Escapes and Recurrence in a Simple Hamiltonian System

Many physical systems can be modeled as scattering problems. For example, the motions of stars escaping from a galaxy can be described using a potential with two or more escape routes. Each escape route is crossed by an unstable Lyapunov orbit. The region between the two Lyapunov orbits is where the particle interacts with the system. We study a simple dynamical system with escapes using a suitably selected surface of section. The surface of section is partitioned in different escape regions which are defined by the intersections of the asymptotic manifolds of the Lyapunov orbits with the surface of section. The asymptotic curves of the other unstable periodic orbits form spirals around various escape regions. These manifolds, together with the manifolds of the Lyapunov orbits, govern the transport between different parts of the phase space. We study in detail the form of the asymptotic manifolds of a central unstable periodic orbit, the form of the escape regions and the infinite spirals of the asymptotic manifolds around the escape regions. We compute the escape rate for different values of the energy. In particular, we give the percentage of orbits that escape after a finite number of iterations. In a system with escapes one cannot define a Poincaré recurrence time, because the available phase space is infinite. However, for certain domains inside the lobes of the asymptotic manifolds there is a finite minimum recurrence time. We find the minimum recurrence time as a function of the energy.This revised version was published online in October 2005 with corrections to the Cover Date.     

Dynamics of Two Planets in the 2/1 Mean-Motion Resonance

The dynamics of two planets near a first-order mean-motion resonance is modeled in the domain of the general three-body planar problem. The system studied is the pair Uranus-Neptune (2/1 resonance). The phase space of the resonance and near-resonance regions is studied by means of surfaces of section and spectral analysis techniques. After a thorough investigation of the topology of the phase space, we find that several regimes of motion are possible for the Uranus-Neptune system, and the regions of transition between the regimes of motion are the seats of chaotic motion. This revised version was published online in July 2006 with corrections to the Cover Date.     

Dynamical behaviour of the primitive asteroid belt

In this paper we consider the dynamical evolution and orbital stability of objects in the asteroid belt. A simple physical model, including full gravitational perturbations from both giant planets, is used to compute the dynamical evolution of 1000 test particles simulating the primitive asteroids. The criterion of planet crossing (or close approach in the case of resonant objects) is used to reject particles from the simulation. 44 per cent of the particles survived for the whole time-span covered by the numerical integration (∼10⁹ yr).
The 4:1, 3:1 and to a lesser extent the 2:1 Kirkwood gaps are formed in ∼10⁷ yr of evolution, representing direct numerical evidence about their gravitational origin.
We found that the rms eccentricity and inclination of the sample experience a fast increase during the first 10⁶ yr. The final rms eccentricity is 0.11, ∼60 per cent smaller than the present rms eccentricity (0.17). Nevertheless, the gravitational action of the giant planets suffices to prevent the formation of large objects, allowing catastrophic collisions and the subsequent depletion of material from this zone of the Solar system. The excited eccentricity by Jupiter and Saturn may favour mutual encounters and the further increase of the relative velocities up to their present values.     

Simulations of spheroidal systems with substructure: trees in fields

We present a hybrid technique of N -body simulation to deal with collisionless stellar systems having an inhomogeneous global structure. We combine a treecode and a self-consistent field code such that each of the codes models a different component of the system being investigated. The treecode is suited to treatment of dynamically cold or clumpy components, which may undergo significant evolution within a dynamically hot system. The hot system is appropriately evolved by the self-consistent field code. This combined code is particularly suited to a number of problems in galactic dynamics. Applications of the code to these problems are briefly discussed.     

Orbit classification in arbitrary 2D and 3D potentials

A method of classifying generic orbits in arbitrary 2D and 3D potentials is presented. It is based on the concept of spectral dynamics introduced by Binney &38; Spergel that uses the Fourier transform of the time series of each coordinate. The method is tested using a number of potentials previously studied in the literature and is shown to distinguish correctly between regular and irregular orbits, to identify the various families of regular orbits (boxes, loops, tubes, boxlets, etc.), and to recognize the second-rank resonances that bifurcate from them. The method returns the position of the potential centre and, for 2D potentials, the orientation of the principal axes as well, should this be unknown. A further advantage of the method is that it has been encoded in a FORTRAN program that does not require user intervention, except for 'fine tuning' of search parameters that define the numerical limits of the code. The automatic character makes the program suitable for classifying large numbers of orbits.     

Frequency map analysis of the orbital structure in elliptical galaxies

We present an application of the frequency map analysis to an elliptical galaxy which is represented by a generalization of a double-power-law spherical mass model. The density distribution of this model varies as r ^−γ close to the centre and as r ⁻⁴ at large radii. We study the case with γ = 1, which is known as the 'weak-cusp' model and which represents well the density profile of the 'core' galaxies observed by the Hubble Space Telescope . The final objective of our work is to improve our understanding of the dynamics of elliptical galaxies in a similar way to Merritt &38; Fridman, finding the regions of stochasticity, looking for resonances that might play an important role in sustaining the triaxial morphology, and analysing the diffusion of orbits. To this end, we use the frequency map analysis of Laskar, which has been applied widely in the field of celestial mechanics but which is a relatively new technique in the area of galactic dynamics. Finally, we show some useful features of this method in understanding the global dynamical structure of the system.     

Stellar dynamics in a galactic centre surrounded by a massive accretion disc — I. Newtonian description

The long-term evolution of stellar orbits bound to a massive centre is studied in order to understand the cores of star clusters in central regions of galaxies. Stellar trajectories undergo tiny perturbations, the origins of which are twofold: (i) the gravitational field of a thin gaseous disc surrounding the galactic centre, and (ii) cumulative drag arising from successive interactions of the stars with the material of the disc. Both effects are closely related because they depend on the total mass of the disc, assumed to be a small fraction of the central mass. It is shown that, in contrast to previous works, most of the retrograde (with respect to the disc) orbits are captured by the central object, presumably a massive black hole. Initially prograde orbits are also affected, so that statistical properties of the central star cluster in quasi-equilibrium may differ significantly from those deduced in previous analyses.     

Evolution of the mass function of the Galactic globular cluster system

In this work we investigate the evolution of the mass function of the Galactic globular cluster system (GCMF) taking into account the effects of stellar evolution, two-body relaxation, disc shocking and dynamical friction on the evolution of individual globular clusters. We have adopted a lognormal initial GCMF and considered a wide range of initial values for the dispersion, σ, and the mean value, 〈log  M 〉. We have studied in detail the dependence on the initial conditions of the final values of σ, 〈log  M 〉, the fraction of the initial number of clusters surviving after one Hubble time and the difference between the properties of the GCMF of clusters closer to the Galactic Centre and those of clusters located in the outer regions of the Galaxy. In most of the cases considered, evolutionary processes alter significantly the initial population of globular clusters and the disruption of a significant number of globular clusters leads to a flattening in the spatial distribution of clusters in the central regions of the Galaxy. The initial lognormal shape of the GCMF is preserved in most cases and if a power-law in M is adopted for the initial GCMF, evolutionary processes tend to modify it into a lognormal GCMF. The difference between initial and final values of σ and 〈log  M 〉 as well as the difference between the final values of these parameters for inner and outer clusters can be positive or negative depending on initial conditions. A significant effect of evolutionary processes does not necessarily give rise to a strong trend of 〈log  M 〉 with the galactocentric distance. The existence of a particular initial GCMF able to keep its initial shape and parameters unaltered during the entire evolution through a subtle balance between disruption of clusters and evolution of the masses of those which survive, suggested by Vesperini, is confirmed.