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.

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.

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.

Third body perturbation using a single averaged model considering elliptic orbits for the disturbing body

In the present work, we expanded the study done by Solorzanol(1) including the eccentricity of the perturbing body. The assumptions used to develop the single-averaged analytical model are the same ones of the restricted elliptic three-body problem. The disturbing function was expanded in Legendre polynomials up to fourth-order. After that, the equations of motion are obtained from the planetary equations and we performed a set of numerical simulations. Different initial eccentricities for the perturbing and perturbed body are considered. The results obtained perform an analysis of the stability of a near-circular orbits and investigate under which conditions this orbit remain near-circular.

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.

The use of the two-body energy to study problems of escape/capture Ernesto Vieira Neto1

The problem of escape/capture is encountered in many problems of the celestial mechanics -the capture of the giants planets irregular satellites, comets capture by Jupiter, and also orbital transfer between two celestial bodies as Earth and Moon. To study these problems we introduce an approach which is based on the numerical integration of a grid of initial conditions. The two-body energy of the particle relative to a celestial body defines the escape/capture. The trajectories are integrated into the past from initial conditions with negative two-body energy. The energy change from negative to positive is considered as an escape. By reversing the time, this escape turns into a capture. Using this technique we can understand many characteristics of the problem, as the maximum capture time, stable regions where the particles cannot escape from, and others. The advantage of this kind of approach is that it can be used out of plane (that is, for any inclination), and with perturbations in the dynamics of the n-body problem. © 2005 International Astronomical Union.

Quantum propagator for some classes of three-dimensional three-body systems

In this work we solve exactly a class of three-body propagators for the most general quadratic interactions in the coordinates, for arbitrary masses and couplings. This is done both for the constant as the time-dependent couplings and masses, by using the Feynman path integral formalism. Finally, the energy spectrum and the eigenfunctions are recovered from the propagators. © 2005 Elsevier Inc. All rights reserved.

Interplanetary natural routes via Moon and Lagrangiam points L1 and L2

Swing-by techniques are extensively used in interplanetary missions to minimize fuel consumption and to raise payloads of spaceships. The effectiveness of this type of maneuver has been proven since the beginning of space exploration. According to this premise, we have explored the existence of a natural and direct links between low Earth orbits and the lunar sphere of influence, to obtain low-energy interplanetary trajectories through swing-bys with the Moon and the Earth. The existence of these links are related to a family of retrograde periodic orbits around the Lagrangian equilibrium point L1 predicted for the circular, planar, restricted three-body Earth-Moon-particle problem. The trajectories in these links are sensitive to small disturbances. This enables them to be conveniently diverted reducing so the cost of the swing-by maneuver. These maneuvers allow us a gain in energy sufficient for the trajectories to escape from the Earth-Moon system and to stabilize in heliocentric orbits between the Earth and Venus or Earth and Mars. On the other hand, still within the Earth sphere of influence, and taking advantage of the sensitivity of the trajectories, is possible to design other swing-bys with the Earth or Moon. This allows the trajectories to have larger reach, until they can reach the orbit of other planets as Venus and Mars.(3σ)Broucke, R.A., Periodic Orbits in the Restricted Three-Body Problem with Earth-Moon Masses, JPL Technical Report 32-1168, 1968.

Solution of two-body bound state problems with confining potentials

The homogeneous Lippmann-Schwinger integral equation is solved in momentum space by using confining potentials. Since the confining potentials are unbounded at large distances, they lead to a singularity at small momentum. In order to remove the singularity of the kernel of the integral equation, a regularized form of the potentials is used. As an application of the method, the mass spectra of heavy quarkonia, mesons consisting from heavy quark and antiquark (Υ(bb̄), ψ(cc̄)), are calculated for linear and quadratic confining potentials. The results are in good agreement with configuration space and experimental results. © 2010 American Institute of Physics.

Nebular gas drag and planetary accretion with eccentric high-mass planets

Aims.We investigate the dynamics of pebbles immersed in a gas disk interacting with a planet on an eccentric orbit. The model has a prescribed gap in the disk around the location of the planetary orbit, as is expected for a giant planet with a mass in the range of 0.1-1 Jupiter masses. The pebbles with sizes in the range of 1 cm to 3 m are placed in a ring outside of the giant planet orbit at distances between 10 and 30 planetary Hill radii. The process of the accumulation of pebbles closer to the gap edge, its possible implication for the planetary accretion, and the importance of the mass and the eccentricity of the planet in this process are the motivations behind the present contribution. Methods. We used the Bulirsch-Stoer numerical algorithm, which is computationally consistent for close approaches, to integrate the Newtonian equations of the planar (2D), elliptical restricted three-body problem. The angular velocity of the gas disk was determined by the appropriate balance between the gravity, centrifugal, and pressure forces, such that it is sub-Keplerian in regions with a negative radial pressure gradient and super-Keplerian where the radial pressure gradient is positive. Results. The results show that there are no trappings in the 1:1 resonance around the L 4 and L5 Lagrangian points for very low planetary eccentricities (e2 < 0.07). The trappings in exterior resonances, in the majority of cases, are because the angular velocity of the disk is super-Keplerian in the gap disk outside of the planetary orbit and because the inward drift is stopped. Furthermore, the semi-major axis location of such trappings depends on the gas pressure profile of the gap (depth) and is a = 1.2 for a planet of 1 MJ. A planet on an eccentric orbit interacts with the pebble layer formed by these resonances. Collisions occur and become important for planetary eccentricity near the present value of Jupiter (e 2 = 0.05). The maximum rate of the collisions onto a planet of 0.1 MJ occurs when the pebble size is 37.5 cm ≤ s < 75 cm; for a planet with the mass of Jupiter, it is15 cm ≤ s < 30 cm. The accretion stops when the pebble size is less than 2 cm and the gas drag dominates the motion. © 2013 ESO.

Stable regions around Pluto

In a previous work, GiuliattiWinter et al. found several stable regions for test particles in orbit around Pluto associated with families of periodic orbits obtained in the circular, restricted three-body problem. They have shown that a possible eccentricity of the Pluto-Charon binary slightly reduces but does not destroy any of these stable regions. In thiswork, we extended their results by analysing the cases with the orbital inclination (I) equal to zero and considering the argument of pericentre (w) equal to 90°, 180° and 270°. We explore the influence of the orbital inclination of the particles in these stable regions. In this case, the initial inclination varies from 10° to 170° in steps of 10°. We also present a sample of results for the longitude of the ascending node Ω = 90°, considering the cases I = 20°, 50°, 130° and 180°. Our results show that stable regions are present in all of the inclined cases, except when the initial inclination of the particles is equal to 110°. A sample of 3D trajectories of quasi-periodic orbits were found related to the periodic orbits obtained in the planar case by Giuliatti Winter et al. © 2013 The Authors. Published by Oxford University Press on behalf of the Royal Astronomical Society.

Desenvolvimento da função perturbadora e aplicações em dinâmica de exoplanetas

Pós-graduação em Matemática Universitária - IGCE