Synchronous motion in the Kinoshita problem

Breiter, S.; Melendo, B.; Bartczak, P.; Wytrzyszczak, I.

doi:10.1051/0004-6361:20053031

Cited by 39 publications

(18 citation statements)

References 26 publications

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“…Stability maps are commonly used to study dynamical systems, and they allow a general understanding of their resonant structure by highlighting chaotic and stable zones. Aside from their use in many other physical domains, stability maps have been used in celestial mechanics to study the dynamics of various objects like asteroids , binary asteroids (Breiter et al 2005), Trojans asteroids (Robutel & Gabern 2006), satellites (Callegari & Yokoyama 2010), planets of the Solar System (Michtchenko & Ferraz-Mello 2001), or extrasolar planetary systems (Érdi et al 2004). …”

Section: Stability Mapsmentioning

confidence: 99%

The long-term dynamics of the Jovian irregular satellites

Frouard

Vienne

Fouchard

2011

A&A

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Context. The dynamical region of the Jovian irregular satellites presents an interesting web of resonances that are not yet fully understood. Of particular interest is the influence of the resonances on the stochasticity of the individual orbits of the satellites, as well as on the long-term chaotic diffusion of the different families of satellites. Aims. We make a systematic numerical study of the satellite region to determine the important resonances for the dynamics, to search for the chaotic zones, and to determine their influences on the dynamics of the satellites. We also compare these numerical results to previous analytical works. Methods. Using extensive numerical integrations of the satellites along with an indicator of chaos (MEGNO), we show global and detailed views of the retrograde and prograde regions for various dynamical models of increasing complexity and indicate the appearance of the different types of resonances and the implied chaos. Results. Along with secular and mean motion resonances that shape the dynamical regions of the satellites, we report a number of resonances involving the Great Inequality, and which are present in the system thanks to the wide range of the values of frequencies of the pericenter available for the satellites. The chaotic diffusion of the satellites is also studied and shows the long-term stability of the Ananke and Carme families, in contrast to the Pasiphae family.

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Section: Stability Mapsmentioning

confidence: 99%

The long-term dynamics of the Jovian irregular satellites

Frouard

Vienne

Fouchard

2011

A&A

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“…The authors prefer the so‐called MEGNO indicator proposed by Cincotta & Simó (2000)– that choice is justified by the successful application of this method in our previous works (e.g. Breiter et al 2005; Goździewski, Konacki & Maciejewski 2006, and the references therein). The definition of MEGNO for a discrete map is (Cincotta et al 2003) where …”

Section: Numerical Toolsmentioning

confidence: 99%

The long-term stability of extrasolar system HD 37124. Numerical study of resonance effects

Goździewski¹,

Breiter

Borczyk

2007

Monthly Notices of the Royal Astronomical Society

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We describe numerical tools for the stability analysis of extrasolar planetary systems. In particular, we consider the relative Poincaré variables and symplectic integration of the equations of motion. We apply the tangent map to derive a numerically efficient algorithm of the fast indicator Mean Exponential Growth factor of Nearby Orbits (MEGNO), a measure of the maximal Lyapunov exponent, that helps to distinguish chaotic and regular configurations. The results concerning the three‐planet extrasolar system HD 37124 are presented and discussed. The best‐fitting solutions found in earlier works are studied more closely. The system involves Jovian planets with similar masses. The orbits have moderate eccentricities, nevertheless the best‐fitting solutions are found in dynamically active region of the phase space. The long‐term stability of the system is determined by a net of low‐order two‐body and three‐body mean motion resonances. In particular, the three‐body resonances may induce strong chaos that leads to self‐destruction of the system after Myr of apparently stable and bounded evolution. In such a case, numerically efficient dynamical maps are useful to resolve the fine structure of the phase space and to identify the sources of unstable behaviour.

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“…Marsden et al (1999) and Marsden et al (2000) introduce discrete Euler-Poincaré and Lie-Poisson equations, where the discrete dynamics on a Lie group are reduced to the dynamics on the corresponding Lie algebra in the absence of potential. Lie-Poisson integrators have been developed by splitting the Hamiltonian into separate integrable terms for an elliptical body (Touma and Wisdom, 1994;Breiter et al 2005a) and for the secular spin dynamics of a rigid body (Breiter et al 2005b).…”

Section: Introductionmentioning

confidence: 99%

Lie group variational integrators for the full body problem in orbital mechanics

Lee

Leok

McClamroch

2007

Celestial Mech Dyn Astr

107

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Equations of motion, referred to as full body models, are developed to describe the dynamics of rigid bodies acting under their mutual gravitational potential. Continuous equations of motion and discrete equations of motion are derived using Hamilton's principle. These equations are expressed in an inertial frame and in relative coordinates. The discrete equations of motion, referred to as a Lie group variational integrator, provide a geometrically exact and numerically efficient computational method for simulating full body dynamics in orbital mechanics; they are symplectic and momentum preserving, and they exhibit good energy behavior for exponentially long time periods. They are also efficient in only requiring a single evaluation of the gravity forces and moments per time step. The Lie group variational integrator also preserves the group structure without the use of local charts, reprojection, or constraints. Computational results are given for the dynamics of two rigid dumbbell bodies acting under their mutual gravity; these computational results demonstrate the superiority of the Lie group variational integrator compared with integrators that are not symplectic or do not preserve the Lie group structure.

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Synchronous motion in the Kinoshita problem

Cited by 39 publications

References 26 publications

The long-term dynamics of the Jovian irregular satellites

The long-term dynamics of the Jovian irregular satellites

The long-term stability of extrasolar system HD 37124. Numerical study of resonance effects

Lie group variational integrators for the full body problem in orbital mechanics

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