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In a double star system one of mass m(1)...

In a double star system one of mass `m_(1)` and another of mass `m_(2)` with a separation d rotate about their common centre of mass. Then rate of sweeps of area of star of mass `m_(1)` to star of mass `m_(2)` about their common centre of mass is

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The gravitational force between the masses provides the necessary centripetal force
`(Gm_1m_2)/d^2 =m_(1)r_(1)omega^2rarr(1)`
The distance of centre of mass from `m_1` is
`r_(1)=(m_(2)d)/(m_1+m_2)rarr(2)`
From (1) and (2) `(Gm_1m_2)/d^2 =(m_1m_2d)/(m_1+m_2).omega^2`
(or) `omega^2=(G(m_1+m_2))/d^3 " (or) " omega=sqrt((G(m_1+m_2))/d^3)`
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