4. The parameter

Since the pioneering paper by Iben (1968) it is known that evolutionary predictions on the evolution of Pop. II stars can be used to constrain the amount of original He in globular cluster stars. Calibrations of the R parameter, i.e., the number ratio between HB stars and RG more luminous than the HB luminosity level have been given by Buzzoni et al. (1983) and, more recently, by Caputo et al. (1987) and by Bono et al. (1995). According to current estimates, observational values for R appear to range around . In terms of the quoted calibrations this implies , which consequently is the value currently adopted in discussing globular cluster stars.

However, the evolutionary results discussed in the previous sections deeply affect such a scenario. We already found that the updated physics moderately increases theoretical expectations for HB luminosities, largely decreasing HB lifetimes. According to such evidence, one expects a decreasing value of R and thus a larger value of Y for any given value of R. Owing to the relevance of the argument, let us derive a quantitative evaluation of R as given by updated predictions about evolutionary times both along the RG and through the HB evolutionary phases. It has been already found that evolutionary times along the upper portion of the RG branch show a negligible dependence on both the chemical composition (within Pop. II limits) and mass of the evolving stars (see e.g. Castellani & Castellani 1993; Bono et al. 1995; Salaris & Cassisi 1997). Now we find a small but not negligible dependence on the efficiency of sedimentation. By best fitting computational results we find in the interval :

No Diffusion

Diffusion

where represents the time (in 10⁶ yr) spent by a RG above the luminosity L. However, when Z=0.006 these relations can be safely used only in clusters with age lower than, about, 13 Gyr. At larger ages, the clump of stars along the RG branch becomes fainter than the HB luminosity level, as shown by data in the previous Table 8 (click here), and the relations would require a correction to properly account for such an occurrence (see Bono et al. 1995 for a discussion on that matter).

According to the procedure adopted by Bono et al. (1995) we will take as reference the luminosity level of the ZAHB at , evaluating the time spent by RG stars above such a luminosity and taking HB evolutionary lifetimes from the models starting HB evolution at that effective temperature.

 

Z 0 . 0002 0 . 0002 0 . 001 0 . 001 0 . 006 0 . 006

Diffusion NO YES NO YES NO YES

. . . . . .

 1 . 759  1 . 744  1 . 701  1 . 687  1 . 594 1 . 574

76 . 16 71 . 33 83 . 70 81 . 13 93 . 00 91 . 30

73 . 77 . 24 79 . 52 85 . 22 68 . 3  83 . 20

R(3.83) 1 . 043 0 . 923 1 . 053 0 . 952 1 . 362 1 . 097

R(3.83)+0.05 1 . 142 1 . 010 1 . 395 1 . 034 1 . 525 1 . 203

R(3.83)+0.10 1 . 248 1 . 102 1 . 522 1 . 120 1 . 690 1 . 308

. . . . . . .

 

Table 10 (click here) gives data for these two ingredients together with the corresponding estimates of R for the labeled choices for the metallicity, with or without allowing for the efficiency of sedimentation. Top to bottom one finds: the luminosity (Log) of the ZAHB model at the time () spent by the same model during the central He burning (until the disappearance of the convective core), the time () spent by RGB stars above Log, the value (R(3.83)) of the corresponding R parameter and the same values when the ZAHB luminosity level is artificially increased by (R(3.83)+0.05) and 0.1 (R(3.83)+0.1). As already recognized, one sees that an increase of the metallicity tends to slightly increase the expectations on R for a given value of Y. Focusing our attention on the case Z=0.001, one finds that when Y=0.23 the theoretical prediction given by Bono et al. (1995), R= 1.19, should now be decreased to R= 1.05 for the model without sedimentation or to
R= 0.95 if sedimentation is taken into account. According to all available calibrations of R one finds . As a consequence, the present evolutionary scenario would predict that our current estimate of original He should be increased by about if sedimentation is neglected, or by about with sedimentation at work. As a result, observational data already interpreted in the literature as an evidence for Y= 0.23 should now lead to the rather unpalatable conclusion .

However, before entering on a discussion of the values in Table 10 (click here), one has to note that the calibration of R depends on He-burning evolutionary times which, in turn, are mainly governed by the poorly determined cross section for the ¹²C + reaction (see also Dorman 1992). Throughout this paper we adopted for He burning reactions the rates given by Caughlan & Fowler (1988) which should improve previous evaluations given by the same authors in 1985. Comparison between these two rates shows a rather negligible difference in the triple alpha rate, but a large decrease in the ¹²C + rate which, in turn, largely contributes to the decrease of HB lifetimes one finds in Table 1 (click here) between steps 4 and 7. As a matter of fact, about 60% of this decrease in HB lifetime (and of the corresponding decrease in the predicted value of R) can be attributed to these new rates. However, error estimates on such a cross section are still as large as a factor of two, including in this range also previous estimates given by Caughlan et al. (1985). Moreover, numerical experiments performed on HB models adopting recent reaction rates by Buchmann (1996), with error estimates still of about 70%, tend to move the lifetimes toward the values estimated in old computations, based on Caughlan et al. (1985). One can only conclude that theoretical calibrations of R in terms of Y are affected by too large errors to be useful for accurate calibrations of such a relevant parameter, and that the values of R given in Table 10 (click here) are still affected by theoretical errors corresponding to an error on Y of about . If one adds the further errors related to the observational procedure, i.e., the errors on the HB luminosity level, on the bolometric correction for the corresponding RG stars and on the star counts (see, e.g., Brocato et al. 1995) one should conclude that R still appears a too risky parameter to allow evaluations of Y with a reasonable accuracy.

The last two rows in Table 10 (click here) finally give theoretical evaluations for R when the adopted luminosity level is artificially increased above the ZAHB level by and 0.1, respectively. These values can be used to evaluate theoretical expectations on R when the mean luminosity of RR Lyrae is taken instead of the ZAHB luminosity as reference luminosity level. In the meantime these values give an estimate of the error on Y produced by observational errors at that level. One easily finds that an overestimate by (0.125 mag) will produce an overestimate of He by about . Note that previous evaluations of R appear only as a lower limit for theoretical expectations for clusters with a blue HB. Less massive, hot HB structures have He burning evolutionary times increased by 20% or more (see Fig. 4 (click here) and Castellani et al. 1994), with a corresponding increase in the theoretical expectation for the parameter R.