Periodic orbits in three-dimensional planetary systems

Ichtiaroglou, S.; Katopodis, Konstantinos P.; Michalodimitrakis,

doi:10.1007/bf02715072

Cited by 13 publications

(10 citation statements)

References 9 publications

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“…In practice the three-dimensional (spatial) elliptic Hill problem can be derived most directly from the elliptic restricted three-body problem in a similar procedure to the circular case. The elliptic Hill Hamiltonian is the following (Szebehely 1967;Ichtiaroglou 1980;Moons et al 1988;Llibre & Pinol 1990;Brumberg & Ivanova 1990;Astakhov & Farrelly 2004;Astakhov et al 2005;Palacian et al 2006):…”

Section: Hamiltonian and Equations Of Motionmentioning

confidence: 99%

Production of trans-Neptunian binaries through chaos-assisted capture

Lee

Astakhov

Farrelly

2007

Monthly Notices of the Royal Astronomical Society

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The recent discovery of binary objects in the Kuiper-belt opens an invaluable window into past and present conditions in the trans-Neptunian part of the Solar System. For example, knowledge of how these objects formed can be used to impose constraints on planetary formation theories. We have recently proposed a binary-object formation model based on the notion of chaos-assisted capture. Here we present a more detailed analysis with calculations performed in the spatial (three-dimensional) three- and four-body Hill approximations. It is assumed that the potential binary partners are initially following heliocentric Keplerian orbits and that their relative motion becomes perturbed as these objects undergo close encounters. First, the mass, velocity, and orbital element distribu- tions which favour binary formation are identified in the circular and elliptical Hill limits. We then consider intruder scattering in the circular Hill four-body problem and find that the chaos-assisted capture mechanism is consistent with observed, apparently randomly distributed, binary mutual orbit inclinations. It also predicts asymmetric distributions of retrograde versus prograde orbits. The time-delay induced by chaos on particle transport through the Hill sphere is analogous to the formation of a resonance in a chemical reaction. Implications for binary formation rates are considered and the 'fine-tuning' problem recently identified by Noll et al. (2007) is also addressed.Comment: submitted to MNRA

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Section: Hamiltonian and Equations Of Motionmentioning

confidence: 99%

Production of trans-Neptunian binaries through chaos-assisted capture

Lee

Astakhov

Farrelly

2007

Monthly Notices of the Royal Astronomical Society

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“…Concerning spatial three-body models, families of symmetric periodic orbits (s.p.o.) have been mainly computed for asteroids and TNOs (Ichtiaroglou et al, 1989;Hadjidemetriou, 1993; or for exoplanetary systems (Antoniadou and Voyatzis, 2013;Antoniadou and Voyatzis, 2014). Generally, families of s.p.o.…”

Section: Introductionmentioning

confidence: 99%

“…also exist and emanate from L 4 for the circular planar restricted (Zagouras et al, 1996) and these families continue in the general planetary problem (Giuppone et al, 2010;Hadjidemetriou and Voyatzis, 2011).Concerning spatial three-body models, families of symmetric periodic orbits (s.p.o.) have been mainly computed for asteroids and TNOs (Ichtiaroglou et al, 1989;Hadjidemetriou, 1993; or for exoplanetary systems (Antoniadou and Voyatzis, 2013;Antoniadou and Voyatzis, 2014). Generally, families of s.p.o.…”

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confidence: 99%

Inclined asymmetric librations in exterior resonances

Voyatzis

Tsiganis

Antoniadou

2018

Celest Mech Dyn Astr

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Librational motion in celestial mechanics is generally associated with the existence of stable resonant configurations and signified by the existence of stable periodic solutions and oscillation of critical (resonant) angles. When such an oscillation takes place around a value different than 0 or π, the libration is called asymmetric. In the context of the planar circular restricted three-body problem (CRTBP), asymmetric librations have been identified for the exterior meanmotion resonances (MMRs) 1 : 2, 1 : 3 etc. as well as for co-orbital motion (1 : 1). In exterior MMRs the massless body is the outer one. In this paper, we study asymmetric librations in the 3-dimensional space. We employ the computational approach of Markellos (1978) and compute families of asymmetric periodic orbits and their stability. Stable, asymmetric periodic orbits are surrounded in phase space by domains of initial conditions which correspond to stable evolution and librating resonant angles. Our computations were focused on the spatial circular restricted three-body model of the Sun-Neptune-TNO system (TNO = trans-Neptunian object). We compare our results with numerical integrations of observed TNOs, which reveal that some of them perform 1 : 2-resonant, inclined asymmetric librations. For the stable 1 : 2 TNOs librators, we find that their libration seems to be related with the vertically stable planar asymmetric orbits of our model, rather than the 3-dimensional ones found in the present study.keywords circular restricted TBP -exterior resonances -spatial asymmetric periodic orbits -trans-Neptunian object dynamics ) used an analytical approach to show the existence of asymmetric librations in 1 : p exterior MMRs and their absence in other exterior MMRs as e.g. the 2 : 3 and 3 : 4. We should remark that i) a.p.o are found also in the elliptic planar restricted and in the general planetary planar problem (Antoniadou et al., 2011) ii) 1 : 1 families of a.p.o. also exist and emanate from L 4 for the circular planar restricted (Zagouras et al., 1996) and these families continue in the general planetary problem (Giuppone et al., 2010;Hadjidemetriou and Voyatzis, 2011).Concerning spatial three-body models, families of symmetric periodic orbits (s.p.o.) have been mainly computed for asteroids and TNOs (Ichtiaroglou et al., 1989;Hadjidemetriou, 1993; or for exoplanetary systems (Antoniadou and Voyatzis, 2013;Antoniadou and Voyatzis, 2014). Generally, families of s.p.o. bifurcate from planar periodic orbits which are critical with respect to their vertical stability (Hénon, 1973) and are called vertical critical orbits (v.c.o.). These families extend up to some critical value of the inclination, i, or terminate at i = 180 • (planar retrograde orbit). With respect to their linear stability, the studies cited above showed that most prograde orbits of moderate or high inclination are unstable. Nevertheless, segments of stable orbits also exist providing restricted phase space domains, where resonant angles librate around 0 or π. Markellos (1978)...

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“…Motion around stable 2D periodic orbits is maintained when there are small deviations out of the plane only if the vertical stability index −2 < k 3 < 2. When k 3 = 2 (vertical critical orbit or vco) a bifurcation into a new family of 3D periodic orbits with the same multiplicity may occur (Hénon, 1973;Ichtiaroglou & Michalodimitrakis, 1980).…”

Section: Computation Of Periodic Orbitsmentioning

confidence: 99%

A study of the 1/2 retrograde resonance: Periodic orbits and resonant capture

Morais,

Namouni,

Voyatzis

et al. 2021

Preprint

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We describe the families of periodic orbits in the 2-dimensional 1/2 retrograde resonance at mass ratio 10 −3 , analyzing their stability and bifurcations into 3-dimensional periodic orbits. We explain the role played by periodic orbits in adiabatic resonance capture, in particular how the proximity between a stable family and an unstable family with a nearly critical segment, associated with Kozai separatrices, determines the transition between distinct resonant modes observed in numerical simulations. Combining the identification of stable, critical and unstable periodic orbits with analytical modeling, resonance capture simulations and computation of stability maps helps to unveil the complex 3-dimensional structure of resonances.

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Periodic orbits in three-dimensional planetary systems

Cited by 13 publications

References 9 publications

Production of trans-Neptunian binaries through chaos-assisted capture

Production of trans-Neptunian binaries through chaos-assisted capture

Inclined asymmetric librations in exterior resonances

A study of the 1/2 retrograde resonance: Periodic orbits and resonant capture

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