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.     

Existence of equilibrium points and their linear stability in the generalized photogravitational Chermnykh-like problem with power-law profile

We consider the modified restricted three body problem with power-law density profile of disk, which rotates around the center of mass of the system with perturbed mean motion. Using analytical and numerical methods, we have found equilibrium points and examined their linear stability. We have also found the zero velocity surface for the present model. In addition to five equilibrium points there exists a new equilibrium point on the line joining the two primaries. It is found that L ₁ and L ₃ are stable for some values of inner and outer radius of the disk while other collinear points are unstable, but L ₄ is conditionally stable for mass ratio less than that of Routh’s critical value. Lastly, we have studied the effects of radiation pressure, oblateness and mass of the disk on the motion and stability of equilibrium points.     

The Photogravitational Hill Problem: Numerical Exploration

We consider the recently introduced version of the classical Lunar Hill problem, the photogravitational Hill problem, and study it's equilibrium points and zero-velocity curves. The full network of families of periodic orbits is numerically explored, their stability is computed and critical orbits are determined. Non-periodic orbits are also computed as points on a surface of section, providing an outlook of the stability regions, chaotic motions and escape.     

Artificial equilibrium points for a generalized sail in the circular restricted three-body problem

This paper introduces a new approach to the study of artificial equilibrium points in the circular restricted three-body problem for propulsion systems with continuous and purely radial thrust. The propulsion system is described by means of a general mathematical model that encompasses the behavior of different systems like a solar sail, a magnetic sail and an electric sail. The proposed model is based on the choice of a coefficient related to the propulsion type and a performance parameter that quantifies the system technological complexity. The propulsion system is therefore referred to as generalized sail. The existence of artificial equilibrium points for a generalized sail is investigated. It is shown that three different families of equilibrium points exist, and their characteristic locus is described geometrically by varying the value of the performance parameter. The linear stability of the artificial points is also discussed.     

On the Chermnykh-Like Problems: II. The Equilibrium Points

Motivated by Papadakis (2005a, b), we study a Chermnykh-like problem, in which an additional gravitational potential from the belt is included. In addition to the usual five equilibrium points (three collinear and two triangular points), there are some new equilibrium points for this system. We studied the conditions for the existence of these new equilibrium points both analytically and numerically.     

Some new Runge-Kutta methods with interpolation properties and their application to the Magnetic-Binary Problem

Explicit Runge-Kutta methods provide a popular way to solve the initial value problem for a system of nonstiff ordinary differential equations. On the other hand, for these methods, there is no a natural way to approximate the solution at any point within a given integration step. Scaled Runge-Kutta methods have been developed recently which determine the solution of the differential system at non-mesh points of a given integration step. We propose some new such algorithms based upon well known explicit Runge-Kutta methods, and we verify their advantages by applying them to the Magnetic-Binary Problem.     

有摄圆型限制性三体问题平动点稳定的充要条件及应用

摘要给出了一个判断有摄圆型限制性三体问题平动点稳定性的充要条件．该条件只依赖于平动点变分方程的特征方程系数的一个简单关系，使用很方便．用所得到的条件，讨论了任意外力摄动对经典圆型限制性三体问题三角平动点稳定性的影响和惯性阻力摄动对Robe圆型限制性三体问题主要平动点的稳定性的影响．     

Restricted four-body problem. The case of a central configuration: Libration points and their stability

We consider the problem of the motion of a zero-mass body in the vicinity of a system of three gravitating bodies forming a central configuration.We study the case where two gravitating bodies of equal mass lie on the same straight line and rotate around the central body with the same angular velocity. Equations for calculating the equilibrium positions in this system have been derived. The stability of the equilibrium points for a system of three gravitating bodies is investigated. We show that, as in the case of libration points for two bodies, the collinear points are unstable; for the triangular points, there exists a ratio of the mass of the central body to the masses of the extreme bodies, 11.720349, at which stability is observed.     

受摄限制性三体问题平动点线性稳定性的判断条件及在Robe问题的应用

给出了受摄限制性三体问题平动点线性稳定性的一个判断条件．条件只与平动点切映像的特征方程系数有关，使用方便．用判断条件，讨论了Robe问题平动点在阻力摄动下的线性稳定性，得到了Hallan等给出的Robe问题平动点在阻力摄动下的线性稳定范围．并改进了Giordanoc等的结果．     

Effects of triaxiality of primaries on oblate infinitesimal in elliptical restricted three body problem

This paper examines the effects of triaxiality of both the primaries on the position and stability of the oblate infinitesimal mass in the neighborhood of triangular equilibrium points in the framework of Elliptical restricted three body problem. We have found the solutions for the locations of triangular equilibrium points. We have investigated the stability of infinitesimal mass around the triangular equilibrium points.It is observed that the infinitesimal motion around triangular equilibrium points are stable under certain condition with respect to triaxiality of primaries. We have applied the method of averaging used by Grebenivok, throughout the analysis of the stability of the infinitesimal mass around the triangular equilibrium points. We have exploited simulation technique using MATLAB 15 to analyze the stability of the system. The critical mass ratio depends on the triaxiality, oblateness, semi- major axis and eccentricity of the elliptical orbits.     

On the Chermnykh-Like Problems: I. the Mass Parameter μ = 0.5

Following Papadakis (2005)'s numerical exploration of the Chermnykh's problem, we here study a Chermnykh-like problem motivated by the astrophysical applications. We find that both the equilibrium points and solution curves become quite different from the ones of the classical planar restricted three-body problem. In addition to the usual Lagrangian points, there are new equilibrium points in our system. We also calculate the Lyapunov Exponents for some example orbits. We conclude that it seems there are more chaotic orbits for the system when there is a belt to interact with.     

THE HILL PROBLEM WITH OBLATE SECONDARY: NUMERICAL EXPLORATION

We introduce a three-dimensional version of Hill’s problem with oblate secondary, determine its equilibrium points and their stability and explore numerically its network of families of simple periodic orbits in the plane, paying special attention to the evolution of this network for increasing oblateness of the secondary. We obtain some interesting results that differentiate this from the classical problem. Among these is the eventual disappearance of the basic family g′ of the classical Hill problem and the existence of out-of-plane equilibrium points and a family of simple-periodic plane orbits non-symmetric with respect to the x-axis.     

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.     

Numerical exploration of the photogravitational restricted five-body problem

We study numerically the restricted five-body problem when some or all the primary bodies are sources of radiation. The allowed regions of motion as determined by the zero-velocity surface and corresponding equipotential curves, as well as the positions of the equilibrium points are given. We found that the number of the collinear equilibrium points of the problem depends on the mass parameter β and the radiation factors q _i , i=0,…,3. The stability of the equilibrium points are also studied. Critical masses associated with the number of the equilibrium points and their stability are given. The network of the families of simple symmetric periodic orbits, vertical critical periodic solutions and the corresponding bifurcation three-dimensional families when the mass parameter β and the radiation factors q _i vary are illustrated. Series, with respect to the mass (and to the radiation) parameter, of critical periodic orbits are calculated.     

Equilibrium points in the photogravitational restricted four-body problem

We study numerically the photogravitational version of the problem of four bodies, where an infinitesimal particle is moving under the Newtonian gravitational attraction of three bodies which are finite, moving in circles around their center of mass fixed at the origin of the coordinate system, according to the solution of Lagrange where they are always at the vertices of an equilateral triangle. The fourth body does not affect the motion of the three bodies (primaries). We consider that the primary body m ₁ is dominant and is a source of radiation while the other two small primaries m ₂ and m ₃ are equal. In this case (photogravitational) we examine the linear stability of the Lagrange triangle solution. The allowed regions of motion as determined by the zero-velocity surface and corresponding equipotential curves, as well as the positions of the equilibrium points on the orbital plane are given. The existence and the number of the collinear and the non-collinear equilibrium points of the problem depends on the mass parameters of the primaries and the radiation factor q ₁. Critical masses m ₃ and radiation q ₁ associated with the existence and the number of the equilibrium points are given. The stability of the relative equilibrium solutions in all cases are also studied. In the last section we investigate the existence and location of the out of orbital plane equilibrium points of the problem. We found that such critical points exist. These points lie in the (x,z) plane in symmetrical positions with respect to (x,y) plane. The stability of these points are also examined.     

A Hill Problem with Oblate Primaries and Effect of Oblateness on Hill Stability of Orbits

We introduce a new version of Hill's problem to include the effect of oblateness of the primaries, and briefly discuss its equilibrium points and zero velocity curves. As a first application we use this to study Hill stability of direct orbits around the small primary. This can be employed to study the stability of a planet's moon perturbed by an oblate Sun, or of a star's planet perturbed by a distant disk-shaped galaxy. Oblateness of the `Sun' is found to decrese the maximum distance of Hill stable direct `moon' orbits. This revised version was published online in July 2006 with corrections to the Cover Date.     

Collinear equilibrium points of Hill’s problem with radiation and oblateness and their fractal basins of attraction

We discuss the existence, location, and stability of the collinear equilibrium points of a generalized Hill problem with radiation of the primary (the Sun) and oblateness of the secondary (the planet), and present some remarkable fractals created as basins of attraction of Newton’s method applied for their computation in several cases of the parameters.     

Stability of Triangular Equilibrium Points in the Photogravitational Restricted Three-Body Problem with Oblateness and Potential from a Belt

We have examined the effects of oblateness up to J ₄ of the less massive primary and gravitational potential from a circum-binary belt on the linear stability of triangular equilibrium points in the circular restricted three-body problem, when the more massive primary emits electromagnetic radiation impinging on the other bodies of the system. Using analytical and numerical methods, we have found the triangular equilibrium points and examined their linear stability. The triangular equilibrium points move towards the line joining the primaries in the presence of any of these perturbations, except in the presence of oblateness up to J ₄ where the points move away from the line joining the primaries. It is observed that the triangular points are stable for 0 < μ < μ _c and unstable for \(\mu_{\mathrm{c}} \le \mu \le \frac {1}{2},\) where μ _c is the critical mass ratio affected by the oblateness up to J ₄ of the less massive primary, electromagnetic radiation of the more massive primary and potential from the belt, all of which have destabilizing tendencies, except the coefficient J₄ and the potential from the belt. A practical application of this model could be the study of motion of a dust particle near a radiating star and an oblate body surrounded by a belt.     

Stability of the photogravitational restricted three-body problem with variable masses

This paper investigates the stability of equilibrium points in the restricted three-body problem, in which the masses of the luminous primaries vary isotropically in accordance with the unified Meshcherskii law, and their motion takes place within the framework of the Gylden–Meshcherskii problem. For the autonomized system, it is found that collinear and coplanar points are unstable, while the triangular points are conditionally stable. It is also observed that, in the triangular case, the presence of a constant κ, of a particular integral of the Gylden–Meshcherskii problem, makes the destabilizing tendency of the radiation pressures strong. The stability of equilibrium points varying with time is tested using the Lyapunov Characteristic Numbers (LCN). It is seen that the range of stability or instability depends on the parameter κ. The motion around the equilibrium points L _i (i=1,2,…,7) for the restricted three-body problem with variable masses is in general unstable.     

Robe’s Circular Restricted Three-Body Problem Under Oblate and Triaxial Primaries

This paper analyzes Robe??s circular restricted three-body problem when the hydrostatic equilibrium figure of the first primary is assumed to be an oblate spheroid, the shape of the second primary is considered as a triaxial rigid body, and the full buoyancy force of the fluid is taken into account. It is found that there is an equilibrium point near the center of the first primary, another equilibrium point exists on the line joining the centers of the primaries and there exist infinite number of equilibrium points on an ellipse in the orbital plane of the second primary. It is also observed that under certain conditions, all these equilibrium points can be stable. The most interesting and distinguishable results of this study are the existence of elliptical points and their stability.