Straight Line Solutions and their Stability of Libration Points with Oblateness Primaries and Circumbinary Disc in the Elliptic R3BP

Abstract

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, M_b=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  ​ 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.