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With enough epicycles, the motions of the planets can be accounted for in a geocentric model, but do the laws of gravity make such motions impossible? Did Newton's laws not only provide a simpler heliocentric alternative, but also completely rule out the possibility of geocentric motion?

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What do "geocentric" and "heliocentric" even mean when a frame of reference in which the sun stands still is just as valid a frame of reference as a frame in which the earth stands still? Granted, these won't be inertial frames. –  ACuriousMind Jan 8 at 21:37
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Note also the geocentrism predates Newton, so such invalidation wasn't really possible. –  Kyle Kanos Jan 8 at 21:42
    
The epicycle-based theory can be dismissed completely in the Popperian sense because Fourier analysis shows that one can always produce any orbit from them, so they have no predictive power whatsoever. –  Danu Mar 18 at 22:29

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They can these days.

Try explaining the behavior of interplanetary spacecraft in terms of a epicycle model. Just be aware that we can change the cycles and epicycles at will by running the rocket on the probe (or deploying a solar sail).

By contrast a model based on Newton's laws of mechanics and gravitation handles these situations smoothly.

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In principle in Newtonian mechanics the rest frame can be any of the bodies in a gravitational complex. The geocentric system is one possible rest frame and a one to one mathematical transformation exists going from a heliocentric to a geocentric system. It is when one introduces the concept gravitation, a theory that explains the orbits, that the heliocentric system simplifies and explains the epicycle motions.

Newtons laws assign forces that describe the motions mathematically. The geocentric model without masses has no explanation for the complicated paths of the celestial bodies except the geometric rules. A theory is by far better than a phenomenological fit to observations. Further more the expressions if one sticks to a geocentric Newtonian rest frame get complicated unnecessarily, as dmckee points out for satellites etc.

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