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Tamires Dos Santos De Moura Estudo Preliminar Do TAMIRES DOS SANTOS DE MOURA ESTUDO PRELIMINAR DO POTENCIAL DE ALGUNS ASTEROIDES Guaratinguetá 2014 TAMIRES DOS SANTOS DE MOURA Estudo Preliminar do Potencial de Alguns Asteroides Trabalho de Graduação apresentado ao Conselho de Curso de Graduação em Licenciatura em Matemática da Faculdade de Engenharia de Guaratinguetá, Universidade Estadual Paulista, como parte dos requisitos para obtenção do diploma de Graduação em Licenciatura em Matemática. Orientador: Prof. Dr. Othon Cabo Winter Guaratinguetá 2014 Moura, Tamires dos Santos de M929e Estudo Preliminar do Potencial de Alguns Asteroides / Tamires dos Santos de Moura. - Guaratinguetá : [s.n.], 2014 113 f .: il. Bibliografia: f. 100-104 Trabalho de Graduação em Licenciatura em Matemática – Universidade Estadual Paulista, Faculdade de Engenharia de Guaratinguetá, 2014. Orientador: Prof. Dr. Othon Cabo Winter 1. Asteroides 2. Potencial gravitacional 3. Curvas em superfícies I. Título CDU 523.03 AGRADECIMENTOS A minha querida mãe, Celina e minha irmã Aline, que me deram muito apoio para que eu chegasse até essa etapa da minha vida. Obrigada por eu poder sempre contar com a ajuda de vocês e por estarem tão presentes na minha vida. Ao meu marido, Marcelo, que tem sido companheiro, amigo e confidente, me proporcionando apoio, calma, conselhos e incentivo nas horas que foram necessárias. As minhas amigas Vanessa e Michele, que sempre me apoiaram para que eu seguisse em frente e realizasse os meus sonhos. Ao meu orientador Professor Dr. Othon Cabo Winter que acreditou em mim, ouvindo pacientemente as minhas considerações e partilhando comigo as suas ideias e conhecimentos. Quero expressar o meu reconhecimento e gratidão. Aos meus antigos professores da Escola Estadual Paulo Virgínio, Cunha - SP, que sempre me incentivaram a nunca parar e sempre ir em busca de novos conhecimentos. Ao Professor André Amarante Luiz que pacientemente ouviu as minhas dúvidas e auxiliou-me. Hoje realizo um sonho e todos vocês de algum modo foram essenciais para que ele se tornasse realidade. EPÍGRAFE “O sucesso nasce do querer, da determinação e persistência em se chegar a um objetivo. Mesmo não atingindo o alvo, quem busca e vence obstáculos, no mínimo fará coisas admiráveis.” José de Alencar MOURA, T.S. Estudo preliminar do potencial de alguns asteroides. Trabalho de Conclusão de Curso (Licenciatura em Matemática) – Faculdade de Engenharia do Campus de Guaratinguetá, Universidade Estadual Paulista, Guaratinguetá, 2014. RESUMO A pesquisa que realizamos, e trazemos expressa neste texto, é um estudo preliminar sobre o formato de asteroides e nos permitiu compreender um pouco mais a dinâmica ao redor desses corpos, tendo em vista que as imagens que temos de asteroides são as mais irregulares possíveis. Neste trabalho, o asteroide é modelado por meio do método dos poliedros, que fornece uma precisão muito boa da forma irregular do corpo. Por meio de estudo dos modelos de potenciais gravitacionais para corpos não esféricos, implementação de rotinas computacionais e simulações numéricas foi realizada uma análise preliminar em relação ao formato dos asteroides 4179 Toutatis, 6489 Golevka, 2063 Bacchus, 1620 Geographos e 1998 ML14, como também regiões de estabilidade/instabilidade, computamos os coeficientes do potencial gravitacional. O trabalho possibilita ampliação não somente para o caso de asteroides, mas também para outros corpos não esféricos, contribuindo para o desenvolvimento de estudos direcionados a origem e evolução do sistema solar, e quem sabe da origem da Terra, e de novas tecnologias para modelagem e mapeamento de corpos não esféricos. Os principais resultados foram obtidos através da análise dos gráficos do formato e da planificação dos asteroides, que confirmou quão irregulares são esses corpos e como apresentam distribuição de massa não homogênea. Observamos o comportamento das curvas de velocidade zero e curvas de equipotencial, bem como suas respectivas superfícies. Também computamos alguns valores do potencial gravitacional e dos coeficientes harmônicos esféricos de cada objeto. Além disso, encontramos possíveis pontos de equilíbrio dos asteroides, exceto 4179 Toutatis, e analisamos sua estabilidade. PALAVRAS-CHAVE: Sistema Solar. Corpos irregulares. Asteroides. Superfície de um asteroide. Potencial gravitacional. MOURA, T.S. Preliminary study of the potential of some asteroids. Trabalho de Conclusão de Curso (Licenciatura em Matemática) – Faculdade de Engenharia do Campus de Guaratinguetá, Universidade Estadual Paulista, Guaratinguetá, 2014. ABSTRACT A survey we conducted, and bring expressed in this text is a preliminary study on the shape of asteroids and allowed us to understand a little more the dynamics around these bodies, in order that the images we have of asteroids are the most irregular possible. In this work, the asteroid is modeled by the method of the polyhedron, which provides a very good accuracy of the irregular shape of the body. Through study of models for non-spherical gravitational potential bodies, implementation of computational algorithms and numerical simulations a preliminary analysis was performed in relation to the shape of asteroids 4179 Toutatis, 6489 Golevka, 2063 Bacchus, 1620 Geographos and 1998 ML14, as well as regions of stability instability, we compute the coefficients of the gravitational potential. The work not only enables expansion for the case of asteroids, but also for other non-spherical bodies, contributing to the development of targeted studies the origin and evolution of the solar system, and perhaps the origin of the earth, and new technologies for modeling and mapping of non-spherical bodies. The main results were obtained by analyzing the graphics format and planning of asteroids, which confirmed how these bodies are irregular and show how distribution of non- homogeneous mass. Observe the behavior of the curves of zero velocity and equipotential curves as well as their respective surfaces. Also, compute some values of the gravitational potential and the spherical harmonic coefficients of each object. Furthermore, we find possible equilibrium points of asteroids except 4179 Toutatis, and analyze its stability. KEYWORDS: Solar system. Irregular bodies. Asteroids. Surface of an asteroid. Gravitational potential. LISTA DE FIGURAS Figura 1.1 - Imagem do asteroide 216 kleopatra. ..................................................................... 16 Figura 1.2 - Imagem do asteroide Ceres tirada pelo telescópio espacial Hubble ..................... 17 Figura 1.3 - Cinturão Principal ................................................................................................. 18 Figura 1.4 - Imagem do asteroide 433 Eros pela sonda NEAR – Shoemaker .......................... 21 Figura 1.5 - Imagem do asteroide Itokawa tirada pela sonda Hayabusa. ................................. 22 Figura 2.1 - Modelo de poliedros para o asteroide Castalia.. ................................................... 25 Figura 2.2 - Ilustração do formato do asteroide 4179 Toutatis ................................................ 27 Figura 2.3 - Gráfico em 3D representando o formato do asteroide 4179 Toutatis ................... 28 Figura 2.4 - Gráfico em 3D representando o formato do asteroide 4179 Toutatis com faces pintadas ..................................................................................................................................... 28 Figura 2.5 - Gráfico representando o formato do asteroide 4179 Toutatis com projeção nos planos xy, yz e xz ....................................................................................................................... 29 Figura 2.6 - Ilustração do formato do asteroide 1620 Geographos .......................................... 30 Figura 2.7 - Gráfico em 3D representando o formato do asteroide 1620 Geographos............. 30 Figura 2.8 - Gráfico em 3D representando o formato do asteroide 1620 Geographos com faces pintadas. .................................................................................................................................... 31 Figura 2.9 - Gráfico representando o formato do asteroide 1620 Geographos com projeção nos planos xy, yz e xz ....................................................................................................................... 31 Figura 2.10 - Ilustração do formato do asteroide 6489 Golevka .............................................. 32 Figura 2.11 - Gráfico em 3D representando o formato do asteroide 6489 Golevka ................ 33 Figura 2.12 - Gráfico em 3D representando o formato do asteroide 6489 Golevka com faces pintadas ..................................................................................................................................... 33 Figura 2.13 - Gráfico representando o formato do asteroide 6489 Golevka com projeção nos planos xy, yz e xz ....................................................................................................................... 34 Figura 2.14 - Ilustração do formato do asteroide 2063 Bacchus .............................................. 35 Figura 2.15 - Gráfico em 3D representando o formato do asteroide 2063 Bacchus ................ 35 Figura 2.16 - Gráfico em 3D representando o formato do asteroide 2063 Bacchus com faces pintadas ..................................................................................................................................... 36 Figura 2.17 - : Gráfico representando o formato do asteroide 2063 Bacchus com projeção nos planos xy, yz e xz ......................................................................................................................
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