Modulating the Stability Boundary: Secular Dynamics of Compact Three-Planet Systems

The final stage of terrestrial planet formation occurs once the protoplanetary disk dissipates, entering a giant impact phase of collisions and gravitational scatterings that determines the planetary masses and orbital architectures we see today. Understanding the chaotic dynamics of these instabilities is thus crucial for connecting disk formation models to observations.

For two closely orbiting planets, the chaotic boundary is set by mean motion resonances (MMRs), which occur when the orbital periods are close to integer ratios. Each MMR has a finite width, and the stability boundary is at the orbital separation where adjacent resonances start to overlap. However, introducing just one additional planet would greatly complicate the theoretical picture, and empirical scaling laws from decades of numerical simulations have had limited success.

Recent work suggests that instabilities in compact systems of three or more planets are still driven by MMR overlap, but one also needs to account for long-term (secular) dynamics. These secular perturbations lead to oscillations in the MMR widths and modulate the location where MMR overlap occurs. While the leading-order Laplace-Lagrange solution for secular dynamics is well known, the requisite matrix diagonalization needs to be solved numerically, defying a closed-form solution for the stability boundary. Yet, in the compact limit, several simplifications enable analytical progress.

We present a novel, approximate solution for the secular dynamics of compact, co-planar three-planet systems. This provides not only analytical expressions, but also geometric intuition into the conserved Laplace-Lagrange modes driving the dynamics. We additionally present simple expressions for the oscillations in MMR widths and for the resulting stability boundary, which are validated against a suite of numerical simulations.