A second fundamental model for resonance

Henrard, Jacques; Lemaître, Anne

doi:10.1007/bf01234306

Cited by 264 publications

(199 citation statements)

References 14 publications

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“…By definition, È 3 : 2 librates for these 92 objects but circulates for the remaining 308. This capture efficiency of f 3 : 2 % 23% reflects (1) the probability of capture into the isolated 3 : 2 resonant potential just prior to resonance encounter (Henrard & Lemaitre 1983;Borderies From simulation Ib we estimate the capture efficiency of the 2 : 1 resonance to be f 2 : 1 % 212/400 = 53%. This value is more than twice as high as f 3 : 2 , reflecting both the lack of competition from other sweeping resonances that lie interior to the 2 : 1 and the lower probability of scattering by Neptune at these greater distances.…”

Section: Capture Efficienciesmentioning

confidence: 99%

On the Plutinos and Twotinos of the Kuiper Belt

Chiang

Jordán

2002

The Astronomical Journal

103

143

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We illuminate dynamical properties of Kuiper belt objects (KBOs) in the 3 : 2 (Plutino) and 2 : 1 ('' Twotino '') Neptunian resonances within the model of resonant capture and migration. We analyze a series of numerical integrations, each involving the four migratory giant planets and 400 test particles distributed throughout trans-Neptunian space, to measure the efficiencies of capture into the 3 : 2 and 2 : 1 resonances, the efficiencies of capture into Kozai-type secular resonances, and the libration centers and amplitudes of resonant particles, all as functions of the migration speed. We synthesize instantaneous snapshots of the spatial distribution of $10 4 resonant KBOs, from which we derive the longitudinal variation of the sky density of each resonant family. Twotinos cluster AE75 away from Neptune's longitude, while Plutinos cluster AE90 away. Such longitudinal clustering persists even for surveys that are not volume limited in their ability to detect resonant KBOs. Remarkably, between À90 and À60 of Neptune's longitude we find the ratio of sky densities of Twotinos to Plutinos to be nearly unity, despite the greater average distance of Twotinos, assuming the two resonant populations are equal in number and share the same size, albedo, and inclination distributions. We couple our findings to observations to crudely estimate that the intrinsic Twotino population is within a factor of $3 of the Plutino population. Most strikingly, the migration model predicts a possible asymmetry in the spatial distribution of Twotinos: more Twotinos may lie at longitudes behind that of Neptune than ahead of it. The magnitude of the asymmetry amplifies dramatically with faster rates of migration and can be as large as $300%. A differential measurement of the sky density of 2 : 1 resonant objects behind and in front of Neptune's longitude would powerfully constrain the migration history of that planet.

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Section: Capture Efficienciesmentioning

confidence: 99%

On the Plutinos and Twotinos of the Kuiper Belt

Chiang

Jordán

2002

The Astronomical Journal

103

143

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“…We recognize the second fundamental model of resonance (Henrard & Lemaitre 1983). This Hamiltonian is also called an Andoyer Hamiltonian (FerrazMello 2007).…”

Section: Andoyer Hamiltonianmentioning

confidence: 99%

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

Petit

Laskar

Boué

2017

A&A

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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.

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“…Analytic celestial mechanics shows that mean motion resonances with a planet on a circular orbit only cause an oscillation of the small body's semi-major axis coupled with a moderate oscillation of the eccentricity and with the libration of the angle kλ − k λ (where λ and λ are the mean longitudes of the small body and of the planet, respectively, and the integer coefficients k and k define the k :k resonance; Henrard & Lemaitre 1983;Lemaitre 1984). However, if the perturbing planet has a finite eccentricity, inside a mean motion resonance there can be a dramatic secular evolution; the eccentricity of the small body can undergo large excursions correlated with the precession of the longitude of perihelion (Wisdom 1985(Wisdom , 1983Henrard & Caranicolas 1990).…”

Section: Introductionmentioning

confidence: 99%

Extreme secular excitation of eccentricity inside mean motion resonance

Pichierri

Morbidelli

Lai

2017

A&A

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Context. It is well known that asteroids and comets fall into the Sun. Metal pollution of white dwarfs and transient spectroscopic signatures of young stars like β-Pic provide growing evidence that extra solar planetesimals can attain extreme orbital eccentricities and fall into their parent stars. Aims. We aim to develop a general, implementable, semi-analytical theory of secular eccentricity excitation of small bodies (planetesimals) in mean motion resonances with an eccentric planet valid for arbitrary values of the eccentricities and including the short-range force due to General Relativity. Methods. Our semi-analytic model for the restricted planar three-body problem does not make use of series expansion and therefore is valid for any eccentricity value and semi-major axis ratio. The model is based on the application of the adiabatic principle, which is valid when the precession period of the longitude of pericentre of the planetesimal is much longer than the libration period in the mean motion resonance. In resonances of order larger than 1 this is true except for vanishingly small eccentricities. We provide prospective users with a Mathematica notebook with implementation of the model allowing direct use. Results. We confirm that the 4:1 mean motion resonance with a moderately eccentric (e < ∼ 0.1) planet is the most powerful one to lift the eccentricity of planetesimals from nearly circular orbits to star-grazing ones. However, if the planet is too eccentric, we find that this resonance is unable to pump the planetesimal's eccentricity to a very high value. The inclusion of the General Relativity effect imposes a condition on the mass of the planet to drive the planetesimals into star-grazing orbits. For a planetesimal at ∼1 AU around a solar mass star (or white dwarf), we find a threshold planetary mass of about 17 Earth masses. We finally derive an analytical formula for this critical mass. Conclusions. Planetesimals can easily fall into the central star even in the presence of a single moderately eccentric planet, but only from the vicinity of the 4:1 mean motion resonance. For sufficiently high planetary masses the General Relativity effect does not prevent the achievement of star-grazing orbits.

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A second fundamental model for resonance

Cited by 264 publications

References 14 publications

On the Plutinos and Twotinos of the Kuiper Belt

On the Plutinos and Twotinos of the Kuiper Belt

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

Extreme secular excitation of eccentricity inside mean motion resonance

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