Study of resonances for the restricted 3‐body problem

Quarles, Billy; Musielak, Z. E.; Cuntz, M.

doi:10.1002/asna.201111704

Cited by 26 publications

(4 citation statements)

References 43 publications

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“…This could provide an alternative explanation for the signal at 417 d that had previously been identified as a planet [4]. Since ν-Octantis A has a nearby companion (ν-Octantis B) on a 2.8 y orbit [4], a planet at 417 d cannot be stable on a coplanar prograde orbit (although it can survive on a coplanar retrograde orbit) [5]. We measured retrograde precession of −0.86 • /y which could be explained if ν-0ctantis B was a double star inclined i > 45 • with respect to the main binary's orbit [3].…”

Section: Discussionmentioning

confidence: 93%

Stellar wobble in triple star systems

Morais¹,

Correia²

2012

Preprint

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Section: Discussionmentioning

confidence: 93%

Stellar wobble in triple star systems

Morais¹,

Correia²

2012

Preprint

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“…Recent results were obtained in the restricted three body problem in identifying systems of absolute stability (Eberle, Cuntz & Musielak 2008), obtaining a criterion for stability of planetary orbits in stellar binary systems (Quarles et al 2011) and studying resonances (Quarles, Musielak & Cuntz 2012).…”

Section: The Largest Lyapunov Characteristic Exponentmentioning

confidence: 99%

Comparison of chaos detection methods in the circular restricted three‐body problem

Racoveanu

2014

Astron Nachr

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In this study, a sample of orbits is considered in the framework of the planar circular restricted three-body problem. In order to separate ordered from chaotic orbits three numerical methods are compared: the Largest Lyapunov Characteristic Exponent (LLCE) and the Smaller Alignment Index (SALI) provide a fairly good characterization of the chaotic motions, while the computational time required is of the same order; the Correlation Dimension (CD) has the advantage of correctly classifying sticky orbits, but at the expense of a longer computational time. In order to classify a given orbit, any pair of the three methods can be considered, but LLCE and SALI are recommended due to their speed.

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“…Note that the restricted three-body problem is often dealt with in a rotating coordinate system where massive bodies (m 0 and m 1 ) always stay on the x-axis (e.g. Quarles et al, 2012). However, we do not adopt the rotating coordinate system in this monograph.…”

Section: Preliminaries: What We Considermentioning

confidence: 99%

The Lidov-Kozai Oscillation and Hugo von Zeipel

Ito,

Ohtsuka

2019

Preprint

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The circular restricted three-body problem, particularly its doubly averaged version, has been very well studied in celestial mechanics. Despite its simplicity, circular restricted three-body systems are suited for modeling the motion of various objects in the solar system, extrasolar planetary systems, and in many other dynamical systems that show up in astronomical studies. In this context, the so-called Lidov-Kozai oscillation is well known and applied to various objects. This makes the orbital inclination and eccentricity of the perturbed body in the circular restricted three-body system oscillate with a large amplitude under certain conditions. It also causes a libration of the perturbed body's argument of pericenter around stationary points. It is widely accepted that the theoretical framework of this phenomenon was established independently in the early 1960s by a Soviet Union dynamicist (Michail L'vovich Lidov) and by a Japanese celestial mechanist (Yoshihide Kozai). Since then, the theory has been extensively studied and developed. A large variety of studies has stemmed from the original works by Lidov and Kozai, now having the prefix of "Lidov-Kozai" or "Kozai-Lidov." However, from a survey of past literature published in late nineteenth to early twentieth century, we have confirmed that there already existed a pioneering work using a similar analysis of this subject established in that period. This was accomplished by a Swedish astronomer, Edvard Hugo von Zeipel. In this monograph, we first outline the basic framework of the circular restricted three-body problem including typical examples where the Lidov-Kozai oscillation occurs. Then, we introduce what was discussed and learned along this line of studies from the early to mid-twentieth century by summarizing the major works of Lidov, Kozai, and relevant authors. Finally, we make a summary of von Zeipel's work, and show that his achievements in the early twentieth century already comprehended most of the fundamental and necessary formulations that the Lidov-Kozai oscillation requires. By comparing the works of Lidov, Kozai, and von Zeipel, we assert that the prefix "von Zeipel-Lidov-Kozai" should be used for designating this theoretical framework, and not just Lidov-Kozai or Kozai-Lidov. This justifiably shows due respect and appropriately commemorates these three major pioneers who made significant contributions to the progress of modern celestial mechanics.

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Study of resonances for the restricted 3‐body problem

Cited by 26 publications

References 43 publications

Stellar wobble in triple star systems

Stellar wobble in triple star systems

Comparison of chaos detection methods in the circular restricted three‐body problem

The Lidov-Kozai Oscillation and Hugo von Zeipel

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