Four-body Problem Research Articles

Linear stability of elliptic relative equilibria of four-body problem with two infinitesimal masses

In this paper, we consider the elliptic relative equilibria of four-body problem with two infinitesimal masses. The most interesting case occurs when the two small masses tend to the same Lagrangian point L4 (or L5). In [20], Xia showed that there exist four central configurations: two of them are non-convex, and the other two are convex. We prove that the elliptic relative equilibria raised from the non-convex central configurations are always linearly unstable, while for the elliptic relative equilibria raised from the convex central configurations, the conditions of linear stability with respect to the parameters are provided.

Equilibrium stability in the triangular restricted four-body problem with non-spherical primaries

This paper investigates the Lagrangian configuration of the restricted four-body problem in which the three primaries are non-spherical, specifically either prolate or oblate. By using various standard numerical methods, the positions of equilibrium points and their linear stability and dynamical type were determined. The impact of mass and shape of the primaries on the system’s equilibrium points and their linear stability were systematically explored by discretizing the parameter space for the non-sphericity parameter within a specified interval. The study revealed that the system always has an even number of equilibrium points, ranging from 8 to 22. Linearly stable points always exist, except for the case where there are 10 equilibrium points, where all the points are unstable.

Energy analysis of the single impulsive Earth–Moon transfer with the temporary lunar capture

The single impulsive planar Earth–Moon transfer strategy is investigated by analyzing the energy change of the lunar capture. A planar bicircular model of the restricted four-body problem is applied to define the single impulsive trajectory to the Moon from the Earth. Then analytical expressions are derived to describe the energy changes of the lunar capture in the Sun–Earth rotating frame. Accordingly, the eccentricity at the perilune of the single impulsive transfer trajectory can be calculated analytically to explore the lunar escape and capture conditions. The error analysis shows that the theoretical solutions serve as good approximations of the energy changes of the single impulsive transfer. Furthermore, it has been found that the perilune angle and the perilune radius are the critical factors for lunar capture. Finally, the theoretical results are successfully employed to find a series of single impulsive planar Earth–Moon transfers.

Unveiling the Attracting Regions in Photogravitational Four-Body Problem Including the Effect of Asteroids Belts

Unveiling the Attracting Regions in Photogravitational Four-Body Problem Including the Effect of Asteroids Belts

Cross sections for single-electron capture from heliumlike targets by fast heavy nuclei

Single charge-exchange in collisions of heavy bare nuclei with the ground state of two-electron atomic targets is described perturbatively as a four-body problem. The employed four-body boundary-corrected continuum intermediate state (BCIS-4B) method considers the correlated and uncorrelated target wave functions ${\ensuremath{\varphi}}_{i}.$ A thorough examination is performed for the formation of any final hydrogen-like $nlm$ state of the captured electron. For arbitrary projectile and target nuclear charges, the nine-dimensional integral in the transition amplitude is reduced to a two-dimensional numerical quadrature. The general analysis is applied to one-electron capture by protons from helium targets beginning with the lower edge (10 keV) of intermediate energies and extending to the higher (12.5 MeV) domain. These include the main peaks (Massey, Thomas) due to single and double scattering, respectively. The results encompass over 70 state-selective and state-summed cross sections $(n\ensuremath{\le}6,\phantom{\rule{0.16em}{0ex}}0\ensuremath{\le}l\ensuremath{\le}n\ensuremath{-}1,\phantom{\rule{0.16em}{0ex}}\ensuremath{-}l\ensuremath{\le}m\ensuremath{\le}l).$ In comparison to measurements, the electronic correlations in ${\ensuremath{\varphi}}_{i}$ greatly improve the overall performance of the BCIS-4B method around the Massey peak, below about 100 keV. Moreover, while largely outperforming the three-body boundary-corrected continuum intermediate state method, the cross sections in the BCIS-4B with the correlated ${\ensuremath{\varphi}}_{i}$ compare excellently overall with the available experimental data at 10 to 12 500 keV. Hence, the BCIS-4B method, with its built-in two main capture mechanisms (one-step Massey and two-step Thomas) is capable of spanning impact energies covering three or more orders of magnitude at which the state-summed cross sections vary over 11 orders of magnitude.

Transfer from Lunar Gateway to Sun-Earth Halo Orbits Using Solar Sails

To extend the usability of solar sails in the sun–Earth–moon system, we analyze the transfer trajectories from the 9:2 Earth–moon near-rectilinear halo orbit (NRHO) to halo orbits around the sun–Earth L1 and L2 points under the assumption of a future mission for a solar sail spacecraft equipped with a solar electric propulsion (SEP) system deployed from the Lunar Orbital Platform-Gateway. The dynamics are modeled using the bicircular restricted four-body problem, where the gravitational forces from the sun, Earth, and moon as well as solar radiation pressure (SRP) are considered. We propose a trajectory design method that utilizes both SRP and SEP. The method consists of initial guess generation and optimization steps. The initial guess generation comprises the forward propagation of the escape trajectory from the NRHO, the backward propagation of the stable manifold of the target halo orbits, and their apoapsis patching process. Optimization is conducted to minimize propellant consumption by effectively controlling SRP. We perform optimizations with various parameters, namely, the sail area-to-mass ratio (), specifications of SEP, target sun–Earth halo orbit, and departure direction. The results validate the proposed trajectory design method and verify that solar sail acceleration can reduce the necessary amount of propellant, which indicates that such missions can be realized by small CubeSats.

Secular orbital dynamics of the innermost exoplanet of the upsilon -Andromedæ system

We introduce a quasi-periodic restricted Hamiltonian to describe the secular motion of a small-mass planet in a multi-planetary system. In particular, we refer to the motion of upsilon -And textbf{b} which is the innermost planet among those discovered in the extrasolar system orbiting around the upsilon -Andromedæ A star. We preassign the orbits of the Super-Jupiter exoplanets upsilon -And textbf{c} and upsilon -And textbf{d} in a stable configuration. The Fourier decompositions of their secular motions are reconstructed by using the well-known technique of the (so-called) frequency analysis and are injected in the equations describing the orbital dynamics of upsilon -And textbf{b} under the gravitational effects exerted by those two external exoplanets (that are expected to be major ones in such an extrasolar system). Therefore, we end up with a Hamiltonian model having 2+3/2 degrees of freedom; its validity is confirmed by the comparison with several numerical integrations of the complete four-body problem. Furthermore, the model is enriched by taking into account also the effects due to the relativistic corrections on the secular motion of the innermost exoplanet. We focus on the problem of the stability of upsilon -And textbf{b} as a function of the parameters that mostly impact on its orbit, that are the initial values of its inclination and the longitude of its node (as they are measured with respect to the plane of the sky). In particular, we study the evolution of its eccentricity, which is crucial to exclude orbital configurations with high probability of (quasi)collision with the central star in the long-time evolution of the system. Moreover, we also introduce a normal form approach, that is based on the complete average of our restricted model with respect to the angles describing the secular motions of the major exoplanets. Therefore, our Hamiltonian model is further reduced to a system with 2 degrees of freedom, which is integrable because it admits a constant of motion that is related to the total angular momentum. This allows us to very quickly preselect the domains of stability for upsilon -And textbf{b}, with respect to the set of the initial orbital configurations that are compatible with the observations.

Geometric Analysis of Sun-Assisted Lunar Transfer Trajectories in the Planar Bicircular Four-Body Model

This research presents a geometric analysis of Sun-assisted low-energy lunar transfers and several convenient tools that enable the systematic trajectory design in the framework of the planar bicircular restricted four-body problem. By analogy with the patched conic approximation approach for high-energy transfers, a Sun-assisted low-energy trajectory is divided into three legs. Two interior legs, departing and arriving, are located inside the Earth–Moon region of prevalence and designed in the Earth–Moon circular restricted three-body problem, whereas the exterior leg lies outside the region of prevalence and is calculated in the Earth–Moon–Sun bicircular restricted four-body model. The whole trajectory is obtained by smoothly patching the three legs on the boundary of the region of prevalence. The arrival conditions are met by targeting a specific point in the L2 lunar gateway. The interior legs are easily adjustable to the four-body dynamics. The database of planar lunar transfer trajectories can be used to select an initial guess for the multiple-shooting procedure of designing a three-dimensional Sun-assisted lunar transfer in high-fidelity dynamical models.

Effect of Stokes drag in the restricted four-body problem with variable mass

The aim of this paper is to study the existence, locations and stability of the equilibrium points as well as zero velocity curves (ZVCs) under the effect of dissipative force (Stokes drag) in the restricted four-body problem (R4BP) with a variable mass. We have studied the dynamics of the infinitesimal body whose mass is variable and changing according to Jeans’ law. We have determined the equations of motion of the infinitesimal body by using the space time transformation of Meshcherskii when the change in mass is non-isotropic. We have determined the locations of equilibrium points in the special case of non-isotropic variation of mass under the effect of the dissipative force and found that all the equilibrium points are non-collinear. The effect of parameter k(0<k<1), which controls the dissipative force, on the locations of the libration points and their stability are investigated. It has been observed that the energy integral is time dependent due to the presence of the Stokes drag force. Further, it has been observed that all the equilibrium points are unstable for all values of the parameters used.

OPTIMAL TWO-IMPULSE INTERPLANETARY TRAJECTORIES: EARTH-VENUS AND EARTH-MARS MISSIONS

This work describes several models to design optimal interplanetary trajectories. The transfer problem consists in transferring a space vehicle from a circular low Earth orbit (LEO) to a circular low orbit around a destiny planet (Venus or Mars). Models based on the two-body, four-body, and five-body problems are considered. Also, several versions of the patched-conic approximation are utilized including a detailed version that designs a lunar swing-by maneuver. The results show that the optimal trajectories for Earth-Mars and Earth-Venus missions collide with the Moon if a lunar swing-by maneuver with an unspecified altitude of the closest approach is included in the trajectory design; however, sub-optimal trajectories that do not collide with the Moon exist, presenting a smaller fuel consumption than the trajectories without lunar swing-by and with no greater changes in the time of flight.

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.

Finding regions of bounded motion in binary asteroid environment using Lagrangian descriptors

Trajectory design in highly-perturbed environments like binary asteroids is challenging. It typically requires using realistic, non-autonomous dynamical models in which periodic solutions derived in autonomous systems vanish. In this work, Lagrangian descriptors are employed in the perturbed planar bi-elliptic restricted four-body problem to find regions of bounded motion over a finite horizon about Dimorphos, the secondary body of the (65803) Didymos binary system. Results show that Lagrangian descriptors successfully reveal phase space organizing structures both in the unperturbed and perturbed planar bi-elliptic restricted four-body problem. With no solar radiation pressure, regions of bounded motion are visually identified, so granting access to a vast selection of bounded orbits about Dimorphos. Conversely, the presence of solar radiation pressure breaks down the majority of structures, leading to a large region of unstable motion with rare exceptions. Lagrangian descriptors are computationally inexpensive dynamical indicators that could be conveniently applied to astrodynamics.

A Comparative study on generalized Manev potential and Newtonian potential in perturbed restricted four-body problem

A Comparative study on generalized Manev potential and Newtonian potential in perturbed restricted four-body problem

Equilibrium points and their stability in a new generalized planar version of the collinear restricted four-body problem

We introduce a new generalized planar version of the collinear restricted four-body problem. In this version, all three main bodies are located on the horizontal x-axis, while they have unequal masses. The dynamical system contains two free parameters related to the masses and positions of the main bodies. We investigate the equilibrium dynamics of the system by determining how these two free parameters affect the dynamical properties of the equilibria, such as their position and linear stability. By deploying some standard numerical methods combined with rigorous and systematic analysis, we studied the equilibrium dynamics of the system and revealed its properties regarding the nature of the libration points and the connections with the parameters of the system.

Design of a Low-Energy Earth-Moon Flight Trajectory Using a Planar Auxiliary Problem

The paper presents a sufficiently simple technique for designing a low-energy flight trajectory of a spacecraft (SC) from the Earth to the Moon with insertion into a low circular orbit of the latter. The proposed technique is based on the solution and subsequent analysis of a special model problem, which is a variant of the restricted circular four-body problem (RC4BP) Earth-Sun-SC-Moon; for which it is assumed that the planes of the orbits of all considered bodies coincides. The planar motion of the center of mass of the SC relative to the Earth is considered as perturbed (Sun, Moon). To describe it, equations in osculating elements are used, obtained by using the method of variation of constants based on the analytical solution of the planar circular restricted problem of two bodies (RC2BP)—Earth-SC, for which the rotation of the main axes of the coordinate system (the main plane) is synchronized with the motion of the Sun. The trajectory problem of designing a SC flight from a low circular near-Earth orbit to a low circular selenocentric one (“full” motion model—a restricted four-body problem (R4BP), an ephemeris model) is considered as an optimization one in the impulse formulation. The solution of the main problem is carried out in few (three) stages, on each the appropriate solution of the current variant of the auxiliary problem is determined, which is subsequently used as the basis of the initial approximation to the main one.

Regularization of the Hill four-body problem with oblate bodies

We consider the Hill four-body problem where three oblate, massive bodies form a relative equilibrium triangular configuration, and the 4th, infinitesimal body orbits in a neighborhood of the smallest of the three massive bodies. We regularize collisions between the infinitesimal body and the smallest massive body, via McGehee coordinate transformation. We describe the corresponding collision manifold and show that it undergoes a bifurcation when the oblateness coefficient of the smallest massive body passes through the zero value.

Action minimizing orbits in the trapezoidal four body problem

&lt;abstract&gt;&lt;p&gt;In this paper, we study the minimizing property for the isosceles trapezoid solutions of the four-body problem. We prove that the minimizers of the action functional restricted to homographic solutions are the Keplerian elliptical solutions, and this functional has a minimum equal to $ \frac{3}{2}(2\pi)^{2/3}T^{1/3}\left(\frac{\xi (a, b)}{\eta (a, b)}\right) ^{2/3} $. Further, we investigate the dynamical behavior in the trapezoidal four-body problem using the Poincaré surface of section method.&lt;/p&gt;&lt;/abstract&gt;

Nonlinear Orbital Stability of Periodic Motions in the Planar Restricted Four-Body Problem

In this work we study the nonlinear orbital stability problem for periodic motions emanating from the stable relative equilibrium. To describe motions of the small body in a neighborhood of its periodic orbit, we introduce the so-called local variables. Then we reduce the orbital stability problem to the stability problem of a stationary point of symplectic mapping generated by the system phase flow on the energy level corresponding to the unperturbed periodic motion. This allows rigorous conclusions to be drawn on orbital stability for both the nonresonant and the resonant cases. We apply this method to investigate orbital stability in the case of third- and fourth-order resonances as well as in the nonresonant case. The results of the study are presented in the form of a stability diagram.

Effects of Albedo and Oblateness in the Bi-Circular Restricted Four-Body Problem

In this paper, a bicircular model describing motion of a spacecraft around the Sun–Earth–Moon system is studied. The impact of important characterizations such as radiation from the Sun, the Earth’s and Moon’s Albedo and oblateness of the Moon are all included and analysed​ in this work. This quantitative investigation shows how significant the reflective property of the Earth and its Moon affect the existence of the system’s libration points as well as the stability of motion of the spacecraft about the libration points. The Albedo phenomenon is seen to have dominant effect on the location of the libration points of the spacecraft than the perturbation due to oblateness. It is revealed by the Poincare Surface of Section that the system exhibits chaotic behaviour as it is sensitive to change in initial conditions.

The study of periodic orbits in the spatial collinear restricted four-body problem with non-spherical primaries

The study of periodic orbits in the spatial collinear restricted four-body problem with non-spherical primaries