Discovery of Rubble-Pile Asteroid Dynamics through Sparse Symbolic Regression

Current evidence shows that most asteroids are rubble piles, which are defined to be aggregates of loosely consolidated material bound by gravity and likely a small amount of cohesive strength. Rubble-pile asteroids are granular systems, which can be reshaped through external excitation like meteoritic impacts, the YORP effect, and planetary encounters. Universal modelling of granular systems is one of the major unsolved topics in physics, as these systems are chaotic, multi-scale, and highly dependent on the non-linear interactions between its constituent particles. These difficulties are exacerbated as the low gravity invalidates some terrestrial observations and scaling laws.

Most analytical models are based on continuum mechanics, like the Mohr-Coulomb or Drucker–Prager criterion, fitted to specific sets of observations. These models are able to explain certain aspects of the asteroid population well, but are not able to accurately describe all their critical properties and are furthermore not dynamical. On the other hand, numerical simulations have shown a great potential to predict the evolution of these systems, down to properties of their individual fragments. However, their high computational burden and sensitive dependency on initial conditions make it harder to generalise conclusions made from them.

This work tries to bridge the gap between numerical and analytical modelling of rubble pile asteroids, by using the data produced by numerical simulations to derive a set of analytical equations of motion. Machine learning based system identification methods like genetic programming or neural networks have been shown to work well in predicting complex non-linear dynamical systems. However, key properties of good analytical models, like interpretability and generalizability, are often neglected by these methods. For this reason the sparse identification of non-linear dynamics (SINDy) method was developed, which avoids this problem by applying a sequential thresholding least-squares algorithm on a set of mathematical functions to obtain a sparse representation of the dynamics. 

In this work, first a set of time series data is obtained from the numerical code GRAINS, which is an N-body code that takes into account the complex shape of the individual particles, as they interact through self gravity and contact. A set of macroscopic state and environment variables are selected, which can either be physical values like the moments of inertia and/or spin-up rate, or numerically derived optimal coordinates (using e.g. proper orthogonal decomposition). This time series data is then used by SINDy to obtain a symbolic representation of the time derivative of the state variables. The thresholding parameter of SINDy can be tuned to obtain either a simpler model that mainly qualitatively describes the system, or a more complex model that also has a good quantitative performance. These analytical models are then used to obtain various dynamical properties of the systems, e.g. equilibrium points, bifurcations, etc.

This research shows how the data obtained from simulations can be used to obtain a parsimonious model for the dynamics of rubble pile asteroids. These models can further improve our understanding of the origin and evolution of rubble-pile asteroids, and help inform and interpret future observations.