State what is meant by gravitational force.
A binary star system contains two stars $S_1$ and $S_2$, each moving in a circular orbit. The period of rotation of each star in the system is $T$. Earth-based observations of the binary star are shown in Fig. 1.1.
The stars $S_1$ and $S_2$ have the same angular velocity $\omega$ as they move about point $P$, as shown in Fig. 1.2. Point $P$ is a distance $x$ from the centre of star $S_1$. The period of rotation of the stars is $44.2$ years. Calculate the angular velocity $\omega$.
Using the forces acting on the two stars, show that the mass ratio is $\dfrac{\text{mass of } S_1}{\text{mass of } S_2} = \dfrac{d-x}{x}$.
The mass $M_1$ of star $S_1$ is given by $G M_1 = d^2(d-x)\omega^2$, where $G$ is the gravitational constant. The ratio in (ii) is $1.5$. Use the information from (b) and your result in (c)(i) to find the mass $M_1$.