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Talk:Astronomy on Mercury

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When is there retrograde motion

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Let T be the orbital period of a planet, e be the eccentricity and k be the spin-orbit resonance. When the planet reaches aphelion at a certain moment, define the longitude facing the sun to be 0. At moment t, the longitude facing the sun is the difference between the orbital revolution angle Θ and the rotation angle θ. We have θ = t/T/k, and Kepler's second law implies (see below), so the longitude facing the sun Δθ has . Retrograde motion occurs when changes its sign, namely when changes its sign. Note that the minimum value (obtained when the planet reaches aphelion) LHS is always negative. Since the maximum of 1−ecos Θ is 1+e (when the planet reaches perihelion), there is retrograde motion if and only if , or e is greater than the unique real root of .

k Minimum eccentricity to see retrograde motion
1:1 0 (always exists)
3:2 0.1910588914... (the eccentricity of Mercury is greater than this value)
2:1 0.3106016499...
5:2 0.3936079089...
3:1 0.4552115689...

Calculating /dt: Suppose that the equation of the planet when orbiting sun is r = 1/1-ecos Θ, then Kepler's second law implies . 129.104.241.193 (talk) 23:08, 17 May 2024 (UTC)[reply]