 Solution

The planetary expansion is a decomposition into secular eigenmodes of conservative linear Laplace-Lagrange secular theory. It assumes small orbital eccentricities, coplanarity, approximately constant semi-major axis values, and averaging over fast orbital phases away from important mean-motion resonances and close encounters. Dissipation, migration, and nonlinear secular effects are absent from this constant-coefficient representation.

The $g_i$ are real eigenfrequencies of the Laplace-Lagrange secular matrix, describing apsidal precession. For mode $i$, the signed real coefficients $e_{ji}$ give its amplitude in planet $j$; their relative magnitudes are fixed by an eigenvector, and their signs distinguish aligned and anti-aligned apsides. The common phase ${\beta }_i$ specifies that mode's orientation at the chosen time origin. A planet's actual orbital eccentricity is $|z_j(t)|$, which generally varies through interference of modes; it is not a sum of the positive mode amplitudes.