Theory of Functional Connections and Nelder-Mead optimization methods applied in satellite characterization

The growing population of man-made objects with the build up of mega-constellations not only increases the potential danger to all space vehicles and in-space infrastructures (including space observatories), but above all poses a serious threat to astronomy and dark skies. Monitoring of this population requires precise satellite characterization, which is is a challenging task that involves analyzing observational data such as position, velocity, and light curves using optimization methods. In this study, we propose and analyze the application of two optimization procedures to determine the parameters associated with the dynamics of a satellite: one based on the Theory of Functional Connections (TFC) and another one based on the Nelder-Mead heuristic optimization algorithm. The TFC performs linear functional interpolation to embed the constraints of the problem into a functional. In this paper, we propose to use this functional to analytically embed the observational data of a satellite into its equations of dynamics. After that, any solution will always satisfy the observational data. The second procedure proposed in this research takes advantage of the Nealder-Mead algorithm, that does not require the gradient of the objective function, as alternative solution. The accuracy, efficiency, and dependency on the initial guess of each method is investigated, analyzed, and compared for several dynamical models. These methods can be used to obtain the physical parameters of a satellite from available observational data and for space debris characterization contributing to follow-up monitoring activities in space and astronomical observatories.Comment: Submitted to Acta Astronautic

Long-term rotation of celestial bodies and application to Ceres and Vesta

Le sujet de cette thèse est l'étude de la rotation à long terme des corps célestes.La première partie est consacrée à l’étude de la rotation à long terme de Cérès et Vesta, les deux corps les plus massifs de la ceinture principale d’astéroïdes. Ils sont l’objet d’étude de la sonde spatiale Dawn, qui a permis de déterminer précisément les caractéristiques physiques et de rotation nécessaires au calcul de leurs rotations. La distribution de glace sous et à la surface de Cérès dépend du mouvement de son axe de rotation par le biais de l’obliquité, inclinaison de l’équateur sur l’orbite. Les rotations de Cérès et Vesta étant rapides, l’évolution à long terme des axes de rotation de Cérès et Vesta a été obtenue à l'aide d'une intégration symplectique des équations de la rotation, où une moyenne a été réalisée sur la rotation propre rapide. La stabilité des axes de rotation de Cérès et Vesta a été étudiée en fonction des paramètres de la rotation avec un modèle séculaire semi-analytique, qui a permis de montrer que les axes de rotation ne présentaient pas de caractère chaotique.La seconde partie concerne le développement d'intégrateurs symplectiques dédiés au corps solide. L'intégration de la rotation propre d'un corps solide nécessite d’intégrer les équations issues du hamiltonien du corps solide libre. Ce hamiltonien est certes intégrable et présente une solution explicite nécessitant l’usage des fonctions elliptiques de Jacobi, cependant le coût numérique de ces fonctions est élevé. Lorsque le hamiltonien du corps solide libre est couplé avec une énergie potentielle, l’orientation du corps doit être calculée à chaque pas d’intégration, ce qui augmente le temps de calcul. Des intégrateurs symplectiques ont ainsi été précédemment proposés pour le corps solide libre. Dans ce travail, des intégrateurs spécifiques au corps solide ont été développés en utilisant les propriétés de l’algèbre de Lie du moment cinétique.This thesis concerns the long-term rotation of celestial bodies.The first part is a study of the long-term rotation of Ceres and Vesta, the two heaviest bodies of the main asteroid belt. The spacescraft Dawn studied these two objects and determined the physical and rotational characteristics, which are necessary for the computation of their rotations. The ice distribution under and on the surface of Ceres depends on the evolution of the obliquity, which is the inclination of the equatorial plane on the orbital plane. As the rotations of Ceres and Vesta are fast, the long-term evolution of the spin axes of Ceres and Vesta was obtained by realizing a symplectic integration of the equations of the rotation averaged on the fast proper rotation. The stability of the spin axes of Ceres and Vesta was studied with respect to the parameters of the rotation with a secular and semi-analytical model, which allowed to show that the spin axes are not chaotic.The second part concerns the development of symplectic integrators dedicated to the rigid body. The integration of the proper rotation of a rigid body needs to integrate the equations given by the Hamiltonian of the free rigid body. This Hamiltonian is integrable and presents an explicit solution using the Jacobi elliptic functions. However, the numerical cost of these functions is high. When the Hamiltonian of the free rigid body is coupled to a potential energy, the orientation of the body is needed at each step, which increases the computation time. Symplectic integrators were then previously proposed for the free rigid body. In this work, symplectic integrators dedicated to the rigid body were developed using the properties of the Lie algebra of the angular momentum

Rotation à long terme des corps célestes et application à Cérès et Vesta

This thesis concerns the long-term rotation of celestial bodies.The first part is a study of the long-term rotation of Ceres and Vesta, the two heaviest bodies of the main asteroid belt. The spacescraft Dawn studied these two objects and determined the physical and rotational characteristics, which are necessary for the computation of their rotations. The ice distribution under and on the surface of Ceres depends on the evolution of the obliquity, which is the inclination of the equatorial plane on the orbital plane. As the rotations of Ceres and Vesta are fast, the long-term evolution of the spin axes of Ceres and Vesta was obtained by realizing a symplectic integration of the equations of the rotation averaged on the fast proper rotation. The stability of the spin axes of Ceres and Vesta was studied with respect to the parameters of the rotation with a secular and semi-analytical model, which allowed to show that the spin axes are not chaotic.The second part concerns the development of symplectic integrators dedicated to the rigid body. The integration of the proper rotation of a rigid body needs to integrate the equations given by the Hamiltonian of the free rigid body. This Hamiltonian is integrable and presents an explicit solution using the Jacobi elliptic functions. However, the numerical cost of these functions is high. When the Hamiltonian of the free rigid body is coupled to a potential energy, the orientation of the body is needed at each step, which increases the computation time. Symplectic integrators were then previously proposed for the free rigid body. In this work, symplectic integrators dedicated to the rigid body were developed using the properties of the Lie algebra of the angular momentum.Le sujet de cette thèse est l'étude de la rotation à long terme des corps célestes.La première partie est consacrée à l’étude de la rotation à long terme de Cérès et Vesta, les deux corps les plus massifs de la ceinture principale d’astéroïdes. Ils sont l’objet d’étude de la sonde spatiale Dawn, qui a permis de déterminer précisément les caractéristiques physiques et de rotation nécessaires au calcul de leurs rotations. La distribution de glace sous et à la surface de Cérès dépend du mouvement de son axe de rotation par le biais de l’obliquité, inclinaison de l’équateur sur l’orbite. Les rotations de Cérès et Vesta étant rapides, l’évolution à long terme des axes de rotation de Cérès et Vesta a été obtenue à l'aide d'une intégration symplectique des équations de la rotation, où une moyenne a été réalisée sur la rotation propre rapide. La stabilité des axes de rotation de Cérès et Vesta a été étudiée en fonction des paramètres de la rotation avec un modèle séculaire semi-analytique, qui a permis de montrer que les axes de rotation ne présentaient pas de caractère chaotique.La seconde partie concerne le développement d'intégrateurs symplectiques dédiés au corps solide. L'intégration de la rotation propre d'un corps solide nécessite d’intégrer les équations issues du hamiltonien du corps solide libre. Ce hamiltonien est certes intégrable et présente une solution explicite nécessitant l’usage des fonctions elliptiques de Jacobi, cependant le coût numérique de ces fonctions est élevé. Lorsque le hamiltonien du corps solide libre est couplé avec une énergie potentielle, l’orientation du corps doit être calculée à chaque pas d’intégration, ce qui augmente le temps de calcul. Des intégrateurs symplectiques ont ainsi été précédemment proposés pour le corps solide libre. Dans ce travail, des intégrateurs spécifiques au corps solide ont été développés en utilisant les propriétés de l’algèbre de Lie du moment cinétique

Dedicated symplectic integrators for rotation motions

35 pages, 10 figures, submittedInternational audienceWe propose to use the properties of the Lie algebra of the angular momentum to build symplectic integrators dedicated to the Hamiltonian of the free rigid body. By introducing a dependence of the coefficients of integrators on the moments of inertia of the integrated body, we can construct symplectic dedicated integrators with fewer stages than in the general case for a splitting in three parts of the Hamiltonian. We perform numerical tests to compare the developed dedicated fourth-order integrators to the existing reference integrators for the water molecule. We also estimate analytically the accuracy of these new integrators for the set of the rigid bodies and conclude that they are more accurate than the existing ones only for very asymmetric bodies

Eviction-like resonances for satellite orbits. Application to Phobos, the main satellite of Mars

International audienceThe motion of a satellite can experience secular resonances between the precession frequencies of its orbit and the mean motion of the host planet around the star. Some of these resonances can significantly modify the eccentricity (evection resonance) and the inclination (eviction resonance) of the satellite. In this paper, we study in detail the secular resonances that can disturb the orbit of a satellite, in particular the eviction-like ones. Although the inclination is always disturbed while crossing one eviction-like resonance, capture can only occur when the semi-major axis is decreasing. This is, for instance, the case of Phobos, the largest satellite of Mars, that will cross some of these resonances in the future because its orbit is shrinking owing to tidal effects. We estimate the impact of resonance crossing in the orbit of the satellite, including the capture probabilities, as a function of several parameters, such as the eccentricity and the inclination of the satellite, and the obliquity of the planet. Finally, we use the method of the frequency map analysis to study the resonant dynamics based on stability maps, and we show that some of the secular resonances may overlap, which leads to chaotic motion for the inclination of the satellite

Reversible time-step adaptation for the integration of few-body systems

The time-step criterion plays a crucial role in direct N-body codes. If not chosen carefully, it will cause a secular drift in the energy error. Shared, adaptive time-step criteria commonly adopt the minimum pairwise time-step, which suffers from discontinuities in the time evolution of the time-step. This has a large impact on the functioning of time-step symmetrization algorithms. We provide new demonstrations of previous findings that a smooth and weighted average over all pairwise time-steps in the N-body system, improves the level of energy conservation. Furthermore, we compare the performance of 27 different time-step criteria, by considering three methods for weighting time-steps and nine symmetrization methods. We present performance tests for strongly chaotic few-body systems, including unstable triples, giant planets in a resonant chain, and the current Solar System. We find that the harmonic symmetrization methods (methods A3 and B3 in our notation) are the most robust, in the sense that the symmetrized time-step remains close to the time-step function. Furthermore, based on our Solar System experiment, we find that our new weighting method based on direct pair-wise averaging (method W2 in our notation), is slightly preferred over the other methods

Dynamics of trans-Neptunian objects near the 3/1 mean-motion resonance with Neptune

The complex classification of trans-Neptunian objects (TNOs) that are captured in mean-motion resonances (MMRs) and the constraint of their multiple origins are two significant open problems concerning the Solar System. The case-by-case study of the different MMRs and their characteristics provide information about their origin and dynamics, which helps us to understand the early stages of the Solar System evolution. In this paper, we study the dynamics of the detected TNOs close to a 3/1 MMR with Neptune. We initially use a semi-analytic three-body model to investigate the coplanar secular dynamics of these objects and find the stationary points. We then use surface sections and stability maps to analyse the non-averaged dynamics. These methods allow us to isolate the different stability regions and determine the extent of the chaotic regions. We show that stability maps are an extremely powerful tool for studying the resonant dynamics when they are computed in terms of the resonant angle. We then use these maps to study the non-planar three-body problem and the full dynamics in the presence of planetary perturbations. We confirm that TNOs near the 3/1 MMR regions can exist at very high inclinations. In the framework of the three-body problem, many of these objects can also be stable outside the 3/1 MMR owing to a Kozai secular resonance. However, when we take into account the perturbations of the four giant planets, the Kozai regions disappear and only the 3/1 MMR region remains, with eccentricities e ≲ 0.5