Four-body Problem Research Articles

Analysis of Linear Stability and Bifurcations of Central Configurations in the Planar Restricted Circular Four-Body Problem

The planar restricted four-body problem is considered. That is, motion of an infinitesimal body P in the gravitational field of three attracting bodies is studied. It is supposed, that the above three body form a stable equilateral Lagrange triangle and all four bodies move in a plane. In this case there are eight central configurations formed by the bodies. An analysis of stability and bifurcation of the central configurations is performed. In particular, it is shown that the bifurcation is only possible in cases of degeneracy, when mass of an attracting body vanishes. It is also established that in non-degenerate cases five central configurations are unstable and three central configurations can be both stable and unstable. Domains of stability in linear approximation are constructed in the plane of parameters.

Mass-spectroscopy of [$$bb][{\bar{b}}{\bar{b}}$$] and [$$bq][{\bar{b}}{\bar{q}}$$] tetraquark states in a diquark–antidiquark formalism

In this article, we utilise the non-relativistic potential model to calculate the mass-spectra of all bottom [ $$bb][{\bar{b}}{\bar{b}}$$ ] and heavy-light bottom [ $$bq][{\bar{b}}{\bar{q}}$$ ] (q = u,d) tetraquark states in diquark–antidiquark approximation. The four-body problem is reduced into two body problem by numerically solving the $$Schr\ddot{o}dinger$$ equation using a Cornell-inspired potential along with relativistic correction term. The splitting structure of the tetraquark spectrum is described using spin-dependent terms (spin-spin, spin-orbit, and tensor). We have successfully calculated and predicted the masses of bottom mesons, diquarks and tetraquarks. The masses of S and P-wave tetraquark states [ $$bb][{\bar{b}}{\bar{b}}$$ ] and [ $$bq][{\bar{b}}{\bar{q}}$$ ], respectively, are found to be between 18.7–19.4 GeV and 10.4–11.3 GeV, in which the masses of S-wave [ $$bb][{\bar{b}}{\bar{b}}$$ ] states are less than the 2 $$\eta _{b}$$ , $$\eta _{b}\Upsilon $$ , and 2 $$\Upsilon $$ threshold. Additionally, we have investigated the $$Z_b(10610)$$ and $$Z_ b(10650)$$ states in the current model and found that they are 150 MeV below the $$BB^{*}$$ and $$B^{*}B^{*}$$ thresholds.

Properties of the lunar gravity assisted transfers from LEO to the retrograde-GEO

The retrograde geostationary earth orbit (retro-GEO) is an Earth’s orbit. It has almost the same orbital altitude with that of a GEO, but an inclination of 180°. A retro-GEO monitor-satellite gives the GEO-assets vicinity space-debris warnings per 12 h. For various reasons, the westward launch direction is not compatible or economical. Thereby the transfer from a low earth orbit (LEO) to the retro-GEO via once lunar swing-by is a priority. The monitor-satellite departures from LEO and inserts into the retro-GEO both using only one tangential maneuver, in this paper, its transfer’s property is investigated. The existence of this transfer is verified firstly in the planar circular restricted three-body problem (CR3BP) model based on the Poincaré-section methodology. Then, the two-impulse values and the perilune altitudes are computed with different transfer durations in the planar CR3BP. Their dispersions are compared with different Sun azimuths in the planar bi-circular restricted four-body problem (BR4BP) model. Besides, the transfer’s inclination changeable capacity via lunar swing-by and the Sun-perturbed inclination changeable capacity are investigated. The results show that the two-impulse fuel-optimal transfer has the duration of 1.76 TU (i.e., 7.65 days) with the minimum values of 4.251 km s−1 in planar CR3BP, this value has a range of 4.249–4.252 km s−1 due to different Sun azimuths in planar BR4BP. Its perilune altitude changes from 552.6 to 621.9 km. In the spatial CR3BP, if the transfer duration is more than or equal to 4.00 TU (i.e., 17.59 days), the lunar gravity assisted transfer could insert the retro-GEO with any inclination. In the spatial BR4BP, the Sun’s perturbation does not affect this conclusion in most cases.

Nonlinear Stability of Equilibrium Points in the Planar Equilateral Restricted Mass-Unequal Four-Body Problem

In this paper, we investigate the stability of equilibrium points for the planar restricted equilateral four-body problem in the case that one particle of negligible mass is moving under the Newtonian gravitational attraction of three positive masses [Formula: see text], [Formula: see text] and [Formula: see text] (called primaries). These always lie at the vertices of an equilateral triangle (Lagrangian configuration) and move with constant angular velocity in circular orbits around their center of masses. We consider the case where all the primaries have unequal masses, and investigate the nonlinear stability (in the sense of Lyapunov) of the elliptic equilibrium for the specific values of the mass [Formula: see text] and [Formula: see text] of the primary, fixed on the horizontal axis. Moreover, the [Formula: see text][Formula: see text]:[Formula: see text][Formula: see text] four-order resonant cases are determined and the stability is investigated. In this study, Markeev’s theorem and Arnold’s theorem become key ingredients.

An impulsive model predictive static programming based station-keeping guidance for quasi-halo orbits

In this paper, a control effort minimizing optimal station-keeping guidance is designed and implemented to regulate a spacecraft around an L1 quasi-halo orbit in the Sun–Earth–Moon elliptic four-body problem. The station-keeping guidance is formulated as a finite time, non-linear optimal control problem with hard terminal output constraints and Impulsive Model Predictive Static Programming (I-MPSP) is used to obtain station-keeping maneuvers. The algorithm is iterative, where the guessed station-keeping maneuvers are optimally updated by a simple closed-form equation until an output terminal constraint is satisfied and an optimal cost function is obtained. The technique involves the calculation of sensitivity matrices that is done in a computationally efficient manner owing to their recursive nature. Through extensive simulations, in the presence of disturbance forces and uncertainties, a closed-loop application of I-MPSP guidance results in a trajectory that remains tightly bound to the reference orbit over a long-duration mission.

Exotic resonances of fully-heavy tetraquarks in a lattice-QCD insipired quark model

Fully-heavy tetraquark states, i.e. $cc\bar{c}\bar{c}$, $bb\bar{b}\bar{b}$, $bb\bar{c}\bar{c}$ ($cc\bar{b}\bar{b}$), $cb\bar{c}\bar{c}$, $cb\bar{b}\bar{b}$, and $cb\bar{c}\bar{b}$, are systematically investigated by means of a non-relativistic quark model based on lattice-QCD studies of the two-body $Q\bar{Q}$ interaction, which exhibits a spin-independent Cornell potential along with a spin-spin term. The four-body problem is solved using the Gaussian expansion method; additionally, the so-called complex scaling technique is employed so that bound, resonance, and scattering states can be treated on the same footing. Moreover, a complete set of four-body configurations, including meson-meson, diquark-antidiquark, and K-type configurations, as well as their couplings, are considered for spin-parity quantum numbers $J^{P(C)}=0^{+(+)}$, $1^{+(\pm)}$, and $2^{+(+)}$ in the $S$-wave channel. Several narrow resonances, with two-meson strong decay widths less than 30 MeV, are found in all of the tetraquark systems studied. Particularly, the fully-charm resonances recently reported by the LHCb Collaboration, at the energy range between 6.2 and 7.2 GeV in the di-$J/\psi$ invariant spectrum, can be well identified in our calculation. Focusing on the fully-bottom tetraquark spectrum, resonances with masses between 18.9 and 19.6 GeV are found. For the remaining charm-bottom cases, the masses are obtained within a energy region from 9.8 GeV to 16.4 GeV. All these predicted resonances can be further examined in future experiments.

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.

On the C 8/3-regularisation of simultaneous binary collisions in the planar four-body problem

The dynamics of the four-body problem allows for two binary collisions to occur simultaneously. It is known that in the collinear four-body problem this simultaneous binary collision (SBC) can be block-regularised, but that the resulting block map is only C 8/3 differentiable. In this paper, it is proved that the C 8/3 differentiability persists for the SBC in the planar four-body problem. The proof uses several geometric tools, namely, blow-up, normal forms, dynamics near normally hyperbolic manifolds of equilibrium points, and Dulac maps.

Equilibria, stability and chaos in photogravitational Bi-Circular restricted four-body problem

A model to investigate the influence of gravitational force and radiation pressure on interstellar dust found within the vicinities of certain stellar systems is presented in this study with the inclusion of potential due to the belt. Semi-analytic approach is adopted to examine the dynamical behaviour of the motion of a test particle in the neighbourhood of the radiating stars, namely, Wolf 630, Formalhaut, Omicron Eradani and 36 Ophiuchi respectively. The Libration points were found dependent on the mass ratios of the systems and the radiation pressure exerted by the star(s) and the motion of the dust particle around them is linearly unstable. In each of the four case studies, two Lyapunov Characteristic Exponents were seen positive which validate the chaotic nature of the system, also the Poincare Surface Section revealed the sensitivity of the dynamical system to change in initial conditions. Several scientific questions of great importance in astrophysics and astronomy; such as the motions of interstellar clouds, proto-nebulae, planetary rings and micrometeorites can be answered by engaging this model.

A Restricted Four-body Problem for the Figure-eight Choreography

A Restricted Four-body Problem for the Figure-eight Choreography

Exclusion of quadruple collisions in minimizers of the planar equal-mass N-body problem

It has been shown by Chen that if all masses are in the same proper vector subspace of Rd(d≥2) on one of the free boundaries, local minimizers connecting the two free boundaries have no collision on that boundary. However, not much is known for other types of boundary configurations.In this paper, we consider minimizers connecting two free boundaries and study possible quadruple collisions under the following four symmetric configurations: isosceles trapezoid configuration, rectangular configuration, double isosceles configuration and diamond configuration. Whenever the boundary configuration of four bodies coincides with one of the four symmetric configurations, we show that these four bodies are free of quadruple collision on that boundary set. New ideas and detailed analysis are introduced in order to construct deformed paths in some of the cases and estimate their action values. As its applications, we show the existence of several sets of periodic orbits in the planar four-body and five-body problems.

On the Uniqueness of Co-circular Four Body Central Configurations

We study central configurations lying on a common circle in the Newtonian four-body problem. Using a topological argument we prove that there is at most one co-circular central configuration for each cyclic ordering of the masses on the circle.

On the Equilibria of the Planar Equilateral Restricted Four-Body Problem with Radiation Pressure

On the Equilibria of the Planar Equilateral Restricted Four-Body Problem with Radiation Pressure

Qualitative Analysis of Some Libration Points in the Restricted Four-body Problem

We consider the situation where three heavy gravitational bodies form the Lagrange configuration rotating in a fixed plane and the fourth body of negligible mass moves in this plane. We present three cases of so-called libration points and we study their stability using linear approximation and KAM theory. In some situations we prove the Lyapunov stability for generic values of some parameter of the problem.

Exploration of Low-Energy Earth-Moon Transfer Orbit Based on Crossing Orbit of Double Three-Body System

The design of low energy lunar transfer orbit and orbit optimization method based on crossing orbit is studied in this paper. The four-body problem of Sun-Earth-Moon-satellite is decoupled into two three-body problems of Sun-Earth-satellite and Earth-Moon-satellite. Low-energy lunar exploration orbit is designed by changing orbit at a certain intersection field through manifold of two Halo orbit of three-body system. A serial optimization design method for global initial value search and local gradient optimization are designed. When a motion state variable is partially modified, the differential correction method combined with adaptive regression algorithm is adopted. For the splicing points of manifolds of different systems, the velocity vector is modified to make the spacecraft get into the target berthing track, and to obtain the splicing points that can form crossing track of the two three-body systems. An example of orbit design is given to show that the Earth-Moon transfer orbit based on invariant manifold splicing has the characteristics of energy saving and long-time consumption, which can be used for the tasks with large loads and low time requirements. This research provides a method for the design of low energy lunar transfer orbit. It will benefit the energy consuming, material design of spacecraft, and so on.

On the uniqueness of trapezoidal four-body central configurations

We study central configurations of the Newtonian four-body problem that form a trapezoid. Using a topological argument we prove that there is at most one trapezoidal central configuration for each cyclic ordering of the masses.

On the equilibria of the restricted four-body problem with triaxial rigid primaries - I. Oblate bodies

• We numerically explore the equilibria of the restricted four-body problem with triaxial rigid primaries. • We compute the linear stability of the libration points. • We determine how the triaxility parameters affect the dynamics of the equilibrium points. The equilateral restricted four-body problem, with triaxial rigid oblate bodies, is investigated. Using numerical methods we examine how the linear stability and the positions of the coplanar libration points are affected by the triaxility parameters of the primaries. In each case, we perform a rigorous and systematic analysis for determining the influence of the triaxility parameters σ 1 and σ 2 on the dynamics of the system. Our computations suggest the strong influence of these parameters by revealing additional cases, regarding the total number of equilibria, thus improving the findings of previous related works.

Impact of radiation pressure and circumstellar dust on motion of a test particle in Manev’s field

In this paper, we present a study on the impact of radiation pressure and circumstellar dust on the motion of a test particle in the framework of the restricted four-body problem under the Manev’s field. We show that the distribution of equilibrium points on the plane of motion is slightly different from that of the classical Newtonian problem. With the aid of the Lyapunov characteristic exponents, we show that the system is sensitive to changes in initial conditions; hence, the orbit of the system is found to be chaotic in the phase space for the given initial conditions. Furthermore, a numerical application of this model to a stellar system (Gliese 667C) is considered, which validates the dependence of the equilibrium points on the mass parameter. We show that the non-collinear equilibrium points of this stellar system are distributed symmetrically about the x-axis, and five of them are linearly stable. The basins of attraction of the system show that the equilibrium points have irregular boundaries, and we use the energy integral and the Manev parameter to illustrate the zero-velocity curves showing the permissible region of motion of the test particle with respect to the Jacobi constant.

The effect of radiation pressure on the basins of convergence in the restricted four-body problem

Abstract The present problem deals with the effect of the radiation pressure on the topology of the basins of convergence in the restricted problem of four bodies in two different configurations: (I) the equilateral four-body configuration (II) the collinear four-body problem. We have illustrated the basins of convergence linked to the in-plane as well as out-of-plane libration points by applying the Newton-Raphson multivariate iterative scheme, in both the configurations. Additionally, the basin entropy is illustrated to estimate the degree of fractality of the basins of convergence.

On the four-body problem in the Born–Oppenheimer approximation

The quantum problem of four particles in Rd (d≥3), with arbitrary masses m1,m2,m3 and m4, interacting through a harmonic oscillator potential is considered. This model allows exact solvability and a critical analysis of the Born–Oppenheimer approximation. The study is restricted to the ground state level. We pay special attention to the case of two equally heavy masses m1=m2=M and two light particles m3=m4=m. It is shown that the sum of the first two terms of the Puiseux series, in powers of the dimensionless parameter σ=mM, of the exact phase Φ of the wave function ψ0=e−Φ and the corresponding ground state energy E0, coincide exactly with the values obtained in the Born–Oppenheimer approximation. A physically relevant rough model of the H2 molecule and of the chemical compound H2O2 (Hydrogen peroxide) is described in detail. The generalization to an arbitrary number of particles n, with d degrees of freedom (d≥n−1), interacting through a harmonic oscillator potential is briefly discussed as well.