Two stars, each of mass $M$, at a distance $d$ from each other are rotating in a circular path around their center of mass. An asteroid with mass $m$ $(m<<M)$ moves along the axis of the system perpendicular to the plane of the stars’ orbit. Assuming that at a certain point on the z-axis, the net gravitational force is zero, for a small displacement of the asteroid from that point, the time period of its simple harmonic motion is $T_p $. The time period of motion of the stars is $T_s$. Determine $\frac{T_s}{T_p} $. Hint: For small $x$, $(1+x)^n \cong 1+nx $.

Dhaka Regional-2023, Category : D