A binary star system consists of two stars A | vedclass

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(C) In a binary star system rotating about their common centre of mass,both stars have the same angular velocity $\omega$ at any instant.The distance

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$5$

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(C) In a binary star system rotating about their common centre of mass,both stars have the same angular velocity $\omega$ at any instant.The distance of star $A$ from the centre of mass is $r_A$ and that of star $B$ is $r_B$.By the definition of the centre of mass,$M_A r_A = M_B r_B$,which implies $r_A / r_B = M_B / M_A$.The angular momentum of a particle about the centre of mass is given by $L = mvr = mr^2\omega$.Thus,the angular momentum of star $A$ is $L_A = M_A r_A^2 \omega$ and for star $B$ is $L_B = M_B r_B^2 \omega$.The ratio of angular momenta is $L_A / L_B = (M_A r_A^2) / (M_B r_B^2) = (M_A r_A) r_A / (M_B r_B) r_B$.Since $M_A r_A = M_B r_B$,the ratio simplifies to $L_A / L_B = r_A / r_B = M_B / M_A$.Given $M_A = 2.2 M_S$ and $M_B = 11 M_S$,the ratio is $L_A / L_B = 11 / 2.2 = 5$.

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