Question

if a planets distance from the sun is tripled, how does its orbital period change?
a it becomes 5.2 times longer.
b it becomes 3 times longer.
c it becomes 27 times longer.
d it becomes 9 times longer.

Explanation:

Step1: Recall Kepler's third law

$T^{2}\propto r^{3}$, where $T$ is the orbital - period and $r$ is the distance from the sun. Let the initial distance be $r_1$ and the initial period be $T_1$, so $T_1^{2}=k r_1^{3}$. When the distance is tripled, $r_2 = 3r_1$. Let the new period be $T_2$, then $T_2^{2}=k r_2^{3}$.

Step2: Substitute $r_2 = 3r_1$ into the new - equation

$T_2^{2}=k(3r_1)^{3}=k\times27r_1^{3}$. Since $T_1^{2}=k r_1^{3}$, we can write $T_2^{2}=27T_1^{2}$. Then $T_2=\sqrt{27}T_1\approx5.2T_1$.

Answer:

A. It becomes 5.2 times longer.