Serway & Jewett — Universal Gravitation: Problem 36

A certain quaternary star system consists of three stars, each of mass \(m\), moving in the same circular orbit of radius \(r\) about a central star of mass \(M\). The stars orbit in the same sense and are positioned one-third of a revolution apart from one another. Show that the period of each of the three stars is given by \[ T = 2\pi \sqrt{\frac{r^3}{G(M + m/\sqrt{3})}} \]