Laplace limit

{{Short description|Maximum eccentricity for which a power series for Kepler's equation converges}}

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:

:$E\; =\; M\; +\; \backslash sin(M)\; \backslash ,\; \backslash varepsilon\; +\; \backslash tfrac12\; \backslash sin(2M)\; \backslash ,\; \backslash varepsilon^2\; +\; \backslash left(\; \backslash tfrac38\; \backslash sin(3M)\; -\; \backslash tfrac18\; \backslash sin(M)\; \backslash right)\; \backslash ,\; \backslash varepsilon^3\; +\; \backslash cdots$

or in generalFinch (2003), §4.8Moulton (1914), §99

:$E\; =\; M\; \backslash ;+\backslash ;\; \backslash sum\_\{n=1\}^\{\backslash infty\}\; \backslash frac\{\backslash varepsilon^n\}\{2^\{n-1\}\backslash ,n!\}\; \backslash sum\_\{k=0\}^\{\backslash lfloor\; n/2\backslash rfloor\}\; (-1)^k\backslash ,\backslash binom\{n\}\{k\}\backslash ,(n-2k)^\{n-1\}\backslash ,\backslash sin((n-2k)\backslash ,M)$

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 the unique real solution of the transcendental equation{{cite web |url=https://mathworld.wolfram.com/LaplaceLimit.html |title=Laplace Limit |first=Eric W. |last=Weisstein |website=MathWorld |access-date=18 June 2024}}

:$\backslash frac\{x\backslash exp(\backslash sqrt\{1+x^2\})\}\{1+\backslash sqrt\{1+x^2\}\}=1$

A closed-form expression in terms of r-Lambert special function and an infinite series representation were given by István Mező.{{cite journal |first=István |last=Mező |title=New expressions for the Laplace Limit Constant |journal=Publ. Math. Debrecen |year=2022 |volume=101 |issue=3–4 |pages=491–496 |doi=10.5486/PMD.2022.9313 |url=https://publi.math.unideb.hu/load_doc.php?p=4074&t=pap |url-access=subscription}}