Location: Maryam Mirzakhani Seminar Room at BCAM

Title:
New mathematical tools for calculating gravitational dynamics in planetary systems and other N-body problems


Abstract:
I describe new mathematical tools I've built to solve different problems in gravitational dynamics. I first describe maps that solve the gravitational system of ordinary differential equations describing asteroids in the Solar System. Enforcing that these maps be time-reversible and symplectic can significantly improve the reliability of the long-term dynamics of these bodies.

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 important studies are affected by numerical chaos, which causes artificial Solar System chaos and instability. This numerical instability arises from resonances between the time step and physical Solar System frequencies, and is an inherent property of symplectic maps. I discuss our current work to calculate Solar System stability, and in particular Mercury's future trajectory, without the effects of numerical chaos.
I next describe a suite of tools, including powerful new Kepler solvers and new symplectic integrators and their tangent equations that are designed to solve for the orbits of planets in exoplanetary systems. Unlike other popular methods, we can solve planetary systems with arbitrary geometries and orbits including moons. We have implemented these tools to solve the transit timing variation problem, and derive the properties and possible compositions of TRAPPIST-1 planets. Some of this work has been incorporated in the popular Rebound code.