Stable or Not: Constraining the Stability of Hidden Super-Short Period Planets

Recent ground and space-based observations show that stars with multiple planets are common in the galaxy. Most of these observational methods are biased toward detecting large planets near to their host stars. Because of these observational biases, these systems can hide small, close-in planets inward of observed planets. These hidden planets are influenced dynamically by their companions. In certain configurations, this influence can destabilize the system; in others, the star’s gravitational influence through both general relativity and the stellar gravitational quadrupole moment (J2) can instead further stabilize (and in some cases, J2 can destabilize) the system. We derive criteria for hidden planets orbiting within known planets that quantify how strongly general relativistic and J2 effects can stabilize systems that would otherwise be unstable. Furthermore, we show the regions of parameter space that possible hidden planets lie in if they are to be stable.