Restricted Five-body Problem Research Articles

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.

A Study of the Axisymmetric Restricted Five-Body Problem within the Frame of Variable Mass: The Concave Case

A Study of the Axisymmetric Restricted Five-Body Problem within the Frame of Variable Mass: The Concave Case

On the rhomboidal restricted five-body problem: Analysis of the basins of convergence

This manuscript aims to investigate numerically the effect of parameter λ on the basins of convergence (BoCs) associated with the equilibrium points (EPs) of the restricted rhomboidal five-body problem (RR5BP). Moreover, the parametric variation of EPs and zero velocity curves (ZVCs) are also illustrated. Firstly, we have scanned the entire interval for λ∈(13,3) to evaluate the critical value of λ where the number of EPs changes. It is observed that there exist either eleven, thirteen or fifteen EPs in total, however the stability analysis suggests that none of the EPs are linearly stable. The effect of the parameter λ and Jacobian constant C on the regions of possible motion are also illustrated. A systematic numerical investigation is performed to unveil the fact that how the parameter λ affects the geometry of the BoCs. Moreover, we have recorded the total number of iterations needed for each of the initial condition (IC) to converge a specific attractor and shown how the BoCs are related to these iterations and the associated probability distributions. Our numerical results strongly suggest that the parameter λ is indeed very influential factor in this dynamical system. The evolution of attracting regions in this dynamical system is very complicated yet an issue of paramount importance.

On the axisymmetric restricted five-body problem within the frame of variable mass: The convex case

The present manuscript reveals the existence and stability of the libration points (LPs) in the axisymmetric restricted five-body problem (AR5BP) in which mass of the fifth particle varies with respect to time as per Jeans’ law. It has been assumed that the four bodies Pi with masses mi,i=1,2,…,4 (when m3=m4=m˜) form an axisymmetric convex configuration. The system of differential equations which govern the motion of the test particle (whose mass is variable) moving under the gravitational influence of the four primaries, have been presented, indeed, these equations are different from those of the test particle with constant mass in the AR5BP. We have determined the in-plane and out-of-plane LPs along with their linear stability. Furthermore, it is observed that the existence of these LPs depends not only on the angle parameters α and β but also on the parameters occur due to variation in mass namely γ(0<γ≤1) and σ(0≤σ≤2.2, a proportionality constant occurs in Jeans’ law). Moreover, we have investigated the evolution of the regions of possible motion as function of variable mass parameters where the fifth body can move freely. The topology of the basins of convergence (BoC) linked with the LPs as function of σ is unveiled by deploying the bivariate version of Newton-Raphson (NR) iterative method, and, in order to measure the uncertainty of the basins, the basin entropy is also evaluated. The co-relations between the domains of basins of convergence (DoBoC) and the required number of iterations to achieve predefined accuracy, and also with the corresponding probability distributions are illustrated.

A Planar Five-body Problem in a Framework of Heterogeneous and Mass Variation Effects

Abstract The aim of the present paper is to study the effects of heterogeneous oblate spheroid and variable mass on the motion of the fifth infinitesimal body in the frame of the circular restricted five-body problem, with the imposition that the three primaries are placed at the vertices of an equilateral triangle and the fourth primary is placed at the center of the equilateral triangle. By using Jeans law and Meshcherskii space transformation, we evaluate the equations of motion and find a quasiJacobian integral. We determine the locations of equilibrium points, the regions of motion, and the attracting domain of in-plane motion. Furthermore, the effects of heterogeneous oblate spheroid and variable mass have been examined with the help of Poincaré surfaces of section. We studied the linear stability of equilibrium points and found that all of them are unstable.

Elliptic Sitnikov five-body problem under radiation pressure

We consider the square configuration of photo-gravitational elliptic restricted five-body problem and study the Sitnikov motions. The four radiating primaries are of equal mass placed at the vertices of square and the fifth body having negligible mass performs oscillations along a straight line perpendicular to the orbital plane of the primaries. The motion of the fifth body is called vertical periodic motion and the main aim of this paper is to study the effect of radiation pressure on these periodic motions in the linear approximation. Moreover, the effects of radiation pressure on the motion of fifth body have been examined with the help of Poincare surfaces of section. By escalating the radiation pressure, surrounding periodic tubes and islands disappear and chaotic motion occurs near the hyperbolic points. Further, by escalating the radiation pressure, the main stochastic region joins the escaping one.

Overview and comparison of approaches towards the planar restricted five-body problem with primaries forming an axisymmetric four-body central configuration

The aim of this paper is to present recent results on the restricted five-body problem where the primaries are located at a known central configuration having the $x$-axis as symmetry axis, so that two bodies with equal masses are situated symmetrically with respect to this axis. We firstly give an overview of the axisymmetric central configurations computed by Erdi and Czirjak (2016), as well as that where three bodies with equal masses are situated at the vertices of an equilateral triangle, while the fourth body lies at the center of the triangle. This last one bridges the gap between convex and concave four-body central configurations. After that, the characterization of the geometry of the restricted five-body problem families with these configurations are described, and the number and the evolution positions of equilibrium points, which depend on the mass parameters of the primaries, are compared.

On the perturbed photogravitational restricted five-body problem: the analysis of fractal basins of convergence

In the framework of photogravitational version of the restricted five-body problem, the existence and stability of the in-plane equilibrium points, the possible regions for motion are explored and analysed numerically, under the combined effect of small perturbations in the Coriolis and centrifugal forces. Moreover, the multivariate version of the Newton-Raphson iterative scheme is applied in an attempt to unveil the topology of the basins of convergence linked with the libration points as function of radiation parameters, and the parameters corresponding to Coriolis and centrifugal forces.

Heterogeneous Oblate Primaries in Photo-gravitational CR5BP with Kite Configuration

This paper presents the investigation of the motion of infinitesimal body in the circular restricted five-body problem in which four bodies are taken as heterogeneous oblate spheroid with different densities in three layers and sources of radiation pressure. These four primaries are moving on the circumference of a circle and form a kite configuration. After evaluating the equations of motion and Jacobi-integral, we study the numerical part of the paper such as equilibria, zero-velocity curves and regions of motion. Finally, we examine the stability of the equilibria and observed that all the equilibria are unstable.

On the Newton–Raphson basins of convergence associated with the libration points in the axisymmetric restricted five-body problem: The concave configuration

The axisymmetric restricted five-body problem with the concave configuration has been studied numerically to reveal the basins of convergence, by exploring the Newton–Raphson iterative scheme, corresponding to the coplanar libration points (which act as attractors). In addition, four primaries are set in axisymmetric central configurations introduced by Érdi and Czirják [13] and the motion is governed by mutual gravitational attraction only. The evolution of the positions of libration points is illustrated, as a function of the value of angle parameters. A systematic and rigorous investigation is performed in an effort to unveil how the angle parameters affect the topology of the basins of convergence. In addition, the relation of the domain of basins of convergence with required number of iterations and the corresponding probability distributions are illustrated.

The effect of small perturbations in the Coriolis and centrifugal forces in the axisymmetric restricted five-body problem

In the framework of the axisymmetric problem of restricted five bodies, the existence and stability of the libration points, the regions of possible motion are illustrated and analyzed numerically, under the effect of small perturbations in the Coriolis and centrifugal forces. It is explored how the parameters influence substantially the positions of the libration points and the possible regions of motion. In an attempt to understand how the parameters involved due to the small perturbations in the Coriolis and centrifugal forces affect the stability of the libration points, we perform a systematic investigation and reveal that some of the collinear and non-collinear libration points are stable under these perturbations, whereas none of these libration points are stable for any combination of the angle parameters when the effects of these forces are neglected.

Orbit classification and networks of periodic orbits in the planar circular restricted five-body problem

The aim of this paper is to numerically investigate the orbital dynamics of the circular planar restricted problem of five bodies. By numerically integrating several large sets of initial conditions of orbits we classify them into three main categories: (i) bounded (regular or chaotic) (ii) escaping and (iii) close encounter orbits. In addition, we determine the influence of the mass parameter on the orbital structure of the system, on the degree of fractality, as well as on the families of symmetric and non-symmetric periodic orbits. The networks and the linear stability of both symmetric and non-symmetric periodic orbits are revealed, while the corresponding critical periodic solutions are also identified. The parametric evolution of the horizontal and the vertical linear stability of the periodic orbits is also monitored, as a function of the mass parameter.

The analysis of restricted five–body problem within frame of variable mass

In the framework of restricted five-body problem, the existence and stability of the libration points are explored and analysed numerically, under the effect of non–isotropic mass variation of the fifth body (test particle or infinitesimal body). The evolution of the positions of these points are illustrated, as a function of the perturbation parameter. We perform a systematic investigation in an attempt to understand how the perturbation parameter due to variable mass of the fifth body affects the positions, movement and stability of the libration points. In addition, we have revealed how the basins domain of convergence associated with the libration points are substantially influenced by the perturbation parameter.

Out-of-plane equilibria in the restricted five-body problem with radiation pressure

Out-of-plane equilibria in the restricted five-body problem with radiation pressure

Numerical Study of the Zero Velocity Surface and Transfer Trajectory of a Circular Restricted Five-Body Problem

We focus on a type of circular restricted five-body problem in which four primaries with equal masses form a regular tetrahedron configuration and circulate uniformly around the center of mass of the system. The fifth particle, which can be regarded as a small celestial body or probe, obeys the law of gravity determined by the four primaries. The geometric configuration of zero-velocity surfaces of the fifth particle in the three-dimensional space is numerically simulated and addressed. Furthermore, a transfer trajectory of the fifth particle skimming over four primaries then is designed.

Basins of attraction of equilibrium points in the planar circular restricted five-body problem

We numerically explore the Newton-Raphson basins of convergence, related to the libration points (which act as attractors), in the planar circular restricted five-body problem (CR5BP). The evolution of the position and the linear stability of the equilibrium points is determined, as a function of the value of the mass parameter. The attracting regions, on several types of two dimensional planes, are revealed by using the multivariate version of the classical Newton-Raphson iterative method. We perform a systematic investigation in an attempt to understand how the mass parameter affects the geometry as well as the degree of fractality of the basins of attraction. The regions of convergence are also related with the required number of iterations and also with the corresponding probability distributions.

Out - of - plane equilibria in the restricted five-body problem with radiation pressure

The photogravitational restricted five-body problem is employed to describe the motion of an infinitesimal test particle in the special case where two of the primaries are radiation sources. The four primaries mi, i = 0, , 3 three of which have equal masses (m1 = m2 = m3 = m) are located at the vertices of an equilateral triangle, while the fourth one with a different mass m0 is located at the centre of this configuration (Ollongren's configuration). The fifth body of negligible mass moves in the resultant force field of the primaries and does not affect the motion of the four bodies (primaries). We consider that the central primary body m0 and one of the peripheral bodies m1 are radiation sources. The equilibrium points lying out of the orbital plane of the primaries as well as the allowed regions of motion as determined by the zero velocity curves are studied numerically. Finally, the stability of these points is also examined.

Equilibrium points and zero velocity surfaces in the axisymmetric restricted five-body problem

In this paper, we have analyzed the motion of an infinitesimal mass in axisymmetric restricted five-body problem. The four primaries are supposed to be in axisymmetric central configurations based on Erdi and Czirjak and the motion is governed by mutual gravity only. We have derived the equations of motion of the infinitesimal mass firstly and then found the number and locations of equilibrium points numerically for various angle coordinates. In addition, the zero velocity surfaces (ZVS) are drawn and regions of motion are analyzed for specific configurations. Finally the linear stability of equilibrium points are discussed preliminarily.

A simple model for the location of Saturn’s F ring

In this paper, we introduce a simplified model to understand the location of Saturn’s F ring. The model is a planar restricted five-body problem defined by the gravitational field of Saturn, including its second zonal harmonic J2, the shepherd moons Prometheus and Pandora, and Titan. We compute accurate long-time numerical integrations of (about 2.5 million) non-interacting test-particles initially located in the region between the orbits of Prometheus and Pandora, and address whether they escape or remain trapped in this region. We obtain a wide region of initial conditions of the test particles that remain confined. We consider a dynamical stability indicator for the test particles’ motion defined by computing the ratio of the standard deviation over the average value of relevant dynamical quantities, in particular, for the mean-motion and the semi-major axis. This indicator separates clearly a subset of trapped initial conditions that appear as very localized stripes in the initial semi-major axis and eccentricity space for the most stable orbits. Retaining only these test particles, we obtain a narrow eccentric ring which displays sharp edges and collective alignment. The semi-major axis of the accumulation stripes of the stable ring-particles can be associated with resonances, mostly involving Prometheus’ outer Lindblad and co-rotation resonances, but not exclusively. Pandora’s inner Lindblad and co-rotation resonances as well as low-order three-body resonances typically coincide with gaps, i.e., regions of instabilities. Comparison of our results with the nominal data for the F ring shows some correspondence.

Optimal station-keeping near Earth–Moon collinear libration points using continuous and impulsive maneuvers

In this study the gravitational perturbations of the Sun and other planets are modeled on the dynamics near the Earth–Moon Lagrange points and optimal continuous and discrete station-keeping maneuvers are found to maintain spacecraft about these points. The most critical perturbation effect near the L1 and L2 Lagrange points of the Earth–Moon is the ellipticity of the Moon’s orbit and the Sun’s gravity, respectively. These perturbations deviate the spacecraft from its nominal orbit and have been modeled through a restricted five-body problem (R5BP) formulation compatible with circular restricted three-body problem (CR3BP). The continuous control or impulsive maneuvers can compensate the deviation and keep the spacecraft on the closed orbit about the Lagrange point. The continuous control has been computed using linear quadratic regulator (LQR) and is compared with nonlinear programming (NP). The multiple shooting (MS) has been used for the computation of impulsive maneuvers to keep the trajectory closed and subsequently an optimized MS (OMS) method and multiple impulses optimization (MIO) method have been introduced, which minimize the summation of multiple impulses. In these two methods the spacecraft is allowed to deviate from the nominal orbit; however, the spacecraft trajectory should close itself. In this manner, some closed or nearly closed trajectories around the Earth–Moon Lagrange points are found that need almost zero station-keeping maneuver.