# The Role of Invariant Manifolds in the Formation of Spiral Arms and Rings in Barred Galaxies

Universitat Politècnica de Catalunya, February 12nd. 2007

Barred galaxies have been the subject of study of many researchers. In this thesis we propose a new formation mechanism that accounts for the spiral arms and the different types of rings present in barred galaxies. Different from other theories that consider rings and spiral arms to be density waves, our models suggest that they are formed by orbits driven by the normally hyperbolic invariant manifolds of objects in the central manifold, for instance of invariant manifolds associated to periodic orbits, which essentially forms the backbone of the dynamics and it is the case we will focus.

A barred galaxy system has five equilibrium points on the galactic plane, of which three of them are linearly stable and two lie symmetric along the bar semi-minor axis and the other at the centre of coordinates. The other two are also symmetric, they lie along the bar semi-major axis, and they are linearly unstable or hyperbolic. We claim that the unstable family of periodic orbits around the hyperbolic equilibrium points act as a gateway between the bar region and the rest of the galaxy. Two sets of asymptotic orbits emanate from each periodic orbit. One approaches the periodic orbit asymptotically defining the stable invariant manifold, while the other one departs asymptotically from the periodic orbit defining the unstable invariant manifold. The invariant manifolds act like tubes that drive the dynamics around the hyperbolic equilibrium points. The orbits that belong to the invariant manifolds can connect periodic orbits of the same equilibrium point or of the opposite equilibrium point, thus being responsible for the global structure of a galaxy.

We have considered different barred galaxy models consisting of the superposition of an axisymmetric and a bar-like component. We have computed the invariant manifolds of the unstable periodic orbits within an energy range. We have checked that, indeed, they are responsible for the galaxy structure. We have chosen reference models with characteristic rotation curves and studied the influence of the model parameters on the global structure. We have also studied the influence of choosing a particular bar model. We have used three different bar potentials, namely a Ferrers ellipsoid and two other bar potentials not associated to a density distribution. They are the Barbanis-Woltjer potential and a generalized Dehnen's potential.

We have observed that only the variation of bar strength and of the corotation radius have an influence on the shape of the invariant manifolds. We also note that when using a Ferrers ellipsoid to model the bar component, the main structure obtained is that of an rR

_{1}ring, i.e. similar to an 8 or Thetashape. We also obtain spiral arms and other types of rings but they are not realistic models, because the bar is either too strong or the corotation radius lies well inside the bar. We can conclude that Ferrers models are not suitable to model spiral arms because the ratio of the nonaxisymmetric to axisymmetric radial forces decreases too abruptly in the outer parts making the influence of the bar not considerable in that region. On the other hand, the other two bar potentials we have considered do not have this disadvantage. In either case, we observe that the bar has a greater influence in the outer parts when we plot the ratio of the nonaxisymmetric to axisymmetric radial forces. However, the Barbanis-Woltjer potential is more suitable to study the bar region because this ratio changes sign and is slightly increasing in the outer parts. For the case of a generalized Dehnen's potential, the ratio decreases smoothly in the outer part. The galaxy presents different morphologies when we vary the shape of the rotation curve in the outer parts, the bar strength, and the corotation radius. We obtain spiral arms for models with strong bars. rR

_{1}rings are obtained in the case we have a rising rotation curve and a weak bar. R

_{1}R

_{2}rings are obtained in the

case we have weak or mild bars. In the case we have a mild bar and the

corotation radius lies in the vicinity of the bar ends, we have also obtained R

_{2}rings.

Finally, our models suggest that spiral arms and rings in barred galaxies are not density waves. They are formed by a set of orbits emanating from the corotation region at the end of the bar. They are not associated with the galaxy resonances as other models have suggested