Keplerian origins of the inverse square law

A more verbose title for this post might be something like “Why Kepler’s 3rd Law implies that gravitation is an inverse square law.” Kepler’s 3rd Law states that, in regards to planetary orbits, the square of the period is proportional to the cube of the radius (or more properly the semi-major axis)

\displaystyle \tau^2 \sim r^3

This proportionality becomes an equivalence when solar units are used, ie. years, solar masses, and astronomical units. Now, the angular velocity of a particle traveling in a circular orbit is given by

\displaystyle \dot{\theta} =\omega = \frac{2 \pi}{\tau} \sim \frac{1}{\tau}

While if you recall from mechanics the acceleration of such a particle is directed towards the center of the circle with magnitude \displaystyle a = \omega^2 r. We know from Newton’s 2nd Law that the force is proportional to the acceleration, and it is the nature of this force that we’d like to know, ie. how does it vary with the radius?

\displaystyle a = \omega ^2 r \sim F(r) \Rightarrow \omega ^2 \sim \frac{ F(r) }{r}

While we also have from before

\displaystyle \omega^2 \sim \frac{1}{\tau^2} \sim \frac{F(r)}{r}

With this and Kepler’s 3rd Law, we clearly see that in order to make the dimensions match we must have

\displaystyle F(r) \sim \frac{1}{r^2}

being the inverse square law of Newtonian gravitation that we desire. We see in summary that Kepler’s 3rd law which relates the period and “radius” of a planetary orbit directly enforces that the law of universal gravitation must be an inverse square law with respect to the “radius” of the gravitational orbit.

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