This paper investigates the motion of an infinitesimal mass (test particle) around Lagrangian collinear equilibrium points L1,2,3 in the frame work of the Elliptic Restricted Three-Body Problem (ER3BP) under the influences of albedo (reflected radiation) of both, an oblate primary and a triaxial secondary surrounding them a belt. It is observed that the positions of the points L1,2,3 depend on the combined effect of albedo, oblateness, triaxiality and the potential due to the belt. An increase in any of these parameters produces shifts towards or away from the origin in the classical locations of points L1,2,3. It is also found that the equilibrium points remain linearly unstable, but stable periodic orbits in their neighbourhood may still exist for certain parameter ranges..

The paper examines the impacts of disk potential on motion of a test particle around the triangular Lagrangian points (TLPs) of the restricted five body problem (R5BP). The setup is such that, the test particle moves under the gravitational attractions of four unequal massive bodies and the configuration is enclosed by a disk. The dynamical equations are deduced and the locations, zero velocity curves (ZVCs) and linear stability of the TLPs are explored numerically when the test particle gravitates in the vicinity of a stellar and planetary systems. Additionally, the impact of the mass parameter shows that the test particle is positioned nearest to the line joining the central star and the secondary body for low mass parameter but drifts away as the mass parameter increases, while the impact of the radii of the disks of both systems shows that for smaller radius of the disk, the locations drifts away more while they drift towards the line joining the central bodies when they are larger. In the case of the ZVCs, it is seen that the increasing mass parameter reduces the region where motion is allowed while the radius of the gas disk increases the region where motion is allowed. The vicinity of the central body is no-travel zone for the test particle for both systems while the second body is also forbidden for the test particle to access in the stellar system. The stability is investigated and the TLPs are all stable in the planetary environment while instability is possible is the stellar system. Finally, the orbital stability of the test particle around the TLPs and the bodies in the setup is investigated using the Poincaré surface of sections (PSS) and it is seen that quasi-periodic orbits around the third and fourth body exists for both systems and the orbits are stable or unstable depending on the mass parameter and the system in which the test particle gravitates. The radii of the disk show that larger radii result in orbital instability, while smaller radius of the disk improves orbital stability. This study will contribute to the sparse available knowledge in the R5BP.

The influence of the zonal harmonics J4 on the positions and stability of the out-of-plane equilibrium points of an infinitesimal mass, in the framework of the photogravitational elliptic restricted three-body problem (ER3BP), has been investigated. The positions change with an increase in the oblateness up to zonal harmonics J4, radiation pressure, eccentricity and semi-major axis of the orbit. The positions and stability of the out-of-plane points are affected by the parameters involved. The effect of these parameters on the positions of the out-of-plane equilibrium points is examined numerically both for the binary system 61 CYGNI and for arbitrary values. The results obtained from this study can be applied to different methods of celestial mechanics, with application to the planetary system.

This paper investigates the dynamical behaviour of an infinitesimal mass body moving under the gravitational influence of a heterogeneous triaxial primary body and a secondary body that produces a modified Newtonian gravitational force in the elliptic restricted three-body problem. In this model, the two primary bodies move around their common center of mass in elliptical orbits, while the mass of the third body is assumed to be sufficiently small so that its effect on the motion of the primary bodies can be neglected. The equations of motion of the infinitesimal body are formulated, and the corresponding mean motion of the system is determined by taking into account the triaxial structure and mass heterogeneity of the primary body as well as the modified gravitational field of the secondary. The equilibrium points of the system are then investigated. Both collinear equilibrium points, which lie along the line joining the two primary bodies, and non-collinear equilibrium points, which are located away from this line, are determined analytically. Their linear stability is examined by studying the characteristic equations associated with small perturbations around these points. In addition, the dynamical characteristics of the system are illustrated numerically through potential surfaces, the locations of equilibrium points, permissible regions of motion, periodic orbits, and basins of attraction. These numerical results provide a clear understanding of how the eccentricity of the motion, triaxiality of the primary body, and the modified gravitational force of the secondary influence the motion of the infinitesimal body. The results may be useful in understanding the dynamics of celestial bodies.

This paper investigates the impacts of varying zonal harmonics up to J4 on dynamics of a satellite around out-of-plane equilibrium points of the restricted three-body problem with variable masses. The motion and mass variations of the primaries are described by the Gylden-Mestschersky problem (GMP) and the unified Mestschersky law (UML), respectively.The non-autonomous differential dynamical equations of the satellite are obtained and transformed to an autonomized system, with the help of the Mestschersky transformation, the UML, the particular integral and solutions of the GMP. Four out-of-plane equilibrium points (OEPs) denoted by and are obtained. Further, the OEPs are seen to be unstable and a numerical application is rendered for a satellite placed in the neighborhood of PQ Pegasi. It is observed that Increasing values of zonal harmonics results in the satellites position at the OEPs drifting towards the binary while the satellite is forced away from the PQ Pegasi at the points. A contrary observation occurs when the mass variation is large. Also, the zero velocity curves (ZVCs) are explored and it is seen that increasing the mass variation parameter increases the forbidden region in the presence or absence of zonal harmonic terms. Also, it is seen that, the zonal harmonic J4 term increases the region where motion is permissible, while the J2 term decreases it. Furthermore, it was seen that the orbits of the satellite around the OEPs could be quasi-periodic, divergent straight line or a chaotic. The study is applicable in space missions and astronomy.

This study numerically explores the dynamics of the photogravitational circular restricted three-body problem, where an infinitesimal particle moves under the gravitational influence of two primary bodies connected by a massless rod. These primary masses revolve in circular orbits around their common center of mass, which remains fixed at the origin of the coordinate system. The distance between the two masses remains constant, independent of their rotation period. The third body, being infinitesimally small compared to the primary masses, has a negligible effect on their motion. The primary mass is considered as a radiating body, while the secondary is modeled as an elongated one comprising two hypothetical point masses separated by a fixed distance. The analysis focuses on determining the number, location, and stability of equilibrium points, as well as examining the structure of zero-velocity curves under the influence of system parameters such as mass and force ratio, radiation pressure and geometric configuration of the secondary body. The system is found to allow up to six equilibria: four collinear and two non-collinear. Their number and positions are significantly affected by variations in the system’s parameters. Stability analysis reveals that the two non-collinear equilibrium points can exhibit stability under specific parameter configurations, while the four collinear points are typically unstable. An exception is the innermost collinear equilibrium point, which can be stable for certain parameter values. Our numerical investigation on periodic orbits around the collinear equilibrium points of the asteroid triple-system 2001SN263 show that a variation, either to the values of radiation or the force ratio parameters, influence their special characteristics such as period and stability. Also, their continuation in the space of initial conditions shows that all families terminate naturally at collision orbits with either the primary or the secondary.

In this study, we numerically investigate the equilibrium dynamics of a rotating system consisting of two masses connected by a massless rod within the framework of the circular restricted three-body problem. The larger primary is modeled as a radiating body and the smaller as an oblate spheroid. We explore the influence of the involved parameters, i.e., mass ratio (μ), force ratio (k), radiation pressure factor (q1), and oblateness coefficient (A2), on the number, positions, and linear stability of equilibrium points. Zero velocity curves are presented in the equatorial plane for varying values of the Jacobi constant. Up to five equilibrium points are identified of which three are collinear (L1, L2, L3) and two are non-collinear (L4, L5). The positions of all equilibria shift under variations in the perturbing parameters. While the collinear points are generally unstable, L1 can exhibit stability for certain combinations of μ, k, and q1. The non-collinear points may also be stable under specific conditions with stability zones expanding with increased parameter values. The model is applied to the irregular, elongated asteroid 951 Gaspra, for which five equilibrium points are found. Despite positional dependence on oblateness and radiation, the perturbations do not significantly affect the equilibrium points’ stability and the motion near them remains linearly unstable. The Lyapunov families of periodic orbits emanating from the collinear equilibria of this particular system are also investigated.

This paper deals with the motion of a body of infinitesimal mass (third body) around collinear equilibrium points in the elliptic restricted three-body problem (ER3BP) whose massive bodies (primaries) are oblate and luminous together with Poynting-Robertson (P-R) drag. The effects of the oblateness, radiation pressure and P-R drag of the primaries and the eccentricity and semi-major axis of the orbits on the positions and stability of collinear points are examined analytically and numerically. It is observed that due to the presence of P-R drags, a new collinear point exists apart from classical collinear points L0 L1 2 3. Various trends of shifts in the positions of these points are observed. The stability of these points is demonstrated by the Routh-Hurwitz criterion and it is found that they are all unstable. The numerical explorations conducted for the binary systems 61 Cygni and Archird also reveal how the involved parameters affect the positions and stability of these points and conformity with the analytical results obtained. (c) 2025 COSPAR. Published by Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

This paper investigates the derivations of the time-dependent equations of motion of a test particle in the frame of the R3BP with variable masses and zonal harmonics. The motion and mass variations of the primaries are described by the Gylden-Mestschersky problem (GMP) and the unified Mestschersky law (UML), respectively, with further assumptions that the oblateness of the bigger primary varies with zonal harmonics coefficients up to J4 terms. The non-autonomous equations of the test mass in a reference frame rotating are derived using the Hamiltonian method. These equations are DE with variable coefficients and are defined by the oblateness of the bigger body with zonal harmonics coefficients up to J4, the angular velocity of revolution and the masses of the primaries. This study will in no doubt expand the knowledge base of celestial mechanics and will allow for more extensions with applications to space missions.

The paper is an investigation of the motion of a spacecraft around triangular libration points of the restricted five-body problem (R5BP) and its stability. With the assumption that the spacecraft moves in gravitational environments of four primaries, the configuration is such that the first primary is located at the origin of the coordinate system, while the second primary is collinear with the first primary, and the third and fourth are located above and below, on the left of the first primary. The equations of motion are established, and the locations of the libration points, zero velocity curve, stability of the libration points, and the Poincaré surfaces of sections are thoroughly investigated analytically and numerically when the mass of the first primary is varied. It is observed that when the mass of the first primary is increasing, the position of the spacecraft drifts away from the first primary. Further, it is seen that the increasing mass of the first primary reduces the region where motion of the spacecraft is allowed around the triangular points. Finally, the Poincaré Surface of section was explored, and it was seen that as the mass of the first primary is increasing, more clusters were noticed around the primaries, and this shows the presence of stable or quasi-periodic orbits, which correspond to regular motion. Consequently, the orbits are stable. The problem can be applied to study the motion of a spacecraft in the environments of Jupiter and its three Moons.

This study investigates the motion of a test particle around triangular equilibrium points in the elliptic restricted three-body problem (ER3BP) under the influence of the two oblate and radiating primaries having Poynting-Robertson (P-R) drag. It is observed that the position of triangular points of the problem is affected by oblateness, radiation pressure, eccentricity, semi-major axis and Poynting-Robertson (P-R) drag. The stability of these points is demonstrated analytically by the Routh-Hurwitz criterion. It is seen that they are unstable under the combined effect of involved parameters. The effect of these parameters on the position of triangular points is examined numerically using the binary systems, 61 Cygni and Archird. The results obtained by these binary systems can be used to broaden the scope of interest in astronomy, astrophysics, space science and celestial mechanics in general.

We have examined the effects of Albedo in the Elliptic Restricted Three-Body Problem (ER3BP) with an oblate primary, a triaxial secondary, and potential due to belt for the Earth-Moon system. We have found that as the perturbed parameters increases, the possible boundary regions of the primary come closer to one other, allowing particles to travel from one region to the next freely and possibly merge the permissible regions. Our study has revealed that the formation of triangular libration points depends on the Albedo effects, semi-major axis, the Eccentricity of the orbits, triaxiality, and the potential due to the belt. As the parameters mentioned above increase, the triangular positions [Formula: see text] and [Formula: see text] move towards the center of origin in cases 1, 2, 3, and 4 and away from the center of the origin in cases 5, 6, and 7. Considering the range of a stable and unstable libration point for the problem under study given as [Formula: see text] for stable libration points and [Formula: see text] for unstable libration points, our study has established that the triangular libration points are respectively stable and unstable for cases 1, 2, and 6 and cases 3, 4, 5, and 7. Our study has also revealed that each set of values has at least one characteristic complex root with a positive real part. Hence, the triangular libration points for the Earth-Moon system are unstable in the sense of Lyapunov. The Earth-Moon system's Poincare Surface of Section (PSS) has demonstrated that a slight change in the initial conditions, the semi-major axis, and the Eccentricity of the orbits have affected the system's behavior dramatically. Further, it is seen that a chaotic dynamical behavior of the system results into either regular or irregular orbits.

In this paper, the existence and stability of the out-of-plane equilibrium points of extra-solar planets using the model of the restricted three-body problem, when the central binaries are surrounded by a clusters of material points and their motion is described by the Gylden-Mestschersky problem, is investigated. The masses of the central binaries and the clusters of material points are assumed to vary with time at the same rate in accordance with the unified Mestschersky law. The governing non-autonomous equations of motion of the model is obtained and for a complete transformation to the autonomized systems to be obtainable, we assume that the mass of the clusters of materials varies at the same rate as the masses of the central binaries while the radius and the parameters which determines the density profile of the clusters vary at the same rate as the distances. Thus, we introduce a transformation which defines the mass variation of the clusters with the help of the unified Mestschersky law and the Mestschersky transformation and obtain the autonomized system with constant coefficients. Next, the coordinates of the out-of-plane points are found using perturbation method, the Newton-Raphson's method and analytical approximations in the form of power series in the cluster's mass coefficient. The existence of these points depends solely on the mass variation parameter κ, although the points are however affected by the total mass of the clusters of material points Md and the mass parameter υ. Two pairs of out-of-plane equilibrium points are found; the first pair of equilibrium points L6,7 exists for κ>1 while the second pair L8,9 exists for κ>1 and ξ<υ(κ−1), where ξ is the abscissa of the out-of-plane equilibrium points. Further, for our numerical evidence, we compute the out-of-plane equilibrium points of two extra-solar planets PSR B1620-26b and Kepler-16b, in the binary systems PSR B1620-26 and Kepler-16, respectively. Our numerical results shows that the equilibrium points for PSR B1620-26b, exist when the mass variation parameter and the mass of the clusters lie in the intervals 1<κ≤4 and 0≤Md≤0.04, respectively, while for Kepler-16b, the points exist for 1<κ≤4.1 and 0≤Md≤0.04. Also, it is seen that in the absence of the clusters more out-of-plane equilibrium points evolve and are located far away from the line joining the binaries, while in the presence of the clusters, less out-of-plane equilibrium points evolve, but the evolved points are located closest to the line joining the binaries. Finally, we analyze the stability of the equilibrium points of the autonomized and non-autonomous systems, and both were found to be unstable equilibrium points.

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 ( L 1 , 2 , 3 , 6 ) which are greatly affected by the parameters under review. The collinear position L 1 move away and closer as the parameters increase and decrease respectively. For the collinear positions L 2 a n d L 3 , we witnessed a uniform space movement away from the origin in the negative direction while L 6 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 ( L 1 , 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 ( L 3 ) is stable for the characteristic roots ( λ 1,2 ) 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.

In this paper, a bicircular model describing motion of a spacecraft around the Sun–Earth–Moon system is studied. The impact of important characterizations such as radiation from the Sun, the Earth’s and Moon’s Albedo and oblateness of the Moon are all included and analysed​ in this work. This quantitative investigation shows how significant the reflective property of the Earth and its Moon affect the existence of the system’s libration points as well as the stability of motion of the spacecraft about the libration points. The Albedo phenomenon is seen to have dominant effect on the location of the libration points of the spacecraft than the perturbation due to oblateness. It is revealed by the Poincare Surface of Section that the system exhibits chaotic behaviour as it is sensitive to change in initial conditions.

The study investigates the stability and velocity sensitivities of libration points in the elliptic restricted synchronous three-body problem under an oblate primary and a dipole secondary for Luhman-16 and HD188753 systems. We have observed that the position of L4 moves away from the centre of origin for both systems as the oblateness and the half mass dipole distance increases. As the oblateness and the half mass dipole distance increase, there is a shift in the position of L5 closer to the centre of the origin for both systems. The Poincare Surfaces of Section (PSS) for both systems have revealed that the behaviour of the system changes significantly with a bit change in the initial conditions, oblateness and the half mass dipole distance. We have observed that the sensitivity of both systems to change in position and velocities results in either regular orbits or irregular orbits. Hence, the dynamical behaviour of the systems is chaotic. Considering the range of a stable and unstable libration points for the problem under study given as 0<ν<νc and νc≤ν≤14respectively, our study has revealed that, the triangular libration points are stable and unstable for some values of oblateness for the binary systems. In the absence of oblateness, the νc* indicates that triangular points are stable for Luhman-16 system. When the parameters are varied differently with order of commensurability k, the critical mass parameters show that the triangular points are stable for Luhman-16 and unstable for HD188753 system. Using binary systems in our study, the results obtained can be used as springboards for broading the scope of interest in Celestial Mechanics and its investigations have shown significant improvement in the study of this longstanding problem.

This paper studies the motion of an infinitesimal particle near the triangular equilibrium points (TEPs) in the elliptic restricted three body problem (ER3BP) when the primaries are radiating- triaxial rigid bodies and are affected by the Poynting-Robertson (P-R) drag force enclosed in a circumbinary disc (disc,belt). We present the equations of motion, obtained the positions of (TEPs) and found that there exist two Lagragian equilibria points L4,5 which lies in the ξη- plane in symmetrical positions with respect to the orbital plane.The parameters involved in the system affect their positions.The position changes with an increase in triaxiality, radiation, P-R drag force and the disc. The positions and linear stability of the TEPs are investigated numerically using the binary systems Archid and it was observed that the effect of P-R drag force of the smaller primary is not sufficient in causing instability at the EPs but the radiation pressure force.We observed that when the triaxiality coefficient, values of the belt and P-R drag force are varied increasingly in the absence of radiation equilibrium points (EPs) are stable in the linear sense but becomes unstable on introducing radiation.Thus radiation is the cause of instability when both primaries are radiating and triaxial with the smaller primary having an effective P-R drag force enclosed in a circumbinary disc and not the P-R drag.

The motion of a test particle within the context of the restricted four-body problem (R4BP) driven by a new kind of potential, called the generalized Manev potential, with perturbations in the Coriolis and centrifugal forces is considered in this study. The system possesses eight libration points which were distributed on its plane of motion in different manner from those of the usual Newtonian potential. Unlike the case of the perturbed R4BP under Newtonian potential, where two of these librations are stable, all of them are unstable in linear sense under Manev potential. We found that a gradual perturbation in the centrifugal force causes the trajectories of motion to drift inward but the Coriolis force was proven to have no effect on the location of the libration points of the system. Using first order Lyapunov characteristic exponents, the dynamical behavior of the system is found irregular. We experimented with a high velocity stellar system (82 G. Eridani) to establish the applicability of the model in astrophysics.

We consider the primaries of the circular restricted three-body problem (CR3BP) to be luminous and study the effects of small perturbations in the Coriolis and centrifugal forces together with Poynting-Robertson (P-R) drag from both primaries on the motion of an infinitesimal body near the out-of-plane equilibrium points (OEPs). It is found that these points appear in pairs and, depending on the values of the parameters of the system, their number may be zero, two, L6,7 or four, L6,7,8,9. It is observed that the positions of these points depend on all the system parameters except small perturbation in the Coriolis force. This has been shown for binary systems RW-Monocerotis and Krüger-60. The linear stability of the out-of-plane equilibria is also studied and it is found that stability of some of these points significantly depends on the perturbing forces. Specifically, the motion of the infinitesimal body around the equilibria is conditionally stable only at points L6 and L7 in the absence of P-R drag effect in both binary systems. However, all the equilibria are unstable in the presence of the P-R drag effect. We may conclude therefore, that P–R effect destroys stability of the out-of-plane equilibria.