The role of chaotic resonances in the Solar System

Murray, Norman; Holman, Matthew J.

doi:10.1038/35071000

Cited by 84 publications

(43 citation statements)

References 47 publications

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“…Of particular importance will be the inclusion of systems with multiple planets per star (Murray & Holman 2001). Full realization of the capabilities of the GRAPE-6 hardware when the 1-TÑop board becomes available will allow larger particle numbers, and consequently more planetary systems, to be studied per simulation.…”

Section: Discussionmentioning

confidence: 99%

Free‐floating Planets in Stellar Clusters: Not So Surprising

Hurley

Shara

2002

ApJ

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We Ðnd that free-Ñoating planets can remain bound to an open cluster for much longer than was previously calculated : of the order of the cluster half-mass relaxation timescale as opposed to the crossing time. This result is based on N-body simulations performed with the new GRAPE-6 special purpose hardware and is important in the context of the preliminary detection of a population of free-Ñoating, substellar objects in the globular cluster M22. The planets in our N-body study are of Jupiter mass and are initially placed in circular orbits of between 0.05 and 50 AU about a parent star whose mass is chosen from a realistic initial mass function. The presence of the free-Ñoating planets is the result of dynamical encounters between planetary systems and the cluster stars. Most planets are liberated from their parent star in, or near, the cluster core, and then drift outward on a timescale of D108È109 yr. This still implies the existence of many (D100) planets per star if the M22 result is conÐrmed.

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Section: Discussionmentioning

confidence: 99%

Free‐floating Planets in Stellar Clusters: Not So Surprising

Hurley

Shara

2002

ApJ

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“…Specific applications of the Lyapunov exponents to the circular restricted 3-body problem were discussed by many authors, including Gonczi & Froeschlé (1981), Jefferys & Yi (1983), Lecar et al (1992), Milani & Nobili (1992), Smith & Szebehely (1993), Murray & Holman (2001), and others. Some of these authors also considered the so-called Lyapunov time, which measures the e-folding time for the divergence of nearby trajectories.…”

Section: Lyapunov Exponentsmentioning

confidence: 99%

“…The focus of this paper is the analysis of orbital stability by Lyapunov exponents, which are among the most commonly used numerical tools for investigating chaotic behaviour of different dynamical systems (e.g., Hilborn 1994). The exponents have already repeatedly been used in orbital mechanics studies of the Solar System (e.g., Lissauer 1999;Murray & Holman 2001). For example, Lissauer (1999) discussed the long-term stability of the eight Solar System planets, while also considering previous studies.…”

Section: Introductionmentioning

confidence: 99%

The instability transition for the restricted 3-body problem

Quarles

Eberle

Musielak

et al. 2011

A&A

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Aims. We establish a criterion for the stability of planetary orbits in stellar binary systems by using Lyapunov exponents and power spectra for the special case of the circular restricted 3-body problem (CR3BP). The criterion augments our earlier results given in the two previous papers of this series where stability criteria have been developed based on the Jacobi constant and the hodograph method.Methods. The centerpiece of our method is the concept of Lyapunov exponents, which are incorporated into the analysis of orbital stability by integrating the Jacobian of the CR3BP and orthogonalizing the tangent vectors via a well-established algorithm originally developed by Wolf et al. The criterion for orbital stability based on the Lyapunov exponents is independently verified by using power spectra. The obtained results are compared to results presented in the two previous papers of this series. Results. It is shown that the maximum Lyapunov exponent can be used as an indicator for chaotic behaviour of planetary orbits, which is consistent with previous applications of this method, particularly studies for the Solar System. The chaotic behaviour corresponds to either orbital stability or instability, and it depends solely on the mass ratio μ of the binary components and the initial distance ratio ρ 0 of the planet relative to the stellar separation distance. Detailed case studies are presented for μ = 0.3 and 0.5. The stability limits are characterized based on the value of the maximum Lyapunov exponent. However, chaos theory as well as the concept of Lyapunov time prevents us from predicting exactly when the planet is ejected. Our method is also able to indicate evidence of quasi-periodicity. Conclusions. For different mass ratios of the stellar components, we are able to characterize stability limits for the CR3BP based on the value of the maximum Lyapunov exponent. This theoretical result allows us to link the study of planetary orbital stability to chaos theory noting that there is a large array of literature on the properties and significance of Lyapunov exponents. Although our results are given for the special case of the CR3BP, we expect that it may be possible to augment the proposed Lyapunov exponent criterion to studies of planets in generalized stellar binary systems, which is strongly motivated by existing observational results as well as results expected from ongoing and future planet search missions.

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“…Along this transition, atypical structures like the first chaotic regions to appear, or the last regular islands to survive, play a crucial role in many physical situations. For instance, resonances and separatrices determine the fate of planetary systems [3,4], and localised objects like solitons and breathers provide mechanisms of energy transport in nonlinear systems such as Bose-Einstein condensates and biological molecules [5,6]. Unfortunately, despite the fundamental progress made in the last years, most of the numerical methods to locate these 'rare' trajectories are confined to low-dimensional or toy models, while the realms of statistical physics, chemical reactions, or astronomy are still hard to reach.…”

Section: Legendre Transformmentioning

confidence: 99%

Probing rare physical trajectories with Lyapunov weighted dynamics

Tailleur¹,

Kurchan²

2007

Nature Phys

122

227

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The transition from order to chaos has been a major subject of research since the work of Poincare, as it is relevant in areas ranging from the foundations of statistical physics to the stability of the solar system. Along this transition, atypical structures like the first chaotic regions to appear, or the last regular islands to survive, play a crucial role in many physical situations. For instance, resonances and separatrices determine the fate of planetary systems, and localised objects like solitons and breathers provide mechanisms of energy transport in nonlinear systems such as Bose-Einstein condensates and biological molecules. Unfortunately, despite the fundamental progress made in the last years, most of the numerical methods to locate these 'rare' trajectories are confined to low-dimensional or toy models, while the realms of statistical physics, chemical reactions, or astronomy are still hard to reach. Here we implement an efficient method that allows one to work in higher dimensions by selecting trajectories with unusual chaoticity. As an example, we study the Fermi-Pasta-Ulam nonlinear chain in equilibrium and show that the algorithm rapidly singles out the soliton solutions when searching for trajectories with low level of chaoticity, and chaotic-breathers in the opposite situation. We expect the scheme to have natural applications in celestial mechanics and turbulence, where it can readily be combined with existing numerical methodsComment: Accepted for publication in Nature Physics. Due to size restrictions, the figures are not of high qualit

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The role of chaotic resonances in the Solar System

Cited by 84 publications

References 47 publications

Free‐floating Planets in Stellar Clusters: Not So Surprising

Free‐floating Planets in Stellar Clusters: Not So Surprising

The instability transition for the restricted 3-body problem

Probing rare physical trajectories with Lyapunov weighted dynamics

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