Stability of equilibrium points in modified elliptic restricted three-body problem with various perturbation sources

TL;DR

This work extends the elliptic restricted three-body problem by incorporating perturbations from radiation pressure, oblateness of the larger primary, elongation of the smaller primary modeled as a finite segment, and an external disk. Using a rotating-pulsating frame and an averaged potential, it locates the five Lagrange points ($L_1$–$L_5$) and analyzes their linear stability via eigenvalue analysis, showing that collinear points are unstable across the studied eccentricities, while non-collinear points are stable only within a critical eccentricity $e_c$ that shifts slightly with perturbations. The study reveals systematic displacements of all equilibrium points from classical positions due to the perturbations, with oblateness and radiation pressure playing major roles in the collinear shifts and the disk/segment terms influencing the non-collinear geometry. The results delineate stability boundaries as a function of eccentricity and perturbation parameters, offering insights applicable to Sun–asteroid–dwarf-planet systems with very small mass ratios and more realistic force models.

Abstract

This study examines the dynamics of the third body in an elliptic restricted three-body problem (ERTBP) framework, taking into account perturbations from radiation pressure, oblateness, and elongation of the primary bodies, as well as disk-like structures. The objectives are to determine the positions and stability of the equilibrium points, asses how these points shift under the influence of perturbations, and evaluate the dependence of their stability on the orbital eccentricity and perturbation parameters. The ERTBP model is modified to include a radiating, oblate primary body and an elongated secondary body modeled as a finite straight segment, alongside perturbations from a surrounding disk. The system's equations of motion are numerically solved using parameters from perturbed and classical cases. Equilibrium positions are computed over a range of eccentricities and perturbation values, and stability is analyzed using linearized equations and eigenvalue methods. In all cases, we have found three collinear ($L_1$, $L_2$, $L_3$) and two non-collinear ($L_4$, $L_5$) equilibrium points solutions. The inclusion of radiations, oblateness, elongation using a finite straight segment, and disk perturbation systematically displaces each equilibrium point from its classical location, with the magnitude and direction of the displacement varying with the perturbation parameter. Stability analysis confirms that the collinear points remain linearly stable under all tested conditions. Meanwhile, non-collinear points are stable under a specific condition. We investigate the stability boundary of these points as a function of orbital eccentricity and we found there is a critical range of eccentricity values within which stability is preserved.

Figures (4)

• Figure 1: Illustration of the ERTBP system used in this work.
• Figure 2: Effect of perturbation parameter variation on the position of equilibrium points. The original range scale for each perturbation parameter is $0$ to $1.6\times10^{-6}$ for $1-q$; $0$ to $2.6\times10^{-7}$ for $A$; $0$ to $7\times10^{-6}$ for $\ell$; and $0$ to $3\times10^{-7}$ for $M_b$. The ranges were then normalized to a scale of 0 to 1 for the $x$-axis. The $y$-axis is the positional difference between equilibrium points in the perturbed ERTBP and their location in the classical case. The last panel (f) is the zoomed-in version of $L_3$, $L_{4,x}$, and $L_{4,y}$ plots around $y=0$. For each variation of the perturbation parameter, the other parameters were set equal to $0$. We used $\mu=1\times10^{-9}$ and $e=0.2$.
• Figure 3: Plot of eccentricity versus characteristic roots ($\lambda_{1,2,3,4}$) for collinear equilibrium points ($L_1$, $L_2$, and $L_3$) for perturbed (a) and classical (b) cases. The details of the parameters used in each subfigure are as follows: (a) $\mu=2\times 10^{-9}$, $A=1-q=\ell=M_b=1\times10^{-5}$, and $T=0.11$; (b) $\mu=2\times10^{-9}$ and $A=1-q=\ell=M_b=T=0$. The bottommost panels show the zoomed-in version of $L_3$ eigenplot. The real and imaginary parts of the characteristic roots are shown by solid and dashed lines, respectively.
• Figure 4: Plot of eccentricity versus characteristic roots ($\lambda_{1,2,3,4}$) for non-collinear equilibrium points ($L_4$ and $L_5$) for perturbed (a) and classical (b) cases. Zoomed-in versions are shown in the bottom subfigures. The details of the parameters used in each subfigure are as follows: (a) $\mu=2\times 10^{-9}$, $A=1-q=\ell=M_b=1\times10^{-5}$, and $T=0.11$; (b) $\mu=2\times10^{-9}$ and $A=1-q=\ell=M_b=T=0$. The real and imaginary parts of the characteristic roots are shown by solid and dashed lines, respectively.