Vis-viva equation - Wikipedia

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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/Equaci%C3%B3_vis-viva" title="Equació vis-viva – Catalan" lang="ca" hreflang="ca" data-title="Equació vis-viva" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Vis-Viva-Gleichung" title="Vis-Viva-Gleichung – German" lang="de" hreflang="de" data-title="Vis-Viva-Gleichung" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Vis-viva_(ecuaci%C3%B3n)" title="Vis-viva (ecuación) – Spanish" lang="es" hreflang="es" data-title="Vis-viva (ecuación)" data-language-autonym="Español" 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<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">Equation to model the motion of orbiting bodies</div> <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 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.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 href="/wiki/Hyperbolic_trajectory" title="Hyperbolic trajectory">Hyperbolic orbit</a></li> <li><a href="/wiki/Radial_trajectory" title="Radial trajectory">Radial orbit</a></li> <li><a href="/wiki/Orbital_decay" title="Orbital decay">Decaying orbit</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible"><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 class="mw-selflink selflink">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>, the <b><i>vis-viva</i> equation</b>, also referred to as <b>orbital-energy-invariance law</b> or <b>Burgas formula</b><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><sup class="noprint Inline-Template noprint noexcerpt Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:NOTRS" class="mw-redirect" title="Wikipedia:NOTRS"><span title="Citation is to a document by Stefan Ivanov, a high-school student who attended the International Astronomy Olympiad, (April 2024)">better source needed</span></a></i>]</sup><b>,</b> is one of the equations that model the <a href="/wiki/Orbital_mechanics" title="Orbital mechanics">motion of orbiting bodies</a>. It is the direct result of the principle of <a href="/wiki/Mechanical_energy#Conservation_of_mechanical_energy" title="Mechanical energy">conservation of mechanical energy</a> which applies when the only force acting on an object is its own weight which is the gravitational force determined by the product of the mass of the object and the strength of the surrounding <a href="/wiki/Gravitational_field" title="Gravitational field">gravitational field</a>. </p><p><i><a href="/wiki/Vis_viva" title="Vis viva">Vis viva</a></i> (Latin for "living force") is a term from the history of mechanics, and it survives in this sole context. It represents the principle that the difference between the total <a href="/wiki/Mechanical_work" class="mw-redirect" title="Mechanical work">work</a> of the <a href="/wiki/Acceleration" title="Acceleration">accelerating</a> <a href="/wiki/Force" title="Force">forces</a> of a <a href="/wiki/System" title="System">system</a> and that of the retarding forces is equal to one half the <i>vis viva</i> accumulated or lost in the system while the work is being done. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Equation">Equation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vis-viva_equation&action=edit&section=1" title="Edit section: Equation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For any <a href="/wiki/Keplerian_orbit" class="mw-redirect" title="Keplerian orbit">Keplerian orbit</a> (<a href="/wiki/Elliptic_orbit" title="Elliptic orbit">elliptic</a>, <a href="/wiki/Parabolic_trajectory" title="Parabolic trajectory">parabolic</a>, <a href="/wiki/Hyperbolic_trajectory" title="Hyperbolic trajectory">hyperbolic</a>, or <a href="/wiki/Radial_trajectory" title="Radial trajectory">radial</a>), the <i>vis-viva</i> equation<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> is as follows:<sup id="cite_ref-lissauer2019_3-0" class="reference"><a href="#cite_note-lissauer2019-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v^{2}=GM\left({2 \over r}-{1 \over a}\right)}"> <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> <mi>G</mi> <mi>M</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>r</mi> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>a</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v^{2}=GM\left({2 \over r}-{1 \over a}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88834a37676cfd29817f318b061e5b2c15faf0dd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.262ex; height:6.176ex;" alt="{\displaystyle v^{2}=GM\left({2 \over r}-{1 \over a}\right)}"></span> where: </p> <ul><li><span class="texhtml mvar" style="font-style:italic;">v</span> is the relative speed of the two bodies</li> <li><span class="texhtml mvar" style="font-style:italic;">r</span> is the distance between the two bodies' centers of mass</li> <li><span class="texhtml mvar" style="font-style:italic;">a</span> is the length of the <a href="/wiki/Semi-major_axis" class="mw-redirect" title="Semi-major axis">semi-major axis</a> (<span class="texhtml"><i>a</i> > 0</span> for <a href="/wiki/Ellipse" title="Ellipse">ellipses</a>, <span class="texhtml"><i>a</i> = ∞</span> or <span class="texhtml">1/<i>a</i> = 0</span> for <a href="/wiki/Parabola" title="Parabola">parabolas</a>, and <span class="texhtml"><i>a</i> < 0</span> for <a href="/wiki/Hyperbola" title="Hyperbola">hyperbolas</a>)</li> <li><span class="texhtml"><i>G</i></span> is the <a href="/wiki/Gravitational_constant" title="Gravitational constant">gravitational constant</a></li> <li><span class="texhtml mvar" style="font-style:italic;">M</span> is the mass of the central body</li></ul> <p>The product of <span class="texhtml"><i>GM</i></span> can also be expressed as the <a href="/wiki/Standard_gravitational_parameter" title="Standard gravitational parameter">standard gravitational parameter</a> using the Greek letter <span class="texhtml mvar" style="font-style:italic;">μ</span>. </p> <div class="mw-heading mw-heading2"><h2 id="Derivation_for_elliptic_orbits_(0_≤_eccentricity_<_1)"><span id="Derivation_for_elliptic_orbits_.280_.E2.89.A4_eccentricity_.3C_1.29"></span>Derivation for elliptic orbits (0 ≤ eccentricity < 1)</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vis-viva_equation&action=edit&section=2" title="Edit section: Derivation for elliptic orbits (0 ≤ eccentricity < 1)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the vis-viva equation the mass <span class="texhtml mvar" style="font-style:italic;">m</span> of the orbiting body (e.g., a spacecraft) is taken to be negligible in comparison to the mass <span class="texhtml mvar" style="font-style:italic;">M</span> of the central body (e.g., the Earth). The central body and orbiting body are also often referred to as the primary and a particle respectively. In the specific cases of an elliptical or circular orbit, the vis-viva equation may be readily derived from conservation of energy and momentum. </p><p><a href="/wiki/Specific_orbital_energy" title="Specific orbital energy">Specific total energy</a> is constant throughout the orbit. Thus, using the subscripts <span class="texhtml"><i>a</i></span> and <span class="texhtml"><i>p</i></span> to denote apoapsis (apogee) and periapsis (perigee), respectively, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon ={\frac {v_{a}^{2}}{2}}-{\frac {GM}{r_{a}}}={\frac {v_{p}^{2}}{2}}-{\frac {GM}{r_{p}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> </mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> </mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon ={\frac {v_{a}^{2}}{2}}-{\frac {GM}{r_{a}}}={\frac {v_{p}^{2}}{2}}-{\frac {GM}{r_{p}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f35a65b5e5636ddc8d9c8d26e70d3427cfc82bd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.26ex; height:6.509ex;" alt="{\displaystyle \varepsilon ={\frac {v_{a}^{2}}{2}}-{\frac {GM}{r_{a}}}={\frac {v_{p}^{2}}{2}}-{\frac {GM}{r_{p}}}}"></span> </p><p>Rearranging, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {v_{a}^{2}}{2}}-{\frac {v_{p}^{2}}{2}}={\frac {GM}{r_{a}}}-{\frac {GM}{r_{p}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> </mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> </mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {v_{a}^{2}}{2}}-{\frac {v_{p}^{2}}{2}}={\frac {GM}{r_{a}}}-{\frac {GM}{r_{p}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87b5f389376943af848ed7631925963d44f7dc9b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.078ex; height:6.509ex;" alt="{\displaystyle {\frac {v_{a}^{2}}{2}}-{\frac {v_{p}^{2}}{2}}={\frac {GM}{r_{a}}}-{\frac {GM}{r_{p}}}}"></span> </p><p>Recalling that for an elliptical orbit (and hence also a circular orbit) the velocity and radius vectors are perpendicular at apoapsis and periapsis, conservation of angular momentum requires specific angular momentum <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=r_{p}v_{p}=r_{a}v_{a}={\text{constant}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>=</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>constant</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h=r_{p}v_{p}=r_{a}v_{a}={\text{constant}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52241188e76d5a81ef7e75916a93e9e17bd1a909" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.977ex; height:2.843ex;" alt="{\displaystyle h=r_{p}v_{p}=r_{a}v_{a}={\text{constant}}}"></span>, thus <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_{p}={\frac {r_{a}}{r_{p}}}v_{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mfrac> </mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{p}={\frac {r_{a}}{r_{p}}}v_{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d25eabbb71a8107a58b98f7729fc00ff9375c213" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:10.501ex; height:5.343ex;" alt="{\displaystyle v_{p}={\frac {r_{a}}{r_{p}}}v_{a}}"></span>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2}}\left(1-{\frac {r_{a}^{2}}{r_{p}^{2}}}\right)v_{a}^{2}={\frac {GM}{r_{a}}}-{\frac {GM}{r_{p}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> </mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> </mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}\left(1-{\frac {r_{a}^{2}}{r_{p}^{2}}}\right)v_{a}^{2}={\frac {GM}{r_{a}}}-{\frac {GM}{r_{p}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25287b9c0496c216bd8b2cc36cbb97690773c8fe" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:31.822ex; height:7.509ex;" alt="{\displaystyle {\frac {1}{2}}\left(1-{\frac {r_{a}^{2}}{r_{p}^{2}}}\right)v_{a}^{2}={\frac {GM}{r_{a}}}-{\frac {GM}{r_{p}}}}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2}}\left({\frac {r_{p}^{2}-r_{a}^{2}}{r_{p}^{2}}}\right)v_{a}^{2}={\frac {GM}{r_{a}}}-{\frac {GM}{r_{p}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> <mo>)</mo> </mrow> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> </mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> </mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}\left({\frac {r_{p}^{2}-r_{a}^{2}}{r_{p}^{2}}}\right)v_{a}^{2}={\frac {GM}{r_{a}}}-{\frac {GM}{r_{p}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0745a7adf406bbd8a70bf7423ec05d407e5e245c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:32.767ex; height:7.509ex;" alt="{\displaystyle {\frac {1}{2}}\left({\frac {r_{p}^{2}-r_{a}^{2}}{r_{p}^{2}}}\right)v_{a}^{2}={\frac {GM}{r_{a}}}-{\frac {GM}{r_{p}}}}"></span> </p><p>Isolating the kinetic energy at apoapsis and simplifying, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {1}{2}}v_{a}^{2}&=\left({\frac {GM}{r_{a}}}-{\frac {GM}{r_{p}}}\right)\cdot {\frac {r_{p}^{2}}{r_{p}^{2}-r_{a}^{2}}}\\{\frac {1}{2}}v_{a}^{2}&=GM\left({\frac {r_{p}-r_{a}}{r_{a}r_{p}}}\right){\frac {r_{p}^{2}}{r_{p}^{2}-r_{a}^{2}}}\\{\frac {1}{2}}v_{a}^{2}&=GM{\frac {r_{p}}{r_{a}(r_{p}+r_{a})}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> </mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> </mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>G</mi> <mi>M</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> <mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>G</mi> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {1}{2}}v_{a}^{2}&=\left({\frac {GM}{r_{a}}}-{\frac {GM}{r_{p}}}\right)\cdot {\frac {r_{p}^{2}}{r_{p}^{2}-r_{a}^{2}}}\\{\frac {1}{2}}v_{a}^{2}&=GM\left({\frac {r_{p}-r_{a}}{r_{a}r_{p}}}\right){\frac {r_{p}^{2}}{r_{p}^{2}-r_{a}^{2}}}\\{\frac {1}{2}}v_{a}^{2}&=GM{\frac {r_{p}}{r_{a}(r_{p}+r_{a})}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7db7131eb318712a31c5eaf0cf4c26e37e7fe7ef" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.505ex; width:34.164ex; height:20.176ex;" alt="{\displaystyle {\begin{aligned}{\frac {1}{2}}v_{a}^{2}&=\left({\frac {GM}{r_{a}}}-{\frac {GM}{r_{p}}}\right)\cdot {\frac {r_{p}^{2}}{r_{p}^{2}-r_{a}^{2}}}\\{\frac {1}{2}}v_{a}^{2}&=GM\left({\frac {r_{p}-r_{a}}{r_{a}r_{p}}}\right){\frac {r_{p}^{2}}{r_{p}^{2}-r_{a}^{2}}}\\{\frac {1}{2}}v_{a}^{2}&=GM{\frac {r_{p}}{r_{a}(r_{p}+r_{a})}}\end{aligned}}}"></span> </p><p>From the geometry of an ellipse, <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 2a=r_{p}+r_{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>a</mi> <mo>=</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2a=r_{p}+r_{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eae29940b03af33d9e123a2f6b12eb71a2230fb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.589ex; height:2.843ex;" alt="{\displaystyle 2a=r_{p}+r_{a}}"></span> where <i>a</i> is the length of the semimajor axis. Thus, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2}}v_{a}^{2}=GM{\frac {2a-r_{a}}{r_{a}(2a)}}=GM\left({\frac {1}{r_{a}}}-{\frac {1}{2a}}\right)={\frac {GM}{r_{a}}}-{\frac {GM}{2a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mi>G</mi> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>a</mi> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> <mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>2</mn> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>G</mi> <mi>M</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>a</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> </mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> </mrow> <mrow> <mn>2</mn> <mi>a</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}v_{a}^{2}=GM{\frac {2a-r_{a}}{r_{a}(2a)}}=GM\left({\frac {1}{r_{a}}}-{\frac {1}{2a}}\right)={\frac {GM}{r_{a}}}-{\frac {GM}{2a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5fce61b78a71b1960da40419684509c13e363e7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:56.195ex; height:6.343ex;" alt="{\displaystyle {\frac {1}{2}}v_{a}^{2}=GM{\frac {2a-r_{a}}{r_{a}(2a)}}=GM\left({\frac {1}{r_{a}}}-{\frac {1}{2a}}\right)={\frac {GM}{r_{a}}}-{\frac {GM}{2a}}}"></span> </p><p>Substituting this into our original expression for specific orbital energy, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon ={\frac {v^{2}}{2}}-{\frac {GM}{r}}={\frac {v_{p}^{2}}{2}}-{\frac {GM}{r_{p}}}={\frac {v_{a}^{2}}{2}}-{\frac {GM}{r_{a}}}=-{\frac {GM}{2a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> </mrow> <mi>r</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> </mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> </mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> </mrow> <mrow> <mn>2</mn> <mi>a</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon ={\frac {v^{2}}{2}}-{\frac {GM}{r}}={\frac {v_{p}^{2}}{2}}-{\frac {GM}{r_{p}}}={\frac {v_{a}^{2}}{2}}-{\frac {GM}{r_{a}}}=-{\frac {GM}{2a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ca0456c19611d7341aea6bbcbca4a46e5c0cc92" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:53.333ex; height:6.509ex;" alt="{\displaystyle \varepsilon ={\frac {v^{2}}{2}}-{\frac {GM}{r}}={\frac {v_{p}^{2}}{2}}-{\frac {GM}{r_{p}}}={\frac {v_{a}^{2}}{2}}-{\frac {GM}{r_{a}}}=-{\frac {GM}{2a}}}"></span> </p><p>Thus, <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 =-{\frac {GM}{2a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> </mrow> <mrow> <mn>2</mn> <mi>a</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon =-{\frac {GM}{2a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a75095b85879536bda260f337dc26daecc9f882f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.095ex; height:5.343ex;" alt="{\displaystyle \varepsilon =-{\frac {GM}{2a}}}"></span> and the vis-viva equation may be written <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {v^{2}}{2}}-{\frac {GM}{r}}=-{\frac {GM}{2a}}}"> <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> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> </mrow> <mi>r</mi> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> </mrow> <mrow> <mn>2</mn> <mi>a</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {v^{2}}{2}}-{\frac {GM}{r}}=-{\frac {GM}{2a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bca77240595cee2044661344f3924efbe43f261" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.975ex; height:5.676ex;" alt="{\displaystyle {\frac {v^{2}}{2}}-{\frac {GM}{r}}=-{\frac {GM}{2a}}}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v^{2}=GM\left({\frac {2}{r}}-{\frac {1}{a}}\right)}"> <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> <mi>G</mi> <mi>M</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>r</mi> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>a</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v^{2}=GM\left({\frac {2}{r}}-{\frac {1}{a}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aa200ed09458b6a3f869ce4e0a5d9b58228cd6c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.262ex; height:6.176ex;" alt="{\displaystyle v^{2}=GM\left({\frac {2}{r}}-{\frac {1}{a}}\right)}"></span> </p><p>Therefore, the conserved <b>angular momentum</b> <span class="texhtml"><i>L</i> = <i>mh</i></span> can be derived 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 r_{a}+r_{p}=2a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{a}+r_{p}=2a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff26d7161f056c8ce9182341bf64dd51428a4eb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.589ex; height:2.843ex;" alt="{\displaystyle r_{a}+r_{p}=2a}"></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 r_{a}r_{p}=b^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{a}r_{p}=b^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f52309c3db69dc29ca079949f5bac00785da09e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.409ex; height:3.343ex;" alt="{\displaystyle r_{a}r_{p}=b^{2}}"></span>, where <span class="texhtml mvar" style="font-style:italic;">a</span> is <b>semi-major axis</b> and <span class="texhtml mvar" style="font-style:italic;">b</span> is <b>semi-minor axis</b> of the elliptical orbit, as follows: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{a}^{2}=GM\left({\frac {2}{r_{a}}}-{\frac {1}{a}}\right)={\frac {GM}{a}}\left({\frac {2a-r_{a}}{r_{a}}}\right)={\frac {GM}{a}}\left({\frac {r_{p}}{r_{a}}}\right)={\frac {GM}{a}}\left({\frac {b}{r_{a}}}\right)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mi>G</mi> <mi>M</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>a</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> </mrow> <mi>a</mi> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>a</mi> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> </mrow> <mi>a</mi> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> </mrow> <mi>a</mi> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{a}^{2}=GM\left({\frac {2}{r_{a}}}-{\frac {1}{a}}\right)={\frac {GM}{a}}\left({\frac {2a-r_{a}}{r_{a}}}\right)={\frac {GM}{a}}\left({\frac {r_{p}}{r_{a}}}\right)={\frac {GM}{a}}\left({\frac {b}{r_{a}}}\right)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23e174d07d9ebeb655e09e8d9e60a11e04742a5a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:72.193ex; height:6.509ex;" alt="{\displaystyle v_{a}^{2}=GM\left({\frac {2}{r_{a}}}-{\frac {1}{a}}\right)={\frac {GM}{a}}\left({\frac {2a-r_{a}}{r_{a}}}\right)={\frac {GM}{a}}\left({\frac {r_{p}}{r_{a}}}\right)={\frac {GM}{a}}\left({\frac {b}{r_{a}}}\right)^{2}}"></span> and alternately, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{p}^{2}=GM\left({\frac {2}{r_{p}}}-{\frac {1}{a}}\right)={\frac {GM}{a}}\left({\frac {2a-r_{p}}{r_{p}}}\right)={\frac {GM}{a}}\left({\frac {r_{a}}{r_{p}}}\right)={\frac {GM}{a}}\left({\frac {b}{r_{p}}}\right)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mi>G</mi> <mi>M</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>a</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> </mrow> <mi>a</mi> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>a</mi> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> </mrow> <mi>a</mi> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> </mrow> <mi>a</mi> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{p}^{2}=GM\left({\frac {2}{r_{p}}}-{\frac {1}{a}}\right)={\frac {GM}{a}}\left({\frac {2a-r_{p}}{r_{p}}}\right)={\frac {GM}{a}}\left({\frac {r_{a}}{r_{p}}}\right)={\frac {GM}{a}}\left({\frac {b}{r_{p}}}\right)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48fb108cd3c2fab1ac349c59f0b8cee6229c8bd9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:72.022ex; height:6.509ex;" alt="{\displaystyle v_{p}^{2}=GM\left({\frac {2}{r_{p}}}-{\frac {1}{a}}\right)={\frac {GM}{a}}\left({\frac {2a-r_{p}}{r_{p}}}\right)={\frac {GM}{a}}\left({\frac {r_{a}}{r_{p}}}\right)={\frac {GM}{a}}\left({\frac {b}{r_{p}}}\right)^{2}}"></span> </p><p>Therefore, specific angular momentum <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=r_{p}v_{p}=r_{a}v_{a}=b{\sqrt {\frac {GM}{a}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>=</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>=</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> </mrow> <mi>a</mi> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h=r_{p}v_{p}=r_{a}v_{a}=b{\sqrt {\frac {GM}{a}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc8b37478ccd75de3f369d559985ebe5856c7643" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.735ex; height:6.343ex;" alt="{\displaystyle h=r_{p}v_{p}=r_{a}v_{a}=b{\sqrt {\frac {GM}{a}}}}"></span>, and </p><p>Total angular momentum <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 L=mh=mb{\sqrt {\frac {GM}{a}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mi>m</mi> <mi>h</mi> <mo>=</mo> <mi>m</mi> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> </mrow> <mi>a</mi> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=mh=mb{\sqrt {\frac {GM}{a}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e16515ef5b749df75d250d1bc507cdeadb36509e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.626ex; height:6.343ex;" alt="{\displaystyle L=mh=mb{\sqrt {\frac {GM}{a}}}}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Practical_applications">Practical applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vis-viva_equation&action=edit&section=3" title="Edit section: Practical applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Given the total mass and the scalars <span class="texhtml mvar" style="font-style:italic;">r</span> and <span class="texhtml mvar" style="font-style:italic;">v</span> at a single point of the orbit, one can compute: </p> <ul><li><span class="texhtml mvar" style="font-style:italic;">r</span> and <span class="texhtml mvar" style="font-style:italic;">v</span> at any other point in the orbit;<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>notes 1<span class="cite-bracket">]</span></a></sup> and</li> <li>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 \varepsilon \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </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/9b55ebcc38150a773c67f7a525004ed964e3efb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.471ex; height:1.676ex;" alt="{\displaystyle \varepsilon \,\!}"></span>, allowing an object orbiting a larger object to be classified as having not enough energy to remain in orbit, hence being "<a href="/wiki/Suborbital" class="mw-redirect" title="Suborbital">suborbital</a>" (a ballistic missile, for example), having enough energy to be "orbital", but without the possibility to complete a full orbit anyway because it eventually collides with the other body, or having enough energy to come from and/or go to infinity (as a meteor, for example).</li></ul> <p>The formula for <a href="/wiki/Escape_velocity" title="Escape velocity">escape velocity</a> can be obtained from the Vis-viva equation by taking the limit as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> approaches <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 \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c26c105004f30c27aa7c2a9c601550a4183b1f21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.676ex;" alt="{\displaystyle \infty }"></span>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{e}^{2}=GM\left({\frac {2}{r}}-0\right)\rightarrow v_{e}={\sqrt {\frac {2GM}{r}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mi>G</mi> <mi>M</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>r</mi> </mfrac> </mrow> <mo>−</mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">→</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn>2</mn> <mi>G</mi> <mi>M</mi> </mrow> <mi>r</mi> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{e}^{2}=GM\left({\frac {2}{r}}-0\right)\rightarrow v_{e}={\sqrt {\frac {2GM}{r}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a511131c8e7b78754731cdbe173028a1932a3ef" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:36.789ex; height:6.509ex;" alt="{\displaystyle v_{e}^{2}=GM\left({\frac {2}{r}}-0\right)\rightarrow v_{e}={\sqrt {\frac {2GM}{r}}}}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vis-viva_equation&action=edit&section=4" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">For the <a href="/wiki/Three-body_problem" title="Three-body problem">three-body problem</a> there is hardly a comparable vis-viva equation: conservation of energy reduces the larger number of <a href="/wiki/Degrees_of_freedom_(physics_and_chemistry)" title="Degrees of freedom (physics and chemistry)">degrees of freedom</a> by only one.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vis-viva_equation&action=edit&section=5" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Ivanov, Stefan: XXV Национална олимпиада по астрономия, Бургас, 06-08.05.2022, Полезни формули и справочни данни (Useful formulas and reference data)</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"><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="CITEREFTom_Logsdon1998" class="citation book cs1">Tom Logsdon (1998). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=C70gQI5ayEAC"><i>Orbital Mechanics: Theory and Applications</i></a>. John Wiley & Sons. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-14636-0" title="Special:BookSources/978-0-471-14636-0"><bdi>978-0-471-14636-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Orbital+Mechanics%3A+Theory+and+Applications&rft.pub=John+Wiley+%26+Sons&rft.date=1998&rft.isbn=978-0-471-14636-0&rft.au=Tom+Logsdon&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DC70gQI5ayEAC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVis-viva+equation" class="Z3988"></span></span> </li> <li id="cite_note-lissauer2019-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-lissauer2019_3-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLissauerde_Pater2019" class="citation book cs1">Lissauer, Jack J.; de Pater, Imke (2019). <i>Fundamental Planetary Sciences : physics, chemistry, and habitability</i>. New York, NY, USA: Cambridge University Press. pp. 29–31. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781108411981" title="Special:BookSources/9781108411981"><bdi>9781108411981</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamental+Planetary+Sciences+%3A+physics%2C+chemistry%2C+and+habitability&rft.place=New+York%2C+NY%2C+USA&rft.pages=29-31&rft.pub=Cambridge+University+Press&rft.date=2019&rft.isbn=9781108411981&rft.aulast=Lissauer&rft.aufirst=Jack+J.&rft.au=de+Pater%2C+Imke&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVis-viva+equation" class="Z3988"></span></span> </li> </ol></div></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=Vis-viva_equation&oldid=1258003424">https://en.wikipedia.org/w/index.php?title=Vis-viva_equation&oldid=1258003424</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">Categories</a>: <ul><li><a href="/wiki/Category:Orbits" title="Category:Orbits">Orbits</a></li><li><a href="/wiki/Category:Conservation_laws" title="Category:Conservation laws">Conservation laws</a></li><li><a href="/wiki/Category:Equations_of_astronomy" title="Category:Equations of astronomy">Equations of astronomy</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><li><a href="/wiki/Category:All_articles_lacking_reliable_references" title="Category:All articles lacking reliable references">All articles lacking reliable references</a></li><li><a href="/wiki/Category:Articles_lacking_reliable_references_from_April_2024" title="Category:Articles lacking reliable references from April 2024">Articles lacking reliable references from April 2024</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 17 November 2024, at 17:10<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. 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Vis-viva equation - Wikipedia