On the extension of the Laplace-Lagrange secular theory to order two in the masses for extrasolar systems

Libert, Anne-Sophie; Sansottera, Marco

doi:10.1007/s10569-013-9501-z

Cited by 31 publications

(43 citation statements)

References 39 publications

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“…The data are obtained from those reported in Table IV of [50] by projecting them on the invariant plane (that is perpendicular to the total angular momentum) in a standard way. [17], [32] and [33]). To this end we follow the approach described in [38], carrying out two "Kolmogorov-like" normalization steps in order to eliminate the main perturbation terms depending on the fast angles λ .…”

Section: The Secular Modelmentioning

confidence: 99%

Secular dynamics of a planar model of the Sun-Jupiter-Saturn-Uranus system; effective stability in the light of Kolmogorov and Nekhoroshev theories

2017

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We investigate the long-time stability of the Sun-Jupiter-Saturn-Uranus system by considering a planar secular model, that can be regarded as a major refinement of the approach first introduced by Lagrange. Indeed, concerning the planetary orbital revolutions, we improve the classical circular approximation by replacing it with a solution that is invariant up to order two in the masses; therefore, we investigate the stability of the secular system for rather small values of the eccentricities. First, we explicitly construct a Kolmogorov normal form, so as to find an invariant KAM torus which approximates very well the secular orbits. Finally, we adapt the approach that is at basis of the analytic part of the Nekhoroshev's theorem, so as to show that there is a neighborhood of that torus for which the estimated stability time is larger than the lifetime of the Solar System. The size of such a neighborhood, compared with the uncertainties of the astronomical observations, is about ten times smaller. *

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Section: The Secular Modelmentioning

confidence: 99%

Secular dynamics of a planar model of the Sun-Jupiter-Saturn-Uranus system; effective stability in the light of Kolmogorov and Nekhoroshev theories

2017

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“…Thanks to the adoption of the Laplace plane, we can further reduce the expansion to two degrees of freedom. It was shown in previous works (see for example Libert & Henrard 2007;Libert & Sansottera 2013) that if the planetary system is far from a mean-motion resonance, the secular approximation at the first order in the masses is accurate enough to describe the evolution of the system. Such an analytical approach is of interest for the present purpose, since, being faster than pure n-body simulations which also consider small-period effects, it allows us to perform an extensive parametric exploration at a reasonable computational cost.…”

Section: Introductionmentioning

confidence: 99%

The 3D secular dynamics of radial-velocity-detected planetary systems

Volpi

Roisin

Libert

2019

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Aims. To date, more than 600 multi-planetary systems have been discovered. Due to the limitations of the detection methods, our knowledge of the systems is usually far from complete. In particular, for planetary systems discovered with the radial velocity (RV) technique, the inclinations of the orbital planes, and thus the mutual inclinations and planetary masses, are unknown. Our work aims to constrain the spatial configuration of several RV-detected extrasolar systems that are not in a mean-motion resonance.Methods. Through an analytical study based on a first-order secular Hamiltonian expansion and numerical explorations performed with a chaos detector, we identified ranges of values for the orbital inclinations and the mutual inclinations, which ensure the long-term stability of the system. Our results were validated by comparison with n-body simulations, showing the accuracy of our analytical approach up to high mutual inclinations (∼ 70 • -80 • ).Results. We find that, given the current estimations for the parameters of the selected systems, long-term regular evolution of the spatial configurations is observed, for all the systems, i) at low mutual inclinations (typically less than 35 • ) and ii) at higher mutual inclinations, preferentially if the system is in a Lidov-Kozai resonance. Indeed, a rapid destabilisation of highly mutually inclined orbits is commonly observed, due to the significant chaos that develops around the stability islands of the Lidov-Kozai resonance.

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“…The secular approximation at order two in the masses is based on a "Kolmogorov-like" normalization step aiming at removing the fast angles from terms that are at most linear in the fast actions L. Again, we modify the standard approach by putting the resonant combinations of the fast angles, the harmonics k * · λ, in the normal form, thus removing them from the generating function. We briefly summarize the averaging procedure here, more details can be found in, e.g., Locatelli & Giorgilli (2007); and Libert & Sansottera (2013). We adopt the standard Lie series algorithm (see, e.g., Henrard (1973) and Giorgilli (1995)) to transform the Hamiltonian (4) into…”

Section: Resonant Hamiltonian At Order Two In the Massesmentioning

confidence: 99%

Resonant Laplace-Lagrange theory for extrasolar systems in mean-motion resonance

Sansottera

Libert

2019

Celest Mech Dyn Astr

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Extrasolar systems with planets on eccentric orbits close to or in meanmotion resonances are common. The classical low-order resonant Hamiltonian expansion is unfit to describe the long-term evolution of these systems. We extend the Laplace-Lagrange secular approximation for coplanar systems with two planets by including (near-)resonant harmonics, and realize an expansion at high order in the eccentricities of the resonant Hamiltonian both at orders one and two in the masses. We show that the expansion at first order in the masses gives a qualitative good approximation of the dynamics of resonant extrasolar systems with moderate eccentricities, while the second order is needed to reproduce more accurately their orbital evolutions. The resonant approach is also required to correct the secular frequencies of the motion given by the Laplace-Lagrange secular theory in the vicinity of a mean-motion resonance. The dynamical evolutions of four (near-)resonant extrasolar systems are discussed, namely GJ 876 (2:1 resonance), HD 60532 (3:1), HD 108874 and GJ 3293 (close to 4:1).

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On the extension of the Laplace-Lagrange secular theory to order two in the masses for extrasolar systems

Cited by 31 publications

References 39 publications

Secular dynamics of a planar model of the Sun-Jupiter-Saturn-Uranus system; effective stability in the light of Kolmogorov and Nekhoroshev theories

Secular dynamics of a planar model of the Sun-Jupiter-Saturn-Uranus system; effective stability in the light of Kolmogorov and Nekhoroshev theories

The 3D secular dynamics of radial-velocity-detected planetary systems

Resonant Laplace-Lagrange theory for extrasolar systems in mean-motion resonance

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