Linear Stability Of Equilibrium Points Research Articles

Impact of a disc and drag forces on the existence linear stability of equilibrium points and Newton- Raphson basins of attraction

Impact of a disc and drag forces on the existence linear stability of equilibrium points and Newton- Raphson basins of attraction

A Planar Five-body Problem in a Framework of Heterogeneous and Mass Variation Effects

Abstract The aim of the present paper is to study the effects of heterogeneous oblate spheroid and variable mass on the motion of the fifth infinitesimal body in the frame of the circular restricted five-body problem, with the imposition that the three primaries are placed at the vertices of an equilateral triangle and the fourth primary is placed at the center of the equilateral triangle. By using Jeans law and Meshcherskii space transformation, we evaluate the equations of motion and find a quasiJacobian integral. We determine the locations of equilibrium points, the regions of motion, and the attracting domain of in-plane motion. Furthermore, the effects of heterogeneous oblate spheroid and variable mass have been examined with the help of Poincaré surfaces of section. We studied the linear stability of equilibrium points and found that all of them are unstable.

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

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.

Equilibrium points and zero velocity surfaces in the axisymmetric restricted five-body problem

In this paper, we have analyzed the motion of an infinitesimal mass in axisymmetric restricted five-body problem. The four primaries are supposed to be in axisymmetric central configurations based on Erdi and Czirjak and the motion is governed by mutual gravity only. We have derived the equations of motion of the infinitesimal mass firstly and then found the number and locations of equilibrium points numerically for various angle coordinates. In addition, the zero velocity surfaces (ZVS) are drawn and regions of motion are analyzed for specific configurations. Finally the linear stability of equilibrium points are discussed preliminarily.

Analytic results for the linear stability of the equilibrium point in Robe's restricted elliptic three-body problem

We study the Robe's restricted three-body problem. Such a motion was firstly studied by A. G. Robe in \cite{Robe}, which is used to model small oscillations of the earth's inner core taking into account the moon attraction. For the linear stability of elliptic equilibrium points of the Robe's restricted three-body problem, earlier results of such linear stability problem depend on a lot of numerical computations, while we give an analytic approach to it. The linearized Hamiltonian system near the elliptic relative equilibrium point in our problem coincides with the linearized system near the Euler elliptic relative equilibria in the classical three-body problem except for the rang of the mass parameter. We first establish some relations from the linear stability problem to symplectic paths and its corresponding linear operators. Then using the Maslov-type $\omega$-index theory of symplectic paths and the theory of linear operators, we compute $\omega$-indices and obtain certain properties of the linear stability of elliptic equilibrium points of the Robe's restricted three-body problem.

Sitnikov cyclic configuration of N+1-body problem

This manuscript deals with the generalisation of all previous works on series solutions and linear stability of equilibrium points of the Sitnikov problem. Following Giacaglia (1967), in Sect. 2 we have derived the equation of motion of the infinitesimal mass moving along the z-axis about which the plane of motion is rotating with unit angular velocity. In Sects. 3, 4 and 5 the series solutions of the Sitnikov problem have been developed by the method of MacMillan, Lindstedt-Poincare and iteration of Green’s function respectively. In Sect. 6 the three series solutions have been compared graphically by putting N=2, 3, 4. In Sect. 7 the coordinates of equilibrium points have been calculated. In Sect. 8 the linear stability of equilibrium points has been examined by the method of Murray and Dermott (Solar System Dynamics, Cambridge University Press, Cambridge, 1999) and it was found that the equilibrium points are stable in Sitnikov problem.

Robe’s circular restricted three-body problem under oblate primaries with perturbations in Coriolis and centrifugal forces

This paper examines the existence and linear stability of equilibrium points in the perturbed Robe’s circular restricted three-body problem under the assumption that the hydrostatic equilibrium figure of the first primary is an oblate spheroid, and the shape of the second primary is also an oblate spheroid. The problem is perturbed in the sense that small perturbations given to the Coriolis and centrifugal forces are being considered. Results of the analysis found two axial equilibrium points on the line joining the centre of both primaries. It is further observed that under certain conditions, points on the circle within the first primary are also equilibrium points. The linear stability of this configuration is examined; it is observed that the first axial point is unstable while the second one is conditionally stable and the circular points are unstable.

Perturbed Robe’s circular restricted three-body problem under an Oblate Primary

This paper examines the existence and linear stability of equilibrium points in the perturbed Robe’s circular restricted three-body problem under the assumption that the hydrostatic equilibrium figure of the first primary is an oblate spheroid. The problem is perturbed in the sense that small perturbations are given to the Coriolis and centrifugal forces are being considered. Results of the analysis found two axial equilibrium points on the line joining the centre of both primaries. It is further observed that under certain conditions, points on the circle within the first primary are also equilibrium points. And a special case where the density of the fluid and that of the infinitesimal mass are equal (D=0) is discussed. The linear stability of this configuration is examined; it is observed that the first axial point is unstable while the second one is conditionally stable and the circular points are unstable.

Robe's Restricted Three-Body Problem with Variable Masses and Perturbing Forces

The linear stability of equilibrium points of a test particle of infinitesimal mass in the framework of Robe's circular restricted three-body problem, as in Hallan and Rana, together with effect of variation in masses of the primaries with time according to the combined Meshcherskii law, is investigated. It is seen that, due to a small perturbation in the centrifugal force and an arbitrary constant of a particular integral of the Gylden-Meshcherskii problem, every point on the line joining the centers of the primaries is an equilibrium point provided they lie within the shell. Further, a number of pairs of equilibrium points lying on the -plane and forming triangles with the centers of the shell and the second primary exist, for some values of . The points collinear with the center of the shell are found to be stable under some conditions and the range of stability depends on the small perturbations and , while the triangular points are unstable. Illustrative numerical exploration is given to indicate significant improvement of the problem in Hallan and Rana.

Existence and Linear Stability of Equilibrium Points in the Robe’s Restricted Three-Body Problem with Oblateness

This paper investigates the positions and linear stability of an infinitesimal body around the equilibrium points in the framework of the Robe’s circular restricted three-body problem, with assumptions that the hydrostatic equilibrium figure of the first primary is an oblate spheroid and the second primary is an oblate body as well. It is found that equilibrium point exists near the centre of the first primary. Further, there can be one more equilibrium point on the line joining the centers of both primaries. Points on the circle within the first primary are also equilibrium points under certain conditions and the existence of two out-of-plane points is also observed. The linear stability of this configuration is examined and it is found that points near the center of the first primary are conditionally stable, while the circular and out of plane equilibrium points are unstable.

Poynting–Robertson effect on the linear stability of equilibrium points in the generalized photogravitational Chermnykh's problem

The Poynting–Robertson (P–R) effect on linear stability of equilibrium points is investigated in the generalized photogravitational Chermnykh's problem when a bigger primary is radiating and a smaller primary is an oblate spheroid. The positions of equilibrium points and their linear stabilities for various values of perturbing parameters are studied. It is found that the positions of the equilibrium points are different from the positions in the classical restricted three body problem. When the P-R effect is taken into account, these points are unstable in a linear sense. It is also found that the equilibrium points are unstable when the mass of the belt Mb ≥ 0.4.

Linear stability of equilibrium points in the generalized photogravitational Chermnykh’s problem

The equilibrium points and their linear stability has been discussed in the generalized photogravitational Chermnykh’s problem. The bigger primary is being considered as a source of radiation and small primary as an oblate spheroid. The effect of radiation pressure has been discussed numerically. The collinear points are linearly unstable and triangular points are stable in the sense of Lyapunov stability provided μ<μ Routh =0.0385201. The effect of gravitational potential from the belt is also examined. The mathematical properties of this system are different from the classical restricted three body problem.

Existence and linear stability of equilibrium points in the Robe's restricted three body problem when the first primary is an oblate spheroid

The existence and linear stability of equilibrium points in the Robe's restricted three body problem have been studied after considering the full buoyancy force as in Plastino and Plastino and by assuming the hydrostatic equilibrium figure of the first primary as an oblate spheroid. The pertinent equations of motion are derived and existence of all equilibrium points is discussed. It is found that there is an equilibrium point near the centre of the first primary. Further there can be one more equilibrium point on the line joining the centre of the first primary and second primary and infinite number of equilibrium points lying on a circle in the orbital plane of the second primary provided the parameters occurring in the problem satisfy certain conditions. So, there can be infinite number of equilibrium points contrary to the classical restricted three body problem.

Linear stability analysis and metastable solutions for a phase-field model

We study the linear stability of equilibrium points of a semilinear phase-field model, giving criteria for stability and instability. In the one-dimensional case, we study the distribution of equilibria and also prove the existence of metastable solutions that evolve very slowly in time.