The small dense stars rotate about their common centre of mass as a binary system, each with a period of `1` year. One star has mass double than that of the other, while mass of the lighter star is one-third the mass of the Sun. The distance between the two stars is `r` and the distance of the earth from the Sun is `R`, find the relation between `r` and `R`.

Answer

R

Explanation

They will rotate about their c.m. centripetal force wilbe provided by gravitational force of attraction.
`((2m_(e ))/(3))((d)/(3))((2pi)/(T'))^(2) = (G(2)/(3)m_(s)xx(m_(s))/(3))/(d^(2))`
`((d)/(3))((2pi)/(T'))^(2) = (Gm_(s))/(3d^(2))`
`T'^(2) = (4pi^(2)R^(3))/(Gm_(s))` ....(i)
For sun and Earth
`m_(e )R ((2pi)/(T,))^(2) = (Gm_(s))/(R^(2))xxm_(e ) rArr T^(2) = (4pi^(2)R^(3))/(Gm_(s))` ...(ii)
Since `T = T'` so `rArr d = R`