Astron. Astrophys. 355, 929-948 (2000)

5. Some "successful" models

In Sect. 4 we have illustrated the qualitative effects of radial inflows using, for the sake of example, models with a Schmidt SF law. We will now consider models with various SF laws (see PC99) and tune the inflow velocity pattern, in each case, so as to match the observational data on the radial profile of the Galactic Disc. The various "successful" models presented here should not be taken as detailed, unique recipes to reproduce the Disc. Rather, they are meant as examples of how inclusion of radial flows in the chemical model can improve the match with the data.

5.1. Models with Schmidt law

Let us first consider again the Schmidt SF law. Inspection of models S15RFb and S15RFd (with and without radial inflows from the outer disc) suggests to proceed as follows.

• Some inflow from the outer gaseous layer is needed to reproduce the shallow decline of the gas distribution observed out of the Solar ring, though the inflow speed from the outer disc should be slower than -1 km sec^-1 otherwise the predicted gas density is too high in the outer parts (as in model S15RFb).

• Models with radial inflows can predict metallicity gradients close to the observed ones even with a Schmidt SF law, provided drift velocities within the star-forming edge are relatively high (of the order of -1 km sec^-1). Inflow patterns decelerating inward are particularly efficient in building the metallicity gradient (Götz & Köppen 1992).

An example of a drift velocity profile shaped according to these considerations is shown in Fig. 4 ; the corresponding model gives indeed a reasonable fit to the data (model S15RFe , Fig. 5). The gas profile keeps increasing inward, yet the model does not reproduce the peak of the molecular ring, for which we need to include the peculiar radial flows induced by the Bar (see the discussion in PC99 and Sect. 6). Model S15RFe shows how radial inflows can in fact allow for metallicity gradients comparable with the observed ones, even with a Schmidt SF law which would be excluded on the sole base of static models (PC99).

Fig. 4. Inflow velocity pattern for the "successful" models S15RFe and O10RFe .

Fig. 5. A "successful" S15RFe model compared to the case with no flows. Data and symbols as in Fig. 2.

5.2. Models with spiral-triggered SF laws

Let us consider now models adopting an Oort-type SF law, with a SF efficiency inversely proportional to the galactocentric radius and a Schmidt-like exponent (Kennicutt 1998, PC99). The static and the "successful" model with this SF law are model O10a from PC99 and model O10RFe , respectively (Table 1).

As in the static case, (see PC99), these models behave somewhat similarly to models adopting the Schmidt SF law. Fig. 6 shows that a good fit is obtained with model O10RFe , where we applied the velocity pattern of Fig. 4 with inflows becoming slower inwar, very close to that used for model S15RFe . Notice that in this case the higher SF efficiency in the inner region and the slowdown of inflows conspire to accumulate gas around  kpc while consuming it at inner radii, generating a peak in the predicted gas distribution which closely reminds the observed molecular ring.

Fig. 6. A "successful" O10RF model compared to the case with no flows.

5.3. Models with gravitational self-regulating SF laws

We now investigate models adopting a self-regulating SF process driven by gravitational settling and feed-back from massive stars, implying a SF efficiency exponentially decaying outward in radius, such as the law by Dopita & Ryder (1994; see PC99). The reference static model in this case is model DRa of PC99, while the corresponding "successful" model with radial flows is DRRFe (see Table 1 for the relevant parameters).

To reproduce the observed metallicity gradient with this SF law, negligible inflow is required in the outer regions where static models already predict a gradient with the right slope (see model DRa), while a moderate inflow velocity (0.3 km sec^-1) in the inner parts is needed to maintain the observed slope also where the predicted gradient would otherwise flatten (see PC99). Such a model is, for example, model DRRFe , obtained with the inflow velocity pattern shown in Fig. 7. In Fig. 8 the "successful" model DRRFe is compared to the original model DRa of PC99 with no radial flows.

Fig. 7. Inflow velocity pattern for the "successful" model DRRF

Fig. 8. A "successful" DRRF model compared to the case with no flows.

5.4. Concluding remarks

The aim of the previous "successful" examples is to remark how, for each SF law considered, it is possible to tune the inflow velocity pattern so as to get a good overall agreement with the observational data. As an indication, suitable inflow profiles have the following characteristics.

• With a Schmidt SF law, relatively fast (but still plausible) flows are needed within the star-forming disc, with velocities of the order of 1 km sec^-1. The required metallicity gradient is easily obtained, especially if the inflow velocity is slowly decreasing inward.
  Inflow from the outer gaseous disc must be less prominent (though present), otherwise the predicted gas density in the outer regions is too high.

• In models adopting an Oort-type SF law with , a good fit to the data is obtained with an inflow profile of the same kind: inflow at  km sec^-1 from the outer gaseous disc, drift velocities raising at  km sec^-1 at the stellar disc edge and declining inward. Curiously enough, in this case the higher SF efficiency in the inner regions combines with the slowdown of inflows to predict a peak in the gas distribution around  kpc, quite reminiscent of the observed molecular ring.

• Models adopting the SF law by Dopita & Ryder give a good description of the outer regions of the Disc already in the absence of flows (PC99). Mild radial inflows can be assumed in the inner regions, where the radial gradient would otherwise flatten, to obtain the right slope throughout the disc. The required drift velocities are around 0.3 km sec^-1, which can be reasonably provided, for instance, by the shocks occurring within the spiral arms (see Sect. 2).

The various "successful models" presented here are not meant to be definitive recipes to reproduce the radial profile of the Galactic Disc. In fact, the adopted inflow velocity profiles are quite arbitrary and ad hoc . These models are rather meant to show how radial flows can be a viable mechanism to interpret the properties of the Disc. In PC99 we showed how none of the various SF laws investigated is able, by itself, to reproduce the observed metallicity gradient throughout the whole extent of the Disc, unless some additional "dynamical" assumption is included. In PC99 we considered the classical case of an inside-out disc formation, namely of an infall time-scale increasing outward. Here, we just show that radial inflows can provide another viable "dynamical" assumption to be combined with any SF law to reproduce the observed gradient. In particular, if we adopt a Schmidt SF law with or an Oort-type SF law with , as recent empirical evidence seems to support (Kennicutt 1998), the required variation of the accretion time-scale is too extreme in the pure inside-out assumption (PC99), and radial inflows are then necessary to explain the metallicity gradient. Of course, the two effects (inside-out formation and radial flows) can also play at the same time. We do not address here their combined outcome, because in our models this would merely translate into increasing the number of parameters which one can tune to fit the observational data. No further insight in the problem would be gained. With this kind of models we can just learn how the various players (different SF laws, radially varying accretion time-scales, radial gas inflows) enter the game of reconstructing the general picture, and analyse their effects one by one.

It is also worth stressing that even slow radial gas flows, with velocities well plausible in terms of the triggering physical mechanisms and within the observational limits (Sect. 2), have non-negligible influence on chemical models, especially on the gas density distribution. It is therefore misleading to seek for a one-to-one relation between gas content and metallicity, or between gas profile and metallicity gradient, like the one predicted by simple models (e.g. Tinsley 1980). When studying the chemical features of galaxies, it is very dangerous to assume that such a relation must hold, since even mild flows can easily alter the overall distribution.

The models with smooth radial inflows presented in this section (with the possible exception of model O10RFe) are still unable to reproduce the gas density peak corresponding to the molecular ring around 4 kpc, since that needs to take into account the peculiar dynamical influence of the Galactic Bar on gas flows. This will be the issue of the next section.

© European Southern Observatory (ESO) 2000

Online publication: March 21, 2000
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