Possibility of collision between co-orbital asteroids and the Earth

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

Algumas considerações sobre o estatuto do psicológico no Projetofreudiano

Embora no Projeto freudiano não seja oferecida, de uma maneira explícita, uma conceituação a respeito do estatuto do psicológico, pensamos que é possível delinear esse estatuto e tentaremos fazê-lo com base em uma maneira de interpretar o modelo do psiquismo que é construído no texto freudiano. Isto levar-nos-á também a situar o Projeto em relação ao clássico problema mente-corpo. Pretendemos dar conta da tarefa proposta apoiando-nos, sobretudo, em uma distinção que estabeleceremos entre o representar, entendido como ato intencional, e a representação, entendida como traço de memória. Também contemplaremos a forma de comportamento do fator quantitativo-energético nas ordens processuais do psicológico e do neurofisiológico.

Distant stable direct orbits around the Moon

In the present work we explore regions of distant direct stable orbits around the Moon. First, the location and size of apparently stable regions are searched for numerically, adopting the approach of temporary capture time presented in Vieira Neto & Winter (2001). The study is made in the framework of the planar, circular, restricted three-body problem, Earth-Moon-particle. Regions of the initial condition space whose trajectories are apparently stable are determined. The criterion adopted was that the trajectories do not escape from the Moon during an integration period of 10(4) days. Using Poincare surface of sections the reason for the existence of the two stable regions found is studied. The stability of such regions proved to be due to two families of simple periodic orbits, h1 and h2, and the associated quasi-periodic orbits that oscillate around them. The robustness of the stability of the larger region, h2, is tested with the inclusion of the solar perturbation. The size of the region decreases, but it is still significant in size and can be useful in spacecraft missions.

Time analysis for temporary gravitational capture - Stable orbits

In a previous work, Vieira Neto & Winter (2001) numerically explored the capture times of particles as temporary satellites of Uranus. The study was made in the framework of the spatial, circular, restricted three-body problem. Regions of the initial condition space whose trajectories are apparently stable were determined. The criterion adopted was that the trajectories do not escape from the planet during an integration of 10(5) years. These regions occur for a wide range of orbital initial inclinations (i). In the present work it is studied the reason for the existence of such stable regions. The stability of the planar retrograde trajectories is due to a family of simple periodic orbits and the associated quasi-periodic orbits that oscillate around them. These planar stable orbits had already been studied (Henon 1970; Huang & Innanen 1983). Their results are reviewed using Poincare surface of sections. The stable non-planar retrograde trajectories, 110 degrees less than or equal to i < 180, are found to be tridimensional quasi-periodic orbits around the same family of periodic orbits found for the planar case (i = 180 degrees). It was not found any periodic orbit out of the plane associated to such quasi-periodic orbits. The largest region of stable prograde trajectories occurs at i = 60 degrees. Trajectories in such region are found to behave as quasi-periodic orbits evolving similarly to the stable retrograde trajectories that occurs at i = 120 degrees.

The Liapunov exponent as a tool for exploring phase space

We have used the Liapunov exponent to explore the phase space of a dynamical system. Considering the planar, circular restricted three-body problem for a mass ratio mu = 10(-3) (close to the Jupiter/Sun case), we have integrated similar to 16,000 starting conditions for orbits started interior to that of the perturber and we have estimated the maximum Liapunov characteristic exponent for each starting condition. Despite the fact that the integrations, in general, are for only a few thousand orbital periods of the secondary, a comparative analysis of the Liapunov exponents for various values of the 'cut-off' gives a good overview of the structure of the phase space. It provides information about the diffusion rates of the various chaotic regions, the location of the regular regions associated with primary resonances and even details such as the location of secondary resonances that produce chaotic regions inside the regular regions of primary resonances.

Renormalization in non-relativistic quantum mechanics

The importance and usefulness of renormalization are emphasized in non-relativistic quantum mechanics. The momentum space treatment of both two-body bound state and scattering problems involving some potentials singular at the origin exhibits ultraviolet divergence. The use of renormalization techniques in these problems leads to finite converged results for both the exact and perturbative solutions. The renormalization procedure is carried out for the quantum two-body problem in different partial waves for a minimal potential possessing only the threshold behaviour and no form factors. The renormalized perturbative and exact solutions for this problem are found to be consistent with each other. The useful role of the renormalization group equations for this problem is also pointed out.

Resonance and chaos .1. First-order interior resonances

Analytical models for studying the dynamical behaviour of objects near interior, mean motion resonances are reviewed in the context of the planar, circular, restricted three-body problem. The predicted widths of the resonances are compared with the results of numerical integrations using Poincare surfaces of section with a mass ratio of 10(-3) (similar to the Jupiter-Sun case). It is shown that for very low eccentricities the phase space between the 2:1 and 3:2 resonances is predominantly regular, contrary to simple theoretical predictions based on overlapping resonance. A numerical study of the 'evolution' of the stable equilibrium point of the 3:2 resonance as a function of the Jacobi constant shows how apocentric libration at the 2:1 resonance arises; there is evidence of a similar mechanism being responsible for the centre of the 4:3 resonance evolving towards 3:2 apocentric libration. This effect is due to perturbations from other resonances and demonstrates that resonances cannot be considered in isolation. on theoretical grounds the maximum libration width of first-order resonances should increase as the orbit of the perturbing secondary is approached. However, in reality the width decreases due to the chaotic effect of nearby resonances.

Effects of the eccentricity on the primaries in the gravitational capture phenomenon

Gravitational capture is a characteristic of some dynamical systems in celestial mechanics, as in the elliptic restricted three-body problem that is considered in this paper. The basic idea is that a spacecraft (or any particle with negligible mass) can change a hyperbolic orbit with a small positive energy around a celestial body into an elliptic orbit with a small negative energy without the use of any propulsive system. The force responsible for this modification in the orbit of the spacecraft is the gravitational force of the third body involved in the dynamics. In this way, this force is used as a zero cost control, equivalent to a continuous thrust applied in the spacecraft. One of the most important applications of this property is the construction of trajectories to the Moon. The objective of the present paper is to study in some detail the effects of the eccentricity of the primaries in this maneuver.

Peculiar trajectories around the Lagrangian equilateral points

In the present work we analyse the behaviour of a particle under the gravitational influence of two massive bodies and a particular dissipative force. The circular restricted three body problem, which describes the motion of this particle, has five equilibrium points in the frame which rotates with the same angular velocity as the massive bodies: two equilateral stable points (L-4, L-5) and three colinear unstable points (L-1, L-2, L-3). A particular solution for this problem is a stable orbital libration, called a tadpole orbit, around the equilateral points. The inclusion of a particular dissipative force can alter this configuration. We investigated the orbital behaviour of a particle initially located near L4 or L5 under the perturbation of a satellite and the Poynting-Robertson drag. This is an example of breakdown of quasi-periodic motion about an elliptic point of an area-preserving map under the action of dissipation. Our results show that the effect of this dissipative force is more pronounced when the mass of the satellite and/or the size of the particle decrease, leading to chaotic, although confined, orbits. From the maximum Lyapunov Characteristic Exponent a final value of gamma was computed after a time span of 10(6) orbital periods of the satellite. This result enables us to obtain a critical value of log y beyond which the orbit of the particle will be unstable, leaving the tadpole behaviour. For particles initially located near L4, the critical value of log gamma is -4.07 and for those particles located near L-5 the critical value of log gamma is -3.96. (c) 2006 Elsevier B.V. All rights reserved.

Irregular Satellites of Jupiter: a study of the capture direction

The irregular satellites of Jupiter are believed to be captured asteroids or planetesimals. In the present work is studied the direction of capture of these objects as a function of their orbital inclination. We performed numerical simulations of the restricted three-body problem, Sun-Jupiter-particle, taking into account the growth of Jupiter. The integration was made backward in time. Initially, the particles have orbits as satellites of Jupiter, which has its present mass. Then, the system evolved with Jupiter losing mass and the satellites escaping from the planet. The reverse of the escape direction corresponds to the capture direction. The results show that the Lagrangian points L1 and L2 mainly guide the direction of capture. Prograde satellites are captured through these two gates with very narrow amplitude angles. In the case of retrograde satellites, these two gates are wider. The capture region increases as the orbital inclination increases. In the case of planar retrograde satellites the directions of capture cover the whole 360 degrees around Jupiter. We also verified that prograde satellites are captured earlier in actual time than retrograde ones.

Resonance and chaos .2. Exterior resonances and asymmetric libration

The motion of a test particle in the vicinity of exterior resonances is examined in the context of the planar, circular, restricted three-body problem. The existence of asymmetric periodic orbits associated with the 1 : n resonances (where n = 2, 3, 4, 5) is confirmed; there is also evidence of asymmetric resonances associated with larger values of n. A detailed examination of the evolution of the family of orbits associated with the 1:2 resonance shows the sequence that leads to asymmetric libration. on the basis of numerical studies of the phase space it is concluded that the existence of asymmetric libration means that the region exterior to the perturbing mass is more chaotic than the interior region. The apparent absence of 'particles' in 1 : n resonances in the solar system may reflect this inherent bias.

Sphere of influence and gravitational capture radius: a dynamical approach

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Controlling the Eccentricity of Polar Lunar Orbits with Low-Thrust Propulsion

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Resonance and chaos: II. Exterior resonances and asymmetric libration

The motion of a test particle in the vicinity of exterior resonances is examined in the context of the planar, circular, restricted three-body problem. The existence of asymmetric periodic orbits associated with the 1 : n resonances (where n = 2, 3, 4, 5) is confirmed; there is also evidence of asymmetric resonances associated with larger values of n. A detailed examination of the evolution of the family of orbits associated with the 1:2 resonance shows the sequence that leads to asymmetric libration. On the basis of numerical studies of the phase space it is concluded that the existence of asymmetric libration means that the region exterior to the perturbing mass is more chaotic than the interior region. The apparent absence of 'particles' in 1 : n resonances in the solar system may reflect this inherent bias.

The stability evolution of a family of simply periodic lunar orbits

The present work deals with a family of simply periodic orbits around the Moon in the rotating Earth Moon-particle system. Taking the framework of the planar, circular, restricted three-body problem, we follow the evolution of this family of periodic orbits using the numerical technique of Poincaré surface of section. The maximum amplitude of oscillation about the periodic orbits are determined and can be used as a parameter to measure the degree of stability in the phase space for such orbits. Despite the fact that the whole family of periodic orbits remain stable, there is a dichotomy in the quasi-periodic ones at the Jacobi constant Cj = 2.85. The quasi-periodic orbits with Cj < 2.85 oscillate around the periodic orbits in a different way from those with Cj > 2.85. © 1999 Elsevier Science Ltd. All rights reserved.