P. A. Patsis

Particle swarm optimization: an efficient method for tracing periodic orbits in three-dimensional galactic potentials

We propose particle swarm optimization (PSO) as an alternative method for locating periodic orbits in a three-dimensional (3D) model of barred galaxies. We develop an appropriate scheme that transforms the problem of finding periodic orbits into the problem of detecting global minimizers of a function, which is defined on the Poincare surface section of the Hamiltonian system. By combining the PSO method with deflection techniques, we succeeded in tracing systematically several periodic orbits of the system. The method succeeded in tracing the initial conditions of periodic orbits in cases where Newton iterative techniques had difficulties. In particular, we found families of two- and three-dimensional periodic orbits associated with the inner 8:1 to 12:1 resonances, between the radial 4:1 and corotation resonances of our 3D Ferrers bar model. The main advantages of the proposed algorithm are its simplicity, its ability to work using function values solely, and its ability to locate many periodic orbits per run at a given Jacobian constant. Ke yw ords: methods: numerical - galaxies: kinematics and dynamics - galaxies: structure.

Boxy/peanut ‘bulges’: comparing the structure of galaxies with the underlying families of periodic orbits

The vertical profiles of disc galaxies are built by the material trapped around stable periodic orbits, which form their ‘skeletons’. Therefore, knowledge of the stability of the main families of periodic orbits in appropriate 3D models enables one to predict possible morphologies for edge-on disc galaxies. In a pilot survey we compare the orbital structures that lead to the appearance of ‘peanut’- and ‘X’-like features with the edge-on profiles of three disc galaxies (IC 2531, NGC 4013 and UGC 2048). The subtraction from the images of a model representing the axisymmetric component of the galaxies reveals the contribution of the non-axisymmetric terms. We find a direct correspondence between the orbital profiles of 3D bars in models and the observed main morphological features of the residuals. We also apply a simple unsharp masking technique in order to study the sharpest features of the images. Our basic conclusion is that the morphology of the boxy ‘bulges’ of these galaxies can be explained by considering disc material trapped around stable 3D periodic orbits. In most models, these building-block periodic orbits are bifurcated from the planar central family of a non-axisymmetric component, usually a bar, at low-order vertical resonances. In such a case, the boxy ‘bulges’ are parts of bars seen edge-on. For the three galaxies we study, the families associated with the ‘peanut’ or ‘X’-shape morphology are probably bifurcations at the vertical 2/1 or 4/1 resonance.

Structures induced by companions in galactic discs

Using N-body simulations we study the structures induced on a galactic disc by repeated flybys of a companion in decaying eccentric orbit around the disc. Our system is composed by a stellar disc, bulge and live dark matter halo, and we study the systems dynamical response to a sequence of a companions flybys, when we vary i) the discs temperature (parameterized by Toomres Q-parameter) and ii) the companions mass and initial orbit. We use a new 3D Cartesian grid code: MAIN (Mesh-adaptive Approximate Inverse N-body solver). The main features of MAIN are reviewed, with emphasis on the use of a new Symmetric Factored Approximate Sparse Inverse (SFASI) matrix in conjunction with the multigrid method that allows the efficient solution of Poissons equation in three space variables. We find that: i) companions need to be assigned initial masses in a rather narrow window of values in order to produce significant and more long-standing non-axisymmetric structures (bars and spirals) in the main galaxys disc by the repeated flyby mechanism. ii) a crucial phenomenon is the antagonism between companion-excited and self-excited modes on the disc. Values of

Morphologies introduced by bistability in barred-spiral galactic potentials

Q >1.5

Boxy Orbital Structures in Rotating Bar Models

are needed in order to allow for the growth of the companion-excited modes to prevail over the the growth of the discs self-excited modes. iii) We give evidence that the companion-induced spiral structure is best represented by a density wave with pattern speed nearly constant in a region extending from the ILR to a radius close to, but inside, corotation.

Boxy isophotes in face–on views of barred galaxies

We investigate the orbital dynamics of a barred-spiral model when the system is rotating slowly and corotation is located beyond the end of the spiral arms. In the characteristic of the central family of periodic orbits we find a “bistable region” . In the response model we observe a ring surrounding the bar and spiral arms starting tangenti al to the ring. This is a morphology resembling barred-spiral systems with inner rings. However, the dynamics associated with this structure in the case we study is different from that of a typical bar ending close to corotation. The ring of our model is round, or rather elongated perpendicular to the bar. It is associated with a folding (an “S” shaped feature) of the characteristic of the central family, which is typical in bistable bifurcations. Along the “S” part of the c haracteristic we have a change in the orientation of the periodic orbits from a x1-type to a x 2-type morphology. The orbits populated in the response model change rather abruptly their orientation when reaching the lowest energy of the “S”. The spirals of the model follow a standard “precessing ellipses flow” and the orbits building them have energies beyond the “S” region. The bar is structured mainly by sticky orbits from regions around the stability islands o f the central family. This leads to the appearance of X-features in the bars on the galactic plane. Such a bar morphology appears in the unsharp-masked images of some moderately inclined galaxies.

Outer dynamics and escapes in barred galaxies

We investigate regular and chaotic two-dimensional (2D) and three-dimensional (3D) orbits of stars in models of a galactic potential consisting in a disk, a halo and a bar, to find the origin of boxy components, which are part of the bar or (almost) the bar itself. Our models originate in snapshots of an N-body simulation, which develops a strong bar. We consider three snapshots of the simulation and for the orbital study we treat each snapshot independently, as an autonomous Hamiltonian system. The calculated corotation-to-bar-length ratios indicate that in all three cases the bar rotates slowly, while the orientation of the orbits of the main family of periodic orbits changes along its characteristic. We characterize the orbits as regular, sticky, or chaotic after integrating them for a 10 Gyr period by using the GALI

The phase space of boxy–peanut and X-shaped bulges in galaxies – I. Properties of non-periodic orbits

_2

On the relation between orbital structure and observed bar morphology

index. Boxiness in the equatorial plane is associated either with quasi-periodic orbits in the outer parts of stability islands, or with sticky orbits around them, which can be found in a large range of energies. We indicate the location of such orbits in diagrams, which include the characteristic of the main family. They are always found about the transition region from order to chaos. By perturbing such orbits in the vertical direction we find a class of 3D non-periodic orbits, which have boxy projections both in their face-on and side-on views.

The backbones of stellar structures in barred-spiral models – the concerted action of various dynamical mechanisms on galactic discs

We study the conditions that favour boxiness of isodensities in the face-on views of orbital 3D models for barred galaxies. in particular, we consider several cases with different mass values for the bar, disk and bulge components of the potential, as well as different values of the bar pattern speed. Using orbital weighted profiles of the basic stable periodic orbits of the x1 tree for every model, we show that boxiness is in general a composite effect due to the presence of stable orbits belonging to several 2D and 3D families.