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Stability Analysis of Collinear Equilibrium Points in a Modified Restricted Three-Body Problem with Dipole and Triaxial Perturbations

This paper analyzes the existence and linear instability of four collinear equilibrium points in a modified restricted three-body problem featuring a dipole primary and triaxial secondary, using analytical, semi-analytical, and numerical methods to model the dynamics of irregular binary asteroid systems.

Original authors: Job Ibrahim Obadaki, Jessica Mrumun Gyegwe, Helen Olaronke Edogbanya

Published 2026-09-24

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CC BY 4.0

Original authors: Job Ibrahim Obadaki, Jessica Mrumun Gyegwe, Helen Olaronke Edogbanya

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast emptiness of space, gravity is the invisible hand that shapes the movement of everything, from the smallest dust grain to the largest planet. When two massive objects, like stars or asteroids, orbit a common center, they create a complex gravitational landscape. A third, much smaller object moving through this field does not simply fall toward one or the other; instead, it is caught in a delicate tug-of-war between the pull of gravity and the outward push created by the rotation of the system. Scientists have long used a simplified model, known as the restricted three-body problem, to map this terrain. This model assumes the two large bodies are perfect spheres, allowing researchers to find specific spots where the forces balance out. These spots, called equilibrium points, are crucial for space exploration because they act as natural gateways where spacecraft can hover with minimal fuel or travel along invisible highways to distant destinations. However, the real universe is rarely perfect. Many celestial bodies are not smooth spheres but are lumpy, elongated, or irregular, and their shapes significantly alter the gravitational field around them.

A team of researchers at the Federal University Lokoja in Nigeria has taken a closer look at this problem by updating the classic model to account for these irregular shapes. They focused on a scenario where the larger primary body is not a single sphere but is instead modeled as a rotating dipole—essentially two equal masses connected by a rigid rod, representing an elongated object like a stretched-out asteroid. The smaller secondary body was modeled as a triaxial ellipsoid, a shape that is stretched differently along three axes, much like a slightly squashed egg. This specific combination of shapes is not just theoretical; it closely mirrors the real-world binary asteroid system known as Didymos and its smaller companion, Dimorphos. The researchers wanted to understand how these specific, non-spherical shapes change the locations of the balance points and whether those points remain stable enough to be useful for future missions.

To investigate this, the team formulated a set of equations that describe the motion of a tiny particle within this modified system. They did not rely on a single method but combined three different approaches to ensure their results were robust. First, they used advanced mathematical techniques to derive approximate solutions for where these balance points would be located. Second, they employed a powerful numerical method, essentially a computerized trial-and-error process, to find the exact coordinates. Finally, they ran direct simulations to verify the findings. Their work confirmed that the three classic balance points still exist in this modified system, but the irregular shapes of the asteroids shift their positions. More significantly, the dipole nature of the larger primary created a brand new, fourth balance point located directly between the two ends of the elongated primary body. This new point has no equivalent in the classical model where the primary is treated as a single sphere.

The researchers then examined the stability of these points, asking a critical question: if a spacecraft were placed at one of these spots, would it stay there, or would it drift away? Their analysis showed that, just like in the classical model, all four of these collinear points are unstable. This means that any object placed at these locations will eventually be pushed away by the slightest disturbance, requiring active thrusters to maintain position. The new point between the dipole masses is particularly unstable, with a tendency to eject objects much faster than the other points. The team also mapped out the "zero-velocity curves," which are boundaries that define the regions where a particle can and cannot travel based on its energy. They found that the new dipole-generated point creates a distinct, isolated region of motion that was previously invisible in standard models.

To test the practical value of their findings, the team applied their model to the Didymos–Dimorphos system, a real binary asteroid pair that has been the subject of intense study following a NASA impact mission. Using the known masses, distances, and shapes of these two asteroids, they calculated the specific locations of the equilibrium points for this real-world system. The results showed that the model successfully predicts the existence of the three classical points and the additional interior point for this specific pair. The calculations revealed that the new point sits very close to the center of the system, surrounded by a high-energy barrier that keeps it distinct from the rest of the space. While the instability of these points means they cannot serve as permanent, passive parking spots for spacecraft, understanding their exact locations and how they shift with the asteroids' shapes is vital for planning trajectories. The study demonstrates that ignoring the elongated, lumpy nature of asteroids can lead to significant errors in predicting where these gravitational gateways are located. By incorporating the dipole and triaxial shapes, the researchers have provided a more accurate map of the gravitational terrain around binary asteroids, offering a better foundation for designing future missions to these complex and fascinating systems.

Technical Summary: Stability Analysis of Collinear Equilibrium Points in a Modified Restricted Three-Body Problem with Dipole and Triaxial Perturbations

Problem Statement
The classical Circular Restricted Three-Body Problem (CR3BP) assumes spherical primaries, an assumption that often fails to accurately model the gravitational environments of irregular binary asteroid systems, such as Didymos–Dimorphos. In such systems, the primary bodies exhibit significant deviations from spherical symmetry due to elongation (dipole characteristics) and triaxiality. This paper addresses the gap in existing literature by formulating a modified CR3BP where the larger primary is modeled as a rotating mass dipole (two equal point masses connected by a rigid rod) and the smaller primary is modeled as a homogeneous triaxial ellipsoid. The study focuses specifically on the existence, location, and linear stability of the collinear equilibrium points ( $L_1, L_2, L_3$ ) within this unified dipole–triaxial framework.

Methodology
The authors formulate the governing equations of motion in a uniformly rotating barycentric frame using dimensionless units. The system is defined by a modified mass parameter ( $\mu_m$ ), a dipole half-separation parameter ( $l$ ), a force-ratio parameter ( $k$ ), and triaxiality parameters ( $\sigma_1, \sigma_2$ ).

Equilibrium Conditions: The effective potential ( $\Omega$ ) and the Jacobi integral are derived. The collinear equilibrium points are identified by solving the stationary condition $\Omega_x = 0$ along the $x$ -axis ( $y=0$ ).
Analytical Approximation: Due to the nonlinearity introduced by the perturbations, closed-form solutions are unavailable. The authors employ asymptotic perturbation techniques. For the points $L_1$ and $L_2$ , where the governing equation exhibits singular inverse-power terms due to triaxiality, a Puiseux expansion in fractional powers of the perturbation parameter is used. For $L_3$ , where the equation remains regular, a conventional integer-power perturbation expansion is applied.
Numerical Verification: The analytical approximations serve as initial estimates for a semi-analytical Newton–Raphson iteration and direct numerical solutions. A multi-start search is utilized to identify potential additional equilibrium points within the dipole region.
Stability Analysis: Linear stability is assessed by introducing small perturbations to the equilibrium positions and deriving the characteristic equation. The eigenvalues of the resulting system determine the stability nature.
Dynamical Visualization: Zero-velocity curves (ZVCs) and Poincaré surfaces of section (PSS) are generated to visualize the topology of admissible regions and verify the stability conclusions.
Case Study: The model is applied to the Didymos–Dimorphos binary asteroid system using physical parameters derived from DART mission data.

Key Contributions and Results

Existence of an Additional Equilibrium Point: The analysis confirms the existence of the three classical collinear equilibrium points ( $L_1, L_2, L_3$ ). Crucially, the dipole configuration introduces a fourth, additional collinear equilibrium point ( $L_6$ ) located strictly between the two dipole masses. This point has no counterpart in the classical CR3BP.
Parametric Influence: Numerical simulations demonstrate that the governing parameters significantly alter the equilibrium configuration:
- The force-ratio parameter ( $k$ ) and dipole half-separation ( $l$ ) produce the most pronounced displacements of the exterior points ( $L_1$ and $L_3$ ).
- The triaxiality parameters primarily influence the interior points ( $L_1$ and $L_2$ ).
- The scaled mass parameter ( $\mu_m$ ) affects the displacement of $L_2$ most significantly.
Linear Instability: The linear stability analysis reveals that all four collinear points ( $L_1, L_2, L_3, L_6$ $L_{1}, L_{2}, L_{3}, L_{6}$ ) are linearly unstable within the investigated parameter ranges. The characteristic equations consistently yield eigenvalues with positive real parts, indicating exponential growth of perturbations.
- The additional point $L_6$ exhibits the highest growth rate (instability measure), attributed to the strong gravitational gradients within the dipole region.
- Among the classical points, $L_2$ generally shows the largest growth rate.
Dynamical Topology: The Zero-velocity curves show that the additional point $L_6$ creates a distinct, highly localized interior region of motion. The Poincaré surfaces of section confirm the linear instability, showing dispersed, non-confined trajectories near the equilibrium points rather than closed invariant curves.
Application to Didymos–Dimorphos: When applied to the Didymos–Dimorphos system, the model successfully recovers the three classical points and identifies the additional $L_6$ point. The calculated Jacobi constants and stability characteristics align with the theoretical predictions, validating the model's utility for irregular binary systems.

Significance and Claims
The paper claims to provide a unified mathematical framework that integrates dipole and triaxial perturbations to better represent the dynamics of irregular binary asteroid systems. The primary significance lies in the identification and characterization of the additional collinear equilibrium point ( $L_6$ ), which arises solely from the dipole representation of the primary.

The authors assert that while the combined perturbations modify the geometry and local dynamics of the system, they do not stabilize the collinear points; all remain linearly unstable. Consequently, the study concludes that while the model offers new insights into equilibrium structures and accessible regions of motion, the additional equilibrium point $L_6$ is unsuitable for passive long-term particle confinement without active station-keeping. The work serves as a foundation for future investigations into periodic orbits, invariant manifolds, and spacecraft trajectory design in binary asteroid environments, extending the applicability of the CR3BP beyond spherical assumptions.

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