## 4. Comparison with observations and analysis## 4.1. -calculationsIn 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) where the field index . For each field k, the whence the The (seven) individually plus the (one) globally best-fitting models are used as pivots in the subsequent analyses. ## 4.2. Determination of optimized parameter valuesUsing 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. -curvesFor each parameter, we determine its
The need for taking full advantage of the information contained in
the
Thus, calculation of optimized parameter values is preferably based
on the -curves, because these curves in turn
are based on model fits to the Examples of -curves for combined data in seven fields are illustrated for two parameters in Figs. 5 and 6.
In order to further explore the statistical significance of the
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 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 modelsWe employ the parent population of ## 4.2.3. Frequency distributionsFor discrete choices of increasing with
increments between and
, we now construct the 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
-scores even for unrealistic combinations of
parameter values, the Note that the calculations are performed on a sub-population of
models drawn from a parent population which includes The significance of the -factor is
illustrated in Figs. 7 and 8, which show that, with increasing
, 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
On the other hand, since in general, the associated dispersion
shows a stronger dependence on
(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 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 implied by the -selection of models. In fact, most of our total predicted counts agree with the observations to within better than if but get rapidly worse if . ## 4.2.4. Parameter couplingAs 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 Presence or absence of parameter coupling are illustrated in Figs. 9 and 10, respectively. In either case the -selection of good models leads to frequency distributions which are strongly peaked at thin-disk dwarf scale height factor and which are actually built from models covering large ranges of the local thick-disk density, or the thick-disk scale height, , respectively.
## 4.2.5. Optimized parameter valuesThe statistical weight of a parameter value,
, occurring in model where and are
calculated for The where n is the number of good models selected by the -criterion. Figs. 11-13 provide representative illustrations of how such
calculations reproduce the observed marginal distributions,
, or , 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
## 4.2.6. Mean metallicities and metallicity gradientsApart from deriving improved structural parameters and constraints,
the principal goal of this project is to determine the As a first attempt at answering these important questions, the thick disk is modelled in either of two basic ways: - as a
*chemically homogeneous*component whose stars all have the same . 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 ; of course in this case, the adopted is identical with the*mean metallicity*, , of the thick disk component; - as a component whose chemical structure is characterizable by a
*vertical metallicity gradient*, , 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 is allowed to range in the interval .
Since the metallicity structure of the Comparison of model calculations with observations is now performed using both two-dimensional distributions, and , to reinforce the metallicity-sensitivity provided by the additional data. Thus, for each model and field we calculate and individually or globally best-fitting values for or 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 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.
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
-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,
, corresponding to the dispersions of the
color-magnitude relations adopted in converting the luminosity
function appropriate to the given mean metallicity from the
system to the system.
We shall see below that while the good models in Fig. 17 suggest that
the thick disk has , the somewhat more detailed
model including a metallicity gradient indicates that the
© European Southern Observatory (ESO) 1998 Online publication: March 3, 1998 |