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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.

Answer and Explanation:

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}


Learn more about this topic:

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Kepler's Three Laws of Planetary Motion

from Basics of Astronomy

Chapter 22 / Lesson 12
43K

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