# How can I use Kepler's law of harmonies to predict the time Mars takes to orbit the sun?

May 22, 2018

$\approx 1.87$ years

#### Explanation:

Kepler's law of harmonies (also known as Kepler's third law) states that the orbital time period $T$ is related to the mean distance $R$ from the sun by

${T}^{2} \propto {R}^{3}$

For Mars ${R}_{\text{mars" = 2.28times 10^11\ "m}}$,
while for earth, ${R}_{\text{earth}} =$1.50times 10^11\ "m"

Thus

(T_"mars"/T_"earth")^2 = (R_"mars"/R_"earth")^3 = (( 2.28times 10^11\ "m")/( 1.50times 10^11\ "m"))^3~~3.51#

Hence

${T}_{\text{mars"/T_"earth}} = \sqrt{3.51} \approx 1.87$