Orbit - Wikipedia

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<a class="vector-toc-link" href="#Newton's_laws_of_motion"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Newton's laws of motion</span> </div> </a> <button aria-controls="toc-Newton's_laws_of_motion-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 Newton's laws of motion subsection</span> </button> <ul id="toc-Newton's_laws_of_motion-sublist" class="vector-toc-list"> <li id="toc-Newton's_law_of_gravitation_and_laws_of_motion_for_two-body_problems" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Newton's_law_of_gravitation_and_laws_of_motion_for_two-body_problems"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Newton's law of gravitation and laws of motion for two-body problems</span> </div> </a> <ul id="toc-Newton's_law_of_gravitation_and_laws_of_motion_for_two-body_problems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Defining_gravitational_potential_energy" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Defining_gravitational_potential_energy"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Defining gravitational potential energy</span> </div> </a> <ul id="toc-Defining_gravitational_potential_energy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Orbital_energies_and_orbit_shapes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Orbital_energies_and_orbit_shapes"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Orbital energies and orbit shapes</span> </div> </a> <ul id="toc-Orbital_energies_and_orbit_shapes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kepler's_laws" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Kepler's_laws"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Kepler's laws</span> </div> </a> <ul id="toc-Kepler's_laws-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Limitations_of_Newton's_law_of_gravitation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Limitations_of_Newton's_law_of_gravitation"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5</span> <span>Limitations of Newton's law of gravitation</span> </div> </a> <ul id="toc-Limitations_of_Newton's_law_of_gravitation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Approaches_to_many-body_problems" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Approaches_to_many-body_problems"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.6</span> <span>Approaches to many-body problems</span> </div> </a> <ul id="toc-Approaches_to_many-body_problems-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Formulation" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Formulation"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Formulation</span> </div> </a> <button aria-controls="toc-Formulation-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 Formulation subsection</span> </button> <ul id="toc-Formulation-sublist" class="vector-toc-list"> <li id="toc-Newtonian_analysis_of_orbital_motion" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Newtonian_analysis_of_orbital_motion"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Newtonian analysis of orbital motion</span> </div> </a> <ul id="toc-Newtonian_analysis_of_orbital_motion-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relativistic_orbital_motion" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relativistic_orbital_motion"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Relativistic orbital motion</span> </div> </a> <ul id="toc-Relativistic_orbital_motion-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Specification" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Specification"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Specification</span> </div> </a> <button aria-controls="toc-Specification-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 Specification subsection</span> </button> <ul id="toc-Specification-sublist" class="vector-toc-list"> <li id="toc-Orbital_planes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Orbital_planes"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Orbital planes</span> </div> </a> <ul id="toc-Orbital_planes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Orbital_period" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Orbital_period"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Orbital period</span> </div> </a> <ul id="toc-Orbital_period-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Perturbations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Perturbations"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Perturbations</span> </div> </a> <button aria-controls="toc-Perturbations-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 Perturbations subsection</span> </button> <ul id="toc-Perturbations-sublist" class="vector-toc-list"> <li id="toc-Radial,_prograde_and_transverse_perturbations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Radial,_prograde_and_transverse_perturbations"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Radial, prograde and transverse perturbations</span> </div> </a> <ul id="toc-Radial,_prograde_and_transverse_perturbations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Orbital_decay" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Orbital_decay"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>Orbital decay</span> </div> </a> <ul id="toc-Orbital_decay-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Oblateness" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Oblateness"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.3</span> <span>Oblateness</span> </div> </a> <ul id="toc-Oblateness-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Multiple_gravitating_bodies" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Multiple_gravitating_bodies"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.4</span> <span>Multiple gravitating bodies</span> </div> </a> <ul id="toc-Multiple_gravitating_bodies-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Light_radiation_and_stellar_wind" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Light_radiation_and_stellar_wind"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.5</span> <span>Light radiation and stellar wind</span> </div> </a> <ul id="toc-Light_radiation_and_stellar_wind-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Strange_orbits" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Strange_orbits"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Strange orbits</span> </div> </a> <ul id="toc-Strange_orbits-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Astrodynamics" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Astrodynamics"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Astrodynamics</span> </div> </a> <ul id="toc-Astrodynamics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Earth_orbits" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Earth_orbits"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Earth orbits</span> </div> </a> <ul id="toc-Earth_orbits-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Scaling_in_gravity" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Scaling_in_gravity"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Scaling in gravity</span> </div> </a> <ul id="toc-Scaling_in_gravity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tidal_locking" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Tidal_locking"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>Tidal locking</span> </div> </a> <ul id="toc-Tidal_locking-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">14</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">15</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">16</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">Orbit</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 108 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-108" 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">108 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Wentelbaan" title="Wentelbaan – Afrikaans" lang="af" hreflang="af" data-title="Wentelbaan" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Umlaufbahn" title="Umlaufbahn – Alemannic" lang="gsw" hreflang="gsw" data-title="Umlaufbahn" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%AF%D8%A7%D8%B1" title="مدار – Arabic" lang="ar" hreflang="ar" data-title="مدار" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Orbita" title="Orbita – Aragonese" lang="an" hreflang="an" data-title="Orbita" data-language-autonym="Aragonés" data-language-local-name="Aragonese" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-arc mw-list-item"><a href="https://arc.wikipedia.org/wiki/%DC%AB%DC%92%DC%9D%DC%A0%DC%90_(%DC%A1%DC%A1%DC%A0%DC%A0%DC%98%DC%AC_%DC%9F%DC%98%DC%9F%DC%92%DC%90)" title="ܫܒܝܠܐ (ܡܡܠܠܘܬ ܟܘܟܒܐ) – Aramaic" lang="arc" hreflang="arc" data-title="ܫܒܝܠܐ (ܡܡܠܠܘܬ ܟܘܟܒܐ)" data-language-autonym="ܐܪܡܝܐ" data-language-local-name="Aramaic" class="interlanguage-link-target"><span>ܐܪܡܝܐ</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/%C3%93rbita" title="Órbita – Asturian" lang="ast" hreflang="ast" data-title="Órbita" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Orbit" title="Orbit – Azerbaijani" lang="az" hreflang="az" data-title="Orbit" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D8%A7%D9%88%D8%B1%D8%A8%DB%8C%D8%AA" title="اوربیت – South Azerbaijani" lang="azb" hreflang="azb" data-title="اوربیت" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%95%E0%A6%95%E0%A7%8D%E0%A6%B7%E0%A6%AA%E0%A6%A5_(%E0%A6%97%E0%A7%8D%E0%A6%B0%E0%A6%B9)" title="কক্ষপথ (গ্রহ) – Bangla" lang="bn" hreflang="bn" data-title="কক্ষপথ (গ্রহ)" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%90%D1%80%D0%B1%D1%96%D1%82%D0%B0" title="Арбіта – Belarusian" lang="be" hreflang="be" data-title="Арбіта" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%90%D1%80%D0%B1%D1%96%D1%82%D0%B0" title="Арбіта – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Арбіта" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bh mw-list-item"><a href="https://bh.wikipedia.org/wiki/%E0%A4%95%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A4%BE_(%E0%A4%96%E0%A4%97%E0%A5%8B%E0%A4%B2)" title="कक्षा (खगोल) – Bhojpuri" lang="bh" hreflang="bh" data-title="कक्षा (खगोल)" data-language-autonym="भोजपुरी" data-language-local-name="Bhojpuri" class="interlanguage-link-target"><span>भोजपुरी</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9E%D1%80%D0%B1%D0%B8%D1%82%D0%B0" title="Орбита – Bulgarian" lang="bg" hreflang="bg" data-title="Орбита" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Planetarna_putanja" title="Planetarna putanja – Bosnian" lang="bs" hreflang="bs" data-title="Planetarna putanja" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Orbitenn" title="Orbitenn – Breton" lang="br" hreflang="br" data-title="Orbitenn" data-language-autonym="Brezhoneg" data-language-local-name="Breton" class="interlanguage-link-target"><span>Brezhoneg</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/%C3%92rbita" title="Òrbita – Catalan" lang="ca" hreflang="ca" data-title="Òrbita" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Ob%C4%9B%C5%BEn%C3%A1_dr%C3%A1ha" title="Oběžná dráha – Czech" lang="cs" hreflang="cs" data-title="Oběžná dráha" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Boterekwa" title="Boterekwa – Shona" lang="sn" hreflang="sn" data-title="Boterekwa" data-language-autonym="ChiShona" data-language-local-name="Shona" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Oml%C3%B8bsbane" title="Omløbsbane – Danish" lang="da" hreflang="da" data-title="Omløbsbane" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Umlaufbahn" title="Umlaufbahn – German" lang="de" hreflang="de" data-title="Umlaufbahn" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Orbiit" title="Orbiit – Estonian" lang="et" hreflang="et" data-title="Orbiit" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/%C3%93rbita" title="Órbita – Spanish" lang="es" hreflang="es" data-title="Órbita" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Orbito" title="Orbito – Esperanto" lang="eo" hreflang="eo" data-title="Orbito" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Orbita" title="Orbita – Basque" lang="eu" hreflang="eu" data-title="Orbita" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%AF%D8%A7%D8%B1_(%D8%B3%DB%8C%D8%A7%D8%B1%D9%87)" title="مدار (سیاره) – Persian" lang="fa" hreflang="fa" data-title="مدار (سیاره)" 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/Orbite" title="Orbite – French" lang="fr" hreflang="fr" data-title="Orbite" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-fy mw-list-item"><a href="https://fy.wikipedia.org/wiki/Baan_(astronomy)" title="Baan (astronomy) – Western Frisian" lang="fy" hreflang="fy" data-title="Baan (astronomy)" data-language-autonym="Frysk" data-language-local-name="Western Frisian" class="interlanguage-link-target"><span>Frysk</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Fithis" title="Fithis – Irish" lang="ga" hreflang="ga" data-title="Fithis" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gv mw-list-item"><a href="https://gv.wikipedia.org/wiki/Cruinlagh" title="Cruinlagh – Manx" lang="gv" hreflang="gv" data-title="Cruinlagh" data-language-autonym="Gaelg" data-language-local-name="Manx" class="interlanguage-link-target"><span>Gaelg</span></a></li><li class="interlanguage-link interwiki-gd mw-list-item"><a href="https://gd.wikipedia.org/wiki/Reul-chuairt" title="Reul-chuairt – Scottish Gaelic" lang="gd" hreflang="gd" data-title="Reul-chuairt" data-language-autonym="Gàidhlig" data-language-local-name="Scottish Gaelic" class="interlanguage-link-target"><span>Gàidhlig</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/%C3%93rbita" title="Órbita – Galician" lang="gl" hreflang="gl" data-title="Órbita" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B6%A4%EB%8F%84" title="궤도 – Korean" lang="ko" hreflang="ko" data-title="궤도" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%88%D6%82%D5%B2%D5%A5%D5%AE%D5%AB%D6%80" title="Ուղեծիր – Armenian" lang="hy" hreflang="hy" data-title="Ուղեծիր" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%95%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A4%BE_(%E0%A4%AD%E0%A5%8C%E0%A4%A4%E0%A4%BF%E0%A4%95%E0%A5%80)" title="कक्षा (भौतिकी) – Hindi" lang="hi" hreflang="hi" data-title="कक्षा (भौतिकी)" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Planetarna_putanja" title="Planetarna putanja – Croatian" lang="hr" hreflang="hr" data-title="Planetarna putanja" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Orbito" title="Orbito – Ido" lang="io" hreflang="io" data-title="Orbito" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-ig mw-list-item"><a href="https://ig.wikipedia.org/wiki/Ebe_a_na-agba_ya" title="Ebe a na-agba ya – Igbo" lang="ig" hreflang="ig" data-title="Ebe a na-agba ya" data-language-autonym="Igbo" data-language-local-name="Igbo" class="interlanguage-link-target"><span>Igbo</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Orbit" title="Orbit – Indonesian" lang="id" hreflang="id" data-title="Orbit" 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-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Orbita" title="Orbita – Interlingua" lang="ia" hreflang="ia" data-title="Orbita" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-zu mw-list-item"><a href="https://zu.wikipedia.org/wiki/Uzungezohlelo" title="Uzungezohlelo – Zulu" lang="zu" hreflang="zu" data-title="Uzungezohlelo" data-language-autonym="IsiZulu" data-language-local-name="Zulu" class="interlanguage-link-target"><span>IsiZulu</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Orbita" title="Orbita – Italian" lang="it" hreflang="it" data-title="Orbita" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A1%D7%9C%D7%95%D7%9C_%D7%9B%D7%91%D7%99%D7%93%D7%AA%D7%99" title="מסלול כבידתי – Hebrew" lang="he" hreflang="he" data-title="מסלול כבידתי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Orbit" title="Orbit – Javanese" lang="jv" hreflang="jv" data-title="Orbit" data-language-autonym="Jawa" data-language-local-name="Javanese" class="interlanguage-link-target"><span>Jawa</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%95%E0%B2%95%E0%B3%8D%E0%B2%B7%E0%B3%86" title="ಕಕ್ಷೆ – Kannada" lang="kn" hreflang="kn" data-title="ಕಕ್ಷೆ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-krc mw-list-item"><a href="https://krc.wikipedia.org/wiki/%D0%A7%D0%BE%D1%80%D1%85" title="Чорх – Karachay-Balkar" lang="krc" hreflang="krc" data-title="Чорх" data-language-autonym="Къарачай-малкъар" data-language-local-name="Karachay-Balkar" class="interlanguage-link-target"><span>Къарачай-малкъар</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9D%E1%83%A0%E1%83%91%E1%83%98%E1%83%A2%E1%83%90" title="ორბიტა – Georgian" lang="ka" hreflang="ka" data-title="ორბიტა" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9E%D1%80%D0%B1%D0%B8%D1%82%D0%B0" title="Орбита – Kazakh" lang="kk" hreflang="kk" data-title="Орбита" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Mzunguko" title="Mzunguko – Swahili" lang="sw" hreflang="sw" data-title="Mzunguko" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/%C3%92bit" title="Òbit – Haitian Creole" lang="ht" hreflang="ht" data-title="Òbit" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/%C3%92rbit" title="Òrbit – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Òrbit" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/Xulgeh" title="Xulgeh – Kurdish" lang="ku" hreflang="ku" data-title="Xulgeh" data-language-autonym="Kurdî" data-language-local-name="Kurdish" class="interlanguage-link-target"><span>Kurdî</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%9E%D1%80%D0%B1%D0%B8%D1%82%D0%B0" title="Орбита – Kyrgyz" lang="ky" hreflang="ky" data-title="Орбита" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Orbita" title="Orbita – Latin" lang="la" hreflang="la" data-title="Orbita" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Orb%C4%ABta" title="Orbīta – Latvian" lang="lv" hreflang="lv" data-title="Orbīta" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/%C3%8Bmlafbunn" title="Ëmlafbunn – Luxembourgish" lang="lb" hreflang="lb" data-title="Ëmlafbunn" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxembourgish" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Orbita" title="Orbita – Lithuanian" lang="lt" hreflang="lt" data-title="Orbita" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Orbita" title="Orbita – Lombard" lang="lmo" hreflang="lmo" data-title="Orbita" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9E%D1%80%D0%B1%D0%B8%D1%82%D0%B0" title="Орбита – Macedonian" lang="mk" hreflang="mk" data-title="Орбита" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%AD%E0%B5%8D%E0%B4%B0%E0%B4%AE%E0%B4%A3%E0%B4%AA%E0%B4%A5%E0%B4%82" title="ഭ്രമണപഥം – Malayalam" lang="ml" hreflang="ml" data-title="ഭ്രമണപഥം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%95%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A4%BE" title="कक्षा – Marathi" lang="mr" hreflang="mr" data-title="कक्षा" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Orbit" title="Orbit – Malay" lang="ms" hreflang="ms" data-title="Orbit" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-min mw-list-item"><a href="https://min.wikipedia.org/wiki/Orbit" title="Orbit – Minangkabau" lang="min" hreflang="min" data-title="Orbit" data-language-autonym="Minangkabau" data-language-local-name="Minangkabau" class="interlanguage-link-target"><span>Minangkabau</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%A2%D0%BE%D0%B9%D1%80%D0%BE%D0%B3_%D0%B7%D0%B0%D0%BC" title="Тойрог зам – Mongolian" lang="mn" hreflang="mn" data-title="Тойрог зам" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%95%E1%80%90%E1%80%BA%E1%80%9C%E1%80%99%E1%80%BA%E1%80%B8%E1%80%80%E1%80%BC%E1%80%B1%E1%80%AC%E1%80%84%E1%80%BA%E1%80%B8" title="ပတ်လမ်းကြောင်း – Burmese" lang="my" hreflang="my" data-title="ပတ်လမ်းကြောင်း" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Baan_(hemellichaam)" title="Baan (hemellichaam) – Dutch" lang="nl" hreflang="nl" data-title="Baan (hemellichaam)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E8%BB%8C%E9%81%93_(%E5%8A%9B%E5%AD%A6)" title="軌道 (力学) – Japanese" lang="ja" hreflang="ja" data-title="軌道 (力学)" 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/Bane" title="Bane – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Bane" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Bane" title="Bane – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Bane" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Orbita" title="Orbita – Occitan" lang="oc" hreflang="oc" data-title="Orbita" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Orbiiti" title="Orbiiti – Oromo" lang="om" hreflang="om" data-title="Orbiiti" data-language-autonym="Oromoo" data-language-local-name="Oromo" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Orbitalar_haqida" title="Orbitalar haqida – Uzbek" lang="uz" hreflang="uz" data-title="Orbitalar haqida" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%AA%E0%A9%B0%E0%A8%A7_(%E0%A8%A4%E0%A8%BE%E0%A8%B0%E0%A8%BE_%E0%A8%B5%E0%A8%BF%E0%A8%97%E0%A8%BF%E0%A8%86%E0%A8%A8)" title="ਪੰਧ (ਤਾਰਾ ਵਿਗਿਆਨ) – Punjabi" lang="pa" hreflang="pa" data-title="ਪੰਧ (ਤਾਰਾ ਵਿਗਿਆਨ)" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D9%85%D8%AF%D8%A7%D8%B1" title="مدار – Western Punjabi" lang="pnb" hreflang="pnb" data-title="مدار" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Aabit" title="Aabit – Jamaican Creole English" lang="jam" hreflang="jam" data-title="Aabit" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/%C3%9Cmloopbahn" title="Ümloopbahn – Low German" lang="nds" hreflang="nds" data-title="Ümloopbahn" data-language-autonym="Plattdüütsch" data-language-local-name="Low German" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Orbita" title="Orbita – Polish" lang="pl" hreflang="pl" data-title="Orbita" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/%C3%93rbita" title="Órbita – Portuguese" lang="pt" hreflang="pt" data-title="Órbita" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Orbit%C4%83_(astronomie)" title="Orbită (astronomie) – Romanian" lang="ro" hreflang="ro" data-title="Orbită (astronomie)" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-rue mw-list-item"><a href="https://rue.wikipedia.org/wiki/%D0%9E%D1%80%D0%B1%D0%B8%D1%82%D0%B0" title="Орбита – Rusyn" lang="rue" hreflang="rue" data-title="Орбита" data-language-autonym="Русиньскый" data-language-local-name="Rusyn" class="interlanguage-link-target"><span>Русиньскый</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9E%D1%80%D0%B1%D0%B8%D1%82%D0%B0" title="Орбита – Russian" lang="ru" hreflang="ru" data-title="Орбита" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%AD%D1%80%D0%B3%D0%B8%D0%B9%D1%8D%D1%80_%D0%B8%D0%B8" title="Эргийэр ии – Yakut" lang="sah" hreflang="sah" data-title="Эргийэр ии" data-language-autonym="Саха тыла" data-language-local-name="Yakut" class="interlanguage-link-target"><span>Саха тыла</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Orbita" title="Orbita – Albanian" lang="sq" hreflang="sq" data-title="Orbita" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/%C3%92rbita" title="Òrbita – Sicilian" lang="scn" hreflang="scn" data-title="Òrbita" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%9A%E0%B6%9A%E0%B7%8A%E0%B7%82" title="කක්ෂ – Sinhala" lang="si" hreflang="si" data-title="කක්ෂ" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Orbit" title="Orbit – Simple English" lang="en-simple" hreflang="en-simple" data-title="Orbit" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Obe%C5%BEn%C3%A1_dr%C3%A1ha" title="Obežná dráha – Slovak" lang="sk" hreflang="sk" data-title="Obežná dráha" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Tir" title="Tir – Slovenian" lang="sl" hreflang="sl" data-title="Tir" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%AE%D9%88%D9%84%DA%AF%DB%95" title="خولگە – Central Kurdish" lang="ckb" hreflang="ckb" data-title="خولگە" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9E%D1%80%D0%B1%D0%B8%D1%82%D0%B0" title="Орбита – Serbian" lang="sr" hreflang="sr" data-title="Орбита" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Orbita" title="Orbita – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Orbita" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Orbit" title="Orbit – Sundanese" lang="su" hreflang="su" data-title="Orbit" data-language-autonym="Sunda" data-language-local-name="Sundanese" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Kiertorata" title="Kiertorata – Finnish" lang="fi" hreflang="fi" data-title="Kiertorata" 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/Omloppsbana" title="Omloppsbana – Swedish" lang="sv" hreflang="sv" data-title="Omloppsbana" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Ligiran" title="Ligiran – Tagalog" lang="tl" hreflang="tl" data-title="Ligiran" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%9A%E0%AF%81%E0%AE%B1%E0%AF%8D%E0%AE%B1%E0%AF%81%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AE%BE%E0%AE%A4%E0%AF%88" title="சுற்றுப்பாதை – Tamil" lang="ta" hreflang="ta" data-title="சுற்றுப்பாதை" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/Timezzit" title="Timezzit – Kabyle" lang="kab" hreflang="kab" data-title="Timezzit" data-language-autonym="Taqbaylit" data-language-local-name="Kabyle" class="interlanguage-link-target"><span>Taqbaylit</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%95%E0%B0%95%E0%B1%8D%E0%B0%B7%E0%B1%8D%E0%B0%AF" title="కక్ష్య – Telugu" lang="te" hreflang="te" data-title="కక్ష్య" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%A7%E0%B8%87%E0%B9%82%E0%B8%84%E0%B8%88%E0%B8%A3" title="วงโคจร – Thai" lang="th" hreflang="th" data-title="วงโคจร" 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%B6r%C3%BCnge" title="Yörünge – Turkish" lang="tr" hreflang="tr" data-title="Yörünge" 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%9E%D1%80%D0%B1%D1%96%D1%82%D0%B0" title="Орбіта – Ukrainian" lang="uk" hreflang="uk" data-title="Орбіта" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%85%D8%AF%D8%A7%D8%B1" title="مدار – Urdu" lang="ur" hreflang="ur" data-title="مدار" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://vec.wikipedia.org/wiki/%C3%92rbita" title="Òrbita – Venetian" lang="vec" hreflang="vec" data-title="Òrbita" data-language-autonym="Vèneto" data-language-local-name="Venetian" class="interlanguage-link-target"><span>Vèneto</span></a></li><li class="interlanguage-link interwiki-vep mw-list-item"><a href="https://vep.wikipedia.org/wiki/Orbit" title="Orbit – Veps" lang="vep" hreflang="vep" data-title="Orbit" data-language-autonym="Vepsän kel’" data-language-local-name="Veps" class="interlanguage-link-target"><span>Vepsän kel’</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Qu%E1%BB%B9_%C4%91%E1%BA%A1o_(thi%C3%AAn_th%E1%BB%83)" title="Quỹ đạo (thiên thể) – Vietnamese" lang="vi" hreflang="vi" data-title="Quỹ đạo (thiên thể)" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Orbita" title="Orbita – Waray" lang="war" hreflang="war" data-title="Orbita" data-language-autonym="Winaray" data-language-local-name="Waray" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E8%BD%A8%E9%81%93%EF%BC%88%E5%8A%9B%E5%AD%A6%EF%BC%89" title="轨道（力学） – Wu" lang="wuu" hreflang="wuu" data-title="轨道（力学）" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E8%BB%8C%E9%81%93" title="軌道 – Cantonese" lang="yue" hreflang="yue" data-title="軌道" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E8%BD%A8%E9%81%93_(%E5%8A%9B%E5%AD%A6)" title="轨道 (力学) – Chinese" lang="zh" hreflang="zh" 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							cdx-button--size-medium cdx-button--icon-only cdx-dialog__header__close-button" aria-label="Close" onclick="document.getElementById("mw-fr-revision-details").style.display = "none";" type="submit"><span class="cdx-icon cdx-icon--medium
							cdx-fr-css-icon--close"></span></button></header><div class="cdx-dialog__body">This is the <a href="/wiki/Wikipedia:Pending_changes" title="Wikipedia:Pending changes">latest accepted revision</a>, <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Special:Log&type=review&page=Special:Badtitle/Message">reviewed</a> on <i>21 November 2024</i>.</div></div><div tabindex="0"></div></div></div></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Curved path of an object around a point</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is about orbits in celestial mechanics, due to gravity. For other uses, see <a href="/wiki/Orbit_(disambiguation)" class="mw-disambig" title="Orbit (disambiguation)">Orbit (disambiguation)</a>.</div> <p class="mw-empty-elt"> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Animation_of_C-2018_Y1_orbit_1600-2500.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Animation_of_C-2018_Y1_orbit_1600-2500.gif/330px-Animation_of_C-2018_Y1_orbit_1600-2500.gif" decoding="async" width="330" height="248" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Animation_of_C-2018_Y1_orbit_1600-2500.gif/495px-Animation_of_C-2018_Y1_orbit_1600-2500.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/f/f4/Animation_of_C-2018_Y1_orbit_1600-2500.gif 2x" data-file-width="560" data-file-height="420" /></a><figcaption>An animation showing a low <a href="/wiki/Orbital_eccentricity" title="Orbital eccentricity">eccentricity</a> orbit (near-circle, in red), and a high eccentricity orbit (ellipse, in purple)</figcaption></figure> <p>In <a href="/wiki/Celestial_mechanics" title="Celestial mechanics">celestial mechanics</a>, an <b>orbit</b> (also known as <b>orbital revolution</b>) is the curved <a href="/wiki/Trajectory" title="Trajectory">trajectory</a> of an <a href="/wiki/Physical_body" class="mw-redirect" title="Physical body">object</a><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> such as the trajectory of a <a href="/wiki/Planet" title="Planet">planet</a> around a star, or of a <a href="/wiki/Natural_satellite" title="Natural satellite">natural satellite</a> around a planet, or of an <a href="/wiki/Satellite" title="Satellite">artificial satellite</a> around an object or position in space such as a planet, moon, asteroid, or <a href="/wiki/Lagrange_point" title="Lagrange point">Lagrange point</a>. Normally, orbit refers to a regularly repeating trajectory, although it may also refer to a non-repeating trajectory. To a close approximation, planets and satellites follow <a href="/wiki/Elliptic_orbit" title="Elliptic orbit">elliptic orbits</a>, with the <a href="/wiki/Barycenter" class="mw-redirect" title="Barycenter">center of mass</a> being orbited at a focal point of the ellipse,<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> as described by <a href="/wiki/Kepler%27s_laws_of_planetary_motion" title="Kepler's laws of planetary motion">Kepler's laws of planetary motion</a>. </p><p>For most situations, orbital motion is adequately approximated by <a href="/wiki/Newtonian_mechanics" class="mw-redirect" title="Newtonian mechanics">Newtonian mechanics</a>, which explains <a href="/wiki/Newton%27s_law_of_universal_gravitation" title="Newton's law of universal gravitation">gravity</a> as a force obeying an <a href="/wiki/Inverse-square_law" title="Inverse-square law">inverse-square law</a>.<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> However, <a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a>'s <a href="/wiki/General_theory_of_relativity" class="mw-redirect" title="General theory of relativity">general theory of relativity</a>, which accounts for gravity as due to curvature of <a href="/wiki/Spacetime" title="Spacetime">spacetime</a>, with orbits following <a href="/wiki/Geodesic" title="Geodesic">geodesics</a>, provides a more accurate calculation and understanding of the exact mechanics of orbital motion. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orbit&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Planisphaerium_Ptolemaicum_siue_machina_orbium_mundi_ex_hypothesi_Ptolemaica_in_plano_disposita_(2709983277).jpg" class="mw-file-description"><img alt="Andreas Cellarius, a Dutch mathematician and geographer in the 17th century, compiled a celestial atlas with theories from astronomers like Ptolemy and Copernicus. This illustration shows the Earth at the center, with the Moon and planets orbiting around it, based on Ptolemy's geocentric model before Copernicus' heliocentric model." src="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/Planisphaerium_Ptolemaicum_siue_machina_orbium_mundi_ex_hypothesi_Ptolemaica_in_plano_disposita_%282709983277%29.jpg/220px-Planisphaerium_Ptolemaicum_siue_machina_orbium_mundi_ex_hypothesi_Ptolemaica_in_plano_disposita_%282709983277%29.jpg" decoding="async" width="220" height="184" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/Planisphaerium_Ptolemaicum_siue_machina_orbium_mundi_ex_hypothesi_Ptolemaica_in_plano_disposita_%282709983277%29.jpg/330px-Planisphaerium_Ptolemaicum_siue_machina_orbium_mundi_ex_hypothesi_Ptolemaica_in_plano_disposita_%282709983277%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/28/Planisphaerium_Ptolemaicum_siue_machina_orbium_mundi_ex_hypothesi_Ptolemaica_in_plano_disposita_%282709983277%29.jpg/440px-Planisphaerium_Ptolemaicum_siue_machina_orbium_mundi_ex_hypothesi_Ptolemaica_in_plano_disposita_%282709983277%29.jpg 2x" data-file-width="1200" data-file-height="1001" /></a><figcaption>The Earth-centered universe according to Ptolemy, illustration by Andreas Cellarius from Harmonia Macrocosmica, 1660</figcaption></figure> <p>Historically, the apparent motions of the planets were described by European and Arabic philosophers using the idea of <a href="/wiki/Celestial_spheres" title="Celestial spheres">celestial spheres</a>. This model posited the existence of perfect moving spheres or rings to which the stars and planets were attached. It assumed the heavens were fixed apart from the motion of the spheres and was developed without any understanding of gravity. After the planets' motions were more accurately measured, theoretical mechanisms such as <a href="/wiki/Deferent_and_epicycle" title="Deferent and epicycle">deferent and epicycles</a> were added. Although the model was capable of reasonably accurately predicting the planets' positions in the sky, more and more epicycles were required as the measurements became more accurate, hence the model became increasingly unwieldy. Originally <a href="/wiki/Geocentric_model" title="Geocentric model">geocentric</a>, it was modified by <a href="/wiki/Copernicus" class="mw-redirect" title="Copernicus">Copernicus</a> to place the Sun at the centre to help simplify the model. The model was further challenged during the 16th century, as comets were observed traversing the spheres.<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><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Moon_apsidal_precession.png" class="mw-file-description"><img alt="Apsidal precession refers to the rotation of the Moon's elliptical orbit over time, with the major axis completing one revolution every 8.85 years." src="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Moon_apsidal_precession.png/220px-Moon_apsidal_precession.png" decoding="async" width="220" height="191" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Moon_apsidal_precession.png/330px-Moon_apsidal_precession.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/53/Moon_apsidal_precession.png/440px-Moon_apsidal_precession.png 2x" data-file-width="1446" data-file-height="1257" /></a><figcaption>Moon's elliptic orbit</figcaption></figure> <p>The basis for the modern understanding of orbits was first formulated by <a href="/wiki/Johannes_Kepler" title="Johannes Kepler">Johannes Kepler</a> whose results are summarised in his three laws of planetary motion. First, he found that the orbits of the planets in our <a href="/wiki/Solar_System" title="Solar System">Solar System</a> are elliptical, not <a href="/wiki/Circle" title="Circle">circular</a> (or <a href="/wiki/Epicycle" class="mw-redirect" title="Epicycle">epicyclic</a>), as had previously been believed, and that the Sun is not located at the center of the orbits, but rather at one <a href="/wiki/Focus_(geometry)" title="Focus (geometry)">focus</a>.<sup id="cite_ref-Kepler's_Laws_of_Planetary_Motion_6-0" class="reference"><a href="#cite_note-Kepler's_Laws_of_Planetary_Motion-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Second, he found that the orbital speed of each planet is not constant, as had previously been thought, but rather that the speed depends on the planet's distance from the Sun. Third, Kepler found a universal relationship between the orbital properties of all the planets orbiting the Sun. For the planets, the cubes of their distances from the Sun are proportional to the squares of their orbital periods. Jupiter and Venus, for example, are respectively about 5.2 and 0.723 <a href="/wiki/Astronomical_unit" title="Astronomical unit">AU</a> distant from the Sun, their orbital periods respectively about 11.86 and 0.615 years. The proportionality is seen by the fact that the ratio for Jupiter, 5.2<sup>3</sup>/11.86<sup>2</sup>, is practically equal to that for Venus, 0.723<sup>3</sup>/0.615<sup>2</sup>, in accord with the relationship. Idealised orbits meeting these rules are known as <a href="/wiki/Kepler_orbits" class="mw-redirect" title="Kepler orbits">Kepler orbits</a>. </p><p><a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> demonstrated that Kepler's laws were derivable from his theory of <a href="/wiki/Gravitation" class="mw-redirect" title="Gravitation">gravitation</a> and that, in general, the orbits of bodies subject to gravity were <a href="/wiki/Conic_section" title="Conic section">conic sections</a> (this assumes that the force of gravity propagates instantaneously). Newton showed that, for a pair of bodies, the orbits' sizes are in inverse proportion to their <a href="/wiki/Mass" title="Mass">masses</a>, and that those bodies orbit their common <a href="/wiki/Center_of_mass" title="Center of mass">center of mass</a>. Where one body is much more massive than the other (as is the case of an artificial satellite orbiting a planet), it is a convenient approximation to take the center of mass as coinciding with the center of the more massive body. </p><p>Advances in Newtonian mechanics were then used to explore variations from the simple assumptions behind Kepler orbits, such as the perturbations due to other bodies, or the impact of spheroidal rather than spherical bodies. <a href="/wiki/Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Joseph-Louis Lagrange</a> developed a <a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">new approach</a> to <a href="/wiki/Newtonian_mechanics" class="mw-redirect" title="Newtonian mechanics">Newtonian mechanics</a> emphasizing energy more than force, and made progress on the <a href="/wiki/Three-body_problem" title="Three-body problem">three-body problem</a>, discovering the <a href="/wiki/Lagrangian_points" class="mw-redirect" title="Lagrangian points">Lagrangian points</a>. In a dramatic vindication of classical mechanics, in 1846 <a href="/wiki/Urbain_Le_Verrier" title="Urbain Le Verrier">Urbain Le Verrier</a> was able to predict the position of <a href="/wiki/Neptune" title="Neptune">Neptune</a> based on unexplained perturbations in the orbit of <a href="/wiki/Uranus" title="Uranus">Uranus</a>. </p><p><a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a> in his 1916 paper <i>The Foundation of the General Theory of Relativity</i> explained that gravity was due to curvature of <a href="/wiki/Space-time" class="mw-redirect" title="Space-time">space-time</a> and removed Newton's assumption that changes in gravity propagate instantaneously. This led astronomers to recognize that Newtonian mechanics did not provide the highest accuracy in understanding orbits. In <a href="/wiki/Relativity_theory" class="mw-redirect" title="Relativity theory">relativity theory</a>, orbits follow geodesic trajectories which are usually approximated very well by the Newtonian predictions (except where there are very strong gravity fields and very high speeds) but the differences are measurable. Essentially all the experimental evidence that can distinguish between the theories agrees with relativity theory to within experimental measurement accuracy. The original vindication of general relativity is that it was able to account for the remaining unexplained amount in <a href="/wiki/Tests_of_general_relativity#Perihelion_precession_of_Mercury" title="Tests of general relativity">precession of Mercury's perihelion</a> first noted by Le Verrier. However, Newton's solution is still used for most short term purposes since it is significantly easier to use and sufficiently accurate. </p> <div class="mw-heading mw-heading2"><h2 id="Planetary_orbits">Planetary orbits</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orbit&action=edit&section=2" title="Edit section: Planetary orbits"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_citations_needed_section plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Orbit" title="Special:EditPage/Orbit">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a> in this section. Unsourced material may be challenged and removed.</span> <span class="date-container"><i>(<span class="date">September 2020</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>Within a <a href="/wiki/Planetary_system" title="Planetary system">planetary system</a>, planets, <a href="/wiki/Dwarf_planet" title="Dwarf planet">dwarf planets</a>, <a href="/wiki/Asteroid" title="Asteroid">asteroids</a> and other <a href="/wiki/Minor_planet" title="Minor planet">minor planets</a>, <a href="/wiki/Comet" title="Comet">comets</a>, and <a href="/wiki/Space_debris" title="Space debris">space debris</a> orbit the system's <a href="/wiki/Barycentric_coordinates_(astronomy)" class="mw-redirect" title="Barycentric coordinates (astronomy)">barycenter</a> in <a href="/wiki/Elliptical_orbit" class="mw-redirect" title="Elliptical orbit">elliptical orbits</a>. A comet in a <a href="/wiki/Parabolic_trajectory" title="Parabolic trajectory">parabolic</a> or <a href="/wiki/Hyperbolic_trajectory" title="Hyperbolic trajectory">hyperbolic</a> orbit about a barycenter is not gravitationally bound to the star and therefore is not considered part of the star's planetary system. Bodies that are gravitationally bound to one of the planets in a planetary system, either <a href="/wiki/Natural_satellite" title="Natural satellite">natural</a> or <a href="/wiki/Satellite" title="Satellite">artificial satellites</a>, follow orbits about a barycenter near or within that planet. </p><p>Owing to mutual <a href="/wiki/Perturbation_(astronomy)" title="Perturbation (astronomy)">gravitational perturbations</a>, the <a href="/wiki/Eccentricity_(orbit)" class="mw-redirect" title="Eccentricity (orbit)">eccentricities</a> of the planetary orbits vary over time. <a href="/wiki/Mercury_(planet)" title="Mercury (planet)">Mercury</a>, the smallest planet in the Solar System, has the most eccentric orbit. At the present <a href="/wiki/Epoch_(astronomy)" title="Epoch (astronomy)">epoch</a>, <a href="/wiki/Mars" title="Mars">Mars</a> has the next largest eccentricity while the smallest orbital eccentricities are seen with <a href="/wiki/Venus" title="Venus">Venus</a> and <a href="/wiki/Neptune" title="Neptune">Neptune</a>. </p><p>As two objects orbit each other, the <a href="/wiki/Periapsis" class="mw-redirect" title="Periapsis">periapsis</a> is that point at which the two objects are closest to each other and the <a href="/wiki/Apoapsis" class="mw-redirect" title="Apoapsis">apoapsis</a> is that point at which they are the farthest. (More specific terms are used for specific bodies. For example, <i>perigee</i> and <i>apogee</i> are the lowest and highest parts of an orbit around Earth, while <i>perihelion</i> and <i>aphelion</i> are the closest and farthest points of an orbit around the Sun.) </p><p>In the case of planets orbiting a star, the mass of the star and all its satellites are calculated to be at a single point called the barycenter. The paths of all the star's satellites are elliptical orbits about that barycenter. Each satellite in that system will have its own elliptical orbit with the barycenter at one focal point of that ellipse. At any point along its orbit, any satellite will have a certain value of kinetic and potential energy with respect to the barycenter, and the sum of those two energies is a constant value at every point along its orbit. As a result, as a planet approaches <a href="/wiki/Periapsis" class="mw-redirect" title="Periapsis">periapsis</a>, the planet will increase in speed as its potential energy decreases; as a planet approaches <a href="/wiki/Apoapsis" class="mw-redirect" title="Apoapsis">apoapsis</a>, its velocity will decrease as its potential energy increases. </p> <div class="mw-heading mw-heading2"><h2 id="Principles">Principles</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orbit&action=edit&section=3" title="Edit section: Principles"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are a few common ways of understanding orbits: </p> <ul><li>A force, such as gravity, pulls an object into a curved path as it attempts to fly off in a straight line.</li> <li>As the object is pulled toward the massive body, it falls toward that body. However, if it has enough <a href="/wiki/Tangential_velocity" class="mw-redirect" title="Tangential velocity">tangential velocity</a> it will not fall into the body but will instead continue to follow the curved trajectory caused by that body indefinitely. The object is then said to be orbiting the body.</li></ul> <p>The velocity relationship of two moving objects with mass can thus be considered in four practical classes, with subtypes: </p> <dl><dt>No orbit</dt> <dd></dd> <dt><a href="/wiki/Sub-orbital_spaceflight" title="Sub-orbital spaceflight">Suborbital trajectories</a></dt> <dd>Range of interrupted elliptical paths</dd> <dt>Orbital trajectories (or simply, orbits)</dt> <dd><div><ul><li>Range of elliptical paths with closest point opposite firing point</li><li>Circular path</li><li>Range of elliptical paths with closest point at firing point</li></ul></div></dd> <dt><a href="/wiki/Escape_orbit" class="mw-redirect" title="Escape orbit">Open (or escape) trajectories</a></dt> <dd><div><ul><li>Parabolic paths</li><li>Hyperbolic paths</li></ul></div></dd></dl> <p>Orbital rockets are launched vertically at first to lift the rocket above the atmosphere (which causes frictional drag), and then slowly pitch over and finish firing the rocket engine parallel to the atmosphere to achieve orbit speed.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (October 2024)">citation needed</span></a></i>]</sup> </p><p>Once in orbit, their speed keeps them in orbit above the atmosphere. If e.g., an elliptical orbit dips into dense air, the object will lose speed and re-enter (i.e. fall). Occasionally a space craft will intentionally intercept the atmosphere, in an act commonly referred to as an aerobraking maneuver. </p> <div class="mw-heading mw-heading3"><h3 id="Illustration">Illustration</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orbit&action=edit&section=4" title="Edit section: Illustration"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Newton%27s_cannonball" title="Newton's cannonball">Newton's cannonball</a></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Newton_Cannon.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Newton_Cannon.svg/300px-Newton_Cannon.svg.png" decoding="async" width="300" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Newton_Cannon.svg/450px-Newton_Cannon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Newton_Cannon.svg/600px-Newton_Cannon.svg.png 2x" data-file-width="240" data-file-height="240" /></a><figcaption><a href="/wiki/Newton%27s_cannonball" title="Newton's cannonball">Newton's cannonball</a>, an illustration of how objects can "fall" in a curve</figcaption></figure> <p>As an illustration of an orbit around a planet, the <a href="/wiki/Newton%27s_cannonball" title="Newton's cannonball">Newton's cannonball</a> model may prove useful (see image below). This is a '<a href="/wiki/Thought_experiment" title="Thought experiment">thought experiment</a>', in which a cannon on top of a tall mountain is able to fire a cannonball horizontally at any chosen muzzle speed. The effects of air friction on the cannonball are ignored (or perhaps the mountain is high enough that the cannon is above the Earth's atmosphere, which is the same thing).<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> </p><p>If the cannon fires its ball with a low initial speed, the trajectory of the ball curves downward and hits the ground (A). As the firing speed is increased, the cannonball hits the ground farther (B) away from the cannon, because while the ball is still falling towards the ground, the ground is increasingly curving away from it (see first point, above). All these motions are actually "orbits" in a technical sense—they are describing a portion of an elliptical path around the center of gravity—but the orbits are interrupted by striking the Earth. </p><p>If the cannonball is fired with sufficient speed, the ground curves away from the ball at least as much as the ball falls—so the ball never strikes the ground. It is now in what could be called a non-interrupted or circumnavigating, orbit. For any specific combination of height above the center of gravity and mass of the planet, there is one specific firing speed (unaffected by the mass of the ball, which is assumed to be very small relative to the Earth's mass) that produces a <a href="/wiki/Circular_orbit" title="Circular orbit">circular orbit</a>, as shown in (C). </p><p>As the firing speed is increased beyond this, non-interrupted elliptic orbits are produced; one is shown in (D). If the initial firing is above the surface of the Earth as shown, there will also be non-interrupted elliptical orbits at slower firing speed; these will come closest to the Earth at the point half an orbit beyond, and directly opposite the firing point, below the circular orbit. </p><p>At a specific horizontal firing speed called <a href="/wiki/Escape_velocity" title="Escape velocity">escape velocity</a>, dependent on the mass of the planet and the distance of the object from the barycenter, an open orbit (E) is achieved that has a <a href="/wiki/Parabolic_trajectory" title="Parabolic trajectory">parabolic path</a>. At even greater speeds the object will follow a range of <a href="/wiki/Hyperbolic_trajectory" title="Hyperbolic trajectory">hyperbolic trajectories</a>. In a practical sense, both of these trajectory types mean the object is "breaking free" of the planet's gravity, and "going off into space" never to return. </p> <div class="mw-heading mw-heading2"><h2 id="Newton's_laws_of_motion"><span id="Newton.27s_laws_of_motion"></span>Newton's laws of motion</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orbit&action=edit&section=5" title="Edit section: Newton's laws of motion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Newton's_law_of_gravitation_and_laws_of_motion_for_two-body_problems"><span id="Newton.27s_law_of_gravitation_and_laws_of_motion_for_two-body_problems"></span>Newton's law of gravitation and laws of motion for two-body problems</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orbit&action=edit&section=6" title="Edit section: Newton's law of gravitation and laws of motion for two-body problems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In most situations, relativistic effects can be neglected, and <a href="/wiki/Newton%27s_laws" class="mw-redirect" title="Newton's laws">Newton's laws</a> give a sufficiently accurate description of motion. The acceleration of a body is equal to the sum of the forces acting on it, divided by its mass, and the gravitational force acting on a body is proportional to the product of the masses of the two attracting bodies and decreases inversely with the square of the distance between them. To this Newtonian approximation, for a system of two-point masses or spherical bodies, only influenced by their mutual gravitation (called a <a href="/wiki/Two-body_problem" title="Two-body problem">two-body problem</a>), their trajectories can be exactly calculated. If the heavier body is much more massive than the smaller, as in the case of a satellite or small moon orbiting a planet or for the Earth orbiting the Sun, it is accurate enough and convenient to describe the motion in terms of a <a href="/wiki/Coordinate_system" title="Coordinate system">coordinate system</a> that is centered on the heavier body, and we say that the lighter body is in orbit around the heavier. For the case where the masses of two bodies are comparable, an exact Newtonian solution is still sufficient and can be had by placing the coordinate system at the center of the mass of the system. </p> <div class="mw-heading mw-heading3"><h3 id="Defining_gravitational_potential_energy">Defining gravitational potential energy</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orbit&action=edit&section=7" title="Edit section: Defining gravitational potential energy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Energy is associated with <a href="/wiki/Gravitational_field" title="Gravitational field">gravitational fields</a>. A stationary body far from another can do external work if it is pulled towards it, and therefore has gravitational <i><a href="/wiki/Potential_energy" title="Potential energy">potential energy</a></i>. Since work is required to separate two bodies against the pull of gravity, their gravitational potential energy increases as they are separated, and decreases as they approach one another. For point masses, the gravitational energy decreases to zero as they approach zero separation. It is convenient and conventional to assign the potential energy as having zero value when they are an infinite distance apart, and hence it has a negative value (since it decreases from zero) for smaller finite distances. </p> <div class="mw-heading mw-heading3"><h3 id="Orbital_energies_and_orbit_shapes">Orbital energies and orbit shapes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orbit&action=edit&section=8" title="Edit section: Orbital energies and orbit shapes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>When only two gravitational bodies interact, their orbits follow a <a href="/wiki/Conic_section" title="Conic section">conic section</a>. The orbit can be open (implying the object never returns) or closed (returning). Which it is depends on the total <a href="/wiki/Energy" title="Energy">energy</a> (<a href="/wiki/Kinetic_energy" title="Kinetic energy">kinetic</a> + <a href="/wiki/Potential_energy" title="Potential energy">potential energy</a>) of the system. In the case of an open orbit, the speed at any position of the orbit is at least the <a href="/wiki/Escape_velocity" title="Escape velocity">escape velocity</a> for that position, in the case of a closed orbit, the speed is always less than the escape velocity. Since the kinetic energy is never negative if the common convention is adopted of taking the potential energy as zero at infinite separation, the bound orbits will have negative total energy, the parabolic trajectories zero total energy, and hyperbolic orbits positive total energy. </p><p>An open orbit will have a parabolic shape if it has the velocity of exactly the escape velocity at that point in its trajectory, and it will have the shape of a <a href="/wiki/Hyperbola" title="Hyperbola">hyperbola</a> when its velocity is greater than the escape velocity. When bodies with escape velocity or greater approach each other, they will briefly curve around each other at the time of their closest approach, and then separate, forever. </p><p>All closed orbits have the shape of an <a href="/wiki/Ellipse" title="Ellipse">ellipse</a>. A circular orbit is a special case, wherein the foci of the ellipse coincide. The point where the orbiting body is closest to Earth is called the <a href="/wiki/Perigee" class="mw-redirect" title="Perigee">perigee</a>, and when orbiting a body other than earth it is called the periapsis (less properly, "perifocus" or "pericentron"). The point where the satellite is farthest from Earth is called the <a href="/wiki/Apogee" class="mw-redirect" title="Apogee">apogee</a>, apoapsis, or sometimes apifocus or apocentron. A line drawn from periapsis to apoapsis is the <i><a href="/wiki/Line_of_apsides" class="mw-redirect" title="Line of apsides">line-of-apsides</a></i>. This is the major axis of the ellipse, the line through its longest part. </p> <div class="mw-heading mw-heading3"><h3 id="Kepler's_laws"><span id="Kepler.27s_laws"></span>Kepler's laws</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orbit&action=edit&section=9" title="Edit section: Kepler's laws"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Bodies following closed orbits repeat their paths with a certain time called the period. This motion is described by the empirical laws of Kepler, which can be mathematically derived from Newton's laws. These can be formulated as follows: </p> <ol><li>The orbit of a planet around the <a href="/wiki/Sun" title="Sun">Sun</a> is an ellipse, with the Sun in one of the focal points of that ellipse. [This focal point is actually the <a href="/wiki/Barycenter" class="mw-redirect" title="Barycenter">barycenter</a> of the <a href="/wiki/Solar_System" title="Solar System">Sun-planet system</a>; for simplicity, this explanation assumes the Sun's mass is infinitely larger than that planet's.] The planet's orbit lies in a plane, called the <b><a href="/wiki/Orbital_plane_(astronomy)" class="mw-redirect" title="Orbital plane (astronomy)">orbital plane</a></b>. The point on the orbit closest to the attracting body is the periapsis. The point farthest from the attracting body is called the apoapsis. There are also specific terms for orbits about particular bodies; things orbiting the Sun have a <a href="/wiki/Perihelion" class="mw-redirect" title="Perihelion">perihelion</a> and <a href="/wiki/Aphelion" class="mw-redirect" title="Aphelion">aphelion</a>, things orbiting the Earth have a <a href="/wiki/Perigee" class="mw-redirect" title="Perigee">perigee</a> and <a href="/wiki/Apogee" class="mw-redirect" title="Apogee">apogee</a>, and things orbiting the <a href="/wiki/Moon" title="Moon">Moon</a> have a <a href="/wiki/Perilune" class="mw-redirect" title="Perilune">perilune</a> and <a href="/wiki/Apolune" class="mw-redirect" title="Apolune">apolune</a> (or <a href="/wiki/Periselene" class="mw-redirect" title="Periselene">periselene</a> and <a href="/wiki/Aposelene" class="mw-redirect" title="Aposelene">aposelene</a> respectively). An orbit around any <a href="/wiki/Star" title="Star">star</a>, not just the Sun, has a <a href="/wiki/Periastron" class="mw-redirect" title="Periastron">periastron</a> and an <a href="/wiki/Apastron" class="mw-redirect" title="Apastron">apastron</a>.</li> <li>As the planet moves in its orbit, the line from the Sun to the planet sweeps a constant area of the <a href="/wiki/Orbital_plane_(astronomy)" class="mw-redirect" title="Orbital plane (astronomy)">orbital plane</a> for a given period of time, regardless of which part of its orbit the planet traces during that period of time. This means that the planet moves faster near its <a href="/wiki/Perihelion" class="mw-redirect" title="Perihelion">perihelion</a> than near its <a href="/wiki/Aphelion" class="mw-redirect" title="Aphelion">aphelion</a>, because at the smaller distance it needs to trace a greater arc to cover the same area. This law is usually stated as "equal areas in equal time."</li> <li>For a given orbit, the ratio of the cube of its <a href="/wiki/Semi-major_axis" class="mw-redirect" title="Semi-major axis">semi-major axis</a> to the square of its period is constant.</li></ol> <div class="mw-heading mw-heading3"><h3 id="Limitations_of_Newton's_law_of_gravitation"><span id="Limitations_of_Newton.27s_law_of_gravitation"></span>Limitations of Newton's law of gravitation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orbit&action=edit&section=10" title="Edit section: Limitations of Newton's law of gravitation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Note that while bound orbits of a point mass or a spherical body with a <a href="/wiki/Newtonian_gravitational_field" class="mw-redirect" title="Newtonian gravitational field">Newtonian gravitational field</a> are closed <a href="/wiki/Ellipse" title="Ellipse">ellipses</a>, which repeat the same path exactly and indefinitely, any non-spherical or non-Newtonian effects (such as caused by the slight oblateness of the <a href="/wiki/Earth" title="Earth">Earth</a>, or by <a href="/wiki/Theory_of_relativity" title="Theory of relativity">relativistic effects</a>, thereby changing the gravitational field's behavior with distance) will cause the orbit's shape to depart from the closed <a href="/wiki/Ellipse" title="Ellipse">ellipses</a> characteristic of Newtonian <a href="/wiki/Two-body_motion" class="mw-redirect" title="Two-body motion">two-body motion</a>. The two-body solutions were published by Newton in <a href="/wiki/Philosophiae_Naturalis_Principia_Mathematica" class="mw-redirect" title="Philosophiae Naturalis Principia Mathematica">Principia</a> in 1687. In 1912, <a href="/wiki/Karl_Fritiof_Sundman" class="mw-redirect" title="Karl Fritiof Sundman">Karl Fritiof Sundman</a> developed a converging infinite series that solves the <a href="/wiki/Three-body_problem" title="Three-body problem">three-body problem</a>; however, it converges too slowly to be of much use. Except for special cases like the <a href="/wiki/Lagrangian_point" class="mw-redirect" title="Lagrangian point">Lagrangian points</a>, no method is known to solve the equations of motion for a system with four or more bodies. </p> <div class="mw-heading mw-heading3"><h3 id="Approaches_to_many-body_problems">Approaches to many-body problems</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orbit&action=edit&section=11" title="Edit section: Approaches to many-body problems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Rather than an exact closed form solution, orbits with many bodies can be approximated with arbitrarily high accuracy. These approximations take two forms: </p> <dl><dd>One form takes the pure elliptic motion as a basis and adds <a href="/wiki/Perturbation_(astronomy)" title="Perturbation (astronomy)">perturbation</a> terms to account for the gravitational influence of multiple bodies. This is convenient for calculating the positions of astronomical bodies. The equations of motion of the moons, planets, and other bodies are known with great accuracy, and are used to generate <a href="/wiki/Ephemeris" title="Ephemeris">tables</a> for <a href="/wiki/Celestial_navigation" title="Celestial navigation">celestial navigation</a>. Still, there are <a href="/wiki/Secular_phenomena" class="mw-redirect" title="Secular phenomena">secular phenomena</a> that have to be dealt with by <a href="/wiki/Parameterized_post-Newtonian_formalism" title="Parameterized post-Newtonian formalism">post-Newtonian</a> methods.</dd> <dd>The <a href="/wiki/Differential_equation" title="Differential equation">differential equation</a> form is used for scientific or mission-planning purposes. According to Newton's laws, the sum of all the forces acting on a body will equal the mass of the body times its acceleration (<i>F = ma</i>). Therefore accelerations can be expressed in terms of positions. The perturbation terms are much easier to describe in this form. Predicting subsequent positions and velocities from initial values of position and velocity corresponds to solving an <a href="/wiki/Initial_value_problem" title="Initial value problem">initial value problem</a>. Numerical methods calculate the positions and velocities of the objects a short time in the future, then repeat the calculation ad nauseam. However, tiny arithmetic errors from the limited accuracy of a computer's math are cumulative, which limits the accuracy of this approach.</dd></dl> <p>Differential simulations with large numbers of objects perform the calculations in a hierarchical pairwise fashion between centers of mass. Using this scheme, galaxies, star clusters and other large assemblages of objects have been simulated.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Formulation">Formulation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orbit&action=edit&section=12" title="Edit section: Formulation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Orbit_modeling" title="Orbit modeling">Orbit modeling</a></div> <div class="mw-heading mw-heading3"><h3 id="Newtonian_analysis_of_orbital_motion">Newtonian analysis of orbital motion</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orbit&action=edit&section=13" title="Edit section: Newtonian analysis of orbital motion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Kepler_orbit" title="Kepler orbit">Kepler orbit</a>, <a href="/wiki/Orbit_equation" title="Orbit equation">orbit equation</a>, and <a href="/wiki/Kepler%27s_first_law" class="mw-redirect" title="Kepler's first law">Kepler's first law</a></div> <p>The following derivation applies to such an elliptical orbit. We start only with the <a href="/wiki/Classical_mechanics" title="Classical mechanics">Newtonian</a> law of gravitation stating that the gravitational acceleration towards the central body is related to the inverse of the square of the distance between them, namely </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{2}=-{\frac {Gm_{1}m_{2}}{r^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </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 F_{2}=-{\frac {Gm_{1}m_{2}}{r^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c3165c7494ba1de97468bdfcaa3291db51684eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:16.308ex; height:5.676ex;" alt="{\displaystyle F_{2}=-{\frac {Gm_{1}m_{2}}{r^{2}}}}"></span></dd></dl> <p>where <i>F</i><sub>2</sub> is the force acting on the mass <i>m</i><sub>2</sub> caused by the gravitational attraction mass <i>m</i><sub>1</sub> has for <i>m</i><sub>2</sub>, <i>G</i> is the universal gravitational constant, and <i>r</i> is the distance between the two masses centers. </p><p>From Newton's Second Law, the summation of the forces acting on <i>m</i><sub>2</sub> related to that body's acceleration: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{2}=m_{2}A_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{2}=m_{2}A_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dba8434884436bb09347c0bd46294e55fc5831ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.539ex; height:2.509ex;" alt="{\displaystyle F_{2}=m_{2}A_{2}}"></span></dd></dl> <p>where <i>A</i><sub>2</sub> is the acceleration of <i>m</i><sub>2</sub> caused by the force of gravitational attraction <i>F</i><sub>2</sub> of <i>m</i><sub>1</sub> acting on <i>m</i><sub>2</sub>. </p><p>Combining Eq. 1 and 2: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {Gm_{1}m_{2}}{r^{2}}}=m_{2}A_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {Gm_{1}m_{2}}{r^{2}}}=m_{2}A_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6924dd36d2f574232731b6ae605a86bd151f01be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:19.651ex; height:5.676ex;" alt="{\displaystyle -{\frac {Gm_{1}m_{2}}{r^{2}}}=m_{2}A_{2}}"></span></dd></dl> <p>Solving for the acceleration, <i>A</i><sub>2</sub>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{2}={\frac {F_{2}}{m_{2}}}=-{\frac {1}{m_{2}}}{\frac {Gm_{1}m_{2}}{r^{2}}}=-{\frac {\mu }{r^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>μ</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 A_{2}={\frac {F_{2}}{m_{2}}}=-{\frac {1}{m_{2}}}{\frac {Gm_{1}m_{2}}{r^{2}}}=-{\frac {\mu }{r^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0b325ec1a8f813c5e6aff5e888dde2a076d8a6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:35.362ex; height:5.676ex;" alt="{\displaystyle A_{2}={\frac {F_{2}}{m_{2}}}=-{\frac {1}{m_{2}}}{\frac {Gm_{1}m_{2}}{r^{2}}}=-{\frac {\mu }{r^{2}}}}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d20addf0d9f04e185714134b97726c4bf17d340" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.789ex; height:2.176ex;" alt="{\displaystyle \mu \,}"></span> is the <a href="/wiki/Standard_gravitational_parameter" title="Standard gravitational parameter">standard gravitational parameter</a>, in this case <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Gm_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Gm_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e8991b9e06ab9fbe4a0f79da48bca1db4364d0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.921ex; height:2.509ex;" alt="{\displaystyle Gm_{1}}"></span>. It is understood that the system being described is <i>m</i><sub>2</sub>, hence the subscripts can be dropped. </p><p>We assume that the central body is massive enough that it can be considered to be stationary and we ignore the more subtle effects of <a href="/wiki/General_relativity" title="General relativity">general relativity</a>. </p><p>When a pendulum or an object attached to a spring swings in an ellipse, the inward acceleration/force is proportional to the distance <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=F/m=-kr.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>m</mi> <mo>=</mo> <mo>−</mo> <mi>k</mi> <mi>r</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=F/m=-kr.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac46d9600b9102182e3449d2edd0558d2ac0dd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.599ex; height:2.843ex;" alt="{\displaystyle A=F/m=-kr.}"></span> Due to the way vectors add, the component of the force in the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\mathbf {x} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {x} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a5451354f837d5d89774e1de386b44a903d929d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:2.343ex;" alt="{\displaystyle {\hat {\mathbf {x} }}}"></span> or in the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\mathbf {y} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {y} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3826fc591cfe42348f58b15e4c23152f30931ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.411ex; height:2.676ex;" alt="{\displaystyle {\hat {\mathbf {y} }}}"></span> directions are also proportionate to the respective components of the distances, <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''_{x}=A_{x}=-kr_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mo>″</mo> </msubsup> <mo>=</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi>k</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r''_{x}=A_{x}=-kr_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f564bb7d3311a1f2159891c6b09acc9118bb6f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.574ex; height:2.509ex;" alt="{\displaystyle r''_{x}=A_{x}=-kr_{x}}"></span>. Hence, the entire analysis can be done separately in these dimensions. This results in the harmonic parabolic equations <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 x=A\cos(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>A</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=A\cos(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7af003d42ac20f5f46c26a433210910e74022c29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.318ex; height:2.843ex;" alt="{\displaystyle x=A\cos(t)}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=B\sin(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>B</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=B\sin(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04291c735b8465cf816d080d78349bc27a8b8da4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.91ex; height:2.843ex;" alt="{\displaystyle y=B\sin(t)}"></span> of the ellipse. </p><p>The location of the orbiting object at the current time <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span> is located in the plane using <a href="/wiki/Vector_calculus" title="Vector calculus">vector calculus</a> in <a href="/wiki/Polar_coordinates" class="mw-redirect" title="Polar coordinates">polar coordinates</a> both with the standard Euclidean basis and with the polar basis with the origin coinciding with the center of force. Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> be the distance between the object and the center 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 \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span> be the angle it has rotated. Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\mathbf {x} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {x} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a5451354f837d5d89774e1de386b44a903d929d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:2.343ex;" alt="{\displaystyle {\hat {\mathbf {x} }}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\mathbf {y} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {y} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3826fc591cfe42348f58b15e4c23152f30931ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.411ex; height:2.676ex;" alt="{\displaystyle {\hat {\mathbf {y} }}}"></span> be the standard <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean</a> bases and let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\mathbf {r} }}=\cos(\theta ){\hat {\mathbf {x} }}+\sin(\theta ){\hat {\mathbf {y} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {r} }}=\cos(\theta ){\hat {\mathbf {x} }}+\sin(\theta ){\hat {\mathbf {y} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e43843800f4be3f5f7be1274b135d5e61ccce974" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.689ex; height:2.843ex;" alt="{\displaystyle {\hat {\mathbf {r} }}=\cos(\theta ){\hat {\mathbf {x} }}+\sin(\theta ){\hat {\mathbf {y} }}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\boldsymbol {\theta }}}=-\sin(\theta ){\hat {\mathbf {x} }}+\cos(\theta ){\hat {\mathbf {y} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\boldsymbol {\theta }}}=-\sin(\theta ){\hat {\mathbf {x} }}+\cos(\theta ){\hat {\mathbf {y} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e24f0578060902f73bb3db70ab39a1e553de2f24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.179ex; height:3.343ex;" alt="{\displaystyle {\hat {\boldsymbol {\theta }}}=-\sin(\theta ){\hat {\mathbf {x} }}+\cos(\theta ){\hat {\mathbf {y} }}}"></span> be the radial and transverse <a href="/wiki/Polar_coordinate_system#Vector_calculus" title="Polar coordinate system">polar</a> basis with the first being the unit vector pointing from the central body to the current location of the orbiting object and the second being the orthogonal unit vector pointing in the direction that the orbiting object would travel if orbiting in a counter clockwise circle. Then the vector to the orbiting object is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\mathbf {O} }}=r\cos(\theta ){\hat {\mathbf {x} }}+r\sin(\theta ){\hat {\mathbf {y} }}=r{\hat {\mathbf {r} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">O</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>r</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi>r</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {O} }}=r\cos(\theta ){\hat {\mathbf {x} }}+r\sin(\theta ){\hat {\mathbf {y} }}=r{\hat {\mathbf {r} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab5e216c6dadf4b37e29952f437adf153732e531" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.716ex; height:3.343ex;" alt="{\displaystyle {\hat {\mathbf {O} }}=r\cos(\theta ){\hat {\mathbf {x} }}+r\sin(\theta ){\hat {\mathbf {y} }}=r{\hat {\mathbf {r} }}}"></span></dd></dl> <p>We use <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 {\dot {r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99b51a39764cf6daa615ef2144ac420ecff01cfc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.292ex; height:2.176ex;" alt="{\displaystyle {\dot {r}}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\theta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f78124b2aea53527d3e053cbbdd9c7ded2c8f05f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.356ex; height:2.843ex;" alt="{\displaystyle {\dot {\theta }}}"></span> to denote the standard derivatives of how this distance and angle change over time. We take the derivative of a vector to see how it changes over time by subtracting its location at time <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span> from that at time <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+\delta t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>+</mo> <mi>δ</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t+\delta t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cf2cc78aea7996c669d0428babe7c88053ab22d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.568ex; height:2.509ex;" alt="{\displaystyle t+\delta t}"></span> and dividing by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb5768e76c1d7222b1b53d613e8622471ef18327" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.888ex; height:2.343ex;" alt="{\displaystyle \delta t}"></span>. The result is also a vector. Because our basis 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 {\hat {\mathbf {r} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {r} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9740464b71653e12932278ee944540be8caa5b96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.343ex;" alt="{\displaystyle {\hat {\mathbf {r} }}}"></span> moves as the object orbits, we start by differentiating it. From time <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span> to <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+\delta t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>+</mo> <mi>δ</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t+\delta t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cf2cc78aea7996c669d0428babe7c88053ab22d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.568ex; height:2.509ex;" alt="{\displaystyle t+\delta t}"></span>, the 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 {\hat {\mathbf {r} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {r} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9740464b71653e12932278ee944540be8caa5b96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.343ex;" alt="{\displaystyle {\hat {\mathbf {r} }}}"></span> keeps its beginning at the origin and rotates from angle <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 \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span> to <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 \theta +{\dot {\theta }}\ \delta t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mtext> </mtext> <mi>δ</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta +{\dot {\theta }}\ \delta t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e43a1beeb2deb404ff0c6489faa0a8d7e20e1f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.756ex; height:3.009ex;" alt="{\displaystyle \theta +{\dot {\theta }}\ \delta t}"></span> which moves its head a distance <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\theta }}\ \delta t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mtext> </mtext> <mi>δ</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\theta }}\ \delta t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ca4f11e634480b3b07ec326cf3964d6e9fd0a9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.825ex; height:2.843ex;" alt="{\displaystyle {\dot {\theta }}\ \delta t}"></span> in the perpendicular direction <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\boldsymbol {\theta }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\boldsymbol {\theta }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca32f6d693b0a183a7728b875f7e2042bb1dbca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.457ex; height:2.843ex;" alt="{\displaystyle {\hat {\boldsymbol {\theta }}}}"></span> giving a derivative 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 {\dot {\theta }}{\hat {\boldsymbol {\theta }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\theta }}{\hat {\boldsymbol {\theta }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89c2050809e39db493da13208ce78fa999d60905" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.813ex; height:2.843ex;" alt="{\displaystyle {\dot {\theta }}{\hat {\boldsymbol {\theta }}}}"></span>. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\hat {\mathbf {r} }}&=\cos(\theta ){\hat {\mathbf {x} }}+\sin(\theta ){\hat {\mathbf {y} }}\\{\frac {\delta {\hat {\mathbf {r} }}}{\delta t}}={\dot {\mathbf {r} }}&=-\sin(\theta ){\dot {\theta }}{\hat {\mathbf {x} }}+\cos(\theta ){\dot {\theta }}{\hat {\mathbf {y} }}={\dot {\theta }}{\hat {\boldsymbol {\theta }}}\\{\hat {\boldsymbol {\theta }}}&=-\sin(\theta ){\hat {\mathbf {x} }}+\cos(\theta ){\hat {\mathbf {y} }}\\{\frac {\delta {\hat {\boldsymbol {\theta }}}}{\delta t}}={\dot {\boldsymbol {\theta }}}&=-\cos(\theta ){\dot {\theta }}{\hat {\mathbf {x} }}-\sin(\theta ){\dot {\theta }}{\hat {\mathbf {y} }}=-{\dot {\theta }}{\hat {\mathbf {r} }}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi>δ</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi>δ</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\hat {\mathbf {r} }}&=\cos(\theta ){\hat {\mathbf {x} }}+\sin(\theta ){\hat {\mathbf {y} }}\\{\frac {\delta {\hat {\mathbf {r} }}}{\delta t}}={\dot {\mathbf {r} }}&=-\sin(\theta ){\dot {\theta }}{\hat {\mathbf {x} }}+\cos(\theta ){\dot {\theta }}{\hat {\mathbf {y} }}={\dot {\theta }}{\hat {\boldsymbol {\theta }}}\\{\hat {\boldsymbol {\theta }}}&=-\sin(\theta ){\hat {\mathbf {x} }}+\cos(\theta ){\hat {\mathbf {y} }}\\{\frac {\delta {\hat {\boldsymbol {\theta }}}}{\delta t}}={\dot {\boldsymbol {\theta }}}&=-\cos(\theta ){\dot {\theta }}{\hat {\mathbf {x} }}-\sin(\theta ){\dot {\theta }}{\hat {\mathbf {y} }}=-{\dot {\theta }}{\hat {\mathbf {r} }}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3baa9934c7e4e62e87b0d07d4d3eb8de62e08e08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.671ex; width:41.508ex; height:18.509ex;" alt="{\displaystyle {\begin{aligned}{\hat {\mathbf {r} }}&=\cos(\theta ){\hat {\mathbf {x} }}+\sin(\theta ){\hat {\mathbf {y} }}\\{\frac {\delta {\hat {\mathbf {r} }}}{\delta t}}={\dot {\mathbf {r} }}&=-\sin(\theta ){\dot {\theta }}{\hat {\mathbf {x} }}+\cos(\theta ){\dot {\theta }}{\hat {\mathbf {y} }}={\dot {\theta }}{\hat {\boldsymbol {\theta }}}\\{\hat {\boldsymbol {\theta }}}&=-\sin(\theta ){\hat {\mathbf {x} }}+\cos(\theta ){\hat {\mathbf {y} }}\\{\frac {\delta {\hat {\boldsymbol {\theta }}}}{\delta t}}={\dot {\boldsymbol {\theta }}}&=-\cos(\theta ){\dot {\theta }}{\hat {\mathbf {x} }}-\sin(\theta ){\dot {\theta }}{\hat {\mathbf {y} }}=-{\dot {\theta }}{\hat {\mathbf {r} }}\end{aligned}}}"></span></dd></dl> <p>We can now find the velocity and acceleration of our orbiting object. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\hat {\mathbf {O} }}&=r{\hat {\mathbf {r} }}\\{\dot {\mathbf {O} }}&={\frac {\delta r}{\delta t}}{\hat {\mathbf {r} }}+r{\frac {\delta {\hat {\mathbf {r} }}}{\delta t}}={\dot {r}}{\hat {\mathbf {r} }}+r\left[{\dot {\theta }}{\hat {\boldsymbol {\theta }}}\right]\\{\ddot {\mathbf {O} }}&=\left[{\ddot {r}}{\hat {\mathbf {r} }}+{\dot {r}}{\dot {\theta }}{\hat {\boldsymbol {\theta }}}\right]+\left[{\dot {r}}{\dot {\theta }}{\hat {\boldsymbol {\theta }}}+r{\ddot {\theta }}{\hat {\boldsymbol {\theta }}}-r{\dot {\theta }}^{2}{\hat {\mathbf {r} }}\right]\\&=\left[{\ddot {r}}-r{\dot {\theta }}^{2}\right]{\hat {\mathbf {r} }}+\left[r{\ddot {\theta }}+2{\dot {r}}{\dot {\theta }}\right]{\hat {\boldsymbol {\theta }}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">O</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">O</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mi>r</mi> </mrow> <mrow> <mi>δ</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi>δ</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi>r</mi> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mo>]</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">O</mi> </mrow> <mo>¨</mo> </mover> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mo>]</mo> </mrow> <mo>+</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>−</mo> <mi>r</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mo>]</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mo>−</mo> <mi>r</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow> <mo>[</mo> <mrow> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\hat {\mathbf {O} }}&=r{\hat {\mathbf {r} }}\\{\dot {\mathbf {O} }}&={\frac {\delta r}{\delta t}}{\hat {\mathbf {r} }}+r{\frac {\delta {\hat {\mathbf {r} }}}{\delta t}}={\dot {r}}{\hat {\mathbf {r} }}+r\left[{\dot {\theta }}{\hat {\boldsymbol {\theta }}}\right]\\{\ddot {\mathbf {O} }}&=\left[{\ddot {r}}{\hat {\mathbf {r} }}+{\dot {r}}{\dot {\theta }}{\hat {\boldsymbol {\theta }}}\right]+\left[{\dot {r}}{\dot {\theta }}{\hat {\boldsymbol {\theta }}}+r{\ddot {\theta }}{\hat {\boldsymbol {\theta }}}-r{\dot {\theta }}^{2}{\hat {\mathbf {r} }}\right]\\&=\left[{\ddot {r}}-r{\dot {\theta }}^{2}\right]{\hat {\mathbf {r} }}+\left[r{\ddot {\theta }}+2{\dot {r}}{\dot {\theta }}\right]{\hat {\boldsymbol {\theta }}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d92049db70580d670e5dd37a305c3ebd3073f2b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.502ex; margin-bottom: -0.336ex; width:40.755ex; height:18.676ex;" alt="{\displaystyle {\begin{aligned}{\hat {\mathbf {O} }}&=r{\hat {\mathbf {r} }}\\{\dot {\mathbf {O} }}&={\frac {\delta r}{\delta t}}{\hat {\mathbf {r} }}+r{\frac {\delta {\hat {\mathbf {r} }}}{\delta t}}={\dot {r}}{\hat {\mathbf {r} }}+r\left[{\dot {\theta }}{\hat {\boldsymbol {\theta }}}\right]\\{\ddot {\mathbf {O} }}&=\left[{\ddot {r}}{\hat {\mathbf {r} }}+{\dot {r}}{\dot {\theta }}{\hat {\boldsymbol {\theta }}}\right]+\left[{\dot {r}}{\dot {\theta }}{\hat {\boldsymbol {\theta }}}+r{\ddot {\theta }}{\hat {\boldsymbol {\theta }}}-r{\dot {\theta }}^{2}{\hat {\mathbf {r} }}\right]\\&=\left[{\ddot {r}}-r{\dot {\theta }}^{2}\right]{\hat {\mathbf {r} }}+\left[r{\ddot {\theta }}+2{\dot {r}}{\dot {\theta }}\right]{\hat {\boldsymbol {\theta }}}\end{aligned}}}"></span></dd></dl> <p>The coefficients 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 {\hat {\mathbf {r} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {r} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9740464b71653e12932278ee944540be8caa5b96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.343ex;" alt="{\displaystyle {\hat {\mathbf {r} }}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\boldsymbol {\theta }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\boldsymbol {\theta }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca32f6d693b0a183a7728b875f7e2042bb1dbca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.457ex; height:2.843ex;" alt="{\displaystyle {\hat {\boldsymbol {\theta }}}}"></span> give the accelerations in the radial and transverse directions. As said, Newton gives this first due to gravity 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 -\mu /r^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\mu /r^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf627c13db72632020419c214834d8d1ecdcca70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.475ex; height:3.176ex;" alt="{\displaystyle -\mu /r^{2}}"></span> and the second is zero. </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><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 {\ddot {r}}-r{\dot {\theta }}^{2}=-{\frac {\mu }{r^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mo>−</mo> <mi>r</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>μ</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 {\ddot {r}}-r{\dot {\theta }}^{2}=-{\frac {\mu }{r^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba52138da59f13ad96d4271343f34a99fd67ee15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:15.436ex; height:5.176ex;" alt="{\displaystyle {\ddot {r}}-r{\dot {\theta }}^{2}=-{\frac {\mu }{r^{2}}}}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(1)</b></td></tr></tbody></table> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><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{\ddot {\theta }}+2{\dot {r}}{\dot {\theta }}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r{\ddot {\theta }}+2{\dot {r}}{\dot {\theta }}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bc0c4f8ba4831ce15c0b313a26ef8146f3d391c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.316ex; height:3.009ex;" alt="{\displaystyle r{\ddot {\theta }}+2{\dot {r}}{\dot {\theta }}=0}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(2)</b></td></tr></tbody></table> <p>Equation (2) can be rearranged using integration by parts. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r{\ddot {\theta }}+2{\dot {r}}{\dot {\theta }}={\frac {1}{r}}{\frac {d}{dt}}\left(r^{2}{\dot {\theta }}\right)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r{\ddot {\theta }}+2{\dot {r}}{\dot {\theta }}={\frac {1}{r}}{\frac {d}{dt}}\left(r^{2}{\dot {\theta }}\right)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cae737e0b7f29cae9b7ae88d1aaeac8b4c2e75c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:27.926ex; height:5.509ex;" alt="{\displaystyle r{\ddot {\theta }}+2{\dot {r}}{\dot {\theta }}={\frac {1}{r}}{\frac {d}{dt}}\left(r^{2}{\dot {\theta }}\right)=0}"></span></dd></dl> <p>We can multiply through by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> because it is not zero unless the orbiting object crashes. Then having the derivative be zero gives that the function is a constant. </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><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^{2}{\dot {\theta }}=h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r^{2}{\dot {\theta }}=h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7084abb4ddbb69a17ace67209fe0682a1944cb5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.896ex; height:2.843ex;" alt="{\displaystyle r^{2}{\dot {\theta }}=h}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(3)</b></td></tr></tbody></table> <p>which is actually the theoretical proof of <a href="/wiki/Kepler%27s_second_law" class="mw-redirect" title="Kepler's second law">Kepler's second law</a> (A line joining a planet and the Sun sweeps out equal areas during equal intervals of time). The constant of integration, <i>h</i>, is the <a href="/wiki/Specific_relative_angular_momentum" class="mw-redirect" title="Specific relative angular momentum">angular momentum per unit mass</a>. </p><p>In order to get an equation for the orbit from equation (1), we need to eliminate time.<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> (See also <a href="/wiki/Binet_equation" title="Binet equation">Binet equation</a>.) In polar coordinates, this would express the distance <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> of the orbiting object from the center as a function of its angle <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 \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span>. However, it is easier to introduce the auxiliary variable <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 u=1/r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u=1/r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a54441342a7391a1d75f9816d6aa16f1b72c150" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.802ex; height:2.843ex;" alt="{\displaystyle u=1/r}"></span> and to express <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 u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}"></span> as a function 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 \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span>. Derivatives 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 r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> with respect to time may be rewritten as derivatives 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 u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}"></span> with respect to angle. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u={1 \over r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u={1 \over r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edb030fe58ea5d65b793046e1fc6503b68f913de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.427ex; height:5.176ex;" alt="{\displaystyle u={1 \over r}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\theta }}={\frac {h}{r^{2}}}=hu^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>h</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mi>h</mi> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\theta }}={\frac {h}{r^{2}}}=hu^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34c3839d0e74d320724bae5e68c218eeb1a0d062" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:14.215ex; height:5.676ex;" alt="{\displaystyle {\dot {\theta }}={\frac {h}{r^{2}}}=hu^{2}}"></span> (reworking (3))</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {\delta u}{\delta \theta }}&={\frac {\delta }{\delta t}}\left({\frac {1}{r}}\right){\frac {\delta t}{\delta \theta }}=-{\frac {\dot {r}}{r^{2}{\dot {\theta }}}}=-{\frac {\dot {r}}{h}}\\{\frac {\delta ^{2}u}{\delta \theta ^{2}}}&=-{\frac {1}{h}}{\frac {\delta {\dot {r}}}{\delta t}}{\frac {\delta t}{\delta \theta }}=-{\frac {\ddot {r}}{h{\dot {\theta }}}}=-{\frac {\ddot {r}}{h^{2}u^{2}}}\ \ \ {\text{ or }}\ \ \ {\ddot {r}}=-h^{2}u^{2}{\frac {\delta ^{2}u}{\delta \theta ^{2}}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mi>u</mi> </mrow> <mrow> <mi>δ</mi> <mi>θ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>δ</mi> <mrow> <mi>δ</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mi>t</mi> </mrow> <mrow> <mi>δ</mi> <mi>θ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>˙</mo> </mover> </mrow> <mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>˙</mo> </mover> </mrow> <mi>h</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>u</mi> </mrow> <mrow> <mi>δ</mi> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>h</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi>δ</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mi>t</mi> </mrow> <mrow> <mi>δ</mi> <mi>θ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>¨</mo> </mover> </mrow> <mrow> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>¨</mo> </mover> </mrow> <mrow> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext> or </mtext> </mrow> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>u</mi> </mrow> <mrow> <mi>δ</mi> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {\delta u}{\delta \theta }}&={\frac {\delta }{\delta t}}\left({\frac {1}{r}}\right){\frac {\delta t}{\delta \theta }}=-{\frac {\dot {r}}{r^{2}{\dot {\theta }}}}=-{\frac {\dot {r}}{h}}\\{\frac {\delta ^{2}u}{\delta \theta ^{2}}}&=-{\frac {1}{h}}{\frac {\delta {\dot {r}}}{\delta t}}{\frac {\delta t}{\delta \theta }}=-{\frac {\ddot {r}}{h{\dot {\theta }}}}=-{\frac {\ddot {r}}{h^{2}u^{2}}}\ \ \ {\text{ or }}\ \ \ {\ddot {r}}=-h^{2}u^{2}{\frac {\delta ^{2}u}{\delta \theta ^{2}}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f09581310b50f8ae8a9ab58043a77b66b83f3c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:59.184ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}{\frac {\delta u}{\delta \theta }}&={\frac {\delta }{\delta t}}\left({\frac {1}{r}}\right){\frac {\delta t}{\delta \theta }}=-{\frac {\dot {r}}{r^{2}{\dot {\theta }}}}=-{\frac {\dot {r}}{h}}\\{\frac {\delta ^{2}u}{\delta \theta ^{2}}}&=-{\frac {1}{h}}{\frac {\delta {\dot {r}}}{\delta t}}{\frac {\delta t}{\delta \theta }}=-{\frac {\ddot {r}}{h{\dot {\theta }}}}=-{\frac {\ddot {r}}{h^{2}u^{2}}}\ \ \ {\text{ or }}\ \ \ {\ddot {r}}=-h^{2}u^{2}{\frac {\delta ^{2}u}{\delta \theta ^{2}}}\end{aligned}}}"></span></dd></dl> <p>Plugging these into (1) gives </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\ddot {r}}-r{\dot {\theta }}^{2}&=-{\frac {\mu }{r^{2}}}\\-h^{2}u^{2}{\frac {\delta ^{2}u}{\delta \theta ^{2}}}-{\frac {1}{u}}\left(hu^{2}\right)^{2}&=-\mu u^{2}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mo>−</mo> <mi>r</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>μ</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>u</mi> </mrow> <mrow> <mi>δ</mi> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>u</mi> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <mi>h</mi> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>μ</mi> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\ddot {r}}-r{\dot {\theta }}^{2}&=-{\frac {\mu }{r^{2}}}\\-h^{2}u^{2}{\frac {\delta ^{2}u}{\delta \theta ^{2}}}-{\frac {1}{u}}\left(hu^{2}\right)^{2}&=-\mu u^{2}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1773ecda9fe1421723149bdfac81d5693ec57bbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:32.216ex; height:11.176ex;" alt="{\displaystyle {\begin{aligned}{\ddot {r}}-r{\dot {\theta }}^{2}&=-{\frac {\mu }{r^{2}}}\\-h^{2}u^{2}{\frac {\delta ^{2}u}{\delta \theta ^{2}}}-{\frac {1}{u}}\left(hu^{2}\right)^{2}&=-\mu u^{2}\end{aligned}}}"></span></dd></dl> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><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 {\frac {\delta ^{2}u}{\delta \theta ^{2}}}+u={\frac {\mu }{h^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>u</mi> </mrow> <mrow> <mi>δ</mi> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>u</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>μ</mi> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\delta ^{2}u}{\delta \theta ^{2}}}+u={\frac {\mu }{h^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4192fa23166823266f19f6bbcf6fcb7e9e73d317" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:14.772ex; height:6.009ex;" alt="{\displaystyle {\frac {\delta ^{2}u}{\delta \theta ^{2}}}+u={\frac {\mu }{h^{2}}}}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(4)</b></td></tr></tbody></table> <p>So for the gravitational force – or, more generally, for <i>any</i> inverse square force law – the right hand side of the equation becomes a constant and the equation is seen to be the <a href="/wiki/Harmonic_oscillator" title="Harmonic oscillator">harmonic equation</a> (up to a shift of origin of the dependent variable). The solution is: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(\theta )={\frac {\mu }{h^{2}}}+A\cos(\theta -\theta _{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>μ</mi> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mi>A</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>−</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u(\theta )={\frac {\mu }{h^{2}}}+A\cos(\theta -\theta _{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fab8ce40c503dcfb8c1009b878938e831b2b1ff3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:26.524ex; height:5.176ex;" alt="{\displaystyle u(\theta )={\frac {\mu }{h^{2}}}+A\cos(\theta -\theta _{0})}"></span></dd></dl> <p>where <i>A</i> and <i>θ</i><sub>0</sub> are arbitrary constants. This resulting equation of the orbit of the object is that of an <a href="/wiki/Ellipse#Polar_form_relative_to_focus" title="Ellipse">ellipse</a> in Polar form relative to one of the focal points. This is put into a more standard form by letting <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 e\equiv h^{2}A/\mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>≡</mo> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e\equiv h^{2}A/\mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1140979616948d79c62f66180b7f249f77dcaeff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.882ex; height:3.176ex;" alt="{\displaystyle e\equiv h^{2}A/\mu }"></span> be the <a href="/wiki/Eccentricity_(orbit)" class="mw-redirect" title="Eccentricity (orbit)">eccentricity</a>, which when rearranged we see: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(\theta )={\frac {\mu }{h^{2}}}(1+e\cos(\theta -\theta _{0}))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>μ</mi> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>e</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>−</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u(\theta )={\frac {\mu }{h^{2}}}(1+e\cos(\theta -\theta _{0}))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2db7995b7efe3bdc9dfb29de01807314a90be16c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:28.836ex; height:5.176ex;" alt="{\displaystyle u(\theta )={\frac {\mu }{h^{2}}}(1+e\cos(\theta -\theta _{0}))}"></span></dd></dl> <p>Note that by letting <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\equiv h^{2}/\mu \left(1-e^{2}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≡</mo> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>μ</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\equiv h^{2}/\mu \left(1-e^{2}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8657654204e2f1f4cf11179f4dc74a12f8e8b336" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.943ex; height:3.343ex;" alt="{\displaystyle a\equiv h^{2}/\mu \left(1-e^{2}\right)}"></span> be the semi-major axis and letting <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 \theta _{0}\equiv 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>≡</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{0}\equiv 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/272d6afef5d598d137d6e645b78f88e9ce5d92fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.406ex; height:2.509ex;" alt="{\displaystyle \theta _{0}\equiv 0}"></span> so the long axis of the ellipse is along the positive <i>x</i> coordinate we yield: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r(\theta )={\frac {a\left(1-e^{2}\right)}{1+e\cos \theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>e</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r(\theta )={\frac {a\left(1-e^{2}\right)}{1+e\cos \theta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/378bfa0603892f4d8a8e1b01ecb17e93cc5f2949" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:17.945ex; height:6.343ex;" alt="{\displaystyle r(\theta )={\frac {a\left(1-e^{2}\right)}{1+e\cos \theta }}}"></span></dd></dl> <p>When the two-body system is under the influence of torque, the angular momentum <i>h</i> is not a constant. After the following calculation: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {\delta r}{\delta \theta }}&=-{\frac {1}{u^{2}}}{\frac {\delta u}{\delta \theta }}=-{\frac {h}{m}}{\frac {\delta u}{\delta \theta }}\\{\frac {\delta ^{2}r}{\delta \theta ^{2}}}&=-{\frac {h^{2}u^{2}}{m^{2}}}{\frac {\delta ^{2}u}{\delta \theta ^{2}}}-{\frac {hu^{2}}{m^{2}}}{\frac {\delta h}{\delta \theta }}{\frac {\delta u}{\delta \theta }}\\\left({\frac {\delta \theta }{\delta t}}\right)^{2}r&={\frac {h^{2}u^{3}}{m^{2}}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mi>r</mi> </mrow> <mrow> <mi>δ</mi> <mi>θ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mi>u</mi> </mrow> <mrow> <mi>δ</mi> <mi>θ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>h</mi> <mi>m</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mi>u</mi> </mrow> <mrow> <mi>δ</mi> <mi>θ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>r</mi> </mrow> <mrow> <mi>δ</mi> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>u</mi> </mrow> <mrow> <mi>δ</mi> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>h</mi> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mi>h</mi> </mrow> <mrow> <mi>δ</mi> <mi>θ</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mi>u</mi> </mrow> <mrow> <mi>δ</mi> <mi>θ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mi>θ</mi> </mrow> <mrow> <mi>δ</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>r</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {\delta r}{\delta \theta }}&=-{\frac {1}{u^{2}}}{\frac {\delta u}{\delta \theta }}=-{\frac {h}{m}}{\frac {\delta u}{\delta \theta }}\\{\frac {\delta ^{2}r}{\delta \theta ^{2}}}&=-{\frac {h^{2}u^{2}}{m^{2}}}{\frac {\delta ^{2}u}{\delta \theta ^{2}}}-{\frac {hu^{2}}{m^{2}}}{\frac {\delta h}{\delta \theta }}{\frac {\delta u}{\delta \theta }}\\\left({\frac {\delta \theta }{\delta t}}\right)^{2}r&={\frac {h^{2}u^{3}}{m^{2}}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb05d20a0100e53cfb5ca965e403eeae50a408d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.671ex; width:37.882ex; height:18.509ex;" alt="{\displaystyle {\begin{aligned}{\frac {\delta r}{\delta \theta }}&=-{\frac {1}{u^{2}}}{\frac {\delta u}{\delta \theta }}=-{\frac {h}{m}}{\frac {\delta u}{\delta \theta }}\\{\frac {\delta ^{2}r}{\delta \theta ^{2}}}&=-{\frac {h^{2}u^{2}}{m^{2}}}{\frac {\delta ^{2}u}{\delta \theta ^{2}}}-{\frac {hu^{2}}{m^{2}}}{\frac {\delta h}{\delta \theta }}{\frac {\delta u}{\delta \theta }}\\\left({\frac {\delta \theta }{\delta t}}\right)^{2}r&={\frac {h^{2}u^{3}}{m^{2}}}\end{aligned}}}"></span></dd></dl> <p>we will get the Sturm-Liouville equation of two-body system.<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> </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><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 {\frac {\delta }{\delta \theta }}\left(h{\frac {\delta u}{\delta \theta }}\right)+hu={\frac {\mu }{h}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>δ</mi> <mrow> <mi>δ</mi> <mi>θ</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mi>u</mi> </mrow> <mrow> <mi>δ</mi> <mi>θ</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>h</mi> <mi>u</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>μ</mi> <mi>h</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\delta }{\delta \theta }}\left(h{\frac {\delta u}{\delta \theta }}\right)+hu={\frac {\mu }{h}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cef5f504d154989901390cc2ab6a9679964c16c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.182ex; height:6.176ex;" alt="{\displaystyle {\frac {\delta }{\delta \theta }}\left(h{\frac {\delta u}{\delta \theta }}\right)+hu={\frac {\mu }{h}}}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(5)</b></td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Relativistic_orbital_motion">Relativistic orbital motion</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orbit&action=edit&section=14" title="Edit section: Relativistic orbital motion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The above classical (<a href="/wiki/Classical_mechanics" title="Classical mechanics">Newtonian</a>) analysis of <a href="/wiki/Orbital_mechanics" title="Orbital mechanics">orbital mechanics</a> assumes that the more subtle effects of <a href="/wiki/General_relativity" title="General relativity">general relativity</a>, such as <a href="/wiki/Frame_dragging" class="mw-redirect" title="Frame dragging">frame dragging</a> and <a href="/wiki/Gravitational_time_dilation" title="Gravitational time dilation">gravitational time dilation</a> are negligible. Relativistic effects cease to be negligible when near very massive bodies (as with the <a href="/wiki/Kepler_problem_in_general_relativity" class="mw-redirect" title="Kepler problem in general relativity">precession of Mercury's orbit</a> about the Sun), or when extreme precision is needed (as with calculations of the <a href="/wiki/Orbital_elements" title="Orbital elements">orbital elements</a> and time signal references for <a href="/wiki/Global_Positioning_System#Relativity" title="Global Positioning System">GPS</a> satellites.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup>). </p> <div class="mw-heading mw-heading2"><h2 id="Specification">Specification</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orbit&action=edit&section=15" title="Edit section: Specification"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Ephemeris" title="Ephemeris">Ephemeris</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Keplerian_elements" class="mw-redirect" title="Keplerian elements">Keplerian elements</a></div> <p>Six parameters are required to specify a <a href="/wiki/Keplerian_orbit" class="mw-redirect" title="Keplerian orbit">Keplerian orbit</a> about a body. For example, the three numbers that specify the body's initial position, and the three values that specify its velocity will define a unique orbit that can be calculated forwards (or backwards) in time. However, traditionally the parameters used are slightly different. </p><p>The traditionally used set of orbital elements is called the set of <a href="/wiki/Orbital_elements" title="Orbital elements">Keplerian elements</a>, after Johannes Kepler and his laws. The Keplerian elements are six: </p> <ul><li><a href="/wiki/Inclination" class="mw-redirect" title="Inclination">Inclination</a> (<i>i</i>)</li> <li><a href="/wiki/Longitude_of_the_ascending_node" title="Longitude of the ascending node">Longitude of the ascending node</a> (Ω)</li> <li><a href="/wiki/Argument_of_periapsis" title="Argument of periapsis">Argument of periapsis</a> (ω)</li> <li><a href="/wiki/Orbital_eccentricity" title="Orbital eccentricity">Eccentricity</a> (<i>e</i>)</li> <li><a href="/wiki/Semimajor_axis" class="mw-redirect" title="Semimajor axis">Semimajor axis</a> (<i>a</i>)</li> <li><a href="/wiki/Mean_anomaly" title="Mean anomaly">Mean anomaly</a> at <a href="/wiki/Epoch_(astronomy)" title="Epoch (astronomy)">epoch</a> (<i>M</i><sub>0</sub>).</li></ul> <p>In principle, once the orbital elements are known for a body, its position can be calculated forward and backward indefinitely in time. However, in practice, orbits are affected or <a href="/wiki/Perturbation_(astronomy)" title="Perturbation (astronomy)">perturbed</a>, by other forces than simple gravity from an assumed point source (see the next section), and thus the orbital elements change over time. </p><p>Note that, unless the eccentricity is zero, <i>a</i> is not the average orbital radius. The time-averaged orbital distance is given by:<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> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {r}}=a\left(1+{\frac {e^{2}}{2}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>a</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {r}}=a\left(1+{\frac {e^{2}}{2}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8eb2dff2276fddcdfdddc6b67d817867530637c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.405ex; height:6.343ex;" alt="{\displaystyle {\bar {r}}=a\left(1+{\frac {e^{2}}{2}}\right)}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Orbital_planes">Orbital planes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orbit&action=edit&section=16" title="Edit section: Orbital planes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Orbital_plane" title="Orbital plane">Orbital plane</a></div> <p>The analysis so far has been two dimensional; it turns out that an <a href="/wiki/Perturbation_theory" title="Perturbation theory">unperturbed</a> orbit is two-dimensional in a plane fixed in space, and thus the extension to three dimensions requires simply rotating the two-dimensional plane into the required angle relative to the poles of the planetary body involved. </p><p>The rotation to do this in three dimensions requires three numbers to uniquely determine; traditionally these are expressed as three angles. </p> <div class="mw-heading mw-heading3"><h3 id="Orbital_period">Orbital period</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orbit&action=edit&section=17" title="Edit section: Orbital period"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Orbital_period" title="Orbital period">Orbital period</a></div> <p>The orbital period is simply how long an orbiting body takes to complete one orbit. </p> <div class="mw-heading mw-heading2"><h2 id="Perturbations">Perturbations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orbit&action=edit&section=18" title="Edit section: Perturbations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Perturbation_(astronomy)" title="Perturbation (astronomy)">Perturbation (astronomy)</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Osculating_orbit#Perturbations" title="Osculating orbit">Osculating orbit § Perturbations</a>, and <a href="/wiki/Orbit_modeling#Perturbations" title="Orbit modeling">Orbit modeling § Perturbations</a></div> <p>An orbital perturbation is when a force or impulse which is much smaller than the overall force or average impulse of the main gravitating body and which is external to the two orbiting bodies causes an acceleration, which changes the parameters of the orbit over time. </p> <div class="mw-heading mw-heading3"><h3 id="Radial,_prograde_and_transverse_perturbations"><span id="Radial.2C_prograde_and_transverse_perturbations"></span>Radial, prograde and transverse perturbations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orbit&action=edit&section=19" title="Edit section: Radial, prograde and transverse perturbations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A small radial impulse given to a body in orbit changes the <a href="/wiki/Eccentricity_(mathematics)" title="Eccentricity (mathematics)">eccentricity</a>, but not the <a href="/wiki/Orbital_period" title="Orbital period">orbital period</a> (to first order). A <a href="/wiki/Direct_motion" class="mw-redirect" title="Direct motion">prograde</a> or <a href="/wiki/Retrograde_motion" class="mw-redirect" title="Retrograde motion">retrograde</a> impulse (i.e. an impulse applied along the orbital motion) changes both the eccentricity and the <a href="/wiki/Orbital_period" title="Orbital period">orbital period</a>. Notably, a prograde impulse at <a href="/wiki/Periapsis" class="mw-redirect" title="Periapsis">periapsis</a> raises the altitude at <a href="/wiki/Apoapsis" class="mw-redirect" title="Apoapsis">apoapsis</a>, and vice versa and a retrograde impulse does the opposite. A transverse impulse (out of the orbital plane) causes rotation of the <a href="/wiki/Orbital_plane_(astronomy)" class="mw-redirect" title="Orbital plane (astronomy)">orbital plane</a> without changing the <a href="/wiki/Orbit_(dynamics)" title="Orbit (dynamics)">period</a> or eccentricity. In all instances, a closed orbit will still intersect the perturbation point. </p> <div class="mw-heading mw-heading3"><h3 id="Orbital_decay">Orbital decay</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orbit&action=edit&section=20" title="Edit section: Orbital decay"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Orbital_decay" title="Orbital decay">Orbital decay</a></div> <p>If an orbit is about a planetary body with a significant atmosphere, its orbit can decay because of <a href="/wiki/Drag_(physics)" title="Drag (physics)">drag</a>. Particularly at each <a href="/wiki/Periapsis" class="mw-redirect" title="Periapsis">periapsis</a>, the object experiences atmospheric drag, losing energy. Each time, the orbit grows less eccentric (more circular) because the object loses kinetic energy precisely when that energy is at its maximum. This is similar to the effect of slowing a pendulum at its lowest point; the highest point of the pendulum's swing becomes lower. With each successive slowing more of the orbit's path is affected by the atmosphere and the effect becomes more pronounced. Eventually, the effect becomes so great that the maximum kinetic energy is not enough to return the orbit above the limits of the atmospheric drag effect. When this happens the body will rapidly spiral down and intersect the central body. </p><p>The bounds of an atmosphere vary wildly. During a <a href="/wiki/Solar_maximum" title="Solar maximum">solar maximum</a>, the Earth's atmosphere causes drag up to a hundred kilometres higher than during a solar minimum. </p><p>Some satellites with long conductive tethers can also experience orbital decay because of electromagnetic drag from the <a href="/wiki/Earth%27s_magnetic_field" title="Earth's magnetic field">Earth's magnetic field</a>. As the wire cuts the magnetic field it acts as a generator, moving electrons from one end to the other. The orbital energy is converted to heat in the wire. </p><p>Orbits can be artificially influenced through the use of rocket engines which change the kinetic energy of the body at some point in its path. This is the conversion of chemical or electrical energy to kinetic energy. In this way changes in the orbit shape or orientation can be facilitated. </p><p>Another method of artificially influencing an orbit is through the use of <a href="/wiki/Solar_sail" title="Solar sail">solar sails</a> or <a href="/wiki/Magnetic_sail" title="Magnetic sail">magnetic sails</a>. These forms of propulsion require no propellant or energy input other than that of the Sun, and so can be used indefinitely. See <a href="/wiki/Statite" title="Statite">statite</a> for one such proposed use. </p><p>Orbital decay can occur due to <a href="/wiki/Tidal_force" title="Tidal force">tidal forces</a> for objects below the <a href="/wiki/Synchronous_orbit" title="Synchronous orbit">synchronous orbit</a> for the body they're orbiting. The gravity of the orbiting object raises <a href="/wiki/Tidal_bulge" class="mw-redirect" title="Tidal bulge">tidal bulges</a> in the primary, and since below the synchronous orbit, the orbiting object is moving faster than the body's surface the bulges lag a short angle behind it. The gravity of the bulges is slightly off of the primary-satellite axis and thus has a component along with the satellite's motion. The near bulge slows the object more than the far bulge speeds it up, and as a result, the orbit decays. Conversely, the gravity of the satellite on the bulges applies <a href="/wiki/Torque" title="Torque">torque</a> on the primary and speeds up its rotation. Artificial satellites are too small to have an appreciable tidal effect on the planets they orbit, but several moons in the Solar System are undergoing orbital decay by this mechanism. Mars' innermost moon <a href="/wiki/Phobos_(moon)" title="Phobos (moon)">Phobos</a> is a prime example and is expected to either impact Mars' surface or break up into a ring within 50 million years. </p><p>Orbits can decay via the emission of <a href="/wiki/Gravitational_wave" title="Gravitational wave">gravitational waves</a>. This mechanism is extremely weak for most stellar objects, only becoming significant in cases where there is a combination of extreme mass and extreme acceleration, such as with <a href="/wiki/Black_hole" title="Black hole">black holes</a> or <a href="/wiki/Neutron_star" title="Neutron star">neutron stars</a> that are orbiting each other closely. </p> <div class="mw-heading mw-heading3"><h3 id="Oblateness">Oblateness</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orbit&action=edit&section=21" title="Edit section: Oblateness"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The standard analysis of orbiting bodies assumes that all bodies consist of uniform spheres, or more generally, concentric shells each of uniform density. It can be shown that such bodies are gravitationally equivalent to point sources. </p><p>However, in the real world, many bodies rotate, and this introduces <a href="/wiki/Oblateness" class="mw-redirect" title="Oblateness">oblateness</a> and distorts the gravity field, and gives a <a href="/wiki/Quadropole#Gravitational_quadrupole" class="mw-redirect" title="Quadropole">quadrupole</a> moment to the gravitational field which is significant at distances comparable to the radius of the body. In the general case, the gravitational potential of a rotating body such as, e.g., a planet is usually expanded in multipoles accounting for the departures of it from spherical symmetry. From the point of view of satellite dynamics, of particular relevance are the so-called even zonal harmonic coefficients, or even zonals, since they induce secular orbital perturbations which are cumulative over time spans longer than the orbital period.<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 id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><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> They do depend on the orientation of the body's symmetry axis in the space, affecting, in general, the whole orbit, with the exception of the semimajor axis. </p> <div class="mw-heading mw-heading3"><h3 id="Multiple_gravitating_bodies">Multiple gravitating bodies</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orbit&action=edit&section=22" title="Edit section: Multiple gravitating bodies"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/N-body_problem" title="N-body problem">n-body problem</a></div> <p>The effects of other gravitating bodies can be significant. For example, the <a href="/wiki/Orbit_of_the_Moon" title="Orbit of the Moon">orbit of the Moon</a> cannot be accurately described without allowing for the action of the Sun's gravity as well as the Earth's. One approximate result is that bodies will usually have reasonably stable orbits around a heavier planet or moon, in spite of these perturbations, provided they are orbiting well within the heavier body's <a href="/wiki/Hill_sphere" title="Hill sphere">Hill sphere</a>. </p><p>When there are more than two gravitating bodies it is referred to as an <a href="/wiki/N-body_problem" title="N-body problem">n-body problem</a>. Most n-body problems have no <a href="/wiki/Closed-form_expression" title="Closed-form expression">closed form solution</a>, although some special cases have been formulated. </p> <div class="mw-heading mw-heading3"><h3 id="Light_radiation_and_stellar_wind">Light radiation and stellar wind</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orbit&action=edit&section=23" title="Edit section: Light radiation and stellar wind"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For smaller bodies particularly, light and <a href="/wiki/Stellar_wind" title="Stellar wind">stellar wind</a> can cause significant perturbations to the <a href="/wiki/Attitude_(geometry)" class="mw-redirect" title="Attitude (geometry)">attitude</a> and direction of motion of the body, and over time can be significant. Of the planetary bodies, the motion of <a href="/wiki/Asteroid" title="Asteroid">asteroids</a> is particularly affected over large periods when the asteroids are rotating relative to the Sun. </p> <div class="mw-heading mw-heading2"><h2 id="Strange_orbits">Strange orbits</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orbit&action=edit&section=24" title="Edit section: Strange orbits"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Mathematicians have discovered that it is possible in principle to have multiple bodies in non-elliptical orbits that repeat periodically, although most such orbits are not stable regarding small perturbations in mass, position, or velocity. However, some special stable cases have been identified, including a planar figure-eight orbit occupied by <a href="/wiki/Three-body_problem" title="Three-body problem">three moving bodies</a>.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> Further studies have discovered that nonplanar orbits are also possible, including one involving 12 masses moving in 4 roughly circular, interlocking orbits <a href="/wiki/Topology" title="Topology">topologically</a> equivalent to the edges of a <a href="/wiki/Cuboctahedron" title="Cuboctahedron">cuboctahedron</a>.<sup id="cite_ref-Peterson_17-0" class="reference"><a href="#cite_note-Peterson-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p><p>Finding such orbits naturally occurring in the universe is thought to be extremely unlikely, because of the improbability of the required conditions occurring by chance.<sup id="cite_ref-Peterson_17-1" class="reference"><a href="#cite_note-Peterson-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Astrodynamics">Astrodynamics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orbit&action=edit&section=25" title="Edit section: Astrodynamics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Orbital_mechanics" title="Orbital mechanics">Orbital mechanics</a></div> <p>Orbital mechanics or astrodynamics is the application of <a href="/wiki/Ballistics" title="Ballistics">ballistics</a> and <a href="/wiki/Celestial_mechanics" title="Celestial mechanics">celestial mechanics</a> to the practical problems concerning the motion of <a href="/wiki/Rocket" title="Rocket">rockets</a> and other <a href="/wiki/Spacecraft" title="Spacecraft">spacecraft</a>. The motion of these objects is usually calculated from <a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</a> and <a href="/wiki/Newton%27s_law_of_universal_gravitation" title="Newton's law of universal gravitation">Newton's law of universal gravitation</a>. It is a core discipline within space mission design and control. Celestial mechanics treats more broadly the orbital dynamics of systems under the influence of <a href="/wiki/Gravity" title="Gravity">gravity</a>, including spacecraft and natural astronomical bodies such as star systems, <a href="/wiki/Planet" title="Planet">planets</a>, <a href="/wiki/Natural_satellite" title="Natural satellite">moons</a>, and <a href="/wiki/Comet" title="Comet">comets</a>. Orbital mechanics focuses on spacecraft <a href="/wiki/Trajectory" title="Trajectory">trajectories</a>, including <a href="/wiki/Orbital_maneuver" title="Orbital maneuver">orbital maneuvers</a>, orbit plane changes, and interplanetary transfers, and is used by mission planners to predict the results of <a href="/wiki/Spacecraft_propulsion" title="Spacecraft propulsion">propulsive maneuvers</a>. <a href="/wiki/General_relativity" title="General relativity">General relativity</a> is a more exact theory than Newton's laws for calculating orbits, and is sometimes necessary for greater accuracy or in high-gravity situations (such as orbits close to the Sun). </p> <div class="mw-heading mw-heading2"><h2 id="Earth_orbits">Earth orbits</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orbit&action=edit&section=26" title="Edit section: Earth orbits"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/List_of_orbits" title="List of orbits">List of orbits</a></div> <ul><li><a href="/wiki/Low_Earth_orbit" title="Low Earth orbit">Low Earth orbit</a> (LEO): <a href="/wiki/Geocentric_orbit" title="Geocentric orbit">Geocentric orbits</a> with altitudes up to 2,000 <a href="/wiki/Km" class="mw-redirect" title="Km">km</a> (0–1,240 <a href="/wiki/Mile" title="Mile">miles</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></li> <li><a href="/wiki/Medium_Earth_orbit" title="Medium Earth orbit">Medium Earth orbit</a> (MEO): <a href="/wiki/Geocentric_orbit" title="Geocentric orbit">Geocentric orbits</a> ranging in altitude from 2,000 <a href="/wiki/Km" class="mw-redirect" title="Km">km</a> (1,240 <a href="/wiki/Mile" title="Mile">miles</a>) to just below <a href="/wiki/Geosynchronous_orbit" title="Geosynchronous orbit">geosynchronous orbit</a> at 35,786 kilometers (22,236 mi). Also known as an <a href="/wiki/Intermediate_circular_orbit" class="mw-redirect" title="Intermediate circular orbit">intermediate circular orbit</a>. These are "most commonly at 20,200 kilometers (12,600 mi), or 20,650 kilometers (12,830 mi), with an orbital period of 12 hours."<sup id="cite_ref-nasa_orbit_definition_19-0" class="reference"><a href="#cite_note-nasa_orbit_definition-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup></li> <li>Both <a href="/wiki/Geosynchronous_orbit" title="Geosynchronous orbit">geosynchronous orbit</a> (GSO) and <a href="/wiki/Geostationary_orbit" title="Geostationary orbit">geostationary orbit</a> (GEO) are orbits around Earth matching Earth's <a href="/wiki/Sidereal_rotation" class="mw-redirect" title="Sidereal rotation">sidereal rotation</a> period. All geosynchronous and geostationary orbits have a <a href="/wiki/Semi-major_axis" class="mw-redirect" title="Semi-major axis">semi-major axis</a> of 42,164 km (26,199 mi).<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> All geostationary orbits are also geosynchronous, but not all geosynchronous orbits are geostationary. A geostationary orbit stays exactly above the equator, whereas a geosynchronous orbit may swing north and south to cover more of the Earth's surface. Both complete one full orbit of Earth per sidereal day (relative to the stars, not the Sun).</li> <li><a href="/wiki/High_Earth_orbit" title="High Earth orbit">High Earth orbit</a>: <a href="/wiki/Geocentric_orbit" title="Geocentric orbit">Geocentric orbits</a> above the altitude of <a href="/wiki/Geosynchronous_orbit" title="Geosynchronous orbit">geosynchronous orbit</a> 35,786 <a href="/wiki/Km" class="mw-redirect" title="Km">km</a> (22,240 <a href="/wiki/Mile" title="Mile">miles</a>).<sup id="cite_ref-nasa_orbit_definition_19-1" class="reference"><a href="#cite_note-nasa_orbit_definition-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading2"><h2 id="Scaling_in_gravity">Scaling in gravity</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orbit&action=edit&section=27" title="Edit section: Scaling in gravity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Gravitational_constant" title="Gravitational constant">gravitational constant</a> <i>G</i> has been calculated as: </p> <ul><li>(6.6742 ± 0.001) × 10<sup>−11</sup> (kg/m<sup>3</sup>)<sup>−1</sup>s<sup>−2</sup>.</li></ul> <p>Thus the constant has dimension density<sup>−1</sup> time<sup>−2</sup>. This corresponds to the following properties. </p><p><a href="/wiki/Scale_factor" class="mw-redirect" title="Scale factor">Scaling</a> of distances (including sizes of bodies, while keeping the densities the same) gives <a href="/wiki/Similarity_(geometry)" title="Similarity (geometry)">similar</a> orbits without scaling the time: if for example distances are halved, masses are divided by 8, gravitational forces by 16 and gravitational accelerations by 2. Hence velocities are halved and orbital periods and other travel times related to gravity remain the same. For example, when an object is dropped from a tower, the time it takes to fall to the ground remains the same with a scale model of the tower on a scale model of the Earth. </p><p>Scaling of distances while keeping the masses the same (in the case of point masses, or by adjusting the densities) gives similar orbits; if distances are multiplied by 4, gravitational forces and accelerations are divided by 16, velocities are halved and orbital periods are multiplied by 8. </p><p>When all densities are multiplied by 4, orbits are the same; gravitational forces are multiplied by 16 and accelerations by 4, velocities are doubled and orbital periods are halved. </p><p>When all densities are multiplied by 4, and all sizes are halved, orbits are similar; masses are divided by 2, gravitational forces are the same, gravitational accelerations are doubled. Hence velocities are the same and orbital periods are halved. </p><p>In all these cases of scaling. if densities are multiplied by 4, times are halved; if velocities are doubled, forces are multiplied by 16. </p><p>These properties are illustrated in the formula (derived from the <a href="/wiki/Orbital_period#Small_body_orbiting_a_central_body" title="Orbital period">formula for the orbital period</a>) </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle GT^{2}\rho =3\pi \left({\frac {a}{r}}\right)^{3},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ρ</mi> <mo>=</mo> <mn>3</mn> <mi>π</mi> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>r</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle GT^{2}\rho =3\pi \left({\frac {a}{r}}\right)^{3},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6ec5a9836f4acf84cfe6b2aebffca080ac9813c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.938ex; height:5.176ex;" alt="{\displaystyle GT^{2}\rho =3\pi \left({\frac {a}{r}}\right)^{3},}"></span></dd></dl> <p>for an elliptical orbit with <a href="/wiki/Semi-major_axis" class="mw-redirect" title="Semi-major axis">semi-major axis</a> <i>a</i>, of a small body around a spherical body with radius <i>r</i> and average density <i>ρ</i>, where <i>T</i> is the orbital period. See also <a href="/wiki/Kepler%E2%80%99s_third_law" class="mw-redirect" title="Kepler’s third law">Kepler’s third law</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Tidal_locking">Tidal locking</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orbit&action=edit&section=28" title="Edit section: Tidal locking"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Tidal_locking" title="Tidal locking">Tidal locking</a></div> <p>Some bodies are tidally locked with other bodies, meaning that one side of the celestial body is permanently facing its host object. This is the case for Earth-<a href="/wiki/Moon" title="Moon">Moon</a> and Pluto-Charon system. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orbit&action=edit&section=29" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col"> <ul><li><a href="/wiki/Ephemeris" title="Ephemeris">Ephemeris</a> is a compilation of positions of naturally occurring astronomical objects as well as artificial satellites in the sky at a given time or times.</li> <li><a href="/wiki/Free_drift" title="Free drift">Free drift</a></li> <li><a href="/wiki/Klemperer_rosette" title="Klemperer rosette">Klemperer rosette</a></li> <li><a href="/wiki/List_of_orbits" title="List of orbits">List of orbits</a></li> <li><a href="/wiki/Molniya_orbit" title="Molniya orbit">Molniya orbit</a></li> <li><a href="/wiki/Orbit_determination" title="Orbit determination">Orbit determination</a></li> <li><a href="/wiki/Orbital_spaceflight" title="Orbital spaceflight">Orbital spaceflight</a></li> <li><a href="/wiki/Perifocal_coordinate_system" title="Perifocal coordinate system">Perifocal coordinate system</a></li> <li><a href="/wiki/Polar_orbit" title="Polar orbit">Polar orbit</a></li> <li><a href="/wiki/Radial_trajectory" title="Radial trajectory">Radial trajectory</a></li> <li><a href="/wiki/Rosetta_orbit" title="Rosetta orbit">Rosetta orbit</a></li> <li><a href="/wiki/VSOP_model" title="VSOP model">VSOP model</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orbit&action=edit&section=30" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation encyclopaedia cs1"><a rel="nofollow" class="external text" href="https://www.britannica.com/EBchecked/topic/431123/orbit">"orbit (astronomy)"</a>. <i>Encyclopædia Britannica</i> (Online ed.). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150505012919/https://www.britannica.com/EBchecked/topic/431123/orbit">Archived</a> from the original on 5 May 2015<span class="reference-accessdate">. Retrieved <span class="nowrap">28 July</span> 2008</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=orbit+%28astronomy%29&rft.btitle=Encyclop%C3%A6dia+Britannica&rft.edition=Online&rft_id=https%3A%2F%2Fwww.britannica.com%2FEBchecked%2Ftopic%2F431123%2Forbit&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrbit" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://spaceplace.nasa.gov/barycenter/">"The Space Place :: What's a Barycenter"</a>. NASA. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130108073405/http://spaceplace.nasa.gov/barycenter/">Archived</a> from the original on 8 January 2013<span class="reference-accessdate">. Retrieved <span class="nowrap">26 November</span> 2012</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+Space+Place+%3A%3A+What%27s+a+Barycenter&rft.pub=NASA&rft_id=http%3A%2F%2Fspaceplace.nasa.gov%2Fbarycenter%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrbit" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Kuhn, <i>The Copernican Revolution</i>, pp. 238, 246–252</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><i>Encyclopædia Britannica</i>, 1968, vol. 2, p. 645</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">M Caspar, <i>Kepler</i> (1959, Abelard-Schuman), at pp.131–140; A Koyré, <i>The Astronomical Revolution: Copernicus, Kepler, Borelli</i> (1973, Methuen), pp. 277–279</span> </li> <li id="cite_note-Kepler's_Laws_of_Planetary_Motion-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-Kepler's_Laws_of_Planetary_Motion_6-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJones" class="citation web cs1">Jones, Andrew. <a rel="nofollow" class="external text" href="http://physics.about.com/od/astronomy/p/keplerlaws.htm">"Kepler's Laws of Planetary Motion"</a>. <a href="/wiki/About.com" class="mw-redirect" title="About.com">about.com</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20161118041151/http://physics.about.com/od/astronomy/p/keplerlaws.htm">Archived</a> from the original on 18 November 2016<span class="reference-accessdate">. Retrieved <span class="nowrap">1 June</span> 2008</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Kepler%27s+Laws+of+Planetary+Motion&rft.pub=about.com&rft.aulast=Jones&rft.aufirst=Andrew&rft_id=http%3A%2F%2Fphysics.about.com%2Fod%2Fastronomy%2Fp%2Fkeplerlaws.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrbit" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">See <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rEYUAAAAQAAJ&pg=PA6">pages 6 to 8 in Newton's "Treatise of the System of the World"</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20161230132051/https://books.google.com/books?id=rEYUAAAAQAAJ&pg=PA6">Archived</a> 30 December 2016 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> (written 1685, translated into English 1728, see <a href="/wiki/Philosophi%C3%A6_Naturalis_Principia_Mathematica#Preliminary_version" title="Philosophiæ Naturalis Principia Mathematica">Newton's 'Principia' – A preliminary version</a>), for the original version of this 'cannonball' thought-experiment.</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCarletonGuoMunshiTremmel2021" class="citation journal cs1">Carleton, Timothy; Guo, Yicheng; Munshi, Ferah; Tremmel, Michael; Wright, Anna (2021). <a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fmnras%2Fstab031">"An excess of globular clusters in Ultra-Diffuse Galaxies formed through tidal heating"</a>. <i>Monthly Notices of the Royal Astronomical Society</i>. <b>502</b>: 398–406. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/2008.11205">2008.11205</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fmnras%2Fstab031">10.1093/mnras/stab031</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Monthly+Notices+of+the+Royal+Astronomical+Society&rft.atitle=An+excess+of+globular+clusters+in+Ultra-Diffuse+Galaxies+formed+through+tidal+heating&rft.volume=502&rft.pages=398-406&rft.date=2021&rft_id=info%3Aarxiv%2F2008.11205&rft_id=info%3Adoi%2F10.1093%2Fmnras%2Fstab031&rft.aulast=Carleton&rft.aufirst=Timothy&rft.au=Guo%2C+Yicheng&rft.au=Munshi%2C+Ferah&rft.au=Tremmel%2C+Michael&rft.au=Wright%2C+Anna&rft_id=https%3A%2F%2Fdoi.org%2F10.1093%252Fmnras%252Fstab031&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrbit" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFitzpatrick2006" class="citation web cs1">Fitzpatrick, Richard (2 February 2006). <a rel="nofollow" class="external text" href="http://farside.ph.utexas.edu/teaching/301/lectures/node155.html">"Planetary orbits"</a>. <i>Classical Mechanics – an introductory course</i>. The University of Texas at Austin. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20010303195257/http://farside.ph.utexas.edu/teaching/301/lectures/node155.html">Archived</a> from the original on 3 March 2001.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Classical+Mechanics+%E2%80%93+an+introductory+course&rft.atitle=Planetary+orbits&rft.date=2006-02-02&rft.aulast=Fitzpatrick&rft.aufirst=Richard&rft_id=http%3A%2F%2Ffarside.ph.utexas.edu%2Fteaching%2F301%2Flectures%2Fnode155.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrbit" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLuo2020" class="citation journal cs1">Luo, Siwei (22 June 2020). <a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F2399-6528%2Fab9c30">"The Sturm-Liouville problem of two-body system"</a>. <i>Journal of Physics Communications</i>. <b>4</b> (6): 061001. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2020JPhCo...4f1001L">2020JPhCo...4f1001L</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F2399-6528%2Fab9c30">10.1088/2399-6528/ab9c30</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Physics+Communications&rft.atitle=The+Sturm-Liouville+problem+of+two-body+system&rft.volume=4&rft.issue=6&rft.pages=061001&rft.date=2020-06-22&rft_id=info%3Adoi%2F10.1088%2F2399-6528%2Fab9c30&rft_id=info%3Abibcode%2F2020JPhCo...4f1001L&rft.aulast=Luo&rft.aufirst=Siwei&rft_id=https%3A%2F%2Fdoi.org%2F10.1088%252F2399-6528%252Fab9c30&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrbit" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">Pogge, Richard W.; <a rel="nofollow" class="external text" href="http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps.html">"Real-World Relativity: The GPS Navigation System"</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20151114135709/http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps.html">Archived</a> 14 November 2015 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. Retrieved 25 January 2008.</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFestouKellerWeaver2004" class="citation book cs1">Festou, M.; Keller, H. Uwe; Weaver, Harold A. (2004). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ehA8EAAAQBAJ&pg=PA157"><i>Comets II</i></a>. 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Retrieved <span class="nowrap">21 July</span> 2017</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Science+News&rft.atitle=Strange+Orbits&rft.date=2013-09-23&rft.aulast=Peterson&rft.aufirst=Ivars&rft_id=https%3A%2F%2Fwww.sciencenews.org%2Farticle%2Fstrange-orbits-1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrbit" class="Z3988"></span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20130215143933/http://orbitaldebris.jsc.nasa.gov/library/NSS1740_14/nss1740_14-1995.pdf">"NASA Safety Standard 1740.14, Guidelines and Assessment Procedures for Limiting Orbital Debris"</a> <span class="cs1-format">(PDF)</span>. Office of Safety and Mission Assurance. 1 August 1995. Archived from <a rel="nofollow" class="external text" href="https://www.orbitaldebris.jsc.nasa.gov/library/NSS1740_14/nss1740_14-1995.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 15 February 2013.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=NASA+Safety+Standard+1740.14%2C+Guidelines+and+Assessment+Procedures+for+Limiting+Orbital+Debris&rft.pub=Office+of+Safety+and+Mission+Assurance&rft.date=1995-08-01&rft_id=http%3A%2F%2Fwww.orbitaldebris.jsc.nasa.gov%2Flibrary%2FNSS1740_14%2Fnss1740_14-1995.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrbit" class="Z3988"></span>, pp. 37–38 (6-1, 6-2); figure 6-1.</span> </li> <li id="cite_note-nasa_orbit_definition-19"><span class="mw-cite-backlink">^ <a href="#cite_ref-nasa_orbit_definition_19-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-nasa_orbit_definition_19-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20130511114407/http://gcmd.nasa.gov/add/ancillaryguide/platforms/orbit.html">"Orbit: Definition"</a>. <i>Ancillary Description Writer's Guide, 2013</i>. National Aeronautics and Space Administration (NASA) Global Change Master Directory. Archived from <a rel="nofollow" class="external text" href="https://gcmd.nasa.gov/add/ancillaryguide/platforms/orbit.html">the original</a> on 11 May 2013<span class="reference-accessdate">. Retrieved <span class="nowrap">29 April</span> 2013</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Ancillary+Description+Writer%27s+Guide%2C+2013&rft.atitle=Orbit%3A+Definition&rft_id=http%3A%2F%2Fgcmd.nasa.gov%2Fadd%2Fancillaryguide%2Fplatforms%2Forbit.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrbit" class="Z3988"></span></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVallado2007" class="citation book cs1">Vallado, David A. (2007). <i>Fundamentals of Astrodynamics and Applications</i>. Hawthorne, CA: Microcosm Press. p. 31.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+of+Astrodynamics+and+Applications&rft.place=Hawthorne%2C+CA&rft.pages=31&rft.pub=Microcosm+Press&rft.date=2007&rft.aulast=Vallado&rft.aufirst=David+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrbit" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orbit&action=edit&section=31" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAbellMorrisonWolff1987" class="citation book cs1">Abell, George O.; Morrison, David & Wolff, Sidney C. (1987). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/explorationofuni0005abel"><i>Exploration of the Universe</i></a></span> (Fifth ed.). Saunders College Publishing. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780030051432" title="Special:BookSources/9780030051432"><bdi>9780030051432</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Exploration+of+the+Universe&rft.edition=Fifth&rft.pub=Saunders+College+Publishing&rft.date=1987&rft.isbn=9780030051432&rft.aulast=Abell&rft.aufirst=George+O.&rft.au=Morrison%2C+David&rft.au=Wolff%2C+Sidney+C.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fexplorationofuni0005abel&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrbit" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLinton2004" class="citation book cs1">Linton, Christopher (2004). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=aJuwFLGWKF8C"><i>From Eudoxus to Einstein: A History of Mathematical Astronomy</i></a>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-139-45379-0" title="Special:BookSources/978-1-139-45379-0"><bdi>978-1-139-45379-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=From+Eudoxus+to+Einstein%3A+A+History+of+Mathematical+Astronomy&rft.pub=Cambridge+University+Press&rft.date=2004&rft.isbn=978-1-139-45379-0&rft.aulast=Linton&rft.aufirst=Christopher&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DaJuwFLGWKF8C&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrbit" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMilaniGronchi2010" class="citation book cs1">Milani, Andrea; Gronchi, Giovanni F. (2010). <i>Theory of Orbit Determination</i>. Cambridge University Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theory+of+Orbit+Determination&rft.pub=Cambridge+University+Press&rft.date=2010&rft.aulast=Milani&rft.aufirst=Andrea&rft.au=Gronchi%2C+Giovanni+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrbit" class="Z3988"></span> Discusses new algorithms for determining the orbits of both natural and artificial celestial bodies.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSwetzFauvelJohanssonKatz1995" class="citation book cs1">Swetz, Frank; Fauvel, John; Johansson, Bengt; Katz, Victor; Bekken, Otto (1995). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=gqGLoh-WYrEC&pg=PA269"><i>Learn from the Masters</i></a>. MAA. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-88385-703-8" title="Special:BookSources/978-0-88385-703-8"><bdi>978-0-88385-703-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Learn+from+the+Masters&rft.pub=MAA&rft.date=1995&rft.isbn=978-0-88385-703-8&rft.aulast=Swetz&rft.aufirst=Frank&rft.au=Fauvel%2C+John&rft.au=Johansson%2C+Bengt&rft.au=Katz%2C+Victor&rft.au=Bekken%2C+Otto&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DgqGLoh-WYrEC%26pg%3DPA269&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrbit" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orbit&action=edit&section=32" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/40px-Wiktionary-logo-en-v2.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/60px-Wiktionary-logo-en-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/80px-Wiktionary-logo-en-v2.svg.png 2x" data-file-width="512" data-file-height="512" /></span></span></div> <div class="side-box-text plainlist">Look up <i><b><a href="https://en.wiktionary.org/wiki/Special:Search/orbit" class="extiw" title="wiktionary:Special:Search/orbit">orbit</a></b></i> in Wiktionary, the free dictionary.</div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237033735"><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <a href="https://commons.wikimedia.org/wiki/Orbits" class="extiw" title="commons:Orbits"><span style="font-style:italic; font-weight:bold;">Orbits</span></a>.</div></div> </div> <ul><li><a rel="nofollow" class="external text" href="http://www.calctool.org/CALC/phys/astronomy/planet_orbit">CalcTool: Orbital period of a planet calculator</a>. Has wide choice of units. Requires JavaScript.</li> <li><a rel="nofollow" class="external text" href="http://www.phy.hk/wiki/englishhtm/Motion.htm">Java simulation on orbital motion</a>. Requires Java.</li> <li><a rel="nofollow" class="external text" href="http://www.ncdc.noaa.gov/paleo/forcing.html">NOAA page on Climate Forcing Data</a> includes (calculated) data on Earth orbit variations over the last 50 million years and for the coming 20 million years</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20100201160808/http://www.bridgewater.edu/~rbowman/ISAW/PlanetOrbit.html">On-line orbit plotter</a>. Requires JavaScript.</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20120204054322/http://www.braeunig.us/space/orbmech.htm">Orbital Mechanics</a> (Rocket and Space Technology)</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20060210012245/http://www.astrobiology.ucla.edu/OTHER/SSO/">Orbital simulations</a> by Varadi, <a href="/wiki/Michael_Ghil" title="Michael Ghil">Ghil</a> and Runnegar (2003) provide another, slightly different series for Earth orbit eccentricity, and also a series for orbital inclination. Orbits for the other planets were also calculated, by <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFF._VaradiB._RunnegarM._Ghil2003" class="citation journal cs1">F. Varadi; B. Runnegar; M. Ghil (2003). <a rel="nofollow" class="external text" href="https://doi.org/10.1086%2F375560">"Successive Refinements in Long-Term Integrations of Planetary Orbits"</a>. <i>The Astrophysical Journal</i>. <b>592</b> (1): 620–630. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2003ApJ...592..620V">2003ApJ...592..620V</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1086%2F375560">10.1086/375560</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Astrophysical+Journal&rft.atitle=Successive+Refinements+in+Long-Term+Integrations+of+Planetary+Orbits&rft.volume=592&rft.issue=1&rft.pages=620-630&rft.date=2003&rft_id=info%3Adoi%2F10.1086%2F375560&rft_id=info%3Abibcode%2F2003ApJ...592..620V&rft.au=F.+Varadi&rft.au=B.+Runnegar&rft.au=M.+Ghil&rft_id=https%3A%2F%2Fdoi.org%2F10.1086%252F375560&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrbit" class="Z3988"></span>, but only the <a rel="nofollow" class="external text" href="https://web.archive.org/web/20041031055131/http://www.astrobiology.ucla.edu/OTHER/SSO/Misc/">eccentricity data for Earth and Mercury</a> are available online.</li> <li><a rel="nofollow" class="external text" href="http://www.lri.fr/~dragice/gravity/">Understand orbits using direct manipulation</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20171108140606/http://www.lri.fr/~dragice/gravity/">Archived</a> 8 November 2017 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. Requires JavaScript and Macromedia</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMerrifield" class="citation web cs1">Merrifield, Michael. <a rel="nofollow" class="external text" href="http://www.sixtysymbols.com/videos/orbit.htm">"Orbits (including the first manned orbit)"</a>. <i>Sixty Symbols</i>. <a href="/wiki/Brady_Haran" title="Brady Haran">Brady Haran</a> for the <a href="/wiki/University_of_Nottingham" title="University of Nottingham">University of Nottingham</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Sixty+Symbols&rft.atitle=Orbits+%28including+the+first+manned+orbit%29&rft.aulast=Merrifield&rft.aufirst=Michael&rft_id=http%3A%2F%2Fwww.sixtysymbols.com%2Fvideos%2Forbit.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrbit" class="Z3988"></span></li></ul> <div class="navbox-styles"><style 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.navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Gravitational_orbits" style="padding:3px"><table class="nowraplinks hlist mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Orbits" title="Template:Orbits"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Orbits" title="Template talk:Orbits"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Orbits" title="Special:EditPage/Template:Orbits"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Gravitational_orbits" style="font-size:114%;margin:0 4em">Gravitational <a class="mw-selflink selflink">orbits</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/List_of_orbits" title="List of orbits">Types</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:6em">General</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Box_orbit" title="Box orbit">Box</a></li> <li><a href="/wiki/Parabolic_trajectory" title="Parabolic trajectory">Capture</a></li> <li><a href="/wiki/Circular_orbit" title="Circular orbit">Circular</a></li> <li><a href="/wiki/Elliptic_orbit" title="Elliptic orbit">Elliptical</a> / <a href="/wiki/Highly_elliptical_orbit" title="Highly elliptical orbit">Highly elliptical</a></li> <li><a href="/wiki/Parabolic_trajectory" title="Parabolic trajectory">Escape</a></li> <li><a href="/wiki/Horseshoe_orbit" title="Horseshoe orbit">Horseshoe</a></li> <li><a href="/wiki/Hyperbolic_trajectory" title="Hyperbolic trajectory">Hyperbolic trajectory</a></li> <li><a href="/wiki/Inclined_orbit" title="Inclined orbit">Inclined</a> / <a href="/wiki/Non-inclined_orbit" class="mw-redirect" title="Non-inclined orbit">Non-inclined</a></li> <li><a href="/wiki/Kepler_orbit" title="Kepler orbit">Kepler</a></li> <li><a href="/wiki/Lagrange_point" title="Lagrange point">Lagrange point</a></li> <li><a href="/wiki/Osculating_orbit" title="Osculating orbit">Osculating</a></li> <li><a href="/wiki/Parabolic_trajectory" title="Parabolic trajectory">Parabolic trajectory</a></li> <li><a href="/wiki/Parking_orbit" title="Parking orbit">Parking</a></li> <li><a href="/wiki/Retrograde_and_prograde_motion" title="Retrograde and prograde motion">Prograde / Retrograde</a></li> <li><a href="/wiki/Synchronous_orbit" title="Synchronous orbit">Synchronous</a> <ul><li><a href="/wiki/Semi-synchronous_orbit" title="Semi-synchronous orbit">semi</a></li> <li><a href="/wiki/Subsynchronous_orbit" title="Subsynchronous orbit">sub</a></li></ul></li> <li><a href="/wiki/Hohmann_transfer_orbit" title="Hohmann transfer orbit">Transfer orbit</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em"><a href="/wiki/Geocentric_orbit" title="Geocentric orbit">Geocentric</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Geosynchronous_orbit" title="Geosynchronous orbit">Geosynchronous</a> <ul><li><a href="/wiki/Geostationary_orbit" title="Geostationary orbit">Geostationary</a></li> <li><a href="/wiki/Geostationary_transfer_orbit" title="Geostationary transfer orbit">Geostationary transfer</a></li></ul></li> <li><a href="/wiki/Graveyard_orbit" title="Graveyard orbit">Graveyard</a></li> <li><a href="/wiki/High_Earth_orbit" title="High Earth orbit">High Earth</a></li> <li><a href="/wiki/Low_Earth_orbit" title="Low Earth orbit">Low Earth</a></li> <li><a href="/wiki/Medium_Earth_orbit" title="Medium Earth orbit">Medium Earth</a></li> <li><a href="/wiki/Molniya_orbit" title="Molniya orbit">Molniya</a></li> <li><a href="/wiki/Near-equatorial_orbit" title="Near-equatorial orbit">Near-equatorial</a></li> <li><a href="/wiki/Orbit_of_the_Moon" title="Orbit of the Moon">Orbit of the Moon</a></li> <li><a href="/wiki/Polar_orbit" title="Polar orbit">Polar</a></li> <li><a href="/wiki/Sun-synchronous_orbit" title="Sun-synchronous orbit">Sun-synchronous</a></li> <li><a href="/wiki/Transatmospheric_orbit" title="Transatmospheric orbit">Transatmospheric</a></li> <li><a href="/wiki/Tundra_orbit" title="Tundra orbit">Tundra</a></li> <li><a href="/wiki/Very_low_Earth_orbit" title="Very low Earth orbit">Very low Earth</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em">About<br />other points</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li>Mars <ul><li><a href="/wiki/Areocentric_orbit" title="Areocentric orbit">Areocentric</a></li> <li><a href="/wiki/Areosynchronous_orbit" title="Areosynchronous orbit">Areosynchronous</a></li> <li><a href="/wiki/Areostationary_orbit" title="Areostationary orbit">Areostationary</a></li></ul></li> <li>Lagrange points <ul><li><a href="/wiki/Distant_retrograde_orbit" title="Distant retrograde orbit">Distant retrograde</a></li> <li><a href="/wiki/Halo_orbit" title="Halo orbit">Halo</a></li> <li><a href="/wiki/Lissajous_orbit" title="Lissajous orbit">Lissajous</a></li> <li><a href="/wiki/Libration_point_orbit" title="Libration point orbit">Libration</a></li></ul></li> <li><a href="/wiki/Lunar_orbit" title="Lunar orbit">Lunar</a></li> <li>Sun <ul><li><a href="/wiki/Heliocentric_orbit" title="Heliocentric orbit">Heliocentric</a> <ul><li><a href="/wiki/Earth%27s_orbit" title="Earth's orbit">Earth's orbit</a></li></ul></li> <li><a href="/wiki/Mars_cycler" title="Mars cycler">Mars cycler</a></li> <li><a href="/wiki/Sun-synchronous_orbit" title="Sun-synchronous orbit">Heliosynchronous</a></li></ul></li> <li>Other <ul><li><a href="/wiki/Lunar_cycler" title="Lunar cycler">Lunar cycler</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Orbital_elements" title="Orbital elements">Parameters</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:6em"><div class="hlist"><ul><li>Shape</li><li>Size</li></ul></div></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><span class="texhtml mvar" style="font-style:italic;">e</span>  <a href="/wiki/Orbital_eccentricity" title="Orbital eccentricity">Eccentricity</a></li> <li><span class="texhtml mvar" style="font-style:italic;">a</span>  <a href="/wiki/Semi-major_and_semi-minor_axes" title="Semi-major and semi-minor axes">Semi-major axis</a></li> <li><span class="texhtml mvar" style="font-style:italic;">b</span>  <a href="/wiki/Semi-major_and_semi-minor_axes" title="Semi-major and semi-minor axes">Semi-minor axis</a></li> <li><span class="texhtml mvar" style="font-style:italic;">Q</span>, <span class="texhtml mvar" style="font-style:italic;">q</span>  <a href="/wiki/Apsis" title="Apsis">Apsides</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em">Orientation</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><span class="texhtml mvar" style="font-style:italic;">i</span>  <a href="/wiki/Orbital_inclination" title="Orbital inclination">Inclination</a></li> <li><span class="texhtml mvar" style="font-style:italic;">Ω</span>  <a href="/wiki/Longitude_of_the_ascending_node" title="Longitude of the ascending node">Longitude of the ascending node</a></li> <li><span class="texhtml mvar" style="font-style:italic;">ω</span>  <a href="/wiki/Argument_of_periapsis" title="Argument of periapsis">Argument of periapsis</a></li> <li><span class="texhtml mvar" style="font-style:italic;">ϖ</span>  <a href="/wiki/Longitude_of_the_periapsis" class="mw-redirect" title="Longitude of the periapsis">Longitude of the periapsis</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em">Position</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><span class="texhtml mvar" style="font-style:italic;">M</span>  <a href="/wiki/Mean_anomaly" title="Mean anomaly">Mean anomaly</a></li> <li><span class="texhtml mvar" style="font-style:italic;">ν</span>, <span class="texhtml mvar" style="font-style:italic;">θ</span>, <span class="texhtml mvar" style="font-style:italic;">f</span>  <a href="/wiki/True_anomaly" title="True anomaly">True anomaly</a></li> <li><span class="texhtml mvar" style="font-style:italic;">E</span>  <a href="/wiki/Eccentric_anomaly" title="Eccentric anomaly">Eccentric anomaly</a></li> <li><span class="texhtml mvar" style="font-style:italic;">L</span>  <a href="/wiki/Mean_longitude" title="Mean longitude">Mean longitude</a></li> <li><span class="texhtml mvar" style="font-style:italic;">l</span>  <a href="/wiki/True_longitude" title="True longitude">True longitude</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em">Variation</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><span class="texhtml mvar" style="font-style:italic;">T</span>  <a href="/wiki/Orbital_period" title="Orbital period">Orbital period</a></li> <li><span class="texhtml mvar" style="font-style:italic;">n</span>  <a href="/wiki/Mean_motion" title="Mean motion">Mean motion</a></li> <li><span class="texhtml mvar" style="font-style:italic;">v</span>  <a href="/wiki/Orbital_speed" title="Orbital speed">Orbital speed</a></li> <li><span class="texhtml"><i>t</i><sub>0</sub></span>  <a href="/wiki/Epoch_(astronomy)" title="Epoch (astronomy)">Epoch</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Orbital_maneuver" title="Orbital maneuver">Maneuvers</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bi-elliptic_transfer" title="Bi-elliptic transfer">Bi-elliptic transfer</a></li> <li><a href="/wiki/Collision_avoidance_(spacecraft)" title="Collision avoidance (spacecraft)">Collision avoidance (spacecraft)</a></li> <li><a href="/wiki/Delta-v" title="Delta-v">Delta-v</a></li> <li><a href="/wiki/Delta-v_budget" title="Delta-v budget">Delta-v budget</a></li> <li><a href="/wiki/Gravity_assist" title="Gravity assist">Gravity assist</a></li> <li><a href="/wiki/Gravity_turn" title="Gravity turn">Gravity turn</a></li> <li><a href="/wiki/Hohmann_transfer_orbit" title="Hohmann transfer orbit">Hohmann transfer</a></li> <li><a href="/wiki/Orbital_inclination_change" title="Orbital inclination change">Inclination change</a></li> <li><a href="/wiki/Low-energy_transfer" title="Low-energy transfer">Low-energy transfer</a></li> <li><a href="/wiki/Oberth_effect" title="Oberth effect">Oberth effect</a></li> <li><a href="/wiki/Orbit_phasing" title="Orbit phasing">Phasing</a></li> <li><a href="/wiki/Tsiolkovsky_rocket_equation" title="Tsiolkovsky rocket equation">Rocket equation</a></li> <li><a href="/wiki/Space_rendezvous" title="Space rendezvous">Rendezvous</a></li> <li><a href="/wiki/Trans-lunar_injection" title="Trans-lunar injection">Trans-lunar injection</a></li> <li><a href="/wiki/Transposition,_docking,_and_extraction" title="Transposition, docking, and extraction">Transposition, docking, and extraction</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Orbital_mechanics" title="Orbital mechanics">Orbital<br />mechanics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Astronomical_coordinate_systems" title="Astronomical coordinate systems">Astronomical coordinate systems</a></li> <li><a href="/wiki/Characteristic_energy" title="Characteristic energy">Characteristic energy</a></li> <li><a href="/wiki/Escape_velocity" title="Escape velocity">Escape velocity</a></li> <li><a href="/wiki/Ephemeris" title="Ephemeris">Ephemeris</a></li> <li><a href="/wiki/Equatorial_coordinate_system" title="Equatorial coordinate system">Equatorial coordinate system</a></li> <li><a href="/wiki/Ground_track" class="mw-redirect" title="Ground track">Ground track</a></li> <li><a href="/wiki/Hill_sphere" title="Hill sphere">Hill sphere</a></li> <li><a href="/wiki/Interplanetary_Transport_Network" title="Interplanetary Transport Network">Interplanetary Transport Network</a></li> <li><a href="/wiki/Kepler%27s_laws_of_planetary_motion" title="Kepler's laws of planetary motion">Kepler's laws of planetary motion</a></li> <li><a href="/wiki/Lagrange_point" title="Lagrange point">Lagrangian point</a></li> <li><a href="/wiki/N-body_problem" title="N-body problem"><i>n</i>-body problem</a></li> <li><a href="/wiki/Orbit_equation" title="Orbit equation">Orbit equation</a></li> <li><a href="/wiki/Orbital_state_vectors" title="Orbital state vectors">Orbital state vectors</a></li> <li><a href="/wiki/Perturbation_(astronomy)" title="Perturbation (astronomy)">Perturbation</a></li> <li><a href="/wiki/Retrograde_and_prograde_motion" title="Retrograde and prograde motion">Retrograde and prograde motion</a></li> <li><a href="/wiki/Specific_orbital_energy" title="Specific orbital energy">Specific orbital energy</a></li> <li><a href="/wiki/Specific_angular_momentum" title="Specific angular momentum">Specific angular momentum</a></li> <li><a href="/wiki/Two-line_element_set" title="Two-line element set">Two-line elements</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="List-Class article"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/16px-Symbol_list_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/23px-Symbol_list_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/31px-Symbol_list_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/List_of_orbits" title="List of orbits">List of orbits</a></li></ul> </div></td></tr></tbody></table></div> <style data-mw-deduplicate="TemplateStyles:r1130092004">.mw-parser-output .portal-bar{font-size:88%;font-weight:bold;display:flex;justify-content:center;align-items:baseline}.mw-parser-output .portal-bar-bordered{padding:0 2em;background-color:#fdfdfd;border:1px solid #a2a9b1;clear:both;margin:1em auto 0}.mw-parser-output .portal-bar-related{font-size:100%;justify-content:flex-start}.mw-parser-output .portal-bar-unbordered{padding:0 1.7em;margin-left:0}.mw-parser-output .portal-bar-header{margin:0 1em 0 0.5em;flex:0 0 auto;min-height:24px}.mw-parser-output .portal-bar-content{display:flex;flex-flow:row wrap;flex:0 1 auto;padding:0.15em 0;column-gap:1em;align-items:baseline;margin:0;list-style:none}.mw-parser-output .portal-bar-content-related{margin:0;list-style:none}.mw-parser-output .portal-bar-item{display:inline-block;margin:0.15em 0.2em;min-height:24px;line-height:24px}@media screen and (max-width:768px){.mw-parser-output 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