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| {{For|specific applications of Kepler's equation|Kepler's laws of planetary motion}}
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| [[File:Kepler's equation solutions.PNG|thumb|right|Kepler's equation solutions for five different eccentricities between 0 and 1]]
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| In [[orbital mechanics]], '''Kepler's equation''' relates various geometric properties of the orbit of a body subject to a [[central force]].
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| It was first derived by [[Johannes Kepler]] in 1609 in Chapter 60 of his ''[[Astronomia nova]]'',<ref>{{cite book |last=Kepler |first=Johannes |year=1609 |title=[[Astronomia Nova|Astronomia Nova Aitiologētos, Seu Physica Coelestis, tradita commentariis De Motibus Stellæ Martis, Ex observationibus G. V. Tychonis Brahe]] |chapter=LX. Methodus, ex hac Physica, hoc est genuina & verissima hypothesi, extruendi utramque partem æquationis, & distantias genuinas: quorum utrumque simul per vicariam fieri hactenus non potuit. argumentum falsæ hypotheseos|language=Latin|pages=299–300|url=http://www.e-rara.ch/zut/content/pageview/162861}}</ref><ref>{{cite book |last=Aaboe |first=Asger |year=2001 |title=Episodes from the Early History of Astronomy |publisher=Springer|pages=146–147|url=http://books.google.com/books?id=yK8Tp0izorMC&pg=PA146|isbn=9780387951362}}</ref> and in book V of his ''Epitome of Copernican Astronomy'' (1621) Kepler proposed an iterative solution to the equation.<ref>{{cite book |last=Kepler |first=Johannes |title=Epitome astronomiæ Copernicanæ usitatâ formâ Quæstionum & Responsionum conscripta, inq; VII. Libros digesta, quorum tres hi priores sunt de Doctrina Sphæricâ |year=1621|chapter=Libri V. Pars altera. |language=Latin|pages=695–696|url=http://www.e-rara.ch/zut/content/pageview/956468}}</ref><ref>{{cite journal|url=http://adsabs.harvard.edu/full/2000JHA....31..339S |title=Kepler's Iterative Solution to Kepler's Equation |first=N. M. |last=Swerdlow|journal=[[Journal for the History of Astronomy]] |volume=31 |pages=339–341 |year=2000 |bibcode=2000JHA....31..339S}}</ref> The equation has played an important role in the history of both physics and mathematics, particularly classical [[celestial mechanics]].
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| ==Equation==
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| '''Kepler's equation''' is
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| {{Equation box 1
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| |indent =:
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| |equation = <math> M = E -\varepsilon \sin E </math>
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| |border colour = #50C878
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| |background colour = #ECFCF4}}
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| where ''M'' is the [[mean anomaly]], ''E'' is the [[eccentric anomaly]], and ε is the [[Eccentricity (mathematics)|eccentricity]]. Kepler's equation is a [[transcendental equation]] because [[sine]] is a [[transcendental function]], meaning it cannot be solved for ''E'' [[algebraic function|algebraically]]. [[Numerical analysis]] and [[Series (mathematics)|series]] expansions are generally required to evaluate ''E''.
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| == Alternate forms ==
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| There are several forms of Kepler's equation. Each form is associated with a specific type of orbit. The standard Kepler equation is used for elliptic orbits (0 ≤ ε < 1). The hyperbolic Kepler equation is used for hyperbolic orbits (ε ≫ 1). The radial Kepler equation is used for linear (radial) orbits (ε = 1). [[Parabolic trajectory#Barker's equation|Barker's equation]] is used for parabolic orbits (ε = 1). When ε = 1, Kepler's equation is not associated with an orbit.
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| When ε = 0, the orbit is circular. Increasing ε causes the circle to flatten into an ellipse. When ε = 1, the orbit is completely flat, and it appears to be a either a segment if the orbit is closed, or a ray if the orbit is open. An infinitesimal increase to ε results in a hyperbolic orbit with a turning angle of 180 degrees, and the orbit appears to be a ray. Further increases reduce the turning angle, and as ε goes to infinity, the orbit becomes a straight line of infinite length.
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| ===Hyperbolic Kepler equation===
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| The Hyperbolic Kepler equation is:
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| {{Equation box 1
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| |indent =:
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| |equation = <math> M = \varepsilon \sinh H - H </math>
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| |border colour|background colour }}
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| where ''H'' is the hyperbolic eccentric anomaly.
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| This equation is derived by multiplying Kepler's equation by the [[Imaginary number|square root of −1]]; ''i'' = √(−1) for [[imaginary unit]], and replacing
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| :<math> E = i H </math>
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| to obtain
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| :<math> M = i \left( E - \varepsilon \sin E \right) </math>
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| ===Radial Kepler equation===
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| The Radial Kepler equation is:
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| {{Equation box 1
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| |indent =:
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| |equation = <math> t(x) = \sin^{-1}( \sqrt{ x } ) - \sqrt{ x ( 1 - x ) } </math>
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| |border colour|background colour}}
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| where ''t'' is time, and ''x'' is the distance along an ''x''-axis.
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| This equation is derived by multiplying Kepler's equation by 1/2 making the replacement
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| :<math> E = 2 \sin^{-1}(\sqrt{ x }) </math>
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| and setting ε = 1 gives
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| :<math> t(x) = \frac{1}{2}\left[ E - \sin( E ) \right]. </math>
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| == Inverse problem ==
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| Calculating ''M'' for a given value of ''E'' is straightforward. However, solving for ''E'' when ''M'' is given can be considerably more challenging.
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| Kepler's equation can be solved for ''E'' [[analytic function|analytically]] by [[Lagrange inversion theorem|Lagrange inversion]]. The solution of Kepler's equation given by two Taylor series below.
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| Confusion over the solvability of Kepler's equation has persisted in the literature for four centuries.<ref>It is often erroneously claimed that Kepler's equation "cannot be solved analytically". http://www.jgiesen.de/kepler/kepler.html</ref><ref>Many authors make the absurd claim that it cannot be solved at all. M. V. K. Chari, Sheppard Joel Salon 2000 Technology & Engineering</ref> Kepler himself expressed doubt at the possibility of finding a general solution.
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| {{cquote|I am sufficiently satisfied that it [Kepler's equation] cannot be solved a priori, on account of the different nature of the arc and the sine. But if I am mistaken, and any one shall point out the way to me, he will be in my eyes the great [[Apollonius of Perga|Apollonius]].|20px|20px|Johannes Kepler <ref>Kepler's Problem, by Asaph Hall 1883 Annals of Mathematics</ref>}}
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| ===Inverse Kepler equation===
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| The inverse Kepler equation is the solution of Kepler's equation for all real values of ε:
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| : <math>
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| E =
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| \begin{cases}
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| \displaystyle \sum_{n=1}^\infty
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| {\frac{M^{\frac{n}{3}}}{n!}} \lim_{\theta \to 0^+} \! \Bigg(
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| \frac{\mathrm{d}^{\,n-1}}{\mathrm{d}\theta^{\,n-1}} \bigg( \bigg(
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| \frac{\theta}{ \sqrt[3]{\theta - \sin(\theta)} } \bigg)^{\!\!\!n} \bigg)
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| \Bigg)
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| , & \varepsilon = 1 \\
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| \displaystyle \sum_{n=1}^\infty
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| { \frac{ M^n }{ n! } }
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| \lim_{\theta \to 0^+} \! \Bigg(
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| \frac{\mathrm{d}^{\,n-1}}{\mathrm{d}\theta^{\,n-1}} \bigg( \Big(
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| \frac{ \theta }{ \theta - \varepsilon \sin(\theta)} \Big)^{\!n} \bigg)
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| \Bigg)
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| , & \varepsilon \ne 1
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| \end{cases}
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| </math>
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| Evaluating this yields:
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| : <math> | |
| E =
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| \begin{cases} \displaystyle
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| x + \frac{1}{60} x^3 + \frac{1}{1400}x^5 + \frac{1}{25200}x^7 + \frac{43}{17248000}x^9 + \frac{ 1213}{7207200000 }x^{11} +
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| \frac{151439}{12713500800000 }x^{13}+ \cdots \ | \ x = ( 6 M )^\frac{1}{3}
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| , & \varepsilon = 1 \\
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| \\
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| \displaystyle
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| \frac{1}{1-\varepsilon} M
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| - \frac{\varepsilon}{( 1-\varepsilon)^4 } \frac{M^3}{3!}
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| + \frac{(9 \varepsilon^2 + \varepsilon)}{(1-\varepsilon)^7 } \frac{M^5}{5!}
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| - \frac{(225 \varepsilon^3 + 54 \varepsilon^2 + \varepsilon ) }{(1-\varepsilon)^{10} } \frac{M^7}{7!}
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| + \frac{ (11025\varepsilon^4 + 4131 \varepsilon^3 + 243 \varepsilon^2 + \varepsilon ) }{(1-\varepsilon)^{13} } \frac{M^9}{9!}+ \cdots
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| , & \varepsilon \ne 1
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| \end{cases} </math>
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| These series can be reproduced in [[Mathematica]] with the InverseSeries operation.
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| : <code>InverseSeries[Series[M - Sin[M], {M, 0, 10}]]</code> | |
| : <code>InverseSeries[Series[M - e Sin[M], {M, 0, 10}]]</code>
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| These functions are simple Taylor series. Taylor series representations of transcendental functions are considered to be definitions of those functions. Therefore this solution is a formal definition of the inverse Kepler equation. While this solution is the simplest in a certain mathematical sense, for values of ε near 1 the convergence is very poor, other solutions are preferable for most applications. Alternatively, Kepler's equation can be solved numerically.
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| The solution for ε ≠ 1 was discovered by [[Karl Stumpff]] in 1968,<ref>Stumpff, Karl (1968b) On The application of Lie-series to the problems of celestial mechanics, NASA Technical Note D-4460</ref> but its significance wasn't recognized.<ref>Colwell, Peter (1993), Solving Kepler's Equation Over Three Centuries, Willmann–Bell. p. 43.</ref>
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| ===Inverse radial Kepler equation===
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| The inverse radial Kepler equation is: | |
| : <math>
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| x( t ) = \sum_{n=1}^{ \infty }
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| \left[
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| \lim_{ r \to 0^+ } \left(
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| {\frac{ t^{ \frac{ 2 }{ 3 } n }}{ n! }}
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| \frac{\mathrm{d}^{\,n-1}}{\mathrm{ d } r ^{\,n-1}} \! \left(
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| r^n \left( \frac{ 3 }{ 2 } \Big( \sin^{-1}( \sqrt{ r } ) - \sqrt{ r - r^2 } \Big)
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| \right)^{ \! -\frac{2}{3} n }
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| \right) \right)
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| \right] </math>
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| Evaluating this yields:
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| :<math>x(t) = p - \frac{1}{5} p^2 - \frac{3}{175}p^3
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| - \frac{23}{7875}p^4 - \frac{1894}{3931875}p^5 - \frac{3293}{21896875}p^6 - \frac{2418092}{62077640625}p^7 - \ \cdots \
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| \bigg| { p = \left( \tfrac{3}{2} t \right)^{2/3} } </math>
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| <br>
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| To obtain this result using [[Mathematica]]:
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| :<code>InverseSeries[Series[ArcSin[Sqrt[t]] - Sqrt[(1 - t) t], {t, 0, 15}]]</code>
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| ==Numerical approximation of inverse problem==
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| For most applications, the inverse problem can be computed numerically by finding the [[Zero of a function|root]] of the function:
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| : <math>
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| f(E) = E - \varepsilon \sin(E) - M(t)
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| </math>
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| This can be done iteratively via [[Newton's method]]:
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| : <math>
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| E_{n+1} = E_{n} - \frac{f(E_{n})}{f'(E_{n})} =
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| E_{n} - \frac{ E_{n} - \varepsilon \sin(E_{n}) - M(t) }{ 1 - \varepsilon \cos(E_{n})}
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| </math>
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| Note that ''E'' and ''M'' are in units of radians in this computation. This iteration is repeated until desired accuracy is obtained (e.g. when ''f''(''E'') < desired accuracy). For most elliptical orbits an initial value of ''E''<sub>0</sub> = ''M''(''t'') is sufficient. For orbits with ''ε'' > 0.8, an initial value of ''E''<sub>0</sub> = ''π'' should be used.<ref>
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| {{cite book
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| | last1 = Montenbruck
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| | first1 = Oliver
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| | last2 = Pfleger
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| | first2 = Thomas
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| | title = Astronomy on the Personal Computer
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| | publisher = Springer-Verlag Berlin Heidelberg
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| | year = 2009
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| | ISBN = 978-3-540-67221-0}}, pp 64–65</ref>
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| A similar approach can be used for the hyperbolic form of Kepler's equation. In the case of a parabolic trajectory, Barker's equation is used.
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| == See also ==
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| *[[Kepler's laws of planetary motion]]
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| *[[Kepler problem]]
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| *[[Kepler problem in general relativity]]
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| *[[Radial trajectory]]
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| == References ==
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| {{Reflist}}
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| == External links ==
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| {{commons category|Kepler's equation}}
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| * [http://mathworld.wolfram.com/KeplersEquation.html Kepler's Equation at Wolfram Mathworld]
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| <!--- Categories --->
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| [[Category:Johannes Kepler]]
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| [[Category:Orbits]]
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