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

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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.

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Notes

  1. The angular momentum deficit is defined as the difference between the total value of the angular momentum and its value in the case of Keplerian circular coplanar orbits having radii equal to the semi-major axes of the planets.

  2. Actually, when the mild hypotheses assumed in Morbidelli and Giorgilli (1995) are satisfied, the diffusion time is estimated to be super-exponentially big. This means that its order of magnitude is given by the exponential of the exponential of the inverse of a fractional power of the distance from a reference KAM torus.

  3. Precisely, being C the total angular momentum, i.e. \(C=\sum _{k=i}^2 \varLambda _k \sqrt{1-e_k^2} \cos i_k\), the angular momentum deficit is defined as \(\hbox {AMD} =\sum _{k=i}^2 \varLambda _k (1-\sqrt{1-e_k^2} \cos i_k)\).

  4. In the Laplace plane frame, the region of the Lidov–Kozai resonance is characterised by the libration of the argument of the pericenter of the inner planet (see Lidov 1962). The implicit adoption of such a frame has been essential in order to perform the reduction of the angular momentum sketched in Sect. 2.1. Therefore, the comparison between our results and those for that resonant region is valid because also our Hamiltonian model is written in the secular canonical coordinates with respect to the Laplace plane.

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Acknowledgements

M.S. has been partially supported by the National Group of Mathematical Physics (GNFM-INdAM), as part of the project “Normal Form techniques in Lattice Dynamics and Celestial Mechanics: perturbed resonant dynamics via resonant normal forms”. U.L. has been financially supported by both GNFM-INdAM and the project “DEXTEROUS - Uncovering Excellence 2014” of the University of Rome “Tor Vergata”. M.V. acknowledges financial support from the FRIA fellowship.

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Authors and Affiliations

  1. naXys, Department of Mathematics, University of Namur, 8 Rempart de la Vierge, 5000, Namur, Belgium

    Mara Volpi

  2. Dipartimento di Matematica, Università degli Studi di Roma “Tor Vergata”, via della Ricerca Scientifica 1, 00133, Rome, Italy

    Ugo Locatelli

  3. Dipartimento di Matematica, UniversitĂ  degli Studi di Milano, via Saldini 50, 20133, Milan, Italy

    Marco Sansottera

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  1. Mara Volpi
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  2. Ugo Locatelli
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Correspondence to Mara Volpi.

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This article is part of the topical collection on Recent advances in the study of the dynamics of N-body problem.

Guest Editors: Giovanni Federico Gronchi, Ugo Locatelli, Giuseppe Pucacco and Alessandra Celletti.

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Volpi, M., Locatelli, U. & Sansottera, M. A reverse KAM method to estimate unknown mutual inclinations in exoplanetary systems. Celest Mech Dyn Astr 130, 36 (2018). https://doi.org/10.1007/s10569-018-9829-5

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