4.7.1. Results from N-body simulations
Another prediction of the analytic work is that the bar pattern speed will decrease with time, as the bar strength increases (Section 4.5). This has been confirmed by a large number of N-body simulations (e.g., Little & Carlberg 1991a, b; Hernquist & Weinberg 1992; Debattista & Sellwood 2000; A03; O'Neil & Dubinski 2003; Martínez-Valpuesta et al. 2006; Villa-Vargas et al. 2009). The amount of this decrease was found to vary considerably from one simulation to another, depending on the mass as well as on the velocity distribution in the disk and the spheroidal (halo plus classical bulge) components, consistent with the fact that these mass and velocity distributions will condition the angular momentum exchange and therefore the bar slowdown.
There is a notable exception to the above very consistent picture. Valenzuela & Klypin (2003) found in their simulations a counter-example to the above, where the pattern speed of a strong bar hardly decreases over a considerable period of time. The code they use, ART, includes adaptive mesh refinement, and thus reaches high resolution in regions where the particle density is high. According to these authors, the difference between their results and those of other simulations are due to the high resolution (20-40 pc) and the large number of particles (up to 107) they use. Sellwood & Debattista (2006) examined cases where the bar pattern speed fluctuates upward. After such a fluctuation, the density of resonant halo particles will have a local inflection created by the earlier exchanges, so that bar slowdown can be delayed for some period of time. They show that this is more likely to occur in simulations using an adaptive refinement and propose that this explains the evolution of the pattern speed in the simulation of Valenzuela & Klypin (2003). Klypin et al. (2009) did not agree and replied that Sellwood & Debattista did not have the same adaptive refinement implementation as ART. Sellwood (2010) stressed that such episodes of non-decreasing pattern speed are disturbed by perturbations, as e.g., a halo substructure, and thus are necessarily short lived. He thus concludes that simulations where the pattern speed does not decrease have simply not been run long enough. At the other extreme, Villa-Vargas et al. (2009) find a similar stalling of the pattern speed for prolonged time periods when the simulation is run so long that the corotation radius gets beyond the edge of the disc.
Dubinski et al. (2009) published a series of simulations, all with the same model but with increasing resolution. They use between 1.8 × 103 and 18 × 106 particles in the disk and between 104 and 108 particles in the halo. They also present results from a multi-mass model with an effective resolution of ~ 1010 particles. They have variable, density dependent softening, with a minimum of the order of 10 pc. Their Fig. 18 shows clearly that the decrease of the pattern speed with time does not depend on the resolution and that it is present for all of their simulations, even the ones with the highest resolution, much higher than the one used by Valenzuela & Klypin. They conclude that `the bar displays a convergent behavior for halo particle numbers between 106 and 107 particles, when comparing bar growth, pattern speed evolution, the DM [dark matter] halo density profile and a nonlinear analysis of the orbital resonances'. This makes it clear that, at least for their model, the pattern speed decreases with time for all reasonable values of particle numbers.
4.7.2. A schematic view
Figure 4.12 shows very schematically an interesting effect of the bar slowdown. The solid line shows the radial profile of (r) for a very simple model with a constant circular velocity, but the following hold for any realistic circular velocity curve. Let us assume that at time t = t1 the pattern speed is given by the dashed horizontal line and drops by t = t2 to a lower value given by the dotted horizontal line, as shown by the vertical arrow. This induces a change in the location of the resonances. For example the CR, which at t1 is located at 5 kpc, as given by vertical dashed line, will move by t2 considerably outwards to a distance beyond 6 kpc, as given by the dotted vertical line and shown by the horizontal arrow. This increases also the region in which the bar is allowed to extend, since, as shown by orbital theory (Contopoulos 1980 and Section 4.3), the bar size is limited by the CR. This schematic plot also makes it easy to understand how a `fast' bar can slow down considerably while remaining `fast'. As we saw in Section 4.3, a bar is defined to be `fast' if the ratio of the corotation radius to the bar length is less than 1.4. Thus, a bar that slows down and whose corotation radius increases can still have < 1.4, provided the bar length increases accordingly. This occurs in a number of simulations, see, e.g., Dubinski et al. (2009).
Figure 4.12. Schematic plot of the effects of the decrease of the bar pattern speed with time for a very simple model with a constant circular velocity. The solid line gives (r) and the horizontal dashed and dotted lines give the pattern speed p at two instants of time. The vertical dashed and dotted lines show the CR radius at these same two instants of time. The vertical arrow indicates the decrease of the bar pattern speed and the horizontal one the increase of the corotation radius.
4.7.3. What sets the pattern speed value?
What sets the value of the pattern speed in a simulation (and thus also presumably in real galaxies)? The value of the pattern speed is set by the value of the corotation radius, which is in fact the borderline between emitters and absorbers. Thus, if the galaxy wants to maximise the amount of angular momentum it pushes outwards (i.e., the amount of angular momentum that it redistributes), it has to set this boundary, and therefore its bar pattern speed, appropriately. If the spheroid is massive, i.e., if it has sufficient mass in the resonant regions, then the bar can lower its pattern speed in order to have more emitters, since the absorbers are anyway strong. On the other hand if the spheroid is not sufficiently massive, then the bar should not lower its pattern speed overly, because it needs the absorption it can get from the outer disk. Thus, indirectly, it could be the capacity of the spheroid resonances to absorb angular momentum that sets the value of the bar pattern speed. This would mean that properties of the dark matter halo and of the classical bulge, such as their mass relative to that of the disk and their velocity dispersion at the resonant regions, will have a crucial role in setting the bar pattern speed.