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Astron. Astrophys. 320, 428-439 (1997)

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3. The density profile in the z direction

At 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- [FORMULA] relation and the SFR history. According to the observed age- [FORMULA] 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- [FORMULA] 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- [FORMULA] relation

The age- [FORMULA] relation is known to vary very rapidly for the youngest stars: [FORMULA] rises from approximately [FORMULA] to 10-13  [FORMULA] 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- [FORMULA] relation and the true level at which this relation saturates. In the model of the vertical distribution of stars, uncertainties in the age- [FORMULA] 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- [FORMULA] relations from Wielen (1974), Mayor (1974), Meusinger et al. (1991), and Strömgren (1987).

[FIGURE] Fig. 2. The 3 age- [FORMULA] relations tested in this study (continuous lines), and data from Mayor (1974) (circles), Wielen (1977) (squares), Strömgren (1987) (open and filled triangles), Meusinger et al. (1991) (diamonds).

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 [FORMULA] +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 [FORMULA] which saturates at 15.3 [FORMULA] km.s-1, but goes to [FORMULA] 21 km.s-1 for stars which may be classified as old disc stars according to their metallicity ([FORMULA]). The Meusinger et al. (1991) relation relies on a sub-sample of the Twarog (1980) data, representing 161 stars.

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- [FORMULA] relation shows a steep increase in the first three Gyrs, and then flattens at [FORMULA] 20 km.s-1, also suggesting that Mayor & Wielen values could be overestimations.

In order to take into account the wide range of possible age- [FORMULA] 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 ([FORMULA]) and ([FORMULA]) bracket the observed measurements of [FORMULA], while [FORMULA] is an intermediate relation.


Table 1. age- [FORMULA] relations used in this work. Designations [FORMULA], [FORMULA] and [FORMULA] come from the larger set of relations given in Table 2 of Paper II.

3.2. Theoretical expectations

The 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- [FORMULA] relation [FORMULA] 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 sech2 law: decreasing SFR (lower thick curve) with [FORMULA] =5 Gyrs gives rise to a density gradient which is not an exponential. With [FORMULA] =5 Gyrs and an age for the disc of 10 Gyrs, the variation of SFR history is about 7. This is within the uncertainties of the star formation history (Sect. 4). On the contrary, the density profile obtained with the increasing SFR (upper thick curve) (exp(t/5)) is very close to an exponential, even when very near the galactic plane.

[FIGURE] Fig. 3. Density profiles in the z direction for different SFR history in the galactic disc, normalized to have the same number of stars at large distance z. The thin continuous curves are exponential with a scale height 180 pc and sech2 profiles. All other profiles have been calculated assuming age- [FORMULA] relation [FORMULA], with exponential SFR of the form exp(-t/ [FORMULA]) and [FORMULA] =-5, + [FORMULA], 5, from top to bottom.

Fig. 4 shows how the density gradient changes with different SFR history and age- [FORMULA] relations. Each plot shows the profile obtained for a given SFR, and for the three age- [FORMULA] relations of Table 1. All these profiles are plausible according to present accuracies in the age- [FORMULA] relation and SFR history. The difference in the densities at a given distance due to uncertainties in the age- [FORMULA] 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 [FORMULA] and [FORMULA] and 170% between profile [FORMULA] and [FORMULA] at the same distance.

[FIGURE] Fig. 4. Density profiles obtained for different SFR and age- [FORMULA] relations. From top to bottom, SFR are respectively increasing, constant and decreasing (defined as in Fig. 3). Each plot shows the profiles for the three age- [FORMULA] relations [FORMULA], [FORMULA], [FORMULA] (respectively bottom, middle and top curves in each plot), and are normalized to 1 at z=0. Dotted lines are exponentials with scale height 200 and 300 pc.

3.3. Questioning the exponential approximation

The 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 review

In 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 [FORMULA] 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 [FORMULA] z [FORMULA] 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 [FORMULA] =4, 10, 13, 15, 20, 24 km.s-1) are very similar to those we find in our model for a constant SFR (for this model, Fig. 4 shows that between 300 and 900 pc, the exponential is a satisfactory approximation). However, there is a significant contradiction between this result of Bahcall (1984) and the star-count model of Bahcall & Soneira (1980) : the disc scale height deduced from his dynamical analysis is around 200 pc, whereas the scale height used in the (star-count) model is 325 pc. We note that this contradiction has not been addressed in subsequent dynamical or star-count analysis. After Bahcall & Soneira (1980), most models have used the value of 325 pc, and have found satisfactory agreement with star count data. It is only recently that an obvious discrepancy has been found between this standard model and observed star-counts (Reid & Majewski, 1993), but without seriously questioning the exponential approximation. In passing, we note that the red dwarfs detected by Majewski (1992) are at typical distances of 1000-2500 pc, whereas Bahcall (1984) specifies that the agreement between exponential and his dynamical model is found to be statisfactory only in the distance range 300 to 900 pc. Outside these limits, the discrepancies are of less than 25% between 900 and 1000 pc and at z [FORMULA] 300 pc, but unspecified for distances beyond 1000 pc. With the availability of more accurate and extreme data, such as Majewski's (1992), this approximation needs to be reconsidered.

3.3.2. Comparison with our model

The 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 [FORMULA] 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 [FORMULA]  500 pc, whatever the SFR history or age- [FORMULA] relation. The most dramatic difference is for the decreasing SFR. For example, with the age- [FORMULA] relation [FORMULA], 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- [FORMULA] relation [FORMULA] and constant or decreasing SFR. For a constant SFR, and age- [FORMULA] relation [FORMULA], which is the model which presents the closest characteristics with Bahcall (1984), the scale height at z [FORMULA] 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 [FORMULA] 500 pc. This is true also for the two other SFR and age- [FORMULA] relation [FORMULA]. However, the scale height here is of about 150 pc, very much lower than the observed 250-350 pc.

[FIGURE] Fig. 5. Scale height (y-axis) of the exponential function best fitting each of the density laws of Fig. 4, between galactic plane and z=2000 pc. Plot a is for increasing, b constant and c decreasing SFR. The value of the best scale height can be read on the y-axis.

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 [FORMULA] 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 [FORMULA] 400-500 pc, (the [FORMULA] shows that the best fit to profile [FORMULA] (constant SFR) for z [FORMULA] 300 pc is found for [FORMULA] 300 pc), while the exponential is an overestimation beyond z [FORMULA] 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 [FORMULA] =300-350 pc, disc stars may be (roughly) correctly estimated at V [FORMULA] 15 and z [FORMULA] 500, while they could be severely overestimated at V [FORMULA] 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 underestimated. This point is discussed in further detail in Paper II. Eventually, KTG (1993) found their data to be compatible with a constant number density at z [FORMULA] 80 pc. Such a feature is clearly present in our simulation (see profile [FORMULA] in Fig. 4b).

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998