Expansion of Kepler's 3rd Law of Planetary Motion

Kepler's 3rd Law of Planetary Motion

Kepler's 3rd law of planetary motion, aka the law of harmony is that the period of a planet in the same solar system, squared, is proportional to the 3rd power of the orbit radius. Kepler proposed this law after years of calculating through Tycho's massive observation data. Newton later derived the formula from his law of gravity. 

 

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* in this article, orbits are assumed to be circles, for the calculations to be simple

 

Kepler's 3rd law of planetary motion

In the image above, the only force applied to the orbiting body(mass m) is gravity. Since the object is moving in a circle, the net force applied to it must equal the centripetal force. Therefore, F = GMm/R^2 = (mv^2)/R. Mulitplying (m*R) to each side results in: v = √(GM/R). The period T = (circumference)/(velocity) = 2πR/(GM/R)^(1/2). Squaring both sides leads to the conclusion that: T^2 = ((4π^2)/GM)*R^3. 

Expanding the fomula

Recently, I wondered how the law of harmony would change with different kinds of force. In the case of gravity, force is proportional to the reciprocal of R squared. Elasticy, on the other hand, is proportional to ∆x. What would the relation of period and orbit radius be with other forces?

 

I started by assuming a force porportional to the nth power of R. Then F = kR^n. (k is the proportional constant). Following the previous steps, I derived a simple equation of T and R:

Substituting (-2) to n and (GMm) to k, we get the law of harmony. For elasticy, n = 1 and T becomes 2π√(m/k), which is the period of an oscilating object attached to a spring.