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

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3. New model calculations and results

As in our previous studies (e.g., Fig. 5 of Paper I), the two-dimensional [FORMULA] data were found to be fairly insensitive to the local density parameter of the thin disk, [FORMULA]. In fact, models computed based on the HIPPARCOS LF with [FORMULA] in the range 0.07-0.12 stars[FORMULA] provide [FORMULA]-fits to our data which are only marginally different, and thus a value [FORMULA] stars[FORMULA] was adopted for all subsequent model calculations, in agreement with the value derived from the HIPPARCOS data for the very local sphere by Jahreiss & Wielen (1997).

Based on the improved input discussed in the preceding section, 28,800 new models have then been computed as follows: optimized mean values for the (remaining) five secondary model parameters were taken from Table 3 in Paper I, while the six primary parameters (to be discussed below), including three for the thick-disk component, were allowed to vary within their adopted original or new extended ranges given in Tables 2 of Paper I and the present paper, respectively. From these new models, star count and color distributions were finally calculated and compared to the observed data for each of the seven fields plus their combined survey.

In order to assess the extent of the improvement and to derive optimized parameter values, we here employ again the analysis tools and procedures developed in Paper I, where [FORMULA] and [FORMULA] are used as the basic estimators of goodness of fit of the model predictions to the data for each of the seven individual fields and for the combined survey in all seven fields, respectively. In particular, the individually or globally best-fitting models are identified by [FORMULA] and [FORMULA], respectively, and serve as pivots for the subsequent selection of models used in calculating optimized parameter values.

3.1. General improvements

The main general result is that [FORMULA] has dropped now to 853, which is almost 25% below the corresponding value obtained for the globally best model in Paper I. (Incidentally, this means that the globally best model of Paper I now does not even make it into the [FORMULA]-selection of good models from which the optimized parameters will be determined below!) While this result stems from the combination of the different improvements to the model input described above, the individual contributions can be roughly traced as follows.

Perhaps the most important effect is traceable to the new luminosity functions (LFs). All the lowest-[FORMULA] models include the LFs shown in Figs. 9-11, which have distinctly different shapes on account of their (adopted) underlying metallicity differences. Calculations for a thick-disk LF whose shape and metallicity are indistinguishable from either the thin-disk or the halo LF invariably result in [FORMULA] scores which exceed those obtained from the three distinct LFs in Fig. 11 by factors larger than 1.5, up to 2.5. This appears most prominently in its effect upon the halo normalization, [FORMULA], and is illustrated in the [FORMULA]-curves of Fig. 12: while the [FORMULA] for all the models involving two different LFs only [i.e., curves (2) and (3)] are in no way competitive with the lowest-[FORMULA] obtained for the models based on three different LFs [i.e., curve (1)], even the best 2LF-models, at the right edge of Fig. 12, tend to accommodate too many stars with the halo, leading to too large a local normalization for that component. In fact, this LF-effect is very similar to the effect produced by the weakening or even annihilation of the thick disk demonstrated in Table 1 and discussed in Sect. 2.1.

[FIGURE] Fig. 12. [FORMULA]-curves for models assuming thick-disk LF to be indistinguishable in shape and metallicity either from the thin-disk LF (2) or from the halo LF (3), compared to result obtained assuming a distinct LF for each component (1). Models (2) and (3) clearly compensate for the implied loss of discrimination by accomodating the metal-poor thick-disk stars preferentially in the halo, leading to an excess local density of halo stars. Note that the best of these latter models have higher [FORMULA]-values by a factor of [FORMULA] over the best models of curve (1)!

We thus conclude that the data of the new Basel survey provide strong evidence for each of the three main Galactic population components 1 having its own (local) LF characterized by a distinctly specific shape and metallicity.

The more detailed transformation equations of Table 3 have basically two effects on the model predictions, since they imply changes of both the absolute magnitudes and the colors of the stars. First, as shown in Fig. 13, the M-star transformations predict fainter absolute G magnitudes than are obtained from the O-K-star transformations, leading to redistribution of apparent magnitudes and, consequently, also of the star counts, [FORMULA]. If applied to a (rising) LF - as, e.g., in the interval [FORMULA] in Fig. 10 - this may eventually also lead to a decrease in predicted numbers of red main sequence stars, because a larger fraction of these may then be pushed beyond the observed apparent magnitude limit of the survey data than are replenished from the brighter magnitude bins.

[FIGURE] Fig. 13. Differences between LFs transformed to RGU from UBV [Fig. 10] with [[FORMULA]] or without [[FORMULA]] taking into account the separate equations applying to the red M-stars in Table 3. Thus, areas above the zero-line measure the number of stars which, upon including the red-star transformations, are subtracted from the corresponding absolute magnitude interval in the original LF used in Paper I [i.e., [FORMULA]], and which are shifted to fainter absolute magnitudes where the LF is accordingly increased [areas below the zero-line].

Second, the M-star transformations also predict redder colors than would be obtained from the O-K-star transformations. Fig. 14 illustrates this effect on the luminosity function, and Fig. 15 gives a typical example (SA 57) of how the effect eventually propagates into the field-star G-R color distribution, whose final shape depends on the relative model-predicted star counts contributed by the individual components, as shown in Fig. 16.

[FIGURE] Fig. 14. Same as Fig. 13, but with LF differences now expressed as functions of G-R color. Note that the nonuniform shifts to redder colors for the different components (insert) may lead to either depletion, compensation, or amplification of resulting total star counts in different color intervals (Fig. 15), depending on the relative count contributions of the thin- and thick-disk components, such as those predicted in the model shown in Fig. 16.

[FIGURE] Fig. 15. Improved match of observed and predicted red wings of G-R color distribution in SA57. Both predicted curves have been computed for the same structural model of the Galaxy; the only difference between the two is that the dashed curve (2) includes the full transformation matrix of Table 3, while the dotted curve (1) does not include the M-star transformations. See the text.

[FIGURE] Fig. 16. Star counts N(G) in the SA57 field, comparing the observed histogram with the predicted total built up by the contributions from the individual components. The thin disk dominates down to [FORMULA] but is overtaken by the thick disk and the halo at fainter magnitudes.

Note that the two predicted color curves in Fig. 15 result from calculations which are identical except for the two different transformation models employed in converting the LFs of Fig. 10 from UBV to the RGU system. The more adequate transformations lead to: (1) significant redistribution of the stars from colors [FORMULA] to the redder interval [FORMULA], (2) a closer match of the predicted color distribution with the observed histogram, and (3) a reduction of [FORMULA] by [FORMULA], as calculated from the fits to the combined two-dimensional [FORMULA] data.

Since in this field the reddest stars predominantly belong to the thick-disk component - which is shown in Fig. 16 to dominate the stellar census at [FORMULA] -, we expect to improve the fit still further in a later paper of this series, when even more appropriate transformations will be available for these lower-abundance stars.

3.2. Optimized parameter values and constraints

Based on the total of 28,800 new models, the improved model fits to the star count and color data described above are now used to analyse their impact on the six primary structural parameters, [FORMULA] (cf. Table 4). For each of these parameters, results for the combined survey of seven fields are given in terms of the [FORMULA]-curve and its associated frequency distributions, [FORMULA], as follows. For each specific value adopted by a particular parameter, its actual minimum value of [FORMULA], [FORMULA], is calculated from all the new models, whose 5 remaining free primary parameters (cf. Table 4) are allowed to vary throughout their original adopted or new extended ranges. Thus, the [FORMULA]-curves trace out the best-fitting model existing at each value of the particular parameter, [FORMULA].


[TABLE]

Table 4. Optimized parameter values [FORMULA] of Galactic model


Obviously, the globally best model defined above is then identified by [FORMULA]. Subsequently, models are selected according to the condition [FORMULA], where [FORMULA], and the frequency distributions of parameter values, [FORMULA], are evaluated as functions of [FORMULA]. Hence, optimized parameter values, [FORMULA], and constraints, [FORMULA], are finally determined from the good models ([FORMULA] and [FORMULA]), using the parameter weighting scheme described in Paper I. Models satisfying this selection were shown in Paper I to be statistically consistent with the estimated external accuracy of the data on the two-sigma level.

3.2.1. Thin-disk and halo parameters

Before discussing the thick-disk parameters, we first check on the impact of the new LF input and photometric calibration (Sect. 2.3) on the thin-disk and halo primary parameters. The new results are reported in the third to fifth columns of Table 4, where the two bottom lines also give the corresponding preliminary results obtained in Paper I for the combined survey in seven fields.

Table 4 shows that the primary parameters of the thin disk, i.e., the optimized values [FORMULA] and [FORMULA] for the scale heights of the old and young dwarfs, respectively, are found to be lower than, but still within one sigma of their corresponding values derived in Paper I. While the mean scale height of the old thin-disk dwarfs, [FORMULA], thus remains essentially unchanged, the somewhat larger dispersion seems to indicate that real deviations from the adopted smooth density model may exist in the data in the different field directions. This conclusion will be supported below by similar results obtained for two parameters of the thick-disk.

No significant changes have been found for the local density [FORMULA] of the halo either. The new models provide an optimized mean value and (formal) constraints for this parameter which are almost the same as derived in Paper I. Thus, the main conclusion of this subsection is that the optimized parameter values of the thin-disk and halo components of our model are essentially robust against the changes in model input and photometric calibration implemented in the present investigation.

3.2.2. Thick-disk parameters

The optimized values and constraints derived for the thick-disk parameters are also summarized in Table 4 for both the individual fields and the combined survey of seven fields. Two principal steps toward these results are illustrated in Figs. 17-22 for each of these parameters, which we shall now briefly discuss.

[FIGURE] Fig. 17. [FORMULA]-curve for [FORMULA], the local density of the thick disk, derived from the combined survey of seven fields and for the new parameter range adopted in this paper. Horizontal lines indicate selection limits of good models having [FORMULA], where [FORMULA], i.e., [FORMULA], and [FORMULA], i.e., [FORMULA], respectively. To illustrate the difference with the [FORMULA]-curve, the [FORMULA]-curve for the the global minimum has also been plotted. Compare with Fig. 3.

[FIGURE] Fig. 18. Frequency distributions [FORMULA] for [FORMULA], the local density of the thick disk, derived from the combined survey of seven fields and for the new parameter range adopted in this paper. 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. Compare with Fig. 4.

[FIGURE] Fig. 19. [FORMULA]-curve for [FORMULA], the scale length of the thick disk, derived from the combined survey of seven fields and for the new parameter range adopted in this paper. Horizontal lines indicate selection limits of good models having [FORMULA], where [FORMULA], i.e., [FORMULA] and [FORMULA], i.e., [FORMULA], respectively. To illustrate the difference with the [FORMULA]-curve, the [FORMULA]-curve for the the global minimum has also been plotted. Compare with Fig. 5.

[FIGURE] Fig. 20. Frequency distributions [FORMULA] for [FORMULA], the scale length of the thick disk, derived from the combined survey of seven fields and for the new parameter range adopted in this paper. 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. Compare with Fig. 6.

[FIGURE] Fig. 21. [FORMULA]-curve for [FORMULA], the scale height of the thick disk, derived from the combined survey of seven fields and for the new parameter range adopted in this paper. Horizontal lines indicate selection limits of good models having [FORMULA], where [FORMULA], i.e., [FORMULA] and [FORMULA], i.e., [FORMULA], respectively. To illustrate the difference with the [FORMULA]-curve, the [FORMULA]-curve for the the global minimum has also been plotted.

[FIGURE] Fig. 22. Frequency distributions [FORMULA] for [FORMULA], the scale height of the thick disk, derived from the combined survey of seven fields and for the new parameter range adopted in this paper. 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. Compare with Figs. 7 and 8.

The local density parameter [FORMULA] is explored in Figs. 17 and 18, which should be compared with its [FORMULA]-curve and frequency distributions previously derived in Paper I and given in Figs. 3 and 4. The most significant new result is that the best models have almost uniform [FORMULA] for parameter values [FORMULA], and good models can thus be found throughout this same range, which is significantly more extended than in the preliminary analysis reported in Figs. 3 and 4.  2 Still, the maximum at [FORMULA] of the (new) frequency distribution for the best models ([FORMULA]) in Fig. 18 corroborates the original result derived in Paper I for the optimized mean value of the thick-disk local density, [FORMULA], while the flatter secondary peak near [FORMULA], which becomes fully pronounced in the frequency distribution for the [FORMULA] models, however demonstrates that the earlier constraint on this parameter, [FORMULA] (Paper I), was too strong. In fact, Table 4 shows that this constraint is now relaxed to [FORMULA], which is a measure of the dispersions exhibited by the optimized parameter values of the individual fields, and which, therefore, reflects considerable large-scale deviations from the adopted smoothness of the global thick-disk density distribution.

Similar results for the scale length parameter [FORMULA] are presented in Figs. 19 and 20. As in Figs. 5 and 6 above, the relatively small amplitudes of the [FORMULA]- and [FORMULA]-curves confirm the low sensitivity of the present data to this parameter, anticipated in the preliminary analysis. This low sensitivity is also evidenced by the very flat frequency distributions obtained for the individual fields, whose weak local maxima at or near [FORMULA] kpc however accumulate to the somewhat more pronouncedly peaked frequency distribution shown in Fig. 20 and the correspondingly low optimized value for the combined survey given in Tab 4. In this case, the larger dispersion primarily measures the uncertainty in derived parameter values [FORMULA].

On the other hand, the [FORMULA]- and [FORMULA]-curves and the frequency distributions displayed in Figs. 21 and 22 again demonstrate convincingly that the present data provide excellent sensitivity to the scale height parameter, [FORMULA]. The sharp minimum of [FORMULA] and peaks of [FORMULA] indicate that an optimized mean value and constraints of this parameter can now be derived unambiguously from the combined data in all seven fields analysed here. Indeed, as shown in Table 4, they result in a lower mean value but also a larger dispersion than was found for the thick-disk scale height in the preliminary analysis of Paper I. In fact, the larger dispersion is in part a natural consequence of the density law, where the intrinsic (anti-)correlation between the local normalization and the scale height provides one of the gauges for matching the models and the data in an individual field . Because an optimized parameter value for the combined survey is calculated from the best models satisfying a field-independent constraint, models for individual fields having parameter values near their (individual) optimum values have usually higher chances of entering the above full-survey selection. Accordingly, the wider range in [FORMULA] found for the good models in Figs. 17 and 18 then also goes along with the broader dispersion in [FORMULA] revealed in Figs. 21 and 22.

3.3. Thick-disk metallicity structure

Although this "ultimate" goal of the present project will be attempted definitively only in a later paper - based on the complete survey data in all 14 fields and the full synthetic calibration of the metallicity-sensitive U-G colors -, for completeness we add a few comments on results obtained beyond Paper I, derived now from the new improved structural model calculations described above. Mean metallicities and metallicity gradients of the thick disk have been determined following the procedure detailed in Paper I, except that both vertical and radial gradients have now been included. Thus, the metallicity of the thick disk component is modelled as

[EQUATION]

where [FORMULA], [FORMULA] are the vertical and radial metallicity gradients, respectively; x and z are the galactocentric cylindrical coordinates of a given point (distance from the Galactic center projected upon the Galactic plane, and height above the Galactic plane, respectively), and [FORMULA] is the distance of the sun from the Galactic center (assumed to be 8.6 kpc).

The main result is that the differences with the preliminary results of Paper I are insignificant. In fact, the radial gradients are found to be [FORMULA] dex/kpc in all seven fields, and the vertical gradient, [FORMULA] dex/kpc, and mean metallicity, [FORMULA] dex, derived for the combined survey, come out almost the same as in Paper I. The [FORMULA]-curve in Fig. 23 shows that the above optimized value for the mean thick-disk metallicity, [FORMULA], is well determined from the best models for the combined survey of seven fields, with a standard error of estimate of only [FORMULA] dex; however note (again) that the actual dispersion [FORMULA] may be (significantly) larger, 3 as this quantity is obtained from the distribution of [FORMULA] pertaining to the individual fields. In fact, [FORMULA] dex, which is again indistinguishable from the result of Paper I.

[FIGURE] Fig. 23. [FORMULA]-curve for [FORMULA], the mean metallicity of the thick disk, derived from the combined survey of seven fields and from the improved structural models calculated in this paper. See the text.

Thus, while the present analysis essentially confirms our earlier results, the available data still - but not unexpectedly - do not provide conclusive evidence of any systematic finer structure, such as a radial and/or a vertical gradient, of the thick disk's larger-scale metallicity distribution.

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© European Southern Observatory (ESO) 1999

Online publication: July 16, 1999
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