Stringent Criteria for Stable and Unstable Planetary Orbits in Stellar Binary Systems

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

doi:10.1086/523873

Cited by 52 publications

(38 citation statements)

References 25 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%

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%

The instability transition for the restricted 3-body problem

Quarles

Eberle

Musielak

et al. 2011

A&A

Self Cite

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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: 62%

“…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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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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“…In the case of the CR3BP, the stability limits previously obtained have recently been updated by Cuntz et al (2007) and (Eberle et al 2008, Paper I) based on an analytical approach. Cuntz et al and Eberle et al were able to derive a stringent (sufficient) criterion for orbital stability based on Jacobi's constant and Jacobi's integral, akin to the criterion of "Hill stability".…”

Section: Introductionmentioning

confidence: 99%

The instability transition for the restricted 3-body problem

Eberle

Cuntz

2010

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Aims. We present a new method that allows identifying the onset of orbital instability, as well as quasi-periodicity and multiperiodicity, for planets in binary systems. This method is given for the special case of the circular restricted 3-body problem (CR3BP). Methods. Our method relies on an approach given by differential geometry that analyzes the curvature of the planetary orbit in the synodic coordinate system. The centerpiece of the method consists in inspecting the effective (instantaneous) eccentricity of the orbit based on the hodograph in rotated coordinates and in calculating the mean and median values of the eccentricity distribution. Results. Orbital stability and instability can be mapped by numerically inspecting the hodograph and/or the effective eccentricity of the orbit in the synodic coordinate system. The behavior of the system 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 noting that the stellar components move on circular orbits. Our study indicates that orbital instability occurs when the median of the effective eccentricity distribution exceeds unity. This instability criterion can be compared to other criteria, including those based on Jacobi's integral and the zero-velocity contour of the planetary orbit. Conclusions. The method can be used during detailed numerical simulations and in contrast to other methods such as methods based on the Lyapunov exponent does not require a piece-wise secondary integration of the planetary orbit. Although the method has been deduced for the CR3BP, it is likely that it can be expanded to more general cases.

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Stringent Criteria for Stable and Unstable Planetary Orbits in Stellar Binary Systems

Cited by 52 publications

References 25 publications

The instability transition for the restricted 3-body problem

The instability transition for the restricted 3-body problem

Orbital stability of Earth-type planets in stellar binary systems

The instability transition for the restricted 3-body problem

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