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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.

Aquest treball ha rebut finançament del programa europeu DISCOVERER (Disruptive Technologies for Very Low Earth Orbit Platforms, EC-H2020-737183) This thesis studies the complementarity of VLEO satellite technologies with existing solutions in the telecommunications market, both in the space and terrestrial sectors. All the characteristics and particularities of VLEO orbits are defined and explained in detail, showing advantages and disadvantages compared to traditional higher orbits. Study of the deployment of the satellite constellation concept within the New Space framework. Next, the business opportunities that can be generated by the introduction of VLEO technologies in the telecommunications market are studied, analysing the possible reshaping of each vertical market and how can they be most benefited. Finally, a business opportunity is proposed and qualitatively analysed in detail using the Value Proposition Canvas methodology.

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.

Treballs Finals de Grau de Física, Facultat de Física, Universitat de Barcelona, Curs: 2022, Tutors: Francesca Figueras, Toni Santana-Ros, This work’s main goal is to study the classical methods of orbit determination developed by Gauss and Laplace. To do so, simulations of objects orbiting the Earth have been programmed to test these methods and compare the results. They show poor accuracy and don’t allow for predictions in the studied cases, but they improve for large enough orbits where they could be a first approach for more complex orbit determination schemes

The lower thermosphere (between 100 and 550 km of height) is known with lesser accuracy than desired. The use of spherical satellites to determine the density could result in a substantial improvement of the present situation. The orbits of satellites at these heights are affected by drag and the inhomogeneities of the terrestrial gravitational potential (among other perturbations). This projects has as main goal to analyse how the atmospheric drag affects the orbit of a real satellite, and then, determine the thermospheric density. Hence, it will be necessary to reconstruct the orbit accounting for the perturbations in the most accurate way possible. Regarding the gravitational potential of the Earth, we have employed three approximations: spherical symmetry, a zonal model truncated at sixth order and a tesseral model truncated at tenth order. This last model has revealed as the most accurate. To achieve our goal, we have analysed the telemetry gathered by one of the satellites of the FSSCat mission. Then, we have determined its orbital elements. Afterwards, we have considered two possible methods for orbit reconstruction: Non-Linear Least Squares and Extended Kalman Filter. After this analysis, we have chosen the Non-Linear Least Squares method for the purpose of orbit reconstruction. The orbital elements determined from a physical analysis of the telemetry and the ones resulting from the non-linear least square method coincide with a remarkable accuracy.

The objective of this project is to propose a passive method for orbit determination (OD) of GEO objects using observational data as position measurements. The proposed method use position measurements expressed as right ascension and declination angles, therefore no need of on-board sensors within a satellite are need it to track them using the methodology to be presented in this thesis. The OD method was split in 3 main parts. The perturbed model for the orbit propagator, the initial orbit determination (IOD) methodology to compute initial conditions and a filter algorithm for orbit estimation. We selected an orbit propagator based on state vector and a force model solved by a Runge-Kutta 7-8 numerical integrator. The acting forces considered in the perturbed model were the earth gravitational potential model EGM96 (n=m=70), the third body perturbation due to the moon and sun gravitational forces and the exerted force due to the solar radiation pressure. Since the case study of this project is focused on GEO objects, the perturbation because of atmospheric drag was neglected. Based on the constraints of our observational data, an angles-only method was necessary for the for initial orbit determination implementation. The Gauss method and the Gibbs-Herrick method are the best fit for the position and velocity vector respectively, when short separated observations are used. To achieve precise orbit determination, a estimation algorithm using a Extended Kalman Filter (EKF) method was implemented. The observational data was used as the sensing measurements and the orbit propagator as the state predictor. The results suggested that the EKF can be used to the reduced the uncertainty associated with the orbit propagator error and to avoid this uncertainty to accumulate over the time (up to 3km per day for GEO objects). Nevertheless, future works should discuss and study how to improve the estimation implementations when observations that are closely separated are used as the measurements (separated by less than 1 minute).

This project is part of a bigger project called Plathon and consists of the study, application and adaptation of NASA’s orbital simulator 42 to the Plathon project. For this purpose, the program has been studied with several test simulations in order to better understand its inner workings and to test the possibilities and limitations of the simulator. After that, a series of modifications and standalone codes, using Matlab, have been implemented to better fit the simulator into the project and to allow for better post-processing of the obtained data. On top of that, the integration of the simulator with the Plathon project will also be discussed.

Rapid developments in the last decade have demonstrated the potential of new techniques within the Artificial Intelligence (AI) field as an effective approach for different classical problems in the area of astrodynamics. Pathfinding strategies in multi-body dynamical regions of space implies some challenges which have traditionally relied on simplifications of the dynamical model or heavy computational loads. However, modern machine learning techniques are likely to provide more lightweight solutions that could be applied to automated on-board guidance systems in future missions traveling though complex environments. This work aims to leverage Artificial Neural Network techniques to study how they can be applied to Libration Point Orbits in the Circular Restricted Three-Body Problem. In particular, new ANN-approaches to compute heteroclinic connections between planar Lyapunov orbits are proposed, and their performance is compared to that of classic methodologies. In order to train, validate and test the ANNs used throughout the work, sufficiently large datasets containing information that accurately represents the dynamics of the Circular Restricted Three-Body Problem are required. To this effect, the datasets are generated by means of solving the problem using well-known classic techniques, since it is the only way to acquire an ensemble of accurate observations of the model. The following steps are taken: first, the heteroclinic connections are computed with arbitrarily high accuracy by numerical propagation of the unstable/stable invariant manifolds of the departure/arrival orbits respectively, using a Runge-Kutta 45 integrator; second, the data from the propagations are tabulated and stored in datasets; third, the datasets are used to train and validate a series of ANNs; and finally, the performance of the obtained ANN models is analysed and compared to the benchmark results. Also, the influence of the sampling technique when generating the datasets and the robustness of the developed method are studied. As a result of this work, it can be concluded that the two new ANN methods developed here can be considered as good as the classical method on terms their fitness to be used as a seed in a realistic ephemeris model, since the Restricted Three Body Problem model used here as a reference is also not an accurate representation of realistic trajectories anyways, but it is an acceptable approximation in early stages of mission design. Objectius de Desenvolupament Sostenible::9 - Indústria, Innovació i Infraestructura

The version of record is available online at: http://dx.doi.org/10.1007/s00332-021-09721-5 In a general setting of a Hamiltonian system with two degrees of freedom and assuming some properties for the undergoing potential, we study the dynamics close and tending to a singularity of the system which in models of N-body problems corresponds to total collision. We restrict to potentials that exhibit two more singularities that can be regarded as two kind of partial collisions when not all the bodies are involved. Regularizing the singularities, the total collision transforms into a 2-dimensional invariant manifold. The goal of this paper is to prove the existence of different types of ejection–collision orbits, that is, orbits that start and end at total collision. Such orbits are regarded as heteroclinic connections between two equilibrium points and are mainly characterized by the partial collisions that the trajectories find on their way. The proof of their existence is based on the transversality of 2-dimensional invariant manifolds and on the behavior of the dynamics on the total collision manifold; both of them are thoroughly described.

Lambert's problem is the two-point boundary-value problem resulting from a two-body orbital transfer between two position vectors in a given time. It lies at the very heart of several fundamental astrodynamics and space engineering problems and, as such, it has attracted the interest of scientists over centuries. In this work, we revisit the solution of Lambert's problem based on Levi-Civita regularization developed by Carles Simó in 1973. We offer an exhaustive derivation of the theory, including proofs of all the results and the formulae employed, and we extend the algorithm to deal with multi-revolution transfers. Then, after investigating a range of initial guess search techniques and testing different numerical methods to approximate the solution, we propose a procedure in which the initial guess is assigned very efficiently by querying a pre-defined interpolating table. Then, in order to achieve both speed and robustness, we combine Newton-Raphson with safety checks to avoid out of boundary deviations to approximate the solution. We validate the method through several tests and applications, and we assess its convergence and performance. The algorithm presents no singularities, converges in all realistic scenarios and its computational cost is comparable with state of the art algorithms.

The effects of quasi-periodicity on the splitting of invariant manifolds are examined. We have found that some harmonics, that could be expected to be dominant in some ranges of the perturbation parameter, actually are nondominant. It is proved that, under reasonable conditions, this is due to the arithmetic properties of the frequencies.

We analyse a dynamical scenario where a constantly charged spacecraft (follower) moves in the vicinity of another one (leader) that follows a circular Keplerian orbit around the Earth and generates a rotating magnetic dipole. The mass of the follower is assumed to be negligible when compared with the one of the leader and they are supposed to be in a high-Earth orbit, so the Lorentz force on the follower due to the geomagnetic field is ignored. With these assumptions, the motion of the leader is not perturbed by the follower and it is only subjected to the Earth’s gravitational force field, while the charged follower is under to both the gravitational force of the Earth and the Lorentz force due to the magnetic dipole of the leader. We focus on the dynamical characteristics of the system as a function of its parameters, with special attention to the ratio of the leader’s mean motion around the Earth to the rotating rate of the dipole. We study the critical points of the model and their stability, the admissible and forbidden regions of motion of the deputy using the zero velocity surfaces and the families of periodic orbits emanating from equilibria. In the normal case we pay special attention to the periodic orbits of elliptic type and to the families of 2D tori surrounding them that are computed by means of a parameterisation method. The result is a fine catalog of orbits together with an accurate dynamical description suitable to researchers interested in potential applications of satellite formation flight using this kind of technology.

We outline some aspects of the dynamics of an infinitesimal mass under the Newtonian attraction of three point masses in a symmetric collinear relative equilibria configuration when a repulsive Manev potential ( $$-1/r +e/r^{2}$$ ), $$e>0$$ , is applied to the central mass. We investigate the relative equilibria of the infinitesimal mass and their linear stability as a function of the mass parameter $$\beta $$ , the ratio of mass of the central body to the mass of one of two remaining bodies, and e. We also prove the nonexistence of binary collisions between the central body and the infinitesimal mass.

International audience

International audience

[eng] This work aims the study of the Rapid Transition Mechanism that explains some properties of orbits of some spatial objects, as for instance, comet 39P/Oterma, which will be the main object of this research. Considering Sun and Jupiter are the masses that more influence the considered object, this mechanism describes a transition which makes the object to change from an orbit which is outside the Jupiter's one from one inside of it or viceversa. This mechanism is observed, in particular, in the phase space of the considered models: the Restricted Three-Body Problem, both Planar Circular and Planar Elliptic. In these models three bodies are considered, two of them, named primaries, have positive mass and their orbits evolve according to the solution of the Two-Body Problem, i.e., they are circles, ellipses, parabolas or hyperbolas, having as focus (or centre) the centre of mass of both masses. The third body (which movement is to be described) is considered to have zero mass, hence it does not influence the movement of the primaries but it is under their gravitational influence. We will present the cases on which the orbit of the primaries is a circle or an ellipse and that the orbit of the third body is confined to the same plane of movement of the primaries. Having chosen the models to study this type of transition, we proceed to the study of the skeletons of these systems, i.e., which invariant objects are the more important and responsible to describe Oterma's dynamics. This methodology is general in the study of the phase space of dynamical system: these objects are equilibrium points, periodic orbits, tori, manifolds, atractors, repulsors, among others, based on each problem. To compute the equilibrium points L1 and L2 in the circular model (which will be also used in the elliptic one) it is enough to numerically solve a polynomial equation of 5th degree, known as Euler's quintic. Afterwards, the periodic orbits around them are computed via two approaches: a semi-analytical one (which also permit the compute a good initial approximation of their stable and unstable invariant manifolds) using Birkhoff Normal Forms at the equilibirum points and a numerical one. In the elliptic model, the tori around L1 and L2 are computed using numerical techniques, approximating a parameterization using Fourier series. In fact, it is considered the mapping as integrating a period of Jupiter and an invariant curve can be computed. Due to the strong instability of the region around the equilibirum points, we consider a representation using more than 1 section in the independent variable and the 1-period-integration is done in smaller steps - this approach is called parallel shooting. Finally, we visualize Oterma in this context. Changes of variable are done in order to fit its real data in both model. This lets us read Oterma's positions and velocities from JPL-Horizons system and represent them in synodical coordinates. Approximating the initial coordinates (projecting them in the primaries plane of movement) and integrating them in the planar elliptic model we obtain a good hint that this model is suitable to reproduce, at least partially, Oterma's dynamics. With this, we are able to visualize Oterma inside the phase space and how it interacts with the considered invariant objects. In particular, making sections in the true anomaly and in the x coordinate at the same time, it is possible to compute invariant tori around L1 and around L2 which invariant manifolds are closer to Oterma's orbit. In addition, still in these double sections we are able to visualize the heteroclinic connections between these tori near Oterma's orbit.

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International audience

Treballs Finals de Grau de Física, Facultat de Física, Universitat de Barcelona, Curs: 2020, Tutor: Alberto Manrique Oliva, Using a modern galactic model that takes into account a dark matter halo, a spherical bulge and a disk, we study the nature of the orbit for star in the galaxy disk. Essentially, we study if the orbits of stars are stable or chaotic depending on the shape of the galaxy, specifically the ratio of the masses between the bulge and the disk of the galaxy. We also study the role of the star specific energy in the nature of the orbits. To analyse the trajectories of the stars we use Poincaré sections, a classical method that takes a qualitative approach to determine if the orbits calculated are regular or chaotic. We also take a more quantitative approach calculating the fraction of the area filled with regular orbits. Our results indicate that the stability of the orbits are highly dependent on the relation of masses between the bulge and the disk and the speci c energy of the star.

We provide conditions to guarantee the occurrence of Shilnikov bifurcations in analytic unfoldings of some Hopf-Zero singularities through a beyond all order phenomenon: the exponentially small breakdown of invariant manifolds which coincide at any order of the normal form procedure. The conditions are computable and satisfied by generic singularities and generic unfoldings. The existence of Shilnikov bifurcations in the C r case was already argued by Guckenheimer in the 80’s. About the same time, endowing the space of C ∞ unfoldings with a convenient topology, persistence and density of the Shilnikov phenomenon was proved by Broer and Vegter in 1984. However, since the proof involves the use of flat perturbations, this approach is not valid in the analytic context. What is more, none of the mentioned approaches provides a computable criteria to decide whether a given unfolding exhibits Shilnikov bifurcations or not. Many people appeals to the appearance of Hopf-Zero singularities to explain the emergence of chaos in a huge number of applications. However, no one can refer to a specific theorem establishing the conditions that a given unfolding should satisfy to ensure that chaotic dynamics are exhibited. We fill this gap by providing an ultimate result about the appearance of Shilnikov bifurcations in analytic unfoldings of a certain class of Hopf-Zero singularities. These conditions are computable and satisfied by generic families. One of these conditions depends on the full jet of the singularity and comes from a beyond all order phenomenon. It can be related with Stokes constants. The other conditions only depend on the 2-jet of the family.

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2019, Director: Antoni Benseny, [en] The extended restricted three body problem studies the motion of three bodies such that their mutual interaction corresponds to a force $F \sim r^{\alpha}$ and that one of their masses is much smaller than the others, which means that this body will not affect the others’ motion. In order to study the motion of this body, the two body problem for the other two will have to be solved and then, the less massive body will be introduced in the two body system without perturbing their motion. A simulator that calculates and draws the three orbits has been built.

This paper intends to give an insight on the geometry of stationkeeping strategies around Libration Point Orbits (LPO). Classically theFloquet modes have been used to determine the¿vrequired to control the trajectory of a spacecraft around LPOs. However, they can alsobe used to describe the dynamics in a neighborhood of a reference LPO. In this paper we use the Floquet reference frame to compare thebehavior of two different stationkeeping strategies:x-axis velocity constraint at plane crossing vs the Floquet Mode. The first approachfinds the¿vrequired to cancel thexcomponent of the velocity vector at the crossing of the trajectory with they= 0plane, in the RotatingLibration Point frame. While the Floquet Mode approach finds the¿vthat cancels the unstable mode in the Floquet reference frame

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We consider planar central configurations of the Newtonian kn-body problem consisting in k groups of regular n-gons of equal masses, called (k , n)-crown. We derive the equations of central configurations for a general ( k, n)-crown. When k = 2 we prove the existence of a twisted (2, n)-crown for any value of the mass ratio. Moreover, for n = 3, 4 and any value of the mass ratio, we give the exact number of twisted (2, n)-crowns, and describe their location. Finally, we conjecture that for any value of the mass ratio there exist exactly three (2; n)-crowns for n>=5 The authors are supported by MINECO grants MTM2016-80117-P and MTM2016-77278-P (FEDER) and Catalan (AGAUR) grants 2017 SGR 1374 and 2017 SGR 1617

En aquest article volem il.lustrar com la comprensió de la dinàmica d’alguns models de la mecànica celeste permet explicar alguns fenòmens astronòmics i dissenyar missions realistes a l’espai. El model paradigmàtic usat és el problema restringit de tres cossos, en el qual els objectes que tenen un paper essencial són les varietats invariants de les anomenades òrbites de libració, és a dir, òrbites periòdiques i quasi periòdiques al voltant dels anomenats punts d’equilibri col.lineals del model.Descriurem alguns d’aquests fenòmens i esmentarem algunes missions concretes.Finalment, comentarem altres models també útils (i més sofisticats) a l’astrodinàmica i acabarem amb algun comentari de com les eines de sistemes dinàmics es poden traslladar del món macroscòpic (celeste) al microscòpic, com per exemple el de la física atòmica clàssica.

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Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2018, Director: Antoni Benseny, [en] This thesis deals with the study of the Planar Circular Restricted Extended Three-Body problem, which is a generalisation of the regular Planar Circular Restricted Three-Body problem in which we replace the classical gravitational field $(F \sim r^{-2})$ with another one such that $F\sim r^{\alpha}$, with arbitrary values of $\alpha$, with the final objective of performing accurate simulations of the trajectory of the third body. We begin by precisely stating the problem as well as introducing some concepts that will be needed throughout this thesis for its development. We will then move onto the study of Hill Regions, which are the regions of the plane in which motion of the third body is possible. These regions are related to the value of the Jacobi constant, the only constant variable known for the restricted three-body problem. Following we will deal with the regularisation of the problem, a useful concept that becomes of utmost importance when computing the trajectory of the third body near the collisions. Finally we outline some important points that arose when computing the trajectory and how did we overcome them.