View source for Kepler's equation

{{Short description|Orbital mechanics term}}
{{For|specific applications of Kepler's equation|Kepler's laws of planetary motion}}
{{Astrodynamics}}
In [[orbital mechanics]], '''Kepler's equation''' relates various geometric properties of the orbit of a body subject to a [[central force]].

It was derived by [[Johannes Kepler]] in 1609 in Chapter 60 of his ''[[Astronomia nova]]'',<ref>{{cite book |last=Kepler |first=Johannes |date=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=la|pages=299–300|chapter-url=http://www.e-rara.ch/zut/content/pageview/162861}}</ref><ref>{{cite book |last=Aaboe |first=Asger |date=2001 |title=Episodes from the Early History of Astronomy |publisher=Springer |pages=146–147 |url=https://books.google.com/books?id=yK8Tp0izorMC&pg=PA146 |isbn=978-0-387-95136-2}}</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 |date=1621|chapter=Libri V. Pars altera. |language=la|pages=695–696|url=http://www.e-rara.ch/zut/content/pageview/956468}}</ref><ref name=kep-iter>{{cite journal|url=http://adsabs.harvard.edu/full/2000JHA....31..339S |title=Kepler's Iterative Solution to Kepler's Equation |first=Noel M. |last=Swerdlow |journal=[[Journal for the History of Astronomy]] |volume=31 |pages=339–341 |date=2000 |issue=4 |bibcode=2000JHA....31..339S |doi=10.1177/002182860003100404 |s2cid=116599258 }}</ref> This equation and its solution, however, first appeared in a 9th-century work by [[Habash al-Hasib al-Marwazi]], which dealt with problems of parallax.<ref>{{Cite book |last=Colwell |first=Peter |url=https://books.google.com/books?id=ynjvAAAAMAAJ&q=What+we+call+Kepler%27s+Equation+and+Kepler%27s+method+of+solving+it+appeared+in |title=Solving Kepler's Equation Over Three Centuries |date=1993 |publisher=Willmann-Bell |isbn=978-0-943396-40-8 |pages=4 |language=en}}</ref><ref>{{Cite journal |last=Dutka |first=J. |date=1997-07-01 |title=A note on "Kepler's equation". |url=https://ui.adsabs.harvard.edu/abs/1997AHES...51...59D |journal=Archive for History of Exact Sciences |volume=51 |issue=1 |pages=59–65|doi=10.1007/BF00376451 |bibcode=1997AHES...51...59D |s2cid=122568981 }}</ref><ref>{{Cite book |last=North |first=John |url=https://books.google.com/books?id=qq8Luhs7rTUC&dq=al+hasib+Kepler%27s+equation&pg=PA357 |title=Cosmos: An Illustrated History of Astronomy and Cosmology |date=2008-07-15 |publisher=University of Chicago Press |isbn=978-0-226-59441-5 |language=en}}</ref><ref>{{Cite book |last=Livingston |first=John W. |url=https://books.google.com/books?id=clcPEAAAQBAJ&dq=al+hasib+Kepler%27s+equation&pg=PA163 |title=The Rise of Science in Islam and the West: From Shared Heritage to Parting of The Ways, 8th to 19th Centuries |date=2017-12-14 |publisher=Routledge |isbn=978-1-351-58926-0 |language=en}}</ref> The equation has played an important role in the history of both physics and mathematics, particularly classical [[celestial mechanics]].

==Equation==
[[File:Kepler_equation_solutions.svg|thumb|right|Kepler's equation solutions for five different eccentricities between 0 and 1]]

'''Kepler's equation''' is

{{Equation box 1
|indent =:
|equation = <math> M = E - e \sin E </math>
|border colour = #50C878
|background colour = #ECFCF4}}

where <math>M</math> is the [[mean anomaly]], <math>E</math> is the [[eccentric anomaly]], and <math>e</math> is the [[Eccentricity (mathematics)|eccentricity]].

The 'eccentric anomaly' <math>E</math> is useful to compute the position of a point moving in a Keplerian orbit. As for instance, if the body passes the periastron at coordinates <math>x = a(1 - e)</math>, <math>y = 0</math>, at time <math>t = t_0</math>, then to find out the position of the body at any time, you first calculate the mean anomaly <math>M</math> from the time and the [[mean motion]] <math>n</math> by the formula <math>M = n(t - t_0)</math>, then solve the Kepler equation above to get <math>E</math>, then get the coordinates from:

{{Equation box 1
|indent =:
|equation = <math> \begin{array}{lcl}
x & = & a (\cos E - e) \\
y & = & b \sin E
\end{array}
 </math>
|border colour = #50C878
|background colour = #ECFCF4}}

where <math>a</math> is the [[semi-major axis]], <math>b</math> the [[semi-minor axis]].

Kepler's equation is a [[transcendental equation]] because [[sine]] is a [[transcendental function]], and it cannot be solved for <math>E</math> [[algebraic function|algebraically]]. [[Numerical analysis]] and [[Series (mathematics)|series]] expansions are generally required to evaluate <math>E</math>.

== Alternate forms ==

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 (<math>0 \le e < 1</math>). The hyperbolic Kepler equation is used for hyperbolic trajectories (<math>e > 1</math>). The radial Kepler equation is used for linear (radial) trajectories (<math>e = 1</math>). [[Parabolic trajectory#Barker's equation|Barker's equation]] is used for parabolic trajectories (<math>e = 1</math>).

When <math>e = 0</math>, the orbit is circular. Increasing <math>e</math> causes the circle to become elliptical. When <math>e = 1</math>, there are four possibilities:
* a parabolic trajectory,
* a trajectory that goes back and forth along a line segment from the centre of attraction to a point at some distance away,
* a trajectory going in or out along an infinite ray emanating from the centre of attraction, with its speed going to zero with distance
* or a trajectory along a ray, but with speed not going to zero with distance.

A value of <math>e</math> slightly above 1 results in a hyperbolic orbit with a turning angle of just under 180 degrees. Further increases reduce the turning angle, and as <math>e</math> goes to infinity, the orbit becomes a straight line of infinite length.

===Hyperbolic Kepler equation===

The Hyperbolic Kepler equation is:
{{Equation box 1
|indent =:
|equation = <math> M = e \sinh H - H </math>
|border colour|background colour }}

where <math>H</math> is the hyperbolic eccentric anomaly.
This equation is derived by redefining M to be the [[imaginary unit|square root of −1]] times the right-hand side of the elliptical equation:

:<math> M = i \left( E - e \sin E \right) </math>

(in which <math>E</math> is now imaginary) and then replacing <math>E</math> by <math>iH</math>.

===Radial Kepler equations===
The Radial Kepler equation for the case where the object does not have enough energy to escape is:
{{Equation box 1
|indent =:
|equation = <math> t(x) =\pm\biggr[ \sin^{-1}( \sqrt{ x } ) - \sqrt{ x ( 1 - x ) } \biggr] </math>
|border colour|background colour}}

where <math>t</math> is proportional to time and <math>x</math> is proportional to the distance from the centre of attraction along the ray and attains the value 1 at the maximum distance. This equation is derived by multiplying Kepler's equation by 1/2 and setting <math>e</math> to 1:

:<math> t(x) = \frac{1}{2}\left[ E - \sin E \right]. </math>

and then making the substitution

:<math> E = 2 \sin^{-1}(\sqrt{ x }).</math>

The radial equation for when the object has enough energy to escape is:
{{Equation box 1
|indent =:
|equation = <math> t(x) = \pm \biggr[\sinh^{-1}( \sqrt{ x } ) - \sqrt{ x ( 1 + x ) } \biggr] </math>
|border colour|background colour}}

When the energy is exactly the minimum amount needed to escape, then the time is simply proportional to the distance to the power 3/2.

== Inverse problem ==

Calculating <math>M</math> for a given value of <math>E</math> is straightforward. However, solving for <math>E</math> when <math>M</math> is given can be considerably more challenging. There is no [[closed-form solution]]. Solving for <math>E</math> is more or less equivalent to solving for the true anomaly, or the difference between the true anomaly and the mean anomaly, which is called the "[[Equation of the center]]".

One can write an [[infinite series]] expression for the solution to Kepler's equation using [[Lagrange inversion theorem|Lagrange inversion]], but the series does not converge for all combinations of <math>e</math> and <math>M</math> (see below).

Confusion over the solvability of Kepler's equation has persisted in the literature for four centuries.<ref>It is often claimed that Kepler's equation "cannot be solved [[Analytical solution|analytically]]"; see for example [http://www.jgiesen.de/kepler/kepler.html here]. Other authors claim that it cannot be solved at all; see for example Madabushi V. K. Chari; Sheppard Joel Salon; ''Numerical Methods in Electromagnetism'', Academic Press, San Diego, CA, USA, 2000, {{ISBN|0-12-615760-X}}, [https://books.google.com/books?id=pEoti_sUxtoC&pg=PA659 p. 659]</ref> Kepler himself expressed doubt at the possibility of finding a general solution:

{{blockquote|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]].|Johannes Kepler<ref>[https://www.e-rara.ch/zut/content/pageview/162862 "Mihi ſufficit credere, ſolvi a priori non poſſe, propter arcus & ſinus ετερογενειαν. Erranti mihi, quicumque viam monſtraverit, is erit mihi magnus Apollonius."] {{Cite journal |title=Kepler's Problem |first=Asaph |last=Hall |author-link=Asaph Hall |url=https://archive.org/details/jstor-2635832 |date=May 1883 |journal=Annals of Mathematics |volume=10 |issue=3 |pages=65–66 |doi=10.2307/2635832|jstor=2635832 }}</ref>}}

[[Fourier series]] expansion (with respect to <math>M</math>) using [[Bessel functions]] is <ref>{{cite book|last1=Fitzpatrick|first1=Philip Matthew|year=1970|title=Principles of celestial mechanics|isbn=0-12-257950-X|publisher=Academic Press}}</ref><ref>{{cite journal | last1 = Colwell | first1 = Peter | title = Bessel Functions and Kepler's Equation | journal = The American Mathematical Monthly | date = January 1992 | volume = 99 | issue = 1 | pages = 45–48 | issn = 0002-9890 | doi = 10.2307/2324547| jstor = 2324547 }}</ref>
<ref>{{cite journal |first1=John P. |last1=Boyd 
  |title=Rootfinding for a transcendental equation without a first guess: Polynomialization of Kepler's equation through Chebyshev polynomial equation of the sine
  |doi=10.1016/j.apnum.2005.11.010 |year=2007 |volume=57 |number=1 
  |journal=Applied Numerical Mathematics |pages=12–18 }}</ref>
:<math>
E = M + \sum_{m=1}^\infty \frac{2}{m} J_m(me) \sin(mM), \quad e\le 1, \quad M \in [-\pi,\pi].
</math>
With respect to <math>e</math>, it is a [[Kapteyn series]].

===Inverse Kepler equation===

The inverse Kepler equation is the solution of Kepler's equation for all real values of <math>e</math>:

: <math>
 E =
\begin{cases}

\displaystyle \sum_{n=1}^\infty
 {\frac{M^{\frac{n}{3}}}{n!}} \lim_{\theta \to 0^+} \! \Bigg(
  \frac{\mathrm{d}^{\,n-1}}{\mathrm{d}\theta^{\,n-1}} \bigg( \bigg(
  \frac{\theta}{ \sqrt[3]{\theta - \sin(\theta)} } \bigg)^{\!\!\!n} \bigg)
\Bigg)
,  & e = 1  \\

\displaystyle \sum_{n=1}^\infty
{ \frac{ M^n }{ n! } }
\lim_{\theta \to 0^+} \! \Bigg(
  \frac{\mathrm{d}^{\,n-1}}{\mathrm{d}\theta^{\,n-1}}  \bigg( \Big(
  \frac{ \theta }{ \theta - e \sin(\theta)} \Big)^{\!n} \bigg)
\Bigg)
, & e \ne  1

\end{cases}
</math>

Evaluating this yields:

: <math>
E =
\begin{cases} \displaystyle
s + \frac{1}{60} s^3 + \frac{1}{1400}s^5 + \frac{1}{25200}s^7 + \frac{43}{17248000}s^9 + \frac{ 1213}{7207200000 }s^{11} +
 \frac{151439}{12713500800000 }s^{13}+ \cdots \text{ with }s = ( 6 M )^{1/3}
 ,  & e = 1\\
\\
\displaystyle
  \frac{1}{1-e} M
- \frac{e}{(1-e)^4 } \frac{M^3}{3!}
+ \frac{(9 e^2 + e)}{(1-e)^7 } \frac{M^5}{5!}
- \frac{(225 e^3 + 54 e^2 + e) }{(1-e)^{10} } \frac{M^7}{7!}
+ \frac{ (11025 e^4 + 4131 e^3 + 243 e^2 + e) }{(1-e)^{13} } \frac{M^9}{9!}+ \cdots
, & e \ne  1

\end{cases} </math>

These series can be reproduced in [[Mathematica]] with the InverseSeries operation.
: <syntaxhighlight lang="mathematica" inline>InverseSeries[Series[M - Sin[M], {M, 0, 10}]]</syntaxhighlight>
: <syntaxhighlight lang="mathematica" inline>InverseSeries[Series[M - e Sin[M], {M, 0, 10}]]</syntaxhighlight>

These functions are simple [[Maclaurin series]]. Such [[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. However, <math>E</math> is not an [[entire function]] of <math>M</math> at a given non-zero <math>e</math>. Indeed, the derivative

:<math>\mathrm{dM}/\mathrm{d}E=1-e\cos E</math>

goes to zero at an infinite set of complex numbers when <math>e < 1,</math> the nearest to zero being at <math>E=\pm i\cosh^{-1}(1/e),</math> and at these two points

:<math>M=E-e\sin E=\pm i\left(\cosh^{-1}(1/e)-\sqrt{1-e^2}\right)</math>

(where inverse cosh is taken to be positive), and <math>\mathrm{d}E/\mathrm{d}M</math> goes to infinity at these values of <math>M</math>. This means that the radius of convergence of the Maclaurin series is <math>\cosh^{-1}(1/e)-\sqrt{1-e^2}</math> and the series will not converge for values of <math>M</math> larger than this. The series can also be used for the hyperbolic case, in which case the radius of convergence is <math>\cos^{-1}(1/e)-\sqrt{e^2-1}.</math> The series for when <math>e = 1</math> converges when <math>M < 2\pi</math>.

While this solution is the simplest in a certain mathematical sense,{{which|date=August 2017}}, other solutions are preferable for most applications. Alternatively, Kepler's equation can be solved numerically.

The solution for <math>e \ne 1</math> was found by [[Karl Stumpff]] in 1968,<ref>{{Cite journal |last=Stumpff |first=Karl |date=1 June 1968 |title=On The application of Lie-series to the problems of celestial mechanics |url=https://ntrs.nasa.gov/search.jsp?R=19680017171 |id=NASA Technical Note D-4460}}</ref> but its significance wasn't recognized.<ref>{{Cite book |last=Colwell |first=Peter |year=1993 |title=Solving Kepler's Equation Over Three Centuries |isbn=0-943396-40-9 |publisher=Willmann–Bell |page=43}}</ref>{{clarify|date=August 2017}}

One can also write a Maclaurin series in <math>e</math>. This series does not converge when <math>e</math> is larger than the [[Laplace limit]] (about 0.66), regardless of the value of <math>M</math> (unless <math>M</math> is a multiple of {{math|2π}}), but it converges for all <math>M</math> if <math>e</math> is less than the Laplace limit. The coefficients in the series, other than the first (which is simply <math>M</math>), depend on <math>M</math> in a periodic way with period {{math|2π}}.

===Inverse radial Kepler equation===

The inverse radial Kepler equation (<math>e = 1</math>) for the case in which the object does not have enough energy to escape can similarly be written as:
: <math>
 x( t ) = \sum_{n=1}^{ \infty }
\left[
\lim_{ r \to 0^+ } \left(
 {\frac{ t^{ \frac{ 2 }{ 3 } n }}{ n! }}
 \frac{\mathrm{d}^{\,n-1}}{\mathrm{ d } r ^{\,n-1}} \! \left(
    r^n \left( \frac{ 3 }{ 2 } \Big( \sin^{-1}( \sqrt{ r } ) - \sqrt{ r - r^2 }  \Big)
    \right)^{ \! -\frac{2}{3} n }
  \right) \right)
 \right] </math>

Evaluating this yields:
:<math>x(t) =  p - \frac{1}{5} p^2 - \frac{3}{175}p^3
 - \frac{23}{7875}p^4 - \frac{1894}{3031875}p^5 - \frac{3293}{21896875}p^6 - \frac{2418092}{62077640625}p^7 - \ \cdots \ 
 \bigg| { p = \left( \tfrac{3}{2} t  \right)^{2/3} }  </math>

To obtain this result using [[Mathematica]]:
:<syntaxhighlight lang="mathematica" inline>InverseSeries[Series[ArcSin[Sqrt[t]] - Sqrt[(1 - t) t], {t, 0, 15}]]</syntaxhighlight>

==Numerical approximation of inverse problem==

=== Newton's method ===

For most applications, the inverse problem can be computed numerically by finding the [[Zero of a function|root]] of the function:

: <math>
f(E) = E - e \sin(E) - M(t)
</math>

This can be done iteratively via [[Newton's method]]:

: <math>
E_{n+1} = E_{n} - \frac{f(E_{n})}{f'(E_{n})} = 
E_{n} - \frac{ E_{n} - e \sin(E_{n}) - M(t) }{ 1 - e \cos(E_{n})}
</math>

Note that <math>E</math> and <math>M</math> are in units of radians in this computation.  This iteration is repeated until desired accuracy is obtained (e.g. when <math>f(E)</math> < desired accuracy). For most elliptical orbits an initial value of <math>E_0 = M(t)</math> is sufficient. For orbits with <math>e > 0.8</math>, a initial value of <math>E_0 = \pi</math> can be used. Numerous works developed accurate (but also more complex) start guesses.<ref name="Odell Gooding 1986 pp. 307–334">{{cite journal | last1=Odell | first1=A. W. | last2=Gooding | first2=R. H. | title=Procedures for solving Kepler's equation | journal=Celestial Mechanics | publisher=Springer Science and Business Media LLC | volume=38 | issue=4 | year=1986 | issn=1572-9478 | doi=10.1007/bf01238923 | pages=307–334 | bibcode=1986CeMec..38..307O| s2cid=120179781 }} </ref> If <math>e</math> is identically 1, then the derivative of <math>f</math>, which is in the denominator of Newton's method, can get close to zero, making derivative-based methods such as Newton-Raphson, secant, or regula falsi numerically unstable. In that case, the bisection method will provide guaranteed convergence, particularly since the solution can be bounded in a small initial interval. On modern computers, it is possible to achieve 4 or 5 digits of accuracy in 17 to 18 iterations.<ref>{{cite web |last1=Keister |first1=Adrian |title=The Numerical Analysis of Finding the Height of a Circular Segment |url=https://www.winemantech.com/blog/the-numerical-analysis-of-finding-the-height-of-a-circular-segment |website=Wineman Technology |publisher=Wineman Technology, Inc. |access-date=28 December 2019}}</ref> A similar approach can be used for the hyperbolic form of Kepler's equation.<ref name=Pfleger1998>{{cite book |last1=Pfleger |first1=Thomas |first2=Oliver |last2=Montenbruck |title=Astronomy on the Personal Computer |date=1998 |publisher=Springer |location=Berlin, Heidelberg |isbn=978-3-662-03349-4 |edition= Third |url=https://archive.org/details/springer_10.1007-978-3-662-03349-4 }}</ref>{{rp|66–67}} In the case of a parabolic trajectory, [[Parabolic trajectory#Barker's equation|Barker's equation]] is used.

=== Fixed-point iteration ===

A related method starts by noting that <math>E = M + e \sin{E}</math>. Repeatedly substituting the expression on the right for the <math>E</math> on the right yields a simple [[fixed-point iteration]] algorithm for evaluating <math>E(e,M)</math>. This method is identical to Kepler's 1621 solution.<ref name="kep-iter"/> In [[pseudocode]]:

<syntaxhighlight lang="mathematica">
function E(e, M, n)
    E = M
    for k = 1 to n
        E = M + e*sin E
    next k
    return E
</syntaxhighlight>

The number of iterations, <math>n</math>, depends on the value of <math>e</math>. The hyperbolic form similarly has <math>H = \sinh^{-1}\left(\frac{H+M}{e}\right)</math>.

This method is related to the [[Newton's method]] solution above in that

: <math>
E_{n+1} = E_{n} - \frac{E_{n} - e \sin(E_{n}) - M(t)}{ 1 - e \cos(E_{n})} = E_{n} + \frac {(M + e \sin{E_{n}} - E_{n})(1 + e \cos{E_{n}})}{1 - e^2 (\cos{E_{n}})^2}
</math>

To first order in the small quantities <math>M-E_{n}</math> and <math>e</math>,

:<math>E_{n+1} \approx M + e \sin{E_{n}}</math>.

== See also ==
*[[Equation of the center]]
*[[Kepler's laws of planetary motion]]
*[[Kepler problem]]
*[[Kepler problem in general relativity]]
*[[Radial trajectory]]

== References ==

{{Reflist}}

== External links ==
{{commons category|Kepler's equation}}

* {{cite journal |first1=John M. |last1=Danby |first2=Thomas M. |last2=Burkardt 
  |journal=Celestial Mechanics |year=1983 |volume=31 |pages=95–107 
  |title=The solution of Kepler's equation. I |issue=2 |bibcode=1983CeMec..31...95D |doi=10.1007/BF01686811 |s2cid=189832421 }}
* {{cite conference |first1=Bruce A. |last1=Conway 
  |title=24th Aerospace Sciences Meeting 
 |chapter=An improved algorithm due to Laguerre for the solution of Kepler's equation |year=1986 
  |doi=10.2514/6.1986-84 }}
* {{cite journal |first1=Seppo |last1=Mikkola |author-link1=Seppo Mikkola 
  |title=A cubic approximation for Kepler's equation |year=1987 |volume=40 |number=3 
  |journal=Celestial Mechanics |pages=329–334 |url=https://articles.adsabs.harvard.edu/pdf/1987CeMec..40..329M.pdf |doi=10.1007/BF01235850 |bibcode=1987CeMec..40..329M |s2cid=122237945 }}
* {{cite journal |first1=Albert |last1=Nijenhuis |author-link1=Albert Nijenhuis
  |title=Solving Kepler's equation with high efficiency and accuracy 
  |journal=Celestial Mechanics and Dynamical Astronomy |doi=10.1007/BF00052925 
  |year=1991 |pages=319–330 |volume=51 |number=4 |bibcode=1991CeMDA..51..319N |s2cid=121845017 }}
* {{cite journal |first1=F. Landis |last1=Markley |title=Kepler equation solver |year=1995 
  |journal=Celestial Mechanics and Dynamical Astronomy |volume=63 |number=1 |pages=101–111 
  |doi=10.1007/BF00691917 |bibcode=1995CeMDA..63..101M |s2cid=120405765 }}
* {{cite journal |first1=Toshio |last1=Fukushima 
  |title=A method solving kepler's equation without transcendental function evaluations 
  |year=1996 |volume=66 |number=3 |pages=309–319 |journal=Celestial Mechanics and Dynamical Astronomy
  |doi=10.1007/BF00049384 |bibcode=1996CeMDA..66..309F |s2cid=120352687 
 }}
* {{cite journal |first1=Edgar D. |last1=Charles |first2=Jeremy B. |last2=Tatum
  |title=The convergence of Newton-Raphson iteration with Kepler's equation
  |year=1997 |journal=Celestial Mechanics and Dynamical Astronomy |doi=10.1023/A:1008200607490
  |volume=69 |number=4 |pages=357–372 |bibcode=1997CeMDA..69..357C |s2cid=118637706 }}
* {{cite journal |first1=Laura |last1=Stumpf 
  |title=Chaotic behaviour in the Newton iterative function associated with Kepler's equation
  |journal=Celestial Mechanics and Dynamical Astronomy |doi=10.1023/A:1008339416143
  |year=1999 |volume=74 |number=2 |pages=95–109 |bibcode=1999CeMDA..74...95S 
 |s2cid=122491746 
 }}
* {{cite journal |first1=Manuel |last1=Palacios |title=Kepler equation and accelerated Newton method
  |doi=10.1016/S0377-0427(01)00369-7 |journal=Journal of Computational and Applied Mathematics |year=2002
  |volume=138 |issue=2 |pages=335–346 |bibcode=2002JCoAM.138..335P |doi-access=free }}
* {{cite journal |first1=John P. |last1=Boyd 
  |title=Rootfinding for a transcendental equation without a first guess: Polynomialization of Kepler's equation through Chebyshev polynomial equation of the sine
  |doi=10.1016/j.apnum.2005.11.010 |year=2007 |volume=57 |number=1 
  |journal=Applied Numerical Mathematics |pages=12–18 }}
* {{cite journal |first1=András |last1=Pál |title=An analytical solution for Kepler's problem
  |doi=10.1111/j.1365-2966.2009.14853.x |journal=Monthly Notices of the Royal Astronomical Society |year=2009 
  |volume=396 |number=3 |pages=1737–1742 |arxiv=0904.0324 |bibcode=2009MNRAS.396.1737P |doi-access=free }}
* {{cite journal |first1=Reza |last1=Esmaelzadeh |first2=Hossein |last2=Ghadiri 
  |title=Appropriate starter for solving the Kepler's equation
  |journal=International Journal of Computer Applications |year=2014 |volume=89 |number=7 |pages=31–38 
  |doi=10.5120/15517-4394 |bibcode=2014IJCA...89g..31E |doi-access=free }}
* {{cite journal |first1=Mathias |last1=Zechmeister |title=CORDIC-like method for solving Kepler's equation
  |journal=Astronomy and Astrophysics |volume=619 |year=2018 |pages=A128 |doi=10.1051/0004-6361/201833162 |arxiv=1808.07062 |bibcode=2018A&A...619A.128Z |doi-access=free }}

* [http://mathworld.wolfram.com/KeplersEquation.html Kepler's Equation at Wolfram Mathworld]

{{Johannes Kepler}}

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[[Category:Eponymous equations of physics]]
[[Category:Johannes Kepler]]
[[Category:Orbits]]