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Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2023, Director: Ernest Fontich, [en] The Sitnikov Problem is the study of the dynamics of a particular system in the restricted Three-Body Problem, where the masses interact with each other through Newton’s law of universal gravitation. This paper studies the motion of the constrained mass of the Sitnikov Problem in order to demonstrate, through the Bernoulli Shift, the existence of periodic orbits.

Este artículo presenta dos formas de hacer una primera aproximación al rendezvous entre dos objetos mediante transferencias de Hohmann. En realidad se muestran las condiciones de posicionamiento en las que se puede realizar, los tiempos necesarios para su transición y los impulsos requeridos para su ejecución. En el caso de transferencias interplanetarias se estudian solo los casos entre órbitas circulares y coplanarias., Leer con atención y practicar con los ejemplos presentados

Treballs Finals de Grau d'Enginyeria Informàtica, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2022, Director: Gerardo Gómez Muntané, [en] In the modern world, artificial satellites orbiting around the Earth are used in a wide variety of areas. The tracking and modeling of their orbits is of utmost importance. In the following text, a mathematical framework for their study as well as a software implementation and the use of a software suite, GMAT, are presented.

The Satellite for Orbital Aerodynamics Research (SOAR) is a 3U CubeSat mission that aims to investigate the gas–surface interactions (GSIs) of different materials in the very low Earth orbit environment (VLEO), i.e. below 450 km. Improving the understanding of these interactions is critical for the development of satellites that can operate sustainably at these lower orbital altitudes, with particular application to future Earth observation and communications missions. SOAR has been designed to perform the characterisation of the aerodynamic coefficients of four different materials at different angles of incidence with respect to the flow and at different altitudes in the VLEO altitude range. Two conventional and erosion-resistant materials (borosilicate glass and sputter-coated gold) have first been selected to support the validation of the ground-based Rarefied Orbital Aerodynamics Research (ROAR) facility. Two further, novel materials have been selected for their potential to reduce the drag experienced in orbit whilst also remaining resistant to the detrimental effects of atomic oxygen erosion in VLEO. In this paper, the uncertainty associated with the experimental method for determining the aerodynamic coefficients of satellite with different configurations of the test materials from on-orbit data is estimated for different assumed gas–surface interaction properties. The presented results indicate that for reducing surface accommodation coefficients the experimental uncertainty on the drag coefficient determination generally increases, a result of increased aerodynamic attitude perturbations. This effect is also exacerbated by the high atmospheric density at low orbital altitude (i.e. 200 km), resulting in high experimental uncertainty. Co-rotated steerable fin configurations are shown to provide generally lower experimental uncertainty than counter-rotated configurations, with the lowest uncertainties expected in the mid-VLEO altitudes ($$\sim$$ ∼ 300 km). For drag coefficient experiments, configurations with two fins oriented at 90$$^{\circ }$$ ∘ were found to allow the best differentiation between surfaces with different GSI performance. In comparison, the determination of the lift coefficient is found to be improve as the altitude is reduced from 400 to 200 km. These experiments were also found to show the best expected performance in determining the GSI properties of different materials. SOAR was deployed into an orbit of 421 km $$\times$$ × 415 km with 51.6$$^{\circ }$$ ∘ inclination on 14 June 2021. This orbit will naturally decay allowing access to different altitudes over the lifetime of the mission. The results presented in this paper will be used to plan the experimental schedule for this mission and to maximise the scientific output.

Mubody is an astrodynamics open-source Python library focused on the libration points. Such points result from the equilibrium of the gravitational forces between two massive bodies as the Sun and Earth, for example. The library is mainly intended for the generation of orbits in these regions, which is not a straightforward process, specially if perturbations are considered. Currently, the library allows to generate Lissajous orbits in the second Lagrange point of the Sun-Earth system under the influence of perturbations such as the Earth orbit eccentricity. The next milestone, as a result of a master student work, is the incorporation of Halo orbits and the expansion to all three collinear libration points from any two massive bodies of the Solar System. This tool has been developed as part of a PhD, motivated by the need of performing mission analysis in libration point regions. Nevertheless, since its creation it has also proven to be an excellent academic tool for both enhancing the library itself and using its results for further studies (collision risk, thermal analysis, formation flight control, etc). As a result, the tool has rapidly evolved, building onto the knowledge and experience that the students gather while working on their academic projects (bachelor’s degree dissertations, master theses, subjects, internships). The participation on the library development provides students with experience in orbital mechanics, software design, version control and it compels them to ensure that their work can be readily used by others as it is properly documented. The project is hosted in GitLab under a MIT licence

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[ES] En los últimos años las misiones espaciales tienden a extenderse cada vez más, hasta llegar a salir del sistema solar. Esto ha llevado a optimizar los sistemas de propulsión actuales de forma que se alcancen mayores velocidades a menor coste económico. Uno de estos sistemas de propulsión son las velas solares que serán objeto de estudio en este trabajo. Varias misiones espaciales con esta tecnología se han desarrollado en esta última década, quizás la más exitosa hasta la fecha sea la misión IKAROS de la Agencia de Exploración Aeroespacial de Japón (JAXA). En este trabajo se desarrolla un simulador de órbitas utilizando el Software de Wolfram Mathematica comprobando su precisión con datos de la sonda Parker en su órbita alrededor del Sol de la web de Horizons/JPL(NASA). Partiendo de esta misión de la sonda Parker se simulará la nueva órbita que resultaría al desplegar una vela solar tras su paso por el perihelio, obteniendo de esta forma un aumento progresivo y significativo de la velocidad. También se estudiará el momento óptimo para el despliegue de la vela en función de velocidad, distancia al Sol, temperatura y tipo de vela entre otros parámetros. Se analizará la posibilidad de alcanzar velocidades cercanas a la velocidad de escape del sistema solar., [EN] In recent years, space missions tend to spread more and more, until they come out of the solar system. This has led to optimizing current propulsion systems so that higher speeds are achieved at lower economic cost. One of these propulsion systems is the solar sails that will be the object of study in this thesis. Several space missions with this technology have been developed in the last decade, perhaps the most successful to date is the IKAROS mission of the Japan Aerospace Exploration Agency (JAXA). In this Master¿s thesis, an orbiting simulator is developed using Mathematica of Wolfram Software, checking its accuracy with data from the Parker probe in its orbit around the Sun from the Horizons / JPL (NASA) website. Starting from this mission of the Parker probe, the new orbit that would result from deploying a solar sail after passing through perihelion will be simulated, thus obtaining a progressive and significant increase in speed. The optimal moment for the deployment of the sail will also be studied according to speed, distance to the Sun, temperature and type of sail, among other parameters. The possibility of reaching speeds close to the escape velocity of the solar system will be analysed.

Aquest treball presenta l'estudi numèric de viabilitat de l'arquitectura de missió proposada per l'equip 4DCube de l'IEEC per a monitorització in situ de fragments petits de brossa espacial (

This paper deals with direct transfers from the Earth to Halo orbits related to the translunar point. The gravitational influence of the Sun as a fourth body is taken under consideration by means of the Bicircular Problem (BCP), which is a periodic time dependent perturbation of the Restricted Three Body Problem (RTBP) that includes the direct effect of the Sun on the spacecraft. In this model, the Halo family is quasi-periodic. Here we show how the effect of the Sun bends the stable manifolds of the quasi-periodic Halo orbits in a way that allows for direct transfers.

En aquest projecte s'estudia i s'avalua el rendiment d'un coet de diverses etapes de propel·lent sòlid durant tot el seu ascens atmosfèric i fins a la seva inserció orbital. Per tant, en primer lloc s'analitza la composició química del fuel i l'oxidant i s'elucida una proposta preliminar d'acord amb els resultats obtinguts del programa CEARUN de la NASA, que empra la termodinàmica química d'equilibri per analitzar problemes de combustions i coets. Posteriorment, es resolen numèricament les equacions diferencials que caracteritzen el moviment del vehicle al llarg de la trajectòria. Per fer l'estudi, s'ha dut a terme diferents simulacions modificant paràmetres de rellevància, com ara la relació d'àrees entre la gola i la sortida de la tovera, el temps d'encesa de cada etapa, la distribució de propel·lent entre fases o la resposta al fet d'incloure una tercera etapa. Finalment, es proposa una alternativa de coet optimitzat, tenint en compte els resultats obtinguts i d'acord amb la inserció orbital proposada. This project studies and evaluates the performance of a multi-stage solid propellant rocket throughout its atmospheric ascent and until its orbital insertion. Therefore, the chemical composition of the fuel and oxidant is first analyzed and, then, a preliminary proposal is elucidated under the results obtained from NASA's CEARUN program, which employs equilibrium chemical thermodynamics to analyze combustion and rocket problems. Subsequently, the differential equations that characterize the movement of the vehicle along the trajectory are solved numerically. To conduct the research, different simulations were executed by modifying parameters of relevance, such as the ratio of areas between the throat and the exit of the nozzle, the ignition time of each stage, the distribution of propellant between phases, or the response of including the third stage. Finally, an optimized rocket alternative is proposed, taking into account the results obtained and following the suggested orbital insertion.

The Bicircular Problem (BCP) is a periodic time dependent perturbation of the Earth-Moon Restricted Three-Body Problem that includes the direct gravitational effect of the Sun. In this paper we use the BCP to study the existence of Halo-like orbits around $$L_2$$ in the Earth-Moon system taking into account the perturbation of the Sun. By means of computing families of 2D invariant tori, we show that there are at least two different families of Halo-like quasi-periodic orbits around $$L_2$$ .

This paper is devoted to a new approach to construct the transfers from the Earth to the Earth-Moon (EM) $L_{4}$ using stable and unstable manifolds of planar quasiperiodic Lyapunov orbits (QPLOs) of the EM $L_{3}$ in the planar bicircular SunEarth-Moon system. Some planar QPLOs have stable manifolds intersecting the Earth parking orbits and unstable manifolds passing through the EM $L_{4}$ region, which gives a skeleton to design such transfers. Tentatively, the stable and unstable manifolds of a planar QPLO are employed to construct transfer segments connecting the Earth and $\mathrm{L}_{4}$ vicinity, respectively. The trajectories near the stable manifolds spent some time moving around $L_{3}$ and then approach the unstable manifolds towards $L_{4}$. To reduce the multi-revolution behavior around EM $L_{3}$, a multiple shooting algorithm is developed to switch from the stable to the unstable manifold, where additional maneuvers are performed at some distance from the planar QPLO to reduce the time spent in the EM $L_{3}$ vicinity. By such construction a spacecraft can visit and park around two high-cost far-away libration points in a single journey. Eliminating most of the loops around EM $L_{3}$, the quickest one among the example transfers requests about 175 days. Furthermore, it is presented how to utilize the stable manifolds of the planar QPLO alone to design faster transfers to the EM $L_{4}$ vicinity. By this construction, the lowest time of flight is about 61 days. The advantages of these two constructions are discussed.

[eng] In this thesis we explored some aspects of the dynamics around the Earth-Moon L1 and L2 points in the context of two Restricted Four Body Problems: the Bicircular Problem (BCP) and the Quasi-bicircular Problem (QBCP). Both the BCP and QBCP model the dynamics of a massless particle moving under the influence of the Sun, Earth, and Moon. Although these two models focus on the same system, it is relevant to study both because their behavior around the L2 is qualitatively different.These two models can be written in the Hamiltonian formalism as periodic time-dependent perturbations of the RTBP. To study these Hamiltonians, we used numerical tools tailored to these type of models to get an insight on the phase space. These two techniques are the reduction to the center manifold, and the computation and continuation of 2D tori.ßor the BCP, the analysis focused around the L2 point. The results obtained showed that the reduction to the center manifold, and the non-autonomous normal form computed in this thesis do not provide useful information about the neutral motion around L2. The approach taken was to compute families of 2D tori, and explore any connections and their stability. As a summary of this effort we identified a total of six families of 2D tori: two Lyapunov-type planar quasi-periodic orbits, and four vertical. One of the vertical families was obtained by direct continuation of Halo orbits from the RTBP. This showed that the family of Halo orbits from the RTBP survive in the BCP, with the understanding that this new family is Cantorian. It was also shown that one of the other vertical families is Halo-like. Hence, members of this family may be potential candidates for future space missions. However, these tori are hyperbolic, as opposed the ones coming directly from the RTBP Halo obits, which are partially elliptic. It was also shown that this family of Halo-like tori comes from a family of quasi-periodic orbits in the RTBP that are resonant with the frequency of the Sun. Hence, these family of Halo-like orbits in the BCP have their counterparts in the RTBP.ßor the QBCP, the focus of the analyses was there Earth-Moon L1 and L2 points. In this model, the reduction to the center manifold provided relevant qualitative information about the dynamics around L1 and L2. The main takeaway was that L1 and L2 had a similar qualitative behavior. In both cases there were two families of quasi-periodic Lyapunov orbits, one planar and one vertical. It was also shown that the quasi-periodic planar Lyapunov family underwent a (quasi-periodic) pitchfork bifurcation, giving rise to two families of quasi-periodic orbits with an out-of-plane component. Between them, there was a family of Lissajous quasi-periodic orbits, with three basic frequencies. Qualitatively, the phase space of the center manifold, as constructed in this thesis, resembled the phase space of the center manifold of the RTBP around L1 and L2. In the QBCP we also continued families of invariant 2D tori, and for both L1 and L2. In these cases, the quasi-periodic planar and vertical families were continued. The bifurcations of the quasi-periodic planar Lyapunov were identified. A conclusion from this numerical experiment was that the family of out-of-plane orbits born from the bifurcation seemed not to be the RTBP Halo counterparts in the QBCP. The RTBP Halo orbits do survive in the QBCP, but do not seem to be connected to the quasi-periodic planar Lyapunov family. ßinally, and also in the context of the BCP and the QBCP, numerical simulations to study transfers from a parking orbit around the Earth to a Halo orbit around the Earth-Moon L2 point were studied. The main conclusion is that the invariant manifolds of the target orbits studied intersect with potential parking orbits around the Earth. The relevance of this result is that it shows that there are one-maneuver transfers from a vicinity of the Earth to Earth-Moon L2 Halo orbits. This is not case when using the RTBP as reference model. Experiments were done for both the BCP and the QBCP, and in all cases is it was shown that the total cost in terms of ∆V and transfer time is comparable to other techniques requiring two or more maneuvers., [cat] En aquesta tesi explorem alguns aspectes de la dinàmica al voltant dels punts L1 i L2 Terra-Lluna en el context de dos problemes restringits de quatre cossos: el problema bicircular (PBC) i el problema quasi-circular (PQBC). Tant el PBC com el PQBC modelen la dinàmica d’una partícula sense massa que es mou sota la influència del Sol, la Terra i la Lluna. Tot i que aquests dos models es centren en el mateix sistema, és rellevant estudiar-los tots dos perquè el seu comportament al voltant de la L2 és qualitativament diferent. Aquests dos models es poden escriure en el formalisme hamiltonià com a pertorbacions periòdiques del problema restringit dels tres cossos (PRTC) dependents del temps. Per estudiar aquests hamiltonians, utlitzem eines numèriques adaptades a aquest tipus de models per obtenir una idea de l’espai de fases. Aquestes dues tècniques són la reducció a la variety central i el càlcul i la continuació de tors 2D. Per al PBC, l'anàlisi es centra al voltant del punt L2. Els resultats obtinguts mostren que la reducció a la varietat central i la forma normal no autònoma calculada en aquesta tesi no proporcionen informació útil sobre el moviment neutre al voltant de L2. L'enfocament adoptat es calcular famílies de tors 2D, i explorar les seves connexions i estabilitat. Com a resum d’aquest esforç, s'identifiquen un total de sis famílies de tors 2D: dues famílies d'òrbites quasi-periòdiques planes tipus Lyapunov i quatre verticals. Una de les famílies verticals s'obté per continuació directa de les òrbites Halo del PRTC. Això demostra que la família de les òrbites Halo del PRTC sobreviuen al PBC, entenent que aquesta nova família és cantoriana. També es demostra que una de les altres famílies verticals és semblant a les Halo. Per tant, els membres d’aquesta família poden ser candidats potencials per a futures missions espacials. No obstant això, aquests tors són hiperbòlics, a diferència dels que provenen directament de les Halo del PRTC, que són parcialment el·líptics. També es mostra que aquesta família de tors semblants a les Halo prové d’una família d’òrbites quasi-periòdiques del PRTC que són ressonants amb la freqüència del Sol. Per tant, aquestes famílies d’òrbites semblants a les Halo al PBC tenen els seus homòlegs al PRTC. Per al PQBC, el focus de les anàlisis es troba en els punts L1 i L2 Terra-Lluna. En aquest model, la reducció a la varietat central proporciona informació qualitativa rellevant sobre la dinàmica al voltant de L1 i L2. El principal resultat és que L1 i L2 tenen un comportament qualitatiu similar. En ambdós casos hi ha dues famílies d’òrbites quasi-periòdiques tipus Lyapunov, una plana i una vertical. També es demostra que la família plana quasi-periòdica tipus Lyapunov sobrevé una bifurcació tipus pitchfork (quasi-periòdica), donant lloc a dues famílies d’òrbites quasi periòdiques amb un component vertical. Entre ells, hi havia una família d’òrbites quasi-periòdiques tipos Lissajous, amb tres freqüències bàsiques. Qualitativament, l’espai de fase de la varietat central, tal com es construeix en aquesta tesi, s’assembla a l’espai de fase del la varietat central del PRTC al voltant de L1 i L2. Al PQBC també es continuen famílies de tors 2D invariants, tant per a L1 com per a L2. En aquests casos, es continuen les famílies planes i verticals quasi-periòdiques. Durant aquest procés es troben bifurcacions a les families d'òrbites quasi-periòdiques planes. Una conclusió d’aquest experiment numèric és que la família d’òrbites amb component vertical nascudes de la bifurcació no són les contraparts de les Halo de PRTC. Les òrbites Halo de PRTC sobreviuen en el PQBC, però no semblen estar connectades a la família plana quasi-periòdica. Finalment, i també en el context del PBC i el PQBC, es s'estudien simulacions numèriques per estudiar les transferències des d’una òrbita d’estacionària al voltant de la Terra fins a una òrbita Halo al voltant del punt L2 Terra-Lluna. La principal conclusió és que les varietats invariants de les òrbites objectiu estudiades passen molt a prop de la Terra. La rellevància d’aquest resultat és que mostra que hi ha transferències d’una maniobra des de d'una òrbita al voltat de Terra a les òrbites L2 Halo Terra-Lluna. No és així quan s’utilitza el PRTC com a model de referència. Es fan experiments tant per al BCP com per al QBCP, i en tots els casos es demostra que el cost total en termes de ∆V i temps de transferència és comparable a altres tècniques que requereixen dues o més maniobres.

Treballs Finals de Grau de Física, Facultat de Física, Universitat de Barcelona, Curs: 2020, Tutora: Mercè Romero-Gómez, In early 2020, studies using Gaia DR2 provided us with information about all the visible clusters existent in the Milky Way. Two of them stood out for having high latitude, namely Berkeley 20 and Berkeley 29. The goal of this study is to shed some light into the clusters origin using orbital analysis. The results indicate that these old, high latitude clusters have almost circular orbits around the Galactic centre which seem to confirm the disc origin. We also conclude that Berkeley 20 value of Z is purely coincidental, given its uniform distribution across all ranges of latitude. Berkeley 29 positive value of Z suggests that the line-of-nodes of the Galactic plane does not correspond to the one imposed in the warped models, given its distribution on the XY projection

This paper focuses on the capture of Near-Earth Asteroids (NEAs) in a neighbourhood of the $\mathrm{L}_{3}$ point of the Earth-Moon system. The dynamical model for the motion of the asteroid is the planar Earth-Moon-Sun Bicircular problem (BCP). It is known that the $\mathrm{L}_{3}$ point of the Restricted Three-Body Problem is replaced, in the BCP, by a periodic orbit of centre $\times$ saddle type, with a family of mildly hyperbolic tori that is born from the elliptic direction of this periodic orbit. It is remarkable that some pieces of the stable manifolds of these tori escape (backward in time) the Earth-Moon system and become nearly circular orbits around the Sun. In this work we compute this family of invariant tori and also high order approximations to their stable/unstable manifolds. We show how to use these manifolds to compute an impulsive transfer of a NEA to an invariant tori near $\mathrm{L}_{3}$. As an example, we study the capture of the asteroid $2006 \mathrm{RH} 120$ in its approach of 2006. We show that there are several opportunities for this capture, with different costs. It is remarkable that one of them requires a $\Delta v$ as low as 20 $\mathrm{m} / \mathrm{s}$.

En este artículo presentamos una aproximación topológica y geométrica al estudio de órbitas periódicas (existencia y localización) tomando como punto de partida problemas de mecánica celeste. Describimos, a grandes rasgos, diversas conjeturas abiertas, y presentamos un enfoque novedoso que incluye problemas de compactificación de trayectorias de escape en astrodinámica.

Renewed interest in Very Low Earth Orbits (VLEO) - i.e. altitudes below 450 km - has led to an increased demand for accurate environment characterisation and aerodynamic force prediction. While the former requires knowledge of the mechanisms that drive density variations in the thermosphere, the latter also depends on the interactions between the gas-particles in the residual atmosphere and the surfaces exposed to the flow. The determination of the aerodynamic coefficients is hindered by the numerous uncertainties that characterise the physical processes occurring at the exposed surfaces. Several models have been produced over the last 60 years with the intent of combining accuracy with relatively simple implementations. In this paper the most popular models have been selected and reviewed using as discriminating factors relevance with regards to orbital aerodynamics applications and theoretical agreement with gas-beam experimental data. More sophisticated models were neglected, since their increased accuracy is generally accompanied by a substantial increase in computation times which is likely to be unsuitable for most space engineering applications. For the sake of clarity, a distinction was introduced between physical and scattering kernel theory based gas-surface interaction models. The physical model category comprises the Hard Cube model, the Soft Cube model and the Washboard model, while the scattering kernel family consists of the Maxwell model, the Nocilla-Hurlbut-Sherman model and the Cercignani-Lampis-Lord model. Limits and assets of each model have been discussed with regards to the context of this paper. Wherever possible, comments have been provided to help the reader to identify possible future challenges for gas-surface interaction science with regards to orbital aerodynamic applications., Journal paper (accepted for publication in "Progress in Aerospace Sciences") Replacement: Corrected typos in Equations

The main goal of this thesis is to deepen the understanding of the influence that the various perturbations exert on the trajectory calculations performed by the JPL ephemeris model to provide high-fidelity ephemeris in support of spacecraft navigation and other activities related to Solar System bodies. One of the drawbacks behind such a complex method are the large machine-oriented technical requirements it has in terms of time and computational cost. The algorithm that integrates the full dynamical model of the N-Body differential equations of motion requires a continuous update of the dynamical states (positions and velocities) of a large number of bodies for each new integration step, regardless of the perturbation model applied or the nature and diversity of the sources of perturbations other than those gravitational considered. This work focuses especially to the gravitational implications of interplanetary trajectories, based on the fact that within its main and longest phase (the cruise phase) a spacecraft is essentially affected by a multi-body attraction. A procedure has been conducted to build a tool that, applied to any given interplanetary trajectory, has the ability of predicting which gravitational perturbations are not relevant and can thus be neglected without a significant loss of accuracy. Validation tests based on the BepiColombo interplanetary ephemeris trajectory planned mission proved that no significant differences are found when comparing the discrepancies exhibited by a numerical propagation of its initial conditions with the N-Body differential equations of motion, including all planetary perturbations or only those indicated as relevant by the developed 2D maps. A comparison is also carried out between two of the most popular special perturbation techniques, Cowell and Encke, which demonstrated that round off errors take its toll even working in double precision and showed that Encke’s method performs much better for interplanetary trajectories where there is a large difference in forces between the central dominant body and the perturbing bodies. Finally, a simple debugging test carried out by MatLab showed that both time and computational resources (amount of data) are sensibly reduced by not considering those negligible bodies suggested by the application of maps and tests. Thus, this result offer the possibility to save time both in the access and in the download of ephemeris data from the platform while avoiding loading times to create large data matrices in the workspace. El objetivo principal de esta tesis es profundizar en la comprensión de la influencia que las diversas perturbaciones ejercen en los cálculos de trayectoria realizados por el modelo de efemérides JPL que proporciona efemérides de alta fidelidad en apoyo de la navegación de naves espaciales y otras actividades relacionadas con los cuerpos del Sistema Solar. Una de las desventajas de un método tan complejo son los grandes requisitos técnicos orientados a la computación que tiene en términos de tiempo y costes de cálculo. El algoritmo que integra el modelo dinámico completo de las ecuaciones diferenciales de movimiento de N cuerpos requiere una actualización continua de los estados dinámicos (posiciones y velocidades) de un gran número de cuerpos celestiales para cada nuevo paso de integración, independientemente del modelo de perturbación aplicado o de la naturaleza y diversidad de las fuentes de perturbaciones distintas de las gravitacionales consideradas. Este trabajo se centra especialmente en las implicaciones gravitatorias de las trayectorias interplanetarias, basadas en el hecho de que dentro de su fase principal y más larga (la fase de crucero) una nave espacial se ve esencialmente afectada por una atracción multicuerpo. Se ha llevado a cabo un procedimiento para construir una herramienta que, aplicada a cualquier trayectoria interplanetaria, tiene la capacidad de predecir qué perturbaciones gravitatorias no son relevantes y, por lo tanto, pueden ser ignoradas sin una pérdida significativa de precisión. Las pruebas de validación basadas en la misión planificada de trayectoria de efemérides interplanetarias de BepiColombo demostraron que no se encuentran diferencias significativas al comparar las discrepancias exhibidas por una propagación numérica de sus condiciones iniciales con las ecuaciones diferenciales de movimiento incluyendo todas las perturbaciones planetarias o sólo aquellas indicadas como relevantes por los mapas 2D desarrollados. También se realiza una comparación entre dos de las técnicas de perturbación especiales más populares, Cowell y Encke, que demostraron que los errores de redondeo se cobran su peaje incluso trabajando con doble precisión y que el método de Encke funciona mucho mejor para trayectorias interplanetarias en las que existe una gran diferencia de fuerzas entre el cuerpo central dominante y los cuerpos perturbadores. Finalmente, una sencilla prueba de rendimiento del código, llevada a cabo por MatLab, mostró que tanto el tiempo como los recursos computacionales (cantidad de datos) se reducen sensiblemente al no considerar aquellos cuerpos previamente sugeridos como poco significativos por la aplicación de mapas y pruebas. De esta manera, este resultado ofrece la posibilidad de ahorrar tiempo tanto en el acceso como en la descarga de datos de efemérides desde la plataforma, evitando tiempos de carga para crear grandes matrices de datos en el espacio de trabajo. Outgoing

[EN] The present work is based on the modelization of two different and already existing ideas. On one hand the maneuver of the Flybys. This technique is used in almost all the interplanetary missions, based on using the big bodies gravity to gain a speed boost without any fuel consuption. On the other hand, is a project carried out in the last sixties. The NERVA project is one of the most important studies about thermal nuclear rockets existing, getting to a point of building and testing real models. The construction of the model is based on the integration of a fictional probe during its trayectory on a flyby. In addition to this, an impulse of one thermal nuclear rocked from the project is added using the Tsiolkovsky ecuation. In order to do that, we first integrate the actual trayectory of the Pioneer11 during its flyby, and later on we simulate the power on of the engines. The following study is based on improving the outter speed of the fictional probe. For that, the better point of power on of the engines is located on the periapsis of the orbit. At this point the probe has the maximun speed, and from this point the probe will go against the Planet´s gravity, trying to pull away from it. The passing altitue is another variable. With all this factor in consider, the fictional probe can reach up to 8 times the speed of a normal flyby, gaining access to a heavy time reduced kind of missions., [ES] Este trabajo se basa en la modelización de la unión de dos ideas ya existentes. La primera es la maniobra de un sobrevuelo. Esta maniobra es la técnica esencial en todas las misiones interplanetarias, basada en impulsar gravitacionalmente las sondas espaciales sin ningún gasto de combustible. La segunda es un proyecto llevado a cabo en los años sesenta. El proyecto NERVA realizó el estudio más importante hasta la fecha sobre los cohetes térmicos nucleares, construyendo y obteniendo sus especificaciones experimentalmente. La modelización consiste en integrar mediante el problema de los tres cuerpos una órbita ficticia que realiza un sobrevuelo y al que se le aplica un impulso térmico nuclear de un motor tipo NERVA. Para ello se integra primero el sobrevuelo de la sonsa Pioneer11 alrededor de Saturno y a través de la ecuación de Tsiolkovsky se añade el efecto del encendido de motores. El estudio del modelo se basa en obtener la máxima velocidad de salida posible para así optimizar el vuelo ficticio. Esto se consigue encendiendo los motores en el periapsis de la órbita, momento en el cual la sonda tiene la máxima velocidad y momento a partir del cual la gravedad del planeta sobrevolado comienza a actuar en contra. La variación de la altura de paso es otro factor influyente. Con todo, se consiguen velocidades de salida de hasta 183 km/s, una velocidad de hasta 8 veces superior a las de un sobrevuelo clásico.

[EN] In Classical Mechanics, the solution of the two body problem with known initialvalues is well established and finds many applications. For instance, it can be applied to approximate the position of an Earth satellite. However, as the Earth-satellite system is in practice affected by numerous other bodies, a differential correction is often necessary.The data we observe to achieve this improvement are the right ascension (α) and declination (δ), that yield the position of the satellite at an specific instant. The objective of this work is to correct the orbit giving a future position, based on a set of observations that define a preliminary orbit. Nevertheless, we aim to correct the orbit, getting an improved position using higher order iterative methods and measuring only angle coordinates., [ES] En Mecánica Clásica, la solución del problema de dos cuerpos con valores iniciales conocidos está bien establecida y encuentra numerosas aplicaciones. Por ejemplo, puede ser usada para aproximar la posición de un satélite artificial en torno a la Tierra. En cambio, el sistema Tierra-sat ́elite está, en la práctica, afectado por otros cuerpos, así que a menudo es necesaria una corrección diferencial de la órbita. Los datos con los que se va a trabajar para conseguir esta mejora son la ascensión recta (α) y la declinación (δ), las cuales definen la posición de un satélite en un instante específico. El objetivo de este trabajo es, partiendo de un conjunto de observaciones que definen una órbita preliminar, corregir la óbita dando una posición futura mejorada. Para ello, usaremos métodos iterativos de alto orden de convergencia libres de evaluación de matrices Jacobianas.

[CA] El present treball versa sobre l’estudi d’una missió de caràcter científic al planeta Mart amb l’objectiu de començar a assentar les bases per a un posterior estudi sobre la seua possible colonització, fent de l’espècie humana una espècie interplanetària. Per a això, es presenta un estudi sobre la trajectòria Terra - Mart seguida per una sonda i la seua posterior fase de descens en el planeta roig. Tot això es realitza des d’un punt de vista teòric pràctic, realitzant una primera aproximació manual als càlculs pertinents i posteriorment fent una sèrie de simulacions per mitjà de computador que permeten afinar estos resultats i simular-los. A més es comprovarà la influència de la resta de cossos celests, principalment Júpiter, sobre la trajectòria; es realitzarà un dimensionamiento inicial del sistema de llançament i s’estudiarà la data de llançament òptima que permeta reduir el temps de viatge i el consum de propel·lent., [EN] The present document is about a study of a scientific mission to the planet Mars with the aim of beginning to lay the foundations for a later study on its possible colonization, making the human species an interplanetary species. To achieve this, a study is presented on the trajectory Earth - Mars followed by a probe and its subsequent phase of descent on the red planet. All this is carried out from a theoretical practical point of view, making a first manual approach to the relevant calculations and then making a series of computer simulations to refine these results and simulate them. In addition, the influence of the rest of the celestial bodies, mainly Jupiter, on the trajectory will be verified; an initial dimensioning of the launch system will be carried out and the optimal launch date will be studied to reduce travel time and propellant consumption., [ES] El presente trabajo versa sobre el estudio de una misión de carácter científico al planeta Marte con el objetivo de comenzar a sentar las bases para un posterior estudio sobre su posible colonización, haciendo de la especie humana una especie interplanetaria. Para ello, se presenta un estudio sobre la trayectoria Tierra - Marte seguida por una sonda y su posterior fase de descenso en el planeta rojo. Todo ello se realiza desde un punto de vista teórico práctico, realizando una primera aproximación manual a los cálculos pertinentes y posteriormente haciendo una serie de simulaciones mediante computador que permitan afinar estos resultados y simularlos. Además se comprobará la influencia del resto de cuerpos celestes, principalmente Júpiter, sobre la trayectoria; se realizará un dimensionamiento inicial del sistema de lanzamiento y se estudiará la fecha de lanzamiento óptima que permita reducir el tiempo de viaje y el consumo de propelente.

[CASTELLÀ] El objetivo principal del trabajo es elaborar una propuesta de misión espacial para explorar el cinturón de Kuiper, una región del sistema solar situada a 30-55 ua del sol. Para la exploración se plantea el uso de una sonda capaz de llegar e investigar el cinturón. Este trabajo hace un estudio del estado del arte actual y se analizan cuatro temas distintos: el primero es el cinturón de Kuiper y las condiciones a las que nos enfrentamos en el espacio exterior. En segundo lugar, los sistemas de instrumentación de la sonda, donde se habla y se mencionan sistemas existentes y que han sido usados por otras sondas espaciales a lo largo de la historia. De esta manera, se propone un equipo de instrumentación preparado para el estudio de los cuerpos que constituyen el cinturón de Kuiper (objetos transneptunianos, cometas, planetoides). Los tipos de propulsores que puede llegar a equipar, haciendo una comparación y análisis de cada uno de ellos. Por último, se hace un estudio de la mecánica orbital para el cálculo aproximado de la trayectoria de la sonda. Se propone como punto de partida la órbita de la estación espacial internacional (ISS) y, con los cambios orbitales adecuados, la llegada a Júpiter, donde se analiza cómo llegar hasta Kuiper mediante asistencia gravitatoria y sin usarla. Se debe tener en cuenta que el diseño de una misión espacial es una tarea que requiere años y muchas personas especializadas en la materia. Por lo tanto, lo que recoge este trabajo es un humilde acercamiento al tema de viajes interplanetarios con sondas espaciales, teniendo en cuenta las limitaciones de tiempo y recursos para resolver los problemas que surjan durante el desarrollo del proyecto. [ANGLÈS] The main objective of this project is to elaborate a proposal of a space mission that aims to explore the Kuiper Belt. A payload that is able to properly investigate the Kuiper belt is presented for this exploration. This Project studies the recent state of the art and analyses four different aspects: In first place, the Kuiper belt and the conditions to be faced in the outer space, the instrumentation systems of the payload, where mentioned the previous systems employed in other payloads along the history. Thus, an instrumentation equipment towards the Kuiper belt objects is suggested. Lastly, a study of orbital mechanics for the approximate calculation of the payload path. The orbit of the International Space Station is suggested as a starting point, and with the right orbital changes, the arrival in Jupiter where we analyse how to arrive to Kuiper with -or without- the gravity assist. It should be taken into account that the work of designing a space mission involves years and many specialized people. Therefore, what the project accomplishes is a humble approach to the topic of interplanetary travelling with payloads, considering the limitations attached to the timing and resources when solving the problems that appear in the project’s development.

The restricted planar elliptic three body problem (RPETBP) describes the motion of a massless particle (a comet or an asteroid) under the gravitational field of two massive bodies (the primaries, say the Sun and Jupiter) revolving around their center of mass on elliptic orbits with some positive eccentricity. The aim of this paper is to show the existence of orbits whose angular momentum performs arbitrary excursions in a large region. In particular, there exist diffusive orbits, that is, with a large variation of angular momentum. The leading idea of the proof consists in analyzing parabolic motions of the comet. By a well-known result of McGehee, the union of future (resp. past) parabolic orbits is an analytic manifold $${\mathcal{P}^+}$$ (resp. $${\mathcal{P}^-}$$ ). In a properly chosen coordinate system these manifolds are stable (resp. unstable) manifolds of a manifold at infinity $${\mathcal{P}_\infty}$$ , which we call the manifold at parabolic infinity. On $${\mathcal{P}_\infty}$$ it is possible to define two scattering maps, which contain the map structure of the homoclinic trajectories to it, i.e. orbits parabolic both in the future and the past. Since the inner dynamics inside $${\mathcal{P}_\infty}$$ is trivial, two different scattering maps are used. The combination of these two scattering maps permits the design of the desired diffusive pseudo-orbits. Using shadowing techniques and these pseudo orbits we show the existence of true trajectories of the RPETBP whose angular momentum varies in any predetermined fashion.

SOAR (Satellite for Orbital Aerodynamics Research) is a CubeSat mission designed to investigate the interaction betweendifferent materials and the atmospheric flow regime in Very Low Earth Orbits (VLEO) and to demonstrate aerodynamicattitude and orbit control manoeuvres. Improving knowledge of the gas-surface interactions is important for the designof future satellites operating in lower altitude orbits and will enable the identification of materials which can minimisedrag or improve aerodynamic control, a key aim of the Horizon 2020 DISCOVERER project. In order to achieve theseobjectives, SOAR features two payloads: i) a set of steerable fins which provide the ability to expose different materialsor surface finishes to the oncoming flow with varying angle of incidence whilst also providing variable geometry toinvestigate aerostability and aerodynamic control; and ii) an Ion and Neutral Mass Spectrometer with Time-of-Flightcapability which enables accurate measurement of the in-situ flow composition, density, and thermospheric wind velocity.Using precise orbit and attitude determination information and the measured atmospheric flow characteristics the dragand side-force experienced by the satellite in orbit can studied and estimates of the aerodynamic coefficients calculated.This paper first presents the scientific design and operational concept of the SOAR mission, focusing on the stabilityand control strategy which enables the spacecraft to maintain the flow-pointing attitude required by the payloads. Themethodology for recovery of the (relative) aerodynamic coefficients from the measured orbit and in-situ atmosphericdata is then presented. Finally, the uncertainty of the resolved aerodynamic coefficients is estimated statistically usingsimulations.

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2017, Director: Antoni Benseny, [ca] Aquest treball tracta sobre la respresentació del àtoms d’hidrogen. A partir de solucions pròpies de l’equació d’Schrödinger independent del temps, es millorarà una aplicació capaç de representar els orbitals de l’hidrogen entesos com superfícies de densitat constant. En el programa s’incorporaran diferents formes de visualització, tant en 2D com en 3D i també s’argumentarà quina pot ser la manera més representativa de visualitzar un orbital. Prèviament al treball computacional s’ha de fer un tractament analı́tic per tal d’adaptar de manera eficient el treball computacional a l’estructura de les funcions que es volen representar.

This article has been accepted for publication in Monthly notices of the Royal Astronomical Society ©: 2017 The Authors. Published by Oxford University Press on behalf of the Royal Astronomical Society. All rights reserved. After a close encounter of two galaxies, bridges and tails can be seen between or around them. A bridge would be a spiral arm between a galaxy and its companion, whereas a tail would correspond to a long and curving set of debris escaping from the galaxy. The goal of this paper is to present a mechanism, applying techniques of dynamical systems theory, that explains the formation of tails and bridges between galaxies in a simple model, the so-called parabolic restricted three-body problem, i.e. we study the motion of a particle under the gravitational influence of two primaries describing parabolic orbits. The equilibrium points and the final evolutions in this problem are recalled,and we showthat the invariant manifolds of the collinear equilibrium points and the ones of the collision manifold explain the formation of bridges and tails. Massive numerical simulations are carried out and their application to recover previous results are also analysed.

Treballs Finals de Grau de Física, Facultat de Física, Universitat de Barcelona, Curs: 2016, Tutor: Marc Ribó Gomis, Binary systems composed of a neutron star or a black hole, and emitting in very high energy (VHE) -rays, have attracted a lot of interest in the last decade. One of these systems, 1FGL J1018.6-5856, is known to be composed of massive star of spectral type O6V((f)) and has an orbital period of 16.58 days, from the periodic emission in both X-rays and radio. Apart from this, the orbital parameters of the system are not well known. Since the VHE -rays interact with the photons from the massive companion, mainly via [ípsilon]-[ípsilon] absorption, the emitted VHE light-curve depends on the orbital configuration. We created a Fortran code to compute the [ípsilon]-[ípsilon] absorption and simulate the emitted light-curve, assuming an intrinsic constant VHE ux, with different orbital parameters. Comparing our simulated results with H.E.S.S observations of 1FGL J1018.6-5856 we are able to fit the best possible parameters within the observation uncertainties, obtaining an eccentricity e = 0:35 +- 0:05, an inclination i = 42º +- 5º and a periastron angle w = 83º +- 10º. The results are coherent with the range of values suggested in the literature. However, these results are not in agreement if the GeV emission is only produced by anisotropic Inverse Compton scattering