The SOPHIE search for northern extrasolar planets: X. Detection and characterization of giant planets by the dozen

We present new radial velocity measurements of eight stars that were secured with the spectrograph SOPHIE at the 193 cm telescope of the Haute-Provence Observatory. The measurements allow detecting and characterizing new giant extrasolar planets. The host stars are dwarfs of spectral types between F5 and K0 and magnitudes of between 6.7 and 9.6; the planets have minimum masses Mp sin i of between 0.4 to 3.8 MJup and orbitalperiods of several days to several months. The data allow only single planets to be discovered around the first six stars (HD 143105, HIP 109600, HD 35759, HIP 109384, HD 220842, and HD 12484), but one of them shows the signature of an additional substellar companion in the system. The seventh star, HIP 65407, allows the discovery of two giant planets that orbit just outside the 12:5 resonance in weak mutual interaction. The last star, HD 141399, was already known to host a four-planet system; our additional data and analyses allow new constraints to be set on it. We present Keplerian orbits of all systems, together with dynamical analyses of the two multi-planet systems. HD 143105 is one of the brightest stars known to host a hot Jupiter, which could allow numerous follow-up studies to be conducted even though this is not a transiting system. The giant planets HIP 109600b, HIP 109384b, and HD 141399c are located in the habitable zone of their host star.

Close-in planets around giant stars: lack of hot-Jupiters and prevalence of multiplanetary systems

Extrasolar planets abound in almost any possible configuration. However, until five years ago, there was a lack of planets orbiting closer than 0.5 au to giant or subgiant stars. Since then, recent detections have started to populated this regime by confirming 13 planetary systems. We discuss the properties of these systems in terms of their formation and evolution off the main sequence. Interestingly, we find that 70.0 ± 6.6% of the planets in this regime are inner components of multiplanetary systems. This value is 4.2σ higher than for main-sequence hosts, which we find to be 42.4 ± 0.1%. The properties of the known planets seem to indicate that the closest-in planets (a< 0.06 au) to main-sequence stars are massive (i.e., hot Jupiters) and isolated and that they are subsequently engulfed by their host as it evolves to the red giant branch, leaving only the predominant population of multiplanetary systems in orbits 0.06

Secular and tidal evolution of circumbinary systems

We investigate the secular dynamics of three-body circumbinary systems under the effect of tides. We use the octupolar non-restricted approximation for the orbital interactions, general relativity corrections, the quadrupolar approximation for the spins, and the viscous linear model for tides. We derive the averaged equations of motion in a simplified vectorial formalism, which is suitable to model the long-term evolution of a wide variety of circumbinary systems in very eccentric and inclined orbits. In particular, this vectorial approach can be used to derive constraints for tidal migration, capture in Cassini states, and stellar spin–orbit misalignment. We show that circumbinary planets with initial arbitrary orbital inclination can become coplanar through a secular resonance between the precession of the orbit and the precession of the spin of one of the stars. We also show that circumbinary systems for which the pericenter of the inner orbit is initially in libration present chaotic motion for the spins and for the eccentricity of the outer orbit. Because our model is valid for the non-restricted problem, it can also be applied to any three-body hierarchical system such as star–planet–satellite systems and triple stellar systems.

Neuronal networks with gap junctions: A study of piece-wise linear planar neuron models

The presence of gap junction coupling among neurons of the central nervous systems has been appreciated for some time now. In recent years there has been an upsurge of interest from the mathematical community in understanding the contribution of these direct electrical connections between cells to large-scale brain rhythms. Here we analyze a class of exactly soluble single neuron models, capable of producing realistic action potential shapes, that can be used as the basis for understanding dynamics at the network level. This work focuses on planar piece-wise linear models that can mimic the firing response of several different cell types. Under constant current injection the periodic response and phase response curve (PRC) is calculated in closed form. A simple formula for the stability of a periodic orbit is found using Floquet theory. From the calculated PRC and the periodic orbit a phase interaction function is constructed that allows the investigation of phase-locked network states using the theory of weakly coupled oscillators. For large networks with global gap junction connectivity we develop a theory of strong coupling instabilities of the homogeneous, synchronous and splay state. For a piece-wise linear caricature of the Morris-Lecar model, with oscillations arising from a homoclinic bifurcation, we show that large amplitude oscillations in the mean membrane potential are organized around such unstable orbits.

Instabilities in threshold-diffusion equations with delay

The introduction of delays into ordinary or partial differential equation models is well known to facilitate the production of rich dynamics ranging from periodic solutions through to spatio-temporal chaos. In this paper we consider a class of scalar partial differential equations with a delayed threshold nonlinearity which admits exact solutions for equilibria, periodic orbits and travelling waves. Importantly we show how the spectra of periodic and travelling wave solutions can be determined in terms of the zeros of a complex analytic function. Using this as a computational tool to determine stability we show that delays can have very different effects on threshold systems with negative as opposed to positive feedback. Direct numerical simulations are used to confirm our bifurcation analysis, and to probe some of the rich behaviour possible for mixed feedback.

Relação de alcance em dígrafos transitivos e a propriedade Z

Dissertação (mestrado)—Universidade de Brasília, Instituto de Ciências Exatas, Departamento de Matemática, 2016.

Software for Designing Custom Orbital-Craniofacial Implant Plans

Purpose: Custom cranio-orbital implants have been shown to achieve better performance than their hand-shaped counterparts by restoring skull anatomy more accurately and by reducing surgery time. Designing a custom implant involves reconstructing a model of the patient's skull using their computed tomography (CT) scan. The healthy side of the skull model, contralateral to the damaged region, can then be used to design an implant plan. Designing implants for areas of thin bone, such as the orbits, is challenging due to poor CT resolution of bone structures. This makes preoperative design time-intensive since thin bone structures in CT data must be manually segmented. The objective of this thesis was to research methods to accurately and efficiently design cranio-orbital implant plans, with a focus on the orbits, and to develop software that integrates these methods. Methods: The software consists of modules that use image and surface restoration approaches to enhance both the quality of CT data and the reconstructed model. It enables users to input CT data, and use tools to output a skull model with restored anatomy. The skull model can then be used to design the implant plan. The software was designed using 3D Slicer, an open-source medical visualization platform. It was tested on CT data from thirteen patients. Results: The average time it took to create a skull model with restored anatomy using our software was 0.33 hours ± 0.04 STD. In comparison, the design time of the manual segmentation method took between 3 and 6 hours. To assess the structural accuracy of the reconstructed models, CT data from the thirteen patients was used to compare the models created using our software with those using the manual method. When registering the skull models together, the difference between each set of skulls was found to be 0.4 mm ± 0.16 STD. Conclusions: We have developed a software to design custom cranio-orbital implant plans, with a focus on thin bone structures. The method described decreases design time, and is of similar accuracy to the manual method.

Autonomous formation flying: unified control and collision avoidance methods for close manoeuvring spacecraft

The idea of spacecraft formations, flying in tight configurations with maximum baselines of a few hundred meters in low-Earth orbits, has generated widespread interest over the last several years. Nevertheless, controlling the movement of spacecraft in formation poses difficulties, such as in-orbit high-computing demand and collision avoidance capabilities, which escalate as the number of units in the formation is increased and complicated nonlinear effects are imposed to the dynamics, together with uncertainty which may arise from the lack of knowledge of system parameters. These requirements have led to the need of reliable linear and nonlinear controllers in terms of relative and absolute dynamics. The objective of this thesis is, therefore, to introduce new control methods to allow spacecraft in formation, with circular/elliptical reference orbits, to efficiently execute safe autonomous manoeuvres. These controllers distinguish from the bulk of literature in that they merge guidance laws never applied before to spacecraft formation flying and collision avoidance capacities into a single control strategy. For this purpose, three control schemes are presented: linear optimal regulation, linear optimal estimation and adaptive nonlinear control. In general terms, the proposed control approaches command the dynamical performance of one or several followers with respect to a leader to asymptotically track a time-varying nominal trajectory (TVNT), while the threat of collision between the followers is reduced by repelling accelerations obtained from the collision avoidance scheme during the periods of closest proximity. Linear optimal regulation is achieved through a Riccati-based tracking controller. Within this control strategy, the controller provides guidance and tracking toward a desired TVNT, optimizing fuel consumption by Riccati procedure using a non-infinite cost function defined in terms of the desired TVNT, while repelling accelerations generated from the CAS will ensure evasive actions between the elements of the formation. The relative dynamics model, suitable for circular and eccentric low-Earth reference orbits, is based on the Tschauner and Hempel equations, and includes a control input and a nonlinear term corresponding to the CAS repelling accelerations. Linear optimal estimation is built on the forward-in-time separation principle. This controller encompasses two stages: regulation and estimation. The first stage requires the design of a full state feedback controller using the state vector reconstructed by means of the estimator. The second stage requires the design of an additional dynamical system, the estimator, to obtain the states which cannot be measured in order to approximately reconstruct the full state vector. Then, the separation principle states that an observer built for a known input can also be used to estimate the state of the system and to generate the control input. This allows the design of the observer and the feedback independently, by exploiting the advantages of linear quadratic regulator theory, in order to estimate the states of a dynamical system with model and sensor uncertainty. The relative dynamics is described with the linear system used in the previous controller, with a control input and nonlinearities entering via the repelling accelerations from the CAS during collision avoidance events. Moreover, sensor uncertainty is added to the control process by considering carrier-phase differential GPS (CDGPS) velocity measurement error. An adaptive control law capable of delivering superior closed-loop performance when compared to the certainty-equivalence (CE) adaptive controllers is finally presented. A novel noncertainty-equivalence controller based on the Immersion and Invariance paradigm for close-manoeuvring spacecraft formation flying in both circular and elliptical low-Earth reference orbits is introduced. The proposed control scheme achieves stabilization by immersing the plant dynamics into a target dynamical system (or manifold) that captures the desired dynamical behaviour. They key feature of this methodology is the addition of a new term to the classical certainty-equivalence control approach that, in conjunction with the parameter update law, is designed to achieve adaptive stabilization. This parameter has the ultimate task of shaping the manifold into which the adaptive system is immersed. The performance of the controller is proven stable via a Lyapunov-based analysis and Barbalat’s lemma. In order to evaluate the design of the controllers, test cases based on the physical and orbital features of the Prototype Research Instruments and Space Mission Technology Advancement (PRISMA) are implemented, extending the number of elements in the formation into scenarios with reconfigurations and on-orbit position switching in elliptical low-Earth reference orbits. An extensive analysis and comparison of the performance of the controllers in terms of total Δv and fuel consumption, with and without the effects of the CAS, is presented. These results show that the three proposed controllers allow the followers to asymptotically track the desired nominal trajectory and, additionally, those simulations including CAS show an effective decrease of collision risk during the performance of the manoeuvre.