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Secular Perturbations (Laplace-Lagrange)

What you are seeing: two coupled planets exchange eccentricity on the secular (long) timescale. The eccentricities oscillate as the eigenmodes beat against each other.

Figure 1. Eccentricity exchange between two coupled planets.
e_00.15
coupling0.30

WHAT TO TRY

  • Raise the coupling between the two planets: they exchange eccentricity faster and the e(t) curves beat against each other more rapidly. The secular interaction conserves a combined angular-momentum deficit.
  • Set the initial eccentricity e_0: the system splits it between the two Laplace-Lagrange eigenmodes, and the beat pattern is those eigenfrequencies interfering.
  • Watch the slow exchange: neither eccentricity is constant, but the pair cycles forever. This secular coupling drives long-term orbital cycles in planetary systems.