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 will be provided by gravitational force of attraction.

`((2m_(s))/3)(d/3)((2pi)/(T'))^(2)=(G.2/3m_(s)xx(m_(s))/3)/(d^(2))`
`d/3((2pi)/(T'))^(2)=(Gm_(s))/(3d^(2))`
`T'^(2)=(4pi^(2)d^(3))/(Gm_(s))....(1)`
For sun and earth
`m_(e)R((rpi)/T)^(2)=(Gm_(s)m_(e))/(R^(2))`
`T^(2)=(4pi^(2)R^(3))/(Gm_(s))...(2)`
since `T=T'` so `impliesd=R`