A&A Supplement series, Vol. 129, April II 1998, 267-279

Received February 27; accepted September 22, 1997

**S. Cassisi ^{,} , V. Castellani ^{,} ,
S. Degl'Innocenti ^{,} , and A. Weiss
**

*Send offprint request: *V. Castellani, Dipartimento di Fisica Università
di Pisa, piazza Torricelli 2, 56126 Pisa, Italy, vittorio@astrte.te.astro.it

Osservatorio Astronomico di Collurania, via Mentore Maggini I-64100 Teramo, Italy

Dipartimento di Fisica, Universitá de L'Aquila, via
Vetoio, 67010 L'Aquila, Italy

Dipartimento di Fisica dell'Universitá di Pisa, piazza Torricelli 2,
I-56126 Pisa, Italy

Istituto Nazionale di Fisica Nucleare,
Sezione di Ferrara, via Paradiso 12, I-44100 Ferrara, Italy

Max Plank Institut for Astrophysics, Karl Schwarzschild
strasse 1, D-85470 Garching b. Munchen, Germany

In the first part of this paper we revisit the history of theoretical
predictions for HB luminosities in old Population II stellar clusters,
starting from the results of "old" evolutionary computations to
introduce in various steps all the available "new" physics. We discuss
the influence of physical ingredients on selected evolutionary
parameters, finally presenting models which incorporate all the most
recent updating of the relevant physics. The evolutionary behavior of
such models is extensively investigated for selected choices about the
cluster metallicity, discussing theoretical predictions concerning
both cluster isochrones and the calibration of the parameter *R* in
terms of the original amount of He in stellar matter. One finds that
the "new" physics has a relevant influence on both these parameters,
moving cluster ages into a much better agreement with current
cosmological evaluations. This scenario is implemented by a further
set of stellar models where element diffusion is taken into
account. The comparison between theoretical scenarios with or without
diffusion is presented and discussed. A discussion of current
observational constraints in the light of the updated theory closes
the paper.

**keywords:**
stars: evolution; general; fundamental parameters; horizontal-branch

Since galaxies were born in an already expanding Universe, the age of the Universe appears as a safe upper limit for the age of any star and any stellar cluster. The fact that several determinations of globular cluster ages yielded values larger than the age of the Universe as based on current evaluations of the Hubble constant (see, e.g., Van den Bergh 1994; Tanvir et al. 1995) has stimulated a renewed interest in the theory of globular cluster Pop. II stars. At the same time, significant improvements in the input physics needed for stellar evolution have been made, such that noticeable changes of the theoretical results could be expected. These improvements initially were motivated by the results of helioseismology, which opened a new window into the interior of the Sun, allowing an extremely accurate determination of the solar structure. The efforts undertaken resulted in a new generation of opacity data (Rogers & Iglesias 1992; Seaton et al.\ 1994; Iglesias & Rogers 1996) and equations of state (Mihalas et al. 1990; Rogers et al.\ 1996), which led to a much better prediction of solar oscillations and also resolved many long-standing problems in our understanding of pulsating stars. In addition, helioseismology required particle diffusion to be taken into account in solar models (see Bahcall et al. 1995 and references therein).

The new opacities and equation of state, along with improvements in low-temperature opacities (e.g. Alexander & Ferguson 1994), nuclear cross-sections and neutrino emission rates, have now been applied to low-mass metal-poor stars in order to investigate the above-mentioned age problem. Several investigations (Chaboyer & Kim 1995; Mazzitelli et al. 1995: MDC; VandenBerg et al. 1996; D'Antona et al. 1997; Salaris et al. 1997: Paper I) have already shown that updated models predict lower cluster ages, thus decreasing the size of the discrepancy, if not resolving it. The new physics still needs to be applied to more massive and more metal-rich stars, although some of it, e.g. opacities, already are in use (Bono et al. 1997a,b) However, the full consequences of all improvements have not yet been evaluated. As an example we mention the evolution and pulsations of Cepheid stars.

In the present paper we are concerned with Pop. II stars only.
We have a twofold purpose. Firstly,
we present stellar models appropriate for globular cluster
studies that include *all* of the improvements listed above. These
models cover the complete relevant mass and metallicity range,
and include all evolutionary stages from the zero-age main sequence
until the end of the helium-burning phase on the horizontal
branch. Our calculations therefore provide the most up-to-date set of
stellar models applicable to all problems of globular cluster dating.
In particular, we show for the first time how particle diffusion
influences the evolution of low-mass stars until the end of the
horizontal-branch phase.

Secondly, we demonstrate how each of the various steps in
improving the input physics influences the models. This is important
because of the variety of calculations available in the literature
that include *some*
but not all of the new physics. In order to compare these results, it
is necessary to be able to translate the differences in physical
assumptions into differences in stellar properties.
In the first part of this paper we will approach this problem,
starting from a suitable set of "old'' evolutionary computations
and introducing, step by step, the available "new'' physics in order
to make
clear the influence of the new assumptions on selected evolutionary
parameters. At the end of Sect. 2, we will finally present our best models
which will incorporate the most recent improvements
in the relevant physics. However, these models will still be
calculated ignoring element diffusion.

In Sect. 3 evolutionary predictions for these best models are
investigated for selected choices of the cluster metallicity, presenting
theoretical predictions for cluster isochrones. This is repeated in Sect. 4
for a set of stellar models where element diffusion is
properly taken into account. The comparison between theoretical
scenarios with and without diffusion is presented and
discussed. Section 5 deals with a discussion of the
influence on the *R*-parameter and the consequences for the inferred
original amount of helium in stellar matter. The theoretical
uncertainties on *R* are critically discussed and final conclusion
given.

- 1. Input physics and population II models
- 2. "Best" canonical models
- 3. Element diffusion
- 4. The parameter
- 5. Conclusions
- References