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%I #41 Jul 15 2021 03:02:59
%S 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,
%T 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,
%U 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
%N Decimal expansion of Gascheau's value, which is defined as the smaller solution of 27*x*(1 - x) = 1.
%C 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.
%C 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
%H B. Sicardy, <a href="https://hal.archives-ouvertes.fr/hal-00552502">Stability of the triangular Lagrange points beyond Gascheau's value</a>, Celest. Mech. Dyn. Astr. (2010) 107:145-155.
%F Equals (1-sqrt(23/27))/2.
%F Equals 1/(25+1/(1+1/(23+1/(1+1/(23+1/(1+1/(23+1/(1+1/(23+..))))))))) - _Peter Luschny_, Jul 14 2021
%e 0.038520896504551397078652...
%p with(NumberTheory): Digits:=200:
%p evalf(Value(ContinuedFraction([[0, 25], [1, 23]]))); # _Peter Luschny_, Jul 10 2021
%t First[RealDigits[N[(1-Sqrt[23/27])/2,106]]] (* _Stefano Spezia_, Jun 20 2021 *)
%K nonn,cons
%O -1,1
%A _Donghwi Park_, Jun 19 2021
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