3. Computation of realistic, synthetic HRDs
To study the effect of overshooting on evolutionary time scales, we first compute synthetic HRDs based on grids of evolutionary tracks, which differ in their description of overshooting. Then, we define characteristic regions of the HRD, where star counts represent certain evolutionary lifetimes and calibrate a realistic mass function.
3.1. Four different grids of evolutionary tracks
Stellar models and evolutionary tracks have been computed by the fast evolutionary code of P.P. Eggleton and colleagues in its most up-to-date version, which can be characterized as follows:
Furthermore, extended mixing in the cores is not prescribed by an overshooting length which is a fixed fraction of (the ` prescription'). Instead, the code uses a ` prescription' for the stability criterion itself: extended mixing occurs in a region with , at all convective stability boundaries - of most significance with H-burning cores, but also including He-burning cores. We define as the product of a specified constant , our overshooting parameter, and a conveniently chosen factor which depends only on the ratio of radiation pressure to gas pressure: . A convenience of the ` prescription' is its capability to produce realistic models over a large mass range without parameter changes (see below).
The two convection parameters ( = mixing length over pressure scale height, and ) have been empirically calibrated in such a way that the resulting evolutionary tracks are precisely consistent with, e.g., well-studied eclipsing binaries (Pols et al. 1997). Very stringent constraints on were derived by Schröder & Eggleton (1996) and Schröder et al. (1997), who used Aur and other well studied binaries which contain a He-burning giant primary. It was found that gives the best match for the whole accessible range of . That corresponds to an overshoot length which increases slightly with mass, from 0.24 to 0.32 pressure scale heights. For smaller masses, evidence for overshooting is less stringent; it appears to diminish for stars with - see Pols et al. (1998), who compare the shape of cluster MSs to overshooting and non-overshooting isochrones computed with the Eggleton code.
Finally, we define 4 grids of evolutionary tracks, which differ in their description of overshooting and in which stellar masses differ by about a factor of 1.1:
All our 4 alternative grids have the same (nearly solar) chemical composition: . In reality, some younger stars may have a somewhat higher metallicity, some low-mass giants a somewhat lower. Such abundance variation can well explain the larger spread in the observed HRD, especially in B-V, when compared with that of our models, which take only observational errors into account.
We also experimented with a fifth grid with a gradual onset of overshooting at lower stellar masses: for , for 1.4 and 1.6 , and otherwise as grid 3. Our computations show that already such a small amount of extended mixing yields a significantly too small number of HG stars with masses around 1.6 , which seems to restrict the onset of overshooting to masses around 1.7 . In the following, we focus on the 4 grids defined above.
3.2. Characteristic HRD regions
The various evolved stellar stages can be well characterised by 4 different regions in the HRD. Star counts taken in those regions are representative of the respective evolutionary time scales. In particular, we focus on the following regions (the exact borderlines are defined below, in Table 1):
The use of characteristic HRD regions as defined above has the additional advantage of their star counts being insensitive (because of their considerable size) to small displacements in luminosity and colour - i.e., through mismatched observational errors (see below) or with stars of other than solar abundances or with a fainter, undetected companion.
3.3. Matching the PDMF
The synthetic HRD were computed supposing a constant star formation rate and a time independent IMF, in agreement with the conclusions of Miller & Scalo (1979) and Scalo (1986). Under such uncomplicated conditions, the presently observed mass distribution, the PDMF, is simply IMF . It is seen to decrease much more steeply towards larger masses than the IMF because the more massive stars have so much shorter lifetimes .
Since the PDMF can be checked sensitively by the stellar distribution on the MS, we start with defining a suitable PDMF. Our PDMF distributes a given number of stars over a suitable interval of stellar masses and is basically of the form , but with three adjustable parameters. These allow adjustment for two different exponents and a mass at which changes. Each of the stars is then given a random age between 0 and its maximum lifetime. Depending on its mass and age, each computed star is then placed on its evolutionary track. For that operation, our code interpolates between the tracks of one of the four grids defined above.
In order to find the matching PDMF, we defined 6 MS regions in the HRD with different sizes so as to obtain large enough star counts, while covering the MS from upwards: , , , , and . Each interval in B-V accommodates the full MS width.
At first we tried a PDMF with a slope chosen to match the empirical PDMF of Miller & Scalo (1979). However, the MS star counts it yields show systematic deviations from the observed MS star counts. We then optimized the PDMF for each grid by variation and verification, until the computed MS star counts would match almost within the statistical fluctuation (i.e. ).
For all grids, the best matching PDMF requires a change of the exponent around a stellar mass of 1.55 . For , the decline of the best matching PDMF is very steep: is required for grids 2 to 4, and 4.0 for grid 1 ( pc), within an estimated range of 0.2. This PDMF is somewhat steeper than that given by Miller & Scalo (1979) for . Within pc, which includes more volume farther away from the galactic plane, we find an even steeper decline of the PDMF, or a relative deficiency of massive stars: for grids 2 to 4, and for grid 1. Only about 1% of the sample stars are involved here, but the difference is statistically significant and appears to be of a galactic origin - such massive, young stars are known to be more concentrated towards the galactic plane.
For , is required with grids 3 and 4, and 2.4 and 2.6 with grids 1 and 2, respectively ( pc). For pc, = 1.8 (grids 3,4), 2.2 (grid 1) and 2.6 (grid 2). Because the mass range in question is small and near the -limit of completeness, these values for should not be interpreted physically.
Finally, a suitable stellar mass interval has to be chosen. The upper limit is not critical and we have set it to , above which the PDMF yields much less than 1 star, even within 100 pc. The lower mass limit is critical, however, since it cuts into large PDMF values and thus demands further thought: MS stars with fall below the minimum luminosity for completeness (), but their evolved counterparts would gain luminosity and then contribute to the LGB, GB, KGC and AGB. On the other hand, low mass giants significantly older than the Sun may well be depleted by diffusion from the galactic disk into the halo. We therefore expect a significant drop of contributions to the LGB, GB and AGB between giants of 1.25 (solar age) and 1.0 (galactic age). We have chosen as a representative mean lower mass limit (corresponding to an upper age limit of yrs).
A good test for the choice of the above boundary condition is provided by the population density on the lower GB (LGB, i.e., below the KGC). The contribution of low mass giants is larger there than in any other part of the HRD and can in fact be matched very well with our choice of lower mass limit (see Sect. 4.1).
3.4. Simulation of an observed HRD
A critical step is the conversion of the theoretical HRD quantities and logL into the classical observed quantities B-V and .
We used the colour (and bolometric correction) tables computed by Kurucz (1991) for solar abundances. These agree fairly well with the colours of the few K and M-type giants, for which empirical values of exist (as from Di Benedetto, 1993), within the uncertainties of their derived log g values. We estimate those conversion-related uncertainties in B-V and to be about 0.05 in most cases, but increasing to about 0.1 towards the red end.
Another critical point is the spreading-out of certain features (like the KGC) in the observed HRD because of the errors in the stellar colours and, especially, the parallax measurements. As discussed in Sect. 2, such errors are non-symmetrical and their distribution is non-gaussian. Our error distribution model displaces the stars in a given error range in a manner which is asymmetric for the larger sample. Guided by the visual appearence of well defined features in the observed HRDs (e.g., ZAMS and KGC), we set the error ranges to -0.06/+0.06 in B-V, -0.2/+0.2 in for pc, and to -0.08/+0.12 (B-V), -0.3/+0.4 () for pc. A distribution of random numbers taken to the third power is used to simulate the error distribution in those intervals. It yields 50% of all synthetic stars being in the inner 1/8 fraction of the given error range, while only 20.6% of the stars are displaced by more than half the maximum error.
That approach appears to be fully adequate for the HRD of the solar neighbourhood within 50 pc. Fig. 2 shows a model of that HRD and it compares well with Fig. 1. With the 100 pc HRD, however, we find that the choice of error distribution has an effect on the star counts in the above defined regions, because the error ranges are already quite large - that degrades the information content anyway. Hence, the interpretation of the larger sample must remain ambiguous to some extent. See Fig. 4 for a matching model, to compare with Fig. 3.
© European Southern Observatory (ESO) 1998
Online publication: June 2, 1998