Question

The distance between two stars of masses $3M_S$ and $6M_s$ is $9R$. Here $R$ is the mean distance between the centers of the Earth and the Sun, and $M_S$ is the mass of the Sun. The two stars orbit around their common centre of mass in circular orbits with period $nT$, where $T$ is the period of Earth's revolution around the Sun. The value of $n$ is ______.

Answer 1

Answer: 9

Both will revolve about common center of mass
$x=\frac{3M_s}{6M_s+3M_s}\times 9R=3R$ ... (i)
$\frac{G6M_s\times 3M_s}{(9R{)}^2}=6M_s\left({\omega }^{2}x\right)$ .... (ii)
Solving equation (i) and (ii) we get
${\omega }^2=\frac{G{M}_S}{81R^3}$
$T^{\prime 2}=\frac{4{\pi }^2{R}^3}{G{M}_S}\times 81$ ... (iii)
For the motion of earth around Sun
$T^2=\frac{4{\pi }^2{R}^3}{G{M}_s}$ .... (iv)
From (iii) and (iv)
$T^{\prime 2}=81T^2$
$T^{\prime }=9T$