Non-Linear Stability of L ₄ in the Restricted Three-Body Problem When the Smaller Primary is a Finite Straight Segment Under Resonance Cases

In this paper, we have studied the non-linear stability of the triangular libration point L₄ in the restricted three-body problem when the smaller primary is a finite straight segment of length 2l under the presence of third and fourth order resonances. It is well known that most of the asteroids are irregularly shaped elongated bodies. We have taken these elongated bodies as a finite straight segment. We have used Markeev's theorem (1978) [1] to examine the non-linear stability for resonance cases 2:1 and 3:1. We have found that the triangular equilibrium solutions are always unstable in the third and fourth order resonance cases for 0 ≤ l < 0.05. We have applied the results to the following solar systems (i) Jupiter - 9 Metis (ii) Jupiter - 22 Kalliope (iii) Jupiter - 216 Kleopatra (iv) Jupiter 243 Ida (v) Jupiter - 433 Eros (vi) Jupiter - 951 Gaspra.