en>Rjwilmsi: Journal cites, added 1 DOI using AWB (9887)

2014-01-25T18:44:22Z

Journal cites, added 1 DOI using AWB (9887)

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In [[mathematics]], the '''Laplace limit''' is the maximum value of the [[eccentricity (mathematics)|eccentricity]] for which the series solution to Kepler's equation converges. It is approximately

: 0.66274 34193 49181 58097 47420 97109 25290.

[[Kepler's equation]] ''M'' = ''E'' − ε sin ''E'' relates the [[mean anomaly]] ''M'' with the [[eccentric anomaly]] ''E'' for a body moving in an [[ellipse]] with eccentricity ε. This equation cannot be solved for ''E'' in terms of [[elementary function]]s, but the [[Lagrange reversion theorem]] yields the solution as a [[power series]] in ε:

:<math> E = M + \sin(M) \, \varepsilon + \tfrac12 \sin(2M) \, \varepsilon^2 + \left( \tfrac38 \sin(3M) - \tfrac18 \sin(M) \right) \, \varepsilon^3 + \cdots </math>

[[Pierre-Simon Laplace|Laplace]] realized that this series converges for small values of the eccentricity, but diverges when the eccentricity exceeds a certain value. The Laplace limit is this value. It is the [[radius of convergence]] of the power series.

==See also==
*[[Orbital eccentricity]]

== References ==
* {{Citation | last1=Finch | first1=Steven R. | title=Mathematical constants | chapter = Laplace limit constant | publisher=Cambridge University Press | isbn=978-0-521-81805-6 | year=2003}}.

== External links ==
* {{MathWorld|urlname=LaplaceLimit|title=Laplace Limit}}
* {{SloanesRef|sequencenumber=A033259}}

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[[Category:Orbits]]
[[Category:Mathematical constants]]
[[Category:Mathematical series]]

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en>Rjwilmsi: Journal cites, added 1 DOI using AWB (9887)