Ugo Locatelli

We study the KAM-stability of several single star two-planet nonresonant extrasolar systems. It is likely that the observed exoplanets are the most massive of the system considered. Therefore, their robust stability is a crucial and necessary condition for the longterm survival of the system when considering potential additional exoplanets yet to be seen. Our study is based on the construction of a combination of lower-dimensional elliptic and KAM tori, so as to better approximate the dynamics in the framework of accurate secular models. For each extrasolar system, we explore the parameter space of both inclinations: the one with respect to the line of sight and the mutual inclination between the planets. Our approach shows that remarkable inclinations, resulting in three-dimensional architectures that are far from being coplanar, can be compatible with the KAM stability of the system. We find that the highest values of the mutual inclinations are comparable to those of the few systems for which the said inclinations are determined by the observations.

We investigate the long-time stability of the Sun-Jupiter-Saturn-Uranus system by considering a planar secular model, which can be regarded as a major refinement of the approach first introduced by Lagrange. Indeed, concerning the planetary orbital revolutions, we improve the classical circular approximation by replacing it with a solution that is invariant up to order two in the masses; therefore, we investigate the stability of the secular system for rather small values of the eccentricities. First, we explicitly construct a Kolmogorov normal form to find an invariant KAM torus which approximates very well the secular orbits. Finally, we adapt the approach that underlies the analytic part of Nekhoroshev’s theorem to show that there is a neighborhood of that torus for which the estimated stability time is larger than the lifetime of the Solar System. The size of such a neighborhood, compared with the uncertainties of the astronomical observations, is about ten times smaller.

We describe an algorithm constructing an invariant KAM torus for a class of planetary systems, such that the mutual attractions, the eccentricities and the inclinations of the planets are small enough. By using computer algebra, we explicitly implement this algorithm for approximating a KAM torus for the problem of three bodies in a case similar to the Sun–Jupiter–Saturn system. We show that, by reducing the masses of the planets by a factor 10 and with a small displacement of the orbits, our semianalytical construction of the torus turns out to be successful.

We investigate the break-down of invariant tori in a four dimensional standard mapping for different values of the coupling parameter. We select various two-dimensional frequency vectors, having eventually one or both components close to a rational value. The dynamics of this model is very reach and depends on two parameters, the perturbing and coupling parameters. Several techniques are introduced to determine the analyticity domain (in the complex perturbing parameter plane) and to compute the break-down threshold of the invariant tori. In particular, the analyticity domain is recovered by means of a suitable implementation of Padé approximants. The break-down threshold is computed through a suitable extension of Greene's method to four dimensional systems. Frequency analysis is implemented and compared with the previous techniques.