Non-collinear 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 shperical 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.

Analysis of albedo and disc effects in the generalized restricted four-body problem

This manuscript explores the Generalized Restricted Four-Body Problem within the context of radiation pressure, the albedo effect, oblateness, and a disc-like structure that orbits around the primary bodies. We have considered that one of the primaries is a bigger primary, and the rest of the two smaller primaries are of equal masses, in which one of the smaller primaries is considered as a black body. Moreover, the perturbing forces by the effect of radiation pressure and oblateness are considered in the bigger primary. The effect of albedo and oblateness due to the second smaller primary is also taken into count. This study yields nine equilibrium points corresponding to various parameter values, including three collinear and six non-collinear equilibria. We conduct an analysis of the equilibrium points, their linear stability, and the motion around them. In the context of the considered parameter set, certain collinear equilibrium points exhibit stability, while others are unstable. Additionally, most of the non-collinear equilibrium points display linear stability. We also investigate variations in Jacobi’s constant and the positions of existing equilibrium points caused by the albedo and disc effects. It is found that the value of Jacobi constant CLj decreases with an increase in the mass parameter μ at the collinear points, which results in an expansion in the prohibited region. The inhibited zone of motion of the infinitesimal body is shown for different sets of parameters.

EQUILIBRIUM POINTS IN THE CR3BP OF THREE OBLATE BODIES UNDER THE EFFECTS OF CIRCUMBINARY DISC AND RADIATING PRIMARIES WITH POYNTING-ROBERTSON DRAG

We study numerically the generalized planar photogravitational circular restricted three-body problem, where an infinitesimal body is moving under the Newtonian gravitational attraction of two bodies which are finite, moving in circles around their center of mass fixed at the origin of the coordinate system, where both bodies are situated on the horizontal -axis. The third body is significantly smaller compared to the masses of the two bodies (primaries) where its influence on them can be neglected. The three participating bodies are modeled as oblate spheroids, under effect of radiation of the two main masses together with effective Poynting-Robertson drag and both of them are enclosed by a belt of homogeneous circular cluster of material points. In this paper, the existence and location of the equilibrium points and their linear stability are explored for various combinations of the model’s parameters. We observe that under constant P–R drag effect, collinear equilibrium solutions cease to exist but there are in the absence of the drag forces. We found that five or seven non-collinear equilibrium points may lie on the plane of primaries motion depends on the particular values of model’s parameters, and it is seen that the perturbing forces have significant effects on their positions and linear stability. In our model, the binary system Kruger 60 is used, and it is found that the positions of the equilibria and their stability are affected by these perturbing forces. In the case where seven critical points exist, all the equilibria are...

Studying the Equilibrium Points of the Modified Circular Restricted Three-body Problem: The Case of Sun–Haumea System

We intend to study a modified version of the planar Circular Restricted Three-Body Problem (CRTBP) by incorporating several perturbing parameters. We consider the bigger primary as an oblate spheroid and emitting radiation while the small primary has an elongated body. We also consider the perturbation from a disk-like structure encompassing this three-body system. First, we develop a mathematical model of this modified CRTBP. We have found there exist five equilibrium points in this modified CRTBP model, where three of them are collinear and the other two are non-collinear. Second, we apply our modified CRTBP model to the Sun–Haumea system by considering several values of each perturbing parameter. Through our numerical investigation, we have discovered that the incorporation of perturbing parameters has resulted in a shift in the equilibrium point positions of the Sun–Haumea system compared to their positions in the classical CRTBP. The stability of equilibrium points is investigated. We have shown that the collinear equilibrium points are unstable and the stability of non-collinear equilibrium points depends on the mass parameter μ of the system. Unlike the classical case, non-collinear equilibrium points have both a maximum and minimum limit of μ for achieving stability. We remark that the stability range of μ in non-collinear equilibrium points depends on the perturbing parameters. In the context of the Sun–Haumea system, we have found that the non-collinear equilibrium points are stable.

The Predictions of Noncollinear Equilibria Positions in ER3BP with Yukawa-Like Corrections

The existence and stability of noncollinear equilibrium points in the elliptic restricted three-body problem under the consideration of Yukawa correction to Newtonian potential are studied in this paper. The effects of various parameters (μ, ê, α, and λ) on the noncollinear equilibrium points are discussed briefly, and it is found that only ordinate of noncollinear equilibria E4,5 is affected by Yukawa correction while abscissa is affected by only mass parameter μ. The noncollinear equilibria was found linearly stable for a critical mass parameter μc. A critical point λ = ½ is also obtained for the critical mass parameter μc, and at this point, the critical mass parameter μc has maximum or minimum values according to α < 0 or α > 0, respectively.

Noncollinear Equilibrium Points in CRTBP with Yukawa-Like Corrections to Newtonian Potential under an Oblate Primary Model

This study is about the effects of Yukawa-like corrections to Newtonian potential on the existence and stability of noncollinear equilibrium points in a circular restricted three-body problem when bigger primary is an oblate spheroid. It is observed that ∂x0/∂λ = 0 = ∂y0/∂λ at λ0 = 1/2, so we have a critical point λ0 = 1/2 at which the maximum and minimum values of x0 and y0 can be obtained, where λ ∈ (0, ∞) is the range of Yukawa force and (x0, y0) are the coordinates of noncollinear equilibrium points. It is found that x0 and y0 are increasing functions in λ in the interval 0 < λ < λ0 and decreasing functions in λ in the interval λ0 < λ < ∞ for all α ∈ α+. On the other hand, x0 and y0 are decreasing functions in λ in the interval 0 < λ < λ0 and increasing functions in λ in the interval λ0 < λ < ∞ for all α ∈ α−, where α ∈ (−1, 1) is the coupling constant of Yukawa force to gravitational force. The noncollinear equilibrium points are found linearly stable for the critical mass parameter β0, and it is noticed that ∂β0/∂λ = 0 at λ ∗ = 1/3; thus, we got another critical point which gives the maximum and minimum values of β0. Also, ∂β0/∂λ > 0 if 0 < λ < λ ∗ and ∂β0/∂λ < 0 if λ ∗ < λ < ∞ for all α ∈ α−, and ∂β0/∂λ < 0 if 0 < λ < λ ∗ and ∂β0/∂λ > 0 if λ ∗ < λ < ∞ for all α ∈ α+. Thus, the local minima for β0 in the interval 0 < λ < λ ∗ can also be obtained.

Analysis of stability of non-collinear equilibrium points: Application to Sun–Mars and Proxima Centauri systems

In the present work, we employ the modified elliptic restricted three-body problem by considering some additional perturbed forces such as: the radiation pressure, the primaries’ oblateness, and the effect of dust belt for studying the dynamical motion of the infinitesimal body. The semi-analytical solutions for the locations of non-collinear equilibrium points are obtained. The proposed model is applied to real astronomical systems. Thus, the Sun–Mars and the Proxima Centauri systems are used to estimate numerically and graphically the locations of these points in both systems at different values of perturbation parameters. In this context, the critical mass values are evaluated for both systems to analyze the stability motion of the infinitesimal body around the locations of equilibrium points and find the stability range of these locations. We demonstrate that the stability of non-collinear points in both systems depends on the parameter of mass ratio values of its own system. It is observed that the locations of these points and their stability will be affected significantly due to the changes in the values of perturbation parameters.

A STUDY ON EQUILIBRIUM POINTS AND STABILITY OF THE ROBES R3BP WITH DENSITY VARIATION

The paper investigates equilibrium points and stability of an infinitesimal mass under the Robe restricted three-body problem with density variation. The equations of motion have been stated and the equilibrium points investigated. It is seen that axial equilibrium point at the centre and axial equilibrium points near the centre of the first primary were found. The axial points near the centre were further explored and it seen that five types of axial equilibrium points namely; minimal axial points, primary axial points of first kind, interior points, primary axial points of second kind and the terminal point, exists. These points are either located inside the first primary or on the points at which the circle bounding the first primary is coincident with the axis. Further a pair of non-collinear equilibrium points and circular equilibrium points which are points on the circle bounding the first primary, exists. The linear stability of these equilibrium points are investigated and it is seen that the axial points at the centre and those near the centre are conditionally stable analytically but our numerical explorations of the stability reveals that these points can be unstable and stable, a condition which depends on the mass parameter and the density variation of the fluid. The circular points are unstable due to double roots of the characteristic equation while the non-collinear equilibrium points are also unstable due to a positive root which induces instability at the non-collinear equilibrium points. For our numerical explorations of the equilibrium points and the stability outcomes, we in particular use the Submarine-Earth and Moon system and the general case for all mass ratios. The Robe’s problem has been useful in discussing the long time stability of the Earth’s core and for describing the motion of underwater vehicles under the Earth’s and Moon’s attraction.

Photogravitational magnetic-binary problem with oblateness and belt of material points

We study the motion of a charged particle in the framework of magnetic-binary problem where the bigger primary is the source of radiation and the smaller primary is the oblate body; and they are enclosed by a homogeneous circular truss of material points centered at the center of mass of the system. We have determined the equations of motion that govern the motion of a charged particle. The coordinates of collinear and non-collinear equilibrium points and their linear stability have been calculated. Numerical results reveal that the ratio of magnetic moment λ has a huge impact on the location, stability and orbital dynamics of the problem. We observed that there exists eight, eleven and thirteen equilibrium points for different values of mass parameter μ and the ratio of magnetic moment λ. Further, we observed that all non-collinear equilibrium points are unstable in the Lyapunov sense. But the collinear points L4 and L6 show a stable behavior for some values of μ and λ, while other collinear equilibrium points are unstable. The geometric configuration of zero velocity curves of the charged particle is numerically simulated and addressed. Moreover, first order approximations to a Lyapunov and Lissajous orbits are summarized near the collinear equilibrium points under the effect of λ.

Dynamics of the restricted three-body problem having elongated smaller primary with disc-like structure

In this paper, we have analysed the dynamical behavior of the restricted three-body problem having elongated smaller primary with disc-like structure. We discuss the zero-velocity curve, equilibrium points, and their stability by considering the different segment-length of the elongated primary and mass ratio of the primary. The impacts of disc-like structure and elongated primary body are observed on the equilibrium points and their stability analysis. We demonstrate that all collinear equilibrium points are unstable by varying the mass parameter μ and the segment length l. Further, the critical point for mass parameter μc at non-collinear equilibrium points is calculated. Furthermore, we emphasize our discussion through an example by taking mass parameter μ=0.01<μc and segment-length l=0.01. Finally, we conclude that the proposed problem is a credible model for describing the infinitesimal body’s motion in a disc-like structure with an elongated smaller primary.

Effect of the Planetesimal Belt on the Dynamics of the Restricted Problem of 2 + 2 Bodies

In this paper, we study the existence and stability of collinear and noncollinear equilibrium points within the frame of the perturbed restricted problem of 2 + 2 bodies by a planetesimal belt. We compare and investigate the corresponding results of the perturbed and unperturbed models. The impact of the planetesimal belt is observed on collinear and noncollinear equilibrium points. We demonstrate that all equilibrium points are unstable, and we numerically investigate the noncollinear equilibrium points. Finally, we emphasize that the proposed problem is a credible model for describing the capture of small bodies by a planet.

STUDY ON EQUILIBRIUM POINTS CHARACTERISTICS OF BINARY ASTEROID SYSTEMS BASED ON THE MASS DIPOLE MODEL

The local dynamics and periodic orbits in the vicinity of equilibrium points in binary asteroid systems are investigated. The dynamical model for synchronous binary asteroid system with one elongated asteroid and one small and regular asteroid is established. For the synchronous binary system, the influences of system parameters on location, stability, and variation trends of the equilibrium points are studied. The linear stability area for non-collinear equilibrium points is given. For the non-synchronous binary system, trajectories of equivalent equilibrium points are found. 1∶1 families of resonant orbits near the collinear equilibrium points of the synchronous binary asteroid system are obtained by the path searching correction method and the pseudo-arclength continuation method. This study can provide theoretical basis for the orbital design in binary asteroid explorations.

On sensitivity of the stability of equilibrium points with respect to the perturbations

The present investigation considers the effect of small perturbations given in the Coriolis and centrifugal forces on the location and stability of the equilibrium points in the Robe’s circular restricted three-body problem with non-spherical primary bodies. The felicitous equations of motion of $$m_3$$ are obtained by taking into account the shapes of primaries $$m_1$$ and $$m_2$$ , the full buoyancy force of the fluid which is filled inside $$m_1$$ of density $$\rho _1$$ , the forces due to the gravitational attraction of the fluid and $$m_2$$ . We assume that the massive body $$m_1$$ is an oblate spheroid and the $$m_2$$ a finite straight segment, and they move under a mutual gravitational attraction described by the Newton’s universal law of gravitation. In the present problem, $$m_3$$ is moving in the fluid and the rotating reference frame is used, its motion is bound to be affected by the perturbed Coriolis and centrifugal forces. In this attempt these effects along with the effects caused by the oblateness and length parameters A and l respectively, on the location and stability of the equilibrium points are observed. A pair of collinear equilibrium points $$L_1$$ and $$L_2$$ and infinite number of non-collinear equilibrium points are obtained. The stability of all the equilibrium points depends on the coefficients of their corresponding characteristic polynomials that are obtained with the help of linear variational equations.

Impact of radiation pressure and circumstellar dust on motion of a test particle in Manev’s field

In this paper, we present a study on the impact of radiation pressure and circumstellar dust on the motion of a test particle in the framework of the restricted four-body problem under the Manev’s field. We show that the distribution of equilibrium points on the plane of motion is slightly different from that of the classical Newtonian problem. With the aid of the Lyapunov characteristic exponents, we show that the system is sensitive to changes in initial conditions; hence, the orbit of the system is found to be chaotic in the phase space for the given initial conditions. Furthermore, a numerical application of this model to a stellar system (Gliese 667C) is considered, which validates the dependence of the equilibrium points on the mass parameter. We show that the non-collinear equilibrium points of this stellar system are distributed symmetrically about the x-axis, and five of them are linearly stable. The basins of attraction of the system show that the equilibrium points have irregular boundaries, and we use the energy integral and the Manev parameter to illustrate the zero-velocity curves showing the permissible region of motion of the test particle with respect to the Jacobi constant.

Stability of generalized elliptic restricted four body problem with radiation and oblateness effects

This paper studies the existence and stability of non-collinear equilibrium points in the elliptic restricted four body problem with bigger primary as a source of radiation and other two primaries having equal masses as oblate spheroid. In the elliptic restricted four body problem, three of the bodies are moving in elliptical orbit around their common centre of mass fixed at the origin of the coordinate system, while the fourth one is infinitesimal. Three pairs of non-collinear points are obtained symmetric with respect to x-axis. We found the equilibrium points are stable in linear sense. We also investigate the pulsating zero velocity surfaces and basin of attraction for varying value of oblateness coefficient and radiation pressure parameter.

Effect of perturbations in the Coriolis and centrifugal forces in the Robe‐finite straight segment model with arbitrary density parameter

AbstractThe present article is a study of the effect of small perturbations ϵ and ϵ′ in the Coriolis and centrifugal forces, respectively, on the existence and stability of the equilibrium points in the Robe's restricted three‐body problem when the smaller primary is a finite straight segment. The effect of small perturbations in the centrifugal force has a substantial effect on the location of the equilibrium points, but a small perturbation in the Coriolis force has no impact on them. The present model has two collinear, two out‐of‐plane, and an infinite number of noncollinear equilibrium points. Furthermore, we have discussed the stability of the equilibrium points analytically. The collinear equilibrium points L1 and L2 are found to be conditionally stable for the parameters μ, k, l, ϵ, and ϵ′. It is perceived that, for any ϵ with ∣ϵ ∣ ≪ 1 and ϵ′ ≤ 0.2, L1 is stable, and for ϵ′ ≥ 0.23, it becomes unstable. Furthermore, for any ϵ with |ϵ| ≪ 1 and ϵ′ ≤ −0.025, L2 is stable, and for ϵ′ ≥ −0.024, it becomes unstable. The noncollinear and out‐of‐plane equilibrium points are always unstable.

Existence and stability of equilibrium points in the photogravitational restricted four-body problem with Stokes drag effect

The restricted four-body problem consists of an infinitesimal particle which is moving under the Newtonian gravitational attraction of three finite bodies, $m_{1}$, $m_{2}$ and $m_{3}$. The three bodies (called primaries) are moving in circular orbits around their common centre of mass fixed at the origin of the coordinate system. Moreover, according to the solution of Lagrange, these primaries are fixed at the vertices of an equilateral triangle. The fourth body does not affect the motion of the three bodies. In this paper, we deal with the photogravitational version of the problem with Stokes drag acting as a dissipative force. We consider the case where all the primaries are sources of radiation and that two of the bodies, $m_{2}$ and $m_{3}$, have equal masses ($m _{2} = m_{3} = \mu $) and equal radiation factors ($q_{2} = q_{3} = q$) while the dominant primary body $m_{1}$ is of mass $1 - 2\mu $. We investigate the dynamical behaviour of an infinitesimal mass in the gravitational field of radiating primaries coupled with the Stokes drag effect. It is found that under constant dissipative force, collinear equilibrium points do not exist (numerically and of course analytically) whereas the existence and positions of the non-collinear equilibrium points depend on the parameters values. The linear stability of the non-collinear equilibrium points ($L_{i},i = 1,2, \ldots ,8$) is also studied and it is found that they are all unstable except $L_{1}$, $L _{7}$ and $L_{8}$ which may be stable for a range of values of $\mu $ and various values of radiation factors. Finally, we justify the relevance of the model in astronomy by applying it to a stellar system (Ross 104-Ross775a-Ross775b), for which all the equilibrium points have been seen to be 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.

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.

Bounded Motions near Contact Binary Asteroids by Hamiltonian Structure-Preserving Control

This paper is devoted to stabilizing motions near a special type of equilibrium point of contact binary asteroids by Hamiltonian structure-preserving control. Aiming at noncollinear equilibrium points of the asteroid 1996 HW1, a Hamiltonian structure-preserving controller is built up using stable and unstable manifolds. Compared with traditional Hamiltonian structure-preserving controllers, it is capable of stabilizing the motion near a special kind of equilibrium point only using stable or unstable manifolds. The analytical solution of controlled motions is approximated by a nonlinear series using the method of variation of parameters. An optimization strategy is proposed including the numerical optimization of control gains by maximum-likelihood and the global optimization of initial deviation analytically. To achieve collision avoidance, the concept of an artificial potential function is introduced to the controller, which still has the capability to change equilibrium points’ topological type to elliptic without breaking the system’s Hamiltonian structure.