An asymptotic solution for the Trojan case of the plane elliptic restricted problem of three bodies

The planar motion of a Trojan asteroid is considered within the framework of the elliptic restricted three-body problem. The solution is derived asymptotically to second order taking the square root of the Jupiter-Sun mass ratio and the orbital eccentricity of Jupiter as first order quantities. The results are given in explicit form for the coordinates as functions of the true anomaly of Jupiter including both short and long periodic terms resulting from the orbital accentricity of Jupiter.