Consider a binary star system of star A and star B with masses m_A and m_B revolving in a circular orbit of radii r_A and r_B, respectively. If T_A and T_B are the time period of star A and star B respectively, then

As per question, a binary star system of two stars A and B with masses m_A and m_B are revolving in a circular orbit of radii r_A and r_B, respectively.

It means both stars A and B will have same angular velocity because to remain perfectly align w.r.t. each other they need to cover equal angular displacement in equal time intervals.

Also, we know that

Time period,   $\mathrm{T}=\frac{2\mathrm{\pi }}{\mathrm{\omega }}$

$\because$ T_A and T_B are the time periods of stars A and B respectively, therefore, we can write

${\mathrm{T}}_{\mathrm{A}}=\frac{2\mathrm{\pi }}{\mathrm{\omega }}$             ...(i)

and ${\mathrm{T}}_{\mathrm{B}}=\frac{2\mathrm{\pi }}{\mathrm{\omega }}$         ...(ii)

$\therefore$ From Eqs. (i) and (ii), we can say

${\mathrm{T}}_{\mathrm{A}}={\mathrm{T}}_{\mathrm{B}}$