A reverse KAM method to estimate unknown mutual inclinations in exoplanetary systems

Abstract

The inclinations of exoplanets detected via radial velocity method are essentially unknown. We aim to provide estimations of the ranges of mutual inclinations that are compatible with the long-term stability of the system. Focusing on the skeleton of an extrasolar system, i.e., considering only the two most massive planets, we study the Hamiltonian of the three-body problem after the reduction of the angular momentum. Such a Hamiltonian is expanded both in Poincaré canonical variables and in the small parameter D 2 , which represents the normalised Angular Momentum Deficit. The value of the mutual inclination is deduced from D 2 and, thanks to the use of interval arithmetic, we are able to consider open sets of initial conditions instead of single values. Looking at the convergence radius of the Kolmogorov normal form, we develop a reverse KAM approach in order to estimate the ranges of mutual inclinations that are compatible with the long-term stability in a KAM sense. Our method is successfully applied to the extrasolar systems HD 141399, HD 143761 and HD 40307. *

Kolmogorov variation: KAM with knobs (à la Kolmogorov)

“…where K and Γ are two positive constants. Furthermore, the classical approach is the only way to directly implement KAM theory in practical applications and it proved advantageous in different contexts, e.g., the construction of lower dimensional elliptic tori in planetary systems in [43,44], the study of the long term dynamics of exoplanets in [24,42,45], the investigation of the effective stability in the spin-orbit problem in [40,41], the design of an a priori control for symplectic maps related to betatronic motion in [39] and the continuation of periodic orbits on resonant tori in [32,33,38]. In the present paper too, we adopt the classical approach, which turns out to be better suited in order to devise a normal form algorithm that introduces a detuning of the initial frequencies that will be determined, step-by-step, along the normalization procedure.…”

Librational KAM tori in the secular dynamics of the $\upsilon$ Andromedæ planetary system

“…Indeed, we focus on the torus corresponding to L = 0. The procedure is quite standard in Celestial Mechanics but, for the sake of completeness, we sketch here the main steps (see for more details [39], that is in turn an adaptation of the approach developed in [26]). The first transformation of coordinates that we define aims at removing the perturbative terms that depend on the angles λ but do not depend on the actions L, being Lj = ∂H/∂λ j for j = 1, 2 .…”

“…Finally, the plots of the eccentricities in the top-right box of Figure 1 highlight that their values oscillate in a range between 0.1 and 0.35. This makes clear that we have to develop an approach substantially different with respect to that described in [39], which has been designed to study planetary models similar to that including Sun, Jupiter and Saturn, where the eccentricities are always smaller than 0.1 and the difference of the perihelion arguments is in a regime of full rotation.…”

“…Then, we apply one last time the classical KAM algorithm, i.e., with the addition of a translation at each normalization step in order to keep the frequency fixed. In Figure 3, we plot the norms of the generating functions χ (r) 2 for both the normalization algorithms, the one without any other translation apart from the initial one χ(q) defined in (39) and the one enqueued with the translation ξ (r) • q at each step r for aiming at 2 introduced by the algorithms for the construction of the normal form for a KAM torus: the one above refers to the algorithm without fixing the frequencies and the one below to the algorithm with the translation for keeping the frequency vector fixed as ω. On the right, the Poincaré sections of the true orbit (in red) and of the orbit with initial conditions on the KAM torus constructed with our algorithm (in black).…”

“…It has been studied within the framework of both complete and secular planetary Hamiltonians (see, e.g., [30], [24] and [40]). In [39], some of the Authors of this article introduced a reverse approach based on KAM theory: values of the mutual inclinations are considered to be acceptable if the subsequent construction of the Kolmogorov normal form is convergent. The effectiveness of that method was limited to extrasolar systems hosting planets with very moderate eccentricities (i.e., smaller than 0.1).…”