Invariant manifolds, phase correlations of chaotic orbits and the spiral structure of galaxies

In the presence of a strong $m=2$ component in a rotating galaxy, the phase space structure near corotation is shaped to a large extent by the {\it invariant manifolds} of the short period family of unstable periodic orbits terminating at L$_1$ or L$_2$. The main effect of these manifolds is to create robust {\it phase correlations} among a number of chaotic orbits large enough to support a {\it spiral} density wave outside corotation. The phenomenon is described theoretically by soliton-like solutions of a Sine-Gordon equation. Numerical examples are given in an N-Body simulation of a barred spiral galaxy. In these examples, we demonstrate how the projection of unstable manifolds in configuration space reproduces essentially the entire observed bar-spiral pattern.Comment: 18 pages, 10 figures, to be published in MNRA

NGC 1300 Dynamics: III. Orbital analysis

We present the orbital analysis of four response models, that succeed in reproducing morphological features of NGC 1300. Two of them assume a planar (2D) geometry with $\Omega_p$=22 and 16 \ksk respectively. The two others assume a cylindrical (thick) disc and rotate with the same pattern speeds as the 2D models. These response models reproduce most successfully main morphological features of NGC 1300 among a large number of models, as became evident in a previous study. Our main result is the discovery of three new dynamical mechanisms that can support structures in a barred-spiral grand design system. These mechanisms are presented in characteristic cases, where these dynamical phenomena take place. They refer firstly to the support of a strong bar, of ansae type, almost solely by chaotic orbits, then to the support of spirals by chaotic orbits that for a certain number of pat tern revolutions follow an n:1 (n=7,8) morphology, and finally to the support of spiral arms by a combination of orbits trapped around L$_{4,5}$ and sticky chaotic orbits with the same Jacobi constant. We have encountered these dynamical phenomena in a large fraction of the cases we studied as we varied the parameters of our general models, without forcing in some way their appearance. This suggests that they could be responsible for the observed morphologies of many barred-spiral galaxies. Comparing our response models among themselves we find that the NGC 130 0 morphology is best described by a thick disc model for the bar region and a 2D disc model for the spirals, with both components rotating with the same pattern speed $\Omega_p$=16 \ksk !. In such a case, the whole structure is included inside the corotation of the system. The bar is supported mainly by regular orbits, while the spirals are supported by chaotic orbits.Comment: 18 pages, 32 figures, accepted for publication in MNRA

Invariant manifolds and the response of spiral arms in barred galaxies

The unstable invariant manifolds of the short-period family of periodic orbits around the unstable Lagrangian points $L_1$ and $L_2$ of a barred galaxy define loci in the configuration space which take the form of a trailing spiral pattern. In the present paper we investigate this association in the case of the self-consistent models of Kaufmann & Contopoulos (1996) which provide an approximation of real barred-spiral galaxies. We also examine the relation of `response' models of barred-spiral galaxies with the theory of the invariant manifolds. Our main results are the following: The invariant manifolds yield the correct form of the imposed spiral pattern provided that their calculation is done with the spiral potential term turned on. We provide a theoretical model explaining the form of the invariant manifolds that supports the spiral structure. The azimuthal displacement of the Lagrangian points with respect to the bar's major axis is a crucial parameter in this modeling. When this is taken into account, the manifolds necessarily develop in a spiral-like domain of the configuration space, delimited from below by the boundary of a banana-like non-permitted domain, and from above either by rotational KAM tori or by cantori forming a stickiness zone. We construct `spiral response' models on the basis of the theory of the invariant manifolds and examine the connection of the latter to the `response' models (Patsis 2006) used to fit real barred-spiral galaxies, explaining how are the manifolds related to a number of morphological features seen in such models.Comment: 16 Page

Asymptotic Orbits in Barred Spiral Galaxies

We study the formation of the spiral structure of barred spiral galaxies, using an $N$-body model. The evolution of this $N$-body model in the adiabatic approximation maintains a strong spiral pattern for more than 10 bar rotations. We find that this longevity of the spiral arms is mainly due to the phenomenon of stickiness of chaotic orbits close to the unstable asymptotic manifolds originated from the main unstable periodic orbits, both inside and outside corotation. The stickiness along the manifolds corresponding to different energy levels supports parts of the spiral structure. The loci of the disc velocity minima (where the particles spend most of their time, in the configuration space) reveal the density maxima and therefore the main morphological structures of the system. We study the relation of these loci with those of the apocentres and pericentres at different energy levels. The diffusion of the sticky chaotic orbits outwards is slow and depends on the initial conditions and the corresponding Jacobi constant.Comment: 17 pages, 24 figure

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.Comment: 17 pages, 12 figures; accepted in MNRA

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

The distinction between chaotic and regular behavior 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 techniques of the Smaller (and Generalized) ALingment Indices, SALI (and GALI), 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 two-degree-of-freedom (DOF) 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 three DOF 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 center of the galaxy, where the relative non-axisymmetric forcing is relatively weak, and at small distances from the equatorial plane, where trapping around the stable periodic orbits is important. We also find that the variation of the bar pattern speed, within a realistic range of values, does not affect much the phase space's 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 behavior. Finally, we find that the fraction of orbits that are chaotic correlates strongly with the bar strength.Comment: 16 pages, 10 figures, MNRAS, in pres

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. We compute the orbits confined by the invariant manifolds of the unstable periodic orbits located at the ends of the bar. We discuss whether the COBE/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 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 main over-densities 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.Comment: 18 pages, 11 figures, accepted in MNRA

Dynamics of Disks and Warps

This chapter reviews theoretical work on the stellar dynamics of galaxy disks. All the known collective global instabilities are identified, and their mechanisms described in terms of local wave mechanics. A detailed discussion of warps and other bending waves is also given. The structure of bars in galaxies, and their effect on galaxy evolution, is now reasonably well understood, but there is still no convincing explanation for their origin and frequency. Spiral patterns have long presented a special challenge, and ideas and recent developments are reviewed. Other topics include scattering of disk stars and the survival of thin disks.Comment: Chapter accepted to appear in Planets, Stars and Stellar Systems, vol 5, ed G. Gilmore. 32 pages, 17 figures. Includes minor corrections made in proofs. Uses emulateapj.st

"Το Κράτος ως διάδικος στην διεθνή διαιτησία"

Η ολοένα και αυξανόμενη διεθνοποίηση των εμπορικών συναλλαγών έχει φέρει στο προσκήνιο την διεθνή διαιτησία ως πρωταγωνιστικό μέσο επίλυσης εμπορικών διαφορών.To Κράτος και οι οντότητες οργανωμένες κατά το δημόσιο δίκαιο καθώς και οι επιχειρήσεις ιδιωτικού δικαίου που ανήκουν στο Κράτος διαδραματίζουν εξίσου πρωταγωνιστικό ρόλο στις διεθνείς αντιδικίες εμπορικής φύσεως. Σε όλες αυτές τις προαναφερθείσες περιπτώσεις το Κράτος συνήθως συνάπτει ρήτρες διαιτησίας για την επίλυση διαφορών που θα προκύψουν. Οι ιδιαιτερότητες όμως, που εμφανίζουν πιθανές αντιδικίες μεταξύ κρατικών και ιδιωτικών φορέων είναι πολυποίκιλες και συγκεντρώνουν εξαιρετικό θεωρητικό αλλά και πρακτικό ενδιαφέρον. Στόχος της παρούσας διπλωματικής εργασίας είναι να αναλυθεί με ευκρίνεια το πλαίσιο που ρυθμίζει τη επίλυση των διαφορών του κράτους με έναν ιδιώτη, από ένα διαιτητικό όργανο και τα πιθανά ζητήματα που μπορούν να ανακύψουν σε κάθε στάδιο της διαδικασίας μέσα από την θεωρία και την σημαντικότερη νομολογία διαιτητικών fora και κρατικών δικαστηρίων.In the current globalized international transactions scene, the State and its entities, either those organized under public law or the companies regarding which the State is the major shareholder, play an equally prominent part along with the private commercial parties in international arbitration. Particularly in disputes related to key business sectors such as construction agreements, energy infrastructure, transports and telecommunications, the State is called upon to lead the economic and business promotion of its imperium with acts that reveal the full capacity of its commercial, jure gestionis activities. For example, development banks will often finance such business projects or the State will transfer the assets and operations from the public to the private sector with the method of privatization. Therefore the interdependence of the public and private commercial sphere is clear and constitutes an integral part and a driving force of modern globalized economic transactions. Theoretical and practical issues may arise at any stage of such a dispute, such as the relationship between a State and its entities, a conjunction of investment and commercial arbitration or matters of State immunity. This juxtaposition between the two spheres generates a plethora of interesting legal issues and the goal of this dissertation is to analyze them through the examination of academic literature and important case law form arbitral tribunals and State Courts