DDA 2015 – End-State Relative Equilibria in the Sphere-Restricted Full Three-Body Problem

This is one of a series of notes taken during the 2015 meeting of the AAS Division on Dynamical Astronomy, 3-7 May, at CalTech. An index to this series (all the papers presented at the meeting) is here.

Travis SJ Gabriel (UC Boulder)


The Sphere-Restricted Full Three-Body Problem studies the motion of three finite density spheres as they interact under surface and gravitational forces. When accounting for the dissipation of energy, full-body systems may achieve minimum energy states that are unatainable in the classic treatment of the N-Body Problem. This serves as a simple model for the mechanics of rubble pile asteroids, interacting grains in a protoplanetary disk, and potentially the interactions of planetary ring particles. Previous studies of this problem have been performed in the case where the three spheres are of equal size and mass, with all possible relative equilibria and their stability having been identified as a function of the total angular momentum of the system. These studies uncovered that at certain levels of angular momentum there exists more than one stable relative equilibrium state. Thus a question of interest is which of these states a dissipative system would preferentially settle in provided some domain of initial conditions, and whether this would be a function of the dissipation parameters. Using perfectly-rigid dynamics, three-equal-sphere systems are simulated in a purpose-written C-based code to uncover these details. Results from this study are relevant to the mechanics and dynamics in small solar system bodies where relative forces are not great enough to compromise the rigidity of the constituents.


  • Sphere-restrictedTBP:
    • $U = -G \dfrac{m_1m_2}{r_{12}}$ singularity
    • $E \ge U + \dfrac{H^2}{2 I_H}$
    • For $N=3$ equal spheres, normalized min. energy function
    • Scheeres 2012: 9 relative equilibria for planar motion case
      • 3 stable
    • Add dissipation
    • $\rightarrow$ 2 min. energy solutions
    • Which solution will the system land on?
  • Numerical simulations
    • randomized ICs in 2-solution regime, vary dissipation
    • brute force statistics
  • Results:
    • More Euler resting states as H increases, regardless of dissipation
    • End state depends heavily on dissipation
    • Hence knowledge of restitution is key

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