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vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Gas_giants"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Gas giants</span> </div> </a> <ul id="toc-Gas_giants-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Non-spherically_symmetric_objects" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Non-spherically_symmetric_objects"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Non-spherically symmetric objects</span> </div> </a> <ul id="toc-Non-spherically_symmetric_objects-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Black_holes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Black_holes"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Black holes</span> </div> </a> <button aria-controls="toc-Black_holes-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Black holes subsection</span> </button> <ul id="toc-Black_holes-sublist" class="vector-toc-list"> <li id="toc-Schwarzschild_solution" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Schwarzschild_solution"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Schwarzschild solution</span> </div> </a> <ul id="toc-Schwarzschild_solution-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kerr_solution" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Kerr_solution"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Kerr solution</span> </div> </a> <ul id="toc-Kerr_solution-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kerr–Newman_solution" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Kerr–Newman_solution"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Kerr–Newman solution</span> </div> </a> <ul id="toc-Kerr–Newman_solution-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dynamical_black_holes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dynamical_black_holes"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Dynamical black holes</span> </div> </a> <ul id="toc-Dynamical_black_holes-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Surface gravity</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 23 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-23" 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">23 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AC%D8%A7%D8%B0%D8%A8%D9%8A%D8%A9_%D8%B3%D8%B7%D8%AD%D9%8A%D8%A9" title="جاذبية سطحية – Arabic" lang="ar" hreflang="ar" data-title="جاذبية سطحية" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Gravetat_superficial" title="Gravetat superficial – Catalan" lang="ca" hreflang="ca" data-title="Gravetat superficial" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%DA%AF%D8%B1%D8%A7%D9%86%D8%B4_%D8%B3%D8%B7%D8%AD%DB%8C" title="گرانش سطحی – Persian" lang="fa" hreflang="fa" data-title="گرانش سطحی" 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/Gravit%C3%A9_de_surface" title="Gravité de surface – French" lang="fr" hreflang="fr" data-title="Gravité de surface" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%91%9C%EB%A9%B4%EC%A4%91%EB%A0%A5" title="표면중력 – Korean" lang="ko" hreflang="ko" data-title="표면중력" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Gravitasi_permukaan" title="Gravitasi permukaan – Indonesian" lang="id" hreflang="id" data-title="Gravitasi permukaan" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Gravit%C3%A0_di_superficie" title="Gravità di superficie – Italian" lang="it" hreflang="it" data-title="Gravità di superficie" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9F%D0%BE%D0%B2%D1%80%D1%88%D0%B8%D0%BD%D1%81%D0%BA%D0%B0_%D0%B3%D1%80%D0%B0%D0%B2%D0%B8%D1%82%D0%B0%D1%86%D0%B8%D1%98%D0%B0" title="Површинска гравитација – Macedonian" lang="mk" hreflang="mk" data-title="Површинска гравитација" 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/Graviti_permukaan" title="Graviti permukaan – Malay" lang="ms" hreflang="ms" data-title="Graviti permukaan" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E8%A1%A8%E9%9D%A2%E9%87%8D%E5%8A%9B" title="表面重力 – Japanese" lang="ja" hreflang="ja" data-title="表面重力" 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/Overflategravitasjon" title="Overflategravitasjon – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Overflategravitasjon" 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-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D0%BE%D0%B2%D0%B5%D1%80%D1%85%D0%BD%D0%BE%D1%81%D1%82%D0%BD%D0%B0%D1%8F_%D0%B3%D1%80%D0%B0%D0%B2%D0%B8%D1%82%D0%B0%D1%86%D0%B8%D1%8F" title="Поверхностная гравитация – Russian" lang="ru" hreflang="ru" data-title="Поверхностная гравитация" 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/Surface_gravity" title="Surface gravity – Simple English" lang="en-simple" hreflang="en-simple" data-title="Surface gravity" 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-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Povr%C5%A1inska_te%C5%BEnost" title="Površinska težnost – Slovenian" lang="sl" hreflang="sl" data-title="Površinska težnost" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9F%D0%BE%D0%B2%D1%80%D1%88%D0%B8%D0%BD%D1%81%D0%BA%D0%B0_%D0%B3%D1%80%D0%B0%D0%B2%D0%B8%D1%82%D0%B0%D1%86%D0%B8%D1%98%D0%B0" title="Површинска гравитација – Serbian" lang="sr" hreflang="sr" data-title="Површинска гравитација" 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-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Pintagravitaatio" title="Pintagravitaatio – Finnish" lang="fi" hreflang="fi" data-title="Pintagravitaatio" 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/Ytgravitation" title="Ytgravitation – Swedish" lang="sv" hreflang="sv" data-title="Ytgravitation" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/Taldayt_n_tjumma" title="Taldayt n tjumma – Kabyle" lang="kab" hreflang="kab" data-title="Taldayt n tjumma" data-language-autonym="Taqbaylit" data-language-local-name="Kabyle" class="interlanguage-link-target"><span>Taqbaylit</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%84%E0%B8%A7%E0%B8%B2%E0%B8%A1%E0%B9%82%E0%B8%99%E0%B9%89%E0%B8%A1%E0%B8%96%E0%B9%88%E0%B8%A7%E0%B8%87%E0%B8%9E%E0%B8%B7%E0%B9%89%E0%B8%99%E0%B8%9C%E0%B8%B4%E0%B8%A7" title="ความโน้มถ่วงพื้นผิว – Thai" lang="th" hreflang="th" data-title="ความโน้มถ่วงพื้นผิว" 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/Y%C3%BCzey_k%C3%BCtle_%C3%A7ekimi" title="Yüzey kütle çekimi – Turkish" lang="tr" hreflang="tr" data-title="Yüzey kütle çekimi" 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%D0%BE%D0%B2%D0%B5%D1%80%D1%85%D0%BD%D0%B5%D0%B2%D0%B0_%D0%B3%D1%80%D0%B0%D0%B2%D1%96%D1%82%D0%B0%D1%86%D1%96%D1%8F" title="Поверхнева гравітація – Ukrainian" lang="uk" hreflang="uk" data-title="Поверхнева гравітація" 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/H%E1%BA%A5p_d%E1%BA%ABn_b%E1%BB%81_m%E1%BA%B7t" title="Hấp dẫn bề mặt – Vietnamese" lang="vi" hreflang="vi" data-title="Hấp dẫn bề mặt" 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href="mw-data:TemplateStyles:r1129693374"><table class="sidebar sidebar-collapse nomobile nowraplinks"><tbody><tr><td class="sidebar-pretitle">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle">Astrodynamics</th></tr><tr><td class="sidebar-image" style="padding-bottom:0.85em;"><span typeof="mw:File"><a href="/wiki/File:Orbit_mechanics_icon.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Orbit_mechanics_icon.svg/60px-Orbit_mechanics_icon.svg.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Orbit_mechanics_icon.svg/90px-Orbit_mechanics_icon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Orbit_mechanics_icon.svg/120px-Orbit_mechanics_icon.svg.png 2x" data-file-width="48" data-file-height="48" /></a></span></td></tr><tr><th class="sidebar-heading" style="padding-bottom:0.55em;"> <div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="/wiki/Orbital_mechanics" title="Orbital mechanics"><span style="font-size:110%;">Orbital mechanics</span></a></div></th></tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="/wiki/Orbital_elements" title="Orbital elements">Orbital elements</a></div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0;"> <ul><li><a href="/wiki/Apsis" title="Apsis">Apsis</a></li> <li><a href="/wiki/Argument_of_periapsis" title="Argument of periapsis">Argument of periapsis</a></li> <li><a href="/wiki/Orbital_eccentricity" title="Orbital eccentricity">Eccentricity</a></li> <li><a href="/wiki/Orbital_inclination" title="Orbital inclination">Inclination</a></li> <li><a href="/wiki/Mean_anomaly" title="Mean anomaly">Mean anomaly</a></li> <li><a href="/wiki/Orbital_node" title="Orbital node">Orbital nodes</a></li> <li><a href="/wiki/Semi-major_and_semi-minor_axes" title="Semi-major and semi-minor axes">Semi-major axis</a></li> <li><a href="/wiki/True_anomaly" title="True anomaly">True anomaly</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Types of <a href="/wiki/Two-body_problem" title="Two-body problem">two-body orbits</a> by <br />eccentricity</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0;"> <ul><li><a href="/wiki/Circular_orbit" title="Circular orbit">Circular orbit</a></li> <li><a href="/wiki/Elliptic_orbit" title="Elliptic orbit">Elliptic orbit</a></li></ul> <div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="/wiki/Transfer_orbit" title="Transfer orbit">Transfer orbit</a> <div class="hlist" style="font-size:90%"><ul><li>(<a href="/wiki/Hohmann_transfer_orbit" title="Hohmann transfer orbit">Hohmann transfer orbit</a></li><li><a href="/wiki/Bi-elliptic_transfer" title="Bi-elliptic transfer">Bi-elliptic transfer orbit</a>)</li></ul></div></div> <ul><li><a href="/wiki/Parabolic_trajectory" title="Parabolic trajectory">Parabolic orbit</a></li> <li><a href="/wiki/Hyperbolic_trajectory" title="Hyperbolic trajectory">Hyperbolic orbit</a></li> <li><a href="/wiki/Radial_trajectory" title="Radial trajectory">Radial orbit</a></li> <li><a href="/wiki/Orbital_decay" title="Orbital decay">Decaying orbit</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Equations</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0;"> <ul><li><a href="/wiki/Dynamical_friction" title="Dynamical friction">Dynamical friction</a></li> <li><a href="/wiki/Escape_velocity" title="Escape velocity">Escape velocity</a></li> <li><a href="/wiki/Kepler%27s_equation" title="Kepler's equation">Kepler's equation</a></li> <li><a href="/wiki/Kepler%27s_laws_of_planetary_motion" title="Kepler's laws of planetary motion">Kepler's laws of planetary motion</a></li> <li><a href="/wiki/Orbital_period" title="Orbital period">Orbital period</a></li> <li><a href="/wiki/Orbital_speed" title="Orbital speed">Orbital velocity</a></li> <li><a class="mw-selflink selflink">Surface gravity</a></li> <li><a href="/wiki/Specific_orbital_energy" title="Specific orbital energy">Specific orbital energy</a></li> <li><a href="/wiki/Vis-viva_equation" title="Vis-viva equation">Vis-viva equation</a></li></ul></div></div></td> </tr><tr><th class="sidebar-heading" style="padding-bottom:0.55em;"> <div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="/wiki/Celestial_mechanics" title="Celestial mechanics"><span style="font-size:110%;">Celestial mechanics</span></a></div></th></tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Gravitational influences</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0;"> <ul><li><a href="/wiki/Barycenter" class="mw-redirect" title="Barycenter">Barycenter</a></li> <li><a href="/wiki/Hill_sphere" title="Hill sphere">Hill sphere</a></li> <li><a href="/wiki/Perturbation_(astronomy)" title="Perturbation (astronomy)">Perturbations</a></li> <li><a href="/wiki/Sphere_of_influence_(astrodynamics)" title="Sphere of influence (astrodynamics)">Sphere of influence</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="/wiki/N-body_problem" title="N-body problem">N-body orbits</a></div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0;"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="/wiki/Lagrange_point" title="Lagrange point">Lagrangian points</a> <div class="hlist" style="font-size:90%"><ul><li>(<a href="/wiki/Halo_orbit" title="Halo orbit">Halo orbits</a>)</li></ul></div></div> <ul><li><a href="/wiki/Lissajous_orbit" title="Lissajous orbit">Lissajous orbits</a></li> <li><a href="/wiki/Lyapunov_stability" title="Lyapunov stability">Lyapunov orbits</a></li></ul></div></div></td> </tr><tr><th class="sidebar-heading" style="padding-bottom:0.55em;"> <div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="/wiki/Aerospace_engineering" title="Aerospace engineering"><span style="font-size:110%;">Engineering and efficiency</span></a></div></th></tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Preflight engineering</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0;"> <ul><li><a href="/wiki/Mass_ratio" title="Mass ratio">Mass ratio</a></li> <li><a href="/wiki/Payload_fraction" title="Payload fraction">Payload fraction</a></li> <li><a href="/wiki/Propellant_mass_fraction" title="Propellant mass fraction">Propellant mass fraction</a></li> <li><a href="/wiki/Tsiolkovsky_rocket_equation" title="Tsiolkovsky rocket equation">Tsiolkovsky rocket equation</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Efficiency measures</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0;"> <ul><li><a href="/wiki/Gravity_assist" title="Gravity assist">Gravity assist</a></li> <li><a href="/wiki/Oberth_effect" title="Oberth effect">Oberth effect</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Propulsive maneuvers</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0;"> <ul><li><a href="/wiki/Orbital_maneuver" title="Orbital maneuver">Orbital maneuver</a></li> <li><a href="/wiki/Orbit_insertion" title="Orbit insertion">Orbit insertion</a></li></ul></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Astrodynamics" title="Template:Astrodynamics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Astrodynamics" title="Template talk:Astrodynamics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Astrodynamics" title="Special:EditPage/Template:Astrodynamics"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>The <b>surface gravity</b>, <i>g</i>, of an <a href="/wiki/Astronomical_object" title="Astronomical object">astronomical object</a> is the <a href="/wiki/Gravitational_acceleration" title="Gravitational acceleration">gravitational acceleration</a> experienced at its surface at the equator, including the effects of rotation. The surface gravity may be thought of as the <a href="/wiki/Acceleration" title="Acceleration">acceleration</a> due to gravity experienced by a hypothetical test particle which is very close to the object's surface and which, in order not to disturb the system, has negligible mass. For objects where the surface is deep in the atmosphere and the radius not known, the surface gravity is given at the 1 bar pressure level in the atmosphere. </p><p>Surface gravity is measured in units of acceleration, which, in the <a href="/wiki/SI_units" class="mw-redirect" title="SI units">SI</a> system, are <a href="/wiki/Metre_per_second_squared" title="Metre per second squared">meters per second squared</a>. It may also be expressed as a multiple of the <a href="/wiki/Earth" title="Earth">Earth</a>'s <a href="/wiki/Standard_gravity" title="Standard gravity">standard surface gravity</a>, which is equal to<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> </p> <style data-mw-deduplicate="TemplateStyles:r996643573">.mw-parser-output .block-indent{padding-left:3em;padding-right:0;overflow:hidden}</style><div class="block-indent" style="padding-left: 1.5em;"><i>g</i> = <span class="nowrap"><span data-sort-value="7000980665000000000♠"></span>9.806<span style="margin-left:.25em;">65</span> m/s<sup>2</sup></span></div> <p>In <a href="/wiki/Astrophysics" title="Astrophysics">astrophysics</a>, the surface gravity may be expressed as <style data-mw-deduplicate="TemplateStyles:r1238216509">.mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}@media screen{html.skin-theme-clientpref-night .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}</style><span class="vanchor"><span id="log_g"></span><span class="vanchor-text"><span class="texhtml">log <i>g</i></span></span></span>, which is obtained by first expressing the gravity in <a href="/wiki/Cgs_unit" class="mw-redirect" title="Cgs unit">cgs units</a>, where the unit of acceleration and surface gravity is <a href="/wiki/Centimeter" class="mw-redirect" title="Centimeter">centimeters</a> per second squared (cm/s<sup>2</sup>), and then taking the base-10 <a href="/wiki/Logarithm" title="Logarithm">logarithm</a> of the cgs value of the surface gravity.<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> Therefore, the surface gravity of Earth could be expressed in cgs units as <span class="nowrap"><span data-sort-value="7000980665000000000♠"></span>980.665 cm/s<sup>2</sup></span>, and then taking the base-10 <a href="/wiki/Logarithm" title="Logarithm">logarithm</a> ("log <i>g</i>") of 980.665, giving 2.992 as "log <i>g</i>". </p><p>The surface gravity of a <a href="/wiki/White_dwarf" title="White dwarf">white dwarf</a> is very high, and of a <a href="/wiki/Neutron_star" title="Neutron star">neutron star</a> even higher. A white dwarf's surface gravity is around 100,000 <i>g</i> (<span class="nowrap"><span data-sort-value="7006100000000000000♠"></span>10<sup>6</sup> m/s<sup>2</sup></span>) whilst the neutron star's compactness gives it a surface gravity of up to <span class="nowrap"><span data-sort-value="7012700000000000000♠"></span>7<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>12</sup> m/s<sup>2</sup></span> with typical values of order <span class="nowrap"><span data-sort-value="7012100000000000000♠"></span>10<sup>12</sup> m/s<sup>2</sup></span> (that is more than 10<sup>11</sup> times that of Earth). One measure of such immense gravity is that neutron stars have an <a href="/wiki/Escape_velocity" title="Escape velocity">escape velocity</a> of around <a href="/wiki/Orders_of_magnitude_(speed)#8" title="Orders of magnitude (speed)">100,000 km/s</a>, about a third of the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a>. For black holes, the surface gravity must be calculated relativistically. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Relationship_of_surface_gravity_to_mass_and_radius">Relationship of surface gravity to mass and radius</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Surface_gravity&action=edit&section=1" title="Edit section: Relationship of surface gravity to mass and radius"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable sortable" style="float:right; clear:right; margin-left:1em"> <caption>Surface gravity of various<br />Solar System bodies<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><br /><div style="font-size:70%; line-height:110%">(1 <i>g</i> = 9.80665 m/s<sup>2</sup>, the average surface gravitational acceleration on Earth)</div> </caption> <tbody><tr> <th scope="col">Name </th> <th scope="col" data-sort-type="number">Surface gravity </th></tr> <tr style="background:#FF8B8B"> <td><a href="/wiki/Sun" title="Sun">Sun</a></td> <td>28.02 <i>g</i> </td></tr> <tr style="background:#EEFFFF"> <td><a href="/wiki/Mercury_(planet)" title="Mercury (planet)">Mercury</a></td> <td><span style="visibility:hidden;color:transparent;">0</span>0.377 <i>g</i> </td></tr> <tr style="background:#FDFFFF"> <td><a href="/wiki/Venus" title="Venus">Venus</a></td> <td><span style="visibility:hidden;color:transparent;">0</span>0.905 <i>g</i> </td></tr> <tr style="background:#FFFFFF"> <td><a href="/wiki/Gravity_of_Earth" title="Gravity of Earth">Earth</a></td> <td><span style="visibility:hidden;color:transparent;">0</span>1 <i>g</i> (midlatitudes) </td></tr> <tr style="background:#E0FFFF"> <td><a href="/wiki/Gravitation_of_the_Moon" title="Gravitation of the Moon">Moon</a></td> <td><span style="visibility:hidden;color:transparent;">0</span>0.165 7 <i>g</i> (average) </td></tr> <tr style="background:#EEFFFF"> <td><a href="/wiki/Gravity_of_Mars" title="Gravity of Mars">Mars</a></td> <td><span style="visibility:hidden;color:transparent;">0</span>0.379 <i>g</i> (midlatitudes) </td></tr> <tr style="background:#7EFFFF"> <td><a href="/wiki/Phobos_(moon)" title="Phobos (moon)">Phobos</a></td> <td><span style="visibility:hidden;color:transparent;">0</span>0.000 581 <i>g</i> </td></tr> <tr style="background:#72FFFF"> <td><a href="/wiki/Deimos_(moon)" title="Deimos (moon)">Deimos</a></td> <td><span style="visibility:hidden;color:transparent;">0</span>0.000 306 <i>g</i> </td></tr> <tr style="background:#BDFFFF"> <td><a href="/wiki/2_Pallas" title="2 Pallas">Pallas</a></td> <td><span style="visibility:hidden;color:transparent;">0</span>0.022 <i>g</i> (equator) </td></tr> <tr style="background:#BFFFFF"> <td><a href="/wiki/4_Vesta" title="4 Vesta">Vesta</a></td> <td><span style="visibility:hidden;color:transparent;">0</span>0.025 <i>g</i> (equator) </td></tr> <tr style="background:#C2FFFF"> <td><a href="/wiki/Ceres_(dwarf_planet)" title="Ceres (dwarf planet)">Ceres</a></td> <td><span style="visibility:hidden;color:transparent;">0</span>0.029 <i>g</i> </td></tr> <tr style="background:#FFDFDF"> <td><a href="/wiki/Jupiter" title="Jupiter">Jupiter</a></td> <td><span style="visibility:hidden;color:transparent;">0</span>2.528 <i>g</i> (midlatitudes) </td></tr> <tr style="background:#E2FFFF"> <td><a href="/wiki/Io_(moon)" title="Io (moon)">Io</a></td> <td><span style="visibility:hidden;color:transparent;">0</span>0.183 <i>g</i> </td></tr> <tr style="background:#DCFFFF"> <td><a href="/wiki/Europa_(moon)" title="Europa (moon)">Europa</a></td> <td><span style="visibility:hidden;color:transparent;">0</span>0.134 <i>g</i> </td></tr> <tr style="background:#DEFFFF"> <td><a href="/wiki/Ganymede_(moon)" title="Ganymede (moon)">Ganymede</a></td> <td><span style="visibility:hidden;color:transparent;">0</span>0.146 <i>g</i> </td></tr> <tr style="background:#DBFFFF"> <td><a href="/wiki/Callisto_(moon)" title="Callisto (moon)">Callisto</a></td> <td><span style="visibility:hidden;color:transparent;">0</span>0.126 <i>g</i> </td></tr> <tr style="background:#FDFFFF"> <td><a href="/wiki/Saturn" title="Saturn">Saturn</a></td> <td><span style="visibility:hidden;color:transparent;">0</span>1.065 <i>g</i> (midlatitudes) </td></tr> <tr style="background:#A7FFFF"> <td><a href="/wiki/Mimas_(moon)" class="mw-redirect" title="Mimas (moon)">Mimas</a></td> <td><span style="visibility:hidden;color:transparent;">0</span>0.006 48 <i>g</i> </td></tr> <tr style="background:#B1FFFF"> <td><a href="/wiki/Enceladus" title="Enceladus">Enceladus</a></td> <td><span style="visibility:hidden;color:transparent;">0</span>0.011 5 <i>g</i> </td></tr> <tr style="background:#B5FFFF"> <td><a href="/wiki/Tethys_(moon)" title="Tethys (moon)">Tethys</a></td> <td><span style="visibility:hidden;color:transparent;">0</span>0.014 9 <i>g</i> </td></tr> <tr style="background:#BDFFFF"> <td><a href="/wiki/Dione_(moon)" title="Dione (moon)">Dione</a></td> <td><span style="visibility:hidden;color:transparent;">0</span>0.023 7 <i>g</i> </td></tr> <tr style="background:#C0FFFF"> <td><a href="/wiki/Rhea_(moon)" title="Rhea (moon)">Rhea</a></td> <td><span style="visibility:hidden;color:transparent;">0</span>0.026 9 <i>g</i> </td></tr> <tr style="background:#DDFFFF"> <td><a href="/wiki/Titan_(moon)" title="Titan (moon)">Titan</a></td> <td><span style="visibility:hidden;color:transparent;">0</span>0.138 <i>g</i> </td></tr> <tr style="background:#BDFFFF"> <td><a href="/wiki/Iapetus_(moon)" title="Iapetus (moon)">Iapetus</a></td> <td><span style="visibility:hidden;color:transparent;">0</span>0.022 8 <i>g</i> </td></tr> <tr style="background:#A1FFFF"> <td><a href="/wiki/Phoebe_(moon)" title="Phoebe (moon)">Phoebe</a></td> <td><span style="visibility:hidden;color:transparent;">0</span>0.003 9–0.005 1 <i>g</i> </td></tr> <tr style="background:#FDFFFF"> <td><a href="/wiki/Uranus" title="Uranus">Uranus</a></td> <td><span style="visibility:hidden;color:transparent;">0</span>0.886 <i>g</i> (equator) </td></tr> <tr style="background:#A9FFFF"> <td><a href="/wiki/Miranda_(moon)" title="Miranda (moon)">Miranda</a></td> <td><span style="visibility:hidden;color:transparent;">0</span>0.007 9 <i>g</i> </td></tr> <tr style="background:#BFFFFF"> <td><a href="/wiki/Ariel_(moon)" title="Ariel (moon)">Ariel</a></td> <td><span style="visibility:hidden;color:transparent;">0</span>0.025 4 <i>g</i> </td></tr> <tr style="background:#BDFFFF"> <td><a href="/wiki/Umbriel" title="Umbriel">Umbriel</a></td> <td><span style="visibility:hidden;color:transparent;">0</span>0.023 <i>g</i> </td></tr> <tr style="background:#C6FFFF"> <td><a href="/wiki/Titania_(moon)" title="Titania (moon)">Titania</a></td> <td><span style="visibility:hidden;color:transparent;">0</span>0.037 2 <i>g</i> </td></tr> <tr style="background:#C5FFFF"> <td><a href="/wiki/Oberon_(moon)" title="Oberon (moon)">Oberon</a></td> <td><span style="visibility:hidden;color:transparent;">0</span>0.036 1 <i>g</i> </td></tr> <tr style="background:#FFFAFA"> <td><a href="/wiki/Neptune" title="Neptune">Neptune</a></td> <td><span style="visibility:hidden;color:transparent;">0</span>1.137 <i>g</i> (midlatitudes) </td></tr> <tr style="background:#A9FFFF"> <td><a href="/wiki/Proteus_(moon)" title="Proteus (moon)">Proteus</a></td> <td><span style="visibility:hidden;color:transparent;">0</span>0.007 <i>g</i> </td></tr> <tr style="background:#D3FFFF"> <td><a href="/wiki/Triton_(moon)" title="Triton (moon)">Triton</a></td> <td><span style="visibility:hidden;color:transparent;">0</span>0.079 4 <i>g</i> </td></tr> <tr style="background:#CFFFFF"> <td><a href="/wiki/Pluto" title="Pluto">Pluto</a></td> <td><span style="visibility:hidden;color:transparent;">0</span>0.063 <i>g</i> </td></tr> <tr style="background:#C2FFFF"> <td><a href="/wiki/Charon_(moon)" title="Charon (moon)">Charon</a></td> <td><span style="visibility:hidden;color:transparent;">0</span>0.029 4 <i>g</i> </td></tr> <tr style="background:#D4FFFF"> <td><a href="/wiki/Eris_(dwarf_planet)" title="Eris (dwarf planet)">Eris</a></td> <td><span style="visibility:hidden;color:transparent;">0</span>0.084 <i>g</i> </td></tr> <tr style="background:#BFFFFF"> <td><a href="/wiki/Haumea" title="Haumea">Haumea</a></td> <td><span style="visibility:hidden;color:transparent;">0</span>0.0247 <i>g</i> (equator) </td></tr> <tr style="background:#40FFFF"> <td><a href="/wiki/67P/Churyumov%E2%80%93Gerasimenko" title="67P/Churyumov–Gerasimenko">67P-CG</a></td> <td><span style="visibility:hidden;color:transparent;">0</span>0.000 017 <i>g</i> </td></tr></tbody></table> <p>In the <a href="/wiki/Newtonian_gravity" class="mw-redirect" title="Newtonian gravity">Newtonian</a> theory of <a href="/wiki/Gravity" title="Gravity">gravity</a>, the <a href="/wiki/Gravitational_force" class="mw-redirect" title="Gravitational force">gravitational force</a> exerted by an object is proportional to its mass: an object with twice the mass-produces twice as much force. Newtonian gravity also follows an <a href="/wiki/Inverse_square_law" class="mw-redirect" title="Inverse square law">inverse square law</a>, so that moving an object twice as far away divides its gravitational force by four, and moving it ten times as far away divides it by 100. This is similar to the intensity of <a href="/wiki/Light" title="Light">light</a>, which also follows an inverse square law: with relation to distance, light becomes less visible. Generally speaking, this can be understood as geometric dilution corresponding to point-source radiation into three-dimensional space. </p><p>A large object, such as a <a href="/wiki/Planet" title="Planet">planet</a> or <a href="/wiki/Star" title="Star">star</a>, will usually be approximately round, approaching <a href="/wiki/Hydrostatic_equilibrium" title="Hydrostatic equilibrium">hydrostatic equilibrium</a> (where all points on the surface have the same amount of <a href="/wiki/Gravitational_potential_energy" class="mw-redirect" title="Gravitational potential energy">gravitational potential energy</a>). On a small scale, higher parts of the terrain are eroded, with eroded material deposited in lower parts of the terrain. On a large scale, the planet or star itself deforms until equilibrium is reached.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> For most celestial objects, the result is that the planet or star in question can be treated as a near-perfect <a href="/wiki/Sphere" title="Sphere">sphere</a> when the rotation rate is low. However, for young, massive stars, the equatorial <a href="/wiki/Azimuth" title="Azimuth">azimuthal</a> velocity can be quite high—up to 200 km/s or more—causing a significant amount of <a href="/wiki/Equatorial_bulge" title="Equatorial bulge">equatorial bulge</a>. Examples of such <a href="/wiki/Stellar_rotation" title="Stellar rotation">rapidly rotating stars</a> include <a href="/wiki/Achernar" title="Achernar">Achernar</a>, <a href="/wiki/Altair" title="Altair">Altair</a>, <a href="/wiki/Regulus" title="Regulus">Regulus A</a> and <a href="/wiki/Vega" title="Vega">Vega</a>. </p><p>The fact that many large celestial objects are approximately spheres makes it easier to calculate their surface gravity. According to the <a href="/wiki/Shell_theorem" title="Shell theorem">shell theorem</a>, the gravitational force outside a spherically symmetric body is the same as if its entire mass were concentrated in the center, as was established by <a href="/wiki/Sir_Isaac_Newton" class="mw-redirect" title="Sir Isaac Newton">Sir Isaac Newton</a>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Therefore, the surface gravity of a planet or star with a given mass will be approximately inversely proportional to the square of its <a href="/wiki/Radius" title="Radius">radius</a>, and the surface gravity of a planet or star with a given average density will be approximately proportional to its radius. For example, the recently discovered planet, <a href="/wiki/Gliese_581_c" class="mw-redirect" title="Gliese 581 c">Gliese 581 c</a>, has at least 5 times the mass of Earth, but is unlikely to have 5 times its surface gravity. If its mass is no more than 5 times that of the Earth, as is expected,<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> and if it is a rocky planet with a large iron core, it should have a radius approximately 50% larger than that of Earth.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-model_8-0" class="reference"><a href="#cite_note-model-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Gravity on such a planet's surface would be approximately 2.2 times as strong as on Earth. If it is an icy or watery planet, its radius might be as large as twice the Earth's, in which case its surface gravity might be no more than 1.25 times as strong as the Earth's.<sup id="cite_ref-model_8-1" class="reference"><a href="#cite_note-model-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>These proportionalities may be expressed by the formula: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\propto {\frac {m}{r^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>∝</mo> <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 g\propto {\frac {m}{r^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/620b77ae24e16c4c459486a02499d1e32bd20f17" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:7.153ex; height:5.009ex;" alt="{\displaystyle g\propto {\frac {m}{r^{2}}}}"></span> where <span class="texhtml"><i>g</i></span> is the surface gravity of an object, expressed as a multiple of the Earth's, <span class="texhtml"><i>m</i></span> is its mass, expressed as a multiple of the <a href="/wiki/Earth" title="Earth">Earth</a>'s mass (<span class="nowrap"><span data-sort-value="7024597600000000000♠"></span>5.976<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>24</sup> kg</span>) and <span class="texhtml"><i>r</i></span> its radius, expressed as a multiple of the Earth's (mean) radius (6,371 km).<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> For instance, <a href="/wiki/Mars" title="Mars">Mars</a> has a mass of <span class="nowrap"><span data-sort-value="7023641850000000000♠"></span>6.4185<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>23</sup> kg</span> = 0.107 Earth masses and a mean radius of 3,390 km = 0.532 Earth radii.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> The surface <a href="/wiki/Gravity_of_Mars" title="Gravity of Mars">gravity of Mars</a> is therefore approximately <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {0.107}{0.532^{2}}}=0.38}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>0.107</mn> <msup> <mn>0.532</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mn>0.38</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {0.107}{0.532^{2}}}=0.38}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e4854f012613388994a4343681233910bfc8ede" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.42ex; height:5.676ex;" alt="{\displaystyle {\frac {0.107}{0.532^{2}}}=0.38}"></span> times that of Earth. Without using the Earth as a reference body, the surface gravity may also be calculated directly from <a href="/wiki/Newton%27s_law_of_universal_gravitation" title="Newton's law of universal gravitation">Newton's law of universal gravitation</a>, which gives the formula <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g={\frac {GM}{r^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</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 g={\frac {GM}{r^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/853ff89e03328fcf0eef4b0a64e9d78a84172c72" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:9.32ex; height:5.676ex;" alt="{\displaystyle g={\frac {GM}{r^{2}}}}"></span> where <span class="texhtml"><i>M</i></span> is the mass of the object, <span class="texhtml"><i>r</i></span> is its radius, and <span class="texhtml"><i>G</i></span> is the <a href="/wiki/Gravitational_constant" title="Gravitational constant">gravitational constant</a>. If <span class="texhtml"><i>ρ</i> = <i>M</i>/<i>V</i></span> denote the mean <a href="/wiki/Density" title="Density">density</a> of the object, this can also be written as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g={\frac {4\pi }{3}}G\rho r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mi>π</mi> </mrow> <mn>3</mn> </mfrac> </mrow> <mi>G</mi> <mi>ρ</mi> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g={\frac {4\pi }{3}}G\rho r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9de94d854fd21e4cf5eb4f2ec76158ffc625e121" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.622ex; height:5.176ex;" alt="{\displaystyle g={\frac {4\pi }{3}}G\rho r}"></span> so that, for fixed mean density, the surface gravity <span class="texhtml"><i>g</i></span> is proportional to the radius <span class="texhtml"><i>r</i></span>. Solving for mass, this equation can be written as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g=G\left({\frac {4\pi \rho }{3}}\right)^{2/3}M^{1/3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>=</mo> <mi>G</mi> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mi>π</mi> <mi>ρ</mi> </mrow> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mrow> </msup> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g=G\left({\frac {4\pi \rho }{3}}\right)^{2/3}M^{1/3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61a47fb2e2fdf9ffce1d71e2c7db51b600a83d42" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.89ex; height:6.676ex;" alt="{\displaystyle g=G\left({\frac {4\pi \rho }{3}}\right)^{2/3}M^{1/3}}"></span> But density is not constant, but increases as the planet grows in size, as they are not incompressible bodies. That is why the experimental relationship between surface gravity and mass does not grow as 1/3 but as 1/2:<sup id="cite_ref-g_vs_M_11-0" class="reference"><a href="#cite_note-g_vs_M-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g=M^{1/2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>=</mo> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g=M^{1/2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/652fe7f5d0cf8eb7afe74dea00281aadb411afb2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.411ex; height:3.176ex;" alt="{\displaystyle g=M^{1/2}}"></span> here with <span class="texhtml"><i>g</i></span> in times Earth's surface gravity and <span class="texhtml"><i>M</i></span> in times Earth's mass. In fact, the exoplanets found fulfilling the former relationship have been found to be rocky planets.<sup id="cite_ref-g_vs_M_11-1" class="reference"><a href="#cite_note-g_vs_M-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> Thus, for rocky planets, density grows with mass as <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 \rho \propto M^{1/4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>∝</mo> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho \propto M^{1/4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9b81508c7ca9dc36735dc92c995f5756f321a02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.497ex; height:3.343ex;" alt="{\displaystyle \rho \propto M^{1/4}}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Gas_giants">Gas giants</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Surface_gravity&action=edit&section=2" title="Edit section: Gas giants"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For gas giant planets such as Jupiter, Saturn, Uranus, and Neptune, the surface gravity is given at the 1 bar pressure level in the atmosphere.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> It has been found that for giant planets with masses in the range up to 100 times Earth's mass, their gravity surface is nevertheless very similar and close to 1<span class="texhtml"><i>g</i></span>, a region named the <i>gravity plateau</i>.<sup id="cite_ref-g_vs_M_11-2" class="reference"><a href="#cite_note-g_vs_M-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Non-spherically_symmetric_objects">Non-spherically symmetric objects</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Surface_gravity&action=edit&section=3" title="Edit section: Non-spherically symmetric objects"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Most real astronomical objects are not perfectly spherically symmetric. One reason for this is that they are often rotating, which means that they are affected by the combined effects of <a href="/wiki/Gravity" title="Gravity">gravitational force</a> and <a href="/wiki/Centrifugal_force" title="Centrifugal force">centrifugal force</a>. This causes stars and planets to be <a href="/wiki/Oblateness" class="mw-redirect" title="Oblateness">oblate</a>, which means that their surface gravity is smaller at the equator than at the poles. This effect was exploited by <a href="/wiki/Hal_Clement" title="Hal Clement">Hal Clement</a> in his SF novel <i><a href="/wiki/Mission_of_Gravity" title="Mission of Gravity">Mission of Gravity</a></i>, dealing with a massive, fast-spinning planet where gravity was much higher at the poles than at the equator. </p><p>To the extent that an object's internal distribution of mass differs from a symmetric model, the measured surface gravity may be used to deduce things about the object's internal structure. This fact has been put to practical use since 1915–1916, when <a href="/wiki/Roland_E%C3%B6tv%C3%B6s" class="mw-redirect" title="Roland Eötvös">Roland Eötvös</a>'s <a href="/wiki/Torsion_balance" class="mw-redirect" title="Torsion balance">torsion balance</a> was used to prospect for <a href="/wiki/Oil" title="Oil">oil</a> near the city of <a href="/wiki/Egbell" class="mw-redirect" title="Egbell">Egbell</a> (now <a href="/wiki/Gbely" title="Gbely">Gbely</a>, <a href="/wiki/Slovakia" title="Slovakia">Slovakia</a>.)<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><sup class="reference nowrap"><span title="Page: 1663">: 1663 </span></sup><sup id="cite_ref-hung_14-0" class="reference"><a href="#cite_note-hung-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 223">: 223 </span></sup> In 1924, the torsion balance was used to locate the <a href="/w/index.php?title=Nash_Dome&action=edit&redlink=1" class="new" title="Nash Dome (page does not exist)">Nash Dome</a> oil fields in <a href="/wiki/Texas" title="Texas">Texas</a>.<sup id="cite_ref-hung_14-1" class="reference"><a href="#cite_note-hung-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 223">: 223 </span></sup> </p><p>It is sometimes useful to calculate the surface gravity of simple hypothetical objects which are not found in nature. The surface gravity of infinite planes, tubes, lines, hollow shells, cones, and even more unrealistic structures may be used to provide insights into the behavior of real structures. </p> <div class="mw-heading mw-heading2"><h2 id="Black_holes">Black holes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Surface_gravity&action=edit&section=4" title="Edit section: Black holes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In relativity, the Newtonian concept of acceleration turns out not to be clear cut. For a black hole, which must be treated relativistically, one cannot define a surface gravity as the acceleration experienced by a test body at the object's surface because there is no surface, although the event horizon is a natural alternative candidate, but this still presents a problem because the acceleration of a test body at the event horizon of a black hole turns out to be infinite in relativity. Because of this, a renormalized value is used that corresponds to the Newtonian value in the non-relativistic limit. The value used is generally the local proper acceleration (which diverges at the event horizon) multiplied by the <a href="/wiki/Gravitational_time_dilation" title="Gravitational time dilation">gravitational time dilation</a> factor (which goes to zero at the event horizon). For the Schwarzschild case, this value is mathematically well behaved for all non-zero values of <span class="texhtml"><i>r</i></span> and <span class="texhtml"><i>M</i></span>. </p><p>When one talks about the surface gravity of a black hole, one is defining a notion that behaves analogously to the Newtonian surface gravity, but is not the same thing. In fact, the surface gravity of a general black hole is not well defined. However, one can define the surface gravity for a black hole whose event horizon is a Killing horizon. </p><p>The surface gravity <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 \kappa }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54ddec2e922c5caea4e47d04feef86e782dc8e6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }"></span> of a static <a href="/wiki/Killing_horizon" title="Killing horizon">Killing horizon</a> is the acceleration, as exerted at infinity, needed to keep an object at the horizon. Mathematically, if <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 k^{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k^{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f616f6d9abc36f4acf71805487e34f584308fe6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.313ex; height:2.343ex;" alt="{\displaystyle k^{a}}"></span> is a suitably normalized <a href="/wiki/Killing_vector" class="mw-redirect" title="Killing vector">Killing vector</a>, then the surface gravity is defined by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k^{a}\,\nabla _{a}k^{b}=\kappa k^{b},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <mspace width="thinmathspace" /> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msup> <mo>=</mo> <mi>κ</mi> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k^{a}\,\nabla _{a}k^{b}=\kappa k^{b},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6e8e04c68e8153cc230ff0593a6a32058c6d2f4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.12ex; height:3.009ex;" alt="{\displaystyle k^{a}\,\nabla _{a}k^{b}=\kappa k^{b},}"></span> where the equation is evaluated at the horizon. For a static and asymptotically flat spacetime, the normalization should be chosen so that <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 k^{a}k_{a}\to -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo stretchy="false">→</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k^{a}k_{a}\to -1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95575c4932d63312072b6ed3ed26e76698b9b2c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.211ex; height:2.676ex;" alt="{\displaystyle k^{a}k_{a}\to -1}"></span> as <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\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\to \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcd3a85ea2e3d6b4027434e502cace4177d7a3e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.986ex; height:1.843ex;" alt="{\displaystyle r\to \infty }"></span>, and so that <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 \kappa \geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa \geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9116cdbe7518638eb7b26515e903573d6f920d79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.6ex; height:2.343ex;" alt="{\displaystyle \kappa \geq 0}"></span>. For the Schwarzschild solution, take <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 k^{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k^{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f616f6d9abc36f4acf71805487e34f584308fe6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.313ex; height:2.343ex;" alt="{\displaystyle k^{a}}"></span> to be the <a href="/wiki/Time_translation" class="mw-redirect" title="Time translation">time translation</a> <a href="/wiki/Killing_vector" class="mw-redirect" title="Killing vector">Killing vector</a> <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 k^{a}\partial _{a}={\frac {\partial }{\partial t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle k^{a}\partial _{a}={\frac {\partial }{\partial t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a79e8690f73550d0dbd25b6fa6007d0001f8b095" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:10.11ex; height:3.843ex;" alt="{\textstyle k^{a}\partial _{a}={\frac {\partial }{\partial t}}}"></span>, and more generally for the <a href="/wiki/Kerr%E2%80%93Newman_solution" class="mw-redirect" title="Kerr–Newman solution">Kerr–Newman solution</a> take <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 k^{a}\partial _{a}={\frac {\partial }{\partial t}}+\Omega {\frac {\partial }{\partial \varphi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle k^{a}\partial _{a}={\frac {\partial }{\partial t}}+\Omega {\frac {\partial }{\partial \varphi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8baa62c9c3baa60378ab9f84a08ff97f3461da3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:17.471ex; height:4.176ex;" alt="{\textstyle k^{a}\partial _{a}={\frac {\partial }{\partial t}}+\Omega {\frac {\partial }{\partial \varphi }}}"></span>, the linear combination of the time translation and axisymmetry Killing vectors which is null at the horizon, 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 \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }"></span> is the angular velocity. </p> <div class="mw-heading mw-heading3"><h3 id="Schwarzschild_solution">Schwarzschild solution</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Surface_gravity&action=edit&section=5" title="Edit section: Schwarzschild solution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Since <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 k^{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k^{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f616f6d9abc36f4acf71805487e34f584308fe6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.313ex; height:2.343ex;" alt="{\displaystyle k^{a}}"></span> is a Killing vector <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 k^{a}\,\nabla _{a}k^{b}=\kappa k^{b}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <mspace width="thinmathspace" /> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msup> <mo>=</mo> <mi>κ</mi> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k^{a}\,\nabla _{a}k^{b}=\kappa k^{b}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a866eb312dd1b2112a477dae55b5c7464f2aed7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.473ex; height:3.009ex;" alt="{\displaystyle k^{a}\,\nabla _{a}k^{b}=\kappa k^{b}}"></span> implies <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 -k^{a}\,\nabla ^{b}k_{a}=\kappa k^{b}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <mspace width="thinmathspace" /> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msup> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>=</mo> <mi>κ</mi> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -k^{a}\,\nabla ^{b}k_{a}=\kappa k^{b}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fa93aca48cdbfed5aba390035d10246015cf7ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.281ex; height:3.009ex;" alt="{\displaystyle -k^{a}\,\nabla ^{b}k_{a}=\kappa k^{b}}"></span>. In <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 (t,r,\theta ,\varphi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>r</mi> <mo>,</mo> <mi>θ</mi> <mo>,</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (t,r,\theta ,\varphi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e79f5224bc4883bd5ac5706182f564d8f946468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.41ex; height:2.843ex;" alt="{\displaystyle (t,r,\theta ,\varphi )}"></span> coordinates <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 k^{a}=(1,0,0,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k^{a}=(1,0,0,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aeb9ace39c00f4373d271070593356e9428b4d0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.972ex; height:2.843ex;" alt="{\displaystyle k^{a}=(1,0,0,0)}"></span>. Performing a coordinate change to the advanced Eddington–Finklestein coordinates <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 v=t+r+2M\ln |r-2M|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mi>t</mi> <mo>+</mo> <mi>r</mi> <mo>+</mo> <mn>2</mn> <mi>M</mi> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>r</mi> <mo>−</mo> <mn>2</mn> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle v=t+r+2M\ln |r-2M|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e30e78eb1e3b0d28d457bf110fd857023522266" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.901ex; height:2.843ex;" alt="{\textstyle v=t+r+2M\ln |r-2M|}"></span> causes the metric to take the form <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ds^{2}=-\left(1-{\frac {2M}{r}}\right)\,dv^{2}+\left(dv\,dr+\,dr\,dv\right)+r^{2}\left(d\theta ^{2}+\sin ^{2}\theta \,d\varphi ^{2}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>M</mi> </mrow> <mi>r</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mi>d</mi> <mi>v</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>r</mi> <mo>+</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>r</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>v</mi> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mi>d</mi> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ds^{2}=-\left(1-{\frac {2M}{r}}\right)\,dv^{2}+\left(dv\,dr+\,dr\,dv\right)+r^{2}\left(d\theta ^{2}+\sin ^{2}\theta \,d\varphi ^{2}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d76f1ce38aeb3ada237f4a7dd1dd20ef4713840c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:66.818ex; height:6.176ex;" alt="{\displaystyle ds^{2}=-\left(1-{\frac {2M}{r}}\right)\,dv^{2}+\left(dv\,dr+\,dr\,dv\right)+r^{2}\left(d\theta ^{2}+\sin ^{2}\theta \,d\varphi ^{2}\right).}"></span> </p><p>Under a general change of coordinates the Killing vector transforms as <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 k^{v}=A_{t}^{v}k^{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msup> <mo>=</mo> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msubsup> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k^{v}=A_{t}^{v}k^{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4b1d7f758673bafbc3f36b5f6239b8a09064641" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.149ex; height:3.176ex;" alt="{\displaystyle k^{v}=A_{t}^{v}k^{t}}"></span> giving the vectors <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 k^{a'}=\delta _{v}^{a'}=(1,0,0,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>a</mi> <mo>′</mo> </msup> </mrow> </msup> <mo>=</mo> <msubsup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>a</mi> <mo>′</mo> </msup> </mrow> </msubsup> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k^{a'}=\delta _{v}^{a'}=(1,0,0,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a78cbd355c05e48fdfad1bb95486d0334e014192" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.29ex; height:3.343ex;" alt="{\displaystyle k^{a'}=\delta _{v}^{a'}=(1,0,0,0)}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle k_{a'}=g_{a'v}=\left(-1+{\frac {2M}{r}},1,0,0\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>a</mi> <mo>′</mo> </msup> </mrow> </msub> <mo>=</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>a</mi> <mo>′</mo> </msup> <mi>v</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>M</mi> </mrow> <mi>r</mi> </mfrac> </mrow> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle k_{a'}=g_{a'v}=\left(-1+{\frac {2M}{r}},1,0,0\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ae39677bcd978a1ed54b441db13546c8ee26b25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:32.176ex; height:4.843ex;" alt="{\textstyle k_{a'}=g_{a'v}=\left(-1+{\frac {2M}{r}},1,0,0\right).}"></span> </p><p>Considering the <span class="texhtml"><i>b</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 v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span></span> entry for <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 k^{a}\,\nabla _{a}k^{b}=\kappa k^{b}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <mspace width="thinmathspace" /> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msup> <mo>=</mo> <mi>κ</mi> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k^{a}\,\nabla _{a}k^{b}=\kappa k^{b}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a866eb312dd1b2112a477dae55b5c7464f2aed7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.473ex; height:3.009ex;" alt="{\displaystyle k^{a}\,\nabla _{a}k^{b}=\kappa k^{b}}"></span> gives the differential equation <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}}{\frac {\partial }{\partial r}}\left(-1+{\frac {2M}{r}}\right)=\kappa .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>M</mi> </mrow> <mi>r</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>κ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle -{\frac {1}{2}}{\frac {\partial }{\partial r}}\left(-1+{\frac {2M}{r}}\right)=\kappa .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4038c2aa22bc132ae411608b63c30fba980efa95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.419ex; height:4.843ex;" alt="{\textstyle -{\frac {1}{2}}{\frac {\partial }{\partial r}}\left(-1+{\frac {2M}{r}}\right)=\kappa .}"></span> </p><p>Therefore, the surface gravity for the <a href="/wiki/Schwarzschild_solution" class="mw-redirect" title="Schwarzschild solution">Schwarzschild solution</a> with 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/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></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 \kappa ={\frac {1}{4M}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>M</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa ={\frac {1}{4M}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c943af396c876f643069ec56dacc11b716dd04d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.878ex; height:5.176ex;" alt="{\displaystyle \kappa ={\frac {1}{4M}}}"></span> (<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 \kappa ={c^{4}}/{4GM}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mi>G</mi> <mi>M</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa ={c^{4}}/{4GM}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5c181ce4298afbcc8a20a8e9804632b860d6de3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.092ex; height:3.176ex;" alt="{\displaystyle \kappa ={c^{4}}/{4GM}}"></span> in SI units).<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> <div class="mw-heading mw-heading3"><h3 id="Kerr_solution">Kerr solution</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Surface_gravity&action=edit&section=6" title="Edit section: Kerr solution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The surface gravity for the uncharged, rotating black hole is, simply <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa =g-k,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> <mo>=</mo> <mi>g</mi> <mo>−</mo> <mi>k</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa =g-k,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12d231ca17771781cfbfbaac40d58b0f657cab18" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.252ex; height:2.509ex;" alt="{\displaystyle \kappa =g-k,}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle g={\frac {1}{4M}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>g</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>M</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle g={\frac {1}{4M}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76231949e8805ea2f0b1273ca37986d24c969d90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.599ex; height:3.509ex;" alt="{\textstyle g={\frac {1}{4M}}}"></span> is the Schwarzschild surface gravity, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k:=M\Omega _{+}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>:=</mo> <mi>M</mi> <msubsup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k:=M\Omega _{+}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d6fe03eb4856e467d4c29a2053088fb6d64171c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.588ex; height:3.176ex;" alt="{\displaystyle k:=M\Omega _{+}^{2}}"></span> is the spring constant of the rotating black hole. <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 \Omega _{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega _{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aece5d68daf75b9115c8bda0d66c4a0c1d462888" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.189ex; height:2.509ex;" alt="{\displaystyle \Omega _{+}}"></span> is the angular velocity at the event horizon. This expression gives a simple Hawking temperature 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 2\pi T=g-k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <mi>T</mi> <mo>=</mo> <mi>g</mi> <mo>−</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi T=g-k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3c9827f76480592360bc636eebf0d9909d3ca0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.397ex; height:2.509ex;" alt="{\displaystyle 2\pi T=g-k}"></span>.<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> </p> <div class="mw-heading mw-heading3"><h3 id="Kerr–Newman_solution"><span id="Kerr.E2.80.93Newman_solution"></span>Kerr–Newman solution</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Surface_gravity&action=edit&section=7" title="Edit section: Kerr–Newman solution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The surface gravity for the <a href="/wiki/Kerr%E2%80%93Newman_solution" class="mw-redirect" title="Kerr–Newman solution">Kerr–Newman solution</a> is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa ={\frac {r_{+}-r_{-}}{2\left(r_{+}^{2}+a^{2}\right)}}={\frac {\sqrt {M^{2}-Q^{2}-J^{2}/M^{2}}}{2M^{2}-Q^{2}+2M{\sqrt {M^{2}-Q^{2}-J^{2}/M^{2}}}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> </mrow> <mrow> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> <mrow> <mn>2</mn> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa ={\frac {r_{+}-r_{-}}{2\left(r_{+}^{2}+a^{2}\right)}}={\frac {\sqrt {M^{2}-Q^{2}-J^{2}/M^{2}}}{2M^{2}-Q^{2}+2M{\sqrt {M^{2}-Q^{2}-J^{2}/M^{2}}}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/366d6fa0bd08522e8b5d5d2234ddfacb830aad73" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:59.857ex; height:7.509ex;" alt="{\displaystyle \kappa ={\frac {r_{+}-r_{-}}{2\left(r_{+}^{2}+a^{2}\right)}}={\frac {\sqrt {M^{2}-Q^{2}-J^{2}/M^{2}}}{2M^{2}-Q^{2}+2M{\sqrt {M^{2}-Q^{2}-J^{2}/M^{2}}}}},}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"></span> is the electric charge, <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 J}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/359e4f407b49910e02c27c2f52e87a36cd74c053" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}"></span> is the angular momentum, define <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 r_{\pm }:=M\pm {\sqrt {M^{2}-Q^{2}-J^{2}/M^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>±</mo> </mrow> </msub> <mo>:=</mo> <mi>M</mi> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle r_{\pm }:=M\pm {\sqrt {M^{2}-Q^{2}-J^{2}/M^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/429593999482dc90175907dc76130de7bb8ea7a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:33.333ex; height:3.343ex;" alt="{\textstyle r_{\pm }:=M\pm {\sqrt {M^{2}-Q^{2}-J^{2}/M^{2}}}}"></span> to be the locations of the two horizons and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a:=J/M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>:=</mo> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a:=J/M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59ef26abe0b96635502bb56f551ebfb0edddb84b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.051ex; height:2.843ex;" alt="{\displaystyle a:=J/M}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Dynamical_black_holes">Dynamical black holes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Surface_gravity&action=edit&section=8" title="Edit section: Dynamical black holes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Surface gravity for stationary black holes is well defined. This is because all stationary black holes have a horizon that is Killing.<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> Recently there has been a shift towards defining the surface gravity of dynamical black holes whose spacetime does not admit a timelike <a href="/wiki/Killing_vector_field" title="Killing vector field">Killing vector (field)</a>.<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> Several definitions have been proposed over the years by various authors, such as peeling surface gravity and Kodama surface gravity.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> As of current, there is no consensus or agreement on which definition, if any, is correct.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Semiclassical_gravity" title="Semiclassical gravity">Semiclassical</a> results indicate that the peeling surface gravity is ill-defined for transient objects formed in finite time of a distant observer.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p> <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=Surface_gravity&action=edit&section=9" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output 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