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Periods of Orbits

We can now explain (at least for circular orbits) Kepler's Third Law. The period  T  of an orbit is the circumference $2 \pi r$ divided by the speed of travel, v. Using the equation above for v in terms of r gives

$$T = {2 \pi r \over \sqrt{GM_E \over {\textstyle r} }}$$

$$\;\; = {2 \pi \over \sqrt{GM_E}} r^{\textstyle {3 \over 2}}$$

$$\hbox{\rm or} \qquad T^2 \propto r^3$$

as observed by Kepler. Newton explained why.