You have discovered the planet Zoltron. It is 20.0 times farther from the sun than earth (20.0...

Question:

You have discovered the planet Zoltron. It is 20.0 times farther from the sun than earth (20.0 au). How long does it take this planet to go once around the sun?

Kepler's Third Law

Kepler's three laws of planetary motion are very important laws to understand the motions of the planets and their satellite around the star, according to Kepler's third law the square of the time period T of a planet around the sun is directly proportional to the cube of the mean distance r from the sun, Mathematically

{eq}\begin{align} T^2 \propto r^3 \end{align} {/eq}

Answer and Explanation:

Data Given

  • Distance between the sun and the Zoltron {eq}r = 20.0 \ \rm AU {/eq}

Using Kepler's third law

{eq}\begin{align} T^2 \propto r^3 \end{align} {/eq}

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

If the time T is measured in Earth Year, Mass is measured in terms of the mass of the sun and distance is measured in astronomical unit then

{eq}\begin{align} \frac{4\pi^2}{GM} = 1 \end{align} {/eq}

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

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

{eq}\begin{align} T = \left( 20.0 \ \rm AU \right ) ^{\frac{3}{2}} \end{align} {/eq}

{eq}\begin{align} \color{blue}{\boxed{ \ T = 89.44 \ \rm Years }} \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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