State Newton’s law of gravitation for point masses.
Jupiter and one of its moons, Io, may each be modelled as a uniform sphere isolated in space. Jupiter has radius $R$ and mean density $\rho$. Io has mass $m$ and follows a circular orbit round Jupiter of radius $nR$, as shown in Fig. 1.1. The time for Io to complete one orbit of Jupiter is $T$. Show that $T$ is related to the mean density $\rho$ of Jupiter by the expression $\rho T^2 = \frac{3\pi n^3}{G}$, where $G$ is the gravitational constant.
Jupiter has radius $R$ of $7.15 \times 10^4\,\text{km}$, and the distance between the centres of Jupiter and Io is $4.32 \times 10^5\,\text{km}$. The period $T$ of Io’s orbit is $42.5\,\text{hours}$. Calculate the mean density $\rho$ of Jupiter.
The Earth has a mean density of $5.5 \times 10^3\,\text{kg m}^{-3}$. It is described as a planet made of rock. By referring to your answer in (i), comment on Jupiter’s possible composition.