Solutions to equilibrium sequences of irrotational binary polytropic stars in Newtonian gravity are expanded in a power of \epsilon = a _ { 0 } / R , where R and a _ { 0 } are the orbital separation of the binary system and the radius of each star for R = \infty .
For each order of \epsilon , we should solve ordinary differential equations for arbitrary polytropic indices n .
We show solutions for polytropic indices n = 0.5 , 1 , 1.5 and 2 up to \epsilon ^ { 6 } orders .
Our semi-analytic solutions can be used to check the validity of numerical solutions .