The reducing transformation and Apocentric Librators

Henrard, Jacques; Lemaître, Anne; Milani, Andrea; Murray, C. D.

doi:10.1007/bf01238924

Cited by 51 publications

(45 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Although first-order resonant motion can be apparently complex, here, greatly aided by the pioneering works of Sessin & Ferraz-Mello (1984) as well as Henrard et al (1986) and Wisdom (1986), we have shown that the essential features of the dynamics is captured within the context of a simple integrable Hamiltonian. The Hamiltonian in question is qualitatively similar to that of a pendulum and more precisely, is related to the second fundamental model for resonance (Henrard & Lemaitre 1983).…”

Section: Resultsmentioning

confidence: 99%

“…Specifically, we shall follow the pioneering work of Peale (1976) and Sessin & Ferraz-Mello (1984). The calculation will be greatly simplified by a reducing transformation (see Henrard et al 1986;Wisdom 1986) and the final Hamiltonian will closely resemble the second fundamental model for resonance (Henrard & Lemaitre 1983).…”

Section: An Integrable Approximationmentioning

confidence: 99%

“…Through a rather involved calculation, utilizing the perturbation method devised by Hori (1966), these authors demonstrated the integrability of the first-order resonant motion. In a subsequent effort, aimed primarily at the elliptic restricted three-body problem, Henrard et al (1986) and Wisdom (1986) greatly simplified the Sessin & Ferraz-Mello (1984) solution by introducing a canonical transformation that explicitly identifies a novel constant of motion. Here, we shall follow the latter approach.…”

Section: An Integrable Approximationmentioning

confidence: 99%

See 2 more Smart Citations

Analytical treatment of planetary resonances

Batygin

Morbidelli

2013

A&A

View full text Add to dashboard Cite

An ever-growing observational aggregate of extrasolar planets has revealed that systems of planets that reside in or near mean-motion resonances are relatively common. While the origin of such systems is attributed to protoplanetary disk-driven migration, a qualitative description of the dynamical evolution of resonant planets remains largely elusive. Aided by the pioneering works of the last century, we formulate an approximate, integrable theory for first-order resonant motion. We utilize the developed theory to construct an intuitive, geometrical representation of resonances within the context of the unrestricted three-body problem. Moreover, we derive a simple analytical criterion for the appearance of secondary resonances between resonant and secular motion. Subsequently, we demonstrate the onset of rapid chaotic motion as a result of overlap among neighboring first-order mean-motion resonances, as well as the appearance of slow chaos as a result of secular modulation of the planetary orbits. Finally, we take advantage of the integrable theory to analytically show that, in the adiabatic regime, divergent encounters with first-order mean-motion resonances always lead to persistent apsidal anti-alignment.

show abstract

Section: Resultsmentioning

confidence: 99%

Section: An Integrable Approximationmentioning

confidence: 99%

Section: An Integrable Approximationmentioning

confidence: 99%

See 1 more Smart Citation

Analytical treatment of planetary resonances

Batygin

Morbidelli

2013

A&A

View full text Add to dashboard Cite

show abstract

“…The problem of two planets close to a first-order MMR on nearly circular and coplanar orbits can be reduced to a one-degree-of-freedom system through a sequence of canonical transformations (Wisdom 1986;Henrard et al 1986;Delisle et al 2012Delisle et al , 2014.…”

Section: The Resonant Hamiltonianmentioning

confidence: 99%

“…The system can be made integrable by a rotation of the coordinates X i (Sessin & Ferraz-Mello 1984;Henrard et al 1986;Delisle et al 2014). We introduce R and φ such that…”

Section: Integrable Hamiltonianmentioning

confidence: 99%

AMD-stability in the presence of first-order mean motion resonances

Petit

Laskar

Boué

2017

A&A

View full text Add to dashboard Cite

The AMD-stability criterion allows to discriminate between a-priori stable planetary systems and systems for which the stability is not granted and needs further investigations. AMD-stability is based on the conservation of the Angular Momentum Deficit (AMD) in the averaged system at all orders of averaging. While the AMD criterion is rigorous, the conservation of the AMD is only granted in absence of mean-motion resonances (MMR). Here we extend the AMD-stability criterion to take into account mean-motion resonances, and more specifically the overlap of first-order MMR. If the MMR islands overlap, the system will experience generalized chaos leading to instability. The Hamiltonian of two massive planets on coplanar quasi-circular orbits can be reduced to an integrable one degree of freedom problem for period ratios close to a first-order MMR. We use the reduced Hamiltonian to derive a new overlap criterion for first-order MMR. This stability criterion unifies the previous criteria proposed in the literature and admits the criteria obtained for initially circular and eccentric orbits as limit cases. We then improve the definition of AMD-stability to take into account the short term chaos generated by MMR overlap. We analyze the outcome of this improved definition of AMD-stability on selected multi-planet systems from the Extrasolar Planets Encyclopaedia.

show abstract