Analytical criteria of Hill stability in the elliptic restricted three body problem

Gong, Shengping; Li, Junfeng

doi:10.1007/s10509-015-2436-y

Cited by 13 publications

(7 citation statements)

References 56 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Consequently, our integrator satisfies the inequalities of the original ER3BP that determine the motion region of the massless point particle. In Section 5, we analytically clarify that this integrator preserves the Jacobi integral of CR3BP and fulfills some Hill stability criteria in ER3BP (Luk'yanov 2005(Luk'yanov , 2010Gong & Li 2015;Gasanov & Mammadli 2016). The criteria represent a sufficient condition that the massless point particle remains inside either the drop-shaped region around the primary mass point or that around the secondary one.…”

Section: Introductionmentioning

confidence: 90%

See 1 more Smart Citation

Quasi-conservative Integration Method for Restricted Three-body Problem

Minesaki

2023

ApJ

View full text Add to dashboard Cite

The simplest restricted three-body problem, in which two massive points and a massless point particle attract one another according to Newton’s law of inverse squares, has pulsating Hill’s regions where the massless particle moves inside the closed regions surrounding only one of the massive points. Until now, no numerical integrator is known to maintain these regions, making it challenging to reproduce the phenomenon of gravitational capture of massless particles. In this article, we propose a second-order integrator that preserves Hill’s regions to accurately simulate this phenomenon. Our integrator is based on a logarithmic Hamiltonian leapfrog method developed by Mikkola and Tanikawa and features a parameter that is adjusted to preserve a second-order approximation of an invariant integration relation of this restricted three-body problem. We analytically and numerically clarify that this integrator has the following properties: (i) it retains the collinear and triangular Lagrangian solutions regardless of the eccentricity of the relative orbit of the two massive points, (ii) it has the same Hill stability criterion for satellite-type motion of the massless point particle as the original problem, and (iii) it conserves the Jacobi integral for zero eccentricity.

show abstract

Section: Introductionmentioning

confidence: 90%

“…This relation reduces to the Jacobi integral in the limit of e → 0, i.e., for CR3BP. Some inequalities (Luk'yanov 2005(Luk'yanov , 2010Gong & Li 2015;Gasanov & Mammadli 2016) determine the motion region of the massless body for different time intervals. These inequalities are derived from this invariant relation.…”

Section: Introductionmentioning

confidence: 99%

Quasi-conservative Integration Method for Restricted Three-body Problem

Minesaki

2023

ApJ

View full text Add to dashboard Cite

show abstract

“…Hence, it is an interesting application based problem, and many scientists have studied different versions of this problem by considering different perturbation forces in the classical Hill problem. This means the primary bodies possess point masses and move in circular orbits around their common centre of mass or in elliptical trajectory, while the third body moves in space under the effect of gravitational forces of the primary bodies without affecting their motions [1][2][3][4].…”

Section: Introductionmentioning

confidence: 99%

Analysis of Equilibrium Points in Quantized Hill System

Ansari¹,

Alhowaity

Abouelmagd

et al. 2022

Mathematics

View full text Add to dashboard Cite

In this work, the quantized Hill problem is considered in order for us to study the existence and stability of equilibrium points. In particular, we have studied three different cases which give all whole possible locations in which two points are emerging from the first case and four points from the second case, while the third case does not provide a realistic locations. Hence, we have obtained four new equilibrium points related to the quantum perturbations. Furthermore, the allowed and forbidden regions of motion of the first case are investigated numerically. We demonstrate that the obtained result in the first case is a generalization to the classical one and it can be reduced to the classical result in the absence of quantum perturbation, while the four new points will disappear. The regions of allowed motions decrease as the value of the Jacobian constant increases, and these regions will form three separate areas. Thus, the infinitesimal body can never move from one allowed region to another, and it will be trapped inside one of the possible regions of motion with the relative large values of the Jacobian constant.

show abstract

“…Anselmo et al (1983); Bhanderi & Bak (2005) have analyzed the effect of Earth's albedo on the Earth bound satellites. Few researcher (McInnes et al 1994;Abdel-Aziz et al 2011;Gong & Li 2015;Idrisi 2017) have studied the RTBP under albedo effect of secondary in addition to the oblateness effect. Del Genio et al (2017) studied the predicted temperature and predicted albedo for the exo-planets such as Kepler-186 f, Proxima Centauri b, etc.…”

Section: Introductionmentioning

confidence: 99%

Effects of the albedo and disc on the zero velocity curves and linear stability of equilibrium points in the generalized restricted three-body problem

Yousuf

Kishor

2019

Monthly Notices of the Royal Astronomical Society

View full text Add to dashboard Cite

The most important aspects of a dynamical system are its stability and the factors which affects the stability property. This paper presents the analysis of the effects of albedo and disc on the zero velocity curves, existence of equilibrium points and on their linear stability in a generalized restricted three body problem that consists of motion of an infinitesimal mass under the uniform gravity field of radiating-oblate primary, oblate secondary and a disc, which is rotating about the common center of the mass of the system. A significant effect of albedo and disc are observed on the zero velocity curves, positions of equilibrium points and on the stability region. Linear stability analysis of collinear equilibrium points is performed with respect to mass ratio µ and albedo parameter of secondary, separately and it is found that these are unstable in both the cases. On the other hand, non-collinear equilibrium point is stable in a certain range of mass ratio. After analyzing individual as well as combined effect of radiation pressure force of the primary, albedo of secondary, oblateness of both the massive bodies and the disc, it is found that these perturbations play a significant on the motion of infinitesimal mass in the vicinity of equilibrium points. These results may be help to analyze more generalized problem of few bodies under the influence of different kind of perturbations such as P-R drag, solar wind drag etc. Present study is limited to the regular symmetric disc which will extend later.

show abstract

Analytical criteria of Hill stability in the elliptic restricted three body problem

Cited by 13 publications

References 56 publications

Quasi-conservative Integration Method for Restricted Three-body Problem

Quasi-conservative Integration Method for Restricted Three-body Problem

Analysis of Equilibrium Points in Quantized Hill System

Effects of the albedo and disc on the zero velocity curves and linear stability of equilibrium points in the generalized restricted three-body problem

Contact Info

Product

Resources

About