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Astron. Astrophys. 323, 363-373 (1997)

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5. Clues from numerical simulations

5.1. Spontaneous bar formation in discs

The most recent ideas concerning the various processes which drive the formation, the evolution and the destruction of bars can be found in the thorough reviews by Sellwood & Wilkinson (1993), Martinet (1995), and Sellwood (1996). Here, only the timescale problem for spontaneous bar formation is briefly discussed.

The details of the formation of galactic discs are still under debate (e.g. Dalcanton et al. 1997), but there are various evidences that their formation could require a non-negligible fraction of Hubble time, or could even be an ongoing process. For instance, various cosmological numerical simulations using the standard CDM scenario indicate timescales of several Gyr for the disc formation (e.g. Steinmetz & Müller 1995). Based on chemical abundance arguments, Sommer-Larsen & Yoshii (1990) also found that continuous infall of proto-galactic material onto the disc over timescales of 4-6 Gyr is necessary. In addition, it is interesting to note that late-type discs appear to be younger than early-type discs (Sommer-Larsen 1996) which suggests that the beginning of the disc assembly might not occur at an universal epoch.

The mechanism of bar formation requires the presence of a well defined disc. The growth of a spontaneous bar may occur in later stages of the disc evolution, i.e. when sufficient gas mass has been accreted onto the disc and, above all, transformed into stars via star formation processes. At some point, the stellar disc will meet the critical conditions necessary for the onset of the bar instability. Star formation works towards this by progressively adding dynamically cool masses in the stellar disc. For instance, in the models of Noguchi (1996), spontaneous bars typically appear only 6-7 Gyr after the beginning of the disc formation. This is about 10 times longer than the growth of a strong bar (see Sect. 5.3). Thus, the existence of young bars among nearby galaxies is highly expected and very likely observed (Martin & Roy 1995; Martin & Friedli 1997). The fact that barred galaxies seem to be scarce in the Hubble Deep Field (van den Bergh et al. 1996) could also be considered as a possible confirmation of the late appearance of such structures in the life of flattened galaxies. However, this latter study only presents preliminary results which clearly have to be confirmed and interpreted with great care.

5.2. Method, previous and present models

In order to try to explain the observed connection between the bar strength and the SFR presented in the previous section, we have performed a new set of 3D self-consistent numerical simulations with stars, gas, and star formation. Technical details can be found in Pfenniger & Friedli (1993), and Friedli & Benz (1993, 1995). Since our galaxy sample is made of non- or weakly-interacting galaxies, and because the complete process of disc formation is clearly beyond the scope of this paper, we here emphasize some steps of possible evolution implying the mechanism of spontaneous bar formation in an already existing disc.

Previous numerical models (Friedli & Benz 1993) have indicated that the intensity of the gas fueling phenomenon strongly depends on the strength of the bar. As the bar axis ratio [FORMULA] decreases, much faster gas accumulation into the center occurs. In less than one Gyr, strong bars have nearly pushed all the gas initially inside the bar region into the center, whereas weak bars have only accreted a small fraction of it. The formation of a spontaneous or induced strong bar in a gas-rich Sc-like disc typically triggers a starburst of intermediate power and duration. Depending on the initial amount of gas and the various parameters of the star formation "recipe" (Friedli & Benz 1995), the peak star formation rate is [FORMULA] and the duration is [FORMULA] Gyr. In particular, stars are formed in gaseous regions unstable with respect to the Toomre parameter (Toomre 1964) [FORMULA], where s is the sound speed, [FORMULA] is the epicyclic frequency, and [FORMULA] is the gas surface density. The star formation parameters used in the numerical simulations are not unique but have carefully been chosen in order to reproduce as accurately as possible the observed star formation properties of barred galaxies.

Below, the computed SFR is the mean value over 100 Myr, i.e. from 50 Myr before to 50 Myr after the time at which the corresponding [FORMULA] is determined. This determination was done by applying a standard ellipse-fitting routine to the stellar surface density distribution. The generic model has an initial Sc-like disc with an initial gas to stars mass ratio [FORMULA], where [FORMULA] is the total gas mass and [FORMULA] is the total stellar mass. The value of the Toomre parameter (Toomre 1964) [FORMULA], where [FORMULA] is the radial stellar velocity dispersion, and [FORMULA] is the stellar surface density. The initial O/H abundance gradient has been set to [FORMULA]. In order to smoothly switch on the star formation, the models are first calculated during 400 Myr in a forced axisymmetric state. This procedure also allows us to compute at [FORMULA] the SFRs for the corresponding unbarred galaxy. After that, the simulations described below evolve in a fully self-consistent way.

5.3. Strong bar case

For two representative simulations, Fig. 4 shows the time evolution of both the total SFR and the maximum bar axis ratio [FORMULA]. As time is evolving, a bar instability progressively develops triggering more star formation. Note that the bar growth and evolution timescales quoted below can be either shorter or longer depending on the instability level of the stellar disc at the beginning of the simulation.

[FIGURE] Fig. 4. Time evolution of the total SFR and the maximum bar axis ratio [FORMULA] for typical numerical simulations forming a strong bar, either with [FORMULA] (full circles; generic model), or with [FORMULA] (open squares). The time in Myr is indicated beside symbols when possible. Dotted lines separate strong bars from weak ones as well as galaxies actively forming stars from more quiescent ones. The dashed ellipses schematically indicate the position of the four observational classes defined in Sect. 6

For the generic model at times [FORMULA]  Myr and [FORMULA]  Myr, the bar is already strong but the SFR is still modest, mainly concentrated along the bar major axis. This is the "pre-starburst" phase where not enough gas mass has been pushed into the center to exceed the critical gas surface density [FORMULA] necessary for the onset of star formation. The bar axis ratio is progressively decreasing, whereas the bar length is gradually increasing up to [FORMULA]  Myr where the bar reaches a quasi-stable state ([FORMULA]). This is the lowest bar axis ratio of the whole simulation and it coincides with the maximum SFR observed ([FORMULA]) with the star formation essentially concentrated at the center. By [FORMULA]  Myr the SFR has become more moderate once more, although the bar is still strong. This is the beginning of the "post-starburst" phase where the gas has been sufficiently consumed to go again below [FORMULA] and nearly stop star formation although large amounts of gas remain near the center.

By [FORMULA]  Myr, the bar axis ratio has increased very slightly ([FORMULA]), but the SFR has dropped by a factor of more than 6 ([FORMULA]). Finally, after one further Gyr, at [FORMULA]  Myr, the SFR has decreased to less than [FORMULA], while the bar has become a little weaker still ([FORMULA]). At this time, the total gas to star mass ratio [FORMULA], and the gas represents less than one percent of the dynamical mass inside 1 kpc.

So, clearly strong bars do not necessarily always host enhanced central star formation. The presence of observed galaxies in the lower left corner of Fig. 1 is thus easily explained. The upper right corner of Fig. 1 cannot be reached and crossed by the generic model. However, one possible way is to strongly increase the gas mass ([FORMULA]) in order to produce a widely over-critical disc of gas with respect to the Toomre criterion. Such discs will form stars at a very high rate whatever the bar strength is, and the formation of a strong bar only results in a moderate increase of the total SFR. As an example, the evolutionary track of a model similar to the generic one but with [FORMULA] is also presented in Fig. 4.

For the generic simulation, the time evolution of the radial O/H abundance gradient [FORMULA] and the maximum bar axis ratio [FORMULA] is presented in Fig. 5. The time evolution of [FORMULA] in the bar (i.e. 2 - 8 kpc) and disc (i.e. 12 - 18 kpc) regions are both shown. A very different behaviour is observed. The disc abundance gradient becomes very quickly very shallow as soon as the strong bar develops. It moves from -0.10 at [FORMULA]  Myr to [FORMULA] at [FORMULA]  Myr. On the contrary, the abundance gradient in the bar region remains first steep. It changes from -0.10 at [FORMULA]  Myr to [FORMULA] at [FORMULA]  Myr, and only becomes shallower (-0.01) around [FORMULA]  Myr. Moreover, the galaxy core becomes very oxygen-rich as well.

[FIGURE] Fig. 5. Time evolution of the O/H abundance gradient [FORMULA] and the maximum bar axis ratio [FORMULA]. Circles corresponds to the strong bar case (generic model) whereas squares are for the weak bar case. Open and full symbols are respectively for the abundance gradients in the bar (i.e. 2 - 8 kpc) and disc (i.e. 12 - 18 kpc) regions. Both models have [FORMULA]. The time in Myr is indicated beside symbols when possible. Dotted lines separate strong bars from weak ones as well as steep abundance gradients from shallow ones. The dashed line corresponds to the observed relation by Martin & Roy (1994)

In the bar region, during the early phase of its existence, the gradient is maintained since the gas dilution (following the significant gas inflow) is compensating for the heavy-element production in the furious star formation then going on in the nuclear vicinity. For more details, see also Friedli et al. (1994), Friedli & Benz (1995), Martin & Friedli (1997). This of course results in the presence of two different radial abundance gradients in young strongly barred galaxies. For instance, at [FORMULA]  Myr, the slope ratio is about 4.3. After [FORMULA], the disc abundance gradient remains essentially flat, whereas the lack of gas inside the bar region prevents there any reliable determination of the abundance gradient. So, there is a "steep-shallow" break in the slope profile of [FORMULA] as already observed in at least two galaxies (see Table 1). A "shallow-steep" break could also be present close to the edge of the optical disc (see e.g. Friedli et al. 1994; Roy & Walsh 1997) but it has not yet been observed.

5.4. Weak bar case

Observed weak bars are either i) asymptotically and intrinsically weak, or ii) transient features progressively turned into strong bars as seen in the previous section. In the latter case, the duration of the weak bar phase depends on the timescale of the bar instability growth. This timescale becomes longer if more of the total mass resides in slow or non-rotating components like massive dark halos or stellar bulges, i.e. the lower values of the Ostriker-Peebles parameter (Ostriker & Peebles 1973). The fact that weak bars are short (see Sect. 4.2) is an indication that they might in fact be growing strong bars. Moreover, it has been proven quite difficult to numerically form permanent, realistic, weak bars. They can however be produced either by putting at the center a "point mass", e.g. a dense cluster or a supermassive black hole with mass [FORMULA] (Friedli 1994; see also Norman et al. 1996), or by increasing the stellar radial velocity dispersion so that [FORMULA] (Athanassoula 1983). The presence of large amounts of highly viscous gas significantly reduces the maximum bar strength as well (see Figs. 4 and   6).

[FIGURE] Fig. 6. The same as for Fig. 4 but for typical numerical simulations forming a weak bar, either with [FORMULA] (full circles), or with [FORMULA] (open squares). Note the different scales for both axes with respect to Fig. 4

We chose the method of disc heating. So, the models presented here are similar to the ones of Sect. 5.3 except for the [FORMULA] value which has been increased by 30%, i.e. [FORMULA]. The time evolution of both the total SFR and [FORMULA] is shown in Fig. 6. The bar growth timescale is much smaller than for the strong bar case. The spiral arms also remain very weak since the transfer of angular momentum is modest.

For the model with [FORMULA], up to [FORMULA]  Myr, the SFR is essentially constant and very low ([FORMULA]), whereas the bar length is gradually increasing and the bar axis ratio is progressively decreasing ([FORMULA]). Then, up to [FORMULA]  Myr, the SFR appreciably increases and the bar becomes a little stronger. The maximum SFR is [FORMULA], the star formation being essentially concentrated along the bar major axis and in the center. The lowest bar axis ratio of the whole simulation is [FORMULA] so that this model corresponds in fact to an intermediate case between weak and strong bars. After that, up to [FORMULA]  Myr, the SFR continues to decrease to reach a quasi-stationary state of [FORMULA] whose major contribution comes from a nuclear ring. The bar axis ratio increases up to [FORMULA]. At the end of the simulation, [FORMULA].

The evolution of the model with [FORMULA] is somewhat surprising. Up to [FORMULA]  Myr, the over-critical gaseous disc forms stars at a relatively high rate ([FORMULA]). Then, the SFR suddenly drops as the disc is progressively becoming self-regulated, i.e. [FORMULA] all over the gaseous disc. At [FORMULA]  Myr, the growth of the weak bar (up to [FORMULA]) starts to influence the SFR which increases. However, although high amounts of gas are still present, the peak SFR of this model is smaller by a factor 2.5 than the one of the model with [FORMULA]. The efficiency of gas fueling is much reduced since the bar remains short, and its maximum axis ratio gradually increases ([FORMULA] at [FORMULA]  Myr). Thus, in our models, the increase of the [FORMULA] ratio results in shorter and weaker bars.

The time evolution of [FORMULA] and [FORMULA] is presented for both bar and disc regions in Fig. 5. Contrary to the strong bar case, a similar behaviour is observed in these two regions although the abundance gradient remains always a little steeper in the bar (at most by 27%). The abundance gradients essentially remain unchanged up to [FORMULA]  Myr where [FORMULA]. Then, when [FORMULA], they relatively quickly flatten to finally reach a nearly constant value around -0.035 at [FORMULA]  Myr. This value is slightly larger than the one derived from the observed relation given by Martin & Roy (1994), i.e. [FORMULA]. The relevant points are that, a) weak bars need much longer timescales to flatten radial abundance gradients, b) weak bars are unable to produce totally flat abundance gradients.

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© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998