Question:

If the mass of each of the members of a binary star were the same as that of the Sun, and their distance apart were equal to the Sun-Earth distance, what would be their period of revolution?

Step-by-step

The gravitational acceleration produced by the Sun at the position of the Earth is the same as that due to one of the stars at the position of the other. This is because, according to Newton’s law of gravitation, g (the gravitational force acting on a unit mass) depends only on the mass at the centre of attraction and the distance of the second body from it. These quantities are identical in the two systems.
Thus, the members of the binary star move with the same acceleration as the Earth but in an orbit of a radius only half that of the Earth’s orbit. This means that since $(a=r{{w}^{2}})$, the square of their angular velocity has to be twice as large as that of the Earth. The period of the binary star therefore equals that of the Earth around the Sun divided by ${\sqrt {2}}$, i.e $8{\frac {1} {2}}$ months.