The elliptic restricted problem at the 3 : 1 resonance

Hadjidemetriou, John D.

doi:10.1007/bf00049463

Cited by 44 publications

(40 citation statements)

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“…For the details of computing periodic orbits in this problem and their stability, see Broucke (1968Broucke ( , 1969. Families of periodic orbits of the planar ER3BP were computed by Kribbel & Dvorak (1988) and Hadjidemetriou (1992Hadjidemetriou ( , 1993.…”

Section: The Elliptic Casementioning

confidence: 99%

The dynamics of the 1:2 resonant motion with Neptune in the 3D elliptic restricted three-body problem

Kotoulas¹

2005

A&A

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Abstract. We study the planar and the three-dimensional 1:2 resonant motion with Neptune (N1:2) in the framework of the elliptic restricted three-body problem. We examine the dynamics at the 1:2 mean motion resonance with Neptune in a hierarchy of models: the planar circular, the planar elliptic, the three-dimensional circular and the three-dimensional elliptic restricted three body problem. We start from the planar circular model in which we compute families of symmetric periodic orbits with stable and unstable segments. Using the circular planar model as the basic model, we proceed to more realistic models, namely the planar elliptic and the 3D circular problem. Families of symmetric periodic orbits of the above models bifurcate from the families of periodic orbits of the planar circular problem. Moreover, we compute families of symmetric periodic orbits in the 3D elliptic restricted three-body problem. These new families bifurcate from the families of periodic orbits of the planar elliptic and the 3D circular problem. The stability of all orbits is studied and the structure of the phase space is discussed in detail.

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Section: The Elliptic Casementioning

confidence: 99%

The dynamics of the 1:2 resonant motion with Neptune in the 3D elliptic restricted three-body problem

Kotoulas¹

2005

A&A

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“…Additionally, bifurcation points (denoted as BP0 in table 1) exist in the circular family C where the period is T = qπ. Such circular orbits are continued for e ′ = 0, providing periodic orbits of period T = 2qπ (Hadjidemetriou 1992). There are two different initial configurations giving rise to these families denoted as E p/q 01 and E p/q 02 :…”

Section: Second Order Resonancesmentioning

confidence: 99%

“…The basic theory and the numerical methods for the computation of periodic orbits in the planar elliptic problem have been described in detail by Broucke (1969a,b) and their application in our Solar system and the asteroidal motion has been shown (e.g. Hadjidemetriou, 1988Hadjidemetriou, ,1992. Applications also can be found for the dynamics of extra-solar systems (Haghighipour et al, 2003).…”

Section: Introductionmentioning

confidence: 99%

Planar periodic orbits in exterior resonances with Neptune

Voyatzis

Kotoulas

2005

Planetary and Space Science

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In the framework of the planar restricted three body problem we study a considerable number of resonances associated to the Kuiper Belt dynamics and located between 30 and 48 a.u. Our study is based on the computation of resonant periodic orbits and their stability. Stable periodic orbits are surrounded by regular librations in phase space and in such domains the capture of trans-Neptunian object is possible. All the periodic orbits found are symmetric and there is evidence for the existence of asymmetric ones only in few cases. In the present work first, second and third order resonances are under consideration. In the planar circular case we found that most of the periodic orbits are stable. The families of periodic orbits are temporarily interrupted by collisions but they continue up to relatively large values of the Jacobi constant and highly eccentric regular motion exists for all cases. In the elliptic problem and for a particular eccentricity value of the primary bodies the periodic orbits are isolated. The corresponding families, where they belong to, bifurcate from specific periodic orbits of the circular problem and seem to continue up to the rectilinear problem. Both stable and unstable orbits are obtained for each case. In the elliptic problem the unstable orbits found are associated with narrow chaotic domains in phase space. The evolution of the orbits, which are located in such chaotic domains, seems to be practically regular and bounded for long time intervals.

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“…However both models have their limitations, as we shall explain: -The averaged model on which these two mapping models are based is valid only for values of the eccentricity smaller than 0.3. This means that the high eccentricity resonances that exist in the real problem (the elliptic restricted three body problem) at eccentricities e = 0.8 are missing (Hadjidemetriou 1992 (Hadjidemetriou 1993). The corrected mapping behaves in a much different way and the eccentricity can jump to very high values of the eccentricity, 0.9 or even larger.…”

Section: Remarks On the Previous Two Mappingsmentioning

confidence: 99%