Semi-analytical model for third-body perturbations including the inclination and eccentricity of the perturbing body

Nie, Tao; Gurfil, Pini; Zhang, Shijie

doi:10.1007/s10569-019-9905-5

Cited by 14 publications

(6 citation statements)

References 22 publications

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“…As seen from fig. 3a, these intermediate-period oscillations (Nie et al, 2019) are significant compared to the variation of the doubly-averaged orbital elements. In particular, their magnitude is sufficient to flip the direction of motion between direct and retrograde five times before the spacecraft eventually collides with the Earth.…”

Section: Superimposing Averaged and Osculating Evolutionsmentioning

confidence: 94%

Lunar close encounters compete with the circumterrestrial Lidov-Kozai effect

Amato,

Malhotra,

Sidorenko

et al. 2020

Preprint

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Luna 3 (or Lunik 3 in Russian sources) was the first spacecraft to perform a flyby of the Moon. Launched in October 1959 on a translunar trajectory with large semi-major axis and eccentricity, it collided with the Earth in late March 1960. The short, 6-month dynamical lifetime has often been explained through an increase in eccentricity due to the Lidov-Kozai effect. However, the classical Lidov-Kozai solution is only valid in the limit of small semi-major axis ratio, a condition that is satisfied only for solar (but not for lunar) perturbations. We undertook a study of the dynamics of Luna 3 with the aim of assessing the principal mechanisms affecting its evolution. We analyze the Luna 3 trajectory by generating accurate osculating solutions, and by comparing them to integrations of singly-and doubly-averaged equations of motion in vectorial form. Lunar close encounters, which cannot be reproduced in an averaging approach, decisively affect the trajectory and break the doubly-averaged dynamics. Solar perturbations induce oscillations of intermediate period that affect the geometry of the close encounters and cause the singly-averaged and osculating inclinations to change quadrants (the orbital plane "flips"). We find that the peculiar evolution of Luna 3 can only be explained by taking into account lunar close encounters and intermediate-period terms; such terms are averaged out in the Lidov-Kozai solution, which is not adequate to describe translunar or cislunar trajectories. Understanding the limits of the Lidov-Kozai solution is of particular significance for the motion of objects in the Earth-Moon environment and of exoplanetary systems.

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Section: Superimposing Averaged and Osculating Evolutionsmentioning

confidence: 94%

Lunar close encounters compete with the circumterrestrial Lidov-Kozai effect

Amato,

Malhotra,

Sidorenko

et al. 2020

Preprint

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“…That is, this model provides chance to study the resonance between the secular precession of the inner orbit and the mean motion of the outer body. On the other hand, the single-averaged model may be significantly more accurate than the double-averaged model in certain conditions (Wang and Gurfil, 2016;Nie et al, 2019;Carvalho et al, 2022).…”

Section: Semi-analytical Modelmentioning

confidence: 99%

“…Actually, only two integrals exist: the semi-major axis and the double-averaged perturbing potential (or the doubleaveraged Hamiltonian). There are only a few studies contributing to studying the secular evolution including the inclination of the perturbing body, and most of them were based on the quadrupole-order approximation (Yokoyama, 1999;Liu et al, 2012;Circi et al, 2017;Nie et al, 2019;Nie and Gurfil, 2021). Yokoyama (1999) pointed out that the high third-body inclination, with J 2 of the central body included, can produce strong chaotic motions and large eccentricities.…”

Section: Introductionmentioning

confidence: 99%

“…Yokoyama (1999) pointed out that the high third-body inclination, with J 2 of the central body included, can produce strong chaotic motions and large eccentricities. Liu et al (2012) and Nie et al (2019) have developed a double-averaged model with the perturbing body in an eccentric and inclined orbit. However, although they assume an inclined eccentric perturbing body, the central body's oblateness is not included, which, actually, can be related to the classical Lidov-Kozai mechanism via a coordinate transformation.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Semi-analytical orbital model around an oblate body with an inclined eccentric perturber

Wang²,

Hu³

2023

Front. Astron. Space Sci.

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The semi-analytical model (based on the averaging technique) for long-term orbital evolution has proven to be useful in both astrophysical and astrodynamical contexts. In this secular approximation, orbits exhibit rich evolutionary behaviors under the effects of various perturbations. For example, in the hierarchical three-body systems, the Lidov-Kozai mechanism based on the quadrupole-level third-body perturbation model can produce large-amplitude oscillations of the eccentricity and inclination. In recent years, the octupole order has been found to induce dramatically new features when the perturbing body’s orbit is eccentric, including extremely high eccentricities and orbit flips between prograde and retrograde. Motivated by the striking effects of the octupole-order terms, we intend to derive a more general dynamical model by incorporating J2 of the central body and the inclined eccentric third-body perturbation to the hexadecapole order with its non-spherical gravity also included. This issue can be relevant for astrophysical and astrodynamical systems such as planets in stellar binaries, irregular satellites in planetary systems, and artificial probes about binary asteroid systems. Applications to the binary asteroid system 4951 Iwamoto and a fictitious exoplanetary system are illustrated as examples to validate our secular model. From these numerical results, we show the high accuracy of our secular model. Also, we show the important role of these high-order terms and the effects of the third-body’s inclination and eccentricity. Besides, we have found several different secular effects that could drive large eccentricities.

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“…Although the single averaging procedure is not very popular now, mostly because they can be replaced for direct numerical integration in many cases, there is still a certain flow of studies along this line in modern celestial mechanics (e.g. Domingos et al, 2013;Elshaboury and Mostafa, 2013;Nie et al, 2019). There are even studies of singly averaged hyperbolic (Sorokovich, 1982) and parabolic (Mamedov, 1989) restricted three-body problems.…”

Section: Later Developmentsmentioning

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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Semi-analytical model for third-body perturbations including the inclination and eccentricity of the perturbing body

Cited by 14 publications

References 22 publications

Lunar close encounters compete with the circumterrestrial Lidov-Kozai effect

Lunar close encounters compete with the circumterrestrial Lidov-Kozai effect

Semi-analytical orbital model around an oblate body with an inclined eccentric perturber

The Lidov-Kozai Oscillation and Hugo von Zeipel

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