  We evaluate the dynamical stability of a selection of outer solar system
objects in the presence of the proposed new Solar System member Planet Nine. We
use a Monte Carlo suite of numerical N-body integrations to construct a variety
of orbital elements of the new planet and evaluate the dynamical stability of
eight Trans-Neptunian objects (TNOs) in the presence of Planet Nine. These
simulations show that some combinations of orbital elements ($a,e$) result in
Planet Nine acting as a stabilizing influence on the TNOs, which can otherwise
be destabilized by interactions with Neptune. These simulations also suggest
that some TNOs transition between several different mean-motion resonances
during their lifetimes while still retaining approximate apsidal anti-alignment
with Planet Nine. This behavior suggests that remaining in one particular orbit
is not a requirement for orbital stability. As one product of our simulations,
we present an {\it a posteriori} probability distribution for the semi-major
axis and eccentricity of the proposed Planet Nine based on TNO stability. This
result thus provides additional evidence that supports the existence of this
proposed planet. We also predict that TNOs can be grouped into multiple
populations of objects that interact with Planet Nine in different ways: one
population may contain objects like Sedna and 2012 VP$_{113}$, which do not
migrate significantly in semi-major axis in the presence of Planet Nine and
tend to stay in the same resonance; another population may contain objects like
2007 TG$_{422}$ and 2013 RF$_{98}$, which may both migrate and transition
between different resonances.

Earth and Planetary Astrophysics