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

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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:

  • It uses a self-adapting mesh; structure and composition are solved simultaneously (Eggleton 1971, 1973).
  • Convective mixing and semiconvection are treated as a diffusion process with a diffusion constant adopted as a function of [FORMULA], while standard mixing-length theory is used to describe the heat transport (Eggleton 1972).
  • Opacities have been taken from OPAL (Rogers & Iglesias 1992), complemented at lower temperatures ([FORMULA] K) by opacities from Alexander & Ferguson (1994). Nuclear rates, neutrino losses and the equation of state have also been up-dated (see Pols et al. 1995 and more references therein).

Furthermore, extended mixing in the cores is not prescribed by an overshooting length which is a fixed fraction of [FORMULA] (the ` [FORMULA] prescription'). Instead, the code uses a ` [FORMULA] prescription' for the stability criterion itself: extended mixing occurs in a region with [FORMULA], at all convective stability boundaries - of most significance with H-burning cores, but also including He-burning cores. We define [FORMULA] as the product of a specified constant [FORMULA], our overshooting parameter, and a conveniently chosen factor which depends only on the ratio [FORMULA] of radiation pressure to gas pressure: [FORMULA]. A convenience of the ` [FORMULA] prescription' is its capability to produce realistic models over a large mass range without parameter changes (see below).

The two convection parameters ([FORMULA] = mixing length over pressure scale height, and [FORMULA]) 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 [FORMULA] were derived by Schröder & Eggleton (1996) and Schröder et al. (1997), who used [FORMULA] Aur and other well studied binaries which contain a He-burning giant primary. It was found that [FORMULA] gives the best match for the whole accessible range of [FORMULA]. 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 [FORMULA] - 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:

  • Grid 1 is computed without any overshooting ([FORMULA] for all stellar masses).
  • Grid 2 uses [FORMULA] for all stellar masses, i.e., it extrapolates the empirical calibration of [FORMULA] by Schröder et al. (1997) well below 2 [FORMULA].
  • Grid 3 is a hybrid grid which incorporates a relatively gradual onset of overshooting: the tracks are computed with [FORMULA] = 0.0 at [FORMULA], 0.12 for [FORMULA], and 0.06 inbetween.
  • Grid 4 is a hybrid grid much like grid 3, but with a sudden onset of overshooting: [FORMULA] for [FORMULA], 0.12 for [FORMULA], and 0.06 at [FORMULA].

All our 4 alternative grids have the same (nearly solar) chemical composition: [FORMULA]. 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: [FORMULA] for [FORMULA], [FORMULA] for 1.4 and 1.6 [FORMULA], 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 [FORMULA], which seems to restrict the onset of overshooting to masses around 1.7 [FORMULA]. 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 HG (Hertzsprung gap), which characterises the onset of H-shell burning and in which the evolutionary speed depends strongly on the mass of the He core, which in turn depends on stellar mass and any core overshooting during the preceding H burning. For [FORMULA], stars are sufficiently frequent to strongly constrain overshooting by their counts, to [FORMULA]. There is no other such sensitive test. Any overshooting effects on the shape of isochrones or evolutionary tracks are almost insignificant at such low masses.
  • The LGB (lower giant branch), which is characteristic of the further stages of H-shell burning and subsequent He-core growth. Since the evolutionary speed is slower with lower mass giants, these are most abundant in that region. In order to reproduce their counts, there must be a strong decline among giants with initial masses between 1 and 1.25 solar masses (i.e. with ages between 5 and 12 billion years).
  • The KGC (K giant clump), which primarily contains those He-core burning stars which went through a He flash. Degeneracy of the He core on the GB and the He flash are avoided with initial masses larger than 2.0 [FORMULA] (if [FORMULA]), or 2.4 [FORMULA] (if there is no overshooting). Stars up to about 2.6 [FORMULA] spend their He-core burning time mostly in this same region but altogether are only a minority (under 20% as computed with the hybrid grids 3 and 4).
  • The cool giants, which are found on the AGB and the upper GB beyond B-V [FORMULA]. They are of particular interest because of their 'cool winds' (CW), through which they inject a lot of gas and dust into the galactic interstellar medium. Their internal structure is described by one (GB) or two (AGB) advanced burning shells.

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 [FORMULA] IMF [FORMULA]. It is seen to decrease much more steeply towards larger masses than the IMF because the more massive stars have so much shorter lifetimes [FORMULA].

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 [FORMULA], but with three adjustable parameters. These allow adjustment for two different exponents [FORMULA] and a mass at which [FORMULA] 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 [FORMULA] upwards: [FORMULA], [FORMULA] [FORMULA], [FORMULA] [FORMULA], [FORMULA] [FORMULA], [FORMULA] [FORMULA] and [FORMULA] [FORMULA]. 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. [FORMULA]).

For all grids, the best matching PDMF requires a change of the exponent [FORMULA] around a stellar mass of 1.55 [FORMULA]. For [FORMULA], the decline of the best matching PDMF is very steep: [FORMULA] is required for grids 2 to 4, and [FORMULA] 4.0 for grid 1 ([FORMULA] pc), within an estimated range of [FORMULA] 0.2. This PDMF is somewhat steeper than that given by Miller & Scalo (1979) for [FORMULA]. Within [FORMULA] 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: [FORMULA] for grids 2 to 4, and [FORMULA] 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 [FORMULA], [FORMULA] is required with grids 3 and 4, and [FORMULA] 2.4 and 2.6 with grids 1 and 2, respectively ([FORMULA] pc). For [FORMULA] pc, [FORMULA] = 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 [FORMULA] -limit of completeness, these values for [FORMULA] 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 [FORMULA], 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 [FORMULA] fall below the minimum luminosity for completeness ([FORMULA]), 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  [FORMULA] (solar age) and 1.0  [FORMULA] (galactic age). We have chosen [FORMULA] as a representative mean lower mass limit (corresponding to an upper age limit of [FORMULA] 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 [FORMULA] and logL into the classical observed quantities B-V and [FORMULA].

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 [FORMULA] 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 [FORMULA] 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 [FORMULA] for [FORMULA] pc, and to -0.08/+0.12 (B-V), -0.3/+0.4 ([FORMULA]) for [FORMULA] 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.

[FIGURE] Fig. 2. Computer simulation of the solar neighbourhood HRD for [FORMULA] pc, based on grid 3, with 1340 stars of random age with [FORMULA]. Compare to Fig. 1.

[FIGURE] Fig. 4. Same as Fig. 2 but for [FORMULA] pc, with 9000 stars and a larger error distribution. Compare to Fig. 3.

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© European Southern Observatory (ESO) 1998

Online publication: June 2, 1998