%I #10 Jun 25 2024 07:33:39
%S 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,
%T 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,
%U 8,2,0,3,1,4,8,4,0,8,7,2,1,7,3,0,9,3,8,5,7,3
%N Decimal expansion of the real root of 9*x^3 - 27*x^2 + 31*x - 5 = 0.
%C 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.
%C The orbital eccentricity of Mercury is 0.205630, which is greater than 0.1910588914..., so there is retrograde motion of the sun on Mercury.
%H Jianing Song, <a href="https://en.wikipedia.org/wiki/Talk:Astronomy_on_Mercury#When_is_there_retrograde_motion">When is there retrograde motion</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Astronomy_on_Mercury">Astronomy on Mercury</a>.
%e Equals 0.19105889149184691057...
%t First[RealDigits[Root[9*#^3 - 27*#^2 + 31*# - 5 &, 1], 10, 100]] (* _Paolo Xausa_, Jun 25 2024 *)
%o (PARI) solve(x=0, 1, 9*x^3 - 27*x^2 + 31*x - 5)
%K nonn,cons,easy
%O 0,2
%A _Jianing Song_, May 17 2024