CINXE.COM

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available" lang="en" dir="ltr"> <head> <meta charset="UTF-8"> <title>Hyperbolic trajectory - Wikipedia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available";var cookie=document.cookie.match(/(?:^|; )enwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":["",""],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"dmy", "wgMonthNames":["","January","February","March","April","May","June","July","August","September","October","November","December"],"wgRequestId":"0e79261f-27ee-4765-a857-3780113fbefc","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Hyperbolic_trajectory","wgTitle":"Hyperbolic trajectory","wgCurRevisionId":1254651146,"wgRevisionId":1254651146,"wgArticleId":996910,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Articles with short description","Short description is different from Wikidata","Orbits"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"Hyperbolic_trajectory","wgRelevantArticleId":996910,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":false,"wgFlaggedRevsParams":{"tags":{"status": {"levels":1}}},"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"en","pageLanguageDir":"ltr","pageVariantFallbacks":"en"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":false,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":10000,"wgRelatedArticlesCompat":[],"wgCentralAuthMobileDomain":false,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q2755058","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={"ext.globalCssJs.user.styles":"ready", "site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.cite.styles":"ready","ext.math.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","jquery.makeCollapsible.styles":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["ext.cite.ux-enhancements","mediawiki.page.media","site","mediawiki.page.ready","jquery.makeCollapsible","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.ReferenceTooltips","ext.gadget.switcher","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents", "ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.growthExperiments.SuggestedEditSession","wikibase.sidebar.tracking"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=en&modules=ext.cite.styles%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cjquery.makeCollapsible.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=en&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=en&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.4"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/8/89/OrbitalEccentricityDemo.svg/1200px-OrbitalEccentricityDemo.svg.png"> <meta property="og:image:width" content="1200"> <meta property="og:image:height" content="1200"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/8/89/OrbitalEccentricityDemo.svg/800px-OrbitalEccentricityDemo.svg.png"> <meta property="og:image:width" content="800"> <meta property="og:image:height" content="800"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/8/89/OrbitalEccentricityDemo.svg/640px-OrbitalEccentricityDemo.svg.png"> <meta property="og:image:width" content="640"> <meta property="og:image:height" content="640"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Hyperbolic trajectory - Wikipedia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//en.m.wikipedia.org/wiki/Hyperbolic_trajectory"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=Hyperbolic_trajectory&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (en)"> <link rel="EditURI" type="application/rsd+xml" href="//en.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://en.wikipedia.org/wiki/Hyperbolic_trajectory"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en"> <link rel="alternate" type="application/atom+xml" title="Wikipedia Atom feed" href="/w/index.php?title=Special:RecentChanges&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="//login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Hyperbolic_trajectory rootpage-Hyperbolic_trajectory skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Jump to content</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Main menu" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-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-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">Main menu</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Main menu</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">hide</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navigation </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Main_Page" title="Visit the main page [z]" accesskey="z"><span>Main page</span></a></li><li id="n-contents" class="mw-list-item"><a href="/wiki/Wikipedia:Contents" title="Guides to browsing Wikipedia"><span>Contents</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Current_events" title="Articles related to current events"><span>Current events</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:Random" title="Visit a randomly selected article [x]" accesskey="x"><span>Random article</span></a></li><li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:About" title="Learn about Wikipedia and how it works"><span>About Wikipedia</span></a></li><li id="n-contactpage" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us" title="How to contact Wikipedia"><span>Contact us</span></a></li> </ul> </div> </div> <div id="p-interaction" class="vector-menu mw-portlet mw-portlet-interaction" > <div class="vector-menu-heading"> Contribute </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/Help:Contents" title="Guidance on how to use and edit Wikipedia"><span>Help</span></a></li><li id="n-introduction" class="mw-list-item"><a href="/wiki/Help:Introduction" title="Learn how to edit Wikipedia"><span>Learn to edit</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Community_portal" title="The hub for editors"><span>Community portal</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Special:RecentChanges" title="A list of recent changes to Wikipedia [r]" accesskey="r"><span>Recent changes</span></a></li><li id="n-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_upload_wizard" title="Add images or other media for use on Wikipedia"><span>Upload file</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Main_Page" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="The Free Encyclopedia" src="/static/images/mobile/copyright/wikipedia-tagline-en.svg" width="117" height="13" style="width: 7.3125em; height: 0.8125em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Special:Search" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Search Wikipedia [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Search</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Search Wikipedia" aria-label="Search Wikipedia" autocapitalize="sentences" title="Search Wikipedia [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Personal tools"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Appearance" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-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-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Appearance</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en" class=""><span>Donate</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:CreateAccount&returnto=Hyperbolic+trajectory" title="You are encouraged to create an account and log in; however, it is not mandatory" class=""><span>Create account</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:UserLogin&returnto=Hyperbolic+trajectory" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o" class=""><span>Log in</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Log in and more options" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Personal tools" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-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-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Personal tools</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en"><span>Donate</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:CreateAccount&returnto=Hyperbolic+trajectory" title="You are encouraged to create an account and log in; however, it is not mandatory"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Create account</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:UserLogin&returnto=Hyperbolic+trajectory" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Log in</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Pages for logged out editors <a href="/wiki/Help:Introduction" aria-label="Learn more about editing"><span>learn more</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:MyContributions" title="A list of edits made from this IP address [y]" accesskey="y"><span>Contributions</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:MyTalk" title="Discussion about edits from this IP address [n]" accesskey="n"><span>Talk</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Contents" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Contents</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Parameters_describing_a_hyperbolic_trajectory" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Parameters_describing_a_hyperbolic_trajectory"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Parameters describing a hyperbolic trajectory</span> </div> </a> <button aria-controls="toc-Parameters_describing_a_hyperbolic_trajectory-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 Parameters describing a hyperbolic trajectory subsection</span> </button> <ul id="toc-Parameters_describing_a_hyperbolic_trajectory-sublist" class="vector-toc-list"> <li id="toc-Semi-major_axis,_energy_and_hyperbolic_excess_velocity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Semi-major_axis,_energy_and_hyperbolic_excess_velocity"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Semi-major axis, energy and hyperbolic excess velocity</span> </div> </a> <ul id="toc-Semi-major_axis,_energy_and_hyperbolic_excess_velocity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Eccentricity_and_angle_between_approach_and_departure" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Eccentricity_and_angle_between_approach_and_departure"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Eccentricity and angle between approach and departure</span> </div> </a> <ul id="toc-Eccentricity_and_angle_between_approach_and_departure-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Impact_parameter_and_the_distance_of_closest_approach" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Impact_parameter_and_the_distance_of_closest_approach"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Impact parameter and the distance of closest approach</span> </div> </a> <ul id="toc-Impact_parameter_and_the_distance_of_closest_approach-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Equations_of_motion" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Equations_of_motion"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Equations of motion</span> </div> </a> <button aria-controls="toc-Equations_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 Equations of motion subsection</span> </button> <ul id="toc-Equations_of_motion-sublist" class="vector-toc-list"> <li id="toc-Position" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Position"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Position</span> </div> </a> <ul id="toc-Position-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Flight_path_angle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Flight_path_angle"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Flight path angle</span> </div> </a> <ul id="toc-Flight_path_angle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Velocity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Velocity"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Velocity</span> </div> </a> <ul id="toc-Velocity-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Radial_hyperbolic_trajectory" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Radial_hyperbolic_trajectory"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Radial hyperbolic trajectory</span> </div> </a> <ul id="toc-Radial_hyperbolic_trajectory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Deflection_with_finite_sphere_of_influence" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Deflection_with_finite_sphere_of_influence"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Deflection with finite sphere of influence</span> </div> </a> <ul id="toc-Deflection_with_finite_sphere_of_influence-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relativistic_two-body_problem" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Relativistic_two-body_problem"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Relativistic two-body problem</span> </div> </a> <ul id="toc-Relativistic_two-body_problem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</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">Hyperbolic trajectory</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 18 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-18" 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">18 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%A7%D9%84%D9%85%D8%B3%D8%A7%D8%B1_%D8%A7%D9%84%D8%B2%D8%A7%D8%A6%D8%AF%D9%8A" title="المسار الزائدي – Arabic" lang="ar" hreflang="ar" data-title="المسار الزائدي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Traject%C3%B2ria_hiperb%C3%B2lica" title="Trajectòria hiperbòlica – Catalan" lang="ca" hreflang="ca" data-title="Trajectòria hiperbòlica" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Trayectoria_hiperb%C3%B3lica" title="Trayectoria hiperbólica – Spanish" lang="es" hreflang="es" data-title="Trayectoria hiperbólica" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%B3%DB%8C%D8%B1_%D9%87%D8%B0%D9%84%D9%88%D9%84%DB%8C" title="مسیر هذلولی – Persian" lang="fa" hreflang="fa" data-title="مسیر هذلولی" 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/Trajectoire_hyperbolique" title="Trajectoire hyperbolique – French" lang="fr" hreflang="fr" data-title="Trajectoire hyperbolique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%8C%8D%EA%B3%A1%EC%84%A0_%EA%B6%A4%EB%8F%84" title="쌍곡선 궤도 – Korean" lang="ko" hreflang="ko" data-title="쌍곡선 궤도" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Traiettoria_iperbolica" title="Traiettoria iperbolica – Italian" lang="it" hreflang="it" data-title="Traiettoria iperbolica" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%8F%8C%E6%9B%B2%E7%B7%9A%E8%BB%8C%E9%81%93" title="双曲線軌道 – Japanese" lang="ja" hreflang="ja" data-title="双曲線軌道" 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-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Orbita_hiperboliczna" title="Orbita hiperboliczna – Polish" lang="pl" hreflang="pl" data-title="Orbita hiperboliczna" 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/Trajet%C3%B3ria_hiperb%C3%B3lica" title="Trajetória hiperbólica – Portuguese" lang="pt" hreflang="pt" data-title="Trajetória hiperbólica" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%93%D0%B8%D0%BF%D0%B5%D1%80%D0%B1%D0%BE%D0%BB%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D1%82%D1%80%D0%B0%D0%B5%D0%BA%D1%82%D0%BE%D1%80%D0%B8%D1%8F" title="Гиперболическая траектория – Russian" lang="ru" hreflang="ru" data-title="Гиперболическая траектория" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Hyperbolic_trajectory" title="Hyperbolic trajectory – Simple English" lang="en-simple" hreflang="en-simple" data-title="Hyperbolic trajectory" 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/Hyperbolick%C3%A1_dr%C3%A1ha" title="Hyperbolická dráha – Slovak" lang="sk" hreflang="sk" data-title="Hyperbolická dráha" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Hyperbelirata" title="Hyperbelirata – Finnish" lang="fi" hreflang="fi" data-title="Hyperbelirata" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%A7%E0%B8%B4%E0%B8%96%E0%B8%B5%E0%B9%82%E0%B8%84%E0%B8%88%E0%B8%A3%E0%B9%84%E0%B8%AE%E0%B9%80%E0%B8%9E%E0%B8%AD%E0%B8%A3%E0%B9%8C%E0%B9%82%E0%B8%9A%E0%B8%A5%E0%B8%B2" title="วิถีโคจรไฮเพอร์โบลา – Thai" lang="th" hreflang="th" data-title="วิถีโคจรไฮเพอร์โบลา" 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/Hiperbolik_y%C3%B6r%C3%BCnge" title="Hiperbolik yörünge – Turkish" lang="tr" hreflang="tr" data-title="Hiperbolik 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%93%D1%96%D0%BF%D0%B5%D1%80%D0%B1%D0%BE%D0%BB%D1%96%D1%87%D0%BD%D0%B0_%D1%82%D1%80%D0%B0%D1%94%D0%BA%D1%82%D0%BE%D1%80%D1%96%D1%8F" title="Гіперболічна траєкторія – Ukrainian" lang="uk" hreflang="uk" data-title="Гіперболічна траєкторія" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Qu%E1%BB%B9_%C4%91%E1%BA%A1o_hyperbol" title="Quỹ đạo hyperbol – Vietnamese" lang="vi" hreflang="vi" data-title="Quỹ đạo hyperbol" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q2755058#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Hyperbolic_trajectory" title="View the content page [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:Hyperbolic_trajectory" rel="discussion" title="Discuss improvements to the content page [t]" accesskey="t"><span>Talk</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Change language variant" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">English</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Views"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Hyperbolic_trajectory"><span>Read</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Hyperbolic_trajectory&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Hyperbolic_trajectory&action=history" title="Past revisions of this page [h]" accesskey="h"><span>View history</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Tools" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Tools</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Tools</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">hide</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="More options" > <div class="vector-menu-heading"> Actions </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Hyperbolic_trajectory"><span>Read</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Hyperbolic_trajectory&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Hyperbolic_trajectory&action=history"><span>View history</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:WhatLinksHere/Hyperbolic_trajectory" title="List of all English Wikipedia pages containing links to this page [j]" accesskey="j"><span>What links here</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:RecentChangesLinked/Hyperbolic_trajectory" rel="nofollow" title="Recent changes in pages linked from this page [k]" accesskey="k"><span>Related changes</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u"><span>Upload file</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Special:SpecialPages" title="A list of all special pages [q]" accesskey="q"><span>Special pages</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Hyperbolic_trajectory&oldid=1254651146" title="Permanent link to this revision of this page"><span>Permanent link</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Hyperbolic_trajectory&action=info" title="More information about this page"><span>Page information</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:CiteThisPage&page=Hyperbolic_trajectory&id=1254651146&wpFormIdentifier=titleform" title="Information on how to cite this page"><span>Cite this page</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FHyperbolic_trajectory"><span>Get shortened URL</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FHyperbolic_trajectory"><span>Download QR code</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Print/export </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=Hyperbolic_trajectory&action=show-download-screen" title="Download this page as a PDF file"><span>Download as PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Hyperbolic_trajectory&printable=yes" title="Printable version of this page [p]" accesskey="p"><span>Printable version</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> In other projects </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q2755058" title="Structured data on this page hosted by Wikidata [g]" accesskey="g"><span>Wikidata item</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Concept in astrodynamics</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:OrbitalEccentricityDemo.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/89/OrbitalEccentricityDemo.svg/220px-OrbitalEccentricityDemo.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/89/OrbitalEccentricityDemo.svg/330px-OrbitalEccentricityDemo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/89/OrbitalEccentricityDemo.svg/440px-OrbitalEccentricityDemo.svg.png 2x" data-file-width="350" data-file-height="350" /></a><figcaption>The blue path in this image is an example of a hyperbolic trajectory.</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Gravity_Wells_Potential_Plus_Kinetic_Energy_-_Circle-Ellipse-Parabola-Hyperbola.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/94/Gravity_Wells_Potential_Plus_Kinetic_Energy_-_Circle-Ellipse-Parabola-Hyperbola.png/250px-Gravity_Wells_Potential_Plus_Kinetic_Energy_-_Circle-Ellipse-Parabola-Hyperbola.png" decoding="async" width="250" height="243" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/94/Gravity_Wells_Potential_Plus_Kinetic_Energy_-_Circle-Ellipse-Parabola-Hyperbola.png/375px-Gravity_Wells_Potential_Plus_Kinetic_Energy_-_Circle-Ellipse-Parabola-Hyperbola.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/94/Gravity_Wells_Potential_Plus_Kinetic_Energy_-_Circle-Ellipse-Parabola-Hyperbola.png/500px-Gravity_Wells_Potential_Plus_Kinetic_Energy_-_Circle-Ellipse-Parabola-Hyperbola.png 2x" data-file-width="1130" data-file-height="1100" /></a><figcaption>A hyperbolic trajectory is depicted in the bottom-right quadrant of this diagram, where the <a href="/wiki/Gravity_well" class="mw-redirect" title="Gravity well">gravitational potential well</a> of the central mass shows potential energy, and the kinetic energy of the hyperbolic trajectory is shown in red. The height of the kinetic energy decreases as the speed decreases and distance increases according to Kepler's laws. The part of the kinetic energy that remains above zero total energy is that associated with the hyperbolic excess velocity.</figcaption></figure> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><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><style data-mw-deduplicate="TemplateStyles:r1246091330">.mw-parser-output .sidebar{width:22em;float:right;clear:right;margin:0.5em 0 1em 1em;background:var(--background-color-neutral-subtle,#f8f9fa);border:1px solid var(--border-color-base,#a2a9b1);padding:0.2em;text-align:center;line-height:1.4em;font-size:88%;border-collapse:collapse;display:table}body.skin-minerva .mw-parser-output .sidebar{display:table!important;float:right!important;margin:0.5em 0 1em 1em!important}.mw-parser-output .sidebar-subgroup{width:100%;margin:0;border-spacing:0}.mw-parser-output .sidebar-left{float:left;clear:left;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-none{float:none;clear:both;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-outer-title{padding:0 0.4em 0.2em;font-size:125%;line-height:1.2em;font-weight:bold}.mw-parser-output .sidebar-top-image{padding:0.4em}.mw-parser-output .sidebar-top-caption,.mw-parser-output .sidebar-pretitle-with-top-image,.mw-parser-output .sidebar-caption{padding:0.2em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-pretitle{padding:0.4em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-title,.mw-parser-output .sidebar-title-with-pretitle{padding:0.2em 0.8em;font-size:145%;line-height:1.2em}.mw-parser-output .sidebar-title-with-pretitle{padding:0.1em 0.4em}.mw-parser-output .sidebar-image{padding:0.2em 0.4em 0.4em}.mw-parser-output .sidebar-heading{padding:0.1em 0.4em}.mw-parser-output .sidebar-content{padding:0 0.5em 0.4em}.mw-parser-output .sidebar-content-with-subgroup{padding:0.1em 0.4em 0.2em}.mw-parser-output .sidebar-above,.mw-parser-output .sidebar-below{padding:0.3em 0.8em;font-weight:bold}.mw-parser-output .sidebar-collapse .sidebar-above,.mw-parser-output .sidebar-collapse .sidebar-below{border-top:1px solid #aaa;border-bottom:1px solid #aaa}.mw-parser-output .sidebar-navbar{text-align:right;font-size:115%;padding:0 0.4em 0.4em}.mw-parser-output .sidebar-list-title{padding:0 0.4em;text-align:left;font-weight:bold;line-height:1.6em;font-size:105%}.mw-parser-output .sidebar-list-title-c{padding:0 0.4em;text-align:center;margin:0 3.3em}@media(max-width:640px){body.mediawiki .mw-parser-output .sidebar{width:100%!important;clear:both;float:none!important;margin-left:0!important;margin-right:0!important}}body.skin--responsive .mw-parser-output .sidebar a>img{max-width:none!important}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><table class="sidebar sidebar-collapse nomobile nowraplinks"><tbody><tr><td class="sidebar-pretitle">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle">Astrodynamics</th></tr><tr><td class="sidebar-image" style="padding-bottom:0.85em;"><span typeof="mw:File"><a href="/wiki/File:Orbit_mechanics_icon.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Orbit_mechanics_icon.svg/60px-Orbit_mechanics_icon.svg.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Orbit_mechanics_icon.svg/90px-Orbit_mechanics_icon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Orbit_mechanics_icon.svg/120px-Orbit_mechanics_icon.svg.png 2x" data-file-width="48" data-file-height="48" /></a></span></td></tr><tr><th class="sidebar-heading" style="padding-bottom:0.55em;"> <div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="/wiki/Orbital_mechanics" title="Orbital mechanics"><span style="font-size:110%;">Orbital mechanics</span></a></div></th></tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="/wiki/Orbital_elements" title="Orbital elements">Orbital elements</a></div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0;"> <ul><li><a href="/wiki/Apsis" title="Apsis">Apsis</a></li> <li><a href="/wiki/Argument_of_periapsis" title="Argument of periapsis">Argument of periapsis</a></li> <li><a href="/wiki/Orbital_eccentricity" title="Orbital eccentricity">Eccentricity</a></li> <li><a href="/wiki/Orbital_inclination" title="Orbital inclination">Inclination</a></li> <li><a href="/wiki/Mean_anomaly" title="Mean anomaly">Mean anomaly</a></li> <li><a href="/wiki/Orbital_node" title="Orbital node">Orbital nodes</a></li> <li><a href="/wiki/Semi-major_and_semi-minor_axes" title="Semi-major and semi-minor axes">Semi-major axis</a></li> <li><a href="/wiki/True_anomaly" title="True anomaly">True anomaly</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Types of <a href="/wiki/Two-body_problem" title="Two-body problem">two-body orbits</a> by <br />eccentricity</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0;"> <ul><li><a href="/wiki/Circular_orbit" title="Circular orbit">Circular orbit</a></li> <li><a href="/wiki/Elliptic_orbit" title="Elliptic orbit">Elliptic orbit</a></li></ul> <div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="/wiki/Transfer_orbit" title="Transfer orbit">Transfer orbit</a> <div class="hlist" style="font-size:90%"><ul><li>(<a href="/wiki/Hohmann_transfer_orbit" title="Hohmann transfer orbit">Hohmann transfer orbit</a></li><li><a href="/wiki/Bi-elliptic_transfer" title="Bi-elliptic transfer">Bi-elliptic transfer orbit</a>)</li></ul></div></div> <ul><li><a href="/wiki/Parabolic_trajectory" title="Parabolic trajectory">Parabolic orbit</a></li> <li><a class="mw-selflink selflink">Hyperbolic orbit</a></li> <li><a href="/wiki/Radial_trajectory" title="Radial trajectory">Radial orbit</a></li> <li><a href="/wiki/Orbital_decay" title="Orbital decay">Decaying orbit</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Equations</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0;"> <ul><li><a href="/wiki/Dynamical_friction" title="Dynamical friction">Dynamical friction</a></li> <li><a href="/wiki/Escape_velocity" title="Escape velocity">Escape velocity</a></li> <li><a href="/wiki/Kepler%27s_equation" title="Kepler's equation">Kepler's equation</a></li> <li><a href="/wiki/Kepler%27s_laws_of_planetary_motion" title="Kepler's laws of planetary motion">Kepler's laws of planetary motion</a></li> <li><a href="/wiki/Orbital_period" title="Orbital period">Orbital period</a></li> <li><a href="/wiki/Orbital_speed" title="Orbital speed">Orbital velocity</a></li> <li><a href="/wiki/Surface_gravity" title="Surface gravity">Surface gravity</a></li> <li><a href="/wiki/Specific_orbital_energy" title="Specific orbital energy">Specific orbital energy</a></li> <li><a href="/wiki/Vis-viva_equation" title="Vis-viva equation">Vis-viva equation</a></li></ul></div></div></td> </tr><tr><th class="sidebar-heading" style="padding-bottom:0.55em;"> <div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="/wiki/Celestial_mechanics" title="Celestial mechanics"><span style="font-size:110%;">Celestial mechanics</span></a></div></th></tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Gravitational influences</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0;"> <ul><li><a href="/wiki/Barycenter" class="mw-redirect" title="Barycenter">Barycenter</a></li> <li><a href="/wiki/Hill_sphere" title="Hill sphere">Hill sphere</a></li> <li><a href="/wiki/Perturbation_(astronomy)" title="Perturbation (astronomy)">Perturbations</a></li> <li><a href="/wiki/Sphere_of_influence_(astrodynamics)" title="Sphere of influence (astrodynamics)">Sphere of influence</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="/wiki/N-body_problem" title="N-body problem">N-body orbits</a></div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0;"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="/wiki/Lagrange_point" title="Lagrange point">Lagrangian points</a> <div class="hlist" style="font-size:90%"><ul><li>(<a href="/wiki/Halo_orbit" title="Halo orbit">Halo orbits</a>)</li></ul></div></div> <ul><li><a href="/wiki/Lissajous_orbit" title="Lissajous orbit">Lissajous orbits</a></li> <li><a href="/wiki/Lyapunov_stability" title="Lyapunov stability">Lyapunov orbits</a></li></ul></div></div></td> </tr><tr><th class="sidebar-heading" style="padding-bottom:0.55em;"> <div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="/wiki/Aerospace_engineering" title="Aerospace engineering"><span style="font-size:110%;">Engineering and efficiency</span></a></div></th></tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Preflight engineering</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0;"> <ul><li><a href="/wiki/Mass_ratio" title="Mass ratio">Mass ratio</a></li> <li><a href="/wiki/Payload_fraction" title="Payload fraction">Payload fraction</a></li> <li><a href="/wiki/Propellant_mass_fraction" title="Propellant mass fraction">Propellant mass fraction</a></li> <li><a href="/wiki/Tsiolkovsky_rocket_equation" title="Tsiolkovsky rocket equation">Tsiolkovsky rocket equation</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Efficiency measures</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0;"> <ul><li><a href="/wiki/Gravity_assist" title="Gravity assist">Gravity assist</a></li> <li><a href="/wiki/Oberth_effect" title="Oberth effect">Oberth effect</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Propulsive maneuvers</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0;"> <ul><li><a href="/wiki/Orbital_maneuver" title="Orbital maneuver">Orbital maneuver</a></li> <li><a href="/wiki/Orbit_insertion" title="Orbit insertion">Orbit insertion</a></li></ul></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Astrodynamics" title="Template:Astrodynamics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Astrodynamics" title="Template talk:Astrodynamics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Astrodynamics" title="Special:EditPage/Template:Astrodynamics"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Astrodynamics" class="mw-redirect" title="Astrodynamics">astrodynamics</a> or <a href="/wiki/Celestial_mechanics" title="Celestial mechanics">celestial mechanics</a>, a <b>hyperbolic trajectory</b> or <b>hyperbolic orbit</b> is the trajectory of any object around a <a href="/wiki/Central_body" class="mw-redirect" title="Central body">central body</a> with more than enough speed to escape the central object's gravitational pull. The name derives from the fact that according to <a href="/wiki/Newton%27s_law_of_universal_gravitation" title="Newton's law of universal gravitation">Newtonian theory</a> such an orbit has the shape of a <a href="/wiki/Hyperbola" title="Hyperbola">hyperbola</a>. In more technical terms this can be expressed by the condition that the <a href="/wiki/Orbital_eccentricity" title="Orbital eccentricity">orbital eccentricity</a> is greater than one. </p><p>Under simplistic assumptions a body traveling along this trajectory will coast towards infinity, settling to a final excess velocity relative to the central body. Similarly to <a href="/wiki/Parabolic_trajectory" title="Parabolic trajectory">parabolic trajectories</a>, all hyperbolic trajectories are also <a href="/wiki/Escape_trajectory" class="mw-redirect" title="Escape trajectory">escape trajectories</a>. The <a href="/wiki/Specific_orbital_energy" title="Specific orbital energy">specific energy</a> of a hyperbolic trajectory orbit is positive. </p><p>Planetary flybys, used for <a href="/wiki/Gravity_assist" title="Gravity assist">gravitational slingshots</a>, can be described within the planet's <a href="/wiki/Sphere_of_influence_(astrodynamics)" title="Sphere of influence (astrodynamics)">sphere of influence</a> using hyperbolic trajectories. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Parameters_describing_a_hyperbolic_trajectory">Parameters describing a hyperbolic trajectory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_trajectory&action=edit&section=1" title="Edit section: Parameters describing a hyperbolic trajectory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Like an elliptical orbit, a hyperbolic trajectory for a given system can be defined (ignoring orientation) by its semi major axis and the eccentricity. However, with a hyperbolic orbit other parameters may be more useful in understanding a body's motion. The following table lists the main parameters describing the path of body following a hyperbolic trajectory around another under standard assumptions and the formula connecting them. </p> <table class="wikitable"> <caption>Hyperbolic trajectory equations <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> </caption> <tbody><tr> <th>Element</th> <th>Symbol</th> <th>Formula</th> <th>using <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26145027656115364f96e031bca0c0c11022986e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.003ex; height:2.009ex;" alt="{\displaystyle v_{\infty }}"></span> (or <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></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 b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> </th></tr> <tr> <td><a href="/wiki/Standard_gravitational_parameter" title="Standard gravitational parameter">Standard gravitational parameter</a></td> <td><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></td> <td><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 {v^{2}}{(2/r-1/a)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>r</mi> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {v^{2}}{(2/r-1/a)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41d3fac216552137c18a8eeb0e27c1b015eec7b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:12.414ex; height:6.509ex;" alt="{\displaystyle {\frac {v^{2}}{(2/r-1/a)}}}"></span></td> <td><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 bv_{\infty }^{2}\cot \theta _{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mi>cot</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle bv_{\infty }^{2}\cot \theta _{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adc483d6ba59cddd0f41b88a5fc2803e3416998f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.84ex; height:2.843ex;" alt="{\displaystyle bv_{\infty }^{2}\cot \theta _{\infty }}"></span> </td></tr> <tr> <td><a href="/wiki/Orbital_eccentricity" title="Orbital eccentricity">Eccentricity</a> (>1)</td> <td><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}"></span></td> <td><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 {\ell }{r_{p}}}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ℓ</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\ell }{r_{p}}}-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4244ee477bdfb5ee7dce312c049c9bb2304d059d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:6.947ex; height:6.009ex;" alt="{\displaystyle {\frac {\ell }{r_{p}}}-1}"></span></td> <td><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 {\sqrt {1+b^{2}/a^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {1+b^{2}/a^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d934d6be5fef8bd35d7261812198c38a25e3459f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:11.825ex; height:4.843ex;" alt="{\displaystyle {\sqrt {1+b^{2}/a^{2}}}}"></span> </td></tr> <tr> <td><a href="/wiki/Semi-major_axis" class="mw-redirect" title="Semi-major axis">Semi-major axis</a> (<0)</td> <td><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\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99a94f96d2455b9d7faf3cec3eb02ab3c455aec1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.617ex; height:1.676ex;" alt="{\displaystyle a\,\!}"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/(2/r-v^{2}/\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>r</mi> <mo>−</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/(2/r-v^{2}/\mu )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c18d09ca96f6e3f55aa85ed461ab525e513e85e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.094ex; height:3.176ex;" alt="{\displaystyle 1/(2/r-v^{2}/\mu )}"></span></td> <td><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 /v_{\infty }^{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> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\mu /v_{\infty }^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46d60d52c72605ce727607c1f4363b040333d3c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.375ex; height:3.009ex;" alt="{\displaystyle -\mu /v_{\infty }^{2}}"></span> </td></tr> <tr> <td>Hyperbolic excess velocity</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26145027656115364f96e031bca0c0c11022986e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.003ex; height:2.009ex;" alt="{\displaystyle v_{\infty }}"></span></td> <td><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 {\sqrt {-\mu /a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>a</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {-\mu /a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/075558f2fcbeae4d97c059fa05b6c290fbb88a81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.926ex; height:4.843ex;" alt="{\displaystyle {\sqrt {-\mu /a}}}"></span></td> <td> </td></tr> <tr> <td>(External) Angle between asymptotes</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\theta _{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\theta _{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc2c0382fddcdf052333ad3530d8c72189d27c1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.128ex; height:2.509ex;" alt="{\displaystyle 2\theta _{\infty }}"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\cos ^{-1}(-1/e)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>e</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\cos ^{-1}(-1/e)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6ea104681d16821e959fe745530ef2a5f6d2c72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.019ex; height:3.176ex;" alt="{\displaystyle 2\cos ^{-1}(-1/e)}"></span></td> <td><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 \pi +2\tan ^{-1}(b/a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>+</mo> <mn>2</mn> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi +2\tan ^{-1}(b/a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6505b727afe556e19c9016dbd2f9df1178739bc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.613ex; height:3.176ex;" alt="{\displaystyle \pi +2\tan ^{-1}(b/a)}"></span><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> </td></tr> <tr> <td>Angle between asymptotes and the conjugate axis <br /> of the hyperbolic path of approach</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db2654d0632cb4b4cd30cfddb80a75ba7d743216" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.395ex; height:2.176ex;" alt="{\displaystyle 2\nu }"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\theta _{\infty }-\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mo>−</mo> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\theta _{\infty }-\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26a7794287a4bbe63abf8b6cf4c3196a9618e74e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.301ex; height:2.509ex;" alt="{\displaystyle 2\theta _{\infty }-\pi }"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\sin ^{-1}{\bigg (}{\frac {1}{(1+r_{p}v_{\infty }^{2}/\mu )}}{\bigg )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>μ</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\sin ^{-1}{\bigg (}{\frac {1}{(1+r_{p}v_{\infty }^{2}/\mu )}}{\bigg )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00eab45e81d8d3a4c8f36805404ed51cca83e54c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:24.869ex; height:6.509ex;" alt="{\displaystyle 2\sin ^{-1}{\bigg (}{\frac {1}{(1+r_{p}v_{\infty }^{2}/\mu )}}{\bigg )}}"></span> </td></tr> <tr> <td><a href="/wiki/Impact_parameter" title="Impact parameter">Impact parameter</a> (<a href="/wiki/Semi-minor_axis" class="mw-redirect" title="Semi-minor axis">semi-minor axis</a>)</td> <td><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 b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span></td> <td><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{\sqrt {e^{2}-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -a{\sqrt {e^{2}-1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f13aadd635a4942a0d61bfcfdd4f276b815da9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.502ex; height:3.509ex;" alt="{\displaystyle -a{\sqrt {e^{2}-1}}}"></span></td> <td><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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0" /> </mrow> <annotation encoding="application/x-tex">{\displaystyle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df4dcd61276328f7c7ec5bdc399b6e11114a2c68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:0; height:0.343ex;" alt="{\displaystyle }"></span> </td></tr> <tr> <td><a href="/wiki/Semi-latus_rectum" class="mw-redirect" title="Semi-latus rectum">Semi-latus rectum</a></td> <td><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 \ell }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f066e981e530bacc07efc6a10fa82deee985929e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\displaystyle \ell }"></span></td> <td><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(1-e^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a(1-e^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbb340cf9657b20eff278ce5989aaae8669a2c07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.18ex; height:3.176ex;" alt="{\displaystyle a(1-e^{2})}"></span></td> <td><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 -b^{2}/a=h^{2}/\mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -b^{2}/a=h^{2}/\mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe2a7453afd57c6d8d72916eb764c328c496680a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.308ex; height:3.176ex;" alt="{\displaystyle -b^{2}/a=h^{2}/\mu }"></span> </td></tr> <tr> <td><a href="/wiki/Apsis" title="Apsis">Periapsis distance</a></td> <td><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_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d02ec163fac8837ac757005d783b375e52808b97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.108ex; height:2.343ex;" alt="{\displaystyle r_{p}}"></span></td> <td><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(e-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>e</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -a(e-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de5e1fba849db9eafd4dcb9442d55bbf1054d555" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.934ex; height:2.843ex;" alt="{\displaystyle -a(e-1)}"></span></td> <td><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 {\sqrt {a^{2}+b^{2}}}+a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>+</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {a^{2}+b^{2}}}+a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6eec7edcaea5112336495fc34b1bcc437b87a7ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.57ex; height:3.509ex;" alt="{\displaystyle {\sqrt {a^{2}+b^{2}}}+a}"></span> </td></tr> <tr> <td><a href="/wiki/Specific_orbital_energy" title="Specific orbital energy">Specific orbital energy</a></td> <td><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 \varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }"></span></td> <td><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 /2a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\mu /2a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2d5dab5f4e467882cc5cebacd0bc5d299ff4d2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.765ex; height:2.843ex;" alt="{\displaystyle -\mu /2a}"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{\infty }^{2}/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{\infty }^{2}/2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c313c46e9af96b3c2552e5d2eaf985957d3db59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:3.009ex;" alt="{\displaystyle v_{\infty }^{2}/2}"></span> </td></tr> <tr> <td><a href="/wiki/Specific_angular_momentum" title="Specific angular momentum">Specific angular momentum</a></td> <td><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 h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span></td> <td><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 {\sqrt {\mu \ell }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>μ</mi> <mi>ℓ</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\mu \ell }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b823f1b6b11cd8964fc1751df3ffd245f384285" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.695ex; height:3.509ex;" alt="{\displaystyle {\sqrt {\mu \ell }}}"></span></td> <td><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 bv_{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle bv_{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1faaf75a1ec3622253cbf73d5fb3212cf6b2846" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.001ex; height:2.509ex;" alt="{\displaystyle bv_{\infty }}"></span> </td></tr> <tr> <td><a href="/wiki/Kepler%27s_laws_of_planetary_motion" title="Kepler's laws of planetary motion">Area swept up per time</a> </td> <td><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 A}{\Delta t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>A</mi> </mrow> <mrow> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\Delta A}{\Delta t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab53d6973f4cf0f37207bda4d73b87e871f7d9d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:4.515ex; height:5.509ex;" alt="{\displaystyle {\frac {\Delta A}{\Delta t}}}"></span> </td> <td><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 {h}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>h</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {h}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75b287bab861f07eda0822c0f28254d92294d9e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.175ex; height:5.343ex;" alt="{\displaystyle {\frac {h}{2}}}"></span> </td> <td> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Semi-major_axis,_energy_and_hyperbolic_excess_velocity"><span id="Semi-major_axis.2C_energy_and_hyperbolic_excess_velocity"></span>Semi-major axis, energy and hyperbolic excess velocity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_trajectory&action=edit&section=2" title="Edit section: Semi-major axis, energy and hyperbolic excess velocity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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">See also: <a href="/wiki/Characteristic_energy" title="Characteristic energy">Characteristic energy</a></div> <p>The semi major axis (<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\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99a94f96d2455b9d7faf3cec3eb02ab3c455aec1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.617ex; height:1.676ex;" alt="{\displaystyle a\,\!}"></span>) is not immediately visible with an hyperbolic trajectory but can be constructed as it is the distance from periapsis to the point where the two asymptotes cross. Usually, by convention, it is negative, to keep various equations consistent with elliptical orbits. </p><p>The semi major axis is directly linked to the <a href="/wiki/Specific_orbital_energy" title="Specific orbital energy">specific orbital energy</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/587ec0778f84a86a9567b2e296aec908d895ff21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.331ex; height:1.676ex;" alt="{\displaystyle \epsilon \,}"></span>) or <a href="/wiki/Characteristic_energy" title="Characteristic energy">characteristic energy</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66e9abeb5057b7afbf88e3169101849354f13c65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.509ex;" alt="{\displaystyle C_{3}}"></span> of the orbit, and to the velocity the body attains at as the distance tends to infinity, the hyperbolic excess velocity (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{\infty }\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{\infty }\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d264d5bc6d12e2dab4f4a622900bcff2e83e3f8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:3.39ex; height:2.009ex;" alt="{\displaystyle v_{\infty }\,\!}"></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 v_{\infty }^{2}=2\epsilon =C_{3}=-\mu /a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mn>2</mn> <mi>ϵ</mi> <mo>=</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{\infty }^{2}=2\epsilon =C_{3}=-\mu /a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95025d12979e6a774da3379219d2796deb45bf12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.723ex; height:3.009ex;" alt="{\displaystyle v_{\infty }^{2}=2\epsilon =C_{3}=-\mu /a}"></span> or <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=-{\mu /{v_{\infty }^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=-{\mu /{v_{\infty }^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90888d9594db46d4889d69022e0191854d6237f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.704ex; height:3.009ex;" alt="{\displaystyle a=-{\mu /{v_{\infty }^{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 =Gm\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <mi>G</mi> <mi>m</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =Gm\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01e8b0603cf6757632316c92dcaa6203fa1a8cbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:8.754ex; height:2.676ex;" alt="{\displaystyle \mu =Gm\,\!}"></span> is the <a href="/wiki/Standard_gravitational_parameter" title="Standard gravitational parameter">standard gravitational parameter</a> 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 C_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66e9abeb5057b7afbf88e3169101849354f13c65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.509ex;" alt="{\displaystyle C_{3}}"></span> is characteristic energy, commonly used in planning interplanetary missions </p><p>Note that the total energy is positive in the case of a hyperbolic trajectory (whereas it is negative for an elliptical orbit). </p> <div class="mw-heading mw-heading3"><h3 id="Eccentricity_and_angle_between_approach_and_departure">Eccentricity and angle between approach and departure</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_trajectory&action=edit&section=3" title="Edit section: Eccentricity and angle between approach and departure"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>With a hyperbolic trajectory the <a href="/wiki/Orbital_eccentricity" title="Orbital eccentricity">orbital eccentricity</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc00de5da9bf6ddc4b0acc12b29025464d03359a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:1.676ex;" alt="{\displaystyle e\,}"></span>) is greater than 1. The eccentricity is directly related to the angle between the asymptotes. With eccentricity just over 1 the hyperbola is a sharp "v" shape. At <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={\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e={\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae37adc9a631c89e1f35d3446bea2b101f4b15a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.28ex; height:3.009ex;" alt="{\displaystyle e={\sqrt {2}}}"></span> the asymptotes are at right angles. With <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>2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>></mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e>2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7d3dc3f453c1b0a0131148080e79ed7e2a1fa3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.344ex; height:2.176ex;" alt="{\displaystyle e>2}"></span> the asymptotes are more than 120° apart, and the periapsis distance is greater than the semi major axis. As eccentricity increases further the motion approaches a straight line. </p><p>The angle between the direction of periapsis and an asymptote from the central body is the <a href="/wiki/True_anomaly" title="True anomaly">true anomaly</a> as distance tends to infinity (<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 _{\infty }\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{\infty }\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93d614df8db9954a974e38e1388a8a763e356bc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.353ex; height:2.509ex;" alt="{\displaystyle \theta _{\infty }\,}"></span>), so <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\theta _{\infty }\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\theta _{\infty }\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/392b33847c6b27715c833c16efc2f87f87e18a07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.515ex; height:2.509ex;" alt="{\displaystyle 2\theta _{\infty }\,}"></span> is the external angle between approach and departure directions (between asymptotes). Then </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 \theta {_{\infty }}=\cos ^{-1}(-1/e)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> </mrow> <mo>=</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>e</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta {_{\infty }}=\cos ^{-1}(-1/e)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44ffae52738a32f8e51fa2ad576acad0fcef6d17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.921ex; height:3.176ex;" alt="{\displaystyle \theta {_{\infty }}=\cos ^{-1}(-1/e)\,}"></span> or <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=-1/\cos \theta {_{\infty }}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e=-1/\cos \theta {_{\infty }}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/641e71fae102e38273b0f1da847398b70a11204e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.553ex; height:2.843ex;" alt="{\displaystyle e=-1/\cos \theta {_{\infty }}\,}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Impact_parameter_and_the_distance_of_closest_approach">Impact parameter and the distance of closest approach <span class="anchor" id="Impact_parameter"></span><span class="anchor" id="Closest_approach"></span></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_trajectory&action=edit&section=4" title="Edit section: Impact parameter and the distance of closest approach"><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:Hyperbolic_trajectories_with_different_impact_parameters.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Hyperbolic_trajectories_with_different_impact_parameters.png/330px-Hyperbolic_trajectories_with_different_impact_parameters.png" decoding="async" width="330" height="275" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Hyperbolic_trajectories_with_different_impact_parameters.png/495px-Hyperbolic_trajectories_with_different_impact_parameters.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Hyperbolic_trajectories_with_different_impact_parameters.png/660px-Hyperbolic_trajectories_with_different_impact_parameters.png 2x" data-file-width="922" data-file-height="768" /></a><figcaption>Hyperbolic trajectories followed by objects approaching central object (small dot) with same hyperbolic excess velocity (and semi-major axis (=1)) and from same direction but with different impact parameters and eccentricities. The yellow line indeed passes around the central dot, approaching it closely.</figcaption></figure> <p>The <a href="/wiki/Impact_parameter" title="Impact parameter">impact parameter</a> is the distance by which a body, if it continued on an unperturbed path, would miss the central body at its <a href="/wiki/Closest_approach" class="mw-redirect" title="Closest approach">closest approach</a>. With bodies experiencing gravitational forces and following hyperbolic trajectories it is equal to the semi-minor axis of the hyperbola. </p><p>In the situation of a spacecraft or comet approaching a planet, the impact parameter and excess velocity will be known accurately. If the central body is known the trajectory can now be found, including how close the approaching body will be at periapsis. If this is less than planet's radius an impact should be expected. The distance of closest approach, or periapsis distance, is given by: </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_{p}=-a(e-1)={\frac {\mu }{v_{\infty }^{2}}}\left({\sqrt {1+\left(b{\frac {v_{\infty }^{2}}{\mu }}\right)^{2}}}-1\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>e</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>μ</mi> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mi>μ</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{p}=-a(e-1)={\frac {\mu }{v_{\infty }^{2}}}\left({\sqrt {1+\left(b{\frac {v_{\infty }^{2}}{\mu }}\right)^{2}}}-1\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee350aac4f84fa337e513fa7b61560e6a6bce5f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:46.173ex; height:8.176ex;" alt="{\displaystyle r_{p}=-a(e-1)={\frac {\mu }{v_{\infty }^{2}}}\left({\sqrt {1+\left(b{\frac {v_{\infty }^{2}}{\mu }}\right)^{2}}}-1\right)}"></span></dd></dl> <p>So if a comet approaching <a href="/wiki/Earth" title="Earth">Earth</a> (effective radius ~6400 km) with a velocity of 12.5 km/s (the approximate minimum approach speed of a body coming from the outer <a href="/wiki/Solar_System" title="Solar System">Solar System</a>) is to avoid a collision with Earth, the impact parameter will need to be at least 8600 km, or 34% more than the Earth's radius. A body approaching <a href="/wiki/Jupiter" title="Jupiter">Jupiter</a> (radius 70000 km) from the outer Solar System with a speed of 5.5 km/s, will need the impact parameter to be at least 770,000 km or 11 times Jupiter radius to avoid collision. </p><p>If the mass of the central body is not known, its standard gravitational parameter, and hence its mass, can be determined by the deflection of the smaller body together with the impact parameter and approach speed. Because typically all these variables can be determined accurately, a spacecraft flyby will provide a good estimate of a body's mass. </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 \mu =bv_{\infty }^{2}\tan \delta /2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <mi>b</mi> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mi>tan</mi> <mo>⁡</mo> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =bv_{\infty }^{2}\tan \delta /2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63f4ee8c740529d90446d69b56bae5e5809d71b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.008ex; height:3.009ex;" alt="{\displaystyle \mu =bv_{\infty }^{2}\tan \delta /2}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta =2\theta _{\infty }-\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo>=</mo> <mn>2</mn> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mo>−</mo> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta =2\theta _{\infty }-\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81627f5751867f0972f5ee75dea232249035d260" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.448ex; height:2.676ex;" alt="{\displaystyle \delta =2\theta _{\infty }-\pi }"></span> is the angle the smaller body is deflected from a straight line in its course.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Equations_of_motion">Equations of motion</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_trajectory&action=edit&section=5" title="Edit section: Equations of motion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Position">Position</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_trajectory&action=edit&section=6" title="Edit section: Position"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In a hyperbolic trajectory the <a href="/wiki/True_anomaly" title="True anomaly">true anomaly</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\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> is linked to the distance between the orbiting bodies (<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> <mspace width="thinmathspace" /> </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/f08ce4d4c86c5b43f36c8435fb598da6471047c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.436ex; height:1.676ex;" alt="{\displaystyle r\,}"></span>) by the <a href="/wiki/Orbit_equation" title="Orbit equation">orbit equation</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 r={\frac {\ell }{1+e\cdot \cos \theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ℓ</mi> <mrow> <mn>1</mn> <mo>+</mo> <mi>e</mi> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r={\frac {\ell }{1+e\cdot \cos \theta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68a1cfc6cc9357a9cde05e7c23625619351f5285" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:16.337ex; height:5.676ex;" alt="{\displaystyle r={\frac {\ell }{1+e\cdot \cos \theta }}}"></span></dd></dl> <p>The relation between the true anomaly <span class="texhtml mvar" style="font-style:italic;">θ</span> and the <a href="/wiki/Eccentric_anomaly" title="Eccentric anomaly">eccentric anomaly</a> <i>E</i> (alternatively the hyperbolic anomaly <i>H</i>) is:<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <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 \cosh {E}={{\cos {\theta }+e} \over {1+e\cdot \cos {\theta }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cosh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> <mo>+</mo> <mi>e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>+</mo> <mi>e</mi> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cosh {E}={{\cos {\theta }+e} \over {1+e\cdot \cos {\theta }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81424e45984386473f663fb5a8c44b3e3a0bb433" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:21.855ex; height:5.676ex;" alt="{\displaystyle \cosh {E}={{\cos {\theta }+e} \over {1+e\cdot \cos {\theta }}}}"></span>     or     <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 \tan {\frac {\theta }{2}}={\sqrt {\frac {e+1}{e-1}}}\cdot \tanh {\frac {E}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>e</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow> <mi>e</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </msqrt> </mrow> <mo>⋅</mo> <mi>tanh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>E</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan {\frac {\theta }{2}}={\sqrt {\frac {e+1}{e-1}}}\cdot \tanh {\frac {E}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1471846f0751c14a1e358f9503b96890d5e4f413" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.42ex; height:6.343ex;" alt="{\displaystyle \tan {\frac {\theta }{2}}={\sqrt {\frac {e+1}{e-1}}}\cdot \tanh {\frac {E}{2}}}"></span>     or   <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 \tanh {\frac {E}{2}}={\sqrt {\frac {e-1}{e+1}}}\cdot \tan {\frac {\theta }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tanh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>E</mi> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>e</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mi>e</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </msqrt> </mrow> <mo>⋅</mo> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tanh {\frac {E}{2}}={\sqrt {\frac {e-1}{e+1}}}\cdot \tan {\frac {\theta }{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b616f39d3f079f11bc1ba911835f8b9b0d3b43c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.42ex; height:6.343ex;" alt="{\displaystyle \tanh {\frac {E}{2}}={\sqrt {\frac {e-1}{e+1}}}\cdot \tan {\frac {\theta }{2}}}"></span></dd></dl> <p>The eccentric anomaly <i>E</i> is related to the <a href="/wiki/Mean_anomaly" title="Mean anomaly">mean anomaly</a> <i>M</i> by <a href="/wiki/Kepler%27s_equation" title="Kepler's equation">Kepler's equation</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 M=e\sinh E-E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <mi>e</mi> <mi>sinh</mi> <mo>⁡</mo> <mi>E</mi> <mo>−</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M=e\sinh E-E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfd0d5a64af778c992db5b46d8b7b56312077e34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:17.938ex; height:2.343ex;" alt="{\displaystyle M=e\sinh E-E}"></span></dd></dl> <p>The mean anomaly is proportional to time </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 M={\sqrt {\frac {\mu }{-a^{3}}}}.(t-\tau ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>μ</mi> <mrow> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </msqrt> </mrow> <mo>.</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M={\sqrt {\frac {\mu }{-a^{3}}}}.(t-\tau ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a034eabdfc9f1f413d345f19c5beb3807cf70e61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:21.165ex; height:6.176ex;" alt="{\displaystyle M={\sqrt {\frac {\mu }{-a^{3}}}}.(t-\tau ),}"></span> where <i>μ</i> is a <a href="/wiki/Standard_gravitational_parameter" title="Standard gravitational parameter">gravitational parameter</a> and <i>a</i> is the <a href="/wiki/Semi-major_axis" class="mw-redirect" title="Semi-major axis">semi-major axis</a> of the orbit.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Flight_path_angle">Flight path angle</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_trajectory&action=edit&section=7" title="Edit section: Flight path angle"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The flight path angle (φ) is the angle between the direction of velocity and the perpendicular to the radial direction, so it is zero at periapsis and tends to 90 degrees at infinity. </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 \tan(\phi )={\frac {e\cdot \sin \theta }{1+e\cdot \cos \theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>e</mi> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>e</mi> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan(\phi )={\frac {e\cdot \sin \theta }{1+e\cdot \cos \theta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d63d905ca69deb9f20b07283197d66956e52736b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:21.843ex; height:5.676ex;" alt="{\displaystyle \tan(\phi )={\frac {e\cdot \sin \theta }{1+e\cdot \cos \theta }}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Velocity">Velocity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_trajectory&action=edit&section=8" title="Edit section: Velocity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Under standard assumptions the <a href="/wiki/Orbital_speed" title="Orbital speed">orbital speed</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b67d1fd725a759a151374b793113d7a78a65da4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.515ex; height:1.676ex;" alt="{\displaystyle v\,}"></span>) of a body traveling along a <b>hyperbolic trajectory</b> can be computed from the <a href="/wiki/Vis-viva_equation" title="Vis-viva equation"><i>vis-viva</i> equation</a> as: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v={\sqrt {\mu \left({2 \over {r}}+{1 \over {a}}\right)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>μ</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v={\sqrt {\mu \left({2 \over {r}}+{1 \over {a}}\right)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/048560f373601cc397868b8099cc931a3e446f0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:18.665ex; height:7.509ex;" alt="{\displaystyle v={\sqrt {\mu \left({2 \over {r}}+{1 \over {a}}\right)}}}"></span><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></dd></dl> <p>where: </p> <ul><li><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 <a href="/wiki/Standard_gravitational_parameter" title="Standard gravitational parameter">standard gravitational parameter</a>,</li> <li><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> <mspace width="thinmathspace" /> </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/f08ce4d4c86c5b43f36c8435fb598da6471047c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.436ex; height:1.676ex;" alt="{\displaystyle r\,}"></span> is radial distance of orbiting body from <a href="/wiki/Central_body" class="mw-redirect" title="Central body">central body</a>,</li> <li><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\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99a94f96d2455b9d7faf3cec3eb02ab3c455aec1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.617ex; height:1.676ex;" alt="{\displaystyle a\,\!}"></span> is the (negative) <a href="/wiki/Semi-major_axis" class="mw-redirect" title="Semi-major axis">semi-major axis</a>.</li></ul> <p>Under standard assumptions, at any position in the orbit the following relation holds for <a href="/wiki/Kinetic_energy" title="Kinetic energy">orbital velocity</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b67d1fd725a759a151374b793113d7a78a65da4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.515ex; height:1.676ex;" alt="{\displaystyle v\,}"></span>), local <a href="/wiki/Escape_velocity" title="Escape velocity">escape velocity</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {v_{esc}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> <mi>s</mi> <mi>c</mi> </mrow> </msub> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {v_{esc}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c8cdbeafd2377f61f5ca6429e0b585644cf2db5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.996ex; height:2.009ex;" alt="{\displaystyle {v_{esc}}\,}"></span>) and hyperbolic excess velocity (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{\infty }\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{\infty }\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d264d5bc6d12e2dab4f4a622900bcff2e83e3f8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:3.39ex; height:2.009ex;" alt="{\displaystyle v_{\infty }\,\!}"></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 v^{2}={v_{esc}}^{2}+{v_{\infty }}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> <mi>s</mi> <mi>c</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v^{2}={v_{esc}}^{2}+{v_{\infty }}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab70113809487acfff2c28160394bf9e34347048" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.841ex; height:3.009ex;" alt="{\displaystyle v^{2}={v_{esc}}^{2}+{v_{\infty }}^{2}}"></span></dd></dl> <p>Note that this means that a relatively small extra <a href="/wiki/Delta-v" title="Delta-v">delta-<i>v</i></a> above that needed to accelerate to the escape speed results in a relatively large speed at infinity. For example, at a place where escape speed is 11.2 km/s, the addition of 0.4 km/s yields a hyperbolic excess speed of 3.02 km/s. </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 {\sqrt {11.6^{2}-11.2^{2}}}=3.02}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mn>11.6</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mn>11.2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>=</mo> <mn>3.02</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {11.6^{2}-11.2^{2}}}=3.02}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3bdb08a5cafa2af0de07a536cffa9e0f291238b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.774ex; height:3.509ex;" alt="{\displaystyle {\sqrt {11.6^{2}-11.2^{2}}}=3.02}"></span></dd></dl> <p>This is an example of the <a href="/wiki/Oberth_effect" title="Oberth effect">Oberth effect</a>. The converse is also true - a body does not need to be slowed by much compared to its hyperbolic excess speed (e.g. by atmospheric drag near periapsis) for velocity to fall below escape velocity and so for the body to be captured. </p> <div class="mw-heading mw-heading2"><h2 id="Radial_hyperbolic_trajectory">Radial hyperbolic trajectory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_trajectory&action=edit&section=9" title="Edit section: Radial hyperbolic trajectory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A radial hyperbolic trajectory is a non-periodic <a href="/wiki/Radial_trajectory" title="Radial trajectory">trajectory on a straight line</a> where the relative speed of the two objects always exceeds the <a href="/wiki/Escape_velocity" title="Escape velocity">escape velocity</a>. There are two cases: the bodies move away from each other or towards each other. This is a hyperbolic orbit with semi-minor axis = 0 and eccentricity = 1. Although the eccentricity is 1 this is not a parabolic orbit. </p> <div class="mw-heading mw-heading2"><h2 id="Deflection_with_finite_sphere_of_influence">Deflection with finite sphere of influence</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_trajectory&action=edit&section=10" title="Edit section: Deflection with finite sphere of influence"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A more accurate formula for the deflection 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 \delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5321cfa797202b3e1f8620663ff43c4660ea03a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }"></span> considering the sphere of influence radius <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\text{SOI}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>SOI</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\text{SOI}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/846d78c95ddd93fe34ee4e6efc823cb4a722cfc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.782ex; height:2.509ex;" alt="{\displaystyle R_{\text{SOI}}}"></span> of the deflecting body, assuming a periapsis <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 p_{e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{e}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea9d1daea5e2e2bca1b08d45ef8f2b10a55184ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.257ex; height:2.009ex;" alt="{\displaystyle p_{e}}"></span> 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 \delta =2\arcsin \left({\frac {{\sqrt {1-{\frac {p_{e}}{R_{\text{SOI}}}}}}{\sqrt {1+{\frac {p_{e}}{R_{\text{SOI}}}}-{\frac {2\mu p_{e}}{v_{\infty }^{2}R_{\text{SOI}}^{2}}}}}}{1+{\frac {v_{\infty }^{2}p_{e}}{\mu }}-{\frac {2p_{e}}{R_{\text{SOI}}}}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo>=</mo> <mn>2</mn> <mi>arcsin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>SOI</mtext> </mrow> </msub> </mfrac> </mrow> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>SOI</mtext> </mrow> </msub> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>μ</mi> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mrow> <mrow> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msubsup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>SOI</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> </msqrt> </mrow> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mrow> <mi>μ</mi> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>SOI</mtext> </mrow> </msub> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta =2\arcsin \left({\frac {{\sqrt {1-{\frac {p_{e}}{R_{\text{SOI}}}}}}{\sqrt {1+{\frac {p_{e}}{R_{\text{SOI}}}}-{\frac {2\mu p_{e}}{v_{\infty }^{2}R_{\text{SOI}}^{2}}}}}}{1+{\frac {v_{\infty }^{2}p_{e}}{\mu }}-{\frac {2p_{e}}{R_{\text{SOI}}}}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba944ec2f51eb986dca178d5868c049171416e41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:47.88ex; height:13.176ex;" alt="{\displaystyle \delta =2\arcsin \left({\frac {{\sqrt {1-{\frac {p_{e}}{R_{\text{SOI}}}}}}{\sqrt {1+{\frac {p_{e}}{R_{\text{SOI}}}}-{\frac {2\mu p_{e}}{v_{\infty }^{2}R_{\text{SOI}}^{2}}}}}}{1+{\frac {v_{\infty }^{2}p_{e}}{\mu }}-{\frac {2p_{e}}{R_{\text{SOI}}}}}}\right)}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Relativistic_two-body_problem">Relativistic two-body problem</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_trajectory&action=edit&section=11" title="Edit section: Relativistic two-body problem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In context of the <a href="/wiki/Two-body_problem_in_general_relativity" title="Two-body problem in general relativity">two-body problem in general relativity</a>, trajectories of objects with enough energy to escape the gravitational pull of the other no longer are shaped like a hyperbola. Nonetheless, the term "hyperbolic trajectory" is still used to describe orbits of this type. </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=Hyperbolic_trajectory&action=edit&section=12" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Orbit" title="Orbit">Orbit</a></li> <li><a href="/wiki/Orbit_equation" title="Orbit equation">Orbit equation</a></li> <li><a href="/wiki/Kepler_orbit" title="Kepler orbit">Kepler orbit</a></li> <li><a href="/wiki/List_of_orbits" title="List of orbits">List of orbits</a></li> <li><a href="/wiki/Planetary_flyby" title="Planetary flyby">Planetary flyby</a></li> <li><a href="/wiki/Hyperbolic_asteroid" title="Hyperbolic asteroid">Hyperbolic asteroid</a></li> <li><a href="/wiki/List_of_hyperbolic_comets" title="List of hyperbolic comets">List of hyperbolic comets</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_trajectory&action=edit&section=13" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFVallado2007" class="citation book cs1">Vallado, David A. (2007). <i>Fundamentals of Astrodynamics and Applications, Third Edition</i>. Hawthorne, CA.: Hawthorne Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-881883-14-2" title="Special:BookSources/978-1-881883-14-2"><bdi>978-1-881883-14-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+of+Astrodynamics+and+Applications%2C+Third+Edition&rft.place=Hawthorne%2C+CA.&rft.pub=Hawthorne+Press&rft.date=2007&rft.isbn=978-1-881883-14-2&rft.aulast=Vallado&rft.aufirst=David+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHyperbolic+trajectory" class="Z3988"></span></li></ul> </div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFS.O.Saraiva2014" class="citation book cs1">S.O., Kepler; Saraiva, Maria de Fátima (2014). <i>Astronomia e Astrofísica</i>. Porto Alegre: Department of Astronomy - Institute of Physics of Federal University of Rio Grande do Sul. pp. 97–106.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Astronomia+e+Astrof%C3%ADsica&rft.place=Porto+Alegre&rft.pages=97-106&rft.pub=Department+of+Astronomy+-+Institute+of+Physics+of+Federal+University+of+Rio+Grande+do+Sul&rft.date=2014&rft.aulast=S.O.&rft.aufirst=Kepler&rft.au=Saraiva%2C+Maria+de+F%C3%A1tima&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHyperbolic+trajectory" 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://web.archive.org/web/20120204054322/http://www.braeunig.us/space/orbmech.htm#hyperbolic">"Basics of Space Flight: Orbital Mechanics"</a>. Archived from <a rel="nofollow" class="external text" href="http://www.braeunig.us/space/orbmech.htm#hyperbolic">the original</a> on 2012-02-04<span class="reference-accessdate">. Retrieved <span class="nowrap">2012-02-28</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Basics+of+Space+Flight%3A+Orbital+Mechanics&rft_id=http%3A%2F%2Fwww.braeunig.us%2Fspace%2Forbmech.htm%23hyperbolic&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHyperbolic+trajectory" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPeet2019" class="citation web cs1">Peet, Matthew M. (13 June 2019). <a rel="nofollow" class="external text" href="http://control.asu.edu/Classes/MAE462/462Lecture05.pdf">"Spacecraft Dynamics and Control"</a> <span class="cs1-format">(PDF)</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Spacecraft+Dynamics+and+Control&rft.date=2019-06-13&rft.aulast=Peet&rft.aufirst=Matthew+M.&rft_id=http%3A%2F%2Fcontrol.asu.edu%2FClasses%2FMAE462%2F462Lecture05.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHyperbolic+trajectory" class="Z3988"></span></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">Orbital Mechanics & Astrodynamics by Bryan Weber: <a rel="nofollow" class="external free" href="https://orbital-mechanics.space/the-orbit-equation/hyperbolic-trajectories.html">https://orbital-mechanics.space/the-orbit-equation/hyperbolic-trajectories.html</a></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_trajectory&action=edit&section=14" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20081008041919/http://homepage.mac.com/sjbradshaw/msc/traject.html">Trajectories</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20050316084931/http://www.go.ednet.ns.ca/~larry/orbits/ellipse.html">Orbits</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20120204054322/http://www.braeunig.us/space/orbmech.htm#hyperbolic">Hyperbolic</a></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><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 href="/wiki/Orbit" title="Orbit">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 class="mw-selflink selflink">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>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Hyperbolic_trajectory&oldid=1254651146">https://en.wikipedia.org/w/index.php?title=Hyperbolic_trajectory&oldid=1254651146</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Category</a>: <ul><li><a href="/wiki/Category:Orbits" title="Category:Orbits">Orbits</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 1 November 2024, at 00:38<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. By using this site, you agree to the <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use" class="extiw" title="foundation:Special:MyLanguage/Policy:Terms of Use">Terms of Use</a> and <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy" class="extiw" title="foundation:Special:MyLanguage/Policy:Privacy policy">Privacy Policy</a>. Wikipedia® is a registered trademark of the <a rel="nofollow" class="external text" href="https://wikimediafoundation.org/">Wikimedia Foundation, Inc.</a>, a non-profit organization.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:About">About Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:General_disclaimer">Disclaimers</a></li> <li id="footer-places-contact"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us">Contact Wikipedia</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Code of Conduct</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Developers</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/en.wikipedia.org">Statistics</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Cookie statement</a></li> <li id="footer-places-mobileview"><a href="//en.m.wikipedia.org/w/index.php?title=Hyperbolic_trajectory&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobile view</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-f69cdc8f6-nhjg5","wgBackendResponseTime":131,"wgPageParseReport":{"limitreport":{"cputime":"0.440","walltime":"0.652","ppvisitednodes":{"value":1513,"limit":1000000},"postexpandincludesize":{"value":65276,"limit":2097152},"templateargumentsize":{"value":3825,"limit":2097152},"expansiondepth":{"value":13,"limit":100},"expensivefunctioncount":{"value":2,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":45312,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 415.290 1 -total"," 25.63% 106.433 1 Template:Astrodynamics"," 25.16% 104.493 1 Template:Sidebar_with_collapsible_lists"," 25.05% 104.045 2 Template:Cite_book"," 17.84% 74.068 1 Template:Short_description"," 16.59% 68.889 3 Template:Navbox"," 14.67% 60.913 1 Template:Orbits"," 11.21% 46.562 5 Template:Longitem"," 10.52% 43.675 2 Template:Pagetype"," 10.50% 43.595 3 Template:Hlist"]},"scribunto":{"limitreport-timeusage":{"value":"0.242","limit":"10.000"},"limitreport-memusage":{"value":5272120,"limit":52428800}},"cachereport":{"origin":"mw-web.codfw.main-f69cdc8f6-dfwpk","timestamp":"20241122142736","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Hyperbolic trajectory","url":"https:\/\/en.wikipedia.org\/wiki\/Hyperbolic_trajectory","sameAs":"http:\/\/www.wikidata.org\/entity\/Q2755058","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q2755058","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2004-09-18T21:35:27Z","dateModified":"2024-11-01T00:38:56Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/8\/89\/OrbitalEccentricityDemo.svg","headline":"trajectory of any object around a central body with more than enough speed to escape the central object's gravitational pull"}</script> </body> </html>