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Introduç˜Ao `A Teoria De Movimentos Ressonantes Em Sistemas

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Introduç˜Ao `A Teoria De Movimentos Ressonantes Em Sistemas Universidade de S~aoPaulo Instituto de Astronomia, Geof´ısicae Ci^enciasAtmosf´ericas Departamento de Astronomia Raphael Alves Silva Introdu¸c~ao`aTeoria de Movimentos Ressonantes em Sistemas Planet´arios S~aoPaulo Raphael Alves Silva Introdu¸c~ao`aTeoria de Movimentos Ressonantes em Sistemas Planet´arios Disserta¸c~aoapresentada ao Departamento de Astronomia do Instituto de Astronomia, Geof´ısica e Ci^enciasAtmosf´ericasda Universidade de S~aoPaulo como requisito parcial para a ob- ten¸c~aodo t´ıtulode Mestre em Ci^encias. Area´ de Concentra¸c~ao:Astronomia Orientador(a): Prof.a Dr.a Tatiana Alexan- drova Michtchenko. Vers~aocorrigida. O original encontra-se dispon´ıvel na Unidade. S~aoPaulo Aos meus pais Agradecimentos A Deus, primeiramente, por me dar a for¸capara continuar, e a sabedoria necess´aria para trilhar o caminho. A` minha fam´ılia,por todo o apoio dedicado diante das escolhas que tomei em minha vida, desde o in´ıciode minha trajet´oriaacad^emica, quando trilhei os primeiros passos da carreira cient´ıfica.Por sempre terem acreditado em minha capacidade, e me estimulado a al¸carv^oosmaiores. A` professora Tatiana A. Michtchenko, por haver aceitado assumir o papel de ser minha orientadora e, ao longo desses dois anos, pela paci^encia,conselhos e aux´ıliosdurante o processo de desenvolvimento deste trabalho; por haver me indicado um caminho a tomar, nos momentos em que pensei haver chegado `abecos sem-sa´ıda. Aos professores Sylvio Ferraz-Mello e Ramachrisna Teixeira, por suas aulas, discuss~oes e ensinamentos, vozes de sabedoria e experi^enciae verdadeiras inspira¸c~oespara a minha carreira acad^emica. Aos colegas do IAG: Ronaldo e Douglas, com quem dividi o espa¸coda sala E-307, e que compartilharam comigo os questionamentos e discuss~oesque permeiam a cabe¸cado jovem pesquisador, seja no que tange os assuntos concernentes `apesquisa que desenvolvemos, seja quanto a assuntos muito mais profundos, perpectivas de carreira e vida profissional. Ao Paulo Jackson, que me acolheu e me deu muitas dicas sobre o Mestrado no IAG. Ao Hugo e ao Eduardo, membros mais experientes do grupo de Din^amicaOrbital, e que sempre se mostraram prontos a me auxiliar, no que fosse poss´ıvel. A` minha segunda fam´ılia,membros da Rep´ublicaTFP (Tradi¸c~ao,Fam´ıliae Proprie- dade), pelo `amor, carinho e dedica¸c~ao',e por haverem compartilhado de minhas ang´ustias acerca de meu trabalho e das dificuldades inerentes, e diante disso, haverem me apoiado e me aconselhado, ao longo de madrugadas regadas a cervejas e risadas. Minha gratid~ao aqui ´ededicada a Alda, Gustavo, Guilherme, Felipe e Irapuan. Ao IAG e seus funcion´arios,por contribuirem para o bom funcionamento da institui¸c~ao, e para a manuten¸c~aode um ambiente de trabalho agrad´avel. Esta tese/disserta¸c~aofoi escrita em LATEX com a classe IAGTESE, para teses e disserta¸c~oesdo IAG. \O pensamento n~aopassa de um clar~aona noite; mas esse clar~aorepresenta tudo." Henri Poincar´e(1854-1912) \For it is the duty of an astronomer to compose the history of the celestial motions through careful and expert study." Nicolaus Copernicus (1473-1543) Resumo Neste trabalho, desenvolvemos um modelo anal´ıtico para o potencial perturbador de um problema de tr^escorpos coplanar, utilizando o formalismo hamiltoniano expressado atrav´esdas coordenadas can^onicasde Jacobi. Consideramos um sistema de tr^esmassas pontuais, dentro de uma configura¸c~aode resson^ancia de movimentos m´ediosgeneralizada do tipo n1=n2 = (p + q)=p. Desenvolver um modelo anal´ıticose mostra vantajoso no sentido de possibilitar o melhor entendimento da din^amica dos fen^omenos que caracterizam um determinado sistema, a identifica¸c~aomais imediata das principais vari´aveis do problema, e a melhor caracteriza¸c~ao dos comportamentos poss´ıveis em fun¸c~aodestas ´ultimas. Aplicamos a metodologia cl´assicade expans~oesem s´eriesde pot^encias das excentrici- dades e semi-eixos maiores. Usando expans~oesem s´eriesde Taylor e Fourier, obtivemos a express~aocompleta para o potencial perturbador, em fun¸c~aodos elementos orbitais λi; $i; ai; ei, i = 1; 2. Os termos de curto per´ıodo foram eliminados por simples inspe¸c~ao visual, removendo as contribui¸c~oesadvindas de argumentos que continham os ^angulos r´apidos λ1 e λ2, exceto aqueles nos quais estavam contidos as combina¸c~oescr´ıticas. Esse processo de m´edia`manual' reduziu o sistema a um problema de dois graus de liberdade. O comportamento da fun¸c~aoanal´ıticadesenvolvida foi testado para o caso espec´ıfico de resson^ancia3:1. Nossos resultados foram comparados com `aqueles gerados atrav´esde um modelo semi-anal´ıticooutro, o qual tamb´emcomputa um Hamiltoniano m´ediopara um sistema ressonante. Abstract In this work we develop an analytical model for the disturbing potential of the pla- nar Three-Body problem, using the Hamiltonian formalism expressed in canonical Jacobi coordinates. We consider a system of massive points in a general mean motion resonant configuration of the general type n1=n2 = (p + q)=p. The development of an analytical model is very important once it allows us to better understand the dynamical phenomena of a system, identifying the main variables and the possible regimes of motion that the system may assume. As a methodology, we apply the classical method of expansions in power series of eccentricities and semi-major axes. Through the application of Taylor and Fourier series expansions, we obtain the exact expression for the disturbing potential as a function of the orbital elements λi; $i; ai; ei, i = 1; 2. The short-period terms were eliminated by removing those terms whose arguments were depending at least on one of the fast angles λ1 and λ2, except for those which contain the critical combination of the fast angles. This average process allows us to reduce the system to a two-degree-of-freedom problem. The behavior of the obtained analytical function is tested for the specific case of a 3:1 MMR. We compare the results with a semi-analytical model, which also gives the averaged Hamiltonian of a resonant system - in Jacobi coordinates. Lista de Figuras 2.1 Sistema exemplificando 4 corpos, com os vetores posi¸c~aodados dentro de um sistema de coordenadas inercial arbitr´ariocom origem em S. 29 2.2 Sistema de 4 corpos, com os vetores posi¸c~aodados em coordenadas referen- ciadas ao baricentro B do sistema. 31 2.3 Sistema de 4 corpos, com os vetores posi¸c~aodados em coordenadas referen- ciadas ao corpo de massa m0. ......................... 35 2.4 Sistema de 4 corpos, com os vetores posi¸c~aodados para um sistema de coordenadas de Jacobi. O ponto B1 ´eo baricentro de m0 e m1. B2 ´eo baricentro de m0, m1 e m2............................ 38 3.1 Representa¸c~aogr´aficado Problema de Tr^esCorpos, referenciado em um sistema inercial arbitr´ariode origem S..................... 53 3.2 Representa¸c~aogr´aficado sistema utilizando as coordenadas de Jacobi. O ponto CM indica a posi¸c~aodo centro de massa de m0 e m1; ´eo ^angulo entre formado entre os vetores posi¸c~aono sistema de Jacobi, r1 e r2; ri;j ´eo vetor de dist^anciaentre os repectivos corpos i e j. (Figura retirada da tese de Doutoramento de Eduardo A. Ines.) . 55 (j) 5.1 (a) Comportamento da s´eriedos coeficientes de Laplace b1=2(~α), 0 ≤ j ≤ 5, em termos do´ındice i; "2 ∼ 1 eα ~ = 0:48. Essas s´eries,que nada mais s~aoque P 2i os somat´oriosnas f´ormulas dos coeficientes i Biα , aparecem para a parte direta do potencial perturbador. (b) Comportamento da fun¸c~aoanal´ıtica dos coef. Laplace, em termos do par^ametro α, para distintos valores do ´ındice j...................................... 89 5.2 Curvas de < R > vs e2, para os valores de e1 conforme indicado. S~aoto- mados diferentes intervalos para os coeficientes n, j e N. K e AM s~ao ◦ ◦ constantes de movimento, σ1 = 90 , ∆$ = 0 . Em linha cont´ınua vermelha tem-se o compotamento do modelo semi-anal´ıticousado como padr~ao.. 91 5.3 Curvas de @ < R > =@e2 vs e2, tomadas para cada um dos casos de excen- tricidades e1 indicados. O modelo anal´ıticofoi computado novamente para diferentes configura¸c~oesde ordens em n, j e N. A linha vermelha cont´ınua novamente indica o comportamento utilizado como padr~ao,neste caso, a curva derivada num´erica.. 93 5.4 Curvas de < R > vs e2, tomadas para e1 = 0:005. Os gr´aficosindicam os valores m´aximosde harm^onicosconsiderados. A linha vermlha cont´ınua apresenta o comportamento do modelo padr~ao semi-anal´ıtico. 95 5.5 Curvas de < R > vs e2, tomadas para e1 = 0:4. Nesse caso, testamos diferentes configura¸c~oespara as ordens de expans~ao.. 96 5.6 Curvas de @ < R > =@e2 vs e2, para e1 = 0:005. 96 5.7 Curvas de @ < R > =@e2 vs e2, para e1 = 0:005. Aqui, mostramos as configura¸c~oesde ordem m´axima que podem ser adotadas, de maneira tal a obter-se a melhor correspond^enciaposs´ıvel com a curva padr~ao(vermelha cont´ınua). 97 5.8 Curvas de @ < R > =@e2 vs e2, para e1 = 0:4. 97 5.9 Curvas para a raz~aoentre os par^ametrosexterno e interno, γext/γint, se- guindo a nota¸c~aoque denominamos no Cap´ıtulo3, Se¸c~ao(3.1.1). 98 5.10 Planos representativos (e1; e2) tomados dentro da resson^anciaexata 3/1, para raz~oesde massa m2=m2: (a)0:1, (b)0:5 e (c)1. Os valores positivos (negativos) de e1 indicam que σ1 = 0 (π=2), enquanto que valores positivos (negativos) de e2 significam ∆$ = 0 (π).
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