\section{Structural changes}

The appearance of a maximum in the melting line of hydrogen near 82~GPa has been explained by a softening of intermolecular interactions in the liquid \cite{bonev_nature_2004}, similar to what was observed in the orientationally-ordered phase of the solid (phase III) \cite{moshary_prb_1993}. This has prompted us to investigate the structure of the molecular liquid. We have discovered a new region in the molecular fluid in the vicinity of the melt line turnover that exhibits short range orientational ordering. It is revealed in the spatial distribution function, $SDF(r,\theta)$. Given an H$_2$ molecule, $SDF(r,\theta)$ is defined as the probability for finding a hydrogen atom at a distance $r$ from the molecular center of mass and at an angle $\theta$ with the molecular axis, normalized by the average H number density.

In Fig.~\ref{order}, we show a graphic of the $SDF$ for two densities along the same 1000~K isotherm; for clarity only the contributions from nearest-neighbors are shown. In the low density case (vertical plane), a molecule's nearest neighbor has an equal probability of being found at any $\theta$. At high pressure (horizontal plane), there is an orientational correlation between neighboring molecules. Particles are less likely to be found at the molecular poles, and more likely to be found near the equators. This can be further seen in the inset of Fig.~\ref{order}, where we have integrated over the radial degree of freedom, $\int \! SDF(r,\theta) \, dr = \overline{SDF}(\theta)$. To describe the evolution of the structural transition under compression, we define an order parameter as $\alpha = \overline{SDF}(\theta=90)/\overline{SDF}(\theta=0)$. A plot of $\alpha$ as a function of $r_s$ along the 1000~K isotherm (Fig.~\ref{asdf}) indicates a rapid increase in the vicinity of the previously reported \cite{bonev_nature_2004, shanti_prl_2008, eremets_jetp_2009} maximum in the melting curve. It is this change in the structure of the liquid, driven by a change in the intermolecular interactions, that is responsible for the turnover in the melting line.

\begin{figure}[tbh]
  \includegraphics[width=0.45\textwidth,clip]{fig3_order.eps}
  \caption{\label{order}Structural order parameter, $\alpha$, and spatial distributions functions, $SDF(r,\theta)$ and $\overline{SDF}(\theta)$ (see text for definitions), along the 1000~K isotherm in molecular H. The density where the melting curve has a maximum is indicated on the plot; it is in the region where $\alpha$ begins to increase rapidly. The graphic shows nearest neighbor contributions to $SDF(r,\theta)$ at low ($r_s= 2.40$, $P=6$~GPa) and high ($r_s = 1.45$, $P=170$~GPa) density. The $\overline{SDF}(\theta)$ further illustrate changes in the angular distribution with density.}
\end{figure}

A detailed quantitative description of the emerging orientational order will be presented in a follow up paper. Such analysis is expected to have relevance for identifying the finite (and possible zero) $T$ phases of the solid; similarities with phases
II and III of H are plausible. Here we mention only that in addition to the described dependence on $\theta$, there is a very strong tendency for neighboring molecules to lie in plane. For $\theta \sim 90^{\circ}$, the angle between molecular axes tends to be about 15$^{\circ}$ (for different $\theta$ it depends slightly on whether the molecules are tilted towards or away from each other).

One of the most prominent features of dense H is the sharp transition from molecular to atomic liquid. At sufficiently high density, the onset of dissociation leads to an EOS with a negative slope, i.e. $dP/dT |_{V=const} < 0$. There has been considerable debate as to whether the abrupt change in $P$ over a narrow $T$ range is due to a discontinuous ($1^{st}$ order) transition in the liquid and how it is affected by the presence of impurities (e.g. He). In what follows, we show that the anomalous features in the EOS, its density and impurity dependence, and its sensitivity to computational parameters can be understood on the basis of the changes in the liquid structure across the dissociation transition.

In order to describe the liquid structure in the presence of both atoms and molecules, we decompose $SDF(r,\theta)$ into atomic and molecular components. In both cases, $SDF(r,\theta)$ is calculated with respect to reference molecules center of mass. For the atomic $SDF(r,\theta)$, statistics for $r$ and $\theta$ are collected for single atoms only, while for the molecular one, the atoms at $r$ and $\theta$ are paired. In Fig.~\ref{asdf} we show such a decomposition at conditions corresponding to a large degree of local angular structure ($\alpha \sim 10$). At these $P$ and $T$, the liquid has just begun to dissociate ($\Pi(\tau) \approx 99\%$). The molecular $SDF$ is similar to that of a purely molecular liquid, with a peak near the reference molecule's equator. However, the atomic $SDF$ shows that unpaired atoms tend to shift towards the molecular poles. The atomic $\overline{SDF}(\theta)$ is peaked at $\theta \approx 50^{\circ}$ (and $130^{\circ}$) versus $90^{\circ}$ for the molecular $\overline{SDF}(\theta)$ [Fig.~\ref{asdf}(b)]. Similar analysis at low density shows no such angular dependence.

\begin{figure}[tbh]
  \includegraphics[width=0.45\textwidth,clip]{fig4_asdf.eps}
  \caption{\label{asdf}Molecular and atomic spatial distribution functions, $SDF(r,\theta)$ (a) and $\overline{SDF}(\theta)$ (b); see text for definitions. For clarity, only contributions from nearest neighbor atoms are included. Atoms forming molecules are more likely to reside near the reference molecule's equator. The distribution of unpaired atoms is more even and indicates that they are most likely to be found at $\theta \approx 50^{\circ}$ and $\theta \approx 130^{\circ}$.}
\end{figure}

At low $P$, where there is no short-range orientational order, the H$_2$ molecules are freely rotating and their packing is similar to that of spheres. Upon dissociation, the problem becomes that of spheres of different sizes, however, there is no apparent optimization of packing. On the other hand, because of covalent bonding, the effective volume of H$_2$ is less than that of two isolated hydrogen atoms. For these reasons, the dissociation transition at low density is characterized by $dP/dT |_{V=const} > 0$.

At high $P$, the molecules are no longer freely rotating and the packing problem becomes that of prolate spheroids (see graphic in Fig.~\ref{asdf}). As seen Fig.~\ref{asdf}, the molecular orientational order creates voids between them, which can be filled in by single atoms after dissociation. There will be a drop $P$ upon dissociation when the optimization in packing is sufficient to compensate for the higher effective volume of single atoms compared to that of molecules. This picture is consistent with the fact that the sharpness in the EOS increases with density \cite{bonev_prb_2004,miguel,caspersen,vorberger_prb_2007} - our order parameter follows the same trend. Furthermore, it has been noted that diluting the system with neutral impurities such as helium \cite{vorberger_prb_2007} softens the transition. We suggest that performing $SDF$ analysis on high density mixtures of molecular
hydrogen and helium (or other noble gases) will reveal similar angular local order, with the dopant atoms residing within the voids of the liquid. Finally, we note that in a one-dimensional hydrogen liquid, where angular order is not possible, $dP/dT |_{V=const}$ must always be positive.

Packing considerations also provide an explanation for the large sensitivity of the transition to the number of atoms, $N$, in the simulation supercell. We find that the degree of dissociation and related properties depend on $N$ in a non-intuitive way (similar observations have been made by Morales \cite{miguel}). At high densities, $\Pi(\tau)$ oscillates as $N$ is varied from 256, 512, 768, 1024. Surprisingly, simulations conducted with 256 atoms are in better agreement with those done with $N=1024$ than with $N=512$ ($\Pi_{N=128}=0.3338$, $\Pi_{N=256}=0.9842$, $\Pi_{N=512}=0.8328$, $\Pi_{N=768}=0.9660$, and $\Pi_{N=1024}=0.9656$ for $r_s=1.45$, $T=1000 K$). This effect \emph{cannot} be removed with a finer $k$-point sampling of the Brillouin zone. With $N=128$ the error in $P$ is as large as 8\% at $r_s$=1.40, $T$=750 K.

The origin of this initially puzzling behavior is that at a given density, $N$ defines the physical dimensions of the simulation box. Depending on how these dimensions relate to the correlation length of the liquid, different sized cells will have a slight bias towards one phase or another. Ideally one should use a value large enough that this effect is no longer an issue. Given the current cost of using large cells, we report values corresponding to $N=256$, as they are in relatively good agreement with $N=1024$ (the oscillation in $\Pi(\tau)$ is quite damped in this larger cell). Simulations performed with $N=256$ tend to favor the molecular phase relative to that of $N=1024$; this should partially correct for the underestimate of the dissociation barrier due to approximations of the DFT exchange-correlation potential.

The existence of short range angular structure in the liquid, specifically the different arrangement of molecules and atoms, suggests that something similar might also occur in the solid phase. In future searches for solid state structures it would be worthwhile to include variations of a two-component solid, comprised of molecules and atoms. Indeed, work by Pickard and Needs \cite{pickard_natphys_2007} indicates that mixed layered structures, comprised of molecules and atomic graphene-like sheets, remain energetically competitive over a wide range of pressures.

We now turn our discussion to the possiblity of a discontinous (\emph{i.e.} $1^{st}$ order) transition in the liquid. Based on our sampling of the phase diagram, the molecular-atomic transition appears to be continuous up to 200~GPa. We do not find a flat region in the $P(V)$ EOS - a signature of a $1^{st}$ order transition. Our data imply one of three possibilities: the transition is not $1^{st}$ order, it takes place outside of the $P$, $T$ space we considered, or it occurs over a range of densities too small to be resolved by our sampling. We can use our results, however, to establish a connection between the microscopic and macroscopic properties characterizing the transition and provide insight for the physical mechanism that could lead to it being $1^{st}$ order. Let $\delta V$ be the reduction in volume (due to packing) that can be achieved by dissociating a single molecule, while keeping $P$ and $T$ constant (the resulting state need not be in equilibrium). If a $1^{st}$ phase transition exists at these $P$ and $T$, we must have $-P\Delta V = \Delta U - T\Delta S$. Here $\Delta V$, $\Delta U$ and $\Delta S$ are the volume, energy and entropy differences across the transition. If it results in the dissociation of $n_d$ molecules, then we define $E_d \equiv \Delta U/n_d$, which has the physical meaning of dissociation energy. A $1^{st}$ order transition will take place if $n_d \delta V = - \Delta V$, which means $\delta V \approx E_d/P$ (assuming that $T\Delta S/P$ can be neglected, especially at high $P$ and low $T$). We note that $E_d$ \emph{decreases} under compression due to screening, while $\delta V$ \emph{increases} due to more pronounced orientational order. Thus, convergence of these two terms, $\delta V$ and $ E_d/P$, is increasingly likely at high $P$ and low $T$. Conclusive determination of the existence of a critical point will require very fine sampling of the phase diagram below 1000~K and above 200~GPa.

We have mapped the molecular-atomic transition of liquid hydrogen over a large pressure range. Our transition line correlates well with changes that can be observed in the electronic properties of the system such as the static dielectric constant. We predict that the liquid demonstrates significant short range orientational ordering. Its development coincides with the turnover in the melt line. The existence of this structural order is responsible for the large dissociation-induced $P$ drop in the EOS. Furthermore, it provides an explanation for the significant finite size effects that are present in this system. A $1^{st}$ order transition in the liquid, if it does exist, will likely occur at $T < $ 1000~K and $P > $ 200~GPa.

Work supported by NSERC, CFI, and Killam Trusts. Computational resources provided by ACEnet, Sharcnet, IRM Dalhousie, and Westgrid. We thank E. Schwegler, K. Caspersen, T. Ogitsu, and M. Morales for discussions and communicating unpublished results.


\section{Introduction}

Evolutionary tracks and isochrones are usually computed taking into
account a fixed law of helium to metal enrichment
$( \Delta Y/\Delta Z )$.   In
fact most of the available tracks follow a linear $Y(Z)$ relation of
the form $Y=Y_P+(\Delta Y/\Delta Z)\,Z$ (e.g. Bertelli et al. 1994
with $Y_{\rm p}=0.23,\Delta Y/\Delta Z=2.5$; or Girardi et al. 2000,
where $Y_{\rm p}=0.23,\Delta Y/\Delta Z=2.25$). Grids of stellar
models in general used $\Delta Y/\Delta Z$ values higher than $\sim 2$
in order to fit both the primordial and the solar initial He content.
On one side we recall that WMAP has provided a value of the primordial
helium ($Y_P\sim 0.248$, Spergel et al. 2003, 2007), significantly higher
than the value assumed in our previous stellar models.
On the other side the determinations of the helium enrichment from nearby
stars and K dwarfs or from Hyades binary systems show a large range of
values for this ratio (Pagel \& Portinari 1998, Jimenez et al. 2003,
Casagrande et al. 2007, Lebreton et al. 2001).
Therefore it is suitable to make available stellar models that can be used
to simulate different enrichment laws.

Several new results suggest that the naive assumption that the helium
enrichment law is universal, might not be correct. In fact there has been
recently evidence of significant variations in the helium content (and
perhaps of the age) in some globular clusters, that were traditionally
considered as formed of a simple stellar population of uniform age and
chemical composition.
 According to Piotto et al. (2005) and Villanova et al. (2007), only
greatly enhanced helium can explain the color
difference between the two main sequences in $\omega$ Cen. Piotto et
al.(2007) attribute the triple main sequence in NGC 2808 to successive
rounds of star formation, with different helium abundances.  In NGC
2808 a helium enhanced population could explain the MS spread and the HB
morphology ( Lee et al. 2005, D'Antona et al. 2005).

 In general the main sequence morphology can allow to
detect differences in helium content only when the helium content is
really very large. Many globular clusters might have a population with enhanced
helium (up to $Y= 0.3 - 0.33$), but only a few have a small population
with very high helium, such as suggested for NGC 2808.
For NGC 6441 (Caloi \& D'Antona 2007) a very
large helium abundance is required to explain part of the horizontal
branch population, that might be interpreted as due to self-enrichment
and multiple star formation episodes in the early evolution of
globular clusters.
The helium self-enrichment in globular clusters has been discussed in
several papers. Among them we recall Karakas et al. (2006) on the AGB
self-pollution scenario, Maeder \& Meynet (2006) on rotating low-metallicity
massive stars, while Prantzos and Charbonnel (2006) discuss the drawbacks
of both hypotheses.

 This short summary of recent papers displays
how much interest there is on the enriched helium content of some stellar
populations. This is the reason why we started these new sets of
evolutionary models allowing to simulate a large range of helium enrichment
laws.
There are a number of other groups producing extended databases of tracks and
isochrones both for scaled solar and for $\alpha$-enhanced chemical
compositions (Pietrinferni et al. 2004, 2006; Yi et al. 2001 and Demarque et
al. 2004; VandenBerg et al. 2006), but their models usually take into account
a fixed helium enrichment law.  Unlike these authors, very recently Dotter et
al. (2007) presented new stellar tracks and isochrones for three initial helium
 abundances in the range of mass between $0.1$ and $1.8 \mbox{$M_{\odot}$} $ for scaled
solar and $\alpha$-enhanced chemical compositions.

In this paper we present new
stellar evolutionary tracks for different values of the initial He
for each metal content. In fact homogeneous  grids of stellar evolution
models and isochrones in an extended region of the Z-Y plane and for a large
range of masses  are required
for the interpretation of photometric and spectroscopic observations of
resolved and unresolved stellar populations and for the  investigation of the
formation and evolution of galaxies.

Section 2 describes the input physics and the coverage of the Z-Y plane and
Section 3  the new synthetic TP-AGB models. Section 4 presents the
evolutionary tracks and relative electronic tables.
Section 5 describes the derived isochrones and the interpolation scheme;
Section 6 presents the comparison with other stellar models databases
and Section 7 the concluding remarks.

\section {Input physics and coverage of the $Z-Y$ plane}

The stellar evolution code adopted in this work is basically the same
as in Bressan et al. (1993), Fagotto et al. (1994a,b) and Girardi et
al. (2000), but for updates in the opacities, in the rates of energy
loss by plasma neutrinos (see Salasnich et al. 2000), and for the
different way the nuclear network is integrated (Marigo et al. 2001).
In the following we will describe the input physics, pointing out the
differences with respect to Girardi et al. (2000).

\subsection{Initial chemical composition}
\label{sec_chemic}

\begin{table}[!ht]
\caption{Combinations of $Z$ and $Y$ of the computed tracks}
\label{tab_comp}
\smallskip
\begin{center}
{\small
\begin{tabular}{ccccccc}
\hline
\noalign{\smallskip}
Z & Y1 & Y2 & Y3 & Y4 & Y5 & Y6\\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
0.0001 & 0.23 & 0.26 & 0.30 &      & 0.40 &      \\
0.0004 & 0.23 & 0.26 & 0.30 &      & 0.40 &      \\
0.001  & 0.23 & 0.26 & 0.30 &      & 0.40 &      \\
0.002  & 0.23 & 0.26 & 0.30 &      & 0.40 &      \\
0.004  & 0.23 & 0.26 & 0.30 &      & 0.40 &      \\
0.008  & 0.23 & 0.26 & 0.30 & 0.34 & 0.40 &      \\
0.017  & 0.23 & 0.26 & 0.30 & 0.34 & 0.40 &      \\
0.040  &      & 0.26 & 0.30 & 0.34 & 0.40 & 0.46 \\
0.070  &      &      & 0.30 & 0.34 & 0.40 & 0.46 \\
\noalign{\smallskip}
\hline
\end{tabular}
}
\end{center}
\end{table}

Stellar models are assumed to be chemically homogeneous when they
settle on the zero age main sequence (ZAMS).  Several grids of stellar
models were computed from the ZAMS with initial mass from $0.15$ to
$20 M_{\odot}$. The first release of the new evolutionary models, presented
in this paper, deals with very low and low mass stars up to
$2.5 M_{\odot}$.  In general for each chemical composition  the separation
in mass is of $0.05 M_{\odot}$ between $0.15$ and $0.6 M_{\odot}$, of $0.1
M_{\odot}$ between $0.6$ and $2.0 M_{\odot}$.  In addition evolutionary
models at $2.20$ and $2.50 M_{\odot}$ are computed in order to make
available isochrones  from the Hubble time down to 1 Gyr for each set of
models.
 The initial chemical composition is in the range $0.0001
\le Z \le 0.070$ for the metal content and for the helium content in
the range $0.23 \le Y \le 0.46$ as shown in Table 1. The
helium content $Y=0.40$ was supplemented at very low metallicities as
useful for simulations of significant helium enrichment in
globular clusters.

For each value of $Z$, the fractions of different metals follow a
scaled solar 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).

\subsection{Opacities}
\label{sec_opac}

The radiative opacities for scaled solar mixtures are from the OPAL
group (Iglesias \& Rogers 1996) for temperatures higher than $\log T =
4$, and the molecular opacities from Alexander \& Ferguson (1994) for
$\log T < 4.0$ as in Salasnich et al. (2000). In the temperature
interval $3.8<\log T < 4.0$, a linear interpolation between the
opacities derived from both sources is adopted. The agreement between
both tables is excellent and the transition is very smooth in this
temperature interval. For very high temperatures ($log T \ge 8.7$)
opacities by Weiss et al. (1990) are used.

The conductive opacities of electron-degenerate matter are from Itoh
et al. (1983), whereas in Girardi et al. (2000) conductive opacities
were from Hubbard \& Lampe (1969). A second difference with respect to
Girardi et al. (2000) lies in the numerical tecnique used to
interpolate within the grids of the opacity tables. In this paper we
used the two-dimensional bi-rational cubic damped-spline algorythm
(see Schlattl \& Weiss 1998, Salasnich et al. 2000 and Weiss \&
Schlattl 2000).

\subsection{Equation of state}
\label{sec_eos}

The equation of state (EOS) for temperatures higher than $10^7$~K is
that of a fully-ionized gas, including electron degeneracy in the way
described by Kippenhahn et al.\ (1967). The effect of Coulomb
interactions between the gas particles at high densities is taken into
account as described in Girardi et al.\ (1996).

For temperatures lower than $10^7$~K, the detailed ``MHD'' EOS of
Mihalas et al.\ (1990, and references therein) is adopted. 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), D\"appen et al.\
(1988), and Mihalas et al.\ (1988). In our cases, we explicitly
calculated EOS tables for all the $Z$ and $Y$ values of our tracks,
using the Mihalas et al.\ (1990) code, as in Girardi et al. (2000).
We recall that the MHD EOS is critical only for stellar models with
masses lower than 0.7~\mbox{$M_{\odot}$}, during their main sequence evolution. In
fact, only in dwarfs with masses lower than 0.7~\mbox{$M_{\odot}$}\ the surface
temperatures are low enough so that the formation of the H$_2$ molecule
dramatically changes the adiabatic temperature gradient. Moreover,
only for masses smaller than $\sim0.4$~\mbox{$M_{\odot}$}\ the envelopes become
dense and cool enough so that the many non-ideal effects included in
the MHD EOS (internal excitation of molecules, atoms, and ions;
translational motions of partially degenerate electrons, and Coulomb
interactions) become important.

\subsection{Reaction rates and neutrino losses}
\label{sec_rates}

The adopted network of nuclear reactions  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 Girardi et al. 2000).

The reaction rates are from the compilation of Caughlan \& Fowler
(1988), but for $^{17}{\rm O}({\rm
p},\alpha)^{14}{\rm N}$ and $^{17}{\rm O}({\rm p},\gamma)^{18}{\rm
F}$, for which we use the determinations by
Landr\'e et al.\ (1990). The
uncertain $^{12}$C($\alpha,\gamma$)$^{16}$O 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 adopted rate is consistent with the recent determination by
Kunz et al. (2002) within the uncertainties.
 The electron screening factors for all reactions are those
from Graboske et al.\ (1973).
The abundances of the various elements are evaluated with the aid of a
semi-implicit extrapolation scheme, as described in Marigo et al. (2001).

The energy losses by neutrinos are from Haft et al. (1994). Compared with the
previous ones by Munakata et al.\ (1985) used in Girardi et al. (2000),
neutrino cooling during the RGB is more efficient.

\subsection{Convection}
\label{sec_conv}

The extension of the convective regions is
determined considering the presence of convective overshoot.
The energy transport in the outer convection zone is described
according to the mixing-length theory (MLT) of B\"ohm-Vitense (1958). The
mixing length parameter $\alpha$ is calibrated by means of the solar model.

{\em Overshoot} The extension of the convective regions takes into account
overshooting from the borders of both core and envelope convective zones.
In the following we adopt the formulation by Bressan et al. (1981) in which
the boundary of the convective core is set at the layer where the velocity
(rather than the acceleration) of convective elements vanishes. Also this
non-local treatment of  convection requires a free parameter related to
 the mean free path $l$ of convective elements through $l=\Lambda_c H_p$,
($H_p$ being the pressure scale height).   The choice of this parameter
determines the extent of the overshooting
{\em across} the border of the classical core (determined by the Schwarzschild
 criterion).
 Other authors fix the extent
of the overshooting zone at the distance $d=\Lambda_c H_p$ from the border
of the convective core (Schwarzschild criterion).

The  $\Lambda_ c$ parameter in Bressan et al.(1981)
formalism is not equivalent to others found in literature. For
instance, the overshooting extension determined by $\Lambda_c =0.5$ with
the Padova formalism roughly corresponds to the   $d=0.25 H_p$
 {\em above} the convective border, adopted by the Geneva group
(Meynet et al.\ 1994 and references therein)
to describe the same physical phenomenum.
This also applies to the extent of overshooting in the case of models
computed by Teramo group (Pietrinferni et al. 2004), in the Yale-Yonsei
database (Yi et al. 2001 and Demarque et al. 2004), as they adopted the value
0.2 for the overshoot parameter {\em above} the border of the classical
convective core. In Victoria-Regina stellar models by
VandenBerg et al. (2006) a free parameter varying with mass is adopted to take
into account overshoot in the context of Roxburgh's equation for the maximum
size of a central convective zone, so the comparison is not so straight.
 The non-equivalency of the parameter
used to describe the extension of convective overshooting by different groups
 has been a recurrent source of misunderstanding in the literature.

We adopt the same prescription as in Girardi et al. (2000) for the
parameter $\Lambda_c$ as a function of stellar mass.
$\Lambda_c$ is set to zero for stellar masses $M\le1.0$~\mbox{$M_{\odot}$}, and
for $M\ge 1.5 \mbox{$M_{\odot}$}$, we adopt $\Lambda_c=0.5$,
i.e.\ a moderate amount of overshoting.
In the range $1.0<M<1.5 \mbox{$M_{\odot}$}$, we adopt a gradual increase of the overshoot
efficiency with mass, i.e.\ $\Lambda_c=M/\mbox{$M_{\odot}$} - 1.0$. This
because the calibration of the overshooting efficiency in this mass
range is still very uncertain and from observations there are indications
that this efficiency should be lower than in intermediate-mass stars.

In the stages of core helium burning (CHeB), the value $\Lambda_c=0.5$ is
used for all stellar masses. This amount of overshooting
significantly 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
height. This value of $\Lambda_e=0.25$ (see Alongi et al.\
1991) was then adopted for the
stars with $0.6\le(M/\mbox{$M_{\odot}$})<2.0$, whereas $\Lambda_e=0$ was
adopted for $M \la 0.6\mbox{$M_{\odot}$}$.
For $M>2.5\mbox{$M_{\odot}$}$ a value of $\Lambda_e=0.7$ was assumed as in
Bertelli et al.\ (1994) and Girardi et al. (2000). Finally, for masses
between 2.0 and $2.5 \mbox{$M_{\odot}$}$, $\Lambda_e$ was let to increase gradually
from 0.25 to 0.7, but for the helium burning evolution, where it is always
set to 0.7.

 The adopted approach to overshooting perhaps does not
represent the complex situation found in real stars. However it represents a
pragmatic choice, supported by comparisons with observations. We recall that
 also  the question of
the overshooting efficiency in stars of different masses is still a matter of
debate (see a recent discussion by Claret 2007, and by Deng and Xiong
2007).

{\em Helium semiconvection} As He-burning proceeds in the convective core of
low mass stars during the early stages of the horizontal branch (HB),
the size of the convective core increases.
Once the central value of helium falls below $Y_c =0.7$, the
temperature gradient reaches a local minimum, so that continued overshoot is
no longer able to restore the neutrality condition at the border of the core.
The core splits into an inner convective
core and an outer convective shell. As further helium is captured by the
convective shell, this latter tends to become stable, leaving behind a region
of varying composition in condition of neutrality.
This zone is called semiconvective.
Shortly, starting from the center and going outwards, the matter
in the radiatively stable region above the formal convective core is mixed
layer by layer until the neutrality condition is achieved. This picture holds
during most of the central He-burning phase.

The extension of the semiconvective region
varies with the stellar mass, being important in the low- and intermediate-mass
stars up to about $5 M_\odot$, and negligible in more massive stars.
We followed the scheme by Castellani et al. (1985), as described in Bressan
et al. (1993).

{\em Breathing convection} As central helium gets as low as $Y_c \simeq 0.1$,
the enrichment of fresh helium caused by semiconvective mixing enhances the
rate of energy produced by the $3\alpha$ reactions in such a way that the
radiative gradient at the convective edge is increased. A very rapid
enlargement followed by an equally rapid decrease of the fully convective core
takes place (pulse of convection). Several pulses may occur before the
complete exhaustion of the central helium content. This convective instability
was called breathing convection by Castellani et al. (1985).
While semiconvection is a true theoretical prediction, the breathing pulses
are most likely an artifact of the idealized algorithm used to describe mixing
(see Bressan et al. 1993 and references therein). In our code breathing pulses
are suppressed by imposing that the core cannot be enriched in helium by mixing
with the outer layers more than a fixed fraction F of the amount burnt by
nuclear reactions. In this way a time dependence of convection is implicitely
taken into account.

\subsection{Calibration of the solar model}
\label{sec_sun}

Recent models of the Sun's evolution and interior show that currently observed
photospheric abundances (relative to hydrogen) must be lower than those of the
proto-Sun because helium and other heavy elements have settled toward the Sun's
interior since the time of its formation about 4.6 Gyr ago.
The recent update of the solar chemical composition (Asplund et al 2005, 2006
and Montalban et al. 2006) has led to a decrease in the CNO and Ne abundances
of the order of $30 \%$. Their new solar chemical composition corresponds to
the values: $ X=0.7393, Y=0.2485, Z=0.0122$ with $Z/X=0.0165$.
The corresponding decrease in opacity increases
dramatically the discrepancies between the sound-speed derived from
helioseismology and the new standard solar model (SSM). Combinations of
increases in opacity and diffusion rates are able to restore part of the
sound-speed profile agreement, but the required changes are larger than the
uncertainties accepted for the opacity and the diffusion uncertainties.
Antia and Basu (2006) investigated the possibility of determining the solar
heavy-element abundances from helioseismic data, used the dimensionless
sound-speed derivative in the solar convection zone and gave a mean value of
$Z=0.0172 \pm 0.002$.

Bahcall et al.(2006) used Monte Carlo simulations to determine the
uncertainties
in solar model predictions of parameters measured by helioseismology (depth of
the convective zone, the surface helium abundance, the profiles of the sound
speed and density versus radius) and provided determinations of the correlation
coefficients of the predicted solar model neutrino fluxes. They incorporated
both the Asplund et al. recently determined heavy element abundances and the
higher abundances by Grevesse \& Sauval (1998). Their Table 7 points out that
the derived characteristic solar model quantities are significantly different
for the two cases.

According to VandenBerg et al. (2007) the new Asplund et al. metallicity for
the Sun presents some difficulties for fits of solar abundance models to the
M67 CMD, in that they do not predict a gap near the turnoff, which however is
observed. If the Asplund et al. solar abundances are correct, only those low-Z
solar models that treat diffusion may be able to reproduce the M67 CMD.

Owing to the many uncertainties,
we adopted for the Sun the initial metallicity of $Z=0.017$, according to
Grevesse \& Sauval (1998). Our choice is a compromise between the previous
solar metal content, Z=0.019 or 0.020, usually considered for evolutionary
models and the significantly lower value supported by Asplund et al.
(2005, 2006).

The usual procedure for the calibration of the solar model is of computing a
number of models while varying the MLT parameter
$\alpha$ and the initial helium content $Y_0$ until the observed solar
radius and luminosity are reproduced within a predetermined range. Several
1~\mbox{$M_{\odot}$}\ models, for different values of the
mixing-length parameter $\alpha$ and helium content $Y_\odot$, 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 $[\alpha, Y_\odot]$ which allows
for a simultaneous match of the present-day solar radius and
luminosity, $R_\odot$ and $L_\odot$.
An additional constraint
for the solar model comes from the helioseismological
determination of the depth of the outer solar convective zone
($R_c$ of the order of $0.710-0.716$ $R_\odot$) and from the surface helium
value.
The envelope overshooting parameter was adopted as $\Lambda_e=0.25$, which
allows a reasonable reproduction of the observed value of $R_c$ (depth of the
solar convective envelope).

Our final solar model reproduces well the solar $R_\odot$, $L_\odot$,
and $R_{\rm c}$ values. The differences with respect to observed values are
smaller than 0.2~\% for $R_\odot$ and $L_\odot$, and $\sim1$~\% for
$R_{\rm c}$. From
this model with initial $Z_{\odot}=0.017$ we derive the value of initial helium
 $Y_{\odot}=0.260$ and  $\alpha = 1.68$ (this value of the mixing-length
parameter was used in all our stellar models as described previously).

\subsection {Mass loss on the RGB}
\label{sec_rgbagb}

 Our evolutionary models were computed at constant mass for all stages
previous to the TP-AGB. However, mass loss by stellar wind during the RGB
of low-mass stars is taken into account at the stage of isochrone
construction. We use the empirical formulation by Reimers (1975), with the
mass loss rate expressed in function of the free parameter $\eta$,
by default assumed as 0.35 in our models (see Renzini \& Fusi-Pecci 1988).
The procedure is basically the following: we integrate the mass loss
rate along the RGB of every single track, in order to estimate the actual
mass of the correspondent model of ZAHB. See Bertelli et al. (1994) for
a more detailed description.

This approximation is a good one since the mass loss does not affect
significantly  the internal structure of models along the luminous part
of the RGB.

\section{New synthetic TP-AGB models}

An important update of the database of evolutionary tracks is the extension
of stellar models and isochrones until the end of the thermal pulses  along
the Asymptotic Giant Branch, particularly relevant for stellar population
analyses in the near-infrared, where the contribution of AGB stars to the
integrated  photometric properties  is significant.

The new synthetic TP-AGB models
have been computed with the aid of a synthetic code that has been
recently revised and updated in many aspects as described in the paper
by Marigo \& Girardi (2007), to which the reader should refer for
all details.
It should be emphasized that the synthetic TP-AGB model in use does not
provide a mere analytic description  of this phase, since one key
ingredient is
a complete static envelope model which is numerically integrated from the
photosphere down to the core. The basic structure of the envelope model
is the same as the one used in the Padova stellar evolution code.

The major improvements are briefly recalled below.
\begin{itemize}
\item
{\em{Luminosity.}} Thanks to high-accuracy formalisms (Wagenhuber
  \& Groenewegen 1998; Izzard et al. 2004) we follow the complex
  behaviour of the luminosity due to the flash-driven variations
 and the over-luminosity effect caused by hot-bottom burning (HBB).\\
\item
{\em{Effective temperature.}} One fundamental improvement is the
adoption of {\em molecular opacities} coupled to the actual surface
C/O ratio (Marigo 2002), in place of tables valid for scaled solar
chemical compositions (e.g. Alexander \& Ferguson 1994). The most
significant effect is the drop of the effective temperature as soon
as C/O increases above unity as a consequence of the third dredge-up,
in agreement with observations of carbon stars.\\
\item
{\em{ The third dredge-up.}} We use a more realistic description
of the process as a function of stellar mass and metallicity, with the
aid of available analytic relations of the characteristic parameters, i.e.
minimum core mass  $M_{\rm c}^{\rm min}(M,Z)$ and efficiency
$\lambda(M,Z)$, as derived from full AGB calculations (Karakas et al. 2002).\\
\item
{\em{ Pulsation properties.}} A first attempt is made
  to account for the switching between different  pulsation modes
  along the AGB evolution. Basing on available pulsation models for
  long period variables (LPV; Fox \& Wood 1982, Ostlie \& Cox 1986)  we
  derive a criterion on the luminosity to predict the transition from
the first overtone mode to the fundamental one (and viceversa).\\
\item
{\em{ Mass-loss rates.}} These are evaluated with the aid
of different formalisms, based on dynamical atmosphere models for
LPVs, depending not only on stellar mass, luminosity, effective
temperature, and  pulsation period, but also on the chemical type, namely:
Bowen \& Willson (1991) for C/O $<1$, and Wachter et al. (2002) for C/O $>1$.
\end{itemize}

It should be also emphasized that in Marigo \& Girardi (2007)
the free parameters
of the synthetic TP-AGB model -- i.e. the minimum core mass
$M_{\rm c}^{\rm  min}$
and efficiency $\lambda$ --  were calibrated as a function of stellar mass
and metallicity, on the base of two basic observables derived for
both Magellanic Clouds (MC), namely:
i) the counts of AGB stars (for both M- and C-types) in stellar
clusters and ii) the carbon star luminosity functions
in galaxy fields.
As illustrated by Girardi \& Marigo (2007), the
former observable quantifies the TP-AGB lifetimes for both
the oxygen- and carbon-rich phase as a function of stellar mass and
metallicity and it helps to determine $M_{\rm c}^{\rm  min}$. The
latter observable provides complementary constraints to both
$M_{\rm c}^{\rm  min}$ and $\lambda$ and their
dependence on stellar mass and metallicity.

The calibration carried out by Marigo \& Girardi (2007)
has been mainly motivated by the aim of
assuring that for that set of TP-AGB tracks,
the predicted luminosities and lifetimes,
hence the nuclear fuel of this phase, are correctly evaluated.

The same dredge-up parameters as derived by Marigo \& Girardi (2007)
have been adopted for computing the present sets
of TP-AGB tracks, despite of the fact that they
are chacterized by
different initial conditions at the first thermal pulse
(e.g. $M_{\rm c},\, T_{\rm eff},\, L$),  and  different
chemical compositions of the envelope. This
fact implies that the new sets of TP-AGB tracks are, to a certain extent,
un-calibrated since no attempt was made to have our basic set of
observables reproduced.

However, the choice to keep the same model paramemeters as derived from
the MG07 calibration is still a meaningful option since the present
TP-AGB models a) include already several important improvements in the
treatment of the TP-AGB phase, as recalled at the beginning of this
section; b) they constitute the reference grid from which we will
start the new calibration procedure. The preliminary step will be to
adopt a reasonable enrichment law, i.e. $Y_P$ (zero point) and the
$\Delta Y/\Delta Z$ (slope) so as to limit the calibration of the free
parameters to TP-AGB models belonging to a particular subset of
initial ($Y,\,Z$) combinations. This work is being undertaken.

\begin{figure*}
\begin{minipage}{0.45\textwidth} \noindent a)
\resizebox{\hsize}{!}{\includegraphics{fig1a.eps}}
\end{minipage}
\hfill
\begin{minipage}{0.45\textwidth} \noindent b)
\resizebox{\hsize}{!}{\includegraphics{fig1b.eps}}
\end{minipage}
\begin{minipage}{0.45\textwidth} \noindent c)
\resizebox{\hsize}{!}{\includegraphics{fig1c.eps}}
\end{minipage}
\hfill
\begin{minipage}{0.45\textwidth} \noindent d)
\resizebox{\hsize}{!}{\includegraphics{fig1d.eps}}
\end{minipage}
\caption{
Evolutionary tracks in the HR diagram, for the composition $[Z=0.001,
Y=0.23]$. For tracks of low-mass stars up to the RGB-tip
 and intermediate-mass ones ($M \le 2.5 M_{\odot}$) up to the bTP-AGB
(panel a),
 and from the ZAHB up to the bTP-AGB (panel b). In panels c) and d) tracks
are displayed for $[Z=0.001,Y=0.40]$
 }
\label{hrd_z001}
\end{figure*}

\begin{figure*}
\begin{minipage}{0.45\textwidth} \noindent a)
\resizebox{\hsize}{!}{\includegraphics{fig2a.eps}}
\end{minipage}
\hfill
\begin{minipage}{0.45\textwidth} \noindent b)
\resizebox{\hsize}{!}{\includegraphics{fig2b.eps}}
\end{minipage}
\begin{minipage}{0.45\textwidth} \noindent c)
\resizebox{\hsize}{!}{\includegraphics{fig2c.eps}}
\end{minipage}
\hfill
\begin{minipage}{0.45\textwidth} \noindent d)
\resizebox{\hsize}{!}{\includegraphics{fig2d.eps}}
\end{minipage}
\caption{
Evolutionary tracks in the HR diagram, for the composition $[Z=0.040,
Y=0.26]$. For tracks of low-mass stars up to the RGB-tip
 and intermediate-mass ones up to the bTP-AGB (panel a),
 and from the ZAHB up to the bTP-AGB (panels b). In panels c) and d) tracks
are displayed for $[Z=0.040,Y=0.40]$
 }
\label{hrd_z040}
\end{figure*}

\begin{figure*}
\begin{minipage}{0.45\textwidth} \noindent a)
\resizebox{\hsize}{!}{\includegraphics{fig3a.eps}}
\end{minipage}
\hfill
\begin{minipage}{0.45\textwidth} \noindent b)
\resizebox{\hsize}{!}{\includegraphics{fig3b.eps}}
\end{minipage}
\caption{
Evolutionary tracks in the HR diagram, for the composition $[Z=0.001,
Y=0.23$ (dot-dashed line) and $ Y=0.40$ (solid line)] in panel a).
 In panel b) tracks are displayed for $[Z=0.040, Y=0.26$ (dot-dashed line)
 and $Y=0.40$ (solid line)].
 }
\label{hrd_y23y40}
\end{figure*}

\section{Stellar tracks}
\label{sec_tracks}

\subsection{Evolutionary stages and mass ranges}
\label{sec_massranges}

Our models are evolved from the ZAMS and
the evolution through the whole
H- and He-burning phases is followed in detail. The
tracks are stopped at the beginning of the TP-AGB phase (bTP-AGB)
in intermediate- and low-mass stars. In the
case of stellar masses lower than 0.6~\mbox{$M_{\odot}$}, the main sequence
evolution takes place on time scales much longer than a Hubble time.
For them, we stopped the computations at an age of about 20~Gyr.

In low-mass stars with $0.6 M_{\odot} \le M \le M_{Hef}$ the evolution is
interrupted at the stage of He-flash in the electron degenerate
hydrogen-exhausted core.
The evolution is
then re-started from a ZAHB model with the same core mass and surface
chemical composition as the last RGB model.
For the core we compute the total energy difference between the last RGB
model and the ZAHB configuration and assume it has been provided by helium
burning during the helium flash phase. In this way a certain amount of the
helium in the core is converted into carbon (about 3 percent in mass, depending
on the stellar mass and initial composition),
and the initial ZAHB model takes into account the
approximate amount of nuclear fuel necessary to lift the core degeneracy
during the He-flash. The evolution is then followed along the HB up to the
bTP-AGB phase. We point out that this procedure is more detailed than
the one adopted in Girardi et al. (2000), where the fraction of He core burnt
during the He-flash was assumed to be constant and equal to 5\%.

In intermediate-mass stars, the evolution goes from the ZAMS up to
either the beginning of the TP-AGB phase, or to the carbon ignition in
our more massive models (in this paper we consider masses up to $2.5~\mbox{$M_{\odot}$}$).

Table 2 gives the values of the transition mass
\mbox{$M\sub{Hef}$}\, as derived from the present tracks. \mbox{$M\sub{Hef}$}\ is
the maximum mass for a star to develop an electron-degenerate core
after the main sequence, and sets the limit between low- and
intermediate-mass stars (see e.g. Chiosi et al.\ 1992).
Given the low mass separation between the tracks we computed,
the \mbox{$M\sub{Hef}$}\ values here presented are uncertain by about 0.05~\mbox{$M_{\odot}$},

\begin{table}
\caption{The transition mass \mbox{$M\sub{Hef}$}}
\label{tab_mhef}
\begin{tabular}{lllllll}
\noalign{\smallskip}\hline\noalign{\smallskip}
 & \multicolumn{6}{c}{$Y$} \\
 \cline{2-7}
\noalign{\smallskip}
$Z$ & 0.23 & 0.26 & 0.30 & 0.34 & 0.40 & 0.46 \\
\noalign{\smallskip}\hline\noalign{\smallskip}
0.0001 & 1.70 & 1.70 & 1.60 &      & 1.40 &      \\
0.0004 & 1.70 & 1.60 & 1.50 &      & 1.40 &      \\
0.001 & 1.60 & 1.60 & 1.50 &      & 1.40 &     \\
0.002 & 1.70 & 1.60 & 1.50 &      & 1.40 &     \\
0.004 & 1.80 & 1.75 & 1.60 &      & 1.40 &     \\
0.008 & 1.90 & 1.85 & 1.75 & 1.60 & 1.40 &     \\
0.017 & 2.10 & 2.00 & 1.90 & 1.70 & 1.60 &     \\
0.040 &      & 2.20 & 2.05 & 1.90 & 1.70 & 1.40\\
0.070 &      &      & 2.05 & 1.80 & 1.60 & 1.40\\
\noalign{\smallskip}
\hline
\end{tabular}
\end{table}

\subsection{Tracks in the HR diagram}
\label{sec_hrd}

The complete set of tracks for very low-mass stars ($M<0.6$~\mbox{$M_{\odot}$}) are
computed starting at
a stage identified with the ZAMS, and end at the age of 20 Gyr. The ZAMS
model is defined to be the stage of minimum \mbox{$T\sub{eff}$}\ along the computed
track; it follows a stage of much faster evolution in which the pp-cycle
is out of equilibrium, and in which gravitation may provide a
non negligible fraction of the radiated energy.
It is well known that these stars evolve very little during the Hubble time.

We display in Figures 1  and 2
only some examples of the sets of
the computed evolutionary tracks (i.e. $[Z=0.001$ for $Y=0.23$ and $Y=0.40]$
and $[Z=0.040$ for $Y=0.26$ and $Y=0.40]$). In these
figures, panel a) and c)  present the tracks for masses between 0.6 and
2.5  $M_{\odot}$ from the ZAMS up to the RGB-tip, or to the bTP-AGB, while in
 panel b) and d) the low-mass tracks from the ZAHB up to the bTP-AGB phase are
plotted from $0.55 M_{\odot}$ to $M_{Hef}$.
Figure 3 is aimed to illustrate the range of temperatures and
luminosities involved for the boundary values of helium content at given
metallicity in the theoretical HR diagram.
The reader can notice, for instance, that at low metallicity and high helium
(Z=0.001, Y=0.040) loops begin to be present during the core He burning phase,
while they are
practically missing in the $Z=0.040$ ones for masses about $2.5 M_{\odot}$.

\subsection{Description of the tables}
\label{sec_tabletrack}

The data tables for the present evolutionary tracks are available only
in electronic format.  A website with the complete data-base
(including additional data and future
extensions) will be mantained at \verb$http://stev.oapd.inaf.it/YZVAR$.

For each evolutionary track, the corresponding data file presents
23 columns with the following information:
	\begin{enumerate}
	\item \verb$n$: row number;
	\item \verb$age/yr$: stellar age in yr;
	\item \verb$logL$: logarithm of surface luminosity (in solar units),
\mbox{$\log(L/L_{\odot})$};
	\item \verb$logTef$: logarithm of effective temperature (in K),
\mbox{$\log T\sub{eff}$};
	\item \verb$grav$: logarithm of surface gravity ( in cgs units);
	\item \verb$logTc$: logarithm of central temperature (in K);
	\item \verb$logrho$: logarithm of central density (in cgs units);
	\item \verb$Xc$: mass fraction of hydrogen in the stellar centre;
	\item \verb$Yc$: mass fraction of helium in the stellar centre;
	\item \verb$Xc_C$: mass fraction of carbon in the stellar centre;
	\item \verb$Xc_O$: mass fraction of oxygen in the stellar centre;
	\item \verb$Q_conv$: fractionary mass of the convective core;
	\item \verb$Q_disc$: fractionary mass of the first mesh point where
the chemical composition differs from the surface value;
	\item \verb$L_H/L$: fraction of the total luminosity provided by
H-burning reactions;
	\item \verb$Q1_H$: fractionary mass of the inner border of the
H-rich region;
	\item \verb$Q2_H$: fractionary mass of the outer border of the
H-burning region;
	\item \verb$L_He/L$: fraction of the total luminosity provided by
He-burning reactions;
	\item \verb$Q1_He$: fractionary mass of the inner border of the
He-burning region;
	\item \verb$Q2_He$: fractionary mass of the outer border of the
He-burning region;
	\item \verb$L_C/L$: fraction of the total luminosity provided by
C-burning reactions;
	\item \verb$L_nu/L$: fraction of the total luminosity lost by
neutrinos;
	\item \verb$Q_Tmax$: fractionary mass of the point with the highest
temperature inside the star;
	\item \verb$stage$: label indicating particular evolutionary stages.
\label{item_stage}
	\end{enumerate}

A number of evolutionary stages are indicated along the tracks
(column 23). They correspond either to: the
beginning/end of main evolutionary stages, local maxima and minima of
$L$ and \mbox{$T\sub{eff}$}, and main changes of slope in the HR diagram. These
particular stages were, in general, detected by an authomated
algorithm. They can be useful for the derivation of physical
quantities (as e.g.\ lifetimes) as a function of mass and metallicity,
and are actually used as equivalent evolutionary points in our
isochrone-making routines.

For TP-AGB tracks during the quiescent stages of evolution preceding He-shell
flashes, our tables provide:
     \begin{enumerate}
     \item\verb$n$: row number;
     \item\verb$age/yr$: stellar age in yr;
     \item\verb$logL$: logarithm of surface luminosity (in solar units),
\mbox{$\log(L/L_{\odot})$};
     \item\verb$logTef$: logarithm of effective temperature (in K),
\mbox{$\log T\sub{eff}$};
     \item\verb$Mact$: the current mass (in solar units);
     \item\verb$Mcore$: the mass of the H-exhausted core (in solar units);
     \item\verb$C/O$: the surface C/O ratio.
     \end{enumerate}

\subsection{Changes in surface chemical composition}
\label{sec_chemical}

The surface chemical composition of the stellar models changes
on two well-defined dredge-up events. The first one occurs at
the first ascent of the RGB for all stellar models (except for
the very-low mass ones which are not evolved out of the main
sequence). The second dredge-up is found after the core
He-exhaustion, being remarkable only in models with
$M\ga3.5$~\mbox{$M_{\odot}$}. We provide tables with the surface chemical
composition of H, $^3$He, $^4$He, and main CNO isotopes, before
and after the first dredge-up, as in this paper we present models
from $0.15$ up to $2.5 M_{\odot}$.
Table 3 shows, as an example, the surface abundances for
the chemical composition Z=0.008 and Y=0.26.

\begin{table*}
\caption{Surface chemical composition (by mass fraction) of
$[Z=0.008, Y=0.26]$ models.}
\label{tab_du}
\begin{tabular}{lllllllllll}
\noalign{\smallskip}\hline\noalign{\smallskip}
$M/\mbox{$M_{\odot}$}$ &	 H  &    $^3$He  &      $^4$He  &  $^{12}$C   &    $^{13}$C   &    $^{14}$N   &    $^{15}$N  &     $^{16}$O   &    $^{17}$O  &     $^{18}$O \\
\noalign{\smallskip}\hline\noalign{\smallskip}
\multicolumn{11}{l}{Initial:}   \\
  all   & 0.732 & 2.78$\:10^{-5}$ &  0.260 & 1.37$\:10^{-3}$ & 1.65$\:10^{-5}$ & 4.24$\:10^{-4}$ & 1.67$\:10^{-6}$ & 3.85$\:10^{-3}$ & 1.56$\:10^{-6}$ & 8.68$\:10^{-6}$ \\
\noalign{\smallskip}\hline\noalign{\smallskip}
\multicolumn{11}{l}{After the first dredge-up:} \\

  0.55 &  0.719 & 2.65$\:10^{-3}$ &  0.270 & 1.27$\:10^{-3}$ & 3.99$\:10^{-5}$ & 5.10$\:10^{-4}$ & 1.34$\:10^{-6}$ & 3.85$\:10^{-3}$ & 1.57$\:10^{-6}$ & 8.55$\:10^{-6}$ \\
  0.60 &  0.719 & 2.94$\:10^{-3}$ &  0.270 & 1.26$\:10^{-3}$ & 2.91$\:10^{-5}$ & 5.44$\:10^{-4}$ & 1.41$\:10^{-6}$ & 3.85$\:10^{-3}$ & 1.66$\:10^{-6}$ & 8.31$\:10^{-6}$ \\
  0.65 &  0.715 & 2.34$\:10^{-3}$ &  0.274 & 1.30$\:10^{-3}$ & 3.49$\:10^{-5}$ & 4.91$\:10^{-4}$ & 1.38$\:10^{-6}$ & 3.85$\:10^{-3}$ & 1.59$\:10^{-6}$ & 8.52$\:10^{-6}$ \\
  0.70 &  0.715 & 2.40$\:10^{-3}$ &  0.274 & 1.29$\:10^{-3}$ & 3.27$\:10^{-5}$ & 4.96$\:10^{-4}$ & 1.40$\:10^{-6}$ & 3.85$\:10^{-3}$ & 1.63$\:10^{-6}$ & 8.45$\:10^{-6}$ \\
  0.80 &  0.712 & 1.83$\:10^{-3}$ &  0.278 & 1.26$\:10^{-3}$ & 4.48$\:10^{-5}$ & 5.26$\:10^{-4}$ & 1.31$\:10^{-6}$ & 3.85$\:10^{-3}$ & 1.61$\:10^{-6}$ & 8.46$\:10^{-6}$ \\
  0.90 &  0.709 & 1.43$\:10^{-3}$ &  0.281 & 1.18$\:10^{-3}$ & 4.52$\:10^{-5}$ & 6.13$\:10^{-4}$ & 1.22$\:10^{-6}$ & 3.85$\:10^{-3}$ & 1.68$\:10^{-6}$ & 8.24$\:10^{-6}$ \\
  1.00 &  0.708 & 1.17$\:10^{-3}$ &  0.283 & 1.12$\:10^{-3}$ & 4.55$\:10^{-5}$ & 6.86$\:10^{-4}$ & 1.15$\:10^{-6}$ & 3.85$\:10^{-3}$ & 1.73$\:10^{-6}$ & 7.92$\:10^{-6}$ \\
  1.10 &  0.708 & 9.56$\:10^{-4}$ &  0.283 & 1.07$\:10^{-3}$ & 4.54$\:10^{-5}$ & 7.40$\:10^{-4}$ & 1.09$\:10^{-6}$ & 3.85$\:10^{-3}$ & 1.84$\:10^{-6}$ & 7.64$\:10^{-6}$ \\
  1.20 &  0.710 & 8.20$\:10^{-4}$ &  0.281 & 1.03$\:10^{-3}$ & 4.56$\:10^{-5}$ & 7.90$\:10^{-4}$ & 1.03$\:10^{-6}$ & 3.85$\:10^{-3}$ & 2.25$\:10^{-6}$ & 7.40$\:10^{-6}$ \\
  1.30 &  0.711 & 7.25$\:10^{-4}$ &  0.280 & 9.92$\:10^{-4}$ & 4.60$\:10^{-5}$ & 8.36$\:10^{-4}$ & 9.80$\:10^{-7}$ & 3.85$\:10^{-3}$ & 2.46$\:10^{-6}$ & 7.19$\:10^{-6}$ \\
  1.40 &  0.712 & 6.41$\:10^{-4}$ &  0.279 & 9.50$\:10^{-4}$ & 4.59$\:10^{-5}$ & 8.85$\:10^{-4}$ & 9.25$\:10^{-7}$ & 3.85$\:10^{-3}$ & 3.14$\:10^{-6}$ & 6.94$\:10^{-6}$ \\
  1.50 &  0.713 & 5.71$\:10^{-4}$ &  0.279 & 9.25$\:10^{-4}$ & 4.56$\:10^{-5}$ & 9.19$\:10^{-4}$ & 9.00$\:10^{-7}$ & 3.84$\:10^{-3}$ & 1.12$\:10^{-5}$ & 6.78$\:10^{-6}$ \\
  1.60 &  0.712 & 5.22$\:10^{-4}$ &  0.280 & 9.05$\:10^{-4}$ & 4.54$\:10^{-5}$ & 9.68$\:10^{-4}$ & 8.81$\:10^{-7}$ & 3.80$\:10^{-3}$ & 1.50$\:10^{-5}$ & 6.68$\:10^{-6}$ \\
  1.70 &  0.711 & 4.46$\:10^{-4}$ &  0.281 & 8.92$\:10^{-4}$ & 4.52$\:10^{-5}$ & 1.03$\:10^{-3}$ & 8.63$\:10^{-7}$ & 3.75$\:10^{-3}$ & 1.83$\:10^{-5}$ & 6.57$\:10^{-6}$ \\
  1.80 &  0.709 & 3.99$\:10^{-4}$ &  0.283 & 8.81$\:10^{-4}$ & 4.51$\:10^{-5}$ & 1.09$\:10^{-3}$ & 8.50$\:10^{-7}$ & 3.70$\:10^{-3}$ & 1.59$\:10^{-5}$ & 6.50$\:10^{-6}$ \\
  1.85 &  0.707 & 3.73$\:10^{-4}$ &  0.284 & 8.77$\:10^{-4}$ & 4.44$\:10^{-5}$ & 1.13$\:10^{-3}$ & 8.49$\:10^{-7}$ & 3.66$\:10^{-3}$ & 1.64$\:10^{-5}$ & 6.46$\:10^{-6}$ \\
  1.90 &  0.706 & 3.54$\:10^{-4}$ &  0.285 & 8.77$\:10^{-4}$ & 4.43$\:10^{-5}$ & 1.14$\:10^{-3}$ & 8.46$\:10^{-7}$ & 3.64$\:10^{-3}$ & 1.81$\:10^{-5}$ & 6.44$\:10^{-6}$ \\
  2.00 &  0.706 & 3.19$\:10^{-4}$ &  0.285 & 8.71$\:10^{-4}$ & 4.42$\:10^{-5}$ & 1.17$\:10^{-3}$ & 8.43$\:10^{-7}$ & 3.62$\:10^{-3}$ & 1.47$\:10^{-5}$ & 6.40$\:10^{-6}$ \\
  2.20 &  0.702 & 2.58$\:10^{-4}$ &  0.289 & 8.55$\:10^{-4}$ & 4.44$\:10^{-5}$ & 1.26$\:10^{-3}$ & 8.22$\:10^{-7}$ & 3.54$\:10^{-3}$ & 1.37$\:10^{-5}$ & 6.31$\:10^{-6}$ \\
  2.50 &  0.699 & 2.00$\:10^{-4}$ &  0.292 & 8.42$\:10^{-4}$ & 4.43$\:10^{-5}$ & 1.33$\:10^{-3}$ & 8.07$\:10^{-7}$ & 3.49$\:10^{-3}$ & 1.06$\:10^{-5}$ & 6.21$\:10^{-6}$ \\
 \noalign{\smallskip}\hline\noalign{\smallskip}
\end{tabular}
\end{table*}

\begin{figure}
 \resizebox{\hsize}{!}{\includegraphics{fig4.eps}}
\caption{Isochrones for a chemical composition intermediate among those of the
computed tracks (Z=0.0003,Y=0.25) at Log Age= 10, 9.7, 9.4, 9.0, 8.9 and 8.7
years.The new synthetic TP-AGB models allow the extension of the isochrones
(red) until the end of the thermal pulses along the AGB (black).
}
\label{int_iso1}
\end{figure}

\begin{figure}
 \resizebox{\hsize}{!}{\includegraphics{fig5.eps}}
\caption{Isochrones for the chemical composition Z=0.019 and Y=0.28,
intermediate among those of the computed tracks at Log Age= 10, 9.7, 9.4,
9.0 years. The same as Figure 4 for the AGB.
}
\label{int_iso2}
\end{figure}

\begin{figure}
 \resizebox{\hsize}{!}{\includegraphics{fig6.eps}}
\caption{Comparison of isochrones for the same Z=0.001 and different helium
content. Solid lines correspond to Y=0.23, dashed ones to Y=0.40.
Also in this case the isochrones are extended until the end of the thermal
pulses along the AGB. }
\label{isozy1y2}
\end{figure}

\section{Isochrones}
\label{sec_isochrones}

From the tracks presented in this paper, we have constructed isochrones
updating and improving the algorithm of ``equivalent evolutionary points''
used in Bertelli et al.\ (1994) and Girardi et al. (2000).
The initial point of each isochrone is the 0.15~\mbox{$M_{\odot}$}\ model in the
lower main sequence. The terminal stage of the isochrones is
the tip of the TP-AGB for the evolutionary tracks presented in this first
paper (up to 2.5 $M_{\odot}$). We recall that usually there are small
differences in the logarithm of the effective temperature between the last
model of the track after the central He-exhaustion and the starting point
of the synthetic TP-AGB model of the correspondent mass. Constructing the
isochrones we removed these small discontinuities with a suitable shift.
 The differences arise as the Alexander \& Ferguson (1994) low-temperature
opacity tables, used for the evolutionary models, are replaced by those
provided by Marigo's (2002) algorithm from the beginning to the end of the
TP-AGB phase.

In Figures 4 and 5 isochrones are shown for the chemical composition
(Z=0.0003,Y=0.25) and (Z=0.019,Y=0.28)  intermediate
among those of the evolutionary tracks.
The interpolation method to obtain the isochrone for a specific composition
intermediate between the values of the computed tracks is described in the
following subsection.
In Figures 4, 5 and 6 the plotted isochrones are extended until the end of
the  TP-AGB phase and a different line color points out this phase
in  the theoretical HR diagram. The flattening of the AGB phase of the
isochrones marks the transition from M stars to Carbon stars ($C/O > 1$).

An increase of the helium content at the same metallicity in stellar
models causes a decrease in the mean opacity and an increase in the
mean molecular weight of the envelope (Vemury and Stothers, 1978), and
in turn higher luminosities, hotter effective temperatures and shorter
hydrogen and helium lifetimes of stellar models.
 Apparently in contradiction with the previous statement of
higher luminosity for helium increase in evolutionary tracks, the evolved
portion of the isochrones with lower helium are more luminous, as shown in
figure 6 where we plot isochrones with Z=0.001 for Y=0.23 and Y=0.40.
This effect is related to the
interplay between the increase in luminosity and the decrease in lifetime
of stellar models with higher helium content (at the same mass and
metallicity).

\subsection{Interpolation scheme}

The program ZVAR, already used in many papers (for instance in Ng et
al. 1995, Aparicio et al. 1996, Bertelli \& Nasi 2001, Bertelli et
al. 2003) has been extended to obtain isochrones and to simulate
stellar populations in a large region of the $Z-Y$ plane (now its
name is YZVAR).
For each of a few discrete values of the metallicity of former
evolutionary tracks there was only one value of the helium content,
derived from the primordial helium with a fixed enrichment law.  In
the present case we deal with 39 sets of stellar tracks in the
plane $Z-Y$ and we aim at obtaining isochrones for whatever $Z-Y$
combination and stellar populations with the required Y(Z)
enrichment laws.  This problem requires a double interpolation in Y and
Z. We try to describe our method with the help of
Figure 7. In this figure the corners of the box
represent 4 different chemical compositions in a generic mesh of the
grid, i.e. $( Y_1,Z_1)$, $( Y_2,Z_1)$, $( Y_2,Z_2)$, $( Y_1,Z_2)$.

\begin{figure}
 \resizebox{\hsize}{!}{\includegraphics{fig7.eps}}
\caption{Interpolation scheme inside the range [Z1,Z2] and [Y1,Y2].}
\label{interpolation}
\end{figure}

The vertical lines departing from the corners represent the sequences of
increasing mass of the tracks computed for the corresponding values of
$(Y,Z)$.  Along each of these lines
the big points represent the separation masses  between tracks with a
different number of {\em characteristic} points ( a different number of
equivalent evolutionary phases), indicated as
 $M_{Y_1Z_1{\rm sep}}^{i}$, $M_{Y_2Z_1{\rm sep}}^{i}$,
$M_{Y_1Z_2{\rm sep}}^{i}$, $M_{Y_2Z_2{\rm sep}}^{i}$.
The separation masses are marked with the indexes $i$ and $i+1$ in
Figure 7.
 Actually three intervals of masses are shown in
this figure, marked with big numbers, identifying the iso-phase intervals,
which means that inside each of them the tracks are
characterized by the same number of equivalent evolutionary phases
\footnote{With phase we mean a particular stage of the evolution identified
by common properties both in the C-M diagram and in the internal structure of
the models.}. The different phases along the tracks are separated by
{\em characteristic} points, in the following indicated by the index $k$.

For a given mass $M$, age $t$ and  chemical composition $(Y,Z)$ our
interpolation scheme has
to determine the luminosity and the effective temperature of the star. This is
equivalent to assert that we look for the corresponding interpolated track.
The interpolation must be done inside  the appropriate iso-phase intervals of
mass.
For this reason our first step is the identification of the separation masses
($M_{{\rm sep}}^{i},i=1,\ldots,n-1$), where $n$ marks the number of
 iso-phase intervals (n=3 for the case of figure 7). To do this we
interpolate in $Y$
between $M_{Y_1Z_1{\rm sep}}^{i}$ and $M_{Y_2Z_1{\rm sep}}^{i}$ obtaining
$M_{Z_1{\rm sep}}^{i}$. Analogously we obtain $M_{Z_2{\rm sep}}^{i}$
 for the $Z_2$ value. Interpolating  in $\log Z$ between $M_{Z_1{\rm sep}}^{i}$
and $M_{Z_2{\rm sep}}^{i}$  we obtain $M_{{\rm sep}}^{i}$. Analogous procedure
is followed for all $i=1,\ldots,n-1$ values.

In the example of Fig. 7 the particular mass $M$ that we are
looking for turns out to be located between $M_{{\rm sep}}^{i}$ and
$M_{{\rm sep}}^{i+1}$ and in the corresponding  iso-phase interval  labelled
{\bf 2}. Reducing to essentials, we proceed according to the following
scheme:

\begin{itemize}
\item
Definition of the adimensional mass

$\tau_m=(M-M_{{\rm sep}}^{i})/(M_{{\rm sep}}^{i+1}-M_{{\rm sep}}^{i})$,

in this way
$\tau_m$ determines the correspondent masses $M_{Y_1Z_1}$, $M_{Y_2Z_1}$,
$M_{Y_1Z_2}$, $M_{Y_2Z_2}$
for each  chemical composition inside the iso-phase interval {\bf 2}.
\item
Interpolation of the tracks $M_{Y_1Z_1}$, $M_{Y_2Z_1}$, $M_{Y_1Z_2}$,
$M_{Y_2Z_2}$ between pairs of computed tracks (for details see Bertelli et al.
 1990), to obtain
the ages of the characteristic points.

\item
Interpolation in $Y$ and $\log Z$ to obtain the ages $t_k$ of the
characteristic points of the mass M.  We recall that all the masses inside
the same iso-phase interval show the same number of {\em characteristic}
points.
\item
Identification of the current phase  (the actual age $t$ is inside the
interval ($t_k$,$t_{k+1}$)) and definition of the adimensional time

 $\tau_t= (t-t_k)/(t_{k+1}-t_k)$

\item
Determination of the luminosities $L_{Y_1Z_1}$, $L_{Y_2Z_1}$,
  etc. relative to the masses $M_{Y_1Z_1}$, $M_{Y_2Z_1}$ etc.,
in correspondence to the adimensional time $\tau_t$ inside the ($k ,k+1$)
phase.

\item
Interpolation in $Y$ between $L_{Y_1Z_1}$ and $L_{Y_2Z_1}$ obtaining $L_{Z_1}$
, between $L_{Y_1Z_2}$ and $L_{Y_2Z_2}$ obtaining $L_{Z_2}$ and finally
interpolating in $\log Z$ between $L_{Z_1}$ and $L_{Z_2}$ obtaining the
luminosity $L$ for the mass M and the age $t$.

\item
Analogous procedure to obtain $\log\mbox{$T\sub{eff}$}$ (or other physical properties).
\end{itemize}

For each knot of the grid YZ the data file, used to compute the isochrones,
 contain the evolutionary tracks (log Age, $log L/L_{\odot}$, $log T_{eff}$)
for the provided range of
masses and four interfaces (one for each of the four quadrants centered
on the knot). The interfaces are such as to assure, inside every mesh of the
grid, the same number of iso-phase intervals.
This method is used to obtain an interpolated track of given mass and
chemical composition (inside the provided range) or to derive isochrones of
given age and chemical composition.

\subsection{Reliability of the interpolation}

\begin{figure}
 \resizebox{\hsize}{!}{\includegraphics{fig8.eps}}
\caption{Comparison between computed evolutionary tracks (green line) for a
chemical composition intermediate among those of the computed grids and the
corresponding interpolated (red points) with the above described method for
 (Z=0.055,Y=0.37)
}
\label{intz055y37}
\end{figure}

\begin{figure}
 \resizebox{\hsize}{!}{\includegraphics{fig9.eps}}
\caption{Comparison between computed evolutionary tracks (green line) for a
chemical composition intermediate among those of the computed grids and the
corresponding interpolated (red points) with the above described method for
 (Z=0.0003,Y=0.28)
}
\label{intz0003y28}
\end{figure}

\begin{figure}
 \resizebox{\hsize}{!}{\includegraphics{fig10.eps}}
\caption{Comparison between a computed evolutionary track for $M = 1.12
M_{\odot}$ (green line) and the correspondent track obtained by interpolation
from the grids (red points), with the chemical composition Z=0.00065 and
Y=0.35, in the more complex case of close tracks with different morphology
}
\label{intm1_12}
\end{figure}

We have done some tests in order to check the reliability of the
method for the interpolation in the Z-Y plane. For a few combinations
Z-Y we have calculated some tracks and then we have interpolated the
same masses from the existing grids, using the new program YZVAR.  The
results of the comparison are shown in Figs.8, 9 and 10.
In Fig. 8 for the chemical composition Z=0.055 and
Y=0.37 (a point at the center of a mesh) we compare the computed
evolutionary tracks of low mass stars of 0.8, 1.0, 1.3, 1.5 $M_{\odot}$
(green line) and those interpolated with YZVAR (red points). The
result on the HR diagram is satisfactory and the ages differ by less
than $1 \%$.  Fig. 9 presents the results as in Fig 8,
but for the chemical composition Z=0.003 and Y=0.28.  In
Fig. 10 we display a critical case as the considered 1.12
$M_{\odot}$ is between two iso-phase intervals. In fact
this mass is comprised between two masses with different morphology in
the HR diagram. It is evident that the interpolation is able to reproduce
the computed track very well.

\subsection{Bolometric corrections}

The present isochrones are provided in the Johnson-Cousins-Glass
system as defined by Bessell (1990) and Bessell \& Brett (1988). As soon as
possible they will be available in the Vegamag systems of ACS and WFPC2 on
board of HST (cf. Sirianni et al. 2005; Holtzman et al. 1995) and in the
SDSS system.
The formalism we follow to derive
bolometric corrections in these systems is described in Girardi et
al. (2002). The definition of zeropoints has been revised and is
detailed in a forthcoming paper by Girardi et al. (in prep.; see also
Marigo et al. 2007) and will not be repeated here.

Suffice it to recall that the bolometric correction tables stand
on an updated and extended library of stellar spectral fluxes. The core
of the library now consists of the ``ODFNEW'' ATLAS9 spectral fluxes
from Castelli \& Kurucz (2003), for $T_{\rm eff}$ between 3500 and
50000~K, $\log g$ between $-2$ and $5$, and scaled-solar metallicities
[M/H] between -2.5 and +0.5. This library is extended at the intervals
of high $T_{\rm eff}$ with pure blackbody spectra.
For lower $T_{\rm eff}$, the library is completed
with the spectral fluxes for M, L and T dwarfs from Allard et
al. (2000), M giants from Fluks et al. (1994), and finally the C star
spectra from Loidl et al. (2001).  Details about the implementation of
this library, and in particular about the C star spectra, are provided
in Marigo et al. (2007) and Girardi et al. (in prep.).

It is also worth mentioning that in the isochrones we apply the
bolometric corrections derived from this library without making any
correction for the enhanced He content. As demonstrated in Girardi et
al. (2007), for a given metal content, an enhancement of He
as high as
$\Delta Y=0.1$ produces changes in the bolometric corrections of just a
few thousandths of magnitude. Just in some very particular situations,
for instance at low $T_{\rm eff}$ and for blue pass-bands, can
He-enhancement produce more sizeable effects on BCs; these situations
however correspond to cases where the emitted stellar flux would
anyway be very small, and therefore are of little interest in
practice.

\subsection{Description of isochrone tables}
\label{sec_tableisoc}

Complete tables with the isochrones can be obtained through the web site
 \verb$http://stev.oapd.inaf.it/YZVAR$.
In this data-base, isochrones are provided at $\Delta\log t=0.05$
intervals; this means that any two consecutive isochrones differ by
only 12 percent in their ages.

For each isochrone table the corresponding data file presents
16 columns with the following information:
	\begin{description}
	\item \verb$1. logAge$: logarithm of the age in years;
	\item \verb$2. log(L/Lo)$: logarithm of surface luminosity (in solar units);
	\item \verb$3. logTef$: logarithm of effective temperature (in K);
	\item \verb$4. logG$: logarithm of surface gravity (in cgs units);
        \item \verb$5. Mi$: initial mass in solar masses;
        \item \verb$6. Mcur$: actual stellar mass in solar masses;
        \item \verb$7. FLUM$: indefinite integral over the initial mass M of
the Salpeter initial mass function by number;
        \item \verb$8. - 15.$: UBVRIJHK absolute magnitudes in the
Johnson-Cousins-Glass system;
        \item \verb$16. C.P.$: index marking the presence of a characteristic
point, when different from zero.
	\end{description}

 We recall that the
initial mass is the useful quantity for population synthesis calculations,
since together with the initial mass function it determines the relative
number of stars in different sections of the isochrones.
 In  column 7  the
indefinite integral over the initial mass $M$ of the initial mass
function (IMF) by number,

i.e.\
	\begin{equation}
\mbox{\sc flum} = \int\phi(M) \mbox{d} M
	\end{equation}
is presented, for the case of the Salpeter IMF, $\phi(M)=AM^{-\alpha}$,
with $\alpha=2.35$. When we assume a normalization constant of $A=1$,
{\sc flum} is simply given by {\sc flum}$ = M^{1-\alpha}/(1-\alpha)$.
This is a useful quantity since the difference between any two
values of {\sc flum} is proportional to the number of stars located in
the corresponding mass interval. It is worth remarking that we
present {\sc flum} values
for the complete mass interval down to 0.15~\mbox{$M_{\odot}$}, always assuming a
Salpeter (1955) IMF, whereas we know that such an IMF cannot be extended to
such low values of the mass. However, the reader can easily derive
{\sc flum} relations for alternative choices of the IMF, by using
the values of the initial mass we present in Column 5 of the
isochrone tables.
In the last column when there appears the value 1, it marks the presence
of a characteristic evolutionary point from the ZAMS to the beginning of
the early AGB phase, while the value 2 is related to the characteristic
points of the TP-AGB phase.
If there is only one characteristic point (marked with 2) at the
end of the isochrone, this means that the TP-AGB phase is very short.
The beginning and the end of the TP-AGB phase are pointed out with index 2,
as well as when C/O increases above unity (transition from M to carbon stars).

\section{Comparison with other databases}

\begin{figure}
 \resizebox{\hsize}{!}{\includegraphics{fig11.eps}}
\caption{Comparison between the new Padova isochrones (PD08) and Girardi et al.
(PD00) ones for Z=0.019 and Y=0.273 and log Age=10., 9.7, 9.5, 9.2, 9. years.
Solid lines correspond to PD00, dash-dotted ones to PD08 isochrones.}
\label{compPD00_07}
\end{figure}

\begin{figure*}
\begin{minipage}{0.45\textwidth} \noindent a)
\resizebox{\hsize}{!}{\includegraphics{fig12a.eps}}
\end{minipage}
\hfill
\begin{minipage}{0.45\textwidth} \noindent b)
\resizebox{\hsize}{!}{\includegraphics{fig12b.eps}}
\end{minipage}
\caption{
Comparison between Padova and Teramo isochrones with overshoot for the
composition $[Z=0.008, Y=0.256]$ in panel a). Dot-dashed lines correspond to
Teramo ones  and solid lines to our new isochrones. The largest difference
between the PD and Te isochrones is met at log Age=9.4.
In panel b) the comparison is between Padova (solid line) and $YY$ isochrones
(dot-dashed) with  Z=0.007 and Y=0.244  }
\label{isoPD_TE_Y2}
\end{figure*}

\begin{figure*}
\begin{minipage}{0.45\textwidth} \noindent a)
\resizebox{\hsize}{!}{\includegraphics{fig13a.eps}}
\end{minipage}
\hfill
\begin{minipage}{0.45\textwidth} \noindent b)
\resizebox{\hsize}{!}{\includegraphics{fig13b.eps}}
\end{minipage}
\caption{
Evolutionary tracks from Girardi et al. (2000)  for the composition $[Z=0.019,
Y=0.273]$ with overshoot (dot-dashed line) and without (solid line) in panel
a).  In panel b) Teramo tracks are displayed for $[Z=0.0198, Y=0.273]$ with
overshoot (dot-dashed line)  and without (solid line).
 The ZAMS location for TE models with and
without overshoot is not coincident.}
\label{PD_TEzsun}
\end{figure*}

\begin{figure}
 \resizebox{\hsize}{!}{\includegraphics{fig14.eps}}
\caption{New Padova tracks and Teramo ones with overshoot from 1 to 2.5
$M_{\odot}$ for Z=0.008.
Solid lines correspond to PD08 models, dash-dotted ones to Teramo tracks.}
\label{PD07_TEz008}
\end{figure}

\begin{table*}
\begin{center}
\caption{Comparison of H-lifetimes for models with overshoot with nearly solar composition and for Z=0.008 }
\label{tab_tH}
\begin{tabular}{lllllllll}
\noalign{\smallskip}\hline\noalign{\smallskip}
\multicolumn{4}{c}{$Z\sim Z_{\odot}$}& &\multicolumn{4}{c}{$Z=0.008$}\\
\noalign{\smallskip}
\cline{1-4}\cline{6-9}
\noalign{\smallskip}
$M/\mbox{$M_{\odot}$}$  & $t_H$ TE $^1$  &  $t_H$ PD08 $^2$ & $t_H$ $YY$ $^3$ &  &
$M/\mbox{$M_{\odot}$}$  & $t_H$ TE $^4$  &  $t_H$ PD08 $^5$ & $t_H$ $YY$ $^6$  \\
\noalign{\smallskip}\cline{1-4}\cline{6-9}\noalign{\smallskip}
  1.00 &  10.889  & 10.547 & 10.855 & & 1.00 & 8.387 & 7.389 & 7.795 \\
  1.20 &   5.679  &  5.109 &  5.359 & & 1.20 & 4.625 & 3.743 & 3.994 \\
  1.40 &   4.182  &  3.390 &  3.424 & & 1.30 & 4.009 & 3.040 & 3.076 \\
  1.60 &   3.067  &  2.347 &  2.390 & & 1.50 & 2.986 & 2.162 & 2.215 \\
  1.80 &   2.136  &  1.647 &  1.671 & & 1.80 & 1.847 & 1.266 & 1.306 \\
  2.00 &   1.439  &  1.198 &  1.229 & & 2.00 & 1.250 & 0.943 & 0.976 \\
  3.00 &   0.375  &  0.394 &  0.392 & & 3.00 & 0.331 & 0.324 & 0.334 \\
  5.00 &   0.094  &  0.107 &  0.104 & & 5.00 & 0.089 & 0.097 & 0.098 \\
\noalign{\smallskip}\hline\noalign{\smallskip}
\end{tabular}

$^1$  Z=0.0198 Y=0.273; $^2$  Z=0.017 Y=0.26; $^3$  Z=0.020 Y=0.27

$^4$  Z=0.008  Y=0.256; $^5$  Z=0.008 Y=0.26; $^6$  Z=0.007 Y=0.244

Ages are in Gyr

\end{center}
\end{table*}

Figure 11 shows the comparison between the isochrones by Girardi
et al. (2000), hereafter PD00, with the new ones, hereafter PD08, in the
age range  between 1 and 14 Gyr, for the chemical composition $Z=0.017$ and
$Y=0.273$. There are
differences both in the MS phase and in the luminous part of the RGB
and/or AGB phase. The main contribution to the differences in the MS
phase is due to the interpolation technique for the opacity tables (see Section
2.2). The amount of the variation depends on mass and chemical
composition. We have verified that low-mass stellar evolutionary tracks
change very little with the method of opacity interpolation. For
intermediate mass stars there is a lowering of the luminosity
and a consequent increase of the lifetime of the new models.
For values of log Age$ \le 9.5-9.7$ (it depends on
the chemical composition) to obtain the same luminosity of PD00
for the MS termination point in the new isochrones (PD08) we must increase
the logarithm of the age  of about 0.05.  The color of the
turn-off of the new isochrones (younger than 9.5 Gyr) is bluer.

We point out that the luminous part of the RGB and/or AGB phases in the new
isochrones
is redder than in PD00 ones and the separation increases for increasing
luminosity. This effect is due to the treatment of the
density inversion in the red giant envelope. In the outermost layers of stars
with $T_{eff} \le  10^4 K$ a convective zone develops caused by the
incomplete ionization of hydrogen. If convection is treated according to the
mixing-length
theory (MLT) with the mixing-length proportional to the pressure scale height
($H_p$), when $T_{eff} \le  10^4 K$ a density inversion occurs. This situation
might be unphysical and it has been the subject of many speculations.
This difficulty can be avoided by adopting the density scale height $H_{\rho}$
instead of $H_p$ in the MLT.
Really in Girardi et al. tracks we avoided the inversion density
imposing that the temperature gradient  $\Delta T$ is such that the density
gradient obeys the condition $\Delta _{\rho} \ge 0$. However the comparison of
the PD00 isochrones with the observations seems to require redder RGB and/or
AGB branches.
For this reason in the present computations we have used the MLT
proportional to the pressure scale height,
allowing the inversion density in the red convective envelope, as already
adopted in Bertelli et al. (1994) isochrones.  The
presence of the inversion density determines RGB and/or AGB phases redder
than in previous isochrones (PD00).

Neutrino cooling is also more efficient during the RGB phase in the new tracks,
as described in Section 2.4 and it contributes to make the tip of the RGB more
luminous than in PD00. We have verified that for Z in the range 0.0001 - 0.001
(low values of Y) and for the solar case the overluminosity is of
the order 0.12 - 0.15 in  units of $log L/L_{\odot}$ (the considered ages are
around  log Age=10.0).
The effect on the I magnitude consists in an overluminosity of the order
of between -0.20 and -0.35 for Z in the range 0.0001 - 0.001. On the other hand
 in  the solar case
the interplay between lowering effective temperature and increasing
bolometric corrections is such that the tip of the PD00 case appears to be
approximately one magnitude more luminous than the new one, inverting
what happens to the I magnitude for low values of Z.

The comparison of some selected tracks and isochrones of the new grids
 of  the Padova database(PD08) with the analogous
counterparts of the publicly available Teramo database (TE) by Pietrinferni et
al. (2004) and the $YY$ database by Yi et al. (2001) and Demarque et al.
(2004) pointed out some differences.

As already evident in the comparison of isochrones
by different groups, shown in Pietrinferni et al. (2004), there is a systematic
difference from their 1.8 Gyr isochrone and the one by Girardi et al. (2000)
and by Yi et
al. (2001), as for the 2.0 Gyr one by VandenBerg (see figures 8, 11 and 12 in
Pietrinferni et al. 2004). In fact the turn-off
luminosity of their 1.8 Gyr isochrone (or 2.0 in the comparison with
VandenBerg) is higher, whereas the 10 Gyr and the 0.5 Gyr
ones are practically the same as by the other groups. They ascribed the
differences to differences in input physics and in the overshooting
treatment relatively to the comparison with Girardi et al. (2000).

In Table 4 we compare the H-lifetimes for models with overshoot in
the mass range between 1 and 5 $M_{\odot}$ and with a chemical composition
approximately solar
We point out that the largest differences can be found in the range
between 1 and 2 solar masses, in the sense that while PD08 and $YY$ tracks
have similar H-lifetimes, TE models are systematically longer up to 0.8 Gyr.
This difference cannot be ascribed to the treatment of overshoot, in fact
both groups (PD and TE) assume a similar prescription in the range between
1 and 2 $M_{\odot}$ (the core overshoot efficiency decreases linearly from
about $1.7-1.5 M_{\odot}$ down to $1 M_{\odot}$, where the efficiency is
zero for stars with a radiative core).

The recent determination of the age of the LMC globular cluster NGC 1978,
studied by Mucciarelli et al. (2007) shows the same anomaly, in this case
for the chemical composition typical of the LMC. The authors find that the
age obtained with Teramo isochrones results significantly older than that
obtained with Pisa and Padova isochrones, even if the same amount of overshoot
is assumed in the considered models.
Comparing the H-lifetimes for $Z=0.008$ we notice that there is a very
satisfactory agreement between the H lifetimes of $YY$ and PD08, whereas TE
ages are older up to about 1 Gyr at about $1.3 M_{\odot}$. The largest
differences are present in the range between 1 and 2 $M_{\odot}$, as shown in
Table~\ref{tab_tH}.

Figure 12 panel a) shows isochrones (at log Age = 10.0, 9.7,
9.4, 9.2, 9.0 years) from
Teramo database with Z=0.008 and Y=0.256, and (interpolated at the same metal
and helium content) from our new isochrones. This figure remarks the difference
in luminosity of the isochrones for the same age from the two databases,
more evident for log Age=9.4 (black solid line for PD08 and black dash-dotted
for Teramo) in the color version of the electronic paper.
In panel b) we compare Padova new isochrones for Z=0.007 and
Y=0.244  with $YY$  and we
find out that the agreement is very good, considering that their
models take into account convective core overshoot (0.2) and helium diffusion
(PD08 and $YY$ isochrones are shown for log Age = 10.0, 9.7, 9.4 and 9.0
years).

The reasons of the above mentioned disagreement cannot be easily disentangled,
but we notice that the ZAMS models for PD00 tracks (Girardi et al. 2000 for
solar chemical composition) with and without overshoot are coincident (as the
ZAMS is the location of chemically homogeneous models for the various
considered masses), while they are different in luminosity and temperature
for Teramo ones between 1 and 2 $M_{\odot}$.
 The difference in the location of ZAMS models can be seen also
in Figure 14 where we plot our new models and Teramo ones
with overshoot for Z=0.008.

In Figure 13 the PD00 and TE tracks are plotted in the range of
mass between 1.2 and 2 $M_{\odot}$. As we did not compute new tracks without
overshoot, we plotted PD00 and TE models with and without overshoot to make
a comparison.
The anomalously longer H-burning lifetimes of TE models with respect
to other authors is probably, at least partly, related to the lower luminosiy
of the beginning of the central H-burning for their models with
overshoot.
We point out that Teramo models with overshoot were computed
starting from the pre-MS phase for masses lower than $3 M_{\odot}$.
 Anyway also $YY$ models were evolved from the pre-MS stellar birthline to
the onset of helium burning (Yi et al. 2001), but there are not significant
differences (comparing PD08 to YY results) in the  H-lifetimes and for the
isochrones in the age range between 1 and 10 Gyr, as shown  in Table 4 and
Figure 12 b).

\section{Concluding remarks}
\label{sec_remarks}

Large grids of homogeneous stellar evolution models and isochrones are a
necessary ingredient for the interpretation of photometric and spectroscopic
observations of resolved and unresolved stellar populations.
Appropriate tools to compute synthetic color-magnitude diagrams for the
analysis of stellar populations require also the selection of the chemical
composition to be used in the simulation. Evolutionary stellar models are
usually computed taking into account a fixed law of helium to metal enrichment.
Determinations of the helium enrichment $\Delta Y / \Delta Z$ from nearby stars
and K dwarfs, or
from Hyades binary systems show a large range of values for this ratio.
Recent results suggest that the naive assumption that the helium enrichment
law is universal, might not be correct.
In fact there has been recently evidence of significant variations in the
helium content (and perhaps of the age) in some globular clusters, like
$\omega$ Cen, NGC 2808 and NGC 6441, while
globular clusters were traditionally considered as formed of a simple
stellar population of uniform age and chemical composition.

These results prompted us to compute new stellar evolutionary tracks covering
an extended region of the plane Z-Y, in order to enable users to analyse
stellar populations with different helium enrichment laws.

 In this paper we present 39 grids of stellar evolutionary tracks up to
$2.5 M_{\odot}$. Next paper will present tracks and isochrones from $2.5$
up to $20 M_{\odot}$ for the same grid of chemical
compositions. The typical mass resolution for low mass stars is of 0.1
$M_{\odot}$, reduced to $0.05$ in the interval of very low masses ($M < 0.6
M_{\odot}$), and occasionally in the vicinity of the separation mass $M_{Hef}$
between low and intermediate mass stars. In this way we provide enough tracks
to allow a very detailed mapping of the HR diagram and the relative theoretical
isochrones are also very detailed. An important update of this database is the
extension of stellar models and isochrones until the end of the TP-AGB
 by means of new synthetic models (cf. Marigo \& Girardi 2007).
The results of these new grids allow the estimate of several astrophysical
quantities as the RGB transition mass, the He core mass at He ignition, the
amount of changes of chemical elements due to the first dredge-up, and the
properties of the models at the beginning of the thermal pulse phase on the
AGB.

The web site (http://stev.oapd.inaf.it/YZVAR), dedicated to make available the
whole theoretical framework to the scientific community, includes:
\begin{itemize}
\item
{Data files with the information relative to each evolutionary track}
\item
{Isochrone files with the chemical composition of the computed grids in
the Z-Y plane  }
\end{itemize}
Every file contains isochrones with $\sim 10.15 \le$ Log Age $ \le \sim 9.0$
years with
time step equal to 0.05. The initial and the final value can be a little more
or a little less, dependent on the chemical composition.
To compute the isochrones we adopted the interpolation scheme described in
Section 5.1. A web interactive interface will be tuned up as soon as possible
to allow users to obtain isochrones of whatever chemical composition inside
the original grid.

A modified version of the program which computes the isochrones is suited to
the purposes of evolutionary population synthesis, either of simple (star
clusters) or complex stellar populations (galaxies). In the latter case it is
possible to select the kind of star formation history, the chemical enrichment
law in the plane Z-Y, the initial mass function and the mass loss rate during
the RGB phase. The previous program ZVAR, in which the helium enrichment law
was implicit in the chemical composition of the various evolutionary sets, is
now updated for being used in the extended region of the Z-Y plane and is named
YZVAR.
A web interactive interface will provide stellar populations for selected input
data for the SFR, IMF, chemical composition and mass loss during the RGB phase.

\begin{acknowledgements}
We thank  C. Chiosi for his continuous interest and support to stellar
evolution computations. We thank A. Weiss and B. Salasnich for help with
opacity tables, and A. Bressan for useful discussions.
The authors acknowledge constructive comments of the referee that helped
to clarify and improve the text.
We acknowledge financial support from INAF COFIN 2005 ``A Theoretical lab
for stellar population studies'' and  from
Padova University (Progetto di Ricerca di Ateneo CPDA 052212).

\end{acknowledgements}