On predicting long-term orbital instability - A relation between the Lyapunov time and sudden orbital transitions

Lecar, M.; Franklin, F. A.; Murison, M. A.

doi:10.1086/116312

Cited by 64 publications

(61 citation statements)

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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 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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Section: Lyapunov Exponentsmentioning

confidence: 99%

The instability transition for the restricted 3-body problem

Quarles

Eberle

Musielak

et al. 2011

A&A

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“…Except in the case of ''bounded chaos,'' orbital parameters for the planet undergo unbounded random walks, leading to ejection on a timescale called the event timescale (T e ). Although T e fluctuates depending on the system, studies have shown that it roughly correlates with T L (see, e.g., Lecar et al 1992).…”

Section: à3mentioning

confidence: 99%

Resonance Overlap Is Responsible for Ejecting Planets in Binary Systems

Mudryk

2006

ApJ

107

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A planet orbiting around a star in a binary system experiences both secular and resonant perturbations from the companion star. It may be dislodged from its host star if it is simultaneously affected by two or more resonances. We find that overlap between subresonances lying within mean-motion resonances (mostly of the j :1 type) can account for the boundary of orbital stability within binary systems first observed in numerical studies (e.g., Holman & Wiegert). Strong secular forcing from the companion displaces the centroids of different subresonances, producing large regions of resonance overlap. Planets lying within these overlapping regions experience chaotic diffusion, which in most cases leads to their eventual ejection. The overlap region extends to shorter period orbits as either the companion's mass or its eccentricity increase, with boundaries largely agreeing with those obtained by Holman & Wiegert. Furthermore, we find the following two results: First, at a given binary mass ratio, the instability boundary as a function of eccentricity appears jagged, with jutting peninsulas and deep inlets corresponding to islands of instability and stability, respectively; as a result, the largest stable orbit could be reduced from the Holman & Wiegert values by as much as 20%. Second, very high-order resonances (e.g., 50:3) do not significantly modify the instability boundary; these weak resonances, while producing slow chaotic diffusion that may be missed by finiteduration numerical integrations, do not contribute markedly to planet instability. We present some numerical evidence for the first result. More extensive experiments are called for to confirm these conclusions. For the special case of circular binaries, we are intrigued to find that the Hill criterion (based on the critical Jacobi integral) yields an instability boundary that is very similar to that obtained by resonance overlap arguments, making the former both a necessary and a sufficient condition for planet instability.

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“…The time required for that to happen is called the "event time" (T E ). Several studies involving the instability of asteroids disturbed by Jupiter and/or other planets (Soper et al 1990; Lecar et al 1992;Franklin et al 1993;Levison & Duncan 1993;Murison et al 1994;Whipple 1995) have claimed that there is a simple relation between the Lyapunov time, T L , and the event time, T E (to become a Jupiter's crosser, for example) given by…”

Section: Introductionmentioning

confidence: 99%

Short Lyapunov time: a method for identifying confined chaos

Winter

Mourão

Winter

2010

A&A

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Context. The orbital instability of minor solar system bodies (asteroids, small satellites, moonlets, and particles) is frequently studied in terms of the Lyapunov characteristic exponent (LCE). Asteroids interior to Jupiter often exihibit very short Lyapunov times, T L , and very large radial variations, becoming Jupiter's crossers and escapers. However, a few cases of asteroids with very short T L and no significant radial variation have been found. These orbits were called "confined chaos" or even "stable chaos". This feature also appeared in the case of moonlets embedded in Saturn's F ring and disturbed by the nearby satellites Prometheus and Pandora. Aims. We present a simple approach to estimating the contribution of the radial component of the LCE to identify trajectories in the "confined chaos" regime. Methods. To estimate the radial contribution to the maximum LCE, we considered a rotating reference system in which one of the axis was aligned with the radial direction of the reference trajectory. Measuring the distance in the phase space between the two nearby orbits then allowed us to separate the contribution of the radial component from the others. We applied the method to two different dynamical systems: (a) an asteroid around the Sun disturbed by Jupiter; (b) a moonlet of Saturn's F-ring disturbed by the satellites Prometheus and Pandora. Results. In all cases, we found that the method of comparing the radial contribution of the LCE to the entire contribution allows us to correctly distinguish between confined chaos and escapers.

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On predicting long-term orbital instability - A relation between the Lyapunov time and sudden orbital transitions

Cited by 64 publications

References 0 publications

The instability transition for the restricted 3-body problem

The instability transition for the restricted 3-body problem

Resonance Overlap Is Responsible for Ejecting Planets in Binary Systems

Short Lyapunov time: a method for identifying confined chaos

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