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

Mara Volpi; Ugo Locatelli; Marco Sansottera

doi:10.1007/s10569-018-9829-5

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

Mara Volpi, Ugo Locatelli, Marco Sansottera

Research output: Contribution to journal › Article › peer-review

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 Poincar\'e 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.

Original language	English
Article number	36
Journal	Celestial Mechanics & Dynamical Astronomy
Volume	130
Issue number	5
DOIs	https://doi.org/10.1007/s10569-018-9829-5
Publication status	Published - 1 May 2018
Externally published	Yes

Funding

Acknowledgements M.S. has been partially supported by the National Group of Mathematical Physics (GNFM-INdAM), as part of the project \u201CNormal Form techniques in Lattice Dynamics and Celestial Mechanics: perturbed resonant dynamics via resonant normal forms\u201D. U.L. has been financially supported by both GNFM-INdAM and the project \u201CDEXTEROUS - Uncovering Excellence 2014\u201D of the University of Rome \u201CTor Vergata\u201D. M.V. acknowledges financial support from the FRIA fellowship.

Funders	Funder number
GNFM-INdAM
Sapienza Università di Roma
Istituto Nazionale di Alta Matematica "Francesco Severi"
Fonds pour la Formation à la Recherche dans l’Industrie et dans l’Agriculture
Not added	0000

Keywords

Celestial Mechanics
Exoplanets
KAM theory
n-Body planetary problem

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10.1007/s10569-018-9829-5

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title = "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 Poincar\'e 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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AU - Locatelli, Ugo

AU - Sansottera, Marco

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N2 - 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 Poincar\'e 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.

AB - 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 Poincar\'e 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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A reverse KAM method to estimate unknown mutual inclinations in exoplanetary systems

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