Laplace limit

In mathematics, the Laplace limit is the maximum value of the 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 functions, but the Lagrange reversion theorem yields the solution as a power series in ε:

:$E\; =\; M\; +\; sin(M)\; ,\; varepsilon\; +\; frac12\; sin(2M)\; ,\; varepsilon^2\; +\; left(\; frac38\; sin(3M)\; -\; frac18\; sin(M)\; ight)\; ,\; varepsilon^3\; +\; cdots$

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.

ee also

*Orbital eccentricity

References

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External links

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