Collinear Equilibrium Points Research Articles

Effects of Variable Mass, Disk-Like Structure, and Radiation Pressure on the Dynamics of Circular Restricted Three-Body Problem

In this paper, we intend to investigate the dynamics of the Circular Restricted Three-Body Problem. Here we assumed the primaries as the source of radiation and have variable mass. The gravitational perturbation from disk-like structure are also considered in this study. There exist five equilibrium points in this system. By considering the combined effect of disk–like structure and the mass transfer, we found that the classical collinear equilibrium points depart from the x–axis. We called these equilibrium points as quasi–collinear equilibrium points. Meanwhile, this combined effect also breaks the symmetry of triangular equilibrium point positions. We noted that the quasi–equilibrium points are unstable whereas the triangular equilibrium points are stable if the mass ratio μ is smaller than critical mass μc. Besides the mass ratio, the stability of triangular equilibrium points depend on time.

Location of the collinear equilibrium points in the elliptic restricted three-body problem with various perturbation effects

Abstract This study aims to examine the elliptic restricted three-body problem (ERTBP) by modifying the classical case and applying various perturbation sources to the three-body system. In this study, the locations of the Lagrange collinear equilibrium points of ERTBP were examined. We consider that the first primary body emits radiation and has an oblate shape. In contrast, the second primary body was considered to be elongated and approximated as a finite straight-segment. In addition, the perturbations from the disk-like structure around the three-body system were also included. The equations of motion of the infinitesimal body are presented in a dimensionless pulsating coordinate system. Three collinear equilibrium points were identified. The locations of the collinear equilibrium points were calculated numerically for several cases of perturbation values and also presented versus eccentricity over its range. We observed that the position of the collinear equilibrium points (L 1, L 2, and L 3) shifted when perturbing parameters were included, as opposed to where they were in the classical ERTBP.

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.

Dynamics of infinitesimal body in the concentric restricted five-body problem

This research presents a model of a specific restricted five-body problem. In this model, there are four main bodies, referred to as primary bodies. The first two primary bodies have the same mass while the third and fourth primary bodies also share an identical mass, which is lesser than the former. These primary bodies are lined up in a straight line along a designated axis. They revolve in circular paths with different distances from a common central point that represents their shared center of mass. The first set of orbits has a shorter radius compared to the second set. Simplifying the motion equations of an infinitesimally small mass reveals a single governing parameter, λ, constrained within λ ∊ (λ0, 1), λ0 = 0.417221. Within the orbital plane of these primaries, seven equilibrium points are identified: four along the x-axis, two on the y-axis, and one at the system's origin. Notably, no equilibrium points were found outside this orbital plane. The study concludes that collinear equilibrium points are linearly unstable, while the non-collinear points maintain stability for values λc < λ < 1, λc = 0.971105.

Zero velocity curves of a dust grain around equilibrium points under effects of radiation, perturbations and variable Kruger 60

The paper unveils zero velocity curves of a dust grain particle around eleven equilibrium points under effects of radiation, perturbations and variable Kruger 60. The study is carried out using the model formulations of the restricted three-body problem in which the primaries are assumed to be radiation sources and vary their masses with time coupled with small perturbations in the Coriolis and centrifugal forces due to circular motion. The equations of motion of the non-autonomous system and those of the autonomized equations with constant coefficients have been stated and the equilibrium points explored. Numerical illustrations have been provided to support the study for a particular case when the dust grain moves under the influence of the central binary Kruger 60. It is seen that there exist a pair of triangular points, five collinear equilibrium points, and four out-of-plane equilibrium points, which are defined by the radiation pressure of the Kruger 60, centrifugal perturbation and the arbitrary constant K which defines the sum of the masses of Kruger 60. Finally, the zero velocity curves of the dust grain around the equilibrium points are explored and the effects of the parameters divulged. In the case of the collinear equilibrium points, as K increases the region where motion is allowed decreases, while in the case of the triangular points, the dust grain is permitted to move around the primaries and the triangular points and is also allowed access to the exterior realm as K increases. In the case of the out-of-plane equilibrium points, as K increases the region where motion is allowed decreases while the presence of the perturbing forces due to radiation and centrifugal perturbation also reduces the region where motion of the dust grain particle is allowed. It is seen that the Coriolis perturbation do not affect the locations and zero velocity curves around all the equilibrium points.

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.

Equilibrium points and Lyapunov families in the circular restricted three-body problem with an oblate primary and a synchronous rotating dipole secondary: Application to Luhman-16 binary system

We numerically study a version of the synchronous circular restricted three-body problem, where an infinitesimal mass body is moving under the Newtonian gravitational forces of two massive bodies. The primary body is an oblate spheroid while the secondary is an elongated asteroid of a combination of two equal masses forming a rotating dipole which is synchronous to the rotation of the primaries of the classic circular restricted three-body problem. In this paper, we systematically examine the existence, positions, and linear stability of the equilibrium points for various combinations of the model's parameters. We observe that the perturbing forces have significant effects on the positions and stability of the equilibrium points as well as the regions where the motion of the particle is allowed. The allowed regions of motion as determined by the zero-velocity surface and the corresponding isoenergetic curves as well as the positions of the equilibrium points are given. Finally, we numerically study the binary system Luhman-16 by computing the positions of the equilibria and their stability as well as the allowed regions of motion of the particle. The corresponding families of periodic orbits emanating from the collinear equilibrium points are computed along with their stability properties.

NUMERICAL INVESTIGATIONS OF THE ORBITAL DYNAMICS AROUND A SYNCHRONOUS BINARY SYSTEM OF ASTEROIDS

In this article, equilibrium points and families of periodic orbits in the vicinity of the collinear equilibrium points of a binary asteroid system are investigated with respect to the angular velocity of the secondary body, the mass ratio of the system and the size of the secondary. We assume that the gravitational fields of the bodies are modeled considering the primary as a mass point and the secondary as a rotating mass dipole. This model allows to compute families of planar and halo periodic orbits that emanate from the equilibrium points L1 and L2. The stability and bifurcations of these families are analyzed and the results are compared with the results obtained with the restricted three-body problem (RTBP). The results provide an overview of the dynamical behavior in the vicinity of a binary asteroid system.

The collinear equilibrium points in the elliptic restricted synchronous three-body problem under an oblate primary and a dipole secondary

The study investigates the collinear positions and stability in the elliptic restricted synchronous three-body problem under an oblate primary and a dipole secondary for Luhman 16 and HD188753 systems. Our study has established four collinear equilibrium points (L1,2,3,6) which are greatly affected by the parameters under review. The collinear position L1 move away and closer as the parameters increase and decrease respectively. For the collinear positions L2andL3, we witnessed a uniform space movement away from the origin in the negative direction while L6 seems to be moving closer to the origin from the negative part of the origin. We observed changes in the movements of the collinear positions (L1,2,3,6) as a result of the half distance between the mass dipoles and the oblateness of the primary for the problem under review. The movements away and closer to the origin from collinear positions do not change the status of the collinear points as they remain unstable and unchanged. It is also found that as the half distance between mass dipoles and oblateness of the primary increase, the region of stability of the collinear positions decreases for the aforementioned binary systems. The collinear equilibrium point (L3) is stable for the characteristic roots (λ1,2) for Luhman 16 system. This is evidenced by at least one characteristic root, a positive real part and a complex root. The stability of collinear points in most cases are unstable for the stated binary systems in Lyapunov.

Families of periodic orbits about Lagrangian points L1, L2 and L3 with continuation method

This study deals the impact of radiation pressure and albedo from the respective primaries in the context of regularized periodic motion of infinitesimal mass under the frame of restricted three body problem with the help of continuation method. The effect of these perturbing parameters on the orbital properties about the collinear Lagrangian points and their stability property are investigated. An expansion in the structure of Lyapunov orbits are found due to radiation pressure effect and albedo whereas crossing of stability bounds (±1) occurs in the continuation process of Lyapunov orbits indicating the presence of halo bifurcation. This occurrence of halo bifurcation leads to the origin of halo periodic orbits from Lyapunov periodic orbits at different values of radiation parameter ( q ) and albedo ( Q A ). From these bifurcating orbits, families of halo periodic orbits are determined using continuation in radiation parameter and albedo, respectively. Due to the existence of one of the stability index ( ν ), which is either greater than 1 or less than ​− ​1, periodic orbits become unstable. • Regularized restricted three body problem with radiation pressure and albedo are discussed. • Numerical computation of periodic orbits about collinear equilibrium points are discussed. • The parametric evolution of periodic orbit as the function of radiation and albedo parameter are presented. • Families of periodic orbits are found with the help of continuation in radiation and albedo parameter.

Short and long period periodic orbits around a stable collinear equilibrium point in the circular restricted three-body problem with a three-body interaction

We deal with a modification of the circular restricted three-body problem (CR3BP), which incorporates an additional effect of a three-body interaction. The parameters determining the configuration of the corresponding dynamical system are the mass ratio μ of the primary bodies and the constant k that scales the contribution of the aforementioned interaction. We are interested in the families of short and long period planar symmetric periodic orbits, which emanate from the collinear equilibrium point L3 in the case that this point is neutrally stable. First, we numerically investigate the correlation between the values of μ and k that result such kinds of families. Subsequently, these families are numerically constructed and presented for an indicative case of these parameters. It is found that both families terminate on some members of the family of short period orbits.

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.

Existence and Stability of Equilibrium Points in the R3BP With Triaxial-Radiating Primaries and an Oblate Massless Body Under the Effect of the Circumbinary Disc

The aim of this article is to study the existence, location, and stability of equilibrium points in a generalized restricted three-body problem (R3BP) that consists of an oblate infinitesimal body when the primaries are radiating sources with triaxiality of the two stars surrounded by a belt (circumbinary disc). The existence, number, location, and stability of the collinear and triangular Lagrangian equilibrium points of the problem depend on the mass parameter and the perturbing forces involved in the equations of motion. We find numerically that four additional collinear equilibrium pointsLni,i= 1, 2, 3, 4, exist, in addition to the three Eulerian pointsLi,i= 1, 2, 3, of the classical case, making up a total of up to seven collinear points.Ln1andLn2result due to the potential from the belt, whileLn3andLn4arise from the effect of triaxiality. The positions of the equilibrium points are affected by the presence of perturbations, since they are deviated from the classical R3BP on thex-axis and out of thex-axis, respectively. The stability of the equilibrium points, for a particular set of the parameters, is analyzed, and it is concluded that all the collinear points are unstable exceptLn1, which is always linearly stable. The range of stability of the Lagrangian pointsL4,5is determined analytically and found that being stable for 0 &amp;lt;μ&amp;lt;μcritand unstable forμcrit≤μ≤ 1/2, whereμcritis the critical mass ratio which depends on the combined effects of the perturbing forces. It is noticed that the critical mass ratio decreases with the increase in the values of the radiation pressure, triaxiality, and oblate infinitesimal body; however, it increases with the increase in the value of mass of the disc. All three of the former and the latter one possess destabilizing and stabilizing behavior, respectively. The net effect is that the size of the region of stability that decreases when the value of these parameters increases. In our model, the binary HD155876 system is used, and it is found that there exists one stable collinear equilibrium pointviz. Ln1.

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.

Elliptic Restricted Three Body Problem Under the Effect of Circumstellar Belt

The dynamics of the infinitesimal for the planar elliptic restricted three body model interacting with circumstellar belt is studied. Analytic exploration of the problem reveals the existence of more than three collinear equilibrium points and a pair of triangular equilibrium points. Stability analysis based on Grebnikov’s averaging method gave the value of critical mass ratio for the stable behaviour of triangular equilibrium point. Also the criteria for stability of collinear points are explored analytically. Applying these analytical results to the Sun–Earth system under the influence of Asteroid belt and the star Epsilon–Eridani-its planet Eridani-b and the circumstellar belt show the presence of four or more collinear and a pair of equilibrium points in the pulsating rotating plane. We found that only the triangular equilibrium point is stable for most of the cases. The zero velocity curves (ZVC) and fractal basin of convergence for the two systems are also plotted. It was found that the ZVC are affected by the mass of the belt. On other hand the basin of convergence for particular equilibrium points are found to be reduced on increasing the value of mass of belt.

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.

Three-Saddle-Foci Chaotic Behavior of a Modified Jerk Circuit with Chua’s Diode

This paper investigates the chaotic behavior of a modified jerk circuit with Chua’s diode. The Chua’s diode considered here is a nonlinear resistor having a symmetric piecewise linear voltage-current characteristic. To describe the system, we apply fundamental laws in electrical circuit theory to formulate a mathematical model in terms of a third-order (jerk) nonlinear differential equation, or equivalently, a system of three first-order differential equations. The analysis shows that this system has three collinear equilibrium points. The time waveform and the trajectories about each equilibrium point depend on its associated eigenvalues. We prove that all three equilibrium points are of type saddle focus, meaning that the trajectory of (x(t),y(t)) diverges in a spiral form but z(t) converges to the equilibrium point for any initial point (x(0),y(0),z(0)). Numerical simulation illustrates that the oscillations are dense, have no period, are highly sensitive to initial conditions, and have a chaotic hidden attractor.

Observational properties of low-energy orbits around icy moons

The moons of the giant planets are getting into the focus of the next exploration missions to the outer solar system. The reason for this interest resides in their dynamical role within their respective systems and the signs of habitability suggested in some cases by surface and subsurface features. Flyby missions offer adequate recognition scenarios for the initial exploration phase of a celestial body, but they must be followed by orbiters if more extended observations are required. The dynamical structures of the Circular Restricted Three-Body Problem pave the way to such opportunities. In particular, libration point orbits around the two collinear equilibrium points L1 and L2 of a planet–moon-spacecraft system are characterized by low speeds relative to the moon. Heteroclinic and homoclinic connections built from these orbits allow close-up views of the moons. The aim of this work is to determine the observational performance of heteroclinic and homoclinic connections of planar Lyapunov orbits around L1 and L2 for three systems of scientific relevance: Jupiter–Europa, Uranus–Titania and Uranus–Oberon. Lunar surface coverage, repeat patterns, lunar distance ranges and speed profiles are derived and discussed for each case.