Geography of the rotational resonances and their stability in the ellipsoidal full two body problem

Nadoushan, Mahdi Jafari; Assadian, Nima

doi:10.1016/j.icarus.2015.10.011

Cited by 14 publications

(13 citation statements)

References 31 publications

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“…This criterion has been widely used to predict the onset of chaotic dynamics in the spin-orbit problem, and was used in the original Wisdom et al (1984) study to predict the tumbling of Hyperion, as well as in Section 4 of Jafari Nadoushan & Assadian (2015). We follow a precedent similar to that presented in Nadoushan & Assadian (2016), who first applied the Chirikov criterion to predict the onset of chaos for the spinspin problem. The criterion for the onset of chaos, or for resonance overlap, will be ( )¯, the criterion for chaos for the spin-spin resonance can be written as…”

Section: Spin-spin Resonances For An Eccentric Orbitmentioning

confidence: 99%

“…This Hamiltonian may be directly compared to the Hamiltonian presented in Equation (9) of Nadoushan & Assadian (2016). This Hamiltonian has the same harmonics, with the exception of the terms resulting from the spin of the primary only, and those to higher order in the asphericity of the satellite.…”

Section: Overlap Of All Resonances and The Onset Of Chaosmentioning

confidence: 99%

“…They showed that, as the satellite's asphericity and eccentricity increased, it was susceptible to strong chaos (where "chaotic regions are large compared to the non-chaotic regions," as defined in the Conclusion section of Jafari Nadoushan & Assadian (2015)). In two studies, Nadoushan & Assadian (2016) and Batygin & Morbidelli (2015) used a Hamiltonian approach to demonstrate that, if not only the secondary but also the primary body can sustain a significant quadrupole moment, then in addition to spin-orbit resonances, the phase space portrait of the system is augmented with spin-spin resonances. This yields an even more sophisticated evolutionary picture.…”

Section: Introductionmentioning

confidence: 99%

“…The analysis of Batygin & Morbidelli (2015), however, is largely analytic, and the nature of its exploratory design restricts its scope. Nadoushan & Assadian (2016) explored spin-orbit and spin-spin resonances in the context of the ellipsoidal full two-body problem, while Hou & Xin (2017) included the variations in the mutual orbit between the two bodies, which may be appropriate for similarly sized and close minor planets. Haumea has one of the largest quadrupole moments for bodies of its size and a fast ∼3.9 hr spin period, which makes its satellites seemingly strong candidates for the exotic spin-spin resonances investigated in Batygin & Morbidelli (2015).…”

Section: Introductionmentioning

confidence: 99%

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The Onset of Chaos in Permanently Deformed Binaries from Spin–Orbit and Spin–Spin Coupling

Seligman

Batygin

2021

ApJ

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Permanently deformed objects in binary systems can experience complex rotation evolution, arising from the extensively studied effect of spin-orbit coupling as well as more nuanced dynamics arising from spin-spin interactions. The ability of an object to sustain an aspheroidal shape largely determines whether or not it will exhibit nontrivial rotational behavior. In this work, we adopt a simplified model of a gravitationally interacting primary and satellite pair, where each body's quadrupole moment is approximated by two diametrically opposed point masses. After calculating the net gravitational torque on the satellite from the primary, as well as the associated equations of motion, we employ a Hamiltonian formalism that allows for a perturbative treatment of the spin-orbit and retrograde and prograde spin-spin coupling states. By analyzing the resonances individually and collectively, we determine the criteria for resonance overlap and the onset of chaos, as a function of orbital and geometric properties of the binary. We extend the 2D planar geometry to calculate the obliquity evolution. This calculation indicates that satellites in spin-spin resonances undergo precession when inclined out of the plane, but they do not tumble. We apply our resonance overlap criteria to the contact binary system (216) Kleopatra, and find that its satellites, Cleoselene and Alexhelios, may plausibly be exhibiting chaotic rotational dynamics from the overlap of the spin-orbit and retrograde spin-spin resonances. While this model is, by construction, generalizable to any binary system, it will be particularly useful for studies of small bodies in the Solar System, whose irregular shapes make them ideal candidates for exotic rotational states.

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Section: Spin-spin Resonances For An Eccentric Orbitmentioning

confidence: 99%

Section: Overlap Of All Resonances and The Onset Of Chaosmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The Onset of Chaos in Permanently Deformed Binaries from Spin–Orbit and Spin–Spin Coupling

Seligman

Batygin

2021

ApJ

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“…The spin-spin problem was introduced in [Mis21] as a planar version of the Full Two-Body problem for ellipsoids (compare with [BM15,BL09]), by using the expansion of the potential up to order 1/r 5 , which results in the coupling of the spins of both bodies. An equivalent model was studied in [JA16] (see also [HX17]).…”

Section: Introductionmentioning

confidence: 99%

The spin-spin problem in Celestial Mechanics

Celletti,

Gimeno,

Misquero

2021

Preprint

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We study the dynamics of two homogeneous rigid ellipsoids subject to their mutual gravitational influence. We assume that the spin axis of each ellipsoid coincides with its shortest physical axis and is perpendicular to the orbital plane. Due to such assumptions, the problem is planar and depends on particular parameters of the ellipsoids, most notably, the equatorial oblateness and the flattening with respect to the shortest physical axes. We consider two models for such configuration: while in the full model, there is a coupling between the orbital and rotational motions, in the Keplerian model, the centers of mass of the bodies are constrained to move on coplanar Keplerian ellipses. The Keplerian case, in the approximation that includes the coupling between the spins of the two ellipsoids, is what we call spin-spin problem, that is a generalization of the classical spin-orbit problem. In this paper we continue the investigations of [Mis21] on the spin-spin problem by comparing it with the spin-orbit problem and also with the full model.Beside detailing the models associated to the spin-orbit and spin-spin problems, we introduce the notions of standard and balanced resonances, which lead us to investigate the existence of periodic and quasi-periodic solutions. We also give a qualitative description of the phase space and provide results on the linear stability of solutions for the spin-orbit and spin-spin problems. We conclude by providing a comparison between the full and the Keplerian models with particular reference to the interaction between the rotational and orbital motions.

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The Spin–Spin Problem in Celestial Mechanics

2022

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We study the dynamics of two homogeneous rigid ellipsoids subject to their mutual gravitational influence. We assume that the spin axis of each ellipsoid coincides with its shortest physical axis and is perpendicular to the orbital plane. Due to such assumptions, the problem is planar and depends on particular parameters of the ellipsoids, most notably, the equatorial oblateness and the flattening with respect to the shortest physical axes. We consider two models for such configuration: while in the full model, there is a coupling between the orbital and rotational motions, in the Keplerian model, the centers of mass of the bodies are constrained to move on coplanar Keplerian ellipses. The Keplerian case, in the approximation that includes the coupling between the spins of the two ellipsoids, is what we call spin–spin problem, that is a generalization of the classical spin–orbit problem. In this paper we continue the investigations of Misquero (Nonlinearity 34:2191–2219, 2021) on the spin–spin problem by comparing it with the spin–orbit problem and also with the full model. Beside detailing the models associated to the spin–orbit and spin–spin problems, we introduce the notions of standard and balanced resonances, which lead us to investigate the existence of periodic and quasi-periodic solutions. We also give a qualitative description of the phase space and provide results on the linear stability of solutions for the spin–orbit and spin–spin problems. We conclude by providing a comparison between the full and the Keplerian models with particular reference to the interaction between the rotational and orbital motions.

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Geography of the rotational resonances and their stability in the ellipsoidal full two body problem

Cited by 14 publications

References 31 publications

The Onset of Chaos in Permanently Deformed Binaries from Spin–Orbit and Spin–Spin Coupling

The Onset of Chaos in Permanently Deformed Binaries from Spin–Orbit and Spin–Spin Coupling

The spin-spin problem in Celestial Mechanics

The Spin–Spin Problem in Celestial Mechanics

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