Astron. Astrophys. 345, 499-504 (1999)

3. Synthetic WD sequences

According to the procedure outlined above, we computed selected WD sequences by randomly distributing stars until 1000 objects were found along the cooling sequence and in the previous He burning phase. Fig. 2 shows the predicted distribution in the logL, logTeff plane for a Salpeter IMF and for the cluster ages 1, 3, 8 and 12 Gyr. Comparison of the sequence with the evolutionary path for selected masses, as given in the same Fig. 2, allows a short discussion of some relevant features.

Fig. 2. Synthetic cooling sequence for the different ages (filled triangle). The adopted WD evolutionary tracks are also plotted.

One finds that a major portion of the sequence always follows with good approximation the cooling line of low mass, i.e. 0.5-0.6 M WD. At the bottom of this dwarf "Main Sequence" (DMS) the more massive WD, as earlier originated from the most massive progenitors, abandon this sequence to move towards larger effective temperatures, making a more or less pronounced "hook" toward larger effective temperatures. Note that, since the nuclear evolutionary times of the more massive progenitors are much shorter than the cooling times, the differences in luminosity of these stars is only due to differences in the cooling law.

The shape of this hook for the two lower cluster ages can be understood if we recall that the more massive stars start cooling earlier but more slowly. However, they undergo crystallization earlier, and this eventually accelerates the cooling. According to the Wood (1992) extrapolation of these final cooling phases, one expects that at 12 Gyr a few (massive) WD have already reached this final stage: one finds one out of these objects in Fig. 2 at logTe3.15, whereas 4 further objects have already faded away at even lower temperature and luminosities.

The history of the various sequences is summarized in Fig. 3, where we report the luminosity of WD along the sequences as a function of their mass or of the original mass of the progenitor (Fig. 4). According to the data in these figures, one finds that in all cases the bottom luminosity of the sequence can be reasonably approximated by the cooling of a 0.5 M model only.

Fig. 3. Mass-luminosity relation for the expected WD sequence. The assumed ages are (from top to bottom) 1, 3, 8 and 12 Gyr.

Fig. 4. As in Fig. 3 but the masses of the WD progenitors are reported.

Data in both Figs. 3 and 4 suggest that the luminosity of the "hook" at the basis of the sequence should represent a relevant observational parameter to be calibrated in terms of the cluster age. This suggestion is reinforced by data in Fig. 5, where we report the predicted luminosity function of the sequence, making clear the contribution given by the more massive dwarfs in the hook. The separation between the two sequences has been conventionally fixed at the point where effective temperatures start to increase again.

Fig. 5. Luminosity function of WD sequences for the 1, 3, 8 and 12 Gyr models. Dashed lines show the contribution of stars in the `hook'.

One finds that dwarfs are indeed accumulating at the bottom of the DMS, making its luminosity an easily detectable observational parameter. Such a feature is mantaineded even varying the assumptions about the IMF. The computations have been repeated assuming either a flat (exponent 0) or an extremely steep IMF (exponent 3.35). For example, Fig. 6 compares the luminosity functions of the WD for the age 8 Gyr and for the three quoted assumptions about the IMF exponent. One finds that the location of the maximum is not affected by the IMF slope, which governs the relative abundance of stars at the bottom of the sequence. As expected, for a given number of WDs, a flat IMF generates more objects in the range of more massive progenitors and, therefore, at the bottom of the sequence.

Fig. 6. Luminosity function for the labelled assumption on the IMF.

In conclusion, the exploration reported above indicates that WD sequences can be used, at least in principle, as indicators of cluster ages, provided that reliable cooling laws are available. From the best fit of data in Fig. 5 one finds a relation between the bottom luminosity of the WD sequence and the cluster age as given by:

According to the discussion given in the previous section this relation can be considered a safe upper limit for the age of clusters older than about 3 Gyr.

Note that the logarithmic representation adopted in Fig. 6 reveals that the slope of the luminous portion of the sequence is scarcely affected by the IMF. This has to be taken as evidence that such a distribution is essentially governed by the evolutionary time, whereas the IMF governs the relative abundance of these stars with respect to WD at the faint end.

However, informations on the IMF can be directly obtained from the abundance of WDs with respect to Helium burning stars. This is shown in Table 1, where we report theoretical predictions on this parameter as a function of both the cluster age and the IMF exponent. Data in Table 1 provide useful informations about the expected abundance of WDs in a field, showing that the number ratio between He burning stars and WD is a rather sensitive function of both the cluster age and the IMF slope.

Table 1. Expected number ratios for different assumption on the age and the IMF.

Finally, let us discuss the influence of metallicity upon the adopted H-burning and, in turn, upon our cooling predictions. All the predictions about the faint luminosity end of the cooling sequence are independent from these lifetimes, since they are a consequence of the fast evolution of the more massive stars. Thus, one expects that modifications in the pre-WD evolutionary lifetimes should only lead to marginal differences in the mass along the upper and intermediate portions of the cooling sequence.

© European Southern Observatory (ESO) 1999

Online publication: April 19, 1999
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