Invariant KAM Tori: From Theory to Applications to Exoplanetary Systems

Locatelli, Ugo; Caracciolo, Chiara; Sansottera, Marco; Volpi, Mara

doi:10.1007/978-3-031-13115-8_1

Cited by 2 publications

(2 citation statements)

References 31 publications

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“…Here, it is convenient to adopt a different version of the classical Kolmogorov normalization algorithm, which is slightly modified in such a way to not keep fixed the angular velocity vector ω (r ) , that is defined at the r -th step of the procedure and corresponds to the quasi-periodic approximation of the motion on the final sought KAM torus. We basically follow the approach described in Locatelli et al (2022), where the normalization procedure introduced by Kolmogorov is adapted in such a way to skip the small translation of the actions performed at every step of that algorithm.…”

Section: Construction Of the Kam Torusmentioning

confidence: 99%

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Existence proof of librational invariant tori in an averaged model of HD60532 planetary system

Danesi

Locatelli

Sansottera

2023

Celest Mech Dyn Astron

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We investigate the long-term dynamics of HD60532, an extrasolar system hosting two giant planets orbiting in a 3:1 mean motion resonance. We consider an average approximation at order one in the masses which results (after the reduction in the constants of motion) in a resonant Hamiltonian with two libration angles. In this framework, the usual algorithms constructing the Kolmogorov normal form approach do not easily apply and we need to perform some untrivial preliminary operations, in order to adapt the method to this kind of problems. First, we perform an average over the fast angle of libration which provides an integrable approximation of the Hamiltonian. Then, we introduce action-angle variables that are adapted to such an integrable approximation. This sequence of preliminary operations brings the Hamiltonian in a suitable form to successfully start the Kolmogorov normalization scheme. The convergence of the KAM algorithm is proved by applying a technique based on a computer-assisted proof. This allows us to reconstruct the quasi-periodic motion of the system, with initial conditions that are compatible with the observations.

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Section: Construction Of the Kam Torusmentioning

confidence: 99%

“…Its easy proof is sketched (for a wider type of classes of functions) in Subsect. 3.1 ofLocatelli et al (2022).…”

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confidence: 99%

Existence proof of librational invariant tori in an averaged model of HD60532 planetary system

Danesi

Locatelli

Sansottera

2023

Celest Mech Dyn Astron

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Secular orbital dynamics of the innermost exoplanet of the $$\upsilon $$-Andromedæ system

Mastroianni

Locatelli

2023

Celest Mech Dyn Astron

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We introduce a quasi-periodic restricted Hamiltonian to describe the secular motion of a small-mass planet in a multi-planetary system. In particular, we refer to the motion of $$\upsilon $$ υ -And $$\textbf{b}$$ b which is the innermost planet among those discovered in the extrasolar system orbiting around the $$\upsilon $$ υ -Andromedæ A star. We preassign the orbits of the Super-Jupiter exoplanets $$\upsilon $$ υ -And $$\textbf{c}$$ c and $$\upsilon $$ υ -And $$\textbf{d}$$ d in a stable configuration. The Fourier decompositions of their secular motions are reconstructed by using the well-known technique of the (so-called) frequency analysis and are injected in the equations describing the orbital dynamics of $$\upsilon $$ υ -And $$\textbf{b}$$ b under the gravitational effects exerted by those two external exoplanets (that are expected to be major ones in such an extrasolar system). Therefore, we end up with a Hamiltonian model having $$2+3/2$$ 2 + 3 / 2 degrees of freedom; its validity is confirmed by the comparison with several numerical integrations of the complete four-body problem. Furthermore, the model is enriched by taking into account also the effects due to the relativistic corrections on the secular motion of the innermost exoplanet. We focus on the problem of the stability of $$\upsilon $$ υ -And $$\textbf{b}$$ b as a function of the parameters that mostly impact on its orbit, that are the initial values of its inclination and the longitude of its node (as they are measured with respect to the plane of the sky). In particular, we study the evolution of its eccentricity, which is crucial to exclude orbital configurations with high probability of (quasi)collision with the central star in the long-time evolution of the system. Moreover, we also introduce a normal form approach, that is based on the complete average of our restricted model with respect to the angles describing the secular motions of the major exoplanets. Therefore, our Hamiltonian model is further reduced to a system with 2 degrees of freedom, which is integrable because it admits a constant of motion that is related to the total angular momentum. This allows us to very quickly preselect the domains of stability for $$\upsilon $$ υ -And $$\textbf{b}$$ b , with respect to the set of the initial orbital configurations that are compatible with the observations.

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Invariant KAM Tori: From Theory to Applications to Exoplanetary Systems

Cited by 2 publications

References 31 publications

Existence proof of librational invariant tori in an averaged model of HD60532 planetary system

Existence proof of librational invariant tori in an averaged model of HD60532 planetary system

Secular orbital dynamics of the innermost exoplanet of the $$\upsilon $$-Andromedæ system

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