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

Abstract

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 = μc and as μ increases further within the range μc < 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 μc < μ < μc1. Finally, these results are applied to the Saturn–Janus–Epimetheus system to illustrate the model’s practical relevance.