A345449 Decimal expansion of Gascheau's value, which is defined as the smaller solution of 27*x*(1 - x) = 1.

3, 8, 5, 2, 0, 8, 9, 6, 5, 0, 4, 5, 5, 1, 3, 9, 7, 0, 7, 8, 6, 5, 2, 0, 6, 9, 7, 2, 7, 3, 6, 1, 5, 5, 4, 9, 8, 7, 0, 9, 9, 2, 0, 8, 3, 9, 1, 3, 5, 2, 4, 5, 6, 6, 9, 8, 2, 1, 1, 7, 5, 7, 2, 7, 5, 6, 8, 9, 7, 2, 0, 3, 6, 5, 3, 8, 0, 4, 6, 8, 1, 1, 8, 4, 7, 7, 8, 6, 0, 6, 5, 3, 7, 5, 7, 9, 4, 1, 6, 5, 1, 9, 4, 3, 6, 6

Offset

-1,1

Comments

Gascheau's value is the maximum mass fraction of the second largest mass in a restricted three-body problem with stable rotating equilateral configuration. Named after French scientist Gabriel Gascheau. Also called Routh's value. If the mass fraction of the second massive object is lower than Gascheau's value, the trojan points L4 and L5 are stable for the zero-mass object.

The Gascheau value G arises in a simplified model of the three body problem. The simplifications are: the mass of the lightest body is negligible ("the restricted three body problem") and the orbits occur on a two-dimensional plane ("the Euler model"). This is "the circular restricted three body problem". In such a system the stability of the Lagrangian points L4 and L5 depends on the mass ratio of the primary masses M1, M2, letting M3 = 0. Assuming M1 > M2 the Lagrangian points are stable only when M2/(M1 + M2 ) < G. - Peter Luschny, Jul 14 2021

References

Christian Marchal, The Three-body Problem, Elsevier, 1990. See pp. 49-51, 62, 227, 322.

Archie E. Roy, Orbital Motion, 4th ed., IOP Publishing, 2005. See p. 130.

Hanspeter Schaub and John L. Junkins, Analytical Mechanics of Space Systems, 4th ed., AIAA, 2018. See p. 611, eq. (10.123).

Links

Table of n, a(n) for n=-1..104.

Gabriel Gascheau, Mouvements relatifs d'un système de corps, Thèse de mécanique, présentée à la Faculté des sciences de Paris, 1843.

Bart Oldeman, Analysis of resonances in the three body problem using planar reduction, Master's Thesis, University of Groningen, 1998. See p. 38.

Edward John Routh, On Laplace's three particles, with a supplement on the stability of steady motion, Proceedings of the London Mathematical Society, Vol. s1-6, Issue 1 (1874), pp. 86-97.

B. Sicardy, Stability of the triangular Lagrange points beyond Gascheau's value, Celest. Mech. Dyn. Astr. (2010) 107:145-155.

Index entries for algebraic numbers, degree 2.

Formula

Equals (1-sqrt(23/27))/2.

Equals 1/(25+1/(1+1/(23+1/(1+1/(23+1/(1+1/(23+1/(1+1/(23+..))))))))) - Peter Luschny, Jul 14 2021

Equals 1/(1 + A230242). - Amiram Eldar, Jan 03 2026

Example

0.038520896504551397078652...

Maple

with(NumberTheory): Digits:=200:
evalf(Value(ContinuedFraction([[0, 25], [1, 23]]))); # Peter Luschny, Jul 10 2021

Mathematica

First[RealDigits[N[(1-Sqrt[23/27])/2, 106]]] (* Stefano Spezia, Jun 20 2021 *)

Crossrefs

Cf. A230242.

Sequence in context: A131653 A011239 A261644 * A348026 A200699 A113813

Adjacent sequences: A345446 A345447 A345448 * A345450 A345451 A345452

Keyword

nonn,cons

Author

Donghwi Park, Jun 19 2021

Status

approved