scholarly journals Stability Analysis of the Rhomboidal Restricted Six-Body Problem

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
M. A. R. Siddique ◽  
A. R. Kahsif ◽  
M. Shoaib ◽  
S. Hussain
Keyword(s):  
Body Problem ◽  
Equilibrium Points ◽  
First Integral ◽  
Small Mass ◽  
Equilibrium Solutions ◽  
Integral Of Motion ◽  
Jacobian Constant ◽  
Zero Velocity Curves ◽  
The Masses ◽  
Uniqueness Of Equilibrium

We discuss the restricted rhomboidal six-body problem (RR6BP), which has four positive masses at the vertices of the rhombus, and the fifth mass is at the intersection of the two diagonals. These masses always move in rhomboidal CC with diagonals 2 a and 2 b . The sixth body, having a very small mass, does not influence the motion of the five masses, also called primaries. The masses of the primaries are m 1 = m 2 = m 0 = m and m 3 = m 4 = m ˜ . The masses m and m ˜ are written as functions of parameters a and b such that they always form a rhomboidal central configuration. The evolution of zero velocity curves is discussed for fixed values of positive masses. Using the first integral of motion, we derive the region of possible motion of test particle m 5 and identify the value of Jacobian constant C for different energy intervals at which these regions become disconnected. Using semianalytical techniques, we show the existence and uniqueness of equilibrium solutions on the axes and off the axes. We show that, for b ∈ 1 / 3 , 1.1394282249562009 , there always exist 12 equilibrium points. We also show that all 12 equilibrium points are unstable.

scholarly journals Dynamical Aspects of an Equilateral Restricted Four-Body Problem

2009 ◽  
Vol 2009 ◽  
pp. 1-23 ◽  
Author(s):  
Martha Álvarez-Ramírez ◽  
Claudio Vidal
Keyword(s):  
Degrees Of Freedom ◽  
Body Problem ◽  
Center Of Mass ◽  
Hamiltonian Function ◽  
First Integral ◽  
Equilibrium Solutions ◽  
Integral Of Motion ◽  
Planar Case ◽  
Four Body Problem ◽  
The Masses

The spatial equilateral restricted four-body problem (ERFBP) is a four body problem where a mass point of negligible mass is moving under the Newtonian gravitational attraction of three positive masses (called the primaries) which move on circular periodic orbits around their center of mass fixed at the origin of the coordinate system such that their configuration is always an equilateral triangle. Since fourth mass is small, it does not affect the motion of the three primaries. In our model we assume that the two masses of the primariesm2andm3are equal toμand the massm1is1−2μ. The Hamiltonian function that governs the motion of the fourth mass is derived and it has three degrees of freedom depending periodically on time. Using a synodical system, we fixed the primaries in order to eliminate the time dependence. Similarly to the circular restricted three-body problem, we obtain a first integral of motion. With the help of the Hamiltonian structure, we characterize the region of the possible motions and the surface of fixed level in the spatial as well as in the planar case. Among other things, we verify that the number of equilibrium solutions depends upon the masses, also we show the existence of periodic solutions by different methods in the planar case.

scholarly journals A note on the existence of invariant punctured tori in the planar circular restricted three-body problem

1988 ◽  
Vol 8 (8) ◽  
pp. 63-72 ◽  
Keyword(s):  
Body Problem ◽  
Small Mass ◽  
Connected Components ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Jacobi Constant ◽  
Hill Region ◽  
The Masses ◽  
Three Body ◽  
Collision Orbits

AbstractThe existence of transversal ejection—collision orbits in the restricted three-body problem is shown to imply, via the KAM theorem, the existence, for certain intervals of (large) values of the Jacobi constant, of an uncountable number of invariant punctured tori in the corresponding (non-compact) energy surface. The proof is based on a comparison between Levi-Civita and McGehee regularizing variables. That these transversal ejection-collision orbits do actually exist was proved in [5] in the case where one of the primaries has a small mass and the zero-mass body revolves around the other (and for all values of the Jacobi constant compatible with the existence of three connected components for the Hill region); it is proved here without any restriction on the masses, well in the spirit of Conley's thesis [3].

Exploring the Location and Linear Stability of the Equilibrium Points in the Equilateral Restricted Four-Body Problem

2020 ◽  
Vol 30 (10) ◽  
pp. 2050155
Author(s):  
Euaggelos E. Zotos
Keyword(s):  
Linear Stability ◽  
Body Problem ◽  
Equilibrium Points ◽  
Libration Points ◽  
The Other ◽  
Classification Method ◽  
Four Body Problem ◽  
The Masses ◽  
Stable Points ◽  
Planar Version

The planar version of the equilateral restricted four-body problem, with three unequal masses, is numerically investigated. By adopting the grid classification method we locate the coordinates, on the plane [Formula: see text], of the points of equilibrium, for all possible values of the masses of the primaries. The linear stability of the libration points is also determined, as a function of the masses. Our analysis indicates that linearly stable points of equilibrium exist only when one of the primaries has a considerably larger mass, with respect to the other two primary bodies, when the triangular configuration of the primaries is also dynamically stable.

EQUILIBRIUM POINTS AND THEIR STABILITY IN THE RESTRICTED FOUR-BODY PROBLEM

2011 ◽  
Vol 21 (08) ◽  
pp. 2179-2193 ◽  
Author(s):  
A. N. BALTAGIANNIS ◽  
K. E. PAPADAKIS
Keyword(s):  
Body Problem ◽  
Equilibrium Points ◽  
Relative Equilibrium ◽  
Center Of Mass ◽  
Basins Of Attraction ◽  
Equilibrium Solutions ◽  
Velocity Surface ◽  
Four Body Problem ◽  
The Stability ◽  
Zero Velocity Surface

We study numerically the problem of four bodies, three of which are finite, moving in circles around their center of mass fixed at the origin of the coordinate system, according to the solution of Lagrange where they are always at the vertices of an equilateral triangle, while the fourth is infinitesimal. The fourth body does not affect the motion of the three bodies (primaries). The allowed regions of motion as determined by the zero-velocity surface and corresponding equipotential curves as well as the positions of the equilibrium points are given. The existence and the number of collinear and noncollinear equilibrium points of the problem depend on the mass parameters of the primaries. For three unequal masses, collinear equilibrium solutions do not exist. Critical masses associated with the existence and the number of equilibrium points, are given. The stability of the relative equilibrium solutions in all cases is also studied. The regions of the basins of attraction for the equilibrium points of the present dynamical model for some values of the mass parameters are illustrated.

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

2019 ◽  
Vol 488 (2) ◽  
pp. 1894-1907
Author(s):  
Saleem Yousuf ◽  
Ram Kishor
Keyword(s):  
Linear Stability ◽  
Radiation Pressure ◽  
Body Problem ◽  
Equilibrium Points ◽  
Mass Parameter ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Zero Velocity ◽  
Zero Velocity Curves ◽  
Three Body

ABSTRACT The important aspects of a dynamical system are its stability and the factors that affect its stability. In this paper, we present an analysis of the effects of the albedo and the disc on the zero velocity curves, the existence of equilibrium points and their linear stability in a generalized restricted three-body problem (RTBP). The proposed problem consists of the motion of an infinitesimal mass under the gravitational field of a radiating-oblate primary, an oblate secondary and a disc that is rotating about the common centre of mass of the system. Significant effects of the albedo and the disc are observed on the zero velocity curves, on the positions of equilibrium points and on the stability region. A linear stability analysis of collinear equilibrium points L1, 2, 3 is performed with respect to the mass parameter μ and albedo parameter QA of the secondary, separately. It is found that L1, 2, 3 are unstable in both cases. However, the non-collinear equilibrium points L4, 5 are stable in a finite range of mass ratio μ. After analysing the individual as well as combined effects of the radiation pressure force of the primary, the albedo force of the secondary, the oblateness of both the primary and secondary and the disc, it is found that these perturbations play a significant role in the design of the trajectories in the vicinity of equilibrium points and in the analysis of their stability property. In the future, the results obtained will improve existing results and will help in the analysis of different space missions. These results are limited to the regular symmetric disc and radiation pressure, which can be extended later.

scholarly journals Existence of Resonance Stability of Triangular Equilibrium Points in Circular Case of the Planar Elliptical Restricted Three-Body Problem under the Oblate and Radiating Primaries around the Binary System

2014 ◽  
Vol 2014 ◽  
pp. 1-17 ◽  
Author(s):  
A. Narayan ◽  
Amit Shrivastava
Keyword(s):  
Binary System ◽  
Body Problem ◽  
Equilibrium Points ◽  
Hamiltonian Function ◽  
Oblate Spheroid ◽  
True Anomaly ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Equilibrium Solutions ◽  
Three Body

This paper analyzes the existence of resonance stability of the triangular equilibrium points of the planar elliptical restricted three-body problem when both the primaries are oblate spheroid as well as the source of radiation under the particular case, whene=0. We have derived Hamiltonian function describing the motion of infinitesimal mass in the neighborhood of the triangular equilibrium solutions taken as a convergent series. Hamiltonian function for the system has been derived and also expanded in powers of the generalized components of momenta. We have used canonical transformation to make the Hamiltonian function independent of true anomaly. The most interesting and distinguishable results of this study are establishing the relation for determining the range of stability at and near the resonanceω2=1/2around the binary system.

scholarly journals On the Stability of Collinear Points of the RTBP with Triaxial and Oblate Primaries and Relativistic Effects

2021 ◽  
pp. 56-73
Author(s):  
S. E. Abd El-Bar
Keyword(s):  
Characteristic Equation ◽  
Body Problem ◽  
Equilibrium Points ◽  
Equations Of Motion ◽  
Relativistic Effects ◽  
Three Body Problem ◽  
Total Potential ◽  
The Mean ◽  
The Stability ◽  
Three Body

Under the influence of some different perturbations, we study the stability of collinear equilibrium points of the Restricted Three Body Problem. More precisely, the perturbations due to the triaxiality of the bigger primary and the oblateness of the smaller primary, in addition to the relativistic effects, are considered. Moreover, the total potential and the mean motion of the problem are obtained. The equations of motion are derived and linearized around the collinear points. For studying the stability of these points, the characteristic equation and its partial derivatives are derived. Two real and two imaginary roots of the characteristic equation are deduced from the plotted figures throughout the manuscript. In addition, the instability of the collinear points is stressed. Finally, we compute some selected roots corresponding to the eigenvalues which are based on some selected values of the perturbing parameters in the Tables 1, 2.

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