Laplace limit

From Wikipedia, the free encyclopedia

In mathematics, the Laplace limit is the maximum value of the eccentricity for which a solution to Kepler's equation, in terms of a power series in the eccentricity, 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 functions, but the Lagrange reversion theorem gives the solution as a power series in ε:

Laplace realized that this series converges for small values of the eccentricity, but diverges for any value of M other than a multiple of π if the eccentricity exceeds a certain value that does not depend on M. The Laplace limit is this value. It is the radius of convergence of the power series.

It is given by the solution to the equation:

See also

References

  • Finch, Steven R. (2003), "Laplace limit constant", Mathematical constants, Cambridge University Press, ISBN 978-0-521-81805-6 .

External links

Retrieved from "https://en.wikipedia.org/w/index.php?title=Laplace_limit&oldid=832695926"
This content was retrieved from Wikipedia : http://en.wikipedia.org/wiki/Laplace_limit
This page is based on the copyrighted Wikipedia article "Laplace limit"; it is used under the Creative Commons Attribution-ShareAlike 3.0 Unported License (CC-BY-SA). You may redistribute it, verbatim or modified, providing that you comply with the terms of the CC-BY-SA