## 3. The density profile in the z directionAt a given height above the plane, the disc in the model is therefore a sum of age-components, and the vertical density depends on both the age- relation and the SFR history. According to the observed age- relation, the oldest components must have the largest vertical extensions. This does not necessarily imply that old stars dominate counts: the relative importance of each component (i.e SFR history) plays a role too. The aim of the present section is to study what z-density profiles should be expected, considering the uncertainties in age- relation, which we briefly review in Sect. 3.1, and in SFR history in our Galaxy. Then, we discuss how these profiles compare with the exponential density laws used in most IMF investigations and galaxy models. ## 3.1. Observational measurements of the age- relationThe age- relation is known to vary very rapidly for the youngest stars: rises from approximately to 10-13 within a few Gyrs (less than 3 Gyrs). Consequently, the young disc in the model cannot be considered isothermal and is divided in 4 different components of ages 0-0.15, 0.15-1.0, 1.0-2.0, and 2.0-3.0 Gyrs. The "old" disc is represented by 3 components of ages 3-5, 5-7 and 7 to 10 Gyrs. Unfortunately, no consensus has been reached on either the shape of the age- relation and the true level at which this relation saturates. In the model of the vertical distribution of stars, uncertainties in the age- relation produce effects which are of the same order of magnitude, or greater, as the effect generated by the SFR that we want to investigate, hence they must be taken into account. A review of the different empirical relations and the mechanism which are invoked to explain the increase of the velocity dispersion with age can be found in Lacey (1991). The present situation is summarized in Fig. 2, where we plotted four empirical age- relations from Wielen (1974), Mayor (1974), Meusinger et al. (1991), and Strömgren (1987).
Some authors suggested that Mayor and Wielen samples could be
contaminated by thick disc stars (Strömgren (1987), Lacey
(1991)). Strömgren (1987) divided his sample in two groups
according to their metallicity, with a "pur disc sample" (-0.15
+0.15) (empty triangles in Fig. 2) and a
second group classified by decreasing metallicity (filled triangles).
Oldest stars in the "pur disc sample" have which
saturates at 15.3 km.s As can be seen from Fig. 2, data from Meusinger et al. (1991)
are in satisfactory agreement with the sample from Strömgren
(1987). A new sample of 189 F & G stars, with accurate age
determinations, is given by Edvardsson et al. (1993). According to
Freeman (1991), the age- relation shows a steep
increase in the first three Gyrs, and then flattens at
20 km.s In order to take into account the wide range of possible age- relations, either on the shape or the level at which these relations saturate, we have studied a set of 3 representative relations, displayed in Fig. 2 as thin lines and in Table 1. Relations () and () bracket the observed measurements of , while is an intermediate relation.
## 3.2. Theoretical expectationsThe dependency of the vertical density laws on the SFR history is
qualitatively illustrated by Fig. 3, which shows the various
profiles we obtained assuming the age- relation
and various exponential SFR. We note that the
profile is very sensitive to the adopted SFR. As expected, the SFRs
that must give greater weights to old stars get closer to a
sech
Fig. 4 shows how the density gradient changes with different SFR history and age- relations. Each plot shows the profile obtained for a given SFR, and for the three age- relations of Table 1. All these profiles are plausible according to present accuracies in the age- relation and SFR history. The difference in the densities at a given distance due to uncertainties in the age- are very large. In the case of the constant SFR (Fig. 4b), this difference amounts to more than 65% at z=500 pc, between profile and and 170% between profile and at the same distance.
## 3.3. Questioning the exponential approximationThe exponential scale height is an unneccessary restrictive concept in the case of the BRC model, however, it is instructive to investigate how exponentials compare with a more sophisticated model, as most pictures of the large scale structure of the Galaxy are based on this assumption. The comparison with our models will show that this approximation is too crude. Paper II will demonstrate that it is probably responsible for misleading interpretation of star count data. ## 3.3.1. Brief reviewIn 1980, Bahcall & Soneira presented a model of the Galaxy with
the galactic disc density law based on the exponential approximation,
justified by a long lasting accepted convention but rather weak
observational facts. In the eighties, mainly two results have provided
more solid support for the exponentials as convenient fitting
functions, concomitantly with the renewal of galactic models. The
first one is the work by Gilmore and Reid (1983), where it was shown
that the observed vertical density distribution is well represented by
two exponentials at z 4000 pc. This fact
is reconsidered in Sect. 6.1.2 of Paper II. The second one is the
numerical calculation by Bahcall (1984) of the disc density law that
comes out from his dynamical model. Bahcall demonstrates that the
numerical integration of the Poisson-Boltzmann equations for the
parameters of the disc given in Bahcall & Soneira (1980) gives a
vertical density distribution at 300 z
900 pc very near to what one would obtain
using an exponential. This result is confirmed by our own calculation.
Examining the Table 4 of Bahcall (1984), the relative weights
attributed to his different kinematic components (with
=4, 10, 13, 15, 20, 24 km.s ## 3.3.2. Comparison with our modelThe inadequacy of exponentials to represent the more realistic profiles of Fig. 4 is shown in Fig. 5. Each curve in Fig. 5 gives the exponential scale height (y-axis) that should best-fit the profiles of Fig. 4, at various distances above the plane (x-axis). Each scale height was obtained by calculating fits between exponentials and our profiles over 300 pc distance ranges, without forcing the density profiles to have the same normalization. Obviously from Fig. 5, a single exponential is not adequate to describe the density variation at z 500 pc, whatever the SFR history or age- relation. The most dramatic difference is for the decreasing SFR. For example, with the age- relation , the exponential scale height varies from 300 and 230 pc between z=100 and 500 pc. It should be noted also that there are very few profiles which, at distances greater than z=300 pc, can be fitted with a scale height of 300 pc or higher. This is the case only for profiles corresponding to age- relation and constant or decreasing SFR. For a constant SFR, and age- relation , which is the model which presents the closest characteristics with Bahcall (1984), the scale height at z 1000 pc is greater than 250 pc, whereas Bahcall found 203 pc. The main difference responsible for this discrepancy is, of course, the huge amount of dark matter in Bahcall's model, equal, in volume density, to the visible matter. The increasing SFR provides profiles which are much closer to exponentials at z 500 pc. This is true also for the two other SFR and age- relation . However, the scale height here is of about 150 pc, very much lower than the observed 250-350 pc.
Now, does the use of exponential profiles explains why star-count
models adopted scale height of 325 pc, while our model has
smaller disc thickness ? Anticipating our discussion on star-count
fitting with Galaxy models of Paper II, we want to make the
following comment. Let us suppose that star-count data at the galactic
pole are adequately described using the profile
on Fig. 4b (constant SFR). Had we used instead an exponential law
with scale height 300 pc, normalized to the local LF (as shown in
Fig. 4), the net effect would be that we would approximately
match the number of disc stars at z
400-500 pc, (the shows that the best fit
to profile (constant SFR) for z
300 pc is found for
300 pc), while the exponential is an overestimation beyond z
400-500 pc. Looking at galactic pole
predictions of our star-count model (Paper II), most disc stars
brighter than V=14 lay below 400 pc while most stars fainter than
V=15 lay beyond 400 pc, but not all of them are disc stars, since
thick disc rapidly increases. Consequently, using an exponential law
with =300-350 pc, disc stars may be
(roughly) correctly estimated at V 15 and z
500, while they could be severely overestimated
at V 15. This is of great consequence for star
counts at faint magnitudes, in particular because, if disc stars are
overestimated, we expect that thick disc stars at the same magnitudes
will be © European Southern Observatory (ESO) 1997 Online publication: June 30, 1998 |