# The Location of Secular Resonances

• Alessandro Morbidelli
Part of the NATO ASI Series book series (NSSB, volume 272)

## Abstract

The importance of the secular resonances for the dynamics in the asteroid belt is well known since Tisserand (1882) and Poincáre (1892). Indeed a Lagrangian linear theory for the secular motion of the planets shows that their elements e w,i,v (eccentricity, longitude of perihelion, inclination and longitude of the ascending node, using the notations of Poincare) are not constants of motion, as they are in the Keplerian problem, but vary with the following law (see Bretagnon (1974)):
$$\begin{array}{*{20}c} {e_k \,\cos \bar \omega _k \, = \mathop \sum \limits_{j = 1,8} M_{k,j} \cos (g_j t + \alpha _j ),} & {e_k \,\sin \bar \omega _k \, = \mathop \sum \limits_{j = 1,8} M_{k,j} \sin (g_j t + \alpha _j )} \\{\sin \frac{{i_k }}{2}\cos v_k = \,\mathop \sum \limits_{j = 1,8} M_{k,j} \cos (s_j t + \beta _j ),\,} & {\sin \frac{{i_k }} {2}\sin v_k = \,\mathop \sum \limits_{j = 1,8} M_{k,j} \sin (s_j t + \beta _j )} \\ \end{array}$$
(1)
Here the indexes j, k refer to each planet, from Mercury (1), to Neptune (8). If one considers only the system formed by the Sun, Jupiter, and Saturn, only the terms with k = 5,6 and j = 5,6 survive.

## Keywords

Circular Orbit Keplerian Problem Stable Equilibrium Point Asteroid Belt Motion Resonance
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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