Study the Non-Linear Stability of Non-Collinear Libration Point in the Restricted Three-Body Configuration When the Shapes of the Primaries are Taken as Heterogeneous and Finite-Straight Segment

The main focus of the present research work is to analyze the non-linear stability of the triangular equilibrium points \({{\mathcal{L}}_{4}}\) and \({{\mathcal{L}}_{5}}\) in the restricted three-body problem (R3BP). The condition of stability has been found out under the influence of the heterogeneous primary and a radiating finite-straight segment secondary and also under the effect by Coriolis as well as Centrifugal forces. This piece of research has been done by doing the normalization of the Hamiltonian in order to attained the Birkhoff’s normal form of the Hamiltonian, since normal forms of Hamiltonian are important to study the non-linear stability of equilibrium points. The conditions of KAM Theorem have been examined in the presence of resonance cases \(\omega _{1}^{'} = 2\omega _{2}^{'}\) and \(\omega _{1}^{'} = 3\omega _{2}^{'}\) and found that these conditions have been failed for three values of mass ratios \({{\mu }_{1}},\) \({{\mu }_{2}}\) and \({{\mu }_{3}}.\) Except these three values, \({{\mathcal{L}}_{4}}\) and \({{\mathcal{L}}_{5}}\) are stable in non-linear sense within the range of linear stability \(0 < \mu < {{\mu }_{c}},\) where \({{\mu }_{c}}\) is the critical value of mass parameter \(\mu .\) Consequently, in the presence of above mentioned purturbations the triangular equilibrium points are unstable for these three values of mass ratios.