does Keplers third law work if one were to, in some way use the semi minor axis instead of the semi major axis? and does the difference between the arithmetic, geometric and harmonic means play any role?
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1If b is semi-minor, it won't be $P \not\propto b^3$ because semi-major and semi-minor are not linearly related. – Kornpob Bhirombhakdi May 05 '18 at 14:00
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Perhaps Wikipedia's page on Semi major and semi minor axes will help. – StephenG - Help Ukraine May 05 '18 at 16:18
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1@KornpobBhirombhakdi I think that should be P squared, but otherwise, spot on. A formula could still probably be worked out though. – userLTK May 05 '18 at 17:12
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No, the relationship that Kepler discovered was between the length of the long axis of the ellipse, and the period of the planet.
It is possible for a comet with a very long skinny orbit, and an asteroid with a nearly circular orbit to have the same minor axis length, but obviously, the comet will have a much longer orbital period. For example, Halley's comet has a semiminor axis of about 3.5AU. An asteroid in a circular orbit with the same semi-minor axis would have a period of about 6 years, compared to Halley's period of 76 years.
If you know the semi-minor axis, and the perihelion distance (for example), then it would be possible to work out the semi-major axis, and the relative orbital period, but you would need more information than just the semi-minor axis.
James K
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With those 3 means the semimajor axis, semiminor axis, and semilatus rectum can be calculated from the perihelion and aphelion distances. See https://en.wikipedia.org/wiki/Kepler%27s_laws_of_planetary_motion#First_law_of_Kepler – PM 2Ring May 05 '18 at 21:20