Normal mode analysis of rubble-pile asteroids using a discrete element method

Abstract

Seismic shaking of small bodies has an important role in revealing information about internal structure and contributing to our understanding of microgravity geology. Since many small asteroids are likely to be rubble piles, it is important to understand their dynamics, which is largely different from that of monolithic asteroids. We introduce a new normal mode analysis method as an approach to seismic study based on a discrete element method (DEM), where a rubble pile is modeled as a group of elastic spheres bound together with gravitational force, with elastic repulsion forces following Hertzian contact theory and Mindlin’s theory. Normal mode analysis is formulated by differentiating the interaction between particles around an equilibrium state. Our results show that normal modes and eigenfrequencies are independent of the size of the particles if their physical properties and geometry are identical. Furthermore, we apply a scaling law, which shows that the shape of normal modes does not change even if the size of a rubble pile is enlarged or contracted, but the eigenfrequency varies in inverse proportion with the scale to the power of two-thirds. We also compare normal modes between a rubble pile and a monolith and show that the shapes of normal modes are similar to each other, but the eigenfrequency is smaller in the rubble pile. Finally, we describe dynamic simulations to compare results from nonlinear DEM and results from linearization of the normal mode. It is found that normal mode analysis gives a good approximation for the frequency distribution when the kinetic energy is sufficiently small, and that the presence of tangential force between particles suppresses vibration motion.