<p dir="ltr">This research considers orbit maintenance strategies for multi-body orbits in the context of the Earth-Moon Circular Restricted Three Body Problem (CR3BP). Dynamical requirements for successful long-term orbit maintenance strategies are highlighted.</p>

Recently there is an increase in interest in the Moon as a destination for space missions. This increased interest is in the composition and geography of the Moon as well as using the Moon to travel beyond the Earth to other planets in the solar system. This thesis explores the mechanics behind a lunar mission and the costs and benefits of different approaches. To constrain this problem, the launch criteria are those of Wallops Flight Facility (WFF), which has expressed interest in launching small spacecraft to the Moon for exploration and study of the lunar surface. The flight from the Earth to the Moon and subsequent lunar orbits, referred to hereafter as the mission, is broken up into three different phases: first the launch and parking orbit around the Earth, second the transfer orbit, and finally the lunar capture and orbit. A launch from WFF constrains the direction of the launch and the possible initial parking orbits. Recently WFF has been offered the use of a Taurus XL launch vehicle whose specifications will be used for all other limitations of the launch and initial parking orbit. The orbit investigated in this part of the mission is a simple circular orbit with limited disturbances. These disturbances are only a major factor for long duration orbits and don't affect the parking orbit significantly. The transfer orbit from the Earth to the Moon is the most complex and interesting part of the mission. To fully describe the dynamics of the Earth-Moon system a three-body model is used. The model is a restricted three-body problem keeping the Earth and Moon orbiting circularly around the system barycenter. This model allows the spacecraft to experience the influence of the Earth and Moon during the entire transfer orbit, making the simulation more closely related to what will actually happen rather than what a patched conic solution would give. This trajectory is examined using Newtonian, Lagrangian, and Hamiltonian mechanics along with using a rotating and non-rotating frame of reference for the equations of motion. The objective of the transfer orbit is to reduce the time and fuel cost of the mission as well as allow for various insertion angles to the Moon. The final phase of the mission is the lunar orbit and the analysis also uses a simple two body model similar to the parking orbit. The analysis investigates how the orbits around the Moon evolve and decay and explores more than just circular orbits, but orbits with different eccentricities. The non-uniform lunar gravity field is investigated to accurately model the lunar orbit. These factors give a proper simulation of what happens to the craft for the duration of the lunar orbit. Tracking the changes in the orbit gives a description of where it will be and how much of the lunar surface it can observe without any active changes to the orbit. The analysis allows for either pursuing a long duration sustained orbit or a more interesting orbit that covers more of the lunar surface. These three phases are numerically simulated using MATLAB, which is a focus of this thesis. In all parts of the mission the simulations are refined and optimized to reduce the time of the simulation. Also this refinement gives a more accurate portrayal of what would really happen in orbit. This reduction in time is necessary to allow for many different orbits and scenarios to be investigated without using an unreasonable amount of time. / Master of Science

La découverte de planètes extrasolaires d'excentricités importantes ravive l'intérêt pour la dynamique des systèmes planétaires. Ce travail a pour objet l'étude analytique du problème séculaire des trois corps, grâce à une généralisation de la théorie de Laplace-Lagrange obtenue en poussant le développement de la perturbation à un ordre largement supérieur en excentricités et en inclinaisons. Nous montrons que cette approche est apte à décrire la dynamique séculaire d'un système planétaire formé de deux planètes hors résonance en moyen mouvement. Une vérification analytique de la proximité du système à une quelconque résonance en moyen mouvement est également entreprise. Tant dans le cas de systèmes coplanaires que de systèmes tridimensionnels, deux optiques sont poursuivies: d'une part, l'analyse des équilibres du problème séculaire et des implications de ces derniers sur la structure de l'espace de phase et d'autre part, le calcul des fréquences fondamentales de ce même problème permettant la reproduction de l'évolution temporelle du système planétaire, grâce à une méthode totalement analytique basée sur les transformées de Lie. Nous disposons ainsi d'un modèle analytique fiable et peu coûteux pouvant prendre en compte un large éventail de paramètres et qui peut être appliqué avec précision aux systèmes extrasolaires hors résonance en moyen mouvement. / The discovery of extrasolar planets with large eccentricities renews interest in the study of the dynamics of planetary systems. This work is concerned with the analytical study of the secular three-body problem by means of a generalization of the Laplace-Lagrange theory based on a high-order expansion of the disturbing potential in the eccentricities and the inclinations. We show that this approach is able to describe the secular dynamics of a two-planets system not close to a mean motion resonance. The proximity of a system to any mean motion resonance is also analytically investigated. For coplanar and tridimensional systems, we pursue a twofold objective: on the one hand, the study of the equilibria of the secular problem and their implications on the structure of the phase space and on the other hand, the computation of both the fundamental frequencies of the problem and the long-term time evolution of the planetary system with a totally analytical method based on Lie transforms. This reliable time-saving analytical model can take into account a large spectrum of parameters and can be applied successfully to non-resonant extrasolar systems.

As a comparative study, the global performance of two cases for a floating harbor system are researched by numerical analysis and compared with results from experiments: one is a two-body case such that a floating quay is placed next to a fixed quay, a normal harbor, and the other is a three-body case such that a container ship is posed in the middle of the floating quay and the fixed quay. The numerical modeling is built based on the experimental cases. Mooring system used in the experiments is simplified to sets of linear springs, and gaps between adjacent bodies are remarkably narrow as 1.3m~1.6m with reference to large scales of the floating structures; a water plane of the fixed quay is 480m×160m, and the ship is 15000 TEU (twenty-foot equivalent unit) class. With the experiment-based models, numerical analysis is implemented on two domains: frequency domain using a three dimensional constant panel method, WAMIT, and time domain using a coupled dynamic analysis program of moored floating structures, CHARM3D/HARP. Following general processes of the two main tools, additional two calibrations are implemented if necessary: revision of external stiffness and estimation of damping coefficients. The revision of the external stiffness is conducted to match natural frequency of the simulation with that of the experiment; to find out natural frequencies RAO comparison is used. The next, estimation of damping coefficients is carried out on time domain to match the responses of the simulation with those of the experiment. After optimization of the numerical analysis, a set of experimental results from regular wave tests is compared with RAO on frequency domain, and results from an irregular wave test of the experiment are compared with response histories of simulation on time domain. In addition, fender forces are compared between the simulation and experiment. Based on response histories relative motions of the floating quay and container ship are compared. And the floating harbor system, the three-body case, is compared with a conventional harbor system, a fixed quay on the portside of the container ship, in terms of motions of the container ship. As an additional simulation, the three-body case is investigated on an operating sea state condition. From the present research, the experimental results are well matched with the numerical results obtained from the simulation tools optimized to the experiments. In addition, the floating harbor system show more stable motions of the container ship than the conventional harbor system, and the floating harbor system in the operating sea state condition have motions even smaller enough to operate in term of relative motions between the floating quay and the container ship.

This study focuses on the Sun's influence on the motion near the triangular libration points of the Earth-Moon system. It is known that there exists a very strong resonant perturbation near those points that produces large deviations from the libration points, with an amplitude of about 250,000 km and a period of 1,500 days. However, it has been shown that it is possible to find initial conditions that negate the effects of that perturbation, even resulting in stable, although very large, periodic orbits. Using two different models, the goal of this research is to determine the initial configurations of the Earth-Moon-Sun system that produce minimal deviations from the libration points, and to provide a better understanding of the dynamics of this highly nonlinear problem. First, the Bicircular Problem (BCP) is considered, which is an idealized model of the Earth-Moon-Sun System. The impact of the initial configuration of the Earth-Moon-Sun system is studied for various propagation times and it is found that there exist two initial configurations that produce minimal deviations from L₄ or L₅. The resulting trajectories are very sensitive to the initial configuration, as the mean deviation from the libration points can decrease by 30,000 km with less than a degree change in the initial configuration. Two critical initial configurations of the system were identified that could allow a particle to remain within 30,000 km of the libration points for as long as desired. A more realistic model, based on JPL ephemerides, is also used, and the influence of the initial epoch on the motion near the triangular points is studied. Through the year 2007, 51 epochs are found that produce apparently stable librational motion near L₄, and 60 near L₅. But the motion observed depends greatly on the initial epoch. Some epochs are even found to significantly reduce the deviation from L₄ and L₅, with the spacecraft remaining within at most 90,000 km from the triangular points for upwards of 3,000 days. Similarly to what was observed in the BCP, these trajectories are found to be extremely sensitive to the initial epoch. / text

L'ensemble des travaux menés au cours de cette thèse concerne l'étude de la résonance co-orbitale. Ce domaine de trajectoires particulières, où un astéroïde et une planète gravitent autour du Soleil avec la même période de révolution, possède une dynamique très riche liée aux célèbres configurations équilatérales de Lagrange, L4 et L5, ainsi qu'aux configurations alignées de Euler, L1, L2 et L3. Un exemple majeur dans le système solaire est donné par les astéroïdes « troyens » qui accompagnent Jupiter au voisinage des équilibres L4 et L5. Une deuxième configuration étonnante est donnée par les satellites Janus et Épiméthée qui gravitent autour de la planète Saturne ; suite à la forme décrite par la trajectoire d’un des satellites dans un repère tournant avec l’autre, la dynamique résultante est appelée « fer-à-cheval ». Un nouveau type de dynamique a été récemment misen évidence dans le contexte de la résonance coorbitale : les « quasi-satellites ». Il s’agit de configurations remarquables où, dans un repère tournant avec la planète, la trajectoire de l’astéroïde correspond à celle d’un satellite rétrograde. Des astéroïdes accompagnant les planètes Venus, Jupiter et la Terre ont notamment été observés dans ces configurations. La dynamique des quasi-satellites possède un grand intérêt, pas seulement parce qu’elle relie les différents domaines de la résonance co-orbitale (voir les travaux de Namouni, 1999) mais aussi parce qu’elle semble faire le pont entre les notions de satellisation et celles de trajectoires héliocentriques. Cependant, bien que le terme « quasi-satellite" soit devenu dominant dans la communauté de mécanique céleste, certains auteurs utilisent plutôt la terminologie « satellite rétrograde » révélant ainsi une ambiguïté sur la définition de ces trajectoires. Les récentes découvertes autour des exo-planètes ont motivé le développement de travaux concernant la résonance co-orbitale dans le problème des trois corps planétaire. Dans ce contexte Giuppone et al. (2010) ont mis en évidence (par une méthode numérique) les quasisatellites ainsi que des nouvelles familles de configurations remarquables : les orbites « anti-Lagrange ». La troisième partie de thèse présente alors une méthode analytique pour l'étude planétaire, permettant de révéler analytiquement les orbites anti-Lagrange ainsi qu'une esquisse d'étude des quasisatellites en adaptant à ce contexte plus général la méthode présentée dans la seconde partie. Pour ces raisons, la première partie de cette thèse a consisté à clarifier la définition de ces orbites en revisitant le cas circulaire-plan (trajectoires coplanaires avec la planète qui gravite sur une orbite circulaire) dans le cadre du problème moyen. Dans la deuxième partie de cette thèse, nous avons développé une méthode analytique apte à explorer le domaine des quasi-satellites dans le cadre du problème moyen. Nous avons réalisé cette exploration dans le cas circulaire-plan et proposé une extension aux cas excentrique-plan et circulaire-spatial. / This work of thesis focuses on the study of the coorbital resonance. This domain of particular trajectories, where an asteroid and a planet gravitate around the Sun with the same period possesses a very rich dynamics connected to the famous Lagrange’s equilateral configurations L4 and L5, as well as to the Eulerian’s configurations L1, L2 and L3. A major example in the solar system is given by the “Trojan” asteroids harboured by Jupiter in the neighborhood of L4 and L5. A second astonishing configuration is given by the system Saturn-Janus-Epimetheus; this peculiar dynamics is known as “horseshoe”. Recently, a new type of dynamics has been highlighted in the context of co-orbital resonance: the quasi-satellites. They correspond to remarkable configurations : in the rotating frame with the planet, the trajectory of the asteroid seems the one of a retrograde satellite. Some asteroids harboured by Venus, Jupiter and the Earth have been observed in this kind of configuration. The quasi-satellite dynamics possesses great interest not only because it connects the different domains of the co-orbital resonance (see works of Namouni, 1999), but also because it seems to bridge the gap between satellization and heliocentric trajectories. However, despite the term quasi-satellite has become dominant in the celestial mechanics community, some authors rather use the term “retrograde satellite”. This reveals an ambiguity on the definition of these trajectories. For these reasons, the first part of this thesis consisted in clarifying the definition of these orbits by revisiting the planar-circular case (planet on circular motion) in the framework of the averaged problem. In the second part of this thesis, we developed an analytic method to explore the quasi-satellite domain in the averaged problem. We realized this exploration in the planar-circular case and proposed an extension to the planar-eccentric and spatial-circular cases. The recent discoveries around the exo-planets motivated some works on the co-orbital resonance in the planetary Three-Body Problem. In this context, Giuppone et al. (2010) highlighted (numerically) the quasi-satellite as well as new families of remarkable configurations: the “anti-Lagrange”. Then the third part of this thesis presents an analytical method for the planetary problem that allows to reveal the anti-Lagrange orbits as well as a sketch of study of quasi-satellite trajectories.

<div> <div> <div> <p>Resonant orbits in a multi-body environment have been investigated in the past to aid the understanding of perceived chaotic behavior in the solar system. The invariant manifolds associated with resonant orbits have also been recently incorporated into the design of trajectories requiring reduced maneuver costs. Poincaré sections are now also extensively utilized in the search for novel, maneuver-free trajectories in various systems. This investigation employs dynamical systems techniques in the computation and characterization of resonant orbits in the higher-fidelity Circular Restricted Three-Body model. Differential corrections and numerical methods are widely leveraged in this analysis in the determination of orbits corresponding to different resonance ratios. The versatility of resonant orbits in the design of low cost trajectories to support exploration for several planet-moon systems is demonstrated. The efficacy of the resonant orbits is illustrated via transfer trajectory design in the Earth-Moon, Saturn-Titan, and the Mars-Deimos systems. Lastly, Poincaré sections associated with different resonance ratios are incorporated into the search for natural, maneuver-free trajectories in the Saturn-Titan system. To that end, homoclinic and heteroclinic trajectories are constructed. Additionally, chains of periodic orbits that mimic the geometries for two different resonant ratios are examined, i.e., periodic orbits that cycle between different resonances are determined. The tools and techniques demonstrated in this investigation are useful for the design of trajectories in several different systems within the CR3BP. </p> </div> </div> </div>

<p>A dynamical understanding of orbits in the Earth-Moon neighborhood that can sustain long-term activities and trajectories that link locations of interest forms a critical foundation for the creation of infrastructure to support a lasting presence in this region of space. In response, this investigation aims to identify and exploit fundamental dynamical motion in the vicinity of a candidate ‘hub’ orbit, the L2 southern 9:2 lunar synodic resonant near rectilinear halo orbit (NRHO), while incorporating realistic mission constraints. The strategies developed in this investigation are, however, not restricted to this particular orbit but are, in fact, applicable to a wide variety of stable and nearly-stable cislunar orbits. Since stable and nearly-stable orbits that may lack useful manifold structures are of interest for long-term activities in cislunar space due to low orbit maintenance costs, strategies to alternatively initiate transfer design into and out of these orbits are necessary. Additionally, it is crucial to understand the complex behaviors in the neighborhood of any candidate hub orbit. In this investigation, a bifurcation analysis is used to identify periodic orbit families in close proximity to the hub orbit that may possess members with favorable stability properties, i.e., unstable orbits. Stability properties are quantified using a metric defined as the stability index. Broucke stability diagrams, a tool in which the eigenvalues of the monodromy matrix are recast into two simple parameters, are used to identify bifurcations along orbit families. Continuation algorithms, in combination with a differential corrections scheme, are used to compute new families of periodic orbits originating at bifurcations. These families possess unstable members with associated invariant manifolds that are indeed useful for trajectory design. Members of the families nearby the L2 NRHOs are demonstrated to persist in a higher-fidelity ephemeris model. </p><p><br></p> <p>Transfers based on the identified nearby dynamical structures and their associated manifolds are designed. To formulate initial guesses for transfer trajectories, a Poincaré mapping technique is used. Various sample trajectory designs are produced in this investigation to demonstrate the wide applicability of the design methodology. Initially, designs are based in the circular restricted three-body problem, however, geometries are demonstrated to persist in a higher-fidelity ephemeris model, as well. A strategy to avoid Earth and Moon eclipse conditions along many-revolution quasi-periodic ephemeris orbits and transfer trajectories is proposed in response to upcoming mission needs. Lunar synodic resonance, in combination with careful epoch selection, produces a simple eclipse-avoidance technique. Additionally, an integral-type eclipse avoidance path constraint is derived and incorporated into a differential corrections scheme as well. Finally, transfer trajectories in the circular restricted three-body problem and higher-fidelity ephemeris model are optimized and the geometry is shown to persist.</p>

According to the standard model, the size of CP violation in the weak decay Λ → pπ- is predicted to be ∼ 10-5. This means that a very high precision is needed if we want to be able to measure it. It turns out that the precision mainly depends on three factors: if the decay already has a large parity-breaking effect, if we take more measurements, or if the Lambda baryon has a large polarization. The uncertainty is inversely proportional to the root mean square of the polarization, √⟨P2⟩, of Λ. Polarization is a measurement of the fraction of particles with parallel spin. In this project, we simulated the kinematic properties of the three-body decay of the charmed lambda baryon Λc+  → Λπ+ π0 by assuming that Λ+ c decays 100% of the times via the ρ+-resonance, i.e. Λc+ decays like Λc+ → Λρ+ , and ρ+ subsequently decays ρ+ → π+π0. The invariant masses of the daughter particles were then plotted in a so-called Dalitz plot. The fact that the ρ+-meson is a vector meson is exploited to compute the helicity amplitudes HλΛ,λρ, where λ is the helicity of respective particle, meaning the projection of the spin in the direction of the ρ+-mesons momentum. The helicity amplitudes describe the helicities of the particles involved before and after the decay. These are then used to compute the polarization of Λ. After doing the simulation, the invariant masses are plotted in the Dalitz plot and the polarization in each point is computed. The root mean square polarization of Λ was √⟨P2⟩ = 0.81 ± 0.08. This result gives a large polarization and points to this process being a good baryon polarizer. When conducting more research into the three-body decay of Λc+ as a potential source of Λ with large polarization, the contributions of other resonances should be taken into account. In light of these results, it would also be interesting to investigate if three-body decays could work as a baryon polarizer for other baryons. / CP brytning i sönderfallet Λ → pπ-, via den svaga växelverkan, kommer enligt standardmodellen att ha en storlek runt ∼ 10-5. Det betyder att en väldigt hög precision är nödvändig om man ska kunna observera denna effekt. Det visar sig att precisionen av mätningarna främst beror på tre faktorer: om sönderfallet redan har stor paritestbrytande effect, om man tar fler mätningar, eller om Lambdabaryonerna har stor polarisation. Osäkerheten är proportionerlig mot inversen av det kvadratiska medelvärdet, √⟨P2⟩ av Lambdabaryonerna. Polarisationen av en mängd partiklar är ett mått på andelen partiklar med parallellt spinn. I detta projekt kommer vi simulera de kinematiska egenskaperna hos trekroppssönderfallet av den charmade Lambdabaryonen Λc+ → Λπ+π0 genom att anta 100% sönderfall via ρ+-resonansen, dvs att Λc+ först sönderfaller som Λc+ → Λρ+, och att ρ+ sedan sönderfaller som ρ+ → π+π0. De invarianta massorna av dotterpartiklarna plottades sedan i en s.k Dalitz-plot. Vi utnyttjade att ρ+-mesonen är en vektormeson för att beräkna helisitetsamplituderna HλΛ,λρ, där λ är helisiteterna av respektive partikel, dvs projektionen av partiklarnas spinn längst ρ+-mesonens rörelsemängdsriktning. Helisitetsamplituderna beskriver helisiteten hos partiklarna före och efter sönderfallet. Dessa användes sedan för att beräkna polarisationen av Λ. Efter att simuleringen gjorts, plottades de invarianta massorna i en Dalitz-plott och polarisationen beräknades i varje punkt. Det kvadratiska medelvärdet av polarisationen hos Λ var √⟨P2⟩ = 0.81 ± 0.08. Detta är en stor polarisation som indikerar att trekroppsönderfall kan vara en bra baryonpolarisator. För vidare forskning om trekroppssönderfallet av Λc+ som en källa av Λ med stor polarisation, borde bidragen från andra resonanser också medräknas. Vidare skulle det även vara intressant att undersöka om trekroppssönderfall kan fungera som en baryonpolarisator för andra baryoner.

[pt] O Modelo Padrão (SM) da física de partículas é atualmente a melhor teoria para descrever as interações entre partículas elementares e suas propriedades. A observação da violação do CP em diversos processos fracos no setor de quarks - permitindo uma distinção absoluta entre partículas e antipartículas - é bem explicada pelo SM (através do chamado Cabibbo-Kobayashi-Maskawa ansatz). A violação do CP é uma das condições necessárias para a bariogênese e pode ser a chave para explicar a assimetria matéria-antimatéria no universo. Esta dissertação apresenta a busca por violação de CP no decaimento D+ → π−π+K+ duplamente suprimido por Cabibbo (DCS), usando dados coletados pelo LHCb de 2016-2018 de colisões pp com uma energia de centro de massa de 13 TeV, correspondendo a uma luminosidade integrada de 5,6 fb1. O objetivo desta análise é a implementação de uma técnica independente de modelo para realizar uma comparação estatística entre o espaço de fase, denominado Dalitz Plot (DP), de partícula e antipartícula no canal de decaimento, buscando diferenças locais na distribuição de eventos entre os dois DPs. Primeiro, foi feito um processo de seleção para remover contribuições de background de outros decaimentos de charme, bem como uma análise multivariada utilizando algoritmos de aprendizado de máquina para reduzir os níveis de background combinatoriais. Após a seleção, foi obtida uma amostra final 6M de candidatos a sinal, que é hoje a maior amostra já obtida para um canal de decaimento D+ DCS, permitindo uma excelente sensibilidade para busca de VCP. Essa é a primeira busca de VCP no canal de decaimento estudado. Esta análise é realizada de forma cega, o que significa que não há resultado para a região do sinal nesta primeira etapa e, para garantir que não haja assimetrias espúrias, como efeitos de produção e detecção, foram realizados testes para a região de background, para o canal de controle D+ → K−π+π+ e para a amostra simulada de Monte Carlo. / [en] The Standard Model (SM) of particle physics is currently the best theory to describe the interactions between elementary particles and their properties. The observation of CP violation in a variety of weak processes in the quark sector — allowing an absolute distinction of particles and antiparticles — is well explained by the SM (through so-called Cabibbo-Kobayashi-Maskawa ansatz). CP violation is one of the necessary conditions for baryogenesis and might be the key to explain the matter-antimatter asymmetry in the universe. This dissertation presents the search for CP violation in the doubly Cabibbo suppressed (DCS) D+ → π−π+K+ decay, using data collected by LHCb from 2016-2018 of pp collisions with a centre of mass energy of 13 TeV, corresponding to an integrated luminosity of 5.6 fb1. The goal of this analysis is the implementation of a model-independent technique to perform a statistical comparison between the phase space, called Dalitz Plot (DP), of particle and antiparticle decay channel, searching for local differences in the distribution of events between the two DPs. First, a selection process was executed to remove background contributions from other charm decays, as well as a multivariate analysis using machine learning algorithms to reduce combinatorial background levels. After the selection, a final sample of ¨6M signal candidates was obtained, which is nowadays the largest sample ever obtained for a DCS D+ decay channel, allowing an outstanding sensitivity for CPV search. This is the first CPV search in the studied decay channel. This analysis is performed blinded, meaning that there is no actual result for the signal region at this first stage and in order to guarantee that there are no nuisance asymmetries, from production and detection effects, tests were performed using the background region, the control channel D+ → K−π+π+ and the Monte Carlo simulated sample.