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 center 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

The time period of earth revolving around the sun is given by

$T=2\pi \sqrt{\frac{R^3}{G{M}_S}}$


Centre of mass of two stars lies at $x$ from $3M_S$, which is

$x=\frac{(3M_S\times 0)+(6M_S\times 9R)}{(3M_S+6M_S)}=6R$

Let $T^{\prime }$ be the time period of star having mass $3M_S$ and $v$ be its velocity.

$\frac{3M_S{v}^2}{6R}=\frac{G\left(3M_S\right)\left(6M_S\right)}{(9R{)}^2}$

$v=\frac{2}{3}\sqrt{\frac{G{M}_S}{R}}$

$\therefore T^{\prime }=\frac{2\pi \left(6R\right)}{v}=\frac{18\pi R\sqrt{R}}{\sqrt{G{M}_S}}=9\times 2\pi \sqrt{\frac{R^3}{G{M}_S}}$

$\Rightarrow T^{\prime }=9T$

$\therefore n=9$