A Dynamical Systems Theory Approach to Circumbinary Orbital Dynamics

Unlike The Solar System, ~20% of stars are found to be gravitationally bound, forming a binary system. A fraction of these stars, composing millions of solar systems across the galaxy, are formed in close binaries, <1 AU. To date, astronomers have confirmed ~20 circumbinary ‘P-type’ planets using transit and imaging methods, thanks largely to data from the NASA Kepler and TESS missions. Planets forming and persisting in these systems are subject to gravitational interactions with two major bodies, forming a three-body dynamics problem. Unlike its two-body counterpart, this dynamical system does not permit analytic solutions and possesses extreme sensitivities to initial conditions.

In this investigation, we have turned to applications of dynamical systems theory inspired by modern advances in multi-body astrodynamics to classify families of orbits and their stability in the Circular Restricted Three-Body Problem (CR3BP). Studying the CR3BP as a nonlinear Hamiltonian dynamical system, we have identified retrograde and prograde families of periodic solutions (limit cycles) using differential corrections algorithms. Via Floquet Theory, we can directly interpret the periodic solution’s linear stability by calculating the Poincaré exponents encoded in eigenvalues of the monodromy matrix. Through this technique, we also record the locations of bifurcations associated with the termination and generation of new periodic families. To complement the study of periodic solutions, we have investigated the evolution of the global circumbinary solution space through Poincaré mapping techniques.