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Numerical Study of Earth-Mars Trajectories with Lunar Gravity Assist Manoeuvres (Versão Corrigida Após Defesa)

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Numerical Study of Earth-Mars Trajectories with Lunar Gravity Assist Manoeuvres (Versão Corrigida Após Defesa) UNIVERSIDADE DA BEIRA INTERIOR Engenharia Numerical study of Earth-Mars trajectories with lunar gravity assist manoeuvres (Versão corrigida após defesa) Rui Pedro Martins Oliveira Dissertação para a obtenção do Grau de Mestre em Engenharia Aeronáutica (Ciclo de estudos integrado) Orientador: Prof. Doutor Kouamana Bousson Covilhã, Dezembro de 2017 ii À minha mãe o meu muito obrigado. iii iv Resumo Um estudo numérico de trajectórias entre Terra e Marte com manobras de assistência gravita- cional é realizado. Os métodos e modelos utilizados para alcançar esse objectivo são descritos e seus resultados são comparados com os valores reais. Todas as trajectórias de transferência são calculadas pela primeira vez com a formulação do problema dos dois corpos e, em seguida, usando um software de código aberto. Este software é o General Mission Analysis Tool desen- volvido pela NASA. As manobras de assistência gravitacional são normalmente utilizadas para missões aos plane- tas exteriores, a Mercúrio ou quando são usados sistemas de propulsão de baixo impulso. As manobras lunares de assistência gravitacional nunca foram usadas com sucesso nas trajectórias Terra-Marte. Sete missões reais, com órbitas de transferência directa, são usadas para comparar os resulta- dos obtidos das trajectórias com manobras de assistência gravitacional lunar. Não foi possível aplicar essa manobra a todas as missões analisadas. Para as restantes missões foi realizada uma passagem pela Lua, que resulta na diminuição da energia de lançamento quando comparada a uma órbita de transferência directa na mesma altura de injecção. No entanto, quando se comparam com as alturas de injecção reais, apenas uma missão diminui a sua energia de lança- mento. Os resultados mostram que, em algumas circunstâncias, esta manobra pode diminuir significa- tivamente a energia de lançamento necessária para alcançar Marte. Palavras-chave Transferência orbital; Lunar gravity assist; Multiple shooting method; Marte v vi Abstract A numerical study of Earth-Mars trajectories with lunar gravity assist manoeuvres is performed. The methods and models used to achieve this goal are described and their results are compared to the actual values. All transfer trajectories are first computed under the two-body problem formulation and then recurring to an open-source software. This software is the General Mission Analysis Tool maintained by NASA. Gravity assist manoeuvres are most used to missions to the outer planets, to Mercury, or when low-thrust propulsion systems are used. Lunar gravity assist manoeuvres were never successfully used in Earth-Mars trajectories. Seven real missions, with direct transfer orbits, are used to compare the results obtained from trajectories with lunar gravity assist manoeuvres. It was not possible to apply this manoeuvre to all missions analysed. For the missions where a lunar flyby was performed, the manoeuvre results in a decrease of the launch energy when comparing to a direct transfer orbit at the same injection epoch. However, when comparing to the actual injection epoch, only one mission decrease its launch energy. The results show that in some circumstances this manoeuvre can decrease significantly the launch energy need to reach Mars. Keywords Transfer orbit; Lunar gravity assist; Multiple shooting method; Mars vii viii Contents 1 Introduction 1 2 Modelling 7 2.1 Astrodynamics models ............................... 7 2.1.1 Two-body problem ............................. 7 2.1.2 Kepler’s equation ............................. 8 2.1.3 Lambert’s problem ............................. 10 2.1.4 Gravity assist model ............................ 10 2.1.5 Patched conic method ........................... 12 2.1.6 B-plane targeting ............................. 12 2.1.7 Keplerian and Cartesian elements ..................... 13 2.1.8 Epoch format and coordinate system ................... 14 2.2 Numerical methods ................................ 14 2.2.1 Differential correction ........................... 15 2.2.2 Optimisation ................................ 18 3 Numerical Simulation 21 3.1 Direct transfer ................................... 21 3.2 Lunar gravity assist trajectory ........................... 23 4 Conclusions 29 References 31 A Appendix 33 ix x List of Figures 1.1 Representations of a segment of the trajectories of STEREO space probes at LGA manoeuvres. Moon’s orbit is represented in green, STEREO-A’s orbit in red and STEREO-B in yellow. At the centre, the blue dot represents the Earth. (credit: NASA) ....................................... 5 2.1 Diagram of a gravity assist manoeuvre. Increase in spacecraft energy after the flyby. ....................................... 10 2.2 Diagram of hyperbolic passage. .......................... 11 2.3 Geometry of B-plane with spacecraft’s trajectory represented in yellow. ..... 13 2.4 Change in partial derivatives for different perturbations magnitudes. ...... 16 2.5 Representation of multiple shooting method. ................... 17 2.6 Representation of an grid pattern on Moon’s SOI. The Moon is illustrated in orange. 18 3.1 Level 1 pruning (left) and level 2 pruning (right). ................. 24 3.2 The feasible region in Earth-centred coordinates. The color bar shows the lunar periapsis radius. The red line represents the Moon’s Orbit. ............ 24 3.3 The velocity magnitude (left) and eccentricity of the pruned orbits with respect to the Earth (right), for the set of feasible epoch determined in level 2 ..... 25 3.4 The parameters BT and BR. The color bar shows the epochs in the Modified Julian Date format. ................................... 25 3.5 Injection epoch vs epoch when the pruning and grid search occurred. ...... 26 3.6 Graphical representation of the lunar gravity assist. ............... 27 A.1 2001 Mars Odyssey ................................. 34 A.2 Mars Exploration Rover-A ............................. 35 A.3 Mars Science Laboratory .............................. 35 A.4 Mars Orbiter Mission ................................ 36 A.5 Mars Atmosphere and Volatile Evolution ...................... 36 A.6 Trace Gas Orbiter ................................. 37 A.7 MER-A LGA: Plane xy ............................... 37 A.8 MER-A LGA: Plane xz ............................... 38 A.9 MER-A LGA: Plane yz ............................... 38 xi xii List of Tables 3.1 Absolute error of two-body problem and GMAT results. .............. 22 3.2 Results from the b-plane targeting scheme (Bp-T) and the multiple shooting method scheme (MSM), where error1 and error2 represent the discontinuity in the patch points in the first iteration and rp the lunar periapsis radius achieved. ...... 26 3.3 Results of the lunar gravity assist manoeuvre for the MER-A mission. ....... 27 3.4 Results of the lunar gravity assist manoeuvre . .................. 27 A.1 Physical data of Earth, Moon, Mars and Sun. ................... 33 A.2 The real injections velocities and those determined with the two-body problem and with the GMAT. ................................ 33 A.3 The injection ephemeris in Earth MJ2000Eq reference frame. .......... 34 A.4 Injection and orbit insertion epochs. ....................... 34 xiii xiv List of Acronyms 2bp Two-body Problem EDL Entry, Descent and Landing ESA European Space Agency GMAT General Mission Analysis Tool ISRO Indian Space Research Organisation ISS International Space Station JD Julian Date LGA Lunar Gravity Assist MAVEN Mars Atmosphere and Volatile Evolution MER-A Mars Exploration Rover-A MER-B Mars Exploration Rover-B MJD Modified Julian Date MOM Mars Orbiter Mission MSL Mars Science Laboratory MSM Multiple Shooting Method NASA National Aeronautics and Space Administration SOI Sphere of Influence STM State-Transition Matrix TCM Trajectory Correction Manoeuvre TGO Trace Gas Orbiter xv xvi Nomenclature a Semi-major axis [km] b Semi-minor axis [km] BT B-Plane parameter [km] BR B-Plane parameter [km] 2 2 C3 Characteristic energy [km /s ] e Orbit eccentricity [−] e Orbit eccentricity vector [−] E Eccentric anomaly [rad] f General mathematical function [−] F Hyperbolic anomaly [rad] h Specific angular momentum [km2/s] h Specific angular momentum vector [km2/s] i Orbital inclination [rad] I Identity matrix [−] J Jacobian matrix [−] m Mass [kg] M Mean anomaly [rad] p Semi-parameter [km] r,R Position [km] r,R Position vector [km] R B-plane axis [−] Rot Rotation matrix [−] rp Periapsis radius [km] S B-plane axis [−] t Time [s] tflight Time of flight [s] T B-plane axis [−] v,V Velocity [km/s] v,V Velocity vector [km/s] w General mathematical variable [−] X State vector [−] Greek Letters α Angle between two vectors [rad] β Angle in flyby [rad] γ Angle in flyby [rad] θ Spherical coordinate [rad] λ Angle between Moon’s orbital plane and the hyperbolic [rad] asymptote µ Gravitational parameter [km3/s2] xvii ν True anomaly [rad] ξ Specific orbit energy [km2/s2] ϕ Spherical coordinate [rad] ! Argument of periapsis [rad] ∆ Variation [−] Ω Right ascension of the ascending node [rad] Subscripts 0 Injection 1 Initial f Final k kth iteration ref Reference state In this work, some symbols may refer to more than one variable. In addition, specific variables can be designated with different units than those here described. These situations are defined by the context in which they are used. xviii Chapter 1 Introduction Sputnik 1 marks the beginning of space exploration. Placed into an elliptical low-Earth orbit by the Soviet Union in 1957, it was the first space vehicle to orbit a celestial body. In 1959 lunar exploration
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