State Newton’s law of gravitation, in full.
A star and a planet are isolated in space. The planet moves in a circular orbit of radius $R$ around the star, as shown in Fig. 1.1. The planet’s angular speed about the star is $\omega$. By considering the circular motion of the planet around the star of mass $M$, show that $\omega$ and $R$ are linked by the expression $R^{3}\omega^{2} = GM$, where $G$ is the gravitational constant. Explain your working.
The Earth travels around the Sun in a circular orbit of radius $1.5 \times 10^8\,\text{km}$. The mass of the Sun is $2.0 \times 10^{30}\,\text{kg}$. A distant star is observed to have a planet moving in a circular orbit around the star. The orbit radius is $6.0 \times 10^8\,\text{km}$ and the orbital period is $2.0\,\text{years}$. Use the expression in part (b) to calculate the mass of the star.