The Ring Problem of (N+1) Bodies: An Overview

Kalvouridis, T. J.

doi:10.1007/978-1-4614-0454-5_6

Cited by 4 publications

(5 citation statements)

References 34 publications

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“…The equations of motion of the N-body ring problem without central body are well known and have been described in many works [10,11,15,26]. This problem deals with the motion of a point particle under the Newtonian influence of a configuration of N bodies (called primaries) with the same mass distributed on a circumference occupying the vertices of a regular polygon of N sides.…”

Section: Equations Of Motion and Curves Of Zero Velocitymentioning

confidence: 99%

“…In [10], Kalvouridis reports the main results of his research concerning the existence of periodic orbits for N > 7, and considering several values of the mass ratio, showing the distribution of these orbits in the configuration space of initial conditions. In [11], Kalvouridis gives a detailed overview of the results obtained on the topic between the years 1997 and 2008.…”

Section: Introductionmentioning

confidence: 99%

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Dependence of the probability of escape on the Jacobi constant in the N-body ring problem without central body

Belgharbi

Navarro

2023

Eur. Phys. J. Plus

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In this work, we perform a numerical exploration of the escape in the N-body ring problem in absence of a central body, for $$4\le N\le 9$$ 4 ≤ N ≤ 9 and ten values of the Jacobi constant. We show how the probability of escape per interval of time varies as a function of the Jacobi constant, finding that, for values of the Jacobi constant smaller than a certain limit value, the probability of escape from the system tends to decrease with time. However, if we consider values of the Jacobi constant larger than this limit value, the probability of escape grows with time, for times of escape smaller than 100 units of time.

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Section: Equations Of Motion and Curves Of Zero Velocitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Dependence of the probability of escape on the Jacobi constant in the N-body ring problem without central body

Belgharbi

Navarro

2023

Eur. Phys. J. Plus

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“…[11] and [8] have analyzed the dynamics of the (N + 1) ring body problem in planar case, Symmetric periodic motion in a ring problem discussed by [18], Periodic solutions in the ring problem studied by [9], [10]. [12] have expressed the fundamentals of the Ring Problem of (N+ 1) Bodies: An Overview. [16,15] has studied the analysis of the distribution of times of escape in the N-body ring problem.…”

Section: Introductionmentioning

confidence: 99%

A study of libration points and analysis of their stability behavior in-ring body problem under albedo effect

Kumar

Kushvah

2022

Preprint

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Our intention in this article is to analyze the fundamental analysis of the zones and their existing libration points under the solar radiation pressure and albedo effects. We can feel about albedos at any Nature’s property wherever the Sun rays exist. As we know that important phase of the dynamical system of any celestial body is its stability and the factors which change its behavior of motion. This research article presents an explicit analysis of the albedo effect for ring body problems for a specific value of v = 6, which explains the behavior of existing zone’s stability, shifting of their libration points with monotonicity and the characteristic roots of δ confirms the behavior of stability condition for existing zones as well as their existing libration points in a synodic frame of references. We observed that in this problem from an experimental point of view, a little change in v, changes the complete behavior of the problem and leaves a nice platform of the proof. In this problem, β plays a significant role, which affects firstly the whole dynamics of peripheral primaries, the mass parameters β = 0 and β ≠ 0 generally confirm that this synodic system wears, an absence of a center of mass and the presence of the center of mass, respectively. We have observed in this article is the libration point L0, which exists in the center and has peripheral primary P0, remains unchanged throughout applying the albedo effect. The stability analysis confirms the location of the libration points, whether it is stable, unstable, etc., but here we have seen the more interesting things i.e., most of the locations of the libration points except exceptionals are found unstable. This article also shows the momentum of the position of all the libration points with their existing zone as well as peripheral primaries, by the effect of the radiation pressure and albedo. It is also analyzed here that this complete problem works in a certain short and very sensitive range set (−1.75, 1.75) because this problem is filled with trigonometric periodic functions. In this problem, we have considered only the real roots of the axis x, y, whereas imaginary roots are not considered. For stability analysis, we have taken both real and complex roots of δ. Results of this article for v = 6 may help us to analyze an imaginary behavior of our solar system’s planets in a blank paper, provided by certain rules and regulations by equations of motion because real behavior may be far away from us. By the present study, it is found that the whole behavior of the system for various v whenever change, which can be extended later in another way.

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“…Later on, independently, Kalvouridis[5] reintroduced the problem, and after then, many papers on this topic have been published. A quite complete list of related publications may be found in [6].What is important to remark is that the N + 1 configuration is a relative configuration, that is, the configuration becomes equilibrium of Newton's differential equations in uniformly rotating coordinates. Two principal facts have to be considered in this problem.…”

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confidence: 99%

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confidence: 99%

“Comments on a Paper of Idrisi, Ullah and Sikkandhar (Effect of Perturbations in Coriolis and Centrifugal Forces on Libration Points in the Restricted Six-Body Problem: JAS (2021) 68:4-25)”

Elipe

2022

J Astronaut Sci

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In a recent paper, Idrisi, Ullah and Sikkandhar [1] considered some aspects of the Restricted Six-Body Problem. The problem consists of the motion of an infinitesimal mass under the gravity field of five bodies in a very special configuration: There are four equal points with equal masses m located at the vertices of a regular square that is rotating on its own plane about its center of mass with constant angular velocity ω. Besides, there is a fifth body of mass m 0 placed at the center of the square.It is not said in Idrisi et al.[1], hereafter IUS Paper, but this a just a particular example of the so-called Maxwell ring or N -gon problem, which consists of an arbitrary number of equal bodies placed at the vertices of a regular N -gon. Maxwell [2] built this model to simulate Saturn's rings and already studied the stability of the discrete particle ring. Soon after, Tisserand [3, Ch. 12] reformulated Maxwell's analysis and obtained a relation between the mass of each ring particle and the number of them for the system be linearly stable. The problem was tackled again by Scheeres [4] who investigated this configuration including Hill stability, invariant transformations, equilibrium points, periodic orbits, and other features. Later on, independently, Kalvouridis[5] reintroduced the problem, and after then, many papers on this topic have been published. A quite complete list of related publications may be found in [6].What is important to remark is that the N + 1 configuration is a relative configuration, that is, the configuration becomes equilibrium of Newton's differential equations in uniformly rotating coordinates. Two principal facts have to be considered in this problem. Firstly, the question of its stability. Moeckel [7] proved that the configuration is linearly stable for N ≥ 7, and unstable for N ≤ 6. This is the explanation why most of the considered cases in the literature are for N = 6 or N = 7. It is worth to mention that when other central forces (different of the pure Newtonian ones) are considered, this condition for the stability changes [8].Besides, because of the symmetry of the configuration, it is sufficient to restrict the study for the equilibria to the line joining the central primary (or the origin)

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The Ring Problem of (N+1) Bodies: An Overview

Cited by 4 publications

References 34 publications

Dependence of the probability of escape on the Jacobi constant in the N-body ring problem without central body

Dependence of the probability of escape on the Jacobi constant in the N-body ring problem without central body

A study of libration points and analysis of their stability behavior in-ring body problem under albedo effect

“Comments on a Paper of Idrisi, Ullah and Sikkandhar (Effect of Perturbations in Coriolis and Centrifugal Forces on Libration Points in the Restricted Six-Body Problem: JAS (2021) 68:4-25)”

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