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Optimization Methods Applied to Stellar Astrophysics

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Optimization Methods Applied to Stellar Astrophysics MSc 2.º CICLO FCUP 2013 Astrophysics Optimization methods applied to Stellar to applied methods Optimization Optimization methods applied to Stellar Astrophysics Sérgio Filipe Assunção Batista Dissertação de Mestrado apresentada à Faculdade de Ciências da Universidade do Porto em Engenharia Matmática 2013 Sérgio Filipe Assunção Batista Filipe Assunção Sérgio Globular Cluster Omega Centauri. Credits: ESO Optimization methods applied to Stellar Astrophysics Sérgio Filipe Assunção Batista Sérgio Filipe Assunção Batista Departamento de Matemática 2013 Supervisors: Sérgio António Gonçalves de Sousa, Investigador, Centro de Astrofísica da Universidade do Porto Nuno Cardoso Santos, Investigador Coordenador, Faculdade de Ciências da Universidade do Porto Todas as correções determinadas pelo júri, e só essas, foram efetuadas. O Presidente do Júri, Porto, ______/______/_________ FCUP 1 Optimization methods applied to Stellar Astrophysics I dedicate this thesis to all my family and to my girlfriend, Carolina Peres. FCUP 2 Optimization methods applied to Stellar Astrophysics Special Thanks I firstly would like to thank my parents, grandparents, uncles and my girlfriend for all the support that they gave me and also for their patience and help. I also would like to thank my supervisor Dr. Sérgio Sousa and my cosupervisor Dr. Nuno Santos for having accepted me as a masters student and for all the support and help given during the work of this thesys. Due to both my supervisors, it was possible during this work to learn how to program in C language and also to learn more about optimization methods and strategies. I am also thankfull to the fellowship grant CAUP2012-04UnF-BI, founded under the FP7 through Starting Grant agreement number 239953, supported by the European Research Council/European Community. I am also thankfull to had the opportunity of giving two talks in conferences, during this work. I also would like to thank Prof. Dr. João Nuno Tavares for the help and disponibility of helping in the search for optimization methods. FCUP 3 Optimization methods applied to Stellar Astrophysics Abstract The analysis of spectroscopic data of solar type stars provides a powerful tool to derive stellar parameters, such as effective temperature, surface gravity, microturbulence or metallicity. This can be done through the measurement of Fe I and Fe II weak absorption lines, which are then used to compute abundances, assuming local thermodynamics equilibrium. In this work, the derivation of stellar parameters is done by forcing excitation and ionization equilibrium of iron. The knowledge of these parameters is of crucial importance in several hot topics in astrophysics, such as the characterization of the solar neighborhood, the study of exoplanets, the composition and kinematics of our Milky Way or even for asteroseismology. The process implemented to derive the stellar parameters is based on the comparison between observations and theoretical models through an iterative process, which stops when the most appropriated model to fit the observations is found. In this work, I aim to identify the best optimization method which can be applied to the described problem. For this purpose, I will implement and test several optimization methods. In addition, I will derive stellar parameters using all the implemented methods to test their efficiency and to compare the obtained results with the tabulated values. Keywords: optimization; minimization; astrophysics; stellar parameters; FCUP 4 Optimization methods applied to Stellar Astrophysics Resumo A análise espectral de estrelas do tipo solar providencia uma ferramenta ponderosa para determinar parâmetros estelares, tais como a temperatura efectiva, gravidade à superfície da estrela, microturbulência ou a metalicidade. Estas determinações são efectuadas a partir da medição das larguras equivalentes das linhas espectrais de ferro neutro e ferro ionizado. Estas medições, são posteriormente utilizadas para computacionar as abundâncias, assumindo um regime de equilíbrio termodinâmico local. Por conseguinte, neste trabalho, a determinação dos parâmetros estelares é efectuada forçando um equilíbrio de excitação e de ionização do ferro. A determinação de parâmetros estelares é de importância crucial em diversas áreas da astrofísica, tais como a caracterização das estrelas presentes na vizinhança do Sol, estudar a composição e cinemática da nossa galáxia, ou até mesmo em áreas como a asterosismologia. O processo implementado para determinar os parâmetros estelares é baseado na comparação entre as observações e modelos teóricos, através de um processo iterativo. Este processo termina quando se encontra o modelo que melhor se ajusta ás obervações. Neste trabalho, pretendo identificar o melhor método de minimização que possa ser aplicado ao problema acima descrito. Para tal, irei implementar e testar alguns métodos de otimização. Igualmente, irei efectuar a determinação de parâmetros estelares de diversas estrelas, utilizando todos os métodos estudados e testar a sua eficiência. Por outro lado, irei também comparar os valores dos parâmetros estelares determinados através dos métodos implementados neste estudo com valores tabelados. Palavras-chave: otimização; minimização; astrofísica; parâmetros estelares; FCUP 5 Optimization methods applied to Stellar Astrophysics Indices Page: Indices of Figures……………………………….………………………………..………..…..9 Indices of Tables……………………………………………………………………………...15 Indices of Abbreviations……………………………………………………………………...17 Introduction……………………………………………………………….…………..……….19 1. Fundaments of stellar spectroscopy…………………………………….………..……..21 1.1. Basic Concepts of Spectroscopy………………….……...…………………..22 1.1.1. The physics of spectral lines ……………………………………….23 1.1.2. Origin of stellar spectra………………………………………………25 1.1.3. Spectral Sequence: The classification of stars……………………29 1.2. A spectroscopic method to derive stellar parameters……………………….30 1.2.1. MOOG…………………………………………………………………32 2. Deterministic Optimization methods……………………………………………………..35 2.1. Direct search methods………………………………………………………….37 2.1.1. Hooke and Jeeves method………………………………………….38 2.1.2. Downhill Simplex method……………………………………………39 2.1.3. Rosenbrock’s method………………………………………………..43 2.2. Conjugate direction methods: Powell’s quadratically convergent method..45 2.3. Conjugate Gradient methods: The Fletcher-Reeves algorithm…………….48 2.4. Quasi-Newton Methods………………………………………………………...50 2.4.1. BFGS method……………………………………………...…………51 FCUP 6 Optimization methods applied to Stellar Astrophysics Page: 3. Stochastic Optimization methods………………………………………………………...53 3.1. Simulated Annealing……………………………………………………...…….54 3.1.1. Modified Downhill Simplex method with a cooling scheme…......56 3.2. Genetic Algorithms……………………………………………………………...57 3.3. Particle Swarm Optimization…………………………………………………..60 4. Implementation procedure of optimization methods and Results…………………….65 4.1. Objective Function and the Amoeba implementation default version……..65 4.2. Sample of spectroscopic parameters for 451 stars in the HARPS GTO planet search program……………………………………………………….………68 4.3. Implementation of deterministic methods and Results……………………...69 4.3.1. Amoeba………………………………………………………….…….70 4.3.2. Amebsa……………………………………………………….……….76 4.4. Implementation of stochastic methods: Particle Swarm Optimization (PSO) and Results……………………………………………………………………………81 4.5. Implementation of a combination between the PSO and some deterministic methods and Results………………………………………………………………...88 4.5.1. PSO and Amoeba……………………………...…………………….90 4.5.2. PSO and Amebsa…………………………………………………….92 4.5.3. PSO and Powell conjugate directions method………………...….94 4.6. Analysis of convergence rates and times…………………………………….94 4.6.1. Performance of the Amoeba (Fortran) and Amoeba (C)…………95 4.6.2. Performances of the implemented optimization methods…….….96 FCUP 7 Optimization methods applied to Stellar Astrophysics Page: 5. Conclusions………………………………..……………………………………………...101 References…………………………………………………………………………………...103 Annexes……………………………………………………………………………….….….107 A. Glossary………………..………………………………………………………...107 B. Tables of Results………………………………………………………..………111 C. Communications in Conferences…………………………………………...…149 C.1. Abstract of the oral communication in the Ist Portuguese Meeting on Mathematics for Indrustry (6-8 of June, 2013)………………………….149 C.2. Abstract of the oral communication in the XXIII ENAA (Encontro Nacional de Astronomia e Astrofísica – 16-17 of July, 2013)…………151 FCUP 8 Optimization methods applied to Stellar Astrophysics (This page was left blank by the author.) FCUP 9 Optimization methods applied to Stellar Astrophysics Indices of Figures Fig.1.1 – Zoom of a high-resolution stellar spectrum of the star HD225097, obtained from the ÉLODIE catalogue, available online at http://atlas.obs-hp.fr/elodie/. The spectral line Hβ can be observed at around 4860Ǻ...……………………………...Page 22 Fig.1.2 – Example of how stellar spectra are formed. An incandescent light source is represented by the sun dot. By directly looking to the light source, a blackbody spectrum is observed (spectrograph 1). Photons of all wavelengths enter in the box filled with hydrogen. Photons that have energies which do not match the energy differences between the orbits of hydrogen, pass through the box. For example, photons which have wavelengths that correspond to the Balmer series of hydrogen have a high probability of being absorbed. This is observed by spectrograph 2. The dotted line represents the original continuum and shows the evidence of some continuous
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