Out-of-Plane Equilibria in the Restricted Eight-Body Problem: Dynamics and Stability

1. Introduction

The restricted eight-body problem is a fascinating extension of classical celestial mechanics, providing insights into the intricate dynamics of systems influenced by gravitational and radiative forces. This model features a radiating central primary surrounded by six peripheral primaries in circular motion, creating a unique framework for studying out-of-plane equilibrium points, which are located along the z-axis and exhibit critical behaviors under varying radiation factors.

Recent work has highlighted the significance of out-of-plane dynamics in multi-body systems. Work in [1] explored these dynamics within the circular restricted eight-body framework, identifying symmetrical equilibrium points and analyzing their stability. Similarly, research in [2] proposed a low-voltage CMOS rectifier for biomedical implantable devices, showcasing advancements in energy harvesting and efficiency optimization. Studies in [3,4,5] investigated light bending in the galactic halo, autonomous Hamiltonian systems, and the size of the galactic halo, respectively, providing insights into gravitational effects and the utility of Hamiltonian methods in celestial mechanics. Additionally, research in [6] utilized SpinalNet for the morphological classification of galaxies, indirectly aiding in understanding the large-scale dynamics of celestial systems. This formulation builds upon the foundational work of Maxwell (1859) and further extensions by Kalvouridis and Hadjifotinou (2011). Building on these findings, this study delves deeper into the existence and alignment of these points at critical radiation values and examines the associated periodic orbits. By investigating the interplay of gravitational and radiative effects, this research contributes to a comprehensive understanding of celestial mechanics in complex systems. These equations originate from the analysis of Maxwell-type n-body systems and are a continuation of studies into the influence of radiation pressure on equilibrium dynamics.

2. Mathematical Framework and Formulas

The restricted eight-body problem involves a radiating central primary (μ0) and six peripheral primaries (μ) revolving in a circular orbit of radius a around the central primary. The equations of motion for a test particle (ε) are given by:

where the potential function includes both gravitational and radiative components. The general form of the potential function is expressed as:where represent the coordinates of the six peripheral primaries.

Critical Radiation Solutions

q	−6<q<0	Radiation factor range for equilibrium points.
${\mathit{q}}_{\mathit{c}}$		Critical radiation value where equilibrium aligns at aaa.
${\mathit{z}}_0$		Position along the z-axis for equilibrium points.
v		Angle in the equilibrium computation.

q	−6<q<0	Radiation factor range for equilibrium points.
${\mathit{q}}_{\mathit{c}}$		Critical radiation value where equilibrium aligns at aaa.
${\mathit{z}}_0$		Position along the z-axis for equilibrium points.
v		Angle in the equilibrium computation.

Periodic Orbits Periodic orbits around the equilibrium points are analyzed for specific values of . The orbits exhibit varying amplitudes and frequencies based on initial conditions and radiation factors. These periodic motions provide insights into the long-term dynamical behavior near the unstable points.

Critical Radiation Value

At the critical radiation value , the equilibrium points align precisely along the z-axis at a distance equal to the orbital radius of the peripheral primaries. This critical value determines the boundary condition where equilibrium points transition in their alignment

Radiation Factor (q)	$\mathbf{Position}\mathbf{Along}\mathbf{Z}-\mathbf{Axis}({\mathit{Z}}_0$)	Observation
−$6$<$\mathit{q}$<${\mathit{q}}_{\mathit{c}}$​	${\mathit{z}}_0>\mathit{a}$	Points lie outside the radius aaa.
$\mathit{q}={\mathit{q}}_{\mathit{c}}$​	${\mathit{z}}_0=\mathit{a}$	Points align exactly at aaa.
${\mathit{q}}_{\mathit{c}}<\mathit{q}<0$	${\mathit{z}}_0<\mathit{a}$	Points lie within the radius aaa.

Radiation Factor (q)	$\mathbf{Position}\mathbf{Along}\mathbf{Z}-\mathbf{Axis}({\mathit{Z}}_0$)	Observation
−$6$<$\mathit{q}$<${\mathit{q}}_{\mathit{c}}$​	${\mathit{z}}_0>\mathit{a}$	Points lie outside the radius aaa.
$\mathit{q}={\mathit{q}}_{\mathit{c}}$​	${\mathit{z}}_0=\mathit{a}$	Points align exactly at aaa.
${\mathit{q}}_{\mathit{c}}<\mathit{q}<0$	${\mathit{z}}_0<\mathit{a}$	Points lie within the radius aaa.

Linear Stability Analysis

The stability of the equilibrium points is analyzed by introducing small perturbations () around the equilibrium positions. Let the perturbations be:

(1)

where λ are the characteristic roots of the system. The variational equations for these perturbations are derived from the linearized equations of motion:

(2)

The determinant of the Jacobian matrix formed by these partial derivatives determines the eigenvalues λ:

(3)

For the range of radiation factors −6<q<0 the eigenvalues satisfy ${\lambda }^2>0$ indicating that the equilibrium points are linearly unstable. This instability implies that small deviations from the equilibrium points grow exponentially over time, leading to chaotic dynamics around these points [1,2].

Origin of the Formula

Equations of Motion: For a test particle in the restricted eight-body problem, the equations of motion are:

(4)

Linearization: Substituting/these perturbed positions into the equations of motion and expanding U(x,y,z)U(x, y, z)U(x,y,z) as a Taylor series around the equilibrium point gives:

(5)

These are the variational equations governing small oscillations around the equilibrium.

Perturbations Around Equilibrium: Introduce small perturbations ($੬$,η,ζ) about the equilibrium position ($x_0,y_0,z_0$​):

(6)

Role of Eigenvalues (λ) in Stability Analysis

Eigenvalues (λλ) are mathematical quantities derived from the system of linearized equations around an equilibrium point. They provide critical information about the stability of the system by describing how small perturbations evolve over time.

Linearized Equations of Motion: Eigenvalues are derived from the linearization of the system around equilibrium points [4]. By approximating the system's potential U(x,y,z) with a second-order Taylor expansion about the equilibrium point, the resulting variational equations form a matrix. The eigenvalues of this matrix determine how perturbations (ξ,η,ζ) evolve over time.

Dynamic Behavior: Eigenvalues dictate whether perturbations grow (instability), decay (stability), or oscillate (neutral stability). This is crucial in identifying whether the system remains near the equilibrium or diverges away:

• Positive Real Eigenvalues: Indicate exponential divergence, leading to an unstable system.
• Negative Real Eigenvalues: Indicate exponential decay, leading to a stable system.
• Complex Eigenvalues: Indicate oscillatory behavior, which may be stable if the real parts are negative.

Mathematical Interpretation

Given the eigenvalue equation:

the stability conditions are determined by the sign of ${\lambda }^2$

If λ² > 0, λ is real, and the system is unstable.

If λ$^2$ is imaginary, and the system exhibits oscillatory stability.