Two stars of masses ${\mathrm{m}}_1 \mathrm{and} {\mathrm{m}}_2$ are parts of a binary star system. The radii of their orbits are ${\mathrm{r}}_1 \mathrm{and} {\mathrm{r}}_2$ respectively, measured from the centre of mass of the system. The magnitude of gravitational force ${\mathrm{m}}_1$ exerts on ${\mathrm{m}}_2$ is

The situation is as shown in the figure.
According to Newton's law of gravitation, gravitational force between two bodies of masses ${\mathrm{m}}_1 \mathrm{and} {\mathrm{m}}_2$ is

$\mathrm{F} = \frac{{\mathrm{Gm}}_1{\mathrm{m}}_2}{{\mathrm{r}}^2}$

where r is the distance between the two masses.

Here, $\mathrm{r} = {\mathrm{r}}_1 + {\mathrm{r}}_2$

$\therefore \mathrm{F} = \frac{{\mathrm{Gm}}_1{\mathrm{m}}_2}{{({\mathrm{r}}_1+{\mathrm{r}}_2)}^2}$