The instability transition for the restricted 3-body problem

Eberle, J.; Cuntz, M.

doi:10.1051/0004-6361/200912500

Cited by 10 publications

(16 citation statements)

References 23 publications

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“…Comparisons to previously established criteria for stability show that our results are consistent with previously obtained stability limits (e.g., Holman & Wiegert 1999;Musielak et al 2005;Cuntz et al 2007;Eberle et al 2008;Eberle & Cuntz 2010). In Eberle et al (2008), Paper I, the onset of instability was related to the topology of the ZVC, whereas in Eberle & Cuntz (2010), Paper II, it was shown that the onset of orbital instability occurs when the median of the effective eccentricity distribution exceeds unity.…”

Section: Discussionsupporting

confidence: 91%

“…Using the method of Lyapunov exponents, we are able to verify and extend the methods described by Eberle et al (2008) and Eberle & Cuntz (2010). Absolute orbital stability can be more rigorously shown through the Lyapunov exponent method.…”

Section: Resultsmentioning

confidence: 94%

“…In Eberle et al (2008), Paper I, the onset of instability was related to the topology of the ZVC, whereas in Eberle & Cuntz (2010), Paper II, it was shown that the onset of orbital instability occurs when the median of the effective eccentricity distribution exceeds unity. Both results are consistent with the findings of Paper III, i.e., the inspection of the maximum Lyapunov exponent.…”

Section: Discussionmentioning

confidence: 99%

“…In Paper II, Eberle & Cuntz (2010) followed another theoretical concept based on the hodograph eccentricity criterion. This method relies on an approach given by differential geometry that analyzes the curvature of the planetary orbit in the synodic coordinate system.…”

Section: Introductionmentioning

confidence: 99%

“…Previously, the work by Eberle & Cuntz (2010) dealt with the assessment of short-time orbital stability, encompassing time scales of 10 3 yr or less. One of the findings was the identification of a quasi-periodic region in the stellar mass and distance (μ, ρ 0 ) parameter space (see Sect.…”

Section: Introductionmentioning

confidence: 99%

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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: Discussionsupporting

confidence: 91%

Section: Resultsmentioning

confidence: 94%

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The instability transition for the restricted 3-body problem

Quarles

Eberle

Musielak

et al. 2011

A&A

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Study of resonances for the restricted 3‐body problem

Quarles¹,

Musielak²,

Cuntz³

2012

Astron Nachr

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Our aim is to identify and classify mean‐motion resonances (MMRs) for the coplanar circular restricted three‐body problem (CR3BP) for mass ratios between 0.10 and 0.50. Our methods include the maximum Lyapunov exponent, which is used as an indicator for the location of the resonances, the Fast Fourier Transform (FFT) used for determining what kind of resonances are present, and the inspection of the orbital elements to classify the periodicity. We show that the 2:1 resonance occurs the most frequently. Among other resonances, the 3:1 resonance is the second most common, and furthermore both 3:2 and 5:3 resonances occur more often than the 4:1 resonance. Moreover, the resonances in the coplanar CR3BP are classified based on the behaviour of the orbits. We show that orbital stability is ensured for high values of resonance (i.e., high ratios) where only a single resonance is present. The resonances attained are consistent with the previously established resonances for the solar system, i.e., specifically, in regards to the asteroid belt. Previous work employed digital filtering and Lyapunov characteristic exponents to determine stochasticity of the eccentricity, which is found to be consistent with our usage of Lyapunov exponents as an alternate approach based on varying the mass ratio instead of the eccentricity. Our results are expected to be of principal interest to future studies, including augmentations to observed or proposed resonances, of extra‐solar planets in binary stellar systems (© 2012 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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Astronomical Applications

Musielak¹,

Quarles²

2017

SpringerBriefs in Astronomy

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The instability transition for the restricted 3-body problem

Cited by 10 publications

References 23 publications

The instability transition for the restricted 3-body problem

The instability transition for the restricted 3-body problem

Study of resonances for the restricted 3‐body problem

Astronomical Applications

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