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

The relative equilibria for the spherical, finite density three-body problem are identified. Specifically, there are 28 distinct relative equilibria in this problem which include the classical five relative equilibria for the point-mass three-body problem. None of the identified relative equilibria exist or are stable over all values of angular momentum. The stability and bifurcation pathways of these relative equilibria are mapped out as the angular momentum of the system is increased. This is done under the assumption that they have equal and constant densities and that the entire system rotates about its maximum moment of inertia. The transition to finite density greatly increases the number of relative equilibria in the three-body problem and ensures that minimum energy configurations exist for all values of angular momentum.

[1]  S. Smale,et al.  Topology and mechanics. I , 1970 .

[2]  Daniel J. Scheeres,et al.  Stability in the Full Two-Body Problem , 2002 .

[3]  Vladimir I. Arnold,et al.  Encyclopaedia of Mathematical Sciences , 1990 .

[4]  Daniel J. Scheeres,et al.  DEM simulation of rotation-induced reshaping and disruption of rubble-pile asteroids , 2012 .

[5]  D. Scheeres,et al.  Minimum energy configurations in the N-body problem and the celestial mechanics of granular systems , 2011, 1111.4245.

[6]  Richard Moeckel.  On central configurations , 1990 .

[7]  Derek C. Richardson,et al.  Numerical experiments with rubble piles: equilibrium shapes and spins , 2005 .

[8]  J. Kawaguchi,et al.  The Rubble-Pile Asteroid Itokawa as Observed by Hayabusa , 2006, Science.

[9]  Daniel J. Scheeres,et al.  Hill Stability of Configurations in the Full N-Body Problem , 2015, Proceedings of the International Astronomical Union.

[10]  Kyrille Popoff,et al.  Sur le problème des trois corps , 1929 .

[11]  Aurel Wintner,et al.  The Analytical Foundations of Celestial Mechanics , 2014 .

[12]  S. Smale,et al.  Topology and mechanics. II , 1970 .

[13]  Daniel J. Scheeres,et al.  Hill Stability in the Full 3-Body Problem , 2014, Proceedings of the International Astronomical Union.

[14]  D. T. Greenwood.  Principles of dynamics , 1965 .

[15]  S. Debei,et al.  Two independent and primitive envelopes of the bilobate nucleus of comet 67P , 2015, Nature.

[16]  Daniel J. Scheeres,et al.  Relative Equilibria in the Full N-Body Problem with Applications to the Equal Mass Problem , 2016 .