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·s

2

) 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.)

Solve both and explain them properly ...

Answer 1

a)

$$F=ma_c=\frac{GmM}{R^2}\\a_c=\frac{GM}{R^2}=\frac{4\pi^2R}{T^2}$$

$$M=\frac{4\pi^2R^3}{GT^2}\\M=\frac{4\pi^2(4.8\cdot10^7)^3}{(6.67\cdot10^{-11})(7\cdot3600)^2}=1.03\cdot10^{26}kg$$

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