Copyright © 2016 by Priscilla Sousa-Silva, Maisa O. Terra, Collin R. McInnes, and Matteo Ceriotti.The problem of finding optimal trajectories is essential for modern space mission design. In order to obtain low-cost solutions, the models employed in mission analysis have become more realistic, considering multi-body gravitational dynamics, enabling orbits that do not exist in two-body dynamics. Besides that, the missions have increased in complexity, exploiting both low-thrust and high-thrust, including alternative forms of propulsion such as solar sailing. In this context, sets of good initial guesses are fundamental for the convergence to local or global optimal solutions, using both direct or indirect methods available to solve the optimal control problem. Thus, this paper deals with producing trajectories in the patched three-body approach that are designed to be good initial guesses as input to search optimal low-energy short-time Earth-Moon transfers with ballistic capture. We introduce a more realistic modelling in which the restricted four-body system Sun-Earth-Moon-Spacecraft is decoupled in two patched Planar Circular Restricted Three-Body Problems (PCRTBP). Contrary to previous works that consider these two systems as coplanar, we take into account the inclination of the orbital plane of the Moon with respect to the ecliptic by considering that the PCRTBPs are tilted with respect to each other, with the line of the nodes of the orbit of the Moon being the intersection of the orbital planes of the pairs of primaries. First, we present a heuristic strategy to obtain ensembles of ballistic capture orbits around the Moon relying on the hyperbolic invariant structures associated to the Lagrangian points of the Earth-Moon system that fulfill specific mission requirements. Then, we propose a patching procedure between the two three-body systems that takes into account that the motion of the primaries is not coplanar, but allows to exploit the fundamental solutions of the planar dynamics as a starting point for the search of optimal fully refined three-dimensional transfer orbits. Finally, we exploit quasi-periodic orbits of the Sun-Earth system to produce good initial guesses for the departing stage aiming optimized solutions, with low, high, and hybrid thrust. Given that setup, we exploit the proposed algorithm in order to obtain a diverse set of good low-cost and short-time initial guesses for different strategies of thrust. An assessment of these solutions will be performed as a function of chosen relevant parameters.

Copyright © 2016 by Sullo, de Sousa-Silva, O. Terra and Ceriotti.The need to drive the Δv down has led, in recent times, to improve the trajectory models, and take into account the forces that were considered just unwanted perturbations in the past. One of the examples is the exploration and exploitation of multi-body dynamics, to replace the Keplerian two-body problem, from the very initial phases of the trajectory design. Multi-body dynamics allows to exploit families of trajectories that simply do not exist in the two-body problem, at the additional cost of considerably increased the complexity of the design problem. One example is in the Earth-Moon transfer trajectories, where high-energy Apollo-like transfers are possible but highly expensive, as opposed to solutions that make use of the gravitational attraction of the sun, such as weak stability boundary transfers, to reduce the Δv. This paper presents an optimization procedure to generate fast and low-Δv Earth-Moon transfer trajectories. Ideal (first-guess) trajectories are generated at first, using two coupled planar circular restricted three-body problems, one representing the Earth-Moon system, and one representing the Sun-Earth. The two systems are rotating with respect to each other and tilted around a common line of the nodes, to account for the obliquity of the Moon over the ecliptic. The trajectories consist of a first ballistic arc in the Sun-Earth system, and a second ballistic arc in the Earth-Moon system. The two are connected at a patching point on the line of the nodes at one end (with an instantaneous Δv), and they are bounded at Earth and Moon respectively at the other end. Families of these trajectories are found by means of an evolutionary optimization method, with patching Δv of the order of a hundred m/s and transfer time of about 10 days. Subsequently, they are used as first-guess for solving an optimal control problem. At this stage, the full-three dimensional problem is introduced and the patching point is set free, and realistic analytical ephemerides are used. The objective of the optimisation, carried out with pseudospectral methods and sequential quadratic programming, is to reduce the total Δv, and the time of flight, together with introducing the constraints of the considered propulsion technology. In this work, we will present a trade-off of different options, including conventional solar-electric low-thrust, chemical high-thrust, and also a hybridisation of the two, envisaging future spacecraft that can carry both systems. A trade-off of the different optimal trajectories and propulsion options will be shown.

Copyright © 2015 by the American Institute Federation of Aeronautics and Astronautics. Inc. All rights reserved.In this paper, we investigate the process of escape of trajectories in the planar circular restricted three-body problem with a scattering region around the small primary, considering several subsystems of the Solar System. Our main goal is to examine the properties of the escape basins and the fractal properties of their boundaries in systems with practical interest in Astrodynamics. Namely, we consider ten subsystems of the Solar System: Sun-Neptune, Sun-Saturn, Sun-Jupiter, Sun-Earth, Jupiter-Callisto, Saturn-Titan, Pluto-Charon, Earth-Moon, Jupiter-Europe, and Jupiter-Io. The process of escape was analyzed through of necks around of the collinear Lagragian points L\ and Z,2, considering also the leaking produced by collisions with the surface of the smaller primary. Three are the relevant parameters for this investigation, i.e., the mass parameter of the mathematical model, the scaled mean radius of the smaller primary, and the Jacobi Constant. Besides escape basin diagrams and Poincare sections, the spatial distribution of escape time values was investigated to elucidate the escape process. Finally, we briefly discuss the implications of the properties of the boundary basins in the context of modern space mission design and analysis of natural bodies.

Copyright © 2015 by the American Institute Federation of Aeronautics and Astronautics. Inc. All rights reserved.Near-Earth Objects are strategic assets to scientific exploration and a possible source of raw materials for future space endeavours. Also they may represent the closest threats to Earth. The growing awarness about the importance of such objects has boosted the interest in sample return missions, mitigation and deflection technologies and in the possibility of capturing and relocating Near-Earth asteroids to allow for scientific investigations of the formation of the Solar System and resource utilization purposes. In this paper, we explore the invariant structures which account for long term confinement in the Earth-Moon system in order to establish how long term capture can be achieved and maintained under the dynamics of the Circular Restricted Three-Body Problem.

© 2014, Springer Science+Business Media Dordrecht.The escape of trajectories of a spacecraft, or comet or asteroid in the presence of the Earth–Moon system is investigated in detail in the context of the planar circular restricted three-body problem, in a scattering region around the Moon. The escape through the necks around the collinear points L1 and L2 as well as the leaking produced by considering collisions with the Moon surface, taking the lunar mean radius into account, were considered. Given that different transport channels are available as a function of the Jacobi constant, four distinct escape regimes are analyzed. Besides the calculation of exit basins and of the spatial distribution of escape time, the qualitative dynamical investigation through Poincaré sections is performed in order to elucidate the escape process. Our analyses reveal the dependence of the properties of the considered escape basins with the energy, with a remarkable presence of fractal basin boundaries along all the escape regimes. Finally, we observe the plentiful presence of stickiness motion near stability islands which plays a remarkable role in the longest escape time behavior. The application of this analysis is important both in space mission design and study of natural systems, given that fractal boundaries are related with high sensitivity to initial conditions, implying in uncertainty between safe and unsafe solutions, as well as between escaping solutions that evolve to different phase space regions.

This paper concerns Earth-to-Moon transfers in the patched-three body approximation, in which the Sun-Earth-Moon-Spacecraft system is modeled by two coupled Restricted Three-Body Problems. The standard transfers in this approach are manifold guided solu-tions, connecting transit and non-transit orbits of each three-body system. These are low-energy solutions but require long transfer time, usually more than 100 days. We present alternative solutions in the Sun-Earth portion of the transfer that reduce total transfer time to about 10 days, while still providing lunar ballistic capture at arrival. Instead of connecting transit and non-transit orbits, these alternative transfer solutions connect a bi-parametric family of quasi-periodic orbits around the Earth in the Sun-Earth system with transit orbits of the Earth-Moon system. We investigate coupling possibilities in view of the Jacobi constant of the three-body systems and illustrate the alternative patching solutions. Due to the quasi-periodic nature of the orbits in the first part of the transfer, these alternative solutions provide new dynamical possibilities in the patched-three body approximation, even when the hyperbolic invariant manifolds of the coupled three-body systems do not intersect properly.

Copyright © 2014 by the authors.In this contribution we present evidences of diffusion of trajectories in the framework of the Spatial Restricted Three-Body Problem (SRTBP) and introduce a methodology to quantify and to examine the diffusion process. In the circular SRTBP, the center manifold of the equilibrium L3,W L3C is four-dimensional and contains vertical Lyapunov orbits, planar Lyapunov orbits, two-dimensional invariant tori, other periodic orbits associated to resonances, and small chaotic zones. We compute the invariant structures inside W L3C, in particular, we obtain Fourier representations of invariant curves in a four-dimensional space, corresponding to Poincare sections of bidimensional invariant tori inside W L3C. Then, we report on the diffusion of trajectories with initial conditions very close to these invariant curves. We introduce a methodology for diffusion analysis which provides statistics of the process and shows that the diffusion rate of trajectories is not constant in the phase space. The diffusion rate increases as trajectories go away from the vertical periodic orbit and then decreases when they approach the planar periodic orbit, wandering across the hyperbolic manifolds of distinct tori. Besides the analysis of a large ensemble of trajectories, specific cases are also studied to illustrate the process. In some cases, stickiness to tridimensional tori is observed. Eventually, the trajectories escape due to approximation to the secondary. Finally, we show that the diffusion mechanism is associated to the existence of transition chains of heteroclinic connections and relate the rate of diffusion with the splitting of the stable and unstable hyperbolic manifolds of the two-dimensional invariant tori. Our analysis corresponds to the investigation for the mass ratio of the Saturn-Titan system, but our methodology can be extended to many other concrete applications in orbital dynamics in the Solar system.

Copyright © 2014 by the authors.This paper deals with Earth-to-Moon transfers in the patched three-body approach, in which the Sun-Earth-Moon-Spacecraft four-body system is approximated by two coupled Circular Restricted Three-Body Problems (CR3BP). This approach provides preliminary solutions that can be be numerically refined into full four-body solutions. The standard transfers in this approach are low-energy manifold guided solutions with long transfer time which connect transit and non-transit orbits of each three-body system. Besides the standard transit-non-transit connections, there are alternative solutions involving a bi-parametric family of quasi-periodic orbits around the Earth. These solutions connect quasi-periodic orbits on two-dimensional tori of the Sun-Earth-Spacecraft system with L<inf>1</inf> or L<inf>2</inf> transit solutions of the Earth-Moon-Spacecraft system to provide transfers with lunar ballistic capture and short flight time. We review the dynamical elements employed to obtain the different classes of transfers and give examples of solutions obtained from sets of initial conditions around the Earth consistent with current infrastructure for space exploration.

Copyright © 2014 by the authors.Heteroclinic connections (HC) play a significant role in the design of complex libration point missions, since they provide natural channels connecting motions that are otherwise unrelated. The Genesis and ARTEMIS missions, for example, have used trajectories that followed closely HC between L<inf>1</inf> and L<inf>2</inf> Lissajous orbits of the Sun-Earth and Earth-Moon systems, respectively. HC are computed by intersecting the stable (unstable) manifold of the arrival orbit with the unstable (stable) manifold of the departure orbit, usually looking for intersections in a strategic hypersurface. When the endpoint trajectories are quasi-periodic and non-planar, continuation strategies are essential to deal with the challenges introduced by the dimensionality of the involved structures. In this paper we elaborate a continuation strategy to compute HC between quasi-periodic solutions in the spatial circular restricted three-body problem. In particular, we show examples of the continuation of HC detected between Lissajous orbits around the L<inf>1</inf> and L<inf>2</inf> libration points of the Earth-Moon system. These results indicate that this strategy can be extensively used as a tool for the elaboration of a map of connections between quasi-periodic trajectories around L<inf>1</inf> and L<inf>2</inf>.

We consider the problem of stability around the triangular libration points in the Restricted Three-Body Problem. The "local stability" is decided thanks to KAM theory and Nekhorosev-like estimates, as it is well known. The question addressed in this note is which is the extend of the domains of "practical stability". © Springer-Verlag Berlin Heidelberg 2013.

This paper is devoted to verify the consistency of the algorithmic Weak Stability Boundary definition concerning the achievement of capture-escape detection, through examining the transitions produced by the implementation of this definition. Our main goal is to show that many types of spurious transitions concerning capture-escape behavior are found besides the expected transitions due to the separatrix role of the hyperbolic invariant manifolds of the central manifold of the collinear equilibria of the Planar Circular Restricted Three-Body Problem. We identify and characterize authentic and spurious transitions and discuss their spatial distribution along the boundary for sets of initial conditions with high eccentricity, showing the frequent occurrence of spurious transitions and of collisional trajectories. Also, we investigate smooth and fractal-like portions of the boundary. Finally, we propose an alternative stability boundary definition based on the effective detection of capture-escape transitions. © 2012 Springer Science+Business Media B.V.

The present contribution investigates the applicability of the associated initial condition sets generated by the algorithmic definition of the weak stability boundary in the lunar sphere of influence in the context of Earth-to-Moon low-energy capture transfers. First, we test the applicability of the initial condition sets in view of two mission strategies, namely, direct inner transfers in the Planar Circular Restricted Three-Body Problem and external transfers based in the Patched Three-Body approach. Then, we check the consistency of the stability classification regarding the selection of captured solutions, identifying the stable subsets which provide temporary capture solutions. Additionally, we show the diversity of behaviors classified as stable by the current stability criteria of the algorithmic definition, which indicates that the algorithmic criteria must be reviewed in order to guarantee that the stable solutions correspond solely to low-energy capture orbits. The combination of our analyses allows the detection of the subsets of stable initial conditions that provide feasible Earth-to-Moon transfer solutions. Specifically, we show an external low-energy Earth- to-Moon transfer with zero midcourse correction at the patching section. © 2012 Springer Science+Business Media B.V.

In this work, the resonance problem in the artificial satellites motion is studied. The development of the geopotential includes the zonal harmonics J 20 and J40 and the tesseral harmonics J22 and J42. Through an averaging procedure and successive Mathieu transformations, the order of dynamical system is reduced and the final system is solved by numerical integration. In the simplified dynamical model, three critical angles are studied. The half-width of the separatrix is calculated through a linearized model which describes the behavior of the dynamical system in a neighborhood of each critical angle. Through the resonance overlap criterion the possible regular and irregular motions are investigated by the time behavior of the semi-major axis, argument of perigee and eccentricity. The largest Lyapunov exponent is used as tool to verify the chaotic motion. © 2012 Elsevier Ltd.

The equilibria bifurcations of the biparametric version of the classical Pierce diode, a one-dimensional plasma-filled device, are analyzed in detail. Our investigation reveals that this spatiotemporal model is not structurally stable in relation to a second control parameter, the ratio of the plasma ion density to the injected electron beam density. For the first time, we relate the existence of one-fluid chaotic regions with specific biparametric equilibria bifurcations, identifying the restricted regions in the parametric plane where they occur. We show that the system presents several biparametric scenarios involving codimension-two transcritical bifurcations. Finally, we provide the spatial profile of the stable and unstable one-fluid equilibria in order to describe their metamorphoses. © 2011 American Institute of Physics.

In this contribution the weak stability boundary algorithmic definition was numerically accomplished with the inclusion of lunar and earth collisional sets and a subclassification of the unstable set. Then, the associated sets to WSB definition were analyzed and characterized according to relevant dynamical properties in order to clarify their applicability in earth-moon transfer orbit design. The obtained stable, unstable, and collisional sets are defined as a function of the osculating ellipse eccentricity for prograde and retrogade initial conditions. The stable sets, candidates to ballistic capture transfers, are subclassified according to chosen specific criteria, namely, the Jacobi constant intervals defined by distinct classes of Hill regions, the location of the final state after a complete cycle with respect to the Hill sphere, the permanence in the lunar sphere of influence in a full cycle around the moon, and exit basins for retrograde evolution. By the first time, with this investigation, elucidative criteria based on three-body problem elements are employed to identify initial condition subsets with required properties to design ballistic capture transfers. © 2010 IOP Publishing Ltd.

Electron beam transport through a quadrupole electrostatic system is investigated by particle-in-cell simulation in the present work, where - at the advantage of easier experimental implementation - the analogous parabolic electrostatic potential replaces the usual neutralizing ion background of bounded plasma systems. Looking at the maximum transported current and the dynamical behavior dependence on the electron beam injection energy, we have found that for a partial neutralizing electrostatic potential (i) the transmitted current significantly increases in relation to other electrostatic devices, due mainly to two-dimensional effects, (ii) the occurrence of stable static solutions with the typical profile of unstable static solutions of the classical Pierce diode, and (iii) a new bifurcation sequence of the steady-state solutions, at which periodic virtual cathode oscillations turn into intermittent spiking oscillations, which ultimately evolve to stable oscillations when increasing the input energy of the injected electron beam.

The second oscillatory window of the classical Pierce diode is explored by a particle-in-cell simulation, and for the first time, results that support the existence of another chaotic region are presented. The classical Pierce model consists of a one-dimensional plasma-filled diode into which a monoenergetic electron beam is injected. This system presents a rich dynamical behavior as a function of a single control parameter α (the electron transit angle) and has four different operating regimes: stable, oscillatory, chaotic and unstable oscillatory with virtual cathode formation. The second oscillatory region, revealed by linear approximation analysis, presents a similar dynamical behavior to the first one, studied by many works and located below α = 3π. By gradually diminishing a from 5π, we report here, through numerical experiments, the existence of a sequence of subharmonic bifurcations leading to chaos, periodic windows, and a crisis from which just unstable oscillations with virtual cathode can be observed.

The complex and chaotic dynamics of a one-dimensional plasma-filled diode is numerically explored by a particle-in-cell simulation. The system has just one control parameter, namely, the electron transit angle, and presents a complex behavior as a function of such a parameter. We report here the large diversity of nonlinear behaviors observed in our simulations when a virtual cathode formation occurs in the system. Particularly, among these interesting situations we stress those where virtual cathode discharges coexist with nonlinear oscillations.

We studied the effects of finite temperature in the magnetic susceptibility of a system of N non-interacting electrons in a homogeneous magnetic field and in a smooth confinement potential: the two-dimensional harmonic oscillator. Different exact ensemble calculations are considered and discussed: canonical (N fixed), canonical via grand-canonical (N average fixed) and partial canonical ensembles. We compute a Gaussian average of the susceptibility over the number of particles and another one over the size of system in order to compare our results with data of mesoscopic systems experiments. We conclude that it is fundamental to consider interactions between the particles in the theoretical analysis to obtain the temperature dependence of the experimental results. © 1998 Elsevier Science B.V. All rights reserved.

We study the magnetic susceptibility of an ensemble of noninteracting electrons confined by parabolic potentials and subjected to a perpendicular magnetic field at finite temperatures. We show that the behavior of the average susceptibility is qualitatively different from that of billiards. When averaged over the Fermi energy the susceptibility exhibits a large paramagnetic response only at certain special field values, corresponding to commensurate classical frequencies, being negligible elsewhere. We derive approximate analytical formulas for the susceptibility and compare the results with numerical calculations. © 1998 The American Physical Society.

In the present work we use nonrelativistic many body physics techniques to generalize the classical limit of quantum systems in such a way as to incorporate statistical mixtures. Finite temperature effects are thus incorporated in a natural way. We give a detailed account of the thermodynamics of theSU(3) Lipkin model and then derive the classical thermal (chaotic) dynamics of the system. The most remarkable features of our analysis are twofold: firstly the appearance of a new degree of freedom essentially connected to thermal effects, i.e., for high enough temperatures. Secondly we give a quantitative characterization of the temperature effects on the chaotic volume of the system. Thermal effects are shown to be responsible for novel nonlinear contributions to the dynamics and to consistently counterbalance the inter- action part of the dynamics. This is the case in the context both of thermodynamics and of the thermal dynamics and we believe it to be true in general. © 1998 Academic Press.

A variational approach is applied to the description of temperature effects in a N-particle exactly soluble schematic model, with SU(3) symmetry. The ground-state energy for different couplings and the strength distribution for a weak coupling versus temperature are compared with the exact results. For a weak coupling, or for a large enough N, the mean-field approximation agrees well with the exact values. For T ≠ 0, new RPA modes appear which did not exist at T = 0. © 1996 IOP Publishing Ltd.

We present a detailed analysis of the thermodynamical properties as well as thermal effects on the classical dynamics of the SU(2) Lipkin model. Particular attention is devoted to the temperature dependence of fixed points and bifurcation of equilibria. We find that qualitatively, temperature effects tend to counterbalance the effects of the two-body interaction.