scholarly journals Effects of Radiation Pressure on the Elliptic Restricted Four-Body Problem

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Sahar H. Younis ◽  
M. N. Ismail ◽  
Ghada F. Mohamdien ◽  
A. H. Ibrahiem
Keyword(s):  
Radiation Pressure ◽  
Body Problem ◽  
Equilibrium Points ◽  
Equations Of Motion ◽  
Numerical Approach ◽  
The Earth ◽  
Families Of Periodic Orbits ◽  
Four Body Problem ◽  
The Stability ◽  
Effects Of Radiation

In this paper, under the effects of the largest primary radiation pressure, the elliptic restricted four-body problem is formulated in Hamiltonian form. Moreover, the canonical equations are obtained which are considered as the equations of motion. The Lagrangian points within the frame of the elliptic restricted four-body problem are obtained. The true anomalies are considered as independent variables. An analytical and numerical approach had been used. A code of Mathematica version 12 is constructed to truncate these considerations and is applied on the Earth-Moon-Sun system. In addition, the stability and periodicity of the motion about the equilibrium points are studied by using the Poincare maps. The motion about the collinear point L2 is presented as an example for the obtained results, and some families of periodic orbits are presented.

scholarly journals Motion of the Infinitesimal Variable Mass in the Generalized Circular Restricted Three-Body Problem under the Effect of Asteroids Belt

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Ferdaous Bouaziz-Kellil
Keyword(s):  
Body Problem ◽  
Equilibrium Points ◽  
Equations Of Motion ◽  
Center Of Mass ◽  
Variable Mass ◽  
Three Body Problem ◽  
Common Center ◽  
The Stability ◽  
Effects Of Radiation ◽  
Three Body

The present paper deals with the study of the motion’s properties of the infinitesimal variable mass body moving in the same orbital plan as two massive bodies (considered as primaries). It is assumed that the massive bodies have radiating effects, have oblate shapes, and are moving in circular orbits around their common center of mass. Using the procedures established by Singh and Abouelmagd, we determined the equations of motion of the infinitesimal body for which we assumed that under the effects of radiation and oblateness of the primaries, its mass varies following Jean’s law. We evaluated analytically and numerically the locations of equilibrium points and examined the stability of these equilibrium points. Finally, we found that all the points are unstable.

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.

Equilibrium Points in the Restricted Four-Body Problem with Radiation Pressure

2015 ◽  
Vol 57 (1) ◽  
pp. 83-91 ◽  
Author(s):  
Jagadish Singh ◽  
Aguda Ekele Vincent
Keyword(s):  
Radiation Pressure ◽  
Body Problem ◽  
Equilibrium Points ◽  
Four Body Problem

scholarly journals Dynamics of Satellites Around Asteroids in Presence of Solar Radiation Pressure

2015 ◽  
Vol 10 (S318) ◽  
pp. 259-264
Author(s):  
Xiaosheng Xin ◽  
Daniel J. Scheeres ◽  
Xiyun Hou ◽  
Lin Liu
Keyword(s):  
Solar Radiation ◽  
Radiation Pressure ◽  
Equilibrium Points ◽  
Equations Of Motion ◽  
Solar Radiation Pressure ◽  
Approximate Solutions ◽  
Triaxial Ellipsoid ◽  
Stability Maps ◽  
The Stability ◽  
Near Earth Asteroids

AbstractDue to the close distance to the Sun, solar radiation pressure (SRP) plays an important role in the dynamics of satellites around near-Earth asteroids (NEAs). In this paper, we focus on the equilibrium points of a satellite orbiting around an asteroid in presence of SRP in the asteroid rotating frame. The asteroid is modelled as a uniformly rotating triaxial ellipsoid. When SRP comes into play, the equilibrium points transformed into periodic orbits termed as``dynamical substitutes". We obtain the analytical approximate solutions of the dynamical substitutes from the linearised equations of motion. The analytical solutions are then used as initial guesses and are numerically corrected to compute the accurate orbits of the dynamical substitutes. The stability of the dynamical substitutes is analysed and the stability maps are obtained by varying parameters of the ellipsoid model as well as the magnitude of SRP.

scholarly journals Heterogeneous Oblate Primaries in Photo-gravitational CR5BP with Kite Configuration

2019 ◽  
Vol 2 (1) ◽  
pp. 1-14
Author(s):  
Abdullah A. Ansari ◽  
Prashant Kumar ◽  
Mehtab Alam
Keyword(s):  
Radiation Pressure ◽  
Body Problem ◽  
Equations Of Motion ◽  
Oblate Spheroid ◽  
Zero Velocity ◽  
Jacobi Integral ◽  
The Stability ◽  
Zero Velocity Curves

This paper presents the investigation of the motion of infinitesimal body in the circular restricted five-body problem in which four bodies are taken as heterogeneous oblate spheroid with different densities in three layers and sources of radiation pressure. These four primaries are moving on the circumference of a circle and form a kite configuration. After evaluating the equations of motion and Jacobi-integral, we study the numerical part of the paper such as equilibria, zero-velocity curves and regions of motion. Finally, we examine the stability of the equilibria and observed that all the equilibria are unstable.

Nonlinear Stability of Equilibrium Points in the Planar Equilateral Restricted Mass-Unequal Four-Body Problem

2021 ◽  
Vol 31 (11) ◽  
pp. 2130031
Author(s):  
José Alejandro Zepeda Ramírez ◽  
Martha Alvarez-Ramírez ◽  
Antonio García
Keyword(s):  
Nonlinear Stability ◽  
Mass Formula ◽  
Body Problem ◽  
Equilibrium Points ◽  
Constant Angular Velocity ◽  
Gravitational Attraction ◽  
Stability Of Equilibrium ◽  
Four Body Problem ◽  
Elliptic Equilibrium ◽  
The Stability

In this paper, we investigate the stability of equilibrium points for the planar restricted equilateral four-body problem in the case that one particle of negligible mass is moving under the Newtonian gravitational attraction of three positive masses [Formula: see text], [Formula: see text] and [Formula: see text] (called primaries). These always lie at the vertices of an equilateral triangle (Lagrangian configuration) and move with constant angular velocity in circular orbits around their center of masses. We consider the case where all the primaries have unequal masses, and investigate the nonlinear stability (in the sense of Lyapunov) of the elliptic equilibrium for the specific values of the mass [Formula: see text] and [Formula: see text] of the primary, fixed on the horizontal axis. Moreover, the [Formula: see text][Formula: see text]:[Formula: see text][Formula: see text] four-order resonant cases are determined and the stability is investigated. In this study, Markeev’s theorem and Arnold’s theorem become key ingredients.

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 Equilibrium Points and Basins of Convergence in the Triangular Restricted Four-Body Problem with a Radiating Body

2020 ◽  
Vol 30 (02) ◽  
pp. 2030003 ◽  
Author(s):  
J. E. Osorio-Vargas ◽  
Guillermo A. González ◽  
F. L. Dubeibe
Keyword(s):  
Body Problem ◽  
Equilibrium Points ◽  
Center Of Mass ◽  
Libration Points ◽  
The Sun ◽  
Radiation Factor ◽  
Trojan Asteroid ◽  
Four Body Problem ◽  
Common Center ◽  
The Stability

In this paper, we extend the basic equilateral four-body problem by introducing the effect of radiation pressure, Poynting–Robertson drag, and solar wind drag. In our setup, three primaries lie at the vertices of an equilateral triangle and move in circular orbits around their common center of mass. Here, one of the primaries is a radiating body and the fourth body (whose mass is negligible) does not affect the motion of the primaries. We show that the existence and the number of equilibrium points of the problem depend on the mass parameters and radiation factor. Consequently, the allowed regions of motion, the regions of the basins of convergence for the equilibrium points, and the basin entropy will also depend on these parameters. The present dynamical model is analyzed for three combinations of mass for the primaries: equal masses, two equal masses, different masses. As the main results, we find that in all cases the libration points are unstable if the radiation factor is larger than 0.01 and hence able to destroy the stability of the libration points in the restricted four-body problem composed by the Sun, Jupiter, Trojan asteroid and a test (dust) particle. Also, we conclude that the number of fixed points decreases with the increase of the radiation factor.

On the Dynamical Properties of the Sun-Earth-Moon Restricted Four Body Problem

2013 ◽  
Vol 834-836 ◽  
pp. 1869-1872
Author(s):  
Yun Yan Zhang ◽  
Nan Li ◽  
Yan Jun Li
Keyword(s):  
Body Problem ◽  
Equilibrium Points ◽  
Integral Relation ◽  
The Sun ◽  
Hill Region ◽  
Calculation Results ◽  
Four Body Problem ◽  
The Stability ◽  
Three Body ◽  
The Hill

The properties of the sun-earth-moon restricted four body problem were investigated. On the base of the earth-moon circular restricted three body problem, taking gravitation effect of the sun into account, a dynamic model of restricted four body problem was established. Equilibrium points were calculated by proper numerical method and linearization procedure was introduced to obtain the stability of these points. An invariant integral relation of this four body problem similar with Jacobi integral was derived, then the Hill region was investigated. Calculation results show that the equilibrium points will keep moving near libration points, also has their stability unchanged, and the Hill region will be non-periodically time varying.

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