Invariant manifolds and transport in a Sun-perturbed Earth-Moon system

[eng] This dissertation is devoted to the analysis of the motion of small bodies, like asteroids, in the neighbourhood of the Earth-Moon system from a celestial mechanics approach. This is an extensive area of research where probably, the most extended simplified mathematical model is the well-known autonomous Hamiltonian system the Restricted Three-Body Problem (RTBP). Many modifications to this model have been proposed, looking for a more accurate description of the system. One of the simplest ways of introducing additional physical effects is through time-periodic perturbations, such that such that the new non-autonomous system is close to the autonomous one, and it has many periodic or quasi-periodic solutions. If these solutions are hyperbolic, they have stable/unstable invariant manifolds, such that stable manifolds approach the quasi-periodic solutions forward in time, meanwhile unstable manifolds do it backward in time, constituting the skeleton for the dynamical transport phenomena we are interested in. Notice that one dimension can be reduced by defining a suitable temporal Poincar´e map. Therefore, our aim is to compute the quasi-periodic solutions and their manifolds in this map. Most of the effort of this dissertation is addressed to the Bicircular Problem (BCP), in which the Earth and Moon are treated as the primaries in the RTBP and the gravitational field of the Sun is introduced as a time-periodic forcing of the RTBP. In particular, we have extensively analysed the horizontal family of two dimensional quasi-periodic solutions in the neighbourhood of the collinear unstable equilibrium point L3. We found that diverse trajectories connecting the Earth, the Moon and the outside Earth-Moon system are governed by L3 dynamics. Big attention is paid to the trajectories coming from the Moon towards the Earth, since they may give an insight of the travel that lunar meteorites perform before landing in our planet. These results have been translated and compared with those of a realistic model based on JPL (Jet Propulsion Laboratory) ephemeris, showing a good agreement between the results obtained. We also have proposed and carried out a strategy for capturing a Near Earth Asteroid (NEA) using the stable invariant manifolds of the horizontal family of quasi-periodic orbits around L3 in the BCP. To this aim the high order parametrization of the stable/unstable invariant manifolds is introduced, for which computation we have employed the jet transport technique. Finally, the strategy is applied to the NEA 2006 RH120. The contributions to the BCP presented in this dissertation include two other applications. The first one is devoted to the study of the unstable behaviour near the triangular points, meanwhile the second is devoted to a family of stable invariant curves around the Moon that are close to a resonance, promoting the appearance of chaotic motion. The last part of the dissertation is focused on the effective computation of the high or- der parametrization of the stable and unstable invariant manifolds associated with reducible invariant tori of any high dimension. To this aim, we resort on the reducible system, that offers a high degree of parallelization of the computations. Besides, we explain how to com- bine the presented methods with multiple shooting techniques to accurately compute highly unstable invariant objects. Finally, we apply the developed algorithms to compute the high order parametrization of the manifolds associated to L1 and L2 in an Earth-Moon system that includes five time-periodic forcings regarded to four physical features of the system, besides the solar gravitational field.[spa] Esta tesis analiza el movimiento de pequeños cuerpos, como asteroides, en el sistema Tierra­ Luna desde el marco de la mecánica celeste. El modelo que hemos empleado en mayor profundidad es el Problema Bicircular (PBC), el cual se puede entender como una perturbación periódica en el tiempo del conocido Problema Restringido de Tres Cuerpos (PRTC), dado que en el PBC se incluye el campo gravitatorio de un tercer cuerpo masivo que rota en movimiento circular alrededor de la configuración del PRTC. El cuerpo que causa la perturbación es para nosotros el Sol de tal forma que los objetos invariantes del PRTC adquieren una dimensión angular debida a la frecuencia del movimiento relativo entre el Sol y el baricentro Tierra-Luna. En el marco del PBC hemos analizado los fenómenos de transporte gobernados por la familia horizontal de soluciones cuasi-periódicas dos dimensionales (toros 2D) alrededor punto inestable colinear L3. Estas soluciones tienen asociadas variedades invariantes estables e inestables que constituyen el esqueleto de los fenómenos que queremos estudiar. Las trayectorias encontradas conectan la Tierra y la Luna y también el exterior/interior del sistema Tierra-Luna. Hemos prestado especial atención a las trayectorias que van de la Luna a la Tierra ya que podrían explicar el viaje que realizan los meteoritos lunares encontrados en nuestro planeta. Estos resultados han sido testeados en un modelo más realista basado en las efemérides del JPL (Jet Propulsion Laboratory). Otra de las aplicaciones propuestas es la de capturar un asteroide cercano a la Tierra usando la parametrización a orden alto de las variedades invariantes asociadas a los toros 2D alrededor de L3. La parte final trata del desarrollo de algoritmos para el cálculo preciso de la parametrización a orden alto de variedades invariantes estables/inestables asociadas a toros reducibles de cualquier dimensión alta. Además, se explica cómo combinar dichos algoritmos con métodos de tiro múltiple para aquellos objetos invariantes que sean muy inestables. Finalmente, aplicamos la metodología al cálculo de las variedades asociadas a L1 y L2 de un sistema Tierra-Luna que incluye cinco perturbaciones periódicas en el tiempo

Using invariant manifolds to capture an asteroid near the L3 point of the Earth-Moon Bicircular model

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 $\mathrm{m} / \mathrm{s}$

Numerical simulation of the shear stress produced by the hot metal jet on the blast furnace runner

During steel casting process a jet of molten metal runs out of the blast furnace hearth and strikes the runner. The continuous impact of hot fluids causes significant damage to its surface, which is made of refractory concrete. In particular, the initial impact on the dry runner is expected to be critical. This work deals with the analysis of the mechanical impact on the runner through the numerical simulation of the process. We propose an incompressible turbulent isothermal Navier-Stokes model, where turbulence is modelled considering two models (standard and SST). The interface dynamics is described by applying the Volume of Fluid (VOF) method, while the surface tension vector is provided by the Continuum Surface model (CSF). Their numerical results are performed in 2D. A comparative analysis of the most suitable transient turbulent multiphase model is presented by simulating benchmark physical experiments. The shear stress arising from the impact of the jet on the runner is also analyzed. An improvement of the classical analytical expression given in [1] is proposed. Both, the chosen turbulence model, and the formulas to compute the shear stress are validated using two benchmark laboratory tests and three numerical experiments. Numerical results are given for the impact of the jet on the dry runner of the blast furnaceThis work was supported by FEDER and Xunta de Galicia [grant number ED431C 2017/60, ED431C 2021/15], the Ministry of Economy, Industry and Competitiveness through the Plan Nacional de I+D+i [grant number MTM2015-68275-R], Agencia Estatal de Investigación [PID2019-105615RB-I00/AEI/10.13039/501100011033] and by the Vicerreitoría de Investigación e Innovación da Universidade de Santiago de Compostela via the Programa de Becas de Colaboración en Investigación 2016S

The evolution of the ventilatory ratio is a prognostic factor in mechanically ventilated COVID-19 ARDS patients

Background: Mortality due to COVID-19 is high, especially in patients requiring mechanical ventilation. The purpose of the study is to investigate associations between mortality and variables measured during the first three days of mechanical ventilation in patients with COVID-19 intubated at ICU admission. Methods: Multicenter, observational, cohort study includes consecutive patients with COVID-19 admitted to 44 Spanish ICUs between February 25 and July 31, 2020, who required intubation at ICU admission and mechanical ventilation for more than three days. We collected demographic and clinical data prior to admission; information about clinical evolution at days 1 and 3 of mechanical ventilation; and outcomes. Results: Of the 2,095 patients with COVID-19 admitted to the ICU, 1,118 (53.3%) were intubated at day 1 and remained under mechanical ventilation at day three. From days 1 to 3, PaO2/FiO2 increased from 115.6 [80.0-171.2] to 180.0 [135.4-227.9] mmHg and the ventilatory ratio from 1.73 [1.33-2.25] to 1.96 [1.61-2.40]. In-hospital mortality was 38.7%. A higher increase between ICU admission and day 3 in the ventilatory ratio (OR 1.04 [CI 1.01-1.07], p = 0.030) and creatinine levels (OR 1.05 [CI 1.01-1.09], p = 0.005) and a lower increase in platelet counts (OR 0.96 [CI 0.93-1.00], p = 0.037) were independently associated with a higher risk of death. No association between mortality and the PaO2/FiO2 variation was observed (OR 0.99 [CI 0.95 to 1.02], p = 0.47). Conclusions: Higher ventilatory ratio and its increase at day 3 is associated with mortality in patients with COVID-19 receiving mechanical ventilation at ICU admission. No association was found in the PaO2/FiO2 variation

Treatment with tocilizumab or corticosteroids for COVID-19 patients with hyperinflammatory state: a multicentre cohort study (SAM-COVID-19)

Objectives: The objective of this study was to estimate the association between tocilizumab or corticosteroids and the risk of intubation or death in patients with coronavirus disease 19 (COVID-19) with a hyperinflammatory state according to clinical and laboratory parameters. Methods: A cohort study was performed in 60 Spanish hospitals including 778 patients with COVID-19 and clinical and laboratory data indicative of a hyperinflammatory state. Treatment was mainly with tocilizumab, an intermediate-high dose of corticosteroids (IHDC), a pulse dose of corticosteroids (PDC), combination therapy, or no treatment. Primary outcome was intubation or death; follow-up was 21 days. Propensity score-adjusted estimations using Cox regression (logistic regression if needed) were calculated. Propensity scores were used as confounders, matching variables and for the inverse probability of treatment weights (IPTWs). Results: In all, 88, 117, 78 and 151 patients treated with tocilizumab, IHDC, PDC, and combination therapy, respectively, were compared with 344 untreated patients. The primary endpoint occurred in 10 (11.4%), 27 (23.1%), 12 (15.4%), 40 (25.6%) and 69 (21.1%), respectively. The IPTW-based hazard ratios (odds ratio for combination therapy) for the primary endpoint were 0.32 (95%CI 0.22-0.47; p < 0.001) for tocilizumab, 0.82 (0.71-1.30; p 0.82) for IHDC, 0.61 (0.43-0.86; p 0.006) for PDC, and 1.17 (0.86-1.58; p 0.30) for combination therapy. Other applications of the propensity score provided similar results, but were not significant for PDC. Tocilizumab was also associated with lower hazard of death alone in IPTW analysis (0.07; 0.02-0.17; p < 0.001). Conclusions: Tocilizumab might be useful in COVID-19 patients with a hyperinflammatory state and should be prioritized for randomized trials in this situatio

Leveraging L3 to transfer to L4 in the Sun-perturbed Earth-Moon system

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