Question

A research team has discovered that a moon is circling a planet of our solar system: The moon orbits the planet once every 7 hours on a nearly circular orbit in a distance R of 48000 km from the centre of the planet. Unfortunately, the mass m of the moon is not known. Use Newton’s law of gravitation with G = 6.67 · 10⁻¹¹ m³ /(kg·s² ) to approach the following questions: F = G · mM / R²

(a) Based on the observations, determine the total mass M of the planet.

(b) Which moon and planet of our solar system is the team observing? (Use literature.)

Answer 1

(a) Determine the total mass M of the planet:

$$F=ma_c=G\frac{Mm}{R^2},\\\space\\ a_c=G\frac{M}{R^2}.\\\space\\ a_c=\frac{v^2}{R}=\frac{(2\pi R/T)^2}{R}=\frac{4\pi^2R}{T^2}.\\\space\\ \frac{4\pi^2R}{T^2}=G\frac{M}{R^2},\\\space\\ M=\frac{4\pi^2R^3}{GT^2}=1.03\cdot10^{26}\text{ kg}.$$

(b) That is Neptune and the moon is Naiad with the semi-major axis of 48224 km.