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Astron. Astrophys. 362, 465-586 (2000)

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4. Discussion

In this section we discuss, based on our simulations, the different aspects related to the formation and the evolution of rings, both in barred and non-barred galaxies. Finally, we discuss the limitations of our modelling approach.

4.1. The relationship of bars, rings and resonances

We determined the major and minor axes of the rings in selected timesteps (3-4 per simulation) in our 15 models. From these we have calculated the ratios of the geometrical mean radii of outer pseudorings of subclasses [FORMULA] and [FORMULA] to the OLR radii of the bar. For [FORMULA] pseudorings the average ratio is 0.892 and for [FORMULA] pseudorings it is 1.143. This clearly shows that the outer rings tend to be located near the outer Lindblad resonance of the bar, the first subclass inside and the second subclass outside the OLR radius. The simulations applying a rigid bar potential in modelling the behaviour of the gas component (Schwarz 1981; Byrd et al. 1994) suggest that the two different subclasses of outer rings can be explained by two major orbit families in the vicinity of the OLR. Which one is occupied depends either on evolutionary stage or on gas distribution. However, N-body simulations provide other possibilities. For example, there can also be a spiral mode in the outer disk with a lower pattern speed than the bar. In such cases the morphology of the outer ring is not steady: it keeps changing between different outer ring subclasses and occasionally the morphology does not resemble either of them (see Fig. 10).

[FIGURE] Fig. 10. The outer ring evolution of the model C1.75. Only the gas component is shown. Time is shown in gigayears. The outer ring morphology is evolving constantly, e.g. [FORMULA]: T = 1.56, 1.88, 3.88; [FORMULA]: T = 1.44, 2.69, 4.19.

This cyclic behaviour suggests that there may not always be a profound difference between galaxies of different outer ring subclasses. This would mean that the evolutionary change from subclass [FORMULA] to [FORMULA], as suggested by Byrd et al. (1994), is not the reason for the existence of two ring subclasses. The cyclic change also disagrees with Buta & Crocker (1991), who found a correlation between the outer and the inner structures. According to their results, galaxies with outer ring subclasses [FORMULA] and [FORMULA] tend to have inner and nuclear rings and dust lanes, whereas galaxies of subclass [FORMULA] usually lack them. Buta and Crocker's original analysis was based on only 22 galaxies but they later enlarged the sample (Crocker & Buta 1993). One should also note that the observed correlation was not exclusive: there are galaxies of subclass [FORMULA] with nuclear rings and those of subclass [FORMULA] without them.

Many outer ring structures in our simulations could not be comfortably classified to subclasses. Pseudorings could have mixed characteristics: either the orientation of the major axis or the winding of the spiral arms was between the values typical for subclasses [FORMULA] and [FORMULA] or the arms where asymmetric, one arm behaving like [FORMULA] pseudoring and the other like [FORMULA]. Such problems exist also with real galaxies: for example, the outer pseudoring of NGC 1300 is elongated about 70o to the bar (Elmegreen et al. 1996). In addition to previous difficulties, it was usually impossible to give a subclass for complete detached outer rings, which formed from pseudorings after the evolution of several gigayears.

Another element of outer ring classifications is provided by different morphologies of the stellar and gas components. Although ring formation usually takes place only in the gas component, there are a few cases in our simulations where the stellar spiral arms also form an outer pseudoring. Stellar pseudorings are much broader than gaseous rings, and usually are not steady because they form and dissolve repeatedly. In a few cases, like model C1.25 in Fig. 11, there is a [FORMULA] morphology, where the gas component forms the [FORMULA] part, whereas the stellar component enhances or solely forms the [FORMULA] part. Interestingly, there are some observed cases where the situation can be rather similar: NGC 6782, which has a partial [FORMULA] pseudoring in [FORMULA], whereas the continuum image has [FORMULA] morphology (Crocker et al. 1996) and IC 1438, which has strong [FORMULA] morphology in a B-band image, but a [FORMULA] morphology in I-band (Byrd et al. 1994; Buta 1995). Byrd et al. (1994) suggested that in IC 1438 the [FORMULA] component formed first in the gas component and left a remnant in the stellar component after evolving to [FORMULA] stage. However, based on our new simulations, we think that at least some stellar [FORMULA] rings are not remnants, but phenomena of self-gravitating stellar component.

[FIGURE] Fig. 11. The double outer ring of model C1.25 as seen in stellar, gas and combined distribution. Reconstructed from Fourier decompositions using [FORMULA] 0,1,2,3 and 4 components.

The inner rings have traditionally been thought to be related to the inner 4/1-resonance of the bar. Indeed, in low amplitude simulations with a rigidly rotating bar potential, they appear as four extremely tightly wound spiral arms precisely in this resonance (see e.g. Fig. 7 in Salo et al. 1999), which firmly supports the hypothesis. When the amplitude is increased, their shape evolves to a complete ring. At the same time, the ring major axis moves towards the corotation resonance and the ring becomes more elongated. The N-body simulations agree well with this picture. The average ratio of the geometrical mean radius of inner rings to inner 4/1-resonance radius was 1.0, whereas the corresponding ratio to the corotation radius was 0.72. The inner ring major axis is often between the previously mentioned resonances, usually closer to the inner 4/1. Few inner rings are completely inside the inner 4/1-resonance. Sometimes, e.g. models A2.25 and C2.25, the density plots give an impression that the bar ends before reaching the inner ring. There are also real galaxies showing the same phenomenon: the bar fills the inner ring major axis incompletely. An example of this is NGC 7098, where this can be seen in the B-band CCD-image (see Fig. 35 in Buta 1995) and also in the second generation red image of the STScI Digitized Sky Survey.

The nuclear rings are usually thought to be related to the inner Lindblad resonance of the main bar or alternatively to a nuclear bar component. Both situations are present in our simulations. When related to the main bar component, the nuclear ring is usually not in either of the ILR radii, but between them. Here one should note that the resonance radius calculated from the epicyclic approximation is not reliable in the case of strong perturbations: the stronger the bar, the smaller are the nuclear rings (see e.g. Fig. 7 in Salo et al. 1999). When the bar is very strong, the [FORMULA] orbits (see Contopoulos & Grosbol 1989) supporting the nuclear rings can become unstable.

Although there are simulations where the nuclear ring clearly follows a nuclear mode with a higher pattern speed than that of the main bar (e.g. model A1.75), the presence of a nuclear mode does not necessarily mean that the nuclear ring follows it. There are also cases where a nuclear bar precedes the main bar by several gigayears (e.g. model C1.75), but a clear nuclear ring does not form until a main bar is pesent. Apparently, the nuclear bar alone is usually insufficient to transport gas to the central area to form a ring. In a few of our models there are two nuclear rings: the outer one following the main bar and the inner one following the nuclear mode. It is not clear if this is the case with the few observed galaxies with double nuclear rings (e.g. NGC 1317; detected in [FORMULA] by Crocker et al. 1996 or M83; detected in near-IR by Elmegreen et al. 1998), the other possibility is that the rings are related to the two inner Lindblad resonances of the main bar. Another interesting phenomenon is the capture of nuclear rings, which happens in some of our simulations: when a mode originally forming the ring fades, another mode can adopt the ring, and possibly change its radius.

The average minor to major axis ratio is 0.80 for [FORMULA] pseudorings and 0.91 for [FORMULA] pseudorings (see Table 1). This means that outer pseudorings of both subclasses are a bit more circular than in the observations by Buta (1995), where the corresponding ratios were 0.74 and 0.87. The average minor to major axis ratio of the inner rings and pseudorings is 0.77, whereas in the Catalogue of Southern Ringed Galaxies it is 0.81. The average minor to major axis ratio of the nuclear rings in our simulations is 0.70, the extreme values being 0.39 and 1.0. Unfortunately, precise statistics about the intrinsic shapes of the observed nuclear rings are not available to be compared with our simulation data (Buta & Combes 1996), but a rough estimate of the average axis ratio is about 0.9, varying from 0.6 to 1.0 (Crocker & Buta 1993; Buta, private communication).


Table 1. The axial ratios of the different ring types in the simulations, compared with the observed sample.

We have also determined the relative sizes of rings. Table 2 shows both the major axis ratios [FORMULA] and the ratios of geometrical mean radii [FORMULA] and compares them with the observations. The average ratio of geometrical mean radii of outer and inner rings is 2.2, very close to what is observed (Buta & Crocker 1993; Buta 1995). However, this accuracy is partly coincidental, the corresponding ratios for model families A, B and C are 1.9, 2.6 and 1.9, respectively. Our selection of models, which shows a wide variety of different morphologies, was composed to study the effect of different mass models and values of [FORMULA]. Thus, it probably does not correspond to any realistic galaxy sample. This is more clearly seen when comparing the relative sizes of the nuclear rings with the observations. For average values of ratios [FORMULA] and [FORMULA] we obtained 14.4 and 5.3, respectively, whereas the observed values are 18.9 and 8.7. Thus, the nuclear rings in our simulations are larger than the observed ones. Part of this discrepancy is caused by few anomalous cases in our selection of models, especially the huge nuclear ring of model A2.5. Omitting such cases would make our results closer to the observed values. Another possible factor is the uncomfortably large value of the softening parameter [FORMULA] when compared to the sizes of most of the nuclear rings in our simulations. A considerably smaller value of [FORMULA] would correspond to an unrealistically thin disk and thus we leave further studies of this problem to forthcoming three-dimensional simulations.


Table 2. Average relative diameters of rings in simulations and observations (Buta & Crocker 1993). For simulations both the ratio of major axes, [FORMULA], and the ratio of geometrical mean diameters, [FORMULA], were determined.

4.2. The absence of rings in barred galaxies

The absence of rings in many barred galaxies also requires an explanation. In the case of outer rings, there are several plausible possibilities. If the bar is relatively young, it has had not enough time to form an outer ring, whose typical formation time scale is one gigayear or more (note however that e.g. in model C1.75 an outer pseudoring forms very quickly, about 0.2 Gyr after the bar formation). On the other hand, the formation of a nuclear ring can coincide with bar formation, or occasionally can happen even before a clear bar component has formed, then related to a mode that later becomes the bar. The inner rings can appear quickly, the formation timescales ranging from practically coinciding with formation of the bar to a few hundred million years, although there are also cases where a pronounced inner ring appears only after several gigayears. Interactions with other galaxies can also destroy outer rings (Elmegreen et al. 1992). The third sometimes-mentioned possibility is that the region near the outer Lindblad resonance of the bar is dominated by a slower spiral mode, which inhibits the outer ring formation. However, we found several examples where an outer ring formed in spite of the presence of a slower spiral mode. There were also outer rings where the strongest [FORMULA] amplitude signal came from the slower mode.

The absence of nuclear rings can be related to the strength of the bar: in high amplitude cases the orbits supporting the nuclear ring become unstable. It is also possible that the pattern speed of the bar is sufficiently high, when compared to the peak of the [FORMULA] curve, that it does not have an inner Lindblad resonance, and thus cannot form a nuclear ring. This was the case for the three models with slowly rising rotation curves.

Also, the inner rings seem to be absent in simulations with the strongest bars, e.g. models A1.25 and C1.25. However, especially in these simulations, the bar formation is a very violent phenomenon, which includes strong outflow of gas particles. This means that when the bar has settled to a slow evolution, there are not enough gas particles left in the inner regions to form an inner ring in simulation. In real galaxies, the gas component can be refreshed via supernova explosions, stellar winds and gas accretion. To verify if the early outflow is the reason for the absence of inner rings, we also performed simulations where we included the gas component only after the bar had formed and its pattern speed change had decreased. In a few models this can indeed lead to the formation of an inner ring. An example of this is C1.25, shown in Fig. 12. In this particular case, there is also an outer pseudoring of subclass [FORMULA], which is absent in the original model where the gas was included from the beginning of the simulation (note that this pseudoring is completely inside outer 4/1-resonance, which is very unusual in these simulations). However, in model A1.25, this procedure did not lead to formation of a clear inner ring.

[FIGURE] Fig. 12. Comparison of three different gas inclusion methods. The frame on the left shows the gas distribution in a simulation (model C1.25) where the gas particles were included from the beginning of the simulation. The middle frame shows the same model, but the gas particles are included only after bar has formed and its pattern speed change has decreased. The frame on the right shows a model were the gas inclusion was done as in the model shown in the middle frame, but the initial gas particle distribution was exponential.

Inspired by the effect of delayed gas inclusion, we also made a few of our simulations with an exponential initial gas surface density distribution (having the same scale length as the stellar component). When combined with the delayed gas inclusion method for model C1.25, the main effect was the lack of the [FORMULA] component while the [FORMULA] component remained (Fig. 12). This behaviour resembles the simulations by Schwarz (1981), where the outer cut-off radius of the gas component determined which of the pseudoring subclasses was produced. However, such an effect was not clearly present in other simulations, where we included the exponential gas distribution from the beginning. Instead, the outer structures became more diffuse, simply due to reduced number density of particles. In two models, B1.75 and C1.75, long-lasting nuclear rings appeared, possibly due to larger amount of particles in the inner region.

A nuclear ring also can be destroyed by consumption of the gas by active star formation. Based on this argument, it has been claimed that the lifetime of a nuclear ring can be as short as [FORMULA] years (see also the discussion in Sect. 4.4). However, there are several galaxies with nuclear rings coexisting with inner and outer rings, both having formation time scales longer than this suggested lifetime (Buta & Combes 1996). Possible explanations for this discrepancy are accretion of gas from a small companion galaxy or recycled gas from the stellar component.

Although we have included a process that randomly changes colliding gas particles into collisionless test particles (this method was actually adopted to avoid excessive cpu-time consumption in giant gas clumps formed in the centres due to inflow in a few of the models), our usual parameter choice makes this process effective only in timescales of gigayears. The result of this process can be seen for example in the erosion of the nuclear ring of models A1.75 and A2.25 (see Fig. 13).

[FIGURE] Fig. 13. Gallery of nuclear rings in models A1.75, A2.25, B2.25 and C1.75. The gas particle distribution is plotted over isodensity contours of the stellar distribution. The shown timesteps are the same as in the figures showing the large scale morphological evolution of these models, except in model C1.75, where the evolution is shown in time steps T = 1.125, 1.375, 1.625 and 1.875 Gyr. The width of the frames is 6 kpc. The nuclear rings in models A1.75 and A2.25 disappear because the high collision frequency leads to transformation of gas particles to collisionless test particles (see text).

Most nuclear rings in simulations shrink during their early evolution. Similar behaviour was seen in the standard model of Piner et al. (1995). There are also models where shrinking can take place later. An extreme example of this is the nuclear ring in model C1.75, shown in the last row of Fig. 13: the nuclear ring finally sinks into the centre. In this case the shrinking of the nuclear ring is not due to [FORMULA]-orbits becoming unstable. This is shown by the orbit intergration of collisionless test particles. Furthermore, because changing the size or the initial distribution of gas particles can make the nuclear ring stable, it is possible that the sticky particle method has met its limits in this high density environment. There are some galaxies (Combes et al. 1992; Buta et al. 1995b) where the gas or dust component of the nuclear ring is located inside the stellar component and it has been suggested that these rings could be shrinking. It is unclear whether the shrinking of the nuclear rings in our simulations has something to do with these observed cases.

It has been suggested that the constancy of the pattern speed of the bar is necessary for the ring formation (Buta & Combes 1996) and that galaxies where the bar slows down fast could not have resonance rings. However, our results show that fairly large slow-down rates can be tolerated. For example, in model C1.75 the pattern speed of the bar decreases about 40% during the first gigayear after the bar formation. At the same time, the inner ring survives and the outer pseudoring forms. During the last 13 gigayears of this simulation, the bar pattern speed further decreases by 30%, half of which happens during the first two gigayears. While this happens, the OLR moves outwards by about 30%, which corresponds closely to the change in the size of the outer ring. However, note that the pattern speed change is much smaller during one bar rotation period, ranging from 5% by [FORMULA] Gyr to 0.5% by the end of the simulation. Larger pattern speed decreases can be present in 3D-simulations with a self-consistent halo component (Debattista & Sellwood 1998).

4.3. Miscellaneous cases

The existence of clearly misaligned ring structures, like in ESO 565-11 (Buta et al. 1995b), can at least in a few cases be explained by multiple modes present in the disks of these systems. An example of this is model B2.25. Four higher resolution frames of its evolution are shown in Fig. 14. It can be seen clearly that the relative position angle of the inner ring to the bar changes constantly. The reconstructed modes in corresponding timesteps are shown in Fig. 15. The reason for the misalignment is that the inner ring is evidently strongly affected by the mode with a lower pattern speed than the bar. In the amplitude spectrum of the gas component in the ring region, the signal is stronger in the lower pattern speed. The shape of the inner ring seems to vary with a period equal to relative period of the two modes. Maciejewski & Sparke (2000) introduced a concept of loop orbits, which exhibit a similar behaviour in their two-barred model. This concept could be useful also with nuclear and sometimes even with outer rings. Although Buta et al. (1999b) modelled ESO 565-11 with a rigidly rotating single pattern speed potential, which was derived from near-IR observations, we believe that the presence of two different modes is a more natural solution. Misaligned inner rings are very rare, only a few clear cases are known. This does not disagree with our results: model B2.25 is the only example that we have found among the approximately one hundred models we have made.

[FIGURE] Fig. 14. The evolution of the misaligned inner ring in model B2.25. The timesteps shown are 10.0. 11.25, 12.5 and 13.75 Gyr. The width of the frames is 36 kpc.

[FIGURE] Fig. 15. A sequence showing the isodensity contours of reconstructed bar (top row) and spiral (bottom row) modes of model B2.25, which has a misaligned inner ring. The timesteps are the same as in Fig. 14.

There are also galaxies which have rings but do not have a bar. It is possible that the bar has been destroyed by the inflow of gas to the nuclear region (e.g. Friedli & Benz 1993; Norman et al. 1996). However, note that in model A2.5 the oval disappears even without the effect of a massive gas component, and circular rings remain. Another possible reason for the existence of non-barred galaxies with rings is that the resonances are induced by a weak oval component (like in model A2.5) which can be hard to observe. Also, a spiral potential can form a ring, as is shown in Fig. 14 of Salo et al. (1999). This potential, which was derived from near-IR observations of IC 4214 (Buta et al. 1999a), was assumed to rotate rigidly. However, it has been suggested that in non-barred galaxies the spiral structures are formed by transient, but recurrent, patterns (Sellwood & Carlberg 1984). This view of a nonsteady spiral structure in non-barred galaxies lead Buta & Combes (1996) to consider the gravity torque of the spiral structure too inefficient to cause ring formation.

Buta and Combes' conclusion seems to be too strict, because the strongest ring structure of model C2.5, where the disk is so hot that formation of a major bar component never appears, is related to the spiral mode. The gas distribution and the contours of the restored [FORMULA] mode, with a spiral shape, at T = 6.25 are shown in Fig. 16. The inner Lindblad resonance radius of the spiral mode is also shown and the position of the ring nicely coincides with it. Although the outer Lindblad resonance of the nuclear bar is also rather close, the strength of the nuclear mode is very weak at this radius and the amplitude spectrum of the gas component clearly shows that the gas ring has the same pattern speed as the spiral component. Thus this ring is a nuclear ring although its size is typical of inner rings. Weak innermost ring structures seen in Fig. 16 are related to the nuclear bar. This simulation shows that a spiral mode also can form a ring, at least if it is long lasting and its pattern speed is almost constant, as in this model. Furthermore, as we discussed in the previous subsection, the spiral modes often have a strong contribution in the amplitude spectra of the outer rings.

[FIGURE] Fig. 16. The gas distribution of model C2.5 at [FORMULA] Gyr and the [FORMULA] spiral mode that supports a ring in its inner Lindblad resonance (shown as a dashed line). The contour levels of the mode are the same as in the figures showing the amplitude spectra.

In many simulations with rigid bar potentials, the regions around Lagrange points [FORMULA] and [FORMULA], near the minor axis of the bar, are populated by gas particles which follow the so called banana orbits. In real galaxies, this region is usually rather empty. The situation is very similar in self-consistent simulations: such structures are very rare (something like this can be seen in the later phases of model C2.25). When the amplitude of the bar increases, this region also usually becomes empty in the analytical simulations. It is unclear why such features are not observed in galaxies with weak bars. One possibility is that in these systems this region is dominated or interfered with by a slower spiral mode. However, there is one possible exception, NGC 4579 (see e.g. the second generation images of the STScI Digitized Sky Survey), where the morphology resembles our simulations with features near the Lagrangian points.

4.4. Limitations of our modelling approach

Some of our simulations have very sharp features in the gas component. This is largely related to the level of velocity dispersion of the gas component: in models with the sharpest features, the average radial velocity dispersion, [FORMULA], is about 4 [FORMULA], whereas in models with more "fuzzy" features [FORMULA] is close to 10 [FORMULA]. Furthermore, during the evolution of an individual model, the change in the sharpness of the features is usually accompanied by a change in the velocity dispersion. For example, models where a bar forms quickly, e.g. model C1.25, can have [FORMULA] close to 20 [FORMULA] during the early evolution, but later the velocity dispersion becomes much lower. At the same time, the sharpness of the morphological features increases. Inspired by these trends, we have tested the effect of gas velocity dispersion on an individual model. We performed these tests by running modified versions of model C2.25, that contains all the ring types and much small scale structure in the gas component.

The energy input by stellar winds and supernovae can increase the velocity dispersion of the gas component. We modelled this by randomly forming massless "OB-particles" in collisions. These particles explode after 20 million years and give velocity pushes [FORMULA] to gas particles nearer than a given radius [FORMULA]. A similar method was used by Noguchi (1988). Also, more elaborate schemes have been constructed (e.g. Elmegreen & Thomasson 1993; Heller & Shlosman 1994; Friedli & Benz 1995). We made simulations of mass model C2.25 varying the probability of OB-particle formation in collisions ([FORMULA] is 0.005 or 0.1) and the effective radius of the explosion (30, 150, or 375 pc). The velocity impact given by a supernova is about 15 [FORMULA], or about 7% of the circular velocity in the peak of the rotation curve.

The effect of the supernova explosions is strongest where the collision frequency is very high, i.e. in nuclear rings. The increased velocity dispersion makes the nuclear rings wider, or if [FORMULA] and [FORMULA] are large enough, the ring can become unstable and collapse. In the outer disk, the increased velocity dispersion makes features less sharp. Adopting a sufficiently high probability of OB-particle formation destroys practically all the fine structure in the gas component (see Fig. 17). Interestingly, Friedli & Benz (1995) determine the upper limit for the energy injected to the gaseous component by requiring that the formation of spiral arms and rings be possible. Although our tests were done with a very crude method and rather arbitrary parameters, they show that the effect of supernovae on rings and other features can be considerable.

[FIGURE] Fig. 17. The effect of the gas velocity dispersion. The top row shows model C2.25 at [FORMULA] Gyr, and otherwise similar models with supernova explosions with [FORMULA] included ([FORMULA] 0.005 or 0.1). The bottom row shows C2.25 but with different values of the coefficient of restitution (the original case C2.25 has [FORMULA] = 0). The average radial velocity dispersion are 6.9, 7.6 and 11.3 [FORMULA] for the top row, and 7.2, 9.2 and 14.5 [FORMULA] for the bottom row.

We also tested the effect of the coefficient of restitution [FORMULA] by repeating the model C2.25 with [FORMULA] ranging from our standard value 0.0 to 0.7. Adopting higher values of [FORMULA] causes the features to be more diffuse, but to avoid the formation of sharp features, the model requires [FORMULA] (see Fig. 17). This agrees well with Noguchi (1999; see also Hänninen & Salo 1994, Fig. 3 for the effect of [FORMULA] on the sharpness of ILR resonance features on planetary ring simulations). As can be seen from Fig. 17, the inclusion of supernovae has a rather similar effect to the adopting of a higher value of the coefficient of restitution. In both test series, most of the morphological changes happen continuously, and the large scale morphology is essentially similar inside a wide range of parameters. When the velocity dispersion is high, the formation of small-scale structure is inhibited in both series, and the formation of a nuclear ring is delayed or the ring can even become unstable.

The gas component is massless in our simulations. The inclusion of the gas mass would change the stability properties of the disk (e.g. Bertin & Romeo 1988; Shlosman & Noguchi 1993). Sellwood & Carlberg (1984) modelled the accretion process by adding stars to a self-gravitating disk on circular orbits, imitating the cooling effect of the young stellar population formed from the accreted gas. They suggested that barred galaxies would be systems with a high initial accretion rate. The opposite conclusion was found by Noguchi (1996). In his numerical model the high gas accretion rate led to formation of massive gas clumps, which scattered the disk stars and made the system stable against bar formation. Another gas related process is the bar-induced gas inflow towards nucleus, which can change the gravitational potential so that the bar would be destroyed, or at least become weaker (Friedli & Benz 1993; Norman et al. 1996).

Three-dimensional simulations of barred galaxies (e.g. Combes & Sanders 1981; Raha et al. 1991) have shown that a bar can become considerably thicker than the original stellar disk, forming a box-like or peanut-shaped bulge component. One would expect that this process would decrease the strength of the non-axisymmetric perturbation, at least in the bar area. In single-mode systems with a very strong bar, this weakening of the non-axisymmetric perturbation could make [FORMULA] orbits stable, and thus lead to the formation of a nuclear ring.

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Online publication: October 24, 2000