Secular Dynamics of Compact Three-Planet Systems

The secular Laplace-Lagrange orbital solution, decomposing eccentricities into a set of uniformly precessing eigenmodes is a classical result that is typically solved numerically. However, in the limit where orbits are closely spaced, several simplifications make it possible to make analytical progress. We derive simple expressions for the eccentricity eigenmodes in a co-planar 3-planet system where the middle planet is massless, and show that these approximate the true eigenmodes of more general systems with 3 massive planets in various limits. These results provide intuition for the secular dynamics of real systems, and have applications for understanding the stability boundary for compact multi-planet systems.