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A372948
Decimal expansion of the real root of 9*x^3 - 27*x^2 + 31*x - 5 = 0.
0
1, 9, 1, 0, 5, 8, 8, 9, 1, 4, 9, 1, 8, 4, 6, 9, 1, 0, 5, 7, 8, 5, 7, 9, 0, 8, 1, 7, 4, 5, 6, 7, 3, 6, 4, 8, 9, 1, 8, 1, 9, 5, 6, 1, 7, 3, 5, 3, 4, 0, 4, 3, 3, 6, 3, 1, 4, 9, 5, 4, 4, 2, 5, 0, 4, 9, 5, 7, 8, 2, 0, 3, 1, 4, 8, 4, 0, 8, 7, 2, 1, 7, 3, 0, 9, 3, 8, 5, 7, 3
OFFSET
0,2
COMMENTS
Minimum orbital eccentricity of a planet orbiting the sun with a 3:2 spin-orbit resonance (like Mercury) that allows one on it to see retrograde motion of the sun. In general, for a planet with a spin-orbit resonance k, there is retrograde motion if and only if the orbital eccentricity is greater than the unique real root of x^3 - 3*x^2 + (3 + 1/k^2)*x - (1 - 1/k^2) = 0.
The orbital eccentricity of Mercury is 0.205630, which is greater than 0.1910588914..., so there is retrograde motion of the sun on Mercury.
EXAMPLE
Equals 0.19105889149184691057...
MATHEMATICA
First[RealDigits[Root[9*#^3 - 27*#^2 + 31*# - 5 &, 1], 10, 100]] (* Paolo Xausa, Jun 25 2024 *)
PROG
(PARI) solve(x=0, 1, 9*x^3 - 27*x^2 + 31*x - 5)
CROSSREFS
Sequence in context: A248414 A176980 A227817 * A102209 A222131 A156546
KEYWORD
nonn,cons,easy
AUTHOR
Jianing Song, May 17 2024
STATUS
approved