Positions Of Equilibrium Points Research Articles

A study of the equilibrium dynamics of the test particle in the collinear circular restricted four-body problem with non-spherical central primary

We consider the collinear restricted four-body problem (CR4BP), where the test particle of infinitesimal mass is moving under the gravitational influence of the three primary bodies. It is further assumed that the central primary is a non-spherical body, particularly either an oblate or prolate spheroid, whereas the peripheral primaries are spherical in shape. A numerical analysis is presented to unveil the effect of the oblateness and prolateness parameters on the position of equilibrium points (EPs) and their linear stability in the CR4BP. Moreover, the permissible regions of possible motion as determined by the zero-velocity surface and associated equipotential curves and the basins of convergence linked with the EPs on the orbital plane are presented. The existence and number of collinear EPs and non-collinear EPs in the problem depend on the combination of the mass parameter of the primaries and the oblateness/prolateness parameter. Additionally, the application of the problem in the Saturn-Moon(1)-Moon(2)-System has been presented.

Equilibrium stability in the triangular restricted four-body problem with non-spherical primaries

This paper investigates the Lagrangian configuration of the restricted four-body problem in which the three primaries are non-spherical, specifically either prolate or oblate. By using various standard numerical methods, the positions of equilibrium points and their linear stability and dynamical type were determined. The impact of mass and shape of the primaries on the system’s equilibrium points and their linear stability were systematically explored by discretizing the parameter space for the non-sphericity parameter within a specified interval. The study revealed that the system always has an even number of equilibrium points, ranging from 8 to 22. Linearly stable points always exist, except for the case where there are 10 equilibrium points, where all the points are unstable.

Equilibrium points of heterogeneous small body in finite element method

ABSTRACT This paper presents a finite element method to search for equilibrium points around a heterogeneous small body, which degenerates from the full two-body problem in finite element form. The gravitational potential, acceleration, gravitational gradient matrix, and the linearized perturbation equation, which are key formulas for solving the equilibrium points and discriminating their stabilities, are interpolated by the tetrahedral model nodes of the small body. The finite element method could capture the complex internal structures of small bodies and provide a uniform and simple formula for various configurations. The method is applied to the Kuiper Belt Object (486958) Arrokoth, which is a contact binary asteroid. Three types of heterogeneous structures are hypothesized to investigate the evolutions of equilibrium points, namely density disparity of two sub-lobes, hardcore structure, and cavity structure. Under the condition of constant Arrokoth mass, the density disparity of the two sub-lobes greatly influences the local gravitational field. It thus has a significant influence on the positions of equilibrium points. The hardcore and cavity structures have less impact on the equilibrium points. These three simulation groups verified that the equilibrium points of a heterogeneous small body are quite different from those of a homogeneous small body. Such investigation of the equilibrium points may give an in-depth understanding of the dynamical environment around the heterogeneous small body, which is significant for future deep-space missions.

Mathematical Model to Determine Motion of Infinitesimal Mass When Primaries Are both Radiating and Triaxial

This model has examined the motion of an infinitesimal particle when both primaries are radiating and triaxial in the framework of the Circular Restricted Three Body Problem (CR3BP). It was observed that the equations of motion were affected by perturbing forces (triaxiality and radiation pressures) of the primary bodies. Analytically, the locations of equilibrium points were obtained and applied to the binary systems Kruger 60 (AB) and Achird to obtain the numerical results with the help of MathLab software. Numerical investigations reveal that locations of equilibrium points of this problem have been determined and the effect of increasing triaxiality on the position of equilibrium point were obtained with graphical results. It reveal that the infinitesimal mass moves in the direction of the bigger primary towards the line joining the primary bodies. The linear stability of the equilibrium points has also been examined and a numerical solution of the analytical result was applied to the binary stars Kruger 60. It was found that the triangular points are unstable which means that the instability exist as a result of the presence of perturbation forces (Radiation and Triaxiality).
 Keywords: Model, Motion, Infinitesimal, Radiating, Triaxial.

Multifarious Chaotic Attractors and Its Control in Rigid Body Attitude Dynamical System

The Euler dynamical equation which describes the attitude motion of a rigid body will exhibit very complex dynamic behaviors under the action of different external torques. Many special types of new chaotic attractors are presented, including hidden attractors, double-body-double-core chaotic attractors, and single-body-three-core-tree-wing chaotic attractors. The position of equilibrium points in several typical cases of the Euler dynamic equation is solved, and the stability of linearized equation at each equilibrium point and its influence on the formation of the chaotic attractor are analyzed. An improved nonlinear relay control law based on Euler angle feedback is developed to stabilize a new chaotic spacecraft attitude motion to an appointed equilibrium point or a periodic orbit.

Overview and comparison of approaches towards the planar restricted five-body problem with primaries forming an axisymmetric four-body central configuration

The aim of this paper is to present recent results on the restricted five-body problem where the primaries are located at a known central configuration having the $x$-axis as symmetry axis, so that two bodies with equal masses are situated symmetrically with respect to this axis. We firstly give an overview of the axisymmetric central configurations computed by Erdi and Czirjak (2016), as well as that where three bodies with equal masses are situated at the vertices of an equilateral triangle, while the fourth body lies at the center of the triangle. This last one bridges the gap between convex and concave four-body central configurations. After that, the characterization of the geometry of the restricted five-body problem families with these configurations are described, and the number and the evolution positions of equilibrium points, which depend on the mass parameters of the primaries, are compared.

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.

Combined effect of Stokes drag, oblateness and radiation pressure on the existence and stability of equilibrium points in the restricted four-body problem

This paper studies numerically the existence of collinear and non-collinear equilibrium points and their linear stability in the frame work of photogravitational circular restricted four-body problem with Stokes drag acting as a dissipative force and considering the first primary as a radiating body and the second primary as an oblate spheroid. The mass of the fourth body is assumed to be infinitesimal and does not affect the motion of the three primaries which are always at the vertices of an equilateral triangle (Lagrangian configuration). It is found that at constant dissipative force, and a simultaneous increase in both radiation pressure and oblateness coefficients, the curves that define the path of the motion of the infinitesimal body are found to shrink due to shifts in the positions of equilibrium points. All the collinear and non-collinear equilibrium points are found to be linearly unstable under the combined effect of radiation pressure, oblateness and Stokes drag. The energy integral is seen to be time dependent due to the presence of the drag force. More so the dynamic property of the system is investigated with the help of Lyapunov characteristic exponents (LCEs). It is found that the system is chaotic as the trajectories locally diverge from each other and the equilibrium points are chaotic attractors. We justified the relevance of the model in astronomy by applying it to a stellar system (Gliese 667C) and another found that both the existence and stability of the equilibrium points of any restricted few body system greatly depend of the value of the mass parameter.

A particle-linkage model for non-axisymmetric elongated asteroids

This paper investigates a simplified model for describing the gravitational fields of non-axisymmetric elongated asteroids. The connection between the simplified model and the target asteroid is built by considering the positions of equilibrium points. To improve the performance of position matching for the equilibrium points associated with these non-axisymmetric asteroids, a non-axisymmetric triple-particle-linkage model is proposed based on two existing axisymmetric particle-linkage models. The unknown parameters of the simplified model are determined by minimizing the matching error using the nonlinear optimization method. The proposed simplified model is applied for three realistic elongated asteroids, 243 Ida, 433 Eros and (8567) 1996 HW1. The simulation results verify that the current particle-linkage model has better matching accuracy than the two existing particle-linkage models. The comparison, between the simplified model and the polyhedral model, on the topological cases of the equilibrium points and the distribution of gravitational potential further validate the rationality and accuracy of the simplified model.

The periodic dynamics of the irregular heterogeneous celestial bodies

In this paper, we develop a methodology to study the periodic dynamics of irregular heterogeneous celestial bodies. Heterogeneous bodies are not scarce in space. It has been found that bodies, such as 4 Vesta, 624 Hektor, 87 Sylvia, 16 Psyche and 25143 Itokawa, may all have varied internal structures. They can be divided into large-scale and small-scale cases. The varied internal structures of large-scale bodies always result from gradient pressure inside, which leads to compactness differences of the inner material. However, the heterogeneity of a small-scale body is always reflected by the different densities of different areas, which may originate from collision formation from multiple objects. We propose a modeling procedure for the heterogeneous bodies derived from the conventional polyhedral method and then compare its dynamical characteristics with those of the homogeneous case. It is found that zero-velocity curves, positions of equilibrium points, types of bifurcations in the continuation of the orbital family and the stabilities of periodic orbits near the heterogeneous body are different from those in the homogeneous case. The suborbicular orbits near the equatorial plane are potential parking orbits for a future mission, so we discuss the switching of the orbital stability of the family because it has fundamental significance to orbit maintenance and operations around actual asteroids.

New Mechanical Feature of Two-Solitary Wave to the KdV Equation

New breather solitary solution and two-solitary solutions depending on constant equilibrium solution to the Korteweg de Vries equation are obtained by using an extended homoclinic test approach. A new mechanical feature of a two-solitary wave, namely, dependence of propagation direction and shape on position of equilibrium point, is investigated.

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.

Combined effects of perturbations, radiation and oblateness on the location of equilibrium points in the restricted three-body problem.

We have studied the effect of small perturbations in the coriolis and the centrifugal forces together with oblateness and radiation pressure forces of the primaries on the locations of equilibrium points in the restricted three-body problem. We have found that oblate-ness and radiation pressure forces affect the locations of equilibrium points. We have further seen that the positions of equilibrium points are not affected by the change in the coriolis force. They are only affected by the change in the centrifugal force. It is also observed that the triangular points form triangles with the primaries and lie on the line joining the primaries. Journal of the Nigerian Association of Mathematical Physics Vol. 8 2004: pp. 19-24

Internal resonant interactions of three free surface-waves in a circular cylindrical basin

The basic equations of free capillary-gravity surface-waves in a circular cylindrical basin were derived from Luke's principle. Taking Galerkin's expansion of the velocity potential and the free surface elevation, the second-order perturbation equations were derived by use of expansion of multiple scale. The nonlinear interactions with the second order internal resonance of three free surface-waves were discussed based on the above. The results include: derivation of the couple equations of resonant interactions among three waves and the conservation laws; analysis of the positions of equilibrium points in phase plane; study of the resonant parameters and the non-resonant parameters respectively in all kinds of circumstances; derivation of the stationary solutions of the second-order interaction equations corresponding to different parameters and analysis of the stability property of the solutions; discussion of the effective solutions only in the limited time range. The analysis makes it clear that the energy transformation mode among three waves differs because of the different initial conditions under nontrivial circumstance. The energy may either exchange among three waves periodically or damp or increase in single waves.