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Modeling the Evolution of the Sphere-Restricted Full Three-Body Problem

Introduction: In Scheeres (2014) the Sphere-Restricted Full Three-Body Problem (RF3BP) is outlined as a particular aspect of a problem set regarding full-body dynamical systems [1]. Full-body problems host particles of finite density, disallowing them to come infinitely close to one another as in classical dynamics. By allowing for the dissipation of energy in the system, a discrete set of equilibrium states can be achieved-all of which are listed in Figure 1. By applying friction and restitution, potential relative equilibria will include resting states where relative motion between objects is zero. The solution space has been described in terms of stable and unstable states assuming spheres of uniform density and mass [2]. Using perfectly-rigid dynamics, we simulate systems with a given angular momentum in a purpose-written C-based code. This investigation explores various restitution and friction coefficients for given initial angular momentum. We aim to describe the relative mechanics and dynamics in rubble-pile bodies where relative forces are not great enough to compromise the rigidity of the constituents. All output is analyzed in terms of normalized units, with the gravitational parameter and the minimum distance between the equal spheres as unity. Computation: A C-based Discrete Element Method (DEM) code has been developed for the study of full-body dynamical systems. The code resolves perfectly rigid-body interaction between spheres with surface friction and of various densities and masses. Impact mechanics are dictated by restitution and friction. Such a formulation is subject to inelastic collapse in the case of a less-than-unity coefficient of restitution, where an infinite number of impacts occur in a finite time, i.e. dissipation results in relative velocities tending to zero, requiring the timestep to do the same in order to resolve the interaction. We implement a variable coefficient of restitution for cases of negligible relative velocity to mitigate this issue – applying a unity restitution condition. In a strict sense, this generates resting solutions that should be described as 'pseudo-resting' since persistent contact is not explicitly handled. Numerically resolving persistent contact requires the employment of soft-body mechanics, whose contact interaction is usually modeled by spring dampeners; however, this method also encounters inelastic collapse in the limit