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%I #15 Jun 18 2024 03:25:00
%S 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,
%T 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,
%U 3,7,7,5,5,0,1,3,4,7,2,7,3,6,0,8,0,1
%N Decimal expansion of (25+3*sqrt(69))/2.
%C 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.
%C A quadratic integer with minimal polynomial x^2 - 25x + 1. - _Charles R Greathouse IV_, Mar 06 2015
%C Note that the L1, L2, and L3 Lagrangian points are unstable regardless of mass ratio. - _Charles R Greathouse IV_, Mar 25 2018
%H Neil J. Cornish, <a href="http://wmap.gsfc.nasa.gov/media/ContentMedia/lagrange.pdf">The Lagrange points</a>, 1998.
%H <a href="/index/Al#algebraic_02">Index entries for algebraic numbers, degree 2</a>
%e 24.959935794377112278876394117361238015348321373433483669148282553556...
%t First[RealDigits[(25 + 3*Sqrt[69])/2, 10, 100]] (* _Paolo Xausa_, Jun 18 2024 *)
%o (PARI) (25+3*sqrt(69))/2 \\ _Charles R Greathouse IV_, Oct 13 2013
%o (PARI) polrootsreal(x^2 - 25*x + 1)[2] \\ _Charles R Greathouse IV_, Jan 05 2016
%K nonn,cons
%O 2,1
%A _Charles R Greathouse IV_, Oct 13 2013
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