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Astron. Astrophys. 348, 98-112 (1999)

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2. Analysis of thick-disk parameters

The purpose of this section is twofold. First, we shall demonstrate that the thick-disk component in our Galactic models is indispensable for explaining the observed star count and color data. And secondly, we shall show that our preliminary knowledge of the thick-disk parameters (as essentially derived in Paper I) can still be made more reliable by allowing for wider ranges of thick-disk parameters and by improving on basic model input data and photometric calibration.

2.1. Evidence for the thick-disk component

Evidence for the presence of a substantial thick-disk component in the new Basel survey data is most prominently illustrated in Figs. 1 and 2. They show how the lowest-[FORMULA] models, calculated with or without a thick-disk component, differ dramatically in their performances of matching the star counts and stellar distributions as functions of apparent G magnitude and G-R color observed in the two fields Praesepe and M101, respectively. These differences correspond to factors larger than 1.5 and up to 2.5 in [FORMULA]-values, clearly indicating that, if acceptable models without a thick disk could be found at all, they would have to have structural parameter values for the thin disk and halo ranging far beyond the adopted canonical limits. This conclusion is corroborated by the results given in Table 1, which summarizes a sample of representative calculations of lowest-[FORMULA] models for three different assumed values of the local thick-disk density, [FORMULA], including the optimized value from Paper I (model A) and a model with zero thick disk (model B). For models with no thick disk at all (i.e., [FORMULA]) or with only a weak thick disk (i.e., [FORMULA]), both the [FORMULA]-values and their minima, [FORMULA], become excessively large; in fact, as given by [FORMULA]) in the last column of Table 1, in all cases there are several hundred up to a few thousand(!) models A (i.e., including a strong thick disk) which have [FORMULA]-values less than [FORMULA] obtained for models B or C (i.e., including a zero thick disk or only a weak thick disk.)

[FIGURE] Fig. 1. Star counts for the Praesepe field. The calculated curves are for the field-specific optimized model with (1) [FORMULA] and for lowest-[FORMULA] models having (2) [FORMULA], (3) [FORMULA].

[FIGURE] Fig. 2. [FORMULA] color distribution for the field M101. The calculated curves are for the field-specific optimized model with (1) [FORMULA] and for the lowest-[FORMULA] model having (2) [FORMULA].


Table 1. Compensation of zero or weak thick disk by thin-disk and halo stars1
1) Thick-disk local densities, [FORMULA], are assumed to be 0.054, 0.000, and 0.010 of the local thin-disk density in models A, B, and C, respectively.
Model A is the globally best model from Paper I.

Table 1 also shows that for the best (i.e., [FORMULA]) models B and C, the scale height of the old thin-disk dwarfs, [FORMULA], and the local density of the halo, [FORMULA], both increase significantly above their optimized values found for model A in Paper I, to compensate for the missing thick-disk stars.

Thus, models with only a weak or even a zero thick disk are plainly ruled out by the present results. The evidence provided here (and in Paper I) leaves no doubt that a substantial thick-disk component exists in the Milky Way Galaxy with a local density (relative to the thin disk) [FORMULA].

2.2. Thick-disk parameters and constraints

We now examine in greater detail each of the three thick-disk parameters included in our models. For this purpose, we use the same procedure as applied in Paper I for determining their optimized values by calculation of the [FORMULA]-curves and the frequency distributions of parameter values for low-[FORMULA] models. Note that the [FORMULA]-curves have been calculated to show how [FORMULA] changes upon variation of a particular parameter throughout its adopted range, while all other parameters (i.e., including the two remaining thick-disk parameters) are fixed at their values found for the globally best-fitting model identified by [FORMULA] in Paper I.

2.2.1. Local density

Figs. 3 and 4 are representative illustrations - i.e., calculated for the combined survey of seven fields - of the general properties and behavior of the [FORMULA]-curves and the frequency distributions for the local density of the thick disk. Good models, i.e., with [FORMULA], all have [FORMULA]! Even for the adopted quality limit, [FORMULA], only [FORMULA] and [FORMULA], respectively, of the models are found to have [FORMULA], for just two (M67 and SA 141) out of all seven fields. (Incidentally, these two fields are those which provide the lowest counts ([FORMULA]) by far of all seven survey fields analysed here.) However, with [FORMULA] increasing from 1.1 to 1.5, the frequency distributions tend to become wider-ranged, with their peaks shifting from [FORMULA] to [FORMULA], indicating that the constraint on this parameter derived in Paper I ([FORMULA]) may in fact be weaker. This will indeed be confirmed by the extended analysis below.

[FIGURE] Fig. 3. [FORMULA]-curve for the local density, [FORMULA], of the thick disk, derived from the combined survey of seven fields and for the parameter range adopted in Paper I.

[FIGURE] Fig. 4. Frequency distributions for the local density, [FORMULA], of the thick disk, derived from the combined survey of seven fields and for the parameter range adopted in Paper I. The sequence of curves from bottom to top corresponds to values of [FORMULA] growing from 1.1 to 1.5; labels indicate the number of models involved.

Thus, from the frequency distributions of Fig. 4, we cannot either exclude that acceptable models may even exist with [FORMULA], at least for a few of the Galactic directions investigated here. Evidence of a thick-disk with such higher local densities in the range [FORMULA] has also previously been found from field-RHB stars (Rose 1985) and, in particular, from radial velocities (Sandage 1987), from proper motions (Casertano et al. 1990), and from combined photometric and proper motion data (Ojha et al. 1994a,b). For a more conclusive comparison with these recent independent results, we shall, therefore, have to extend the parameter range in our model calculations, which will also allow us - in Sect. 3 below - to obtain a safer upper limit or at least a more reliable constraint on the optimized mean value [FORMULA]. Before taking this step, however, we shall now determine if similar extensions of the variation ranges are required for the scale length and scale height parameters of the thick disk by examining more closely the general properties and behavior of their [FORMULA]-curves and frequency distributions.

2.2.2. Scale length

Adopting a local density [FORMULA], found for the thick disk from the parameter optimization calculations in Paper I, the (representative) [FORMULA]-curve and frequency distributions for the scale length ([FORMULA]) parameter as derived from the combined survey of seven fields are shown in Figs. 5 and 6.

[FIGURE] Fig. 5. [FORMULA]-curve for the scale length, [FORMULA], of the thick disk, derived from the combined survey of seven fields and for the parameter range adopted in Paper I.

[FIGURE] Fig. 6. Frequency distributions for the scale length, [FORMULA], of the thick disk, derived from the combined survey of seven fields and for the parameter range adopted in Paper I. The sequence of curves from bottom to top corresponds to values of [FORMULA] growing from 1.1 to 1.5.

Due to the generally high Galactic latitudes of the survey fields, the present data are naturally expected to be rather insensitive to the scale length parameter ([FORMULA]). This is borne out by the relatively low amplitude of the [FORMULA]-curve, [FORMULA], within most of the variation range in Fig. 5, and by the very flat frequency distributions for the good models ([FORMULA]) in Fig. 6. These results for the combined survey are very similar to those obtained for each individual field. In Paper I, a correspondingly weak constraint, [FORMULA] kpc, on the optimized value [FORMULA] kpc was derived for this parameter. The accuracy of this value is rather low, however, because the position of [FORMULA] in Fig. 5 and the resulting [FORMULA] turned out to be close to the lower limit of the variation range (3.5 kpc), and uniform extrapolations of the [FORMULA]- and the frequency distribution curves beyond the above limit at [FORMULA] kpc had to be assumed in estimating the constraint. Obviously, further calculations are required now to confirm (or reject) this assumption and to derive more reliable optimized values for this parameter and its constraints.

2.2.3. Scale height

On the other hand, the [FORMULA]-curves and, particularly, the frequency distributions for the thick-disk scale height ([FORMULA]) are not similarly uniform among the individual fields, as they are for the scale length. Although all the individual values of [FORMULA] occur at scale heights [FORMULA] kpc, the actual [FORMULA]-curves depend sensitively on the Galactic direction, the count completeness magnitude limit, and the correspondingly different star counts in each field, leading to very different [FORMULA]-growth rates as functions of [FORMULA]. Thus, for a slow [FORMULA]-growth rate, acceptably good models satisfying [FORMULA] may still be found far from the parameter value characterizing the actual [FORMULA], and even at (or beyond) the lower boundary (0.9 kpc) of the adopted variation range of [FORMULA].

This case is illustrated in Fig. 7, where we show the frequency distribution for the scale height of the thick disk, [FORMULA], derived from the data in the field near M67 (i.e., in the outer Galaxy at [FORMULA]). Obviously, the most frequent value occurring among the best models (dotted and short-dashed curves in lower part of figure) is [FORMULA] kpc, and a clear trend to the same also persists with the lower-quality models ([FORMULA]). Thus, even though the weighted mean, or optimized, value [FORMULA] kpc determined for this individual field in Paper I is still well within [FORMULA] kpc of the optimized parameter value, [FORMULA] kpc, derived from the all-field survey, it is also evident from the present data that the best results for an individual field like M67 should very likely be obtained from models having a thick-disk scale height [FORMULA] kpc.

[FIGURE] Fig. 7. Frequency distributions for the scale height, [FORMULA], of the thick disk, derived for the field M67 and for the parameter range adopted in Paper I. The sequence of curves from bottom to top corresponds to values of [FORMULA] growing from 1.1 to 1.5.

This conclusion also holds for at least three other fields of the present survey (M101, Praesepe, and SA141), whose frequency distributions exhibit essentially the same behavior as the M67 field discussed above, providing strong indications that the thick-disk scale height is lower than suggested by the preliminary results derived in Paper I. On the other hand, the frequency distributions for the fields SA54 and M5 are similar to that shown in Fig. 8 for the polar field SA57, with best-model curves ([FORMULA]) having flat but well defined peaks at [FORMULA] and which become significantly more strongly pronounced at [FORMULA].

[FIGURE] Fig. 8. Frequency distributions for the scale height, [FORMULA], of the thick disk, derived for the field SA57 and for the parameter range adopted in Paper I. The sequence of curves from bottom to top corresponds to values of [FORMULA] growing from 1.1 to 1.5.

To summarize, Figs. 5-8 tell us that, in order to derive more reliable optimized parameter values and constraints for both individual fields and the all-survey data, we have to extend our analysis beyond the variation limits adopted originally for the thick-disk parameter ranges. Hence, new model calculations will be carried out according to Table 2.


Table 2. Extended variation ranges of thick-disk density law parameters.
1) [FORMULA] stars/[FORMULA] is the optimized value found in Paper I for the local density of the old thin disk, while the new optimized value derived below is [FORMULA] stars/[FORMULA] (see the text).

2.3. Luminosity functions and photometric transformations

Of course, the structural parameter values derived from the new Basel data critically depend on the reliability of basic model input and calibration data. In the first place, these are given by the luminosity functions (LFs) adopted for the different population components and their subsequent transformation from the original Johnson-standard UBV to the actual survey-standard RGU systems. In this subsection, we discuss improvements over the preliminary results of Paper I that we should expect from new luminosity function and color transformation data.

2.3.1. Luminosity functions

It has been well known for quite some time that LF features, such as the so-called "globular cluster feature" (Bahcall et al. 1985), the "Wielen dip" (Bahcall & Soneira 1983), or the changes in LF shape implied by limited resolution of binary stars and multiple systems (Buser & Kaeser 1985), have important effects on predicted star count and color field-survey data in different ranges of apparent magnitude. Thus, the initial strategy followed in Paper I has been to use component-specific LFs which account for perhaps the crudest of real differences that we should expect to exist between the LFs pertaining to the different components.

Accordingly, all calculations have so far been performed assuming the component-specific LFs discussed in Paper I. Three properties of these functions should be recalled here: (1) the thin-disk LF has been based on the Gliese (1969) catalog of nearby stars and was derived by Buser & Kaeser (1985), anticipating that a sizeable fraction of the individual members of binary and multiple systems in the source catalog would remain unresolved if seen from the larger distances sampled in the present RGU data; (2) the thick-disk and halo LFs have been assumed to have the same shapes (though not the same local normalizations, of course), given by Da Costa's (1982) LF derived for the intermediately metal-poor ([M/H]=-0.85) globular cluster 47 Tuc, supplemented by Wielen et al.'s (1983) thin-disk LF for the fainter stars; (3) for each of these LFs, the local normalization has been left as a free parameter whose (optimized) value is to be determined from the actual analysis of the observed RGU data.

The results of our preliminary analysis of Paper I have now led us to subject these original precepts to revision, as follows.

1. HIPPARCOS data.  The effect of low spatial resolution on the (apparent) local LF is less significant than anticipated, essentially because the relevant resolution limit is set by stars seen at distances [FORMULA] kpc, which is true for a small fraction of thin-disk stars only in most fields of the new Basel high-latitude survey. Thus, for the resolution-corrected LF originally calculated from the Gliese (1969) catalog by Buser & Kaeser (1985), we now substitute the thin-disk LF derived by Jahreiss & Wielen (1997) from the superior HIPPARCOS data for the local stellar individuals. As can be seen in Figs. 10 and 11 below, this new LF is not quite as smooth as the original Buser & Kaeser LF but exhibits several rather strong features. Note however that the most pronounced of these, at [FORMULA], is irrelevant to the present analysis, since such stars are much too faint to be sampled in any significant number in the new Basel survey.

2. Metallicity dependence.  McClure et al. (1986) were among the first to provide observational evidence for a possible dependence of the luminosity functions on stellar metallicity in Galactic globular clusters. Although, or even just because, the reality of such a metallicity dependence has been extremely difficult to establish and is, in fact, still a matter of debate, these cluster-LFs given by McClure et al. may at least provide a more appropriate representation of possible LF variations which are to be expected among the different observed field-star population components in the Galaxy. In particular, they can be used for a more continuous and consistent interpolation between the LFs of the local thin disk and the extreme halo than is possible from the approach employed thus far: while the 47 Tuc LF used in Paper I for the thick disk and the halo may just suffice to account for the major differences between two discrete populations separated by order of [FORMULA] dex or more in mean metallicities, it is almost certainly inadequate for matching at least three real populations that also have large and even mutually overlapping metallicity dispersions which altogether cover (essentially) the full observed metallicity range of more than 2 dex. If there should indeed be a metallicity dependence of the stellar LF of the kind suggested by the globular cluster study of McClure et al., the improved power to measure photometric metallicity effects by well-calibrated broad-band systems (such as the Basel RGU or the Washington [FORMULA] systems) provides hope for answering the important question of whether this metallicity dependence is also present in the LFs of the field-star populations. If successful, the result will (more or less) directly translate into the corresponding (now suspected) metallicity dependence of the underlying IMF (McClure et al. 1986) and/or mass-luminosity relation (von Hippel et al. 1996), and will eventually allow us to provide a more physical interpretation of the present empirical structural models in terms of luminosity functions derived from theoretical calculations of the evolution of star formation and chemical enrichment of the large-scale Galactic field-star populations.

Thus, in the present paper, we expand the LF options in the following way: for the thick-disk and halo main sequence stars, i.e, with absolute magnitudes [FORMULA], we substitute the LFs given by McClure et al. (1986) for globular clusters of different metallicities, shown in normalized form in Fig. 9. These LFs are interpolated to the metallicities [M/H] = (-0.625, -1.75) initially adopted for the thick-disk and the halo components. Fig. 10 illustrates the resulting new set of LFs adopted in the present Paper for all three main components, while Fig. 11 gives these same LFs as transformed (from Fig. 10) to RGU, and as actually used in calculating the predicted star count and color distributions of the present survey.

[FIGURE] Fig. 9. Normalized luminosity functions for metal-poor populations. They were taken from the globular-cluster luminosity functions published by McClure et al (1986) for [FORMULA] primarily to provide a possibly more adequate representation of the thick-disk and halo dwarf stars in the present survey. The extensions to the brighter and fainter absolute magnitudes are the high-metallicity luminosity functions taken from Da Costa (1982) and Wielen et al. (1983), and scaled to the lower metallicities.

[FIGURE] Fig. 10. Luminosity functions for the Galactic components investigated in the present analysis. The thin-disk LF provided by the HIPPARCOS data is assumed to represent solar metallicity ([M/H]=0.0) and has been taken from Jahreiss & Wielen (1997); the thick-disk and halo LFs have been interpolated from the original data of Fig. 9 to metallicities [FORMULA] initially adopted for these components.

[FIGURE] Fig. 11. Same as Fig. 10, but transformed from the Johnson-V to the Basel-G system of absolute magnitudes (see text).

2.3.2. Color-magnitude transformations for red stars

As anticipated in Paper I (Figs. 12-15) and evidenced here again in Fig. 2, almost all the model calculations still fail to reproduce reliably the red tails of the observed color distributions. One of the likely reasons is lying with the limited range of application of the photometric transformations used between the calibration standard-UBV and the survey standard-RGU systems.

In the exploratory calculations of Paper I, as the simplest approximation for converting the luminosity functions from extant UBV data to the RGU system, a single set of transformation equations were used for the full ranges of stellar types O through M. However, due to their strong atomic and molecular absorption features, the reddest dwarfs and giants, of spectral type M at [FORMULA], do not share the same transformation properties with the bluer stars, which have smoother spectral flux distributions. In fact, O-K-star transformations will lead to derived RGU colors and magnitudes for M-stars with systematic errors of up to 0.4 magnitudes bluer and brighter than observed (Buser 1988). It is obvious, then, that the calculations should be substantially improved by introducing separate transformation equations for these coolest objects in the new Basel survey catalog.

Thus, in the present new model calculations we have implemented Table 3, which gives the full matrix of coefficients applying to the transformation equations


where C is one of the color indices (G-V), (G-R), or (U-G). This matrix was derived by Buser (1988) from observed spectrophotometric flux distributions of normal Population I stars, and explicitly accounts for the spectral differences and their implied differences in transformation properties existing between O-K- and M-type stars and also between dwarfs and giants of either of these two varieties. For stars having colors in the interval [FORMULA], simple averages obtained from the relevant "blue" and "red" equations provide a smooth transition between the different transformations.


Table 3. Transformation coefficients for [FORMULA]

In general, these equations allow us to derive transformed colors and magnitudes with a systematic accuracy on the level of a few hundredths of magnitude. Note that the above equations have been used in calculating the one- and two-dimensional star count distributions [FORMULA], [FORMULA], and [FORMULA] for all the stellar population components, irrespective of their different metallicities adopted in the model. Although not strictly correct, this approximation can be (temporarily) justified by the fact (Buser 1988, Güngör Ak 1995) that metallicity should be expected to have a significant effect on the transformations to the ultraviolet color index U-G only, but to be of considerably lower importance for G and G-R. However, for a fully consistent analysis, a complete set of transformation equations as functions of stellar metallicity [M/H] is being worked out (Buser et al. 1998b) from synthetic photometry of a new comprehensive library of theoretical stellar spectra (Lejeune et al. 1997, 1998), and will be applied to the all-survey RGU data in later papers of this series.

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© European Southern Observatory (ESO) 1999

Online publication: July 16, 1999