Question

A certain triple-star system consists of two stars, each of mass m, revolving in the same circular orbit of radius r around a central star of mass m (fig. 13-54).The two orbiting stars are always at opposite ends of a diameter of the orbit. Derive an expression for the period of revolution of the stars

Answer 1

The centripetal force is obtained from the net gravitational force acting on the system.

Thus,

$F_{net} = F_c$

Substitute,

$\frac {Gm}{r^2}[2M+\frac {m}{4}] = \frac {4\pi^2 mr }{T^2}$

Where G = Universal Gravitational Constant = $6.67408 \times 10^{-11}m^3 kg^{-1}s^{-2}$

T = Period of revolution of the stars

So,

$T = \frac {\sqrt {16\pi^2 r^3}}{G[8M+m]}$

Therefore, the expression for the period of revolution of stars is $\frac {\sqrt {16\pi^2 r^3}}{G[8M+m]}$