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 each other. Show that the period of each of the
three stars is given by
T = 2pi* sqrt(r^3/g(M+m/sqrt(3)))

Question

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 each other. Show that the period of each of the
three stars is given by
T = 2pi* sqrt(r^3/g(M+m/sqrt(3)))

anonymous · Answer

Consider that the vector sum of gravitational forces on any of the orbiting stars is always pointing radially inward. Use the superposition principle to add the radial components and to show that the tangential components cancel.