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(D) Let the masses of the two stars be $m_1 = m$ and $m_2 = 2m$,separated by a distance $l$.The distance of the centre of mass from the star of mass $

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$m^{-1/2}$

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(D) Let the masses of the two stars be $m_1 = m$ and $m_2 = 2m$,separated by a distance $l$.The distance of the centre of mass from the star of mass $m$ is $r_1 = \frac{m_2 l}{m_1 + m_2} = \frac{2ml}{3m} = \frac{2l}{3}$.The gravitational force between the stars provides the necessary centripetal force for rotation.$\frac{G(m)(2m)}{l^2} = m \omega^2 r_1 = m \left(\frac{2\pi}{T}\right)^2 \left(\frac{2l}{3}\right)$.$\frac{2Gm^2}{l^2} = m \frac{4\pi^2}{T^2} \frac{2l}{3}$.$\frac{Gm}{l^2} = \frac{4\pi^2 l}{3T^2}$.$T^2 = \frac{4\pi^2 l^3}{3Gm}$.$T = \sqrt{\frac{4\pi^2 l^3}{3Gm}} = 2\pi \sqrt{\frac{l^3}{3Gm}}$.Thus,$T \propto m^{-1/2}$.

$A$ double star is a system of two stars rotating about their centre of mass only under their mutual gravitational attraction. Let the stars have masses $m$ and $2m$ and their separation be $l$. Their time period of rotation about their centre of mass will be proportional to:

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