= 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.
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