Hill’s problem plays an important role in analyzing the local dynamics of an infinitesimal body under the gravitational influence of a distant massive primary and a nearby secondary body of smaller mass. When radiation pressure is included, the resulting model becomes particularly relevant for studying the motion of dust particles and solar-sail spacecraft in the vicinity of minor celestial bodies, such as planets or asteroids. This inclusion breaks the symmetry with respect to the Oy axis that characterizes the configurations of motion in the classical Hill’s problem. Thus, the location of the collinear equilibrium points, and the evolution of the Lyapunov families must be studied independently. Although the planar dynamics of the photogravitational Hill’s problem have been extensively investigated, its three-dimensional structure remains largely unexplored. The present study undertakes a systematic numerical investigation of branches of spatial periodic orbits that bifurcate from the planar Lyapunov families. Specifically, we compute all three-dimensional bifurcations up to multiplicity four and classify them according to their symmetry properties. The analysis reveals that these families exhibit distinct evolutionary patterns in the space of initial conditions, with most of them terminating in collision orbits with the secondary body.

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.

In this study, we consider an extension of the classical restricted three-body problem in which an additional three-body interaction is incorporated and investigate the resulting periodic orbits. Specifically, we analyze the Lyapunov families of planar periodic orbits that emerge from the collinear equilibrium points, along with their vertical stability characteristics. Furthermore, we delve into the three-dimensional periodic orbits that bifurcate from these planar families, focusing on spatial bifurcations with periods that are equal to, double or triple the period of the associated Lyapunov orbits. Our findings reveal the presence of various symmetry types, such as plane-plane, axis-axis or combination of both. Our analysis has been conducted for a set of parameter values associated with binary systems of a relatively large mass ratio.

We numerically study a version of the synchronous circular restricted three-body problem, where an infinites-imal 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 signifi-cant 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 corre-sponding 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.

This paper investigates the movement of a negligible mass body (third body) in the vicinity of the out-of-plane equilibrium points of the Hill three-body problem under the effect of radiation pressure of the primaries. We study the effect of the radiation parameters through the factors qi,i=1,2 on the existence, position, zero-velocity curves and stability of the out-of-plane equilibrium points. These equilibrium positions are derived analytically under the action of radiation pressure exerted by the radiating primary bodies. We determined that these points emerge in symmetrical pairs, and based on the values of the radiation parameters, there may be two along the Oz axis and either none or two on the Oxz plane (outside the axes). A thorough numerical investigation found that both radiation factors have a strong influence on the position of the out-of-plane equilibrium points. Our results also reveal that the parameters have impact on the geometry of the zero-velocity curves. Furthermore, the stability of these points is examined in the linear sense. To do so, the spatial distribution of the eigenvalues on the complex plane of the linearized system is visualized for a wide range of radiation parameter combinations. By a numerical investigation, it is found that all equilibrium points are unstable in general.

In the framework of the planar circular restricted three-body problem (R3BP), we explore the effects of oblateness of the infinitesimal mass body as well as radiation pressure and triaxiality of the two primaries on the position and stability of the triangular equilibrium points (TEPs). It is found that all the involved parameters affect the positions and stability of these points. Specifically, it has been shown that TEPs are stable for 0 < μ < μc and unstable for $$\mu _c \leqslant \mu \leqslant 1/2$$ , where μc denotes the critical mass parameter which depends on system’s parameters. In addition, all the parameters of the bigger primary, except that of triaxiality, have destabilizing tendencies resulting in a decrease in the size of the region of stability. Finally, we justify the relevance of the model in astronomy by applying it to the binary Lalande 21258 system for which the equilibrium points have been seen to be unstable.

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 <μ<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.

The present work performs a semi–analytical solution for the orbit of an infinitesimal particle in the framework of the bicircular Sun–Earth–Moon system. In particular, Lie series technique is applied to find the solution of the equations of motion of bicircular Sun–Earth–Moon system with radiating bigger primary. To apply Lie-series technique, the second order system of ordinary differential equations has been reduced to the corresponding first order system. Then, a set of recurrence relations is obtained in the Lie series solutions of the bicircular model (BCM) and graphical representations of the orbit for short, intermediate and long time are shown. Moreover, we study the effect of the radiation parameter on the orbit of the massless body and demonstrate that this parameter as well as the initial conditions affect its size. Specifically, it is observed that the trajectory enlarges further when the values of the radiation parameter increase while additionally its size enlarges or compacts according to the selected set of initial conditions.

In this paper, we present a modified version of Hill’s dynamical system that is called the quantized Hill’s three-body problem in the sense that the equations of motion for the classical Hill’s problem are now derived under the effects of quantum corrections. To do so, the position variables and the parameters that correspond to the quantum corrections of the respective quantized three-body problem are scaled appropriately, and then by taking the limit when the parameter of mass ratio tends to zero, we obtain the relevant equations of motion for the spatial quantized Hill’s problem. Furthermore, the Hamiltonian formula and related equations of motion are also derived.

In the framework of the circular restricted three-body problem, the dynamical effects of Stokes and Poynting–Robertson (P–R) drag forces on the existence, location, and stability of equilibrium points are investigated. It is found that under constant effects of P–R and/or Stokes drags, collinear equilibrium points cease to exist, but there are in the absence of the perturbing forces. The problem admits five non-collinear equilibrium points, and it is seen that the perturbing forces have significant effects on their positions. The linear stability of the equilibrium points is also studied in certain cases, and it is found that the stability of some of these points significantly depends on the perturbing forces. More precisely, the motion of the infinitesimal body near the non-collinear equilibrium points is unstable under the effect of both kinds of perturbing forces except from the equilibria L4 and L5 for which is stable only for Stokes drag effect, namely, the remaining parameter that corresponds to P–R drag is fixed to zero. We may conclude, therefore, that the P–R effect destroys stability of the equilibrium points.

We examine the dynamical effects of Poynting–Robertson (P–R) drag and oblateness together with small perturbations in the Coriolis and centrifugal forces on the existence, location and stability of equilibrium points in the photogravitational restricted three-body problem. It is found that under constant P–R drag effect, collinear equilibrium points cease to exist numerically and of course analytically. The problem admits five non-collinear equilibrium points and it is found that the positions of these points depend on all the system parameters except small perturbation in the Coriolis force. Finally, we justify the relevance of the model in astronomy by applying it to Cen X-4 binary system, for which all the equilibrium points have been seen to be unstable.

The aim of this paper is to numerically investigate the orbital dynamics of a test particle, in the planar circular restricted Pluto-Charon system. By numerically integrating the equations of motion, forward in time, with several large sets of initial conditions of orbits, we manage to classify them into three main categories: (i) bounded (regular or chaotic) (ii) escaping and (iii) collision orbits. The SALI method is used for safely identifying the chaotic or regular nature of the orbits. Furthermore, we determine the influence of the value of the total energy (or equivalently the value of the Jacobi constant) on the orbital structure of the system. In addition, the network as well as the stability of the symmetric periodic orbits are also revealed. In our analysis, we consider a large variety of symmetric periodic orbits, regarding their multiplicity.

The elliptic restricted three-body problem when the primary is a source of radiation and the secondary is an oblate spheroid is considered and the stability of the triangular equilibrium points is studied. The transition curves separating stable from unstable regions are determined in the parametric space both analytically and numerically. Our results show that the oblateness and radiation parameters do not cause significant changes on the topology of the stability regions in the parametric plane defined by the mass parameter and eccentricity. However, in the remaining parametric planes, we observe that by increasing the values of the parameters which are kept fixed stability gives place to instability.

The collinear equilibrium points and periodic motion around them are studied in the framework of the restricted three-body problem where the two primaries are triaxial rigid bodies which emit radiation. Firstly, the positions and stability of the collinear equilibria are studied for the HD 191408, Kruger 60 and HD 155876 binary systems. Then, the planar and three-dimensional periodic motion about these points is considered. Our study includes both semi-analytical and numerical determination of these motions. It is found that all families of planar periodic orbits emanating from these points terminate with asymptotic periodic orbits at the triangular equilibrium points while the corresponding families of three-dimensional periodic orbits terminate with planar periodic orbits. Families of Halo orbits bifurcating from the first vertical critical periodic orbit of the three planar Lyapunov families were also considered.

A modification of the Hill problem when the larger primary is a source of radiation is considered and asymptotic motions around the collinear equilibrium points are studied. Our work focuses on the computation of homoclinic orbits to the collinear equilibrium points themselves or to the Lyapunov orbits emanating from each equilibrium point. These orbits depart asymptotically from an equilibrium point (or a Lyapunov orbit) and return to the same point (or orbit) asymptotically. In both cases, semi-analytical solutions have been obtained in order to determine appropriate initial conditions which have been used as suitable seed for the numerical computation of the asymptotic orbits with a predetermined accuracy. In addition, for homoclinic orbits to the Lyapunov periodic orbits, transversality is achieved by the construction of appropriate surface of section portraits of the unstable manifolds.

An investigation of three-dimensional periodic orbits and their stability emanating from the collinear equilibrium points of the restricted three-body problem with oblate and radiating primaries is presented. A simulation is done by using five binary systems: Sirius, Procyon, Luhman 16, α-Centuari and Luyten 726-8. Firstly, based on the topological degree theory, the total number of the collinear equilibrium points for the five binary systems were obtained and then, their positions were determined numerically. The linear stability of these equilibrium points was also examined and found to be unstable in the Lyapunov sense. An analytical approximation of three-dimensional periodic solutions around them was established via the Lindstedt–Poincaré local analysis. Finally, using the analytical solution to obtain starting orbits, the families of three-dimensional periodic orbits emanating from these equilibria have been continued numerically.

ABSTRACT We consider a modification of the restricted three-body problem where the primary (more massive body) is a triaxial rigid body and the secondary (less massive body) is an oblate spheroid and study periodic motions around the collinear equilibrium points. The locations of these points are first determined for 10 combinations of the parameters of the problem. In all 10 cases, the collinear equilibrium points are found to be unstable, as in the classical problem, and the Lyapunov periodic orbits around them have been computed accurately by applying known corrector–predictor algorithms. An extensive study on the families of three-dimensional periodic orbits emanating from these points has also been done. To find suitable starting points, for all the computed families, semianalytical solutions have been obtained, for both two- and three-dimensional cases, around the collinear equilibrium points using the Lindstedt–Poincaré method. Finally, the stability of all computed periodic orbits has been studied.

The paper deals with a modification of the restricted three-body problem in which the angular velocity variation is considered in the case where the primaries are sources of radiation. In particular, the existence and stability of its equilibrium points in the plane of motion of the primaries are studied. We find that this problem admits the well-known five planar equilibria of the classical problem with the difference that the corresponding collinear points may be stable depending on the parameters of the problem. For all planar equilibria, sufficient parametric conditions for their stability have been established which are used for the numerical determination of the stability regions in various parametric planes. Also, for certain values of the parameters of the problem for which the equilibrium points are stable, the short and long period families have been computed. To do so, semianalytical expressions have been found for the determination of appropriate initial conditions. Special attention has been given to the continuation of the long period family, in the case of the classical restricted three-body problem, where we show numerically that periodic orbits of the short period family, which are bifurcation points with the long period family, are connected through the characteristic curve of the long period family.

Three-dimensional motions in the Chermnykh restricted three-body problem are studied. Specifically, families of three-dimensional periodic orbits are determined through bifurcations of the family of straight-line periodic oscillations of the problem which exists for equal masses of the primaries. These rectilinear oscillations are perpendicular to the plane of the primaries and give rise to an infinite number of families consisting entirely of periodic orbits which belong to the three-dimensional space except their respective one-dimensional bifurcations as well as their planar terminations. Many of the computed branch families are continued in all mass range that they exist.