A230242 Decimal expansion of (25+3*sqrt(69))/2.

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

Offset

2,1

Comments

Minimum mass ratio required for stable L4 and L5 Lagrange points. Because the mass of the sun is about 333060 times the mass of the earth which is greater than 24.95993..., the sun-earth Lagrange points L4 and L5 are stable. Similarly, since the earth is about 81.3 times more massive than the moon, the earth-moon L4 and L5 points are stable.

A quadratic integer with minimal polynomial x^2 - 25x + 1. - Charles R Greathouse IV, Mar 06 2015

Note that the L1, L2, and L3 Lagrangian points are unstable regardless of mass ratio. - Charles R Greathouse IV, Mar 25 2018

References

Michel Capderou, Satellites: Orbits and Missions, Springer-Verlag France, 2005. See p. 120.

Marianne Freiberger and Rachel Thomas, Math Squared: 100 Concepts You Should Know, Metro Books, New York, 2016. See p. 197.

Hanspeter Schaub and John L. Junkins, Analytical Mechanics of Space Systems, 4th ed., AIAA, 2018. See p. 611, eq. (10.125).

Links

Table of n, a(n) for n=2..87.

Neil J. Cornish, The Lagrange points, 1998.

Minute Physics, The Trojan Test (2025).

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

Wikipedia, Lagrange point.

Index entries for algebraic numbers, degree 2.

Formula

Equals 1/A345449 - 1. - Amiram Eldar, Jan 03 2026

Example

24.959935794377112278876394117361238015348321373433483669148282553556...

Mathematica

First[RealDigits[(25 + 3*Sqrt[69])/2, 10, 100]] (* Paolo Xausa, Jun 18 2024 *)

Prog

(PARI) (25+3*sqrt(69))/2 \\ Charles R Greathouse IV, Oct 13 2013
(PARI) polrootsreal(x^2 - 25*x + 1)[2] \\ Charles R Greathouse IV, Jan 05 2016

Crossrefs

Cf. A345449.

Sequence in context: A377970 A269063 A161360 * A104654 A381494 A011182

Adjacent sequences: A230239 A230240 A230241 * A230243 A230244 A230245

Keyword

nonn,cons

Author

Charles R Greathouse IV, Oct 13 2013

Status

approved