<p>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, assess 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 (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">$L_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$L_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$L_3$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mn>3</mn> </msub> </math></EquationSource> </InlineEquation>) and two non-collinear (<InlineEquation ID="IEq4"> <EquationSource Format="TEX">$L_4$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mn>4</mn> </msub> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$L_5$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mn>5</mn> </msub> </math></EquationSource> </InlineEquation>) equilibrium points solutions. The inclusion of radiation, 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 unstable 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.</p>

Stability of Equilibrium Points in Modified Elliptic Restricted Three-Body Problem with Various Perturbation Sources

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发表日期：

2026年1月19日

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Muhammad Bayu Saputra,
Handhika Satrio Ramadhan,
Ibnu Nurul Huda,
Leonardus Brahmantyo Putra

摘要

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, assess 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 ( $L_1$ $L_{1}$ , $L_2$ $L_{2}$ , $L_3$ $L_{3}$ ) and two non-collinear ( $L_4$ $L_{4}$ , $L_5$ $L_{5}$ ) equilibrium points solutions. The inclusion of radiation, 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 unstable 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.