# New parsec data base of α-enhanced stellar evolutionary tracks and isochrones – I. Calibration with 47 Tuc (NGC 104) and the improvement on RGB bump

New parsec data base of α-enhanced stellar evolutionary tracks and isochrones – I.... Abstract Precise studies on the Galactic bulge, globular cluster, Galactic halo, and Galactic thick disc require stellar models with α enhancement and various values of helium content. These models are also important for extra-Galactic population synthesis studies. For this purpose, we complement the existing parsec models, which are based on the solar partition of heavy elements, with α-enhanced partitions. We collect detailed measurements on the metal mixture and helium abundance for the two populations of 47 Tuc (NGC 104) from the literature, and calculate stellar tracks and isochrones with these α-enhanced compositions. By fitting the precise colour–magnitude diagram with HST ACS/WFC data, from low main sequence till horizontal branch (HB), we calibrate some free parameters that are important for the evolution of low mass stars like the mixing at the bottom of the convective envelope. This new calibration significantly improves the prediction of the red giant branch bump (RGBB) brightness. Comparison with the observed RGB and HB luminosity functions also shows that the evolutionary lifetimes are correctly predicted. As a further result of this calibration process, we derive the age, distance modulus, reddening, and the RGB mass-loss for 47 Tuc. We apply the new calibration and α-enhanced mixtures of the two 47 Tuc populations ([α/Fe] ∼ 0.4 and 0.2) to other metallicities. The new models reproduce the RGB bump observations much better than previous models. This new parsec data base, with the newly updated α-enhanced stellar evolutionary tracks and isochrones, will also be a part of the new stellar products for Gaia. stars: evolution, Hertzsprung-Russell and colour-magnitude diagrams, stars: interiors, stars: low-mass 1 INTRODUCTION parsec  (PAdova-TRieste Stellar Evolution Code) is widely used in the astronomical community. It provides input for population synthesis models to study resolved and unresolved star clusters and galaxies (e.g. Perren, Vázquez & Piatti 2015; Chevallard & Charlot 2016; Gutkin, Charlot & Bruzual 2016), and offers reliable models for many other field of studies, such as to derive black hole mass when observing gravitational waves (e.g. Spera, Mapelli & Bressan 2015; Belczynski et al. 2016), to get host star parameters for exoplanets (Santos et al. 2013; Maldonado et al. 2015, etc.), to explore the mysterious ‘cosmological lithium problem’ (Fu et al. 2015), to derive the main parameters of star clusters (for instance, Borissova et al. 2014; Donati et al. 2014; Goudfrooij et al. 2015; San Roman et al. 2015) and Galactic structure (e.g. Küpper et al. 2015; Balbinot et al. 2016; Li et al. 2016; Ramya et al. 2016), to study dust formation (e.g. Nanni et al. 2013; Nanni et al. 2014), to constrain dust extinction (e.g. Schlafly et al. 2014; Schultheis et al. 2015; Bovy et al. 2016), and to understand the stars themselves (e.g. Kalari et al. 2014; Casey et al. 2016; Gullikson, Kraus & Dodson-Robinson 2016; Reddy & Lambert 2016; Smiljanic et al. 2016), etc. There are now four versions of parsec isochrones available online.1 The very first version parsec v1.0 (Bressan et al. 2012) provides isochrones for 0.0005 ≤ Z ≤ 0.07 (−1.5 ≤ [M/H] ≤ +0.6) with the mass range 0.1 ≤ M < 12  M⊙ from pre-main-sequence (pre-MS) to the thermally pulsing asymptotic giant branch (TP-AGB). In parsec v1.1 (based on Bressan et al. 2012), we expanded the metallicity range down to Z = 0.0001 ([M/H] = −2.2). parsec v1.2S included big improvements both on the very low mass stars and massive stars: Chen et al. (2014) improve the surface boundary conditions for stars with mass M ≲ 0.5  M⊙ in order to fit the mass–radius relation of dwarf stars; Tang et al. (2014) introduce mass-loss for massive star M ≥ 14  M⊙; Chen et al. (2015) improve the mass-loss rate when the luminosity approaches the Eddington luminosity and supplement the model with new bolometric corrections till M = 350 M⊙. In a later version (parsec v1.2S + COLIBRI PR16), we describe improved isochrones with the addition of COLIBRI (Marigo et al. 2013) evolutionary tracks of TP-AGB stars (Rosenfield et al. 2016; Marigo et al. 2017). All previous versions of parsec models are calculated assuming solar-scaled metal mixtures, in which the initial partition of heavy elements keeps always the same relative number density as that in the Sun. It is now well established that the solar-scaled metal mixture is not universally applicable for all types of stars. In fact, one of the most important group of elements, the so called α-elements group, is not always observed in solar proportions. Many studies have confirmed the existence of an ‘enhancement’ of α-elements in the Milky Way halo (e.g. Zhao & Magain 1990; Nissen et al. 1994; McWilliam et al. 1995; Venn et al. 2004), globular clusters (GCs; e.g. Carney 1996; Sneden 2004; Pritzl, Venn & Irwin 2005), the Galactic Bulge (Gonzalez et al. 2011; Johnson et al. 2014), and thick disc (e.g. Fulbright 2002; Reddy, Lambert & Prieto 2006; Ruchti et al. 2010). Stars in the dwarf spheroidal Milky Way satellite galaxies show different α-abundance trends compared to the Galactic halo stars, possibly indicating different star formation paths (Kirby et al. 2011). The α-elements (O, Ne, Mg, Si, S, Ar, Ca, and Ti) are mainly produced by core collapse (mostly Type II) supernovae (SNe) on short time-scale, whereas the iron-peak elements (V, Cr, Mn, Fe, Co and Ni) are mainly synthesized in Type Ia SNe on longer time-scales. Therefore, the evolution profile of [α/Fe] records the imprint of the star formation history of the system. An alternative explanation could be that the initial mass function (IMF) of the α-enhanced stellar populations was much richer in massive stars than the one from which our Sun was born (Chiosi et al. 1998). However, there is no clear evidence in support of this alternative possibility. In order to model star clusters, galaxies and Galactic components more precisely, the previous Padova isochrone data base offered a few sets of α-enhanced models for four relatively high metallicities (Salasnich et al. 2000), other stellar evolution groups also published isochrones that allow for α enhancement (e.g. VandenBerg et al. 2000, 2014; Pietrinferni et al. 2006; Valcarce, Catelan & Sweigart 2012). Now, with the thorough revision and update input physics, we introduce α-enhanced metal mixtures in parsec . In this paper, we first calibrate the new parsec α-enhanced stellar evolutionary tracks and isochrones with the well-studied GC 47 Tucanae (NGC 104), then we apply the calibrated parameters to obtain models for other metallicities. Section 2 briefly describes the input physics. Section 3 introduces the comparison with 47 Tuc data in details, including the isochrone fitting and luminosity function (LF), envelope overshooting calibration with red giant branch bump (RGBB), and mass-loss in the RGB from horizontal branch (HB) morphology. Section 4 compares the new parsec models with other stellar models and shows its improvement on RGB bump prediction. A summary of this paper and the discussion are in Section 5. 2 INPUT PHYSICS The main difference with respect to the previous versions of parsec is the adoption of new nuclear reaction rates, α-enhanced opacities, and various helium contents. We update the nuclear reaction rates from JINA REACLIB data base (Cyburt et al. 2010) with their 2015 April 6 new recommendation. In addition, more reactions – 52 instead of the 47 described in Bressan et al. (2012) for the previous versions of parsec – are taken into account. They are all listed in Table 1 together with the reference from which we take the reaction energy Q value. In the updated reactions, more isotopic abundances are considered, in total Nel = 35: 1H, D, 3He, 4He,7Li, 7Be, 8B, 12C, 13C, 13N, 14N, 15N, 15O, 16O, 17O, 18O, 17F, 18F, 19F, 20Ne, 21Ne, 22Ne, 21Na, 22Na, 23Na, 23Mg, 24Mg, 25Mg, 26Mg, 25Al, 26Alm, 26Alg, 27Al, 27Si, and 28Si. Table 1. Nuclear reaction rates adopted in this work and the reference from which we take their reaction energy Q. Reaction  Reference  $$\rm \,{}^{}\hspace{-0.8pt}{p}\,({p}\,,{\beta ^+\,\nu }) \,{}^{}\hspace{-0.8pt}{D}\,$$  Betts, Fortune & Middleton (1975)  $$\rm \,{}^{}\hspace{-0.8pt}{p}\,({D}\,,{\gamma }) \,{}^{3}\hspace{-0.8pt}{He}\,$$  Descouvemont et al. (2004)  $$\rm \,{}^{3}\hspace{-0.8pt}{He}\,({^{3}He}\,,{\gamma }) \,{}^{}\hspace{-0.8pt}{2\,p + ^{4}\hspace{-0.8pt}{He}}\,$$  Angulo et al. (1999)  $$\rm \,{}^{4}\hspace{-0.8pt}{He}\,({^{3}He}\,,{\gamma }) \,{}^{7}\hspace{-0.8pt}{Be}\,$$  Cyburt & Davids (2008)  $$\rm \,{}^{7}\hspace{-0.8pt}{Be}\,({e^-}\,,{\gamma }) \,{}^{7}\hspace{-0.8pt}{Li}\,$$  Cyburt et al. (2010)  $$\rm \,{}^{7}\hspace{-0.8pt}{Li}\,({p}\,,{\gamma }) \,{}^{}\hspace{-0.8pt}{^{4}\hspace{-2.0pt}{He} + ^{4}\hspace{-2.0pt}{He}}\,$$  Descouvemont et al. (2004)  $$\rm \,{}^{7}\hspace{-0.8pt}{Be}\,({p}\,,{\gamma }) \,{}^{8}\hspace{-0.8pt}{B}\,$$  Angulo et al. (1999)  $$\rm \,{}^{12}\hspace{-0.8pt}{C}\,({p}\,,{\gamma }) \,{}^{13}\hspace{-0.8pt}{N}\,$$  Li et al. (2010)  $$\rm \,{}^{13}\hspace{-0.8pt}{C}\,({p}\,,{\gamma }) \,{}^{14}\hspace{-0.8pt}{N}\,$$  Angulo et al. (1999)  $$\rm \,{}^{14}\hspace{-0.8pt}{N}\,({p}\,,{\gamma }) \,{}^{15}\hspace{-0.8pt}{O}\,$$  Imbriani et al. (2005)  $$\rm \,{}^{15}\hspace{-0.8pt}{N}\,({p}\,,{\gamma }) \,{}^{}\hspace{-0.8pt}{^4He + ^{12}\hspace{-2.0pt}{C}}\,$$  Angulo et al. (1999)  $$\rm \,{}^{15}\hspace{-0.8pt}{N}\,({p}\,,{\gamma }) \,{}^{16}\hspace{-0.8pt}{O}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{16}\hspace{-0.8pt}{O}\,({p}\,,{\gamma }) \,{}^{17}\hspace{-0.8pt}{F}\,$$  Iliadis et al. (2008)  $$\rm \,{}^{17}\hspace{-0.8pt}{O}\,({p}\,,{\gamma }) \,{}^{}\hspace{-0.8pt}{\,^4He + ^{14}\hspace{-2.0pt}{N}}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{17}\hspace{-0.8pt}{O}\,({p}\,,{\gamma }) \,{}^{18}\hspace{-0.8pt}{F}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{18}\hspace{-0.8pt}{O}\,({p}\,,{\gamma }) \,{}^{}\hspace{-0.8pt}{\,^4He + ^{15}\hspace{-2.0pt}{N}}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{18}\hspace{-0.8pt}{O}\,({p}\,,{\gamma }) \,{}^{19}\hspace{-0.8pt}{F}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{19}\hspace{-0.8pt}{F}\,({p}\,,{\gamma }) \,{}^{}\hspace{-0.8pt}{\,^4He + ^{16}\hspace{-2.0pt}{O}}\,$$  Angulo et al. (1999)  $$\rm \,{}^{19}\hspace{-0.8pt}{F}\,({p}\,,{\gamma }) \,{}^{20}\hspace{-0.8pt}{Ne}\,$$  Angulo et al. (1999)  $$\rm \,{}^{4}\hspace{-0.8pt}{He}\,({2\,^{4}He}\,,{\gamma }) \,{}^{12}\hspace{-0.8pt}{C}\,$$  Fynbo et al. (2005)  $$\rm \,{}^{12}\hspace{-0.8pt}{C}\,({^{4}He}\,,{\gamma }) \,{}^{16}\hspace{-0.8pt}{O}\,$$  Cyburt, Hoffman & Woosley (2012)  $$\rm \,{}^{14}\hspace{-0.8pt}{N}\,({^{4}He}\,,{\gamma }) \,{}^{18}\hspace{-0.8pt}{F}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{15}\hspace{-0.8pt}{N}\,({^{4}He}\,,{\gamma }) \,{}^{19}\hspace{-0.8pt}{F}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{16}\hspace{-0.8pt}{O}\,({^{4}He}\,,{\gamma }) \,{}^{20}\hspace{-0.8pt}{Ne}\,$$  Constantini & LUNA Collaboration (2010)  $$\rm \,{}^{18}\hspace{-0.8pt}{O}\,({^{4}He}\,,{\gamma }) \,{}^{22}\hspace{-0.8pt}{Ne}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{20}\hspace{-0.8pt}{Ne}\,({^{4}He}\,,{\gamma }) \,{}^{24}\hspace{-0.8pt}{Mg}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{22}\hspace{-0.8pt}{Ne}\,({^{4}He}\,,{\gamma }) \,{}^{26}\hspace{-0.8pt}{Mg}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{24}\hspace{-0.8pt}{Mg}\,({^{4}He}\,,{\gamma }) \,{}^{28}\hspace{-0.8pt}{Si}\,$$  Strandberg et al. (2008)  $$\rm \,{}^{13}\hspace{-0.8pt}{C}\,({^{4}He}\,,{n}) \,{}^{16}\hspace{-0.8pt}{O}\,$$  Heil et al. (2008)  $$\rm \,{}^{17}\hspace{-0.8pt}{O}\,({^{4}He}\,,{n}) \,{}^{20}\hspace{-0.8pt}{Ne}\,$$  Angulo et al. (1999)  $$\rm \,{}^{18}\hspace{-0.8pt}{O}\,({^{4}He}\,,{n}) \,{}^{21}\hspace{-0.8pt}{Ne}\,$$  Angulo et al. (1999)  $$\rm \,{}^{21}\hspace{-0.8pt}{Ne}\,({^{4}He}\,,{n}) \,{}^{24}\hspace{-0.8pt}{Mg}\,$$  Angulo et al. (1999)  $$\rm \,{}^{22}\hspace{-0.8pt}{Ne}\,({^{4}He}\,,{n}) \,{}^{25}\hspace{-0.8pt}{Mg}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{25}\hspace{-0.8pt}{Mg}\,({^{4}He}\,,{n}) \,{}^{28}\hspace{-0.8pt}{Si}\,$$  Angulo et al. (1999)  $$\rm \,{}^{20}\hspace{-0.8pt}{Ne}\,({p}\,,{\gamma }) \,{}^{21}\hspace{-0.8pt}{Na}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{21}\hspace{-0.8pt}{Ne}\,({p}\,,{\gamma }) \,{}^{22}\hspace{-0.8pt}{Na}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{22}\hspace{-0.8pt}{Ne}\,({p}\,,{\gamma }) \,{}^{23}\hspace{-0.8pt}{Na}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{23}\hspace{-0.8pt}{Na}\,({p}\,,{\gamma }) \,{}^{}\hspace{-0.8pt}{\,^4He + ^{20}\hspace{-2.0pt}{Ne}}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{23}\hspace{-0.8pt}{Na}\,({p}\,,{\gamma }) \,{}^{24}\hspace{-0.8pt}{Mg}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{24}\hspace{-0.8pt}{Mg}\,({p}\,,{\gamma }) \,{}^{25}\hspace{-0.8pt}{Al}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{25}\hspace{-0.8pt}{Mg}\,({p}\,,{\gamma }) \,{}^{26}\hspace{-0.8pt}{Al^g}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{25}\hspace{-0.8pt}{Mg}\,({p}\,,{\gamma }) \,{}^{26}\hspace{-0.8pt}{Al^m}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{26}\hspace{-0.8pt}{Mg}\,({p}\,,{\gamma }) \,{}^{27}\hspace{-0.8pt}{Al}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{26}\hspace{-0.8pt}{Al^g}\,({p}\,,{\gamma }) \,{}^{27}\hspace{-0.8pt}{Si}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{27}\hspace{-0.8pt}{Al}\,({p}\,,{\gamma }) \,{}^{}\hspace{-0.8pt}{\,^4He + ^{24}\hspace{-2.0pt}{Mg}}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{27}\hspace{-0.8pt}{Al}\,({p}\,,{\gamma }) \,{}^{28}\hspace{-0.8pt}{Si}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{26}\hspace{-0.8pt}{Al}\,({p}\,,{\gamma }) \,{}^{27}\hspace{-0.8pt}{Si}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{26}\hspace{-0.8pt}{Al}\,({n}\,,{p}) \,{}^{26}\hspace{-0.8pt}{Mg}\,$$  Tuli (2012)  $$\rm \,{}^{12}\hspace{-0.8pt}{C}\,({^{12}C}\,,{n}) \,{}^{23}\hspace{-0.8pt}{Mg}\,$$  Caughlan & Fowler (1988)  $$\rm \,{}^{12}\hspace{-0.8pt}{C}\,({^{12}C}\,,{p}) \,{}^{23}\hspace{-0.8pt}{Na}\,$$  Caughlan & Fowler (1988)  $$\rm \,{}^{12}\hspace{-0.8pt}{C}\,({^{12}C}\,,{^{4}He}) \,{}^{20}\hspace{-0.8pt}{Ne}\,$$  Caughlan & Fowler (1988)  $$\rm \,{}^{20}\hspace{-0.8pt}{Ne}\,({\gamma }\,,{^{4}He}) \,{}^{16}\hspace{-0.8pt}{O}\,$$  Constantini & LUNA Collaboration (2010)  Reaction  Reference  $$\rm \,{}^{}\hspace{-0.8pt}{p}\,({p}\,,{\beta ^+\,\nu }) \,{}^{}\hspace{-0.8pt}{D}\,$$  Betts, Fortune & Middleton (1975)  $$\rm \,{}^{}\hspace{-0.8pt}{p}\,({D}\,,{\gamma }) \,{}^{3}\hspace{-0.8pt}{He}\,$$  Descouvemont et al. (2004)  $$\rm \,{}^{3}\hspace{-0.8pt}{He}\,({^{3}He}\,,{\gamma }) \,{}^{}\hspace{-0.8pt}{2\,p + ^{4}\hspace{-0.8pt}{He}}\,$$  Angulo et al. (1999)  $$\rm \,{}^{4}\hspace{-0.8pt}{He}\,({^{3}He}\,,{\gamma }) \,{}^{7}\hspace{-0.8pt}{Be}\,$$  Cyburt & Davids (2008)  $$\rm \,{}^{7}\hspace{-0.8pt}{Be}\,({e^-}\,,{\gamma }) \,{}^{7}\hspace{-0.8pt}{Li}\,$$  Cyburt et al. (2010)  $$\rm \,{}^{7}\hspace{-0.8pt}{Li}\,({p}\,,{\gamma }) \,{}^{}\hspace{-0.8pt}{^{4}\hspace{-2.0pt}{He} + ^{4}\hspace{-2.0pt}{He}}\,$$  Descouvemont et al. (2004)  $$\rm \,{}^{7}\hspace{-0.8pt}{Be}\,({p}\,,{\gamma }) \,{}^{8}\hspace{-0.8pt}{B}\,$$  Angulo et al. (1999)  $$\rm \,{}^{12}\hspace{-0.8pt}{C}\,({p}\,,{\gamma }) \,{}^{13}\hspace{-0.8pt}{N}\,$$  Li et al. (2010)  $$\rm \,{}^{13}\hspace{-0.8pt}{C}\,({p}\,,{\gamma }) \,{}^{14}\hspace{-0.8pt}{N}\,$$  Angulo et al. (1999)  $$\rm \,{}^{14}\hspace{-0.8pt}{N}\,({p}\,,{\gamma }) \,{}^{15}\hspace{-0.8pt}{O}\,$$  Imbriani et al. (2005)  $$\rm \,{}^{15}\hspace{-0.8pt}{N}\,({p}\,,{\gamma }) \,{}^{}\hspace{-0.8pt}{^4He + ^{12}\hspace{-2.0pt}{C}}\,$$  Angulo et al. (1999)  $$\rm \,{}^{15}\hspace{-0.8pt}{N}\,({p}\,,{\gamma }) \,{}^{16}\hspace{-0.8pt}{O}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{16}\hspace{-0.8pt}{O}\,({p}\,,{\gamma }) \,{}^{17}\hspace{-0.8pt}{F}\,$$  Iliadis et al. (2008)  $$\rm \,{}^{17}\hspace{-0.8pt}{O}\,({p}\,,{\gamma }) \,{}^{}\hspace{-0.8pt}{\,^4He + ^{14}\hspace{-2.0pt}{N}}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{17}\hspace{-0.8pt}{O}\,({p}\,,{\gamma }) \,{}^{18}\hspace{-0.8pt}{F}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{18}\hspace{-0.8pt}{O}\,({p}\,,{\gamma }) \,{}^{}\hspace{-0.8pt}{\,^4He + ^{15}\hspace{-2.0pt}{N}}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{18}\hspace{-0.8pt}{O}\,({p}\,,{\gamma }) \,{}^{19}\hspace{-0.8pt}{F}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{19}\hspace{-0.8pt}{F}\,({p}\,,{\gamma }) \,{}^{}\hspace{-0.8pt}{\,^4He + ^{16}\hspace{-2.0pt}{O}}\,$$  Angulo et al. (1999)  $$\rm \,{}^{19}\hspace{-0.8pt}{F}\,({p}\,,{\gamma }) \,{}^{20}\hspace{-0.8pt}{Ne}\,$$  Angulo et al. (1999)  $$\rm \,{}^{4}\hspace{-0.8pt}{He}\,({2\,^{4}He}\,,{\gamma }) \,{}^{12}\hspace{-0.8pt}{C}\,$$  Fynbo et al. (2005)  $$\rm \,{}^{12}\hspace{-0.8pt}{C}\,({^{4}He}\,,{\gamma }) \,{}^{16}\hspace{-0.8pt}{O}\,$$  Cyburt, Hoffman & Woosley (2012)  $$\rm \,{}^{14}\hspace{-0.8pt}{N}\,({^{4}He}\,,{\gamma }) \,{}^{18}\hspace{-0.8pt}{F}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{15}\hspace{-0.8pt}{N}\,({^{4}He}\,,{\gamma }) \,{}^{19}\hspace{-0.8pt}{F}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{16}\hspace{-0.8pt}{O}\,({^{4}He}\,,{\gamma }) \,{}^{20}\hspace{-0.8pt}{Ne}\,$$  Constantini & LUNA Collaboration (2010)  $$\rm \,{}^{18}\hspace{-0.8pt}{O}\,({^{4}He}\,,{\gamma }) \,{}^{22}\hspace{-0.8pt}{Ne}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{20}\hspace{-0.8pt}{Ne}\,({^{4}He}\,,{\gamma }) \,{}^{24}\hspace{-0.8pt}{Mg}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{22}\hspace{-0.8pt}{Ne}\,({^{4}He}\,,{\gamma }) \,{}^{26}\hspace{-0.8pt}{Mg}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{24}\hspace{-0.8pt}{Mg}\,({^{4}He}\,,{\gamma }) \,{}^{28}\hspace{-0.8pt}{Si}\,$$  Strandberg et al. (2008)  $$\rm \,{}^{13}\hspace{-0.8pt}{C}\,({^{4}He}\,,{n}) \,{}^{16}\hspace{-0.8pt}{O}\,$$  Heil et al. (2008)  $$\rm \,{}^{17}\hspace{-0.8pt}{O}\,({^{4}He}\,,{n}) \,{}^{20}\hspace{-0.8pt}{Ne}\,$$  Angulo et al. (1999)  $$\rm \,{}^{18}\hspace{-0.8pt}{O}\,({^{4}He}\,,{n}) \,{}^{21}\hspace{-0.8pt}{Ne}\,$$  Angulo et al. (1999)  $$\rm \,{}^{21}\hspace{-0.8pt}{Ne}\,({^{4}He}\,,{n}) \,{}^{24}\hspace{-0.8pt}{Mg}\,$$  Angulo et al. (1999)  $$\rm \,{}^{22}\hspace{-0.8pt}{Ne}\,({^{4}He}\,,{n}) \,{}^{25}\hspace{-0.8pt}{Mg}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{25}\hspace{-0.8pt}{Mg}\,({^{4}He}\,,{n}) \,{}^{28}\hspace{-0.8pt}{Si}\,$$  Angulo et al. (1999)  $$\rm \,{}^{20}\hspace{-0.8pt}{Ne}\,({p}\,,{\gamma }) \,{}^{21}\hspace{-0.8pt}{Na}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{21}\hspace{-0.8pt}{Ne}\,({p}\,,{\gamma }) \,{}^{22}\hspace{-0.8pt}{Na}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{22}\hspace{-0.8pt}{Ne}\,({p}\,,{\gamma }) \,{}^{23}\hspace{-0.8pt}{Na}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{23}\hspace{-0.8pt}{Na}\,({p}\,,{\gamma }) \,{}^{}\hspace{-0.8pt}{\,^4He + ^{20}\hspace{-2.0pt}{Ne}}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{23}\hspace{-0.8pt}{Na}\,({p}\,,{\gamma }) \,{}^{24}\hspace{-0.8pt}{Mg}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{24}\hspace{-0.8pt}{Mg}\,({p}\,,{\gamma }) \,{}^{25}\hspace{-0.8pt}{Al}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{25}\hspace{-0.8pt}{Mg}\,({p}\,,{\gamma }) \,{}^{26}\hspace{-0.8pt}{Al^g}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{25}\hspace{-0.8pt}{Mg}\,({p}\,,{\gamma }) \,{}^{26}\hspace{-0.8pt}{Al^m}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{26}\hspace{-0.8pt}{Mg}\,({p}\,,{\gamma }) \,{}^{27}\hspace{-0.8pt}{Al}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{26}\hspace{-0.8pt}{Al^g}\,({p}\,,{\gamma }) \,{}^{27}\hspace{-0.8pt}{Si}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{27}\hspace{-0.8pt}{Al}\,({p}\,,{\gamma }) \,{}^{}\hspace{-0.8pt}{\,^4He + ^{24}\hspace{-2.0pt}{Mg}}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{27}\hspace{-0.8pt}{Al}\,({p}\,,{\gamma }) \,{}^{28}\hspace{-0.8pt}{Si}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{26}\hspace{-0.8pt}{Al}\,({p}\,,{\gamma }) \,{}^{27}\hspace{-0.8pt}{Si}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{26}\hspace{-0.8pt}{Al}\,({n}\,,{p}) \,{}^{26}\hspace{-0.8pt}{Mg}\,$$  Tuli (2012)  $$\rm \,{}^{12}\hspace{-0.8pt}{C}\,({^{12}C}\,,{n}) \,{}^{23}\hspace{-0.8pt}{Mg}\,$$  Caughlan & Fowler (1988)  $$\rm \,{}^{12}\hspace{-0.8pt}{C}\,({^{12}C}\,,{p}) \,{}^{23}\hspace{-0.8pt}{Na}\,$$  Caughlan & Fowler (1988)  $$\rm \,{}^{12}\hspace{-0.8pt}{C}\,({^{12}C}\,,{^{4}He}) \,{}^{20}\hspace{-0.8pt}{Ne}\,$$  Caughlan & Fowler (1988)  $$\rm \,{}^{20}\hspace{-0.8pt}{Ne}\,({\gamma }\,,{^{4}He}) \,{}^{16}\hspace{-0.8pt}{O}\,$$  Constantini & LUNA Collaboration (2010)  View Large The α-enhanced opacities and equation of state (EOS) are derived for our best estimate of the metal mixture of 47 Tuc, which is described in Section 3.1. Details about the preparation of the opacity tables are provided in Bressan et al. (2012). Suffice it to recall that the Rosseland mean opacities come from two sources: from the Opacity Project At Livermore (OPAL, Iglesias & Rogers 1996, and references therein)2 team at high temperatures (4.0 < log (T/K) < 8.7), and from AESOPUS (Marigo & Aringer 2009)3 at low temperatures (3.2 < log (T/K) < 4.1), with a smooth transition being adopted in the 4.0 < log (T/K) < 4.1 interval. Conductive opacities are provided by Itoh et al. (2008) routines. As for the EOS, we choose the widely used FreeEOS code (version 2.2.1 in the EOS4 configuration)4 developed by Alan W. Irwin for its computational efficiency. It is worth noting here that when we change the heavy element number fractions (Ni/NZ) to obtain a new metal partition in parsec, their fractional abundances by mass (Zi/Ztot) are re-normalized in such a way that the global metallicity, Z, is kept constant. Hence, compared to the solar partition at the same total metallicity Z, a model with enhanced α-elements shows a depression of Fe and the related elements, because the total metallicity remains unchanged by construction. The Hertzsprung–Russell diagram (HRD) in the left-hand panel of Fig. 1 shows that, with the same total metallicity Z and helium content Y, the α-enhanced star (orange solid line) is slightly hotter than the solar-scaled one (blue dashed line) both on the MS and on the RGB because of the net effect of changes to the opacity. Higher temperature leads to a faster evolution, as illustrated in the right-hand panel of Fig. 1. It is also interesting to compare the α-enhanced star to a solar-scaled one with the same [Fe/H] (black dotted line in Fig. 1). With the same [Fe/H] but higher total metallicity Z, the α-enhanced star is cooler. Indeed, VandenBerg et al. (2012) report that if keeping [Fe/H] constant, the giant branch is shifted to a cooler temperature with increased Mg or Si, whereas O, Ne, S abundances mainly affect the temperatures of MS and turn-off phases. Figure 1. View largeDownload slide A comparison between the α-enhanced evolutionary track (orange solid line) and the solar-scaled one with the same metallicity Z (blue dashed line). For comparison, a solar-scaled evolutionary track with the same [Fe/H] value (black dotted line) is also displayed. The helium content and the stellar mass of the three stars are the same (Y = 0.276, M = 0.85 M⊙). The left-hand panel is HRD with sub-figure zoom-in around the RGBB region. The right-hand panel shows how the luminosity of the star evolve with time. Figure 1. View largeDownload slide A comparison between the α-enhanced evolutionary track (orange solid line) and the solar-scaled one with the same metallicity Z (blue dashed line). For comparison, a solar-scaled evolutionary track with the same [Fe/H] value (black dotted line) is also displayed. The helium content and the stellar mass of the three stars are the same (Y = 0.276, M = 0.85 M⊙). The left-hand panel is HRD with sub-figure zoom-in around the RGBB region. The right-hand panel shows how the luminosity of the star evolve with time. Various initial helium abundance values, for a given metallicity, are allowed in the new version of parsec. In the previous versions, the initial helium mass fraction of the stars was obtained from the helium-to-metals enrichment law:   $$Y = Y_{\rm p} + \frac{\Delta Y}{\Delta Z}Z = 0.2485 + 1.78 \times Z$$ (1)where Yp is the primordial helium abundance (Komatsu et al. 2011), and ΔY/ΔZ is the helium-to-metal enrichment ratio. Because of differences in the adopted primordial and solar calibration He and metallicity values by different authors, the above two parameters are slightly different in different stellar evolution codes. The latest YY isochrone (Spada et al. 2013) adopts the relation Y = 0.25 + 1.48Z; dsep (Dotter 2007; Dotter et al. 2007, 2008) uses Y = 0.245 + 1.54Z; mist (Choi et al. 2016) gives Y = 0.249 + 1.5Z; and BaSTI (Pietrinferni et al. 2006) adopts Y = 0.245 + 1.4Z. However, observations reveal that the helium content does not always follow a single relation. Differences in helium abundance have been widely confirmed in GCs between stellar populations with very similar metallicity. The evidence includes the direct He i measurement on the blue HB star (Villanova, Piotto & Gratton 2009; Marino et al. 2014; Mucciarelli et al. 2014; Gratton et al. 2015, for instance), on giant stars (Dupree, Strader & Smith 2011; Pasquini et al. 2011), and the splitting of sequences in the colour–magnitude diagram (CMD) of both GCs in the Milky Way (e.g. Bedin et al. 2004; Piotto et al. 2007; Villanova et al. 2007; Milone et al. 2008; Di Criscienzo et al. 2010) and of clusters in the Magellanic Clouds (Milone et al. 2015a, 2016). Bragaglia et al. (2010) found that the brightness of the RGB bump, which should increase with He abundance, is fainter in the first generation than the second generation in 14 GCs. Indeed, He variation is considered one of the key parameters (and problems) to understand multiple populations in GCs (see the review by Gratton, Carretta & Bragaglia 2012, and the references therein). In the new version of parsec, we allow different helium contents at any given metallicity Z. 3 CALIBRATION WITH 47 TUC GCs have been traditionally considered as the paradigm of a single stellar population, a coeval and chemically homogeneous population of stars covering a broad range of evolutionary phases, from the low-mass MS to the HB and white dwarf sequences. For this reason, they were considered the ideal laboratory to observationally study the evolution of low-mass stars and to check and calibrate the stellar evolution theory. This picture has been challenged during the last two decades by photometric and spectroscopic evidence of the presence of multiple populations in most, if not all, GCs [for instance NGC 6397 (Gratton et al. 2001; Milone et al. 2012), NGC 6752 (Gratton et al. 2001; Milone et al. 2010), NGC 1851 (Carretta et al. 2014), NGC 2808 (D'Antona et al. 2005; Carretta et al. 2006; Piotto et al. 2007; Milone et al. 2015b), NGC 6388 (Carretta et al. 2007), NGC 6139 (Bragaglia et al. 2015), M22 (Marino et al. 2011), etc.). Nevertheless, GCs remain one of the basic workbenches for the stellar model builders, besides their importance for dynamical studies and, given the discovery of multiple populations, also for the early chemo-dynamical evolution of stellar systems. 47 Tuc, a relatively metal-rich Galactic GC, also shows evidence of the presence of at least two different populations: (i) bimodality in the distribution of CN-weak and CN-strong targets, not only in red giant stars (Briley 1997; Norris & Freeman 1979; Harbeck, Smith & Grebel 2003) but also in MS members (Cannon et al. 1998); (ii) luminosity dispersion in the sub-giant branch, low MS and HB (Anderson et al. 2009; Di Criscienzo et al. 2010; Nataf et al. 2011; Salaris, Cassisi & Pietrinferni 2016), indicating a dispersion in He abundance; and (iii) anticorrelation of Na–O in RGB and HB stars (Carretta et al. 2009b, 2013; Gratton et al. 2013) and also in main-sequence turn-off (MSTO) ones (D'Orazi et al. 2010; Dobrovolskas et al. 2014). The presence of at least two different populations with different chemical compositions seems irrefutable (even if their origin is still under debate). Particularly convincing is the photometric study by Milone et al. (2012, and references therein), which concludes, in good agreement with other works (Carretta et al. 2009b, 2013), that for each evolutionary phase, from MS to HB, the stellar content of 47 Tuc belongs to two different populations, ‘first generation’ and ‘second generation’ ones (thereafter, FG and SG, respectively). The FG population represents ∼30 per cent of the stars, and it is more uniformly spatially distributed than the SG population, which is more concentrated in the central regions of the cluster. Choosing 47 Tuc as a reference to calibrate parsec stellar models requires therefore computation of stellar models with metal mixtures corresponding to the two identified populations. In the next section, we describe the sources to derive the two different metal mixtures that will be used for the opacity and EOS tables in the stellar model computations, and in the follow-up isochrone fitting. 3.1 Metal mixtures Chemical element abundances are given in the literature as the absolute values A(X),5 or as [X/Fe],6 the abundance with respect to the iron content and referred to the same quantity in the Sun. Since the solar metal mixture has changed lately and since there is still a hot debate about the chemical composition of the Sun, it is important to translate all the available data to absolute abundances, taking into account the solar mixture considered in each source. We follow that procedure to derive the metal mixtures for the first and second generations in 47 Tuc. The separation between the two populations based on photometric colours done by Milone et al. (2012) agrees with the separation based on Na–O anticorrelation by Carretta et al. (2009b) and Gratton et al. (2013). We decide hence to use the same criteria to classify the star as FG or SG member. Concerning He mass fraction Y, the scatter in luminosity seen in some evolutionary phases has been attributed to different amounts of He in the stellar plasma (see references above). The analyses presented in Milone et al. (2012) suggest that the best fitting of the colour difference between the two populations is obtained with a combination of different C, N, and O abundances, plus a small increase of He content in the SG (ΔY = 0.015 − 0.02). These results agree with those presented in Di Criscienzo et al. (2010, ΔY = 0.02 − −0.03), and rule out the possibility of explaining the 47 Tuc CMD only with the variation of He abundance. Table 2 lists the elemental abundances we adopt for the two generations of 47 Tuc together with the corresponding references. The abundances of some elements, like carbon, nitrogen, and oxygen, may change during the evolution because of standard (convection) and non-standard (i.e. rotational mixing) transport processes. Therefore, CNO abundances are compiled from available measurements for MS/TO stars, and their sum abundances are nearly the same in both populations. Other elements are not expected to be affected by mixing processes during stellar evolution, so we use the values measured mainly in the red giant phase where hundreds of stars are observed. If available, abundance determinations that take into account NLTE and 3D effects are adopted. There is no clear abundance difference of elements Mg, Al, Si, Ca, and Ti between the two populations of 47 Tuc. Since their abundances show large scatter from different literature sources and are sensitive to the choice of measured lines, we use the mean values of the literature abundances for both FG and SG. The iron abundance [Fe/H] = −0.76 dex is adopted from Carretta et al. (2009a) and Gratton et al. (2013), who measure the largest giant sample and HB sample of 47 Tuc, respectively, with internal fitting errors less than 0.02 dex. The [Fe/H] values derived from giants are much less dependent on the effects of microscopic diffusion than in the case of MS stars. We notice that in literature some authors suggest an [Fe/H] dispersion of ∼0.1 dex for 47 Tuc (Alves-Brito et al. 2005). On the other hand, Anderson et al. (2009) conclude that for 47 Tuc an He dispersion of ∼0.026 has an equal effect on the MS as an [Fe/H] dispersion of ∼0.1 dex. Since we consider different He contents for the FG and SG stars, we do not apply any further [Fe/H] dispersion. Other elements that are not displayed in Table 2 keep the solar abundances ([X/Fe] = 0). The anticorrelation between the abundances of C and N, as well as O and Na, contributes to the main difference of the metal mixtures between FG and SG. The difference in the final [α/Fe] values between the two generations is due to the difference in O abundance. The resulting metallicities are ZFG = 0.0056 for the first generation and ZSG = 0.0055 for the second generation. Following Milone et al. (2012), the assumed He abundances are Y = 0.256 and 0.276 for FG and SG, respectively. Table 3 lists the general metal-mixture information for the two stellar populations, including Z, Y, [M/H], [Fe/H], and [α/Fe] . The referred solar abundance is derived from Caffau et al. (2011), as described in Bressan et al. (2012). We consider eight α elements when calculating the total α enrichment [α/Fe] : O, Ne, Mg, Si, S, Ca, Ar, and Ti. Thus, for the FG stars of 47 Tuc [Z = 0.0056, Y = 0.256], [α/Fe] = 0.4057 dex, and, for the SG stars [Z = 0.0055, Y = 0.276], [α/Fe] = 0.2277 dex. These two values, approximately ∼ 0.4 and 0.2, respectively, are the typical [α/Fe] values observed in α-enriched stars. Finally, we note that we will adopt the metal partitions of these two α-enriched generations to calculate stellar evolutionary tracks and isochrones also for other metallicities. Table 2. Chemical element abundances of 47 Tuc two stellar populations (FG and SG). The abundances are written in the format of [X/Fe], their corresponding references are also listed.   FG  SG  Reference  Note  [C/Fe]  0.12  −0.09  Cannon et al. (1998); Milone et al. (2012)  MS  [N/Fe]  0.32  1.17  Cannon et al. (1998); Milone et al. (2012)  MS  [O/Fe]  0.42  0.17  Dobrovolskas et al. (2014)  TO, NLTE+3D  [Ne/Fe]  0.40  0.40  –  estimated  [Na/Fe]  −0.12  0.10  Dobrovolskas et al. (2014)  TO, NLTE+3D  [Mg/Fe]  0.32  0.32  Carretta et al. (2009b, 2013); Gratton et al. (2013); Cordero et al. (2014); Thygesen et al. (2014)  mean value of RGB/HB  [Al/Fe]  0.20  0.20  Cordero et al. (2014); Thygesen et al. (2014)  mean value of RGB  [Si/Fe]  0.27  0.27  Gratton et al. (2013); Thygesen et al. (2014); Carretta et al. (2009b); Cordero et al. (2014)  mean value of RGB/HB  [S/Fe]  0.40  0.40  –  estimated  [Ca/Fe]  0.27  0.27  Carretta et al. (2009b); Gratton et al. (2013); Cordero et al. (2014); Thygesen et al. (2014)  mean value of RGB/HB  [Ti/Fe]  0.20  0.20  Cordero et al. (2014); Thygesen et al. (2014)  mean value of RGB/HB    FG  SG  Reference  Note  [C/Fe]  0.12  −0.09  Cannon et al. (1998); Milone et al. (2012)  MS  [N/Fe]  0.32  1.17  Cannon et al. (1998); Milone et al. (2012)  MS  [O/Fe]  0.42  0.17  Dobrovolskas et al. (2014)  TO, NLTE+3D  [Ne/Fe]  0.40  0.40  –  estimated  [Na/Fe]  −0.12  0.10  Dobrovolskas et al. (2014)  TO, NLTE+3D  [Mg/Fe]  0.32  0.32  Carretta et al. (2009b, 2013); Gratton et al. (2013); Cordero et al. (2014); Thygesen et al. (2014)  mean value of RGB/HB  [Al/Fe]  0.20  0.20  Cordero et al. (2014); Thygesen et al. (2014)  mean value of RGB  [Si/Fe]  0.27  0.27  Gratton et al. (2013); Thygesen et al. (2014); Carretta et al. (2009b); Cordero et al. (2014)  mean value of RGB/HB  [S/Fe]  0.40  0.40  –  estimated  [Ca/Fe]  0.27  0.27  Carretta et al. (2009b); Gratton et al. (2013); Cordero et al. (2014); Thygesen et al. (2014)  mean value of RGB/HB  [Ti/Fe]  0.20  0.20  Cordero et al. (2014); Thygesen et al. (2014)  mean value of RGB/HB  View Large Table 3. General metal mixture of 47 Tuc two stellar populations (FG and SG).   FG  SG  Z  0.0056  0.0055  Y  0.256  0.276  [M/H]  −0.43  −0.41  [Fe/H]  −0.76  −0.76  [α/Fe] a  0.41  0.23    FG  SG  Z  0.0056  0.0055  Y  0.256  0.276  [M/H]  −0.43  −0.41  [Fe/H]  −0.76  −0.76  [α/Fe] a  0.41  0.23  Notes.aLabelled as [α/Fe] ∼ 0.4 and 0.2. Difference in the [α/Fe] values is due to O abundance differences. View Large 3.2 Isochrones fitting and luminosity function With the detailed metal mixture and helium abundance of 47 Tuc, we calculate new sets of evolutionary tracks and isochrones, and transform them into the observational CMD in order to fit the data. This fitting procedure, based on our adopted model prescriptions (e.g. mixing length, atmospheric boundary condition, bolometric corrections), aims to calibrate other parameters (e.g. extra mixing) in the model as described below. 3.2.1 Low MS to turn-off Kalirai et al. (2012) provide deep images of 47 Tuc taken with the Advanced Camera for Surveys (ACS) on Hubble Space Telescope (HST). The corresponding CMDs cover the whole MS of this cluster, till the faintest stars. Fig. 2 shows our isochrone fitting of their photometric data, i.e. in the F606W and F814W bands. In order to display the relative density of stars on CMD, the data are plotted with the Hess diagram (binsize 0.025 mag in colour and 0.1 mag in I magnitude). By assuming a standard extinction law (Cardelli et al. 1989), we derive, from the isochrone fitting, an age of 12.00 Gyr, a distance modulus (m − M)0 of 13.22 (apparent distance modulus (m-M)F606W ∼ 13.32), and a reddening of E(F606W − F814W) = 0.035 (hereafter named E(6 − 8)). The fitting is performed not only to the MS but also to the giant branch and HB phase as we will show later in Sections 3.2.3 and 3.2.4. The values of age, (m − M)0, and E(6 − 8) are adjusted with visual inspection with the priority of improving the turn-off and HB fittings. Figure 2. View largeDownload slide Isochrone fitting with Hess diagram of 47 Tuc data for the low MS (Kalirai et al. 2012). The bin size of Hess diagram is 0.025 mag in colour and 0.1 mag in F814W magnitude. The fitting parameters [age, η, (m − M)0, and E(6 − 8)] are listed in the legend. Figure 2. View largeDownload slide Isochrone fitting with Hess diagram of 47 Tuc data for the low MS (Kalirai et al. 2012). The bin size of Hess diagram is 0.025 mag in colour and 0.1 mag in F814W magnitude. The fitting parameters [age, η, (m − M)0, and E(6 − 8)] are listed in the legend. The distance modulus of 47 Tuc has been determined by many other authors, however with different results. For instance, using HST proper motion, Watkins et al. (2015) derive a distance of 4.15 kpc [(m − M)0 ∼13.09], which is lower than the values in the Harris catalogue (4.5 kpc [(m − M)0 ∼ 13.27] and (m − M)V = 13.37, Harris 1996, 2010 edition); the eclipsing binary distance measurement ((m − M)V = 13.35, Thompson et al. 2010); the result based on the white dwarf cooling sequence ((m − M)0 ∼ 13.32, Hansen et al. 2013); and that derived from isochrone fitting to BVI photometry ((m − M)V = 13.375, Bergbusch & Stetson 2009). Our best-fitting distance lies in between them, and agrees with other recent distance modulus determinations (e.g. Brogaard et al. 2017, (m − M)0 = 13.21 ± 0.06 based on the eclipsing binary). Gaia will release the parallaxes and proper motions including stars in 47 Tuc in its DR2 in early 2018, and will help to solve the distance problem. However, we will show in the following section that our best estimate result offers a very good global fitting, from the very low MS till the red giant and HBs. 3.2.2 RGB bump and envelope overshooting calibration Some GC features in CMD are very sensitive to stellar model parameters, which are, otherwise, hardly constrained from observations directly. This is the case of the efficiency of mixing below the convective envelope (envelope overshooting), that is known to affect the luminosity of the RGBB. In this section, we will use the 47 Tuc data to calibrate the envelope overshooting to be used in low-mass stars by parsec . The RGB bump is one of the most intriguing features in the CMD. When a star evolves to the ‘first dredge-up’ in the red giant phase, its surface convective zone deepens while the burning hydrogen shell moves outwards. When the hydrogen burning shell encounters the chemical composition discontinuity left by the previous penetration of the convective zone, the sudden increase of H affects the efficiency of the burning shell and the star becomes temporarily fainter. Soon after a new equilibrium is reached, the luminosity of the star raises again. Since the evolutionary track crosses the same luminosity three times in a short time, there is an excess of star counts in a small range of magnitudes, making a ‘bump’ in the star number distribution (LF) along the RGB. This is because the number of stars in the post-MS phases is proportional to the evolutionary time of the stars in these phases. The longer the crossing time of the chemical composition discontinuity by the burning shell, the more the stars accumulate in that region of the RGB. The properties of RGBB, including the brightness and the extent, are important to study the stellar structure and to investigate the nature of GCs. 47 Tuc was the first GC where the existence of the RGBB was confirmed (King, Da Costa & Demarque 1985). Since then, many works, both theoretical and observational (for instance, Alongi et al. 1991; Cassisi & Salaris 1997; Zoccali et al. 1999; Bono et al. 2001; Cassisi, Salaris & Bono 2002; Bjork & Chaboyer 2006; Salaris et al. 2006; Bragaglia et al. 2010; Cecco et al. 2010; Cassisi et al. 2011; Nataf et al. 2013), have studied the features of RGBB. The intrinsic brightness and extent of the RGBB are sensitive to the following: Total metallicity and metal partition: Nataf et al. (2013) propose an empirical function of RGBB extent to metallicity: the more metal-poor the GC is, the smaller is the extent of the RGBB. From the theoretical point of view, stars with lower total metallicity are brighter compared to the higher metallicity stars, causing their hydrogen burning shell to move outwards faster. Since they are also hotter, the surface convective envelope is thinner and the chemical composition discontinuity is smaller and less deep. As a consequence, their RGBB is very brief and covers a small range of magnitudes at higher luminosity. This is why RGBB in metal-poor GCs is very difficult to be well sampled. The metal partition also affects the features of RGBB, even when the total metallicity remains the same. As already shown in Fig. 1 and in Section 2, a stellar track with α-enhancement is hotter than the solar-scaled one with the same total metallicity Z because of different opacity, leading to a brighter RGBB. Since CNO are the most affected elements in the giant branch and they are important contributors to the opacities, their varying abundances have an important impact on the location of RGBB. For example, Rood & Crocker (1985) show that enhancing CNO by a factor of 10 has larger effect on the RGBB luminosity than enhancing Fe by a factor of 10 over the same metallicity Z. More recently, VandenBerg (2013) shows that higher oxygen abundance leads to a fainter RGBB if the [Fe/H] is fixed. Helium content: A larger helium content renders the star hotter and brighter (Fagotto et al. 1994). Bragaglia et al. (2010) studied the RGBB of 14 GCs and found that the more He-rich second generation shows brighter RGBB than the first generation. Similar to the mechanism in metal-poor stars, hot He-rich stars have less deep convective envelopes and their high luminosity makes the hydrogen burning shells to move faster across the discontinuity. Hence, with the same total metallicity Z and stellar mass, the RGBB of the He-rich star is brighter, less extended, and more brief. In Table 4, we take an M = 0.85 M⊙ star as an example to show how the RGBB luminosity and evolution time vary with different helium contents. Table 4. RGBB parameters of stars with a constant mass (M = 0.85 M⊙) and metallicity (Z = 0.0055) but different helium contents. Mean luminosity $$\bar{\log (L/L_{\odot })}_{{\rm RGBB}}$$, luminosity extent Δ log (L)RGBB, RGBB beginning time t0, RGBB and RGBB lifetime Δ tRGBB are listed. Z  Y  $$\bar{\log (L/L_{\odot })}_{\rm RGBB}$$  Δlog (L)RGBB  t0, RGBB (Gyr)  Δ tRGBB (Myr)  0.0055  0.276  1.5443  0.03557  12.062  27.159  0.0055  0.296  1.5887  0.03177  10.584  22.637  Z  Y  $$\bar{\log (L/L_{\odot })}_{\rm RGBB}$$  Δlog (L)RGBB  t0, RGBB (Gyr)  Δ tRGBB (Myr)  0.0055  0.276  1.5443  0.03557  12.062  27.159  0.0055  0.296  1.5887  0.03177  10.584  22.637  Age: Stars with younger age are hotter, and with their thinner convective envelope, their RGBB are brighter. In principle, multiple populations born in different ages spread the GC RGBB luminosity. However, considering that the age variation of the multiple populations is usually small (∼a few Myr), it contributes little to the GC RGBB luminosity spread compared to the He variation (see e.g. Nataf et al. 2011). Mixing efficiency: The mixing efficiency of the star, both mixing length and envelope overshooting (EOV), determines the maximum depth of the convective envelope and affects the brightness and evolutionary time of RGBB. The more efficient the mixing is, the deeper the convective envelope is, the earlier the hydrogen-burning shell meets the discontinuity left by the penetration of the surface convective zone, and the fainter the RGBB is. For the mixing length, we adopt the solar-calibrated value $$\alpha _{\rm MLT}^{{\odot }}=1.74$$ in parsec as described in Bressan et al. (2012). The EOV is calibrated with the new stellar tracks against the observations of the RGBB of 47 Tuc. Overshooting is the non-local mixing that may occur at the borders of any convectively unstable region (i.e. Bressan et al. 2015, and references therein). The extent of the overshooting at the base of the convective envelope is called envelope overshooting, and the one above the stellar convective core is called core overshooting. There are observations that can be better explained with envelope overshooting, for instance, the blue loops of intermediate and massive stars (Alongi et al. 1991; Tang et al. 2014), and the carbon stars LFs in the Magellanic Clouds, which require a more efficient third dredge-up in AGB stars (Herwig 2000; Marigo & Girardi 2007). At the base of the convective envelope of the Sun, models with an envelope overshooting of Λe ≈ 0.3 ∼ 0.5Hp (where Hp is the pressure scaleheight) provide a better agreement with the helioseismology data (Christensen-Dalsgaard et al. 2011). The envelope overshooting also affects the surface abundance of light elements (Fu et al. 2015), and asteroseismic signatures in stars (Lebreton & Goupil 2012). In Fig. 3, we compare the RGBB evolution of models computed with different EOV values, Λe, at the same stellar mass and composition. Every pair of filled dots marks the brightness extent of the RGBB. The figure shows that a larger envelope overshooting not only makes the RGBB fainter, but also of longer duration, leading to a more populated RGBB. A larger EOV value leads to a deeper surface convective zone, and the hydrogen burning shell encounters the chemical discontinuity earlier. Figure 3. View largeDownload slide The RGBB luminosity as a function of stellar age for a 0.85 M⊙star but with different EOV. The black, red, green, and blue line from the top to bottom represent tracks with Λe = 0.05, 0.3, 0.4, and 0.5. The filled dots mark the minimum and maximum luminositities of RGBB for each track, and ΔtRGBB is the evolution time from the minimum luminosity to the maximum one. Figure 3. View largeDownload slide The RGBB luminosity as a function of stellar age for a 0.85 M⊙star but with different EOV. The black, red, green, and blue line from the top to bottom represent tracks with Λe = 0.05, 0.3, 0.4, and 0.5. The filled dots mark the minimum and maximum luminositities of RGBB for each track, and ΔtRGBB is the evolution time from the minimum luminosity to the maximum one. The LF is a useful tool to compare the observed morphology of RGBB with that predicted by the theory. Taking into account that the 47 Tuc population contribution is 30 per cent from the FG and 70 per cent from the SG as suggested by Milone et al. (2012) and Carretta et al. (2009b), we simulated the LF of 47 Tuc with our isochrones with different EOV values. The comparison between the observed and predicted LFs is shown in Fig. 4. For the observed LF, we have used data from the HST/ACS survey of GCs (Sarajedini et al. 2007). Both observations and models are sampled in bins of 0.05 magnitudes. The fitting parameters are the same as those we used in Fig. 2. The model LF (orange histogram) is calculated with envelope overshooting Λe = 0.3 in the upper panel and with Λe = 0.5 in the lower panel. It is evident that the LF computed with the small envelope overshooting value Λe = 0.3 has RGBB too bright compared to data (black histogram filled with oblique lines). We find that the agreement between observations and models is reached when one adopts a value of Λe = 0.5Hp below the convective border, with our adopted metal mixtures and best-fitting isochrone. This provides a robust calibration of the envelope overshooting parameter. This envelope overshooting calibration will be applied to all other stellar evolution calculations of low-mass stars. Figure 4. View largeDownload slide Comparison between LF of 47 Tuc data (Sarajedini et al. 2007) and the new parsec isochrone with different EOV. The fitting parameters [age, (m − M)0, and E(6 − 8)] are the same as in Fig. 2. The black histogram filled with oblique lines is the data LF, whereas orange histogram is LF derived from new parsec isochrones with 30 per cent contribution from the FG of 47 Tuc and 70 per cent from the SG. The upper panel isochrones of each sub-figure are calculated with EOV value Λe = 0.3, and the lower panels are the ones with Λe = 0.5. Orange and black arrows mark the location of RGBB in model and in data, respectively. The bin size of the LF is 0.05 mag. Figure 4. View largeDownload slide Comparison between LF of 47 Tuc data (Sarajedini et al. 2007) and the new parsec isochrone with different EOV. The fitting parameters [age, (m − M)0, and E(6 − 8)] are the same as in Fig. 2. The black histogram filled with oblique lines is the data LF, whereas orange histogram is LF derived from new parsec isochrones with 30 per cent contribution from the FG of 47 Tuc and 70 per cent from the SG. The upper panel isochrones of each sub-figure are calculated with EOV value Λe = 0.3, and the lower panels are the ones with Λe = 0.5. Orange and black arrows mark the location of RGBB in model and in data, respectively. The bin size of the LF is 0.05 mag. 3.2.3 Red Giant Branch Although Kalirai et al. (2012) focus on the faint part of the MS as shown in Fig. 2, another data set of 47 Tuc, the HST/ACS survey of GCs (Sarajedini et al. 2007), is devoted to the HB (Anderson et al. 2008) with the same instrument. In Fig. 5, we show the global fitting of ACS data of 47 Tuc, from MS up to the RGB and HB. The best-fitting parameters we derived are the same as those we used to fit the lower MS data in Fig. 2. The HESS diagram is used for the global fitting (the left-hand panel) with bin size 0.025 mag in colour and 0.1 mag in F814W magnitude. The HB region and the turn-off region are zoomed in with scatter plots in the two right-hand panels. Thanks to the detailed composition derived from the already quoted observations for the two main populations of 47 Tuc and the new computed models, by assuming a standard extinction law (Cardelli et al. 1989) and using the adopted EOV and mixing length parameters, we are able to perform a global fit to the CMD of 47 Tuc covering almost every evolutionary phase over a range of about 13 mag. This must be compared with other fittings that can be found in literature and that usually are restricted to only selected evolutionary phases (Kim et al. 2002; Salaris et al. 2007; VandenBerg et al. 2013, 2014; Chen et al. 2014; McDonald & Zijlstra 2015, etc.). However, it is worth noting that the distance of this cluster, as already discussed in Section 3.2.1, together with the cluster age, has varied over the years in many careful studies. We look forward to Gaia DR2 to put more constrains to this problem. Figure 5. View largeDownload slide Isochrone fitting with Hess diagram (left-hand panel) of 47 Tuc data (Sarajedini et al. 2007) for all the evolutionary phases, and with scatter plots highlighting the HB region (upper right-hand panel) and the turn-off region (lower right-hand panel). The red and blue lines represent isochrones of the first and second generations, respectively, as the legend shows. The fitting parameters are: age = 12.00 Gyr, (m − M)0 = 13.22, E(6 − 8) = 0.035. Figure 5. View largeDownload slide Isochrone fitting with Hess diagram (left-hand panel) of 47 Tuc data (Sarajedini et al. 2007) for all the evolutionary phases, and with scatter plots highlighting the HB region (upper right-hand panel) and the turn-off region (lower right-hand panel). The red and blue lines represent isochrones of the first and second generations, respectively, as the legend shows. The fitting parameters are: age = 12.00 Gyr, (m − M)0 = 13.22, E(6 − 8) = 0.035. As the upper right-hand panel of Fig. 5 show, the isochrones corresponding to both of the two stellar generations run on the red side of the data in the RGB phase. Part of the discrepancy could be explained by the bolometric correction used. Here, we are using bolometric correction from phoenix atmosphere models as described in Chen et al. (2015) for parsec v1.2S, where only the total metallicity is considered in the transformation of log (L) versus log (Teff) into F814W versus (F606W − F814W). As the metallicities of the two 47 Tuc populations (Z = 0.0056 and Z = 0.0055) show only a marginal difference, we adopt for the two populations the same bolometric corrections. Thus, Fig. 5 reflects basically the difference of the two populations in the theoretical log (L) versus log (Teff) HRD. This ‘RGB-too-red’ problem also exists in Dotter et al. (2007), when they fit the same set of data using dsep models (see their fig. 12), as they apply bolometric correction from phoenix as well. To minimize this discrepancy, we use the atlas12 code (Kurucz 2005), which considers not only the total metallicity Z but also log (g) and detailed chemical compositions for the colour transformation, to compute new atmosphere models with our best estimate chemical compositions of the two 47 Tuc populations. We adopt these atlas12 models for the new fits to 47 Tuc, but only for models with Teff hotter than 4000 K ((F606W − F814W)∼1.3). For lower Teff, we still use phoenix because atlas12 models may be not reliable at cooler temperatures (Chen et al. 2014). Here, we show the fit obtained with atlas12+phoenix bolometric correction in Fig. 6. We see that with the same fitting parameters as in Fig. 5, the prediction of the RGB colours is improved by applying new atlas12 bolometric correction. The two stellar generations are split on RGB phase in Fig. 6. We see that the SG (Z = 0.0055), which is the main contributor as suggested by Milone et al. (2012) and Carretta et al. (2009b), is consistent with the denser region of the RGB data. In other evolutionary phases, the new atlas12 bolometric corrections do not bring noticeable changes. Figure 6. View largeDownload slide The same isochrone fitting with Hess diagram and scatter plots of 47 Tuc data (Sarajedini et al. 2007) as in Fig. 5, but with atmosphere models fromatlas12 for Teff hotter than 4000 K. Figure 6. View largeDownload slide The same isochrone fitting with Hess diagram and scatter plots of 47 Tuc data (Sarajedini et al. 2007) as in Fig. 5, but with atmosphere models fromatlas12 for Teff hotter than 4000 K. Since atlas12 only slightly affects the colour of the RGB base, and the remainder of this paper deals with the LF of the bump and of the HB, in the following discussion, we will continue to use the standard atmosphere models of parsec v1.2S. ATLAS$$\scriptstyle{12}$$ atmosphere models for PARSEC alpha-enhanced isochrones will be discussed in detail in another following work (Chen et al., in preparation). Mass-loss by stellar winds during the RGB phase has been considered for low-mass stars, using the empirical formula by Reimers (1975) multiplied by an efficiency factor η. In Fig. 7, we show the mass lost by RGB stars in unit of  M⊙ for the FG of 47 Tuc (the plot for SG is very similar). Different efficiency factors (η) and ages are applied. ΔM in the figure is the difference between the initial mass and current mass of the tip RGB star: ΔM = Minitial − Mcurrent. The lost mass, which is greater with larger η, is an increasing function of the cluster age. It is very difficult to derive observationally the mass lost in RGB stars directly since an accurate mass is not easy to derive and the RGB tip is hard to identify. However, the RGB mass-loss characterizes the HB morphology, and this will be discussed in next section. Figure 7. View largeDownload slide RGB mass lost in unit of  M⊙ for FG of 47 Tuc (Z = 0.0056, Y = 0.256). The X axis is the initial mass of the tip RGB star, and the Y axis shows the mass lost in this star during RGB phase. Five different efficiency factors η are illustrated, from the top to bottom η = 0.40 (filled diamond), η = 0.35 (filled triangle), η = 0.30 (filled square), η = 0.25 (filled star), and η = 0.20 (filled dots). The colour code displays the age, as shown in the colour bar. Figure 7. View largeDownload slide RGB mass lost in unit of  M⊙ for FG of 47 Tuc (Z = 0.0056, Y = 0.256). The X axis is the initial mass of the tip RGB star, and the Y axis shows the mass lost in this star during RGB phase. Five different efficiency factors η are illustrated, from the top to bottom η = 0.40 (filled diamond), η = 0.35 (filled triangle), η = 0.30 (filled square), η = 0.25 (filled star), and η = 0.20 (filled dots). The colour code displays the age, as shown in the colour bar. 3.2.4 Horizontal branch morphology The morphology of the HB in GCs is widely studied since the ‘second parameter problem’ (that is, the colour of the HB is determined not only by metallicity, van den Bergh 1967; Sandage & Wildey 1967) was introduced. Aside from metallicity as the ‘first parameter’, age, He content, mass-loss, and cluster central density have been suggested as candidates to be the second, or even third, parameter affecting the morphology of the HB (Fusi Pecci & Bellazzini 1997; D'Antona et al. 2002; Caloi & D'Antona 2005; Catelan 2008; Dotter et al. 2010; Gratton et al. 2010; McDonald & Zijlstra 2015, etc.). Most of these parameters involve an effect on the mass of the stars that populate the cluster HB. Stars with smaller stellar mass are hotter in temperature and bluer in colour. The HB stellar mass decreases as the cluster ages. At a given age, He-rich star evolves faster and reach the zero-age horizontal branch (ZAHB) with lower mass. If the age and He content are the same, the mass of HB stars is fixed by the mass-loss along the RGB (here the mass-loss driven by the helium flash is not considered). Although the RGB mass-loss does not significantly affect the RGB evolutionary tracks, it determines the location of the stars on the HB, by tuning the stellar mass. Here, we illustrate how helium content and the RGB mass-loss affect the HB morphology in the case of 47 Tuc. The HB morphology with five different values of η is displayed in Fig. 8 for our best-fitting parameters derived in Section 3.2.3. Different metal/helium abundances ([Z = 0.0056, Y = 0.256], [Z = 0.0055, Y = 0.276], [Z = 0.0056, Y = 0.276], and [Z = 0.0055 Y = 0.296]) are displayed. The isochrones with Z = 0.0056 are calculated with [α/Fe] ∼ 0.4 and those with Z = 0.0055 are calculated with [α/Fe] ∼ 0.2. The 47 Tuc data (Sarajedini et al. 2007) are also plotted for comparison. The differences between the isochrone with [Z = 0.0055, Y = 0.276] (blue solid line) and the one with [Z = 0.0056, Y = 0.276] (orange dashed line) are negligible on the HB, even though they refer to a different α-enhanced mixture. With the same RGB mass-loss factor η, He-rich stars have their HB more extended (because of smaller stellar mass), bluer (due to both the smaller stellar mass and the He-rich effect on radiative opacity), and more luminous (because of larger He content in the envelope). For stars with larger mass-loss efficiency η during their RGB phase, their HB is bluer, fainter, and more extended, because of smaller stellar mass (hence smaller envelope mass, since the core mass does not vary significantly with the mass-loss rate). Indeed, the effects of a higher He content and of a lower mass (no matter if it is the result of an older age or a larger RGB mass-loss) on HB stars are difficult to distinguished by means of the colour, but can be disentangled because the larger helium content makes the He-rich star slightly more luminous. Figure 8. View largeDownload slide HB morphology for different RGB mass-loss parameters (η) and metal/helium abundances, with the same isochrone fitting parameters [age, (m − M)0, and E(6 − 8)] as in Fig. 5. The red solid line, blue solid line, orange dashed line, and green dash–dotted line represent isochrones of [Z = 0.0056, Y = 0.256], [Z = 0.0055, Y = 0.276], [Z = 0.0056, Y = 0.276], and [Z = 0.0055, Y = 0.296], respectively. The mass lost during the RGB in unit of  M⊙ for each η and metal/helium abundance is listed in Table 5. Figure 8. View largeDownload slide HB morphology for different RGB mass-loss parameters (η) and metal/helium abundances, with the same isochrone fitting parameters [age, (m − M)0, and E(6 − 8)] as in Fig. 5. The red solid line, blue solid line, orange dashed line, and green dash–dotted line represent isochrones of [Z = 0.0056, Y = 0.256], [Z = 0.0055, Y = 0.276], [Z = 0.0056, Y = 0.276], and [Z = 0.0055, Y = 0.296], respectively. The mass lost during the RGB in unit of  M⊙ for each η and metal/helium abundance is listed in Table 5. Table 5 lists the current mass, MZAHB, of the first HB star and the corresponding mass that has been lost ΔMRGB, in unit of  M⊙. In the table, we also show the HB mass range δMHB that produces the corresponding colour extent of HB. All cases displayed in Fig. 8 are itemized. Table 5. The mass lost during the RGB in unit of  M⊙ for different η and metal/helium abundance. The current mass of the first HB star is MZAHB, and ΔMRGB represents its RGB mass-loss in unit of  M⊙. The HB mass range is itemized in the last column δMHB. All values listed here are derived from isochrones with age = 12.0 Gyr, (m − M)0 = 13.22, and E(V − I) = 0.035, as shown on Fig. 8. Z  Y  η  MZAHB ( M⊙)  ΔMRGB ( M⊙)  δMHB ( M⊙)  0.0056  0.256  0.20  0.795 832  0.0946  0.0023      0.25  0.770 375  0.1201  0.0033      0.30  0.744 052  0.1464  0.0044      0.35  0.716 765  0.1737  0.0053      0.40  0.688 402  0.2020  0.0059  0.0055  0.276  0.20  0.758 027  0.0953  0.0029      0.25  0.732 270  0.1211  0.0039      0.30  0.705 582  0.1478  0.0051      0.35  0.677 852  0.1755  0.0061      0.40  0.648 949  0.2044  0.0067  0.0056  0.276  0.20  0.763 649  0.0948  0.0028      0.25  0.738 049  0.1204  0.0039      0.30  0.711 533  0.1470  0.0050      0.35  0.683 996  0.1745  0.0059      0.40  0.655 309  0.2032  0.0066  0.0055  0.296  0.20  0.726 732  0.0954  0.0035      0.25  0.700 873  0.1213  0.0047      0.30  0.674 033  0.1481  0.0058      0.35  0.646 090  0.1760  0.0066      0.40  0.616 896  0.2052  0.0067  Z  Y  η  MZAHB ( M⊙)  ΔMRGB ( M⊙)  δMHB ( M⊙)  0.0056  0.256  0.20  0.795 832  0.0946  0.0023      0.25  0.770 375  0.1201  0.0033      0.30  0.744 052  0.1464  0.0044      0.35  0.716 765  0.1737  0.0053      0.40  0.688 402  0.2020  0.0059  0.0055  0.276  0.20  0.758 027  0.0953  0.0029      0.25  0.732 270  0.1211  0.0039      0.30  0.705 582  0.1478  0.0051      0.35  0.677 852  0.1755  0.0061      0.40  0.648 949  0.2044  0.0067  0.0056  0.276  0.20  0.763 649  0.0948  0.0028      0.25  0.738 049  0.1204  0.0039      0.30  0.711 533  0.1470  0.0050      0.35  0.683 996  0.1745  0.0059      0.40  0.655 309  0.2032  0.0066  0.0055  0.296  0.20  0.726 732  0.0954  0.0035      0.25  0.700 873  0.1213  0.0047      0.30  0.674 033  0.1481  0.0058      0.35  0.646 090  0.1760  0.0066      0.40  0.616 896  0.2052  0.0067  View Large If one considers a uniform mass-loss parameter η for the two populations of 47 Tuc ([Z = 0.0056, Y = 0.256] and [Z = 0.0055, Y = 0.276]), η = 0.35 is the value that fits better the HB morphology in our best-fitting case, as Fig. 8 illustrates. As shown in Table 5, an RGB mass-loss parameter of η = 0.35 leads to a value of the mass lost during RGB between 0.1737  M⊙–0.1755  M⊙. In the literature, there is a discrepancy among the results on RGB mass-loss derived with different approaches, namely: cluster dynamics, infrared (IR) excess due to dust, and HB modelling for this cluster. Heyl et al. (2015) study the dynamics of white dwarf in 47 Tuc, and concluded that the mass lost by stars at the end of the RGB phase should be less than about 0.2 M⊙. Origlia et al. (2007) observe the circumstellar envelopes around RGB stars in this cluster from mid-IR photometry and find the total mass lost on the RGB is ≈0.23 ± 0.07 M⊙. McDonald & Zijlstra (2015) use HB star mass from literature to study the RGB mass-loss and derive a Reimers factor η = 0.452 (corresponding to an RGB mass-loss greater than ∼0.20 M⊙). Most recently, Salaris et al. (2016) assume a distribution of the initial He abundance to simulate the observed HB of 47 Tuc. They derive a lower limit to the RGB mass-loss of about 0.17  M⊙, but larger values are also possible, up to 0.30  M⊙, with younger age, higher metallicity, and reddening. Our RGB mass-loss results, based on a uniform mass-loss parameter and our best-fitting case of the two populations of this cluster, are consistent with the lower and upper limit values from the literature. However, the real situations of the RGB mass-loss in GCs, as discussed in the references above, could be much more complicated. In our final data base of the new parsec isochrones, we will provide different choices of He contents and mass-loss parameters for the users’ science purpose. The LFs from the turn-off to the HB with an RGB mass-loss parameter η = 0.35 are displayed in Fig. 9 for our best-fitting parameters. For comparison, the LF of HST GC survey data (Sarajedini et al. 2007) is also plotted (black histogram filled with oblique lines) with the same bin size 0.05 mag. We adopt the Salpeter IMF (Salpeter 1955) to generate LFs though, as discussed in Section 3.2.2, LFs in this phase are not sensitive to IMF because the stellar mass varies very little. LFs are instead sensitive to the evolution time along the phase. All model LFs are normalized to the total number of observed RGB stars within a range of F814W magnitude between 14 and 16 mag. The left-hand panels of Fig. 9 show the LFs from the turn-off to the HB, for a 100 per cent FG (red histogram), a 100 per cent SG (blue histogram), and the percentage adopted in Section 3.2.2, 30 per cent from FG and 70 per cent from SG (orange histogram), respectively. With our best isochrone fitting parameters, age = 12.00 Gyr, (m − M)0 = 13.22, E(6 − 8) = 0.035, η = 0.35, and the population percentage obtained from literature (Carretta et al. 2009b; Milone et al. 2012), the model LF (orange histogram in each figure) shows a very good agreement with the observed LF. The three right-hand panels in Fig. 9 are zoomed in on the HB and RGBB regions. The total number of HB stars within 12.9–13.3 mag in the observations and in the models are listed in the figure. Since the LF is directly proportional to the evolution time, the good agreement of LF between model and observation in Fig. 9 indicates that the hydrogen shell burning lifetime is correctly predicted in parsec . Figure 9. View largeDownload slide Comparison between the LF of 47 Tuc data (Sarajedini et al. 2007) and that derived from the new parsec models, from the turn-off to the HB. The Y axis represent the star counts in magnitude F814W. The black histogram filled with oblique lines is the data LF, whereas the red histogram in the upper panel, blue histogram in the middle panel, and orange histogram in the lower panel, represent 100 per cent FG of 47 Tuc [Z = 0.0056, Y = 0.256], 100 per cent SG [Z = 0.0055, Y = 0.276], and their mix with 30 per cent from the FG and 70 per cent from the SG, respectively. The three panels on the right-hand side show the LF of the RGBB and the HB region, for each population mixture. The fitting parameters are: η = 0.35, age = 12 Gyr, (m − M)0 = 13.22, and E(6 − 8) = 0.035. Figure 9. View largeDownload slide Comparison between the LF of 47 Tuc data (Sarajedini et al. 2007) and that derived from the new parsec models, from the turn-off to the HB. The Y axis represent the star counts in magnitude F814W. The black histogram filled with oblique lines is the data LF, whereas the red histogram in the upper panel, blue histogram in the middle panel, and orange histogram in the lower panel, represent 100 per cent FG of 47 Tuc [Z = 0.0056, Y = 0.256], 100 per cent SG [Z = 0.0055, Y = 0.276], and their mix with 30 per cent from the FG and 70 per cent from the SG, respectively. The three panels on the right-hand side show the LF of the RGBB and the HB region, for each population mixture. The fitting parameters are: η = 0.35, age = 12 Gyr, (m − M)0 = 13.22, and E(6 − 8) = 0.035. 4 COMPARISON WITH OTHER MODELS AND GC DATA The new parsec α-enhanced isochrones provide a very good fit of the CMD of 47 Tuc in all evolutionary stages from the lower MS to the HB. The location of the RGB bump shows that the efficiency of the envelope overshoot is quite significant, requiring EOV of Λe = 0.5Hp. This can be considered as a calibration of this phenomenon. We now use the calibrated EOV value to obtain α-enhanced isochrones of different metallicities. For this purpose, we adopt the partition of heavy elements of the two stellar generations of 47 Tuc ([α/Fe] ∼ 0.4 and 0.2). In this section, we compare our new α-enhanced models with isochrones from other stellar evolution groups and GC data of different metallicities. 4.1 Comparison with other models The RGBB of GCs, as already said in Section 3.2.2, has been studied over 30 years since the 47 Tuc RGBB was observed in 1985 (King et al. 1985). However, there is a discrepancy between the observed brightness of RGBB and the model predictions: The model RGBB magnitude is about 0.2–0.4 mag brighter than the observed one (Fusi Pecci et al. 1990; Cecco et al. 2010; Troisi et al. 2011). This discrepancy becomes more pronounced in metal-poor GCs (Cassisi et al. 2011). Here, we compare the RGBB magnitude of our newly calibrated parsec models with other α-enhanced stellar tracks. Since the BaSTI (Pietrinferni et al. 2006, 2013) and dsep (Dotter 2007; Dotter et al. 2008) isochrones are publicly available online, we download the [α/Fe] = 0.4 isochrones at 13 Gyr from BaSTI canonical models data base and dsep web tool 2012 version. We then compare the mean values of their absolute RGBB magnitude in the F606W (HST ACS/WFC) band, with our models. Fig. 10 shows this comparison as a function of total metallicity [M/H] and iron abundance [Fe/H]. The model [M/H] is approximated by   $$[M/H] \approx \log \frac{Z/X}{Z_{\odot }/X_{\odot }}.$$ (2)And for both of the two new parsec models with α enhancement, [Fe/H] ≈ [M/H] −0.33. The solar metallicity in parsec is Z⊙ = 0.01524 and Z⊙/X⊙ = 0.0207. Since dsep models do not provide [M/H] directly but only [Fe/H] in their isochrones, we calculate [M/H] following equation (2) with total metallicity Z, He content Y, and solar Z⊙/X⊙ taken from their models. Additionally, two parsec models with solar-scaled metal mixture ([α/Fe] = 0), parsec v1.2S and parsec with EOV calibration from this work Λe = 0.5Hp, are also plotted. Compared with the new set of solar-scaled parsec model with Λe = 0.5Hp (dark blue line with diamond), the α-enhanced one (red line with triangle) at the same [M/H] (thus same Z and Y) is slightly brighter as we have already discussed in Section 2. We notice that the RGBB behaviour of parsec v1.2S in this figure is different from fig. 3 of Joyce & Chaboyer (2015), which compares parsec v1.2S with other models. The reason for this disagreement is unclear to us. Figure 10. View largeDownload slide Comparison of the RGBB magnitude of different evolutionary tracks at 13 Gyr as a function of [M/H] (left-hand panel) and [Fe/H] (right-hand panel). There are three different α-enhanced models ([α/Fe]   = 0.4) in the figure: parsec (red solid line with triangle), BaSTI (yelow green solid line with dot), and dsep model (green solid line with star). Other two sets of solar-scaled parsec models ([α/Fe]   = 0) are plotted for comparison: parsec v1.2S with negligible overshoot (light blue dotted line with cross) and parsec with EOV calibration Λe = 0.5Hp (dark blue dotted line with diamond). The Y axis is the mean value of the absolute F606W magnitude of the RGBB (MV, RGBB). Figure 10. View largeDownload slide Comparison of the RGBB magnitude of different evolutionary tracks at 13 Gyr as a function of [M/H] (left-hand panel) and [Fe/H] (right-hand panel). There are three different α-enhanced models ([α/Fe]   = 0.4) in the figure: parsec (red solid line with triangle), BaSTI (yelow green solid line with dot), and dsep model (green solid line with star). Other two sets of solar-scaled parsec models ([α/Fe]   = 0) are plotted for comparison: parsec v1.2S with negligible overshoot (light blue dotted line with cross) and parsec with EOV calibration Λe = 0.5Hp (dark blue dotted line with diamond). The Y axis is the mean value of the absolute F606W magnitude of the RGBB (MV, RGBB). Among the factors that may affect the brightness of the RGBB, as summarized in Section 3.2.2, we list the mixing efficiency and He contents. The helium-to-metal enrichment law of the different models is different, as discussed in Section 2. parsec (Y = 0.2485 + 1.78Z) uses a slightly higher He abundance (∼0.002) than the other two models (BaSTI: Y = 0.245 + 1.4Z, dsep: Y = 0.245 + 1.54Z). Different model also adopts different mixing length parameters. The parsec mixing length parameter is αMLT = 1.74, BaSTI uses αMLT = 1.913, and dsep adopts αMLT = 1.938. If all other parameters are the same, a higher He content and a smaller mixing length parameter lead to a brighter RGBB (Fu 2006). This can explain why in the left-hand panel of Fig. 10 the solar-scaled parsec v1.2S shows nearly the same MV, RGBB independent of [M/H] as BaSTI and dsep models. parsec v1.2S has slightly higher He content and smaller mixing length parameter, which make the RGBB brighter as already discussed above, whereas its solar-scaled metal mixture leads to a fainter RGBB at the same metallicity. The combined effects make the three models to show similar RGBB magnitude. BaSTI and dsep RGBB have almost the same performance and are eventually brighter than parsec [α/Fe] ∼ 0.4 models no matter as a function of [M/H] or [Fe/H]. We remind that a fainter RGBB magnitude can be produced by a more efficient EOV, our new α-enhanced models are computed with the calibrated EOV parameter, whereas BaSTI and dsep do not consider envelope overshooting. Also, compared to parsec v1.2S (Λe = 0.05Hp), the new solar-scaled model with Λe = 0.5Hp shifts MV, RGBB down by about 0.35 mag. This brightness change is consistent with the work of Cassisi et al. (2002), who conclude that the difference should be of about 0.8 mag/Hp. Since the RGBB brightness difference between the new parsec α-enhanced models and the solar-scaled models with the same Λe is much smaller than the difference between the two solar-scaled parsec models with different Λe, we conclude that the mixing efficiency has much stronger impact on the RGBB performance than the metal partition. 4.2 Comparison with other GC data Comparing the location of the RGB bump predicted by the models with the observed one in GCs with different metallicity is a good way to test the models. In Fig. 11, we compare our new α-enhanced models with HST data from Nataf et al. (2013, 55 clusters) and Cassisi et al. (2011, 12 clusters). The models extend till [M/H] ∼ −2 ([Fe/H] ∼ −2.3). For comparison, two sets of models with solar-scaled metal partition, [α/Fe] = 0 (parsec v1.2S and parsec with Λe = 0.5Hp) are also plotted. Figure 11. View largeDownload slide F606W magnitude difference between the MSTO and the RGBB ($$\Delta V^{\rm MSTO}_{\rm RGBB}$$) as a function of the total metallicity [M/H] (left-hand panel) and iron abundance [Fe/H]. Four different sets of theoretical $$\Delta V^{\rm MSTO}_{\rm RGBB}$$ value are plotted, at both 13 Gyr (solid line) and 11 Gyr (dashed line). Three of them are with new calibrated EOV Λe = 0.5Hp: [α/Fe] ∼ 0.4 (red lines with triangle), [α/Fe] ∼ 0.2 (green lines with square), and [α/Fe] = 0 (dark blue lines with diamond). Another one is from the standard parsec v1.2S (light blue lines with cross). The data are 55 clusters from Nataf et al. (2013, grey dots with error bar) and 12 clusters from Cassisi et al. (2011, black dots with error bar). Figure 11. View largeDownload slide F606W magnitude difference between the MSTO and the RGBB ($$\Delta V^{\rm MSTO}_{\rm RGBB}$$) as a function of the total metallicity [M/H] (left-hand panel) and iron abundance [Fe/H]. Four different sets of theoretical $$\Delta V^{\rm MSTO}_{\rm RGBB}$$ value are plotted, at both 13 Gyr (solid line) and 11 Gyr (dashed line). Three of them are with new calibrated EOV Λe = 0.5Hp: [α/Fe] ∼ 0.4 (red lines with triangle), [α/Fe] ∼ 0.2 (green lines with square), and [α/Fe] = 0 (dark blue lines with diamond). Another one is from the standard parsec v1.2S (light blue lines with cross). The data are 55 clusters from Nataf et al. (2013, grey dots with error bar) and 12 clusters from Cassisi et al. (2011, black dots with error bar). Here, we use the magnitude difference between the RGBB and the main-sequence turn-off (MSTO), $$\Delta V^{\rm MSTO}_{\rm RGBB}$$, as a reference for comparison between the theoretical magnitude of RGBB and the observed one. Unlike the absolute magnitude MV, RGBB, $$\Delta V^{\rm MSTO}_{\rm RGBB}$$ is not affected by uncertainties in the distance modulus (m − M)0 and extinction AV of the cluster. There are also works using the magnitude difference between HB and RGBB ($$\Delta V^{\rm RGBB}_{\rm HB} = M_{V,{\rm RGBB}} - M_{V,{\rm HB}}$$, e.g. Fusi Pecci et al. 1990; Cassisi & Salaris 1997; Cecco et al. 2010) or the one between HB and MSTO ($$\Delta V^{HB}_{TO} = M_{V,TO} - M_{V,{\rm HB}}$$, e.g. VandenBerg et al. 2013) as a way to avoid distance and extinction uncertainties, but, as we have elaborated in Section 3.2.4, the RGB mass-loss together with different metal mixture and He content may affect the HB magnitude and thus make $$\Delta V^{\rm RGBB}_{\rm HB}$$ difficult to be interpreted. The only free parameter of the $$\Delta V^{\rm MSTO}_{\rm RGBB}$$ method is the age, if the composition of the cluster is fixed. In Fig. 11, we compare the theoretical $$\Delta V^{\rm MSTO}_{\rm RGBB}$$ value at typical GC ages of 11 and 13 Gyr, with the observed value from Nataf et al. (2013) and (Cassisi et al. 2011). The comparisons are displayed both in the [M/H] frame and [Fe/H] frame. The MSTO in Nataf et al. (2013) is defined by taking the bluest point of a polynomial fit to the upper MS of each GC in the (F606W, F606W − F814W). Cassisi et al. (2011) derive the MSTO magnitude by fitting isochrones to the MS. To obtain the theoretical MSTO F606W magnitude in our model, we select the bluest point of the isochrone in the MS. The models of 13 Gyr show larger difference between RGBB and MSTO $$\Delta V^{\rm MSTO}_{\rm RGBB}$$ than those at 11 Gyr. Models with [α/Fe] ∼ 0.2 show a slightly greater $$\Delta V^{\rm MSTO}_{\rm RGBB}$$ value than the models computed with [α/Fe] ∼ 0.4. In the right-hand panel of the figure, we see that the α-enhanced models show greater $$\Delta V^{\rm MSTO}_{\rm RGBB}$$ than the solar-scaled ones with the same [Fe/H]. This said, if one has [Fe/H] measurement of a GC with α enhancement and takes $$\Delta V^{\rm MSTO}_{\rm RGBB}$$ as an age indicator, choosing the solar-scaled models will lead to an underestimated cluster age. Compared to the previous parsec version v1.2S, the new models significantly improve the $$\Delta V^{\rm MSTO}_{\rm RGBB}$$ prediction in both the [M/H] frame and [Fe/H] frame. At the most metal-poor side, around [M/H] ∼ −2.0 ([Fe/H] ∼ −2.3), the new models are consistent with Cassisi et al. (2011) data (black dots), but are higher than the values derived by Nataf et al. (2013) (grey dots) by ∼0.1 mag. We will discuss the possible reasons in the next section. 5 SUMMARY AND DISCUSSION Studies on GCs, Galactic bulge, halo, and thick disc call for stellar models with α enhancement because stars residing in them have α-to-iron number ratio larger than the solar value. This ratio, [α/Fe] , not only affects the stellar features like the luminosities and effective temperature, but also echoes the formation history of the cluster/structure the stars are in. To investigate such stars, and to trace back their formation history, we have now extend the parsec models to include α-enhanced mixtures. In this paper, we check the α-enhanced models with the nearby GC 47 Tuc (NGC 104). The chemical compositions including the helium abundances of 47 Tuc are studied by many works. We collect detailed elemental abundances of this cluster and derive absolute metal mixtures for two populations: first generation [Z = 0.0056, Y = 0.256], and second generation [Z = 0.0055, Y = 0.276]. The α-to-iron ratios of them are [α/Fe] = 0.4057 ([α/Fe] ∼ 0.4) and [α/Fe] = 0.2277 ([α/Fe] ∼ 0.2), respectively. We calculate evolutionary tracks and isochrones with these two α-enhanced metal mixtures, and fit CMD to HST/ACS data. The model envelope overshooting is then calibrated to the value Λe = 0.5Hp in order to reproduce the RGB bump morphology in 47 Tuc. After the calibration, the new α-enhanced isochrones nicely fit the data from the low MS to the turn-off, giant branch, and the HB with age of 12.00 Gyr, absolute distance modulus (m-M)0 = 13.22 (apparent distance modulus (m-M)F606W = 13.32), and reddening E(6 − 8) = 0.035. These results compare favourably with many other determinations in the literature. The LFs inform us that the lifetime of hydrogen burning shell appears to be correctly predicted. By studying the morphology of the HB, we conclude that the mean mass lost by stars during the RGB phase is around 0.17 M⊙. There are also other methods to estimate the age of this cluster in the literature. For instance, mass-radius constraints of the detached eclipsing binary stars V69 in 47 Tuc have also been used (Weldrake et al. 2004; Dotter, Kaluzny & Thompson 2008; Thompson et al. 2010; Brogaard et al. 2017). This approach, which considers the two components of the detached binary as single stars, is much less affected by the uncertainties arising from the unknown distance, reddening, and transformation from the theoretical to the observational plane. Thompson et al. (2010) derive an age of 11.25 ± 0.21(random) ± 0.85(systematic) for 47 Tuc, and Brogaard et al. (2017) give 11.8 Gyr as their best estimate with 3σ limits from 10.4 to 13.4 Gyr. We examined the mass–radius and mass–Teff constraints provided by V69 with our best-fitting FG and SG models at 12.0 Gyr, as shown in Fig. 12. We have adopted for the two components of V69 the following data (Thompson et al. 2010): current mass Mp = 0.8762 ± 0.0048 M⊙, Ms = 0.8588 ± 0.0060 M⊙, radius Rp = 1.3148 ± 0.0051 R⊙, Rs = 1.1616 ± 0.0062 R ⊙ and effective temperature Teffp = 5945 ± 150 K, Teffs = 5959 ± 150 K of the primary and secondary star, respectively. Differences between the FG and SG isochrones are due to the difference of He abundances and [α/Fe]. The comparison with our models indicates that V69 cannot belong to the SG population. Concerning the mass–radius relation, which provides the most stringent constraints on the two stars, we see that our best-fitting FG isochrone of 12 Gyr is only marginally able to reproduce the secondary component (filled black star), within 3σ, whereas there is a tension with the radius of the primary component (empty star). A better match can be obtained for both components by considering a slightly higher metallicity, as also suggested by Brogaard et al. (2017). For example, the green dashed isochrones in Fig. 12 illustrate the effects of assuming [Fe/H] = −0.6 and [α/Fe] = 0.4, which correspond to Z = 0.008, and an He content of Y = 0.263. In this example, the He content of the isochrone follows equation (1) and the [α/Fe] value is chosen to be 0.4 without any special consideration. We also note that, to bring the radius discrepancy of the primary component within 3σ or 1σ, with the abundances assumed for the FG population, an age of 11.7 or 11.2 Gyr would be required, respectively. These ages, in particularly the lower one, would be quite different from the one obtained by the best fit presented in this paper. Up to now, we have assumed that effects of binary interaction are negligible so that V69 can be analysed with single star evolution models. However, considering other possible causes for the discrepancy, we note that the primary component of V69 is among the bluest stars past the turn-off of the CMD (see fig. 4 of Thompson et al. 2010) so that the fitting isochrone would likely fit the bluer envelope of the cluster CMD. We thus suspect that its position in the CMD could partly be due to the effects of the binary dynamical interaction with the companion star. In fact, tidal effects in close binary stars may change the structure and evolution of stars even before any possible mass transfer (de Mink et al. 2009; Song et al. 2016). In particular, the primary star of V69 matches our model that shows a thin surface convective envelope (∼2 per cent of total mass) and, following the simple approximation of Zahn (1977, equation 6.1), its tidal synchronization time should be roughly ∼7.2 Gyr, comparable to its current age. Tidal friction during the previous evolution may have introduced shear mixing and extra turbulence (Lanza & Mathis 2016) at the base of the external convective region, whose effects are primarily those of mitigating helium and heavy element diffusion away from the convective region. The net effect will be a lower growth of the surface hydrogen abundance with a corresponding decrease of the current surface opacity, which should result in a smaller current radius, in the observed direction. Of course, a more sophisticated theoretical analysis is necessary to assess if the location of the primary star of V69 in the CMD might be affected by previous dynamical interaction, but this is beyond the scope of this paper. Figure 12. View largeDownload slide Mass–radius and Mass–Teff diagrams for binary V69. The observed values of the two components of V69 are marked with black stars, their corresponding 1σ and 3σ uncertanties are indicated with dashed line boxes and shaded boxes, respectively (data from Thompson et al. 2010). Isochrones of 47 Tuc FG (red solid lines) and SG (blue dotted lines) are overlaid. To illustrate the effects of a higher metallicity, isochrones of [Fe/H] =−0.6, [α/Fe] =0.4 (Z = 0.008, Y = 0.263) are also displayed (green dashed lines). All isochrones are with our best estimate age 12.0 Gyr. Figure 12. View largeDownload slide Mass–radius and Mass–Teff diagrams for binary V69. The observed values of the two components of V69 are marked with black stars, their corresponding 1σ and 3σ uncertanties are indicated with dashed line boxes and shaded boxes, respectively (data from Thompson et al. 2010). Isochrones of 47 Tuc FG (red solid lines) and SG (blue dotted lines) are overlaid. To illustrate the effects of a higher metallicity, isochrones of [Fe/H] =−0.6, [α/Fe] =0.4 (Z = 0.008, Y = 0.263) are also displayed (green dashed lines). All isochrones are with our best estimate age 12.0 Gyr. The envelope overshooting calibration together with the α-enhanced metal mixtures of 47 Tuc is applied to other metallicities till Z = 0.0001. The RGB bump magnitudes of the new α-enhanced isochrones are compared with other stellar models and GC observations. We take $$\Delta ^{\rm MSTO}_{\rm RGBB}$$, the magnitude difference between the MSTO and the RGB bump, as the reference to compare with the observation in order to avoid uncertainties from distance and extinction. Our new models fit the data quite well and significantly improve the prediction of RGB bump magnitude compared to previous models. However, we notice that in Fig. 11 around [M/H] = −2.0 ([Fe/H] ∼ −2.3) our model predicts $$\Delta V^{\rm MSTO}_{\rm RGBB}$$ about ∼0.1 mag greater than the data points obtained by Nataf et al. (2013). If we consider a more He-rich model with the same metallicity Z, the discrepancy will become even larger. There are works arguing that diffusion also affects the brightness of RGBB (eg. Michaud, Richer & Richard 2010; Cassisi et al. 2011; Joyce & Chaboyer 2015). Michaud et al. (2010) conclude that without atomic diffusion the RGBB luminosity is about 0.02 dex brighter (∼0.05 mag). In parsec , we always take diffusion into account. To see the effect of diffusion on RGBB morphology, we calculate a 0.80 M⊙ model without diffusion and compare it with the one with standard parsec diffusion in Fig. 13. The two stars have the same metallicity, He content, stellar mass, and [α/Fe] . As we can see from the right-hand panel, diffusion shorten the MS lifetime. The left-hand panel of this figure is HRD, similar to that in fig. 1 of Michaud et al. (2010), evolutionary track with diffusion shows redder MSTO and slightly fainter RGBB. For isochrones obtained from these two sets of evolutionary tracks, at 13 Gyr, the RGBB without diffusion is 0.072 mag (in F606W) brighter than the one with diffusion, and $$\Delta V^{\rm MSTO}_{\rm RGBB}$$ value is 0.008 mag (in F606W) larger. This result confirms that inhibiting the diffusion during H-burning phase will eventually makes the discrepancy more severe. Pietrinferni, Cassisi & Salaris (2010) conclude that the updated nuclear reaction rate for $$\rm \,{}^{14}\hspace{-0.8pt}{N}\,({p}\,,{\gamma }) \,{}^{15}\hspace{-0.8pt}{O}\,$$ makes RGBB brighter by ∼0.06 mag compared to the old rate. However, we remind that we are already adopting the new rate (Imbriani et al. 2005) for this reaction (Table 1). Assuming that all other input physics, in particular opacities, is correct, the only possible solution to cover this ∼0.1 mag discrepancy is that the mixing at the bottom of the convective envelope is even higher than that assumed here. Either EOV in metal-poor stars is larger than our adopted value (e.g. Λe = 0.7Hp suggested by Alongi et al. 1991) or another kind of extra mixing is responsible. Figure 13. View largeDownload slide Comparison of [Z = 0.0002, Y = 0.249] tracks for an M = 0.80 M⊙ star with (orange line) and without (blue line) diffusion. The left-hand panel shows the HRD of these two tracks, with the RGBB region zoomed in in the subfigure. The right-hand panel illustrates the luminosity evolution as the star ages. Figure 13. View largeDownload slide Comparison of [Z = 0.0002, Y = 0.249] tracks for an M = 0.80 M⊙ star with (orange line) and without (blue line) diffusion. The left-hand panel shows the HRD of these two tracks, with the RGBB region zoomed in in the subfigure. The right-hand panel illustrates the luminosity evolution as the star ages. We have shown in Section 4.2 that, comparing the absolute magnitude MV, RGBB between model and observation directly, would introduce uncertainties from distance and extinction. However, putting the MV, RGBB and $$\Delta V^{\rm MSTO}_{\rm RGBB}$$ comparison together could help us to constrain the distance of the clusters. Fig. 14 displays the differences between the theoretical MV, RGBB and those of data from Nataf et al. (2013) and Cassisi et al. (2011). The absolute magnitude MV, RGBB of the data takes into account the apparent distance modulus, hence extinction effects are excluded. The four sets of models in Fig. 14 are the same as those in Fig. 11. The discrepancy of MV, RGBB between models and the data at the metal-poor end in Fig. 14 is larger than that of $$\Delta V^{\rm MSTO}_{\rm RGBB}$$ in Fig. 11 in both [M/H] and [Fe/H] frames. Since $$\Delta V^{\rm MSTO}_{\rm RGBB}$$ is a reference without distance effect, this larger discrepancy indicates that the apparent distance modulus (m − M)V used in Fig. 14 (Nataf et al. 2013) for metal-poor GCs are underestimated. Figure 14. View largeDownload slide Absolute magnitude in F606W band of RGBB (MV, RGBB) as a function of metallicity [M/H] (left panel) and iron abundance [Fe/H] (right panel). Three sets of EOV-calibrated parsec models, [α/Fe] =[0.0, 0.2, 0.4], and parsec v1.2S are shown both at age 13 Gyr (solid line) and 11 Gyr (dashed line). For comparison, Nataf et al. (2013, grey filled dots with error bar) and Cassisi et al. (2011, black filled dots with error bar) RGBB data are plotted. See the text for the details. Figure 14. View largeDownload slide Absolute magnitude in F606W band of RGBB (MV, RGBB) as a function of metallicity [M/H] (left panel) and iron abundance [Fe/H] (right panel). Three sets of EOV-calibrated parsec models, [α/Fe] =[0.0, 0.2, 0.4], and parsec v1.2S are shown both at age 13 Gyr (solid line) and 11 Gyr (dashed line). For comparison, Nataf et al. (2013, grey filled dots with error bar) and Cassisi et al. (2011, black filled dots with error bar) RGBB data are plotted. See the text for the details. In a following paper of the ‘parsec α-enhanced stellar evolutionary tracks and isochrones’ series, we will provide other [α/Fe] choices based on metal mixtures derived from atlas9 APOGEE atmosphere models (Mészáros et al. 2012). The full set of isochrones with chemical compositions suitable for GCs and Galactic bulge/thick disc stars will be available online after the full calculation and calibration are performed. Acknowledgements XF thanks Angela Bragaglia and Francesca Primas for the useful discussion, and Fiorella Castelli for the guide on ATLAS12 calculation. AB acknowledges INAF-PRIN-2014-14 ‘Star formation and evolution in galactic nuclei’, and thanks Michela Mapelli for the useful discussion on binary interaction. PM, JM, YC, and AN acknowledge support from the ERC Consolidator Grant funding scheme (project STARKEY, G. A. n. 615604). Footnotes 1 CMD input form: http://stev.oapd.inaf.it/cgi-bin/cmd/. 2 http://opalopacity.llnl.gov/ 3 http://stev.oapd.inaf.it/aesopus 4 http://freeeos.sourceforge.net/ 5 $$\rm {A(X)}=\log (\rm {N}_{\rm X}/\rm {N}_{\rm H}) + 12$$, with NX is the abundance in number for the element X. 6 [X/Fe] = log (NX/NFe) − log (NX/NFe)⊙ REFERENCES Alongi M., Bertelli G., Bressan A., Chiosi C., 1991, A&A , 244, 95 Alves-Brito A. et al.  , 2005, A&A , 667, 657 CrossRef Search ADS   Anderson J. et al.  , 2008, AJ , 135, 2055 https://doi.org/10.1088/0004-6256/135/6/2055 CrossRef Search ADS   Anderson J., Piotto G., King I. R., Bedin L. R., Guhathakurta P., 2009, ApJ , 697, L58 https://doi.org/10.1088/0004-637X/697/1/L58 CrossRef Search ADS   Angulo C. et al.  , 1999, Nucl. Phys. A , 656, 3 https://doi.org/10.1016/S0375-9474(99)00030-5 CrossRef Search ADS   Balbinot E. et al.  , 2016, ApJ , 820, 58 https://doi.org/10.3847/0004-637X/820/1/58 CrossRef Search ADS   Bedin L. R., Piotto G., Anderson J., Cassisi S., King I. R., Momany Y., Carraro G., 2004, ApJ , 605, L125 https://doi.org/10.1086/420847 CrossRef Search ADS   Belczynski K., Holz D. E., Bulik T., O'Shaughnessy R., 2016, Nature , 534, 512 https://doi.org/10.1038/nature18322 CrossRef Search ADS PubMed  Bergbusch P. A., Stetson P. B., 2009, AJ , 138, 1455 https://doi.org/10.1088/0004-6256/138/5/1455 CrossRef Search ADS   Betts R. R., Fortune H. T., Middleton R., 1975, Phys. Rev. C , 11, 19 https://doi.org/10.1103/PhysRevC.11.19 CrossRef Search ADS   Bjork S. R., Chaboyer B., 2006, ApJ , 641, 1102 https://doi.org/10.1086/500505 CrossRef Search ADS   Bono G., Cassisi S., Zoccali M., Piotto G., 2001, ApJ , 546, L109 https://doi.org/10.1086/318866 CrossRef Search ADS   Borissova J. et al.  , 2014, A&A , 569, A24 CrossRef Search ADS   Bovy J., Rix H.-W., Green G. M., Schlafly E. F., Finkbeiner D. P., 2016, ApJ , 818, 130 https://doi.org/10.3847/0004-637X/818/2/130 CrossRef Search ADS   Bragaglia A., Carretta E., Gratton R., D'Orazi V., Cassisi S., Lucatello S., 2010, A&A , 519, A60 CrossRef Search ADS   Bragaglia A., Carretta E., Sollima A., Donati P., D'Orazi V., Gratton R. G., Lucatello S., Sneden C., 2015, A&A , 583, A69 CrossRef Search ADS   Bressan A., Marigo P., Girardi L., Salasnich B., Dal Cero C., Rubele S., Nanni A., 2012, MNRAS , 427, 127 https://doi.org/10.1111/j.1365-2966.2012.21948.x CrossRef Search ADS   Bressan A. Girardi L. Marigo P. Rosenfield P. Tang J., 2015, in Miglio A. Eggenberger P. Girardi L. Montalbán J., eds., Astrophysics and Space Science Proceedings, Vol. 39, Asteroseismology of Stellar Populations in the Milky Way . Springer, Cham, p. 25 Briley M. M., 1997, AJ , 114, 1051 https://doi.org/10.1086/118535 CrossRef Search ADS   Brogaard K., VandenBerg D. A., Bedin L. R., Milone A. P., Thygesen A., Grundahl F., 2017, MNRAS , 468, 645 https://doi.org/10.1093/mnras/stx378 CrossRef Search ADS   Caffau E., Ludwig H.-G., Steffen M., Freytag B., Bonifacio P., 2011, Sol. Phys. , 268, 255 https://doi.org/10.1007/s11207-010-9541-4 CrossRef Search ADS   Caloi V., D'Antona F., 2005, A&A , 435, 987 CrossRef Search ADS   Cannon R. D., Croke B. F. W., Bell R. A., Hesser J. E., Stathakis R. A., 1998, MNRAS , 298, 601 https://doi.org/10.1046/j.1365-8711.1998.01671.x CrossRef Search ADS   Cardelli J. A., Clayton G. C., Mathis J. S., 1989, ApJ , 345, 245 https://doi.org/10.1086/167900 CrossRef Search ADS   Carney B. W., 1996, Publ. Astron. Soc. Pac. , 108, 900 https://doi.org/10.1086/133811 CrossRef Search ADS   Carretta E., Bragaglia A., Gratton R. G., Leone F., Recio-Blanco A., Lucatello S., 2006, A&A , 450, 523 CrossRef Search ADS   Carretta E. et al.  , 2007, A&A , 464, 967 CrossRef Search ADS   Carretta E., Bragaglia A., Gratton R., D'Orazi V., Lucatello S., 2009a, A&A , 508, 695 CrossRef Search ADS   Carretta E., Bragaglia A., Gratton R., Lucatello S., 2009b, A&A , 505, 139 CrossRef Search ADS   Carretta E., Gratton R. G., Bragaglia A., D'Orazi V., Lucatello S., 2013, A&A , 550, A34 CrossRef Search ADS   Carretta E., D'Orazi V., Gratton R. G., Lucatello S., 2014, A&A , 563, A32 CrossRef Search ADS   Casey A. R. et al.  , 2016, MNRAS , 461, 3336 https://doi.org/10.1093/mnras/stw1512 CrossRef Search ADS   Cassisi S., Salaris M., 1997, MNRAS , 285, 593 https://doi.org/10.1093/mnras/285.3.593 CrossRef Search ADS   Cassisi S., Salaris M., Bono G., 2002, ApJ , 565, 1231 https://doi.org/10.1086/324695 CrossRef Search ADS   Cassisi S., Marín-Franch A., Salaris M., Aparicio A., Monelli M., Pietrinferni A., 2011, A&A , 527, A59 CrossRef Search ADS   Catelan M., 2008, Proc. Int. Astron. Union , 4, 209 https://doi.org/10.1017/S174392130903186X CrossRef Search ADS   Caughlan G. R., Fowler W. A., 1988, At. Data Nucl. Data Tables , 40, 283 https://doi.org/10.1016/0092-640X(88)90009-5 CrossRef Search ADS   Cecco A. D. et al.  , 2010, ApJ , 712, 527 https://doi.org/10.1088/0004-637X/712/1/527 CrossRef Search ADS   Chen Y., Girardi L., Bressan A., Marigo P., Barbieri M., Kong X., 2014, MNRAS , 444, 2525 https://doi.org/10.1093/mnras/stu1605 CrossRef Search ADS   Chen Y., Bressan A., Girardi L., Marigo P., Kong X., Lanza A., 2015, MNRAS , 452, 1068 https://doi.org/10.1093/mnras/stv1281 CrossRef Search ADS   Chevallard J., Charlot S., 2016, MNRAS , 462, 1415 https://doi.org/10.1093/mnras/stw1756 CrossRef Search ADS   Chiosi C., Bressan A., Portinari L., Tantalo R., 1998, A&A , 381, 355 Choi J., Dotter A., Conroy C., Cantiello M., Paxton B., Johnson B. D., 2016, ApJ , 823, 102 https://doi.org/10.3847/0004-637X/823/2/102 CrossRef Search ADS   Christensen-Dalsgaard J., Monteiro M. J. P. F. G., Rempel M., Thompson M. J., 2011, MNRAS , 414, 1158 https://doi.org/10.1111/j.1365-2966.2011.18460.x CrossRef Search ADS   Constantini H. LUNA Collaboration, 2010, Proc. Sci., Reaction Rate Measurements in Underground Laboratories . SISSA, Trieste, PoS(NIC XI)014 Cordero M. J., Pilachowski C. A., Johnson C. I., McDonald I., Zijlstra A. A., Simmerer J., 2014, ApJ , 780, 94 https://doi.org/10.1088/0004-637X/780/1/94 CrossRef Search ADS   Cyburt R. H., Hoffman R., Woosley S., 2012, Re-evaluation of Buchmann 1996 C12(a,g) and Hoffman Ca40(a,g) and Ti44(a,p) rates . REACLIB Cyburt R. H., Davids B., 2008, Phys. Rev. C , 78, 064614 https://doi.org/10.1103/PhysRevC.78.064614 CrossRef Search ADS   Cyburt R. H. et al.  , 2010, ApJS , 189, 240 https://doi.org/10.1088/0067-0049/189/1/240 CrossRef Search ADS   D'Antona F., Caloi V., Montalbán J., Ventura P., Gratton R., 2002, A&A , 395, 69 CrossRef Search ADS   D'Antona F., Bellazzini M., Caloi V., Pecci F. F., Galleti S., Rood R. T., 2005, ApJ , 631, 868 https://doi.org/10.1086/431968 CrossRef Search ADS   D'Orazi V., Lucatello S., Gratton R., Bragaglia A., Carretta E., Shen Z., Zaggia S., 2010, ApJ , 713, L1 https://doi.org/10.1088/2041-8205/713/1/L1 CrossRef Search ADS   de Mink S. E., Cantiello M., Langer N., Pols O. R., Brott I., Yoon S.-C., 2009, A&A , 497, 243 CrossRef Search ADS   Descouvemont P., Adahchour A., Angulo C., Coc A., Vangioni-Flam E., 2004, Atomic Data and Nuclear Data Tables , 88, 203 https://doi.org/10.1016/j.adt.2004.08.001 CrossRef Search ADS   Di Criscienzo M., Ventura P., D'Antona F., Milone a., Piotto G., 2010, MNRAS , 408, 999 https://doi.org/10.1111/j.1365-2966.2010.17168.x CrossRef Search ADS   Dobrovolskas V. et al.  , 2014, A&A , 565, A121 CrossRef Search ADS   Donati P. et al.  , 2014, A&A , 561, A94 CrossRef Search ADS   Dotter A. L., 2007, PhD thesis , Dartmouth College Dotter A., Chaboyer B., Jevremović D., Baron E., Ferguson J. W., Sarajedini A., Anderson J., 2007, AJ , 134, 376 https://doi.org/10.1086/517915 CrossRef Search ADS   Dotter A., Kaluzny J., Thompson I. B., 2008, Proc. Int. Astron. Union , 4, 171 -176 https://doi.org/10.1017/S1743921309031822 CrossRef Search ADS   Dotter A., Chaboyer B., Jevremović D., Kostov V., Baron E., Ferguson J. W., 2008, ApJS , 178, 89 https://doi.org/10.1086/589654 CrossRef Search ADS   Dotter A. et al.  , 2010, ApJ , 708, 698 https://doi.org/10.1088/0004-637X/708/1/698 CrossRef Search ADS   Dupree A. K., Strader J., Smith G. H., 2011, ApJ , 728, 155 https://doi.org/10.1088/0004-637X/728/2/155 CrossRef Search ADS   Fagotto F. Bressan A. Bertelli G. Chiosi C., 1994, A&AS , 104 Fu X., 2006, PhD thesis , SISSA - International School for Advanced Studies Fu X., Bressan A., Molaro P., Marigo P., 2015, MNRAS , 452, 3256 https://doi.org/10.1093/mnras/stv1384 CrossRef Search ADS   Fulbright J. P., 2002, AJ , 123, 404 https://doi.org/10.1086/324630 CrossRef Search ADS   Fusi Pecci F. Bellazzini M., 1997, in Philip A. G. D. Liebert J. Saffer R. Hayes D. S., eds., The Third Conference on Faint Blue Stars . Davis Press, Schenectady, p. 255 Fusi Pecci F., Ferraro F. R., Crocker D. A., Rood R. T., Buonanno R., 1990, A&A , 238, 95 Fynbo H. O. U. et al.  , 2005, Nature , 433, 136 https://doi.org/10.1038/nature03219 CrossRef Search ADS PubMed  Gonzalez O. A. et al.  , 2011, A&A , 530, A54 CrossRef Search ADS   Goudfrooij P., Girardi L., Rosenfield P., Bressan A., Marigo P., Correnti M., Puzia T. H., 2015, MNRAS , 450, 1693 https://doi.org/10.1093/mnras/stv700 CrossRef Search ADS   Gratton R. G. et al.  , 2001, A&A , 369, 87 CrossRef Search ADS   Gratton R. G., Carretta E., Bragaglia A., Lucatello S., D'Orazi V., 2010, A&A , 517, A81 CrossRef Search ADS   Gratton R. G., Carretta E., Bragaglia A., 2012, A&AR , 20, 50 CrossRef Search ADS   Gratton R. G. et al.  , 2013, A&A , 549, A41 CrossRef Search ADS   Gratton R. G. et al.  , 2015, A&A , 573, A92 CrossRef Search ADS   Gullikson K., Kraus A., Dodson-Robinson S., 2016, AJ , 152, 40 https://doi.org/10.3847/0004-6256/152/2/40 CrossRef Search ADS   Gutkin J., Charlot S., Bruzual G., 2016, MNRAS , 462, 1757 https://doi.org/10.1093/mnras/stw1716 CrossRef Search ADS   Hansen B. M. S. et al.  , 2013, Nature , 500, 51 https://doi.org/10.1038/nature12334 CrossRef Search ADS PubMed  Harbeck D., Smith G. H., Grebel E. K., 2003, AJ , 125, 197 https://doi.org/10.1086/345570 CrossRef Search ADS   Harris W. E., 1996, AJ , 112, 1487 https://doi.org/10.1086/118116 CrossRef Search ADS   Heil M. et al.  , 2008, Phys. Rev. C , 78, 025803 https://doi.org/10.1103/PhysRevC.78.025803 CrossRef Search ADS   Herwig F., 2000, A&A , 360, 952 Heyl J., Kalirai J., Richer H. B., Marigo P., Antolini E., Goldsbury R., Parada J., 2015, ApJ , 810, 127 https://doi.org/10.1088/0004-637X/810/2/127 CrossRef Search ADS   Iglesias C. A., Rogers F. J., 1996, ApJ , 464, 943 https://doi.org/10.1086/177381 CrossRef Search ADS   Iliadis C., Angulo C., Descouvemont P., Lugaro M., Mohr P., 2008, Phys. Rev. C , 77, 045802 https://doi.org/10.1103/PhysRevC.77.045802 CrossRef Search ADS   Iliadis C., Longland R., Champagne A. E., Coc A., 2010, Nucl. Phys. A , 841, 251 https://doi.org/10.1016/j.nuclphysa.2010.04.010 CrossRef Search ADS   Imbriani G. et al.  , 2005, Eur. Phys. J. A , 25, 455 https://doi.org/10.1140/epja/i2005-10138-7 CrossRef Search ADS   Itoh N., Uchida S., Sakamoto Y., Kohyama Y., Nozawa S., 2008, ApJ , 677, 495 https://doi.org/10.1086/529367 CrossRef Search ADS   Johnson C. I., Rich R. M., Kobayashi C., Kunder A., Koch A., 2014, AJ , 148, 67 https://doi.org/10.1088/0004-6256/148/4/67 CrossRef Search ADS   Joyce M., Chaboyer B., 2015, ApJ , 814, 142 https://doi.org/10.1088/0004-637X/814/2/142 CrossRef Search ADS   Kalari V. M. et al.  , 2014, A&A , 564, L7 CrossRef Search ADS   Kalirai J. S. et al.  , 2012, AJ , 143, 11 CrossRef Search ADS   Kim Y., Demarque P., Yi S. K., Alexander D. R., 2002, ApJS , 143, 499 CrossRef Search ADS   King C. R., Da Costa G. S., Demarque P., 1985, ApJ , 299, 674 CrossRef Search ADS   Kirby E. N., Cohen J. G., Smith G. H., Majewski S. R., Sohn S. T., Guhathakurta P., 2011, ApJ , 727, 79 CrossRef Search ADS   Komatsu E. et al.  , 2011, ApJS , 192, 18 CrossRef Search ADS   Küpper A. H. W., Balbinot E., Bonaca A., Johnston K. V., Hogg D. W., Kroupa P., Santiago B. X., 2015, ApJ , 803, 80 CrossRef Search ADS   Kurucz R. L., 2005, Mem. Soc. Astron. Ital. , 8, 14 Lanza A. F., Mathis S., 2016, Celest. Mech. Dyn. Astron. , 126, 249 CrossRef Search ADS   Lebreton Y., Goupil M. J., 2012, A&A , 544, L13 CrossRef Search ADS   Li Z. et al.  , 2010, Sci. China Phys., Mech. Astron. , 53, 658 CrossRef Search ADS   Li J. et al.  , 2016, ApJ , 823, 59 CrossRef Search ADS   Maldonado J., Eiroa C., Villaver E., Montesinos B., Mora A., 2015, A&A , 579, A20 CrossRef Search ADS   Marigo P., Aringer B., 2009, A&A , 508, 1539 CrossRef Search ADS   Marigo P., Girardi L., 2007, A&A , 469, 239 CrossRef Search ADS   Marigo P., Bressan A., Nanni A., Girardi L., Pumo M. L., 2013, MNRAS , 434, 488 CrossRef Search ADS   Marigo P. et al.  , 2017, ApJ , 835, 77 CrossRef Search ADS   Marino A. F. et al.  , 2011, A&A , 532, A8 CrossRef Search ADS   Marino A. F. et al.  , 2014, MNRAS , 437, 1609 CrossRef Search ADS   McDonald I., Zijlstra A. A., 2015, MNRAS , 448, 502 CrossRef Search ADS   McWilliam A., Preston G. W., Sneden C., Searle L., 1995, AJ , 109, 2757 CrossRef Search ADS   Mészáros S. et al.  , 2012, AJ , 144, 120 CrossRef Search ADS   Michaud G., Richer J., Richard O., 2010, A&A , 510, A104 CrossRef Search ADS   Milone a. P. et al.  , 2012, ApJ , 744, 58 CrossRef Search ADS   Milone A. P. et al.  , 2008, ApJ , 673, 241 CrossRef Search ADS   Milone A. P. et al.  , 2010, ApJ , 709, 1183 CrossRef Search ADS   Milone A. P., Marino A. F., Piotto G., Bedin L. R., Anderson J., Aparicio A., Cassisi S., Rich R. M., 2012, ApJ , 745, 27 CrossRef Search ADS   Milone A. P. et al.  , 2015a, MNRAS , 450, 3750 CrossRef Search ADS   Milone A. P. et al.  , 2015b, ApJ , 808, 51 CrossRef Search ADS   Milone A. P., Marino A. F., D'Antona F., Bedin L. R., Da Costa G. S., Jerjen H., Mackey A. D., 2016, MNRAS , 458, 4368 CrossRef Search ADS   Mucciarelli A., Lovisi L., Lanzoni B., Ferraro F. R., 2014, ApJ , 786, 14 CrossRef Search ADS   Nanni A., Bressan A., Marigo P., Girardi L., 2013, MNRAS , 434, 2390 CrossRef Search ADS   Nanni A., Bressan A., Marigo P., Girardi L., 2014, MNRAS , 438, 2328 CrossRef Search ADS   Nataf D. M., Udalski A., Gould A., Pinsonneault M. H., 2011, ApJ , 730, 118 CrossRef Search ADS   Nataf D. M., Gould A., Pinsonneault M. H., Stetson P. B., 2011, ApJ , 736, 94 CrossRef Search ADS   Nataf D. M., Gould A. P., Pinsonneault M. H., Udalski A., 2013, ApJ , 766, 77 CrossRef Search ADS   Nissen P. E., Gustafsson B., Edvardsson B., Gilmore G., 1994, A&A , 285, 440 Norris J., Freeman K. C., 1979, ApJ , 230, L179 CrossRef Search ADS   Origlia L., Rood R. T., Fabbri S., Ferraro F. R., Fusi Pecci F., Rich R. M., 2007, ApJ , 667, L85 CrossRef Search ADS   Pasquini L., Mauas P., Käufl H. U., Cacciari C., 2011, A&A , 531, A35 CrossRef Search ADS   Perren G. I., Vázquez R. A., Piatti A. E., 2015, A&A , 576, A6 CrossRef Search ADS   Pietrinferni A., Cassisi S., Salaris M., Castelli F., 2006, ApJ , 642, 797 CrossRef Search ADS   Pietrinferni A., Cassisi S., Salaris M., 2010, A&A , 522, A76 CrossRef Search ADS   Pietrinferni A., Cassisi S., Salaris M., Hidalgo S., 2013, A&A , 558, A46 CrossRef Search ADS   Piotto G. et al.  , 2007, ApJ , 661, L53 CrossRef Search ADS   Pritzl B. J., Venn K. A., Irwin M., 2005, AJ , 130, 2140 CrossRef Search ADS   Ramya P., Reddy B. E., Lambert D. L., Musthafa M. M., 2016, MNRAS , 460, 1356 CrossRef Search ADS   Reddy A. B. S., Lambert D. L., 2016, A&A , 589, A57 CrossRef Search ADS   Reddy B. E., Lambert D. L., Prieto C. A., 2006, MNRAS , 367, 1329 CrossRef Search ADS   Reimers D., 1975, Mem. Soc. R. Sci. Liege , 8, 369 Rood R. T. Crocker D. A., 1985, in Danziger I. J. Matteucci F. Kjar K., eds., European Southern Observatory Conference and Workshop Proceedings , Vol. 21. ESO, Garching, p. 61 Rosenfield P., Marigo P., Girardi L., Dalcanton J. J., Bressan A., Williams B. F., Dolphin A., 2016, ApJ , 822, 73 CrossRef Search ADS   Ruchti G. R. et al.  , 2010, ApJ , 721, L92 CrossRef Search ADS   Salaris M., Weiss A., Ferguson J. W., Fusilier D. J., 2006, ApJ , 645, 1131 CrossRef Search ADS   Salaris M., Held E. V., Ortolani S., Gullieuszik M., Momany Y., 2007, A&A , 476, 243 CrossRef Search ADS   Salaris M., Cassisi S., Pietrinferni A., 2016, A&A , 590, A64 CrossRef Search ADS   Salasnich B., Girardi L., Weiss A., Chiosi C., 2000, A&A , 361, 1023 Salpeter E. E., 1955, ApJ , 121, 161 CrossRef Search ADS   San Roman I. et al.  , 2015, A&A , 579, A6 CrossRef Search ADS   Sandage A., Wildey R., 1967, ApJ , 150, 469 CrossRef Search ADS   Santos N. C. et al.  , 2013, A&A , 556, A150 CrossRef Search ADS   Sarajedini A. et al.  , 2007, AJ , 133, 1658 CrossRef Search ADS   Schlafly E. F. et al.  , 2014, ApJ , 789, 15 CrossRef Search ADS   Schultheis M. et al.  , 2015, A&A , 577, A77 CrossRef Search ADS   Smiljanic R. et al.  , 2016, A&A , 589, A115 CrossRef Search ADS   Sneden C., 2004, Mem. Soc. Astron. Italiana , 75, 267 Song H. F., Meynet G., Maeder A., Ekström S., Eggenberger P., 2016, A&A , 585, A120 CrossRef Search ADS   Spada F., Demarque P., Kim Y. C., Sills A., 2013, ApJ , 776, 87 CrossRef Search ADS   Spera M., Mapelli M., Bressan A., 2015, MNRAS , 451, 4086 CrossRef Search ADS   Strandberg E. et al.  , 2008, Phys. Rev. C , 77, 055801 CrossRef Search ADS   Tang J., Bressan A., Rosenfield P., Slemer A., Marigo P., Girardi L., Bianchi L., 2014, MNRAS , 445, 4287 CrossRef Search ADS   Thompson I. B., Kaluzny J., Rucinski S. M., Krzeminski W., Pych W., Dotter A., Burley G. S., 2010, AJ , 139, 329 CrossRef Search ADS   Thygesen A. O. et al.  , 2014, A&A , 572, A108 CrossRef Search ADS   Troisi F. et al.  , 2011, Publ. Astron. Soc. Pac. , 123, 879 CrossRef Search ADS   Tuli J. K., 2012, Weak Rates from the Nuclear Wallet Cards . National Nuclear Data Center, Brookhaven Valcarce A. A. R., Catelan M., Sweigart A. V., 2012, A&A , 547, A5 CrossRef Search ADS   van den Bergh S., 1967, AJ , 72, 70 CrossRef Search ADS   VandenBerg D. A., Bergbusch P. A., Dotter A., Ferguson J. W., Michaud G., Richer J., Proffitt C. R., 2012, ApJ , 755, 15 CrossRef Search ADS   VandenBerg D. A., Brogaard K., Leaman R., Casagrande L., 2013, ApJ , 775, 134 CrossRef Search ADS   VandenBerg D. A., Bergbusch P. A., Ferguson J. W., Edvardsson B., 2014, ApJ , 794, 72 CrossRef Search ADS   VandenBerg D. A., 2013, in de Grijs R., ed., IAU Symp., Vol. 289, Advancing the Physics of Cosmic Distances . Kluwer, Dordrecht, p. 161 VandenBerg D. A., Swenson F. J., Rogers F. J., Iglesias C. A., Alexander D. R., 2000, ApJ , 532, 430 CrossRef Search ADS   Venn K. A., Irwin M., Shetrone M. D., Tout C. A., Hill V., Tolstoy E., 2004, AJ , 128, 1177 CrossRef Search ADS   Villanova S. et al.  , 2007, ApJ , 663, 296 CrossRef Search ADS   Villanova S., Piotto G., Gratton R. G., 2009, A&A , 499, 755 CrossRef Search ADS   Watkins L. L., van der Marel R. P., Bellini A., Anderson J., 2015, ApJ , 812, 149 CrossRef Search ADS   Weldrake D. T. F., Sackett P. D., Bridges T. J., Freeman K. C., 2004, AJ , 128, 736 CrossRef Search ADS   Zahn J.-P., 1977, A&A , 57, 383 Zhao G., Magain P., 1990, A&A , 238, 242 Zoccali M., Cassisi S., Piotto G., Bono G., Salaris M., 1999, ApJ , 518, L49 CrossRef Search ADS   © 2018 The Author(s) Published by Oxford University Press on behalf of the Royal Astronomical Society http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Monthly Notices of the Royal Astronomical Society Oxford University Press

# New parsec data base of α-enhanced stellar evolutionary tracks and isochrones – I. Calibration with 47 Tuc (NGC 104) and the improvement on RGB bump

16 pages

/lp/ou_press/new-parsec-data-base-of-enhanced-stellar-evolutionary-tracks-and-gycjBFrvov
Publisher
Oxford University Press
ISSN
0035-8711
eISSN
1365-2966
D.O.I.
10.1093/mnras/sty235
Publisher site
See Article on Publisher Site

### Abstract

Abstract Precise studies on the Galactic bulge, globular cluster, Galactic halo, and Galactic thick disc require stellar models with α enhancement and various values of helium content. These models are also important for extra-Galactic population synthesis studies. For this purpose, we complement the existing parsec models, which are based on the solar partition of heavy elements, with α-enhanced partitions. We collect detailed measurements on the metal mixture and helium abundance for the two populations of 47 Tuc (NGC 104) from the literature, and calculate stellar tracks and isochrones with these α-enhanced compositions. By fitting the precise colour–magnitude diagram with HST ACS/WFC data, from low main sequence till horizontal branch (HB), we calibrate some free parameters that are important for the evolution of low mass stars like the mixing at the bottom of the convective envelope. This new calibration significantly improves the prediction of the red giant branch bump (RGBB) brightness. Comparison with the observed RGB and HB luminosity functions also shows that the evolutionary lifetimes are correctly predicted. As a further result of this calibration process, we derive the age, distance modulus, reddening, and the RGB mass-loss for 47 Tuc. We apply the new calibration and α-enhanced mixtures of the two 47 Tuc populations ([α/Fe] ∼ 0.4 and 0.2) to other metallicities. The new models reproduce the RGB bump observations much better than previous models. This new parsec data base, with the newly updated α-enhanced stellar evolutionary tracks and isochrones, will also be a part of the new stellar products for Gaia. stars: evolution, Hertzsprung-Russell and colour-magnitude diagrams, stars: interiors, stars: low-mass 1 INTRODUCTION parsec  (PAdova-TRieste Stellar Evolution Code) is widely used in the astronomical community. It provides input for population synthesis models to study resolved and unresolved star clusters and galaxies (e.g. Perren, Vázquez & Piatti 2015; Chevallard & Charlot 2016; Gutkin, Charlot & Bruzual 2016), and offers reliable models for many other field of studies, such as to derive black hole mass when observing gravitational waves (e.g. Spera, Mapelli & Bressan 2015; Belczynski et al. 2016), to get host star parameters for exoplanets (Santos et al. 2013; Maldonado et al. 2015, etc.), to explore the mysterious ‘cosmological lithium problem’ (Fu et al. 2015), to derive the main parameters of star clusters (for instance, Borissova et al. 2014; Donati et al. 2014; Goudfrooij et al. 2015; San Roman et al. 2015) and Galactic structure (e.g. Küpper et al. 2015; Balbinot et al. 2016; Li et al. 2016; Ramya et al. 2016), to study dust formation (e.g. Nanni et al. 2013; Nanni et al. 2014), to constrain dust extinction (e.g. Schlafly et al. 2014; Schultheis et al. 2015; Bovy et al. 2016), and to understand the stars themselves (e.g. Kalari et al. 2014; Casey et al. 2016; Gullikson, Kraus & Dodson-Robinson 2016; Reddy & Lambert 2016; Smiljanic et al. 2016), etc. There are now four versions of parsec isochrones available online.1 The very first version parsec v1.0 (Bressan et al. 2012) provides isochrones for 0.0005 ≤ Z ≤ 0.07 (−1.5 ≤ [M/H] ≤ +0.6) with the mass range 0.1 ≤ M < 12  M⊙ from pre-main-sequence (pre-MS) to the thermally pulsing asymptotic giant branch (TP-AGB). In parsec v1.1 (based on Bressan et al. 2012), we expanded the metallicity range down to Z = 0.0001 ([M/H] = −2.2). parsec v1.2S included big improvements both on the very low mass stars and massive stars: Chen et al. (2014) improve the surface boundary conditions for stars with mass M ≲ 0.5  M⊙ in order to fit the mass–radius relation of dwarf stars; Tang et al. (2014) introduce mass-loss for massive star M ≥ 14  M⊙; Chen et al. (2015) improve the mass-loss rate when the luminosity approaches the Eddington luminosity and supplement the model with new bolometric corrections till M = 350 M⊙. In a later version (parsec v1.2S + COLIBRI PR16), we describe improved isochrones with the addition of COLIBRI (Marigo et al. 2013) evolutionary tracks of TP-AGB stars (Rosenfield et al. 2016; Marigo et al. 2017). All previous versions of parsec models are calculated assuming solar-scaled metal mixtures, in which the initial partition of heavy elements keeps always the same relative number density as that in the Sun. It is now well established that the solar-scaled metal mixture is not universally applicable for all types of stars. In fact, one of the most important group of elements, the so called α-elements group, is not always observed in solar proportions. Many studies have confirmed the existence of an ‘enhancement’ of α-elements in the Milky Way halo (e.g. Zhao & Magain 1990; Nissen et al. 1994; McWilliam et al. 1995; Venn et al. 2004), globular clusters (GCs; e.g. Carney 1996; Sneden 2004; Pritzl, Venn & Irwin 2005), the Galactic Bulge (Gonzalez et al. 2011; Johnson et al. 2014), and thick disc (e.g. Fulbright 2002; Reddy, Lambert & Prieto 2006; Ruchti et al. 2010). Stars in the dwarf spheroidal Milky Way satellite galaxies show different α-abundance trends compared to the Galactic halo stars, possibly indicating different star formation paths (Kirby et al. 2011). The α-elements (O, Ne, Mg, Si, S, Ar, Ca, and Ti) are mainly produced by core collapse (mostly Type II) supernovae (SNe) on short time-scale, whereas the iron-peak elements (V, Cr, Mn, Fe, Co and Ni) are mainly synthesized in Type Ia SNe on longer time-scales. Therefore, the evolution profile of [α/Fe] records the imprint of the star formation history of the system. An alternative explanation could be that the initial mass function (IMF) of the α-enhanced stellar populations was much richer in massive stars than the one from which our Sun was born (Chiosi et al. 1998). However, there is no clear evidence in support of this alternative possibility. In order to model star clusters, galaxies and Galactic components more precisely, the previous Padova isochrone data base offered a few sets of α-enhanced models for four relatively high metallicities (Salasnich et al. 2000), other stellar evolution groups also published isochrones that allow for α enhancement (e.g. VandenBerg et al. 2000, 2014; Pietrinferni et al. 2006; Valcarce, Catelan & Sweigart 2012). Now, with the thorough revision and update input physics, we introduce α-enhanced metal mixtures in parsec . In this paper, we first calibrate the new parsec α-enhanced stellar evolutionary tracks and isochrones with the well-studied GC 47 Tucanae (NGC 104), then we apply the calibrated parameters to obtain models for other metallicities. Section 2 briefly describes the input physics. Section 3 introduces the comparison with 47 Tuc data in details, including the isochrone fitting and luminosity function (LF), envelope overshooting calibration with red giant branch bump (RGBB), and mass-loss in the RGB from horizontal branch (HB) morphology. Section 4 compares the new parsec models with other stellar models and shows its improvement on RGB bump prediction. A summary of this paper and the discussion are in Section 5. 2 INPUT PHYSICS The main difference with respect to the previous versions of parsec is the adoption of new nuclear reaction rates, α-enhanced opacities, and various helium contents. We update the nuclear reaction rates from JINA REACLIB data base (Cyburt et al. 2010) with their 2015 April 6 new recommendation. In addition, more reactions – 52 instead of the 47 described in Bressan et al. (2012) for the previous versions of parsec – are taken into account. They are all listed in Table 1 together with the reference from which we take the reaction energy Q value. In the updated reactions, more isotopic abundances are considered, in total Nel = 35: 1H, D, 3He, 4He,7Li, 7Be, 8B, 12C, 13C, 13N, 14N, 15N, 15O, 16O, 17O, 18O, 17F, 18F, 19F, 20Ne, 21Ne, 22Ne, 21Na, 22Na, 23Na, 23Mg, 24Mg, 25Mg, 26Mg, 25Al, 26Alm, 26Alg, 27Al, 27Si, and 28Si. Table 1. Nuclear reaction rates adopted in this work and the reference from which we take their reaction energy Q. Reaction  Reference  $$\rm \,{}^{}\hspace{-0.8pt}{p}\,({p}\,,{\beta ^+\,\nu }) \,{}^{}\hspace{-0.8pt}{D}\,$$  Betts, Fortune & Middleton (1975)  $$\rm \,{}^{}\hspace{-0.8pt}{p}\,({D}\,,{\gamma }) \,{}^{3}\hspace{-0.8pt}{He}\,$$  Descouvemont et al. (2004)  $$\rm \,{}^{3}\hspace{-0.8pt}{He}\,({^{3}He}\,,{\gamma }) \,{}^{}\hspace{-0.8pt}{2\,p + ^{4}\hspace{-0.8pt}{He}}\,$$  Angulo et al. (1999)  $$\rm \,{}^{4}\hspace{-0.8pt}{He}\,({^{3}He}\,,{\gamma }) \,{}^{7}\hspace{-0.8pt}{Be}\,$$  Cyburt & Davids (2008)  $$\rm \,{}^{7}\hspace{-0.8pt}{Be}\,({e^-}\,,{\gamma }) \,{}^{7}\hspace{-0.8pt}{Li}\,$$  Cyburt et al. (2010)  $$\rm \,{}^{7}\hspace{-0.8pt}{Li}\,({p}\,,{\gamma }) \,{}^{}\hspace{-0.8pt}{^{4}\hspace{-2.0pt}{He} + ^{4}\hspace{-2.0pt}{He}}\,$$  Descouvemont et al. (2004)  $$\rm \,{}^{7}\hspace{-0.8pt}{Be}\,({p}\,,{\gamma }) \,{}^{8}\hspace{-0.8pt}{B}\,$$  Angulo et al. (1999)  $$\rm \,{}^{12}\hspace{-0.8pt}{C}\,({p}\,,{\gamma }) \,{}^{13}\hspace{-0.8pt}{N}\,$$  Li et al. (2010)  $$\rm \,{}^{13}\hspace{-0.8pt}{C}\,({p}\,,{\gamma }) \,{}^{14}\hspace{-0.8pt}{N}\,$$  Angulo et al. (1999)  $$\rm \,{}^{14}\hspace{-0.8pt}{N}\,({p}\,,{\gamma }) \,{}^{15}\hspace{-0.8pt}{O}\,$$  Imbriani et al. (2005)  $$\rm \,{}^{15}\hspace{-0.8pt}{N}\,({p}\,,{\gamma }) \,{}^{}\hspace{-0.8pt}{^4He + ^{12}\hspace{-2.0pt}{C}}\,$$  Angulo et al. (1999)  $$\rm \,{}^{15}\hspace{-0.8pt}{N}\,({p}\,,{\gamma }) \,{}^{16}\hspace{-0.8pt}{O}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{16}\hspace{-0.8pt}{O}\,({p}\,,{\gamma }) \,{}^{17}\hspace{-0.8pt}{F}\,$$  Iliadis et al. (2008)  $$\rm \,{}^{17}\hspace{-0.8pt}{O}\,({p}\,,{\gamma }) \,{}^{}\hspace{-0.8pt}{\,^4He + ^{14}\hspace{-2.0pt}{N}}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{17}\hspace{-0.8pt}{O}\,({p}\,,{\gamma }) \,{}^{18}\hspace{-0.8pt}{F}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{18}\hspace{-0.8pt}{O}\,({p}\,,{\gamma }) \,{}^{}\hspace{-0.8pt}{\,^4He + ^{15}\hspace{-2.0pt}{N}}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{18}\hspace{-0.8pt}{O}\,({p}\,,{\gamma }) \,{}^{19}\hspace{-0.8pt}{F}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{19}\hspace{-0.8pt}{F}\,({p}\,,{\gamma }) \,{}^{}\hspace{-0.8pt}{\,^4He + ^{16}\hspace{-2.0pt}{O}}\,$$  Angulo et al. (1999)  $$\rm \,{}^{19}\hspace{-0.8pt}{F}\,({p}\,,{\gamma }) \,{}^{20}\hspace{-0.8pt}{Ne}\,$$  Angulo et al. (1999)  $$\rm \,{}^{4}\hspace{-0.8pt}{He}\,({2\,^{4}He}\,,{\gamma }) \,{}^{12}\hspace{-0.8pt}{C}\,$$  Fynbo et al. (2005)  $$\rm \,{}^{12}\hspace{-0.8pt}{C}\,({^{4}He}\,,{\gamma }) \,{}^{16}\hspace{-0.8pt}{O}\,$$  Cyburt, Hoffman & Woosley (2012)  $$\rm \,{}^{14}\hspace{-0.8pt}{N}\,({^{4}He}\,,{\gamma }) \,{}^{18}\hspace{-0.8pt}{F}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{15}\hspace{-0.8pt}{N}\,({^{4}He}\,,{\gamma }) \,{}^{19}\hspace{-0.8pt}{F}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{16}\hspace{-0.8pt}{O}\,({^{4}He}\,,{\gamma }) \,{}^{20}\hspace{-0.8pt}{Ne}\,$$  Constantini & LUNA Collaboration (2010)  $$\rm \,{}^{18}\hspace{-0.8pt}{O}\,({^{4}He}\,,{\gamma }) \,{}^{22}\hspace{-0.8pt}{Ne}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{20}\hspace{-0.8pt}{Ne}\,({^{4}He}\,,{\gamma }) \,{}^{24}\hspace{-0.8pt}{Mg}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{22}\hspace{-0.8pt}{Ne}\,({^{4}He}\,,{\gamma }) \,{}^{26}\hspace{-0.8pt}{Mg}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{24}\hspace{-0.8pt}{Mg}\,({^{4}He}\,,{\gamma }) \,{}^{28}\hspace{-0.8pt}{Si}\,$$  Strandberg et al. (2008)  $$\rm \,{}^{13}\hspace{-0.8pt}{C}\,({^{4}He}\,,{n}) \,{}^{16}\hspace{-0.8pt}{O}\,$$  Heil et al. (2008)  $$\rm \,{}^{17}\hspace{-0.8pt}{O}\,({^{4}He}\,,{n}) \,{}^{20}\hspace{-0.8pt}{Ne}\,$$  Angulo et al. (1999)  $$\rm \,{}^{18}\hspace{-0.8pt}{O}\,({^{4}He}\,,{n}) \,{}^{21}\hspace{-0.8pt}{Ne}\,$$  Angulo et al. (1999)  $$\rm \,{}^{21}\hspace{-0.8pt}{Ne}\,({^{4}He}\,,{n}) \,{}^{24}\hspace{-0.8pt}{Mg}\,$$  Angulo et al. (1999)  $$\rm \,{}^{22}\hspace{-0.8pt}{Ne}\,({^{4}He}\,,{n}) \,{}^{25}\hspace{-0.8pt}{Mg}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{25}\hspace{-0.8pt}{Mg}\,({^{4}He}\,,{n}) \,{}^{28}\hspace{-0.8pt}{Si}\,$$  Angulo et al. (1999)  $$\rm \,{}^{20}\hspace{-0.8pt}{Ne}\,({p}\,,{\gamma }) \,{}^{21}\hspace{-0.8pt}{Na}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{21}\hspace{-0.8pt}{Ne}\,({p}\,,{\gamma }) \,{}^{22}\hspace{-0.8pt}{Na}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{22}\hspace{-0.8pt}{Ne}\,({p}\,,{\gamma }) \,{}^{23}\hspace{-0.8pt}{Na}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{23}\hspace{-0.8pt}{Na}\,({p}\,,{\gamma }) \,{}^{}\hspace{-0.8pt}{\,^4He + ^{20}\hspace{-2.0pt}{Ne}}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{23}\hspace{-0.8pt}{Na}\,({p}\,,{\gamma }) \,{}^{24}\hspace{-0.8pt}{Mg}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{24}\hspace{-0.8pt}{Mg}\,({p}\,,{\gamma }) \,{}^{25}\hspace{-0.8pt}{Al}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{25}\hspace{-0.8pt}{Mg}\,({p}\,,{\gamma }) \,{}^{26}\hspace{-0.8pt}{Al^g}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{25}\hspace{-0.8pt}{Mg}\,({p}\,,{\gamma }) \,{}^{26}\hspace{-0.8pt}{Al^m}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{26}\hspace{-0.8pt}{Mg}\,({p}\,,{\gamma }) \,{}^{27}\hspace{-0.8pt}{Al}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{26}\hspace{-0.8pt}{Al^g}\,({p}\,,{\gamma }) \,{}^{27}\hspace{-0.8pt}{Si}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{27}\hspace{-0.8pt}{Al}\,({p}\,,{\gamma }) \,{}^{}\hspace{-0.8pt}{\,^4He + ^{24}\hspace{-2.0pt}{Mg}}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{27}\hspace{-0.8pt}{Al}\,({p}\,,{\gamma }) \,{}^{28}\hspace{-0.8pt}{Si}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{26}\hspace{-0.8pt}{Al}\,({p}\,,{\gamma }) \,{}^{27}\hspace{-0.8pt}{Si}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{26}\hspace{-0.8pt}{Al}\,({n}\,,{p}) \,{}^{26}\hspace{-0.8pt}{Mg}\,$$  Tuli (2012)  $$\rm \,{}^{12}\hspace{-0.8pt}{C}\,({^{12}C}\,,{n}) \,{}^{23}\hspace{-0.8pt}{Mg}\,$$  Caughlan & Fowler (1988)  $$\rm \,{}^{12}\hspace{-0.8pt}{C}\,({^{12}C}\,,{p}) \,{}^{23}\hspace{-0.8pt}{Na}\,$$  Caughlan & Fowler (1988)  $$\rm \,{}^{12}\hspace{-0.8pt}{C}\,({^{12}C}\,,{^{4}He}) \,{}^{20}\hspace{-0.8pt}{Ne}\,$$  Caughlan & Fowler (1988)  $$\rm \,{}^{20}\hspace{-0.8pt}{Ne}\,({\gamma }\,,{^{4}He}) \,{}^{16}\hspace{-0.8pt}{O}\,$$  Constantini & LUNA Collaboration (2010)  Reaction  Reference  $$\rm \,{}^{}\hspace{-0.8pt}{p}\,({p}\,,{\beta ^+\,\nu }) \,{}^{}\hspace{-0.8pt}{D}\,$$  Betts, Fortune & Middleton (1975)  $$\rm \,{}^{}\hspace{-0.8pt}{p}\,({D}\,,{\gamma }) \,{}^{3}\hspace{-0.8pt}{He}\,$$  Descouvemont et al. (2004)  $$\rm \,{}^{3}\hspace{-0.8pt}{He}\,({^{3}He}\,,{\gamma }) \,{}^{}\hspace{-0.8pt}{2\,p + ^{4}\hspace{-0.8pt}{He}}\,$$  Angulo et al. (1999)  $$\rm \,{}^{4}\hspace{-0.8pt}{He}\,({^{3}He}\,,{\gamma }) \,{}^{7}\hspace{-0.8pt}{Be}\,$$  Cyburt & Davids (2008)  $$\rm \,{}^{7}\hspace{-0.8pt}{Be}\,({e^-}\,,{\gamma }) \,{}^{7}\hspace{-0.8pt}{Li}\,$$  Cyburt et al. (2010)  $$\rm \,{}^{7}\hspace{-0.8pt}{Li}\,({p}\,,{\gamma }) \,{}^{}\hspace{-0.8pt}{^{4}\hspace{-2.0pt}{He} + ^{4}\hspace{-2.0pt}{He}}\,$$  Descouvemont et al. (2004)  $$\rm \,{}^{7}\hspace{-0.8pt}{Be}\,({p}\,,{\gamma }) \,{}^{8}\hspace{-0.8pt}{B}\,$$  Angulo et al. (1999)  $$\rm \,{}^{12}\hspace{-0.8pt}{C}\,({p}\,,{\gamma }) \,{}^{13}\hspace{-0.8pt}{N}\,$$  Li et al. (2010)  $$\rm \,{}^{13}\hspace{-0.8pt}{C}\,({p}\,,{\gamma }) \,{}^{14}\hspace{-0.8pt}{N}\,$$  Angulo et al. (1999)  $$\rm \,{}^{14}\hspace{-0.8pt}{N}\,({p}\,,{\gamma }) \,{}^{15}\hspace{-0.8pt}{O}\,$$  Imbriani et al. (2005)  $$\rm \,{}^{15}\hspace{-0.8pt}{N}\,({p}\,,{\gamma }) \,{}^{}\hspace{-0.8pt}{^4He + ^{12}\hspace{-2.0pt}{C}}\,$$  Angulo et al. (1999)  $$\rm \,{}^{15}\hspace{-0.8pt}{N}\,({p}\,,{\gamma }) \,{}^{16}\hspace{-0.8pt}{O}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{16}\hspace{-0.8pt}{O}\,({p}\,,{\gamma }) \,{}^{17}\hspace{-0.8pt}{F}\,$$  Iliadis et al. (2008)  $$\rm \,{}^{17}\hspace{-0.8pt}{O}\,({p}\,,{\gamma }) \,{}^{}\hspace{-0.8pt}{\,^4He + ^{14}\hspace{-2.0pt}{N}}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{17}\hspace{-0.8pt}{O}\,({p}\,,{\gamma }) \,{}^{18}\hspace{-0.8pt}{F}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{18}\hspace{-0.8pt}{O}\,({p}\,,{\gamma }) \,{}^{}\hspace{-0.8pt}{\,^4He + ^{15}\hspace{-2.0pt}{N}}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{18}\hspace{-0.8pt}{O}\,({p}\,,{\gamma }) \,{}^{19}\hspace{-0.8pt}{F}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{19}\hspace{-0.8pt}{F}\,({p}\,,{\gamma }) \,{}^{}\hspace{-0.8pt}{\,^4He + ^{16}\hspace{-2.0pt}{O}}\,$$  Angulo et al. (1999)  $$\rm \,{}^{19}\hspace{-0.8pt}{F}\,({p}\,,{\gamma }) \,{}^{20}\hspace{-0.8pt}{Ne}\,$$  Angulo et al. (1999)  $$\rm \,{}^{4}\hspace{-0.8pt}{He}\,({2\,^{4}He}\,,{\gamma }) \,{}^{12}\hspace{-0.8pt}{C}\,$$  Fynbo et al. (2005)  $$\rm \,{}^{12}\hspace{-0.8pt}{C}\,({^{4}He}\,,{\gamma }) \,{}^{16}\hspace{-0.8pt}{O}\,$$  Cyburt, Hoffman & Woosley (2012)  $$\rm \,{}^{14}\hspace{-0.8pt}{N}\,({^{4}He}\,,{\gamma }) \,{}^{18}\hspace{-0.8pt}{F}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{15}\hspace{-0.8pt}{N}\,({^{4}He}\,,{\gamma }) \,{}^{19}\hspace{-0.8pt}{F}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{16}\hspace{-0.8pt}{O}\,({^{4}He}\,,{\gamma }) \,{}^{20}\hspace{-0.8pt}{Ne}\,$$  Constantini & LUNA Collaboration (2010)  $$\rm \,{}^{18}\hspace{-0.8pt}{O}\,({^{4}He}\,,{\gamma }) \,{}^{22}\hspace{-0.8pt}{Ne}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{20}\hspace{-0.8pt}{Ne}\,({^{4}He}\,,{\gamma }) \,{}^{24}\hspace{-0.8pt}{Mg}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{22}\hspace{-0.8pt}{Ne}\,({^{4}He}\,,{\gamma }) \,{}^{26}\hspace{-0.8pt}{Mg}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{24}\hspace{-0.8pt}{Mg}\,({^{4}He}\,,{\gamma }) \,{}^{28}\hspace{-0.8pt}{Si}\,$$  Strandberg et al. (2008)  $$\rm \,{}^{13}\hspace{-0.8pt}{C}\,({^{4}He}\,,{n}) \,{}^{16}\hspace{-0.8pt}{O}\,$$  Heil et al. (2008)  $$\rm \,{}^{17}\hspace{-0.8pt}{O}\,({^{4}He}\,,{n}) \,{}^{20}\hspace{-0.8pt}{Ne}\,$$  Angulo et al. (1999)  $$\rm \,{}^{18}\hspace{-0.8pt}{O}\,({^{4}He}\,,{n}) \,{}^{21}\hspace{-0.8pt}{Ne}\,$$  Angulo et al. (1999)  $$\rm \,{}^{21}\hspace{-0.8pt}{Ne}\,({^{4}He}\,,{n}) \,{}^{24}\hspace{-0.8pt}{Mg}\,$$  Angulo et al. (1999)  $$\rm \,{}^{22}\hspace{-0.8pt}{Ne}\,({^{4}He}\,,{n}) \,{}^{25}\hspace{-0.8pt}{Mg}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{25}\hspace{-0.8pt}{Mg}\,({^{4}He}\,,{n}) \,{}^{28}\hspace{-0.8pt}{Si}\,$$  Angulo et al. (1999)  $$\rm \,{}^{20}\hspace{-0.8pt}{Ne}\,({p}\,,{\gamma }) \,{}^{21}\hspace{-0.8pt}{Na}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{21}\hspace{-0.8pt}{Ne}\,({p}\,,{\gamma }) \,{}^{22}\hspace{-0.8pt}{Na}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{22}\hspace{-0.8pt}{Ne}\,({p}\,,{\gamma }) \,{}^{23}\hspace{-0.8pt}{Na}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{23}\hspace{-0.8pt}{Na}\,({p}\,,{\gamma }) \,{}^{}\hspace{-0.8pt}{\,^4He + ^{20}\hspace{-2.0pt}{Ne}}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{23}\hspace{-0.8pt}{Na}\,({p}\,,{\gamma }) \,{}^{24}\hspace{-0.8pt}{Mg}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{24}\hspace{-0.8pt}{Mg}\,({p}\,,{\gamma }) \,{}^{25}\hspace{-0.8pt}{Al}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{25}\hspace{-0.8pt}{Mg}\,({p}\,,{\gamma }) \,{}^{26}\hspace{-0.8pt}{Al^g}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{25}\hspace{-0.8pt}{Mg}\,({p}\,,{\gamma }) \,{}^{26}\hspace{-0.8pt}{Al^m}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{26}\hspace{-0.8pt}{Mg}\,({p}\,,{\gamma }) \,{}^{27}\hspace{-0.8pt}{Al}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{26}\hspace{-0.8pt}{Al^g}\,({p}\,,{\gamma }) \,{}^{27}\hspace{-0.8pt}{Si}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{27}\hspace{-0.8pt}{Al}\,({p}\,,{\gamma }) \,{}^{}\hspace{-0.8pt}{\,^4He + ^{24}\hspace{-2.0pt}{Mg}}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{27}\hspace{-0.8pt}{Al}\,({p}\,,{\gamma }) \,{}^{28}\hspace{-0.8pt}{Si}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{26}\hspace{-0.8pt}{Al}\,({p}\,,{\gamma }) \,{}^{27}\hspace{-0.8pt}{Si}\,$$  Iliadis et al. (2010)  $$\rm \,{}^{26}\hspace{-0.8pt}{Al}\,({n}\,,{p}) \,{}^{26}\hspace{-0.8pt}{Mg}\,$$  Tuli (2012)  $$\rm \,{}^{12}\hspace{-0.8pt}{C}\,({^{12}C}\,,{n}) \,{}^{23}\hspace{-0.8pt}{Mg}\,$$  Caughlan & Fowler (1988)  $$\rm \,{}^{12}\hspace{-0.8pt}{C}\,({^{12}C}\,,{p}) \,{}^{23}\hspace{-0.8pt}{Na}\,$$  Caughlan & Fowler (1988)  $$\rm \,{}^{12}\hspace{-0.8pt}{C}\,({^{12}C}\,,{^{4}He}) \,{}^{20}\hspace{-0.8pt}{Ne}\,$$  Caughlan & Fowler (1988)  $$\rm \,{}^{20}\hspace{-0.8pt}{Ne}\,({\gamma }\,,{^{4}He}) \,{}^{16}\hspace{-0.8pt}{O}\,$$  Constantini & LUNA Collaboration (2010)  View Large The α-enhanced opacities and equation of state (EOS) are derived for our best estimate of the metal mixture of 47 Tuc, which is described in Section 3.1. Details about the preparation of the opacity tables are provided in Bressan et al. (2012). Suffice it to recall that the Rosseland mean opacities come from two sources: from the Opacity Project At Livermore (OPAL, Iglesias & Rogers 1996, and references therein)2 team at high temperatures (4.0 < log (T/K) < 8.7), and from AESOPUS (Marigo & Aringer 2009)3 at low temperatures (3.2 < log (T/K) < 4.1), with a smooth transition being adopted in the 4.0 < log (T/K) < 4.1 interval. Conductive opacities are provided by Itoh et al. (2008) routines. As for the EOS, we choose the widely used FreeEOS code (version 2.2.1 in the EOS4 configuration)4 developed by Alan W. Irwin for its computational efficiency. It is worth noting here that when we change the heavy element number fractions (Ni/NZ) to obtain a new metal partition in parsec, their fractional abundances by mass (Zi/Ztot) are re-normalized in such a way that the global metallicity, Z, is kept constant. Hence, compared to the solar partition at the same total metallicity Z, a model with enhanced α-elements shows a depression of Fe and the related elements, because the total metallicity remains unchanged by construction. The Hertzsprung–Russell diagram (HRD) in the left-hand panel of Fig. 1 shows that, with the same total metallicity Z and helium content Y, the α-enhanced star (orange solid line) is slightly hotter than the solar-scaled one (blue dashed line) both on the MS and on the RGB because of the net effect of changes to the opacity. Higher temperature leads to a faster evolution, as illustrated in the right-hand panel of Fig. 1. It is also interesting to compare the α-enhanced star to a solar-scaled one with the same [Fe/H] (black dotted line in Fig. 1). With the same [Fe/H] but higher total metallicity Z, the α-enhanced star is cooler. Indeed, VandenBerg et al. (2012) report that if keeping [Fe/H] constant, the giant branch is shifted to a cooler temperature with increased Mg or Si, whereas O, Ne, S abundances mainly affect the temperatures of MS and turn-off phases. Figure 1. View largeDownload slide A comparison between the α-enhanced evolutionary track (orange solid line) and the solar-scaled one with the same metallicity Z (blue dashed line). For comparison, a solar-scaled evolutionary track with the same [Fe/H] value (black dotted line) is also displayed. The helium content and the stellar mass of the three stars are the same (Y = 0.276, M = 0.85 M⊙). The left-hand panel is HRD with sub-figure zoom-in around the RGBB region. The right-hand panel shows how the luminosity of the star evolve with time. Figure 1. View largeDownload slide A comparison between the α-enhanced evolutionary track (orange solid line) and the solar-scaled one with the same metallicity Z (blue dashed line). For comparison, a solar-scaled evolutionary track with the same [Fe/H] value (black dotted line) is also displayed. The helium content and the stellar mass of the three stars are the same (Y = 0.276, M = 0.85 M⊙). The left-hand panel is HRD with sub-figure zoom-in around the RGBB region. The right-hand panel shows how the luminosity of the star evolve with time. Various initial helium abundance values, for a given metallicity, are allowed in the new version of parsec. In the previous versions, the initial helium mass fraction of the stars was obtained from the helium-to-metals enrichment law:   $$Y = Y_{\rm p} + \frac{\Delta Y}{\Delta Z}Z = 0.2485 + 1.78 \times Z$$ (1)where Yp is the primordial helium abundance (Komatsu et al. 2011), and ΔY/ΔZ is the helium-to-metal enrichment ratio. Because of differences in the adopted primordial and solar calibration He and metallicity values by different authors, the above two parameters are slightly different in different stellar evolution codes. The latest YY isochrone (Spada et al. 2013) adopts the relation Y = 0.25 + 1.48Z; dsep (Dotter 2007; Dotter et al. 2007, 2008) uses Y = 0.245 + 1.54Z; mist (Choi et al. 2016) gives Y = 0.249 + 1.5Z; and BaSTI (Pietrinferni et al. 2006) adopts Y = 0.245 + 1.4Z. However, observations reveal that the helium content does not always follow a single relation. Differences in helium abundance have been widely confirmed in GCs between stellar populations with very similar metallicity. The evidence includes the direct He i measurement on the blue HB star (Villanova, Piotto & Gratton 2009; Marino et al. 2014; Mucciarelli et al. 2014; Gratton et al. 2015, for instance), on giant stars (Dupree, Strader & Smith 2011; Pasquini et al. 2011), and the splitting of sequences in the colour–magnitude diagram (CMD) of both GCs in the Milky Way (e.g. Bedin et al. 2004; Piotto et al. 2007; Villanova et al. 2007; Milone et al. 2008; Di Criscienzo et al. 2010) and of clusters in the Magellanic Clouds (Milone et al. 2015a, 2016). Bragaglia et al. (2010) found that the brightness of the RGB bump, which should increase with He abundance, is fainter in the first generation than the second generation in 14 GCs. Indeed, He variation is considered one of the key parameters (and problems) to understand multiple populations in GCs (see the review by Gratton, Carretta & Bragaglia 2012, and the references therein). In the new version of parsec, we allow different helium contents at any given metallicity Z. 3 CALIBRATION WITH 47 TUC GCs have been traditionally considered as the paradigm of a single stellar population, a coeval and chemically homogeneous population of stars covering a broad range of evolutionary phases, from the low-mass MS to the HB and white dwarf sequences. For this reason, they were considered the ideal laboratory to observationally study the evolution of low-mass stars and to check and calibrate the stellar evolution theory. This picture has been challenged during the last two decades by photometric and spectroscopic evidence of the presence of multiple populations in most, if not all, GCs [for instance NGC 6397 (Gratton et al. 2001; Milone et al. 2012), NGC 6752 (Gratton et al. 2001; Milone et al. 2010), NGC 1851 (Carretta et al. 2014), NGC 2808 (D'Antona et al. 2005; Carretta et al. 2006; Piotto et al. 2007; Milone et al. 2015b), NGC 6388 (Carretta et al. 2007), NGC 6139 (Bragaglia et al. 2015), M22 (Marino et al. 2011), etc.). Nevertheless, GCs remain one of the basic workbenches for the stellar model builders, besides their importance for dynamical studies and, given the discovery of multiple populations, also for the early chemo-dynamical evolution of stellar systems. 47 Tuc, a relatively metal-rich Galactic GC, also shows evidence of the presence of at least two different populations: (i) bimodality in the distribution of CN-weak and CN-strong targets, not only in red giant stars (Briley 1997; Norris & Freeman 1979; Harbeck, Smith & Grebel 2003) but also in MS members (Cannon et al. 1998); (ii) luminosity dispersion in the sub-giant branch, low MS and HB (Anderson et al. 2009; Di Criscienzo et al. 2010; Nataf et al. 2011; Salaris, Cassisi & Pietrinferni 2016), indicating a dispersion in He abundance; and (iii) anticorrelation of Na–O in RGB and HB stars (Carretta et al. 2009b, 2013; Gratton et al. 2013) and also in main-sequence turn-off (MSTO) ones (D'Orazi et al. 2010; Dobrovolskas et al. 2014). The presence of at least two different populations with different chemical compositions seems irrefutable (even if their origin is still under debate). Particularly convincing is the photometric study by Milone et al. (2012, and references therein), which concludes, in good agreement with other works (Carretta et al. 2009b, 2013), that for each evolutionary phase, from MS to HB, the stellar content of 47 Tuc belongs to two different populations, ‘first generation’ and ‘second generation’ ones (thereafter, FG and SG, respectively). The FG population represents ∼30 per cent of the stars, and it is more uniformly spatially distributed than the SG population, which is more concentrated in the central regions of the cluster. Choosing 47 Tuc as a reference to calibrate parsec stellar models requires therefore computation of stellar models with metal mixtures corresponding to the two identified populations. In the next section, we describe the sources to derive the two different metal mixtures that will be used for the opacity and EOS tables in the stellar model computations, and in the follow-up isochrone fitting. 3.1 Metal mixtures Chemical element abundances are given in the literature as the absolute values A(X),5 or as [X/Fe],6 the abundance with respect to the iron content and referred to the same quantity in the Sun. Since the solar metal mixture has changed lately and since there is still a hot debate about the chemical composition of the Sun, it is important to translate all the available data to absolute abundances, taking into account the solar mixture considered in each source. We follow that procedure to derive the metal mixtures for the first and second generations in 47 Tuc. The separation between the two populations based on photometric colours done by Milone et al. (2012) agrees with the separation based on Na–O anticorrelation by Carretta et al. (2009b) and Gratton et al. (2013). We decide hence to use the same criteria to classify the star as FG or SG member. Concerning He mass fraction Y, the scatter in luminosity seen in some evolutionary phases has been attributed to different amounts of He in the stellar plasma (see references above). The analyses presented in Milone et al. (2012) suggest that the best fitting of the colour difference between the two populations is obtained with a combination of different C, N, and O abundances, plus a small increase of He content in the SG (ΔY = 0.015 − 0.02). These results agree with those presented in Di Criscienzo et al. (2010, ΔY = 0.02 − −0.03), and rule out the possibility of explaining the 47 Tuc CMD only with the variation of He abundance. Table 2 lists the elemental abundances we adopt for the two generations of 47 Tuc together with the corresponding references. The abundances of some elements, like carbon, nitrogen, and oxygen, may change during the evolution because of standard (convection) and non-standard (i.e. rotational mixing) transport processes. Therefore, CNO abundances are compiled from available measurements for MS/TO stars, and their sum abundances are nearly the same in both populations. Other elements are not expected to be affected by mixing processes during stellar evolution, so we use the values measured mainly in the red giant phase where hundreds of stars are observed. If available, abundance determinations that take into account NLTE and 3D effects are adopted. There is no clear abundance difference of elements Mg, Al, Si, Ca, and Ti between the two populations of 47 Tuc. Since their abundances show large scatter from different literature sources and are sensitive to the choice of measured lines, we use the mean values of the literature abundances for both FG and SG. The iron abundance [Fe/H] = −0.76 dex is adopted from Carretta et al. (2009a) and Gratton et al. (2013), who measure the largest giant sample and HB sample of 47 Tuc, respectively, with internal fitting errors less than 0.02 dex. The [Fe/H] values derived from giants are much less dependent on the effects of microscopic diffusion than in the case of MS stars. We notice that in literature some authors suggest an [Fe/H] dispersion of ∼0.1 dex for 47 Tuc (Alves-Brito et al. 2005). On the other hand, Anderson et al. (2009) conclude that for 47 Tuc an He dispersion of ∼0.026 has an equal effect on the MS as an [Fe/H] dispersion of ∼0.1 dex. Since we consider different He contents for the FG and SG stars, we do not apply any further [Fe/H] dispersion. Other elements that are not displayed in Table 2 keep the solar abundances ([X/Fe] = 0). The anticorrelation between the abundances of C and N, as well as O and Na, contributes to the main difference of the metal mixtures between FG and SG. The difference in the final [α/Fe] values between the two generations is due to the difference in O abundance. The resulting metallicities are ZFG = 0.0056 for the first generation and ZSG = 0.0055 for the second generation. Following Milone et al. (2012), the assumed He abundances are Y = 0.256 and 0.276 for FG and SG, respectively. Table 3 lists the general metal-mixture information for the two stellar populations, including Z, Y, [M/H], [Fe/H], and [α/Fe] . The referred solar abundance is derived from Caffau et al. (2011), as described in Bressan et al. (2012). We consider eight α elements when calculating the total α enrichment [α/Fe] : O, Ne, Mg, Si, S, Ca, Ar, and Ti. Thus, for the FG stars of 47 Tuc [Z = 0.0056, Y = 0.256], [α/Fe] = 0.4057 dex, and, for the SG stars [Z = 0.0055, Y = 0.276], [α/Fe] = 0.2277 dex. These two values, approximately ∼ 0.4 and 0.2, respectively, are the typical [α/Fe] values observed in α-enriched stars. Finally, we note that we will adopt the metal partitions of these two α-enriched generations to calculate stellar evolutionary tracks and isochrones also for other metallicities. Table 2. Chemical element abundances of 47 Tuc two stellar populations (FG and SG). The abundances are written in the format of [X/Fe], their corresponding references are also listed.   FG  SG  Reference  Note  [C/Fe]  0.12  −0.09  Cannon et al. (1998); Milone et al. (2012)  MS  [N/Fe]  0.32  1.17  Cannon et al. (1998); Milone et al. (2012)  MS  [O/Fe]  0.42  0.17  Dobrovolskas et al. (2014)  TO, NLTE+3D  [Ne/Fe]  0.40  0.40  –  estimated  [Na/Fe]  −0.12  0.10  Dobrovolskas et al. (2014)  TO, NLTE+3D  [Mg/Fe]  0.32  0.32  Carretta et al. (2009b, 2013); Gratton et al. (2013); Cordero et al. (2014); Thygesen et al. (2014)  mean value of RGB/HB  [Al/Fe]  0.20  0.20  Cordero et al. (2014); Thygesen et al. (2014)  mean value of RGB  [Si/Fe]  0.27  0.27  Gratton et al. (2013); Thygesen et al. (2014); Carretta et al. (2009b); Cordero et al. (2014)  mean value of RGB/HB  [S/Fe]  0.40  0.40  –  estimated  [Ca/Fe]  0.27  0.27  Carretta et al. (2009b); Gratton et al. (2013); Cordero et al. (2014); Thygesen et al. (2014)  mean value of RGB/HB  [Ti/Fe]  0.20  0.20  Cordero et al. (2014); Thygesen et al. (2014)  mean value of RGB/HB    FG  SG  Reference  Note  [C/Fe]  0.12  −0.09  Cannon et al. (1998); Milone et al. (2012)  MS  [N/Fe]  0.32  1.17  Cannon et al. (1998); Milone et al. (2012)  MS  [O/Fe]  0.42  0.17  Dobrovolskas et al. (2014)  TO, NLTE+3D  [Ne/Fe]  0.40  0.40  –  estimated  [Na/Fe]  −0.12  0.10  Dobrovolskas et al. (2014)  TO, NLTE+3D  [Mg/Fe]  0.32  0.32  Carretta et al. (2009b, 2013); Gratton et al. (2013); Cordero et al. (2014); Thygesen et al. (2014)  mean value of RGB/HB  [Al/Fe]  0.20  0.20  Cordero et al. (2014); Thygesen et al. (2014)  mean value of RGB  [Si/Fe]  0.27  0.27  Gratton et al. (2013); Thygesen et al. (2014); Carretta et al. (2009b); Cordero et al. (2014)  mean value of RGB/HB  [S/Fe]  0.40  0.40  –  estimated  [Ca/Fe]  0.27  0.27  Carretta et al. (2009b); Gratton et al. (2013); Cordero et al. (2014); Thygesen et al. (2014)  mean value of RGB/HB  [Ti/Fe]  0.20  0.20  Cordero et al. (2014); Thygesen et al. (2014)  mean value of RGB/HB  View Large Table 3. General metal mixture of 47 Tuc two stellar populations (FG and SG).   FG  SG  Z  0.0056  0.0055  Y  0.256  0.276  [M/H]  −0.43  −0.41  [Fe/H]  −0.76  −0.76  [α/Fe] a  0.41  0.23    FG  SG  Z  0.0056  0.0055  Y  0.256  0.276  [M/H]  −0.43  −0.41  [Fe/H]  −0.76  −0.76  [α/Fe] a  0.41  0.23  Notes.aLabelled as [α/Fe] ∼ 0.4 and 0.2. Difference in the [α/Fe] values is due to O abundance differences. View Large 3.2 Isochrones fitting and luminosity function With the detailed metal mixture and helium abundance of 47 Tuc, we calculate new sets of evolutionary tracks and isochrones, and transform them into the observational CMD in order to fit the data. This fitting procedure, based on our adopted model prescriptions (e.g. mixing length, atmospheric boundary condition, bolometric corrections), aims to calibrate other parameters (e.g. extra mixing) in the model as described below. 3.2.1 Low MS to turn-off Kalirai et al. (2012) provide deep images of 47 Tuc taken with the Advanced Camera for Surveys (ACS) on Hubble Space Telescope (HST). The corresponding CMDs cover the whole MS of this cluster, till the faintest stars. Fig. 2 shows our isochrone fitting of their photometric data, i.e. in the F606W and F814W bands. In order to display the relative density of stars on CMD, the data are plotted with the Hess diagram (binsize 0.025 mag in colour and 0.1 mag in I magnitude). By assuming a standard extinction law (Cardelli et al. 1989), we derive, from the isochrone fitting, an age of 12.00 Gyr, a distance modulus (m − M)0 of 13.22 (apparent distance modulus (m-M)F606W ∼ 13.32), and a reddening of E(F606W − F814W) = 0.035 (hereafter named E(6 − 8)). The fitting is performed not only to the MS but also to the giant branch and HB phase as we will show later in Sections 3.2.3 and 3.2.4. The values of age, (m − M)0, and E(6 − 8) are adjusted with visual inspection with the priority of improving the turn-off and HB fittings. Figure 2. View largeDownload slide Isochrone fitting with Hess diagram of 47 Tuc data for the low MS (Kalirai et al. 2012). The bin size of Hess diagram is 0.025 mag in colour and 0.1 mag in F814W magnitude. The fitting parameters [age, η, (m − M)0, and E(6 − 8)] are listed in the legend. Figure 2. View largeDownload slide Isochrone fitting with Hess diagram of 47 Tuc data for the low MS (Kalirai et al. 2012). The bin size of Hess diagram is 0.025 mag in colour and 0.1 mag in F814W magnitude. The fitting parameters [age, η, (m − M)0, and E(6 − 8)] are listed in the legend. The distance modulus of 47 Tuc has been determined by many other authors, however with different results. For instance, using HST proper motion, Watkins et al. (2015) derive a distance of 4.15 kpc [(m − M)0 ∼13.09], which is lower than the values in the Harris catalogue (4.5 kpc [(m − M)0 ∼ 13.27] and (m − M)V = 13.37, Harris 1996, 2010 edition); the eclipsing binary distance measurement ((m − M)V = 13.35, Thompson et al. 2010); the result based on the white dwarf cooling sequence ((m − M)0 ∼ 13.32, Hansen et al. 2013); and that derived from isochrone fitting to BVI photometry ((m − M)V = 13.375, Bergbusch & Stetson 2009). Our best-fitting distance lies in between them, and agrees with other recent distance modulus determinations (e.g. Brogaard et al. 2017, (m − M)0 = 13.21 ± 0.06 based on the eclipsing binary). Gaia will release the parallaxes and proper motions including stars in 47 Tuc in its DR2 in early 2018, and will help to solve the distance problem. However, we will show in the following section that our best estimate result offers a very good global fitting, from the very low MS till the red giant and HBs. 3.2.2 RGB bump and envelope overshooting calibration Some GC features in CMD are very sensitive to stellar model parameters, which are, otherwise, hardly constrained from observations directly. This is the case of the efficiency of mixing below the convective envelope (envelope overshooting), that is known to affect the luminosity of the RGBB. In this section, we will use the 47 Tuc data to calibrate the envelope overshooting to be used in low-mass stars by parsec . The RGB bump is one of the most intriguing features in the CMD. When a star evolves to the ‘first dredge-up’ in the red giant phase, its surface convective zone deepens while the burning hydrogen shell moves outwards. When the hydrogen burning shell encounters the chemical composition discontinuity left by the previous penetration of the convective zone, the sudden increase of H affects the efficiency of the burning shell and the star becomes temporarily fainter. Soon after a new equilibrium is reached, the luminosity of the star raises again. Since the evolutionary track crosses the same luminosity three times in a short time, there is an excess of star counts in a small range of magnitudes, making a ‘bump’ in the star number distribution (LF) along the RGB. This is because the number of stars in the post-MS phases is proportional to the evolutionary time of the stars in these phases. The longer the crossing time of the chemical composition discontinuity by the burning shell, the more the stars accumulate in that region of the RGB. The properties of RGBB, including the brightness and the extent, are important to study the stellar structure and to investigate the nature of GCs. 47 Tuc was the first GC where the existence of the RGBB was confirmed (King, Da Costa & Demarque 1985). Since then, many works, both theoretical and observational (for instance, Alongi et al. 1991; Cassisi & Salaris 1997; Zoccali et al. 1999; Bono et al. 2001; Cassisi, Salaris & Bono 2002; Bjork & Chaboyer 2006; Salaris et al. 2006; Bragaglia et al. 2010; Cecco et al. 2010; Cassisi et al. 2011; Nataf et al. 2013), have studied the features of RGBB. The intrinsic brightness and extent of the RGBB are sensitive to the following: Total metallicity and metal partition: Nataf et al. (2013) propose an empirical function of RGBB extent to metallicity: the more metal-poor the GC is, the smaller is the extent of the RGBB. From the theoretical point of view, stars with lower total metallicity are brighter compared to the higher metallicity stars, causing their hydrogen burning shell to move outwards faster. Since they are also hotter, the surface convective envelope is thinner and the chemical composition discontinuity is smaller and less deep. As a consequence, their RGBB is very brief and covers a small range of magnitudes at higher luminosity. This is why RGBB in metal-poor GCs is very difficult to be well sampled. The metal partition also affects the features of RGBB, even when the total metallicity remains the same. As already shown in Fig. 1 and in Section 2, a stellar track with α-enhancement is hotter than the solar-scaled one with the same total metallicity Z because of different opacity, leading to a brighter RGBB. Since CNO are the most affected elements in the giant branch and they are important contributors to the opacities, their varying abundances have an important impact on the location of RGBB. For example, Rood & Crocker (1985) show that enhancing CNO by a factor of 10 has larger effect on the RGBB luminosity than enhancing Fe by a factor of 10 over the same metallicity Z. More recently, VandenBerg (2013) shows that higher oxygen abundance leads to a fainter RGBB if the [Fe/H] is fixed. Helium content: A larger helium content renders the star hotter and brighter (Fagotto et al. 1994). Bragaglia et al. (2010) studied the RGBB of 14 GCs and found that the more He-rich second generation shows brighter RGBB than the first generation. Similar to the mechanism in metal-poor stars, hot He-rich stars have less deep convective envelopes and their high luminosity makes the hydrogen burning shells to move faster across the discontinuity. Hence, with the same total metallicity Z and stellar mass, the RGBB of the He-rich star is brighter, less extended, and more brief. In Table 4, we take an M = 0.85 M⊙ star as an example to show how the RGBB luminosity and evolution time vary with different helium contents. Table 4. RGBB parameters of stars with a constant mass (M = 0.85 M⊙) and metallicity (Z = 0.0055) but different helium contents. Mean luminosity $$\bar{\log (L/L_{\odot })}_{{\rm RGBB}}$$, luminosity extent Δ log (L)RGBB, RGBB beginning time t0, RGBB and RGBB lifetime Δ tRGBB are listed. Z  Y  $$\bar{\log (L/L_{\odot })}_{\rm RGBB}$$  Δlog (L)RGBB  t0, RGBB (Gyr)  Δ tRGBB (Myr)  0.0055  0.276  1.5443  0.03557  12.062  27.159  0.0055  0.296  1.5887  0.03177  10.584  22.637  Z  Y  $$\bar{\log (L/L_{\odot })}_{\rm RGBB}$$  Δlog (L)RGBB  t0, RGBB (Gyr)  Δ tRGBB (Myr)  0.0055  0.276  1.5443  0.03557  12.062  27.159  0.0055  0.296  1.5887  0.03177  10.584  22.637  Age: Stars with younger age are hotter, and with their thinner convective envelope, their RGBB are brighter. In principle, multiple populations born in different ages spread the GC RGBB luminosity. However, considering that the age variation of the multiple populations is usually small (∼a few Myr), it contributes little to the GC RGBB luminosity spread compared to the He variation (see e.g. Nataf et al. 2011). Mixing efficiency: The mixing efficiency of the star, both mixing length and envelope overshooting (EOV), determines the maximum depth of the convective envelope and affects the brightness and evolutionary time of RGBB. The more efficient the mixing is, the deeper the convective envelope is, the earlier the hydrogen-burning shell meets the discontinuity left by the penetration of the surface convective zone, and the fainter the RGBB is. For the mixing length, we adopt the solar-calibrated value $$\alpha _{\rm MLT}^{{\odot }}=1.74$$ in parsec as described in Bressan et al. (2012). The EOV is calibrated with the new stellar tracks against the observations of the RGBB of 47 Tuc. Overshooting is the non-local mixing that may occur at the borders of any convectively unstable region (i.e. Bressan et al. 2015, and references therein). The extent of the overshooting at the base of the convective envelope is called envelope overshooting, and the one above the stellar convective core is called core overshooting. There are observations that can be better explained with envelope overshooting, for instance, the blue loops of intermediate and massive stars (Alongi et al. 1991; Tang et al. 2014), and the carbon stars LFs in the Magellanic Clouds, which require a more efficient third dredge-up in AGB stars (Herwig 2000; Marigo & Girardi 2007). At the base of the convective envelope of the Sun, models with an envelope overshooting of Λe ≈ 0.3 ∼ 0.5Hp (where Hp is the pressure scaleheight) provide a better agreement with the helioseismology data (Christensen-Dalsgaard et al. 2011). The envelope overshooting also affects the surface abundance of light elements (Fu et al. 2015), and asteroseismic signatures in stars (Lebreton & Goupil 2012). In Fig. 3, we compare the RGBB evolution of models computed with different EOV values, Λe, at the same stellar mass and composition. Every pair of filled dots marks the brightness extent of the RGBB. The figure shows that a larger envelope overshooting not only makes the RGBB fainter, but also of longer duration, leading to a more populated RGBB. A larger EOV value leads to a deeper surface convective zone, and the hydrogen burning shell encounters the chemical discontinuity earlier. Figure 3. View largeDownload slide The RGBB luminosity as a function of stellar age for a 0.85 M⊙star but with different EOV. The black, red, green, and blue line from the top to bottom represent tracks with Λe = 0.05, 0.3, 0.4, and 0.5. The filled dots mark the minimum and maximum luminositities of RGBB for each track, and ΔtRGBB is the evolution time from the minimum luminosity to the maximum one. Figure 3. View largeDownload slide The RGBB luminosity as a function of stellar age for a 0.85 M⊙star but with different EOV. The black, red, green, and blue line from the top to bottom represent tracks with Λe = 0.05, 0.3, 0.4, and 0.5. The filled dots mark the minimum and maximum luminositities of RGBB for each track, and ΔtRGBB is the evolution time from the minimum luminosity to the maximum one. The LF is a useful tool to compare the observed morphology of RGBB with that predicted by the theory. Taking into account that the 47 Tuc population contribution is 30 per cent from the FG and 70 per cent from the SG as suggested by Milone et al. (2012) and Carretta et al. (2009b), we simulated the LF of 47 Tuc with our isochrones with different EOV values. The comparison between the observed and predicted LFs is shown in Fig. 4. For the observed LF, we have used data from the HST/ACS survey of GCs (Sarajedini et al. 2007). Both observations and models are sampled in bins of 0.05 magnitudes. The fitting parameters are the same as those we used in Fig. 2. The model LF (orange histogram) is calculated with envelope overshooting Λe = 0.3 in the upper panel and with Λe = 0.5 in the lower panel. It is evident that the LF computed with the small envelope overshooting value Λe = 0.3 has RGBB too bright compared to data (black histogram filled with oblique lines). We find that the agreement between observations and models is reached when one adopts a value of Λe = 0.5Hp below the convective border, with our adopted metal mixtures and best-fitting isochrone. This provides a robust calibration of the envelope overshooting parameter. This envelope overshooting calibration will be applied to all other stellar evolution calculations of low-mass stars. Figure 4. View largeDownload slide Comparison between LF of 47 Tuc data (Sarajedini et al. 2007) and the new parsec isochrone with different EOV. The fitting parameters [age, (m − M)0, and E(6 − 8)] are the same as in Fig. 2. The black histogram filled with oblique lines is the data LF, whereas orange histogram is LF derived from new parsec isochrones with 30 per cent contribution from the FG of 47 Tuc and 70 per cent from the SG. The upper panel isochrones of each sub-figure are calculated with EOV value Λe = 0.3, and the lower panels are the ones with Λe = 0.5. Orange and black arrows mark the location of RGBB in model and in data, respectively. The bin size of the LF is 0.05 mag. Figure 4. View largeDownload slide Comparison between LF of 47 Tuc data (Sarajedini et al. 2007) and the new parsec isochrone with different EOV. The fitting parameters [age, (m − M)0, and E(6 − 8)] are the same as in Fig. 2. The black histogram filled with oblique lines is the data LF, whereas orange histogram is LF derived from new parsec isochrones with 30 per cent contribution from the FG of 47 Tuc and 70 per cent from the SG. The upper panel isochrones of each sub-figure are calculated with EOV value Λe = 0.3, and the lower panels are the ones with Λe = 0.5. Orange and black arrows mark the location of RGBB in model and in data, respectively. The bin size of the LF is 0.05 mag. 3.2.3 Red Giant Branch Although Kalirai et al. (2012) focus on the faint part of the MS as shown in Fig. 2, another data set of 47 Tuc, the HST/ACS survey of GCs (Sarajedini et al. 2007), is devoted to the HB (Anderson et al. 2008) with the same instrument. In Fig. 5, we show the global fitting of ACS data of 47 Tuc, from MS up to the RGB and HB. The best-fitting parameters we derived are the same as those we used to fit the lower MS data in Fig. 2. The HESS diagram is used for the global fitting (the left-hand panel) with bin size 0.025 mag in colour and 0.1 mag in F814W magnitude. The HB region and the turn-off region are zoomed in with scatter plots in the two right-hand panels. Thanks to the detailed composition derived from the already quoted observations for the two main populations of 47 Tuc and the new computed models, by assuming a standard extinction law (Cardelli et al. 1989) and using the adopted EOV and mixing length parameters, we are able to perform a global fit to the CMD of 47 Tuc covering almost every evolutionary phase over a range of about 13 mag. This must be compared with other fittings that can be found in literature and that usually are restricted to only selected evolutionary phases (Kim et al. 2002; Salaris et al. 2007; VandenBerg et al. 2013, 2014; Chen et al. 2014; McDonald & Zijlstra 2015, etc.). However, it is worth noting that the distance of this cluster, as already discussed in Section 3.2.1, together with the cluster age, has varied over the years in many careful studies. We look forward to Gaia DR2 to put more constrains to this problem. Figure 5. View largeDownload slide Isochrone fitting with Hess diagram (left-hand panel) of 47 Tuc data (Sarajedini et al. 2007) for all the evolutionary phases, and with scatter plots highlighting the HB region (upper right-hand panel) and the turn-off region (lower right-hand panel). The red and blue lines represent isochrones of the first and second generations, respectively, as the legend shows. The fitting parameters are: age = 12.00 Gyr, (m − M)0 = 13.22, E(6 − 8) = 0.035. Figure 5. View largeDownload slide Isochrone fitting with Hess diagram (left-hand panel) of 47 Tuc data (Sarajedini et al. 2007) for all the evolutionary phases, and with scatter plots highlighting the HB region (upper right-hand panel) and the turn-off region (lower right-hand panel). The red and blue lines represent isochrones of the first and second generations, respectively, as the legend shows. The fitting parameters are: age = 12.00 Gyr, (m − M)0 = 13.22, E(6 − 8) = 0.035. As the upper right-hand panel of Fig. 5 show, the isochrones corresponding to both of the two stellar generations run on the red side of the data in the RGB phase. Part of the discrepancy could be explained by the bolometric correction used. Here, we are using bolometric correction from phoenix atmosphere models as described in Chen et al. (2015) for parsec v1.2S, where only the total metallicity is considered in the transformation of log (L) versus log (Teff) into F814W versus (F606W − F814W). As the metallicities of the two 47 Tuc populations (Z = 0.0056 and Z = 0.0055) show only a marginal difference, we adopt for the two populations the same bolometric corrections. Thus, Fig. 5 reflects basically the difference of the two populations in the theoretical log (L) versus log (Teff) HRD. This ‘RGB-too-red’ problem also exists in Dotter et al. (2007), when they fit the same set of data using dsep models (see their fig. 12), as they apply bolometric correction from phoenix as well. To minimize this discrepancy, we use the atlas12 code (Kurucz 2005), which considers not only the total metallicity Z but also log (g) and detailed chemical compositions for the colour transformation, to compute new atmosphere models with our best estimate chemical compositions of the two 47 Tuc populations. We adopt these atlas12 models for the new fits to 47 Tuc, but only for models with Teff hotter than 4000 K ((F606W − F814W)∼1.3). For lower Teff, we still use phoenix because atlas12 models may be not reliable at cooler temperatures (Chen et al. 2014). Here, we show the fit obtained with atlas12+phoenix bolometric correction in Fig. 6. We see that with the same fitting parameters as in Fig. 5, the prediction of the RGB colours is improved by applying new atlas12 bolometric correction. The two stellar generations are split on RGB phase in Fig. 6. We see that the SG (Z = 0.0055), which is the main contributor as suggested by Milone et al. (2012) and Carretta et al. (2009b), is consistent with the denser region of the RGB data. In other evolutionary phases, the new atlas12 bolometric corrections do not bring noticeable changes. Figure 6. View largeDownload slide The same isochrone fitting with Hess diagram and scatter plots of 47 Tuc data (Sarajedini et al. 2007) as in Fig. 5, but with atmosphere models fromatlas12 for Teff hotter than 4000 K. Figure 6. View largeDownload slide The same isochrone fitting with Hess diagram and scatter plots of 47 Tuc data (Sarajedini et al. 2007) as in Fig. 5, but with atmosphere models fromatlas12 for Teff hotter than 4000 K. Since atlas12 only slightly affects the colour of the RGB base, and the remainder of this paper deals with the LF of the bump and of the HB, in the following discussion, we will continue to use the standard atmosphere models of parsec v1.2S. ATLAS$$\scriptstyle{12}$$ atmosphere models for PARSEC alpha-enhanced isochrones will be discussed in detail in another following work (Chen et al., in preparation). Mass-loss by stellar winds during the RGB phase has been considered for low-mass stars, using the empirical formula by Reimers (1975) multiplied by an efficiency factor η. In Fig. 7, we show the mass lost by RGB stars in unit of  M⊙ for the FG of 47 Tuc (the plot for SG is very similar). Different efficiency factors (η) and ages are applied. ΔM in the figure is the difference between the initial mass and current mass of the tip RGB star: ΔM = Minitial − Mcurrent. The lost mass, which is greater with larger η, is an increasing function of the cluster age. It is very difficult to derive observationally the mass lost in RGB stars directly since an accurate mass is not easy to derive and the RGB tip is hard to identify. However, the RGB mass-loss characterizes the HB morphology, and this will be discussed in next section. Figure 7. View largeDownload slide RGB mass lost in unit of  M⊙ for FG of 47 Tuc (Z = 0.0056, Y = 0.256). The X axis is the initial mass of the tip RGB star, and the Y axis shows the mass lost in this star during RGB phase. Five different efficiency factors η are illustrated, from the top to bottom η = 0.40 (filled diamond), η = 0.35 (filled triangle), η = 0.30 (filled square), η = 0.25 (filled star), and η = 0.20 (filled dots). The colour code displays the age, as shown in the colour bar. Figure 7. View largeDownload slide RGB mass lost in unit of  M⊙ for FG of 47 Tuc (Z = 0.0056, Y = 0.256). The X axis is the initial mass of the tip RGB star, and the Y axis shows the mass lost in this star during RGB phase. Five different efficiency factors η are illustrated, from the top to bottom η = 0.40 (filled diamond), η = 0.35 (filled triangle), η = 0.30 (filled square), η = 0.25 (filled star), and η = 0.20 (filled dots). The colour code displays the age, as shown in the colour bar. 3.2.4 Horizontal branch morphology The morphology of the HB in GCs is widely studied since the ‘second parameter problem’ (that is, the colour of the HB is determined not only by metallicity, van den Bergh 1967; Sandage & Wildey 1967) was introduced. Aside from metallicity as the ‘first parameter’, age, He content, mass-loss, and cluster central density have been suggested as candidates to be the second, or even third, parameter affecting the morphology of the HB (Fusi Pecci & Bellazzini 1997; D'Antona et al. 2002; Caloi & D'Antona 2005; Catelan 2008; Dotter et al. 2010; Gratton et al. 2010; McDonald & Zijlstra 2015, etc.). Most of these parameters involve an effect on the mass of the stars that populate the cluster HB. Stars with smaller stellar mass are hotter in temperature and bluer in colour. The HB stellar mass decreases as the cluster ages. At a given age, He-rich star evolves faster and reach the zero-age horizontal branch (ZAHB) with lower mass. If the age and He content are the same, the mass of HB stars is fixed by the mass-loss along the RGB (here the mass-loss driven by the helium flash is not considered). Although the RGB mass-loss does not significantly affect the RGB evolutionary tracks, it determines the location of the stars on the HB, by tuning the stellar mass. Here, we illustrate how helium content and the RGB mass-loss affect the HB morphology in the case of 47 Tuc. The HB morphology with five different values of η is displayed in Fig. 8 for our best-fitting parameters derived in Section 3.2.3. Different metal/helium abundances ([Z = 0.0056, Y = 0.256], [Z = 0.0055, Y = 0.276], [Z = 0.0056, Y = 0.276], and [Z = 0.0055 Y = 0.296]) are displayed. The isochrones with Z = 0.0056 are calculated with [α/Fe] ∼ 0.4 and those with Z = 0.0055 are calculated with [α/Fe] ∼ 0.2. The 47 Tuc data (Sarajedini et al. 2007) are also plotted for comparison. The differences between the isochrone with [Z = 0.0055, Y = 0.276] (blue solid line) and the one with [Z = 0.0056, Y = 0.276] (orange dashed line) are negligible on the HB, even though they refer to a different α-enhanced mixture. With the same RGB mass-loss factor η, He-rich stars have their HB more extended (because of smaller stellar mass), bluer (due to both the smaller stellar mass and the He-rich effect on radiative opacity), and more luminous (because of larger He content in the envelope). For stars with larger mass-loss efficiency η during their RGB phase, their HB is bluer, fainter, and more extended, because of smaller stellar mass (hence smaller envelope mass, since the core mass does not vary significantly with the mass-loss rate). Indeed, the effects of a higher He content and of a lower mass (no matter if it is the result of an older age or a larger RGB mass-loss) on HB stars are difficult to distinguished by means of the colour, but can be disentangled because the larger helium content makes the He-rich star slightly more luminous. Figure 8. View largeDownload slide HB morphology for different RGB mass-loss parameters (η) and metal/helium abundances, with the same isochrone fitting parameters [age, (m − M)0, and E(6 − 8)] as in Fig. 5. The red solid line, blue solid line, orange dashed line, and green dash–dotted line represent isochrones of [Z = 0.0056, Y = 0.256], [Z = 0.0055, Y = 0.276], [Z = 0.0056, Y = 0.276], and [Z = 0.0055, Y = 0.296], respectively. The mass lost during the RGB in unit of  M⊙ for each η and metal/helium abundance is listed in Table 5. Figure 8. View largeDownload slide HB morphology for different RGB mass-loss parameters (η) and metal/helium abundances, with the same isochrone fitting parameters [age, (m − M)0, and E(6 − 8)] as in Fig. 5. The red solid line, blue solid line, orange dashed line, and green dash–dotted line represent isochrones of [Z = 0.0056, Y = 0.256], [Z = 0.0055, Y = 0.276], [Z = 0.0056, Y = 0.276], and [Z = 0.0055, Y = 0.296], respectively. The mass lost during the RGB in unit of  M⊙ for each η and metal/helium abundance is listed in Table 5. Table 5 lists the current mass, MZAHB, of the first HB star and the corresponding mass that has been lost ΔMRGB, in unit of  M⊙. In the table, we also show the HB mass range δMHB that produces the corresponding colour extent of HB. All cases displayed in Fig. 8 are itemized. Table 5. The mass lost during the RGB in unit of  M⊙ for different η and metal/helium abundance. The current mass of the first HB star is MZAHB, and ΔMRGB represents its RGB mass-loss in unit of  M⊙. The HB mass range is itemized in the last column δMHB. All values listed here are derived from isochrones with age = 12.0 Gyr, (m − M)0 = 13.22, and E(V − I) = 0.035, as shown on Fig. 8. Z  Y  η  MZAHB ( M⊙)  ΔMRGB ( M⊙)  δMHB ( M⊙)  0.0056  0.256  0.20  0.795 832  0.0946  0.0023      0.25  0.770 375  0.1201  0.0033      0.30  0.744 052  0.1464  0.0044      0.35  0.716 765  0.1737  0.0053      0.40  0.688 402  0.2020  0.0059  0.0055  0.276  0.20  0.758 027  0.0953  0.0029      0.25  0.732 270  0.1211  0.0039      0.30  0.705 582  0.1478  0.0051      0.35  0.677 852  0.1755  0.0061      0.40  0.648 949  0.2044  0.0067  0.0056  0.276  0.20  0.763 649  0.0948  0.0028      0.25  0.738 049  0.1204  0.0039      0.30  0.711 533  0.1470  0.0050      0.35  0.683 996  0.1745  0.0059      0.40  0.655 309  0.2032  0.0066  0.0055  0.296  0.20  0.726 732  0.0954  0.0035      0.25  0.700 873  0.1213  0.0047      0.30  0.674 033  0.1481  0.0058      0.35  0.646 090  0.1760  0.0066      0.40  0.616 896  0.2052  0.0067  Z  Y  η  MZAHB ( M⊙)  ΔMRGB ( M⊙)  δMHB ( M⊙)  0.0056  0.256  0.20  0.795 832  0.0946  0.0023      0.25  0.770 375  0.1201  0.0033      0.30  0.744 052  0.1464  0.0044      0.35  0.716 765  0.1737  0.0053      0.40  0.688 402  0.2020  0.0059  0.0055  0.276  0.20  0.758 027  0.0953  0.0029      0.25  0.732 270  0.1211  0.0039      0.30  0.705 582  0.1478  0.0051      0.35  0.677 852  0.1755  0.0061      0.40  0.648 949  0.2044  0.0067  0.0056  0.276  0.20  0.763 649  0.0948  0.0028      0.25  0.738 049  0.1204  0.0039      0.30  0.711 533  0.1470  0.0050      0.35  0.683 996  0.1745  0.0059      0.40  0.655 309  0.2032  0.0066  0.0055  0.296  0.20  0.726 732  0.0954  0.0035      0.25  0.700 873  0.1213  0.0047      0.30  0.674 033  0.1481  0.0058      0.35  0.646 090  0.1760  0.0066      0.40  0.616 896  0.2052  0.0067  View Large If one considers a uniform mass-loss parameter η for the two populations of 47 Tuc ([Z = 0.0056, Y = 0.256] and [Z = 0.0055, Y = 0.276]), η = 0.35 is the value that fits better the HB morphology in our best-fitting case, as Fig. 8 illustrates. As shown in Table 5, an RGB mass-loss parameter of η = 0.35 leads to a value of the mass lost during RGB between 0.1737  M⊙–0.1755  M⊙. In the literature, there is a discrepancy among the results on RGB mass-loss derived with different approaches, namely: cluster dynamics, infrared (IR) excess due to dust, and HB modelling for this cluster. Heyl et al. (2015) study the dynamics of white dwarf in 47 Tuc, and concluded that the mass lost by stars at the end of the RGB phase should be less than about 0.2 M⊙. Origlia et al. (2007) observe the circumstellar envelopes around RGB stars in this cluster from mid-IR photometry and find the total mass lost on the RGB is ≈0.23 ± 0.07 M⊙. McDonald & Zijlstra (2015) use HB star mass from literature to study the RGB mass-loss and derive a Reimers factor η = 0.452 (corresponding to an RGB mass-loss greater than ∼0.20 M⊙). Most recently, Salaris et al. (2016) assume a distribution of the initial He abundance to simulate the observed HB of 47 Tuc. They derive a lower limit to the RGB mass-loss of about 0.17  M⊙, but larger values are also possible, up to 0.30  M⊙, with younger age, higher metallicity, and reddening. Our RGB mass-loss results, based on a uniform mass-loss parameter and our best-fitting case of the two populations of this cluster, are consistent with the lower and upper limit values from the literature. However, the real situations of the RGB mass-loss in GCs, as discussed in the references above, could be much more complicated. In our final data base of the new parsec isochrones, we will provide different choices of He contents and mass-loss parameters for the users’ science purpose. The LFs from the turn-off to the HB with an RGB mass-loss parameter η = 0.35 are displayed in Fig. 9 for our best-fitting parameters. For comparison, the LF of HST GC survey data (Sarajedini et al. 2007) is also plotted (black histogram filled with oblique lines) with the same bin size 0.05 mag. We adopt the Salpeter IMF (Salpeter 1955) to generate LFs though, as discussed in Section 3.2.2, LFs in this phase are not sensitive to IMF because the stellar mass varies very little. LFs are instead sensitive to the evolution time along the phase. All model LFs are normalized to the total number of observed RGB stars within a range of F814W magnitude between 14 and 16 mag. The left-hand panels of Fig. 9 show the LFs from the turn-off to the HB, for a 100 per cent FG (red histogram), a 100 per cent SG (blue histogram), and the percentage adopted in Section 3.2.2, 30 per cent from FG and 70 per cent from SG (orange histogram), respectively. With our best isochrone fitting parameters, age = 12.00 Gyr, (m − M)0 = 13.22, E(6 − 8) = 0.035, η = 0.35, and the population percentage obtained from literature (Carretta et al. 2009b; Milone et al. 2012), the model LF (orange histogram in each figure) shows a very good agreement with the observed LF. The three right-hand panels in Fig. 9 are zoomed in on the HB and RGBB regions. The total number of HB stars within 12.9–13.3 mag in the observations and in the models are listed in the figure. Since the LF is directly proportional to the evolution time, the good agreement of LF between model and observation in Fig. 9 indicates that the hydrogen shell burning lifetime is correctly predicted in parsec . Figure 9. View largeDownload slide Comparison between the LF of 47 Tuc data (Sarajedini et al. 2007) and that derived from the new parsec models, from the turn-off to the HB. The Y axis represent the star counts in magnitude F814W. The black histogram filled with oblique lines is the data LF, whereas the red histogram in the upper panel, blue histogram in the middle panel, and orange histogram in the lower panel, represent 100 per cent FG of 47 Tuc [Z = 0.0056, Y = 0.256], 100 per cent SG [Z = 0.0055, Y = 0.276], and their mix with 30 per cent from the FG and 70 per cent from the SG, respectively. The three panels on the right-hand side show the LF of the RGBB and the HB region, for each population mixture. The fitting parameters are: η = 0.35, age = 12 Gyr, (m − M)0 = 13.22, and E(6 − 8) = 0.035. Figure 9. View largeDownload slide Comparison between the LF of 47 Tuc data (Sarajedini et al. 2007) and that derived from the new parsec models, from the turn-off to the HB. The Y axis represent the star counts in magnitude F814W. The black histogram filled with oblique lines is the data LF, whereas the red histogram in the upper panel, blue histogram in the middle panel, and orange histogram in the lower panel, represent 100 per cent FG of 47 Tuc [Z = 0.0056, Y = 0.256], 100 per cent SG [Z = 0.0055, Y = 0.276], and their mix with 30 per cent from the FG and 70 per cent from the SG, respectively. The three panels on the right-hand side show the LF of the RGBB and the HB region, for each population mixture. The fitting parameters are: η = 0.35, age = 12 Gyr, (m − M)0 = 13.22, and E(6 − 8) = 0.035. 4 COMPARISON WITH OTHER MODELS AND GC DATA The new parsec α-enhanced isochrones provide a very good fit of the CMD of 47 Tuc in all evolutionary stages from the lower MS to the HB. The location of the RGB bump shows that the efficiency of the envelope overshoot is quite significant, requiring EOV of Λe = 0.5Hp. This can be considered as a calibration of this phenomenon. We now use the calibrated EOV value to obtain α-enhanced isochrones of different metallicities. For this purpose, we adopt the partition of heavy elements of the two stellar generations of 47 Tuc ([α/Fe] ∼ 0.4 and 0.2). In this section, we compare our new α-enhanced models with isochrones from other stellar evolution groups and GC data of different metallicities. 4.1 Comparison with other models The RGBB of GCs, as already said in Section 3.2.2, has been studied over 30 years since the 47 Tuc RGBB was observed in 1985 (King et al. 1985). However, there is a discrepancy between the observed brightness of RGBB and the model predictions: The model RGBB magnitude is about 0.2–0.4 mag brighter than the observed one (Fusi Pecci et al. 1990; Cecco et al. 2010; Troisi et al. 2011). This discrepancy becomes more pronounced in metal-poor GCs (Cassisi et al. 2011). Here, we compare the RGBB magnitude of our newly calibrated parsec models with other α-enhanced stellar tracks. Since the BaSTI (Pietrinferni et al. 2006, 2013) and dsep (Dotter 2007; Dotter et al. 2008) isochrones are publicly available online, we download the [α/Fe] = 0.4 isochrones at 13 Gyr from BaSTI canonical models data base and dsep web tool 2012 version. We then compare the mean values of their absolute RGBB magnitude in the F606W (HST ACS/WFC) band, with our models. Fig. 10 shows this comparison as a function of total metallicity [M/H] and iron abundance [Fe/H]. The model [M/H] is approximated by   $$[M/H] \approx \log \frac{Z/X}{Z_{\odot }/X_{\odot }}.$$ (2)And for both of the two new parsec models with α enhancement, [Fe/H] ≈ [M/H] −0.33. The solar metallicity in parsec is Z⊙ = 0.01524 and Z⊙/X⊙ = 0.0207. Since dsep models do not provide [M/H] directly but only [Fe/H] in their isochrones, we calculate [M/H] following equation (2) with total metallicity Z, He content Y, and solar Z⊙/X⊙ taken from their models. Additionally, two parsec models with solar-scaled metal mixture ([α/Fe] = 0), parsec v1.2S and parsec with EOV calibration from this work Λe = 0.5Hp, are also plotted. Compared with the new set of solar-scaled parsec model with Λe = 0.5Hp (dark blue line with diamond), the α-enhanced one (red line with triangle) at the same [M/H] (thus same Z and Y) is slightly brighter as we have already discussed in Section 2. We notice that the RGBB behaviour of parsec v1.2S in this figure is different from fig. 3 of Joyce & Chaboyer (2015), which compares parsec v1.2S with other models. The reason for this disagreement is unclear to us. Figure 10. View largeDownload slide Comparison of the RGBB magnitude of different evolutionary tracks at 13 Gyr as a function of [M/H] (left-hand panel) and [Fe/H] (right-hand panel). There are three different α-enhanced models ([α/Fe]   = 0.4) in the figure: parsec (red solid line with triangle), BaSTI (yelow green solid line with dot), and dsep model (green solid line with star). Other two sets of solar-scaled parsec models ([α/Fe]   = 0) are plotted for comparison: parsec v1.2S with negligible overshoot (light blue dotted line with cross) and parsec with EOV calibration Λe = 0.5Hp (dark blue dotted line with diamond). The Y axis is the mean value of the absolute F606W magnitude of the RGBB (MV, RGBB). Figure 10. View largeDownload slide Comparison of the RGBB magnitude of different evolutionary tracks at 13 Gyr as a function of [M/H] (left-hand panel) and [Fe/H] (right-hand panel). There are three different α-enhanced models ([α/Fe]   = 0.4) in the figure: parsec (red solid line with triangle), BaSTI (yelow green solid line with dot), and dsep model (green solid line with star). Other two sets of solar-scaled parsec models ([α/Fe]   = 0) are plotted for comparison: parsec v1.2S with negligible overshoot (light blue dotted line with cross) and parsec with EOV calibration Λe = 0.5Hp (dark blue dotted line with diamond). The Y axis is the mean value of the absolute F606W magnitude of the RGBB (MV, RGBB). Among the factors that may affect the brightness of the RGBB, as summarized in Section 3.2.2, we list the mixing efficiency and He contents. The helium-to-metal enrichment law of the different models is different, as discussed in Section 2. parsec (Y = 0.2485 + 1.78Z) uses a slightly higher He abundance (∼0.002) than the other two models (BaSTI: Y = 0.245 + 1.4Z, dsep: Y = 0.245 + 1.54Z). Different model also adopts different mixing length parameters. The parsec mixing length parameter is αMLT = 1.74, BaSTI uses αMLT = 1.913, and dsep adopts αMLT = 1.938. If all other parameters are the same, a higher He content and a smaller mixing length parameter lead to a brighter RGBB (Fu 2006). This can explain why in the left-hand panel of Fig. 10 the solar-scaled parsec v1.2S shows nearly the same MV, RGBB independent of [M/H] as BaSTI and dsep models. parsec v1.2S has slightly higher He content and smaller mixing length parameter, which make the RGBB brighter as already discussed above, whereas its solar-scaled metal mixture leads to a fainter RGBB at the same metallicity. The combined effects make the three models to show similar RGBB magnitude. BaSTI and dsep RGBB have almost the same performance and are eventually brighter than parsec [α/Fe] ∼ 0.4 models no matter as a function of [M/H] or [Fe/H]. We remind that a fainter RGBB magnitude can be produced by a more efficient EOV, our new α-enhanced models are computed with the calibrated EOV parameter, whereas BaSTI and dsep do not consider envelope overshooting. Also, compared to parsec v1.2S (Λe = 0.05Hp), the new solar-scaled model with Λe = 0.5Hp shifts MV, RGBB down by about 0.35 mag. This brightness change is consistent with the work of Cassisi et al. (2002), who conclude that the difference should be of about 0.8 mag/Hp. Since the RGBB brightness difference between the new parsec α-enhanced models and the solar-scaled models with the same Λe is much smaller than the difference between the two solar-scaled parsec models with different Λe, we conclude that the mixing efficiency has much stronger impact on the RGBB performance than the metal partition. 4.2 Comparison with other GC data Comparing the location of the RGB bump predicted by the models with the observed one in GCs with different metallicity is a good way to test the models. In Fig. 11, we compare our new α-enhanced models with HST data from Nataf et al. (2013, 55 clusters) and Cassisi et al. (2011, 12 clusters). The models extend till [M/H] ∼ −2 ([Fe/H] ∼ −2.3). For comparison, two sets of models with solar-scaled metal partition, [α/Fe] = 0 (parsec v1.2S and parsec with Λe = 0.5Hp) are also plotted. Figure 11. View largeDownload slide F606W magnitude difference between the MSTO and the RGBB ($$\Delta V^{\rm MSTO}_{\rm RGBB}$$) as a function of the total metallicity [M/H] (left-hand panel) and iron abundance [Fe/H]. Four different sets of theoretical $$\Delta V^{\rm MSTO}_{\rm RGBB}$$ value are plotted, at both 13 Gyr (solid line) and 11 Gyr (dashed line). Three of them are with new calibrated EOV Λe = 0.5Hp: [α/Fe] ∼ 0.4 (red lines with triangle), [α/Fe] ∼ 0.2 (green lines with square), and [α/Fe] = 0 (dark blue lines with diamond). Another one is from the standard parsec v1.2S (light blue lines with cross). The data are 55 clusters from Nataf et al. (2013, grey dots with error bar) and 12 clusters from Cassisi et al. (2011, black dots with error bar). Figure 11. View largeDownload slide F606W magnitude difference between the MSTO and the RGBB ($$\Delta V^{\rm MSTO}_{\rm RGBB}$$) as a function of the total metallicity [M/H] (left-hand panel) and iron abundance [Fe/H]. Four different sets of theoretical $$\Delta V^{\rm MSTO}_{\rm RGBB}$$ value are plotted, at both 13 Gyr (solid line) and 11 Gyr (dashed line). Three of them are with new calibrated EOV Λe = 0.5Hp: [α/Fe] ∼ 0.4 (red lines with triangle), [α/Fe] ∼ 0.2 (green lines with square), and [α/Fe] = 0 (dark blue lines with diamond). Another one is from the standard parsec v1.2S (light blue lines with cross). The data are 55 clusters from Nataf et al. (2013, grey dots with error bar) and 12 clusters from Cassisi et al. (2011, black dots with error bar). Here, we use the magnitude difference between the RGBB and the main-sequence turn-off (MSTO), $$\Delta V^{\rm MSTO}_{\rm RGBB}$$, as a reference for comparison between the theoretical magnitude of RGBB and the observed one. Unlike the absolute magnitude MV, RGBB, $$\Delta V^{\rm MSTO}_{\rm RGBB}$$ is not affected by uncertainties in the distance modulus (m − M)0 and extinction AV of the cluster. There are also works using the magnitude difference between HB and RGBB ($$\Delta V^{\rm RGBB}_{\rm HB} = M_{V,{\rm RGBB}} - M_{V,{\rm HB}}$$, e.g. Fusi Pecci et al. 1990; Cassisi & Salaris 1997; Cecco et al. 2010) or the one between HB and MSTO ($$\Delta V^{HB}_{TO} = M_{V,TO} - M_{V,{\rm HB}}$$, e.g. VandenBerg et al. 2013) as a way to avoid distance and extinction uncertainties, but, as we have elaborated in Section 3.2.4, the RGB mass-loss together with different metal mixture and He content may affect the HB magnitude and thus make $$\Delta V^{\rm RGBB}_{\rm HB}$$ difficult to be interpreted. The only free parameter of the $$\Delta V^{\rm MSTO}_{\rm RGBB}$$ method is the age, if the composition of the cluster is fixed. In Fig. 11, we compare the theoretical $$\Delta V^{\rm MSTO}_{\rm RGBB}$$ value at typical GC ages of 11 and 13 Gyr, with the observed value from Nataf et al. (2013) and (Cassisi et al. 2011). The comparisons are displayed both in the [M/H] frame and [Fe/H] frame. The MSTO in Nataf et al. (2013) is defined by taking the bluest point of a polynomial fit to the upper MS of each GC in the (F606W, F606W − F814W). Cassisi et al. (2011) derive the MSTO magnitude by fitting isochrones to the MS. To obtain the theoretical MSTO F606W magnitude in our model, we select the bluest point of the isochrone in the MS. The models of 13 Gyr show larger difference between RGBB and MSTO $$\Delta V^{\rm MSTO}_{\rm RGBB}$$ than those at 11 Gyr. Models with [α/Fe] ∼ 0.2 show a slightly greater $$\Delta V^{\rm MSTO}_{\rm RGBB}$$ value than the models computed with [α/Fe] ∼ 0.4. In the right-hand panel of the figure, we see that the α-enhanced models show greater $$\Delta V^{\rm MSTO}_{\rm RGBB}$$ than the solar-scaled ones with the same [Fe/H]. This said, if one has [Fe/H] measurement of a GC with α enhancement and takes $$\Delta V^{\rm MSTO}_{\rm RGBB}$$ as an age indicator, choosing the solar-scaled models will lead to an underestimated cluster age. Compared to the previous parsec version v1.2S, the new models significantly improve the $$\Delta V^{\rm MSTO}_{\rm RGBB}$$ prediction in both the [M/H] frame and [Fe/H] frame. At the most metal-poor side, around [M/H] ∼ −2.0 ([Fe/H] ∼ −2.3), the new models are consistent with Cassisi et al. (2011) data (black dots), but are higher than the values derived by Nataf et al. (2013) (grey dots) by ∼0.1 mag. We will discuss the possible reasons in the next section. 5 SUMMARY AND DISCUSSION Studies on GCs, Galactic bulge, halo, and thick disc call for stellar models with α enhancement because stars residing in them have α-to-iron number ratio larger than the solar value. This ratio, [α/Fe] , not only affects the stellar features like the luminosities and effective temperature, but also echoes the formation history of the cluster/structure the stars are in. To investigate such stars, and to trace back their formation history, we have now extend the parsec models to include α-enhanced mixtures. In this paper, we check the α-enhanced models with the nearby GC 47 Tuc (NGC 104). The chemical compositions including the helium abundances of 47 Tuc are studied by many works. We collect detailed elemental abundances of this cluster and derive absolute metal mixtures for two populations: first generation [Z = 0.0056, Y = 0.256], and second generation [Z = 0.0055, Y = 0.276]. The α-to-iron ratios of them are [α/Fe] = 0.4057 ([α/Fe] ∼ 0.4) and [α/Fe] = 0.2277 ([α/Fe] ∼ 0.2), respectively. We calculate evolutionary tracks and isochrones with these two α-enhanced metal mixtures, and fit CMD to HST/ACS data. The model envelope overshooting is then calibrated to the value Λe = 0.5Hp in order to reproduce the RGB bump morphology in 47 Tuc. After the calibration, the new α-enhanced isochrones nicely fit the data from the low MS to the turn-off, giant branch, and the HB with age of 12.00 Gyr, absolute distance modulus (m-M)0 = 13.22 (apparent distance modulus (m-M)F606W = 13.32), and reddening E(6 − 8) = 0.035. These results compare favourably with many other determinations in the literature. The LFs inform us that the lifetime of hydrogen burning shell appears to be correctly predicted. By studying the morphology of the HB, we conclude that the mean mass lost by stars during the RGB phase is around 0.17 M⊙. There are also other methods to estimate the age of this cluster in the literature. For instance, mass-radius constraints of the detached eclipsing binary stars V69 in 47 Tuc have also been used (Weldrake et al. 2004; Dotter, Kaluzny & Thompson 2008; Thompson et al. 2010; Brogaard et al. 2017). This approach, which considers the two components of the detached binary as single stars, is much less affected by the uncertainties arising from the unknown distance, reddening, and transformation from the theoretical to the observational plane. Thompson et al. (2010) derive an age of 11.25 ± 0.21(random) ± 0.85(systematic) for 47 Tuc, and Brogaard et al. (2017) give 11.8 Gyr as their best estimate with 3σ limits from 10.4 to 13.4 Gyr. We examined the mass–radius and mass–Teff constraints provided by V69 with our best-fitting FG and SG models at 12.0 Gyr, as shown in Fig. 12. We have adopted for the two components of V69 the following data (Thompson et al. 2010): current mass Mp = 0.8762 ± 0.0048 M⊙, Ms = 0.8588 ± 0.0060 M⊙, radius Rp = 1.3148 ± 0.0051 R⊙, Rs = 1.1616 ± 0.0062 R ⊙ and effective temperature Teffp = 5945 ± 150 K, Teffs = 5959 ± 150 K of the primary and secondary star, respectively. Differences between the FG and SG isochrones are due to the difference of He abundances and [α/Fe]. The comparison with our models indicates that V69 cannot belong to the SG population. Concerning the mass–radius relation, which provides the most stringent constraints on the two stars, we see that our best-fitting FG isochrone of 12 Gyr is only marginally able to reproduce the secondary component (filled black star), within 3σ, whereas there is a tension with the radius of the primary component (empty star). A better match can be obtained for both components by considering a slightly higher metallicity, as also suggested by Brogaard et al. (2017). For example, the green dashed isochrones in Fig. 12 illustrate the effects of assuming [Fe/H] = −0.6 and [α/Fe] = 0.4, which correspond to Z = 0.008, and an He content of Y = 0.263. In this example, the He content of the isochrone follows equation (1) and the [α/Fe] value is chosen to be 0.4 without any special consideration. We also note that, to bring the radius discrepancy of the primary component within 3σ or 1σ, with the abundances assumed for the FG population, an age of 11.7 or 11.2 Gyr would be required, respectively. These ages, in particularly the lower one, would be quite different from the one obtained by the best fit presented in this paper. Up to now, we have assumed that effects of binary interaction are negligible so that V69 can be analysed with single star evolution models. However, considering other possible causes for the discrepancy, we note that the primary component of V69 is among the bluest stars past the turn-off of the CMD (see fig. 4 of Thompson et al. 2010) so that the fitting isochrone would likely fit the bluer envelope of the cluster CMD. We thus suspect that its position in the CMD could partly be due to the effects of the binary dynamical interaction with the companion star. In fact, tidal effects in close binary stars may change the structure and evolution of stars even before any possible mass transfer (de Mink et al. 2009; Song et al. 2016). In particular, the primary star of V69 matches our model that shows a thin surface convective envelope (∼2 per cent of total mass) and, following the simple approximation of Zahn (1977, equation 6.1), its tidal synchronization time should be roughly ∼7.2 Gyr, comparable to its current age. Tidal friction during the previous evolution may have introduced shear mixing and extra turbulence (Lanza & Mathis 2016) at the base of the external convective region, whose effects are primarily those of mitigating helium and heavy element diffusion away from the convective region. The net effect will be a lower growth of the surface hydrogen abundance with a corresponding decrease of the current surface opacity, which should result in a smaller current radius, in the observed direction. Of course, a more sophisticated theoretical analysis is necessary to assess if the location of the primary star of V69 in the CMD might be affected by previous dynamical interaction, but this is beyond the scope of this paper. Figure 12. View largeDownload slide Mass–radius and Mass–Teff diagrams for binary V69. The observed values of the two components of V69 are marked with black stars, their corresponding 1σ and 3σ uncertanties are indicated with dashed line boxes and shaded boxes, respectively (data from Thompson et al. 2010). Isochrones of 47 Tuc FG (red solid lines) and SG (blue dotted lines) are overlaid. To illustrate the effects of a higher metallicity, isochrones of [Fe/H] =−0.6, [α/Fe] =0.4 (Z = 0.008, Y = 0.263) are also displayed (green dashed lines). All isochrones are with our best estimate age 12.0 Gyr. Figure 12. View largeDownload slide Mass–radius and Mass–Teff diagrams for binary V69. The observed values of the two components of V69 are marked with black stars, their corresponding 1σ and 3σ uncertanties are indicated with dashed line boxes and shaded boxes, respectively (data from Thompson et al. 2010). Isochrones of 47 Tuc FG (red solid lines) and SG (blue dotted lines) are overlaid. To illustrate the effects of a higher metallicity, isochrones of [Fe/H] =−0.6, [α/Fe] =0.4 (Z = 0.008, Y = 0.263) are also displayed (green dashed lines). All isochrones are with our best estimate age 12.0 Gyr. The envelope overshooting calibration together with the α-enhanced metal mixtures of 47 Tuc is applied to other metallicities till Z = 0.0001. The RGB bump magnitudes of the new α-enhanced isochrones are compared with other stellar models and GC observations. We take $$\Delta ^{\rm MSTO}_{\rm RGBB}$$, the magnitude difference between the MSTO and the RGB bump, as the reference to compare with the observation in order to avoid uncertainties from distance and extinction. Our new models fit the data quite well and significantly improve the prediction of RGB bump magnitude compared to previous models. However, we notice that in Fig. 11 around [M/H] = −2.0 ([Fe/H] ∼ −2.3) our model predicts $$\Delta V^{\rm MSTO}_{\rm RGBB}$$ about ∼0.1 mag greater than the data points obtained by Nataf et al. (2013). If we consider a more He-rich model with the same metallicity Z, the discrepancy will become even larger. There are works arguing that diffusion also affects the brightness of RGBB (eg. Michaud, Richer & Richard 2010; Cassisi et al. 2011; Joyce & Chaboyer 2015). Michaud et al. (2010) conclude that without atomic diffusion the RGBB luminosity is about 0.02 dex brighter (∼0.05 mag). In parsec , we always take diffusion into account. To see the effect of diffusion on RGBB morphology, we calculate a 0.80 M⊙ model without diffusion and compare it with the one with standard parsec diffusion in Fig. 13. The two stars have the same metallicity, He content, stellar mass, and [α/Fe] . As we can see from the right-hand panel, diffusion shorten the MS lifetime. The left-hand panel of this figure is HRD, similar to that in fig. 1 of Michaud et al. (2010), evolutionary track with diffusion shows redder MSTO and slightly fainter RGBB. For isochrones obtained from these two sets of evolutionary tracks, at 13 Gyr, the RGBB without diffusion is 0.072 mag (in F606W) brighter than the one with diffusion, and $$\Delta V^{\rm MSTO}_{\rm RGBB}$$ value is 0.008 mag (in F606W) larger. This result confirms that inhibiting the diffusion during H-burning phase will eventually makes the discrepancy more severe. Pietrinferni, Cassisi & Salaris (2010) conclude that the updated nuclear reaction rate for $$\rm \,{}^{14}\hspace{-0.8pt}{N}\,({p}\,,{\gamma }) \,{}^{15}\hspace{-0.8pt}{O}\,$$ makes RGBB brighter by ∼0.06 mag compared to the old rate. However, we remind that we are already adopting the new rate (Imbriani et al. 2005) for this reaction (Table 1). Assuming that all other input physics, in particular opacities, is correct, the only possible solution to cover this ∼0.1 mag discrepancy is that the mixing at the bottom of the convective envelope is even higher than that assumed here. Either EOV in metal-poor stars is larger than our adopted value (e.g. Λe = 0.7Hp suggested by Alongi et al. 1991) or another kind of extra mixing is responsible. Figure 13. View largeDownload slide Comparison of [Z = 0.0002, Y = 0.249] tracks for an M = 0.80 M⊙ star with (orange line) and without (blue line) diffusion. The left-hand panel shows the HRD of these two tracks, with the RGBB region zoomed in in the subfigure. The right-hand panel illustrates the luminosity evolution as the star ages. Figure 13. View largeDownload slide Comparison of [Z = 0.0002, Y = 0.249] tracks for an M = 0.80 M⊙ star with (orange line) and without (blue line) diffusion. The left-hand panel shows the HRD of these two tracks, with the RGBB region zoomed in in the subfigure. The right-hand panel illustrates the luminosity evolution as the star ages. We have shown in Section 4.2 that, comparing the absolute magnitude MV, RGBB between model and observation directly, would introduce uncertainties from distance and extinction. However, putting the MV, RGBB and $$\Delta V^{\rm MSTO}_{\rm RGBB}$$ comparison together could help us to constrain the distance of the clusters. Fig. 14 displays the differences between the theoretical MV, RGBB and those of data from Nataf et al. (2013) and Cassisi et al. (2011). The absolute magnitude MV, RGBB of the data takes into account the apparent distance modulus, hence extinction effects are excluded. The four sets of models in Fig. 14 are the same as those in Fig. 11. The discrepancy of MV, RGBB between models and the data at the metal-poor end in Fig. 14 is larger than that of $$\Delta V^{\rm MSTO}_{\rm RGBB}$$ in Fig. 11 in both [M/H] and [Fe/H] frames. Since $$\Delta V^{\rm MSTO}_{\rm RGBB}$$ is a reference without distance effect, this larger discrepancy indicates that the apparent distance modulus (m − M)V used in Fig. 14 (Nataf et al. 2013) for metal-poor GCs are underestimated. Figure 14. View largeDownload slide Absolute magnitude in F606W band of RGBB (MV, RGBB) as a function of metallicity [M/H] (left panel) and iron abundance [Fe/H] (right panel). Three sets of EOV-calibrated parsec models, [α/Fe] =[0.0, 0.2, 0.4], and parsec v1.2S are shown both at age 13 Gyr (solid line) and 11 Gyr (dashed line). For comparison, Nataf et al. (2013, grey filled dots with error bar) and Cassisi et al. (2011, black filled dots with error bar) RGBB data are plotted. See the text for the details. Figure 14. View largeDownload slide Absolute magnitude in F606W band of RGBB (MV, RGBB) as a function of metallicity [M/H] (left panel) and iron abundance [Fe/H] (right panel). Three sets of EOV-calibrated parsec models, [α/Fe] =[0.0, 0.2, 0.4], and parsec v1.2S are shown both at age 13 Gyr (solid line) and 11 Gyr (dashed line). For comparison, Nataf et al. (2013, grey filled dots with error bar) and Cassisi et al. (2011, black filled dots with error bar) RGBB data are plotted. See the text for the details. In a following paper of the ‘parsec α-enhanced stellar evolutionary tracks and isochrones’ series, we will provide other [α/Fe] choices based on metal mixtures derived from atlas9 APOGEE atmosphere models (Mészáros et al. 2012). The full set of isochrones with chemical compositions suitable for GCs and Galactic bulge/thick disc stars will be available online after the full calculation and calibration are performed. Acknowledgements XF thanks Angela Bragaglia and Francesca Primas for the useful discussion, and Fiorella Castelli for the guide on ATLAS12 calculation. AB acknowledges INAF-PRIN-2014-14 ‘Star formation and evolution in galactic nuclei’, and thanks Michela Mapelli for the useful discussion on binary interaction. PM, JM, YC, and AN acknowledge support from the ERC Consolidator Grant funding scheme (project STARKEY, G. A. n. 615604). Footnotes 1 CMD input form: http://stev.oapd.inaf.it/cgi-bin/cmd/. 2 http://opalopacity.llnl.gov/ 3 http://stev.oapd.inaf.it/aesopus 4 http://freeeos.sourceforge.net/ 5 $$\rm {A(X)}=\log (\rm {N}_{\rm X}/\rm {N}_{\rm H}) + 12$$, with NX is the abundance in number for the element X. 6 [X/Fe] = log (NX/NFe) − log (NX/NFe)⊙ REFERENCES Alongi M., Bertelli G., Bressan A., Chiosi C., 1991, A&A , 244, 95 Alves-Brito A. et al.  , 2005, A&A , 667, 657 CrossRef Search ADS   Anderson J. et al.  , 2008, AJ , 135, 2055 https://doi.org/10.1088/0004-6256/135/6/2055 CrossRef Search ADS   Anderson J., Piotto G., King I. R., Bedin L. R., Guhathakurta P., 2009, ApJ , 697, L58 https://doi.org/10.1088/0004-637X/697/1/L58 CrossRef Search ADS   Angulo C. et al.  , 1999, Nucl. Phys. A , 656, 3 https://doi.org/10.1016/S0375-9474(99)00030-5 CrossRef Search ADS   Balbinot E. et al.  , 2016, ApJ , 820, 58 https://doi.org/10.3847/0004-637X/820/1/58 CrossRef Search ADS   Bedin L. R., Piotto G., Anderson J., Cassisi S., King I. R., Momany Y., Carraro G., 2004, ApJ , 605, L125 https://doi.org/10.1086/420847 CrossRef Search ADS   Belczynski K., Holz D. E., Bulik T., O'Shaughnessy R., 2016, Nature , 534, 512 https://doi.org/10.1038/nature18322 CrossRef Search ADS PubMed  Bergbusch P. A., Stetson P. B., 2009, AJ , 138, 1455 https://doi.org/10.1088/0004-6256/138/5/1455 CrossRef Search ADS   Betts R. R., Fortune H. T., Middleton R., 1975, Phys. Rev. C , 11, 19 https://doi.org/10.1103/PhysRevC.11.19 CrossRef Search ADS   Bjork S. R., Chaboyer B., 2006, ApJ , 641, 1102 https://doi.org/10.1086/500505 CrossRef Search ADS   Bono G., Cassisi S., Zoccali M., Piotto G., 2001, ApJ , 546, L109 https://doi.org/10.1086/318866 CrossRef Search ADS   Borissova J. et al.  , 2014, A&A , 569, A24 CrossRef Search ADS   Bovy J., Rix H.-W., Green G. M., Schlafly E. F., Finkbeiner D. P., 2016, ApJ , 818, 130 https://doi.org/10.3847/0004-637X/818/2/130 CrossRef Search ADS   Bragaglia A., Carretta E., Gratton R., D'Orazi V., Cassisi S., Lucatello S., 2010, A&A , 519, A60 CrossRef Search ADS   Bragaglia A., Carretta E., Sollima A., Donati P., D'Orazi V., Gratton R. G., Lucatello S., Sneden C., 2015, A&A , 583, A69 CrossRef Search ADS   Bressan A., Marigo P., Girardi L., Salasnich B., Dal Cero C., Rubele S., Nanni A., 2012, MNRAS , 427, 127 https://doi.org/10.1111/j.1365-2966.2012.21948.x CrossRef Search ADS   Bressan A. Girardi L. Marigo P. Rosenfield P. Tang J., 2015, in Miglio A. Eggenberger P. Girardi L. Montalbán J., eds., Astrophysics and Space Science Proceedings, Vol. 39, Asteroseismology of Stellar Populations in the Milky Way . Springer, Cham, p. 25 Briley M. M., 1997, AJ , 114, 1051 https://doi.org/10.1086/118535 CrossRef Search ADS   Brogaard K., VandenBerg D. A., Bedin L. R., Milone A. P., Thygesen A., Grundahl F., 2017, MNRAS , 468, 645 https://doi.org/10.1093/mnras/stx378 CrossRef Search ADS   Caffau E., Ludwig H.-G., Steffen M., Freytag B., Bonifacio P., 2011, Sol. Phys. , 268, 255 https://doi.org/10.1007/s11207-010-9541-4 CrossRef Search ADS   Caloi V., D'Antona F., 2005, A&A , 435, 987 CrossRef Search ADS   Cannon R. D., Croke B. F. W., Bell R. A., Hesser J. E., Stathakis R. A., 1998, MNRAS , 298, 601 https://doi.org/10.1046/j.1365-8711.1998.01671.x CrossRef Search ADS   Cardelli J. A., Clayton G. C., Mathis J. S., 1989, ApJ , 345, 245 https://doi.org/10.1086/167900 CrossRef Search ADS   Carney B. W., 1996, Publ. Astron. Soc. Pac. , 108, 900 https://doi.org/10.1086/133811 CrossRef Search ADS   Carretta E., Bragaglia A., Gratton R. G., Leone F., Recio-Blanco A., Lucatello S., 2006, A&A , 450, 523 CrossRef Search ADS   Carretta E. et al.  , 2007, A&A , 464, 967 CrossRef Search ADS   Carretta E., Bragaglia A., Gratton R., D'Orazi V., Lucatello S., 2009a, A&A , 508, 695 CrossRef Search ADS   Carretta E., Bragaglia A., Gratton R., Lucatello S., 2009b, A&A , 505, 139 CrossRef Search ADS   Carretta E., Gratton R. G., Bragaglia A., D'Orazi V., Lucatello S., 2013, A&A , 550, A34 CrossRef Search ADS   Carretta E., D'Orazi V., Gratton R. G., Lucatello S., 2014, A&A , 563, A32 CrossRef Search ADS   Casey A. R. et al.  , 2016, MNRAS , 461, 3336 https://doi.org/10.1093/mnras/stw1512 CrossRef Search ADS   Cassisi S., Salaris M., 1997, MNRAS , 285, 593 https://doi.org/10.1093/mnras/285.3.593 CrossRef Search ADS   Cassisi S., Salaris M., Bono G., 2002, ApJ , 565, 1231 https://doi.org/10.1086/324695 CrossRef Search ADS   Cassisi S., Marín-Franch A., Salaris M., Aparicio A., Monelli M., Pietrinferni A., 2011, A&A , 527, A59 CrossRef Search ADS   Catelan M., 2008, Proc. Int. Astron. Union , 4, 209 https://doi.org/10.1017/S174392130903186X CrossRef Search ADS   Caughlan G. R., Fowler W. A., 1988, At. Data Nucl. Data Tables , 40, 283 https://doi.org/10.1016/0092-640X(88)90009-5 CrossRef Search ADS   Cecco A. D. et al.  , 2010, ApJ , 712, 527 https://doi.org/10.1088/0004-637X/712/1/527 CrossRef Search ADS   Chen Y., Girardi L., Bressan A., Marigo P., Barbieri M., Kong X., 2014, MNRAS , 444, 2525 https://doi.org/10.1093/mnras/stu1605 CrossRef Search ADS   Chen Y., Bressan A., Girardi L., Marigo P., Kong X., Lanza A., 2015, MNRAS , 452, 1068 https://doi.org/10.1093/mnras/stv1281 CrossRef Search ADS   Chevallard J., Charlot S., 2016, MNRAS , 462, 1415 https://doi.org/10.1093/mnras/stw1756 CrossRef Search ADS   Chiosi C., Bressan A., Portinari L., Tantalo R., 1998, A&A , 381, 355 Choi J., Dotter A., Conroy C., Cantiello M., Paxton B., Johnson B. D., 2016, ApJ , 823, 102 https://doi.org/10.3847/0004-637X/823/2/102 CrossRef Search ADS   Christensen-Dalsgaard J., Monteiro M. J. P. F. G., Rempel M., Thompson M. J., 2011, MNRAS , 414, 1158 https://doi.org/10.1111/j.1365-2966.2011.18460.x CrossRef Search ADS   Constantini H. LUNA Collaboration, 2010, Proc. Sci., Reaction Rate Measurements in Underground Laboratories . SISSA, Trieste, PoS(NIC XI)014 Cordero M. J., Pilachowski C. A., Johnson C. I., McDonald I., Zijlstra A. A., Simmerer J., 2014, ApJ , 780, 94 https://doi.org/10.1088/0004-637X/780/1/94 CrossRef Search ADS   Cyburt R. H., Hoffman R., Woosley S., 2012, Re-evaluation of Buchmann 1996 C12(a,g) and Hoffman Ca40(a,g) and Ti44(a,p) rates . REACLIB Cyburt R. H., Davids B., 2008, Phys. Rev. C , 78, 064614 https://doi.org/10.1103/PhysRevC.78.064614 CrossRef Search ADS   Cyburt R. H. et al.  , 2010, ApJS , 189, 240 https://doi.org/10.1088/0067-0049/189/1/240 CrossRef Search ADS   D'Antona F., Caloi V., Montalbán J., Ventura P., Gratton R., 2002, A&A , 395, 69 CrossRef Search ADS   D'Antona F., Bellazzini M., Caloi V., Pecci F. F., Galleti S., Rood R. T., 2005, ApJ , 631, 868 https://doi.org/10.1086/431968 CrossRef Search ADS   D'Orazi V., Lucatello S., Gratton R., Bragaglia A., Carretta E., Shen Z., Zaggia S., 2010, ApJ , 713, L1 https://doi.org/10.1088/2041-8205/713/1/L1 CrossRef Search ADS   de Mink S. E., Cantiello M., Langer N., Pols O. R., Brott I., Yoon S.-C., 2009, A&A , 497, 243 CrossRef Search ADS   Descouvemont P., Adahchour A., Angulo C., Coc A., Vangioni-Flam E., 2004, Atomic Data and Nuclear Data Tables , 88, 203 https://doi.org/10.1016/j.adt.2004.08.001 CrossRef Search ADS   Di Criscienzo M., Ventura P., D'Antona F., Milone a., Piotto G., 2010, MNRAS , 408, 999 https://doi.org/10.1111/j.1365-2966.2010.17168.x CrossRef Search ADS   Dobrovolskas V. et al.  , 2014, A&A , 565, A121 CrossRef Search ADS   Donati P. et al.  , 2014, A&A , 561, A94 CrossRef Search ADS   Dotter A. L., 2007, PhD thesis , Dartmouth College Dotter A., Chaboyer B., Jevremović D., Baron E., Ferguson J. W., Sarajedini A., Anderson J., 2007, AJ , 134, 376 https://doi.org/10.1086/517915 CrossRef Search ADS   Dotter A., Kaluzny J., Thompson I. B., 2008, Proc. Int. Astron. Union , 4, 171 -176 https://doi.org/10.1017/S1743921309031822 CrossRef Search ADS   Dotter A., Chaboyer B., Jevremović D., Kostov V., Baron E., Ferguson J. W., 2008, ApJS , 178, 89 https://doi.org/10.1086/589654 CrossRef Search ADS   Dotter A. et al.  , 2010, ApJ , 708, 698 https://doi.org/10.1088/0004-637X/708/1/698 CrossRef Search ADS   Dupree A. K., Strader J., Smith G. H., 2011, ApJ , 728, 155 https://doi.org/10.1088/0004-637X/728/2/155 CrossRef Search ADS   Fagotto F. Bressan A. Bertelli G. Chiosi C., 1994, A&AS , 104 Fu X., 2006, PhD thesis , SISSA - International School for Advanced Studies Fu X., Bressan A., Molaro P., Marigo P., 2015, MNRAS , 452, 3256 https://doi.org/10.1093/mnras/stv1384 CrossRef Search ADS   Fulbright J. P., 2002, AJ , 123, 404 https://doi.org/10.1086/324630 CrossRef Search ADS   Fusi Pecci F. Bellazzini M., 1997, in Philip A. G. D. Liebert J. Saffer R. Hayes D. S., eds., The Third Conference on Faint Blue Stars . Davis Press, Schenectady, p. 255 Fusi Pecci F., Ferraro F. R., Crocker D. A., Rood R. T., Buonanno R., 1990, A&A , 238, 95 Fynbo H. O. U. et al.  , 2005, Nature , 433, 136 https://doi.org/10.1038/nature03219 CrossRef Search ADS PubMed  Gonzalez O. A. et al.  , 2011, A&A , 530, A54 CrossRef Search ADS   Goudfrooij P., Girardi L., Rosenfield P., Bressan A., Marigo P., Correnti M., Puzia T. H., 2015, MNRAS , 450, 1693 https://doi.org/10.1093/mnras/stv700 CrossRef Search ADS   Gratton R. G. et al.  , 2001, A&A , 369, 87 CrossRef Search ADS   Gratton R. G., Carretta E., Bragaglia A., Lucatello S., D'Orazi V., 2010, A&A , 517, A81 CrossRef Search ADS   Gratton R. G., Carretta E., Bragaglia A., 2012, A&AR , 20, 50 CrossRef Search ADS   Gratton R. G. et al.  , 2013, A&A , 549, A41 CrossRef Search ADS   Gratton R. G. et al.  , 2015, A&A , 573, A92 CrossRef Search ADS   Gullikson K., Kraus A., Dodson-Robinson S., 2016, AJ , 152, 40 https://doi.org/10.3847/0004-6256/152/2/40 CrossRef Search ADS   Gutkin J., Charlot S., Bruzual G., 2016, MNRAS , 462, 1757 https://doi.org/10.1093/mnras/stw1716 CrossRef Search ADS   Hansen B. M. S. et al.  , 2013, Nature , 500, 51 https://doi.org/10.1038/nature12334 CrossRef Search ADS PubMed  Harbeck D., Smith G. H., Grebel E. K., 2003, AJ , 125, 197 https://doi.org/10.1086/345570 CrossRef Search ADS   Harris W. E., 1996, AJ , 112, 1487 https://doi.org/10.1086/118116 CrossRef Search ADS   Heil M. et al.  , 2008, Phys. Rev. C , 78, 025803 https://doi.org/10.1103/PhysRevC.78.025803 CrossRef Search ADS   Herwig F., 2000, A&A , 360, 952 Heyl J., Kalirai J., Richer H. B., Marigo P., Antolini E., Goldsbury R., Parada J., 2015, ApJ , 810, 127 https://doi.org/10.1088/0004-637X/810/2/127 CrossRef Search ADS   Iglesias C. A., Rogers F. J., 1996, ApJ , 464, 943 https://doi.org/10.1086/177381 CrossRef Search ADS   Iliadis C., Angulo C., Descouvemont P., Lugaro M., Mohr P., 2008, Phys. Rev. C , 77, 045802 https://doi.org/10.1103/PhysRevC.77.045802 CrossRef Search ADS   Iliadis C., Longland R., Champagne A. E., Coc A., 2010, Nucl. Phys. A , 841, 251 https://doi.org/10.1016/j.nuclphysa.2010.04.010 CrossRef Search ADS   Imbriani G. et al.  , 2005, Eur. Phys. J. A , 25, 455 https://doi.org/10.1140/epja/i2005-10138-7 CrossRef Search ADS   Itoh N., Uchida S., Sakamoto Y., Kohyama Y., Nozawa S., 2008, ApJ , 677, 495 https://doi.org/10.1086/529367 CrossRef Search ADS   Johnson C. I., Rich R. M., Kobayashi C., Kunder A., Koch A., 2014, AJ , 148, 67 https://doi.org/10.1088/0004-6256/148/4/67 CrossRef Search ADS   Joyce M., Chaboyer B., 2015, ApJ , 814, 142 https://doi.org/10.1088/0004-637X/814/2/142 CrossRef Search ADS   Kalari V. M. et al.  , 2014, A&A , 564, L7 CrossRef Search ADS   Kalirai J. S. et al.  , 2012, AJ , 143, 11 CrossRef Search ADS   Kim Y., Demarque P., Yi S. K., Alexander D. R., 2002, ApJS , 143, 499 CrossRef Search ADS   King C. R., Da Costa G. S., Demarque P., 1985, ApJ , 299, 674 CrossRef Search ADS   Kirby E. N., Cohen J. G., Smith G. H., Majewski S. R., Sohn S. T., Guhathakurta P., 2011, ApJ , 727, 79 CrossRef Search ADS   Komatsu E. et al.  , 2011, ApJS , 192, 18 CrossRef Search ADS   Küpper A. H. W., Balbinot E., Bonaca A., Johnston K. V., Hogg D. W., Kroupa P., Santiago B. X., 2015, ApJ , 803, 80 CrossRef Search ADS   Kurucz R. L., 2005, Mem. Soc. Astron. Ital. , 8, 14 Lanza A. F., Mathis S., 2016, Celest. Mech. Dyn. Astron. , 126, 249 CrossRef Search ADS   Lebreton Y., Goupil M. J., 2012, A&A , 544, L13 CrossRef Search ADS   Li Z. et al.  , 2010, Sci. China Phys., Mech. Astron. , 53, 658 CrossRef Search ADS   Li J. et al.  , 2016, ApJ , 823, 59 CrossRef Search ADS   Maldonado J., Eiroa C., Villaver E., Montesinos B., Mora A., 2015, A&A , 579, A20 CrossRef Search ADS   Marigo P., Aringer B., 2009, A&A , 508, 1539 CrossRef Search ADS   Marigo P., Girardi L., 2007, A&A , 469, 239 CrossRef Search ADS   Marigo P., Bressan A., Nanni A., Girardi L., Pumo M. L., 2013, MNRAS , 434, 488 CrossRef Search ADS   Marigo P. et al.  , 2017, ApJ , 835, 77 CrossRef Search ADS   Marino A. F. et al.  , 2011, A&A , 532, A8 CrossRef Search ADS   Marino A. F. et al.  , 2014, MNRAS , 437, 1609 CrossRef Search ADS   McDonald I., Zijlstra A. A., 2015, MNRAS , 448, 502 CrossRef Search ADS   McWilliam A., Preston G. W., Sneden C., Searle L., 1995, AJ , 109, 2757 CrossRef Search ADS   Mészáros S. et al.  , 2012, AJ , 144, 120 CrossRef Search ADS   Michaud G., Richer J., Richard O., 2010, A&A , 510, A104 CrossRef Search ADS   Milone a. P. et al.  , 2012, ApJ , 744, 58 CrossRef Search ADS   Milone A. P. et al.  , 2008, ApJ , 673, 241 CrossRef Search ADS   Milone A. P. et al.  , 2010, ApJ , 709, 1183 CrossRef Search ADS   Milone A. P., Marino A. F., Piotto G., Bedin L. R., Anderson J., Aparicio A., Cassisi S., Rich R. M., 2012, ApJ , 745, 27 CrossRef Search ADS   Milone A. P. et al.  , 2015a, MNRAS , 450, 3750 CrossRef Search ADS   Milone A. P. et al.  , 2015b, ApJ , 808, 51 CrossRef Search ADS   Milone A. P., Marino A. F., D'Antona F., Bedin L. R., Da Costa G. S., Jerjen H., Mackey A. D., 2016, MNRAS , 458, 4368 CrossRef Search ADS   Mucciarelli A., Lovisi L., Lanzoni B., Ferraro F. R., 2014, ApJ , 786, 14 CrossRef Search ADS   Nanni A., Bressan A., Marigo P., Girardi L., 2013, MNRAS , 434, 2390 CrossRef Search ADS   Nanni A., Bressan A., Marigo P., Girardi L., 2014, MNRAS , 438, 2328 CrossRef Search ADS   Nataf D. M., Udalski A., Gould A., Pinsonneault M. H., 2011, ApJ , 730, 118 CrossRef Search ADS   Nataf D. M., Gould A., Pinsonneault M. H., Stetson P. B., 2011, ApJ , 736, 94 CrossRef Search ADS   Nataf D. M., Gould A. P., Pinsonneault M. H., Udalski A., 2013, ApJ , 766, 77 CrossRef Search ADS   Nissen P. E., Gustafsson B., Edvardsson B., Gilmore G., 1994, A&A , 285, 440 Norris J., Freeman K. C., 1979, ApJ , 230, L179 CrossRef Search ADS   Origlia L., Rood R. T., Fabbri S., Ferraro F. R., Fusi Pecci F., Rich R. M., 2007, ApJ , 667, L85 CrossRef Search ADS   Pasquini L., Mauas P., Käufl H. U., Cacciari C., 2011, A&A , 531, A35 CrossRef Search ADS   Perren G. I., Vázquez R. A., Piatti A. E., 2015, A&A , 576, A6 CrossRef Search ADS   Pietrinferni A., Cassisi S., Salaris M., Castelli F., 2006, ApJ , 642, 797 CrossRef Search ADS   Pietrinferni A., Cassisi S., Salaris M., 2010, A&A , 522, A76 CrossRef Search ADS   Pietrinferni A., Cassisi S., Salaris M., Hidalgo S., 2013, A&A , 558, A46 CrossRef Search ADS   Piotto G. et al.  , 2007, ApJ , 661, L53 CrossRef Search ADS   Pritzl B. J., Venn K. A., Irwin M., 2005, AJ , 130, 2140 CrossRef Search ADS   Ramya P., Reddy B. E., Lambert D. L., Musthafa M. M., 2016, MNRAS , 460, 1356 CrossRef Search ADS   Reddy A. B. S., Lambert D. L., 2016, A&A , 589, A57 CrossRef Search ADS   Reddy B. E., Lambert D. L., Prieto C. A., 2006, MNRAS , 367, 1329 CrossRef Search ADS   Reimers D., 1975, Mem. Soc. R. Sci. Liege , 8, 369 Rood R. T. Crocker D. A., 1985, in Danziger I. J. Matteucci F. Kjar K., eds., European Southern Observatory Conference and Workshop Proceedings , Vol. 21. ESO, Garching, p. 61 Rosenfield P., Marigo P., Girardi L., Dalcanton J. J., Bressan A., Williams B. F., Dolphin A., 2016, ApJ , 822, 73 CrossRef Search ADS   Ruchti G. R. et al.  , 2010, ApJ , 721, L92 CrossRef Search ADS   Salaris M., Weiss A., Ferguson J. W., Fusilier D. J., 2006, ApJ , 645, 1131 CrossRef Search ADS   Salaris M., Held E. V., Ortolani S., Gullieuszik M., Momany Y., 2007, A&A , 476, 243 CrossRef Search ADS   Salaris M., Cassisi S., Pietrinferni A., 2016, A&A , 590, A64 CrossRef Search ADS   Salasnich B., Girardi L., Weiss A., Chiosi C., 2000, A&A , 361, 1023 Salpeter E. E., 1955, ApJ , 121, 161 CrossRef Search ADS   San Roman I. et al.  , 2015, A&A , 579, A6 CrossRef Search ADS   Sandage A., Wildey R., 1967, ApJ , 150, 469 CrossRef Search ADS   Santos N. C. et al.  , 2013, A&A , 556, A150 CrossRef Search ADS   Sarajedini A. et al.  , 2007, AJ , 133, 1658 CrossRef Search ADS   Schlafly E. F. et al.  , 2014, ApJ , 789, 15 CrossRef Search ADS   Schultheis M. et al.  , 2015, A&A , 577, A77 CrossRef Search ADS   Smiljanic R. et al.  , 2016, A&A , 589, A115 CrossRef Search ADS   Sneden C., 2004, Mem. Soc. Astron. Italiana , 75, 267 Song H. F., Meynet G., Maeder A., Ekström S., Eggenberger P., 2016, A&A , 585, A120 CrossRef Search ADS   Spada F., Demarque P., Kim Y. C., Sills A., 2013, ApJ , 776, 87 CrossRef Search ADS   Spera M., Mapelli M., Bressan A., 2015, MNRAS , 451, 4086 CrossRef Search ADS   Strandberg E. et al.  , 2008, Phys. Rev. C , 77, 055801 CrossRef Search ADS   Tang J., Bressan A., Rosenfield P., Slemer A., Marigo P., Girardi L., Bianchi L., 2014, MNRAS , 445, 4287 CrossRef Search ADS   Thompson I. B., Kaluzny J., Rucinski S. M., Krzeminski W., Pych W., Dotter A., Burley G. S., 2010, AJ , 139, 329 CrossRef Search ADS   Thygesen A. O. et al.  , 2014, A&A , 572, A108 CrossRef Search ADS   Troisi F. et al.  , 2011, Publ. Astron. Soc. Pac. , 123, 879 CrossRef Search ADS   Tuli J. K., 2012, Weak Rates from the Nuclear Wallet Cards . National Nuclear Data Center, Brookhaven Valcarce A. A. R., Catelan M., Sweigart A. V., 2012, A&A , 547, A5 CrossRef Search ADS   van den Bergh S., 1967, AJ , 72, 70 CrossRef Search ADS   VandenBerg D. A., Bergbusch P. A., Dotter A., Ferguson J. W., Michaud G., Richer J., Proffitt C. R., 2012, ApJ , 755, 15 CrossRef Search ADS   VandenBerg D. A., Brogaard K., Leaman R., Casagrande L., 2013, ApJ , 775, 134 CrossRef Search ADS   VandenBerg D. A., Bergbusch P. A., Ferguson J. W., Edvardsson B., 2014, ApJ , 794, 72 CrossRef Search ADS   VandenBerg D. A., 2013, in de Grijs R., ed., IAU Symp., Vol. 289, Advancing the Physics of Cosmic Distances . Kluwer, Dordrecht, p. 161 VandenBerg D. A., Swenson F. J., Rogers F. J., Iglesias C. A., Alexander D. R., 2000, ApJ , 532, 430 CrossRef Search ADS   Venn K. A., Irwin M., Shetrone M. D., Tout C. A., Hill V., Tolstoy E., 2004, AJ , 128, 1177 CrossRef Search ADS   Villanova S. et al.  , 2007, ApJ , 663, 296 CrossRef Search ADS   Villanova S., Piotto G., Gratton R. G., 2009, A&A , 499, 755 CrossRef Search ADS   Watkins L. L., van der Marel R. P., Bellini A., Anderson J., 2015, ApJ , 812, 149 CrossRef Search ADS   Weldrake D. T. F., Sackett P. D., Bridges T. J., Freeman K. C., 2004, AJ , 128, 736 CrossRef Search ADS   Zahn J.-P., 1977, A&A , 57, 383 Zhao G., Magain P., 1990, A&A , 238, 242 Zoccali M., Cassisi S., Piotto G., Bono G., Salaris M., 1999, ApJ , 518, L49 CrossRef Search ADS   © 2018 The Author(s) Published by Oxford University Press on behalf of the Royal Astronomical Society

### Journal

Monthly Notices of the Royal Astronomical SocietyOxford University Press

Published: May 1, 2018

## You’re reading a free preview. Subscribe to read the entire article.

### DeepDyve is your personal research library

It’s your single place to instantly
that matters to you.

over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month ### Explore the DeepDyve Library ### Search Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly ### Organize Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place. ### Access Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals. ### Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. DeepDyve ### Freelancer DeepDyve ### Pro Price FREE$49/month
\$360/year

Save searches from
PubMed

Create lists to

Export lists, citations