Comparative Study of the 2:3 and 3:4 Resonant Motion with Neptune: An Application of Symplectic Mappings and Low Frequency Analysis

Kotoulas, Thomas; Voyatzis, George

doi:10.1023/b:cele.0000023391.85690.31

Cited by 30 publications

(23 citation statements)

References 31 publications

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“…This means that our guiding lines are actually guiding lines for the problem with the Hamiltonian given by Equations (9-12) which can differ from the full Hamiltonian. This approximation is used in many studies of external resonances recently by Kotoulas and Voyatzis (2004) or Hadjifontinou and Hadjidemetriou (2002).…”

Section: Limitations Of the Method Conclusionmentioning

confidence: 99%

A Non-Planar Circular Model for the 4/7 Resonance

Sidlichovsky

2005

Celestial Mech Dyn Astr

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An adiabatic approximation for the non-planar, circular, restricted 3BP is presented for the external resonance 4/7. It can be used as a model for resonant Kuiper belt objects. The Hamiltonian is truncated at the fourth order in eccentricities and inclinations. After averaging, we have a system of two degrees of freedom with two frequencies.Numerical calculations show that the ratio of these frequencies is ∼10 2 . Having introduced suitable canonical variables, we used the adiabatic approach introduced by Wisdom in a different context. We left slow variables frozen and after solving the pendulum problem for fast variables, we used the averaged effect of fast variables on slow variables. In this way we obtained the guiding trajectories for slow variables as contour lines of adiabatic invariant. We discuss the existence of a chaotic region which is formed by trajectories crossing a critical curve which corresponds to the separatrix of fast pendulum motion, where the assumption of sharp division between fast and slow frequencies is not correct and the adiabatic theory fails. The model works well for e ∼ 0.1 and can be used for finding the chaotic regions, but for e ∼ 0.17 it becomes unsatisfactory due to truncation and bad convergence of the Laplace expansion. Qualitatively it can, however, help us to understand how the protective mechanism works as the interplay of mean motion and Kozai-Lidov resonance.

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Section: Limitations Of the Method Conclusionmentioning

confidence: 99%

A Non-Planar Circular Model for the 4/7 Resonance

Sidlichovsky

2005

Celestial Mech Dyn Astr

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“…Nevertheless, the localization of a stable periodic orbit in this case is associated with the existence of regular orbits in the neighborhood of such a periodic orbit. Additionally, the numerical simulations show that the eccentricity of Neptune introduce small perturbations to the circular problem that do not affect significantly the long term stability of the resonant librating motion near the stable periodic orbits of the circular problem (Malhotra, 1996;Kotoulas and Voyatzis, 2004a).…”

Section: Introductionmentioning

confidence: 93%

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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“…Since that time, it has been used to probe the dynamics of many different astrophysical systems [Goździewski et al, 2001, Goździewski, 2002, Laughlin et al, 2002, Pilat-Lohinger et al, 2003. For examples using the MEGNO method and other chaos indicator methods, we recommend Pavlov and Maciejewski [2003], Kotoulas and Voyatzis [2004], Barnes and Greenberg [2006], Funk et al [2009], , Hinse et al [2010], Mestre et al [2011], and Satyal et al [2013].…”

Section: Mean Exponential Growth Factor Of Nearby Orbitsmentioning

confidence: 99%

The three-body problem

2014

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Abstract. The three-body problem, which describes three masses interacting through Newtonian gravity without any restrictions imposed on the initial positions and velocities of these masses, has attracted the attention of many scientists for more than 300 years. In this paper, we present a review of the three-body problem in the context of both historical and modern developments. We describe the general and restricted (circular and elliptic) three-body problems, different analytical and numerical methods of finding solutions, methods for performing stability analysis, search for periodic orbits and resonances, and application of the results to some interesting astronomical and space dynamical settings. We also provide a brief presentation of the general and restricted relativistic three-body problems, and discuss their astronomical applications.

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Comparative Study of the 2:3 and 3:4 Resonant Motion with Neptune: An Application of Symplectic Mappings and Low Frequency Analysis

Cited by 30 publications

References 31 publications

A Non-Planar Circular Model for the 4/7 Resonance

A Non-Planar Circular Model for the 4/7 Resonance

Planar periodic orbits in exterior resonances with Neptune

The three-body problem

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