The collinear equilibrium points in the elliptic restricted synchronous three-body problem under an oblate primary and a dipole secondary

The study investigates the collinear positions and stability in the elliptic restricted synchronous three-body problem under an oblate primary and a dipole secondary for Luhman 16 and HD188753 systems. Our study has established four collinear equilibrium points [Formula: see text] which are greatly affected by the parameters under review. The collinear position [Formula: see text] move away and closer as the parameters increase and decrease respectively. For the collinear positions [Formula: see text] , we witnessed a uniform space movement away from the origin in the negative direction while [Formula: see text] seems to be moving closer to the origin from the negative part of the origin. We observed changes in the movements of the collinear positions [Formula: see text] as a result of the half distance between the mass dipoles and the oblateness of the primary for the problem under review. The movements away and closer to the origin from collinear positions do not change the status of the collinear points as they remain unstable and unchanged. It is also found that as the half distance between mass dipoles and oblateness of the primary increase, the region of stability of the collinear positions decreases for the aforementioned binary systems. The collinear equilibrium point [Formula: see text] is stable for the characteristic roots [Formula: see text] for Luhman 16 system. This is evidenced by at least one characteristic root, a positive real part and a complex root. The stability of collinear points in most cases are unstable for the stated binary systems in Lyapunov.