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Astron. Astrophys. 331, 934-948 (1998)

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4. Comparison with observations and analysis

4.1. [FORMULA] -calculations

In order to compare calculations and observations for a given field k, both model-generated and observed data are first sampled into apparent magnitude and color bins of sizes 0.5 mag and 0.2 mag, respectively. From these, the (computed or observed) star counts per square degree, [FORMULA], falling into successive apparent magnitude or/and color bins are established, and for each model, [FORMULA] -estimates of goodness-of-fit to the observed one- or two-dimensional distributions are then calculated as follows:

[EQUATION]

where the field index [FORMULA].

For each field k, the individually best-fitting model is determined by its lowest- [FORMULA] value, [FORMULA], while measures of each model's simultaneous fitting to the combined all-survey data in seven fields are thus given by the global [FORMULA] -values

[EQUATION]

whence the globally best-fitting model is identified by the lowest- [FORMULA] value, [FORMULA].

The (seven) individually plus the (one) globally best-fitting models are used as pivots in the subsequent analyses.

4.2. Determination of optimized parameter values

Using the above results for a total of nearly 17,000 models for each of 7 fields plus their combination, we now determine the optimized values for each of the 12 primary and secondary parameters.

4.2.1. [FORMULA] -curves

For each parameter, we determine its [FORMULA] -curves by allowing the parameter to vary within its assigned range while keeping all other parameters fixed at their values adopted by the appropriate lowest- [FORMULA] model. Of course, such curves not only allow us to identify a formally best fitting model and parameter value from their minima, but their specific shapes also provide a first idea of the sensitivity of the data to the variations of a particular parameter, whence a more quantitative estimate of the permissible parameter range can be obtained. For example, Fig. 3 illustrates how the two-dimensional (G,G-R) and the one-dimensional (G or G-R) observations for the field SA 54 are reproduced in different but mutually consistent ways by models calculated for variable local densities of the thick disk, [FORMULA]. The close coincidence of the minima near [FORMULA] for all three [FORMULA] -curves demonstrates that the observed data are internally consistent to a high degree, and that an optimized parameter value and associated permissible range can be reliably derived from the [FORMULA] estimate, if due account is taken of the [FORMULA] -growth rates with changing [FORMULA] exhibited by the one-dimensional [FORMULA] - and [FORMULA] -curves. This will be done below by applying an appropriate weighting scheme.

[FIGURE] Fig. 3. [FORMULA] -curves for the local density parameter, [FORMULA], of the thick disk as measured by the SA 54 field data. Labels indicate i.

The need for taking full advantage of the information contained in the two-color data is further illustrated in Fig. 4. Here, the three minima are dispersed over a larger range in [FORMULA], indicating a somewhat lesser degree of internal consistency of the data for the field M67 - as opposed to the nearly ideal example shown for the field SA 54 in Fig. 3. On the other hand, with changing [FORMULA] all the [FORMULA] -growth rates are flatter in Fig. 4 than they are in Fig. 3, which shows that the sensitivity of the data to variations of the particular parameter [FORMULA] is lower for the M67 field than for the higher-latitude field SA 54.

[FIGURE] Fig. 4. [FORMULA] -curves for the local density parameter, [FORMULA], of the thick disk, as measured by the M67 field data. Labels indicate i.

Thus, calculation of optimized parameter values is preferably based on the [FORMULA] -curves, because these curves in turn are based on model fits to the two-dimensional [FORMULA] data, and therefore provide stronger parameter constraints than each of the two one-dimensional [FORMULA] and [FORMULA] estimates; but since the individual patterns exhibited by the different [FORMULA] -curves actually depend sensitively on both the particular Galactic direction and the particular model parameter, this important source of information provided by the [FORMULA] and [FORMULA] estimates cannot be neglected in deriving optimized parameter values for both individual fields and the combined data in seven fields.

Examples of [FORMULA] -curves for combined data in seven fields are illustrated for two parameters in Figs. 5 and 6.

[FIGURE] Fig. 5. Global [FORMULA] -curves for the local density of the thin disk, [FORMULA], in units of [FORMULA]. Labels indicate i.

[FIGURE] Fig. 6. Global [FORMULA] -curves for the scale height of the thick disk, [FORMULA]. Labels indicate i.

In order to further explore the statistical significance of the [FORMULA] model toward determining optimized parameter values and constraints, we proceed along the following line of reasoning: (1) for each parameter, by mathematical design its optimum value and constraints must be found from the low- [FORMULA] models; (2) the optimum value and constraints of a parameter are then given by the frequency distribution of its values occurring among the low- [FORMULA] models; (3) using this frequency distribution, the final determination follows by calculating a weighted average value and dispersion which account for the (particular) parameter's specific [FORMULA] -curves.

In more detail, this procedure is realized in the following operational sequence (cf. also Pritchet 1983; Buser & Kaeser 1985):

4.2.2. Selection of acceptable models

We employ the parent population of all the models calculated in this work to define the sub-population of acceptable models by requiring that, for a model to be considered acceptable, it must satisfy the condition [FORMULA], where [FORMULA] is an (as yet) arbitrary numerical factor that determines the maximum allowed [FORMULA] -value of the acceptable model selection. We shall see below that, at least for the present preliminary analysis, a value of [FORMULA] as an a priori constraint on the goodness of fit appears to be reasonably justified by both the accuracy of available star count data in general and the sensitivities of the model predictions with parameter variations.

4.2.3. Frequency distributions

For discrete choices of [FORMULA] increasing with increments [FORMULA] between [FORMULA] and [FORMULA], we now construct the frequency distributions of parameter values by counting the number of acceptable models where a given parameter value occurs. Intuitively, for each choice of [FORMULA], a larger number of models, i.e., a higher frequency of occurrence of a given parameter value, indicates a higher probability for this value to be true.

The idea behind this principle is the following. Because the large number (12) of parameters allow many models to reproduce the observed star counts and color distributions by chance, i.e., to provide low [FORMULA] -scores even for unrealistic combinations of parameter values, the truly good models can be identified via negationis by arguing that a parameter value that does not appear among the low- [FORMULA] models cannot be the true one. Quantitatively, a slightly relaxed formulation of this criterion then leads one to investigate the above frequency distributions of parameter values among the whole sub-population of acceptable models defined in the previous step above.

Note that the calculations are performed on a sub-population of models drawn from a parent population which includes all possible combinations of parameter values ranging within their adopted variation limits. Therefore, the present selection of acceptable models not only includes models differing on account of the permissible variation of a particular parameter, but automatically also allows for both uncoupled and coupled variations of all other parameters. A fortiori, the same holds for the frequency distributions that are hence derived for each individual parameter. Thus, the subsequent determination of optimized parameter values and constraints does not suffer from selection bias due to the neglect of the effects of parameter coupling - which would be the case if, as usual, only the [FORMULA] -curves were employed instead for this purpose.

The significance of the [FORMULA] -factor is illustrated in Figs. 7 and 8, which show that, with increasing [FORMULA], increasing numbers of acceptable models exist for growing ranges of parameter values. However, the very existence and relative stability of pronounced peaks along with the similarities of the shapes of the successive distributions also suggest that reliable optimized parameter values, [FORMULA], may indeed be derived from the representative [FORMULA] -selections of the (up to several hundred) good models, i.e., those which satisfy the condition [FORMULA].

[FIGURE] Fig. 7. Frequency distributions for the local density of the thin disk, [FORMULA], derived from the combined survey of seven fields. The sequence of curves from bottom to top corresponds to growing values of [FORMULA]: labels indicate the number of models involved.

[FIGURE] Fig. 8. Frequency distributions for the scale height of the thick disk, [FORMULA], derived from the combined survey of seven fields. The sequence of curves from bottom to top corresponds to growing values of [FORMULA]: labels indicate the number of models involved.

On the other hand, since in general, the associated dispersion [FORMULA] shows a stronger dependence on [FORMULA] (as visualised by the increasing half-widths of the frequency distributions in Figs. 7 and 8), it may be more difficult to derive similarly reliable constraints on [FORMULA] . Indeed, for this purpose the calculation of [FORMULA] itself may reasonably be constrained independently by choosing [FORMULA] in such a way as to provide a maximum allowed [FORMULA] which matches the external accuracy of the star count data.

Thus, for the purpose of the present paper, we have used 20% as a rough number for the external accuracy of available star count data, as judged from independent observations of several fields studied by different authors and techniques (Bahcall & Soneira 1984). As it turns out, this number is fairly well accommodated in the present calculations using [FORMULA] implied by the [FORMULA] -selection of models. In fact, most of our total predicted counts agree with the observations to within better than [FORMULA] if [FORMULA] but get rapidly worse if [FORMULA].

4.2.4. Parameter coupling

As anticipated in the previous Subsect. 4.2.3, the present scheme has been explicitly designed to cope with the unavoidable presence of coupling among many parameters. Since each Galactic direction has its own pattern of sensitivities to the various parameter variations, it is important that the selection of good models invariably also include those for which parameter couples exist where allowed variation of the first parameter happens to compensate for the effect brought about by the (corresponding) allowed variation of a second parameter. This may lead to a frequency distribution of low- [FORMULA] models which is different from the corresponding univariate distribution (where only one parameter is allowed to vary) and, consequently, to different derived optimum parameter values and constraints.

Presence or absence of parameter coupling are illustrated in Figs. 9 and 10, respectively. In either case the [FORMULA] -selection of good models leads to frequency distributions which are strongly peaked at thin-disk dwarf scale height factor [FORMULA] and which are actually built from models covering large ranges of the local thick-disk density, [FORMULA] or the thick-disk scale height, [FORMULA], respectively.

[FIGURE] Fig. 9. Frequency distribution for old thin-disk dwarf scale height factor, [FORMULA], among good models, illustrating the presence of parameter coupling with the local normalization of the thick disk, [FORMULA] (labels in insert): for small/large values of [FORMULA], the maximum of the frequency distribution preferentially occurs at large/small values of [FORMULA], while the optimum value, [FORMULA], is associated with intermediate values of [FORMULA] - as indicated by the curve for the overall total. These results were derived from the combined survey of seven fields.

[FIGURE] Fig. 10. Frequency distribution for old thin-disk dwarf scale height factor [FORMULA] for the lowest- [FORMULA] models, illustrating the absence of parameter coupling with thick-disk scale height, [FORMULA] (labels in insert): for all values of [FORMULA], the frequency distributions are similar, having stable maxima at the same value of [FORMULA]. These results were derived from the combined survey of seven fields.

4.2.5. Optimized parameter values

The statistical weight of a parameter value, [FORMULA], occurring in model i from the selection of good models is calculated taking into account this model's [FORMULA] - and [FORMULA] -estimates derived (above) from the independent fits of the model to the G- and G-R-data:

[EQUATION]

where [FORMULA] and [FORMULA] are calculated for an individually or the globally best-fitting model.

The optimized parameter value, [FORMULA] , either for an individual field or for the combined G, G-R survey data in seven fields, is then computed from

[EQUATION]

where n is the number of good models selected by the [FORMULA] -criterion.

Figs. 11-13 provide representative illustrations of how such calculations reproduce the observed marginal distributions, [FORMULA], or [FORMULA], for the fields Praesepe, SA 57, or M 101, respectively. In each figure, the observed histogram is compared with two model-generated distributions that are based on the optimized parameter values derived for either the particular individual field (dotted line) or the combined survey of seven fields (dashed line). While, of course, the individual model-fits are superior to the global fits by definition, the rather non-systematic and relatively weak differences between the two kinds - which are typical for all seven fields - clearly show that the survey data are highly homogeneous and that the structural parameters hence derived have a high probability of being realistic. This conclusion will be further supported by the following discussions of the metallicity structure derived from the three-color data.

[FIGURE] Fig. 11. Star counts in the Praesepe field. The observed histogram is compared with model predictions based on parameter values optimized for either this individual field (dotted line) or for the combined survey of seven fields (dashed line).

[FIGURE] Fig. 12. [FORMULA] color distribution in the field SA 57. The observed histogram is compared with models based on parameter values optimized for either this individual field (dotted line) or for the combined survey of seven fields (dashed line).

[FIGURE] Fig. 13. [FORMULA] color distribution in the field M 101. The observed histogram is compared with models based on parameter values optimized for either this individual field (dotted line) or for the combined survey of seven fields (dashed line).

4.2.6. Mean metallicities and metallicity gradients

Apart from deriving improved structural parameters and constraints, the principal goal of this project is to determine the metallicity structure of the thick disk component. In fact, one important way of addressing the yet unsolved question concerning the cosmogonic status of the thick disk consists in assessing its large-scale metallicity distribution: is it spatially uniform or variable; if it is variable, is there or isn't there uniformity even in its variation, as the presence or absence of one (or several) metallicity gradient(s) might reveal?

As a first attempt at answering these important questions, the thick disk is modelled in either of two basic ways:

  1. as a chemically homogeneous component whose stars all have the same [FORMULA]. In order to derive the most probable value consistent with our survey data, a large number of models are calculated where the constant is allowed to range in the interval [FORMULA] ; of course in this case, the adopted [FORMULA] is identical with the mean metallicity, [FORMULA], of the thick disk component;
  2. as a component whose chemical structure is characterizable by a vertical metallicity gradient, [FORMULA], giving rise to a metallicity distribution with wider (and possibly also non-gaussian) dispersion about a field-specific "mean" metallicity; again, in order to derive the most probable values consistent with our survey data, a large number of models are calculated where [FORMULA] is allowed to range in the interval [FORMULA].

Since the metallicity structure of the thin disk has been better known from spectroscopic investigations, calculations are run for only a few representative values of its mean metallicity and gradient to ascertain consistency with the canonical literature. As we shall see, the present data do not allow us to determine whether or not a metallicity gradient exists in the field halo. Thus, calculations are made for a few discrete values of the halo's mean metallicity only in order to ascertain consistency with canonical results from the literature.

Comparison of model calculations with observations is now performed using both two-dimensional distributions, [FORMULA] and [FORMULA], to reinforce the metallicity-sensitivity provided by the additional [FORMULA] data. Thus, for each model and field [FORMULA] we calculate

[EQUATION]

and individually or globally best-fitting values for [FORMULA] or [FORMULA] are derived following a similar statistical analysis as before.

The sensitivities of the predicted color distributions to changes of the mean metallicity in the model of the thick disk component are illustrated in Figs. 14 and 15. Note that both the predicted blue edges and peaks of the distributions are significantly different in either color for an assumed difference of order [FORMULA] in the mean metallicity of the thick disk. Although the high metallicity sensitivity persists into the red wings of the distributions, the somewhat poorer quality of the model fits is due to the increasing uncertainty of the color-magnitude and two-color calibrations for the reddest stars (Buser & Fenkart 1990, Buser & Kurucz 1992). However, a special effort is being made to improve the temperature and metallicity calibrations of the stellar library underlying the synthetic photometry employed in this work (Lejeune et al. 1997a,b), which will also improve our treatment of the coolest stars in the impending analysis of the full-survey data in 14 fields.

[FIGURE] Fig. 14. [FORMULA] color distribution in the field M101. The observed histogram is compared with models that differ only in the mean metallicity adopted for the thick disk component: [FORMULA] (dotted line) and [FORMULA] (dashed line).

[FIGURE] Fig. 15. [FORMULA] color distribution in the field M5. The observed histogram is compared with models that differ only in the mean metallicity adopted for the thick disk component: [FORMULA] (dotted line) and [FORMULA] (dashed line).

For the present first-half sample of the survey, typical results obtained for the mean metallicity of the thick disk component are illustrated in Figs. 16 and 17, which show the [FORMULA] -curves derived from the data in the South Galactic Pole field SA 141 and from the combined data in seven fields. Note that these results are based on the assumption that the thick disk has uniform mean metallicity throughout its range, with a (gaussian) dispersion, [FORMULA], corresponding to the dispersions of the color-magnitude relations adopted in converting the luminosity function appropriate to the given mean metallicity from the [FORMULA] system to the [FORMULA] system. We shall see below that while the good models in Fig. 17 suggest that the thick disk has [FORMULA], the somewhat more detailed model including a metallicity gradient indicates that the true dispersion may be significantly larger.

[FIGURE] Fig. 16. [FORMULA] -curve for the thick-disk mean metallicity derived in the South Galactic Pole field SA 141. The sharp minimum suggests that in this field, [FORMULA].

[FIGURE] Fig. 17. [FORMULA] -curve for the thick-disk mean metallicity derived from the all-survey data in seven fields. The minimum suggests that globally, the data are consistent with a model that assumes a thick disk with homogeneous metallicity [FORMULA].

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© European Southern Observatory (ESO) 1998

Online publication: March 3, 1998
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