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

3. Methodology

3.1. Concept

Analysis of the catalog is done essentially following the approach taken by Buser & Kaeser (1985) and summarized by Buser & Fenkart (1990): we adopt a parameterized model of the density distributions, luminosity functions, and color-magnitude relations assumed to represent the various stellar population components of the Galaxy, whence we calculate the expected star counts and color distributions for each field. Systematic variation of parameter values then allows us to evaluate, by least-squares techniques, those models which best fit the observed distributions for each individual field or for a number of fields. Eventually, we determine the most stringent constraints to the model parameters imposed by the all-survey data.

3.2. Models: ingredients

3.2.1. Galactic population components

Our models comprise four distinct population components of luminous stars, as follows: a thin disk, which is composed of a young and an old component, a thick disk, and a halo. The primary characteristic of each of these components is its density law, which we assume to be given by one of the parameterized analytical functions discussed in 3.2.2 and 3.2.3 below. The secondary characteristic of each component is its luminosity function, which we assume to be given by the luminosity function derived from observations of the nearby and/or other component tracer stars described in 3.2.4. Finally, since we also want to determine the chemical signature of each population component, the characteristic (or template) color-absolute magnitude and two-color diagrams are given as functions of stellar metallicity, and are discussed in more detail in subsection 3.2.5 below.

3.2.2. Density laws

The density laws adopted for the disk population components are the usual double-exponentials

with local densities , scale lengths , and scale heights . For the (spheroidal) halo, we assume a de Vaucouleurs density distribution,

where is the local density and is the effective radius, and where

being the flattening factor of the halo. In the above equations, l and b are the Galactic coordinates, and 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); r is the given point's distance from the sun, and is the distance of the sun from the Galactic center (assumed to be 8.6 kpc).

3.2.3. Density law parameters

Table 2 gives a summary of the parameter ranges which we adopt for the systematic exploration of parameter space by calculating a large number of models. They are chosen such as to provide fair coverage of published parameter values suggested by modern surveys (cf. Bahcall 1986, Gilmore et al. 1990, Majewski 1993, and Blitz & Teuben 1996 for recent reviews). In the actual calculations, a uniform distribution of between four and eight discrete values are employed to cover the range for each parameter. According to the different anticipated (or supposed) sensitivities of the present survey data to variations of the different parameters, the latter are divided into primary and secondary categories: the majority of primary parameters are expected to be well determined from the present data, while the majority of secondary parameters are expected to provide indispensable but only less-well determined constraints to the models.

Table 2. Variation ranges of density law parameters

3.2.4. Luminosity functions

We have used two basic luminosity functions (LFs) to define the appropriate input data for the different population components: (1) the LF derived by Buser & Kaeser (1985) from the Gliese (1969) nearby star catalog, and (2) the LF derived by Da Costa (1982) for the globular cluster 47 Tuc. Note that the Buser & Kaeser LF was derived for star systems rather than for individual stars, counting all the components of a binary or multiple system with separations as a single object of combined absolute magnitude. This selection thus refers to those Gliese systems which are unresolved on the Palomar-Schmidt plates if observed at distances , as is appropriate for the majority of the present survey field stars. This accounts for the low value of for the local density of old thin disk dwarfs (i.e., systems), as opposed to the corresponding obtained for the single-star LF by Wielen et al. (1983). Thus, for the thin-disk dwarfs and giants, we have used the LFs given in Buser & Kaeser (1985).

For the thick-disk and halo stars, we adopted the following LFs. For magnitudes , the shape of the LF is given by the 47 Tuc LF of Da Costa (1982), while for the fainter stars, the shape given by the old thin-disk LF of Wielen et al. was joined smoothly with the brighter LF at . Thus, while we assume the thick-disk and halo LFs to have the same shapes, their local normalizations relative to the thin disk were left as independent parameters - to be determined by the present analysis of the new Basel survey data.

Thus in this paper, we adopt a unique LF for each individual component, which we also assume to be the same throughout the range of the survey, and which are illustrated in Fig. 1. Effects of canonical uncertainties and intrinsic errors in the adopted LFs have been shown to be of minor importance in the analysis of the original star count and color survey data (Buser & Kaeser 1985), and are confirmed here. Later on (in Paper II), we shall relax the constraint of uniqueness by examining a number of alternatives. Eventually, we shall introduce analytical luminosity functions derived from mass functions and evolutionary calculations in order to explore the sensitivity of photometric star count and color surveys on luminosity function variations more deeply - with an eye on the important question of age-dating the population components.

Fig. 1. Luminosity functions for the different population components, with adopted local densities

3.2.5. Color-magnitude and two-color relations, and transformations from UBV to RGU

For the calculation of star counts as functions of apparent magnitude and colors, the above luminosity functions - which are given in the UBV-standard system of the absolute -magnitudes - were converted to functions by using the intrinsic color-absolute magnitude diagrams derived from observations of tracer stars for the chemically different population components. For the thin-disk solar-abundance dwarfs, we have adopted the relation established by Buser & Fenkart (1990), which in turn was based on the compilations published by Schmidt-Kaler (1965, 1982), Gliese (1982), and Jahreiss & Gliese (1989), while for the thin-disk (solar-abundance) giants we used Table 2 given by Buser & Kaeser (1985), which was derived from the Gliese (1969) catalog and which was found to be very similar to the semi-empirical relation obtained by Neckel (1975) from a large sample of giants with spectroscopic parallaxes. On the other hand, the lower-abundance stars of the thick disk and halo have been represented by Sandage's (1982) template color-absolute magnitude diagrams for the globular clusters 47 Tuc, M5, and M92, whose metallicities were adopted to be and -2.05, respectively (Pilachowski 1984).

Finally, the resulting were combined with the appropriate metallicity-dependent vs. color-color relations constructed from extant data by Johnson (1966), FitzGerald (1970), Carney (1979), Arimoto (1986), and Yoshii et al.(1987), whence detailed luminosity functions, and , have been derived for both dwarfs and giants covering the metallicity range via the corresponding transformation equations based on synthetic UBV and RGU photometry (Buser & Fenkart 1990, Buser & Kurucz 1992, Güngör 1996, Buser et al. 1997c).

A sample two-color diagram which results from the transformation and interpolation processes applied to the UBV data for dwarfs is shown in Fig. 2; also note that the component-specific luminosity functions calculated above provide the final input in the star-count Eq. (1) below.

Fig. 2. The RGU two-color diagram for dwarf stars of different metallicities, as transformed from the corresponding UBV diagram.

3.2.6. Interstellar reddening and extinction

As shown in Table 1, significant interstellar reddening has been measured from the two-color diagrams in only two fields, SA107 and NGC6171, which belong to the inner halo/bulge part of the survey and which will be analysed including appropriate modelling in Paper V of this series. Zero reddenings are also deduced from the maps of Burstein & Heiles (1982) for six out of the fourteen Basel fields; for the remaining eight directions, this same source gives mag, which translates to mag. While these minor interstellar reddening values would not change the conclusions of the present paper, we prefer the values of Table 1 since they are specific for the small areas investigated here.

3.3. Models: general outline of calculations

Of course, the purpose of this project is to determine those specific characteristics of the Galactic population components whose combination provides both a physically plausible picture and the best possible consistency with the observed survey data, , where N is the number of stars counted as a function of apparent G-magnitude and/or and/or color. Thus, simulated star counts and color distributions are calculated from a Galactic model specified by a set of parameter values selected from the ranges adopted in Table 2. Variation of parameter values then leads to a large number of models providing full coverage and allowing a systematic exploration of the adopted parameter space.

In rough outline, the sequence of calculations is as follows:

1. Initially, we assume the (mean) metallicity of each component to be fixed at for the old and young thin disks, the thick disk, and the halo, respectively.
2. Adopting a fixed value for each of the secondary parameters, models are calculated for all combinations of primary parameter values covering the full ranges adopted in Table 2. Results are then used to derive optimum primary parameter values and constraints by comparison with the observed two-color (G, G-R) data and doing the statistical analysis detailed in Sect. 4.
3. Keeping the optimum primary parameter values fixed, models are calculated for all combinations of secondary parameter values covering the full ranges adopted in Table 2. Results are then used to derive optimum secondary parameter values and constraints in the same way as for the primary parameters before.
4. A second iteration between steps (2) and (3) is performed to establish convergence toward stable results - which turns out to be achievable without further iterations in most cases. Thus, for each iteration a total of nearly 17,000 models are calculated for the systematic determination of optimum parameter values and constraints for both individual fields and the all-field survey.
5. Adopting the above optimum values and constraints for the full dozen of primary and secondary (structural) parameters, new models are then calculated for ranges of mean metallicities and metallicity gradients of the Galactic population components, whence optimized values for the mean metallicities and metallicity gradients are derived by comparison with the observed three-color (G, G-R, U-G) data and doing a similar statistical analysis as before.
6. Finally, a second iteration through the full sequence of calculations is performed by propagating the improved values for the mean metallicities from step 1 through step 5.

3.4. Models: basic calculations

For each model, the basic calculation provides differential star counts (i.e., for the apparent magnitude interval and the color index interval ) for a field subtending solid angle in Galactic direction , according to the fundamental equation of stellar statistics:

where and are given by the component-specific density laws and luminosity functions, respectively, and

is the absolute magnitude of the star, is the total Galactic extinction along the line of sight out to distance r from the sun, and is the integration cutoff.

© European Southern Observatory (ESO) 1998

Online publication: March 3, 1998
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