A trinary star system which is a system of three orbiting around centre of mass of system has time peroid of 3 years, while the distance between any two stars of the system is 2 astronomical unit. All the three stars are identical and mass of the sun is M and total mass of this multiple star system is `(a)/(b)xxM` . Then find `(b)/(3)`? (a and b are lowest possible integers (I astronomical unit is equal to distance between sun and earth, time period of rotation of earth around sun is 1year)

Answer

3

Explanation

`T_(0)^(2)=(4pi^(2)R^(3))/(GM_s)`

`(2Gm^(2)sqrt(3))/(2a^(2))=(mw^(2)a)/sqrt(3)`
`w^(2)=(3Gm)/(a^(3))" "....(i)`
`T^(2)=(4pi^(2))/(w^(2))=(4pi^(2)a^(3))/(3Gm)`
Given`T=3T_(0) " " a = 2R`
`9T_(0)^(2)=(4pi^(2))/(3Gm)xx8R^(3)`
`(9xx4pi^(2)R^(3))/(GM_(s))=(8)/(3)xx(4pi^(2)R^(3))/(Gm)`
`m=(8M_(s))/(27)`
Total `mass=3m=(8)/(9)M_(s)`