Relative Equilibria in the Spherical, Finite Density Three-Body Problem.

Area of Science:

• Celestial Mechanics
• Astrophysics
• N-body Problem Dynamics

Background:

• The classical three-body problem, particularly the point-mass model, has limited known relative equilibria.
• Understanding the dynamics of systems with extended bodies and finite density is crucial for astrophysical modeling.

Purpose of the Study:

• To identify and characterize all distinct relative equilibria for the spherical, finite density three-body problem.
• To analyze the stability and bifurcation pathways of these equilibria as a function of angular momentum.
• To investigate the impact of finite density on the existence of minimum energy configurations.

Main Methods:

• Mathematical identification of relative equilibria under the assumption of equal, constant densities.
• Analysis of system rotation about the maximum moment of inertia.
• Mapping stability and bifurcation pathways by systematically varying angular momentum.

Main Results:

• Discovery of 28 distinct relative equilibria, including the 5 classical configurations.
• None of the equilibria are stable across all angular momentum values.
• Finite density significantly increases the number of relative equilibria and guarantees minimum energy states for all angular momenta.

Conclusions:

• The transition from point-mass to finite density models dramatically expands the possible configurations in the three-body problem.
• Angular momentum plays a critical role in the stability and existence of these relative equilibria.
• Finite density ensures the existence of stable, minimum energy configurations across the full spectrum of angular momentum.