Towards an analytical theory of the third-body problem for highly elliptical orbits

When dealing with satellites orbiting a central body on a highly elliptical orbit, it is necessary to consider the effect of gravitational perturbations due to external bodies. Indeed, these perturbations can become very important as soon as the altitude of the satellite becomes high, which is the case around the apocentre of this type of orbit. For several reasons, the traditional tools of celestial mechanics are not well adapted to the particular dynamic of highly elliptical orbits. On the one hand, analytical solutions are quite generally expanded into power series of the eccentricity and therefore limited to quasi-circular orbits [17, 25]. On the other hand, the time-dependency due to the motion of the third-body is often neglected. We propose several tools to overcome these limitations. Firstly, we have expanded the disturbing function into a finite polynomial using Fourier expansions of elliptic motion functions in multiple of the satellite's eccentric anomaly (instead of the mean anomaly) and involving Hansen-like coefficients. Next, we show how to perform a normalization of the expanded Hamiltonian by means of a time-dependent Lie transformation which aims to eliminate periodic terms. The difficulty lies in the fact that the generator of the transformation must be computed by solving a partial differential equation involving variables which are linear with time and the eccentric anomaly which is not time linear. We propose to solve this equation by means of an iterative process.Comment: Proceedings of the International Symposium on Orbit Propagation and Determination - Challenges for Orbit Determination and the Dynamics of Artificial Celestial Bodies and Space Debris, Lille, France, 201

Dynamique des orbites fortement elliptiques

Most of the orbits of artificial satellites around the Earth have relatively low eccentricities. The calculation of their trajectories is very well under control, either by means of numerical methods when it comes to focus on accuracy and comparing observations, or either through analytical or semi-analytical theories to optimize the speed of calculations. This second category is used, in particular, for computing many long-term trajectories that could help to ensure the security and safety of outer space activities. Furthermore, there is also satellites operating in highly elliptical orbits for which the computations of the trajectories can be significantly improved, especially in analytical theories. Due to the fact that such orbits cover a wide range of altitudes, the hierarchy of the perturbations acting on the satellite changes with the position on the orbit. At low altitude, the oblateness of the Earth (the so-called J2 effect) is the dominant perturbation while on high-altitude the lunisolar perturbation acceleration can reach or exceed the order of the J2 acceleration. Moreover, the traditional analytical theories of Celestial Mechanics are not well adapted to this particular dynamic. Analytical solutions are generally quite developed in Fourier series of the mean anomaly. These infinite series converge very slowly in the general case, but in the case of small eccentricities convergence is accelerated thanks to the d’Alembert characteristic which ensures that a Fourier coefficient of order n keeps a degree of cancellation n with respect to the eccentricity e. This argument is no longer effective for large eccentricities. As is often the case, the solutions are more degraded in the case where the Fourier coefficients are developed in power series of the eccentricity. However, there are also theories whose the developments are made with angular variables other than the mean anomaly and well suited to the considered problem: the true anomaly for the inner potential and the eccentric anomaly for the perturbation due to external body attraction (Moon or Sun, often called third body). However, in this latter case, the methods for developing theories always use another type of approximation. Indeed, the explicit time dependence of the Hamiltonian is neglected for solving the partial differential equations that give the generator of the change of variables.This PhD thesis is devoted to propose several tools to overcome these limitations. Firstly, we expand the third-body disturbing function using Fourier series in multiples of the satellite’s eccentric anomaly (instead of the mean anomaly). We then perform a normalization of the expanded Hamiltonian, which aims to eliminate all periodic terms. To this end, we apply a change of canonical variables based on time-dependent Lie transforms. The construction of the generating function of the change of variables requires solving a partial differential equations (PDE) with respect to the angular variables of the third body and the satellite. To our knowledge this PDE has no exact solution. An approximation is usually done at this step to solve it by neglecting the terms related to the third body. We show how this approximation can be avoided by providing an iterative method for solving the PDE. This amounts to carrying out a power series expansion of a small ratio of frequencies 0.7).Moreover, since the traditional numerical integration methods are not very effective for highly elliptical orbits, even with adaptive variable-step size, we show the benefits of the so-called geometric integrators, especially the variational integrators. To this end, we present some high-order numerical schemes and we test their performance.La plupart des orbites de satellites artificiels autour de la Terre ont des excentricités relativement faibles. Le calcul de leurs trajectoires est très bien maitrisé, soit au moyen de méthodes numériques quand il s’agit de privilégier la précision et la comparaison à des observations, soit au moyen de théories analytiques ou semi-analytiques pour optimiser la vitesse des calculs. Cette seconde catégorie est en particulier utilisée pour le calcul de nombreuses trajectoires à long terme dans le cadre de la sécurité de l’espace. Par ailleurs, il existe également des satellites sur des orbites à forte excentricité pour lesquels le calcul des trajectoires peut encore être très nettement amélioré, en particulier en ce qui concerne les théories analytiques. Comme ces orbites couvrent un large spectre d’altitudes, la hiérarchie des perturbations agissant sur le satellite change selon sa position sur l’orbite ; à basse altitude, la perturbation dominante est due à l’aplatissement de la Terre (appelé l’effet du J2 ) tandis que pour les altitudes élevées, les perturbations lunisolaires peuvent être aussi importantes et même supérieures à la perturbation de J2 . Par ailleurs, les théories analytiques classiques ne sont pas adaptées pour étudier la dynamique de ces orbites particulières. Les solutions analytiques sont assez généralement développées en série de Fourier de l’anomalie moyenne. Ces séries infinies convergent assez lentement dans la cas général, mais dans le cas des faibles excentricités la convergence est accélérée grâce à la caractéristique de d’Alembert qui assure qu’un coefficient de Fourier d’ordre n possède un degré d’annulation n par rapport à l’excentricité e. Cet argument perd toute son efficacité pour les fortes excentricités. Les solutions sont encore plus dégradées dans le cas où les coefficients de Fourier sont développés en séries entières de l’excentricité, comme c’est souvent le cas. Néanmoins, il existe aussi des théories dont les développements sont réalisés avec des variables angulaires, autres que l’anomalie moyenne, plus adaptées au problème étudié : l’anomalie vraie pour le potentiel intérieur et l’anomalie excentrique pour la perturbation d’un corps extérieur (Lune ou Soleil, souvent appelé troisième corps). Cependant, dans ce dernier cas, les méthodes de développement des théories utilisent toujours un autre type d’approximation : la dépendance temporelle explicite de l’Hamiltonien est négligée dans la résolution de l’équation aux dérivées partielles qui engendre le générateur du changement de variables.Cette thèse est consacrée au développement d’outils permettant de surmonter ces limitations. Dans un premier temps, nous développons la fonction perturbatrice de troisième corps en utilisant des séries de Fourier en multiples de l’anomalie excentrique du satellite (à la place de l’anomalie moyenne). Nous procédons ensuite à une normalisation du Hamiltonien ainsi développé, dans le but d’éliminer tous les termes périodiques. Pour y parvenir, nous appliquons un changement de variables canoniques construit à l’aide des transformées de Lie dépendantes du temps. La construction de la fonction génératrice du changement de variables nécessite la résolution d’une équation aux dérivées partielles (EDP) par rapport aux variables angulaires du troisième corps et du satellite. A notre connaissance cette EDP n’a pas de solution exacte. Habituellement, à cette étape, on fait une approximation qui consiste à négliger les termes de l’EDP liés au troisième corps afin de la résoudre. Nous montrons comment cette approximation peut être évitée, en proposant une méthode de résolution de l’EDP itérative, qui revient à effectuer un développement en série de puissances d’un rapport de fréquences petit devant 1. De plus, pour une cohérence globale de la théorie nous avons dû reconsidérer la solution classique de potentiel central et en particulier du J2 . Finalement nous obtenons une théorie qui permet d’extrapoler le mouvement osculateur (et pas seulement le mouvement moyen) sur de longues durées (des dizaines d’années) de façon efficace et avec une excellente précision y-compris pour des orbites très excentriques (e > 0.7).Par ailleurs, les méthodes d’intégration numérique classiques (même à pas variable) étant peu efficaces pour ces problèmes à très forte excentricité, nous montrons les avantages que procurent les intégrateurs dits géométriques, et particulièrement les intégrateurs variationnels. A cette fin, nous présenterons des schémas numériques d’ordre élevé dont nous testons les performances

Capnocytophaga canimorsus endocarditis with root abscess in a patient with a bicuspid aortic valve

Infective endocarditis caused by a zoonotic micro organism is a rare clinical condition. Capnocytophaga canimorsus is a commensal bacterium living in the saliva of dogs and cats which produces rarely reported endocarditis whose incidence may be underestimated, considering its failure to grow on standard media. We reported the case of a 65-year-old man with bicuspid aortic valve endocarditis and multiple abscesses of the aortic wall caused by the canine bacteria C. canimorsus

The EU Center of Excellence for Exascale in Solid Earth (ChEESE): Implementation, results, and roadmap for the second phase

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COVID-19 symptoms at hospital admission vary with age and sex: results from the ISARIC prospective multinational observational study

Background: The ISARIC prospective multinational observational study is the largest cohort of hospitalized patients with COVID-19. We present relationships of age, sex, and nationality to presenting symptoms. Methods: International, prospective observational study of 60 109 hospitalized symptomatic patients with laboratory-confirmed COVID-19 recruited from 43 countries between 30 January and 3 August 2020. Logistic regression was performed to evaluate relationships of age and sex to published COVID-19 case definitions and the most commonly reported symptoms. Results: ‘Typical’ symptoms of fever (69%), cough (68%) and shortness of breath (66%) were the most commonly reported. 92% of patients experienced at least one of these. Prevalence of typical symptoms was greatest in 30- to 60-year-olds (respectively 80, 79, 69%; at least one 95%). They were reported less frequently in children (≤ 18 years: 69, 48, 23; 85%), older adults (≥ 70 years: 61, 62, 65; 90%), and women (66, 66, 64; 90%; vs. men 71, 70, 67; 93%, each P < 0.001). The most common atypical presentations under 60 years of age were nausea and vomiting and abdominal pain, and over 60 years was confusion. Regression models showed significant differences in symptoms with sex, age and country. Interpretation: This international collaboration has allowed us to report reliable symptom data from the largest cohort of patients admitted to hospital with COVID-19. Adults over 60 and children admitted to hospital with COVID-19 are less likely to present with typical symptoms. Nausea and vomiting are common atypical presentations under 30 years. Confusion is a frequent atypical presentation of COVID-19 in adults over 60 years. Women are less likely to experience typical symptoms than men