Resonance transitions associated to weak capture in the restricted three-body problem

An interesting dynamics is studied in the restricted three-body problem where a particle abruptly transitions between resonance states, called a resonance hop. It occurs in a region about the secondary mass point which supports weak capture. This region, called a weak stability boundary, was recently proven to give rise to chaotic dynamics. Although it was numerically known that the resonance hop was associated with this boundary, this process was not well understood. In addition, the dynamical structure of the weak stability boundary has not been well understood. In this paper, we give a way to reveal the global structure of the weak stability boundary associated to resonance motions. This structure is shown to be surprisingly rich in resonant periodic motions interconnected by invariant manifolds. In this case, nearly all the motions are approximately resonant in nature where resonance hops can occur. The correlation dimension of orbits undergoing resonant motions, associated to the weak stability boundary, is also examined. The dynamics analyzed in the present paper is related to that studied by J. Marsden et al. under the perspective of Lyapunov orbits and the associated invariant manifolds. Applications are discussed.     

Resonant quasi-periodic near-rectilinear Halo orbits in the Elliptic-Circular Earth-Moon-Sun Problem

Motivated by the near-future re-exploration of the cislunar space, this paper investigates dynamical substitutes of the Earth-Moon’s resonant Near-Rectilinear Halo Orbits (NRHOs) under the Elliptic-Circular Restricted Four-Body Problem formulation of the Earth-Moon-Sun system. This model considers that the Earth and Moon move in elliptical orbits about each other and that a third body, the Sun, moves in a circular orbit about the Earth-Moon barycenter. By making use of this higher-fidelity dynamical model, we are able to incorporate the Sun’s influence and the Moon’s eccentricity, two of the most significant perturbations of the cislunar environment. As a result of these perturbations, resonant periodic NRHOs of the Earth-Moon Circular Restricted Three-Body Problem (CR3BP) are hereby replaced by two-dimensional quasi-periodic tori that better represent the dynamical evolution of satellites near the vicinity of the Moon. We present the steps and algorithms needed to compute these dynamical structures in the Elliptic-Circular model and subsequently assess their utility for spacecraft missions. We focus on the planned orbit for the NASA-led Lunar Gateway mission, a 9:2 synodic resonant L₂ southern NRHO, as well as on the 4:1 synodic and 4:1 sidereal resonances, due to the proximity to the nominal orbit and their advantageous dynamical properties. We verify that the dynamical equivalents of these orbits preserve key dynamical attributes such as eclipse avoidance and near-linear stability. Furthermore, we find that the higher dimensionality of quasi-periodic solutions offers interesting alternatives to mission designers in terms of phasing maneuvers and low-altitude scientific observations.     

地月空间的远距离逆行轨道族及其分岔研究

地月系统中存在着一类绕月逆行、高度稳定的轨道族,称为远距离逆行轨道族(DRO)。以圆型限制性三体问题(CR3BP)为动力学模型研究了DRO轨道族周边的动力系统结构。利用Broucke稳定性图寻找分叉点,判断分叉类型,基于数值延拓计算分岔后产生的一系列新轨道分支。分叉类型主要有切分叉与多倍周期分叉(从3倍周期开始),轨道维度包含平面轨道族与三维轨道族。计算新轨道族的特征,包括形状、周期、能量、稳定性、双曲流形结构等。探讨周期轨道的轨道周期与能量的关系,以几何化的方式展现分叉结构、多周期轨道的双曲流形结构等。该动力结构将为基于DRO轨道族的地月空间任务提供重要的理论支持。      

Four-body cislunar quasi-periodic orbits and their application to ballistic lunar transfer design

Examining the properties of quasi-periodic orbits provides insight into the Sun-perturbed environment in cislunar space. In this investigation, quasi-periodic trajectories and their properties are explored in the Sun-Earth-Moon four-body problem. Computation and the stability characteristics of families of invariant tori are detailed. Furthermore, this investigation offers a framework for construction of ballistic lunar transfer trajectories in the four-body problem. The framework leverages manifold trajectories to supply a set of initial conditions for construction of periapsis Poincaré maps. Periapsis maps reduce the dimensionality of the space and illuminate solutions of interest as a basis to produce feasible families of transfers. Through a continuation process, families of ballistic transfers and families of transfers that include powered lunar flybys are constructed. Ultimately, these solutions supply an initial guess for transition to the Sun-Earth-Moon ephemeris model.     

Low-thrust transfer to Backflip orbits

The aim of the work is to design a low-thrust transfer from a Low Earth Orbit to a “useful” periodic orbit in the Earth–Moon Circular Restricted Three Body Model (CR3BP). A useful periodic orbit is here intended as one that moves both in the Earth–Moon plane and out of this plane without any requirements of propellant mass. This is achieved by exploiting a particular class of periodic orbits named Backflip orbits, enabled by the CR3BP. The unique characteristics of this class of periodic solutions allow the design of an almost planar transfer from a geocentric orbit and the use of the Backflip intrinsic characteristics to explore the geospace out of the Earth–Moon plane. The main advantage of this approach is that periodic plane changes can be obtained by performing an almost planar transfer. In order to save propellant mass, so as to increase the scientific payload of the mission, a low-powered transfer is considered. This foresees a thrusting phase to gain energy from a departing circular geocentric orbit and a second thrusting phase to match the state of the target Backflip orbit, separated by an intermediate ballistic phase. This results in a combined application of a low-thrust manoeuvre and of a periodical solution in the CR3BP to realize a new class of missions to explore the Earth–Moon neighbourhoods in a quite inexpensive way. In addition, a low-thrust transit between two different Backflip orbits is analyzed and considered as a possible extension of the proposed mission. Thus, also a Backflip-to-Backflip transfer is addressed where a low-powered probe is able to experience periodic excursions above and below the Earth–Moon plane only performing almost planar and very short transfers.     

Charged particle dynamics in the combined field of two rotating magnetic dipoles: A numerical investigation of the parametric effect on the equatorial motions

In this contribution we study some aspects of the dynamics of a charged particle in the field produced by two magnetic dipoles that rotate in circular orbits around their common center of mass under their mutual Newtonian attraction. More precisely, we numerically investigate the evolution of the regions of the particle motion on the equatorial plane of each dipole and we explore their parametric variation. We focus our investigation on those problem parameters that are related with the magnetic fields and the masses of the primaries. The obtained preliminary results show that for each particular set of the above parameters and for certain values of energy these regions are bounded, and therefore the motion of the particle is confined inside them.     

Periodic attitude motions along planar orbits in the elliptic restricted three-body problem

A way to improve the accuracy of the three-body problem model is taking into account the eccentricity of primary attractors. Elliptic Restricted Three-Body Problem (ER3BP) is a model for studying spacecraft trajectory within the three-body problem such that the orbital eccentricity of primaries is reflected in it. As the principal cause of perturbation in the employed dynamical model, the primaries eccentricity changes the structure of orbits compared to the ideal Circular Restricted Three-Body Problem (CR3BP). It also changes the attitude behavior of a spacecraft revolving along periodic orbits in this regime. In this paper, the coupled orbit-attitude dynamics of a spacecraft in the ER3BP are exploited to find precise periodic solutions as the spacecraft is considered to be in planar orbits around Lagrangian points and Distant Retrograde Orbits (DRO). Periodic solutions are repetitious behaviors in which spacecraft whole dynamics are repeated periodically, these periodic behaviors are the main interest of this study because they are beneficial for future mission designs and allow delineation of the system’s governing dynamics. Previous studies laid the foundation for spacecraft stability analysis or studying pitch motion of spacecraft in the ER3BP regime. While in this paper, at first, initial guesses for correction algorithms were derived through verified search methods, then correction algorithms were used to refine calculated orbit-attitude periodic behaviors. Periodic orbits and full periodic solutions are portrayed and compared to previous studies and simpler models. Natural periodic solutions are valuable information eventuate in the longer functional lifetime of spacecraft. Since the problem assumption considered in this paper is much closer to real mission conditions, these results may be the means to use natural bounded motions in the actual operational environment.     

小天体环的轨道动力学

随着天文观测的深入,发现不仅土星、木星等气态巨行星环绕有光环,某些小天体也环绕有光环。针对目前发现环的半人马型小天体女凯龙星（10199 Chariklo）,研究其引力场中粒子（视为质点）的轨道动力学,分析赤道椭率和小天体自转对环中粒子轨道运动的影响,通过庞加莱截面中的KAM环迭代,得到第一类周期轨道和1∶3共振周期轨道,并通过对比和分析得到小天体光环位置与小天体自转平运动共振的关系。研究结果表明:女凯龙星内环中的粒子最可能处在满足径向振荡幅值范围的第一类周期轨道及其附近的准周期轨道上,但不能排除处在1∶3共振周期轨道及其附近的准周期轨道上;外环中的粒子不可能处在1∶3共振周期轨道上,只可能处在第一类周期轨道及其附近的准周期轨道上。      

Dynamical evolution of high area-to-mass ratio objects in Molniya orbits

For special demands, some notable orbit types have been developed by human, including the Molniya orbits, which have a relatively high eccentricity up to about 0.7, and a period of 12 h. Considering that space debris with high area-to-mass ratio (A/M) has been discovered, such objects may also exist in Molniya orbits due to spacecraft and upper stages fragmentation events. However, there are not sufficient studies of the complex dynamical phenomena of such orbits. These studies can enrich the knowledge about the long-term evolution of these orbits, be helpful to propose uncatalogued objects observation and identification, and also set the protected region as well as active debris removal. In this paper, the characteristics of 2:1 resonance of Molniya satellite orbits are studied. A large set of numerical simulations, including all the relevant perturbations, is carried out to further investigate the main characteristics, and special attention is payed to the dynamical evolution of objects with high A/M, particularly affected by the direct solar radiation pressure. The long-term dynamical evolution of orbital elements, as well as the dependency of lifetime on the A/M value, is discussed.     

Low altitude dose measurements from APEX, CRRES and DMSP

Dosimeter data taken on the APEX (1994–1996), CRRES (1990–1991) and DMSP (1984–1987) satellites have been used to study the low altitude (down to 350 km) radiation environment. Of special concern has been the inner edge of the inner radiation belt due to its steep gradient. We have constructed dose models of the inner edge of the belt from all three spacecraft and put them into a personal computer utility, called APEXRAD, that calculates dose for user-selected orbits. The variation of dose for low altitude, circular orbits is given as a function of altitude, inclination and particle type. Dose-depth curves show that shielding greater than 1/4 in Al is largely ineffectual for low altitude orbits. The contribution of outer zone electrons to low altitude dose is shown to be important only for thin shields and to have significant variation with magnetic activity and solar cycle.     

Multiobjective genetic optimization of Earth–Moon trajectories in the restricted four-body problem

In this paper, optimal trajectories of a spacecraft traveling from Earth to Moon using impulsive maneuvers (ΔV maneuvers) are investigated. The total flight time and the summation of impulsive maneuvers ΔV are the objective functions to be minimized. The main celestial bodies influencing the motion of the spacecraft in this journey are Sun, Earth and Moon. Therefore, a three-dimensional restricted four-body problem (R4BP) model is utilized to represent the motion of the spacecraft in the gravitational field of these celestial bodies. The total ΔV of the maneuvers is minimized by eliminating the ΔV required for capturing the spacecraft by Moon. In this regard, only a mid-course impulsive maneuver is utilized for Moon ballistic capture. To achieve such trajectories, the optimization problem is parameterized with respect to the orbital elements of the ballistic capture orbits around Moon, the arrival date and a mid-course maneuver time. The equations of motion are solved backward in time with three impulsive maneuvers up to a specified low Earth parking orbit. The results show high potential and capability of this type of parameterization in finding several Pareto-optimal trajectories. Using the non-dominated sorting genetic algorithm with crowding distance sorting (NSGA-II) for the resulting multiobjective optimization problem, several trajectories are discovered. The resulting trajectories of the presented scheme permit alternative trade-off studies by designers incorporating higher level information and mission priorities.     

Phasing with near rectilinear Halo orbits: Design and comparison

In the next future, space agencies are planning to return to the Moon. The objective is to assemble an orbiting space station, called Gateway, on a Near Rectilinear Halo Orbit around the Moon as a base for future Moon and deep space missions. Within this framework, multiple side missions will be planned to sustain the Gateway (Artemis mission). The proposed work is thought in framework of the preliminary design of future cargo missions, in particular on the design of an efficient phasing trajectory, under the Circular Restricted Three body problem hypotheses, to bring a cargo vehicle from the end of the Earth-Moon transfer to the beginning of the proximity operations such as rendezvous and docking with the space station. The work aims covering the lack of literature in phasing trajectories with the NRHO by proposing three different strategies to connect the Earth-Moon transfer trajectory with the proximity operations. The three strategies are classified based on the choice of the parking orbits or the choice of the manifolds. Two strategies use butterfly and Halo orbits to park the vehicle before transferring to the target orbit. The third strategy, instead, uses manifolds to allow a direct phasing. In the paper, the three innovative strategies are designed and compare in a specific scenario.     

Semi-analytical investigations of high area-to-mass ratio geosynchronous space debris including Earth’s shadowing effects

This paper investigates the long-term perturbations of the orbits of geosynchronous space debris influenced by direct radiation pressure including the Earth’s shadowing effects. For this purpose, we propose an extension of our homemade semi-analytical theory [Valk, S., Lemaître, A., Deleflie, F. Semi-analytical theory of mean orbital motion for geosynchronous space debris under gravitational influence. Adv. Space Res., submitted for publication], based on the method developed by Aksnes [Aksnes, K. Short-period and long-period perturbations of a spherical satellite due to direct solar radiation. Celest. Mech. Dyn. Astron. 13, 89–104, 1976] and generalized into a more convenient non-singular formalism. The perturbations accounting for the direct radiation pressure with the Earth’s shadow are computed on a revolution-by-revolution basis, retaining the original osculating Hamiltonian disturbing function. In this framework, we compute the non-singular mean longitude at shadow entry and shadow exit at every orbital revolution in opposition to classical approaches where the singular eccentric anomalies at shadow entry and shadow exit are computed. This new algorithm is developed using non-singular variables. Consequently, it is particularly suitable for both near-circular and near-equatorial orbits as well as orbits which transit periodically around null eccentricities and null inclinations.The algorithm is tested by means of numerical integrations of the equations, averaged over the short periods, including radiation pressure, J₂, the combined Moon and Sun third body attraction as well as the long-term effects of the 1:1 resonance occurring for geosynchronous objects. As an extension of [Valk, S., Lemaître, A., Anselmo, L. Analytical and semi-analytical investigations of geosynchronous space debris with high area-to-mass ratios influenced by solar radiation pressure. Adv. Space Res., doi:10.1016/j.asr.2007.10.025, 2007b], we especially apply our analysis to space debris with area-to-mass as high as 20 m²/kg. This paper provides numerical and semi-analytical investigations leading to a deep understanding of the long-term evolution of the semi-major axis. Finally, these semi-analytical investigations are compared with accurate numerical integrations of the osculating equations of motion over time scales as high as 25 years.     

Analytic and numerical treatment of motion of dust grain particle around triangular equilibrium points with post-AGB binary star and disc

This paper investigates the motion around the triangular equilibrium points, of a passively gravitating dust particle in the gravitational field of a low-mass post-AGB binary system, surrounded by circumbinary disc. The two bodies of the binary are modeled as a triaxial star and a radiating-oblate star. Due to small deviation of disc stars on circular orbits, we have assumed that the Coriolis and centrifugal forces of the stars are slightly perturbed. The triangular equilibrium points of the particle are found. These points are defined by, triaxiality of the primary star, oblateness and radiation of the secondary one and the gravitational potential from the disc mass. Further, when the disc mass increases, the particle moves nearer to the stars and farther away from the disc. In general, these equilibrium points are linearly stable when μ < μ_C; where μ is the mass ratio and μ_C is the critical mass function, defined by the parameters of the system. The effects of each of these parameters on the size of the stability region are stated, and the periodic motion around the stable points is examined. It is seen that the orbits are ellipses, and the orientation, eccentricities, lengths of the semi-major and semi-minor axes are influenced by the parameters of the problem. In particular, for our numerical linear stability analysis, we have taken an extremely depleted pulsating star, IRAS 11472-0800 as the post-AGB triaxial star, with a weakly-radiating young white dwarf star; G29-38 as the secondary. For this system, the stability result of the triangular points comes out different. Here, μ_C < μ throughout the entire range of the mass ratio and the critical mass function. Hence, the triangular equilibrium points are unstable. The stability of the orbits is tested using the Poincaré surfaces of section (Pss). The region of stability is controlled by the introduced parameters and the Jacobi constant.     

DRO计算及其在地月系中的摄动力研究

针对远距逆行轨道（DRO）的航天工程应用问题，研究了DRO的计算方法以及轨道特性，分析了DRO在实际力环境中的主要摄动因素，为DRO的精确建模和标称轨道设计奠定一定的理论基础。首先，利用仿真算例验证流函数法在计算DRO周期轨道族中的有效性。然后，利用该方法，通过改变雅可比常数，延拓计算DRO周期轨道族，获得不同共振比的DRO，仿真结果表明整数共振比的DRO在地月惯性坐标系中的轨迹是封闭的曲线，而共振比非整数的DRO则不封闭。最后，通过轨道外推分析影响DRO稳定性的主要摄动因素，仿真结果表明太阳引力和月球轨道偏心率是影响DRO稳定性的主要摄动因素。在动力学模型中，使用标准星历表示行星的运动状态，当积分时间多于10天时模型误差为km量级，因此在地月系这样大尺度的空间范围内，可以使用星历模型近似的分析DRO在真实力环境中的运动状态，为任务轨道设计提供依据。      

Earth–Moon low energy trajectory optimization in the real system

The problem of the Earth–Moon low energy trajectory optimization in the real system (the model defined by the JPL ephemeris DE405) is considered in this paper. First, this problem is investigated in the model of circular restricted three-body problem, since the fuel consumption is closely related to the Jacobi integral of the transfer trajectory, a method based on Jacobi integral is proposed and eight optimal trajectories are obtained. These optimal trajectories provide initial information (the flight time and the braking velocity impulse) to search the optimal low energy trajectories in the real system through optimization techniques. Considering the merit and drawback of particle swarm optimization and differential evolution algorithm in solving the space trajectory problem, an improved cooperative evolutionary algorithm is put forward. Result shows that the low energy trajectories in the real system are more fuel-efficient than the corresponding ones under the circular restricted three-body problem.     

Semi-analytical theory of mean orbital motion for geosynchronous space debris under gravitational influence

This paper provides a hamiltonian formulation of the equations of motion of an artificial satellite or space debris orbiting the geostationary ring. This theory of order 1 has been formulated using canonical and non-singular elements for eccentricity and inclination. The analysis is based on an expansion in powers of the eccentricity and of the inclination. The theory accounts for the influence of the Earth gravity field expanded in spherical harmonics, paying a particular attention to the resonance occurring for geosynchronous objects. The luni-solar perturbations are also taken into account. We present the resonant motion and its main characteristics: equilibria, stability, fundamental frequencies and width of the resonant area by comparison with a basic analytical model. Finally, we show some results concerning the long term dynamics of a typical space debris under the influence of the gravitational field of the Earth and the luni-solar interactions.     

Effect of evaporation and solutocapillary-driven flow upon motion and resultant deposition of suspended particles in volatile droplet on solid substrate

Particle motion in a volatile droplet on a solid surface, especially the behavior of particles depositing in the vicinity of a solid–liquid–gas boundary line (contact line) is focused. This phenomenon is called the ‘coffee stain problem’. Motion and deposition of the particles suspended in distilled water droplets and distilled water–ethanol mixture droplets are discussed. The spatio-temporal particle motion is analyzed by the three-dimensional particle tracking velocimetry (3D PTV). A discussion of the morphology of the particles stuck to the solid surface after the dryout of the droplet is also given.     

常值径向推力下的飞行器运动轨迹

研究了初始停泊轨道为椭圆时,空间飞行器在常值径向推力下运动的有界性和周期性.首先建立了飞行器运动的动力学方程,并通过能量积分和角动量积分进行了简化.然后将有界性的研究转化为一个一元三次不等式的求解,并在此基础上针对不同的初始真近点角分别进行了研究,得到了运动的边界和有界性条件.接下来利用椭圆积分研究了运动的周期性,分别研究了径向运动、极角转动以及整体运动的周期性.最后用数值算法得到了运动的周期轨道.