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

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

or in general<ref>Finch (2003), §4.8</ref><ref>Moulton (1914), §99</ref> <!-- maybe copy this formula in the article on Kepler's equation? -->

:<math> E = M \;+\; \sum_{n=1}^{\infty} \frac{\varepsilon^n}{2^{n-1}\,n!} \sum_{k=0}^{\lfloor n/2\rfloor} (-1)^k\,\binom{n}{k}\,(n-2k)^{n-1}\,\sin((n-2k)\,M) </math>

[[Pierre-Simon Laplace|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]]<ref>{{cite web |url=https://mathworld.wolfram.com/LaplaceLimit.html |title=Laplace Limit |first=Eric W. |last=Weisstein |website=MathWorld |access-date=18 June 2024}}</ref>
:<math>\frac{x\exp(\sqrt{1+x^2})}{1+\sqrt{1+x^2}}=1</math> 

A [[closed-form expression]] in terms of [[r-Lambert special function]] and an infinite series representation were given by István Mező.<ref>{{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}}</ref>

==History==
Laplace calculated the value 0.66195 in 1827. The Italian astronomer [[Francesco Carlini]] found the limit 0.66 five years before Laplace. [[Augustin-Louis Cauchy|Cauchy]] in the 1829 gave the precise value 0.66274.<ref>{{cite journal |first=Andrea |last=Sacchetti |date=November 2020 |title=Francesco Carlini: Kepler's equation and the asymptotic solution to singular differential equations |journal=Historia Mathematica |volume=53 |pages=1–32  |doi=10.1016/j.hm.2020.06.001 |arxiv=2002.02679 }}</ref>

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

== References ==
{{Reflist}}
* {{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 | url-access=registration | url=https://archive.org/details/mathematicalcons0000finc }}.
* {{Citation | last1=Moulton | first1=Forest R. | title=An Introduction to Celestial Mechanics | chapter=V. The Problem of Two Bodies | publisher=MacMillan | year=1914 | edition=2d | url=https://archive.org/details/introcelestial00moulrich }}.

== External links ==
* {{MathWorld|urlname=LaplaceLimit|title=Laplace Limit}}
* {{OEIS el|sequencenumber=A033259|name=Decimal expansion of Laplace's limit constant}}

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


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