Example

Question:

The planet Mars has two moons, phobos and delmos.
(i) Phobos has a period 7 hours, 39 minutes and an orbital radius of \(9.4 \times 10^3\,\text{km}\). Calculate the mass of Mars.
(ii) Assume that earth and mars move in circular orbits around the sun, with the martian orbit being 1.52 times the orbital radius of the earth. What is the length of the martian year in days?

Solution:

(i) Using \[ T^2 = \frac{4\pi^2}{GM} R^3 \implies M = \frac{4\pi^2 R^3}{G T^2} \] Substitute, \[ M_m = \frac{4 \times (3.14)^2 \times (9.4)^3 \times 10^{18}}{6.67 \times 10^{-11} \times (459 \times 60)^2} \] \[ = \frac{4 \times (3.14)^2 \times (9.4)^3 \times 10^{18}}{6.67 \times (4.59 \times 6)^2 \times 10^{-5}} \] \[ = 6.48 \times 10^{23}\,\text{kg} \] (ii) By Kepler's third law: \[ \frac{T_M^2}{T_E^2} = \frac{R_{MS}^3}{R_{ES}^3} \] \[ T_M = (1.52)^{3/2} \times 365 = 684\,\text{days} \]