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

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3. Simulations

We performed self-consistent simulations with a wide range of initial parameters. When compared to models which use rigid bar potentials, these simulations provide new elements, especially concerning the long-term evolution of the system and the possibility of more than one mode being present in the disk (e.g. Sellwood & Sparke 1988; Rautiainen & Salo 1999).

We studied three basic families of mass models, each producing a fairly similar initial rotation curve in the inner parts of the disk. In family A the rotation curve is disk dominated, except in the central area where the bulge dominates. Family B has a strong halo and the rotation curve is dominated by the spheroidal component (including both the halo and the bulge), except in a small area where the disk provides an equal contribution. Family C is a mixture where the halo starts to dominate only in the outer disk. The rotation curves of these models are shown in Fig. 1.

[FIGURE] Fig. 1. The initial rotation curves of the mass models used. The continuous line shows the rotation curves. Contributions of the disk (dotted line), the bulge (dashed line) and the halo (dash-dotted line) are also shown.

In all models, the bulge component was modelled as an analytical Plummer sphere. The bulge scale length is 0.6 kpc and mass is [FORMULA]. We also performed simulations where we changed the bulge scale length to 7.5 kpc to produce a more slowly rising rotation curve (these models are denoted by the superscript ' in the following paragraphs). The disks are exponential disks with scale lengths of 3 kpc and a cut-off radius of six scale lengths. In families A and C (and also in A' and C'), the disk mass is [FORMULA], and in family B (and B') it is [FORMULA]. In model families B and C (and B' and C') we use an analytical halo potential, which is an isothermal sphere with a smooth transition to a constant core density. Inside five disk scale lengths, the halo mass in family B is [FORMULA] and in family C is [FORMULA].

One of the most important initial parameters that determines the further evolution of the models is the Toomre parameter (Toomre 1964):

[EQUATION]

where [FORMULA] is the radial velocity dispersion of the disk, [FORMULA] the epicycle frequency, G the constant of gravity and [FORMULA] the disk surface density. Although the value of [FORMULA] is enough to stabilise the disk against local axisymmetric instabilities, values typically larger than two are required to inhibit bar formation (see e.g. Athanassoula & Sellwood 1986). We carried out simulations with several values of the Toomre parameter from 1.25 to 2.5, for different degrees of halo-to-disk mass-ratio. Note that in simulations with softened gravity, the value of the softening parameter also affects the stability properties. Thus, the effective value of the Toomre parameter is higher than that implied by [FORMULA] (Romeo 1994). One should also note that since we use epicycle approximation in the construction of the initial state, the models with the highest values of [FORMULA] are a bit problematic: although they can avoid bar instability, they can suffer considerable mass redistribution in their early evolution.

3.1. Disk dominated models

Fig. 2 shows the amplitude spectra of the stellar component for four different simulations using the disk dominated mass model A. The values of [FORMULA] are 1.25, 1.75, 2.25 and 2.5. The corresponding morphologies of the stellar and gas components are shown in Fig. 3. For both the stellar and gas components, the densities are indicated as shades of grey (in a logarithmic scale) in the same frames, but the gas number density was multiplied by twenty for clarity. The effect of the Toomre parameter is clear. Bar formation occurs faster in the models with a cooler initial disk, in agreement with previous studies (see e.g. Athanassoula & Sellwood 1986).

[FIGURE] Fig. 2. The [FORMULA] amplitude spectra of the stellar component for family A models, differing in initial value of [FORMULA]. The contour levels are 0.025, 0.05 (drawn with a thicker line), 0.1, 0.2 and 0.4. The curves showing [FORMULA] and [FORMULA] are plotted with a continuous line and [FORMULA] with a dotted line. The length of the used time interval is 2.5 Gyr. The middle points of the time intervals are indicated in gigayears in the top of each frame. Note that the vertical axis has a logarithmic scale.

[FIGURE] Fig. 3. The evolution of the family A models with [FORMULA] 1.25, 1.75, 2.25 and 2.5, from the uppermost to the lowermost row, respectively. The morphology is shown at T = 2.5, 5.0, 10.0 and 15.0 gigayears from the beginning of the simulation. The gas density is multiplied by 20 to show it more clearly. The width of the frames is 45 kpc, except for model A2.5 where it is 60 kpc.

In the simulation with [FORMULA] (hereafter model A1.25) bar forms very quickly, by [FORMULA] Gyr. This bar mode has a spiral component with the same pattern speed, and there are also two spiral modes with lower pattern speeds. Later, the middle mode weakens but the outer mode becomes stronger. The amplitude spectrum suggests that there could be a nonlinear coupling between the bar and the stronger spiral mode such that the corotation resonance (CR) of the bar mode is roughly at the same distance as the inner Lindblad resonance (ILR) of the spiral mode. This is exactly the coupling suggested by Tagger et al. (1987) and the same as was present in the simulation by Masset & Tagger (1997). The gas morphology at this stage is a two-armed spiral, which occasionally forms short-lived pseudorings of both subclasses. A more robust outer pseudoring of subclass [FORMULA] forms by [FORMULA] Gyr and evolves into a complete detached outer ring in a few gigayears. The amplitude spectra of the gas component shows that both of the previously discussed modes affect the ring, although the contribution of the bar mode increases with time. This is not surprising since at [FORMULA] Gyr the slower modes have weakened below the lowest contour level of Fig. 3. The major axis of this ring changes from about 20 kpc to about 23 kpc during the simulation, following the change in the bar pattern speed. The axial ratio is about 0.9. Thus the whole ring is considerably beyond the outer Lindblad resonance (OLR) radius, 14-17 kpc, calculated from the epicycle approximation. The closest resonance to the ring is the inner 4/1 resonance of the slower spiral mode. This model develops neither inner nor nuclear rings.

The next model with a higher value of the Toomre parameter, A1.75, evolves more slowly than model A1.25: the formation of the bar takes place by [FORMULA] Gyr. However, the bar is preceded by an oval (here we define the oval - bar boundary to be at a major to minor axis ratio [FORMULA]), which has formed by [FORMULA] Gyr. The outer slowly-rotating spiral mode is much weaker than in model A1.25, but there is a nuclear bar (forms by [FORMULA] Gyr) that rotates with a higher pattern speed than the main bar. Note that this nuclear bar forms without the presence of a massive dissipative component, which is not in accordance with the results of Friedli & Martinet (1993). The nuclear bar is associated with a spectacular elongated nuclear ring (forms by [FORMULA] Gyr): the position angle of the ring with respect to the main bar changes constantly. The major axis of the nuclear ring changes from 1.6 kpc to 0.75 kpc during the simulation. At the same time, the axial ratio changes from 0.73 to 0.5. This model has only a transient inner ring appearing just after the formation of the bar. The outer pseudoring (subclass [FORMULA]) forms soon after bar formation, and is very close to the outer Lindblad resonance of the bar. During next few gigayears it becomes larger and reaches the stage of a detached ring, although the stellar spiral occasionally restores the connection to the central bar.

In model A2.25 a large scale bar does not appear until [FORMULA] Gyr. However, there is a large-scale oval that forms by [FORMULA] Gyr. A nuclear ring and weak inner and outer pseudorings emerge in the gas component during the era of the oval. Although there is also a rapidly rotating nuclear bar present, the nuclear ring is related to the oval, being just inside the outer of its inner Lindblad resonances. Rather round inner and outer pseudorings are close to the corotation and OLR of this mode, respectively. When the bar forms (see Fig. 2), its pattern speed is lower than that of the previously discussed oval, and considerable rearrangement of the gas morphology takes place. An inner ring forms in about 700 million years between the inner 4/1- and the corotation resonances of the bar, and an outer pseudoring forms about 400 million years later, very near the OLR. The spiral arms composing this pseudoring start outside the inner ring. The isodensity curves of the stellar component become almost circular near the inner 4/1-resonance, giving the impression that the bar does not fill the major axis of the inner ring completely. The evolution of the nuclear ring is very curious after the bar forms: the nuclear ring is captured by the nuclear bar and becomes smaller (from 1.3 kpc to 0.7 kpc) and less elongated (from 0.67 to 0.78).

The model with the highest Toomre parameter, A2.5, does not develop a clear bar but rather an oval that forms by [FORMULA] Gyr. Because the pattern speed of the oval is rather slow, so that its OLR is much farther in the outer disk than in the previous models, we found it necessary to use a wider gas distribution with 40 000 particles. The oval is surrounded by an inner ring, which forms by [FORMULA] Gyr. This model also has a nuclear bar and a huge nuclear ring (radius 6.4 kpc), which forms at the same time as the inner ring and is also related to the oval. In the early stages of the simulation, features resembling leading offset dustlanes connect the nuclear and the inner rings. The outer pseudoring, which forms by [FORMULA] Gyr, evolves from subclass [FORMULA] to [FORMULA]. Later, when the oval practically disappears, the rings become almost circular, and the system resembles the galaxy NGC 7217, studied by Buta et al. (1995a) and Verdes-Montenegro et al. (1995). It is remarkable that the oval dissolves, even though the often mentioned processes leading to bar destruction (namely massive gas inflow (see e.g. Friedli & Benz. 1993) or an encounter with another galaxy (e.g. Athanassoula 1996)) are not present. In this model, the cause of the disappearance of the oval is probably the strengthening of the nuclear bar. All the major rings in this model have the same pattern speed as the oval.

3.2. Models with a strong halo

Models with a strong halo (the amplitude spectra are shown in Fig. 4 and the morphological evolution in Fig. 5) do not form a bar as quickly as the disk dominated models. The early phases in the evolution of these systems typically show multiarmed or even flocculent spiral structures. Another difference compared to disk dominated systems is that the importance of the bar mode is smaller: the slower modes are usually strong, or even dominating, near the outer Lindblad resonance of the bar. The dominance can start from the corotation radius of the bar, as in model B1.25.

[FIGURE] Fig. 4. The [FORMULA] amplitude spectra of the stellar component for the model family B.

[FIGURE] Fig. 5. The evolution of the family B models with [FORMULA] 1.25, 1.75, 2.25 and 2.5. The morphology is shown at the same timesteps as for family A.

The first six gigayears of model B1.25 are characterised by a multiarmed spiral structure (three- and four-armed spirals are also present in the stellar component), even after the bar formation at [FORMULA] Gyr. The small bar component is surrounded by a rather round inner pseudoring that forms about 200 million years later. When the bar strength increases, the pseudoring becomes a more elongated (reaching an axial ratio of 0.73 by [FORMULA] Gyr) continuous ring. By [FORMULA] Gyr, the outer disk forms a detached outer ring, which seems to consist of two nested structures. The inner ring is very close to the inner 4/1 resonance of the bar, but the outer ring is not at any specific resonance of the two slower modes, which have peaks in the amplitude spectrum of the gas near the ring radius. For several gigayears, this model has a four-mode-chain of CR - inner 4/1 resonance overlappings, resembling several models in Rautiainen & Salo (1999).

Model B1.75 forms a large scale bar more slowly than model B1.25, by [FORMULA] Gyr, although there is an oval for about one gigayear before that. However, there is a nuclear bar that forms by [FORMULA] Gyr, and is captured by the main bar by [FORMULA] Gyr. A short-lived nuclear ring forms by [FORMULA] Gyr: it disappears about the same time as the large scale bar forms. By [FORMULA] Gyr, the multiarmed structure disappears as the model develops an outer pseudoring and a weak inner ring (not clearly visible in Fig. 4). At the early stage, the morphology of the outer pseudoring is somewhere between subclasses [FORMULA] and [FORMULA], but later the ring becomes detached from the bar. Occasionally the evolving stellar spiral has clear [FORMULA] morphology, even when the gas component lacks it. The inner ring is completely inside the corotation resonance and very close to the inner 4/1-resonance. The strongest peak in the amplitude spectrum of the gas in the outer ring is related to a slower mode, although the bar mode also gives a strong signal. The outer ring, with a radius of about 14 kpc, is almost exactly midway between the OLR of the bar and the corotation of the slower mode, 12.5 and 15.5 kpc, respectively.

In model B2.25 the evolution is again slower than in the previous model; an oval forms by [FORMULA] Gyr and becomes a bar about 500 million years later. The bar formation coincides with the formation of a nuclear ring near the inner Lindblad resonance of the bar. Although there is a fast rotating nuclear bar from [FORMULA] Gyr, a nuclear ring forms by [FORMULA] Gyr and the amplitude spectrum clearly shows that the ring follows the main bar/oval. The inner ring is a totally different story: it follows mostly an outer oval (forms by [FORMULA] Gyr) with a slower pattern speed than the bar, which makes the ring often misaligned with the bar. A more detailed discussion of this is given in Sect. 4.3. The outer structure of this model has a changing morphology, evidently due to the effect of several modes.

Model B2.5 develops an oval at [FORMULA] Gyr, but the formation of a clear bar does not take place until [FORMULA] Gyr. Although there is a nuclear bar from [FORMULA] Gyr, a nuclear ring forms by [FORMULA] Gyr, and is close to the ILR of the oval with a pattern speed [FORMULA]. Another nuclear ring develops temporarily inside this ring, but it is related to a faster mode. At [FORMULA] Gyr there is also an outer pseudoring. When the bar finally forms (with [FORMULA]), it develops an inner ring between its inner 4/1- and corotation resonances. It also captures or adopts the nuclear ring formed by the now-disappeared [FORMULA] mode: although the ring is located well inside the ILR radius calculated from the epicycle approximation and although there is a nuclear mode present, the nuclear ring follows the main bar after [FORMULA] Gyr. The outer pseudoring with an evolving structure is close to the OLR of the bar, but shows also the effect of a slower mode in its amplitude spectrum.

3.3. Models with a moderate halo component

Debattista & Sellwood (1998) demonstrated that systems in which the halo dominated inside the region of the optical disk (as is the case in our model family B) are not necessarily very realistic models for barred galaxies. Namely, when a massive self-consistent halo component was included in their 3D-models, the pattern speed of the bar decreased so much that the bar ended well before the corotation resonance, which disagrees with most of the modelling results for individual observed galaxies (e.g. Hunter et al. 1988; Lindblad et al. 1996). For this reason, we also made simulations with a model where the halo component is moderate and its contribution to the rotation curve rises above the disk contribution only in the outermost part of the disk. Fig. 6 shows the amplitude spectra and Fig. 7 the morphological evolution of these models.

[FIGURE] Fig. 6. The [FORMULA] amplitude spectra of the stellar component for the model family C.

[FIGURE] Fig. 7. The evolution of the family C models with [FORMULA] 1.25, 1.75, 2.25 and 2.5. The morphology is shown in the same timesteps as for family A.

The model with the coolest disk, C1.25, develops a bar at [FORMULA] Gyr. An outer ring structure forms about 1 Gyr later. This evolves from a pseudoring very close to the OLR of the bar to a detached outer ring outside the OLR radius. The morphology of the ring keeps changing throughout the simulation and a subclass cannot usually be determined. Occasionally there is a combined outer ring of subclass [FORMULA], where the [FORMULA] component is present practically only in the stellar component, whereas the [FORMULA] part dominates the gas component. Although the ring is rather close to the OLR of the bar, the strongest peak in the [FORMULA] amplitude spectrum of the gas in the ring area is related to a mode with lower pattern speed. There are no inner or nuclear rings in this model.

Changing outer ring morphology is also typical for model C1.75, where the outer pseudoring forms very quickly, only about 200 hundred million years after the bar formation, at [FORMULA] Gyr. The outer pseudoring can often be classified as either [FORMULA] or [FORMULA] (where the [FORMULA] component can be enhanced by the stellar component), but sometimes we found detailed classification impossible: we could only say that it was a pseudoring ([FORMULA]). A more detailed coverage of the early evolution of the gas component in this system is shown in Fig. 10. The time scale of considerable changes in the outer ring morphology can be as short as 100 million years, when the typical orbital period near the ring radius is about 500 million years. The amplitude spectrum of the gas component shows that the outer ring is affected by both the bar mode and the mode with a slower pattern speed, the latter being the dominating one. The ring is located between the OLR of the bar and the corotation of this slower mode. An inner ring or a pseudoring forms at about the same time as the outer pseudoring and it is located between the inner 4/1-resonance and the corotation of the bar. This model also has a shrinking nuclear ring that is destroyed in few hundred million years. A better resolution plot of the evolution of this and a few other nuclear rings is shown in Fig. 13.

Model C2.25 develops an oval and a nuclear bar by [FORMULA] Gyr and a large scale bar by [FORMULA] Gyr. There are short-lived inner and outer pseudorings during the oval stage, but a more robust outer structure forms by [FORMULA] Gyr. This evolves continuously, and cannot usually be classified into outer ring subclasses. In later phases of the simulation, the outer morphology becomes almost flocculent. On the other hand, there are very steady inner and nuclear rings in this model. The inner ring forms by [FORMULA] Gyr and is located between the inner 4/1- and corotation resonances of the bar, being closer to the former one of the resonances. At [FORMULA], the bar does not fill the major axis of the inner ring completely, thus resembling model A2.25. Although there is a nuclear mode inside the nuclear ring, the amplitude spectrum shows that the ring follows the main bar. The nuclear ring forms by [FORMULA] Gyr, initially exactly to the (outer) ILR of the oval, at a radius of 1.7 kpc. Then the ring shrinks so that its major axis is about 1.4 kpc, while the ILR moves outwards to radius of 2.3 kpc, following the deceleration of the bar rotation.

The model with the hottest disk, [FORMULA], lacks the main bar component, but it still has ring structures. Although there is a nuclear bar present, the most spectacular ring is related to the spiral component (see Fig. 16, and discussion in Sect. 4.3), being near its ILR. The outer disk has a flocculent gas morphology and is separated by a gap from the inner region.

3.4. Models with slowly rising rotation curves

We also made simulations where we modified the bulge model so that the rotation curves rise more slowly. Fig. 8 shows the amplitude spectra of three such simulations, all with [FORMULA], and Fig. 9 presents the corresponding evolution. In all these models, the bar is longer than in the corresponding model with a steeper rotation curve. Excluding the last stages of Model A'1.75, these models do not have an inner Lindblad resonance, and thus it is not surprising that they do not develop nuclear rings. They also lack nuclear bars.

[FIGURE] Fig. 8. The amplitude spectra of the stellar component for models A'1.75, B'1.75 and C'1.75, which have slowly rising rotation curves.

[FIGURE] Fig. 9. The evolution of model A'1.75, B'1.75 and C'1.75. In model A'1.75, the width of the frame is 66 kpc, in others it is 45 kpc.

In model A'1.75, we found it necessary to widen the initial gas distribution to be able to cover the OLR region. When doing so, we also increased the gas particle number to 40 000. This model develops an oval by [FORMULA] Gyr, and a bar about 200 million years later. A detached outer ring, whose radius (about 30 kpc) is about twice the bar major axis, but it is not related to any resonance induced by the bar: the OLR of the bar is at 20 kpc. What makes this ring even more strange is that the amplitude spectrum of the gas component has peaks at pattern speeds which do not correspond to any clear modes in the stellar component.

Model B'1.75 develops a bar by [FORMULA] Gyr, and it has an inner ring (from [FORMULA] Gyr) that is first very elongated, but becomes more circular towards the end of the simulation. For several gigayears, there is an additional elongated ring component just inside the main part of the inner ring. The inner ring is very close to the inner 4/1-resonance. The outer morphology changes, but can be occasionally classified using the outer ring subclasses. Both the corotation of a slower mode and the outer Lindblad resonance of the bar are close to the ring radius. According to the amplitude spectra, the bar mode dominates in the ring radius.

Model C'1.75 develops first an oval ([FORMULA] Gyr) and then a bar ([FORMULA] Gyr). There is a rather similar two-component inner ring (since [FORMULA] Gyr) as in model B'1.75, but now the inner, very elongated component is more pronounced. The major axis of the more elongated component is close to the inner 4/1-resonance of the bar, whereas the outer component is between the inner 4/1- and corotation resonances. This model also has an outer pseudoring (from [FORMULA] Gyr), which is of subclass [FORMULA], although its shape is often quite unusual: at early stages there is a secondary pair of spiral arms and later the ring is lopsided. The major axis of the outer pseudoring is close to the outer 4/1-resonance of the bar, which is a rare situation in our simulations. Both the inner and the outer rings follow the bar mode.

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Online publication: October 24, 2000
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