A generalized framework for the collinear restricted four-body problem with a central dominant mass

This study extends the classical circular restricted three-body problem (CR3BP) by introducing a dominant central primary, forming a collinear restricted four-body problem (CR4BP) that better reflects the dynamics of real planetary systems. The model remains dynamically consistent and non-degenerate when the central mass parameter μ₀ lies in (½, 1) and the peripheral mass μ satisfies 0 < μ < ½ (1 - μ₀). It generalizes to the CR3BP by setting μ₀ = 0, recovering classical results. The system exhibits six libration points: four collinear and two symmetric non-collinear points forming an isosceles triangle with the peripheral primaries. Non-collinear points emerge via a saddle-node bifurcation at a critical μ = μ_c and as μ increases further within the range μ_c < μ < ½ (1 - μ₀), these points move away from the x-axis and gradually align closer to the y-axis, while remaining symmetric with respect to the x-axis. The stability analysis reveals that collinear libration points L₁, L₃ and L₄ are linearly unstable under all conditions while L₂ is stable in the interval 0 < μ < μ^* where μ^* is a critical threshold for L₂. The non-collinear points are linearly stable within a defined interval μ_c < μ < μ_c1. Finally, these results are applied to the Saturn-Janus-Epimetheus system to illustrate the model's practical relevance.