Dynamics of a deforming planetary body

Abstract

Self-gravitating planetary bodies possess non-spherical shapes and various density distributions. The dynamics of such bodies are treated using a rigid-body assumption, where bodies do not deform throughout their motion, or simplified force models and shapes like a sphere. These assumptions make problems simple, though it essentially ignores the response of deformation to translation and rotation, and vice versa, limiting its applicability. Describing the coupling of three dynamics modes (deformation, rotation, and translation) is essential to detailing the dynamical evolution of planetary bodies. However, a continuum mechanics approach such as finite element modeling (FEM) needs additional cautions because ill-defined body forces may cause unrealistic stress and displacement conditions. Here, we propose a theoretical framework for characterizing the dynamics of a deforming body, applicable to the numerical implementation of continuum mechanics models. We introduce the kinetics and mechanics to derive the governing equations representing all the dynamics modes. The advantage of this technique is that the equation of deformation no longer deals with translation and rotation included in body forces, which removes difficulties in computing stress and displacement fields around boundary conditions. We demonstrate this technique by considering a system that consists of two masses connected by a spring and a damper. An example of this application discusses the dynamical response of the Cold Classical Kuiper Belt Object (CCKBO) 486958 Arrokoth to the Sky Crater-forming impact event on its small lobe. When the dissipation quality factor, known as the Q factor, is about 100, a typical value for a rubble pile object, the deformation dissipation timescale is just a few days. Considering the yield stress and strain to be ∼10 kPa and ∼0.01, respectively, we derive a displacement of up to tens of meters and normal stress of up to ∼5 kPa if the impact direction consists of the components along the two lobes. Without it, the rotational state change gives no significant displacement and normal stress; however, the major part of the imparted momentum may be converted to shear.