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3 Two-color diagrams and absolute magnitude determination

A sample of observed (U-G,G-R) two-color distributions as functions of successively fainter apparent G magnitudes is given in Fig. 2, which shows the classification of the stars into the different categories that determine their subsequent treatment in the statistical analysis, most notably the assignment of absolute magnitudes (MG) required for the distance and density calculations. The classification is based on the comparison of the observed distributions with the intrinsic loci of the various stellar types provided by the synthetic photometry calibrations. The most important steps of the process are, briefly, as follows.

\centering\includegraphics[]{1475f2.eps}\end{figure} Figure 2: Sample two-color diagrams for three intervals of successively fainter apparent magnitudes. The superposed grid of iso-abundance lines for dwarf stars is based on theoretical model atmosphere calculations (Buser & Fenkart 1990) and has been used for initially classifying the stars into coarse metallicity classes: $(\bullet)$ population I main sequence; $(\triangle)$intermediate population II main sequence; $(\square)$ extreme population II main sequence; $(\circ)$ white objects; $(\times)$ population I late-type red giants; (+) unclassified

In the absence of systematic errors in the data, unreddened normal population I main sequence stars are expected to scatter uniformly about the theoretical zero-abundance line ([M/H]=0.00) (Buser & Fenkart 1990, hereafter BF90) throughout their observed two-color ranges and with a dispersion in metallicity of $\sigma_{[M/H]} \sim 0.3$ dex. Thus, the observed distributions of probable normal main sequence stars (full dots) suggest that in this field, interstellar foreground reddening is absent (i.e., EG-R = 0.00), and an absolute magnitude can be assigned to each star by entering its observed G-R in the color-magnitude diagram (hereafter CMD) given in Fig. 7 of BF90 and reading MG off the appropriate standard main sequence for thin-disk dwarfs ($\langle[M/H]\rangle = 0.00$).
Most other objects within or near the limits of the grid of theoretical iso-abundance lines for dwarf stars cannot be classified unambiguously from their two-color positions only. In particular, metallicity and luminosity effects on both the U-G and the G-R colors are hardly separable for field stars in the color range $\sim 0.8 < G{-}R < 1.5$ (cf. Buser 1978a (Fig. 7), Buser 1982 (Fig. 6), BF90 (Fig. 1 and Table 4)). This means that, e.g., a star's UV-excess, $\delta_{U{-}G}$, relative to the solar-abundance main sequence may be due to either its advanced evolutionary stage (F-G-type subgiant or G-K-type red giant) or its lower metallicity (intermediate or extreme population II type), or both. Because the assignment of an absolute MG magnitude to an individual star critically depends on its previous assignment of evolutionary stage and population type, we here adopt the following iterative procedure, suggested by BF90.

The grid of theoretical iso-abundance lines is used to provisionally categorize the objects into two lower-metallicity population types, as follows: stars within the metallicity ranges $-0.25 \geq [M/H] \gt -1.00$ or $-1.00 \geq [M/H] \gt -10.00$are considered (unevolved) main sequence dwarfs of the intermediate (open triangles) or the extreme population II (open squares), respectively, and their absolute MG magnitudes are then given by the corresponding theoretical main sequences (for $\langle[M/H]\rangle = -0.76$ ["thick disk"] or $\langle[M/H]\rangle = -1.76$ ["halo"]) in the CMD by BF90.

If a discontinuity is apparent in the above two-color distributions near $U{-}G \sim 1.8-2$, the stars redder than this gap are considered normal red giants (crosses) (Becker 1962, Becker & Fang 1982). Similarly, stars outside the theoretical grid for dwarfs (U-G > 3.0), but whose colors match the predicted synthetic loci for normal population I red giants (Buser 1978a), are classified as such. In either case, these stars are assigned absolute magnitudes $\langle M_{G} \rangle = 1.0$ (Buser 1981).

Objects far outside the theoretical calibrations in the two-color diagrams are either classified as "white objects" (open circles), or are left unclassified (plus signs); both these categories are excluded from the subsequent analysis of the density and luminosity functions.

Based on the above absolute magnitude determinations, density functions are derived for each of three population types (i.e., main sequence stars of population I, intermediate population II, and extreme population II, respectively), plus their combination. Using a least-squares method, best-fitting model density gradients predicted by Gilmore & Wyse (1985)[*] are then matched to the observed density profiles in order to extrapolate hence the local stellar space densities as functions of absolute MG magnitude (del Rio & Fenkart 1987; Fenkart & Karaali 1987; Fenkart 1989a-d).

Comparison with the local luminosity function based on the trigonometric parallax catalog of nearby stars by Gliese (1969) then provides the crucial consistency check: if the two disagree substantially and systematically, an iteration cycle is started by reassigning permissible alternative absolute magnitudes to an appropriately chosen number of suitable stars in such a way as to assimilate the resulting local luminosity function to the Gliese standard[*].

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