A Semi-Analytic Algorithm for Constructing Lower Dimensional Elliptic Tori in Planetary Systems

TL;DR

This paper presents a semi-analytic algorithm based on Kolmogorov's normalization to construct invariant elliptic tori in planetary systems, verified through frequency analysis in a four-body model.

Contribution

The authors adapt Kolmogorov's normalization algorithm for constructing elliptic tori and provide explicit analytic expansions for motions on these tori in planetary models.

Findings

Successfully applied to a four-body planetary model

Verified the accuracy of initial conditions via frequency analysis

Provides explicit analytic expansions of motions on elliptic tori

Abstract

We adapt the Kolmogorov's normalization algorithm (which is the key element of the original proof scheme of the KAM theorem) to the construction of a suitable normal form related to an invariant elliptic torus. As a byproduct, our procedure can also provide some analytic expansions of the motions on elliptic tori. By extensively using algebraic manipulations on a computer, we explicitly apply our method to a planar four-body model not too different with respect to the real Sun--Jupiter--Saturn--Uranus system. The frequency analysis method allows us to check that our location of the initial conditions on an invariant elliptic torus is really accurate.