Eccentric anomaly - Wikipedia

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anomaly</span> </div> </a> <ul id="toc-From_the_true_anomaly-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-From_the_mean_anomaly" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#From_the_mean_anomaly"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>From the mean anomaly</span> </div> </a> <ul id="toc-From_the_mean_anomaly-sublist" class="vector-toc-list"> </ul> </li> </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">3</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes_and_references" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes_and_references"> <div class="vector-toc-text"> <span 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mw-list-item"><a href="https://ca.wikipedia.org/wiki/Anomalia_exc%C3%A8ntrica" title="Anomalia excèntrica – Catalan" lang="ca" hreflang="ca" data-title="Anomalia excèntrica" 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/Exzentrische_Anomalie" title="Exzentrische Anomalie – German" lang="de" hreflang="de" data-title="Exzentrische Anomalie" 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/Anomal%C3%ADa_exc%C3%A9ntrica" title="Anomalía excéntrica – Spanish" lang="es" hreflang="es" data-title="Anomalía excéntrica" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Anomalie_excentrique" title="Anomalie excentrique – French" lang="fr" hreflang="fr" data-title="Anomalie excentrique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%8E%B8%EC%8B%AC%EC%9D%B4%EA%B0%81" title="편심이각 – Korean" lang="ko" hreflang="ko" data-title="편심이각" 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/Anomalia_eccentrica" title="Anomalia eccentrica – Italian" lang="it" hreflang="it" data-title="Anomalia eccentrica" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Ekscentrin%C4%97_anomalija" title="Ekscentrinė anomalija – Lithuanian" lang="lt" hreflang="lt" data-title="Ekscentrinė anomalija" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E9%9B%A2%E5%BF%83%E8%BF%91%E7%82%B9%E8%A7%92" title="離心近点角 – Japanese" lang="ja" hreflang="ja" data-title="離心近点角" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Eksentrisk_anomali" title="Eksentrisk anomali – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Eksentrisk anomali" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Anomalia_mimo%C5%9Brodowa" title="Anomalia mimośrodowa – Polish" lang="pl" hreflang="pl" data-title="Anomalia mimośrodowa" 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/Anomalia_exc%C3%AAntrica" title="Anomalia excêntrica – Portuguese" lang="pt" hreflang="pt" data-title="Anomalia excêntrica" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Anomalie_excentric%C4%83" title="Anomalie excentrică – Romanian" lang="ro" hreflang="ro" data-title="Anomalie excentrică" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%AD%D0%BA%D1%81%D1%86%D0%B5%D0%BD%D1%82%D1%80%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D0%B0%D0%BD%D0%BE%D0%BC%D0%B0%D0%BB%D0%B8%D1%8F" title="Эксцентрическая аномалия – Russian" lang="ru" hreflang="ru" data-title="Эксцентрическая аномалия" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Ekscentri%C4%8Dna_anomalija" title="Ekscentrična anomalija – Slovenian" lang="sl" hreflang="sl" data-title="Ekscentrična anomalija" 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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">Angle defining a position in an orbit</div> <p>In <a href="/wiki/Orbital_mechanics" title="Orbital mechanics">orbital mechanics</a>, the <b>eccentric anomaly</b> is an <a href="/wiki/Keplerian_element" class="mw-redirect" title="Keplerian element">angular parameter</a> that defines the position of a body that is moving along an <a href="/wiki/Ellipse" title="Ellipse">elliptic</a> <a href="/wiki/Kepler_orbit" title="Kepler orbit">Kepler orbit</a>. The eccentric anomaly is one of three angular parameters ("anomalies") that define a position along an orbit, the other two being the <a href="/wiki/True_anomaly" title="True anomaly">true anomaly</a> and the <a href="/wiki/Mean_anomaly" title="Mean anomaly">mean anomaly</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Graphical_representation">Graphical representation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Eccentric_anomaly&action=edit&section=1" title="Edit section: Graphical representation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:EccentricAnomaly.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/EccentricAnomaly.svg/324px-EccentricAnomaly.svg.png" decoding="async" width="324" height="316" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/EccentricAnomaly.svg/486px-EccentricAnomaly.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/EccentricAnomaly.svg/648px-EccentricAnomaly.svg.png 2x" data-file-width="437" data-file-height="426" /></a><figcaption>The eccentric anomaly of point <i>P</i> is the angle <i>E</i>. The center of the ellipse is point O, and the focus is point <i>F</i>.</figcaption></figure> <p>Consider the ellipse with equation 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 {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e04cef1c6af3e391a7fe772f38ce56bd0a71cc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:14.019ex; height:6.009ex;" alt="{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1,}"></span></dd></dl> <p>where <i>a</i> is the <i>semi-major</i> axis and <i>b</i> is the <i>semi-minor</i> axis. </p><p>For a point on the ellipse, <i>P</i> = <i>P</i>(<i>x</i>, <i>y</i>), representing the position of an orbiting body in an elliptical orbit, the eccentric anomaly is the angle <i>E</i> in the figure. The eccentric anomaly <i>E</i> is one of the angles of a right triangle with one vertex at the center of the ellipse, its adjacent side lying on the <i>major</i> axis, having hypotenuse <i>a</i> (equal to the <i>semi-major</i> axis of the ellipse), and opposite side (perpendicular to the <i>major</i> axis and touching the point <i>P′</i> on the auxiliary circle of radius <i>a</i>) that passes through the point <i>P</i>. The eccentric anomaly is measured in the same direction as the true anomaly, shown in the figure 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 \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>. The eccentric anomaly <i>E</i> in terms of these coordinates is given by:<sup id="cite_ref-Wentworth_1-0" class="reference"><a href="#cite_note-Wentworth-1"><span class="cite-bracket">[</span>1<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 \cos E={\frac {x}{a}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>E</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>a</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos E={\frac {x}{a}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d5e13f0efc3d68c32b3814b695074f0984abd54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.185ex; height:4.676ex;" alt="{\displaystyle \cos E={\frac {x}{a}},}"></span></dd></dl> <p>and </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 \sin E={\frac {y}{b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>E</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>y</mi> <mi>b</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin E={\frac {y}{b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12963a1a95db03eb5a98d0f14ddf804681fb8840" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.108ex; height:5.009ex;" alt="{\displaystyle \sin E={\frac {y}{b}}}"></span></dd></dl> <p>The second equation is established using the relationship </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 \left({\frac {y}{b}}\right)^{2}=1-\cos ^{2}E=\sin ^{2}E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>y</mi> <mi>b</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>E</mi> <mo>=</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {y}{b}}\right)^{2}=1-\cos ^{2}E=\sin ^{2}E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b0ffa2d0f1819c4f29fa3828b912981f5fe4196" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:28.422ex; height:5.509ex;" alt="{\displaystyle \left({\frac {y}{b}}\right)^{2}=1-\cos ^{2}E=\sin ^{2}E}"></span>,</dd></dl> <p>which implies that <span class="nowrap">sin <i>E</i> = ±<style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num"><i>y</i></span><span class="sr-only">/</span><span class="den"><i>b</i></span></span>⁠</span></span>. The equation <span class="nowrap">sin <i>E</i> = −<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>y</i></span><span class="sr-only">/</span><span class="den"><i>b</i></span></span>⁠</span></span> is immediately able to be ruled out since it traverses the ellipse in the wrong direction. It can also be noted that the second equation can be viewed as coming from a similar triangle with its opposite side having the same length <i>y</i> as the distance from <i>P</i> to the <i>major</i> axis, and its hypotenuse <i>b</i> equal to the <i>semi-minor</i> axis of the ellipse. </p> <div class="mw-heading mw-heading2"><h2 id="Formulas">Formulas</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Eccentric_anomaly&action=edit&section=2" title="Edit section: Formulas"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Radius_and_eccentric_anomaly">Radius and eccentric anomaly</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Eccentric_anomaly&action=edit&section=3" title="Edit section: Radius and eccentric anomaly"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Eccentricity_(mathematics)" title="Eccentricity (mathematics)">eccentricity</a> <i>e</i> is defined 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 e={\sqrt {1-\left({\frac {b}{a}}\right)^{2}}}\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mi>a</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e={\sqrt {1-\left({\frac {b}{a}}\right)^{2}}}\ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec668e40bb9bc51695231b621bb8a5cf549b635a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.277ex; height:7.676ex;" alt="{\displaystyle e={\sqrt {1-\left({\frac {b}{a}}\right)^{2}}}\ .}"></span></dd></dl> <p>From <a href="/wiki/Pythagoras%27s_theorem" class="mw-redirect" title="Pythagoras's theorem">Pythagoras's theorem</a> applied to the triangle with <i>r</i> (a distance <i>FP</i>) as hypotenuse: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}r^{2}&=b^{2}\sin ^{2}E+(ae-a\cos E)^{2}\\&=a^{2}\left(1-e^{2}\right)\left(1-\cos ^{2}E\right)+a^{2}\left(e^{2}-2e\cos E+\cos ^{2}E\right)\\&=a^{2}-2a^{2}e\cos E+a^{2}e^{2}\cos ^{2}E\\&=a^{2}\left(1-e\cos E\right)^{2}\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>E</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>e</mi> <mo>−</mo> <mi>a</mi> <mi>cos</mi> <mo>⁡</mo> <mi>E</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>E</mi> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>e</mi> <mi>cos</mi> <mo>⁡</mo> <mi>E</mi> <mo>+</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>E</mi> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>e</mi> <mi>cos</mi> <mo>⁡</mo> <mi>E</mi> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>E</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mi>e</mi> <mi>cos</mi> <mo>⁡</mo> <mi>E</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}r^{2}&=b^{2}\sin ^{2}E+(ae-a\cos E)^{2}\\&=a^{2}\left(1-e^{2}\right)\left(1-\cos ^{2}E\right)+a^{2}\left(e^{2}-2e\cos E+\cos ^{2}E\right)\\&=a^{2}-2a^{2}e\cos E+a^{2}e^{2}\cos ^{2}E\\&=a^{2}\left(1-e\cos E\right)^{2}\\\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2798c4a01fa73220dba6caf26ee6bad642193ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.171ex; width:59.437ex; height:13.509ex;" alt="{\displaystyle {\begin{aligned}r^{2}&=b^{2}\sin ^{2}E+(ae-a\cos E)^{2}\\&=a^{2}\left(1-e^{2}\right)\left(1-\cos ^{2}E\right)+a^{2}\left(e^{2}-2e\cos E+\cos ^{2}E\right)\\&=a^{2}-2a^{2}e\cos E+a^{2}e^{2}\cos ^{2}E\\&=a^{2}\left(1-e\cos E\right)^{2}\\\end{aligned}}}"></span></dd></dl> <p>Thus, the radius (distance from the focus to point <i>P</i>) is related to the eccentric anomaly by the formula </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=a\left(1-e\cos {E}\right)\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mi>a</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mi>e</mi> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=a\left(1-e\cos {E}\right)\ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3b195f90917cbe45973dcba5141225b20979c33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.935ex; height:2.843ex;" alt="{\displaystyle r=a\left(1-e\cos {E}\right)\ .}"></span></dd></dl> <p>With this result the eccentric anomaly can be determined from the true anomaly as shown next. </p> <div class="mw-heading mw-heading3"><h3 id="From_the_true_anomaly">From the true anomaly</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Eccentric_anomaly&action=edit&section=4" title="Edit section: From the true anomaly"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <i><a href="/wiki/True_anomaly" title="True anomaly">true anomaly</a></i> is the angle labeled <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> in the figure, located at the focus of the ellipse. It is sometimes represented by <span class="texhtml mvar" style="font-style:italic;">f</span> or <span class="texhtml mvar" style="font-style:italic;">v</span>. The true anomaly and the eccentric anomaly are related as follows.<sup id="cite_ref-Tsui_2-0" class="reference"><a href="#cite_note-Tsui-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Using the formula for <span class="texhtml mvar" style="font-style:italic;">r</span> above, the sine and cosine of <span class="texhtml mvar" style="font-style:italic;">E</span> are found in terms of <span class="texhtml mvar" style="font-style:italic;">f</span> : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\cos E&={\frac {\,x\,}{a}}={\frac {\,ae+r\cos f\,}{a}}=e+(1-e\cos E)\cos f\\\Rightarrow \cos E&={\frac {\,e+\cos f\,}{1+e\cos f}}\\\sin E&={\sqrt {\,1-\cos ^{2}E\;}}={\frac {\,{\sqrt {\,1-e^{2}\;}}\,\sin f\,}{1+e\cos f}}~.\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> <mi>cos</mi> <mo>⁡</mo> <mi>E</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="thinmathspace" /> <mi>x</mi> <mspace width="thinmathspace" /> </mrow> <mi>a</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="thinmathspace" /> <mi>a</mi> <mi>e</mi> <mo>+</mo> <mi>r</mi> <mi>cos</mi> <mo>⁡</mo> <mi>f</mi> <mspace width="thinmathspace" /> </mrow> <mi>a</mi> </mfrac> </mrow> <mo>=</mo> <mi>e</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>e</mi> <mi>cos</mi> <mo>⁡</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mi>f</mi> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">⇒</mo> <mi>cos</mi> <mo>⁡</mo> <mi>E</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="thinmathspace" /> <mi>e</mi> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mi>f</mi> <mspace width="thinmathspace" /> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>e</mi> <mi>cos</mi> <mo>⁡</mo> <mi>f</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>E</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mspace width="thinmathspace" /> <mn>1</mn> <mo>−</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>E</mi> <mspace width="thickmathspace" /> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mspace width="thinmathspace" /> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thickmathspace" /> </msqrt> </mrow> <mspace width="thinmathspace" /> <mi>sin</mi> <mo>⁡</mo> <mi>f</mi> <mspace width="thinmathspace" /> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>e</mi> <mi>cos</mi> <mo>⁡</mo> <mi>f</mi> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\cos E&={\frac {\,x\,}{a}}={\frac {\,ae+r\cos f\,}{a}}=e+(1-e\cos E)\cos f\\\Rightarrow \cos E&={\frac {\,e+\cos f\,}{1+e\cos f}}\\\sin E&={\sqrt {\,1-\cos ^{2}E\;}}={\frac {\,{\sqrt {\,1-e^{2}\;}}\,\sin f\,}{1+e\cos f}}~.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aff44d934fd04a99f2dc5e14821741c0c729a8cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.671ex; width:55.851ex; height:18.509ex;" alt="{\displaystyle {\begin{aligned}\cos E&={\frac {\,x\,}{a}}={\frac {\,ae+r\cos f\,}{a}}=e+(1-e\cos E)\cos f\\\Rightarrow \cos E&={\frac {\,e+\cos f\,}{1+e\cos f}}\\\sin E&={\sqrt {\,1-\cos ^{2}E\;}}={\frac {\,{\sqrt {\,1-e^{2}\;}}\,\sin f\,}{1+e\cos f}}~.\end{aligned}}}"></span></dd></dl> <p>Hence, </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 E={\frac {\,\sin E\,}{\cos E}}={\frac {\,{\sqrt {\,1-e^{2}\;}}\,\sin f\,}{e+\cos f}}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mi>E</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="thinmathspace" /> <mi>sin</mi> <mo>⁡</mo> <mi>E</mi> <mspace width="thinmathspace" /> </mrow> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>E</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mspace width="thinmathspace" /> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thickmathspace" /> </msqrt> </mrow> <mspace width="thinmathspace" /> <mi>sin</mi> <mo>⁡</mo> <mi>f</mi> <mspace width="thinmathspace" /> </mrow> <mrow> <mi>e</mi> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mi>f</mi> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan E={\frac {\,\sin E\,}{\cos E}}={\frac {\,{\sqrt {\,1-e^{2}\;}}\,\sin f\,}{e+\cos f}}~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f27a3aa3a72b850371cb76046416cdc1dd7610b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:35.978ex; height:6.676ex;" alt="{\displaystyle \tan E={\frac {\,\sin E\,}{\cos E}}={\frac {\,{\sqrt {\,1-e^{2}\;}}\,\sin f\,}{e+\cos f}}~.}"></span></dd></dl> <p>where the correct quadrant for <span class="texhtml mvar" style="font-style:italic;">E</span> is given by the signs of numerator and denominator, so that <span class="texhtml mvar" style="font-style:italic;">E</span> can be most easily found using an <a href="/wiki/Atan2" title="Atan2">atan2</a> function. </p><p>Angle <span class="texhtml mvar" style="font-style:italic;">E</span> is therefore the adjacent angle of a right triangle with hypotenuse <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+e\cos f\;,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mn>1</mn> <mo>+</mo> <mi>e</mi> <mi>cos</mi> <mo>⁡</mo> <mi>f</mi> <mspace width="thickmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;1+e\cos f\;,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68d68fef2bb7f8e61a64a06f3cd9f9c7af2459f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.187ex; height:2.509ex;" alt="{\displaystyle \;1+e\cos f\;,}"></span> adjacent side <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+\cos f\;,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mi>e</mi> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mi>f</mi> <mspace width="thickmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;e+\cos f\;,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6661bf07f5136a4a739cef5f16fe88b1d935bb12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.638ex; height:2.509ex;" alt="{\displaystyle \;e+\cos f\;,}"></span> and opposite side <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-e^{2}\;}}\,\sin f\;.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mspace width="thinmathspace" /> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thickmathspace" /> </msqrt> </mrow> <mspace width="thinmathspace" /> <mi>sin</mi> <mo>⁡</mo> <mi>f</mi> <mspace width="thickmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;{\sqrt {\,1-e^{2}\;}}\,\sin f\;.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9963581e8ef81432c609401086439266ea915b44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.729ex; height:3.509ex;" alt="{\displaystyle \;{\sqrt {\,1-e^{2}\;}}\,\sin f\;.}"></span> </p><p>Also, </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 {\frac {\,f\,}{2}}={\sqrt {{\frac {\,1+e\,}{1-e}}\,}}\,\tan {\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> <mrow> <mspace width="thinmathspace" /> <mi>f</mi> <mspace width="thinmathspace" /> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="thinmathspace" /> <mn>1</mn> <mo>+</mo> <mi>e</mi> <mspace width="thinmathspace" /> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>e</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> </msqrt> </mrow> <mspace width="thinmathspace" /> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="thinmathspace" /> <mi>E</mi> <mspace width="thinmathspace" /> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan {\frac {\,f\,}{2}}={\sqrt {{\frac {\,1+e\,}{1-e}}\,}}\,\tan {\frac {\,E\,}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98d232fc9f187c9407c8f69aab49f1febb3f285e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.048ex; height:6.343ex;" alt="{\displaystyle \tan {\frac {\,f\,}{2}}={\sqrt {{\frac {\,1+e\,}{1-e}}\,}}\,\tan {\frac {\,E\,}{2}}}"></span></dd></dl> <p>Substituting <span class="texhtml">cos</span> <span class="texhtml mvar" style="font-style:italic;">E</span> as found above into the expression for <span class="texhtml mvar" style="font-style:italic;">r</span>, the radial distance from the focal point to the point <span class="texhtml">P</span>, can be found in terms of the true anomaly as well:<sup id="cite_ref-Tsui_2-1" class="reference"><a href="#cite_note-Tsui-2"><span class="cite-bracket">[</span>2<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 r={\frac {a\left(\,1-e^{2}\,\right)}{\,1+e\cos f\,}}={\frac {p}{\,1+e\cos f\,}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mrow> <mo>(</mo> <mrow> <mspace width="thinmathspace" /> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mspace width="thinmathspace" /> <mn>1</mn> <mo>+</mo> <mi>e</mi> <mi>cos</mi> <mo>⁡</mo> <mi>f</mi> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mrow> <mspace width="thinmathspace" /> <mn>1</mn> <mo>+</mo> <mi>e</mi> <mi>cos</mi> <mo>⁡</mo> <mi>f</mi> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r={\frac {a\left(\,1-e^{2}\,\right)}{\,1+e\cos f\,}}={\frac {p}{\,1+e\cos f\,}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d85b71027b7955e72d01a3b25dfc1c877f6d2fce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:31.354ex; height:6.509ex;" alt="{\displaystyle r={\frac {a\left(\,1-e^{2}\,\right)}{\,1+e\cos f\,}}={\frac {p}{\,1+e\cos f\,}}\,}"></span></dd></dl> <p>where </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 \,p\equiv a\left(\,1-e^{2}\,\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>p</mi> <mo>≡</mo> <mi>a</mi> <mrow> <mo>(</mo> <mrow> <mspace width="thinmathspace" /> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,p\equiv a\left(\,1-e^{2}\,\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d2af0fead22124ae785048cd0cb1ed516d117dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.316ex; height:3.343ex;" alt="{\displaystyle \,p\equiv a\left(\,1-e^{2}\,\right)}"></span></dd></dl> <p>is called <i>"the semi-latus rectum"</i> in classical geometry. </p> <div class="mw-heading mw-heading3"><h3 id="From_the_mean_anomaly">From the mean anomaly</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Eccentric_anomaly&action=edit&section=5" title="Edit section: From the mean anomaly"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <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>:<sup id="cite_ref-Capderou_3-0" class="reference"><a href="#cite_note-Capderou-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 M=E-e\sin E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <mi>E</mi> <mo>−</mo> <mi>e</mi> <mi>sin</mi> <mo>⁡</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M=E-e\sin E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92f49670f292ad45dd2311f04742ed4c790a385a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.646ex; height:2.343ex;" alt="{\displaystyle M=E-e\sin E}"></span></dd></dl> <p>This equation does not have a <a href="/wiki/Closed-form_solution" class="mw-redirect" title="Closed-form solution">closed-form solution</a> for <i>E</i> given <i>M</i>. It is usually solved by <a href="/wiki/Numerical_methods" class="mw-redirect" title="Numerical methods">numerical methods</a>, e.g. the <a href="/wiki/Newton%27s_method" title="Newton's method">Newton–Raphson method</a>. It may be expressed in a <a href="/wiki/Fourier_series" title="Fourier series">Fourier series</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 E=M+2\sum _{n=1}^{\infty }{\frac {J_{n}(ne)}{n}}\sin(nM)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mi>M</mi> <mo>+</mo> <mn>2</mn> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mi>e</mi> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>M</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=M+2\sum _{n=1}^{\infty }{\frac {J_{n}(ne)}{n}}\sin(nM)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa2f185ecf88e4c9f40f3ddaa38e39601e5c44e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:31.97ex; height:6.843ex;" alt="{\displaystyle E=M+2\sum _{n=1}^{\infty }{\frac {J_{n}(ne)}{n}}\sin(nM)}"></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 J_{n}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{n}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3462eec70865e8698d6ae2f4a576a3bb949661e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.648ex; height:2.843ex;" alt="{\displaystyle J_{n}(x)}"></span> is the <a href="/wiki/Bessel_function" title="Bessel function">Bessel function</a> of the first kind. </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=Eccentric_anomaly&action=edit&section=6" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Eccentricity_vector" title="Eccentricity vector">Eccentricity vector</a></li> <li><a href="/wiki/Orbital_eccentricity" title="Orbital eccentricity">Orbital eccentricity</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes_and_references">Notes and references</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Eccentric_anomaly&action=edit&section=7" title="Edit section: Notes and references"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-Wentworth-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Wentworth_1-0">^</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="CITEREFGeorge_Albert_Wentworth1914" class="citation book cs1"><a href="/wiki/George_A._Wentworth" title="George A. Wentworth">George Albert Wentworth</a> (1914). "The ellipse §126". <a rel="nofollow" class="external text" href="https://archive.org/details/elementsanalyti02wentgoog"><i>Elements of analytic geometry</i></a> (2nd ed.). Ginn & Co. p. <a rel="nofollow" class="external text" href="https://archive.org/details/elementsanalyti02wentgoog/page/n159">141</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+ellipse+%C2%A7126&rft.btitle=Elements+of+analytic+geometry&rft.pages=141&rft.edition=2nd&rft.pub=Ginn+%26+Co.&rft.date=1914&rft.au=George+Albert+Wentworth&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Felementsanalyti02wentgoog&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEccentric+anomaly" class="Z3988"></span></span> </li> <li id="cite_note-Tsui-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Tsui_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Tsui_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTsui2000" class="citation book cs1">Tsui, James Bao-yen (2000). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=jPRCxNDZqDQC&pg=PA48"><i>Fundamentals of Global Positioning System receivers: A software approach</i></a> (3rd ed.). <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>. p. 48. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-38154-3" title="Special:BookSources/0-471-38154-3"><bdi>0-471-38154-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+of+Global+Positioning+System+receivers%3A+A+software+approach&rft.pages=48&rft.edition=3rd&rft.pub=John+Wiley+%26+Sons&rft.date=2000&rft.isbn=0-471-38154-3&rft.aulast=Tsui&rft.aufirst=James+Bao-yen&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DjPRCxNDZqDQC%26pg%3DPA48&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEccentric+anomaly" class="Z3988"></span></span> </li> <li id="cite_note-Capderou-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-Capderou_3-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMichel_Capderou2005" class="citation book cs1">Michel Capderou (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=BAihdjtLZXcC&pg=PA21">"Definition of the mean anomaly, Eq. 1.68"</a>. <i>Satellites: orbits and missions</i>. Springer. p. 21. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/2-287-21317-1" title="Special:BookSources/2-287-21317-1"><bdi>2-287-21317-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Definition+of+the+mean+anomaly%2C+Eq.+1.68&rft.btitle=Satellites%3A+orbits+and+missions&rft.pages=21&rft.pub=Springer&rft.date=2005&rft.isbn=2-287-21317-1&rft.au=Michel+Capderou&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DBAihdjtLZXcC%26pg%3DPA21&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEccentric+anomaly" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Sources">Sources</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Eccentric_anomaly&action=edit&section=8" title="Edit section: Sources"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Murray, Carl D.; & Dermott, Stanley F. (1999); <i>Solar System Dynamics</i>, Cambridge University Press, Cambridge, GB</li> <li>Plummer, Henry C. K. (1960); <i>An Introductory Treatise on Dynamical Astronomy</i>, Dover Publications, New York, NY (Reprint of the 1918 Cambridge University Press edition)</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": 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.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"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Orbits" title="Template:Orbits"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Orbits" title="Template talk:Orbits"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Orbits" title="Special:EditPage/Template:Orbits"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Gravitational_orbits" style="font-size:114%;margin:0 4em">Gravitational <a 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 href="/wiki/Hyperbolic_trajectory" title="Hyperbolic trajectory">Hyperbolic trajectory</a></li> <li><a href="/wiki/Inclined_orbit" title="Inclined orbit">Inclined</a> / <a href="/wiki/Non-inclined_orbit" class="mw-redirect" title="Non-inclined orbit">Non-inclined</a></li> <li><a href="/wiki/Kepler_orbit" title="Kepler orbit">Kepler</a></li> <li><a href="/wiki/Lagrange_point" title="Lagrange point">Lagrange point</a></li> <li><a href="/wiki/Osculating_orbit" title="Osculating orbit">Osculating</a></li> <li><a href="/wiki/Parabolic_trajectory" title="Parabolic trajectory">Parabolic trajectory</a></li> <li><a href="/wiki/Parking_orbit" title="Parking orbit">Parking</a></li> <li><a href="/wiki/Retrograde_and_prograde_motion" title="Retrograde and prograde motion">Prograde / Retrograde</a></li> <li><a href="/wiki/Synchronous_orbit" title="Synchronous orbit">Synchronous</a> <ul><li><a href="/wiki/Semi-synchronous_orbit" title="Semi-synchronous orbit">semi</a></li> <li><a href="/wiki/Subsynchronous_orbit" title="Subsynchronous orbit">sub</a></li></ul></li> <li><a href="/wiki/Hohmann_transfer_orbit" title="Hohmann transfer orbit">Transfer orbit</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em"><a href="/wiki/Geocentric_orbit" title="Geocentric orbit">Geocentric</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Geosynchronous_orbit" title="Geosynchronous orbit">Geosynchronous</a> <ul><li><a href="/wiki/Geostationary_orbit" title="Geostationary orbit">Geostationary</a></li> <li><a href="/wiki/Geostationary_transfer_orbit" title="Geostationary transfer orbit">Geostationary transfer</a></li></ul></li> <li><a href="/wiki/Graveyard_orbit" title="Graveyard orbit">Graveyard</a></li> <li><a href="/wiki/High_Earth_orbit" title="High Earth orbit">High Earth</a></li> <li><a href="/wiki/Low_Earth_orbit" title="Low Earth orbit">Low Earth</a></li> <li><a href="/wiki/Medium_Earth_orbit" title="Medium Earth orbit">Medium Earth</a></li> <li><a href="/wiki/Molniya_orbit" title="Molniya orbit">Molniya</a></li> <li><a href="/wiki/Near-equatorial_orbit" title="Near-equatorial orbit">Near-equatorial</a></li> <li><a href="/wiki/Orbit_of_the_Moon" title="Orbit of the Moon">Orbit of the Moon</a></li> <li><a href="/wiki/Polar_orbit" title="Polar orbit">Polar</a></li> <li><a href="/wiki/Sun-synchronous_orbit" title="Sun-synchronous orbit">Sun-synchronous</a></li> <li><a href="/wiki/Transatmospheric_orbit" title="Transatmospheric orbit">Transatmospheric</a></li> <li><a href="/wiki/Tundra_orbit" title="Tundra orbit">Tundra</a></li> <li><a href="/wiki/Very_low_Earth_orbit" title="Very low Earth orbit">Very low Earth</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em">About<br />other points</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li>Mars <ul><li><a href="/wiki/Areocentric_orbit" title="Areocentric orbit">Areocentric</a></li> <li><a href="/wiki/Areosynchronous_orbit" title="Areosynchronous orbit">Areosynchronous</a></li> <li><a href="/wiki/Areostationary_orbit" title="Areostationary orbit">Areostationary</a></li></ul></li> <li>Lagrange points <ul><li><a href="/wiki/Distant_retrograde_orbit" title="Distant retrograde orbit">Distant retrograde</a></li> <li><a href="/wiki/Halo_orbit" title="Halo orbit">Halo</a></li> <li><a href="/wiki/Lissajous_orbit" title="Lissajous orbit">Lissajous</a></li> <li><a href="/wiki/Libration_point_orbit" title="Libration point orbit">Libration</a></li></ul></li> <li><a href="/wiki/Lunar_orbit" title="Lunar orbit">Lunar</a></li> <li>Sun <ul><li><a href="/wiki/Heliocentric_orbit" title="Heliocentric orbit">Heliocentric</a> <ul><li><a href="/wiki/Earth%27s_orbit" title="Earth's orbit">Earth's orbit</a></li></ul></li> <li><a href="/wiki/Mars_cycler" title="Mars cycler">Mars cycler</a></li> <li><a href="/wiki/Sun-synchronous_orbit" title="Sun-synchronous orbit">Heliosynchronous</a></li></ul></li> <li>Other <ul><li><a href="/wiki/Lunar_cycler" title="Lunar cycler">Lunar cycler</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Orbital_elements" title="Orbital elements">Parameters</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:6em"><div class="hlist"><ul><li>Shape</li><li>Size</li></ul></div></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><span class="texhtml mvar" style="font-style:italic;">e</span>  <a href="/wiki/Orbital_eccentricity" title="Orbital eccentricity">Eccentricity</a></li> <li><span class="texhtml mvar" style="font-style:italic;">a</span>  <a href="/wiki/Semi-major_and_semi-minor_axes" title="Semi-major and semi-minor axes">Semi-major axis</a></li> <li><span class="texhtml mvar" style="font-style:italic;">b</span>  <a href="/wiki/Semi-major_and_semi-minor_axes" title="Semi-major and semi-minor axes">Semi-minor axis</a></li> <li><span class="texhtml mvar" style="font-style:italic;">Q</span>, <span class="texhtml mvar" style="font-style:italic;">q</span>  <a href="/wiki/Apsis" title="Apsis">Apsides</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em">Orientation</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><span class="texhtml mvar" style="font-style:italic;">i</span>  <a href="/wiki/Orbital_inclination" title="Orbital inclination">Inclination</a></li> <li><span class="texhtml mvar" style="font-style:italic;">Ω</span>  <a href="/wiki/Longitude_of_the_ascending_node" title="Longitude of the ascending node">Longitude of the ascending node</a></li> <li><span class="texhtml mvar" style="font-style:italic;">ω</span>  <a href="/wiki/Argument_of_periapsis" title="Argument of periapsis">Argument of periapsis</a></li> <li><span class="texhtml mvar" style="font-style:italic;">ϖ</span>  <a href="/wiki/Longitude_of_the_periapsis" class="mw-redirect" title="Longitude of the periapsis">Longitude of the periapsis</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em">Position</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><span class="texhtml mvar" style="font-style:italic;">M</span>  <a href="/wiki/Mean_anomaly" title="Mean anomaly">Mean anomaly</a></li> <li><span class="texhtml mvar" style="font-style:italic;">ν</span>, <span class="texhtml mvar" style="font-style:italic;">θ</span>, <span class="texhtml mvar" style="font-style:italic;">f</span>  <a href="/wiki/True_anomaly" title="True anomaly">True anomaly</a></li> <li><span class="texhtml mvar" style="font-style:italic;">E</span>  <a class="mw-selflink selflink">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, 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Eccentric anomaly - Wikipedia