To first approximation, the structure of large scale astronomical systems is governed by gravity, which is described by time-reversible ordinary differential equations. To solve these equations and study the formation and evolution of gravitational systems like the Galaxy, we use numerical integration methods which may not be time-reversible. I present novel integration methods which respect this time-symmetry and show they can be orders of magnitude more accurate than non-reversible numerical methods in studying gravitational dynamics. One of these methods is being implemented as TRACE in the REBOUND software package, and we find for interesting planetary dynamics problems it offers a speed advantage of an order magnitude or more compared to other codes. I then show how, similarly, constructing numerical methods preserving the Poincare invariants of Hamiltonian dynamics leads to more correct orbits. I describe a code in the REBOUND software package, MERCURIUS, that preserves this mathematical structure, in contrast to previous iterations of the method.
I then tackle the problem of the stability of the Solar System. Although great progress has been made in the last decades towards an understanding of chaos and stability of the Solar System due to the development of modern computers, I show that some studies are affected by numerical chaos, which causes artificial Solar System chaos and instability. The physical mechanism behind Mercury’s orbital instability has been traditionally described by a diffusive process in a secular frequency, but our current work shows a sub-diffusive process fits simulated data far better. An explanation for this sub-diffusion remains elusive.
