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The presence of a bar induces not only the redistribution of angular momentum within the host galaxy (Section 4.5 and 4.6), but also the redistribution of the material within it. The torques it exerts are such that material within the CR is pushed inwards, while material outside the CR is pushed outwards. As a result, there is a considerable redistribution of the disk mass.

4.9.1. Redistribution of the disk mass: formation of the disky bulge

It is well known that gas will concentrate to the inner parts of the disk under the influence of the gravitational torque of a bar, thus forming an inner disk whose extent is of the order of a kpc (Athanassoula 1992b; Wada & Habe 1992, 1995; Friedli & Benz 1993; Heller & Shlosman 1994; Sakamoto et al. 1999; Sheth et al. 2003; Regan & Teuben 2004). When this gaseous disk becomes sufficiently massive it will form stars, which should be observable as a young population in the central part of disks. Kormendy & Kennicutt (2004) estimate that the star formation rate density in this region is 0.1-1 Modot yr-1 kpc-2, i.e., one to three orders of magnitude higher than the star formation rate averaged over the whole disk. Such disks can harbour a number of substructures, such as spirals, rings, bright star-forming knots, dust lanes and even (inner) bars, as discussed, e.g., in Kormendy (1993), Carollo et al. (1998) and Kormendy & Kennicutt (2004). Furthermore, a considerable amount of old stars is pushed inwards so that this inner disk will also contain a considerable fraction of old stars (Grosbøl et al. 2004). Such disks are thus formed in N-body simulations even when the models have no gas, as seen, e.g., in AM02, or Athanassoula (2005b).

Such inner disks are evident in projected surface luminosity radial profiles, as extra light in the central part of the disk, above the exponential profile fitting the remaining (non-central) part. Since this is one of the definitions for bulges, such inner disks have been linked to bulges. When fitting them with an r1/n law - commonly known as Sérsic's law (Sérsic 1968) - the values found for n are of the order of or less than 2 (Kormendy & Kennicutt 2004 and references therein). They are thus often called disky bulges, or disk-like bulges (Athanassoula 2005b), or pseudobulges (Kormendy 1993, Kormendy & Kennicutt 2004).

4.9.2. Redistribution of the disk mass: the disk scale-length and extent

Due to the bar torques and the resulting mass redistribution, the parts of the disk beyond corotation become more extended and the disk scale length increases considerably (e.g., Hohl 1971; Athanassoula & Misiriotis 2002; O'Neil & Dubinski 2003; Valenzuela & Klypin 2003; Debattista et al. 2006; Minchev et al. 2011). Debattista et al. 2006 showed that the value of Toomre Q parameter (Toomre 1964) of the disk can strongly influence how much this increase will be.

Important extensions of the disk can also be brought about by flux-tube manifold spiral arms (Romero-Gómez et al. 2006, 2007; Athanassoula et al. 2009a, 2009b, 2010), as shown by Athanassoula (2012) who reported a strong extension of the disk size, by as much as 50% after two or three episodes of spiral arm formation within a couple of Gyrs.

4.9.3. Redistribution of the disk mass: maximum versus sub-maximum disks

Sackett (1997) and Bosma (2000) discuss a simple, straightforward criterion allowing us to distinguish maximum from sub-maximum disks. Consider the ratio S = Vd,max / Vtot, where Vd,max is the circular velocity due to the disk component and Vtot is the total circular velocity, both calculated at a radius equal to 2.2 disk scalelengths. According to Sackett (1997), this ratio has to be at least 0.75 for the disk to be considered maximum. Of course in the case of strongly barred galaxies the velocity field is non-axisymmetric and one should consider azimuthally averaged rotation curves, or `circular velocity' curves. Furthermore, in the case of strongly barred galaxies it is not easy to define a disk scalelength, so it is better to calculate S at the radius at which the disk rotation curve is maximum, which is a well-defined radius and is roughly equal to 2.2 disk scalelengths in the case of an axisymmetric exponential disk. After these small adjustments, we can apply this criterion to our simulations.

In Section we saw that the disks in MH models are sub-maximum in the beginning of the simulation and in Section 4.9.1 that the bar can redistribute the disk material and in particular push material inwards and create a disky bulge. Is this redistribution sufficient to change sub-maximum disks? The answer is that this can indeed be true in some cases, as was shown in Athanassoula (2002b) and is illustrated in Fig. 4.15. This shows the Sackett parameter S and the bar strength as a function of time for one such simulation. Note that the disk is initially sub-maximum and that it stays so during the bar growth phase. Then the value of S increases very abruptly to a value larger than 0.75, so that the disk becomes maximum. After this abrupt increase the S-parameter hardly changes, although the bar strength increases considerably.

Figure 15

Figure 4.15. Evolution of the Sackett parameter, S, as a function of time for an MH-type simulation (solid line). The two horizontal dotted lines give the limits within which S must lie for the disk to be considered as maximum. The dashed line gives a measure of the bar strength as a function of time, for the same simulation.

4.9.4. Secular evolution of the halo component

The halo also undergoes some secular evolution, albeit not as strong as that of the disk. The most notable feature is that an initially axisymmetric halo becomes elongated in its innermost parts and forms what is usually called the `halo bar', or the `dark matter bar', although the word `bar' in this context is rather exaggerated, and `oval' would have been more appropriate. This structure was already observed in a number of simulations (e.g., Debattista & Sellwood 2000; O'Neil & Dubinski 2003; Holley-Bockelmann et al. 2005; Berentzen & Shlosman 2006) and its properties have been studied in detail by Hernquist & Weinberg (1992), Athanassoula (2005a, 2007) and Colin et al. (2006). It is considerably shorter and its ellipticity is much smaller than the disk bar, while rotating with roughly the same angular velocity. It is due to the particles in the halo ILR (Athanassoula 2003, 2007).

A less clear-cut and certainly much more debated issue concerns the question whether secular evolution due to a strong bar could erase the cusp predicted by cosmological simulations and turn them into cores, which would lead to an agreement with observations. A few authors (e.g., Hernquist & Weinberg 1992; Weinberg & Katz 2002; Holley-Bockelmann et al. 2005; Weinberg & Katz 2007a, 2007b) argued that indeed such a flattening was possible, while a larger consensus was reached for the opposite conclusion (e.g., Sellwood 2003; McMillan & Dehnen 2005; Colin et al. 2006; Sellwood 2008). We refer the reader to these papers for more information.

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