Dynamic portrait of the planetary 2/1 mean-motion resonance - II. Systems with a more massive inner planet

This paper presents the second part in our study of the global structure of the planar phase space of the planetary three-body problem, when both planets lie in the vicinity of a 2/1 mean-motion resonance. While Paper I was devoted to cases where the outer planet is the more massive body, the present work is devoted to the cases where the more massive body is the inner planet. As before, outside the well-known Apsidal Corotation Resonances (ACR), the phase space shows a complex picture marked by the presence of several distinct regimes of resonant and non-resonant motion, crossed by families of periodic orbits and separated by chaotic zones. When the chosen values of the integrals of motion lead to symmetric ACR, the global dynamics are generally similar to the structure presented in Paper I. However, for asymmetric ACR the resonant phase space is strikingly different and shows a galore of distinct dynamical states. This structure is shown with the help of dynamical maps constructed on two different representative planes, one centred on the unstable symmetric ACR and the other on the stable asymmetric equilibrium solution. Although the study described in the work may be applied to any mass ratio, we present a detailed analysis for mass values similar to the Jupiter-Saturn case. Results give a global view of the different dynamical states available to resonant planets with these characteristics. Some of these dynamical paths could have marked the evolution of the giant planets of our Solar system, assuming they suffered a temporary capture in the 2/1 resonance during the latest stages of the formation of our Solar system.

Dynamics of two satellites in the 2/1 Mean-Motion resonance: application to the case of Enceladus and Dione

The dynamics of a pair of satellites similar to Enceladus-Dione is investigated with a two-degrees-of-freedom model written in the domain of the planar general three-body problem. Using surfaces of section and spectral analysis methods, we study the phase space of the system in terms of several parameters, including the most recent data. A detailed study of the main possible regimes of motion is presented, and in particular we show that, besides the two separated resonances, the phase space is replete of secondary resonances.

Collisions with the Earth: the Moon's contribution

In the present work is analyzed the contribution of the Moon on the collisional process of the Earth with asteroids (NEOs). The dynamical system adopted is the restricted four-body problem Sun-Earth-Moon-particle. Using a simple analytical approach one can verify that, the orbit of an object can be significantly affected by the Moon's gravitational field when their relative velocity is smaller than 5 km/s. Therefore, the present work is based on hypothetical asteroids whose velocities relative to Moon are of the order of 1 km/s. In fact, there are several real objects (NEOs) with such velocities at the point they cross the Earth's orbit. The net results obtained indicate that the Moon helps to avoid collisions (2.6%) more than it contributes to extra collisions (0.6%). (C) 2003 COSPAR. Published by Elsevier Ltd. All rights reserved.

Desenvolvimento da função perturbadora e aplicações em dinâmica de exoplanetas

Pós-graduação em Matemática Universitária - IGCE

Records of migration in the exoplanet configurations

When compared to our Solar System, many exoplanet systems exhibit quite unusual planet configurations; some of these are hot Jupiters, which orbit their central stars with periods of a few days, others are resonant systems composed of two or more planets with commensurable orbital periods. It has been suggested that these configurations can be the result of a migration processes originated by tidal interactions of the planets with disks and central stars. The process known as planet migration occurs due to dissipative forces which affect the planetary semi-major axes and cause the planets to move towards to, or away from, the central star. In this talk, we present possible signatures of planet migration in the distribution of the hot Jupiters and resonant exoplanet pairs. For this task, we develop a semi-analytical model to describe the evolution of the migrating planetary pair, based on the fundamental concepts of conservative and dissipative dynamics of the three-body problem. Our approach is based on an analysis of the energy and the orbital angular momentum exchange between the two-planet system and an external medium; thus no specific kind of dissipative forces needs to be invoked. We show that, under assumption that dissipation is weak and slow, the evolutionary routes of the migrating planets are traced by the stationary solutions of the conservative problem (Birkhoff, Dynamical systems, 1966). The ultimate convergence and the evolution of the system along one of these modes of motion are determined uniquely by the condition that the dissipation rate is sufficiently smaller than the roper frequencies of the system. We show that it is possible to reassemble the starting configurations and migration history of the systems on the basis of their final states, and consequently to constrain the parameters of the physical processes involved.

Solution of optimal continuous low-thrust transfer using Lie transforms

This paper addresses the problem of optimal constant continuous low-thrust transfer in the context of the restricted two-body problem (R2BP). Using the Pontryagin’s principle, the problem is formulated as a two point boundary value problem (TPBVP) for a Hamiltonian system. Lie transforms obtained through the Deprit method allow us to obtain the canonical mapping of the phase flow as a series in terms of the order of magnitude of the thrust applied. The reachable set of states starting from a given initial condition using optimal control policy is obtained analytically. In addition, a particular optimal transfer can be computed as the solution of a non-linear algebraic equation. Se investiga el uso de series y transformadas de Lie en problemas de optimización de trayectorias de satélites impulsados por motores de bajo empuje

On the rotation of co-orbital bodies in eccentric orbits

We investigate the resonant rotation of co-orbital bodies in eccentric and planar orbits. We develop a simple analytical model to study the impact of the eccentricity and orbital perturbations on the spin dynamics. This model is relevant in the entire domain of horseshoe and tadpole orbit, for moderate eccentricities. We show that there are three different families of spin-orbit resonances, one depending on the eccentricity, one depending on the orbital libration frequency, and another depending on the pericenter's dynamics. We can estimate the width and the location of the different resonant islands in the phase space, predicting which are the more likely to capture the spin of the rotating body. In some regions of the phase space the resonant islands may overlap, giving rise to chaotic rotation.

DROMO formulation for planar motions: Solution to the Tsien Problem

The two-body problem subject to a constant radial thrust is analyzed as a planar motion. The description of the problem is performed in terms of three perturbation methods: DROMO and two others due to Deprit. All of them rely on Hansen?s ideal frame concept. An explicit, analytic, closed-form solution is obtained for this problem when the initial orbit is circular (Tsien problem), based on the DROMO special perturbation method, and expressed in terms of elliptic integral functions. The analytical solution to the Tsien problem is later used as a reference to test the numerical performance of various orbit propagation methods, including DROMO and Deprit methods, as well as Cowell and Kustaanheimo?Stiefel methods.

Brunnian and Efimov N-Body States

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

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.

Non-reductive Physicalism, Irreducibility of the Mental and the Problem of Mental Causation : A study of Donald Davidson's and Georg Henrik von Wright's positions in the philosophy of mind

In this study I consider what kind of perspective on the mind body problem is taken and can be taken by a philosophical position called non-reductive physicalism. Many positions fall under this label. The form of non-reductive physicalism which I discuss is in essential respects the position taken by Donald Davidson (1917-2003) and Georg Henrik von Wright (1916-2003). I defend their positions and discuss the unrecognized similarities between their views. Non-reductive physicalism combines two theses: (a) Everything that exists is physical; (b) Mental phenomena cannot be reduced to the states of the brain. This means that according to non-reductive physicalism the mental aspect of humans (be it a soul, mind, or spirit) is an irreducible part of the human condition. Also Davidson and von Wright claim that, in some important sense, the mental aspect of a human being does not reduce to the physical aspect, that there is a gap between these aspects that cannot be closed. I claim that their arguments for this conclusion are convincing. I also argue that whereas von Wright and Davidson give interesting arguments for the irreducibility of the mental, their physicalism is unwarranted. These philosophers do not give good reasons for believing that reality is thoroughly physical. Notwithstanding the materialistic consensus in the contemporary philosophy of mind the ontology of mind is still an uncharted territory where real breakthroughs are not to be expected until a radically new ontological position is developed. The third main claim of this work is that the problem of mental causation cannot be solved from the Davidsonian - von Wrightian perspective. The problem of mental causation is the problem of how mental phenomena like beliefs can cause physical movements of the body. As I see it, the essential point of non-reductive physicalism - the irreducibility of the mental - and the problem of mental causation are closely related. If mental phenomena do not reduce to causally effective states of the brain, then what justifies the belief that mental phenomena have causal powers? If mental causes do not reduce to physical causes, then how to tell when - or whether - the mental causes in terms of which human actions are explained are actually effective? I argue that this - how to decide when mental causes really are effective - is the real problem of mental causation. The motivation to explore and defend a non-reductive position stems from the belief that reductive physicalism leads to serious ethical problems. My claim is that Davidson's and von Wright's ultimate reason to defend a non-reductive view comes back to their belief that a reductive understanding of human nature would be a narrow and possibly harmful perspective. The final conclusion of my thesis is that von Wright's and Davidson's positions provide a starting point from which the current scientistic philosophy of mind can be critically further explored in the future.

Optimal explicit guidance of multistage launch vehicle along three-dimensional trajectory

Details of an efficient optimal closed-loop guidance algorithm for a three-dimensional launch are presented with simulation results. Two types of orbital injections, with either true anomaly or argument of perigee being free at injection, are considered. The resulting steering-angle profile under the assumption of uniform gravity lies in a canted plane which transforms a three-dimensional problem into an equivalent two-dimensional one. Effects of thrust are estimated using a series in a recursive way. Encke's method is used to predict the trajectory during powered flight and then to compute the changes due to actual gravity using two gravity-related vectors. Guidance parameters are evaluated using the linear differential correction method. Optimality of the algorithm is tested against a standard ground-based trajectory optimization package. The performance of the algorithm is tested for accuracy, robustness, and efficiency for a sun-synchronous mission involving guidance for a multistage vehicle that requires large pitch and yaw maneuver. To demonstrate applicability of the algorithm to a range of missions, injection into a geostationary transfer orbit is also considered. The performance of the present algorithm is found to be much better than others.

Novel correlations in two dimensions: Some exact solutions

We construct a new many-body Hamiltonian with two- and three-body interactions in two space dimensions and obtain its exact many-body ground state for an arbitrary number of particles. This ground state has a novel pairwise correlation. A class of exact solutions for the excited states is also found. These excited states display an energy spectrum similar to the Calogero-Sutherland model in one dimension. The model reduces to an analog of the well-known trigonometric Sutherland model when projected on to a circular ring.

Two and Three Finger Caging of Polygons and Polyhedra

Multi-finger caging offers a rigorous and robust approach to robot grasping. This thesis provides several novel algorithms for caging polygons and polyhedra in two and three dimensions. Caging refers to a robotic grasp that does not necessarily immobilize an object, but prevents it from escaping to infinity. The first algorithm considers caging a polygon in two dimensions using two point fingers. The second algorithm extends the first to three dimensions. The third algorithm considers caging a convex polygon in two dimensions using three point fingers, and considers robustness of this cage to variations in the relative positions of the fingers.

This thesis describes an algorithm for finding all two-finger cage formations of planar polygonal objects based on a contact-space formulation. It shows that two-finger cages have several useful properties in contact space. First, the critical points of the cage representation in the hand’s configuration space appear as critical points of the inter-finger distance function in contact space. Second, these critical points can be graphically characterized directly on the object’s boundary. Third, contact space admits a natural rectangular decomposition such that all critical points lie on the rectangle boundaries, and the sublevel sets of contact space and free space are topologically equivalent. These properties lead to a caging graph that can be readily constructed in contact space. Starting from a desired immobilizing grasp of a polygonal object, the caging graph is searched for the minimal, intermediate, and maximal caging regions surrounding the immobilizing grasp. An example constructed from real-world data illustrates and validates the method.

A second algorithm is developed for finding caging formations of a 3D polyhedron for two point fingers using a lower dimensional contact-space formulation. Results from the two-dimensional algorithm are extended to three dimension. Critical points of the inter-finger distance function are shown to be identical to the critical points of the cage. A decomposition of contact space into 4D regions having useful properties is demonstrated. A geometric analysis of the critical points of the inter-finger distance function results in a catalog of grasps in which the cages change topology, leading to a simple test to classify critical points. With these properties established, the search algorithm from the two-dimensional case may be applied to the three-dimensional problem. An implemented example demonstrates the method.

This thesis also presents a study of cages of convex polygonal objects using three point fingers. It considers a three-parameter model of the relative position of the fingers, which gives complete generality for three point fingers in the plane. It analyzes robustness of caging grasps to variations in the relative position of the fingers without breaking the cage. Using a simple decomposition of free space around the polygon, we present an algorithm which gives all caging placements of the fingers and a characterization of the robustness of these cages.

Effects of body size and water temperature on food consumption and growth in the sea cucumber Apostichopus japonicus (Selenka) with special reference to aestivation

To investigate the effects of body size and water temperature on feeding and growth in the sea cucumber Apostichopus japonicus (Selenka), the maximum rate of food consumption in terms of energy (C-maxe; J day(-1)) and the specific growth rate in terms of energy (SGRe; % day(-1)) in animals of three body sizes (mean +/- SE) - large (134.0 +/- 3.5 g), medium (73.6 +/- 2.2 g) and small (36.5 +/- 1.2 g) - were determined at water temperatures of 10, 15, 20, 25 and 30 degrees C. Maximum rate of food consumption in terms of energy increased and SGRe decreased with increasing body weight at 10, 15 and 20 degrees C. This trend, however, was not apparent at 25 and 30 degrees C, which could be influenced by aestivation. High water temperatures (above 20 degrees C) were disadvantageous to feeding and growth of this animal; SGRe of A. japonicus during aestivation was negative. The optimum temperatures for food consumption and for growth were similar and were between 14 and 15 degrees C, and body size seemed to have a slight effect on the optimal temperature for food consumption or growth. Because aestivation of A. japonicus was temperature dependent, the present paper also documented the threshold temperatures to aestivation as indicated by feeding cessation. Deduced from daily food consumption of individuals, the threshold temperature to aestivation for large and medium animals (73.3-139.3 g) was 24.5-25.5 degrees C, while that for small animals (28.9-40.7 g) was between 25.5 and 30.5 degrees C. These values are higher than previous reports; differences in sign of aestivation, experimental condition and dwelling district of test animals could be the reasons.