Orientadores: Marco Antonio Teixeira, Alain Guy Jacquemard / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-05T23:50:08Z (GMT). No. of bitstreams: 1 Pereira_WeberFlavio_D.pdf: 1832947 bytes, checksum: 58bb202e90151fc6830fbc0cd1cf430e (MD5) Previous issue date: 2006 / Resumo: De acordo com a classificação feita por Anosov em 1959, obtemos diferentes tipos topológicos de sistemas "semi-lineares" descontínuos em JR4. Esta pré-classificação é feita através da apresentação das respectivas formas normais. Neste trabalho, consideramos perturbações não lineares de tais formas normais. As singularidades típicas são genericamente classificadas e o comportamento dos sistemas em torno destes pontos é analisado. Nosso foco é encontrar condições para a existência de uma família a l-parâmetro de órbitas periódicas terminando em singularidades no sentido do Teorema Centro de Lyapounov. As técnicas principais usadas são elementos do cálculo simbólico e da Teorida das Singularidades de Aplicações / Abstract: According to the classification made by Anosov in 1959, we derive several different topological types of semi-linear"discontinuous systems in R4. This pre-classification is done via pre-sentation of the respective normal forms. In this work, we consider non-linear perturbations of such normal forms. The typical singularities are generically classified and the behavior of the systems around these points is analyzed. Our focus is find conditions for the existence of 1-parameter family of periodic orbit terminating at the singularities in the sense of Lya- pounov Center Theorem. The main techniques used are elements of Symbolic Computation and Theory of Singularities of Mappings / Doutorado / Doutor em Matemática

Depuis la découverte de la première planète orbitant une étoile de la séquence principale autre que le Soleil en 1995, ce champ de recherche a connu une croissance vertigineuse, tant au niveau des observations, que des modèles théoriques développés en parallèle. Même si la formation et l’évolution des systèmes planétaires restent encore mal comprises dans leur globalité, Il est à peu près certain que les planètes se forment dans des disques protoplanétaires et interagissent avec ces derniers durant la phase primordiale de leur évolution. Cette thèse s’attache à décrire certains aspects de ces interactions. Parmi les problèmes soulevés par les nombreuses observations d’exoplanètes, on peut citer l’existence des Jupiter chaudes, géantes gazeuses dont la révolution autour de leur étoile s’effectue en quelques jours à peine. Il est communément admis qu’elles se sont formées dans les parties externes du disque, pour ensuite migrer vers l’intérieur. Cependant , les processus de migration restent encore débattus. On pourra aussi noter qu’un nombre important de planètes détectées, notamment par la méthode des vitesses radiales, présentent de fortes excentricités. Cette observation contraste avec celle de notre propre Système Solaire, où les planètes géantes ont des orbites quasi-circulaires. Cette distribution d’excentricités témoigne probablement d’une certaine richesse dans les interactions dynamiques entre les planètes d’un même système. Un autre résultat majeur des quelques dernières années est l’observation de planètes à faible période orbitale dont l’orbite n’est pas alignée avec l’axe de rotation de leur étoile. Cette observation pourrait potentiellement remettre en question l’idée selon laquelle ces planètes acquièrent leur faible période par le biais de la migration au sein du disque. Par conséquent, il est important de pouvoir différencier quelles sont les caractéristiques observationelles des exoplanètes qui sont le fruit de leurs interactions mutuelles, et celles qui peuvent être expliquées lors de la phase d’interaction avec le disque protoplanétaire. D’une part, cela permet d’imposer des contraintes sur la physique des disques protoplanétaires. D’autre part, il est intéressant de savoir à quoi ressemble le système de planètes une fois que le disque se dissipe, et à quelles conditions intiales peut-on s’attendre lorsque les planètes commencent à interagir entre elles sans la présence du disque. Par exemple, est-il possible pour une ou des planètes d’acquérir de l’excentricité et de l’inclinaison au sein du disque, et de les maintenir par la suite. De plus, il est certain que le disque domine l’évolution des planètes au stage primordial de leur vie, mais jusqu’à quel point cela limite-t-il les interactions entre les planètes ? / Since the discovery of the first planet orbiting a main-sequence star outside the solar system in 1995, the field of exoplanet studies has grown rapidly, both from the observational and theoretical sides. Despite the fact that we are still lacking a global picture for the formation and evolution of planetary systems, it is now commonly accepted that planets form in protoplanetary discs and interact with them in the early stages of their evolution. This thesis aims at studying some of these interactions. The observations of extrasolar planets have brought several puzzling results to the attention of the community. One of them is the existence of hot Jupiters, giant gaseous planets which orbit their parent star with a period of a few days only. The commonly accepted scenario is that they formed in the outer parts of the disc and migrated inward. Furthermore, a significant number of planets detected so far, especially by the method of radial velocities, have high eccentricities. This is in contrast with our own solar system where giant planets have quasi-circular orbits. Such a distribution of eccentricities may be the signature of strong dynamical interactions between the different components of a same planetary system. Finally, there are short-period planets whose orbits is misaligned with the axis of rotation of their host star, which could possibly argue against the smooth migration of planets in their disc. Therefore, it is important to disentangle between the orbital characteristics that planets acquired through mutual dynamical interactions, and the ones they acquired when they interacted with the disc. Firstly, it gives constraints on the physical parameters of protoplanetary discs. Secondly, it is interesting to know the properties of the system of planets after the disc has dissipated, and what sort of initial conditions one can expect when planets start to interact freely one with each other. For instance, one can ask if it is possible for planets to reach large eccentricities and inclinations when the disc was still present, and whether they could maintain them or not.

Let G be a finite permutation group acting on a finite set Ω. Then we denote by σk(G,Ω) the number of G-orbits on the set Ωk, consisting of all k-subsets of Ω. In this thesis we develop methods for calculating the values for σk(G,Ω) and produce formulae for the cases that G is a doubly-transitive simple rank one Lie type group. That is G ∼ = PSL(2,q),Sz(q),PSU(3,q) or R(q). We also give reduced functions for the calculation of the number of orbits of these groups when k = 3 and go on to consider the numbers of orbits, when G is a finite abelian group in its regular representation. We then consider orbit lengths and examine groups with “large” G-orbits on subsetsof size 3.

The work presented here is a continuation of Spacecraft Trajectory Optimization Suite (STOpS), a master’s thesis written by Timothy Fitzgerald at California Polytechnic State University, San Luis Obispo. Low-thrust spacecraft engines are becoming much more common due to their high efficiency, especially for interplanetary trajectories. The version of STOpS presented here optimizes low-thrust trajectories using the Island Model Paradigm with three stochastic evolutionary algorithms: the genetic algorithm, differential evolution, and particle swarm optimization. While the algorithms used here were designed for the original STOpS, they were modified for this work. The low-thrust STOpS was successfully validated with two trajectory problems and their known near-optimal solutions. The first verification case was a constant-thrust, variable-time Earth orbit to Mars orbit transfer where the thrust was 3.787 Newtons and the time was approximately 195 days. The second verification case was a variable-thrust, constant-time Earth orbit to Mercury orbit transfer with the thrust coming from a solar electric propulsion model equation and the time being 355 days. Low-thrust STOpS found similar near-optimal solutions in each case. The final result of this work is a versatile MATLAB tool for optimizing low-thrust interplanetary trajectories.

With the growth of human interest in the Lunar region, methods of enabling Lunar access including surface and Low Lunar Orbit (LLO) from periodic orbit in the Lunar region is becoming more important. The current investigation explores the Lunar access capabilities of these periodic orbits. Impact trajectories originating from the 9:2 Lunar Synodic Resonant (LSR) Near Rectilinear Halo Orbit (NRHO) are determined through explicit propagation and mapping of initial conditions formed by applying small maneuvers at locations across the orbit. These trajectories yielding desirable Lunar impact final conditions are then used to converge impacting transfers from the NRHO to Shackleton crater near the Lunar south pole. The stability of periodic orbits in the Lunar region is analyzed through application of a stability index and time constant. The Lunar access capabilities of the Lunar region periodic orbits found to be sufficiently unstable are then analyzed through impact and periapse maps. Using the impact data, candidate periodic orbits are incorporated in the the NRHO to Shackleton crater mission design to control mission geometry. Finally, the periapse map data is used to determine periodic orbits with desirable apse conditions that are then used to design NRHO to LLO transfer trajectories.

In this dissertation, we develop two novel methods for studying the nature of the Milky Way's dark matter halo. In both cases, we rely on the relationship between the dark matter halo's gravitational potential and the orbital structure it supports. The first method explores the morphology of stellar streams orbiting in non-spherical gravitational potentials. When globular clusters or dwarf galaxies fall into the Milky Way, tidal forces shred them into long filaments of stars called stellar streams. We show that in non-spherical potentials, stream morphologies are heavily dependent on the characteristics of the progenitor's orbit. Flattened axisymmetric galactic potentials, for example, are known to host minor orbit families surrounding special orbits with commensurable frequencies. The behavior of orbits that belong to these orbit families is fundamentally different from that of typical orbits with non-commensurable frequencies. We show that streams evolving near the boundaries, or separatrices, between orbit families, may become fanned out, develop a bifurcation, or both. We utilize perturbation theory to estimate the timescale of this effect and the likelihood of a stream evolving close enough to a separatrix to be affected by it. Next, we study the dynamical reasons for stream fanning and bifurcations near resonances, and find that each morphological outcome has a slightly different dynamical cause. Using a novel numerical approach for measuring the libration frequencies of resonant and near-resonant orbits, we reveal that fans come about due to a large spread in the libration frequencies near a separatrix, whereas bifurcations arise when a separatrix splits the orbital distribution of the stellar stream between two (or more) distinct orbit families. We then demonstrate how these features can arise in streams on realistic galactic orbits, in realistic galactic potentials, over timescales as short as 2-3 Gyr, and discuss how this might be used to constrain the global shape of the Milky Way's gravitational potential. The second method studied in this dissertation enables dynamical tests of a dark matter candidate known as Fuzzy (or Ultra-Light) Dark Matter. Our method relies on a wave generalization of the classic Schwarzschild approach for constructing self-consistent halos -- such a halo consists of a suitable superposition of waves instead of particle orbits, chosen to yield a desired mean density profile. As an illustration, we apply the method to spherically symmetric halos. We derive an analytic relation between the particle distribution function and the wave superposition amplitudes, and show how it simplifies in the high energy (WKB) limit. We verify the stability of such constructed halos by numerically evolving the Schrodinger-Poisson system. The proposed algorithm provides an efficient and accurate way to simulate the time-dependent halo substructures from wave interference, and to test how they will affect dynamical tracers or other observables in a galaxy. The dissertation concludes with a brief discussion of the future prospects of these two methods, especially in the context of upcoming ground- and space-based missions like Rubin LSST and the Roman Space Telescope.

The amount of light an exoplanet reflects and emits towards an observer waxes and wanes as the planet orbits through its phases. The amplitude and profile of reflection phase curves constrain the albedo of planetary surfaces and atmospheres, while the thermal amplitude and profile reveal temperature distributions and heat transport efficiencies, all providing valuable insight into the nature of exoplanet surfaces and atmospheres. In this dissertation I highlight the usefulness of utilizing full orbital phase curves in addition to occultation measurements, which provides a higher sensitivity to planetary photons at the expense of a more challenging data reduction. In the first few chapters of this dissertation, I introduce a novel non-parametric algorithm to produce clean, robust exoplanet phase curves, and apply it to separate ensembles of 115 Neptunian and 50 Terran exoplanets observed by the Kepler satellite to measure an upper limit on the average albedo of Kepler’s Neptunian planets, and make the first constraint on the average albedo of Terran worlds. In the fourth chapter, I present the full orbital phase curve and occultation of the ultra-hot Jupiter WASP-100b observed by the Transiting Exoplanet Survey Satellite (TESS), and with the use of Bayesian methods, present the first measurement of a phase shift of the thermal maximum among the phase curves observed by TESS, the degree of which challenges the predicted efficiency of heat transport in the atmospheres of ultra-hot Jupiters. In the final chapter, I present an example of how the NASA ROCKE-3D general circulation model can be used to explore the physical mechanisms that influence the habitability of terrestrial exoplanets, and then show how I generated phase curves from the 3-dimensional models to study the signals produced by simulated TRAPPIST-1 habitable-zone worlds. The work in this dissertation contributes valuable new information to the astronomical literature and provides avenues for further research on the nature of short-period exoplanets.

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京都大学 / 新制・課程博士 / 博士(情報学) / 甲第24328号 / 情博第812号 / 新制||情||137(附属図書館) / 京都大学大学院情報学研究科数理工学専攻 / (主査)准教授 柴山 允瑠, 教授 矢ヶ崎 一幸, 教授 山下 信雄, 教授 田口 智清 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

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