  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.
