Consider a binary star system of star A and star B with masses m _...

1) From the centre of mass, we get

${M}_{1}{R}_{1} = {M}_{2}{R}_{2} - - - - - - 1$

2) Force acting on both starts i.e. gravitational force and centripetal force are balanced.

$\dfrac{{GM}_{1}{M}_{2}}{{R}^{2}} = \dfrac{{M}_{1}{V}_{1}^{2}}{{R}_{1}} = \dfrac{{M}_{1}{V}_{1}^{2}}{{R}_{1}} - - - - - - 2$

As we know, ${V}_{1} = \dfrac{2{\pi R}_{1}}{{T}_{1}} \text{ and } {V}_{2} = \dfrac{2{\pi R}_{2}}{{T}_{2}}$

Substituting equation 3 in equation 2, we get

$\dfrac{{M}_{1}{R}_{1}}{{\left({T}_{1}\right)}^{2}} = \dfrac{{M}_{1}{R}_{2}}{{\left({T}_{2}\right)}^{2}} - - - - - - 4$

From equation 1, we conclude in eqution 4

${\left({T}_{1}\right)}^{2} = {\left({T}_{2}\right)}^{2}or{T}_{1} = {T}_{2}$