1. M51, the Whirlpool Galaxy

2. Lord Rosse discovered the spiral structure in M51 in 1850 The explanation of this beautiful form has been one of the outstanding problems in astronomy. Jeans tried to identify the arms with pieces of material that would be shed equatorially as a uniformly rotating centrally-condensed mass slowly shrank. Lindblad attempted to give an explanation of arms in term of orbits and then in terms of self-gravitating perturbations of a stellar system

3. If the arm structure rotates differentially, then the pitch must diminish, and in a tipical time scale of 10^8 years the arms will become tightly wound. but the proportion of spiral with tightly wound arms is small and galaxies are typically 10^10 years olds. We deduce that: The spiral structure rotates nearly uniformly although the material rotates differentially. The most promising theory to explain this property is the density wave theory

4. The density wave analysis is a complicate procedure There is an important limit in which this Analysis is much simpler: the tightly wound or WKB approximation (tightly wound: the radial wavelenght is much less than the radius) (WKB: Wentzel-Kramers-Brillouin) In this framework is possible to deduce the dispersion relations for stellar disks: they establish the relation between wavenumber and frequency for a traveling wave as it propagates across the disk.

5. An important application of it is to determine whether a given disk is locally stable to axisymmetric perturbations (m=0) This study lead to the famous Toomre stability criterium: Where σ is the radial velocity dispersion and Σ The surface density (for our Galaxy: Q=1.7) and to to the critical lenght: (the longest wavelenght taht could be unstable: it provides a useful yardstick for Jeans-type instabilities of all kinds)

6. Unfortunately WKB analysis does not give a complete picture of disk dynamics, because it does not apply to loosely wound structures. There are no analytic method that can determine the stability of a general galactic disk to arbitrary perturbations.

7. NUMERICAL WORK ON DISK STABILITY One of the earliest studies was carried on by Hohl (1971) The evolution of a rotating disk of stars, with an initial velocity dispersion given by Toomre’s locally criterium shows that the system is unstable against very large-scale modes.

8. Uniformly rotating disk of 100000 stars Moving under a purely radial gravitational field

9. Non axisymmetric evolution of the disk

10. These experiments show that: Toomre criterium is sufficient against global instability in axysymmetric modes (m=0) But is not sufficient against non axysymmetric modes (m=1,2) Hohl noticed that Q >2.5 stabilizes the disk against bar instability (too high value for real galaxies)

11. Resonances LIR and OLR

12. Two important disk parameters: Q and J Toomre stability parameter: And the parameter J where is the selfgravity parameter

13. Swing amplification (Julian and Toomre, Goldreich and Lynden Bell, 1965) Toomre argues that the bar instability was driven by a positive feedback to the swing amplification mechanism Remarkably, the most features of global instability can be understood by augmenting the WKB dispersion relations with the swing amplification.

14. A mode is a standing wave, by definition, But Toomre showed that the bar instability is more easily understood in terms of a propagating wave packet : A leading spiral disturbance originating near the disk center propagates outward toward corotation, where the swing amplification causes it to shear into a trailing wave of much greater amplitude

15. The transfer of angular momentum toward the outer regions is accompanied by the amplification of the incoming wave: OVERREFLECTION. Overreflection operates inside the corotation radius as a resonant cavity. Overreflection can occur in two different forms: In the regime of low J, operating on trailing wave and in the regime of Higher J, in which overreflection operates Converting a leading wave into a pair of trailing waves. This later form of overreflection corresponds to the so called swing amplification

16. Two different regimes for galaxies: Light disks (low J), in which all the relevant cycle can be all trailing, and gives rise to self excited normal (unbarred) galaxies (Spirals are generally trailing) Heavy disks (high J) : the relevant cycle is based on a leading and a trailing wave and generates barred spiral modes (two blobs structures inside the CR, due to the superposition of the leading with the trailing wave

17. The bar mode is simply the standing wave resulting from an endless wave train propagating trough this cycle Wave action is conserved by an outwardingly propagating wave beyond corotation. Toomre’s mechanism suggests three different ways in which the bar mode can be stabilized

18. Swing amplification, according to Toomre is a strong cooperative effect that inhibits interarm travel It results from a three fold conspiracy: Shear, shaking and self gravity Shear flow and epicyclic vibrations share the same sense in any normal disk having angular speed Ω decreasing outward. Both types of motion occur in a direction opposite to Ω itself It is precisely this agreement that makes it possible for a wide-open pattern of epiciclic vibrations to resonate with the shear flow. The only extra-need is for stellar communication and this bring us to self gravity

19. These three ingredients suggest three different ways in which the bar mode can be stabilized The first is to embed the whole disk in a massive unresponsive halo (decrease self gravity) This solution is effective only if sufficient mass lies interior to corotation radius

20. The second way is to raise the level of random motion in the disk (heat the disk, inhibit collective behaviour) The third is to breack the feedback loop inserting an inner Lindblad resonance between corotation and disk center.

21. Some combination of these three mechanism (e.g. a massive, dense bulge,a responsive dark halo) is presumably responsive for the stability of most galaxies. In spite of the encouraging results of the modal description in the interpretation of spiral structures in galaxies, we are at only the beginning in our understanding of galaxy evolution. This is largely due to our general lack of tools to describe the nonlinear evolution of a dynamical sistem, even when at the linear level its dynamics is dominated by a few modes

22. Dynamical classification of spiral morfologies An extensive survey of realistic models of galaxy disks has shown that the morphology types of the global spiral modes that can be generated in a disk match the general morphological categories that are found along the Hubble sequence.

23. Depending on the parameter regime of a given galaxy disk, the dominant mode may be of the A Or B type. Different excitation mechanisms operate for the two Classes of modes. Moreover a mode rely on a combined support of gas And stars. SB0 SB S moderateS violent Superposition of a Bar mode onto its axysim etric density distribution

24. Responses of a V=const. disk of stars to transient gravity forces from the imposed masses The top tow shows the excess densities

25. These transient imposed forces (1% of the galacto centric force on particle A and 0.25% on particle B) soon yield an evolving Spiral pattern of impressive severity among the disk stars

26. Bertin and Lin (1996):

27. Numerical work on bar models Orbit families in frozen bar-like potentials (Lindblad resonances, Lagrangian points..) Orbital structure of a bar formed in an N- body simulation (2d and 3d) Origin of bars (global instability followed by Nbody simulations) Controlling bar instability

28. Numerical work on the dissipative component Gas behaviour : inside a frozen potential or in a Nbody-SPH simulation: Gravitational coupling between stellar bar and interstellar medium. Star formation in SPH bar simulations: Coupling between stars and ISM via STF

29. Qui dovrebbe stare l’immagine che mi deve scannerizzare Giuseppe

30. Bar forming modes The type of behaviour illustrated is typical of almost every two dimensional simulation for which the underlying model is unstable to global bisymmetric distorsions As the instability runs, the transient features in the surrounding disk fade and the only non axisymmetric feature to survive is the steadly tumbling bar

31. The bar ends just inside corotation The axis ratio of the bar depends upon the degree o random motions in the original disks: the cooler the initial disk the narrower the resulting bar. When the initial bar is short, it continues to interact with the outer disk through spiral activity: the trailing spirals remove angular momentum from particles at their inner end. This enable more stars to be trapped into the bar, increasing its length and lowering its pattern speed. These changes in both bar length and pattern speed conspire to keep co-rotation just beyon the end of the bar.

34. Controlling the bar instability using Dark Matter haloes Spherical halo triaxial halo Non rotating halo spinning halo Analytical passive halo live halo Static halo dynamical halo HOW THE BAR INSTABILITY IS REACTING TO SUCH MORE REALISTIC HALOES MODELS?

36. What happens if we include gas in the disk? Dissipation triggers significant gas fueling of the central regiones once the bar has formed This leads to a high central mass concentration wihich is in the end responsible for the destruction of the bar. (as soon as the mass accreted by the central regions represents a non negligible part of the galaxy mass (1-2%) a strong ILR appears)

37. What happens if we include star formation in the disk? Stronger bars tends to form inside more massive non relaxed haloes If star formation is included, it seems to favour bar formation, lenghtening the bar lifetime (SF works against strong mass concentration in the center of the disk) Since stronger bursts of star formation are triggered in more massive and concentrated haloes, stronber bars develop in more concentrated massive haloes

38. Thus the star formation, which depend on local conditions Is however governed by the local dynamics of the galaxy. And Vice-versa, local Star Formation can modify the global Dynamics, resulting in a highly non linear feed back mechanism. Moreover SF efficiency, IMF, cooling function can tehmselves Be dependent on the metallicity. Since clearly this metallicity is Related to the previous SF hustory, this add another feed back Mechanism. All this seems to suggest that global self regulated non stationary Processes could take place in disk galaxies

39. Controlling bar instability through Cosmological DM haloes We adopt fully cosmological DM haloes, inside a real cosmological scenario, to imbed our stellar disk. We therefore can investigate the role of the infall, the influence of the matter outside the system…. The cosmological expansion and so on…. The aim is to get the disk evolution as a redshift function