Librational KAM tori in the secular dynamics of the υ Andromedæ planetary system

Abstract

ABSTRACT The υ Andromedæ system is the first extrasolar system where the mutual inclination between exoplanets has been determined using astrometric methods. We study a model of the υ Andromedæ planetary system considering the three-body problem formed by the central star and the two largest planets, υ And c and υ And 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. Thus, we are able to prove the stability of the secular dynamics for a set of values of the orbital elements that are in agreement with the observed ones. As a by-product, we can also extract a numerical indicator of robustness for the constructed invariant KAM tori. This allows us to identify ranges of the inclinations that are the most likely candidates according to the KAM stability prescription. In this context, we conclude that the most robust orbital configurations are those with large values of υ And c’s mass, that is about 16 time bigger than Jupiter’s one.