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Although the formation of rings by gas accumulation at bar resonances has gained strong observational support, and is now widely accepted, there remain peculiar cases for which the standard model encounters some difficulties in its application. Let us summarize here the predictions of resonance theory: in a barred spiral disk, gas is subject to strong gravity torques from the total potential, and accumulates at one of the main resonances, where its trajectory follows the main periodic orbit family there. These are symmetrically oriented with respect to the potential, and subject to no net torque. Outer rings correspond to OLR, and the shape of the ring can be perpendicular to the bar, with dimples (R1), if slightly inside OLR, or parallel to the bar slightly outside (R2). The inner ring encircling the bar corresponds to the ultraharmonic resonance, the 4 / 1, slightly inside corotation, while nuclear rings correspond to the ILR region. The relative positions of the rings are therefore constrained by the theory, once the rotation curve is known.

The theory also predicts that at equilibrium, rings are aligned with the bar; the rings are essentially composed of gas, and after star formation, of young stars. If the bar is long-lived, with the same pattern speed, the ring can survive also, although the old stars will diffuse out of the ring, after acquiring velocity dispersion through scattering by perturbations (giant molecular clouds, or gravitational instabilities in the rings themselves). When all the gas from the spiral arms has been depleted to the benefit of the rings, and when enhanced star formation has consumed all the gas in the rings, the latter should age and turn to ``dead'' (or quiescent) rings, with diffuse boundaries. Many diffuse and detached outer rings could be in such a phase, or individual rings within a galaxy, such as the inner and outer rings of NGC 7702 and 1326, the nuclear ring of NGC 5850, or the R1 component of IC 1438.

Let us remark first that the presence of barred galaxies without any ring is not actually a problem for the theory. Ring formation requires gas, which might not be present (for instance in lenticular galaxies). Outer rings require at least a few Gyrs to form, and the bar might be quite recent. Nuclear rings are the manifestation of the inner Lindblad resonance, which is present only in highly concentrated galaxies, and inner rings at UHR are favored in rather squarish potentials near corotation, possessing a strong m = 4 component.

14.1. Time-scales and the Co-existence of Rings

The first problem encountered is the frequent observation of simultaneous outer, inner, and nuclear rings in the same galaxy, in spite of their very different formation time-scales and lifetimes. An excellent example of this conundrum is NGC 1326 (see Figure 45), which has a beautifully developed stellar outer R1 ring and an intense nuclear ring of young stars and Hgamma emission. Other examples include NGC 3081, NGC 5728, NGC 6782, IC 1438, and IC 4214. The outer rings require, in a reasonably strong bar, at least 3 x 109 yrs to form, and can be maintained over a Hubble time if the galaxy has no close companion, while the nuclear rings form in typically 108 yrs, and their life-time can be as short as 108 yrs because of consumption by star formation and flow to the center through self-gravitating instabilities, dynamical friction, or dissipation. This problem has been encountered in simulations (e.g. Gerin et al. 1991; Byrd et al. 1994), but underestimated, since star formation was not taken into account.

A possible solution to this problem is to take into account some gas accretion: either the gas locked up in stars, and then released by stellar winds, explosions, etc., or gas accreted from outside, from the atomic hydrogen clouds outside the optical disk, or from companions. The latter, however, will perturb the dynamics of the disk, modify the bar pattern speed (Gerin et al. 1990, Sundin et al. 1993), or even destroy the bar (Pfenniger 1991). Only mild distortions are welcome, to replenish the gas content in the disk without destroying the outer ring. In any case, there should not be a true outer ring co-existing with a strong bar, or with a self-gravitating gaseous nuclear ring. Outer pseudorings do not need such a long time to form, and a non-self-gravitating gaseous nuclear ring in a weak bar is much more long-lived. Most cases observed with several rings indeed satisfy these conditions (e.g., NGC 4736 with a weak bar (Gerin et al. 1991), or IC 1438 with a low gas content (Fisher & Tully 1981). Also, some dead inner rings have been identified from their colors (e.g., that in NGC 7702, Buta 1991). Some galaxies with very detached outer rings possess secondary lenses and secondary bars, rather than blue nuclear rings (Buta & Crocker 1993).

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