An attempt to constrain Planet Nine’s orbit and position via resonant confinement of distant TNOs

We considered four TNOs on elongated orbits with small semimajor axis uncertainties: Sedna, 2004 VN112, 2012 VP113, and 2000 CR105. We found two sets of simultaneous near commensurabilities for these objects with a putative Planet Nine that are compatible with the current uncertainties in the objects’ orbital periods. We conducted a large number of numerical simulations of quasi-coplanar simulations (i.e. inclinations of Planet Nine and TNOs set to zero but not the giant planets) to find which values of Planet Nine’s mean anomaly and longitude of perihelion could put these objects in stable mean motion resonance (MMR) librations. We found no cases of simultaneous stable librations for multiple TNOs for more than 800 My, with most librations lasting much shorter than this time-scale. The objects 2004 VN112 and 2000 CR105 are the most unstable. Being in an MMR is not a strict requirement for long-term survival in 3D simulations, so our result cannot be used to refute Planet Nine’s existence. Nevertheless, it casts doubt and shows that theoretical attempts to constrain the position of the planet on the sky are not possible.