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Does Kepler's law only apply to planets? If so why doesn't it apply to other objects undergoing circular motion?

By Kepler's law I'm referring to $T^2 \propto r^3$

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you write: "By Kepler's law I'm referring to $T^2 \propto r^2$". Do you mean $t^2 \propto r^3$ (Keppler's third law)? – Johannes Jan 3 '13 at 15:51
Yes sorry, that was a typo. I really should triple check my questions when I'm on an iPad... – Todd Davies Jan 3 '13 at 15:58
up vote 4 down vote accepted

Kepler's third law, the so-called harmonic law, was published by Johannes Kepler in 1619, ten year after he published his first two laws. Not long thereafter, in 1643, the Flemish astronomer Godefroy Wendelin noted that Kepler's third law not only applies to the planets, but also to the moons of Jupiter.

Now we know that this law describes the motion of any two bodies in gravitational orbit around each other. In fact, all you need really is an inverse square central attractive force between the two bodies. This law still holds approximately also if there are other bodies present, as long as their gravitational influence on the smaller of the two bodies is minor compared to the gravity of the larger of the two bodies.

Stretching this law to cover Coulombic systems is perfectly ok. Kepler's third law is also observed in Rydberg atoms.

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Thank you, I just didn't know if it applied to other instances circular motion, such as a plane doing loop the loop etc. – Todd Davies Jan 3 '13 at 16:18
Since all you need is an inverse square central attractive potential between two bodies, then it applies to any two oppositely charged particles as well, because the Coulomb Force is essentially the same form as the Gravitational Force. – QEntanglement Jan 3 '13 at 16:29
Correct. In Rydberg atoms one observes the square of the Kepler frequency to be inversely proportional to the cube of the radius of the orbit. Have added a remark to that effect. – Johannes Jan 3 '13 at 16:44
Does this apply to parabolic and hyperbolic orbits as well? – user3932000 May 4 '15 at 13:22

Kepler's second law (equal areas in equal times) is a consequence of angular momentum conservation, and so can be generalised to many other systems. Kepler's first and third laws (that the orbits are ellipses and the period vs. semi-major axis relationship) are consequences of the inverse square law for gravity and the two-body approximation. They would apply to any two bodies with an inverse square central force where disturbing influences can be neglected.

EDIT: I should add that you need to generalise the ellipses to arbitrary conic sections to cover all cases of the inverse-square law. Ellipses describe bound states, but you can also get parabolic and hyperbolic "fly-by" trajectories. There is also the degenerate case of straight lines when there is no lateral velocity at all and the bodies just fall straight towards each other.

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It is a result of Newton's Laws of Motion that works under a specific set of conditions and would apply to any object put under similar conditions. The key is that the moving body is subject to one major attractive force that is much larger than any other force and there is enough initial velocity tangent to that force to prevent collision. Planetary motion is the most obvious example of this motion as most moving objects you see day to day are subject to alot more forces than just one and definitely are not above earth's escape velocity.

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