2. The theoretical scenario
According to current stellar evolution theories, all stars with mass inferior to a critical value (M) develop an electron-degenerate core at the end of their evolution. Among these objects, stars with an original mass below the Chandrasekhar limit (M1.4 M) end their life as cooling WDs, whereas the final fate of more massive stars depends upon the amount of mass lost during their evolution. If mass loss succeeds in lowering the original mass below the Chandrasekhar limit, the star will become a cooling white dwarf. Otherwise, C will ignite explosively, leading to the disruption of the structure (see, e.g., Iben & Renzini 1981 for a general discussion on the matter).
The upper mass limit for the production of WD is a widely debated topic in the literature, as well as the relation between the mass of the progenitor and the mass of the final cooling structures. Weidemann & Koester (1983, 1984), while discussing open cluster WDs, have estimated the upper mass limit of the WD progenitors at 8 M. For the exploratory investigation presented in this paper, we shall assume M 8 M as upper limit for the mass of our WD progenitors (see also Castellani et al. 1985). Later on, we shall find that this assumption has rather minor consequences on the investigated cooling scenario.
Theoretical ingredients to compute synthetic WD sequences can be found in the current literature. We shall take H-burning evolutionary lifetimes for metal poor stars with mass M from Chieffi & Straniero (1989), whereas for larger masses we will refer to the results of evolutionary computations by Cassisi et al. (1994). As shown in Fig. 1, the H-burning lifetimes can be arranged according to rather simple relations. By adopting two extreme assumptions about metallicity, H lifetimes are fitted within few percent by the relations:
where is the age (in years) at the RGB tip for a star of mass M (in solar masses). Note that the data in Fig. 1 imply that a cluster takes only about 30 Myr before starting to produce WDs. He burning lifetimes deserve a bit more attention. Following a well established theoretical scenario, for each given chemical composition one can define a critical mass Mcrit separating stellar structures which experience He ignition in the electron degenerate core of an H shell burning giant from more massive stars where He is quiescently ignited at the center. Below this limit He burning structures have rather constant He burning lifetimes (), as a consequence of the fairly constant mass of the He cores. According to the evaluation of Mcrit given in Cassisi et al. (1997), we shall assume =100 Myr for these masses, whereas for larger masses we take as a good approximation of data in the literature (see, e.g., Castellani et al. 1992). One finds that a similar estimate of reproduces theoretical evolutionary results, typically within 10 Myr.
For each given cluster age, one can compute synthetic WD sequences by distributing a sample of stars according to the assumed IMF, and evaluating the cooling time as given by the cluster age minus the sum of both H and He burning times. We shall assume the IMF to be a power law, characterized by its slope , where is the classical Salpeter (1955) value. According to the above discussion, one finds that the adopted procedure will be precise within 1% for cooling times of the order of 1 Gyr, the precision increasing for larger cooling times. Since WD cooling times depend on the mass of the cooling structure, one needs a further ingredient, as given by a relation between the mass of the MS progenitors and the final mass of the cooling WD structures. For this purpose, we shall adopt the relation presented by Wood (1992).
Throughout this paper, we shall adopt the cooling laws for the various masses as computed by Wood (1992). However, as Wood argues, cooling times are affected by uncertainties on both the chemical composition of the degenerated C-O cores and the depth of the external He layers. As a result, one can estimate that below log L= -4 for each given luminosity the cooling time is uncertain within, at least, 20%. Moreover, Wood's computations do not include nuclear shell sources, whereas Castellani et al. (1994) have found that these sources can substantially slow down the cooling above the quoted luminosity, depending on the amount of residual H-rich envelope.
These uncertainties on the cooling laws introduce corresponding uncertainties in theoretical predictions about the distribution of WD in actual clusters. Bearing in mind this occurrence, in the next section we shall examine the scenario disclosed by theoretical WD synthetic sequences as computed adopting pure carbon models from Wood (1992) with He envelopes as thin as M. With this choice we selected models with the longest cooling time, so that, for each given luminosity of the faint end of the WD sequence (see below), the predicted age can be regarded as an upper limit for the corresponding cluster age. This is not the case only for clusters in the age range 1-2 Gyr, where nuclear burnings can further slow the cooling and the predicted ages could be underestimated by up to 1 Gyr.
© European Southern Observatory (ESO) 1999
Online publication: April 19, 1999