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.

19 pages, 3 figures

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Cited by in corpus (8)

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• The phase-space architecture in extrasolar systems with two planets in orbits of high mutual inclination
• A numerical criterion evaluating the robustness of planetary architectures; applications to the $\upsilon$ Andromedæ system
• Existence proof of librational invariant tori in an averaged model of HD60532 planetary system
• Exponential stability of fast driven systems, with an application to celestial mechanics