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Astron. Astrophys. 334, 901-910 (1998)

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4. Results

4.1. Evolved star counts: theory versus observation

Characteristic star counts can now be computed from our synthetic HRDs and compared with those of the HIPPARCOS HRDs. They are listed in Table 1, matching entries being emphasized. Table 1 also specifies, how the characteristic region borderlines are defined in the HRD.

The choices of border lines are a compromise between the need to include all relevant stars from the evolutionary stage, and minimum confusion with stars that would, in the absence of observational errors, populate adjacent HRD areas. That has to be considered for the computed and observed HRDs at the same time.

[FIGURE] Fig. 5. Same as Fig. 2, but based on grid 1 (no overshooting). Some of the region boundaries (see Table 1) are indicated.

[FIGURE] Fig. 6. Same as Fig. 2, but based on grid 2 (overshooting for all stars). Note the significantly lower population density in the Hertzsprung gap when compared to Figs. 5, 2 and 1.

In order to assess the stochastic count variation of the synthetic HRDs, star counts have each been averaged from 10 (stochastically different) synthetic samples. There is a remarkable agreement of both hybrid grids (no. 3 and 4, marginally less good for grid 3) with observed counts, while there are some significant differences found with grids 1 and 2, i.e., larger than a statistical fluctuation [FORMULA]). The different stellar frequency in the HG is already noticeable by visual inspection of Figs. 2, 5 and 6, which have been computed on the basis of grids 3, 1 and 2, respectively.

When respective results for the 50 pc and 100 pc HRDs are compared, it becomes obvious (except for the low number of CW giants) that excellent agreement with theoretical star counts (grids 3, 4) is only achieved for the 50 pc HRD. As mentioned before, it is much more delicate to model the deviations of stars in the 100 pc HRD (shown in Fig. 4, computed on the basis of grid 3). That, in addition to a larger contamination by spectroscopic binaries (see Sect. 2), should explain why its numbers of observed HG and LGB stars are somewhat larger than computed. With the KGC, a larger choice of region, to account for the larger scatter, solved all the problems related to the less good data quality. The 7-times larger star counts of the 100 pc HRD become an advantage with the few very late type giants in the CW region. It becomes clear that grids 3 and 4 do yield reasonably good numbers here as well: [FORMULA], which is 13% with the 100 pc HRD but as much as 35% with the 50 pc HRD.

Another group of rare stars, for which the larger sample of the 100 pc HRD is of advantage, are the more luminous blue loop (He-core burning) giants, i.e., with [FORMULA] and [FORMULA]. Here, the comparison between Figs. 3 and 4 reveals a problem with the stellar models when taking into account the amount of overshooting required to match observed cluster isochrones and evolved eclipsing binaries: they yield about the right number of He-core burning (blue loop) giants, but these are less hot than the majority of their observed counterparts. The same discrepancy occurs with evolutionary tracks from other codes (and similar parameterization, see, e.g., Schaller et al. 1992). The reduced effective blue loop temperature stems from the changes which overshooting (during H-core burning) brings to the hydrogen profile around the core, as already pointed out by Weigert (1975). Lower metallicities would give hotter blue loops, but for a majority of those stars this is rather unlikely because they are much younger than the sun ([FORMULA]). The temperature to colour conversion, on the other hand, appears to be reliable within [FORMULA] for B-V in this temperature range. Hence, there could be a more general problem with the codes - it might be related to the approximate treatment of mixing.

4.2. A by-product: the IMF

Although this is not our main intention, our approach leads to an approximate IMF for single stars in the solar neighbourhood - at least for masses from about 1.6 to 4 [FORMULA], where the empirically derived PDMF is well defined (see Sect. 3.3) and the assumption of a time-independent stellar birth-rate and IMF seems to be good enough. For larger masses, the number of stars in the solar neighbourhood becomes insufficient to define the PDMF. For lower masses, this simple aproach becomes inadequate: Diffusion away from the galactic disk during the longer lifetimes becomes involved, as well as, among other possible complications, a probably different stellar birthrate in the earlier history of the galactic disk and the non-completeness of the stellar sample on the lower MS.

For the assumed, strictly random age distribution, the IMF is simply proportional to the PDMF devided by the lifetimes [FORMULA]. With the hybrid grids and for the solar neighbourhood within 50 pc, we find [FORMULA] (roughly, for [FORMULA]) and PDMF [FORMULA] (Sect. 3.3), which yields an IMF = d [FORMULA]. The estimated uncertainty of the exponent is 0.2 to 0.3, mainly related to some ambiguity in the choice of a well matching PDMF. Our IMF is close to the one suggested by Scalo (1986) for [FORMULA], i.e., d [FORMULA].

The total, time-independent stellar birth-rate, [FORMULA], is strongly dominated by stars with less than [FORMULA] and can therfore not be discussed here. For [FORMULA], we count 417 stars in our synthetic HRD (Fig. 2), and a local birth-rate of 1 such star (within [FORMULA] pc) per 6 Gyrs is required.

4.3. Properties of different evolved stages

We have shown that with grids 3 and 4 realistic star counts can be computed for all HRD regions, which are characteristic of the main evolved stellar stages - provided that the best matching PDMF and the right evolutionary history (i.e. overshooting) come together. Thus, observation can test and verify our synthetic HRD on the basis of characteristic star counts. Once this has been done, it is interesting to look at the theoretical results in more detail. We can use them as a statistical approach to give the theoretical mass distribution, or a distribution of any structural properties, for single giants in different regions of the HRD (the KGC, e.g.) or in different evolutionary stages (e.g., the AGB and GB giants in the CW field).

More detailed results are obtained by simply enlarging the number of synthetic stars. As an example, Fig. 7 shows the expected mass distribution function of the giants found in the KGC in Figs. 1 - computed from a [FORMULA] enlarged sample, a total of 13000 stars with [FORMULA] with the same solar neighbourhood PDMF as used for Fig. 2, and an evolution based on grid 3 (grid 4 yields very similar results). Figs. 8 and 9 show the mass distributions ([FORMULA]) for the corresponding GB and AGB giants, respectively, which together populate the "cool wind" (CW) region in Fig. 1.

[FIGURE] Fig. 7. Initial mass distribution for the K clump giants of a sample of 13000 stars ([FORMULA]) with a solar neighbourhood PDMF.

[FIGURE] Fig. 8. Initial mass distribution for GB stars with B-V [FORMULA], from the same sample as for Fig. 7.

[FIGURE] Fig. 9. Initial mass distribution for AGB stars with B-V [FORMULA], otherwise same as Fig. 8.

It is remarkable to see that even the cool wind region of the HRD is predominantly populated by stars with low (1.5 to 2.5 [FORMULA]) initial masses.

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Online publication: June 2, 1998