Question

Two planets (planet X and planet Y) orbit the same star with uniform circular motion. Planet X is three times as far from the star as planet Y.
How many times longer does it take planet X to orbit the star compared to how long it take planet Y to orbit the star?

• A. $3$ times as long
• B. $9$ times as long
• C. $27$ times as long
• D. More than $27$ times as long
• E. The relationship between period and orbital radius is such that it will not give us an integer result

Answer 1

The correct option is E The relationship between period and orbital radius is such that it will not give us an integer result

From Kepler's 3rd law of motion, $T\propto R^{\left(3/2\right)}$

$⟹$ $\frac{T_X}{T_Y}=(\frac{R_X}{R_Y}{)}^{\left(3/2\right)}$

Given : $R_X=3R_Y$

$\therefore$ $\frac{T_X}{T_Y}=(\frac{3R_Y}{R_Y}{)}^{\left(3/2\right)}=3^{\left(3/2\right)}=5.2$ (Not an integer)