(a)[2]
State Newton’s law of gravitation.
(b)[3]
Several planets in the Solar System have multiple moons (satellites) in circular orbits around the planet. The planet and each moon may be treated as point masses. Show that the orbit radius $x$ of a moon is connected to the orbital period $T$ by $GM = \frac{4\pi^2 x^3}{T^2}$, where $G$ denotes the gravitational constant and $M$ is the mass of the planet. Explain your working.
(c)[4]
Neptune has eight moons, each moving in a circular orbit of radius $x$ and period $T$. The relationship with $T^2$ of $x^3$ for some of the moons is displayed in Fig. 1.1. Use Fig. 1.1 together with the expression in (b) to find the mass of Neptune.