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The effect of the atmosphere on near-earth satellites is evaluated by consideration of the drag perturbation and the associated dispersion parameters. Recommendations are made for each of these dispersion parameters. The recommendation concerning the density is that a dynamic density model be utilized instead of a static model. Included are numerical comparisons which quantify the error in predicted satellite positions which can occur due to an inferior density model alone. These comparisons are made for a variety of satellite orbits. / Master of Science

Being able to identify fuel efficient transfers between orbits is critical to planning and executing missions involving spacecraft. With a renewed focus on missions in cislunar space, identifying efficient transfers in the dynamical environment characterized by the Circular Restricted Three-Body Problem (CR3BP) will be especially important, both now and in the immediate future. The focus of this thesis is to develop a methodology that can be used to identify a valid low-cost transfer between a variety of orbits in the CR3BP. The approach consists of two distinct parts. First, tools related to dynamical systems theory and manifolds are used to create an initial set of possible transfers. An optimization scheme is then applied to the initial transfers to obtain an optimized set of transfers. Code was developed in MATLAB to implement and test this approach. The methodology and its implementation were evaluated by using the code to identify a low-cost transfer in three different transfer cases. For each transfer case, the best transfers from each set were compared, and important characteristics of the transfers in the first and final sets were examined. The results from those transfer cases were analyzed to determine the overall efficacy of the approach and effectiveness of the implementation code. In all three cases, in terms of cost and continuity characteristics, the best optimized transfers were noticeably different compared to the best manifold transfers. In terms of the transfer path identified, the best optimized and best manifold transfers were noticeably different for two of the three cases. Suggestions for improvements and other possible applications for the developed methodology were then identified and presented. / Master of Science / Being able to identify fuel efficient transfers between orbits is critical to planning and executing missions involving spacecraft. With a renewed focus on lunar missions, identifying efficient transfers between orbits in the space around the Moon will be especially important, both now and in the immediate future. The focus of this thesis is to develop a methodology that can be used to identify a valid low-cost transfer between a variety of orbits in the space around the Moon. The approach was evaluated by using the code to identify a low-cost transfer in three different transfer cases. The results from those transfer cases were analyzed to determine the overall efficacy of the approach and effectiveness of the implementation code. Suggestions for improvements and other possible applications for the developed methodology were then identified and presented.

Dr. Schumacher can be emailed at paul.schumacher@us.af.mil / Presented herein are some analytical results available from regularization of the differential equations of satellite motion. True-anomaly regularization is developed as a special case of a more general Sundman-type transformation of the independent variable (time) in the equations of motion. Constants of the unperturbed motion are introduced as extra state variables, and regularization with several types of coordinates is considered. Because analytical results are sought, those regularizing transformations which produce rigorously linear governing equations are of main interest. When solutions of the linear regular equations in the true-anomaly domain are examined, it is found that the initial value and boundary value problems of unperturbed motion, typically requiring iterative solutions of the time equation, can be solved with only a single transcendental function evaluation per iteration cycle. Various means are described which can accelerate the evaluation of this function. The time equation developed in this study is a new universal relation between time of flight and true anomaly, and applies uniformly to all types of orbits, including rectilinear ones. It is a well-behaved function, the zero of which can be found reliably by Newton's method or other typical iteration methods. Once this time equation has been solved, the initial and final state vector on the transfer arc can be related to each other by rational algebraic formulae; no other transcendental function is needed. When the two problems are generalized by variation of parameters to the case of oblate-gravity perturbed motion, it is found that, to first order, the corrections of the unperturbed solution can be obtained by direct, noniterative formulae valid for all types of orbits. Moreover, it is possible to compute these corrections with only a single extra evaluation of the same transcendental function used in the unperturbed problem. Additional results are also presented, including exact solutions of the first-order averaged differential equations governing secular variations of the regular orbital elements in the true-anomaly domain. Complete universal expressions are given for the Keplerian state transition matrix in terms of the orbital transfer angle, and a simple midcourse guidance scheme is rederived in terms of universal variables valid for all non-rectilinear transfer orbits. / Ph. D. / incomplete_metadata

In this thesis we employ two recent analytical approaches to investigate the possible classes of traveling wave solutions of some members of recently derived integrable family of generalized Camassa-Holm (GCH) equations. In the first part, a novel application of phase-plane analysis is employed to analyze the singular traveling wave equations of four GCH equations, i.e. the possible non-smooth peakon, cuspon and compacton solutions. Two of the GCH equations do no support singular traveling waves. We generalize an existing theorem to establish the existence of peakon solutions of the third GCH equation. This equation is found to also support four segmented, non-smooth M-wave solutions. While the fourth supports both solitary (peakon) and periodic (cuspon) cusp waves in different parameter regimes. In the second part of the thesis, smooth traveling waves of the four GCH equations are considered. Here, we use a recent technique to derive convergent multi-infinite series solutions for the homoclinic and heteroclinic orbits of their traveling-wave equations, corresponding to pulse and front (kink or shock) solutions respectively of the original PDEs. Unlike the majority of unaccelerated convergent series, high accuracy is attained with relatively few terms. Of course, the convergence rate is not comparable to typical asymptotic series. However, asymptotic solutions for global behavior along a full homoclinic/heteroclinic orbit are currently not available.

La memòria consta de dues parts. En la primera d'elles s'estudien les òrbites homoclíniques i heteroclíniques associades als punts d'equilibri triangulars del problema restringit circular i pla per valors del paràmetre de masses compresos entre 0.1 i 0.5, es donen resultats referents a la seva forma i nombre. En la segona part òrbita halo al voltant del punt l1 del sistema terra-sol, utilitzant les idees geomètriques que proporciona la teoria dels sistemes dinàmics. Es comença l'estudi per models senzills a fi de veure l'essencial de la geometria del problema i la influencia de la lluna, per finalitzar utilitzant el model de sistema solar real donat per les efemèrides del JPL.

This Bachelor's thesis deals with fractals and orbits on Sierpinski carpets. We present the fundamental theory regarding fractals and some illustrative examples together with fractal billiards. In the latter part of the thesis we use elementary methods to present an original proof concerning the closure of some billiard orbits on Sierpinski carpets. A survey of the article Periodic Billiard orbits of self-similar Sierpinski Carpets, see [8], has been done, in which we make a discussion about one open question regarding reflections on the carpet. Furthermore, we state and prove some propositions related to this open question.

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.

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\'