and <a href="/wiki/Saturn" title="Saturn">Saturn</a> (ν6 = g − g6) helps shape the asteroid belt (the subscript "6" identifies Saturn as the sixth planet from the Sun). Asteroids which approach it have their eccentricity slowly increased until they become <a href="/wiki/Mars-crossing_asteroid" class="mw-redirect" title="Mars-crossing asteroid">Mars-crossers</a>, at which point they are usually ejected from the <a href="/wiki/Asteroid_belt" title="Asteroid belt">asteroid belt</a> by a close pass to <a href="/wiki/Mars" title="Mars">Mars</a>. This resonance forms the inner and "side" boundaries of the <a href="/wiki/Asteroid_belt" title="Asteroid belt">asteroid belt</a> around 2 <a href="/wiki/Astronomical_unit" title="Astronomical unit">AU</a>, and at inclinations of about 20°. Numerical simulations have suggested that the eventual formation of a perihelion secular resonance between <a href="/wiki/Mercury_(planet)" title="Mercury (planet)">Mercury</a> and Jupiter (g1 = g5) has the potential to greatly increase Mercury's eccentricity and possibly destabilize the inner Solar System several billion years from now.<a href="#cite_note-Laskar2008-23">[23]</a><a href="#cite_note-Laskar2009-24">[24]</a> The <a href="/wiki/Rings_of_Saturn#Colombo_Gap_and_Titan_Ringlet" title="Rings of Saturn">Titan Ringlet</a> within Saturn's <a href="/wiki/Rings_of_Saturn#C_Ring" title="Rings of Saturn">C Ring</a> represents another type of resonance in which the rate of <a href="/wiki/Apsidal_precession" title="Apsidal precession">apsidal precession</a> of one orbit exactly matches the speed of revolution of another. The outer end of this eccentric ringlet always points towards Saturn's major moon <a href="/wiki/Titan_(moon)" title="Titan (moon)">Titan</a>.<a href="#cite_note-Porco1984-2">[2]</a> A <a href="/wiki/Kozai_resonance" class="mw-redirect" title="Kozai resonance">Kozai resonance</a> occurs when the inclination and eccentricity of a <a href="/wiki/Perturbation_theory" title="Perturbation theory">perturbed</a> orbit oscillate synchronously (increasing eccentricity while decreasing inclination and vice versa). This resonance applies only to bodies on highly inclined orbits; as a consequence, such orbits tend to be unstable, since the growing eccentricity would result in small <a href="/wiki/Apsis" title="Apsis">pericenters</a>, typically leading to a collision or (for large moons) destruction by <a href="/wiki/Tidal_forces" class="mw-redirect" title="Tidal forces">tidal forces</a>. In an example of another type of resonance involving orbital eccentricity, the eccentricities of Ganymede and Callisto vary with a common period of 181 years, although with opposite phases.<a href="#cite_note-Musotto2002-25">[25]</a> <div class="mw-heading mw-heading2"><h2 id="Mean-motion_resonances_in_the_Solar_System">Mean-motion resonances in the Solar System</h2>[<a href="/w/index.php?title=Orbital_resonance&action=edit&section=3" title="Edit section: Mean-motion resonances in the Solar System">edit</a>]</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Haumea.GIF" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Haumea.GIF/300px-Haumea.GIF" decoding="async" width="300" height="261" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Haumea.GIF/450px-Haumea.GIF 1.5x, //upload.wikimedia.org/wikipedia/commons/3/3e/Haumea.GIF 2x" data-file-width="590" data-file-height="514" /></a><figcaption>Depiction of <a href="/wiki/Haumea_(dwarf_planet)" class="mw-redirect" title="Haumea (dwarf planet)">Haumea</a>'s presumed 7:12 resonance with <a href="/wiki/Neptune" title="Neptune">Neptune</a> in a <a href="/wiki/Rotating_frame" class="mw-redirect" title="Rotating frame">rotating frame</a>, with Neptune (blue dot at lower right) held stationary. Haumea's shifting orbital alignment relative to Neptune periodically reverses (<a href="/wiki/Libration" title="Libration">librates</a>), preserving the resonance.</figcaption></figure> There are only a few known mean-motion resonances (MMR) in the <a href="/wiki/Solar_System" title="Solar System">Solar System</a> involving planets, <a href="/wiki/Dwarf_planet" title="Dwarf planet">dwarf planets</a> or larger <a href="/wiki/Natural_satellite" title="Natural satellite">satellites</a> (a much greater number involve <a href="/wiki/Asteroid" title="Asteroid">asteroids</a>, <a href="/wiki/Planetary_ring" class="mw-redirect" title="Planetary ring">planetary rings</a>, <a href="/wiki/Inner_satellite" class="mw-redirect" title="Inner satellite">moonlets</a> and smaller <a href="/wiki/Kuiper_belt" title="Kuiper belt">Kuiper belt</a> objects, including many <a href="/wiki/Possible_dwarf_planets" class="mw-redirect" title="Possible dwarf planets">possible dwarf planets</a>). <ul><li>2:3 <a href="/wiki/Pluto" title="Pluto">Pluto</a>–<a href="/wiki/Neptune" title="Neptune">Neptune</a> (also <a href="/wiki/Orcus_(dwarf_planet)" title="Orcus (dwarf planet)">Orcus</a> and other <a href="/wiki/Plutino" title="Plutino">plutinos</a>)</li> <li>2:4 <a href="/wiki/Tethys_(moon)" title="Tethys (moon)">Tethys</a>–<a href="/wiki/Mimas_(moon)" class="mw-redirect" title="Mimas (moon)">Mimas</a> (Saturn's moons). Not simplified, because the libration of the nodes must be taken into account.</li> <li>1:2 <a href="/wiki/Dione_(moon)" title="Dione (moon)">Dione</a>–<a href="/wiki/Enceladus" title="Enceladus">Enceladus</a> (Saturn's moons)</li> <li>3:4 <a href="/wiki/Hyperion_(moon)" title="Hyperion (moon)">Hyperion</a>–<a href="/wiki/Titan_(moon)" title="Titan (moon)">Titan</a> (Saturn's moons)</li> <li>1:2:4 <a href="/wiki/Ganymede_(moon)" title="Ganymede (moon)">Ganymede</a>–<a href="/wiki/Europa_(moon)" title="Europa (moon)">Europa</a>–<a href="/wiki/Io_(moon)" title="Io (moon)">Io</a> (Jupiter's moons, ratio of orbits).</li></ul> Additionally, <a href="/wiki/Haumea" title="Haumea">Haumea</a> is thought to be in a 7:12 resonance with Neptune,<a href="#cite_note-Brown_2007-26">[26]</a><a href="#cite_note-Ragozzine-27">[27]</a> and <a href="/wiki/Gonggong_(dwarf_planet)" title="Gonggong (dwarf planet)">Gonggong</a> is thought to be in a 3:10 resonance with Neptune.<a href="#cite_note-Buie-28">[28]</a> The simple integer ratios between periods hide more complex relations: <ul><li>the point of <a href="/wiki/Astronomical_conjunction" class="mw-redirect" title="Astronomical conjunction">conjunction</a> can oscillate (<a href="/wiki/Libration" title="Libration">librate</a>) around an equilibrium point defined by the resonance.</li> <li>given non-zero <a href="/wiki/Eccentricity_(orbit)" class="mw-redirect" title="Eccentricity (orbit)">eccentricities</a>, the <a href="/wiki/Orbital_node" title="Orbital node">nodes</a> or <a href="/wiki/Perihelion" class="mw-redirect" title="Perihelion">periapsides</a> can drift (a resonance related, short period, not secular precession).</li></ul> As illustration of the latter, consider the well-known 2:1 resonance of Io-Europa. If the orbiting periods were in this relation, the <a href="/wiki/Mean_motion" title="Mean motion">mean motions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\,\!}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddf4a520ae4c1cdb7467e78a29509510ea61a08f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.782ex; height:1.676ex;" alt="{\displaystyle n\,\!}"> (inverse of periods, often expressed in degrees per day) would satisfy the following <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{\rm {Io}}-2\cdot n_{\rm {Eu}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">I</mi> <mi mathvariant="normal">o</mi> </mrow> </mrow> </msub> <mo>−</mo> <mn>2</mn> <mo>⋅</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> <mi mathvariant="normal">u</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{\rm {Io}}-2\cdot n_{\rm {Eu}}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c2401b24937a5d727cc7034be71fbb532dec9c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.646ex; height:2.509ex;" alt="{\displaystyle n_{\rm {Io}}-2\cdot n_{\rm {Eu}}=0}"></dd></dl> Substituting the data (from Wikipedia) one will get −0.7395° day−1, a value substantially different from zero. Actually, the resonance is perfect, but it involves also the precession of <a href="/wiki/Perihelion" class="mw-redirect" title="Perihelion">perijove</a> (the point closest to Jupiter), <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\omega }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\omega }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acf8369d9915a384686bd5947a0e43c46bccd982" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:2.176ex;" alt="{\displaystyle {\dot {\omega }}}">. The correct equation (part of the Laplace equations) is: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{\rm {Io}}-2\cdot n_{\rm {Eu}}+{\dot {\omega }}_{\rm {Io}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">I</mi> <mi mathvariant="normal">o</mi> </mrow> </mrow> </msub> <mo>−</mo> <mn>2</mn> <mo>⋅</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> <mi mathvariant="normal">u</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">I</mi> <mi mathvariant="normal">o</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{\rm {Io}}-2\cdot n_{\rm {Eu}}+{\dot {\omega }}_{\rm {Io}}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5ec6d54875cc71e289939f48deaa4621146c51d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.58ex; height:2.509ex;" alt="{\displaystyle n_{\rm {Io}}-2\cdot n_{\rm {Eu}}+{\dot {\omega }}_{\rm {Io}}=0}"></dd></dl> In other words, the mean motion of Io is indeed double of that of Europa taking into account the precession of the perijove. An observer sitting on the (drifting) perijove will see the moons coming into conjunction in the same place (elongation). The other pairs listed above satisfy the same type of equation with the exception of Mimas-Tethys resonance. In this case, the resonance satisfies the equation <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4\cdot n_{\rm {Te}}-2\cdot n_{\rm {Mi}}-{\dot {\Omega }}_{\rm {Te}}-{\dot {\Omega }}_{\rm {Mi}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mo>⋅</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mo>−</mo> <mn>2</mn> <mo>⋅</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">i</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">Ω</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">Ω</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">i</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4\cdot n_{\rm {Te}}-2\cdot n_{\rm {Mi}}-{\dot {\Omega }}_{\rm {Te}}-{\dot {\Omega }}_{\rm {Mi}}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e4ebcb07836873c52c0f0590fd7a4b0d13cdda3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:33.301ex; height:3.176ex;" alt="{\displaystyle 4\cdot n_{\rm {Te}}-2\cdot n_{\rm {Mi}}-{\dot {\Omega }}_{\rm {Te}}-{\dot {\Omega }}_{\rm {Mi}}=0}"></dd></dl> The point of conjunctions librates around the midpoint between the <a href="/wiki/Orbital_node" title="Orbital node">nodes</a> of the two moons. <div class="mw-heading mw-heading3"><h3 id="Laplace_resonance">Laplace resonance</h3>[<a href="/w/index.php?title=Orbital_resonance&action=edit&section=4" title="Edit section: Laplace resonance">edit</a>]</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:TheLaplaceResonance2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/TheLaplaceResonance2.png/300px-TheLaplaceResonance2.png" decoding="async" width="300" height="105" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/TheLaplaceResonance2.png/450px-TheLaplaceResonance2.png 1.5x, //upload.wikimedia.org/wikipedia/commons/0/05/TheLaplaceResonance2.png 2x" data-file-width="470" data-file-height="165" /></a><figcaption>Illustration of Io–Europa–Ganymede resonance. From the centre outwards: Io (yellow), Europa (gray), and Ganymede (dark)</figcaption></figure> The Laplace resonance involving Io–Europa–Ganymede includes the following relation locking the orbital phase of the moons: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{L}=\lambda _{\rm {Io}}-3\cdot \lambda _{\rm {Eu}}+2\cdot \lambda _{\rm {Ga}}=180^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">I</mi> <mi mathvariant="normal">o</mi> </mrow> </mrow> </msub> <mo>−</mo> <mn>3</mn> <mo>⋅</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> <mi mathvariant="normal">u</mi> </mrow> </mrow> </msub> <mo>+</mo> <mn>2</mn> <mo>⋅</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">G</mi> <mi mathvariant="normal">a</mi> </mrow> </mrow> </msub> <mo>=</mo> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{L}=\lambda _{\rm {Io}}-3\cdot \lambda _{\rm {Eu}}+2\cdot \lambda _{\rm {Ga}}=180^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52bdf5d416c5209a28f2f93b28feccc84e773e6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:35.455ex; height:2.676ex;" alt="{\displaystyle \Phi _{L}=\lambda _{\rm {Io}}-3\cdot \lambda _{\rm {Eu}}+2\cdot \lambda _{\rm {Ga}}=180^{\circ }}"></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"> are <a href="/wiki/Mean_longitude" title="Mean longitude">mean longitudes</a> of the moons (the second equals sign ignores libration). This relation makes a triple conjunction impossible. (A Laplace resonance in the <a href="/wiki/Gliese_876" title="Gliese 876">Gliese 876</a> system, in contrast, is associated with one triple conjunction per orbit of the outermost planet, ignoring libration.) The graph illustrates the positions of the moons after 1, 2, and 3 Io periods. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{L}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{L}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5153ef8598101c48c82f65b8f22bc428c98591b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.03ex; height:2.509ex;" alt="{\displaystyle \Phi _{L}}"> librates about 180° with an amplitude of 0.03°.<a href="#cite_note-Sinclair1975-29">[29]</a> Another "Laplace-like" resonance involves the <a href="/wiki/Moons_of_Pluto" title="Moons of Pluto">moons</a> <a href="/wiki/Styx_(moon)" title="Styx (moon)">Styx</a>, <a href="/wiki/Nix_(moon)" title="Nix (moon)">Nix</a>, and <a href="/wiki/Hydra_(moon)" title="Hydra (moon)">Hydra</a> of Pluto:<a href="#cite_note-ShowalterHamilton2015-16">[16]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi =3\cdot \lambda _{\rm {S}}-5\cdot \lambda _{\rm {N}}+2\cdot \lambda _{\rm {H}}=180^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> <mo>=</mo> <mn>3</mn> <mo>⋅</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> </mrow> </msub> <mo>−</mo> <mn>5</mn> <mo>⋅</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">N</mi> </mrow> </mrow> </msub> <mo>+</mo> <mn>2</mn> <mo>⋅</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> </mrow> </mrow> </msub> <mo>=</mo> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi =3\cdot \lambda _{\rm {S}}-5\cdot \lambda _{\rm {N}}+2\cdot \lambda _{\rm {H}}=180^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f296c5780b585b184aae18c13f6c8138cb545c71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:34.764ex; height:2.676ex;" alt="{\displaystyle \Phi =3\cdot \lambda _{\rm {S}}-5\cdot \lambda _{\rm {N}}+2\cdot \lambda _{\rm {H}}=180^{\circ }}"></dd></dl> This reflects orbital periods for Styx, Nix, and Hydra, respectively, that are close to a ratio of 18:22:33 (or, in terms of the near resonances with Charon's period, 3+3/11:4:6; see <a class="mw-selflink-fragment" href="#Coincidental_'near'_ratios_of_mean_motion">below</a>); the respective ratio of orbits is 11:9:6. Based on the ratios of <a href="/wiki/Synodic_period" class="mw-redirect" title="Synodic period">synodic periods</a>, there are 5 conjunctions of Styx and Hydra and 3 conjunctions of Nix and Hydra for every 2 conjunctions of Styx and Nix.<a href="#cite_note-ShowalterHamilton2015-16">[16]</a><a href="#cite_note-Witze2015-30">[30]</a> As with the Galilean satellite resonance, triple conjunctions are forbidden. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aed80a2011a3912b028ba32a52dfa57165455f24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Phi }"> librates about 180° with an amplitude of at least 10°.<a href="#cite_note-ShowalterHamilton2015-16">[16]</a> <div class="center" style="width:auto; margin-left:auto; margin-right:auto;"> <div class="center"><div class="noresize thumb tnone" style=";"> <div class="thumbinner" style="overflow:hidden;width:627px;"> <div class="thumbimage" style="height:126px; overflow:hidden; position:relative; background-color:white;"> <div style=";left:0px; top:0px; width:625px; position:absolute;"> <a href="/wiki/File:Hydra,_Nix,_Styx_conjunctions_cycle.png" class="mw-file-description" title="Sequence of conjunctions of Hydra (blue), Nix (red), and Styx (black) over one third of their resonance cycle. Movements are counterclockwise and orbits completed are tallied at upper right of diagrams (click on image to see the whole cycle)."><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Hydra%2C_Nix%2C_Styx_conjunctions_cycle.png/625px-Hydra%2C_Nix%2C_Styx_conjunctions_cycle.png" decoding="async" width="625" height="250" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Hydra%2C_Nix%2C_Styx_conjunctions_cycle.png/938px-Hydra%2C_Nix%2C_Styx_conjunctions_cycle.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Hydra%2C_Nix%2C_Styx_conjunctions_cycle.png/1250px-Hydra%2C_Nix%2C_Styx_conjunctions_cycle.png 2x" data-file-width="1870" data-file-height="748" /></a></div> <div style="text-align:left; background-color:transparent; line-height:110%;"> </div> <div style="visibility:hidden"><a href="/wiki/File:Hydra,_Nix,_Styx_conjunctions_cycle.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Hydra%2C_Nix%2C_Styx_conjunctions_cycle.png/625px-Hydra%2C_Nix%2C_Styx_conjunctions_cycle.png" decoding="async" width="625" height="250" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Hydra%2C_Nix%2C_Styx_conjunctions_cycle.png/938px-Hydra%2C_Nix%2C_Styx_conjunctions_cycle.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Hydra%2C_Nix%2C_Styx_conjunctions_cycle.png/1250px-Hydra%2C_Nix%2C_Styx_conjunctions_cycle.png 2x" data-file-width="1870" data-file-height="748" /></a></div> </div> <div class="thumbcaption"><div class="magnify"><a href="/wiki/File:Hydra,_Nix,_Styx_conjunctions_cycle.png" title="File:Hydra, Nix, Styx conjunctions cycle.png"> </a></div>Sequence of conjunctions of Hydra (blue), Nix (red), and Styx (black) over one third of their resonance cycle. Movements are counterclockwise and orbits completed are tallied at upper right of diagrams (click on image to see the whole cycle).</div> </div></div></div> </div> <div class="mw-heading mw-heading3"><h3 id="Plutino_resonances">Plutino resonances</h3>[<a href="/w/index.php?title=Orbital_resonance&action=edit&section=5" title="Edit section: Plutino resonances">edit</a>]</div> The dwarf planet <a href="/wiki/Pluto" title="Pluto">Pluto</a> is following an orbit trapped in a web of resonances with <a href="/wiki/Neptune" title="Neptune">Neptune</a>. The resonances include: <ul><li>A mean-motion resonance of 2:3</li> <li>The resonance of the <a href="/wiki/Perihelion" class="mw-redirect" title="Perihelion">perihelion</a> (<a href="/wiki/Libration" title="Libration">libration</a> around 90°), keeping the perihelion above the <a href="/wiki/Ecliptic" title="Ecliptic">ecliptic</a></li> <li>The resonance of the longitude of the perihelion in relation to that of Neptune</li></ul> One consequence of these resonances is that a separation of at least 30 AU is maintained when Pluto crosses Neptune's orbit. The minimum separation between the two bodies overall is 17 AU, while the minimum separation between Pluto and <a href="/wiki/Uranus" title="Uranus">Uranus</a> is just 11 AU<a href="#cite_note-31">[31]</a> (see <a href="/wiki/Pluto#Orbit" title="Pluto">Pluto's orbit</a> for detailed explanation and graphs). The next largest body in a similar 2:3 resonance with Neptune, called a <a href="/wiki/Plutino" title="Plutino">plutino</a>, is the probable dwarf planet <a href="/wiki/90482_Orcus" class="mw-redirect" title="90482 Orcus">Orcus</a>. Orcus has an orbit similar in inclination and eccentricity to Pluto's. However, the two are constrained by their mutual resonance with Neptune to always be in opposite phases of their orbits; Orcus is thus sometimes described as the "anti-Pluto".<a href="#cite_note-MBP-32">[32]</a> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Naiad-Thalassa_73-69_orbital_resonance.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/16/Naiad-Thalassa_73-69_orbital_resonance.jpg/300px-Naiad-Thalassa_73-69_orbital_resonance.jpg" decoding="async" width="300" height="168" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/16/Naiad-Thalassa_73-69_orbital_resonance.jpg/450px-Naiad-Thalassa_73-69_orbital_resonance.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/16/Naiad-Thalassa_73-69_orbital_resonance.jpg/600px-Naiad-Thalassa_73-69_orbital_resonance.jpg 2x" data-file-width="968" data-file-height="543" /></a><figcaption>Depiction of the resonance between Neptune's moons <a href="/wiki/Naiad_(moon)" title="Naiad (moon)">Naiad</a> (whose orbital motion is shown in red) and <a href="/wiki/Thalassa_(moon)" title="Thalassa (moon)">Thalassa</a>, in a view that co-rotates with the latter</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Naiad:Thalassa_73:69_resonance">Naiad:Thalassa 73:69 resonance</h3>[<a href="/w/index.php?title=Orbital_resonance&action=edit&section=6" title="Edit section: Naiad:Thalassa 73:69 resonance">edit</a>]</div> Neptune's innermost moon, <a href="/wiki/Naiad_(moon)" title="Naiad (moon)">Naiad</a>, is in a 73:69 fourth-order resonance with the next outward moon, <a href="/wiki/Thalassa_(moon)" title="Thalassa (moon)">Thalassa</a>. As it orbits Neptune, the more inclined Naiad successively passes Thalassa twice from above and then twice from below, in a cycle that repeats every ~21.5 Earth days. The two moons are about 3540 km apart when they pass each other. Although their orbital radii differ by only 1850 km, Naiad swings ~2800 km above or below Thalassa's orbital plane at closest approach. As is common, this resonance stabilizes the orbits by maximizing separation at conjunction, but it is unusual for the role played by orbital inclination in facilitating this avoidance in a case where eccentricities are minimal.<a href="#cite_note-JPLnews2019-33">[33]</a><a href="#cite_note-Brozovic2019-34">[34]</a><a href="#cite_note-36">[note 1]</a> <div class="mw-heading mw-heading2"><h2 id="Mean-motion_resonances_among_extrasolar_planets">Mean-motion resonances among extrasolar planets</h2>[<a href="/w/index.php?title=Orbital_resonance&action=edit&section=7" title="Edit section: Mean-motion resonances among extrasolar planets">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Resonant_planetary_system.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Resonant_planetary_system.gif/220px-Resonant_planetary_system.gif" decoding="async" width="220" height="122" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Resonant_planetary_system.gif/330px-Resonant_planetary_system.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Resonant_planetary_system.gif/440px-Resonant_planetary_system.gif 2x" data-file-width="454" data-file-height="252" /></a><figcaption>Resonant planetary system of two planets with a 1:2 orbit ratio</figcaption></figure> While most <a href="/wiki/Extrasolar_planet" class="mw-redirect" title="Extrasolar planet">extrasolar planetary</a> systems discovered have not been found to have planets in mean-motion resonances, chains of up to five resonant planets<a href="#cite_note-Shale2017-37">[36]</a> and up to seven at least near resonant planets<a href="#cite_note-Luger_etal_2017-38">[37]</a> have been uncovered. Simulations have shown that during <a href="/wiki/Nebular_hypothesis" title="Nebular hypothesis">planetary system formation</a>, the appearance of resonant chains of planetary embryos is favored by the presence of the <a href="/wiki/Protoplanetary_disk" title="Protoplanetary disk">primordial gas disc</a>. Once that gas dissipates, 90–95% of those chains must then become unstable to match the low frequency of resonant chains observed.<a href="#cite_note-Izidoro2017-39">[38]</a> <ul><li>As mentioned above, <a href="/wiki/Gliese_876" title="Gliese 876">Gliese 876</a> e, b and c are in a Laplace resonance, with a 4:2:1 ratio of periods (124.3, 61.1 and 30.0 days).<a href="#cite_note-rivera2010-13">[13]</a><a href="#cite_note-Laughlin2013-40">[39]</a><a href="#cite_note-Marcy_2001-41">[40]</a> In this case, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{L}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{L}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5153ef8598101c48c82f65b8f22bc428c98591b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.03ex; height:2.509ex;" alt="{\displaystyle \Phi _{L}}"> librates with an amplitude of 40° ± 13° and the resonance follows the time-averaged relation:<a href="#cite_note-rivera2010-13">[13]</a></li></ul> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{L}=\lambda _{\rm {c}}-3\cdot \lambda _{\rm {d}}+2\cdot \lambda _{\rm {e}}=0^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> <mo>−</mo> <mn>3</mn> <mo>⋅</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mo>+</mo> <mn>2</mn> <mo>⋅</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mo>=</mo> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{L}=\lambda _{\rm {c}}-3\cdot \lambda _{\rm {d}}+2\cdot \lambda _{\rm {e}}=0^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9dae1ddf92cb7de4a5b4527ac9c2287325bf8380" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:29.944ex; height:2.676ex;" alt="{\displaystyle \Phi _{L}=\lambda _{\rm {c}}-3\cdot \lambda _{\rm {d}}+2\cdot \lambda _{\rm {e}}=0^{\circ }}"></dd></dl> <ul><li><a href="/wiki/Kepler-223" title="Kepler-223">Kepler-223</a> has four planets in a resonance with an 8:6:4:3 orbit ratio, and a 3:4:6:8 ratio of periods (7.3845, 9.8456, 14.7887 and 19.7257 days).<a href="#cite_note-EPE-KOI730-42">[41]</a><a href="#cite_note-Beatty-43">[42]</a><a href="#cite_note-Lissauer_2011-44">[43]</a><a href="#cite_note-Mills2016-45">[44]</a> This represents the first confirmed 4-body orbital resonance.<a href="#cite_note-Koppes2016-46">[45]</a> The librations within this system are such that close encounters between two planets occur only when the other planets are in distant parts of their orbits. Simulations indicate that this system of resonances must have formed via <a href="/wiki/Planetary_migration" title="Planetary migration">planetary migration</a>.<a href="#cite_note-Mills2016-45">[44]</a></li> <li><a href="/wiki/Kepler-80" title="Kepler-80">Kepler-80</a> d, e, b, c and g have periods in a ~ 1.000: 1.512: 2.296: 3.100: 4.767 ratio (3.0722, 4.6449, 7.0525, 9.5236 and 14.6456 days). However, in a frame of reference that rotates with the conjunctions, this reduces to a period ratio of 4:6:9:12:18 (an orbit ratio of 9:6:4:3:2). Conjunctions of d and e, e and b, b and c, and c and g occur at relative intervals of 2:3:6:6 (9.07, 13.61 and 27.21 days) in a pattern that repeats about every 190.5 days (seven full cycles in the rotating frame) in the inertial or nonrotating frame (equivalent to a 62:41:27:20:13 orbit ratio resonance in the nonrotating frame, because the conjunctions circulate in the direction opposite orbital motion). Librations of possible three-body resonances have amplitudes of only about 3 degrees, and modeling indicates the resonant system is stable to perturbations. Triple conjunctions do not occur.<a href="#cite_note-MacDonald2016-47">[46]</a><a href="#cite_note-Shale2017-37">[36]</a></li> <li><a href="/wiki/TOI-178" title="TOI-178">TOI-178</a> has 6 confirmed planets, of which the outer 5 planets form a similar resonant chain in a rotating frame of reference, which can be expressed as 2:4:6:9:12 in period ratios, or as 18:9:6:4:3 in orbit ratios. In addition, the innermost planet b with period of 1.91d orbits close to where it would also be part of the same Laplace resonance chain, as a 3:5 resonance with the planet c would be fulfilled at period of ~1.95d, implying that it might have evolved there but pulled out of resonance, possibly by tidal forces.<a href="#cite_note-48">[47]</a></li> <li><a href="/wiki/TRAPPIST-1" title="TRAPPIST-1">TRAPPIST-1</a>'s seven approximately Earth-sized planets are in a chain of near resonances (the longest such chain known), having an orbit ratio of approximately 24, 15, 9, 6, 4, 3 and 2, or nearest-neighbor period ratios (proceeding outward) of about 8/5, 5/3, 3/2, 3/2, 4/3 and 3/2 (1.603, 1.672, 1.506, 1.509, 1.342 and 1.519). They are also configured such that each triple of adjacent planets is in a Laplace resonance (i.e., b, c and d in one such Laplace configuration; c, d and e in another, etc.).<a href="#cite_note-Gillon2016-49">[48]</a><a href="#cite_note-Luger_etal_2017-38">[37]</a> The resonant configuration is expected to be stable on a time scale of billions of years, assuming it arose during planetary migration.<a href="#cite_note-Tamayo2017-50">[49]</a><a href="#cite_note-NYT-20170510-51">[50]</a> A musical interpretation of the resonance has been provided.<a href="#cite_note-NYT-20170510-51">[50]</a></li> <li><a href="/wiki/Kepler-29" title="Kepler-29">Kepler-29</a> has a pair of planets in a 7:9 resonance (ratio of 1/1.28587).<a href="#cite_note-Lissauer_2011-44">[43]</a></li> <li><a href="/wiki/Kepler-36" title="Kepler-36">Kepler-36</a> has a pair of planets close to a 6:7 resonance.<a href="#cite_note-Carter2012-52">[51]</a></li> <li><a href="/wiki/Kepler-37" title="Kepler-37">Kepler-37</a> d, c and b are within one percent of a resonance with an 8:15:24 orbit ratio and a 15:8:5 ratio of periods (39.792187, 21.301886 and 13.367308 days).<a href="#cite_note-BarclayRowe2013-53">[52]</a></li> <li>Of <a href="/wiki/Kepler-90" title="Kepler-90">Kepler-90</a>'s eight known planets, the period ratios b:c, c:i and i:d are close to 4:5, 3:5 and 1:4, respectively (4:4.977, 3:4.97 and 1:4.13) and d, e, f, g and h are close to a 2:3:4:7:11 period ratio (2: 3.078: 4.182: 7.051: 11.102; also 7: 11.021).<a href="#cite_note-54">[53]</a><a href="#cite_note-Shale2017-37">[36]</a> f, g and h are also close to a 3:5:8 period ratio (3: 5.058: 7.964).<a href="#cite_note-Cabrera2013-55">[54]</a> Relevant to systems like this and that of <a href="/wiki/Kepler-36" title="Kepler-36">Kepler-36</a>, calculations suggest that the presence of an outer gas giant planet facilitates the formation of closely packed resonances among inner super-Earths.<a href="#cite_note-Hands2016-56">[55]</a></li> <li><a href="/wiki/HD_41248" title="HD 41248">HD 41248</a> has a pair of <a href="/wiki/Super-Earth" title="Super-Earth">super-Earths</a> within 0.3% of a 5:7 resonance (ratio of 1/1.39718).<a href="#cite_note-Jenkins13-57">[56]</a></li> <li><a href="/wiki/K2-138" title="K2-138">K2-138</a> has 5 confirmed planets in an unbroken near-3:2 resonance chain (with periods of 2.353, 3.560, 5.405, 8.261 and 12.758 days). The system was discovered in the <a href="/wiki/Citizen_science" title="Citizen science">citizen science</a> project Exoplanet Explorers, using K2 data.<a href="#cite_note-58">[57]</a> K2-138 could host <a href="/wiki/Co-orbital_configuration" title="Co-orbital configuration">co-orbital bodies</a> (in a 1:1 mean-motion resonance).<a href="#cite_note-:0-59">[58]</a> Resonant chain systems can stabilize co-orbital bodies<a href="#cite_note-60">[59]</a> and a dedicated analysis of the K2 light curve and radial-velocity from <a href="/wiki/High_Accuracy_Radial_Velocity_Planet_Searcher" title="High Accuracy Radial Velocity Planet Searcher">HARPS</a> might reveal them.<a href="#cite_note-:0-59">[58]</a> Follow-up observations with the <a href="/wiki/Spitzer_Space_Telescope" title="Spitzer Space Telescope">Spitzer Space Telescope</a> suggest a sixth planet continuing the 3:2 resonance chain, while leaving two gaps in the chain (its period is 41.97 days). These gaps could be filled by smaller non-transiting planets.<a href="#cite_note-61">[60]</a><a href="#cite_note-62">[61]</a> Future observations with <a href="/wiki/CHEOPS" title="CHEOPS">CHEOPS</a> will measure <a href="/wiki/Transit-timing_variation" title="Transit-timing variation">transit-timing variations</a> of the system to further analyse the mass of the planets and could potentially find other planetary bodies in the system.<a href="#cite_note-63">[62]</a></li> <li><a href="/wiki/K2-32" title="K2-32">K2-32</a> has four planets in a near 1:2:5:7 resonance (with periods of 4.34, 8.99, 20.66 and 31.71 days). Planet e has a radius almost identical to that of the Earth. The other planets have a size between Neptune and Saturn.<a href="#cite_note-64">[63]</a></li> <li><a href="/wiki/V1298_Tauri" title="V1298 Tauri">V1298 Tauri</a> has four confirmed planets of which planets c, d and b are near a 1:2:3 resonance (with periods of 8.25, 12.40 and 24.14 days). Planet e only shows a single transit in the K2 light curve and has a period larger than 36 days. Planet e might be in a low-order resonance (of 2:3, 3:5, 1:2, or 1:3) with planet b. The system is very young (23±4 <a href="/wiki/Myr" class="mw-redirect" title="Myr">Myr</a>) and might be a precursor of a compact multiplanet system. The 2:3 resonance suggests that some close-in planets may either form in resonances or evolve into them on timescales of less than 10 Myr. The planets in the system have a size between Neptune and Saturn. Only planet b has a size similar to Jupiter.<a href="#cite_note-65">[64]</a></li> <li><a href="/wiki/HD_158259" title="HD 158259">HD 158259</a> contains four planets in a 3:2 near resonance chain (with periods of 3.432, 5.198, 7.954 and 12.03 days, or period ratios of 1.51, 1.53 and 1.51, respectively), with a possible fifth planet also near a 3:2 resonance (with a period of 17.4 days). The exoplanets were found with the <a href="/wiki/SOPHIE_%C3%A9chelle_spectrograph" title="SOPHIE échelle spectrograph">SOPHIE échelle spectrograph</a>, using the <a href="/wiki/Doppler_spectroscopy" title="Doppler spectroscopy">radial velocity</a> method.<a href="#cite_note-66">[65]</a></li> <li><a href="/wiki/Kepler-1649" title="Kepler-1649">Kepler-1649</a> contains two Earth-size planets close to a 9:4 resonance (with periods of 19.53527 and 8.689099 days, or a period ratio of 2.24825), including one (<a href="/wiki/Kepler-1649c" title="Kepler-1649c">"c"</a>) in the habitable zone. An undetected planet with a 13.0-day period would create a 3:2 resonance chain.<a href="#cite_note-Vanderburg2020-67">[66]</a></li> <li><a href="/wiki/Kepler-88" title="Kepler-88">Kepler-88</a> has a pair of inner planets close to a 1:2 resonance (period ratio of 2.0396), with a mass ratio of ~22.5, producing very large <a href="/wiki/Transit_timing_variation" class="mw-redirect" title="Transit timing variation">transit timing variations</a> of ~0.5 days for the innermost planet. There is a yet more massive outer planet in a ~1400 day orbit.<a href="#cite_note-Weiss2020-68">[67]</a></li> <li><a href="/wiki/HD_110067" title="HD 110067">HD 110067</a> has six known planets, in a 54:36:24:16:12:9 resonance ratio.<a href="#cite_note-69">[68]</a></li></ul> Cases of extrasolar planets close to a 1:2 mean-motion resonance are fairly common. Sixteen percent of systems found by the <a href="/wiki/Transit_method" class="mw-redirect" title="Transit method">transit method</a> are reported to have an example of this (with period ratios in the range 1.83–2.18),<a href="#cite_note-Lissauer_2011-44">[43]</a> as well as one sixth of planetary systems characterized by <a href="/wiki/Doppler_spectroscopy" title="Doppler spectroscopy">Doppler spectroscopy</a> (with in this case a narrower period ratio range).<a href="#cite_note-Wright_2011-70">[69]</a> Due to incomplete knowledge of the systems, the actual proportions are likely to be higher.<a href="#cite_note-Lissauer_2011-44">[43]</a> Overall, about a third of radial velocity characterized systems appear to have a pair of planets close to a <a href="/wiki/Commensurability_(astronomy)" title="Commensurability (astronomy)">commensurability</a>.<a href="#cite_note-Lissauer_2011-44">[43]</a><a href="#cite_note-Wright_2011-70">[69]</a> It is much more common for pairs of planets to have orbital period ratios a few percent larger than a mean-motion resonance ratio than a few percent smaller (particularly in the case of first order resonances, in which the integers in the ratio differ by one).<a href="#cite_note-Lissauer_2011-44">[43]</a> This was predicted to be true in cases where <a href="/wiki/Tidal_acceleration" title="Tidal acceleration">tidal interactions</a> with the star are significant.<a href="#cite_note-Terquem_2007-71">[70]</a> <div class="mw-heading mw-heading2"><h2 id="Coincidental_'near'_ratios_of_mean_motion">Coincidental 'near' ratios of mean motion</h2>[<a href="/w/index.php?title=Orbital_resonance&action=edit&section=8" title="Edit section: Coincidental 'near' ratios of mean motion">edit</a>]</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:PallasJupiter.GIF" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5b/PallasJupiter.GIF/300px-PallasJupiter.GIF" decoding="async" width="300" height="261" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5b/PallasJupiter.GIF/450px-PallasJupiter.GIF 1.5x, //upload.wikimedia.org/wikipedia/commons/5/5b/PallasJupiter.GIF 2x" data-file-width="590" data-file-height="514" /></a><figcaption>Depiction of asteroid <a href="/wiki/2_Pallas" title="2 Pallas">Pallas'</a> 18:7 near resonance with Jupiter in a rotating frame (click for animation). Jupiter (pink loop at upper left) is held nearly stationary. The shift in Pallas' orbital alignment relative to Jupiter increases steadily over time; it never reverses course (i.e., there is no libration).</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Venus_pentagram.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Venus_pentagram.png/300px-Venus_pentagram.png" decoding="async" width="300" height="279" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Venus_pentagram.png/450px-Venus_pentagram.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Venus_pentagram.png/600px-Venus_pentagram.png 2x" data-file-width="853" data-file-height="792" /></a><figcaption>Depiction of the <a href="/wiki/Earth" title="Earth">Earth</a>:<a href="/wiki/Venus" title="Venus">Venus</a> 8:13 near resonance. With Earth held stationary at the center of a nonrotating frame, the successive <a href="/wiki/Inferior_conjunction" class="mw-redirect" title="Inferior conjunction">inferior conjunctions</a> of Venus over eight Earth years trace a <a href="/wiki/Pentagram" title="Pentagram">pentagrammic</a> pattern (reflecting the difference between the numbers in the ratio).</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Moons_of_Pluto.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/Moons_of_Pluto.png/300px-Moons_of_Pluto.png" decoding="async" width="300" height="240" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/Moons_of_Pluto.png/450px-Moons_of_Pluto.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/69/Moons_of_Pluto.png/600px-Moons_of_Pluto.png 2x" data-file-width="2750" data-file-height="2200" /></a><figcaption>Diagram of the orbits of <a href="/wiki/Pluto" title="Pluto">Pluto</a>'s small outer four moons, which follow a 3:4:5:6 sequence of near resonances relative to the period of its large inner satellite <a href="/wiki/Charon_(moon)" title="Charon (moon)">Charon</a>. The moons Styx, Nix and Hydra are also involved in a true <a class="mw-selflink-fragment" href="#Laplace_resonance">3-body resonance</a>.</figcaption></figure> A number of near-<a href="/wiki/Integer" title="Integer">integer</a>-ratio relationships between the orbital frequencies of the planets or major moons are sometimes pointed out (see list below). However, these have no dynamical significance because there is no appropriate precession of <a href="/wiki/Perihelion" class="mw-redirect" title="Perihelion">perihelion</a> or other libration to make the resonance perfect (see the detailed discussion in the <a class="mw-selflink-fragment" href="#Mean-motion_resonances_in_the_Solar_System">section above</a>). Such near resonances are dynamically insignificant even if the mismatch is quite small because (unlike a true resonance), after each cycle the relative position of the bodies shifts. When averaged over astronomically short timescales, their relative position is random, just like bodies that are nowhere near resonance. For example, consider the orbits of Earth and Venus, which arrive at almost the same configuration after 8 Earth orbits and 13 Venus orbits. The actual ratio is 0.61518624, which is only 0.032% away from exactly 8:13. The mismatch after 8 years is only 1.5° of Venus' orbital movement. Still, this is enough that Venus and Earth find themselves in the opposite relative orientation to the original every 120 such cycles, which is 960 years. Therefore, on timescales of thousands of years or more (still tiny by astronomical standards), their relative position is effectively random. The presence of a near resonance may reflect that a perfect resonance existed in the past, or that the system is evolving towards one in the future. Some orbital frequency coincidences include: <table class="wikitable" style="vertical-align:center;text-align:center;"> <caption>Table of some orbital frequency coincidences in the Solar system </caption> <tbody><tr> <th>Ratio </th> <th>Bodies </th> <th>Mismatch after one cycle<a href="#cite_note-72">[a]</a> </th> <th>Randmztn. time<a href="#cite_note-73">[b]</a> </th> <th>Probability<a href="#cite_note-74">[c]</a><a href="#cite_note-75">[d]</a> </th></tr> <tr> <th colspan="5">Trans-planetary resonances </th></tr> <tr> <td>9:23</td> <td><a href="/wiki/Venus" title="Venus">Venus</a>–<a href="/wiki/Mercury_(planet)" title="Mercury (planet)">Mercury</a></td> <td>4.0°</td> <td>200 <a href="/wiki/Year" title="Year">y</a></td> <td>19% </td></tr> <tr> <td>1:4 </td> <td>Earth-Mercury </td> <td>54.8° </td> <td>3 y </td> <td>0.3% </td></tr> <tr> <td>8:13</td> <td><a href="/wiki/Earth" title="Earth">Earth</a>–<a href="/wiki/Venus" title="Venus">Venus</a><a href="#cite_note-Langford-76">[71]</a><a href="#cite_note-Bazsó-77">[72]</a><a href="#cite_note-79">[e]</a> </td> <td>1.5°</td> <td>1000 <a href="/wiki/Year" title="Year">y</a></td> <td>6.5% </td></tr> <tr> <td>243:395</td> <td><a href="/wiki/Earth" title="Earth">Earth</a>–<a href="/wiki/Venus" title="Venus">Venus</a><a href="#cite_note-Langford-76">[71]</a><a href="#cite_note-Shortt-78">[73]</a></td> <td>0.8°</td> <td>50,000 <a href="/wiki/Year" title="Year">y</a></td> <td>68% </td></tr> <tr> <td>1:3</td> <td><a href="/wiki/Mars" title="Mars">Mars</a>–<a href="/wiki/Venus" title="Venus">Venus</a></td> <td>20.6°</td> <td>20 y</td> <td>11% </td></tr> <tr> <td>1:2</td> <td><a href="/wiki/Mars" title="Mars">Mars</a>–<a href="/wiki/Earth" title="Earth">Earth</a></td> <td>42.9°</td> <td>8 y</td> <td>24% </td></tr> <tr> <td>193:363 </td> <td>Mars-Earth </td> <td>0.9° </td> <td>70,000 y </td> <td>0.6% </td></tr> <tr> <td>1:12</td> <td><a href="/wiki/Jupiter" title="Jupiter">Jupiter</a>–<a href="/wiki/Earth" title="Earth">Earth</a><a href="#cite_note-80">[f]</a> </td> <td>49.1°</td> <td>40 y</td> <td>28% </td></tr> <tr> <td>3:19 </td> <td>Jupiter-Mars </td> <td>28.7° </td> <td>200 y </td> <td>0.4% </td></tr> <tr> <td>2:5</td> <td><a href="/wiki/Saturn" title="Saturn">Saturn</a>–<a href="/wiki/Jupiter" title="Jupiter">Jupiter</a><a href="#cite_note-81">[g]</a> </td> <td>12.8°</td> <td>800 y</td> <td>13% </td></tr> <tr> <td>1:7</td> <td><a href="/wiki/Uranus" title="Uranus">Uranus</a>–<a href="/wiki/Jupiter" title="Jupiter">Jupiter</a></td> <td>31.1°</td> <td>500 y</td> <td>18% </td></tr> <tr> <td>7:20</td> <td><a href="/wiki/Uranus" title="Uranus">Uranus</a>–<a href="/wiki/Saturn" title="Saturn">Saturn</a></td> <td>5.7°</td> <td>20,000 y</td> <td>20% </td></tr> <tr> <td>5:28</td> <td><a href="/wiki/Neptune" title="Neptune">Neptune</a>–<a href="/wiki/Saturn" title="Saturn">Saturn</a></td> <td>1.9°</td> <td>80,000 y</td> <td>5.2% </td></tr> <tr> <td>1:2</td> <td><a href="/wiki/Neptune" title="Neptune">Neptune</a>–<a href="/wiki/Uranus" title="Uranus">Uranus</a></td> <td>14.0°</td> <td>2000 y</td> <td>7.8% </td></tr> <tr> <th colspan="5">Mars' satellite system </th></tr> <tr> <td>1:4</td> <td><a href="/wiki/Deimos_(moon)" title="Deimos (moon)">Deimos</a>–<a href="/wiki/Phobos_(moon)" title="Phobos (moon)">Phobos</a><a href="#cite_note-83">[h]</a> </td> <td>14.9°</td> <td>0.04 y</td> <td>8.3% </td></tr> <tr> <th colspan="5">Major asteroids' resonances </th></tr> <tr> <td>1:1</td> <td><a href="/wiki/2_Pallas" title="2 Pallas">Pallas</a>–<a href="/wiki/Ceres_(dwarf_planet)" title="Ceres (dwarf planet)">Ceres</a><a href="#cite_note-Goffin2001-84">[75]</a><a href="#cite_note-Kovacevic-85">[76]</a> </td> <td>0.7°</td> <td>1000 y</td> <td>0.39%<a href="#cite_note-86">[i]</a> </td></tr> <tr> <td>7:18</td> <td><a href="/wiki/Jupiter" title="Jupiter">Jupiter</a>–<a href="/wiki/2_Pallas" title="2 Pallas">Pallas</a><a href="#cite_note-Taylor1982-87">[77]</a> </td> <td>0.10°</td> <td>100,000 y</td> <td>0.4%<a href="#cite_note-88">[j]</a> </td></tr> <tr> <th colspan="5"><a href="/wiki/87_Sylvia" title="87 Sylvia">87 Sylvia</a>'s satellite system<a href="#cite_note-89">[k]</a> </th></tr> <tr> <td>17:45</td> <td><a href="/wiki/Romulus_(moon)" title="Romulus (moon)">Romulus</a>–<a href="/wiki/Remus_(moon)" title="Remus (moon)">Remus</a></td> <td>0.7°</td> <td>40 y</td> <td>6.7% </td></tr> <tr> <th colspan="5">Jupiter's satellite system </th></tr> <tr> <td>1:6</td> <td><a href="/wiki/Io_(moon)" title="Io (moon)">Io</a>–<a href="/wiki/Metis_(moon)" title="Metis (moon)">Metis</a></td> <td>0.6°</td> <td>2 y</td> <td>0.31% </td></tr> <tr> <td>3:5</td> <td><a href="/wiki/Amalthea_(moon)" title="Amalthea (moon)">Amalthea</a>–<a href="/wiki/Adrastea_(moon)" title="Adrastea (moon)">Adrastea</a></td> <td>3.9°</td> <td>0.2 y</td> <td>6.4% </td></tr> <tr> <td>3:7</td> <td><a href="/wiki/Callisto_(moon)" title="Callisto (moon)">Callisto</a>–<a href="/wiki/Ganymede_(moon)" title="Ganymede (moon)">Ganymede</a><a href="#cite_note-Goldreich_1965-90">[78]</a> </td> <td>0.7°</td> <td>30 y</td> <td>1.2% </td></tr> <tr> <th colspan="5">Saturn's satellite system </th></tr> <tr> <td>2:3</td> <td><a href="/wiki/Enceladus" title="Enceladus">Enceladus</a>–<a href="/wiki/Mimas_(moon)" class="mw-redirect" title="Mimas (moon)">Mimas</a></td> <td>33.2°</td> <td>0.04 y</td> <td>33% </td></tr> <tr> <td>2:3</td> <td><a href="/wiki/Dione_(moon)" title="Dione (moon)">Dione</a>–<a href="/wiki/Tethys_(moon)" title="Tethys (moon)">Tethys</a><a href="#cite_note-92">[l]</a> </td> <td>36.2°</td> <td>0.07 y</td> <td>36% </td></tr> <tr> <td>3:5</td> <td><a href="/wiki/Rhea_(moon)" title="Rhea (moon)">Rhea</a>–<a href="/wiki/Dione_(moon)" title="Dione (moon)">Dione</a></td> <td>17.1°</td> <td>0.4 y</td> <td>26% </td></tr> <tr> <td>2:7</td> <td><a href="/wiki/Titan_(moon)" title="Titan (moon)">Titan</a>–<a href="/wiki/Rhea_(moon)" title="Rhea (moon)">Rhea</a></td> <td>21.0°</td> <td>0.7 y</td> <td>22% </td></tr> <tr> <td>1:5</td> <td><a href="/wiki/Iapetus_(moon)" title="Iapetus (moon)">Iapetus</a>–<a href="/wiki/Titan_(moon)" title="Titan (moon)">Titan</a></td> <td>9.2°</td> <td>4 y</td> <td>5.1% </td></tr> <tr> <th colspan="5">Major <a href="/wiki/Centaur_(minor_planet)" class="mw-redirect" title="Centaur (minor planet)">centaurs</a>' resonances<a href="#cite_note-93">[m]</a> </th></tr> <tr> <td>3:4</td> <td><a href="/wiki/Uranus" title="Uranus">Uranus</a>–<a href="/wiki/10199_Chariklo" title="10199 Chariklo">Chariklo</a></td> <td>4.5°</td> <td>10,000 y</td> <td>7.3% </td></tr> <tr> <th colspan="5">Uranus' satellite system </th></tr> <tr> <td>3:5</td> <td><a href="/wiki/Rosalind_(moon)" title="Rosalind (moon)">Rosalind</a>–<a href="/wiki/Cordelia_(moon)" title="Cordelia (moon)">Cordelia</a><a href="#cite_note-Murray_1990-94">[80]</a> </td> <td>0.22°</td> <td>4 y</td> <td>0.37% </td></tr> <tr> <td>1:3</td> <td><a href="/wiki/Umbriel" title="Umbriel">Umbriel</a>–<a href="/wiki/Miranda_(moon)" title="Miranda (moon)">Miranda</a><a href="#cite_note-96">[n]</a> </td> <td>24.5°</td> <td>0.08 y</td> <td>14% </td></tr> <tr> <td>3:5</td> <td><a href="/wiki/Umbriel" title="Umbriel">Umbriel</a>–<a href="/wiki/Ariel_(moon)" title="Ariel (moon)">Ariel</a><a href="#cite_note-98">[o]</a> </td> <td>24.2°</td> <td>0.3 y</td> <td>35% </td></tr> <tr> <td>1:2</td> <td><a href="/wiki/Titania_(moon)" title="Titania (moon)">Titania</a>–<a href="/wiki/Umbriel" title="Umbriel">Umbriel</a></td> <td>36.3°</td> <td>0.1 y</td> <td>20% </td></tr> <tr> <td>2:3</td> <td><a href="/wiki/Oberon_(moon)" title="Oberon (moon)">Oberon</a>–<a href="/wiki/Titania_(moon)" title="Titania (moon)">Titania</a></td> <td>33.4°</td> <td>0.4 y</td> <td>34% </td></tr> <tr> <th colspan="5">Neptune's satellite system </th></tr> <tr> <td>1:20</td> <td><a href="/wiki/Triton_(moon)" title="Triton (moon)">Triton</a>–<a href="/wiki/Naiad_(moon)" title="Naiad (moon)">Naiad</a></td> <td>13.5°</td> <td>0.2 y</td> <td>7.5% </td></tr> <tr> <td>1:2</td> <td><a href="/wiki/Proteus_(moon)" title="Proteus (moon)">Proteus</a>–<a href="/wiki/Larissa_(moon)" title="Larissa (moon)">Larissa</a><a href="#cite_note-ZhangHamilton2007-99">[83]</a><a href="#cite_note-ZhangHamilton2008-100">[84]</a> </td> <td>8.4°</td> <td>0.07 y</td> <td>4.7% </td></tr> <tr> <td>5:6</td> <td><a href="/wiki/Proteus_(moon)" title="Proteus (moon)">Proteus</a>–<a href="/wiki/Hippocamp_(moon)" title="Hippocamp (moon)">Hippocamp</a></td> <td>2.1°</td> <td>1 y</td> <td>5.7% </td></tr> <tr> <th colspan="5">Pluto's satellite system </th></tr> <tr> <td>1:3</td> <td><a href="/wiki/Styx_(moon)" title="Styx (moon)">Styx</a>–<a href="/wiki/Charon_(moon)" title="Charon (moon)">Charon</a><a href="#cite_note-Matson-101">[85]</a> </td> <td>58.5°</td> <td>0.2 y</td> <td>33% </td></tr> <tr> <td>1:4</td> <td><a href="/wiki/Nix_(moon)" title="Nix (moon)">Nix</a>–<a href="/wiki/Charon_(moon)" title="Charon (moon)">Charon</a><a href="#cite_note-Matson-101">[85]</a><a href="#cite_note-WardCanup2006-102">[86]</a> </td> <td>39.1°</td> <td>0.3 y</td> <td>22% </td></tr> <tr> <td>1:5</td> <td><a href="/wiki/Kerberos_(moon)" title="Kerberos (moon)">Kerberos</a>–<a href="/wiki/Charon_(moon)" title="Charon (moon)">Charon</a><a href="#cite_note-Matson-101">[85]</a> </td> <td>9.2°</td> <td>2 y</td> <td>5% </td></tr> <tr> <td>1:6</td> <td><a href="/wiki/Hydra_(moon)" title="Hydra (moon)">Hydra</a>–<a href="/wiki/Charon_(moon)" title="Charon (moon)">Charon</a><a href="#cite_note-Matson-101">[85]</a><a href="#cite_note-WardCanup2006-102">[86]</a> </td> <td>6.6°</td> <td>3 y</td> <td>3.7% </td></tr> <tr> <th colspan="5">Haumea's satellite system </th></tr> <tr> <td>3:8</td> <td><a href="/wiki/Hi%CA%BBiaka_(moon)" title="Hiʻiaka (moon)">Hiʻiaka</a>–<a href="/wiki/Namaka_(moon)" title="Namaka (moon)">Namaka</a><a href="#cite_note-104">[p]</a> </td> <td>42.5°</td> <td>2 y</td> <td>55% </td></tr></tbody></table> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-72"><a href="#cite_ref-72">^</a> Mismatch in orbital longitude of the inner body, as compared to its position at the beginning of the cycle (with the cycle defined as n orbits of the outer body – see below). Circular orbits are assumed (i.e., precession is ignored). </li> <li id="cite_note-73"><a href="#cite_ref-73">^</a> The randomization time is the amount of time needed for the mismatch from the initial relative longitudinal orbital positions of the bodies to grow to 180°. The listed number is rounded to the nearest first <a href="/wiki/Significant_digit" class="mw-redirect" title="Significant digit">significant digit</a>. </li> <li id="cite_note-74"><a href="#cite_ref-74">^</a> Estimated <a href="/wiki/Probability" title="Probability">probability</a> of obtaining by chance an orbital coincidence of equal or smaller mismatch, at least once in n attempts, where n is the integer number of orbits of the outer body per cycle, and the mismatch is assumed to randomly vary between 0° and 180°. The value is calculated as 1 − ( 1 − <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>⁠ mismatch / 180° ⁠ ) n . This is a crude calculation that only attempts to give a rough idea of relative probabilities. </li> <li id="cite_note-75"><a href="#cite_ref-75">^</a> Smaller is better: The smaller the probability of an apparently resonant relationship arising as a mere chance alignment of random numbers, the more credible the proposal that gravitational interaction causes persistence of the relationship, or prolongs it / delays its ultimate dissolution by other, disruptive perturbations. </li> <li id="cite_note-79"><a href="#cite_ref-79">^</a> The two near <a href="/wiki/Commensurability_(astronomy)" title="Commensurability (astronomy)">commensurabilities</a> listed for Earth and Venus are reflected in the timing of <a href="/wiki/Transit_of_Venus" title="Transit of Venus">transits of Venus</a>, which occur in pairs 8 years apart, in a cycle that repeats every 243 years.<a href="#cite_note-Langford-76">[71]</a><a href="#cite_note-Shortt-78">[73]</a> </li> <li id="cite_note-80"><a href="#cite_ref-80">^</a> The near 1:12 resonance between Jupiter and Earth has the coincidental side-effect of making the <a href="/wiki/Alinda_family" class="mw-redirect" title="Alinda family">Alinda asteroids</a>, which occupy (or are close to) the 3:1 resonance with Jupiter, to be close to a 1:4 resonance with Earth. </li> <li id="cite_note-81"><a href="#cite_ref-81">^</a> The long-known near resonance between Jupiter and Saturn has traditionally been called the <a href="/wiki/Great_Inequality" class="mw-redirect" title="Great Inequality">Great Inequality</a>. It was first described by <a href="/wiki/Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Laplace</a> in a series of papers published 1784–1789. </li> <li id="cite_note-83"><a href="#cite_ref-83">^</a> Resonances with a now-vanished inner moon are likely to have been involved in the formation of Phobos and Deimos.<a href="#cite_note-Rosenblatt2016-82">[74]</a> </li> <li id="cite_note-86"><a href="#cite_ref-86">^</a> Based on the <a href="/wiki/Proper_orbital_elements" title="Proper orbital elements">proper orbital periods</a>, 1684.869 and 1681.601 days, for Pallas and Ceres, respectively. </li> <li id="cite_note-88"><a href="#cite_ref-88">^</a> Based on the <a href="/wiki/Proper_orbital_elements" title="Proper orbital elements">"proper" orbital period</a> of Pallas, 1684.869 days, and 4332.59 days for Jupiter. </li> <li id="cite_note-89"><a href="#cite_ref-89">^</a> <a href="/wiki/87_Sylvia" title="87 Sylvia">87 Sylvia</a> is the first asteroid discovered to have more than one moon. </li> <li id="cite_note-92"><a href="#cite_ref-92">^</a> This resonance may have been occupied in the past.<a href="#cite_note-Chen2008-91">[79]</a> </li> <li id="cite_note-93"><a href="#cite_ref-93">^</a> Some <a href="/wiki/Centaur_(minor_planet)#Classification" class="mw-redirect" title="Centaur (minor planet)">definitions of centaurs</a> require that they not be resonant. </li> <li id="cite_note-96"><a href="#cite_ref-96">^</a> This resonance may have been occupied in the past.<a href="#cite_note-Tittemore_Wisdom_1990-95">[81]</a> </li> <li id="cite_note-98"><a href="#cite_ref-98">^</a> This resonance may have been occupied in the past.<a href="#cite_note-Tittemore1988-97">[82]</a> </li> <li id="cite_note-104"><a href="#cite_ref-104">^</a> The results for the Haumea system aren't very meaningful because, contrary to the assumptions implicit in the calculations, Namaka has an eccentric, <a href="/wiki/Osculating_orbit" title="Osculating orbit">non-Keplerian</a> orbit that precesses rapidly (see below). Hiʻiaka and Namaka are much closer to a 3:8 resonance than indicated, and may actually be in it.<a href="#cite_note-Ragozzine&Brown2009-103">[87]</a> </li> </ol></div></div> The least probable orbital correlation in the list – meaning the relationship that seems most likely to have not just be by random chance – is that between Io and Metis, followed by those between Rosalind and Cordelia, Pallas and Ceres, Jupiter and Pallas, Callisto and Ganymede, and Hydra and Charon, respectively. <div class="mw-heading mw-heading2"><h2 id="Possible_past_mean-motion_resonances">Possible past mean-motion resonances</h2>[<a href="/w/index.php?title=Orbital_resonance&action=edit&section=9" title="Edit section: Possible past mean-motion resonances">edit</a>]</div> A past resonance between Jupiter and Saturn may have played a dramatic role in early Solar System history. A 2004 <a href="/wiki/Nice_model" title="Nice model">computer model</a> by <a href="/wiki/Alessandro_Morbidelli_(astronomer)" title="Alessandro Morbidelli (astronomer)">Alessandro Morbidelli</a> of the <a href="/wiki/C%C3%B4te_d%27Azur_Observatory" title="Côte d'Azur Observatory">Observatoire de la Côte d'Azur</a> in <a href="/wiki/Nice" title="Nice">Nice</a> suggested the formation of a 1:2 resonance between Jupiter and Saturn due to interactions with <a href="/wiki/Planetesimal" title="Planetesimal">planetesimals</a> that caused them to migrate inward and outward, respectively. In the model, this created a gravitational push that propelled both Uranus and Neptune into higher orbits, and in some scenarios caused them to switch places, which would have doubled Neptune's distance from the Sun. The resultant expulsion of objects from the proto-Kuiper belt as Neptune moved outwards could explain the <a href="/wiki/Late_Heavy_Bombardment" title="Late Heavy Bombardment">Late Heavy Bombardment</a> 600 million years after the Solar System's formation and the origin of Jupiter's <a href="/wiki/Trojan_asteroid" class="mw-redirect" title="Trojan asteroid">Trojan asteroids</a>.<a href="#cite_note-105">[88]</a> An outward migration of Neptune could also explain the current occupancy of some of its resonances (particularly the 2:5 resonance) within the Kuiper belt. While Saturn's mid-sized moons Dione and Tethys are not close to an exact resonance now, they may have been in a 2:3 resonance early in the Solar System's history. This would have led to orbital eccentricity and <a href="/wiki/Tidal_heating" title="Tidal heating">tidal heating</a> that may have warmed Tethys' interior enough to form a subsurface ocean. Subsequent freezing of the ocean after the moons escaped from the resonance may have generated the extensional stresses that created the enormous <a href="/wiki/Graben" title="Graben">graben</a> system of <a href="/wiki/Ithaca_Chasma" title="Ithaca Chasma">Ithaca Chasma</a> on Tethys.<a href="#cite_note-Chen2008-91">[79]</a> The satellite system of Uranus is notably different from those of Jupiter and Saturn in that it lacks precise resonances among the larger moons, while the majority of the larger moons of Jupiter (3 of the 4 largest) and of Saturn (6 of the 8 largest) are in mean-motion resonances. In all three satellite systems, moons were likely captured into mean-motion resonances in the past as their orbits shifted due to <a href="/wiki/Tidal_acceleration" title="Tidal acceleration">tidal dissipation</a>, a process by which satellites gain orbital energy at the expense of the primary's rotational energy, affecting inner moons disproportionately. In the Uranian system, however, due to the planet's lesser degree of <a href="/wiki/Oblate_spheroid" class="mw-redirect" title="Oblate spheroid">oblateness</a>, and the larger relative size of its satellites, escape from a mean-motion resonance is much easier. Lower oblateness of the primary alters its gravitational field in such a way that different possible resonances are spaced more closely together. A larger relative satellite size increases the strength of their interactions. Both factors lead to more chaotic orbital behavior at or near mean-motion resonances. Escape from a resonance may be associated with capture into a secondary resonance, and/or tidal evolution-driven increases in <a href="/wiki/Orbital_eccentricity" title="Orbital eccentricity">orbital eccentricity</a> or <a href="/wiki/Inclination" class="mw-redirect" title="Inclination">inclination</a>. Mean-motion resonances that probably once existed in the Uranus System include (3:5) Ariel-Miranda, (1:3) Umbriel-Miranda, (3:5) Umbriel-Ariel, and (1:4) Titania-Ariel.<a href="#cite_note-Tittemore1988-97">[82]</a><a href="#cite_note-Tittemore_Wisdom_1990-95">[81]</a> Evidence for such past resonances includes the relatively high eccentricities of the orbits of Uranus' inner satellites, and the anomalously high orbital inclination of Miranda. High past orbital eccentricities associated with the (1:3) Umbriel-Miranda and (1:4) Titania-Ariel resonances may have led to tidal heating of the interiors of Miranda and Ariel,<a href="#cite_note-Tittemore_1990-106">[89]</a> respectively. Miranda probably escaped from its resonance with Umbriel via a secondary resonance, and the mechanism of this escape is believed to explain why its orbital inclination is more than 10 times those of the other <a href="/wiki/Regular_moon" title="Regular moon">regular</a> Uranian moons (see <a href="/wiki/Uranus%27_natural_satellites" class="mw-redirect" title="Uranus' natural satellites">Uranus' natural satellites</a>).<a href="#cite_note-Tittemore1989-107">[90]</a><a href="#cite_note-108">[91]</a> Similar to the case of Miranda, the present inclinations of Jupiter's moonlets Amalthea and <a href="/wiki/Thebe_(moon)" title="Thebe (moon)">Thebe</a> are thought to be indications of past passage through the 3:1 and 4:2 resonances with Io, respectively.<a href="#cite_note-Burns2004-109">[92]</a> Neptune's regular moons Proteus and Larissa are thought to have passed through a 1:2 resonance a few hundred million years ago; the moons have drifted away from each other since then because Proteus is outside a <a href="/wiki/Synchronous_orbit" title="Synchronous orbit">synchronous orbit</a> and Larissa is within one. Passage through the resonance is thought to have excited both moons' eccentricities to a degree that has not since been entirely damped out.<a href="#cite_note-ZhangHamilton2007-99">[83]</a><a href="#cite_note-ZhangHamilton2008-100">[84]</a> In the case of <a href="/wiki/Pluto" title="Pluto">Pluto</a>'s satellites, it has been proposed that the present near resonances are relics of a previous precise resonance that was disrupted by tidal damping of the eccentricity of Charon's orbit (see <a href="/wiki/Pluto%27s_natural_satellites" class="mw-redirect" title="Pluto's natural satellites">Pluto's natural satellites</a> for details). The near resonances may be maintained by a 15% local fluctuation in the Pluto-Charon gravitational field. Thus, these near resonances may not be coincidental. The smaller inner moon of the <a href="/wiki/Dwarf_planet" title="Dwarf planet">dwarf planet</a> <a href="/wiki/Haumea_(dwarf_planet)" class="mw-redirect" title="Haumea (dwarf planet)">Haumea</a>, <a href="/wiki/Namaka_(moon)" title="Namaka (moon)">Namaka</a>, is one tenth the mass of the larger outer moon, <a href="/wiki/Hi%CA%BBiaka_(moon)" title="Hiʻiaka (moon)">Hiʻiaka</a>. Namaka revolves around Haumea in 18 days in an eccentric, <a href="/wiki/Osculating_orbit" title="Osculating orbit">non-Keplerian</a> orbit, and as of 2008 is inclined 13° from Hiʻiaka.<a href="#cite_note-Ragozzine&Brown2009-103">[87]</a> Over the timescale of the system, it should have been tidally damped into a more circular orbit. It appears that it has been disturbed by resonances with the more massive Hiʻiaka, due to converging orbits as it moved outward from Haumea because of tidal dissipation. The moons may have been caught in and then escaped from orbital resonance several times. They probably passed through the 3:1 resonance relatively recently, and currently are in or at least close to an 8:3 resonance. Namaka's orbit is strongly <a href="/wiki/Perturbation_(astronomy)" title="Perturbation (astronomy)">perturbed</a>, with a current precession of about −6.5° per year.<a href="#cite_note-Ragozzine&Brown2009-103">[87]</a> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Orbital_resonance&action=edit&section=10" title="Edit section: See also">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 20em;"> <ul><li><a href="/wiki/1685_Toro" title="1685 Toro">1685 Toro</a>, an asteroid in 5:8 resonance with the Earth</li> <li><a href="/wiki/3753_Cruithne" title="3753 Cruithne">3753 Cruithne</a>, an asteroid in 1:1 resonance with the Earth</li> <li><a href="/wiki/Arnold_tongue" title="Arnold tongue">Arnold tongue</a></li> <li><a href="/wiki/Commensurability_(astronomy)" title="Commensurability (astronomy)">Commensurability (astronomy)</a></li> <li><a href="/wiki/Dermott%27s_law" title="Dermott's law">Dermott's law</a></li> <li><a href="/wiki/Horseshoe_orbit" title="Horseshoe orbit">Horseshoe orbit</a>, followed by an object in another type of 1:1 resonance</li> <li><a href="/wiki/Kozai_mechanism" title="Kozai mechanism">Kozai resonance</a></li> <li><a href="/wiki/Lagrange_point" title="Lagrange point">Lagrange points</a></li> <li><a href="/wiki/Mercury_(planet)#Spin-orbit_resonance" title="Mercury (planet)">Mercury</a>, which has a 3:2 spin-orbit resonance</li> <li><a href="/wiki/Musica_universalis" title="Musica universalis">Musica universalis</a> ("music of the spheres")</li> <li><a href="/wiki/Resonant_interaction" title="Resonant interaction">Resonant interaction</a></li> <li><a href="/wiki/Resonant_trans-Neptunian_object" title="Resonant trans-Neptunian object">Resonant trans-Neptunian object</a></li> <li><a href="/wiki/Tidal_locking" title="Tidal locking">Tidal locking</a></li> <li><a href="/wiki/Tidal_resonance" title="Tidal resonance">Tidal resonance</a></li> <li><a href="/wiki/Titius%E2%80%93Bode_law" title="Titius–Bode law">Titius–Bode law</a></li> <li><a href="/wiki/Transfer_operator" title="Transfer operator">Transfer operator</a></li> <li><a href="/wiki/Trojan_(celestial_body)" title="Trojan (celestial body)">Trojan (celestial body)</a>, a body in a type of 1:1 resonance</li> <li><a href="/wiki/Venus#Orbit_and_rotation" title="Venus">Venus</a>, whose Earth conjunction period (584 Earth days) is close to 5 times its <a href="/wiki/Solar_day" class="mw-redirect" title="Solar day">solar day</a> (116.75 days)</li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Orbital_resonance&action=edit&section=11" title="Edit section: Notes">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-36"><a href="#cite_ref-36">^</a> The nature of this resonance (ignoring subtleties like libration and precession) can be crudely obtained from the orbital periods as follows. From Showalter et al., 2019,<a href="#cite_note-Showalter2019-35">[35]</a> the periods of Naiad (Pn) and Thalassa (Pt) are 0.294396 and 0.311484 days, respectively. From these, the period between conjunctions can be calculated as 5.366 days (1/[1/Pn – 1/Pt]), which is 18.23 (≈ 18.25) orbits of Naiad and 17.23 (≈ 17.25) orbits of Thalassa. Thus, after four conjunction periods, 73 orbits of Naiad and 69 orbits of Thalassa have elapsed, and the original configuration will be restored. </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Orbital_resonance&action=edit&section=12" title="Edit section: References">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> <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 class="citation news cs1"><a rel="nofollow" class="external text" href="http://www.iau.org/static/resolutions/Resolution_GA26-5-6.pdf">"IAU 2006 General Assembly: Resolutions 5 and 6"</a> (PDF). 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The Astronomical Journal. 110: 420. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/astro-ph/9504036">astro-ph/9504036</a>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1995AJ....110..420M">1995AJ....110..420M</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1086%2F117532">10.1086/117532</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:10622344">10622344</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Astronomical+Journal&rft.atitle=The+Origin+of+Pluto%27s+Orbit%3A+Implications+for+the+Solar+System+Beyond+Neptune&rft.volume=110&rft.pages=420&rft.date=1995&rft_id=info%3Aarxiv%2Fastro-ph%2F9504036&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A10622344%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1086%2F117532&rft_id=info%3Abibcode%2F1995AJ....110..420M&rft.aulast=Malhotra&rft.aufirst=Renu&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrbital+resonance" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLemaître2010" class="citation book cs1"><a href="/wiki/Anne_Lema%C3%AEtre" title="Anne Lemaître">Lemaître, A.</a> (2010). <a rel="nofollow" class="external text" href="https://cds.cern.ch/record/1339552">"Resonances: Models and Captures"</a>. 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Vol. 790. <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer</a>. pp. 1–62. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-04458-8">10.1007/978-3-642-04458-8</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-04457-1" title="Special:BookSources/978-3-642-04457-1"><bdi>978-3-642-04457-1</bdi></a>.</cite><span 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