Orientador : Rodolpho Vilhena de Moraes / Banca: Antonio Fernando Bertachini de Almeida Prado / Banca: Hans-Ulrich Pilchowski / Resumo: Aplicações recentes de satélites artificiais, principalmente aquelas com finalidades geodinâmicas e altimétricas, requerem órbitas determinadas com bastante precisão. Em particular as marés terrestres alteram o geopotencial, causando perturbações adicionais no movimento do satélite. Tais perturbações, apesar de pequenas, têm sido detectadas. O presente trabalho trata de perturbações de órbitas de satélites artificiais devidas às marés terrestres. Ênfase é dada aos termos seculares e de longo período. O potencial foi desenvolvido em termos dos elementos orbitais e substituídos nas equações planetárias de Lagrange. Soluções analíticas estão apresentadas para casos particulares considerando os números de Love constantes. Um programa foi elaborado, e colocado a disposição do usuário, permitindo calcular, para um dado satélite, a amplitude e o período dos termos perturbadores mais significativos. / Abstract: Recent applications of artificial satellites, mainly those of geodynamics and altimetric purposes, requires high precise orbit determination. Particularly, Earth tides change the geopotencial causing additional perturbation in the satellite orbital motion. Inspite of being very small such perturbations, has been detected. The present work concerns about orbit perturbations of artificial satellites due do terrestrial tides. Treatment of secular and long period terms is emphasized. The potencial was developed in terms of the orbital elements and substituted in the Lagrange equations. Analytical solutions are presented for particular cases considering the Love's number as constant. A computer enabling to compute, for a given satellite, the amplitude and period of the more significant disturbing terms, was constructed and it is, at the disposal for users. / Mestre

Nesta dissertação, estudamos as órbitas de uma família D de campos vetoriais suaves em uma variedade suave M. O objetivo é demonstrar dois teoremas de Sussmann: o primeiro teorema diz que as órbitas são subvariedades integrais de uma certa distribuição \'P IND. D\' de vetores tangentes em M. O segundo teorema dá condições necessárias e suficientes para que \'P IND. D\' seja igual à distribuição gerada pelos campos de D. Como aplicação, estudamos uma caracterização da condição (P) de Nirenberg-Treves para campos vetoriais complexos em \'R POT. 2\' / In this dissertation, we study the orbits of a family D of smooth vector fields on a smooth manifold M. The goal is to demonstrate two theorems of Sussmann: the first theorem says that the orbits are integral submanifolds of a certain distribution \'P IND. D\' of tangent vectors of M. The second theorem gives necessary and sufficient conditions for \'P IND. D\' to be the same as the distribution generated by the vector fields of D: As an application, we study a characterization of the condition (P) of Nirenberg and Treves for complex vector fields on \'R POT. 2\'

Thesis (MTech (Electrical Engineering))--Cape Peninsula University of Technology, 2015. / Traditional satellite systems employed for use with in-situ monitoring systems are large satellites that have a long development time, high cost and complex sub-systems. The end use of relaying data for in-situ monitoring becomes a costly application for the end user. Shifting this application to make use of nano-satellites, such as CubeSats, for data relaying will make the application more attractive to the end user when measurements are required outside existing ground based communications infrastructure. CubeSats are small, simple satellites that yield a short development time and very low cost compared to conventional satellites. Their sub-systems are generally built from off the shelf components. This keeps the designs simple, manufacture cost low with the potential for flying the latest technologies. This research set out to analyse various scenarios related to in-situ monitoring governed by parameters such as the maximum revisit time, satellite orbit altitude, quantity of sensor nodes and data quantity relayed in the system. A systems level approach is used to analyse each designed scenario using a simulation tool called Systems Tool Kit by Agilent Graphics Incorporated. The data acquired for each scenario through simulation was validated using theoretical approximation methods, which included parameters such as coverage potential, total node access time, communication link performance, power resources, memory resources, access time and number of ground stations. The focus was put on these parameters since they are the main constraints when designing a system using nano-satellites. The outcome of the research was to create an analysis reference for designing an in-situ monitoring system using nano-satellites. It outlines the methods used to calculate and simulate each of the constraints governing the system. Each designed scenario showed satisfactory performance within the defined parameters and can be practically implemented as a reference for designing similar systems. / National Research Foundation / South African National Space Agency

Orientador: Marco Antonio Teixeira / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-13T09:38:10Z (GMT). No. of bitstreams: 1 Mereu_AnaCristinadeOliveira_D.pdf: 1463250 bytes, checksum: 9bbe3e5b625f68effb7acc05409359ea (MD5) Previous issue date: 2009 / Resumo: Este trabalho é voltado ao estudo de existência e persistência de órbitas periódicas e órbitas homoclínicas em perturbações de sistemas dinamicos reversíveis. Primeiramente, rompemos a reversibilidade de centros no plano e em dimensões superiores e detectamos condições para a existência de ciclos limites e toros invariantes. A seguir, estudamos a existência de soluções periódicas simétricas de perturbações de uma família de equações diferencias reversíveis. A existência e persistência de órbitas homoclínicas em tais equações também foram discutidas. / Abstract: In this work we study the existence and persistence of some minimal sets in perturbations of reversible systems. First we make non reversible perturbations of centers in R2 and R4 and we detect conditions for the existence of limit cycles and invariant tori. We study the existence of periodic solutions of the perturbations of a family of di_erential equations expressed by x(2n) + a (2n-2)/2 +¿+ a1x(2) + x = 0 ; for n = 2; 3. The existence and persistence of homoclinic orbits in such equations are also discussed. / Doutorado / Geometria e Topologia / Doutor em Matemática

Uma órbita bilhar em um triângulo é uma poligonal cujos segmentos começam e terminam nos lados do triângulo e que se refletem elasticamente nestes lados. É como o movimento de uma bola numa mesa de bilhar sem atrito (logo a bola tem velocidade constante e jamais para) cujas laterais formam um triângulo. Esta órbita é periódica se ela retorna infinitas vezes ao mesmo ponto com a mesma direção. A existência de órbitas bilhares periódicas em polígonos é uma questão aberta da matemática. Mesmo para um triângulo ainda não há resposta. Para triângulos agudos, a resposta é bem conhecida, pois o triângulo formato pelos pés das alturas do triângulo é uma órbita periódica. Para triângulos obtusos, em geral, pouco se sabe. O objetivo desta dissertação é coletar resultados e técnicas sobre órbitas bilhares periódicas em triângulos obtusos. Começamos introduzindo o trabalho de Vorobets, Galperin e Stepin, que no início dos anos 90 unificaram os casos conhecidos de triângulos que possuem órbita bilhar periódica, introduziram o conceito de estabilidade e mostraram novos resultados, como uma família infinita de órbitas estáveis. Temos também o teorema de 2000 de Halbeisen e Hungerbühler que estende as famílias de órbitas estáveis. Mencionamos em seguida os trabalhos de Schwartz de 2006 e 2009 que utilizam auxílio computacional para mostrar que todo triângulo com ângulos menores que $100\\degree$ possui órbita bilhar periódica. Depois temos os resultados de 2008 de Hooper e Schwartz sobre órbitas bilhares periódicas em triângulos quase isósceles e sobre estabilidade de órbitas em triângulos de Veech. Todos os casos abordados neste trabalho incluem uma vasta variedade de triângulos, mas a questão de existência de órbitas bilhares periódicas para todo triângulo está longe de ser totalmente contemplada. / A billiard orbit in a triangle is a polygonal with vertices at the boundary of the triangle such that its angles reflect elastically. It is similar to a moving ball on a billiard table without friction (so the ball has constant speed and never stops) whose sides form a triangle. This orbit is periodic if it returns infinitely to the same point with the same direction. The existence of periodic billiard orbits in polygons is an open problem in mathematics. Even for a triangle there is still no answer. For acute triangles the answer is well known since the triangle whose vertices are the base points of the three altitudes of the triangle is a periodic orbit. For obtuse triangles, in general, little is known. The aim of this thesis is to collect results and techniques on periodic billiard orbits in obtuse triangles. We start by introducing the work of Vorobets, Gal\'perin and Stepin, who unified in the early 1990s the known cases of triangles that have periodic billiard orbits, introduced the concept of stability and proved new results, such as an infinite family of stable orbits. We also have the theorem of Halbeisen and Hungerbühler of 2000 extending the families of stable orbits. Next, we mention the works of Schwartz of 2006 and 2009 that use computational assistance to prove that every triangle whose angles are at most $100\\degree$ have periodic billiard orbits. Then, we have the results of 2008 by Hooper and Schwartz on periodic billiard orbits in nearly isosceles triangles and on stability of billiard orbits in Veech triangles. All cases covered in this work include a wide variety of triangles, but the question of the existence of periodic billiard orbits for all triangles is far from being fully contemplated.

Exoplanets and debris disks are examples of solar systems other than our own. As the dusty reservoirs of colliding planetesimals, debris disks provide indicators of planetary system evolution on orbital distance scales beyond those probed by the most prolific exoplanet detection methods, and on timescales $\sim$10 Myr to 10 Gyr. The Solar System possesses both planets and small bodies, and through studying the gravitational interactions between both, we gain insight into the Solar System's past. As we enter the era of resolved observations of debris disks residing around other stars, I add to our theoretical understanding of the dynamical interactions between debris, planets, and combinations thereof. I quantify how single planets clear material in their vicinity and how long this process takes for the entire planetary mass regime. I use these relationships to assess the lowest mass planet that could clear a gap in observed debris disks over the system's lifetime. In the distant outer reaches of gaps in young debris systems, this minimum planet mass can exceed Neptune's. To complement the discoveries of wide-orbit, massive, exoplanets by direct imaging surveys, I assess the dynamical stability of high mass multi-planet systems to estimate how many high mass planets could be packed into young, gapped debris disks. I compare these expectations to the planet detection rates of direct imaging surveys and find that high mass planets are not the primary culprits for forming gaps in young debris disk systems. As an alternative model for forming gaps in planetesimal disks with planets, I assess the efficacy of creating gaps with divergently migrating pairs of planets. I find that migrating planets could produce observed gaps and elude detection. Moreover, the inferred planet masses when neglecting migration for such gaps could be expected to be observable by direct imaging surveys for young, nearby systems. Wide gaps in young systems would likely still require more than two planets even with plantesimal-driven migration. These efforts begin to probe the types of potential planets carving gaps in disks of different evolutionary stages and at wide orbit separations on scales similar to our outer Solar System.

Conjugacy classes of nilpotent elements in complex semisimple Lie algebras are classified using the Bala-Carter theory. In this theory, nilpotent orbits in g are parametrized by the conjugacy classes of pairs (l,pl) of Levi subalgebras of g and distinguished parabolic subalgebras of [l,l]. In this thesis we present this theory and use it to give a list of representatives for nilpotent orbits in so(8) and from there we give a partition-type parametrization of them.

The equations of motion for a satellite with a rigid central body and a pair of appendages deforming due to thermal effects of the solar radiation are derived. The dynamics of the system is studied in two stages: (i) librational dynamics of the central body with quasi-steady thermally flexed appendages; (ii) coupled librational/vibrational dynamics of the spacecraft. Response of the system is investigated numerically over a range of system parameters and effect of the thermal deformations assessed. The study indicates that for a circular orbit, the flexible system can become unstable under critical combinations of system parameters and initial conditions although the corresponding rigid system continues to be stable. However, in eccentric orbits, depending on the initial conditions, thermally flexed appendages can stabilize or destabliIize the system. Attempt is also made to obtain an approximate closed-form (analytical) solution of the problem to quickly assess trends and gain better physical appreciation of response characteristics during the preliminary design. Comparisons with numerical results show approximate analysis to be of an acceptable accuracy for the intended objective. The closed-form solution can be used with a measure of confidence thus promising a substantial saving in time, effort, and computational cost. / Applied Science, Faculty of / Mechanical Engineering, Department of / Graduate

On-orbit targeting, guidance, and navigation relies on state vector propagation algorithms that must strike a balance between accuracy and computational efficiency. To better understand this balance, the relative position accuracy and computational requirements of numerical and analytical propagation methods are analyzed for a variety of orbits. For numerical propagation, several differential equation formulations (Cowell, Encke-time, Encke-beta, and Equinoctial Elements) are compared over a range of integration step sizes for a given set of perturbations and numerical integration methods. This comparison is repeated for two numerical integrators: a Runge-Kutta 4th order and a NLZD4/4. For analytical propagation, SGP4, which relies on mean orbital elements, is compared for element sets averaged with different amounts of orbit data.

The information presented in the thesis is a continuation of the Spacecraft Trajectory Optimization Suite (STOpS). This suite was originally designed and developed by Timothy Fitzgerald and further developed by Shane Sheehan, both graduate students at California Polytechnic State University, San Luis Obispo. Spacecraft utilizing low-thrust transfers are becoming more and more common due to their efficiency on interplanetary trajectories, and as such, finding the most optimal trajectory between two planets is something of interest. The version of STOpS presented in this thesis uses Multiple Gravity-Assist Low-Thrust (MGALT) trajectories paired with the island model paradigm to accomplish this goal. The island model utilizes four different global search algorithms: a Genetic Algorithm, Differential Evolution, Particle Swarm Optimization, and Monotonic Basin Hopping. The first three algorithms were featured in the initial version of STOpS written by Fitzgerald [1], and were subsequently modified by Sheehan [2] to work with a low-thrust adaptation of STOpS. For this work, Monotonic Basin Hopping was added to aid the suite with the MGALT trajectory search. Monotonic Basin Hopping was successfully validated against four different test functions which had been used to validate the other three algorithms. The purpose of this validation was to ensure Monotonic Basin Hopping would work as intended, ensuring it would work in cooperation with the other three algorithms to produce a near optimal solution. After verifying the addition of Monotonic Basin Hopping, all four algorithms were used in the island model paradigm to verify MGALT STOpS’ ability to solve two known orbital transfer problem. The first verification case involved an Earth to Mars transfer with fixed thruster parameters and a predetermined time of flight. The second verification case involved a transfer from Earth to Jupiter via a Mars gravity assist; two different versions of the verification case were solved against trajectories produced by industry optimization software, the Satellite Tour Design Program Low-Thrust Gravity Assist and the Gravity Assisted Low-thrust Local Optimization Program. In the first verification case, MGALT STOpS successfully validated the Earth to Mars trajectory problem and found results agreeable to literature. In the second verification case, MGALT STOpS was partially successful in validating the Earth to Mars to Jupiter trajectory problems, and found results similar to literature. The final software produced for this work is a trajectory optimization suite implemented in MATLAB, which can solve interplanetary low-thrust trajectories with or without the inclusion of gravity assists.