The Location of Secular Resonances

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