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Schematic diagram of the Lagrange Points of the Earth-Moon (LL 1 , …), and Sun-Earth Systems (EL 1 , …).
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The origin of the universe and of life itself have been central to human inquiries since the dawn of consciousness. To develop and use the technologies to answer these timeless and profound questions is the mission of NASA's Origins Program. The newly discovered "InterPlanetary Superhighway" (IPS) by Lo and Ross (1997, 2001) is a significant and co...
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... were discovered by Euler (L 1 , L 2 , L 3 ) and Lagrange (L 4 , L 5 ). Figure 2 shows schematically the Lagrange points of the Earth-Moon System and their geometric relationship with the Sun-Earth's L 1 and L 2 . For clarity, we will refer to the lunar Lagrange Points as LL 1 , etc., and the Earth Lagrange Points as EL 1 , etc. ...
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... comparing the resonances of Figure 12 with the eccen- tricity plot of Figure 11, we show that the 3:2 Hilda Re- sonance tends to collect asteroids because its nearly circular orbits do not cross the path of Jupiter. Where as at the 2:1 ...
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... Mueller (author of the Nemesis Star Theory) and Walter Alvarez [17] noted there is evidence that the asteroid which impacted the Earth and wiped out the dinosaurs may have followed a Genesis-like orbit. Figure 12. Poincaré Section of Jupiter's L 1 manifolds plotted in SEMIMAJOR AXIS vs. LONGITUDE OF PERIHELION reveals the resonance structures of the dynamics between Jupiter and Mars. ...
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... of course, is no accident. To understand how this works, we must go back to Figure 12, the Swiss cheese plot showing how the invariant manifolds wend their way through all of the resonances of the three body system. We must also recall our approach to simulating the zodiacal dust with PR drag in Section 6. Putting all this together we have the following picture. ...
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... we are working with the energy of the restricted three body problem, the Jacobi constant. At some point, it reaches an energy surface where there is chaotic motion much like the dynamics of Figure 12, the Swiss cheese diagram. The sensitivity of the chaos in this layer will move the trajectory great distances with little propulsion. ...
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... has stable and unstable manifold tubes winding on and off of itself just like the halo orbit. The intermingling of all these invariant manifolds is what creates the chaotic sea in Figure 12. The fact that we can use the Hamiltonian structure comes from the point of view we adopted to simulating the zodiacal dust. ...
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Microlensing is the only method that can detect and measure mass of wide orbit, low mass, solar system analog exoplanets. Mass measurements of such planets would yield massive science on planet formation, exoplanet demographics, free floating planets, planet frequencies towards the galaxy. High res follow-up observations of past microlens targets p...
Microlensing is the only method that can detect and measure mass of wide orbit, low mass, solar system analog exoplanets. Mass measurements of such planets would yield massive science on planet formation, exoplanet demographics, free floating planets, planet frequencies towards the galaxy. High res follow-up observations of past microlens targets p...
Citations
... This process commonly involves the use of dynamical systems theory and calculations of periodic orbits, quasi-periodic orbits, and trajectories along the hyperbolic invariant manifolds of either. Early work involving the use of these manifolds greatly changed our understanding of how to move about the solar system [2][3][4][5][6]. By linking together the unstable invariant manifolds of one system with the stable invariant manifolds of another, heteroclinic connections can be formed and used as very low-energy transport mechanisms across the solar system, or even around the moon(s) of a single planet [7][8][9][10][11][12]. ...
... For a spacecraft traveling in the CR3BP, the state of the spacecraft with respect to the system barycenter is defined in Eq. (2), the equations of motion are defined in Eqs. (3,4,5), the system potential in Eq. (6), and the Jacobi constant in Eq. (7). In these equations, r 1 and r 2 refer to the distances between the spacecraft and the primary and secondary bodies, respectively. ...
Trajectories at low energies with respect to the secondary body in the Circular Restricted Three-Body Problem are important for the future of orbital and landing missions. However, at these low energies, we demonstrate that high-latitude regions are unreachable for ballistic trajectories traveling through the \(L_1\) and \(L_2\) necks. This study explores the upper bound of vertical motion for these ballistic trajectories at a particular low energy. An initial grid-search across the \(L_1\) and \(L_2\) necks at Europa reveal that certain patterns appear in the trajectories that maximize inclination and out-of-plane velocity—metrics which are used to quantify vertical motion. These patterns are traced back to two nearby families of planar periodic orbits. These full families are generated and points of bifurcation are located. Certain families that originate as out-of-plane bifurcations from the two planar families are generated, and their unique members that match the energy of the grid-search are isolated. The manifolds of these particular periodic orbits exist along the simulated upper bound of vertical motion from the grid-search results. These results suggest that multiple families of stable and near-stable periodic orbits govern the flow of motion near this boundary.
... These factors have even led to the selection of a halo orbit as the placement for the upcoming Lunar Gateway space station. The invariant stable and unstable manifolds of periodic orbits have also become very popular in mission design for nearly free transfers between equilibria within systems such as heteroclinic and homoclinic connections (Koon et al. 2000;Gómez et al. 2004;Haapala 2014), reaching particular science orbits (Lara et al. 2005Russell and Lam 2007;Anderson and Lo 2009), patching together moon tours (Koon et al. 2001;Restrepo 2018;Anderson 2021), and exploring the solar system (Lo 1999(Lo , 2002Lo and Ross 2001). ...
Periodic orbits and their invariant manifolds are known to be useful for transportation in space, but a large portion of the related research goes toward a small number of periodic orbit families that are relatively simple to compute. In this study, motivated by a search for new and lesser-known families of useful periodic orbits, the bifurcation diagram near Europa is explored and 400 bifurcation points are found. Families are generated for 74 of these and provided in a publicly accessible database. Of these 74 generated families, those that also appear to exist in a model perturbed by certain zonal harmonics of Jupiter and Europa are identified. Differential corrections techniques are discussed, and a new method for natural parameter continuation in the three-body problem is presented. Periodic orbits with particularly useful geometric and stability properties for science purposes are highlighted.
... The focus of this paper is to present and demonstrate a systematic motion primitive construction process to summarize dynamical structures in the CR3BP. Generating a set of motion primitives to summarize families of periodic orbits and hyperbolic invariant manifolds supports analysis of the natural transport mechanisms that are often used in trajectory design and govern the natural motion of small celestial bodies Gómez et al. 2004;Lo 2002). This paper presents an automated approach for motion primitive construction that limits the burden on a human analyst and may be applied to a variety of periodic and nonperiodic trajectories in the CR3BP. ...
Rapid trajectory design in multi-body systems often leverages individual arcs along natural dynamical structures that exist in an approximate dynamical model. To reduce the complexity of this analysis in a chaotic gravitational environment, a motion primitive set is constructed to represent the finite geometric, stability, and/or energetic characteristics exhibited by a set of trajectories and, therefore, support the construction of initial guesses for complex trajectories. In the absence of generalizable analytical criteria for extracting these representative solutions, a data-driven procedure is presented. Specifically, k-means and agglomerative clustering are used in conjunction with weighted evidence accumulation clustering, a form of consensus clustering, to construct sets of motion primitives in an unsupervised manner. This data-driven procedure is used to construct motion primitive sets that summarize a variety of periodic orbit families and natural trajectories along hyperbolic invariant manifolds in the Earth–Moon circular restricted three-body problem.
... Heteroclinic connections are formed when an unstable manifold of one structure is patched seamlessly with the stable manifold of another structure. A heteroclinic connection between a pair of L 1 and L 2 periodic orbits was first demonstrated by Koon et al. [60], and the same group used this methodology to demonstrate its utility in designing a moon-tour at Jupiter [62] (a process that has continued to see development, such as Anderson [7] and Canales, Howell, and Fantino [30] [72,73,71] on how it can be used to explore the solar system for very low fuel compared to traditional methods. Many studies go on to further emphasize the role that heteroclinic connections between periodic orbits can play and enabling mission design and for serving as initial guesses for low-thrust missions [43,8,47,78,88]. ...
Outer planets such as Jupiter and Saturn are ripe with moons that contain all the ingredients for life as we know it (water, organic molecules, energy, and time), however, these destinations are very difficult to reach, especially land on. Located deep within the gravitational wells of their host planets and lacking sufficient atmospheres to allow for the use of parachutes, these moons are a unique challenge in mission design. Any spacecraft intending to land on one of these moons must approach with a low relative energy, but motion at low energies in the three-body problem is chaotic and dominated by local dynamical structures. The field of applied dynamical systems theory which studies the utility of these structures, such as periodic orbits and invariant manifolds, has grown much over recent decades, but there has been a limited focus on using these tools to help design landing trajectories.
In this dissertation, a framework for understanding low-energy landing trajectories to the secondary body in the three-body problem is established. Bounds of possible motion and their relationship to nearby dynamical structures are explored. The utility of using the invariant mani- folds of common families of periodic orbits is analyzed, and maps of feasible landing locations are generated. A study of periodic orbit bifurcations results in a publicly accessible database of over 70 families near Jupiter’s moon Europa, and over 400 bifurcations points are identified. Certain members from the various families are labeled as possible “abort solutions” due to their desirable stability and landing-opportunity repeatability. Many of the identified solutions are studied under the lens of a perturbed three-body problem that accounts for zonal harmonics. The z-axis symmetry of these perturbations allows the problem to remain autonomous, making them ideal for higher-fidelity study of the three-body problem without a significant increase in model complexity.
... Researchers throughout the astrodynamics community have demonstrated the capability for dynamical systems theory to rapidly generate an initial guess for a low-thrust trajectory in a multi-body system. Early observations of the low-thrust solutions obtained using the Mystic optimization software note a resemblance to the hyperbolic invariant manifolds of unstable periodic orbits in the CR3BP [9][10][11][12]. Anderson and Lo also demonstrated that optimal trajectories for tour design using the Jovian moons closely follow the hyperbolic invariant manifolds of unstable resonant periodic orbits [13]. ...
Placing a small satellite into a high-inclination orbit with respect to the ecliptic plane may offer a low-cost option for opportunistic and targeted observations of the polar regions of the Sun or the zodiacal dust cloud of the solar system. In this paper, dynamical systems theory and hybrid optimization techniques are integrated into a cohesive framework to design low-thrust trajectories for a small satellite to reach a highly out-of-ecliptic science orbit near the Sun–Earth L2 equilibrium point. Propellant-optimal low-thrust trajectories with a specific geometry are designed and studied across a variety of engine and power models within a low-thrust-enabled circular restricted three-body problem. The geometry of the trajectories is then varied during the initial guess construction process to support a preliminary study of the tradeoff between flight time and propellant mass usage.
... Lo [1], Ross, Koon [2,3], Marsden and Gomez [4] found that the celestial space is filled with invariant manifolds like pipes, and it can be divided into stable manifolds and unstable manifolds. The spacecraft can stabilize the stationary flow enters the Halo orbital and exits the Halo orbital along the unstable manifold. ...
... The dynamic equation of small object P is (1) U is the equivalent potential energy of the rotating coordinate system 2 1 ...
The design of low energy lunar transfer orbit and orbit optimization method based on crossing orbit is studied in this paper. The four-body problem of Sun-Earth-Moon-satellite is decoupled into two three-body problems of Sun-Earth-satellite and Earth-Moon-satellite. Low-energy lunar exploration orbit is designed by changing orbit at a certain intersection field through manifold of two Halo orbit of three-body system. A serial optimization design method for global initial value search and local gradient optimization are designed. When a motion state variable is partially modified, the differential correction method combined with adaptive regression algorithm is adopted. For the splicing points of manifolds of different systems, the velocity vector is modified to make the spacecraft get into the target berthing track, and to obtain the splicing points that can form crossing track of the two three-body systems. An example of orbit design is given to show that the Earth-Moon transfer orbit based on invariant manifold splicing has the characteristics of energy saving and long-time consumption, which can be used for the tasks with large loads and low time requirements. This research provides a method for the design of low energy lunar transfer orbit. It will benefit the energy consuming, material design of spacecraft, and so on.
... Secondly, the safe approach strategy will be defined, and validated keeping into account potential errors due to the propagation of the dynamics using CR3BP equations. Since manifolds exist only for periodic orbits [12], [13], it was necessary to compute the best-fit periodic orbit to the target's orbit. This was achieved using a differential correction procedure [14] to retrieve the closest periodic orbit, with the propagation equations, based on the CR3BP, written with respect to a synodic reference system. ...
... Asteroids have been at the forefront of space exploration for many years, with the proposal and completion of missions such as JAXA's Hayabusa [8], NASA's Dawn [54] and ESA's Rosetta [55]. There are many reasons for the current interest in these bodies: from the fact that their study may answer questions about the formation and evolution of the Solar System [56], to their profusion of potentially valuable resources and useful materials for space manufacturing [57,58]. Furthermore, although main-belt asteroids are known since the early 19 th century, near-Earth asteroids (NEAs) have only been discovered by its end [59]. ...
The proposal of increasingly complex and innovative space endeavours poses growing demands for mission designers. In order to meet the established requirements and constraints while maintaining a low fuel cost, the use of low-energy trajectories is particularly interesting. These allow spacecraft to change orbits and move with little to no fuel, but they are computed using motion models of a higher fidelity than the commonly used two-body problem (2BP). For this purpose, perturbation methods that explore the third-body effect are especially attractive, since they can accurately convey the system dynamics of a three-body configuration with a lower computational cost, by employing mapping techniques or exploring analytical approximations. The focus of this work is to broaden the knowledge of low-energy trajectories by developing new mathematical tools to assist in mission design applications. In particular, novel motion models based on the third-body effect are conceived. One application of this study focuses on the trajectory design for missions to near-Earth asteroids. Two different projects are explored: one is based on the preliminary design of separate rendezvous and capture missions to the invariant manifolds of libration point $L_2$. This is achieved by studying two recently discovered asteroids and determining dates, fuel cost and final control history for each trajectory. The other covers a larger study on asteroid capture missions, where several bodies are regarded as potential targets. The candidates are considered using a multi-fidelity design framework that filters through the trajectory options using models of motion of increasing accuracy, so that a final refined, low-thrust solution is obtained. The trajectory design hinges on harnessing Earth's gravity by exploiting encounters outside its sphere of influence, the named Earth-resonant encounters.
... The concept of stable and unstable invariant manifolds, associated with orbits around libration points, allows one to construct the regular procedures for searching the trajectories between various three-body systems: for example, between the Earth-Moon and Sun-Earth systems. In the early 2000s, the concept of an Interplanetary Superhighway, or Interplanetary Transport Network, appeared in this manner [3]. The potential accessibility of deep space from the environment of libration points of the Earth-Moon system, as well as the urgency of exploring and studying the Moon, gave rise to a new international project named Lunar Orbital Platform-Gateway (former name-Deep Space Gateway) on constructing and introducing into use, a habitable lunar station. ...
... In the vicinity of and there exist the families of periodic orbits, the most famous of which are the planar Lyapunov orbits and halo orbits (see Figs. [3][4]. Formally, halo orbits can be defined as spatial periodic orbits, branching off from planar Lyapunov orbits at a certain value of the energy integral. ...
... (Lo, 2002) ...
La pure réalité mathématique au travers d'une matérialisation Visible et Audible (Seeables™ and Audibles). Mon travail développe une nouvelle traduction au-delà de la théorie pour entrer des visualisations, développer une méthodologie latérale pour l'examen des phénomènes scientifiques. Je me concentre sur le pont entre la réalité mathématique immuable plus grande et la réalité physique humaine que Connes (AC) discute dans ses conversations avec Jean-Pierre Changeux et ailleurs significativement aussi bien. J'explore la « capture » des images géométriques, en plaçant le contexte entre deux parties opposées d'un problème non concilié. Ma recherche pionnière et championne une action transdisciplinaire croisant la technologie, la philosophie, les mathématiques, pour « mettre en lumière/lentillage» des idées mathématiques difficiles pour la société à travers la conception/design, la vulgarisation et la dissémination de ces données intrigantes et architecturalement cachées / multidimensionnelles (quantiques) / théoriques domaines scientifiques. Arts influençant la science, les sciences influençant l'art, interagissent de manière fluide ; avancées de pointe, données et équations restent précises et non exploitées, des visuels géométriques émergent, créant une technique presque « vase de Rubin » pour voir un autre aspect des mêmes phénomènes. Mon concept depuis sa création est de trouver de nouvelles façons visuelles de cartographier qui vont au-delà de l'ancienne forme hypergraphique et au-delà d'une forme géométrique déjà connue en réplication simple (habillage / défilé comme Thurston / Miyake (Paris, 2010, catwalk, Hopf Diagrams), de proposer une nouvelle façon de concevoir la mode / et de ressentir la profondeur de la musique (par exemple la musique de nombres premiers de AC), et des objets visuels (design technologique et abstrait simple), qui pourront montrer les plus beaux, les plus cachés, les plus irrécupérables, dans la beauté des maths possibles. Je me concentre sur la traduction de phénomènes scientifiques théoriques abstraits en une forme de tangibilité qui, jusqu'à présent, ne s'est manifestée que dans l'ingénierie biologiquement inspirée et la biologie synthétique. Je recherche la question « pouvons-nous rechercher la cohérence par le biais de la conception synthétique ? », Sur les structures géométriques cosmologiques et quantiques. Je concrétise des visuels géométriques en une fraction de seconde ainsi que des données et des équations d'avancées théoriques et hypothétiques, en créant des outils visuels sous la forme de « Seeables™ », et « Audibles ». Enfin, des mathématiciens comme Stephan Smale ont reconnu l'importance de la technologie. Dans le cas de mon entreprise et de son concept original, un focus direct est mis sur le visuel (par exemple celui qui a plané dans l'esprit de Connes), puis de créer une nouvelle cartographie par synthèse sur la cosmologie et les structures quantiques. Je remettrai 1) une dissertation et 2) une présentation. »
