Searching for Simpler Models of Astrophysical Pattern Formation

While theories of synchronization in two- or three-body astronomical systems are well understood, a generalization to many-bodied systems remains largely unexplored. Historically, problems of resonant capture among astronomical bodies have been treated primarily using methods from conservative classical mechanics. We investigate the possibility of using nonconservative models together with perturbation theory and numerical methods to understand the phenomenon of resonant capture in large-scale structures such as rings, planetary systems and galactic spiral arms. In particular, we focus on N-body dissipative systems such as circumplanetary discs and use methods drawn from the study of coupled oscillators. One such method is inspired by the Kuramoto model, which describes mean-field behavior in large ensembles of coupled nonlinear oscillators. The Kuramoto model can be modified to allow for non-mean-field coupling, leading to the existence of chimera states, in which most of the oscillators synchronize. These chimera states can appear as clusters or spirals of synced oscillators, and may be suggestive of objects in astronomical contexts. As an illustrative example, we develop a mean-field model for N small particles in a dust ring around a massive planet and integrate it numerically using code developed in MATLAB and Python. Preliminary results show promise that this approach will yield new insight into astronomical synchronization phenomena across a wide range of length scales.