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A345449
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Decimal expansion of Gascheau's value, which is defined as the smaller solution of 27*x*(1 - x) = 1.
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0
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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
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OFFSET
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-1,1
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COMMENTS
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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
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LINKS
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FORMULA
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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
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EXAMPLE
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0.038520896504551397078652...
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MAPLE
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with(NumberTheory): Digits:=200:
evalf(Value(ContinuedFraction([[0, 25], [1, 23]]))); # Peter Luschny, Jul 10 2021
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MATHEMATICA
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First[RealDigits[N[(1-Sqrt[23/27])/2, 106]]] (* Stefano Spezia, Jun 20 2021 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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