A372948 Decimal expansion of the real root of 9*x^3 - 27*x^2 + 31*x - 5 = 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.

Links

Table of n, a(n) for n=0..89.

Jianing Song, When is there retrograde motion.

Wikipedia, Astronomy on Mercury.

Index entries for algebraic numbers, degree 3

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

Adjacent sequences: A372945 A372946 A372947 * A372949 A372950 A372951

Keyword

nonn,cons,easy

Author

Jianing Song, May 17 2024

Status

approved