Question

The average distance of the planet mercury from the sun is 0.39 times the average distance of the earth from the sun. How long is a year on mercury in units of earth years?

Answer 1

Answer:

$T_1=0.24y$

Explanation:

Using Kepler's third law, we can relate the orbital periods of the planets and their average distances from the Sun, as follows:

$(\frac{T_1}{T_2})^2=(\frac{D_1}{D_2})^3$

Where $T_1$ and $T_2$ are the orbital periods of Mercury and Earth respectively. We have $D_1=0.39D_2$ and $T_2=1y$. Replacing this and solving for

$T_1^2=T_2^2(\frac{D_1}{D_2})^3\\T_1^2=(1y)^2(\frac{0.39D_2}{D_2})^3\\T_1^2=1y^2(0.39)^3\\T_1^2=0.059319y^2\\T_1=0.24y$