Observed bars come in a wide variety of strengths, ranging from very strong bars, like NGC 4608 (Gadotti & de Souza 2003) or NGC 1365, to short bars, like our own Galaxy, or ovals, like in NGC 1566. Furthermore, a few discs have no bar at all (Grösbol, Pompei & Patsis 2002). How does that compare with the N-body results?
Haloes were initially thought to stabilise discs (e.g. Ostriker & Peebles 1973). More recent work (AM02, A02, A03) shows that this statement is not necessarily correct, since haloes can stimulate bar growth by taking angular momentum from the inner disc. Nevertheless, bars forming in disc galaxies with a substantial halo take longer to form than bars forming in galaxies with less halo, so that, in a sense, a halo can be considered as having a stabilising influence, although the bars that grow in a more massive and responsive halo may finally become stronger. A03 showed that this can happen for a wide range of halo to disc mass ratios, provided of course the halo is responsive. This, however, will not necessarily extend to very low relative disc masses (A03). Simulations to test this limit would be very CPU intensive, since they would necessitate a very large number of particles, and also because the bar growth would be exceedingly slow. Moreover, if the bar grows in time scales much longer than the Hubble time, the problem is of little astronomical interest and the corresponding galaxies can be considered bar-stable.
Haloes can have a further stabilising influence either if their mass distribution is such that not much material is at resonance with the bar, or, more likely, if the resonant halo material is very hot and thus can not absorb much angular momentum. Unfortunately, not much is known about the velocity distribution within the halo component in real galaxies.
Similarly, a bar forming in a hot disc will take longer to grow than in a cold one (Athanassoula & Sellwood 1986, A03). In very hot discs, when the bar eventually forms, it has the form of an oval (see Fig. 4).
Thus, with the help of N-body simulations, we can account for the large variety of bar strengths observed in real galaxies. As discussed also in the previous section, this will be determined by the total amount of angular momentum exchanged and also by the specific amount by which each of the partners (inner disc, outer disc, halo, bulge) enters in the exchange. In this picture, galaxies with no bars should have a very low relative disc mass and/or a very hot disc and/or a rather unresponsive halo, so that the bar takes very long to grow and so that the halo can not help its secular evolution.
This picture is not complete. There is still the question of black holes, and other `extreme' central concentrations, which should have a stabilising influence. The interplay between this and the responsiveness of the disc and halo will be discussed elsewhere.