Secular eigenmode

A secular eigenmode is a normal mode in which all complex eccentricities precess at one eigenfrequency with fixed amplitude ratios and relative apsidal phases.

For a real two-planet Laplace-Lagrange secular matrix with negative off-diagonal entries, the upper secular eigenmode has anti-aligned apsides and the lower has aligned apsides. Their positive planet-1 to planet-2 amplitude ratios obey $r_p{r}_m=A_{12}/A_{21}$. Under the conventional angular momentum deficit normalization this is $L_2/L_1$.