Relative Equilibria in the Spherical, Finite Density Three-Body Problem
J Nonlinear Sci
Relative Equilibria in the Spherical, Finite Density Three-Body Problem
D. J. Scheeres 0
B D. J. Scheeres
0 Department of Aerospace Engineering Sciences, The University of Colorado at Boulder , Boulder, CO , USA
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. Communicated by Anthony Bloch.
3-body problem; Relative equilibria; Granular mechanics
1 Introduction
The three-body problem is one of the most fundamental and well-studied problems
in celestial mechanics. A key result for this problem is that there exist only five
relative equilibria and that these exist for all levels of angular momentum (Euler 1767;
Lagrange 1772). The properties of these special solutions have been deeply studied
and have motivated significant research in mechanics and dynamics. A hallmark of the
classical problem is that the bodies are considered to be point masses, with no
restrictions on how close they can come to each other. A recent variation of this problem has
been posed that removes this one restriction (Scheeres 2012) and supposes that these
bodies can be rigid bodies with finite density, and hence have limits on their proximity.
Such “full-body” systems inherit the fundamental symmetries of the N -body problem
(Scheeres 2002); however, they also demand that the rotational angular momentum,
energy and dynamics of these rigid bodies be incorporated in the theory as well.
This paper studies the spherical three-body problem under the assumption that the
bodies are rigid and have finite density, and thus the separation between the bodies
is constrained to be positive. This one change completely alters the character of the
problem, and while the traditional Euler and Lagrange solutions still exist for large
enough angular momentum values, a full 23 additional relative equilibria emerge from
the analysis at all values of angular momentum, with a complex and rich bifurcation
scheme.
The celestial mechanics of bodies with finite density and fixed shape can have
dynamical evolution and relative equilibria that are quite distinct from that found in the
classical Newtonian point-mass N -body problem. These differences were previously
explored in Scheeres (2012) where several results were proven for the so-called
fullbody problem, in which the individual bodies are treated as rigid bodies with finite
densities. Specifically, it was shown that, in opposition to the point-mass N -body
problem, the full-body problem will always have a minimum energy configuration.
Further, the number and variety of relative equilibria for that problem are greatly
enhanced and now include configurations where the bodies can rest on each other
and configurations where different collections of resting bodies orbit each other, as
well as the classical central configurations. One important aspect of this problem is
that the existence and stability of configurations become a function of the total angular
momentum of the system, a dependance that does not exist for the classical point-mass
N -body problem.
This paper studies the relative equilibria of one particular problem in the
fullbody problem (FBP) to completeness. Specifically, all relative equilibria of the planar
spherical full three-body problem, which consists of three spheres of equal density but
arbitrary size, located in the plane perpendicular to the angular momentum. The explicit
methodology used was developed in Scheeres (2012, in press), and is fundamentally
based on analysis of the amended potential as developed by Smale (1970a, b) and
motivated by observations from Arnold et al. (1988). The main theorem is stated and
described at first, the problem is technically defined, then several results used to make
the proof are listed, and finally all the detailed computations for the proof are given.
Following the proof, a summary of the proof is provided, indicating how it establishes
the theorem.
A main application of this result is to identify the stable states that can be physically
achieved by a collection of self-gravitating bodies that can sustain contact. This
situation happens in solar system dynamics when considering the physical nature of rubble
pile asteroids (Fujiwara et al. 2006), a (...truncated)