  Observations of clustering among the orbits of the most distant
trans-Neptunian objects (TNOs) has inspired interest in the possibility of an
undiscovered ninth planet lurking in the outskirts of the solar system.
Numerical simulations by a number of authors have demonstrated that, with
appropriate choices of planet mass and orbit, such a planet can maintain
clustering in the orbital elements of the population of distant TNOs, similar
to the observed sample. However, many aspects of the rich underlying dynamical
processes induced by such a distant eccentric perturber have not been fully
explored. We report the results of our investigation of the dynamics of
coplanar test-particles that interact with a massive body on an circular orbit
(Neptune) and a massive body on a more distant, highly eccentric orbit (the
putative Planet Nine). We find that a detailed examination of our idealized
simulations affords tremendous insight into the rich test-particle dynamics
that are possible. In particular, we find that chaos and resonance overlap
plays an important role in particles' dynamical evolution. We develop a simple
mapping model that allows us to understand, in detail, the web of overlapped
mean-motion resonances explored by chaotically evolving particles. We also
demonstrate that gravitational interactions with Neptune can have profound
effects on the orbital evolution of particles. Our results serve as a starting
point for a better understanding of the dynamical behavior observed in more
complicated simulations that can be used to constrain the mass and orbit of
Planet 9.
