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Angular momentum can be redistributed within a barred galaxy. It is emitted from the (near-)resonant stars in the bar region and absorbed by the (near-)resonant material in the spheroid and the outer disk. By following the orbits in a simulation and measuring their frequencies, it is possible to determine whether they are (near-)resonant or not, and, if so, at which resonance. For strong bar cases, the most populated disk resonance is the inner Lindblad resonance. Simulations confirm the theoretical prediction that this emits angular momentum, and that the corotation and outer Lindblad resonances absorb it. In the spheroid the three most populated resonances are the corotation, the outer Lindblad and the inner Lindblad resonance, and, in many cases, it is corotation that is the most populated. Again simulations confirm the theoretical prediction that angular momentum is absorbed at the spheroid resonances.

In order for bars to evolve uninhibited in a simulation, it is necessary that the angular momentum exchange is not artificially restrained, as would be the case if the halo in the simulation was rigid, e.g., represented by an axisymmetric force incapable of emitting or absorbing angular momentum. It is thus necessary to work with live haloes in simulations, and, more generally, to avoid the use of any rigid component.

Note also that the effect of the spheroid on bar growth is different in the early and in the late phases of the evolution. During the initial phases of the evolution, the spheroid, due to the strong axisymmetric force it exerts, delays and slows down the bar growth. Thus, bars will take longer to form in galaxies with a large ratio of spheroid-to-disk mass. On the other hand, at later stages, after the secular evolution has started, the spheroid can increase the bar strength by absorbing a large fraction of the angular momentum emitted from the bar region. Thus, stronger bars will be found in galaxies with a larger spheroid-to-disk mass ratio.

Contrary to spheroid mass, the velocity dispersion in the disk has always the same effect on the bar growth. During the initial phases it slows down the bar growth. Thus, bars will take longer to form in galaxies with hot disks. During the secular evolution phase, a higher velocity dispersion in the disk component will make its resonances less active, since it decreases the amount of angular momentum that a resonance can emit or absorb. A similar comment can be made about the velocity dispersion of the spheroid (near-)resonant material. Thus, increasing the velocity dispersion in the disk and/or the spheroid will lead to less angular momentum redistribution and therefore weaker bars.

As the bar loses angular momentum, its pattern speed decreases, so that the resonant radii will move outwards with time. Since the corotation radius provides an absolute limit to the bar length, this increase implies that the bar can become longer. Indeed, this occurs in simulations. It is thus possible for the pattern speed to decrease while the bar stays `fast', provided the bar becomes longer in such a way that the ratio R of corotation radius to bar length stays within the bracket 1.2 ± 0.2.

As the bar loses angular momentum it also becomes stronger, so that there is a correlation between the bar strength and the amount of angular momentum absorbed by the spheroid. In general, as bars become stronger they become also longer and their shape gets more rectangular-like. They redistribute mass within the disk and create the disky bulge (more often referred to as pseudo-bulge) in the central region. They also increase the disk scalelength. All these changes brought about by the evolution can also strongly influence the form of the rotation curve and change an initially sub-maximum disk to a maximum one.

The strongest bars will be found in cases where the maximum amount of angular momentum has been redistributed within the galaxy, and not when the spheroid mass is maximum. A further parameter which is crucial in trying to maximise the angular momentum redistribution is the bar pattern speed. Indeed, this is set by the location of the corotation radius and therefore by the balance between emitters and absorbers in the disk.

When bars form they are vertically thin, but soon their inner parts puff up and form what is commonly known as the boxy/peanut bulge. This is well understood with the help of orbital structure theory. It gives a complex and interesting shape to the bar - i.e., vertically extended only over a radial extent from the centre to a maximum radius of the order of (0.7 ± 0.3)aB, where aB is the bar length, and then very thin outside that range. This shape explains a number of observations and also argues that the COBE / DIRBE bar and the Long bar in our Galaxy are, respectively, the thin and the thick part of a single bar.

From the above it is thus possible to conclude that there is a continuous redistribution of angular momentum in disks with strong bars and that this drives a secular evolution. It is secular because the timescales involved are long, contrary to, e.g., a merging, which occurs in a very short time interval.


I thank the school organisers, J. Falcón-Barroso and J. H. Knapen for inviting me to give a series of lectures at the XXIIIrd Canary Islands Winter School on Secular Evolution of Galaxies, as well as for their patient nudging when the time came to write up the proceedings. I acknowledge financial support from the CNES and from the People Programme (Marie Curie Actions) of the European Union's FP7/2007-2013/ to the DAGAL network under REA grant agreement number PITN-GA-2011-289313.

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