Restricted Three-body System Research Articles

Statics and dynamics of a lunar anchored tethered system

Space tethered system has various attractive potential applications and now serves as promising lunar infrastructures hopefully. In this paper, statics and dynamics of a lunar anchored space tethered system in the restricted Earth-Moon three-body system will be studied to enable future potential applications. Nonlinear dynamics and potential applications of a space tether system extending from the lunar surface to the Earth’s side are investigated. The static equilibrium equations of the system with a distributed mass tether are established at the L1 side without considering small perturbations caused by eccentricity to analyze internal stress, detailed cross-sectional profiles, and counterweight mass. The elastic deformations are discretized using the assumed modes method, and the Lagrangian approach is used to derive the dynamical equations of the system with a viscoelastic tether considering the elastic elongation within the elliptical Earth-Moon restricted three-body system (EEMRTB). The oscillations of the tether as well as the distribution of stress and strain are addressed using numerical analysis considering both construction difficulty and power generation capacity, the ideal design parameter ranges are provided based on the nonlinear dynamic analysis. Additionally, a brief analysis of the accessible regions within the Earth-Moon system is also conducted. The validity, accuracy and advantage of the established dynamics are clarified.

Cosymplectic Geometry, Reductions, and Energy-Momentum Methods with Applications

Classical energy-momentum methods study the existence and stability properties of solutions of t-dependent Hamilton equations on symplectic manifolds whose evolution is given by their Hamiltonian Lie symmetries. The points of such solutions are called relative equilibrium points. This work devises a new cosymplectic energy-momentum method providing a new and more general framework to study t-dependent Hamilton equations. In fact, cosymplectic geometry allows for using more types of distinguished Lie symmetries (given by Hamiltonian, gradient, or evolution vector fields), relative equilibrium points, and reduction methods, than symplectic techniques. To make our work more self-contained and to fill some gaps in the literature, a review of the cosymplectic formalism and the cosymplectic Marsden–Weinstein reduction is included. Known and new types of relative equilibrium points are characterised and studied. Our methods remove technical conditions used in previous energy-momentum methods, like the Ad∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{Ad}^*$$\\end{document}-equivariance of momentum maps. Eigenfunctions of t-dependent Schrödinger equations are interpreted in terms of relative equilibrium points in cosymplectic manifolds. A new cosymplectic-to-symplectic reduction is developed and a new associated type of relative equilibrium points, the so-called gradient relative equilibrium points, are introduced and applied to study the Lagrange points and Hill spheres of a restricted circular three-body system by means of a not Hamiltonian Lie symmetry of the system.

Lagrange points and regionally conserved quantities

Lagrange points are the equilibrium points within a restricted three-body system, epitomized by the Trojan asteroids near the L4 and L5 points of the Sun–Jupiter system. They also play a crucial role in some space missions, including the James Webb Space Telescope which is located at the Sun–Earth L2 point. While the existence of five Lagrange points is a well-known feature of the restricted three-body problem, the equations describing the precise location of all five points are not extensively documented. This work presents a derivation of all Lagrange points using polar coordinates and a new normalization scheme that offers a simpler interpretation of solutions compared to prior analyses. A subtle issue concerning the treatment of angular momentum in the potential formulation of this problem is addressed and resolved. The supplementary material to this work contains additional mathematical details and discussion.

Dynamical Structures under Nonrestricted Hierarchical Planetary Systems with Different Mass Ratios

Secular dynamics have been extensively studied in both the inner and outer restricted hierarchical three-body systems. In the inner restricted problem, the quadrupole-order resonance (i.e., the well-known Kozai resonance) causes large coupled oscillations of eccentricity and inclination when the maximum inclination is higher than 39.2°, and the octupole-order resonance leads to the behavior of orbital flips. In the outer restricted problem, the behavior of orbital flips is due to the quadrupole-order resonance. Secular dynamics under the inner and outer restricted systems are distinctly different. The mass ratio of inner and outer bodies could change the ratio of circular orbital angular momenta β, which significantly influences dynamical structures of the system. But this influence is still unclear. In this paper, we focus on nonrestricted hierarchical planetary systems where β > 1 and investigate the secular dynamics by changing mass ratios. Dynamical structures are systematically explored from four aspects: periodic orbits, secular resonances, orbital flips, and chaos detection. We find that (a) it tends to lead to more bifurcations in the host family of prograde periodic orbits associated with Kozai resonance with smaller β; (b) with the decrease of β, fewer orbits inside the octupole-order resonance can realize flip; (c) for given initial conditions, the forbidden region appears in the retrograde region and becomes larger as β decreases, meaning that the mutual inclination cannot reach a very high value if β is small; and (d) chaotic orbits are distributed in the low-eccentricity, high-inclination region when β > 1.

Different effects of perturbations (solar sail) on The motion of the test particles in cr3bp

"The idea of solar sail is used to investigate the motion properties of the test particle which is varying its mass according to the Jeans law and moving under the gravitational forces of the primaries, the Coriolis and centrifugal forces in the circular restricted three-body system (CRTBS). The equations of motion of the test particle are determined under the above said perturbations. And hence, the dynamical properties like the locations of equilibrium points, their stability, periodic orbits, Poincare surfaces of section and basins of attracting domain are investigated. This problem will help those researchers who are interested in studying the problem related to solar sail for space Missions."

White dwarf–white dwarf collisions in AGN discs via close encounters

ABSTRACT White dwarfs (WDs) in active galactic nucleus (AGN) discs might migrate to the inner radii of the discs and form restricted three-body systems with two WDs moving around the central supermassive black hole (SMBH) in close orbits. These systems could be dynamically unstable, which can lead to very close encounters or direct collisions. In this work, we use N-body simulations to study the evolution of such systems with different initial orbital separation p, relative orbital inclination Δi, and SMBH mass M. It is found that close encounters of WDs occur mainly at $1.1R_{\rm H} \lesssim p \lesssim 2\sqrt{3}R_{\rm H}$, where RH is the mutual Hill radius. For p < 1.1RH, the majority of WDs move in horseshoe or tadpole orbits, and only few of them with small initial orbital phase difference undergo close encounters. For p = 3.0RH, WD–WD collisions occur in most of the samples within a time 105P1, and considerable collisions occur within a time t < 62P1 for small orbital radii, where P1 is the orbital period. The peak of the closest separation distribution increases and the WD–WD collision fraction decreases with an increase in relative inclination. The closest separation distribution is similar in cases with different SMBH mass, but the WD–WD collision fraction decreases as the mass of SMBHs increases. According to our estimation, the event rate of cosmic WD–WD collisions in AGN discs is about 300 Gpc−3 yr−1, roughly 1 per cent of that of observed Type Ia supernovae. The corresponding electromagnetic emission signals can be observed through large surveys of AGNs.

Conceptual Design and Mechanical Analysis of a Lunar Anchored Cislunar Tether

A long tether anchored to the lunar surface and extended to position tens thousands of kilometers above the Earth is put forward. It could be used as an auxiliary cislunar traffic track to transfer cargos along it to the Moon without rocket landing and ascending, saving fuels. The configuration and statics of the tether is examined carefully within the Earth–Moon circular restricted three-body system. Firstly, the tensile stress in the tether is regarded as a main restriction factor, which is calculated for the case with a constant cross section. By then, the allowable endpoint position is proposed. So far, only materials available in laboratory, such as carbon nanotube, can sustain such a long and thin tether. Secondly, the constant tensile stress tether with variant cross section is taken into account. Then the key restriction factor is becoming the variation ratio of the tether’s cross section, which is determined by the material’s density and the tensile stress. In this condition, available materials expand to carbon fiber and some mass-produced composite materials. As a verdict, such a lunar anchored cislunar tether is more than science fiction in the sense of mechanics and material.

Orbital Flips Caused by the Eccentric Von Zeipel–Lidov–Kozai Effect in Nonrestricted Hierarchical Planetary Systems

The eccentric von Zeipel–Lidov–Kozai effect is widely applied to diverse astrophysical settings. In the restricted hierarchical three-body system, the topic of orbital flips has been extensively studied. However, it is far from being understood for nonrestricted circumstances. In this work, the dynamics of orbital flips are investigated under the Hamiltonian model at the octupole-level approximation for nonrestricted hierarchical planetary systems, where the outer planet is more massive than the inner one. Numerical distribution of flipping orbits shows that there are three major flipping regions, distributed in low-, intermediate-, and high-eccentricity spaces. Poincaré sections indicate that those islands of libration centered near i tot = 90° can lead to orbital flips. Thus, we refer to the behavior of orbital flips as a resonance phenomenon. From this viewpoint, dynamical models of orbital flips can be described by a separable Hamiltonian, which can be treated by a means of perturbation theory. The resonant model for orbital flips is formulated based on the adiabatic invariant approximation and then phase portraits are generated by plotting level curves of adiabatic invariants with the given Hamiltonian. By analyzing phase portraits, analytical boundaries of libration and circulation zones causing orbital flips are obtained. As expected, the numerical and analytical conditions that allow orbits to flip agree well with each other. The phenomenon of orbital flips in nonrestricted hierarchical problems can be well understood with the help of dynamical structures of secular resonance.

Study of Lagrange Points in the Earth–Moon System with Continuation Fractional Potential

In this work, the restricted three-body system is studied in the framework of the continuation fractional potential with its application on the Earth–Moon system. With the help of a numerical technique, we obtained thirteen equilibrium points, such that nine of them are collinear while the remaining four are non-collinear points. We found that the collinear points near the smaller primary were shifted outward from the Moon, whereas the points near the bigger primary were shifted towards the Earth as the value of the continuation fractional parameter increased. We analyzed the zero-velocity curves and discussed the perturbation of the continuation fractional potential effect on the possible regions of the motion. We also discussed the linear stability of all the equilibrium points and found that out of thirteen only two were stable. Due to such a prevalence, the continuation fractional potential is a source of significant perturbation, which embodies the lack of sphericity of the body in the restricted three-body problem

Time dependent nature of the Neptune’s arc configuration

By distinguishing the main arc Fraternite from the minor arcs Egalite (2,1), Liberte, Courage, the conservative restricted three-body system is extended to a non-conservative restricted four-body system with the central body Neptune S, the primary body Galatea X, a minor body Fraternite F, and a test body s. Through the equations of motion, it is shown that the locations where the force is null (null points) along the orbit of s correspond to the locations of Egalite (2,1), Liberte, and Courage. Even if all the arcs were captured by three-body CER sites initially, the orbits over the CER potential maxima would be unstable for the minor arcs due to the disturbing force of Fraternite with finite mass, allowing them to be relocated to the four-body null points. On the other hand, the minor arcs do not have the mass to destabilize Fraternite from its three-body CER site, which is enlarged from 8.37 degrees to 9.7 degrees thus accounting for the Fraternite’s span. In this non-conservative restricted four-body system, s is under the effects of the potentials of X and F, therefore the null points are not absolutely stable as the Lagrangian points. The potential of X drives a long period harmonic pendulum oscillation through Φ s x = [ ( n + 1 ) θ s − n θ x − ϕ s ] centered over a four-body null point with T = 1 , 000 d , while the potential of F drives a much longer period singular pendulum oscillation through Δ θ s f centered over Fraternite with T f = 40 , 000 d . In this four-body system, the dynamics of these two oscillations generates a time alternating symmetric arc configuration about Fraternite with a century long time scale. The non-conservative nature of this system could account for the time varying intensity, configuration, and disappearance of the minor arcs.

Conflict and agreement in the exact solutions of Schrodinger equation in a two and restricted three-body systems

Using the undetermined coefficients method, we obtained the exact solution of Schrodinger equation of a three-body system, H2+ ion, in a restricted form. We found that its ground state energy when the nuclear distance becomes zero can be the same as that of a He+ ion. However, its radial wavefunction is e-r while the corresponding wavefunction is e-2r for a He+ ion, where r is the electron-nucleus distance. We tentatively discussed clues leading to the interesting difference in a hope to find exact solutions of Schrodinger equation for systems of three or more bodies.

Station-keeping of a solar weather detector by a tethered-sailcraft within elliptic Sun-Earth restricted three-body system

To detect and forecast solar weather in advance is of great importance for humankind to protect power infrastructure on ground and communication satellites in orbit from damaging. In this regard, a large space platform consisting of a solar weather detector, long space tether and a sailcraft with actively adjustable lightness number is constructed. Within the elliptic Sun-Earth restricted three-body system (ESERTBS), merely the motion of the platform along the Sun-Earth line is focused on, full nonlinear dimensional and dimensionless dynamics of the system is established with a massless, taut and rigid/flexible tether. A sliding mode controller is developed to make the closed-loop dynamics asymptotically stable and the prescribed constant dimensional distance between the Sun and detector is maintained thereby. The time histories of some critical variables of the closed-loop system are presented to verify the performance of the developed controller, to check the lightness number of the sailcraft required, to show the internal tension/strain within the tether etc. Moreover, the steady-state responses of the closed-loop dynamic system are also presented to give more information. The lightness number of a single sailcraft used for performing the mission is compared to the lightness number of the sailcraft of the proposed platform. Some discussions and suggestions for selecting the tether’s length and elasticity are presented from the aspect of the lightness number required, the internal tension and strain based on numerical simulations.

Bifurcation analysis and approximate analytical periodic solution of ER3BP with radiation and albedo effects

The nonlinear dynamics of an elliptic restricted three-body system with perturbations such as radiation and albedo are studied in this paper. The effects of different perturbation parameters on the nonlinear dynamic behavior of the third-body are analyzed through the system’s bifurcation diagram. The second- and third-order approximate periodic solutions near the collinear libration points in the plane are obtained using the Lindstedt-Poincare perturbation method.

Hierarchical global search for fuel-optimal impulsive transfers between lunar libration orbits

The attention to the periodic orbit in the Earth-Moon restricted three-body system continues to grow due to its special environment and locations. This research investigates the feasibility of constructing fuel-optimal single and multiple impulse transfers between unstable periodic orbits at L1 and L2 points. Invariant manifolds, which could provide the appropriate initial trajectories for optimization, are analyzed deeply to enable previously unknown orbit options and potentially to reduce mission cost. A global search strategy based on comparing the orbital state of the unstable and stable manifolds, incorporated with low-thrust techniques, is performed to seek a suitable matching point for maneuver application. Then the sequential quadratic programming (SQP) is adopted to further optimize the velocity increment and obtain the single/multiple impulse optimal transfers. The associated constraint gradients are derived to achieve higher accuracy and rapidity of the algorithm. To highlight the effectivity of the transfer scheme, three-dimensional low-energy transfers between different types and spatial regions of performing single and multiple impulses are explored. The total Delta-V required varies between a few meters per second and tens of meters per second, and the related flight time is about several weeks, mainly depending on the energy of periodic orbits and the invariant manifold structure. The results obtained in this paper can provide a useful reference for the selection of escape and capture site along the manifolds, maneuver magnitude and transfer time.

Explicit symplectic-like integration with corrected map for inseparable Hamiltonian

ABSTRACT Symplectic algorithms are widely used for long-term integration of astrophysical problems. However, this technique can only be easily constructed for separable Hamiltonian, as preserving the phase-space structure. Recently, for inseparable Hamiltonian, the fourth-order extended phase-space explicit symplectic-like methods have been developed by using the Yoshida’s triple product with a mid-point map, where the algorithm is more effective, stable and also more accurate, compared with the sequent permutations of momenta and position coordinates, especially for some chaotic case. However, it has been found that, for the cases such as with chaotic orbits of spinning compact binary or circular restricted three-body system, it may cause secular drift in energy error and even more the computation break down. To solve this problem, we have made further improvement on the mid-point map with a momentum-scaling correction, which turns out to behave more stably in long-term evolution and have smaller energy error than previous methods. In particular, it could obtain a comparable phase-space distance as computing from the eighth-order Runge–Kutta method with the same time-step.

Dynamics and potential applications of a lunar space tethered system

Nonlinear dynamics and potential applications of a lunar space tether system, with one end connected to the Moon's surface and the other with a large floating counterweight, are investigated in the paper. Dynamical equations of the system with a massless viscoelastic tether considering the planar libration and elastic elongation within the elliptical Earth-Moon restricted three-body (EEMRTB) system are established. The equilibria and stability of the dynamical system are explored for the system at L1 side without considering small perturbations caused by eccentricity to facilitate the rough selection of the length and elastic coefficient of the tether. The potential applications including harvesting the Rayleigh Damping Dissipation power and accessible regions within the Earth-Moon system will be studied based on the dynamic analysis of the full nonlinear dynamics. The allowed parameter regions (the tether system will neither collide the Moon/Earth nor rotate about the Moon) are identified exactly using numerical analysis, moreover, the performance of the potential applications and other related references for building a tether system are explored at L1 and L2 numerically. It is suggested that one should design a space tether based energy harvester at L1 side, release an unpowered spacecraft with high mechanical energy at L2 side respectively. Some conclusions on selecting the parameters and evaluating the performance of the space tether based platform are obtained based on numerical calculations. The research will be useful for the Moon infrastructure constructions.

The Lidov-Kozai Oscillation and Hugo von Zeipel

The so-called oscillation is very well known and applied to various problems in solar system dynamics. This mechanism makes the orbital inclination and eccentricity of the perturbed body in the circular restricted three-body system oscillate with a large amplitude under certain conditions. It is widely accepted that the theoretical framework of this phenomenon was established independently in the early 1960s by a Soviet Union dynamicist (Michail L'vovich Lidov) and by a Japanese celestial mechanist (Yoshihide Kozai). A large variety of studies has stemmed from the original works by Lidov and Kozai, now having the prefix of Lidov-Kozai or Kozai- Lidov. However, from a survey of past literature published in late nineteenth to early twentieth century, we have confirmed that there already existed a pioneering work using a similar analysis of this subject established in that period. This was accomplished by a Swedish astronomer, Edvard Hugo Zeipel. In this monograph, we first outline the basic framework of the circular restricted three-body problem including typical examples where the oscillation occurs. Then, we introduce what was discussed and learned along this line of studies from the early to mid-twentieth century by summarizing the major works of Lidov, Kozai, and relevant authors. Finally, we make a summary of Zeipel's work, and show that his achievements in the early twentieth century already comprehended most of the fundamental and necessary formulations that the oscillation requires. By comparing the works of Lidov, Kozai, and Zeipel, we assert that the prefix von Zeipel-Lidov-Kozai should be used for designating this theoretical framework, and not just or Kozai-Lidov.

Orbital evolution of a planet with tidal dissipation in a restricted three-body system

The angle between planetary spin and the normal direction of an orbital plane is supposed to reveal a range of information about the associated planetary formation and evolution. Since the orbit’s eccentricity and inclination oscillate periodically in a hierarchical triple body and tidal friction makes the spin parallel to the normal orientation of the orbital plane with a short timescale in an isolated binary system, we focus on the comprehensive effect of third body perturbation and tidal mechanism on the angle. Firstly, we extend the Hut tidal model (1981) to the general spatial case, adopting the equilibrium tide and weak friction hypothesis with constant delay time, which is suitable for arbitrary eccentricity and any angle ϑ between the planetary spin and normal orientation of the orbital plane. Furthermore, under the constraint of angular momentumconservation, the equations of orbital and ratational motion are given. Secondly, considering the coupled effects of tidal dissipation and third body perturbation, and adopting the quadrupole approximation as the third body perturbation effect, a comprehensivemodel is established by this work. Finally, we find that the ultimate evolution depends on the timescales of the third body and tidal friction. When the timescale of the third body is much shorter than that of tidal friction, the angle ϑ will oscillate for a long time, even over the whole evolution; when the timescale of the third body is observably larger than that of the tidal friction, the system may enter stable states, with the angle ϑ decaying to zero ultimately, and some cases may have a stable inclination beyond the critical value of Lidov-Kozai resonance. In addition, these dynamical evolutions depend on the initial values of the orbital elements and may aid in understanding the characteristics of the orbits of exoplanets.

Nonlinear Dynamics of a Space Tethered System in the Elliptic Earth-Moon Restricted Three-Body System

AbstractThis paper focuses on nonlinear oscillation of a space tether system connected to the Moon’s surface, elongating along the Earth-Moon line at L1 and L2 sides respectively. The full nonlinea...

Resonant space tethered system for lunar orbital energy harvesting

Using a space tether system attached to the Moon's surface can in principle lead to energy harvesting from the mechanical damping of the tether as it experiences elongational motion due to time-varying tidal forces. It is shown that such a tether system can in principle provide electricity generation for lunar infrastructure, although the power generated is modest relative to the scale of engineering required. First, the dynamics of the coupled planar elongation and librational motion is established for a massless, elastic and damped tether with a large tip-mass in the frame of the elliptic Earth-moon restricted three-body (EEMRTB) system. Equilibria at the natural L2 point are obtained as reference positions to perform the analysis. The method of multiple scales is then used to obtain the steady state amplitude-frequency response by ordering key variables and parameters appropriately. The steady-state power output determined by the elasticity, length and damping of the tether is presented. Specific resonances for peaks of power output, together with corresponding resonance regions, are also investigated along with suggestions for tradeoffs among the key system parameters. Finally, the optimal damping for maximum power output is determined, together with the corresponding natural length and elasticity of the tether.