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.
LINKS
Jianing Song, When is there retrograde motion.
Wikipedia, Astronomy 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
KEYWORD
AUTHOR
Jianing Song, May 17 2024
STATUS
approved