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id="toc-Effects-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Formulation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Formulation"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Formulation</span> </div> </a> <ul id="toc-Formulation-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"> <span class="vector-toc-numb">5</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" 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for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Tidal force</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 44 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-44" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">44 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Gezeitenkraft" title="Gezeitenkraft – Alemannic" lang="gsw" hreflang="gsw" data-title="Gezeitenkraft" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%82%D9%88%D8%A9_%D8%A7%D9%84%D9%85%D8%AF_%D9%88%D8%A7%D9%84%D8%AC%D8%B2%D8%B1" title="قوة المد والجزر – Arabic" lang="ar" hreflang="ar" data-title="قوة المد والجزر" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9F%D1%80%D0%B8%D0%BB%D0%B8%D0%B2%D0%BD%D0%B0_%D1%81%D0%B8%D0%BB%D0%B0" title="Приливна сила – Bulgarian" lang="bg" hreflang="bg" data-title="Приливна сила" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/For%C3%A7a_de_marea" title="Força de marea – Catalan" lang="ca" hreflang="ca" data-title="Força de marea" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A2%D1%83%D0%BB%C4%83%D0%BC%D0%BB%D0%B0_%D0%B2%C4%83%D0%B9%D1%81%D0%B5%D0%BC" title="Тулăмла вăйсем – Chuvash" lang="cv" hreflang="cv" data-title="Тулăмла вăйсем" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Slapov%C3%A1_s%C3%ADla" title="Slapová síla – Czech" lang="cs" hreflang="cs" data-title="Slapová síla" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Tidevandskraft" title="Tidevandskraft – Danish" lang="da" hreflang="da" data-title="Tidevandskraft" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Gezeitenkraft" title="Gezeitenkraft – German" lang="de" hreflang="de" data-title="Gezeitenkraft" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A0%CE%B1%CE%BB%CE%B9%CF%81%CF%81%CE%BF%CF%8A%CE%BA%CE%AD%CF%82_%CE%B4%CF%85%CE%BD%CE%AC%CE%BC%CE%B5%CE%B9%CF%82" title="Παλιρροϊκές δυνάμεις – Greek" lang="el" hreflang="el" data-title="Παλιρροϊκές δυνάμεις" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Fuerza_de_marea" title="Fuerza de marea – Spanish" lang="es" hreflang="es" data-title="Fuerza de marea" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Tajda_forto" title="Tajda forto – Esperanto" lang="eo" hreflang="eo" data-title="Tajda forto" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Itsasaldiaren_indar" title="Itsasaldiaren indar – Basque" lang="eu" hreflang="eu" data-title="Itsasaldiaren indar" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%86%DB%8C%D8%B1%D9%88%DB%8C_%DA%A9%D8%B4%D9%86%D8%AF%DB%8C" title="نیروی کشندی – Persian" lang="fa" hreflang="fa" data-title="نیروی کشندی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Force_de_mar%C3%A9e" title="Force de marée – French" lang="fr" hreflang="fr" data-title="Force de marée" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A1%B0%EC%84%9D%EB%A0%A5" title="조석력 – Korean" lang="ko" hreflang="ko" data-title="조석력" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%9C%E0%A5%8D%E0%A4%B5%E0%A4%BE%E0%A4%B0%E0%A4%AD%E0%A4%BE%E0%A4%9F%E0%A4%BE_%E0%A4%AC%E0%A4%B2" title="ज्वारभाटा बल – Hindi" lang="hi" hreflang="hi" data-title="ज्वारभाटा बल" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Plimna_sila" title="Plimna sila – Croatian" lang="hr" hreflang="hr" data-title="Plimna sila" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-ig mw-list-item"><a href="https://ig.wikipedia.org/wiki/Ike_ebili_mmiri" title="Ike ebili mmiri – Igbo" lang="ig" hreflang="ig" data-title="Ike ebili mmiri" data-language-autonym="Igbo" data-language-local-name="Igbo" class="interlanguage-link-target"><span>Igbo</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Forza_di_marea" title="Forza di marea – Italian" lang="it" hreflang="it" data-title="Forza di marea" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9B%D7%95%D7%97_%D7%92%D7%90%D7%95%D7%AA" title="כוח גאות – Hebrew" lang="he" hreflang="he" data-title="כוח גאות" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Vis_aestualis" title="Vis aestualis – Latin" lang="la" hreflang="la" data-title="Vis aestualis" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Potvynin%C4%97s_j%C4%97gos" title="Potvyninės jėgos – Lithuanian" lang="lt" hreflang="lt" data-title="Potvyninės jėgos" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9F%D0%BB%D0%B8%D0%BC%D0%BD%D0%B0_%D1%81%D0%B8%D0%BB%D0%B0" title="Плимна сила – Macedonian" lang="mk" hreflang="mk" data-title="Плимна сила" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Daya_pasang_surut" title="Daya pasang surut – Malay" lang="ms" hreflang="ms" data-title="Daya pasang surut" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Getijdenveld" title="Getijdenveld – Dutch" lang="nl" hreflang="nl" data-title="Getijdenveld" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%BD%AE%E6%B1%90%E5%8A%9B" title="潮汐力 – Japanese" lang="ja" hreflang="ja" data-title="潮汐力" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Tidekraft" title="Tidekraft – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Tidekraft" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Tidvasskraft" title="Tidvasskraft – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Tidvasskraft" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Si%C5%82a_p%C5%82ywowa" title="Siła pływowa – Polish" lang="pl" hreflang="pl" data-title="Siła pływowa" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/For%C3%A7a_de_mar%C3%A9" title="Força de maré – Portuguese" lang="pt" hreflang="pt" data-title="Força de maré" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/For%C8%9B%C4%83_mareic%C4%83" title="Forță mareică – Romanian" lang="ro" hreflang="ro" data-title="Forță mareică" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D1%80%D0%B8%D0%BB%D0%B8%D0%B2%D0%BD%D1%8B%D0%B5_%D1%81%D0%B8%D0%BB%D1%8B" title="Приливные силы – Russian" lang="ru" hreflang="ru" data-title="Приливные силы" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Tidal_force" title="Tidal force – Simple English" lang="en-simple" hreflang="en-simple" data-title="Tidal force" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Slapy_(astrofyzika)" title="Slapy (astrofyzika) – Slovak" lang="sk" hreflang="sk" data-title="Slapy (astrofyzika)" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Plimska_sila" title="Plimska sila – Slovenian" lang="sl" hreflang="sl" data-title="Plimska sila" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Plimska_sila" title="Plimska sila – Serbian" lang="sr" hreflang="sr" data-title="Plimska sila" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Plimna_sila" title="Plimna sila – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Plimna sila" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Vuorovesivoima" title="Vuorovesivoima – Finnish" lang="fi" hreflang="fi" data-title="Vuorovesivoima" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Tidvattenkrafter" title="Tidvattenkrafter – Swedish" lang="sv" hreflang="sv" data-title="Tidvattenkrafter" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%81%E0%B8%A3%E0%B8%87%E0%B8%99%E0%B9%89%E0%B8%B3%E0%B8%82%E0%B8%B6%E0%B9%89%E0%B8%99%E0%B8%A5%E0%B8%87" title="แรงน้ำขึ้นลง – Thai" lang="th" hreflang="th" data-title="แรงน้ำขึ้นลง" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Gelgit_kuvveti" title="Gelgit kuvveti – Turkish" lang="tr" hreflang="tr" data-title="Gelgit kuvveti" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D1%80%D0%B8%D0%BF%D0%BB%D0%B8%D0%B2%D0%BD%D1%96_%D1%81%D0%B8%D0%BB%D0%B8" title="Припливні сили – Ukrainian" lang="uk" hreflang="uk" data-title="Припливні сили" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/L%E1%BB%B1c_th%E1%BB%A7y_tri%E1%BB%81u" title="Lực thủy triều – Vietnamese" lang="vi" hreflang="vi" data-title="Lực thủy triều" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%BD%AE%E6%B1%90%E5%8A%9B" title="潮汐力 – Chinese" lang="zh" hreflang="zh" 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id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">(Redirected from <a href="/w/index.php?title=Tidal_Force&redirect=no" class="mw-redirect" title="Tidal Force">Tidal Force</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">A gravitational effect also known as the differential force and the perturbing force</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Arp_282.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Arp_282.png/300px-Arp_282.png" decoding="async" width="300" height="219" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Arp_282.png/450px-Arp_282.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/33/Arp_282.png/600px-Arp_282.png 2x" data-file-width="3453" data-file-height="2517" /></a><figcaption>Figure 1: Tidal interaction between the <a href="/wiki/Spiral_galaxy" title="Spiral galaxy">spiral galaxy</a> <a href="/wiki/NGC_169" title="NGC 169">NGC 169</a> and a smaller companion<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></figcaption></figure> <p>The <b>tidal force</b> or <b>tide-generating force</b> is a <a href="/wiki/Gravitational" class="mw-redirect" title="Gravitational">gravitational</a> effect that stretches a body along the line towards and away from the <a href="/wiki/Center_of_mass" title="Center of mass">center of mass</a> of another body due to spatial variations in strength in <a href="/wiki/Gravitational_field" title="Gravitational field">gravitational field</a> from the other body. It is responsible for the <a href="/wiki/Tide" title="Tide">tides</a> and related phenomena, including <a href="/wiki/Solid-earth_tide" class="mw-redirect" title="Solid-earth tide">solid-earth tides</a>, <a href="/wiki/Tidal_locking" title="Tidal locking">tidal locking</a>, breaking apart of celestial bodies and formation of <a href="/wiki/Ring_system" title="Ring system">ring systems</a> within the <a href="/wiki/Roche_limit" title="Roche limit">Roche limit</a>, and in extreme cases, <a href="/wiki/Spaghettification" title="Spaghettification">spaghettification</a> of objects. It arises because the gravitational field exerted on one body by another is not constant across its parts: the nearer side is attracted more strongly than the farther side. The difference is positive in the near side and negative in the far side, which causes a body to get stretched. Thus, the tidal force is also known as the differential force, residual force, or secondary effect of the gravitational field. </p><p>In <a href="/wiki/Celestial_mechanics" title="Celestial mechanics">celestial mechanics</a>, the expression <i>tidal force</i> can refer to a situation in which a body or material (for example, tidal water) is mainly under the gravitational influence of a second body (for example, the Earth), but is also perturbed by the gravitational effects of a third body (for example, the Moon). The perturbing force is sometimes in such cases called a tidal force<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> (for example, the <a href="/wiki/Lunar_theory#Newton" title="Lunar theory">perturbing force on the Moon</a>): it is the difference between the force exerted by the third body on the second and the force exerted by the third body on the first.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>Tidal forces have also been shown to be fundamentally related to <a href="/wiki/Gravitational_wave" title="Gravitational wave">gravitational waves</a>.<sup id="cite_ref-arXiv_2019_p440_4-0" class="reference"><a href="#cite_note-arXiv_2019_p440-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Explanation">Explanation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tidal_force&action=edit&section=1" title="Edit section: Explanation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Tidal_field_and_gravity_field.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Tidal_field_and_gravity_field.svg/290px-Tidal_field_and_gravity_field.svg.png" decoding="async" width="290" height="171" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Tidal_field_and_gravity_field.svg/435px-Tidal_field_and_gravity_field.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Tidal_field_and_gravity_field.svg/580px-Tidal_field_and_gravity_field.svg.png 2x" data-file-width="985" data-file-height="581" /></a><figcaption>Figure 2: Shown in red, the Moon's gravity <i>residual</i> field at the surface of the Earth is known (along with another and weaker differential effect due to the Sun) as the <i>tide generating force</i>. This is the primary mechanism driving tidal action, explaining two simultaneous tidal bulges.<br /><br /> Earth's rotation accounts further for the occurrence of two high tides per day on the same location. In this figure, the Earth is the central black circle while the Moon is far off to the right. It shows both the tidal field (thick red arrows) and the gravity field (thin blue arrows) exerted on Earth's surface and center (label O) by the Moon (label S). The <i>outward</i> direction of the arrows on the right and left of the Earth indicates that where the Moon is at <a href="/wiki/Zenith" title="Zenith">zenith</a> or at <a href="/wiki/Nadir" title="Nadir">nadir</a>.</figcaption></figure> <p>When a body (body 1) is acted on by the gravity of another body (body 2), the field can vary significantly on body 1 between the side of the body facing body 2 and the side facing away from body 2. Figure 2 shows the differential force of gravity on a spherical body (body 1) exerted by another body (body 2). </p><p>These <i>tidal forces</i> cause strains on both bodies and may distort them or even, in extreme cases, break one or the other apart.<sup id="cite_ref-Penrose_5-0" class="reference"><a href="#cite_note-Penrose-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/Roche_limit" title="Roche limit">Roche limit</a> is the distance from a planet at which tidal effects would cause an object to disintegrate because the differential force of gravity from the planet overcomes the attraction of the parts of the object for one another.<sup id="cite_ref-Blanc_6-0" class="reference"><a href="#cite_note-Blanc-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> These strains would not occur if the gravitational field were uniform, because a uniform <a href="/wiki/Field_(physics)" title="Field (physics)">field</a> only causes the entire body to accelerate together in the same direction and at the same rate. </p> <div class="mw-heading mw-heading2"><h2 id="Size_and_distance">Size and distance</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tidal_force&action=edit&section=2" title="Edit section: Size and distance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The relationship of an astronomical body's size, to its distance from another body, strongly influences the magnitude of tidal force.<sup id="cite_ref-Tyson_7-0" class="reference"><a href="#cite_note-Tyson-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> The tidal force acting on an astronomical body, such as the Earth, is directly proportional to the diameter of the Earth and inversely proportional to the cube of the distance from another body producing a gravitational attraction, such as the Moon or the Sun. Tidal action on bath tubs, swimming pools, lakes, and other small bodies of water is negligible.<sup id="cite_ref-Sawicki1999_8-0" class="reference"><a href="#cite_note-Sawicki1999-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Inverse_x_squaired.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Inverse_x_squaired.png/220px-Inverse_x_squaired.png" decoding="async" width="220" height="136" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Inverse_x_squaired.png/330px-Inverse_x_squaired.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Inverse_x_squaired.png/440px-Inverse_x_squaired.png 2x" data-file-width="850" data-file-height="525" /></a><figcaption>Figure 3: Graph showing how gravitational attraction drops off with increasing distance from a body</figcaption></figure> <p>Figure 3 is a graph showing how gravitational force declines with distance. In this graph, the attractive force decreases in proportion to the square of the distance (<span class="nowrap"><i>Y</i> = 1/<i>X</i><sup>2</sup></span>), while the slope (<span class="nowrap"><i>Y</i><span class="nowrap" style="padding-left:0.15em;">′</span> = −2/<i>X</i><sup>3</sup></span>) is inversely proportional to the cube of the distance. </p><p>The tidal force corresponds to the difference in Y between two points on the graph, with one point on the near side of the body, and the other point on the far side. The tidal force becomes larger, when the two points are either farther apart, or when they are more to the left on the graph, meaning closer to the attracting body. </p><p>For example, even though the Sun has a stronger overall gravitational pull on Earth, the Moon creates a larger tidal bulge because the Moon is closer. This difference is due to the way gravity weakens with distance: the Moon's closer proximity creates a steeper decline in its gravitational pull as you move across Earth (compared to the Sun's very gradual decline from its vast distance). This steeper gradient in the Moon's pull results in a larger difference in force between the near and far sides of Earth, which is what creates the bigger tidal bulge. </p><p>Gravitational attraction is inversely proportional to the square of the distance from the source. The attraction will be stronger on the side of a body facing the source, and weaker on the side away from the source. The tidal force is proportional to the difference.<sup id="cite_ref-Sawicki1999_8-1" class="reference"><a href="#cite_note-Sawicki1999-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Sun,_Earth,_and_Moon"><span id="Sun.2C_Earth.2C_and_Moon"></span>Sun, Earth, and Moon</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tidal_force&action=edit&section=3" title="Edit section: Sun, Earth, and Moon"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Earth is 81 times more massive than the Moon, the Earth has roughly 4 times the Moon's radius. As a result, at the same distance, the tidal force of the Earth at the surface of the Moon is about 20 times stronger than that of the Moon at the Earth's surface.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <table class="wikitable"> <tbody><tr> <th colspan="2">Gravitational body causing tidal force</th> <th colspan="3">Body subjected to tidal force</th> <th>Tidal acceleration </th></tr> <tr> <th>Body</th> <th>Mass (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span>)</th> <th>Body</th> <th>Radius (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>)</th> <th>Distance (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span>)</th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Gm~{\frac {2r}{d^{3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mi>m</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>r</mi> </mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Gm~{\frac {2r}{d^{3}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98df85688892043e3ea0388d2a284a059492392c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:7.556ex; height:5.509ex;" alt="{\displaystyle Gm~{\frac {2r}{d^{3}}}}"></span> </th></tr> <tr> <td style="text-align: center"><b>Sun</b></td> <td style="text-align: center"><span class="nowrap"><span data-sort-value="7030199000000000000♠"></span>1.99<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>30</sup> kg</span></td> <td><b>Earth</b></td> <td style="text-align: center"><span class="nowrap"><span data-sort-value="7006637000000000000♠"></span>6.37<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>6</sup> m</span></td> <td style="text-align: center"><span class="nowrap"><span data-sort-value="7011150000000000000♠"></span>1.50<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>11</sup> m</span></td> <td style="text-align: center"><span class="nowrap"><span data-sort-value="6993504999999999999♠"></span>5.05<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>−7</sup> m⋅s<sup>−2</sup></span> </td></tr> <tr> <td style="text-align: center"><b>Moon</b></td> <td style="text-align: center"><span class="nowrap"><span data-sort-value="7022734000000000000♠"></span>7.34<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>22</sup> kg</span></td> <td><b>Earth</b></td> <td style="text-align: center"><span class="nowrap"><span data-sort-value="7006637000000000000♠"></span>6.37<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>6</sup> m</span></td> <td style="text-align: center"><span class="nowrap"><span data-sort-value="7008384000000000000♠"></span>3.84<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>8</sup> m</span></td> <td style="text-align: center"><span class="nowrap"><span data-sort-value="6994110000000000000♠"></span>1.10<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>−6</sup> m⋅s<sup>−2</sup></span> </td></tr> <tr> <td style="text-align: center"><b>Earth</b></td> <td style="text-align: center"><span class="nowrap"><span data-sort-value="7024597000000000000♠"></span>5.97<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>24</sup> kg</span></td> <td><b>Moon</b></td> <td style="text-align: center"><span class="nowrap"><span data-sort-value="7006174000000000000♠"></span>1.74<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>6</sup> m</span></td> <td style="text-align: center"><span class="nowrap"><span data-sort-value="7008384000000000000♠"></span>3.84<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>8</sup> m</span></td> <td style="text-align: center"><span class="nowrap"><span data-sort-value="6995244000000000000♠"></span>2.44<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>−5</sup> m⋅s<sup>−2</sup></span> </td></tr> <tr> <td colspan="6" style="background: #ffffff; font-size: small; font-weight: normal;"><i>G</i> is the <a href="/wiki/Gravitational_constant" title="Gravitational constant">gravitational constant</a> = <span class="nowrap"><span data-sort-value="6989667400000000000♠"></span>6.674<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>−11</sup> m<sup>3</sup>⋅kg<sup>−1</sup>⋅s<sup>−2</sup></span><span style="visibility:hidden; color:transparent; padding-left:2px">‍</span><sup id="cite_ref-physconst-G_10-0" class="reference"><a href="#cite_note-physconst-G-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Effects">Effects</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tidal_force&action=edit&section=4" title="Edit section: Effects"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Saturn_PIA06077.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/25/Saturn_PIA06077.jpg/220px-Saturn_PIA06077.jpg" decoding="async" width="220" height="164" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/25/Saturn_PIA06077.jpg/330px-Saturn_PIA06077.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/25/Saturn_PIA06077.jpg/440px-Saturn_PIA06077.jpg 2x" data-file-width="1824" data-file-height="1360" /></a><figcaption>Figure 4: <a href="/wiki/Saturn" title="Saturn">Saturn</a>'s rings are inside the orbits of its principal moons. Tidal forces oppose gravitational coalescence of the material in the rings to form moons.<sup id="cite_ref-MacKay_11-0" class="reference"><a href="#cite_note-MacKay-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup></figcaption></figure> <p>In the case of an infinitesimally small elastic sphere, the effect of a tidal force is to distort the shape of the body without any change in volume. The sphere becomes an <a href="/wiki/Ellipsoid" title="Ellipsoid">ellipsoid</a> with two bulges, pointing towards and away from the other body. Larger objects distort into an <a href="/wiki/Ovoid" class="mw-redirect" title="Ovoid">ovoid</a>, and are slightly compressed, which is what happens to the Earth's oceans under the action of the Moon. All parts of the Earth are subject to the Moon's gravitational forces, causing the water in the oceans to redistribute, forming bulges on the sides near the Moon and far from the Moon.<sup id="cite_ref-Americana_12-0" class="reference"><a href="#cite_note-Americana-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>When a body rotates while subject to tidal forces, internal friction results in the gradual dissipation of its rotational kinetic energy as heat. In the case for the Earth, and Earth's Moon, the loss of rotational kinetic energy results in a gain of about 2 milliseconds per century. If the body is close enough to its primary, this can result in a rotation which is <a href="/wiki/Tidal_locking" title="Tidal locking">tidally locked</a> to the orbital motion, as in the case of the Earth's moon. <a href="/wiki/Tidal_heating" title="Tidal heating">Tidal heating</a> produces dramatic volcanic effects on Jupiter's moon <a href="/wiki/Io_(moon)" title="Io (moon)">Io</a>. <span class="anchor" id="Stress"></span><a href="/wiki/Stress_(mechanics)" title="Stress (mechanics)">Stresses</a> caused by tidal forces also cause a regular monthly pattern of <a href="/wiki/Moonquake" class="mw-redirect" title="Moonquake">moonquakes</a> on Earth's Moon.<sup id="cite_ref-Tyson_7-1" class="reference"><a href="#cite_note-Tyson-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>Tidal forces contribute to ocean currents, which moderate global temperatures by transporting heat energy toward the poles. It has been suggested that variations in tidal forces correlate with cool periods in the global temperature record at 6- to 10-year intervals,<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> and that <a href="/wiki/Beat_(acoustics)" title="Beat (acoustics)">harmonic beat</a> variations in tidal forcing may contribute to millennial climate changes. No strong link to millennial climate changes has been found to date.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Shoemaker-levy-tidal-forces.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Shoemaker-levy-tidal-forces.jpg/280px-Shoemaker-levy-tidal-forces.jpg" decoding="async" width="280" height="109" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Shoemaker-levy-tidal-forces.jpg/420px-Shoemaker-levy-tidal-forces.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/71/Shoemaker-levy-tidal-forces.jpg/560px-Shoemaker-levy-tidal-forces.jpg 2x" data-file-width="1548" data-file-height="600" /></a><figcaption>Figure 5: <a href="/wiki/Comet_Shoemaker-Levy_9" class="mw-redirect" title="Comet Shoemaker-Levy 9">Comet Shoemaker-Levy 9</a> in 1994 after breaking up under the influence of <a href="/wiki/Jupiter" title="Jupiter">Jupiter</a>'s tidal forces during a previous pass in 1992.</figcaption></figure> <p>Tidal effects become particularly pronounced near small bodies of high mass, such as <a href="/wiki/Neutron_star" title="Neutron star">neutron stars</a> or <a href="/wiki/Black_hole" title="Black hole">black holes</a>, where they are responsible for the "<a href="/wiki/Spaghettification" title="Spaghettification">spaghettification</a>" of infalling matter. Tidal forces create the oceanic <a href="/wiki/Tide" title="Tide">tide</a> of <a href="/wiki/Earth" title="Earth">Earth</a>'s oceans, where the attracting bodies are the <a href="/wiki/Moon" title="Moon">Moon</a> and, to a lesser extent, the <a href="/wiki/Sun" title="Sun">Sun</a>. Tidal forces are also responsible for <a href="/wiki/Tidal_locking" title="Tidal locking">tidal locking</a>, <a href="/wiki/Tidal_acceleration" title="Tidal acceleration">tidal acceleration</a>, and tidal heating. <a href="/wiki/Tidal_triggering_of_earthquakes" title="Tidal triggering of earthquakes">Tides may also induce seismicity</a>. </p><p>By generating conducting fluids within the interior of the Earth, tidal forces also affect the <a href="/wiki/Earth%27s_magnetic_field" title="Earth's magnetic field">Earth's magnetic field</a>.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><span><video id="mwe_player_0" poster="//upload.wikimedia.org/wikipedia/commons/thumb/7/7f/Supermassive_black_hole_rips_star_apart_%28simulation%29.webm/220px--Supermassive_black_hole_rips_star_apart_%28simulation%29.webm.jpg" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="220" height="124" data-durationhint="51" data-mwtitle="Supermassive_black_hole_rips_star_apart_(simulation).webm" data-mwprovider="wikimediacommons" resource="/wiki/File:Supermassive_black_hole_rips_star_apart_(simulation).webm"><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/7/7f/Supermassive_black_hole_rips_star_apart_%28simulation%29.webm/Supermassive_black_hole_rips_star_apart_%28simulation%29.webm.480p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="480p.vp9.webm" data-width="854" data-height="480" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/7/7f/Supermassive_black_hole_rips_star_apart_%28simulation%29.webm/Supermassive_black_hole_rips_star_apart_%28simulation%29.webm.720p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="720p.vp9.webm" data-width="1280" data-height="720" /><source src="//upload.wikimedia.org/wikipedia/commons/7/7f/Supermassive_black_hole_rips_star_apart_%28simulation%29.webm" type="video/webm; codecs="vp8, opus"" data-width="1920" data-height="1080" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/7/7f/Supermassive_black_hole_rips_star_apart_%28simulation%29.webm/Supermassive_black_hole_rips_star_apart_%28simulation%29.webm.1080p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="1080p.vp9.webm" data-width="1920" data-height="1080" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/7/7f/Supermassive_black_hole_rips_star_apart_%28simulation%29.webm/Supermassive_black_hole_rips_star_apart_%28simulation%29.webm.240p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="240p.vp9.webm" data-width="426" data-height="240" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/7/7f/Supermassive_black_hole_rips_star_apart_%28simulation%29.webm/Supermassive_black_hole_rips_star_apart_%28simulation%29.webm.360p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="360p.vp9.webm" data-width="640" data-height="360" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/7/7f/Supermassive_black_hole_rips_star_apart_%28simulation%29.webm/Supermassive_black_hole_rips_star_apart_%28simulation%29.webm.360p.webm" type="video/webm; codecs="vp8, vorbis"" data-transcodekey="360p.webm" data-width="640" data-height="360" /></video></span><figcaption>Figure 6: This simulation shows a <a href="/wiki/Star" title="Star">star</a> getting torn apart by the gravitational tides of a <a href="/wiki/Supermassive_black_hole" title="Supermassive black hole">supermassive black hole</a>.</figcaption></figure> <div class="mw-heading mw-heading2"><h2 id="Formulation">Formulation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tidal_force&action=edit&section=5" title="Edit section: Formulation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Inseparable_galactic_twins.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Inseparable_galactic_twins.jpg/220px-Inseparable_galactic_twins.jpg" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Inseparable_galactic_twins.jpg/330px-Inseparable_galactic_twins.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/19/Inseparable_galactic_twins.jpg/440px-Inseparable_galactic_twins.jpg 2x" data-file-width="2797" data-file-height="2797" /></a><figcaption>Figure 7: Tidal force is responsible for the merge of galactic pair <a href="/wiki/MRK_1034" class="mw-redirect" title="MRK 1034">MRK 1034</a>.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Tidal-forces.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/56/Tidal-forces.svg/220px-Tidal-forces.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/56/Tidal-forces.svg/330px-Tidal-forces.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/56/Tidal-forces.svg/440px-Tidal-forces.svg.png 2x" data-file-width="500" data-file-height="500" /></a><figcaption>Figure 8: Graphic of tidal forces. The top picture shows the gravity field of a body to the right (not shown); the lower shows their residual gravity once the field at the centre of the sphere is subtracted; this is the tidal force. For visualization purposes, the top arrows may be assumed as equal to 1 N, 2 N, and 3 N (from left to right); the resulting bottom arrows would equal, respectively, -1 N (negative, thus 180-degree rotated), 0 N (invisible), and 1 N. See Figure 2 for a more detailed version</figcaption></figure> <p>For a given (externally generated) gravitational field, the <b>tidal acceleration</b> at a point with respect to a body is obtained by <a href="/wiki/Euclidean_vector#Addition_and_subtraction" title="Euclidean vector">vector subtraction</a> of the gravitational acceleration at the center of the body (due to the given externally generated field) from the gravitational acceleration (due to the same field) at the given point. Correspondingly, the term <i>tidal force</i> is used to describe the forces due to tidal acceleration. Note that for these purposes the only gravitational field considered is the external one; the gravitational field of the body (as shown in the graphic) is not relevant. (In other words, the comparison is with the conditions at the given point as they would be if there were no externally generated field acting unequally at the given point and at the center of the reference body. The externally generated field is usually that produced by a perturbing third body, often the Sun or the Moon in the frequent example-cases of points on or above the Earth's surface in a geocentric reference frame.) </p><p>Tidal acceleration does not require rotation or orbiting bodies; for example, the body may be <a href="/wiki/Freefall" class="mw-redirect" title="Freefall">freefalling</a> in a straight line under the influence of a gravitational field while still being influenced by (changing) tidal acceleration. </p><p>By <a href="/wiki/Newton%27s_law_of_universal_gravitation" title="Newton's law of universal gravitation">Newton's law of universal gravitation</a> and laws of motion, a body of mass <i>m</i> at distance <i>R</i> from the center of a sphere of mass <i>M</i> feels a force <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}_{g}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {F}}_{g}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bc28ca0912bb273b4b3cc527f0ecd5989c64b6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.792ex; height:3.509ex;" alt="{\displaystyle {\vec {F}}_{g}}"></span>, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}_{g}=-{\hat {r}}~G~{\frac {Mm}{R^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mtext> </mtext> <mi>G</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>M</mi> <mi>m</mi> </mrow> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {F}}_{g}=-{\hat {r}}~G~{\frac {Mm}{R^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2eef05cce2be6d8cceb9e7d3882e53305eb11320" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:17.297ex; height:5.509ex;" alt="{\displaystyle {\vec {F}}_{g}=-{\hat {r}}~G~{\frac {Mm}{R^{2}}}}"></span></dd></dl> <p>equivalent to an acceleration <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {a}}_{g}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}_{g}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3375926289f9003ab56dd9da049f42b1a2af31c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.251ex; height:3.009ex;" alt="{\displaystyle {\vec {a}}_{g}}"></span>, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {a}}_{g}=-{\hat {r}}~G~{\frac {M}{R^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mtext> </mtext> <mi>G</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>M</mi> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}_{g}=-{\hat {r}}~G~{\frac {M}{R^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/390b9b018d48e1129f78f7588bebad6fda59cd20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:15.092ex; height:5.509ex;" alt="{\displaystyle {\vec {a}}_{g}=-{\hat {r}}~G~{\frac {M}{R^{2}}}}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1009619964ce33a4a02aaa7cf82adc0fb0a50f23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.292ex; height:2.176ex;" alt="{\displaystyle {\hat {r}}}"></span> is a <a href="/wiki/Unit_vector" title="Unit vector">unit vector</a> pointing from the body <i>M</i> to the body <i>m</i> (here, acceleration from <i>m</i> towards <i>M</i> has negative sign). </p><p>Consider now the acceleration due to the sphere of mass <i>M</i> experienced by a particle in the vicinity of the body of mass <i>m</i>. With <i>R</i> as the distance from the center of <i>M</i> to the center of <i>m</i>, let ∆<i>r</i> be the (relatively small) distance of the particle from the center of the body of mass <i>m</i>. For simplicity, distances are first considered only in the direction pointing towards or away from the sphere of mass <i>M</i>. If the body of mass <i>m</i> is itself a sphere of radius ∆<i>r</i>, then the new particle considered may be located on its surface, at a distance (<i>R</i> ± <i>∆r</i>) from the centre of the sphere of mass <i>M</i>, and <i>∆r</i> may be taken as positive where the particle's distance from <i>M</i> is greater than <i>R</i>. Leaving aside whatever gravitational acceleration may be experienced by the particle towards <i>m</i> on account of <i>m</i><span class="nowrap" style="padding-left:0.1em;">'</span>s own mass, we have the acceleration on the particle due to gravitational force towards <i>M</i> as: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {a}}_{g}=-{\hat {r}}~G~{\frac {M}{(R\pm \Delta r)^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mtext> </mtext> <mi>G</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>M</mi> <mrow> <mo stretchy="false">(</mo> <mi>R</mi> <mo>±</mo> <mi mathvariant="normal">Δ</mi> <mi>r</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}_{g}=-{\hat {r}}~G~{\frac {M}{(R\pm \Delta r)^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47c4be3f43a3f5760dba9f5a641c20b97d634376" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:22.726ex; height:6.009ex;" alt="{\displaystyle {\vec {a}}_{g}=-{\hat {r}}~G~{\frac {M}{(R\pm \Delta r)^{2}}}}"></span></dd></dl> <p>Pulling out the <i>R</i><sup>2</sup> term from the denominator gives: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {a}}_{g}=-{\hat {r}}~G~{\frac {M}{R^{2}}}~{\frac {1}{\left(1\pm {\frac {\Delta r}{R}}\right)^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mtext> </mtext> <mi>G</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>M</mi> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>r</mi> </mrow> <mi>R</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}_{g}=-{\hat {r}}~G~{\frac {M}{R^{2}}}~{\frac {1}{\left(1\pm {\frac {\Delta r}{R}}\right)^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70c1873be814ff9e6d34d4c19c49a79dab3d3f7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:27.288ex; height:8.343ex;" alt="{\displaystyle {\vec {a}}_{g}=-{\hat {r}}~G~{\frac {M}{R^{2}}}~{\frac {1}{\left(1\pm {\frac {\Delta r}{R}}\right)^{2}}}}"></span></dd></dl> <p>The <a href="/wiki/Maclaurin_series" class="mw-redirect" title="Maclaurin series">Maclaurin series</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/(1\pm x)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>±</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/(1\pm x)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9cf3fed770ac2b49b1956f74dae63664920a26d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.521ex; height:3.176ex;" alt="{\displaystyle 1/(1\pm x)^{2}}"></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\mp 2x+3x^{2}\mp \cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>∓</mo> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mn>3</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>∓</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\mp 2x+3x^{2}\mp \cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db4a6e82e81069695666a62a5aa0dfe70f78cb0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.445ex; height:3.009ex;" alt="{\displaystyle 1\mp 2x+3x^{2}\mp \cdots }"></span> which gives a series expansion of: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {a}}_{g}=-{\hat {r}}~G~{\frac {M}{R^{2}}}\pm {\hat {r}}~G~{\frac {2M}{R^{2}}}~{\frac {\Delta r}{R}}+\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mtext> </mtext> <mi>G</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>M</mi> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mtext> </mtext> <mi>G</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>M</mi> </mrow> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>r</mi> </mrow> <mi>R</mi> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}_{g}=-{\hat {r}}~G~{\frac {M}{R^{2}}}\pm {\hat {r}}~G~{\frac {2M}{R^{2}}}~{\frac {\Delta r}{R}}+\cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bc66029667404fa2ed864117f28de9f2e56bee7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:36.618ex; height:5.676ex;" alt="{\displaystyle {\vec {a}}_{g}=-{\hat {r}}~G~{\frac {M}{R^{2}}}\pm {\hat {r}}~G~{\frac {2M}{R^{2}}}~{\frac {\Delta r}{R}}+\cdots }"></span></dd></dl> <p>The first term is the gravitational acceleration due to <i>M</i> at the center of the reference body <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span>, i.e., at the point where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc4f36cb1759b5dbc5003f91e305d51f58205200" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.985ex; height:2.176ex;" alt="{\displaystyle \Delta r}"></span> is zero. This term does not affect the observed acceleration of particles on the surface of <i>m</i> because with respect to <i>M</i>, <i>m</i> (and everything on its surface) is in free fall. When the force on the far particle is subtracted from the force on the near particle, this first term cancels, as do all other even-order terms. The remaining (residual) terms represent the difference mentioned above and are tidal force (acceleration) terms. When ∆<i>r</i> is small compared to <i>R</i>, the terms after the first residual term are very small and can be neglected, giving the approximate tidal acceleration <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {a}}_{t,{\text{axial}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>axial</mtext> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}_{t,{\text{axial}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab9486b805308cf472cbfe467c6951ad310e1d71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.94ex; height:3.009ex;" alt="{\displaystyle {\vec {a}}_{t,{\text{axial}}}}"></span> for the distances ∆<i>r</i> considered, along the axis joining the centers of <i>m</i> and <i>M</i>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {a}}_{t,{\text{axial}}}\approx \pm {\hat {r}}~2\Delta r~G~{\frac {M}{R^{3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>axial</mtext> </mrow> </mrow> </msub> <mo>≈</mo> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mtext> </mtext> <mn>2</mn> <mi mathvariant="normal">Δ</mi> <mi>r</mi> <mtext> </mtext> <mi>G</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>M</mi> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}_{t,{\text{axial}}}\approx \pm {\hat {r}}~2\Delta r~G~{\frac {M}{R^{3}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5612235c9322a5e113bbb3354d426b2dfa66fb54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:23.508ex; height:5.509ex;" alt="{\displaystyle {\vec {a}}_{t,{\text{axial}}}\approx \pm {\hat {r}}~2\Delta r~G~{\frac {M}{R^{3}}}}"></span></dd></dl> <p>When calculated in this way for the case where ∆<i>r</i> is a distance along the axis joining the centers of <i>m</i> and <i>M</i>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {a}}_{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}_{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ee300b5066a01bd0a37a25563cdbf83f4736e8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.056ex; height:2.676ex;" alt="{\displaystyle {\vec {a}}_{t}}"></span> is directed outwards from to the center of <i>m</i> (where ∆<i>r</i> is zero). </p><p>Tidal accelerations can also be calculated away from the axis connecting the bodies <i>m</i> and <i>M</i>, requiring a <a href="/wiki/Euclidean_vector" title="Euclidean vector">vector</a> calculation. In the plane perpendicular to that axis, the tidal acceleration is directed inwards (towards the center where ∆<i>r</i> is zero), and its magnitude is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {1}{2}}\left|{\vec {a}}_{t,{\text{axial}}}\right|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>|</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>axial</mtext> </mrow> </mrow> </msub> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {1}{2}}\left|{\vec {a}}_{t,{\text{axial}}}\right|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4159c168824841fc5caac0c4257106014f883f77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:9.279ex; height:3.509ex;" alt="{\textstyle {\frac {1}{2}}\left|{\vec {a}}_{t,{\text{axial}}}\right|}"></span> in linear approximation as in Figure 2. </p><p>The tidal accelerations at the surfaces of planets in the Solar System are generally very small. For example, the lunar tidal acceleration at the Earth's surface along the Moon–Earth axis is about <span class="nowrap"><span data-sort-value="6993110000000000000♠"></span>1.1<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>−7</sup> <i>g</i></span>, while the solar tidal acceleration at the Earth's surface along the Sun–Earth axis is about <span class="nowrap"><span data-sort-value="6992520000000000000♠"></span>0.52<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>−7</sup> <i>g</i></span>, where <i>g</i> is the <a href="/wiki/Standard_gravity" title="Standard gravity">gravitational acceleration</a> at the Earth's surface. Hence the tide-raising force (acceleration) due to the Sun is about 45% of that due to the Moon.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> The solar tidal acceleration at the Earth's surface was first given by Newton in the <i><a href="/wiki/Philosophi%C3%A6_Naturalis_Principia_Mathematica" title="Philosophiæ Naturalis Principia Mathematica">Principia</a></i>.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tidal_force&action=edit&section=6" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Amphidromic_point" title="Amphidromic point">Amphidromic point</a></li> <li><a href="/wiki/Disrupted_planet" title="Disrupted planet">Disrupted planet</a></li> <li><a href="/wiki/Galactic_tide" title="Galactic tide">Galactic tide</a></li> <li><a href="/wiki/Tidal_resonance" title="Tidal resonance">Tidal resonance</a></li> <li><a href="/wiki/Tidal_stripping" title="Tidal stripping">Tidal stripping</a></li> <li><a href="/wiki/Tidal_tensor" title="Tidal tensor">Tidal tensor</a></li> <li><a href="/wiki/Spacetime_curvature" class="mw-redirect" title="Spacetime curvature">Spacetime curvature</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tidal_force&action=edit&section=7" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.nasa.gov/image-feature/goddard/2022/hubble-views-a-cosmic-interaction">"Hubble Views a Cosmic Interaction"</a>. <i>nasa.gov</i>. NASA. February 11, 2022<span class="reference-accessdate">. Retrieved <span class="nowrap">2022-07-09</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=nasa.gov&rft.atitle=Hubble+Views+a+Cosmic+Interaction&rft.date=2022-02-11&rft_id=https%3A%2F%2Fwww.nasa.gov%2Fimage-feature%2Fgoddard%2F2022%2Fhubble-views-a-cosmic-interaction&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATidal+force" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://adsabs.harvard.edu/full/1977SvAL....3...96A">"On the tidal force"</a>, I. N. Avsiuk, in "Soviet Astronomy Letters", vol. 3 (1977), pp. 96–99.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">See p. 509 in <a rel="nofollow" class="external text" href="https://archive.org/details/astronomyphysica00kutn/page/509">"Astronomy: a physical perspective"</a>, M. L. Kutner (2003).</span> </li> <li id="cite_note-arXiv_2019_p440-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-arXiv_2019_p440_4-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFarXiv2019" class="citation web cs1">arXiv, Emerging Technology from the (2019-12-14). <a rel="nofollow" class="external text" href="https://www.technologyreview.com/2019/12/14/131574/tidal-forces-carry-the-mathematical-signature-of-gravitational-waves/">"Tidal forces carry the mathematical signature of gravitational waves"</a>. <i>MIT Technology Review</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2023-11-12</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MIT+Technology+Review&rft.atitle=Tidal+forces+carry+the+mathematical+signature+of+gravitational+waves&rft.date=2019-12-14&rft.aulast=arXiv&rft.aufirst=Emerging+Technology+from+the&rft_id=https%3A%2F%2Fwww.technologyreview.com%2F2019%2F12%2F14%2F131574%2Ftidal-forces-carry-the-mathematical-signature-of-gravitational-waves%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATidal+force" class="Z3988"></span></span> </li> <li id="cite_note-Penrose-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-Penrose_5-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFR_Penrose1999" class="citation book cs1">R Penrose (1999). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/emperorsnewmindc1999penr"><i>The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics</i></a></span>. <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>. p. <a rel="nofollow" class="external text" href="https://archive.org/details/emperorsnewmindc1999penr/page/264">264</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-286198-6" title="Special:BookSources/978-0-19-286198-6"><bdi>978-0-19-286198-6</bdi></a>. <q>tidal force.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Emperor%27s+New+Mind%3A+Concerning+Computers%2C+Minds%2C+and+the+Laws+of+Physics&rft.pages=264&rft.pub=Oxford+University+Press&rft.date=1999&rft.isbn=978-0-19-286198-6&rft.au=R+Penrose&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Femperorsnewmindc1999penr&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATidal+force" class="Z3988"></span></span> </li> <li id="cite_note-Blanc-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-Blanc_6-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThérèse_EncrenazJ_-P_BibringM_Blanc2003" class="citation book cs1"><a href="/wiki/Th%C3%A9r%C3%A8se_Encrenaz" title="Thérèse Encrenaz">Thérèse Encrenaz</a>; J -P Bibring; M Blanc (2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Je61Y7UbqWgC&q=tidal+force&pg=PA16"><i>The Solar System</i></a>. Springer. p. 16. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-00241-3" title="Special:BookSources/978-3-540-00241-3"><bdi>978-3-540-00241-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Solar+System&rft.pages=16&rft.pub=Springer&rft.date=2003&rft.isbn=978-3-540-00241-3&rft.au=Th%C3%A9r%C3%A8se+Encrenaz&rft.au=J+-P+Bibring&rft.au=M+Blanc&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DJe61Y7UbqWgC%26q%3Dtidal%2Bforce%26pg%3DPA16&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATidal+force" class="Z3988"></span></span> </li> <li id="cite_note-Tyson-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-Tyson_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Tyson_7-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.haydenplanetarium.org/tyson/read/1995/11/01/the-tidal-force">"The Tidal Force | Neil deGrasse Tyson"</a>. <i>www.haydenplanetarium.org</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2016-10-10</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.haydenplanetarium.org&rft.atitle=The+Tidal+Force+%7C+Neil+deGrasse+Tyson&rft_id=http%3A%2F%2Fwww.haydenplanetarium.org%2Ftyson%2Fread%2F1995%2F11%2F01%2Fthe-tidal-force&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATidal+force" class="Z3988"></span></span> </li> <li id="cite_note-Sawicki1999-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-Sawicki1999_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Sawicki1999_8-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSawicki1999" class="citation journal cs1">Sawicki, Mikolaj (1999). "Myths about gravity and tides". <i><a href="/wiki/The_Physics_Teacher" title="The Physics Teacher">The Physics Teacher</a></i>. <b>37</b> (7): 438–441. <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/1999PhTea..37..438S">1999PhTea..37..438S</a>. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.695.8981">10.1.1.695.8981</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1119%2F1.880345">10.1119/1.880345</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0031-921X">0031-921X</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Physics+Teacher&rft.atitle=Myths+about+gravity+and+tides&rft.volume=37&rft.issue=7&rft.pages=438-441&rft.date=1999&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.695.8981%23id-name%3DCiteSeerX&rft.issn=0031-921X&rft_id=info%3Adoi%2F10.1119%2F1.880345&rft_id=info%3Abibcode%2F1999PhTea..37..438S&rft.aulast=Sawicki&rft.aufirst=Mikolaj&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATidal+force" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchutz2003" class="citation book cs1">Schutz, Bernard (2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=P_T0xxhDcsIC"><i>Gravity from the Ground Up: An Introductory Guide to Gravity and General Relativity</i></a> (illustrated ed.). Cambridge University Press. p. 45. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-45506-0" title="Special:BookSources/978-0-521-45506-0"><bdi>978-0-521-45506-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Gravity+from+the+Ground+Up%3A+An+Introductory+Guide+to+Gravity+and+General+Relativity&rft.pages=45&rft.edition=illustrated&rft.pub=Cambridge+University+Press&rft.date=2003&rft.isbn=978-0-521-45506-0&rft.aulast=Schutz&rft.aufirst=Bernard&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DP_T0xxhDcsIC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATidal+force" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=P_T0xxhDcsIC&pg=PA45">Extract of page 45</a></span> </li> <li id="cite_note-physconst-G-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-physconst-G_10-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://physics.nist.gov/cgi-bin/cuu/Value?bg">"2022 CODATA Value: Newtonian constant of gravitation"</a>. <i>The NIST Reference on Constants, Units, and Uncertainty</i>. <a href="/wiki/National_Institute_of_Standards_and_Technology" title="National Institute of Standards and Technology">NIST</a>. May 2024<span class="reference-accessdate">. Retrieved <span class="nowrap">2024-05-18</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+NIST+Reference+on+Constants%2C+Units%2C+and+Uncertainty&rft.atitle=2022+CODATA+Value%3A+Newtonian+constant+of+gravitation&rft.date=2024-05&rft_id=https%3A%2F%2Fphysics.nist.gov%2Fcgi-bin%2Fcuu%2FValue%3Fbg&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATidal+force" class="Z3988"></span></span> </li> <li id="cite_note-MacKay-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-MacKay_11-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFR._S._MacKayJ._D._Meiss1987" class="citation book cs1">R. S. MacKay; J. D. Meiss (1987). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=uTeqNsyj86QC&q=tidal+force&pg=PA36"><i>Hamiltonian Dynamical Systems: A Reprint Selection</i></a>. <a href="/wiki/CRC_Press" title="CRC Press">CRC Press</a>. p. 36. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-85274-205-1" title="Special:BookSources/978-0-85274-205-1"><bdi>978-0-85274-205-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Hamiltonian+Dynamical+Systems%3A+A+Reprint+Selection&rft.pages=36&rft.pub=CRC+Press&rft.date=1987&rft.isbn=978-0-85274-205-1&rft.au=R.+S.+MacKay&rft.au=J.+D.+Meiss&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DuTeqNsyj86QC%26q%3Dtidal%2Bforce%26pg%3DPA36&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATidal+force" class="Z3988"></span></span> </li> <li id="cite_note-Americana-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-Americana_12-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRollin_A_Harris1920" class="citation book cs1">Rollin A Harris (1920). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=r8BPAAAAMAAJ&q=tidal+force&pg=PA612"><i>The Encyclopedia Americana: A Library of Universal Knowledge</i></a>. Vol. 26. Encyclopedia Americana Corp. pp. 611–617.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Encyclopedia+Americana%3A+A+Library+of+Universal+Knowledge&rft.pages=611-617&rft.pub=Encyclopedia+Americana+Corp.&rft.date=1920&rft.au=Rollin+A+Harris&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dr8BPAAAAMAAJ%26q%3Dtidal%2Bforce%26pg%3DPA612&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATidal+force" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKeelingWhorf1997" class="citation journal cs1">Keeling, C. D.; Whorf, T. P. (5 August 1997). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC33744">"Possible forcing of global temperature by the oceanic tides"</a>. <i>Proceedings of the National Academy of Sciences</i>. <b>94</b> (16): 8321–8328. <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/1997PNAS...94.8321K">1997PNAS...94.8321K</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1073%2Fpnas.94.16.8321">10.1073/pnas.94.16.8321</a></span>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC33744">33744</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/11607740">11607740</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+National+Academy+of+Sciences&rft.atitle=Possible+forcing+of+global+temperature+by+the+oceanic+tides&rft.volume=94&rft.issue=16&rft.pages=8321-8328&rft.date=1997-08-05&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC33744%23id-name%3DPMC&rft_id=info%3Apmid%2F11607740&rft_id=info%3Adoi%2F10.1073%2Fpnas.94.16.8321&rft_id=info%3Abibcode%2F1997PNAS...94.8321K&rft.aulast=Keeling&rft.aufirst=C.+D.&rft.au=Whorf%2C+T.+P.&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC33744&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATidal+force" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMunkDzieciuchJayne2002" class="citation journal cs1">Munk, Walter; Dzieciuch, Matthew; Jayne, Steven (February 2002). <a rel="nofollow" class="external text" href="https://doi.org/10.1175%2F1520-0442%282002%29015%3C0370%3AMCVITA%3E2.0.CO%3B2">"Millennial Climate Variability: Is There a Tidal Connection?"</a>. <i>Journal of Climate</i>. <b>15</b> (4): 370–385. <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/2002JCli...15..370M">2002JCli...15..370M</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1175%2F1520-0442%282002%29015%3C0370%3AMCVITA%3E2.0.CO%3B2">10.1175/1520-0442(2002)015<0370:MCVITA>2.0.CO;2</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Climate&rft.atitle=Millennial+Climate+Variability%3A+Is+There+a+Tidal+Connection%3F&rft.volume=15&rft.issue=4&rft.pages=370-385&rft.date=2002-02&rft_id=info%3Adoi%2F10.1175%2F1520-0442%282002%29015%3C0370%3AMCVITA%3E2.0.CO%3B2&rft_id=info%3Abibcode%2F2002JCli...15..370M&rft.aulast=Munk&rft.aufirst=Walter&rft.au=Dzieciuch%2C+Matthew&rft.au=Jayne%2C+Steven&rft_id=https%3A%2F%2Fdoi.org%2F10.1175%252F1520-0442%25282002%2529015%253C0370%253AMCVITA%253E2.0.CO%253B2&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATidal+force" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation journal cs1"><a rel="nofollow" class="external text" href="https://books.google.com/books?id=FPEUAQAAMAAJ">"Hungry for Power in Space"</a>. <i>New Scientist</i>. <b>123</b>: 52. 23 September 1989<span class="reference-accessdate">. Retrieved <span class="nowrap">14 March</span> 2016</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=New+Scientist&rft.atitle=Hungry+for+Power+in+Space&rft.volume=123&rft.pages=52&rft.date=1989-09-23&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DFPEUAQAAMAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATidal+force" class="Z3988"></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation news cs1"><a rel="nofollow" class="external text" href="http://www.spacetelescope.org/images/potw1325a/">"Inseparable galactic twins"</a>. <i>ESA/Hubble Picture of the Week</i><span class="reference-accessdate">. Retrieved <span class="nowrap">12 July</span> 2013</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=ESA%2FHubble+Picture+of+the+Week&rft.atitle=Inseparable+galactic+twins&rft_id=http%3A%2F%2Fwww.spacetelescope.org%2Fimages%2Fpotw1325a%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATidal+force" class="Z3988"></span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThe_Admiralty1987" class="citation book cs1">The Admiralty (1987). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=GCgXCxG4VLcC"><i>Admiralty manual of navigation</i></a>. Vol. 1. <a href="/wiki/The_Stationery_Office" title="The Stationery Office">The Stationery Office</a>. p. 277. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-11-772880-6" title="Special:BookSources/978-0-11-772880-6"><bdi>978-0-11-772880-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Admiralty+manual+of+navigation&rft.pages=277&rft.pub=The+Stationery+Office&rft.date=1987&rft.isbn=978-0-11-772880-6&rft.au=The+Admiralty&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DGCgXCxG4VLcC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATidal+force" class="Z3988"></span>, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=GCgXCxG4VLcC&pg=PA277">Chapter 11, p. 277</a></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNewton1729" class="citation book cs1">Newton, Isaac (1729). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=6EqxPav3vIsC"><i>The mathematical principles of natural philosophy</i></a>. Vol. 2. p. 307. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-11-772880-6" title="Special:BookSources/978-0-11-772880-6"><bdi>978-0-11-772880-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+mathematical+principles+of+natural+philosophy&rft.pages=307&rft.date=1729&rft.isbn=978-0-11-772880-6&rft.aulast=Newton&rft.aufirst=Isaac&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D6EqxPav3vIsC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATidal+force" class="Z3988"></span>, <a rel="nofollow" class="external text" href="https://archive.org/details/bub_gb_6EqxPav3vIsC/page/307">Book 3, Proposition 36, Page 307</a> Newton put the force to depress the sea at places 90 degrees distant from the Sun at "1 to 38604600" (in terms of <i>g</i>), and wrote that the force to raise the sea along the Sun-Earth axis is "twice as great" (i.e., 2 to 38604600) which comes to about 0.52 × 10<sup>−7</sup> <i>g</i> as expressed in the text.</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tidal_force&action=edit&section=8" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://geomatix.net/tides/analyzer.htm">Analysis and Prediction of Tides: GeoTide</a></li> <li><a rel="nofollow" class="external text" href="http://burro.astr.cwru.edu/Academics/Astr221/Gravity/tides.html">Gravitational Tides</a> by J. Christopher Mihos of <a href="/wiki/Case_Western_Reserve_University" title="Case Western Reserve University">Case Western Reserve University</a></li> <li><a rel="nofollow" class="external text" href="http://www.astronomycast.com/solar-system/episode-47-tidal-forces/">Audio: Cain/Gay – Astronomy Cast</a> Tidal Forces – July 2007.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrayMerrifield,_Michael" class="citation web cs1">Gray, Meghan; Merrifield, Michael. <a rel="nofollow" class="external text" href="http://www.sixtysymbols.com/videos/tides.htm">"Tidal Forces"</a>. <i>Sixty Symbols</i>. <a href="/wiki/Brady_Haran" title="Brady Haran">Brady Haran</a> for the <a href="/wiki/University_of_Nottingham" title="University of Nottingham">University of Nottingham</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Sixty+Symbols&rft.atitle=Tidal+Forces&rft.aulast=Gray&rft.aufirst=Meghan&rft.au=Merrifield%2C+Michael&rft_id=http%3A%2F%2Fwww.sixtysymbols.com%2Fvideos%2Ftides.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATidal+force" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPau_Amaro_Seoane" class="citation web cs1">Pau Amaro Seoane. <a rel="nofollow" class="external text" href="http://astro-gr.org/stellar-collisions-tidal-disruption-star-massive-black-hole/">"Stellar collisions: Tidal disruption of a star by a massive black hole"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2018-12-28</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Stellar+collisions%3A+Tidal+disruption+of+a+star+by+a+massive+black+hole&rft.au=Pau+Amaro+Seoane&rft_id=http%3A%2F%2Fastro-gr.org%2Fstellar-collisions-tidal-disruption-star-massive-black-hole%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATidal+force" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20130921060553/http://www.jal.cc.il.us/~mikolajsawicki/tides_new2.pdf">Myths about Gravity and Tides</a> by Mikolaj Sawicki of John A. Logan College and the University of Colorado.</li> <li><a rel="nofollow" class="external text" href="https://dsimanek.vialattea.net/scenario/tides.htm">Tidal Misconceptions</a> by Donald E. Simanek</li> <li><a rel="nofollow" class="external text" href="https://www.vialattea.net/content/tides-and-centrifugal-force/">Tides and centrifugal force</a> by Paolo Sirtoli</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist 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title="Longshore drift">Longshore drift</a></li> <li><a href="/wiki/Luke%27s_variational_principle" title="Luke's variational principle">Luke's variational principle</a></li> <li><a href="/wiki/Mild-slope_equation" title="Mild-slope equation">Mild-slope equation</a></li> <li><a href="/wiki/Radiation_stress" title="Radiation stress">Radiation stress</a></li> <li><a href="/wiki/Rogue_wave" title="Rogue wave">Rogue wave</a></li> <li><a href="/wiki/Rossby_wave" title="Rossby wave">Rossby wave</a></li> <li><a href="/wiki/Rossby-gravity_waves" title="Rossby-gravity waves">Rossby-gravity waves</a></li> <li><a href="/wiki/Sea_state" title="Sea state">Sea state</a></li> <li><a href="/wiki/Seiche" title="Seiche">Seiche</a></li> <li><a href="/wiki/Significant_wave_height" title="Significant wave height">Significant wave height</a></li> <li><a href="/wiki/Soliton" title="Soliton">Soliton</a></li> <li><a href="/wiki/Stokes_drift" title="Stokes drift">Stokes drift</a></li> <li><a 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title="Wave power">Wave power</a></li> <li><a href="/wiki/Wave_radar" title="Wave radar">Wave radar</a></li> <li><a href="/wiki/Wave_setup" title="Wave setup">Wave setup</a></li> <li><a href="/wiki/Wave_shoaling" title="Wave shoaling">Wave shoaling</a></li> <li><a href="/wiki/Wave_turbulence" title="Wave turbulence">Wave turbulence</a></li> <li><a href="/wiki/Wave%E2%80%93current_interaction" title="Wave–current interaction">Wave–current interaction</a></li> <li><a href="/wiki/Waves_and_shallow_water" title="Waves and shallow water">Waves and shallow water</a> <ul><li><a href="/wiki/One-dimensional_Saint-Venant_equations" class="mw-redirect" title="One-dimensional Saint-Venant equations">one-dimensional Saint-Venant equations</a></li> <li><a href="/wiki/Shallow_water_equations" title="Shallow water equations">shallow water equations</a></li></ul></li> <li><a href="/wiki/Wind_fetch" title="Wind fetch">Wind fetch</a></li> <li><a href="/wiki/Wind_setup" title="Wind setup">Wind setup</a></li> <li><a href="/wiki/Wind_wave" title="Wind wave">Wind wave</a> <ul><li><a href="/wiki/Wind_wave_model" title="Wind wave model">model</a></li></ul></li></ul> </div></td><td class="noviewer navbox-image" rowspan="10" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/File:Upwelling.svg" class="mw-file-description" title="Upwelling"><img alt="Upwelling" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Upwelling.svg/120px-Upwelling.svg.png" decoding="async" width="120" height="80" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Upwelling.svg/180px-Upwelling.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Upwelling.svg/240px-Upwelling.svg.png 2x" data-file-width="365" data-file-height="242" /></a></span><br /><br /><br /><br /><br /><br /><span typeof="mw:File"><a href="/wiki/File:Antarctic_bottom_water.svg" class="mw-file-description" title="Antarctic bottom water"><img alt="Antarctic bottom water" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Antarctic_bottom_water.svg/120px-Antarctic_bottom_water.svg.png" decoding="async" width="120" height="76" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Antarctic_bottom_water.svg/180px-Antarctic_bottom_water.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Antarctic_bottom_water.svg/240px-Antarctic_bottom_water.svg.png 2x" data-file-width="745" data-file-height="470" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Ocean_current" title="Ocean current">Circulation</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Atmospheric_circulation" title="Atmospheric circulation">Atmospheric circulation</a></li> <li><a href="/wiki/Baroclinity" title="Baroclinity">Baroclinity</a></li> <li><a href="/wiki/Boundary_current" title="Boundary current">Boundary current</a></li> <li><a href="/wiki/Coriolis_force" title="Coriolis force">Coriolis force</a></li> <li><a href="/wiki/Coriolis%E2%80%93Stokes_force" title="Coriolis–Stokes force">Coriolis–Stokes force</a></li> <li><a href="/wiki/Craik%E2%80%93Leibovich_vortex_force" title="Craik–Leibovich vortex force">Craik–Leibovich vortex force</a></li> <li><a href="/wiki/Downwelling" title="Downwelling">Downwelling</a></li> <li><a href="/wiki/Eddy_(fluid_dynamics)" title="Eddy (fluid dynamics)">Eddy</a></li> <li><a href="/wiki/Ekman_layer" title="Ekman layer">Ekman layer</a></li> <li><a href="/wiki/Ekman_spiral" title="Ekman spiral">Ekman spiral</a></li> <li><a href="/wiki/Ekman_transport" title="Ekman transport">Ekman transport</a></li> <li><a href="/wiki/El_Ni%C3%B1o%E2%80%93Southern_Oscillation" title="El Niño–Southern Oscillation">El Niño–Southern Oscillation</a></li> <li><a href="/wiki/General_circulation_model" title="General circulation model">General circulation model</a></li> <li><a href="/wiki/Geochemical_Ocean_Sections_Study" title="Geochemical Ocean Sections Study">Geochemical Ocean Sections Study</a></li> <li><a href="/wiki/Geostrophic_current" title="Geostrophic current">Geostrophic current</a></li> <li><a href="/wiki/Global_Ocean_Data_Analysis_Project" title="Global Ocean Data Analysis Project">Global Ocean Data Analysis Project</a></li> <li><a href="/wiki/Gulf_Stream" title="Gulf Stream">Gulf Stream</a></li> <li><a href="/wiki/Humboldt_Current" title="Humboldt Current">Humboldt Current</a></li> <li><a href="/wiki/Hydrothermal_circulation" title="Hydrothermal circulation">Hydrothermal circulation</a></li> <li><a href="/wiki/Langmuir_circulation" title="Langmuir circulation">Langmuir circulation</a></li> <li><a href="/wiki/Longshore_drift" title="Longshore drift">Longshore drift</a></li> <li><a href="/wiki/Loop_Current" title="Loop Current">Loop Current</a></li> <li><a href="/wiki/Modular_Ocean_Model" title="Modular Ocean Model">Modular Ocean Model</a></li> <li><a href="/wiki/Ocean_current" title="Ocean current">Ocean current</a></li> <li><a href="/wiki/Ocean_dynamical_thermostat" title="Ocean dynamical thermostat">Ocean dynamical thermostat</a></li> <li><a href="/wiki/Ocean_dynamics" title="Ocean dynamics">Ocean dynamics</a></li> <li><a href="/wiki/Ocean_gyre" title="Ocean gyre">Ocean gyre</a></li> <li><a href="/wiki/Overflow_(oceanography)" title="Overflow (oceanography)">Overflow</a></li> <li><a href="/wiki/Princeton_Ocean_Model" title="Princeton Ocean Model">Princeton Ocean Model</a></li> <li><a href="/wiki/Rip_current" title="Rip current">Rip current</a></li> <li><a href="/wiki/Subsurface_ocean_current" title="Subsurface ocean current">Subsurface ocean current</a></li> <li><a href="/wiki/Sverdrup_balance" title="Sverdrup balance">Sverdrup balance</a></li> <li><a href="/wiki/Thermohaline_circulation" title="Thermohaline circulation">Thermohaline circulation</a> <ul><li><a href="/wiki/Shutdown_of_thermohaline_circulation" class="mw-redirect" title="Shutdown of thermohaline circulation">shutdown</a></li></ul></li> <li><a href="/wiki/Upwelling" title="Upwelling">Upwelling</a></li> <li><a href="/wiki/Whirlpool" title="Whirlpool">Whirlpool</a></li> <li><a href="/wiki/Wind_generated_current" title="Wind generated current">Wind generated current</a></li> <li><a href="/wiki/World_Ocean_Circulation_Experiment" title="World Ocean Circulation Experiment">World Ocean Circulation Experiment</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Tide" title="Tide">Tides</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Amphidromic_point" title="Amphidromic point">Amphidromic point</a></li> <li><a href="/wiki/Earth_tide" title="Earth tide">Earth tide</a></li> <li><a href="/wiki/Head_of_tide" title="Head of tide">Head of tide</a></li> <li><a href="/wiki/Internal_tide" title="Internal tide">Internal tide</a></li> <li><a href="/wiki/Lunitidal_interval" title="Lunitidal interval">Lunitidal interval</a></li> <li><a href="/wiki/Perigean_spring_tide" title="Perigean spring tide">Perigean spring tide</a></li> <li><a href="/wiki/Rip_tide" title="Rip tide">Rip tide</a></li> <li><a href="/wiki/Rule_of_twelfths" title="Rule of twelfths">Rule of twelfths</a></li> <li><a href="/wiki/Slack_tide" title="Slack tide">Slack tide</a></li> <li><a href="/wiki/Theory_of_tides" title="Theory of tides">Theory of tides</a></li> <li><a href="/wiki/Tidal_bore" title="Tidal bore">Tidal bore</a></li> <li><a class="mw-selflink selflink">Tidal force</a></li> <li><a href="/wiki/Tidal_power" title="Tidal power">Tidal power</a></li> <li><a href="/wiki/Tidal_race" title="Tidal race">Tidal race</a></li> <li><a href="/wiki/Tidal_range" title="Tidal range">Tidal range</a></li> <li><a href="/wiki/Tidal_resonance" title="Tidal resonance">Tidal resonance</a></li> <li><a href="/wiki/Tide_gauge" title="Tide gauge">Tide gauge</a></li> <li><a href="/wiki/Tideline" title="Tideline">Tideline</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Landform" title="Landform">Landforms</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abyssal_fan" title="Abyssal fan">Abyssal fan</a></li> <li><a href="/wiki/Abyssal_plain" title="Abyssal plain">Abyssal plain</a></li> <li><a href="/wiki/Atoll" title="Atoll">Atoll</a></li> <li><a href="/wiki/Bathymetric_chart" title="Bathymetric chart">Bathymetric chart</a></li> <li><a href="/wiki/Carbonate_platform" title="Carbonate platform">Carbonate platform</a></li> <li><a href="/wiki/Coastal_geography" title="Coastal geography">Coastal geography</a></li> <li><a href="/wiki/Cold_seep" title="Cold seep">Cold seep</a></li> <li><a href="/wiki/Continental_margin" title="Continental margin">Continental margin</a></li> <li><a href="/wiki/Continental_rise" title="Continental rise">Continental rise</a></li> <li><a href="/wiki/Continental_shelf" title="Continental shelf">Continental shelf</a></li> <li><a href="/wiki/Contourite" title="Contourite">Contourite</a></li> <li><a href="/wiki/Guyot" title="Guyot">Guyot</a></li> <li><a href="/wiki/Hydrography" title="Hydrography">Hydrography</a></li> <li><a href="/wiki/Knoll_(oceanography)" title="Knoll (oceanography)">Knoll</a></li> <li><a href="/wiki/Ocean_bank" title="Ocean bank">Ocean bank</a></li> <li><a href="/wiki/Oceanic_basin" title="Oceanic basin">Oceanic basin</a></li> <li><a href="/wiki/Oceanic_plateau" title="Oceanic plateau">Oceanic plateau</a></li> <li><a href="/wiki/Oceanic_trench" title="Oceanic trench">Oceanic trench</a></li> <li><a href="/wiki/Passive_margin" title="Passive margin">Passive margin</a></li> <li><a href="/wiki/Seabed" title="Seabed">Seabed</a></li> <li><a href="/wiki/Seamount" title="Seamount">Seamount</a></li> <li><a href="/wiki/Submarine_canyon" title="Submarine canyon">Submarine canyon</a></li> <li><a href="/wiki/Submarine_volcano" title="Submarine volcano">Submarine volcano</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Plate_tectonics" title="Plate tectonics">Plate<br />tectonics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Convergent_boundary" title="Convergent boundary">Convergent boundary</a></li> <li><a href="/wiki/Divergent_boundary" title="Divergent boundary">Divergent boundary</a></li> <li><a href="/wiki/Fracture_zone" title="Fracture zone">Fracture zone</a></li> <li><a href="/wiki/Hydrothermal_vent" title="Hydrothermal vent">Hydrothermal vent</a></li> <li><a href="/wiki/Marine_geology" title="Marine geology">Marine geology</a></li> <li><a href="/wiki/Mid-ocean_ridge" title="Mid-ocean ridge">Mid-ocean ridge</a></li> <li><a href="/wiki/Mohorovi%C4%8Di%C4%87_discontinuity" title="Mohorovičić discontinuity">Mohorovičić discontinuity</a></li> <li><a href="/wiki/Oceanic_crust" title="Oceanic crust">Oceanic crust</a></li> <li><a href="/wiki/Outer_trench_swell" title="Outer trench swell">Outer trench swell</a></li> <li><a href="/wiki/Ridge_push" title="Ridge push">Ridge push</a></li> <li><a href="/wiki/Seafloor_spreading" title="Seafloor spreading">Seafloor spreading</a></li> <li><a href="/wiki/Slab_pull" title="Slab pull">Slab pull</a></li> <li><a href="/wiki/Slab_suction" title="Slab suction">Slab suction</a></li> <li><a href="/wiki/Slab_window" title="Slab window">Slab window</a></li> <li><a href="/wiki/Subduction" title="Subduction">Subduction</a></li> <li><a href="/wiki/Transform_fault" title="Transform fault">Transform fault</a></li> <li><a href="/wiki/Vine%E2%80%93Matthews%E2%80%93Morley_hypothesis" title="Vine–Matthews–Morley hypothesis">Vine–Matthews–Morley hypothesis</a></li> <li><a href="/wiki/Volcanic_arc" title="Volcanic arc">Volcanic arc</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Ocean zones</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Benthic_zone" title="Benthic zone">Benthic</a></li> <li><a href="/wiki/Deep_ocean_water" title="Deep ocean water">Deep ocean water</a></li> <li><a href="/wiki/Deep_sea" title="Deep sea">Deep sea</a></li> <li><a href="/wiki/Littoral_zone" title="Littoral zone">Littoral</a></li> <li><a href="/wiki/Mesopelagic_zone" title="Mesopelagic zone">Mesopelagic</a></li> <li><a href="/wiki/Oceanic_zone" title="Oceanic zone">Oceanic</a></li> <li><a href="/wiki/Pelagic_zone" title="Pelagic zone">Pelagic</a></li> <li><a href="/wiki/Photic_zone" title="Photic zone">Photic</a></li> <li><a href="/wiki/Surf_zone" title="Surf zone">Surf</a></li> <li><a href="/wiki/Swash" title="Swash">Swash</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Sea_level" title="Sea level">Sea level</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Deep-ocean_Assessment_and_Reporting_of_Tsunamis" title="Deep-ocean Assessment and Reporting of Tsunamis">Deep-ocean Assessment and Reporting of Tsunamis</a></li> <li><a href="/wiki/Global_Sea_Level_Observing_System" title="Global Sea Level Observing System">Global Sea Level Observing System</a></li> <li><a href="/wiki/North_West_Shelf_Operational_Oceanographic_System" title="North West Shelf Operational Oceanographic System">North West Shelf Operational Oceanographic System</a></li> <li><a href="/wiki/Sea-level_curve" title="Sea-level curve">Sea-level curve</a></li> <li><a href="/wiki/Sea_level_drop" title="Sea level drop">Sea level drop</a></li> <li><a href="/wiki/Sea_level_rise" title="Sea level rise">Sea level rise</a></li> <li><a href="/wiki/World_Geodetic_System" title="World Geodetic System">World Geodetic System</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Acoustical_oceanography" class="mw-redirect" title="Acoustical oceanography">Acoustics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Deep_scattering_layer" title="Deep scattering layer">Deep scattering layer</a></li> <li><a href="/wiki/Ocean_acoustic_tomography" title="Ocean acoustic tomography">Ocean acoustic tomography</a></li> <li><a href="/wiki/Sofar_bomb" title="Sofar bomb">Sofar bomb</a></li> <li><a href="/wiki/SOFAR_channel" title="SOFAR channel">SOFAR channel</a></li> <li><a href="/wiki/Underwater_acoustics" title="Underwater acoustics">Underwater acoustics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Satellites</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Jason-1" title="Jason-1">Jason-1</a></li> <li><a href="/wiki/OSTM/Jason-2" title="OSTM/Jason-2">OSTM/Jason-2</a></li> <li><a href="/wiki/Jason-3" title="Jason-3">Jason-3</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Ocean_acidification" title="Ocean acidification">Acidification</a></li> <li><a href="/wiki/Argo_(oceanography)" title="Argo (oceanography)">Argo</a></li> <li><a href="/wiki/Benthic_lander" title="Benthic lander">Benthic lander</a></li> <li><a href="/wiki/Color_of_water" title="Color of water">Color of water</a></li> <li><a href="/wiki/DSV_Alvin" title="DSV Alvin">DSV <i>Alvin</i></a></li> <li><a href="/wiki/Marginal_sea" class="mw-redirect" title="Marginal sea">Marginal sea</a></li> <li><a href="/wiki/Marine_energy" title="Marine energy">Marine energy</a></li> <li><a href="/wiki/Marine_pollution" title="Marine pollution">Marine pollution</a></li> <li><a href="/wiki/Mooring_(oceanography)" title="Mooring (oceanography)">Mooring</a></li> <li><a href="/wiki/National_Oceanographic_Data_Center" title="National Oceanographic Data Center">National Oceanographic Data Center</a></li> <li><a href="/wiki/Ocean" title="Ocean">Ocean</a></li> <li><a href="/wiki/Ocean_exploration" title="Ocean exploration">Explorations</a></li> <li><a href="/wiki/Ocean_observations" title="Ocean observations">Observations</a></li> <li><a href="/wiki/Ocean_reanalysis" title="Ocean reanalysis">Reanalysis</a></li> <li><a href="/wiki/Ocean_surface_topography" title="Ocean surface topography">Ocean surface topography</a></li> <li><a href="/wiki/Ocean_temperature" title="Ocean temperature">Ocean temperature</a></li> <li><a href="/wiki/Ocean_thermal_energy_conversion" title="Ocean thermal energy conversion">Ocean thermal energy conversion</a></li> <li><a href="/wiki/Oceanography" title="Oceanography">Oceanography</a> <ul><li><a href="/wiki/Outline_of_oceanography" title="Outline of oceanography">Outline of oceanography</a></li></ul></li> <li><a href="/wiki/Pelagic_sediment" title="Pelagic sediment">Pelagic sediment</a></li> <li><a href="/wiki/Sea_surface_microlayer" title="Sea surface microlayer">Sea surface microlayer</a></li> <li><a href="/wiki/Sea_surface_temperature" title="Sea surface temperature">Sea surface temperature</a></li> <li><a href="/wiki/Seawater" title="Seawater">Seawater</a></li> <li><a href="/wiki/Science_On_a_Sphere" title="Science On a Sphere">Science On a Sphere</a></li> <li><a href="/wiki/Ocean_stratification" 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