Question

A planet has two small satellites in circular orbits around the planet. The first satellite has a period 12.0 hours and an orbital radius 6.00 x 10⁷. The second planet has a period of 16.0 hours. What is the orbital radius of the second satellite?

Answer 1

The expression for the time period of the satellite is

$$ T=2 \pi \sqrt{\frac{r^{3}}{G M}} $$

From this equation, we can written as

$$ \left(\frac{T^{\prime}}{T}\right)^{2}=\left(\frac{r^{\prime}}{r}\right)^{3} $$

The orbital radius of second planet is

$$ r^{\prime}=r\left(\frac{T^{\prime}}{T}\right)^{\frac{2}{3}}=\left(6.00 \times 10^{7}\right)\left(\frac{16.0 \mathrm{~h}}{12.0 \mathrm{~h}}\right)^{\frac{2}{3}}=7.27 \times 10^{7} $$