The instability transition for the restricted 3-body problem

Eberle, J.; Cuntz, M.; Musielak, Z. E.

doi:10.1051/0004-6361:200809758

Cited by 39 publications

(59 citation statements)

References 32 publications

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“…A comparison with the zero-velocity contour of the planetary orbit shows that this criterion, which is a sufficient criterion for orbital instability, closely coincides with the opening of the zero-velocity contour at the Lagrange point L3, located to the right of the primary star, as discussed by Cuntz et al (2007) and Eberle et al (2008). If instability occurs, extremely large effective eccentricities are found, indicating a highly hyperbolic planetary orbit in the synodic coordinate system, a stark indication of planetary escape.…”

Section: Discussionsupporting

confidence: 63%

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Orbital stability of Earth-type planets in stellar binary systems

2009

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Abstract. An important factor in estimating the likelihood of life elsewhere in the Universe is determining the stability of a planet's orbit. A significant fraction of stars like the Sun occur in binary systems which often has a considerable effect on the stability of any planets in such a system. In an effort to determine the stability of planets in binary star systems, we conducted a numerical simulation survey of several mass ratios and initial conditions. We then estimated the stability of the planetary orbit using a method that utilizes the hodograph to determine the effective eccentricity of the planetary orbit. We found that this method can serve as an orbital stability criterion for the planet.

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Section: Discussionsupporting

confidence: 63%

“…Previous results have been obtained by, e.g., Musielak et al (2005), Cuntz et al (2007), and Eberle et al (2008). In the following, we present a new method that relies on a differential geometrical approach based on the analysis of the curvature of the hodograph in the synodic coordinate system.…”

Section: Introduction and Methodsmentioning

confidence: 99%

Orbital stability of Earth-type planets in stellar binary systems

2009

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“…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: 96%

“…Following the conventions described by Eberle et al (2008), we write the equations of motion in terms of the parameters μ and ρ 0 with μ and α = 1 − μ being related to the ratio of the two stellar masses m 1 and m 2 (see below). Moreover, R 0 denotes the initial distance of the planet from its host star, the more massive of the two stars with mass m 1 , whereas D denotes the distance between the two stars, allowing us to define ρ 0 .…”

Section: Basic Equationsmentioning

confidence: 99%

“…It has also been the focus of the previous papers in this series. In Paper I, Eberle et al (2008) obtained the planetary stability limits using a criterion based on Jacobi's integral and Jacobi's constant. The method, also related to the concept of Hill stability (Roy 2005), showed that orbital stability can be guaranteed only if the initial position of the planet lies within a welldefined limit determined by the mass ratio of the stellar components.…”

Section: Introductionmentioning

confidence: 99%

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The instability transition for the restricted 3-body problem

Quarles

Eberle

Musielak

et al. 2011

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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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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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The instability transition for the restricted 3-body problem

Cited by 39 publications

References 32 publications

Orbital stability of Earth-type planets in stellar binary systems

Orbital stability of Earth-type planets in stellar binary systems

The instability transition for the restricted 3-body problem

Study of resonances for the restricted 3‐body problem

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