Circular Restricted Four-body Problem Research Articles

The Lyapunov Stability of Central Configurations of the Planar Circular Restricted Four-Body Problem

The Lyapunov Stability of Central Configurations of the Planar Circular Restricted Four-Body Problem

A study of the equilibrium dynamics of the test particle in the collinear circular restricted four-body problem with non-spherical central primary

We consider the collinear restricted four-body problem (CR4BP), where the test particle of infinitesimal mass is moving under the gravitational influence of the three primary bodies. It is further assumed that the central primary is a non-spherical body, particularly either an oblate or prolate spheroid, whereas the peripheral primaries are spherical in shape. A numerical analysis is presented to unveil the effect of the oblateness and prolateness parameters on the position of equilibrium points (EPs) and their linear stability in the CR4BP. Moreover, the permissible regions of possible motion as determined by the zero-velocity surface and associated equipotential curves and the basins of convergence linked with the EPs on the orbital plane are presented. The existence and number of collinear EPs and non-collinear EPs in the problem depend on the combination of the mass parameter of the primaries and the oblateness/prolateness parameter. Additionally, the application of the problem in the Saturn-Moon(1)-Moon(2)-System has been presented.

Computer assisted proof of homoclinic chaos in the spatial equilateral restricted four-body problem

We develop constructive, computer assisted arguments for proving the existence of transverse homoclinic connections for hyperbolic periodic orbits in Hamiltonian systems. These arguments are applied at a number of non-perturbative parameter and energy values in the spatial equilateral circular restricted four-body problem: a six dimensional vector field. The idea is to formulate the desired connecting orbits as solutions of certain two point boundary value problems for orbit segments which originate and terminate on the local stable/unstable manifolds attached to a periodic orbit. These boundary value problems are studied via a Newton-Kantorovich argument in an appropriate Cartesian product of Banach algebras of rapidly decaying sequences of Chebyshev coefficients. Perhaps the most delicate part of the problem is controlling the boundary conditions, which must lie on the local stable/unstable manifolds of the periodic orbit. For this portion of the problem we use a parameterization method to develop Fourier-Taylor approximations equipped with mathematically rigorous a-posteriori error bounds. This requires validated computation of a finite number of Fourier-Taylor coefficients via Newton-Kantorovich arguments in appropriate Cartesian product of rapidly decaying sequences of Fourier coefficients, followed by a fixed point argument to bound the tail terms of the Taylor expansion. After some geometric considerations in the energy manifold, transversality of the connection follows as a direct consequence of the Newton-Kantorovich argument.

Equilateral Triangular Configuration in the Perturbed Circular Restricted Four-Body Problem with Kerr-like Primaries and Variable Mass Test Particle

Equilateral Triangular Configuration in the Perturbed Circular Restricted Four-Body Problem with Kerr-like Primaries and Variable Mass Test Particle

On the basins of convergence in the beyond-Newtonian spatial collinear circular restricted four-body problem with spinning primaries

In this paper, we present the analysis of the Newton–Raphson basins of convergence (NR-BoCs) linked with the relative equilibria (LPs) in the Eulerian configuration of a pseudo-Newtonian planar circular restricted problem of four bodies (SCCR4BP) with spinning primaries. In this configuration of the SCCR4BP, the primaries are laid out in a symmetrical collinear shape in which the two peripheral primaries are of equal mass. We derive the equations of motion by assuming that the system consists of three Kerr-like primaries to examine how the geometry of the basins of convergence (BoCs) associated with the LPs of SCCR4BP are affected by the first-order non-Newtonian contribution to the gravitational field. Additionally, we analyze the correlation among the regions of convergence, the required number of iterations, and the associated probability distributions. Finally, for the quantitative study of the basins of convergence, we measure the uncertainty of the respective basins via the basin entropy and the basin boundary entropy.

Periodic solution of circular Sitnikov restricted four-body problem using multiple scales method

Periodic solution of circular Sitnikov restricted four-body problem using multiple scales method

On bifurcations and stability of central configurations in the planar circular restricted four-body problem

The restricted four-body problem is considered. That is, we consider motion of an infinitely small body (particle) under the Newtonian gravitational attraction of three bodies (primaries). It is assumed that the primaries move in circular orbits, forming a stable equilateral Lagrange triangle. It is supposed that four bodies move in a plane. There exist relative equilibriums of the particle in the rotating with the primaries coordinate system. In such an equilibrium the particle forms a central configuration with the primaries. In the case of small mass ofa primary the bifurcations ofthe central configurations are investigated, as well as conditions oflinear stability for the central configurations is obtained. The study is performed analytically by using small parameter method.

Spatial periodic orbits in the equilateral circular restricted four-body problem: computer-assisted proofs of existence

We use validated numerical methods to prove the existence of spatial periodic orbits in the equilateral restricted four-body problem. We study each of the vertical Lyapunov families (up to symmetry) in the triple Copenhagen problem, as well as some halo and axial families bifurcating from planar Lyapunov families. We consider the system with both equal and non-equal masses. Our method is constructive and non-perturbative, being based on a posteriori analysis of a certain nonlinear operator equation in the neighborhood of a suitable approximate solution. The approximation is via piecewise Chebyshev series with coefficients in a Banach space of rapidly decaying sequences. As by-product of the proof, we obtain useful quantitative information about the location and regularity of the solution.

Combined effect of Stokes drag, oblateness and radiation pressure on the existence and stability of equilibrium points in the restricted four-body problem

This paper studies numerically the existence of collinear and non-collinear equilibrium points and their linear stability in the frame work of photogravitational circular restricted four-body problem with Stokes drag acting as a dissipative force and considering the first primary as a radiating body and the second primary as an oblate spheroid. The mass of the fourth body is assumed to be infinitesimal and does not affect the motion of the three primaries which are always at the vertices of an equilateral triangle (Lagrangian configuration). It is found that at constant dissipative force, and a simultaneous increase in both radiation pressure and oblateness coefficients, the curves that define the path of the motion of the infinitesimal body are found to shrink due to shifts in the positions of equilibrium points. All the collinear and non-collinear equilibrium points are found to be linearly unstable under the combined effect of radiation pressure, oblateness and Stokes drag. The energy integral is seen to be time dependent due to the presence of the drag force. More so the dynamic property of the system is investigated with the help of Lyapunov characteristic exponents (LCEs). It is found that the system is chaotic as the trajectories locally diverge from each other and the equilibrium points are chaotic attractors. We justified the relevance of the model in astronomy by applying it to a stellar system (Gliese 667C) and another found that both the existence and stability of the equilibrium points of any restricted few body system greatly depend of the value of the mass parameter.

Revealing the existence and stability of equilibrium points in the circular autonomous restricted four-body problem with variable mass

We have numerically investigated the circular autonomous restricted four-body problem where the fourth particle of variable mass is moving under the gravitational influence of three bodies known as primaries. Moreover, these primaries move in circular orbit around their common center of mass in such a way that their configuration remains an equilateral triangle configuration. The effect of the parameter α on the existence as well as on the locations of the libration points are investigated. The parametric variation of the positions of the libration points and zero velocity curves are also revealed when the parameter α (which occurs in Jeans’ law) increases. Moreover, the Newton–Raphson basins of convergence corresponding to the libration points are unveiled numerically when the parameter α increases. The obtained results strongly suggest that the study of the evolution of the attracting domains of the proposed dynamical system is worth studying in spite of their complexity.

Chaotic motions in the restricted four body problem via Devaney's saddle-focus homoclinic tangle theorem

We prove the existence of chaotic motions in an equilateral planar circular restricted four body problem (CRFBP), establishing that the system is not integrable. The proof works by verifying the hypotheses of a topological forcing theorem for Hamiltonian vector fields on R4 which hypothesizes the existence of a transverse homoclinic orbit in the energy manifold of a saddle focus equilibrium. We develop mathematically rigorous computer assisted arguments for verifying these hypotheses, and provide an implementation for CRFBP. Due to the Hamiltonian structure, this also establishes the existence of a “blue sky catastrophe”, and hence an analytic family of periodic orbits of arbitrarily long period at nearby energy levels.Our method works far from any perturbative regime and requires no mass symmetry. Additionally, the method is constructive and yields additional byproducts such as the locations of transverse connecting orbits, quantitative information about the invariant manifolds, and bounds on transport times.

Unveiling the basins of convergence in the pseudo-Newtonian planar circular restricted four-body problem

The dynamics of the pseudo-Newtonian restricted four-body problem has been studied in the present paper, where the primaries have equal masses. The parametric variation of the existence as well as the position of the libration points are determined, when the value of the transition parameter ϵ ∈(0, 1]. The stability of these libration points has also been discussed. Our study reveals that the Jacobi constant as well as transition parameter ϵ have substantial effect on the regions of possible motion, where the fourth body is free to move. The multivariate version of Newton-Raphson iterative scheme is introduced for determining the basins of attraction in the configuration (x, y) plane. A systematic numerical investigation is executed to reveal the influence of the transition parameter on the topology of the basins of convergence. In parallel, the required number of iterations is also noted to show its correlations to the corresponding basins of convergence. It is unveiled that the evolution of the attracting regions in the pseudo-Newtonian restricted four-body problem is a highly complicated yet worth studying problem.

NUMERICAL-SYMBOLIC METHODS FOR SEARCHING RELATIVE EQUILIBRIA IN THE RESTRICTED PROBLEM OF FOUR BODIES

We discuss here the problem of solving the system of two nonlinear algebraic equations determining the relative equilibrium positions in the planar circular restricted four-body problem formulated on the basis of the Euler collinear solution of the three-body problem. The system contains two parameters $\mu_1$, $\mu_2$ and all its solutions coincide with the corresponding solutions in the three-body problem if one of the parameters equals to zero. For small values of one parameter the solutions are found in the form of power series in terms of this parameter, and they are used for separation of different solutions and choosing the starting point in the numerical procedure for the search of equilibria. Combining symbolic and numerical computation, we found all the equilibrium positions and proved that there are 18 different equilibrium configurations of the system for any reasonable values of the two system parameters $\mu_1$, $\mu_2$. All relevant symbolic and numerical calculations are performed with the aid of the computer algebra system Wolfram Mathematica.

Horseshoe orbits in the restricted four-body problem

The circular restricted four-body problem studies the dynamics of a massless particle under the gravitational force produced by three point masses that follow circular orbits with constant angular velocity, the configuration of these circular orbits forms an equilateral triangle for all time; this problem can be considered as an extension of the celebrated restricted three-body problem. In this work we investigate the orbits which emanate from some equilibrium points. In particular, we focus on the horseshoe shaped orbits (rotating frame), which are well known in the restricted three-body problem. We study some families of symmetric horseshoe orbits and show their relation with a family of the so called Lyapunov orbits.

The effect of perturbations on the circular restricted four-body problem with variable masses

The effect of perturbations on the circular restricted four-body problem with variable masses

Periodic Orbits of the First Kind in the CR4BP When the Second Primary Is a Triaxial Rigid Body

The present paper deals with the existence of periodic orbits in the Circular Restricted Four-Body Problem (CR4BP) in two-dimensional co-ordinate system when the second primary is a triaxial rigid body and the third primary of inferior mass (in comparison of the other primaries) is placed at triangular libration point L4 of the Circular Restricted Three-Body Problem (CR3BP). With the help of generating solutions, we formed a basis for the existence of periodic orbits, then an analytical approach given by Hassan et al. [1], was applied to our model of equilateral triangular configuration. It is found that in general solution also; the character of periodic orbits is conserved. For verification of the existence of periodic orbits, we have applied the criterion of Duboshin [2] and found satisfied.

Families of three-dimensional periodic solutions in the circular restricted four-body problem

In this paper we study the 3D symmetric periodic orbits of the circular restricted four-body problem, through their bifurcation from plane orbits. To this end, we calculate the special generating planar periodic orbits, i.e. the vertical-critical orbits, from five basic planar families of symmetric periodic orbits of the problem. We study the simplest 3D periodic orbits and so we restrict our calculations to simple vertical bifurcation orbits leading to families of one or two revolutions three-dimensional symmetric periodic orbits. We found 21 such families of simple 3D symmetric periodic orbits and typical orbits of all symmetry type 3D orbits are presented. The stability of each calculated 3D periodic orbit is also studied. Characteristic curves as well as stability diagrams of families of 3D periodic orbits are illustrated.

The Photogravitational Circular Restricted Four-body Problem with Variable Masses

The Photogravitational Circular Restricted Four-body Problem with Variable Masses

New family of iterative methods based on the Ermakov–Kalitkin scheme for solving nonlinear systems of equations

A new one-parameter family of iterative methods for solving nonlinear equations and systems is constructed. It is proved that their order of convergence is three for both equations and systems. An analysis of the dynamical behavior of the methods shows that they have a larger domain of convergence than previously known iterative schemes of the second to fourth orders. Numerical results suggest that the methods are also preferable in terms of their relative stability and the number of iteration steps. The methods are compared with previously known techniques as applied to a system of two nonlinear equations describing the dynamics of a passively gravitating mass in the Newtonian circular restricted four-body problem formulated on the basis of Lagrange’s triangular solutions to the threebody problem.

Dynamical study while searching equilibrium solutions in [formula omitted]-body problem

The theory of complex dynamics is usually applied to compare the global convergence properties of different iterative methods, by obtaining the attraction basins for simple polynomial equations in the complex domain. However, in this work, we use it in quite another context: the study of a nontrivial nonlinear system that describes the motion of interacting bodies in celestial mechanics, namely, Newtonian planar circular restricted four-body problem and its relative equilibrium solutions. These have been investigated from a dynamical point of view. New properties of the solutions of this system have been obtained. Practical guidelines for efficient search of relative equilibrium solutions of N-body problem have been given.