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

M Javed Idrisi; Md Sanam Suraj; S Ershkov; Elbaz I Abouelmagd; Dawit Beza

doi:10.1038/s41598-025-20510-y

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

Sci Rep. 2025 Oct 17;15(1):36440. doi: 10.1038/s41598-025-20510-y.

Authors

M Javed Idrisi¹, Md Sanam Suraj², S Ershkov^{3

4

5}, Elbaz I Abouelmagd^{6

7}, Dawit Beza⁸

Affiliations

¹ Department of Sciences (Mathematics), Manav Rachna University, Faridabad, 121004, India.
² Department of Applied Science and Humanities, Government Engineering College, Khagaria, Bihar, 848203, India.
³ Plekhanov Russian University of Economics, Scopus number 60030998, Moscow, Russia.
⁴ Sternberg Astronomical Institute, M.V. Lomonosov's Moscow State University, 13 Universitetskij Prospect, Moscow, 119992, Russia.
⁵ MIREA-Russian Technological University, 78 Vernadsky Avenue, Moscow, 119454, Russia.
⁶ Faculty of Engineering, King Salman International University (KSIU), Eltour, South Sinai, Egypt.
⁷ Celestial Mechanics and Space Dynamics Research Group (CMSDRG), Astronomy Department, National Research Institute of Astronomy and Geophysics (NRIAG), Helwan, 11421, Cairo, Egypt.
⁸ Department of Construction, Technology and Management, College of Engineering, Mizan-Tepi University, Tepi, Ethiopia. dawitalemu@mtu.edu.et.

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 = 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.

Keywords: Collinear restricted four-body problem; Libration points; Linear stability; Saddle-node bifurcation; Saturn–Janus–Epimetheus system.