Secular perturbation

A secular perturbation survives averaging over fast orbital phases and drives slow precession and exchange of eccentricity or inclination.

Laplace-Lagrange secular theory linearizes the orbit-averaged disturbing function to second order in eccentricities and inclinations. It represents planetary eccentricity evolution by a constant matrix.

The eigenvalues of the Laplace-Lagrange secular matrix are apsidal precession frequencies, and its eigenvectors give the relative eccentricities and apsidal phases of the corresponding normal modes.

The complex eccentricity packages eccentricity and longitude of periapsis as $z=e{e}^{i\varpi }$, turning linear secular dynamics into a complex linear system.

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

The forced eccentricity is the particular secular response of a small body to planetary eigenmodes, as distinct from its freely precessing homogeneous eccentricity.

A secular resonance occurs when a small body's free precession frequency equals a planetary secular eigenfrequency. Undamped linear theory then predicts resonant eccentricity growth.

Linear eccentricity damping contributes $\dot{z}=-z/\tau$ to the complex-eccentricity equation, exponentially suppressing the free mode and regularizing a secular resonance.

A secular torque is the orbit-averaged gravitational torque that changes orbital orientation or shape after fast orbital phases have been averaged out.

In a hierarchical orbital system, the quadrupole approximation keeps the first nontrivial term beyond the monopole in the orbit-averaged expansion in the ratio of inner to outer separation.

Nodal precession is the gradual rotation of an orbit's line of nodes, usually driven by a nonspherical or external gravitational torque.

Differential nodal precession occurs when neighboring orbits precess at different rates, winding a flat collection of rings into a warp or a phase-mixed thick structure.

The Laplace plane is the local equilibrium orbital plane about which a satellite or disk annulus precesses under competing secular torques.

The Kozai–Lidov mechanism exchanges eccentricity and inclination in a hierarchical gravitational system while preserving a component of angular momentum in the quadrupole limit.