Straight line solutions and their stability of libration points with oblateness primaries and circumbinary disc in the elliptic R3BP

This study investigates the dynamical behavior and linear stability of the collinear Lagrangian point L3L_3L3 in the elliptic restricted three-body problem under the influence of perturbative parameters including the semi-major axis aaa, orbital eccentricity eee, and the oblateness A1A_1A1 of the more massive primary. For fixed values of A= 0 .01, A2 = 0.022, Mb=0.01 and T=0.01, the location and stability of L3 were examined numerically over varying conditions. Results show that as the semi-major axis decreases from 0.90 to 0.60, the coordinate L3 shifts from –1.06706 to –0.964238, with the corresponding increase in the instability measure Ωξξo Ωηηo from –1.76868 to –2.25071. A similar trend is observed with increasing eccentricity from 0.10 to 0.40, where the coordinate of L3 shifts from –1.08021 to –1.02960, while the instability measure increases from –1.58050 to –2.08276. Additionally, increasing the oblateness A1 from 0.0001 to 0.2 causes the position of L3 to shift from –1.04760 to –1.09594, while the instability product increases significantly from –1.79718 to –2.19480. These results confirm that L3 is inherently unstable and increasingly sensitive to variations in orbital and physical parameters. The findings offer essential insights for mission planning and the stability analysis of celestial systems influenced by such perturbations.