Laplace limit

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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