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In this paper, we have studied the effect of oblateness of the Earth on the resonant curve in synchronous satellite under the gravitational effect of the Sun, the Moon and the Earth. By following the procedure of Frick and Garber (1962), we obtained the equations of motion of the synchronous satellite in the form of a second order linear differential equation by using the perturbation technique. From the solution, we observed that oscillatory amplitude becomes infinitely large when the ratio of two frequencies (angular velocity of the bary-center system around the Sun) and (spin rate of Earth) are commensurable. With the help of different graphs, we have shown the effect of and (coefficient of Earth's oblateness) on the resonant curves. It is observed that oscillatory amplitude increases when (orbital angle of the bary-center system around the Sun) increases and the value of is very close to zero. We also analyzed the phase portrait and phase space by using the method of Poincare section when the system is free from all the forces. Finally, we have obtained the expression for the energy integral, motion of mean longitude near the synchronous satellite. [ABSTRACT FROM AUTHOR]

In non-linear stability analysis, the effects of non-linear terms in the equations of motion are considered. The presence of these non-linear terms can cause the results of linear stability to change dramatically. Therefore, the Arnold–Moser theorem (Kolmogorov–Arnold–Moser theory) has been used to study the non-linear stability of the equilibrium point in the planar Robe's restricted three-body problem when the second primary is a finite straight segment of length . The density parameter is considered zero. The equilibrium point has been found to be stable in non-linear sense for all mass ratios in the range of linear stability except possibly for six critical values of the mass ratio . The critical values depend on the length parameter , which shows that the length parameter has significant effect on the non-linear stability of the equilibrium point. The obtained results are applied to predict the stability of equilibrium point for Jupiter–Amalthea system. It is observed that the equilibrium point is unstable for Jupiter‒Amalthea system. [ABSTRACT FROM AUTHOR]

In the framework of axisymmetric problem of restricted five bodies with variable mass, our goal is to study the existence and stability of libration points. In the axisymmetric restricted five-body problem, we have assumed that the mass of the fifth particle varies according to Jeans' law. We have further supposed that the four bodies having masses and (with ) form an axisymmetric concave configuration. The equations of motion of a test particle of infinitesimal mass have been illustrated, which is similar to the axisymmetric restricted five-body problem when the problem of variable mass evolves with the problem of constant mass. In this paper, we have determined in-plane as well as out-of-plane libration points along with their stability. Further, we have shown the regions of motion where the fifth body can move freely. We have also shown that the angle parameters and and the parameters, arising due to variation of mass and , affect the existence and number of these libration points. Moreover, the bivariate version of the Newton-Raphson iterative scheme is applied in an attempt to unveil the analysis of the basins of convergence linked with the libration points as a function of the mass parameter. [ABSTRACT FROM AUTHOR]

This study analyzes the motion of two infinitesimal bodies in the restricted problem of bodies, including an oblate spheroid and straight segment as primary bodies, along with the photogravitational effect of the more massive primary. The pertinent equations of motion of the infinitesimal bodies in the dimensionless variables are obtained by incorporating the effect caused by the oblateness, radiation pressure, and straight segment. On close examination of the parameters involved in the problem, a study of the equilibrium points of the dynamical system reveals that six of them are collinear with primary bodies' centers, while the remaining eight are not. An in-depth study is done on how the position of the equilibrium points changes with respect to the radiation parameter. The permissible and forbidden regions of motion are shown, and it is inferred that when the Jacobian constant is decreased, the permissible regions of motion expand. Furthermore, a linear stability analysis of the generated equilibrium points is done, and we observe that all the equilibrium points are unstable for a set of values of the model's parameters. [ABSTRACT FROM AUTHOR]

In the present paper, we have determined the equations of motion of the Moon in spherical coordinate system using the gravitational potential of Earth. Using perturbation, equations of motion are reduced to a second order differential equation. From the solution, two types of resonance are observed: (i) due to the frequencies-rate of change of Earth's equatorial ellipticity parameter and Earth's rotation rate, and (ii) due to the frequencies-angular velocity of the bary-center around the sun and Earth's rotation rate. Resonant curves are drawn where oscillatory amplitude becomes infinitely large at the resonant points. The effect of Earth's equatorial ellipticity parameter and resistive force on the resonant curve is analyzed. From the graphs it is observed that the effect of Earth's equatorial ellipticity on the resonant curve is very small while the effect of resistive force is significant. It is also observed that oscillatory amplitude decreases when the magnitude of resistive force increases. Finally, the phase portrait is analyzed when the system is free from forces. Orbits in the phase space are also studied by applying the method of Poincare section. A necessary condition for the bifurcation is derived at the end. [ABSTRACT FROM AUTHOR]

Associating the influences of viscosity and oblateness in the finite straight segment model of the Robe's problem, the linear stability of the collinear and non-collinear equilibrium points for a small solid sphere are analyzed. This small solid sphere is moving inside the first primary which is a homogeneous incompressible viscous fluid whose hydrostatic equilibrium figure is an oblate spheroid. The second primary is a finite straight segment. The existence of the equilibrium points is discussed after deriving the pertinent equations of motion of the small solid sphere. It is found that viscosity does not affect the location and number of equilibrium points but affects the stability as it converts the marginal stability to asymptotic stability. However, oblateness affects the locations of the equilibrium points. Applicability of the results of this study to an astrophysical problem is discussed, and we have calculated a lower bound on the ratio of the orbital radius and the total mass of the primaries of an astrophysical problem to which the results obtained may be applied. This ratio is called the scaling factor. [ABSTRACT FROM AUTHOR]

In this article, we investigated the resonant curves of geo-centric satellite due to the rate of change in earth's equatorial ellipticity parameter (γ ˙) , moon's angular velocity around the earth (ϑ ˙) , and angular velocity of bary-center (Θ ˙) around the sun. We expressed the equations of the motion of the satellite in a spherical coordinate system using the earth's potential. The system of equations are reduced into a second-order ordinary differential equation and determined its solution. From the solution, it is noted that resonance appears due to the frequencies δ s 0 ˙ , γ ˙ , Θ 0 ˙ and ϑ 0 ˙ at ten resonant points δ ˙ s 0 = 2 γ ˙ , δ ˙ s 0 = γ ˙ , δ ˙ s 0 = 2 Θ ˙ 0 , δ ˙ s 0 = Θ ˙ 0 , δ ˙ s 0 = 2 Θ ˙ 0 3 , δ ˙ s 0 = Θ ˙ 0 2 , δ ˙ s 0 = 2 ϑ 0 ˙ , δ ˙ s 0 = ϑ 0 ˙ , δ ˙ s 0 = 2 ϑ 0 ˙ 3 , and δ ˙ s 0 = d o t ϑ 0 2 , where δ ˙ s 0 , Θ ˙ 0 and ϑ 0 ˙ are the steady-state value of δ ˙ s , Θ ˙ and ϑ ˙ respectively. We have drawn the resonant curves corresponding to the resonant points. From the graph, it is observed that the effect of earth's equatorial ellipticity parameter on the resonant curves are very small. Also, we have shown the effect of eccentricity (e) and semi-major axes (a) on the oscillatory amplitudes (A i) , where i = 1 , 2 , 3 ... 21 . Finally, we have analyzed the phase portrait by using Poincaré section method when the system is free from forces. [ABSTRACT FROM AUTHOR]

The present investigation deals with the outcomes of the small perturbations σ 1 and σ 2 in the Coriolis and centrifugal forces respectively, in the oblate straight segment setup of the restricted 2 + 2 body problem. We describe the dynamical system that consists of the primaries of masses m 1 and m 2 as oblate spheroid and straight segment, respectively. With this setup, we obtain the equations of motion of the infinitesimal bodies having masses m 3 and m 4 under the gravitational attraction of the primary bodies, their mutual attraction and the small perturbations in the Coriolis and centrifugal forces. We determine the equilibrium points by using the perturbation technique in the equilibrium points for the case m 4 = 0 . We get fourteen equilibrium points, six of which are on the line connecting the centers of the primaries and eight of which lie on the x y - plane but not on the x - axis. The stability of the equilibrium points for a particular set of the parameters is analyzed and it is concluded that the collinear equilibrium points are unstable. The four non-collinear equilibrium points are stable for some combinations of σ 1 and σ 2 and rest are unstable for all values of σ 1 and σ 2 . The permissible and forbidden regions of motion of the infinitesimal body are also studied for the different values of the Jacobian constant and it is perceived that the permissible regions of motion expands on decreasing the value of the Jacobian constant. These regions are subsequently affected by the involvement of the mass, oblateness, length and perturbations in the Coriolis and centrifugal forces parameters. [ABSTRACT FROM AUTHOR]

This research article presents the effect of the Earth's equatorial ellipticity and the resistive force on a geocentric satellite's resonant curves. The resonance occurs in the motion of the satellite due to the frequencies, the angular velocity of the satellite, the rate of change of Earth's equatorial ellipticity parameter (γ ̇) , the steady-state angular velocity of the Moon around the Earth ( ϑ 0 ̇) , and the steady-state angular velocity of barycenter (Θ ̇) around the Sun. We express the equations of motion of the satellite in a spherical coordinate system by using the Earth's potential. By applying the unperturbed solution, we reduced the system of equations into a second-order differential equation and obtained its solution. From the solution it is observed that the resonance appears between the frequencies δ s 0 ̇ , γ ̇ , Θ 0 ̇ and ϑ 0 ̇ at ten resonant points. We analyze the resonant curves corresponding to the resonant points and show the effect of resistive force and latitude of a satellite. Also, We show the effect of eccentricity (e) and semi-major axis (a) on the oscillatory amplitudes (A i) , for i = 1 , 2 , 3 ... 21. [ABSTRACT FROM AUTHOR]

In this paper, we present the analysis of the Newton–Raphson basins of convergence (NR-BoCs) linked with the relative equilibria (LPs) in the Eulerian configuration of a pseudo-Newtonian planar circular restricted problem of four bodies (SCCR4BP) with spinning primaries. In this configuration of the SCCR4BP, the primaries are laid out in a symmetrical collinear shape in which the two peripheral primaries are of equal mass. We derive the equations of motion by assuming that the system consists of three Kerr-like primaries to examine how the geometry of the basins of convergence (BoCs) associated with the LPs of SCCR4BP are affected by the first-order non-Newtonian contribution to the gravitational field. Additionally, we analyze the correlation among the regions of convergence, the required number of iterations, and the associated probability distributions. Finally, for the quantitative study of the basins of convergence, we measure the uncertainty of the respective basins via the basin entropy and the basin boundary entropy. • On the beyond-Newtonian collinear circular restricted (3 + 1)-body problem with spinning primaries is discussed. • The Newton–Raphson basins of convergence are discussed. • The basin entropy and basin boundary entropy are calculated to measure the uncertainty of the basins. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the existence, locations and stability of the equilibrium points as well as zero velocity curves (ZVCs) under the effect of dissipative force (Stokes drag) in the restricted four-body problem (R4BP) with a variable mass. We have studied the dynamics of the infinitesimal body whose mass is variable and changing according to Jeans' law. We have determined the equations of motion of the infinitesimal body by using the space time transformation of Meshcherskii when the change in mass is non-isotropic. We have determined the locations of equilibrium points in the special case of non-isotropic variation of mass under the effect of the dissipative force and found that all the equilibrium points are non-collinear. The effect of parameter k (0 < k < 1) , which controls the dissipative force, on the locations of the libration points and their stability are investigated. It has been observed that the energy integral is time dependent due to the presence of the Stokes drag force. Further, it has been observed that all the equilibrium points are unstable for all values of the parameters used. • The effect of Stoke's drag in the circular restricted four-body problem with variable mass has been studied. • The locations of equilibrium points and their stability have been discussed. • Dissipative constant k has more impact on the locations of the equilibrium points. • The effect of Stokes drag also has been studied by the zero velocity curves on the regions of possible motion. • The effects of variable mass and Stokes drag have significant impacts on the regions of possible motion. [ABSTRACT FROM AUTHOR]