Stellar models are assumed to be chemically homogeneous when they
settle on the zero age main sequence (ZAMS). The helium and metal mass
fractions, Y and Z, are chosen according to some fixed Y(Z)relation. In the present models, the values of
[Z=0.0004, Y=0.23],
[Z=0.001, Y=0.23],
[Z=0.004, Y=0.24], and
[Z=0.008, Y=0.25], were chosen in order to
coincide with the choices previously adopted in Bertelli et al. (1994, and references therein). On the other hand, the set with solar
composition was computed with
[Z=0.019, Y=0.273], since this
value of Y was fixed by the calibration of the solar model (see
Sect. 2.6 below). From these 5 pairs of [Z, Y] values, we
obtain an helium-to-metal enrichment relation which is
.
We then decide to adopt this mean relation for
super-solar metallicities. It gives origin to the values of
[Z=0.03,
Y=0.30] adopted in our set of highest metallicity.
For each value of Z, the fractions of different metals follow a solar-scaled distribution, as compiled by Grevesse & Noels (1993) and adopted in the OPAL opacity tables. The ratio between abundances of different isotopes is according to Anders & Grevesse (1989).
The radiative opacities are from the OPAL group
(Rogers & Iglesias 1992; Iglesias & Rogers 1993) for temperatures
higher than
,
and from
Alexander & Ferguson (1994) for
.
In the temperature interval
,
a
linear interpolation between the opacities derived from both
sources is adopted. We remind the reader that both opacities sources
provide values in good agreement in this temperature interval; the
relative differences in opacities are typically lower than 5 percent
(see Alexander & Ferguson 1994).
The conductive opacities of electron-degenerate matter are from Hubbard & Lampe (1969). We compared the tracks obtained with this prescription with more recent ones (Salasnich et al., in preparation) which use the Itoh et al. (1983) formulas. No significant differences in the evolutionary features turned out.
The equation of state (EOS) for temperatures higher than 107 K is that of a fully-ionized gas, including electron degeneracy in the way described by Kippenhahn et al. (1965). The effect of Coulomb interactions between the gas particles at high densities is introduced following the prescription by Straniero (1988). The latter was however adapted to the general case of a multiple-component plasma, as described in the Appendix of Girardi et al. (1996a).
For temperatures lower than 107 K, we adopt the detailed "MHD'' EOS of Mihalas et al. (1990, and references therein). The free-energy minimization technique used to derive thermodynamical quantities and derivatives for any input chemical composition, is described in detail by Hummer & Mihalas (1988), Däppen et al. (1988), and Mihalas et al. (1988). In our cases, we explicitly calculated EOS tables for all the Z values of our tracks, using the Mihalas et al. (1990) code. To save computer time, we consider only the four most abundant metal species, i.e. C, N, O, and Ne. For each Z, EOS tables for several closely spaced values of Y were computed, in order to cover the range of surface helium composition found in the evolutionary models before and after the first and second dredge-up events.
Alternatively, we computed some tracks using a much simpler EOS, where
the ionization equilibrium and thermo-dynamical quantities were derived
by solving a simple set of Saha equations for a H+He mixture (Baker &
Kippenhahn 1962). Comparison of these tracks with those obtained with
the MHD EOS revealed that no significant differences in effective
temperature or luminosity arise for dwarf stars of mass higher than
about 0.7 ,
or for giant stars of any mass. This is so because
only dwarf stars of lower mass present the dense and cold envelopes in
which the non-ideal effects included in the MHD EOS become
important. Moreover, only in the lowest-mass stars the surface
temperatures are low enough so that the H2 molecule is formed,
which has dramatic consequences for their internal structures (see
e.g. Copeland et al. 1970). Therefore, the use of the MHD EOS is
essential for our aim of computing stellar models with masses lower
than 0.6
(see also Girardi et al. 1996b).
The network of nuclear reactions we use involves all the important reactions of the pp and CNO chains, and the most important alpha-capture reactions for elements as heavy as Mg (see Bressan et al. 1993 and Maeder 1983 for details).
The reaction rates are from the compilation of
Caughlan & Fowler (1988), but for
and
,
for which we use the more recent determinations by
Landré et al. (1990). The
uncertain 12C(
)16O rate was set to 1.7 times
the values given by Caughlan & Fowler (1988), as indicated by the
study of Weaver & Woosley (1993) on the nucleosynthesis by massive
stars. The electron screening factors for all reactions are those
from Graboske et al. (1973).
The energy losses by pair, plasma, and bremsstrahlung neutrinos, important in the electron degenerate stellar cores, are taken from Munakata et al. (1985) and Itoh & Kohyama (1983).
The energy transport in the outer convection zone is described
according to the mixing-length theory of Böhm-Vitense (1958). The
mixing length parameter
is calibrated
by means of the solar model (see Sect. 2.6 below).
The extension of convective boundaries is estimated by means of an
algorithm which takes into account overshooting from the borders of
both core and envelope convective zones. The formalism is fully
described in Bressan et al. (1981) and Alongi et al. (1991).
The main parameter describing
overshooting is its extent
across
the border of the convective zone, expressed in units of pressure
scale heigth. Importantly, this parameter in the Bressan et al. (1981)
formalism is not equivalent to others found in literature. For
instance, the overshooting scale defined by
in
the Padova formalism roughly corresponds to the 0.25 pressure scale
heigth above the convective border, adopted by the Geneva group
(Meynet et al. 1994 and references therein)
to describe the same physical phenomenum, i.e.
.
The non-equivalency of the parameters
used to describe convective overshooting by different groups, has been
a recurrent source of misunderstanding in the literature.
We adopt the following prescription for the
parameter
as a function of stellar mass:
In the stages of core helium burning (CHeB), the value
is used for all stellar masses. This amount of overshooting
dramatically reduces the extent of the breathing pulses of convection
found in the late phases of CHeB (see Chiosi et al. 1992).
Overshooting at the lower boundary of
convective envelopes is also considered.
The calibration of the solar model required
an envelope overshooting not higher than 0.25 pressure scale
heigths. This value of
(see Alongi et al. 1991, for a description of the formalism) was then adopted for the
stars with
,
whereas
was
adopted for M<0.6
.
On the other hand, low values of
lead to the almost complete suppression of the
Cepheid loops in intermediate-mass models (Alongi et al. 1991).
Therefore, for
M>2.5
a value of
was assumed as in
Bertelli et al. (1994). Finally, for masses between 2.0 and
2.5
,
was let to increase gradually from 0.25
to 0.7.
We are well aware that the present prescription for the overshooting parameters seems not to fit on the ideals of simplicity and homogeneity one would like to find in such large sets of evolutionary tracks. However, they represent a pragmatic choice, since the prescriptions previously adopted were not satisfactory in many details.
The calibration of the solar model is an essential step in the computation of our evolutionary tracks. Some of the parameters found in the solar model are subsequentely adopted in all the stellar models of our data-base.
We adopt for the Sun the metallicity of Z=0.019, i.e. a value
almost identical to the
favoured by
Anders & Grevesse (1989). Several 1
models, for different values of
mixing-length parameter
and helium content
,
are let
to evolve up to the age of 4.6 Gyr. From this series of models, we are
able to single out the pair of
which allows
for a simultaneous match of the present-day solar radius and
luminosity,
and
.
An additional constraint
for the solar model comes from the helioseismological
determination of the depth of the solar convection zone, of
(Christensen-Dalsgaard et al. 1991).
It corresponds to a radius of
for
the lower boundary of the convective envelope. The
adoption of an overshooting parameter of
,
like in
Bertelli et al. (1994), would lead to a solar model with too a deep
convection zone, namely with
.
This parameter was then reduced to
,
which allows a reasonable reproduction of the observed value
of
.
Our final solar model reproduces well the Sun ,
,
and
values. From
this model, we derive the values of
,
,
and
,
used in other stellar models as described
in the previous subsections.
![]() |
Figure 3: The same as Fig. 2, but for [Z=0.030, Y=0.300] |
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