File:SolarSystem Radii and Period (math).svg

English: Log-log chart showing relation period and radius of planets in the solar system.

From the graph it can be deduced that

${\displaystyle \log _{10}T={\frac {3}{2}}\log _{10}R+c}$ where ${\displaystyle c}$ is a constant and ${\displaystyle R}$ and ${\displaystyle T}$ are shown on the chart.

This can be rearranged to give

${\displaystyle T^{2}=kR^{3}}$ where ${\displaystyle k}$ is a constant

By equating centripetal force and gravitiational force, it can be shown that

${\displaystyle T^{2}={\frac {4\pi ^{2}}{GM}}R^{3}}$ where ${\displaystyle G}$ is the universal gravitational constant and ${\displaystyle M}$ is the mass of the sun

thereby giving a theoretical expression for ${\displaystyle k}$.