E. Athanassoula

Rings and spirals in barred galaxies – I. Building blocks

In this paper we present building blocks which can explain the formation and properties both of spirals and of inner and outer rings in barred galaxies. We first briefly summarize the main results of the full theoretical description we have given elsewhere, presenting them in a more physical way, aimed to an understanding without the requirement of extended knowledge of dynamical systems or of orbital structure. We introduce in this manner the notion of manifolds, which can be thought of as tubes guiding the orbits. The dynamics of these manifolds can govern the properties of spirals and of inner and outer rings in barred galaxies. We find that the bar strength affects how unstable the L 1 and L 2 Lagrangian points are, the motion within the manifold tubes and the time necessary for particles in a manifold to make a complete turn around the galactic centre. We also show that the strength of the bar, or, to be more precise, of the non-axisymmetric forcing at and somewhat beyond the corotation region, determines the resulting morphology. Thus, less strong bars give rise to R 1 rings or pseudo-rings, while stronger bars drive R 2 , R 1 R 2 and spiral morphologies. We examine the morphology as a function of the main parameters of the bar and present descriptive two-dimensional plots to that avail. We also derive how the manifold morphologies and properties are modified if the L 1 and L 2 Lagrangian points become stable. Finally, we discuss how dissipation affects the manifold properties and compare the manifolds in gas like and in stellar cases. Comparison with observations as well as clear predictions to be tested by observations will be given in an accompanying paper.

Rings and spirals in barred galaxies III. Further comparisons and links to observations

In a series of papers, we propose a theory to explain the formation and properties of rings and spirals in barred galaxies. The building blocks of these structures are orbits guided by the manifolds emanating from the unstable Lagrangian points located near the ends of the bar. In this paper, the last of the series, we present more comparisons of our theoretical results to observations and also give new predictions for further comparisons. Our theory provides the right building blocks for the rectangular-like bar outline and for ansae. We consider how our results can be used to give estimates for the pattern speed values, as well as their effect on abundance gradients in barred galaxies. We present the kinematics along the manifold loci, to allow comparisons with the observed kinematics along the ring and spiral loci. We consider gaseous arms and their relations to stellar ones. We discuss several theoretical aspects and stress that the orbits that constitute the building blocks of the spirals and rings are chaotic. They are, nevertheless, spatially well confined by the manifolds and are thus able to outline the relevant structures. Such chaos can be termed ‘confined chaos’ and can play a very important role in understanding the formation and evolution of galaxy structures and in galactic dynamics in general. This work, in agreement with several others, argues convincingly that galactic dynamic studies should not be limited to the study of regular motions and orbits.

An iterative method for constructing equilibrium phase models of stellar systems

We present a new method for constructing equilibrium phase models for stellar systems, which we call the iterative method. It relies on constrained, or guided evolution, so that the equilibrium solution has a number of desired parameters and/or constraints. This method is very powerful, to a large extent due to its simplicity. It can be used for mass distributions with an arbitrary geometry and a large variety of kinematical constraints. We present several examples illustrating it. Applications of this method include the creation of initial conditions for N-body simulations and the modelling of galaxies from their photometric and kinematic observations.

Regular and chaotic orbits in barred galaxies - I. Applying the SALI/GALI method to explore their distribution in several models

The distinction between the chaotic and regular behaviour of orbits in galactic models is an important issue and can help our understanding of galactic dynamical evolution. In this paper, we deal with this issue by applying the Smaller ALingment Index (SALI) and Generalized ALingment Index (GALI) techniques to extensive samples of orbits obtained by integrating numerically the equations of motion in a barred galaxy potential. We estimate first the fraction of chaotic and regular orbits for the 2 degrees of freedom (d.o.f.) case [where the galaxy extends only in the (x, y) space] and show that it is a non-monotonic function of the energy. For the 3-d.o.f. extension of this model (in the z-direction), we give similar estimates, both by exploring different sets of initial conditions and by varying the model parameters, like the mass, size and pattern speed of the bar. We find that regular motion is more abundant at small radial distances from the centre of the galaxy, where the relative non-axisymmetric forcing is relatively weak, and at small distances from the equatorial plane, where the trapping around the stable periodic orbits is important. We also find that the variation in the bar pattern speed, within a realistic range of values, does not affect much the phase-space fraction of regular and chaotic motions. Using different sets of initial conditions, we show that chaotic motion is dominant in galaxy models whose bar component is more massive, while models with a fatter or thicker bar present generally more regular behaviour. Finally, we find that the fraction of orbits that are chaotic correlates strongly with the bar strength.

Modelling the inner disc of the Milky Way with manifolds - I. A first step

We study the bar-driven dynamics in the inner part of the Milky Way by using invariant manifolds. This theory has been successfully applied to describe the morphology and kinematics of rings and spirals in external galaxies, and now, for the first time, we apply it to the Milky Way. In particular, we compute the orbits confined by the invariant manifolds of the unstable periodic orbits located at the ends of the bar. We start by discussing whether the COBE/Diffuse Infrared Background Experiment (DIRBE) bar and the Long bar compose a single bar or two independent bars and perform a number of comparisons which, taken together, argue strongly in favour of the former. More specifically, we favour the possibility that the so-called COBE/DIRBE bar is the boxy/peanut bulge of a bar whose outer thin parts are the so-called Long bar. This possibility is in good agreement both with observations of external galaxies, with orbital structure theory and with simulations. We then analyse in detail the morphology and kinematics given by five representative Galactic potentials. Two of these have a Ferrers bar, two have a quadrupole bar and the last one a composite bar. We first consider only the COBE/DIRBE bar and then extend it to include the effect of the Long bar. We find that the large-scale structure given by the manifolds describes an inner ring, whose size is similar to the near and far 3-kpc arm, and an outer ring, whose properties resemble those of the Galactic Molecular Ring. We also analyse the kinematics of these two structures, under the different Galactic potentials, and find they reproduce the relevant overdensities found in the galactic longitude-velocity CO diagram. Finally, we consider for what model parameters, the global morphology of the manifolds may reproduce the two outer spiral arms. We conclude that this would necessitate either more massive and more rapidly rotating bars, or including in the potential an extra component describing the spiral arms.

Loss of halo triaxiality due to bar formation

Cosmological N-body simulations indicate that the dark matter haloes of galaxies should be generally triaxial. Yet, the presence of a baryonic disc is believed to alter the shape of the haloes. Here we aim to study how bar formation is affected by halo triaxiality and how, in turn, the presence of the bar influences the shape of the halo. We perform a set of collisionless N-body simulations of disc galaxies with triaxial dark matter haloes, using elliptical discs as initial conditions. Such discs are much closer to equilibrium with their haloes than circular ones, and the ellipticity of the initial disc depends on the ellipticity of the halo gravitational potential. For comparison, we also consider models with initially circular discs, and find that the differences are very important. In all cases, the mass of the disc is grown quasi-adiabatically within the haloes, but the time-scale of growth is not very important. We study models of different halo triaxialities and, to investigate the behaviour of the halo shape in the absence of bar formation, we run models with different disc masses, halo concentrations, disc velocity dispersions and also models where the disc shape is kept artificially axisymmetric. We find that the introduction of a massive disc, even if this is not circular, causes the halo triaxiality to be partially diluted. Once the disc is fully grown, a strong stellar bar develops within the halo that is still non-axisymmetric, causing it to lose its remaining non-axisymmetry. In triaxial haloes in which the parameters of the initial conditions are such that a bar does not form, the halo is able to remain triaxial and the circularization of its shape on the plane of the disc is limited to the period of disc growth. We conclude that part of the circularization of the halo is due to disc growth, but part must be attributed to the formation of a bar. Bars in the halo component, which have already been found in axisymmetric haloes, are also found in triaxial ones. We find that initially circular discs respond excessively to the triaxial potential and become highly elongated. They also lose more angular momentum than the initially elliptical discs and thus form stronger bars. Because of that, the circularization that their bars induce on their haloes is also more rapid. We also analyse halo vertical shapes and observe that their vertical flattenings remain considerable, meaning that the haloes become approximately oblate by the end of the simulations. Finally, we also analyse the kinematics of a subset of halo particles that rotate in disc-like manner. These particles occupy a layer around the plane of the disc and their rotation is more important in the spherical halo than in triaxial ones. We also find that, even though the final shape of the halo is roughly independent of the initial shape, the initially triaxial ones are able to retain the anisotropy of their velocity dispersions.

Adventures of a tidally induced bar

Using N-body simulations, we study the properties of a bar induced in a discy dwarf galaxy as a result of tidal interaction with the Milky Way. The bar forms at the first pericentre passage and survives until the end of the evolution at 10 Gyr. Fourier decomposition of the bar reveals that only even modes are significant and preserve a hierarchy so that the bar mode is always the strongest. They show a characteristic profile with a maximum, similar to simulated bars forming in isolated galaxies and observed bars in real galaxies. We adopt the maximum of the bar mode as a measure of the bar strength and we estimate the bar length by comparing the density profiles along the bar and perpendicular to it. The bar strength and the bar length decrease with time, mainly at pericentres, as a result of tidal torques acting at those times and not to secular evolution. The pattern speed of the bar varies significantly on a time-scale of 1 Gyr and is controlled by the orientation of the tidal torque from the Milky Way. The bar is never tidally locked, but we discover a hint of a 5/2 orbital resonance between the third and fourth pericentre passage. The speed of the bar decreases in the long run so that the bar changes from initially rather fast to slow in the later stages. The boxy/peanut shape is present for some time and its occurrence is preceded by a short period of buckling instability.

Regular motions in double bars – II. Survey of trajectories and 23 models

We show that stable double-frequency orbits form the backbone of double bars, because they trap around themselves regular orbits, as stable closed periodic orbits do in single bars, and in both cases the trapped orbits occupy similar volume of phase space. We perform a global search for such stable double-frequency orbits in a model of double bars by constructing maps of trajectories with initial conditions well sampled over the available phase space. We use the width of a ring sufficient to enclose a given map as the indicator of how tightly the trajectory is trapped around a double-frequency orbit. We construct histograms of these ring-widths in order to determine the fraction of phase space occupied by ordered motions. We build 22 further models of double bars, and we construct histograms showing the fraction of the phase space occupied by regular orbits in each model. Our models indicate that resonant coupling between the bars may not be the dominant factor reducing chaos in the system.

TIDALLY INDUCED BARS OF GALAXIES IN CLUSTERS

Using N-body simulations, we study the formation and evolution of tidally induced bars in disky galaxies in clusters. Our progenitor is a massive, late-type galaxy similar to the Milky Way, composed of an exponential disk and a Navarro-Frenk-White dark matter halo. We place the galaxy on four different orbits in a Virgo-like cluster and evolve it for 10 Gyr. As a reference case, we also evolve the same model in isolation. Tidally induced bars form on all orbits soon after the first pericenter passage and survive until the end of the evolution. They appear earlier, are stronger and longer, and have lower pattern speeds for tighter orbits. Only for the tightest orbit are the properties of the bar controlled by the orientation of the tidal torque from the cluster at pericenter. The mechanism behind the formation of the bars is the angular momentum transfer from the galaxy stellar component to its halo. All of the bars undergo extended periods of buckling instability that occur earlier and lead to more pronounced boxy/peanut shapes when the tidal forces are stronger. Using all simulation outputs of galaxies at different evolutionary stages, we construct a toy model of the galaxy population in the cluster and measure the average bar strength and bar fraction as a function of clustercentric radius. Both are found to be mildly decreasing functions of radius. We conclude that tidal forces can trigger bar formation in cluster cores, but not in the outskirts, and thus can cause larger concentrations of barred galaxies toward the cluster center.

Star cluster evolution in barred disc galaxies – I. Planar periodic orbits

The dynamical evolution of stellar clusters is driven to a large extent by their environment. Several studies so far have considered the effect of tidal fields and their variations, for example, from giant molecular clouds, galactic discs or spiral arms. In this paper, we will concentrate on a tidal field whose effects on star clusters have not yet been studied, namely that of bars. We present a set of direct N-body simulations of star clusters moving in an analytic potential representing a barred galaxy. We compare the evolution of the clusters moving both on different planar periodic orbits in the barred potential and on circular orbits in a potential obtained by axisymmetrizing its mass distribution. We show that both the shape of the underlying orbit and its stability have strong impact on the cluster evolution as well as the morphology and orientation of the tidal tails and the substructures therein. We find that the dissolution time-scale of the cluster in our simulations is mainly determined by the tidal forcing along the orbit and, for a given tidal forcing, only very little by the exact shape of the gravitational potential in which the cluster is moving.