# Uranus makes one complete orbit of the Sun every 83.8 years. Calculate the radius of the orbit of...

## Question:

Uranus makes one complete orbit of the Sun every 83.8 years. Calculate the radius of the orbit of Uranus.

## Kepler's Third Law

Kepler's third law of planetary motion relates the orbital radius of a planet and the time period of the planet around its star and is mathematically stated as

{eq}\begin{align} T^2 = \frac{4\pi^2}{GM}r^3 \end{align} {/eq}

Where M is the mass of the star. and {eq}\frac{4 \pi^2}{GM} = 1 {/eq} if the time period is measured in earth years and radius is measured in the astronomical unit.

Data Given

• Time Period of the Uranus {eq}T = 83.8 \ \rm years {/eq}

Let R be the radius of the orbit, so using Kepler's third law

{eq}\begin{align} T^2 = \frac{4\pi^2}{GM}R^3 \end{align} {/eq}

As {eq}\frac{4 \pi^2}{GM} = 1 {/eq}

{eq}\begin{align} T^2=R^3 \end{align} {/eq}

{eq}\begin{align} R = \left ( T \right ) ^{\frac{2}{3}} \end{align} {/eq}

{eq}\begin{align} R = \left ( 83.8 \ \rm year \right ) ^{\frac{2}{3}} \end{align} {/eq}

{eq}\begin{align} \color{blue}{\boxed{ R = 19.15 \ \rm AU }} \end{align} {/eq}