Manifold dynamics in the Earth–Moon system via isomorphic mapping with application to spacecraft end-of-life strategies

Recently, manifold dynamics has assumed an increasing relevance for analysis and design of low-energy missions, both in the Earth–Moon system and in alternative multibody environments. With regard to lunar missions, exterior and interior transfers, based on the transit through the regions where the collinear libration points L₁ and L₂ are located, have been studied for a long time and some space missions have already taken advantage of the results of these studies. This paper is focused on the definition and use of a special isomorphic mapping for low-energy mission analysis. A convenient set of cylindrical coordinates is employed to describe the spacecraft dynamics (i.e. position and velocity), in the context of the circular restricted three-body problem, used to model the spacecraft motion in the Earth–Moon system. This isomorphic mapping of trajectories allows the identification and intuitive representation of periodic orbits and of the related invariant manifolds, which correspond to tubes that emanate from the curve associated with the periodic orbit. Heteroclinic connections, i.e. the trajectories that belong to both the stable and the unstable manifolds of two distinct periodic orbits, can be easily detected by means of this representation. This paper illustrates the use of isomorphic mapping for finding (a) periodic orbits, (b) heteroclinic connections between trajectories emanating from two Lyapunov orbits, the first at L₁, and the second at L₂, and (c) heteroclinic connections between trajectories emanating from the Lyapunov orbit at L₁ and from a particular unstable lunar orbit. Heteroclinic trajectories are asymptotic trajectories that travels at zero-propellant cost. In practical situations, a modest delta-v budget is required to perform transfers along the manifolds. This circumstance implies the possibility of performing complex missions, by combining different types of trajectory arcs belonging to the manifolds. This work studies also the possible application of manifold dynamics to defining suitable, convenient end-of-life strategies for spacecraft orbiting the Earth. Seven distinct options are identified, and lead to placing the spacecraft into the final disposal orbit, which is either (a) a lunar capture orbit, (b) a lunar impact trajectory, (c) a stable lunar periodic orbit, or (d) an outer orbit, never approaching the Earth or the Moon. Two remarkable properties that relate the velocity variations with the spacecraft energy are employed for the purpose of identifying the optimal locations, magnitudes, and directions of the velocity impulses needed to perform the seven transfer trajectories. The overall performance of each end-of-life strategy is evaluated in terms of time of flight and propellant budget.     

Control of satellite imaging formations in multi-body regimes

Libration point orbits may be ideal locations for satellite imaging formations. Therefore, control of these arrays in multi-body regimes is critical. A continuous feedback control algorithm is developed that maintains a formation of satellites in motion that is bounded relative to a halo orbit. This algorithm is derived based on the dynamic characteristics of the phase space near periodic orbits in the circular restricted three-body problem (CR3BP). By adjusting parameters of the control algorithm appropriately, satellites in the formation follow trajectories that are particularly advantageous to imaging arrays. Image reconstruction and coverage of the (u, v) plane are simulated for interferometric satellite configurations, demonstrating potential applications of the algorithm and the resulting motion.     

Applications of multi-body dynamical environments: The ARTEMIS transfer trajectory design

The application of forces in multi-body dynamical environments to permit the transfer of spacecraft from Earth orbit to Sun–Earth weak stability regions and then return to the Earth–Moon libration (L₁ and L₂) orbits has been successfully accomplished for the first time. This demonstrated that transfer is a positive step in the realization of a design process that can be used to transfer spacecraft with minimal Delta-V expenditures. Initialized using gravity assists to overcome fuel constraints; the ARTEMIS trajectory design has successfully placed two spacecrafts into Earth–Moon libration orbits by means of these applications.     

Single-impulse transfers from orbits of artificial satellites to orbits around <Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript> or <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript> libration points

In the context of the restricted circular three-body problem a method for constructing families of periodic orbits is described. Each orbit contains a segment of transfer from artificial satellite orbit of a smaller body to an orbit around L ₁ or L ₂ points of the Sun-Earth and Earth-Moon systems, a segment of multiple flyby of this libration point, and a segment of return to the artificial satellite orbit. Dependences of velocities at the pericenter on the pericenter radius are given.     

Symmetric periodic solutions of the Hill’s problem. II

An algorithm for studying the families of symmetric periodic orbits using their generating solutions, whose structure was presented in the first part of this paper [1], is described. The algorithm is essentially based on symmetry of the generating solution and on its initial approximation. More than 20 new families of symmetric periodic solutions of the Hill’s problem have been found and investigated with the use of this algorithm. The families including trajectories for orbital injection into the vicinity of collinear libration points L _1,2 are described.     

Collisions of stars by oscillating orbits of the second kind

Ordinary estimations of the number of star collisions in our galaxy—by simple kinematic considerations—lead to a very small number of such collisions: about one or even less every millions of years. However star collisions can occur through the following indirect way which has a much higher probability. (a) Binary stars are very common in our galaxy, about 30–50% of the stars. (b) If two binary stars meet a triple system can be formed by an ordinary exchange type motion. (c) A triple system is generally decomposed into the “inner orbit” (i.e. the relative orbit of the two nearest stars) and the “outer orbit” (i.e. the relative orbit of the third star with respect to the center of mass of the two nearest stars). The major axes of these two orbits have generally small perturbations and it is the same for the eccentricity of the outer orbit. On the contrary, if the relative inclination of the two orbits is large, the perturbations of the eccentricity of the inner orbit are important and can even in some cases lead to an eccentricity equal to one, that is to a collision of the two stars of the inner orbit.Such orbits can be called “oscillating orbits of the second kind”, indeed the first oscillating orbits—conceived by Khilmi and described for the first time in an example by Sitnikov—have unbounded mutual distances r_ij, but the system always come back to small sizes, it has an infinite number of very large expansions followed by strong contractions and, in the three-body case, an upper bound of lim inf (r_1.2 + r_1.3 + r_2.3) can be given in terms of the three masses and the integrals of motion. For the oscillating orbits of the second kind the mutual distances r_ij are bounded, but the velocities are unbounded (i.e. lim inf r_ij = 0 for at least one r_ij) and the system goes to a collision if the bodies have non-zero radius even small. The analytical study of the oscillating orbits of the second kind is a part of the general analytical study of the three-body problem, a part which must be valid for large eccentricities and large inclinations. The use of Delaunay's variables and of a Von Zeipel transformation lead to a first order integrable approximation, valid for any eccentricities and any inclinations, and giving the following results: (a) The oscillating orbits of the second kind occur when the angular momentum of the outer orbit has a modulus sufficiently close to the modulus of the total angular momentum of the three-body system. Hence these orbits occur for inclinations in the vicinity of 90°. (b) The oscillating orbits represent a set of positive measure of phase space and the first order study allows to give a rough estimation of the probability of collisions—even for stars of infinitely small radius. This probability, for given initial major axes and eccentricities and for isotropic arbitrary initial orientations, is generally of the order of m₃$\text{R}\text{M}$ (m₃ being the mass of the outer star, M the total mass and R the ratio of the period of the inner orbit to that of outer orbit).One question remains to be solved: how many collisions of stars are due to that phenomenon? That question is difficult because the probability of formation of a triple system by a random meeting of two binaries is very uneasy to estimate. However it seems that, compared to the usual evaluations based on pure kinematic considerations without gravitational effects, the number of collisions must be multiplied by a factor between one thousand and one million.     

Design of transfer trajectories between resonant orbits in the Earth–Moon restricted problem

The application of dynamical systems techniques to mission design has demonstrated that employing invariant manifolds and resonant flybys enables previously unknown trajectory options and potentially reduces the ΔV$\mathrm{\Delta }V$ requirements. In this investigation, planar and three-dimensional resonant orbits are analyzed and cataloged in the Earth–Moon system and the associated invariant manifold structures are computed and visualized with the aid of higher-dimensional Poincaré maps. The relationship between the manifold trajectories associated with multiple resonant orbits is explored through the maps with the objective of constructing resonant transfer arcs. As a result, planar and three-dimensional homoclinic- and heteroclinic-type trajectories between unstable periodic resonant orbits are identified in the Earth–Moon system. To further illustrate the applicability of 2D and 3D resonant orbits in preliminary trajectory design, planar transfers to the vicinity of L₅ and an out-of-plane transfer to a 3D periodic orbit, one that tours the entire Earth–Moon system, are constructed. The design process exploits the invariant manifolds associated with orbits in resonance with the Moon as transfer mechanisms.     

Definition and analysis of properties of optimal three-impulse point-to-orbit transfers under time constraint

This paper deals with energetically optimal multi-impulse transfers of a spacecraft in the central Newtonian gravitational field near a planet. The transfer from a point on initial orbit to the final orbit with the given angular momentum and energy constants is considered. The transfer time is bounded above.With the distance from spacecraft to planet limited and the time free, such parameters of given orbits are chosen that the 3-impulse apsidal transfer T_r is optimal with an intermediate impulse at the maximum distance. On the basis of necessary optimality conditions an algorithm is developed to numerically determine the desired optimal transfer trajectory T_t under time constraint, the apsidal trajectory T_r being taken as initial approach. From the geometry and energy viewpoints, both trajectories T_t and T_r are close to each other. The trajectory T_t is also 3-impulsive, all impulses on it are nonapsidal. The distance from the planet is larger and the sum of impulses is less for this trajectory than for the initial trajectory T_r with the same transfer time.The simplified solution of the problem is constructed producing good approximation to the exact numerical optimization results. The solution asymptotics is found when the transfer time tends to infinity.     

Stable spatial orbits around collinear libration points

A technique of generation of spatial periodic solutions to the restricted circular three-body problem from periodic orbits of the planar problem has been used for the families of orbits around collinear libration points L ₁ and L ₂. Developing the families obtained at the 1: 1 resonance, we have obtained stable solutions both in the Earth-Moon system and in the Sun-Earth system. Of course, the term “around the libration point” is rather conventional; the obtained orbits become more similar to the orbits around the smaller attracting body. The further development of the family of orbits “around” the libration point L ₂ in the Sun-Earth system made it possible to find the orbits satisfying the new, much more rigorous constraints on cooling the spacecraft of the Millimetron project.     

Calculation and Study of Limited Orbits around the <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript> Libration Point of the Sun–Earth System

A procedure has been proposed for calculating limited orbits around the L₂ libration points of the Sun–Earth system. The motion of a spacecraft in the vicinity of the libration point has been considered a superposition of three components, i.e., decreasing (stable), increasing (unstable), and limited. The proposed procedure makes it possible to correct the state vector of the spacecraft so as to neutralize the unstable component of the motion. Using this procedure, the calculation of orbits around various types of libration points has been carried out and the dependence on the orbit type on the initial conditions has been studied.     

Gravitation effect on a flux of sporadic micrometeoroids in the vicinity of near-Earth orbits

The new approach to gravitation effect determination in calculating the flux of sporadic micrometeoroids in the near-Earth space is proposed. The technique is based on integration of the equations of motion of sporadic micrometeoroids with accounting for bending their trajectories when particles are approaching the Earth. The technique and results of calculation of the gravitational focusing factor k_g for various conditions are presented. The feature of the proposed technique for calculating coefficient k_g consists in the fact that this coefficient does not explicitly depend on the values of particles velocity at the last point. The results of investigation of coefficient k_g have shown that, for the given initial velocity of micrometeoroids, the values of this coefficient depend on deflection of its direction from the direction to the Earth center. It is shown that for low-altitude orbits the flux density can increase up to 60%. The distribution of probabilities of various directions of particles flying to spacecraft structural elements is found to be non-uniform.     

Two-craft tether formation relative equilibria about circular orbits and libration points

The relative equilibria of a two spacecraft tether formation connected by line-of-sight elastic forces moving in the context of a restricted two-body system and a circularly restricted three-body system are investigated. For a two spacecraft formation moving in a central gravitational field, a common assumption is that the center of the circular orbit is located at the primary mass and the center of mass of the formation orbits around the primary in a great-circle orbit. The relative equilibrium is called great-circle if the center of mass of the formation moves on the plane with the center of the gravitational field residing on it; otherwise, it is called a nongreat-circle orbit. Previous research shows that nongreat-circle equilibria in low Earth orbits exhibit a deflection of about a degree from the great-circle equilibria when spacecraft with unequal masses are separated by 350 km. This paper studies these equilibria (radial, along-track and orbit-normal in circular Earth orbit and Earth–Moon Libration points) for a range of inter-craft distances and semi-major axes of the formation center of mass. In the context of a two-spacecraft Coulomb formation with separation distances on the order of dozens of meters, this paper shows that the equilibria deflections are negligible (less than 10^?6°) even for very heterogeneous mass distributions. Furthermore, the nongreat-circle equilibria conditions for a two spacecraft tether structure at the Lagrangian libration points are developed.     

Transfer orbits to/from the Lagrangian points in the restricted four-body problem

The well-known Lagrangian points that appear in the planar restricted three-body problem are very important for astronautical applications. They are five points of equilibrium in the equations of motion, what means that a particle located at one of those points with zero velocity will remain there indefinitely. The collinear points (L₁, L₂ and L₃) are always unstable and the triangular points (L₄ and L₅) are stable in the present case studied (Earth–Sun system). They are all very good points to locate a space-station, since they require a small amount of ΔV (and fuel), the control to be used, for station-keeping. The triangular points are especially good for this purpose, since they are stable equilibrium points.In this paper, the planar restricted four-body problem applied to the Sun–Earth–Moon–Spacecraft is combined with numerical integration and gradient methods to solve the two-point boundary value problem. This combination is applied to the search of families of transfer orbits between the Lagrangian points and the Earth, in the Earth–Sun system, with the minimum possible cost of the control used. So, the final goal of this paper is to find the magnitude of the two impulses to be applied in the spacecraft to complete the transfer: the first one when leaving/arriving at the Lagrangian point and the second one when arriving/living at the Earth.The dynamics given by the restricted four-body problem is used to obtain the trajectory of the spacecraft, but not the position of the equilibrium points. Their position is taken from the restricted three-body model. The goal to use this model is to evaluate the perturbation of the Sun in those important trajectories, in terms of fuel consumption and time of flight. The solutions will also show how to apply the impulses to accomplish the transfers under this force model.The results showed a large collection of transfers, and that there are initial conditions (position of the Sun with respect to the other bodies) where the force of the Sun can be used to reduce the cost of the transfers.     

Evolution of the out-of-plane amplitude for quasi-periodic trajectories in the Earth–Moon system

The out-of-plane amplitude along quasi-periodic trajectories in the Earth–Moon system is highly sensitive to perturbations in position and/or velocity as underscored recently by the ARTEMIS spacecraft. Controlling the evolution of the out-of-plane amplitude is non-trivial, but can be critical to satisfying mission requirements. The sensitivity of the out-of-plane amplitude evolution to perturbations due to lunar eccentricity, solar gravity, and solar radiation pressure is explored and a strategy for designing low-cost deterministic maneuvers to control the amplitude history is also examined. The method is sufficiently general and is applied to the L₁ quasi-periodic orbit that serves as a baseline for the ARTEMIS P2 trajectory.     

A method of calculation of vertical cutoff rigidity in the geomagnetic field

On the basis of generalization of the results of extensive trajectory calculations for trial charged particles moving in the geomagnetic field the method of calculation of effective vertical cutoff rigidity, taking into account the values of K _p -index and local time, is developed. The IGRF and Tsyganenko-89 models are used for the geomagnetic field. A comparison of the results of model simulations with the experimental data on penetration of charged particles into near-Earth space is made, and penetration functions for typical spacecraft orbits are calculated.     

Autonomous optical navigation for orbits around Earth–Moon collinear libration points

The analysis of optical navigation in an Earth–Moon libration point orbit is examined. Missions to libration points have been winning momentum during the last decades. Its unique characteristics make it suitable for a number of operational and scientific goals. Literature aimed to study dynamics, guidance and control of unstable orbits around collinear libration points is vast. In particular, several papers deal with the optimisation of the Δv budget associated to the station-keeping of these orbits. One of the results obtained in literature establishes the critical character of the Moon–Earth system in this aspect. The reason for this behaviour is twofold: high Δv cost and short optimal manoeuvre spacing. Optical autonomous navigation can address the issue of allowing a more flexible manoeuvre design. This technology has been selected to overcome similar difficulties in other critical scenarios. This paper analyses in detail this solution. A whole GNC system is defined to meet the requirements imposed by the unstable dynamic environment. Finally, a real simulation of a spacecraft following a halo orbit of the L2 Moon–Earth system is carried out to assess the actual capabilities of the optical navigation in this scenario.     

自由返回轨道的P平面参数快速设计方法

针对自由返回轨道求解过程中地心轨道类型变化造成的B平面参数方法计算失败问题,提出一种基于P平面参数的自由返回轨道快速设计方法。首先,基于轨道半通径参数的普适性,给出了不同轨道类型的P平面参数定义,建立了以P平面参数为求解目标量的自由返回轨道求解模型。其次,给出了基于P平面参数的自由返回轨道快速设计方法,在构建的瞬时地月惯性系下,以平面双二体自由返回轨道作为初值,实现了高精度力模型下的自由返回轨道快速求解。对8种构型自由返回轨道的设计结果表明,P平面参数具有类似于B平面参数的大收敛域,且有效解决了轨道类型变化对计算的影响,可直接应用于中国后续月球探测任务轨道设计。     

Missions to the asteroid anteros and the space of true anomalies

Three opportunities for missions to rendezvous ballistically with the Earth-crossing asteroid Anteros are studied to illustrate the requirements for a trip to a near-Earth minor planet. The rationale, sample payload, spacecraft requirements and trajectory characteristics of these opportunities are typical of a rendezvous mission to an accessible near-Earth object. Round trip ballistic trajectories to return small samples of the asteroid with launch dates between 1985 and 2000 are also presented. Contours of minimum total ΔV drawn in the space of launch and arrival true anomalies, given the designation Prime Rib curves, are introduced as a useful tool for mission design.     

Periodic solutions in three-dimensional restricted three-body problem: 2. Loss of symmetry at the 1:1 resonance

Within the framework of the circular restricted three-body problem a family of inverse periodic orbits around the two attracting bodies (the Egorov’s family) and families generated by it at the 1:1, 2:1, and 3:1 resonances for three-dimensional orbits in the Sun-Earth and Earth-Moon systems are considered. Their relationship with families generated by orbits around the libration points L ₁, L ₂ and L ₃ is investigated. One of the families contains periodic solutions that seem promising as possible orbits for the space radio telescope of the Millimetron project.