Question

The distance between two stars of masses $3 M _{ S }$ and $6 M _{ s }$ is $9 R$. 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

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