The Updated BaSTI Stellar Evolution Models and Isochrones. I. Solar-scaled Calculations

The solar heavy element distribution sets the zero point of the metallicity scale and is also a critical input entering the calibration of the Solar Standard Model (SSM; Vinyoles et al. 2017), which in turn serves as a calibrator of the mixing-length parameter (see Section 2.7), the initial solar He abundance and metallicity, and the dY/dZ He-enrichment ratio.

"Classical" estimates of the solar heavy element distribution such as those by Grevesse & Sauval (1998) used in our previous BaSTI models did allow SSMs to match very closely the constraints provided by helioseismology (e.g., Pietrinferni et al. 2004 and references therein). Recent reassessments by Asplund et al. (2005, 2009) have led to a downward revision of the solar metal abundances by up to 40% for important elements such as oxygen. SSMs employing these new metal distributions produce a worse match to helioseismic constraints such as the sound speed at the bottom of the convective envelope, as well as the location of the bottom boundary of the surface convection and the surface He abundance (see, e.g., Serenelli et al. 2009). This evidence has raised the so-called "solar metallicity problem." A reanalysis of the Asplund et al. (2009) results and the use of an independent set of solar model atmospheres (see, e.g., Caffau et al. 2011 for a detailed discussion) has provided a solar heavy element distribution intermediate between those by Grevesse & Sauval (1998) and Asplund et al. (2009).

Although the problem is still unsettled and different solutions are under scrutiny (see, e.g., Vinyoles et al. 2017), we decided to adopt the solar metal mixture by Caffau et al. (2011), supplemented when necessary by the abundances given by Lodders (2010). The reference solar metal mixture adopted in our calculations is listed in Table 1. The actual solar metallicity is Z_⊙ = 0.0153, while the corresponding actual (Z/X)_⊙ is equal to 0.0209.

In our models—apart from the case of core He burning in low- and intermediate-mass stars—we use the Schwarzschild criterion to fix the formal convective boundary, plus instantaneous mixing in the convective regions. In the case of models of massive stars, where layers left behind by shrinking convective cores during the MS have a hydrogen abundance that increases with increasing radius, formally requiring a semiconvective treatment of mixing, we still use the Schwarzschild criterion and instantaneous mixing to determine the boundaries of the mixed region. This follows recent results from 3D hydrodynamic simulations of layered semiconvective regions (Wood et al. 2013) that show how in stellar conditions, mixing in MS semiconvective regions is very fast and essentially equivalent to calculations employing the Schwarzschild criterion and instantaneous mixing (Moore & Garaud 2016).

Theoretical simulations (see, e.g., Andrássy & Spruit 2013, 2015; Viallet et al. 2015 and references therein), observations of open clusters and eclipsing binaries (see, e.g., Demarque et al. 1994; Magic et al. 2010; Stancliffe et al. 2015; Claret & Torres 2016, 2017; Valle et al. 2016, and references therein), as well as asteroseismic constraints (see, e.g., Silva Aguirre et al. 2013) show that in real stars, chemical mixing beyond the formal convective boundary is required and most likely results from the interplay of several physical processes, grouped in stellar evolution modeling under the generic terms overshooting or convective boundary mixing.

In our calculations, overshooting beyond the Schwarzschild boundary of MS convective cores is included as an instantaneous mixing between the formal convective border and layers at a distance λ_ovH_P from this boundary, keeping the radiative temperature gradient in this region. Here, H_P is the pressure scale height at the Schwarzschild boundary, and λ_OV a free parameter that we set equal to 0.2, decreasing to zero when the mass decreases below a certain value. This decrease is required because for increasingly small convective cores, the Schwarzschild boundary moves progressively closer to the center, and the local H_P increases quickly, formally diverging when the core shrinks to zero mass. Keeping λ_OV constant would produce increasingly large overshooting regions for shrinking convective cores.

How to decrease the overshooting efficiency is still somewhat arbitrary (see, e.g., Claret & Torres 2016; Salaris & Cassisi 2017 for a review of the different choices found in the literature). As shown by Pietrinferni et al. (2004), the approach used to decrease the overshooting efficiency in the critical mass range between ∼1.0 and ∼1.5 M_⊙ has a potentially large effect on the isochrone morphology for ages around ∼4–5 Gyr (see Figure 1 in Pietrinferni et al. 2004).

In these new calculations, we have chosen the following procedure to decrease λ_OV with decreasing initial mass of the model. For each chemical composition, we have sampled the mass range between 1.0 ≤ M/M_⊙ ≤ 1.5 with a very fine mass spacing, and determined the initial mass (${M}_{{\rm{o}}{\rm{v}}}^{{\rm{i}}{\rm{n}}{\rm{f}}}$) that develops a convective core reaching, at its maximum extension, a mass ${M}_{{\rm{c}}{\rm{c}}}^{min}=0.04\,{M}_{\odot }$ during core H burning. This initial mass is considered to be the maximum mass for models calculated with λ_OV = 0. We have then determined the minimum initial mass that develops a convective core that is always larger than ${M}_{{\rm{c}}{\rm{c}}}^{{\rm{m}}{\rm{i}}{\rm{n}}}$ during the entire MS. This value of the initial mass is denoted as ${M}_{{\rm{o}}{\rm{v}}}^{{\rm{s}}{\rm{u}}{\rm{p}}}$. For models with initial masses equal to or larger than ${M}_{{\rm{o}}{\rm{v}}}^{{\rm{s}}{\rm{u}}{\rm{p}}}$, we use λ_OV = 0.2, whereas between ${M}_{{\rm{o}}{\rm{v}}}^{{\rm{i}}{\rm{n}}{\rm{f}}}$ and ${M}_{{\rm{o}}{\rm{v}}}^{{\rm{s}}{\rm{u}}{\rm{p}}}$, the free parameter λ_OV increases linearly from 0 to 0.2. An example of how we fix the values of ${M}_{{\rm{o}}{\rm{v}}}^{{\rm{i}}{\rm{n}}{\rm{f}}}$ and ${M}_{{\rm{o}}{\rm{v}}}^{{\rm{s}}{\rm{u}}{\rm{p}}}$ is shown in Figure 1: for the selected metallicity, ${M}_{{\rm{o}}{\rm{v}}}^{{\rm{i}}{\rm{n}}{\rm{f}}}$ is equal to 1.08 M_⊙, while ${M}_{{\rm{o}}{\rm{v}}}^{{\rm{s}}{\rm{u}}{\rm{p}}}$ is equal to 1.42 M_⊙.

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This criterion is obviously somehow arbitrary. It is based on numerical experiments we performed comparing the model predictions with empirical benchmarks such as eclipsing binaries and intermediate-age star clusters, as shown in Section 6. Our choice indirectly introduces a dependence of ${M}_{{\rm{o}}{\rm{v}}}^{{\rm{i}}{\rm{n}}{\rm{f}}}$ and ${M}_{{\rm{o}}{\rm{v}}}^{{\rm{s}}{\rm{u}}{\rm{p}}}$ on the initial metallicity (see Table 2). This is because the relationship between ${M}_{{\rm{c}}{\rm{c}}}^{{\rm{m}}{\rm{i}}{\rm{n}}}$ and the total mass of the model depends on the efficiency of H burning via the CNO cycle, which in turn is affected by a change of the absolute value of the total CNO abundance.

Table 2.  Grid of Initial Chemical Abundances and Corresponding Values (in Solar Masses) of ${M}_{{\rm{o}}{\rm{v}}}^{{\rm{i}}{\rm{n}}{\rm{f}}}$ and ${M}_{{\rm{o}}{\rm{v}}}^{{\rm{s}}{\rm{u}}{\rm{p}}}$ (See the Text for Details)

Z	Y	[Fe/H]	${M}_{{\rm{o}}{\rm{v}}}^{{\rm{i}}{\rm{n}}{\rm{f}}}$	${M}_{{\rm{o}}{\rm{v}}}^{{\rm{s}}{\rm{u}}{\rm{p}}}$
0.00001	0.2470	−3.20	1.30	2.09
0.00005	0.2471	−2.50	1.30	1.78
0.00010	0.2471	−2.20	1.30	1.68
0.00020	0.2472	−1.90	1.30	1.59
0.00031	0.2474	−1.70	1.30	1.54
0.00044	0.2476	−1.55	1.30	1.50
0.00062	0.2478	−1.40	1.32	1.47
0.00079	0.2480	−1.30	1.32	1.45
0.00099	0.2483	−1.20	1.24	1.44
0.00140	0.2488	−1.05	1.21	1.43
0.00197	0.2496	−0.90	1.17	1.42
0.00311	0.2511	−0.70	1.13	1.42
0.00390	0.2521	−0.60	1.10	1.42
0.00614	0.2550	−0.40	1.09	1.42
0.00770	0.2571	−0.30	1.08	1.42
0.00964	0.2596	−0.20	1.08	1.42
0.01258	0.2635	−0.08	1.08	1.43
0.01721	0.2695	0.06	1.09	1.43
0.02081	0.2742	0.15	1.11	1.47
0.02865	0.2844	0.30	1.10	1.42
0.03905	0.2980	0.45	1.09	1.40

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The values of ${M}_{{\rm{o}}{\rm{v}}}^{{\rm{i}}{\rm{n}}{\rm{f}}}$ and ${M}_{{\rm{o}}{\rm{v}}}^{{\rm{s}}{\rm{u}}{\rm{p}}}$ for each initial chemical composition of our model grid are listed in Table 2. This approach is different from the previous BaSTI release where, regardless of the chemical composition, we fixed the overshoot efficiency to its maximum value (λ_OV = 0.2) for initial masses larger than or equal to 1.7 M_⊙, decreasing linearly down to zero when the initial mass is equal to 1.1 M_⊙.

Before closing this discussion, it is interesting to compare our recipe for decreasing λ_OV with decreasing initial mass, with the results of a recent calibration by Claret & Torres (2016). These authors compared their own model grid with the effective temperatures and radii of a sample of detached double-lined eclipsing binaries with well-determined masses, in the [Fe/H] range between about solar and ∼ −1.01. They determined λ_OV equal to zero for masses lower than about 1.2 M_⊙, increasing to 0.2 in the mass range between 1.2 M_⊙ and 2 M_⊙. For masses larger than 2 M_⊙, λ_OV is equal to ∼0.2, as in our calculations. In the same metallicity range, the value we adopt for ${M}_{{\rm{o}}{\rm{v}}}^{{\rm{i}}{\rm{n}}{\rm{f}}}$ ranges between ∼1.1 M_⊙ and ∼1.2 M_⊙, whereas ${M}_{{\rm{o}}{\rm{v}}}^{{\rm{s}}{\rm{u}}{\rm{p}}}$ is always equal to ∼1.4 M_⊙, about 0.6 M_⊙ smaller than the Claret & Torres (2016) result. It is, however, very difficult to compare the two sets of results. Apart from possible intrinsic differences in the models, Claret & Torres (2016) also determine from their fits the individual values of the mixing length for each component and the initial metallicity Z of each system, and allowed age differences of up to 5% between the components of each system. They derived often systematically lower metallicities than the corresponding spectroscopic measurements. In Section 6, we will see that our models fit well the mass–radius relationship of the systems KIC 8410637 and OGLE-LMC-ECL-15260 (this latter also studied by Claret & Torres 2016), whose masses are in the 1.3 M_⊙–1.5 M_⊙ range, bracketing the upper limit where λ_OV reached 0.2 with our calibration. We have imposed in our comparisons equal ages for both systems and no variation of the mixing length, and used models with chemical composition consistent with the spectroscopic measurements.

In case of core He burning of low- and intermediate-mass stars, we model core mixing with the semiconvective formalism by Castellani et al. (1985) and breathing pulses inhibited following Caputo et al. (1989). During core He burning in massive stars, we use the Schwarzschild criterion without overshooting to fix the boundary of the mixed region.

We do not include overshooting from the lower boundaries of convective envelopes.

The sources for the radiative Rosseland opacity are the same as for the previous BaSTI calculations. In more detail, opacities are from the OPAL calculations (Iglesias & Rogers 1996) for temperatures larger than $\mathrm{log}(T)=4.0$, whereas calculations by Ferguson et al. (2005)—including contributions from molecules and grains—have been adopted for lower temperatures. Both high- and low-temperature opacity tables have been computed for the solar-scaled heavy element distribution listed in Table 1.

As for the electron conduction opacities, which are at variance with the models presented in Pietrinferni et al. (2004, 2006), we have now adopted the results by Cassisi et al. (2007). As shown by Cassisi et al. (2007), these opacity calculations affect only slightly (small decrease) the He-core mass at He ignition for low-mass models, and the luminosity of the following horizontal branch (HB) phase (small decrease), compared to the BaSTI calculations that were based on the Potekhin (1999) conductive opacities. For more details on this issue, we refer the reader to the quoted reference as well as to Serenelli et al. (2017).

As in Pietrinferni et al. (2004), we use the detailed EOS by A. Irwin.¹⁰ A brief discussion of the characteristics of this EOS can be found in Cassisi et al. (2003). We recomputed all required EOS tables for the heavy element distribution in Table 1, adopting the option "EOS1" in Irwin's code. This option—recommended by A. Irwin (see also the discussion in Cassisi et al. 2003)—provides the best match to the OPAL EOS (Rogers & Nayfonov 2002) and to the Saumon et al. (1995) EOS in the low-temperature and high-density regime.

The nuclear reaction rates are from the NACRE compilation (Angulo et al. 1999), with the exception of the three following reactions, whose rates come from recent re-evaluations:

• 1.  
  ${}^{3}\mathrm{He}{{(}^{4}\mathrm{He},\gamma )}^{7}\mathrm{Be}$—Cyburt & Davids (2008),
• 2.  
  ${}^{14}{\rm{N}}{({\rm{p}},\gamma )}^{15}{\rm{O}}$—Formicola et al. (2004), and
• 3.  
  ${}^{12}{\rm{C}}{(\alpha ,\gamma )}^{16}{\rm{O}}$—Hammer et al. (2005).

The previous BaSTI calculations employed the NACRE rates (Angulo et al. 1999) for all reactions with the exceptions of the ¹²C(α, γ)¹⁶O rate taken from Kunz et al. (2002)

The first two reaction rates are important for H burning; indeed, the ¹⁴N(p, γ)¹⁵O reaction is crucial among those involved in the CNO cycle, because it is the slowest one. The impact of this recent ¹⁴N(p, γ)¹⁵O rate on stellar evolution models has been investigated by Imbriani et al. (2004), Weiss et al. (2005), and Pietrinferni et al. (2010). However, we have repeated the analysis here to verify the expected variation with respect to the previous BaSTI calculations, due to the combined effects of using the new rates for both ${}^{3}\mathrm{He}{{(}^{4}\mathrm{He},\gamma )}^{7}\mathrm{Be}$ and ${}^{14}{\rm{N}}{({\rm{p}},\gamma )}^{15}{\rm{O}}$ nuclear reactions. When all other physics inputs are kept fixed, we have found that:

• 1.  
  for a 0.8 M_⊙, Z = 0.0003 model, the luminosity at the MS turnoff (TO) increases by ${\rm{\Delta }}\mathrm{log}(L/{L}_{\odot })\sim 0.02$, while the age increases by about 210 Myr when passing from the NACRE reaction rates used in the previous BaSTI calculations to the ones adopted for the new models. For the same mass but with a metallicity Z = 0.008, the effects are smaller, with the MS TO luminosity increased by about 0.01 dex and the age increased by ∼30 Myr;
• 2.  
  as for the evolution along the red giant branch (RGB), the effect of the new rates on the RGB bump luminosity is completely negligible at Z = 0.008, while the RGB bump luminosity increases by ${\rm{\Delta }}\mathrm{log}(L/{L}_{\odot })\sim 0.04$ at Z = 0.0003. Regardless of the metallicity, the use of the new rates decreases the RGB tip brightness by ${\rm{\Delta }}\mathrm{log}(L/{L}_{\odot })\sim 0.02,$ in agreement with the results by Pietrinferni et al. (2010) and Serenelli et al. (2017).

The ¹²C(α, γ)¹⁶O reaction is one of the most critical nuclear processes in stellar astrophysics because of its impact on a number of astrophysical problems (see, e.g., Cassisi et al. 2003; Cassisi & Salaris 2013, and references therein). The more recent assessment of this reaction rate is not significantly different from Kunz et al. (2002) as used by Pietrinferni et al. (2004). As a consequence, the use of this new rate has a small impact on the models: for instance, the core He-burning lifetime is decreased by a negligible ∼0.2% when using this new rate compared to models calculated with the older Kunz et al. (2002) rate.

As in the previous BaSTI calculations, electron screening is calculated according to the appropriate choice among strong, intermediate, and weak, following Dewitt et al. (1973) and Graboske et al. (1973).

Neutrino energy losses are included with the same prescriptions as in the previous BaSTI calculations. For plasma neutrinos, we use the rates by Haft et al. (1994), supplemented by the Munakata et al. (1985) rates for the other relevant neutrino production processes.

The combined effect of the treatment of the superadiabatic layers of convective envelopes and the method to determine the outer boundary conditions of the models has a major impact on the effective temperature scale of stellar models with deep convective (or fully convective) envelopes.

As in the previous BaSTI models, we treat the superadiabatic convective layers according to the Böhm-Vitense (1958) flavor of the mixing-length theory, using the formalism by Cox & Giuli (1968). The value of the mixing-length parameter α_ML is fixed by the solar model calibration to 2.006 (see Section 3 for more details) and kept the same for all masses, initial chemical compositions, and evolutionary phases.

In the previous BaSTI models, the outer boundary conditions were obtained by integrating the atmospheric layers employing the T(τ) relation provided by Krishna Swamy (1966). In this new release, we decided to employ the alternative solar semi-empirical T(τ) by Vernazza et al. (1981). More specifically, we implemented in our evolutionary code the following fit to the tabulation provided by Vernazza et al. (1981):

$$\begin{eqnarray}&&{T}^{4}=0.75\ {T}_{\mathrm{eff}}^{4}\ (\tau +1.017-0.3{e}^{-2.54\tau }-0.291{e}^{-30\tau }).\end{eqnarray} \tag{ 1 }$$

As shown by Salaris & Cassisi (2015), model tracks computed with this T(τ) relation approximate well the results obtained using the hydro-calibrated T(τ) relationships determined from the 3D radiation hydrodynamics calculations by Trampedach et al. (2014) for the solar chemical composition. Figure 2 shows the Hertzsprung–Russell diagram (HRD) of 0.85 M_⊙ evolutionary tracks from the pre-MS to the tip of the RGB, computed for the three labelled initial metallicities. The physics inputs are kept the same as in the old BaSTI calculations, but for the T(τ) relation, they are either from Krishna Swamy (1966) or Vernazza et al. (1981). For both choices, the value of α_ML has been fixed by an appropriate solar calibration.

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The two sets of models overlap almost perfectly along the MS at all Z, whereas some differences in T_eff at fixed luminosity appear along the RGB (and the pre-MS). Differences are of about 60 K at the lowest metallicity, reaching ∼90 K at solar metallicity. Tracks calculated with the Vernazza et al. (1981) T(τ) are always the cooler ones. For a more detailed discussion on the impact of different T(τ) relations on the T_eff scale of RGB stellar models, we refer to Salaris & Cassisi (2015) and references therein.

In the first release of BaSTI, the minimum stellar mass was set to 0.50 M_⊙ for all chemical compositions, while these new calculations include the mass range below 0.50 M_⊙, down to 0.10 M_⊙. As extensively discussed in the literature (see, e.g., Baraffe et al. 1995; Allard et al. 1997; Brocato et al. 1998; Chabrier & Baraffe 2000, and references therein), in this regime of so-called very low-mass (VLM) stars, i.e., M ≤ 0.45 M_⊙, outer boundary conditions provided by accurate non-gray model atmospheres are required. Therefore, for the VLM model calculations, we employed boundary conditions (pressure and temperature at a Rosseland optical depth τ = 100) taken from the PHOENIX model atmosphere library¹¹ (Allard et al. 2012 and references therein), more precisely the BT-Settl model set. These model atmospheres properly cover the required parameter space in terms of effective temperature, surface gravity, and metallicity range. However, this set of models has been computed for the Asplund et al. (2009) solar heavy element distribution, which is different from the one adopted in our calculations (see Section 2.1).

One could argue that this difference in the heavy element mixture may have an impact on the predicted spectral energy distribution, but it should have only a minor effect on the model atmosphere structure, hence on the derived outer boundary conditions. We have verified this latter point as follows. The PHOENIX model atmosphere repository contains a subset of models—labelled CIFIST2011—computed with the same solar heavy element distribution as in our calculations (Caffau et al. 2011), for a few selected metallicities. We have calculated sets of VLM models using, alternatively, the PHOENIX boundary conditions for the Asplund et al. (2009) mixture and the Caffau et al. (2011) one. Figure 3 shows the result of such a comparison for one selected metallicity. As expected, the two sets of VLM calculations provide very similar HRDs. Differences in bolometric luminosity and effective temperature are vanishingly small for masses larger than ∼0.12 M_⊙, while they are equal to just ${\rm{\Delta }}\mathrm{log}(L/{L}_{\odot })\,\sim 0.007$ and ΔT_eff ∼ 16 K for smaller masses.

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We close this section with more details on the transition from VLM models with outer boundary conditions determined from PHOENIX model atmospheres to models calculated with the T(τ) relation in Equation (1). To achieve a smooth transition in the log(L/L_⊙)—T_eff diagram between the two regimes, for each chemical composition, we computed models with mass up to 0.70 M_⊙ with the PHOENIX boundary conditions, and models with mass down to 0.4 M_⊙ using the T(τ) relation. In the overlapping mass range, we selected a specific transition mass corresponding to the pair of models—which happens to fall in the range between ∼0.5M_⊙ and ∼0.65M_⊙, depending on the initial composition—showing negligible differences in both bolometric luminosity and effective temperature, typically ΔT_eff ≤ 25 K and ${\rm{\Delta }}{\rm{l}}{\rm{o}}{\rm{g}}(L/{L}_{\odot })\leqslant 0.004$. For masses equal to and lower than this mass, we keep the calculations with PHOENIX boundary conditions, and above this limit the models with T(τ) integration. This allows the isochrones displaying a smooth transition between the two boundary condition regimes to be calculated.

Mass loss is included in the Reimers (1975) formula, as in the previous BaSTI models. The free parameter η entering this mass-loss prescription has been set equal to 0.3, following the Kepler observational constraints discussed in Miglio et al. (2012). We also provide stellar models computed without mass loss (η = 0). The previous BaSTI calculations included three options, η = 0, 0.2, and 0.4.¹²

As with the first release of the BaSTI database, we have calculated several model grids by varying one at a time some modeling assumptions. A schematic overview of all grids made available in the new BaSTI repository is provided in Table 3. Our reference set of models is set (a) in Table 3, which includes MS convective core overshooting, mass loss with η = 0.3, and atomic diffusion of He and metals.

For each chemical composition (and choice of modeling assumptions), we have computed 56 evolutionary sequences. The minimum initial mass is 0.1 M_⊙, while the maximum value is 15 M_⊙). For initial masses below 0.2 M_⊙, we computed evolutionary tracks for masses equal to 0.10, 0.12, 0.15, and 0.18 M_⊙. In the range between 0.2 and 0.7 M_⊙ a mass step equal to 0.05 M_⊙ has been adopted. Mass steps equal to 0.1 M_⊙, 0.2 M_⊙, 0.5 M_⊙, and 1 M_⊙ have been adopted for the mass ranges 0.7–2.6 M_⊙, 2.6–3.0 M_⊙, and 3.0–10.0 M_⊙, and masses larger than 10.0 M_⊙, respectively.

Models less massive than 4.0 M_⊙ have been computed from the pre-MS, whereas more massive models have been computed starting from a chemically homogeneous configuration on the MS. Relevant to pre-MS calculations, the adopted mass fractions for D, ³He, and ⁷Li are equal to 3.9 $\times$ 10⁻⁵, 2.3 $\times$ 10⁻⁵, and 2.6 $\times$ 10⁻⁹, respectively.

All stellar models—except for the less massive ones whose core H-burning lifetime is longer than the Hubble time—have been calculated until the start of the thermal pulses (TPs)¹³ on the AGB, or C-ignition for the more massive ones. For the long-lived low-mass models, we stopped the calculations when the central H mass fraction is ∼0.3 (corresponding to ages already much older than the Hubble time).

For each initial chemical composition, we also provide an extended set of core He-burning models suitable for the study of the HB in old stellar populations. We have considered different values of the total mass (with fine mass spacing, as in Pietrinferni et al. 2004) but the same mass for the He core and the same envelope chemical stratification, corresponding to an RGB progenitor at the He flash for an age of ∼12.5 Gyr.

All evolutionary tracks presented in this work have been reduced to the same number of points ("normalized") to calculate isochrones (see, e.g., Dotter 2016 for a discussion of this issue) and for ease of interpolation, by adopting the same approach extensively discussed in Pietrinferni et al. (2004) and updated in Pietrinferni et al. (2006). This method is based on the identification of some characteristic homologous points (key points) corresponding to well-defined evolutionary stages along each individual track (see Pietrinferni et al. 2004 for more details on this issues). Given that almost all evolutionary tracks now include the pre-MS stage, we added three additional key points compared to the previous BaSTI calculations. The first one is taken at an age of 10⁴ yr, the second one corresponds to the end of the deuterium-burning stage, while the third key point is set at the first minimum of the surface luminosity for all models but the VLM ones. For these latter masses, this point corresponds to the stage when the energy produced by the p–p chain starts to dominate the energy budget. The fourth key point corresponds to the zero-age MS (ZAMS), defined as the model fully sustained by nuclear reactions, with all secondary elements at their equilibrium abundances.¹⁴ However, for VLM models that attain nuclear equilibrium of the secondary elements involved in the p–p chain over extremely long timescales, this key point corresponds to the first minimum of the bolometric luminosity. All subsequent key points are fixed exactly as in the previous BaSTI database. Table 4 lists the correspondence between key points and evolutionary stages as well as the corresponding line number in the normalized evolutionary track, while Figure 4 shows the location of a subset of key points (the first 10) on selected evolutionary tracks.

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Table 4.  Correspondence between Evolutionary Stage, Key Point, and Line Number of the Normalized Tracks

Key Point	Line	Evolutionary Phase
1	1	Age equal to 1000 years
2	20	End of deuterium burning
3	60	The first minimum in the surface luminosity, or when nuclear energy starts to dominate the energy budget
4	100	Zero-age main sequence or minimum in bolometric luminosity for VLM models
5	300	First minimum of Teff for high-mass or central H mass fraction Xc=0.30 for low-mass and VLM models
6	360	Maximum in Teff along the MS (TO point)
7	420	Maximum in $\mathrm{log}(L/{L}_{\odot })$ for high-mass or Xc=0.0 for low-mass models
8	490	Minimum in $\mathrm{log}(L/{L}_{\odot })$ for high-mass or base of the RGB for low-mass models
9	860	Maximum luminosity along the RGB bump
10	890	Minimum luminosity along the RGB bump
11	1290	Tip of the RGB
12	1300	Start of quiescent core He burning
13	1450	Central abundance of He equal to 0.55
14	1550	Central abundance of He equal to 0.50
15	1650	Central abundance of He equal to 0.40
16	1730	Central abundance of He equal to 0.20
17	1810	Central abundance of He equal to 0.30
18	1950	Central abundance of He equal to 0.00
19	2100	Energy associated with the CNO cycle becomes larger than that provided by He burning

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For each chemical composition, these normalized evolutionary tracks are used to compute extended sets of isochrones for ages between 20 Myr and 14.5 Gyr (older isochrones can also be computed on request).

Figure 5 shows an example of the full set of reference tracks and isochrones calculated for one chemical composition (Y = 0.2695, Z = 0.01721). Panel (a) displays the full grid of tracks for masses ranging from 0.1 M_⊙ to 15 M_⊙, while panel (c) focuses on the RGB region for a subset of models with mass between 0.4 M_⊙ and 4.5 M_⊙ (dotted lines denote the pre-MS evolution of the same models). The set of HB tracks is shown in panel (d) for an RGB progenitor mass equal to 1.0M_⊙ and minimum HB mass equal to 0.4727 M_⊙, while panel (e) displays a subset of pre-MS, MS, and RGB tracks with mass between 0.1M_⊙ and 1.0M_⊙. Finally, panel (b) displays a set of isochrones with ages equal to 20 Myr, 100 Myr, 500 Myr, 1 Gyr, 4 Gyr, and 14 Gyr, respectively (solid lines), overlaid onto the full set of tracks (dashed lines).

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Bolometric luminosities and effective temperatures along evolutionary tracks and isochrones need to be translated to magnitudes and colors in sets of photometric filters for comparison with observed color–magnitude diagrams (CMDs) and to predict integrated fluxes of unresolved stellar populations. This requires sets of stellar spectra covering the relevant parameter space in terms of metallicity, surface gravity, and effective temperature of the models. To such aim, a new grid of model atmospheres has been computed using the latest version of the ATLAS9 code¹⁵ originally developed by Kurucz (1970). ATLAS9 allows one-dimensional, plane-parallel model atmospheres to be calculated under the assumption of local thermodynamical equilibrium for all species. The method of the opacity distribution function (ODF; Kurucz et al. 1974) is employed to handle the line opacity by pretabulating the line opacity as a function of gas pressure and temperature in a given number of wavelength bins. ODFs and Rosseland mean opacity tables are calculated for a given metallicity (fixing the chemical mixture) and for a given value of microturbulent velocity. Even if the computation of ODFs can be time consuming, the calculation of any model atmosphere (defined by its effective temperature and gravity) for the metallicity and microturbulent velocity corresponding to the adopted ODF turns out to be very fast.

Grids of ATLAS9 model atmospheres based on suitable ODFs are freely available but based on different solar chemical abundances compared to the one used in our calculations. The grid by Castelli & Kurucz (2004) adopted the solar abundances by Grevesse & Sauval (1998), which were computed by Kirby (2011) using the abundances of Anders & Grevesse (1989), while the recent one by Mészáros et al. (2012) for the APOGEE survey used the abundances by Asplund et al. (2005). For the new grid presented here, we adopted the same solar metal distribution of the stellar evolution calculations. For the computation of the new ODFs, Rosseland opacity tables, and model atmospheres, we followed the scheme described in Mészáros et al. (2012).

For each [Fe/H] and microturbulent velocity, one ODF and one Rosseland opacity table are calculated using the codes DFSYNTHE and KAPPA9 (Castelli 2005), respectively. The [Fe/H] grid ranges from −4.0 to +0.5 dex in steps of 0.5 dex from −4.0 to −3.0 dex, and in steps of 0.25 dex for the other metallicities, assuming solar-scaled abundances for all elements. The adopted values for the microturbulent velocities are 0, 1, 2, 4, and 8 km s⁻¹. In the calculation of the ODFs, we included all atomic and molecular transitions listed in F. Castelli's web site¹⁶ ; in particular, the line list for TiO is from Schwenke (1998) and that for H₂O is from Langhoff et al. (1997).

For each [Fe/H] (but adopting only the microturbulent velocity of 2 km s⁻¹) a grid of ATLAS9 model atmospheres has been computed, covering the effective temperature–surface gravity parameter space summarized in Table 5, for a total of 475 models.

Table 5.  Effective Temperature and Surface Gravity Ranges Covered by Our New Grid of ATLAS9 Model Atmospheres and Spectra, Together with the Grid Spacings ${{\rm{\Delta }}}_{{T}_{\mathrm{eff}}}$ and Δ_log(g)

Teff	${{\rm{\Delta }}}_{{T}_{\mathrm{eff}}}$	log(g)	Δlog(g)
(K)	(K)	(cgs)	(cgs)
3500–6000	250	0.0–5.0	0.5
6250–7500	250	0.5–5.0	0.5
7750–8250	250	1.0–5.0	0.5
8500–9000	250	1.5–5.0	0.5
9250–11,750	250	2.0–5.0	0.5
12,000–13,000	250	2.5–5.0	0.5
13,000–19,000	1000	2.5–5.0	0.5
20,000–26,000	1000	3.0–5.0	0.5
27,000–31,000	1000	3.5–5.0	0.5
32,000–39,000	1000	4.0–5.0	0.5
40,000–49,000	1000	4.5–5.0	0.5
50,000	⋯	5.0	⋯

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Similarly to those computed by Castelli & Kurucz (2004), these new model atmospheres include 72 plane-parallel layers ranging from $\mathrm{log}\tau =-6.875$ (where τ is the Rosseland optical depth) to +2.00, in steps of 0.125, and have been computed with the overshooting option switched off, adopting a mixing-length equal to 1.25 as in previous calculations. For each model atmosphere, the corresponding emerging flux has then been computed.

The ATLAS9 grid of spectra is complemented by two additional spectral libraries to cover the parameter space of cool giants and low-mass dwarfs. At low T_eff and surface gravities, we use the BaSeL WLBC99 results (Westera et al. 1999, 2002). This is a semi-empirical library, built from a grid of theoretical spectra that have been later calibrated to match empirical color–T_eff relations from neighborhood stars. These templates are available in the metallicity range −2.0 ≤ [Fe/H] ≤ 0.5, in steps of 0.5 dex. For the low T_eff and high gravity regime, we use spectra from the Göttingen Spectral Library (Husser et al. 2013). These have been calculated using the code PHOENIX (Hauschildt & Baron 1999), which is particularly suited to model atmospheres of cool dwarfs. The PHOENIX configuration used for this library employs a variable parametrization of microturbulence and mixing length, depending on the properties of the modeled atmosphere. The metallicity coverage is −4.0 ≤ [Fe/H] ≤ 1.0, in steps of 0.5 dex. Figure 6 shows the range of effective temperature and surface gravity covered by our adopted spectral libraries.

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We have computed tables of BCs for several popular photometric systems (the complete list is found in Table 6), following the prescription by Girardi et al. (2002) for photon-counting defined systems:

$$\begin{eqnarray}{\mathrm{BC}}_{{S}_{\lambda }} & = & {M}_{\mathrm{bol},\odot }-2.5\mathrm{log}[4\pi {(10\mathrm{pc})}^{2}{F}_{\mathrm{bol}}/{L}_{\odot }]\\ & & +2.5\mathrm{log}\left(\displaystyle \frac{{\displaystyle \int }_{{\lambda }_{1}}^{{\lambda }_{2}}\lambda {F}_{\lambda }{S}_{\lambda }d\lambda }{{\displaystyle \int }_{{\lambda }_{1}}^{{\lambda }_{2}}\lambda {f}_{\lambda }^{0}{S}_{\lambda }d\lambda }\right)-{m}_{{S}_{\lambda }}^{0},\end{eqnarray} \tag{ 2 }$$

where S_λ is a generic filter response curve, defined between λ₁ and λ₂, F_bol = σ T_eff⁴ is the total emerging flux at the stellar surface, F_λ is the stellar emerging flux at a given wavelength, f⁰_λ is the wavelength-dependent flux of a reference spectrum and ${m}_{{S}_{\lambda }}^{0}$ is the magnitude of the reference spectrum in the filter S_λ (denoted as zero, point). We adopt M_bol,⊙ = 4.74, following the IAU B2 resolution of 2015 (Mamajek et al. 2015).

The reference spectra are either the spectrum of Vega (α Lyr), for systems that use Vega for the magnitude zero-points (Vegamag systems), or a spectrum with constant flux density per unit frequency ${f}_{\nu }^{0}=3.631\cdot {10}^{-20}\,\mathrm{erg}\,\ {{\rm{s}}}^{-1}\,{\mathrm{cm}}^{-2}\,{\mathrm{Hz}}^{-1}$, for ABmag systems. For older photometric systems, such as the Johnson–Cousins–Glass UBVRIJHKLM, we use the energy-integration equivalent of Equation (2).

Due to the differences between the adopted sets of spectral libraries, the resulting BCs display non-negligible differences in the overlapping T_eff and surface gravity regimes. To eliminate discontinuities in the final merged BC set, the different sets were matched smoothly in the overlapping regions by applying some suitable ramping at the edges of the various tables. After several tests, we adopted the following combination of BC libraries:

• 1.  
  at metallicities equal to or lower than solar, we employ the BCs from our Atlas9 grid, supplemented at gravities lower than $\mathrm{log}({\rm{g}})=0.0$ and ${T}_{{\rm{eff}}}\lt 3700$ K by WLBC99 results; at ${T}_{\mathrm{eff}}$ lower than about $3700$ K and $\mathrm{log}({\rm{g}})\geqslant 4.5$ we switch from our atlas9 BCs to Husser et al. (2013) BCs;
• 2.  
  at supersolar metallicities, we adopt our atlas9 BCs for the V band (or equivalent) as well as for bluer photometric passbands, extrapolating linearly in log(g) and ${T}_{{\rm{eff}}}$ when necessary. For redder photometric passbands we use atlas9 BCs for gravities lower than $\mathrm{log}({\rm{g}})=0.0$ (extrapolated when necessary) and Husser et al. (2013) BCs for gravities larger or equal than $\mathrm{log}({\rm{g}})=4.5$, and ${T}_{{\rm{eff}}}$ lower than about $3700$ K.

Figure 7 shows examples of our adopted composite BC library.

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Asteroseismology has experienced a revolution thanks to past and present space missions such as CoRoT (Baglin et al. 2009), Kepler (Gilliland et al. 2010), and K2 (Chaplin et al. 2015), which have provided high-precision photometric data for hundreds of main-sequence and subgiant stars and for thousands of red giants.

Future satellites like TESS (Ricker et al. 2014) and PLATO (Rauer et al. 2014) hold promise to expand the current sample greatly and thus further extend the impact of asteroseismology in the fields of stellar physics (e.g., Beck et al. 2011; Verma et al. 2014), exoplanet studies (e.g., Huber et al. 2013; Silva Aguirre et al. 2015, and Galactic archaeology (e.g., Casagrande et al. 2016; Silva Aguirre et al. 2017). Given the availability of high-quality oscillation data, we provide the corresponding theoretical quantities to fully exploit their potential.

We have computed adiabatic oscillation frequencies for all of the models using the Aarhus aDIabatic PuLSation package (ADIPLS; Christensen-Dalsgaard 2008). We provide the radial, dipole, quadrupole, and octupole mode frequencies for the models with central hydrogen mass fraction >10⁻⁴ but only the radial mode frequencies for more evolved models. The power spectrum of the solar-like oscillators have several global characteristic features that can be used to constrain the stellar properties. Some of these features do not require very high signal-to-noise data for their determinations—in contrast to the individual oscillation frequencies that need long time-series data with high signal-to-noise ratio for their measurements—and play a crucial role in ensemble studies. We also provide three such global asteroseismic quantities for the models, viz., the frequency of maximum power (ν_max), the large frequency separation for the radial mode frequencies (Δν₀), and the asymptotic period spacing for the dipole mode frequencies (ΔP₁).

The value of ν_max was determined using the well-known scaling relation (Kjeldsen & Bedding 1995)

$$\begin{eqnarray}&&\displaystyle \frac{{\nu }_{\max }}{{\nu }_{\max ,\odot }}=\left(\displaystyle \frac{M}{{M}_{\odot }}\right){\left(\displaystyle \frac{R}{{R}_{\odot }}\right)}^{-2}{\left(\displaystyle \frac{{T}_{\mathrm{eff}}}{{T}_{\mathrm{eff},\odot }}\right)}^{-1/2},\end{eqnarray} \tag{ 3 }$$

where M, R, and T_eff are the model mass, radius, and effective temperature, respectively. We adopted ν_max,⊙ = 3090 μHz from Huber et al. (2011), T_eff,⊙ = 5777 K, and M_⊙ = 1.9891 × 10³³ gm and R_⊙ = 6.9599 × 10¹⁰ cm as used in the corresponding stellar tracks. We extracted Δν₀ following White et al. (2011), i.e., by performing a weighted linear least-squares fit to the radial mode frequencies as a function of the radial order, with a Gaussian weighting function centered around ν_max, with 0.25 ν_max FWHM. The large frequency separation and frequency of maximum power, together with the measurement of the stellar T_eff, have been used to determine the masses and radii of large samples of isolated stars, independent of modeling, thus providing strong constraints on stellar evolution models and on models of Galactic stellar populations (see, e.g., Kallinger et al. 2010; Chaplin et al. 2011; Miglio et al. 2012).

We determined the period spacing ΔP₁ using the asymptotic expression

$$\begin{eqnarray}&&{\rm{\Delta }}{P}_{1}=\sqrt{2}{\pi }^{2}{\left(\int \displaystyle \frac{N}{r}{dr}\right)}^{-1},\end{eqnarray} \tag{ 4 }$$

where N and r are the Brunt–Väisälä frequency and radial coordinate, respectively. The integration is performed over the radiative interior. Since N is weighted with r⁻¹ in the integral, ΔP₁ is very sensitive to the Brunt–Väisälä frequency profile in the core. Hence, the measurement of ΔP₁ offers a unique opportunity to constrain the uncertain aspects of the physical processes taking place in stellar cores. As an example, Degroote et al. (2010) used the measurement of the period spacing for the star HD 50230 observed using the CoRoT satellite to constrain the mixing in its core (see also, Montalbán et al. 2013). Figure 8 illustrates the evolution of models in the Δν₀–ΔP₁ diagram (evolution proceeds from right to left). This is an interesting diagram because Δν₀ contains mostly information about the envelope, whereas ΔP₁ contains mostly information about the core. The hook-like feature on the right (beyond the displayed range for M = 1.0 M_⊙ and 1.5 M_⊙) corresponds to the base of the RGB. The sudden jump at the lowest Δν₀ for M = 1.0 M_⊙, 1.5 M_⊙, and 2.0 M_⊙ is due to the helium flash, which causes the stellar structure to change rapidly in a short period of time. This diagram has been used successfully to distinguish the shell hydrogen-burning red giant stars with those that are fusing helium in the core along with the hydrogen in the shell (e.g., Bedding et al. 2011; Mosser et al. 2011).

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We have compared our new isochrones with independent calculations, considering separately pre-MS isochrones for low- and very low-mass stars, that can be calculated for a minimum age of just 4 Myr with our grid of models, whereas complete isochrones reaching the AGB phase or C-ignition start from an age of 20 Myr.

The pre-MS isochrones have been compared to results from the extensive database of Tognelli et al. (2011) and the "classic" models by Siess et al. (2000), as shown in Figure 11. These latter two calculations differ from ours with regard to some physics inputs. In particular, the Tognelli et al. (2011) isochrones have been calculated by adopting a different EOS and boundary conditions, while the Siess et al. (2000) isochrones have been computed with different low-temperature radiative opacities, EOS, and boundary conditions, and the initial deuterium abundance is about half the value used in our calculations. The reference solar metal mixture is different for each of the three sets of isochrones shown in the figure. The minimum evolving mass along the isochrones is equal to 0.1 M_⊙ for our and Siess et al. (2000) calculations, while it is equal to 0.2 M_⊙ for the Tognelli et al. (2011) models.

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For the comparison, we have selected the Tognelli et al. (2011) calculations (which at fixed Z allow for various choices of Y, the deuterium mass fraction X_D, and mixing length) for Z = 0.0175, Y = 0.265, X_D = 4 $\times$ 10⁻⁵, α_ML = 1.9—very close to our initial solar chemical composition, the adopted initial deuterium mass fraction, and solar-calibrated mixing length—and the Z = 0.02 Siess et al. (2000) isochrones. We have considered ages equal to 4, 10, 15, 30, 50, and 100 Myr. The upper age limit is fixed by the largest age available for the Tognelli et al. (2011) calculations.

The agreement between our Z = 0.0172 ([Fe/H] = 0.06) and the Tognelli et al. (2011) isochrones is remarkable. They are almost indistinguishable, with appreciable differences appearing only for the lowest masses in common and the two youngest ages, where the Tognelli et al. (2011) isochrones are more luminous than ours at a given T_eff. Differences with respect to the Siess et al. (2000) calculations are larger and more systematic, their isochrones being almost always brighter at fixed T_eff for stellar masses between ∼2.0–2.5 M_⊙ and ∼0.4 M_⊙.

Our complete isochrones have been compared with results from the recent PARSEC and MIST isochrones. We considered the nonrotating MIST isochrones and the PARSEC isochrones with VLM stellar models calculated with the "calibrated" boundary conditions, as described in Chen et al. (2014).

We considered our isochrones including convective core overshooting during the MS and atomic diffusion (the reference set (a) described in Table 3), since both effects are included in the MIST and PARSEC isochrones, although with varying implementations. Compared to our models, the nonrotating MIST isochrones have been calculated with different implementations of convective mixing (and include thermohaline mixing during the RGB), as well as different choices for the solar metal distribution, EOS, reaction rates, boundary conditions, mixing length theory formalism, and a lower value of the Reimers η parameter. Radiative levitation is neglected, and the efficiency of atomic diffusion during the MS is moderated by including a competing turbulent diffusive coefficient (see Choi et al. 2016 for details).

The PARSEC calculations have employed, compared to our new models, different choices for the low-temperature radiative opacities, electron conduction opacities, reaction rates, implementation of overshooting, boundary conditions, and a lower value of the Reimers parameter η. Atomic diffusion without radiative levitation is included, but switched off when the mass size of the outer convective region decreases below a given threshold (see Bressan et al. 2012 for details).

Figures 12 and 13 show selected isochrones for 30 Myr, 100 Myr, 1 Gyr, 3 Gyr, 5 Gyr, and 12 Gyr, and [Fe/H] = 0.06 and [Fe/H] = −1.55, respectively. They are shown together with PARSEC isochrones for the same ages, [Fe/H] = 0.07 and −1.59,¹⁷ and MIST isochrones for the same ages and [Fe/H] as our isochrones.¹⁸

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The comparison with PARSEC isochrones displays a remarkable general agreement especially at lower [Fe/H], whereas at higher metallicity, the lower masses (which are still evolving along the pre-MS phase in the two youngest isochrones) are systematically discrepant compared to our models. The TO luminosities are only slightly different, especially at the three lowest ages, where the effect of different core overshooting prescriptions may play a role. The core He-burning phase is slightly overluminous compared to our models, and RGBs are slightly cooler compared to our [Fe/H] = 0.06 isochrones and slightly hotter compared to the [Fe/H] = −1.55 ones. Figures 14 and 15 enlarge the core He-burning portion of the isochrones for ages between 1 and 12 Gyr. The RGB of the PARSEC isochrones is cooler by less than 100 K compared to our models for [Fe/H] = 0.06, and hotter by less than 100 K at lower metallicity. The luminosity of the He-burning phase is only slightly larger (by a few hundredth dex) at both metallicities. Notice that at 12 Gyr the start of quiescent core He burning in our isochrones is at a hotter T_eff than in the PARSEC results, due to our choice of a larger Reimers parameter η.

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The comparison with MIST isochrones yields similar results. There is an overall good agreement for the MS, TO, and subgiant-branch (SGB) phases, and also in the regime of the lowest masses, still evolving along the pre-MS at the youngest ages. The He-burning phase of MIST isochrones is generally overluminous, and RGBs are systematically redder at [Fe/H] = 0.06, and with a different slope at [Fe/H] = −1.55. Figures 14 and 15 show RGBs over 100 K cooler than our models at [Fe/H] = 0.06, and slightly larger core He-burning luminosities, like in the comparison with PARSEC. Also, in comparison with MIST isochrones, at 12 Gyr, the start of quiescent core He burning in our isochrones is at a hotter T_eff, again due to our choice of a larger Reimers parameter η.

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We first consider masses and radii for the pre-MS detached eclipsing binary (DEB) systems compiled by Stassun et al. (2014) and Simon & Toraskar (2017), covering a mass range between 0.2 M_⊙ and 4.0 M_⊙. We assume an initial [Fe/H] = 0.06 (overall consistent with the few available spectroscopic estimates; see Stassun et al. 2014), and consider a minimum age of 4 Myr, the lowest possible value with our model grid. We do not aim to find a best-fit solution for all of the systems, just at least one isochrone that matches simultaneously the mass and radius of both components for each system within the errors, to denote a general consistency between models and observations.

This test is relevant for the general adequacy of both boundary conditions and α_ML values employed in the calculations, given the extreme sensitivity of pre-MS tracks to the combination of these two inputs. It is, however, worth noticing that the lack of model-independent age estimates prevent this type of test from providing very stringent constraints on the models.

We found 13 systems in the age range covered by our pre-MS isochrones, displayed in a mass–radius (MR) diagram in Figure 16. In the case of all these systems, our isochrones can match both components within the quoted 1σ error bars with a single age value, varying between 5 and 60 Myr within the entire sample of DEBs.

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The next test involves low-mass MS models. The existence of a disagreement between theoretical and observational MR relationships for low-mass stars has been recognized for some time now, with model radii typically 10%–20% smaller than observations for a fixed mass (see, e.g., Torres et al. 2010 for a review). Here we examine first the level of agreement between the observed and theoretical MS low-mass MR relationship by comparing our grid of models with data from DEB systems that host components with M < 0.8 M_⊙, as compiled by Feiden & Chaboyer (2012). This compilation includes systems with quoted random uncertainties in both mass and radius below 3%. As for the pre-MS case, the requirement for the models is that they are able to match the position of both system components in the MR diagram for a single value of the age.

We assume that all DEBs have metallicity around solar (see also Feiden & Chaboyer 2012 for spectroscopic metallicity estimates for a few of the systems in their compilation) and split the sample into two subsamples. The first one is made of systems with both components essentially on the ZAMS, displayed in Figure 17, together with isochrones of ages equal to 1 Gyr and 12 Gyr respectively, and [Fe/H] = 0.06. We also show a 12 Gyr isochrone for [Fe/H] = −0.40 to highlight the insensitivity of the theoretical MR relationship to metallicity, when the mass is below ∼0.7 M_⊙.

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Isochrones appear to match all systems reasonably well, without a clear major systematic discrepancy in radius at fixed mass. The effect of age is very small for this mass range. If we denote with ΔR the difference R_obs – R_theory between observed and predicted radii for an object with observed mass M, we find an average ΔR/R_obs = 0.02 ± 0.03 assuming an age of 12 Gyr for all systems, and an average ΔR/R_obs = 0.04 ± 0.03 for an age of 1 Gyr (see the inset of Figure 17). These average differences are consistent with typical systematic errors on empirical radius estimates—of the order of 2%–3%—as determined by Windmiller et al. (2010) in their reanalysis of the DEB system Gu Boo.

The second subsample (see Figure 18) includes systems with one or both components evolved off the ZAMS. We impose an upper limit of 13.5 Gyr to their ages, to match the cosmological constraint. The major discrepancy here is for UV Psc, whereby one component is matched by the 9 Gyr isochrone, whereas the less massive one appears older than 13.5 Gyr. A minor discrepancy also affects IM Vir, with the lower-mass component appearing slightly older than the companion.

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On the whole, there is no major systematic discrepancy between models and observed MR relationships, although there are clear mismatches for a few cases, as found also by Feiden & Chaboyer's (2012) analysis. Another example of a mismatch is the M-dwarf system (both components with masses around 0.4 M_⊙) KELT J041621–620046 studied very recently by Lubin et al. (2017) and shown in Figure 17. Our isochrones give radii systematically lower by ∼20% than observed for both components (as with all other models employed by Lubin et al. 2017). The commonly accepted explanation for these mismatches (see, e.g., Feiden & Chaboyer 2012; Lubin et al. 2017, and references therein) involves the effects of large-scale magnetic fields that suppress convective motions and increase the total surface coverage of starspots. This causes a reduction in the total energy flux across a given surface within the star, forcing the stellar radius to inflate and ensuring flux conservation. For the sake of comparison, we also show in Figure 17 a 50 Myr, [Fe/H] = 0.06 pre-MS isochrone that would match within the error bars the position of the KELT J041621–620046 components in the MR diagram, in case these objects were actually pre-MS stars.

The next comparison involves the DEB system KIC 8410637, studied by Frandsen et al. (2013). It contains an MS and an RGB star, and is another good test for the calibration of convection in the models. Figure 19 compares observations and isochrones in the MR diagram. When considering isochrones for [Fe/H] = 0.06, consistent with the spectroscopic estimate by Frandsen et al. (2013), we find that an age of 2.5 Gyr matches very well the position of the two components (a similar result was found by Frandsen et al. 2013 using PARSEC isochrones).

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The last DEB systems compared to our models are four objects from the Claret & Torres (2017) compilation. Their spectroscopic metallicity is consistent within errors with [Fe/H] = −0.40; the masses of the various components range between ∼1.4 M_⊙ and ∼4.2 M_⊙, and they are evolving along either the RGB or core He-burning phase. Figure 20 shows the comparison of their MR diagrams with theoretical isochrones for [Fe/H] = −0.40, which are able to match simultaneously both components (within their mass and radius error bars) in all four systems for the labelled ages, with our choices of MS core overshooting efficiency and mixing length.

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Finally, we compare the mass–luminosity relationship predicted by our low-mass models in the V and K bands, with the data presented by Delfosse et al. (2000), based mainly on visual and interferometric pairs. We display in Figure 21 the observational data together with three isochrones with [Fe/H] = 0.06 and ages equal to 300 Myr, 1 Gyr, and 10 Gyr respectively (solid lines), plus two 10 Gyr isochrones with [Fe/H] = −0.40 and [Fe/H] = 0.45, respectively (dashed lines).

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First of all, as also noted by Delfosse et al. (2000), the V-band data show a large dispersion at fixed M, with models matching a sort of upper envelope of the data. The K-band data are much tighter and in very good general agreement with the [Fe/H] = 0.06 models, even though the sample is smaller than for the V band. It is interesting to consider the two objects highlighted by the dotted lines. They have estimates of both V and K absolute magnitudes; in the K band, the agreement with theory for [Fe/H] = 0.06 (or higher) is essentially perfect, whereas in the V band, the data are clearly underluminous compared to the models. The fact that the V-band diagram is much more sensitive to the exact metallicity of the sample (as shown by the dashed lines in the figure) suggests that [Fe/H] may play a role in explaining this dispersion. The [Fe/H] = 0.45 isochrone is underluminous at fixed mass compared to the [Fe/H] = 0.06 one, but still cannot explain the full dispersion of the data.

Figure 22 displays a similar comparison with the more recent mass–luminosity empirical data by Benedict et al. (2016). Also, in this case, the dispersion in the V band is larger than that in the K band. In the K band (which is weakly sensitive to chemical composition), the agreement with theory is again generally quite good, apart from the cluster of objects with mass around 0.6 M_⊙ that appear somewhat underluminous with respect to the models, irrespective of the adopted metallicity between [Fe/H] = −0.4 and 0.45.

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A recent study by Tayar et al. (2017) has provided a sample of over 3000 RGB stars with T_eff, mass (determined from asteroseismic scaling relations), surface gravity, [Fe/H], and α-enhancement ([α/Fe]) determinations from the updated APOGEE-Kepler catalog. These stars cover a log(g) range between ∼3.3 and 1.1 (in cgs units), and T_eff between ∼5200 and 3900 K, with the bulk of the stars having [Fe/H] between ∼−0.7 and ∼+0.4 dex, and a maximum α-enhancement typically around 0.25 dex. This sample allows empirically determined T_eff values (calibrated on the González Hernández & Bonifacio 2009 temperature scale) to be compared with theoretical models of the appropriate chemical composition, which are very sensitive to the treatment of the superadiabatic layers, hence to the calibration of α_ML.

In our comparison, we have considered only stars with [α/Fe] < 0.07 (this upper limit corresponds to approximately three to five times the quoted 1σ error on [α/Fe]), but an upper limit closer to zero does not change our results. We have calculated the differences ΔT ≡ T_obs – T_models between the observed and theoretical T_eff for each individual star by interpolating linearly in mass, [Fe/H], and log(g) among our models to determine the corresponding theoretical T_eff. The ΔT values for [Fe/H] larger than ∼−0.7 dex have been collected in 10 [Fe/H] bins with a total width of 0.10 dex, apart from the most metal-poor one, which has a width of 0.20 dex, due to the smaller number of stars populating that metallicity range. We have then performed a linear fit to the mean ΔT values of each bin, and derived a slope equal to 14 ± 11 K dex⁻¹, which is statistically different from zero at much less than 2σ (see Figure 23). The average ΔT is equal to just −14 K, with a 1σ dispersion of 34 K. This small offset between models and observations is well within the error on the González Hernández & Bonifacio (2009) T_eff calibration (the quoted average error on their RGB T_eff scale is $\leqslant 76\,K$).

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The following comparisons with CMDs of a sample of Galactic open clusters and one globular cluster (with solar-scaled initial metal distribution) provide additional tests of the reliability of our evolutionary tracks/isochrones plus the adopted BCs. In all of these comparisons, we have included the effect of extinction according to the standard Cardelli et al. (1989) reddening law, with R_V ≡ A_V/E(B − V) = 3.1.

Figure 24 displays the BVJHK_s CMDs (JHK_s from 2MASS photometry) for Hyades members taken from Röser et al. (2011) and Kopytova et al. (2016), which reach the VLM star regime, down to ∼0.2M_⊙. We have calculated absolute magnitudes by applying the secular parallaxes determined by Röser et al. (2011). The average parallax of these objects is in agreement with the average value of 103 probable members of the Hyades from the Gaia data release 1, as given by Gaia Collaboration et al. (2017), within the quoted errors. We also display color and absolute magnitude error bars (the error bars on the absolute magnitudes also account for the contribution of the parallax errors), given that color errors often are non-negligible along the MS.

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The cluster CMDs are compared with our t = 600 and 800 Myr, [Fe/H] = 0.06 isochrones—close to spectroscopic estimates [Fe/H] = 0.14 ± 0.05 (Cayrel de Strobel et al. 1997) and [Fe/H] = 0.10 ± 0.01 (Taylor & Joner 2005)—assuming E(B − V) = 0, consistent with the results of Taylor (2006). The age range bracketed by these two isochrones is representative of the range of ages estimated for this cluster, as recently debated in the literature (see, e.g., Perryman et al. 1998; Brandt & Huang 2015 and references therein).

The theoretical isochrones follow well the observed MS down to the faintest limit, apart from the JH diagram, which shows a systematic offset due to the H-band BCs, although the models are still consistent with the data within the associated error bars.

Optical CMDs of NGC 2420 (Anthony-Twarog et al. 1990) and M67 (Sandquist 2004) are shown in Figure 25 compared to isochrones with t = 2.5 Gyr and [Fe/H] = −0.40 in the case of NGC 2420, and t = 4 Gyr and [Fe/H] = 0.06 for M67, respectively. These metallicities are consistent with [Fe/H] = −0.44 ± 0.06 (NGC 2420) and [Fe/H] = 0.02 ± 0.06 (M67) quoted by Gratton (2000). The isochrones have been shifted to account for distance moduli and reddenings (m − M)₀ = 11.95, E(B − V) = 0.06 for NGC 2420, and (m − M)₀ = 9.64, E(B − V) = 0.02 for M67. These pairs of values are consistent with the reddening estimates by Twarog et al. (1997) and the MS-fitting distance moduli (using dwarfs with accurate Hipparcos parallaxes) by Percival & Salaris (2003), within their error bars.

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The values of the mass evolving at the TO for the NGC 2420 and M67 isochrones are ∼1.3 M_⊙ and ∼1.2 M_⊙, respectively, in the mass range where the size of the overshooting region is decreased down to zero from the standard value of 0.2H_p. The shape of the TO region, which is sensitive to the extent of the overshooting region, is well traced by the isochrones for both clusters, lending some support to our prescription for the reduction of size of the overshooting region with mass.

One can notice also how, in addition to the MS (apart from the faintest end of the NGC 2420 MS), the RGB, SGB and red clump sequences are also nicely matched by the isochrones.

The next object compared to our isochrones is the old and super metal-rich open cluster NGC 6791. At the supersolar metallicity of this object, BCs are bound to be more uncertain, because inaccuracies in atomic and molecular opacity data entering the model spectra calculations are greatly enhanced in this metallicity regime.

For this cluster, we take advantage of the analysis by Brogaard et al. (2011, 2012) of two DEB systems, which provide estimates of E(B − V) = 0.16 ± 0.025, (m − M)_V = 13.51 ± 0.06, and [Fe/H] = +0.29 ± 0.03(random) ± 0.07(systematic), with this last value in agreement, within the errors, with spectroscopic estimates by Origlia et al. (2006) and Carraro et al. (2006), but lower than [Fe/H] = +0.47 ± 0.04 determined by Gratton et al. (2006).

Figure 26 displays the MR diagram for the four components (the primary component of V20 is in the TO region of the CMD; the other components are increasingly fainter MS stars) of these two DEB systems (named V18 and V20 in Brogaard et al. 2011) including the 1σ and 3σ error bars, together with two theoretical isochrones for [Fe/H] = 0.30, with and without the inclusion of atomic diffusion,¹⁹ and ages of 8.5 Gyr and 9.0 Gyr, respectively.

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As for the isochrones discussed by Brogaard et al. (2012), it is not possible to match perfectly the MR diagram of these EBs with theoretical isochrones. Those shown in Figure 26 represent the best compromise to match the data for the four DEB components, within their errors.

Figure 27 places the same isochrones of the DEB comparison in optical BVI CMDs together with the cluster photometry, corrected for differential reddening, by Brogaard et al. (2012). We have displayed only stars with good quality photometry, i.e., we have considered only objects with photometric reduction yielding a sharp index between −0.4 and +0.4, and a chi index between 0.9 and 1.2. The isochrones have been shifted in color for a reddening E(B − V) = 0.16, and vertically for (m − M)_V = 13.52 (the isochrone with diffusion) and (m − M)_V = 13.54 (the isochrone without diffusion), respectively. These distance moduli, both consistent with the result from the DEB analyses, allow the V-band magnitude of the observed red clump stars to be matched with the core He-burning portion of the isochrones. The overall comparison is better in the BV CMD, where the RGB location and slope are reasonably reproduced, as well as the TO−SGB−upper-MS sequence. The TO region is matched better by the isochrone including atomic diffusion. In VI, the match is worse overall. The RGB of the isochrones is redder than observed; the TO−SGB region is less well reproduced than in BV, although a lower MS is better matched.

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As a last object, we have considered the low-mass (total actual mass lower than 10⁵ M_⊙; see Villanova et al. 2013) outer halo Galactic globular cluster Rup 106, whose stars display a solar-scaled metal distribution, without the O–Na and C–N abundance anticorrelations common in other Galactic globular clusters (Villanova et al. 2013). Figure 28 shows the cluster optical CMD in the HST/ACS camera photometric system (Dotter et al. 2011), together with isochrones for [Fe/H] = −1.55—close to the mean value [Fe/H] = −1.47 ± 0.02 determined spectroscopically by Villanova et al. (2013)—t = 12.5 Gyr (without atomic diffusion) and t = 11.5 Gyr (including atomic diffusion), and ZAHB sequences (obtained from models with and without diffusion, respectively) for the same metallicity. A reddening E(B − V) = 0.18 and distance modulus (m − M)₀ = 16.66, for isochrones and ZAHB models with diffusion, and (m − M)₀ =16.69, for isochrones and ZAHB models without diffusion, have been applied to the models. The distance moduli have been fixed by matching the theoretical ZAHB sequences to the lower envelope of the observed HB.

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The isochrones follow well the observed CMD. The TO region is better matched by the isochrone including atomic diffusion. Increasing the age of the isochrone without diffusion to make its TO redder does not improve the match with observations, because the model SGB would become fainter than the observed one.