Librational KAM tori in the secular dynamics of the $\upsilon$ Andromed{\ae} planetary system

TL;DR

This paper analyzes the secular dynamics of the $$ Andromedae planetary system, demonstrating the existence of a stable KAM torus through numerical and computer-assisted proofs, revealing long-term stability in the system.

Contribution

The study introduces a novel algorithm for constructing KAM tori in the secular model of the $$ Andromedae system, with a rigorous convergence proof using computer assistance.

Findings

The system orbits around a stable elliptic KAM torus.

The algorithm accurately constructs the shape of the KAM torus.

The convergence of the method is rigorously proven.

Abstract

We study the planetary system of $\upsilon$~Andromed{\ae}, considering the three-body problem formed by the central star and the two largest planets, $\upsilon$~And~\emph{c} and $\upsilon$~And~\emph{d}. We adopt a secular, three-dimensional model and initial conditions within the range of the observed values. The numerical integrations highlight that the system is orbiting around a one-dimensional elliptic torus (i.e., a periodic orbit that is linearly stable). This invariant object is used as a seed for an algorithm based on a sequence of canonical transformations. The algorithm determines the normal form related to a KAM torus, whose shape is in excellent agreement with the orbits of the secular model. We rigorously prove that the algorithm constructing the final KAM invariant torus is convergent, by adopting a suitable technique based on a computer-assisted proof.