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A secular resonance is a type of orbital resonance between two bodies with synchronized precessional frequencies. In celestial mechanics, secular refers to the long-term motion of a system and resonance is when two periods or frequencies are a simple numerical ratio of small integers. Typically, the synchronized precessions in secular resonances are between the rates of change of the argument of the periapses or the rates of change of the longitude of the ascending nodes of two system bodies. Secular resonances can be used to study the long-term orbital evolution of asteroids and their families within the asteroid belt (see the &#x3BD;6 resonance below).

Description
Secular resonances occur when the precession of two orbits are synchronized (a precession of the periapsis, usually denoted with frequency $$g$$, or the ascending node, usually denoted with frequency $$s$$, or both). A small body (such as an asteroid or other small Solar System body) in secular resonance with a much larger one (e.g. a planet) will precess at the same rate as the large body. Over relatively short time periods (a million years, or so) a secular resonance will change the eccentricity and inclination of the small body.

One can distinguish between:


 * linear secular resonances between a body (no subscript) and a single other large perturbing body (e.g. a planet, subscript as numbered from the Sun), such as the &#x3BD;6 = g − g6 secular resonance between asteroids and Saturn; and
 * nonlinear secular resonances, which are higher-order resonances, usually combination of linear resonances such as the z1 = (g − g6) + (s − s6), or the ν6 + ν5 = 2g − g6 − g5 resonances.

&#x3BD;6 resonance
A prominent example of a linear resonance is the &#x3BD;6 secular resonance between asteroids and Saturn. Asteroids which approach it have their eccentricity slowly increased until they become Mars-crossers, at which point they are usually ejected from the asteroid belt due to a close encounter with Mars. This resonance forms the inner and "side" boundaries of the asteroid belt around 2 AU, and at inclinations of about 20°.

Secular Frequencies of the Solar System
The fundamental frequencies of the Solar System are the eigenfrequencies of the planets (both the precession of the periapses and the precession of the ascending nodes). They can be computed using Laplace-Lagrange theory or by applying frequency analysis after numerically integrating the equations of motion.

Computing the eigenfrequencies with Laplace-Lagrange theory involves computing the eigenvalues of a perturbation matrix describing how the inner and outer objects affect each other. This perturbation matrix uses the Disturbing Function (a series expansion using Legendre Polynomials to describe the deviations each planet has from its Keplerian orbit) and Laplace coefficients. If the matrix is defined so that the first row/column corresponds to the perturbing influence of Mercury, the second row/column corresponds to the perturbing influence of Venus, and so on until the last row/column corresponds to the perturbing influence of Neptune, then the eigenvalues describe either the precessions of the periapses (denoted by $$g_i$$) or the precessions of the ascending nodes (denoted by $$s_i$$). Where $$i=1$$ refers to Mercury, $$i=2$$ refers to Venus, etc.

Computing the eigenfrequencies by applying frequency analysis after numerically integrating the equations of motion involves tracking the evolution of the orbital elements of each planet and performing a modified Fourier transform (MFT)