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Kepler problem

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</style><div role="note" class="hatnote navigation-not-searchable">This article is about a special case of the <a href="Two-body_problem" title="Two-body problem">two-body problem</a> in <a href="Classical_mechanics" title="Classical mechanics">classical mechanics</a>. For the problem of <a href="Sphere_packing" title="Sphere packing">finding the densest packing of spheres</a> in three-dimensional Euclidean space, see <a href="Kepler_conjecture" title="Kepler conjecture">Kepler conjecture</a>.</div>

In <a href="Classical_mechanics" title="Classical mechanics">classical mechanics</a>, the Kepler problem is a special case of the <a href="Two-body_problem" title="Two-body problem">two-body problem</a>, in which the two bodies interact by a <a href="Central_force" title="Central force">central force</a> that varies in strength as the <a href="Inverse-square_law" title="Inverse-square law">inverse square</a> of the distance between them. The force may be either attractive or repulsive. The problem is to find the position or speed of the two bodies over time given their <a href="Mass" title="Mass">masses</a>, <a href="Position_(geometry)" title="Position (geometry)">positions</a>, and <a href="Velocity" title="Velocity">velocities</a>. Using classical mechanics, the solution can be expressed as a <a href="Kepler_orbit" title="Kepler orbit">Kepler orbit</a> using six <a href="Orbital_elements" title="Orbital elements">orbital elements</a>.

The Kepler problem is named after <a href="Johannes_Kepler" title="Johannes Kepler">Johannes Kepler</a>, who proposed <a href="Kepler's_laws_of_planetary_motion" title="Kepler's laws of planetary motion">Kepler's laws of planetary motion</a> (which are part of <a href="Classical_mechanics" title="Classical mechanics">classical mechanics</a> and solved the problem for the orbits of the planets) and investigated the types of forces that would result in orbits obeying those laws (called Kepler's inverse problem).<a href="#cite_note-goldstein_1980-1">[1]</a>

For a discussion of the Kepler problem specific to radial orbits, see <a href="Radial_trajectory" title="Radial trajectory">Radial trajectory</a>. <a href="General_relativity" title="General relativity">General relativity</a> provides more accurate solutions to the two-body problem, especially in strong <a href="Gravitational_field" title="Gravitational field">gravitational fields</a>.

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

The inverse square law behind the Kepler problem is the most important central force law.<a href="#cite_note-goldstein_1980-1">[1]</a>: 92

The Kepler problem is important in <a href="Celestial_mechanics" title="Celestial mechanics">celestial mechanics</a>, since <a href="Gravitation" class="mw-redirect" title="Gravitation">Newtonian gravity</a> obeys an <a href="Inverse_square_law" class="mw-redirect" title="Inverse square law">inverse square law</a>. Examples include a satellite moving about a planet, a planet about its sun, or two binary stars about each other. The Kepler problem is also important in the motion of two charged particles, since <a href="Coulomb%E2%80%99s_law" class="mw-redirect" title="Coulomb’s law">Coulomb’s law</a> of <a href="Electrostatics" title="Electrostatics">electrostatics</a> also obeys an <a href="Inverse_square_law" class="mw-redirect" title="Inverse square law">inverse square law</a>.

The Kepler problem and the <a href="Simple_harmonic_oscillator" class="mw-redirect" title="Simple harmonic oscillator">simple harmonic oscillator</a> problem are the two most fundamental problems in <a href="Classical_mechanics" title="Classical mechanics">classical mechanics</a>. They are the only two problems that have closed orbits for every possible set of initial conditions, i.e., return to their starting point with the same velocity (<a href="Bertrand's_theorem" title="Bertrand's theorem">Bertrand's theorem</a>).<a href="#cite_note-goldstein_1980-1">[1]</a>: 92

The Kepler problem also conserves the <a href="Laplace%E2%80%93Runge%E2%80%93Lenz_vector" title="Laplace–Runge–Lenz vector">Laplace–Runge–Lenz vector</a>, which has since been generalized to include other interactions. The solution of the Kepler problem allowed scientists to show that planetary motion could be explained entirely by classical mechanics and <a href="Gravitation" class="mw-redirect" title="Gravitation">Newton’s law of gravity</a>; the scientific explanation of planetary motion played an important role in ushering in the <a href="Age_of_Enlightenment" title="Age of Enlightenment">Enlightenment</a>.

<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

The Kepler problem begins with the empirical results of <a href="Johannes_Kepler" title="Johannes Kepler">Johannes Kepler</a> arduously derived by analysis of the astronomical observations of <a href="Tycho_Brahe" title="Tycho Brahe">Tycho Brahe</a>. After some 70 attempts to match the data to circular orbits, Kepler hit upon the idea of the <a href="Elliptic_orbit" title="Elliptic orbit">elliptic orbit</a>. He eventually summarized his results in the form of <a href="Kepler's_laws_of_planetary_motion" title="Kepler's laws of planetary motion">three laws of planetary motion</a>.<a href="#cite_note-2">[2]</a>

What is now called the Kepler problem was first discussed by <a href="Isaac_Newton" title="Isaac Newton">Isaac Newton</a> as a major part of his <a href="Philosophi%C3%A6_Naturalis_Principia_Mathematica" title="Philosophiæ Naturalis Principia Mathematica"> Principia</a>. His "Theorema I" begins with the first two of his three axioms or <a href="Newton's_laws_of_motion" title="Newton's laws of motion">laws of motion</a> and results in <a href="Kepler's_laws_of_planetary_motion#Second_law" title="Kepler's laws of planetary motion"> Kepler's second law</a> of planetary motion. Next Newton proves his "Theorema II" which shows that if Kepler's second law results, then the force involved must be along the line between the two bodies. In other words, Newton proves what today might be called the "inverse Kepler problem": the orbit characteristics require the force to depend on the inverse square of the distance.<a href="#cite_note-Speiser1996-3">[3]</a>: 107

<div class="mw-heading mw-heading2"><h2 id="Mathematical_definition">Mathematical definition</h2></div>

The <a href="Central_force" title="Central force">central force</a> F between two objects varies in strength as the <a href="Inverse_square_law" class="mw-redirect" title="Inverse square law">inverse square</a> of the distance r between them:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {F} ={\frac {k}{r^{2}}}\mathbf {\hat {r}} }</annotation>

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</math><img src="./c70be66b259e66a56858774fb1f056a6b283eee6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:9.057ex; height:5.676ex;" alt="{\displaystyle \mathbf {F} ={\frac {k}{r^{2}}}\mathbf {\hat {r}} }" loading="lazy"></dd></dl>

where k is a constant and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\hat {r}} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {\hat {r}} }</annotation>

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</math><img src="./7fe52dfe80c9a6604b3a46b24d65eb02c92c59e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.343ex;" alt="{\displaystyle \mathbf {\hat {r}} }" loading="lazy"> represents the <a href="Unit_vector" title="Unit vector">unit vector</a> along the line between them.<a href="#cite_note-4">[4]</a> The force may be either attractive (k < 0) or repulsive (k > 0). The corresponding <a href="Scalar_potential" title="Scalar potential">scalar potential</a> is:

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<annotation encoding="application/x-tex">{\displaystyle V(r)={\frac {k}{r}}}</annotation>

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</math><img src="./e8c47e9f451e93f7de4601ac4a5357c2d1f564c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.791ex; height:5.343ex;" alt="{\displaystyle V(r)={\frac {k}{r}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Solution_of_the_Kepler_problem">Solution of the Kepler problem</h2></div>

The equation of motion for the radius <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

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<annotation encoding="application/x-tex">{\displaystyle r}</annotation>

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</math><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"> of a particle

of mass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

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<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

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</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> moving in a <a href="Central_force" title="Central force">central potential</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(r)}">

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<annotation encoding="application/x-tex">{\displaystyle V(r)}</annotation>

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</math><img src="./114fdc48547ee60d02d7a2f4765d52a6ee3507d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.645ex; height:2.843ex;" alt="{\displaystyle V(r)}" loading="lazy"> is given by <a href="Euler%E2%80%93Lagrange_equation" title="Euler–Lagrange equation">Lagrange's equations</a>

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<annotation encoding="application/x-tex">{\displaystyle m{\frac {d^{2}r}{dt^{2}}}-mr\omega ^{2}=m{\frac {d^{2}r}{dt^{2}}}-{\frac {L^{2}}{mr^{3}}}=-{\frac {dV}{dr}}}</annotation>

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</math><img src="./2ff714fec126e699c9ed186e5add7435c04c290c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:40.489ex; height:6.009ex;" alt="{\displaystyle m{\frac {d^{2}r}{dt^{2}}}-mr\omega ^{2}=m{\frac {d^{2}r}{dt^{2}}}-{\frac {L^{2}}{mr^{3}}}=-{\frac {dV}{dr}}}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \omega \equiv {\frac {d\theta }{dt}}}</annotation>

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</math><img src="./18f311da033a579fa6bc787a9e2d359414f3296f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.687ex; height:5.509ex;" alt="{\displaystyle \omega \equiv {\frac {d\theta }{dt}}}" loading="lazy"> and the <a href="Angular_momentum" title="Angular momentum">angular momentum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=mr^{2}\omega }">

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<annotation encoding="application/x-tex">{\displaystyle L=mr^{2}\omega }</annotation>

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</math><img src="./ed62d8b22c4b89aac7b494a04c2effae5ef4d1a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.27ex; height:2.676ex;" alt="{\displaystyle L=mr^{2}\omega }" loading="lazy"> is conserved. For illustration, the first term on the left-hand side is zero for circular orbits, and the applied inwards force <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dV}{dr}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {dV}{dr}}}</annotation>

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</math><img src="./bd3429d5e2b65c86fbbf803506e2963203bd9fc1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.839ex; height:5.509ex;" alt="{\displaystyle {\frac {dV}{dr}}}" loading="lazy"> equals the <a href="Centripetal_force" title="Centripetal force">centripetal force requirement</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle mr\omega ^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle mr\omega ^{2}}</annotation>

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</math><img src="./91d774009a5649ff7c2dd830b2cb25aee0c7a639.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.589ex; height:2.676ex;" alt="{\displaystyle mr\omega ^{2}}" loading="lazy">, as expected.

If L is not zero the definition of <a href="Angular_momentum" title="Angular momentum">angular momentum</a> allows a change of independent variable from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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<annotation encoding="application/x-tex">{\displaystyle \theta }</annotation>

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</math><img src="./6e5ab2664b422d53eb0c7df3b87e1360d75ad9af.svg" 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 }" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dt}}={\frac {L}{mr^{2}}}{\frac {d}{d\theta }}}</annotation>

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</math><img src="./897e442a52d16ec06c4f088c0aa5b91ee3340cbb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:14.112ex; height:5.676ex;" alt="{\displaystyle {\frac {d}{dt}}={\frac {L}{mr^{2}}}{\frac {d}{d\theta }}}" loading="lazy"></dd></dl>

giving the new equation of motion that is independent of time

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<annotation encoding="application/x-tex">{\displaystyle {\frac {L}{r^{2}}}{\frac {d}{d\theta }}\left({\frac {L}{mr^{2}}}{\frac {dr}{d\theta }}\right)-{\frac {L^{2}}{mr^{3}}}=-{\frac {dV}{dr}}}</annotation>

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</math><img src="./debdd487497983bbff8722867b824ea1541f972d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.577ex; height:6.343ex;" alt="{\displaystyle {\frac {L}{r^{2}}}{\frac {d}{d\theta }}\left({\frac {L}{mr^{2}}}{\frac {dr}{d\theta }}\right)-{\frac {L^{2}}{mr^{3}}}=-{\frac {dV}{dr}}}" loading="lazy"></dd></dl>

The expansion of the first term is

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<annotation encoding="application/x-tex">{\displaystyle {\frac {L}{r^{2}}}{\frac {d}{d\theta }}\left({\frac {L}{mr^{2}}}{\frac {dr}{d\theta }}\right)=-{\frac {2L^{2}}{mr^{5}}}\left({\frac {dr}{d\theta }}\right)^{2}+{\frac {L^{2}}{mr^{4}}}{\frac {d^{2}r}{d\theta ^{2}}}}</annotation>

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</math><img src="./8df70f469aa337297a525c095bd8e754dea0b602.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:47.532ex; height:6.509ex;" alt="{\displaystyle {\frac {L}{r^{2}}}{\frac {d}{d\theta }}\left({\frac {L}{mr^{2}}}{\frac {dr}{d\theta }}\right)=-{\frac {2L^{2}}{mr^{5}}}\left({\frac {dr}{d\theta }}\right)^{2}+{\frac {L^{2}}{mr^{4}}}{\frac {d^{2}r}{d\theta ^{2}}}}" loading="lazy"></dd></dl>

This equation becomes quasilinear on making the change of variables <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\equiv {\frac {1}{r}}}">

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<annotation encoding="application/x-tex">{\displaystyle u\equiv {\frac {1}{r}}}</annotation>

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</math><img src="./d0df87fbc24756c06789e164a2b7f6691a1f0356.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.427ex; height:5.176ex;" alt="{\displaystyle u\equiv {\frac {1}{r}}}" loading="lazy"> and multiplying both sides by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {mr^{2}}{L^{2}}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {mr^{2}}{L^{2}}}}</annotation>

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</math><img src="./880eb4e85f2eed09872d5af5a6b33f032d0f08ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:4.979ex; height:5.843ex;" alt="{\displaystyle {\frac {mr^{2}}{L^{2}}}}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {du}{d\theta }}={\frac {-1}{r^{2}}}{\frac {dr}{d\theta }}}</annotation>

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</math><img src="./2d6a7e97b8e1f496d20a9bd2122a79e877f17642.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:13.429ex; height:5.676ex;" alt="{\displaystyle {\frac {du}{d\theta }}={\frac {-1}{r^{2}}}{\frac {dr}{d\theta }}}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {d^{2}u}{d\theta ^{2}}}={\frac {2}{r^{3}}}\left({\frac {dr}{d\theta }}\right)^{2}-{\frac {1}{r^{2}}}{\frac {d^{2}r}{d\theta ^{2}}}}</annotation>

</semantics>

</math><img src="./9c57527f5708653224ac4bb24c62febd52992b58.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.07ex; height:6.509ex;" alt="{\displaystyle {\frac {d^{2}u}{d\theta ^{2}}}={\frac {2}{r^{3}}}\left({\frac {dr}{d\theta }}\right)^{2}-{\frac {1}{r^{2}}}{\frac {d^{2}r}{d\theta ^{2}}}}" loading="lazy"></dd></dl>

After substitution and rearrangement:

<mfrac>

<mrow>

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</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {d^{2}u}{d\theta ^{2}}}+u=-{\frac {m}{L^{2}}}{\frac {d}{du}}V\left({\frac {1}{u}}\right)}</annotation>

</semantics>

</math><img src="./55d750f79ec57bdceb1024f9ee9aa6853d23ad1f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.131ex; height:6.343ex;" alt="{\displaystyle {\frac {d^{2}u}{d\theta ^{2}}}+u=-{\frac {m}{L^{2}}}{\frac {d}{du}}V\left({\frac {1}{u}}\right)}" loading="lazy"></dd></dl>

For an inverse-square force law such as the <a href="Gravity" title="Gravity">gravitational</a> or <a href="Electrostatics" title="Electrostatics">electrostatic potential</a>, the <a href="Scalar_potential" title="Scalar potential">scalar potential</a> can be written

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V(\mathbf {r} )={\frac {k}{r}}=ku}</annotation>

</semantics>

</math><img src="./0173dfe98f8202768090821893a0a4d041f800bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.484ex; height:5.343ex;" alt="{\displaystyle V(\mathbf {r} )={\frac {k}{r}}=ku}" loading="lazy"></dd></dl>

The orbit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(\theta )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u(\theta )}</annotation>

</semantics>

</math><img src="./78cddddbe22b2caa5a72f46c555998a874b31b86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.229ex; height:2.843ex;" alt="{\displaystyle u(\theta )}" loading="lazy"> can be derived from the general equation

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<mfrac>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {d^{2}u}{d\theta ^{2}}}+u=-{\frac {m}{L^{2}}}{\frac {d}{du}}V\left({\frac {1}{u}}\right)=-{\frac {km}{L^{2}}}}</annotation>

</semantics>

</math><img src="./f007706c3c6790237bf760d6edf83eab6f5d4f1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.125ex; height:6.343ex;" alt="{\displaystyle {\frac {d^{2}u}{d\theta ^{2}}}+u=-{\frac {m}{L^{2}}}{\frac {d}{du}}V\left({\frac {1}{u}}\right)=-{\frac {km}{L^{2}}}}" loading="lazy"></dd></dl>

whose solution is the constant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {km}{L^{2}}}}">

<mfrac>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -{\frac {km}{L^{2}}}}</annotation>

</semantics>

</math><img src="./c71331f954928460706435b0f068cd3dcfbae9ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:5.896ex; height:5.509ex;" alt="{\displaystyle -{\frac {km}{L^{2}}}}" loading="lazy"> plus a simple sinusoid

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u\equiv {\frac {1}{r}}=-{\frac {km}{L^{2}}}\left[1+e\cos(\theta -\theta _{0})\right]}</annotation>

</semantics>

</math><img src="./c09ebc5a46ca54c09bc6f71aa0cdd3895e8a33ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:33.571ex; height:5.509ex;" alt="{\displaystyle u\equiv {\frac {1}{r}}=-{\frac {km}{L^{2}}}\left[1+e\cos(\theta -\theta _{0})\right]}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e}</annotation>

</semantics>

</math><img src="./cd253103f0876afc68ebead27a5aa9867d927467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" loading="lazy"> (the eccentricity) and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta _{0}}</annotation>

</semantics>

</math><img src="./18b67de6bf25dba7a24e66967ff6319858798734.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.145ex; height:2.509ex;" alt="{\displaystyle \theta _{0}}" loading="lazy"> (the phase offset) are constants of integration.

This is the general formula for a <a href="Conic_section" title="Conic section">conic section</a> that has one focus at the origin; <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e=0}</annotation>

</semantics>

</math><img src="./d9850169d70a5ab7df71c2126441a86cec93eec8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.344ex; height:2.176ex;" alt="{\displaystyle e=0}" loading="lazy"> corresponds to a <a href="Circle" title="Circle">circle</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e<1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e<1}</annotation>

</semantics>

</math><img src="./2c702b141b56e60fdcb2c72832b2dcf0484607af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.344ex; height:2.176ex;" alt="{\displaystyle e<1}" loading="lazy"> corresponds to an ellipse, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e=1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e=1}</annotation>

</semantics>

</math><img src="./9c2f5932668126c63c844dc00ca187bc58a29e5a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.344ex; height:2.176ex;" alt="{\displaystyle e=1}" loading="lazy"> corresponds to a <a href="Parabola" title="Parabola">parabola</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e>1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e>1}</annotation>

</semantics>

</math><img src="./9605ca17e3915b659685c0326fbbcbfb522f11b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.344ex; height:2.176ex;" alt="{\displaystyle e>1}" loading="lazy"> corresponds to a <a href="Hyperbola" title="Hyperbola">hyperbola</a>. The eccentricity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e}</annotation>

</semantics>

</math><img src="./cd253103f0876afc68ebead27a5aa9867d927467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" loading="lazy"> is related to the total <a href="Energy" title="Energy">energy</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math><img src="./4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"> (cf. the <a href="Laplace%E2%80%93Runge%E2%80%93Lenz_vector" title="Laplace–Runge–Lenz vector">Laplace–Runge–Lenz vector</a>)

<msqrt>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e={\sqrt {1+{\frac {2EL^{2}}{k^{2}m}}}}}</annotation>

</semantics>

</math><img src="./ead74535565317eb2aa67ddf48e914da08c0ec13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.92ex; height:7.509ex;" alt="{\displaystyle e={\sqrt {1+{\frac {2EL^{2}}{k^{2}m}}}}}" loading="lazy"></dd></dl>

Comparing these formulae shows that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E<0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E<0}</annotation>

</semantics>

</math><img src="./426999fbda3c3fb19c4ef94ac9f3b4f0c19d9087.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.037ex; height:2.176ex;" alt="{\displaystyle E<0}" loading="lazy"> corresponds to an ellipse (all solutions which are <a href="Orbit_(dynamics)" title="Orbit (dynamics)">closed orbits</a> are ellipses), <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E=0}</annotation>

</semantics>

</math><img src="./26b9ec424bcc94d232be40bb53ebac3b8d5e9059.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.037ex; height:2.176ex;" alt="{\displaystyle E=0}" loading="lazy"> corresponds to a <a href="Parabola" title="Parabola">parabola</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E>0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E>0}</annotation>

</semantics>

</math><img src="./8f04631216d2e15429be4a35d29eeca15ade4fe7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.037ex; height:2.176ex;" alt="{\displaystyle E>0}" loading="lazy"> corresponds to a <a href="Hyperbola" title="Hyperbola">hyperbola</a>. In particular, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=-{\frac {k^{2}m}{2L^{2}}}}">

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E=-{\frac {k^{2}m}{2L^{2}}}}</annotation>

</semantics>

</math><img src="./a67f7973d1148856916be0e10fc43ba040f2ead9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.824ex; height:5.843ex;" alt="{\displaystyle E=-{\frac {k^{2}m}{2L^{2}}}}" loading="lazy"> for perfectly <a href="Circle" title="Circle">circular</a> orbits (the central force exactly equals the <a href="Centripetal_force" title="Centripetal force">centripetal force requirement</a>, which determines the required angular velocity for a given circular radius).

For a repulsive force (k > 0) only e > 1 applies.

<ul><li><a href="Action-angle_coordinates" title="Action-angle coordinates">Action-angle coordinates</a></li>

<li><a href="Bertrand's_theorem" title="Bertrand's theorem">Bertrand's theorem</a></li>

<li><a href="Binet_equation" title="Binet equation">Binet equation</a></li>

<li><a href="Hamilton%E2%80%93Jacobi_equation" title="Hamilton–Jacobi equation">Hamilton–Jacobi equation</a></li>

<li><a href="Laplace%E2%80%93Runge%E2%80%93Lenz_vector" title="Laplace–Runge–Lenz vector">Laplace–Runge–Lenz vector</a></li>

<li><a href="Kepler_orbit" title="Kepler orbit">Kepler orbit</a></li>

<li><a href="Kepler_problem_in_general_relativity" class="mw-redirect" title="Kepler problem in general relativity">Kepler problem in general relativity</a></li>

<li><a href="Kepler's_equation" title="Kepler's equation">Kepler's equation</a></li>

<li><a href="Kepler's_laws_of_planetary_motion" title="Kepler's laws of planetary motion">Kepler's laws of planetary motion</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFGoldstein1980" class="citation book cs1"><a href="Herbert_Goldstein" title="Herbert Goldstein">Goldstein, Herbert</a> (1980). <a href="Classical_Mechanics_(Goldstein)" title="Classical Mechanics (Goldstein)">Classical mechanics</a>. Addison-Wesley series in physics (2. ed.). Reading, Mass.: <a href="Addison-Wesley" title="Addison-Wesley">Addison-Wesley</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-201-02969-7</bdi>.</cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFCooper1981" class="citation book cs1"><a href="Leon_Cooper" title="Leon Cooper">Cooper, Leon N.</a> (1981). <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontom00coop">An introduction to the meaning and structure of physics</a>. Peleus Press. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/15205048">15205048</a>.</cite>

</li>

<li id="cite_note-Speiser1996-3"><a href="#cite_ref-Speiser1996_3-0">^</a> <cite id="CITEREFSpeiser1996" class="citation journal cs1">Speiser, David (August 1996). "The Kepler Problem from Newton to Johann Bernoulli". <a href="Archive_for_History_of_Exact_Sciences" title="Archive for History of Exact Sciences">Archive for History of Exact Sciences</a>. 50 (2): 103–116. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02327155">10.1007/BF02327155</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0003-9519">0003-9519</a>.</cite>

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFArnolʹd2009" class="citation book cs1"><a href="Vladimir_Arnold" title="Vladimir Arnold">Arnolʹd, V. I.</a> (2009). <a rel="nofollow" class="external text" href="https://archive.org/details/mathematicalmeth0000arno">Mathematical methods of classical mechanics</a>. Graduate texts in mathematics (2. ed.). New York, NY: <a href="Springer_Publishing" title="Springer Publishing">Springer</a>. p. <a rel="nofollow" class="external text" href="https://archive.org/details/mathematicalmeth0000arno/page/38">38</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-96890-2</bdi>.</cite>

</li>

</ol></div></div><div><div class="zim-footer">

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