NuGrid Stellar Data Set. II. Stellar Yields from H to Bi for Stellar Models with to and to
We provide here a significant extension of the NuGrid Set 1 models in mass coverage and toward lower metallicity, adopting the same physics assumptions. The combined data set now includes the initial masses = 1, 1.65, 2, 3, 4, 5, 6, 7, 12, 15, 20, 25 for with -enhanced composition for the lowest three metallicities. These models are computed with the MESA stellar evolution code and are evolved up to the AGB, the white dwarf stage, or until core collapse. The nucleosynthesis was calculated for all isotopes in post-processing with the NuGrid mppnp code. Explosive nucleosynthesis is based on semi-analytic 1D shock models. Metallicity-dependent mass loss, convective boundary mixing in low- and intermediate mass models and H and He core burning massive star models is included. Convective O-C shell mergers in some stellar models lead to the strong production of odd-Z elements P, Cl, K and Sc. In AGB models with hot dredge-up the convective boundary mixing efficiency is reduced to accommodate for its energetic feedback. In both low-mass and massive star models at the lowest metallicity H-ingestion events are observed and lead to i-process nucleosynthesis and substantial production. Complete yield data tables, derived data products and online analytic data access are provided.
keywords:stars: abundances — evolution — interiors
Stellar yields data are a fundamental input for galactic chemical evolution models (e.g. Romano et al., 2010; Nomoto et al., 2013; Mollá et al., 2015), hydrodynamic models (e.g. Scannapieco et al., 2005) and chemodynamic models (e.g. Few et al., 2012; Côté et al., 2013; Schaye et al., 2015). Gibson (2002) and Romano et al. (2010) showed that results of chemical evolution models are strongly affected by uncertainties related to the choice of the yield set: for example, yield sets lead to differences in [C/O] ratio and for [C/Fe] in their galaxy models. These yield studies couple separate yield sets for massive and low-mass stars. These two separate sets often use different stellar evolution codes and different nuclear networks. In this paper, we present yields based on stellar models of a range of initial masses and metallicities calculated with the MESA (Paxton et al., 2011) stellar evolution code and post-processed with the NuGrid post-processing network (Pignatari et al., 2016b, P16).
This work builds upon the study by P16, and includes important improvements over this study. In this work, the same stellar code MESA (Paxton et al., 2011) is used for the full stellar set, while the yields set from P16 are calculated with different stellar evolution codes: MESA for AGB star models and the Geneva stellar evolution code (Eggenberger et al., 2008, GENEC) for massive star models. In this work, we have extended the set of models by adding more low-mass, intermediate-mass and massive stars: we provide models also for , , , and stars, including now low-mass supernova progenitors and super-AGB models, not included in the P16 set. In particular, a finer grid for intermediate-mass stars is important for galactic chemical evolution applications of the yield set, since these stars are important producers of and , in particular at low metallicity (e.g. Siess, 2010; Ventura & D’Antona, 2011; Karakas et al., 2012; Ventura et al., 2013; Gil-Pons et al., 2013; Doherty et al., 2014). Finally, in addition to new masses the yield set is extended by adding models with three lower metallicities for all initial masses. Below an -enhanced initial abundance is adopted which leads to [Fe/H] = -1.24, -2.03 and -3.03 for and .
The yields of massive AGB stars and super-AGB (S-AGB) stars depend on the nucleosynthesis during hot-bottom burning (HBB, Sackmann & Boothroyd, 1992; Lattanzio et al., 1996; Doherty et al., 2010; García-Hernández et al., 2013; Ventura et al., 2015). There are two options to resolve HBB in stellar models: either to couple the mixing and burning operators or choose time steps smaller than the convective turnover timescale of the envelope (e.g. for the , model). The difficulty in modeling the HBB process is that the large networks required for the heavy element nucleosynthesis in HBB require considerable computing time. But post-processing codes which decouple mixing and burning operators need to resolve the extremely short mixing time scale when HBB convective-reactive conditions are relevant. In this work we present a nested-network post-processing approach in which mixing and burning operators are coupled. With this approach we accurately calculate stellar yields also for isotopes affected by HBB conditions.
Ingestion events are common at low and zero-metallicity in AGB models of low mass (e.g. Fujimoto et al., 2000; Cristallo et al., 2009), in He-core flash in low-metallicity low-mass models (e.g. Campbell et al., 2010), and in S-AGB models in a wide range of metallicities (e.g. Gil-Pons & Doherty, 2010; Jones et al., 2016). The energy release as well as nuclear burning on the convective turn-over time scale due to H ingestion might violate the treatment of convection via mixing-length theory (MLT) (Herwig, 2001b) and/or the assumption of hydrostatic equilibrium (e.g. in S-AGB models Jones et al., 2016). The three-dimensional (3D) hydrodynamic simulations of H ingestion of the post-AGB star Sakurai’s object show that global and non-radial instabilities can be triggered in such convective-reactive phases can not be simulated in 1D stellar evolution (Herwig et al., 2014). Herwig et al. (2011) and Herwig (2001b) also reported that observational abundances and light curve of Sakurai’s object can not be explained with 1D models based on the MLT. Thus, the predictive power of 1D stellar evolution models to describe H ingestion events might be limited. The models nevertheless provide information about the frequency of such events as well as their potential impact on the production of elements.
Yield tables are typically provided in the literature but in order to trace back the underlying reasons for certain abundance features in yield tables it is important to have access to the full stellar models. In this paper we provide full web access of the stellar evolution and post-processing data including yield tables and an interactive interface to analyze and retrieve data.
The paper is organized as follows: in Sect. 2 we describe the methods used to perform the stellar evolution simulations, the semi-analytic models of the core-collapse supernova (CCSN) shock and post-processing. In Sect. 3 we introduce the general properties of stellar models and features related to low metallicity. In Sect. 4 we analyze the final yields at low metallicity. The latter are grouped by nucleosynthesis process. We discuss our assumptions in Sect. 5 and compare the results with available literature. In Sect. 6 we summarize the results.
The yields presented in this paper have been produced using 1D stellar evolution calculations and a semi-analytic prescription for CCSN shock propagation together with a post-processing nuclear reaction network. The details of three steps are described in this section.
2.1 Stellar evolution
The stellar evolution calculations were performed using the MESA stellar evolution code (Paxton et al., 2011), rev. 3709. The AGB models in NuGrid Set 1 (Pignatari et al., 2016b) were computed using rev. 3372 of MESA. The AGB models in this work adopt the same opacities as P16, in which case the two revisions produce similar results. For example, the time-evolution of H-free core masses agree to within . A comparison of AGB models of newer MESA revisions with the P16 models is presented in Battino et al. (2016). MESA rev. 3709 was also used for the massive star models. This is in contrast with P16, who used GENEC (Eggenberger et al., 2008). A detailed comparison of GENEC and MESA (and KEPLER) massive star models at solar metallicity was performed by Jones et al. (2015), who found that the CO core masses are within to of one another and the elemental abundances produced in the He core by the weak s-process agree within 30 %. The physics assumptions up to the end of core He burning in the massive star models are as in Jones et al. (2015).
2.1.1 Initial composition and nuclear reaction network
We use solar-scaled initial abundance at and as in P16, based on Grevesse & Noels (1993) and with the isotopic ratios from Lodders (2003). At and below we enhance the isotopes , , , , , , , and . The enhancements were derived from fits of halo and disk stars from Reddy et al. (2006) and references therein. For each enhanced isotope we apply Eq. 1 where and were derived from the fits for metallicities (Reddy et al., 2006). For we assume a constant of .
For isotopes of Ne, S and Ar values from Kobayashi et al. (2006) were adopted. The resulting and mass fractions for are shown in Table 1. The fit result gives which is close to the top of the [O/Fe] distribution but within the maximum given in Reddy et al. (2006). For the initial abundance of Li in AGB models with we choose as a lower limit the Li plateau (Sbordone et al., 2010). In other stellar models the initial Li abundance was unintentionally scaled down with metallicity as other light elements and unrealistic values were adopted.
In the low-mass stellar models up to we use the same network in MESA as in P16 (agb.net). For the massive AGB and super-AGB models () we use the network agbtomassive.net which includes an extended network for C, O and Ne burning and relevant electron-capture reactions. The nuclear reaction network for stellar models with masses is the same as in Jones et al. (2015, their Table 2) from the pre-main sequence until the depletion of oxygen in the core, at which point the network is reduced to approx21.net to follow Si burning and deleptonisation in the Fe core.
2.1.2 Mass loss
Semi-empirical prescriptions for mass loss (e.g. Vassiliadis & Wood, 1993; van Loon et al., 2005) are still commonly used in stellar evolution. In order to stay consistent with P16 we apply the mass loss prescription by Reimers (1975) for the red giant branch phase and the prescription of Blöcker & Schönberner (1995) for the AGB phase. Both prescription are functions of the mass, luminosity and radius of the stellar model. The efficiency parameter is increased to mimic the effect of the C-rich dust-driven phase as described in P16. A more realistic hydrodynamic approach to mass loss models (e.g. Mattsson et al., 2010) in combination with observational calibrations taking into account better data now available (e.g. Rosenfield et al., 2014) should ultimately be deployed for yield calculations.
Our approach here aims to bridge the mass loss choice of P16 with that of Herwig (2004a, H04) who adopted a metallicity dependent mass loss based on van Loon (2000). Since the H04 and these MESA models are slightly different we derive values of to be used in the MESA models to obtain the same mass loss as in H04. We then fit in the mass-metallicity plane to be constrained by the mass loss adopted in P16 for and and by H04 for . The resulting spline fit of in the mass-metallicity plane is shown in Fig. 1. We have added ad-hoc values for stellar models of = 4, 6, 7, 8 for solar and half-solar metallicity to extrapolate the general trend of decreasing at higher initial mass. The fit corresponds to the general notion that , and with it the mass loss, decreases for low-mass AGB stars with decreasing metallicity (Willson, 2000). This contrasts with the observational findings of shorter AGB lifetimes with lower metallicity in low-mass AGB stars (Rosenfield et al., 2014).
The mass loss prescription adopted in the massive star models depends on the effective temperature and the surface hydrogen mass fraction X(H) as in Glebbeek et al. (2009). For and we adopt the mass loss rate of Vink et al. (2001). At lower temperatures the Vink et al. (2001) rate transits into the de Jager et al. (1988) rate and the latter is adopted below . If we adopt either Nugis & Lamers (2000) when , otherwise de Jager et al. (1988). The Nugis & Lamers (2000) and Vink et al. (2001) rates depend explicitly on metallicity. See Glebbeek et al. (2009) for further details. A correction factor of 0.8 is adopted for mass loss rates of massive star models as deduced for MS OB stars in Maeder & Meynet (2001).
2.1.3 Hot-bottom burning
HBB is the activation of the CNO cycle at the bottom of the convective envelope in massive AGB and S-AGB stars (Scalo et al., 1975; Sackmann & Boothroyd, 1992). Higher temperatures in the AGB envelopes at lower metallicity lead to the activation of HBB at lower initial mass compared to AGB models of higher metallicity. This increases the number of stars which experience HBB with decreasing metallicity.
During HBB the mixing timescale of the convective envelope and nuclear timescales of CNO p-capture reactions become similar as shown for the , model in Fig. 2. is calculated as where is the diffusion coefficient and is the mixing length according to MLT. The coupling of mixing and burning operators in stellar evolution codes allow to resolve HBB correctly. Typically, post-processing codes decouple mixing and burning in order to solve differential equations for large reaction networks including heavy elements. To model HBB in the decoupled approach it is necessary to resolve the mixing timescale at the bottom of the convective envelope. This is just hours, for example in this , model (Fig. 2), which is short compared to the interpulse phases of tens of thousands of years. Cristallo et al. (2015) calculate heavy elements with a large network in their stellar evolution code and approximate CNO production due to HBB with a burn-mix-burn step. Our solution is to solve the coupled reaction and diffusion equations for a subset of important isotopes (see Sect. 2.3).
2.1.4 Convective boundary mixing treatment
We apply convective boundary mixing (CBM) at all convective boundaries of the AGB models. CBM is modeled with an exponential-diffusive convective boundary mixing model (Freytag et al., 1996; Herwig, 2000). A CBM efficiency of is used at all convective boundaries of AGB models except for the bottom of the pulse-driven convective zone (PDCZ) and during the third dredge-up (TDUP) of the thermal-pulse (TP)-AGB stage. Motivated by 2D and 3D simulations of Herwig et al. (2007) a lower CBM efficiency of is applied at the PDCZ bottom boundary. An increased mixing efficiency of is applied at the bottom of the convective envelope during the TDUP which is calibrated for low-mass stellar models to produce the pocket (Herwig et al., 2003). This approach is the same as in P16.
CBM is only accounted for in the massive star models from the pre-main sequence up to the end of core He burning. It is implemented as the exponential diffusion model of Freytag et al. (1996) with at all convective boundaries except for the bottom of convective shells in which nuclear fuel is burning, where was used. From the extinction of core He burning, which we have defined as the time when the central mass fraction of helium falls below , is set to zero, equivalent to assuming no CBM.
Corrosive H-burning during TDUP in low-metallicity massive AGB stars leads to an increase of the TDUP efficiency and is referred to as hot dredge-up (HDUP, Herwig, 2004a). The application of CBM at the bottom of the convective envelope results in strong burning of the mixed protons below the envelope and extreme TDUP efficiencies in these massive AGB models at low metallicity. In a , test model with CBM parameter used for the -pocket formation in low-mass AGB stars the TDUP penetrates into the C/O core after the sixth TP as shown in the Kippenhahn diagram in Fig. 3. This finding is in agreement with Herwig (2004b) who found that the HDUP can penetrate into the C/O core and terminate the AGB phase (see also Goriely & Siess, 2004). The abundance profile during the TDUP at the bottom of the convective envelope shows the peak of nuclear burning in the CBM region which steepens the radiative gradient and hence leads to a deeper penetration of the envelope into the He intershell (Fig. 3). Karakas (2010) models do not experience HDUP because the authors do not model CBM in the stellar evolution simulation. Instead, they introduce an ad-hoc partial mixing zone for the formation of the -pocket in the post-processing simulations.
One way to reduce the vigour of H burning during the HDUP is the reduction of . The efficiency of CBM at the lower boundary of the convective envelope in massive and S-AGB is not known. Investigations of the impact of CBM efficiency on structure and nucleosynthesis such as for S-AGB models by Jones et al. (2016) are required. A physical interpretation of the assumption of a reduced CBM is based on the buoyancy of the mixed and burning material which hinders boundary mixing. The situation is similar to the bottom of convective burning shells in the late stage of massive stars where the energy release leads to a lower CBM and a stiffer boundary (e.g. Cristini et al., 2016; Jones et al., 2017). Following Herwig (2004a), we limit CBM by reducing here to for models if the dredge-up after a thermal pulse is hot (Table 2). With this approach we prevent the termination of the AGB phase due to too extreme H burning during the TDUP.
2.2 Semi-analytic CCSN explosions
We use a semi-analytic approach for core-collapse supernova explosions as described in P16. The method drives a shock off the proto-neutron star based on a mass cut derived from Fryer et al. (2012, F12). The mass cuts are mass- and metallicity dependent and are provided for delayed and a rapid explosion prescription. The mass coordinates based on these models are shown in Table 3. For some massive star models such as the , model the mass cut is deeper located than than the outer edge of the Fe core as visible from the Fe-core masses in Table 4.
One of the big uncertainties in the yields is the position of the mass cut. The data from F12 were based on fits to the stellar structures produced by comparing the models from a range of stellar evolution codes (Woosley et al., 2002; Limongi & Chieffi, 2006; Young et al., 2009). These mass-cut prescriptions were then validated against the compact remnant mass distribution (Belczynski et al., 2012). For these stellar evolution models, the mass cut is fairly similar for models with . However, in particular for the model, the core from the MESA model is much larger than that produced by the Kepler code. This corresponds to much higher densities in the inner and, based on the F12 results, we expect the MESA models to collapse down to a black hole rather than explode to produce a low-mass neutron star. In this case, the stars would not provide SN yields and would contribute to the chemical evolution of the Galaxy only by stellar winds. In part, these results for the stellar progenitors are caused by the use of a small nuclear network in the MESA code during Si burning.
At earlier times, the MESA models with and GENEC models with of P16 are very similar. Therefore, for the MESA models with we also use the mass cut prescription of F12 under the assumption of as adopted for these GENEC models. This allows to provide a SN yield set of these MESA models at all metallicities. For more massive MESA models, we use the mass cut prescription of F12 as in P16.
2.3 Nucleosynthesis code and processed data
The temperature, density and diffusion coefficient (from the mixing length theory of convection along with the convective boundary mixing model) and in the MESA stellar evolution models are saved every time step and post-processed with the multi-zone NuGrid code mppnp using and the same reaction network as in P16. To summarize: every stellar evolution time step, the 1097-isotope nuclear reaction network is solved using a first-order Newton-Raphson backward Euler integration, which is followed by an implicit diffusion solve. The network adapts the problem size every time step (and every computational grid cell) depending upon the reaction flux of each isotope at current state.
In Sect. 2.1.3 we described issues that arise with such an operator-splitting method during hot bottom burning in models of massive AGB and super-AGB stars. To predict realistic abundances in these conditions we have implemented a nested-network method to solve the coupled mixing and burning equations for a small network which includes species which are affected by HBB. We solve the small network for zones of the convective envelope and a large decoupled network for the whole stellar model. After each time step the abundances from the coupled solution replace the abundances from the large network. The coupled solution is merged into the large network by normalizing the total abundance of isotopes of the small network to be equal to the total abundance of the corresponding isotopes of the large network. The small network models the Cameron-Fowler transport mechanism and production (Cameron & Fowler, 1971), CNO, NeNa and MgAl cycles and includes isotopes up to similar to Siess (2010). Heavier isotopes, which are only included in the large network, do not take part in HBB nucleosynthesis according to the present state-of-the-art (e.g. review by Herwig, 2005). As such we don’t expect the heavier isotopes to be affected by our choice of decoupling of burning and mixing.
We compare of the surface C/O ratio of the , model of the coupled solution with the nested-network solution and the decoupled solution (Fig. 2). Our nested-network method results in the same evolution of the surface C/O ratio. The decoupled solution based time steps as given for the coupled solution of MESA strongly overestimates the surface C/O ratio compared to the coupled solution from MESA. We find good agreement of the surface abundance of CNO isotopes based on our nested-network method in comparison with predictions from MESA (Fig. 2).
The final stellar yields of CNO isotopes based on the nested-network method are similar to Herwig (2004a, H04) and Karakas (2010, K10) who couple mixing and burning (Table 5). Neither study includes s-process species, although more recent work by Karakas & Lugaro (2016) for does now include heavy elements. The high / ratio of the decoupled solution shows that HBB is not properly resolved. Even larger values of Cristallo et al. (2015, C15) could be due to resolution issues during HBB with the mix-burn-mix approximation. The nested-network approach predicts Li production via HBB as well because Cameron-Fowler mechanism is resolved. In summary, the nested-network method allows to predict Li, CNO isotopes and heavy elements in these HBB stellar models.
The total stellar yield of element/isotope of a stellar model with initial mass includes the yield from stellar winds and the SN explosion as in P16. The yield ejected by stellar winds is calculated as
where is the mass loss rate, is the mass fraction of the element/isotope at the surface and is the stellar lifetime. The yield from the SN ejecta is derived as
where is the mass fraction of element/isotope at mass coordinate and is the remnant mass. Pre-SN yields are calculated as but without taking into account the nucleosynthesis from the SN shock. Instead, the ejecta of matter at the point of collapse is considered. The overproduction factor of element/isotope of the stellar model with initial mass is calculated as
where and is the total ejected mass and initial mass fraction of element/isotope respectively. is the total ejected mass.
3 Results of stellar evolution and explosion
3.1 General properties
3.1.1 The mass and metallicity grid
The new set of models and stellar yields are all calculated with the same stellar evolution code MESA. We calculate massive star models with , and at and as an alternative to the massive star GENEC models from P16. Stellar models with are added at all metallicities to cover the lower-mass end of the massive star mass range. Côté et al. (2016) show that based on our assumption of the remnant mass distribution (cf. Sect. 2.2) adding more masses to the grid would not significantly improve galactic chemical evolution models. Côté et al. (2016) find that the metallicity range covered is more important than the number of metallicities within that range. In addition to the models in P16 we are adding intermediate and S-AGB models at all metallicities (, and ). We also add a models at all metallicities.
3.1.2 Stellar evolution tracks
The influence of metallicity on the stellar evolution is visible in the Hertzsprung-Russell diagram (HRD) with the stellar models with and shown in Fig. 4. The shift of the tracks of lower metallicity to higher luminosities and higher surface temperatures is the result of the larger core masses and lower opacities of the envelopes (Herwig, 2004a). The central temperature-density tracks of models are separated from models. The central densities depend on stellar mass as under the assumption of constant temperature during each burning phase. Lower metallicity models behave as models with higher initial masses which is visible in the approach of the tracks at low metallicities towards the tracks in the central temperature-density diagram (Fig. 4).
Stellar models with for , and exhibit He-core flashes. First dredge-up appears at and in all the AGB models but at only in stellar models with . Second dredge-up occurs in models with and at and respectively. Core flash, first dredge-up and second dredge-up at show the same initial-mass dependence as the AGB models at in P16.
The average luminosity of low-mass, non-HBB stellar models follows the linear core-mass luminosity relation of Blöcker (1993) that was originally derived for models (Fig. 5). AGB models with higher initial masses which experience HBB agree with the exponential core-mass luminosity relationship of Herwig (2004a).
AGB models with for , and ignite C and reach the S-AGB stage. For S-AGB models with initial mass below at and below at and a convective C-burning flame does not appear as in stellar models of higher initial mass. In these models C burning takes place under radiative conditions. For models the maximum temperatures in the C/O core do not exceed which is far below the ignition temperature of found by (Siess, 2007). Farmer et al. (2015) has provided a recent, detailed study of the onset of C burning, which also depends sensitively on the still very uncertain + reaction rate (see also Chen et al., 2014).
Model properties of the TP-AGB phase for each initial mass and metallicity are shown in Table 6. We present in Table 7 the detailed TP properties for stellar models of . The structure evolution of models , and at are shown in the Kippenhahn diagrams in Fig. 6.The final core mass and lifetimes for AGB models are shown in Table 8.
We compare stellar models with and at with models of Karakas (2003, K03) and Weiss & Ferguson (2009, W09) who calculated models with and based on -enhanced initial abundances and models of and of solar-scaled abundance respectively. The core mass of these two stellar models at the first TP are and while K03 obtain and and W09 get and . As P16 we find larger core masses compared to K03 and W09. Our number of thermal pulses of the stellar models are 14 and 32 while K03 have 16 and 83 and W09 have 10 and 38.
The final surface C/O ratio of these stellar models is 3.243 and 3.379 compared to 8.18 and 4.48 of K03 and 3.449 and 0.772 of W09. The latter value of W09 is taken when the envelope mass is and their simulation stops. It differs from ours because in our simulation the dominance of the 3DUP over weakening HBB as described in Frost et al. (1998) increases the C/O ratio from to the large final value over the last two thermal pulses when the stellar model loses the last of envelope mass. The , simulation still has of envelope when it stops and the C/O ratio is . This case does not experience the final thermal pulse where TDUP could have significantly increased the C/O ratio.
The surface C/O ratio increases due to TDUP and decreases during the interpulse HBB in massive AGB models (Lattanzio et al., 1996, 1997; Lattanzio & Boothroyd, 1997). The surface C/O ratios for stellar models of presented is complex (Fig. 7). While at low mass stellar models steadily increase their surface C/O ratio (see Fig. 4 in P16), at low metallicity the first pulses can lead to a surface enhancement close to or even above the He-intershell C/O ratio as shown in Fig. 7. At low metallicity the envelope C/O ratio quickly represents that of the intershell because the total initial amount of O and C in the envelope is smaller due to the low initial metallicity. Due to a steady decrease of the C/O ratio in the He intershell over time the TDUP leads to a decline in the surface C/O ratio. Stellar models at higher metallicity such as the , model experience only an increase of the surface C/O ratio during their evolution. For models with higher initial mass a higher C/O intershell ratio is reached which leads to a higher C/O surface enhancement in the non-HBB models.
The TDUP strength is described by the dredge-up parameter defined as where is the amount of mass dredged-up into the envelope and is the increase in mass of the H-free core during the previous interpulse phase. shows a strong dependence behaviour on metallicity. In Fig. 8 we compare the stellar models with and at and . The low-mass AGB star model with lower metallicity has higher then the higher metallicity model, while the S-AGB model has higher at higher metallicity. This can be understood by considering that the dredge-up efficiency has a maximum for a core mass , which at corresponds to an initial mass of (P16), and in combination with the metallicity dependence of the initial to final mass relation (Fig. 9). The lower metallicity model as a higher core mass and therefore lower . The low-mass model has a also a higher core mass at lower metallicity, but here this implies larger .
While the , model is very similar to the , model shown in Fig. 5 of P16, the , model reaches , similar to the stellar model of the same initial mass and metallicity in Herwig (2004b). The maximum of total dredged-up mass increase up to for and up to for and . For low-mass models both quantities decline towards higher initial masses (Table 7). For comparison, Fishlock et al. (2014) found that the and models at dredge-up the most material. In intermediate-mass stellar models we find lower total mass dredged up compared to Fishlock et al. (2014) who reach another maximum at .
The final core masses are larger at lower metallicity for most stellar models. This implies a steeper initial-final mass relation (IFMR, Fig. 9). The core masses of models from P16 are added for comparison. The IFMRs in Weiss & Ferguson (2009) which spans from to and covers down to show in general a smaller final core mass than the present stellar models and those by P16. The spread in metallicity is more pronounced for these models. Our IFMR covers the upper part of the compiled data of observed open cluster objects shown in Fig. 10 of Weiss & Ferguson (2009). The AGB phase of the , model is terminated due to a H-ingestion event which prevents further core growth.
The massive star models used the same MESA code version and input parameters used by Jones et al. (2015, J15). J15 conducted a resolution study of the time steps at the end of core helium burning and we use the coarsest time step resolution that reproduced the He-free and C/O core masses. The impact of metallicity in the HRD evolution (e.g. models, Fig. 4) is similar to that shown in low-mass models. There is little impact of metallicity on central temperature and density.
P16 found the final fate of massive stellar models in the mass range to to be the red super giant phase which is in agreement with other non-rotating models (e.g. Hirschi et al., 2004). All these massive star models experience the same phase except stellar models of and at . The latter move from the blue region of the HRD into the region of yellow supergiants but not further, similar to models of Pop III stars of Heger & Woosley (2010). Due to their low metallicity these stellar models experience negligible mass loss and their intermediate convective zones are largest among all models. This leads to higher compactness which favours the blue region of the HRD (Hirschi, 2007; Peters & Hirschi, 2013).
The stellar models with at all metallicities and the , model burn C under radiative conditions consistent with solar-metallicity and PopIII models (Heger & Woosley, 2010, P16). The occurrence of convective core C burning in the , model results from the higher luminosity of C core burning present in stellar models of higher metallicity (Hirschi, 2007; Rauscher et al., 2002; El Eid et al., 2004). Convective core C burning is present in all massive star models of lower initial mass as in P16.
The lifetimes of the core-burning stages are given in Table 9, using the definition of the lifetimes as in P16. Most burning stages are shorter for higher initial masses and lower metallicities, as expected. The final masses and the masses of the He, CO and Si cores are shown in Fig. 10, using the definitions of the core masses as in P16. The final mass increases towards lower metallicity at each initial mass. The He core masses and CO core masses show only a mild metallicity dependence compared to the clear metallicity dependence of the final mass. For some initial masses the core mass does not increase with decreasing metallicity such as the CO core masses of the stellar models with . The Si cores do not increase with initial mass as found for the He and CO cores. Instead, we find large variations in the metallicity of similar magnitude at different initial masses and no clear trend with metallicity (Fig. 10). This is due to the non-monotonicity for the Si core (e.g. Ugliano et al., 2012; Sukhbold & Woosley, 2014)
We compare the core masses of these stellar models with initial mass of and for with those of Meynet & Maeder (2002, M02) at and P16 at in Table 10. Our He core masses are in better agreement with P16 who got larger values than M02 in spite of the metallicity difference. This is because we adopt a similar convective overshooting strength for the H-burning cores as P16 while M02 do not adopt any overshooting. More precisely, for core H and He-burning phases, in MESA models an exponentially-decaying diffusion coefficient with is used whereas in GENEC, an instantaneous penetrative overshoot with H is used. The different treatment of convective boundary mixing explains the differences in core masses between this work and P16. The CO core masses show larger differences between this work, P16 and M02 than found for the He core masses. The mass of the Si core is in better agreement with P16 than the CO core mass (Table 10).
The structural differences of stellar models with at and are shown in the Kippenhahn diagram in Fig. 11. Contacts between convective burning shells occur in different advanced burning stages and can have, in particular for a complete shell merger, a profound impact on stellar structure and nucleosynthesis (see Sect. 4) The contact between the convective H-burning shell and convective He-burning shell leads in the , model to a H-ingestion event. The occurrence of shell merger is affected by considerable uncertainties (Woosley et al., 2002) and requires studies with 3D hydrodynamic simulations (e.g. Meakin & Arnett, 2007; Jones et al., 2017). This point is discussed in more details below.
3.1.3 Core-collapse supernovae
The explosion energy and remnant mass of a progenitor depends strongly on the pre-SN structure Fryer (1999); Müller (2016); Janka et al. (2007). The explosion properties determine the layers of the star that are ejected and the shock conditions. We compare the maximum temperatures and densities reached during the shock passage for massive star models of and obtained with the delayed explosion prescription (Fig. 12). The shock temperature for stellar models with and at are the largest of all metallicities. Up to stellar models with reach the highest shock temperatures and densities followed by the models but at higher metallicity the trend is reversed.
The pre-SN structure of these stellar models do not always show trends with metallicity (Fig. 13) and the same counts for the shock temperatures. There is no trend in the Fe-core mass with mass and metallicity and instead the stellar models with show the largest Fe core masses (Fig. 13). Recent studies show that there is no monotonous compactness trend with initial mass and metallicity (Ugliano et al., 2012; Sukhbold & Woosley, 2014; Sukhbold et al., 2016).
Under the convective engine paradigm (Herant et al., 1994), whether or not the model explodes depends sensitively on the ram pressure of the stellar material falling onto the outer edge of the convective region (Fryer, 1999). To drive an explosion, the energy in the convective region must overcome this ram pressure and the energy in the convective region when this occurs determines the explosion energy of the supernova. Typically, the energy in the convective region required to overcome an accretion rate of is . The Fryer et al. (2012) formalism assumes that the energy in the convective region increases over time (either on rapid or delayed timescales) and assumed that, when the pressure in the convective region exceeded the ram pressure, an explosion was launched.
To determine this pressure and, hence, the likelihood of the star exploding, we calculate the accretion rate as a function of time (Fig. 14). Just based on these accretion rates, we see that the convective engine is likely to explode both the and progenitors at z=0.02 and z=0.01. Although the will explode at all metallicities, it becomes increasingly difficult to drive explosions in the at lower metallicities. Typically, the high accretion rates for the and models make them difficult to explode and we expect no or weak explosions from these models. The exception is the z=0.01 metallicity star where a shell merger occurred. This altered the density profile at collapse sufficiently to make this star more-likely to explode, but the high accretion rates at late times is indicative of a large density that may lead to considerable fallback.
The mass at the launch of the explosion can be estimated by looking at the accretion rate as a function of accreted mass. The unique feature of our MESA progenitor is evident here. Its core is larger than other progenitors in the literature and it is more likely to make more massive neutron stars. When the accretion rate falls below , the accreted baryonic mass is to , corresponding to a gravitational mass of to . In the sequence the , and models experience O-C shell mergers (Sect. 4.4). The consequence is a rapid decline of the mass accretion rate at the location of the bottom of the merged shell. At least at the launch of the explosion, the stellar model can produce a smaller remnant than our and models.
The maximum temperatures and densities of these delayed explosions at are similar to those shown in Fig. 31 of P16 for stellar models with = 15, 20, 25. We find qualitatively the same increase with initial mass but lower explosion temperatures except for the model with . We attribute the different explosion conditions to the different pre-SN structures which were calculated with different stellar evolution codes. The density in Fe core layers at collapse of our , model is more than larger than in the model of P16 that were calculated with the GENEC code and for which the pre-collapse phase was not modeled (Fig. 13). The densities of the O shell layers are in better agreement.
3.2 Features at low metallicity
3.2.1 H ingestion
H-ingestion episodes are found in many phases of stellar evolution particularly in low and zero-metallicity AGB and He-core flash models (e.g. Fujimoto et al., 2000; Cristallo et al., 2009; Campbell et al., 2010), in very late thermal pulses in models of post-AGB stars (Herwig et al., 1999), in S-AGB stars (cf. Sect. 3.2.3, Jones et al., 2016). The mixing between the H-burning shell and He-burning shell in massive stars have been reported for models at low metallicity in Woosley & Weaver (1995); Hirschi (2007) and for Pop III models in Heger & Woosley (2010).
At the first TP of the , model the PDCZ penetrates slightly into the H-rich envelope. The protons from the envelope are mixed into the PDCZ and react with and form . The latter decays to which activates the (,n) neutron source. This leads to the production of heavy elements. In the following TP the convective He-burning zone penetrates again into the envelope which leads to the ingestion of much larger amounts of H than previously and stronger surface enrichment of He intershell material. A H-ingestion flash (HIF) with a peak luminosity of occurs. This HIF terminates the AGB phase and is shown in the Kippenhahn diagram in Fig. 15. The conditions are similar to those found in Iwamoto et al. (2004).
The , model experiences a He-shell flash when it leaves its horizontal post-AGB evolution towards the white dwarf (WD) cooling track (Iben et al., 1983; Iben & MacDonald, 1995). This Very-late Thermal Pulse (VLTP, Herwig, 2001a) causes the PDCZ to reach into the H-rich envelope, leading to H ingestion, and a born-again phase (Herwig et al., 1999, 2011). The calculation is terminated six years after the H ingestion due to convergence problems. The H ingestion leads to the production of heavy elements up to the first s-process peak in the He intershell which are mixed to the surface. Due to the energy release of H burning a connected stable layer forms within the PDCZ and the convective zone splits into two. VLTP events like this are not expected to influence significantly the composition of the stellar ejecta and the total yields, because the remaining envelope is small and the cool born-again evolution phase is short. VLTP events have been shown to posses significant non-radial, global oscillations (Herwig et al., 2014) which make their one-dimensional stellar evolution modelling unreliable. This applies equally to H-ingestion flashes in low-metallicity AGB stars (Woodward et al. in prep.).
In the S-AGB models the time between TP and TDUP becomes shorter for lower metallicity, and this may lead to H-ingestion events (Jones et al., 2016). Due to the choice of convective boundary mixing parameters this happens only occasionally in these models, below . For example, H ingestion happens during the 29th TP of the , model. For this thermal pulse we obtain neutron densities of up to in the deepest layers of the PDCZ, for about five days. The splitting of the PDCZ due to H burning prevents the transport of material from the deep layers to the surface. Since these events are not frequent in these models, the nucleosynthesis of HIFs does not contribute significantly to the stellar yields presented here.
Stellar models of and of experience H ingestion at the beginning of convective C shell burning and during O shell burning respectively. At higher metallicity we find H ingestion in the , model and in the , model. In both models H ingestion events occur during Si shell burning. These H-ingestion, or sometimes H/He-shell mixing events happen without the application of CBM at the boundaries of the convective He shell (as all other convective boundaries post-He core burning). The penetration into the convective He-burning layer is visible for the , model in Fig. 11. The resulting energy release leads to the formation of two extended convective regions which persist until collapse. We find at the bottom of the He-shell convective zone neutron densities close to which remain for days until core collapse. There is only a minor production of heavy elements but lighter elements such as F are effectively produced and contribute a relevant fraction of the total stellar yields of this stellar model.
Detailed investigations of the nucleosynthesis and 3D stellar hydrodynamics of the H-ingestion event in the post-AGB star Sakurai’s object (Herwig et al., 2011; Herwig et al., 2014) have shown that the assumption of spherical symmetry and the approximation of mixing via mixing length theory of convection are not appropriate. This is consistent with the failure of such 1D models to reproduce several key observables, such as light-curve and heavy-element abundance patterns of Sakurai’s object. This suggests that the properties of H-ingestion events in this stellar yield grid are indicative at best, and need to be investigated further through 3D hydrodynamics simulations.
3.2.2 Hot bottom burning
The temperature at the bottom of the convective envelope increases with increasing initial mass and decreasing metallicity as shown in Fig. 5 and reaches up to . In S-AGB models with at and temperatures reach more than which allows the activation of the NeNa and MgAl cycles. The , model reaches which leads to HBB. Models of the same mass but of higher metallicity do not experience HBB (Table 6). The threshold initial mass for HBB in Ventura et al. (2013) was found to be at and at which is similar to our findings. HBB is active in stellar models of masses as low as in agreement with models at of Fishlock et al. (2014).
3.2.3 Effects of hot dredge-up and dredge-out
Herwig (2004b) find that hot dredge-up is characterized by extreme H-burning luminosities of for their , model. For stellar models with and often exceeds the peak He-burning luminosities of the TP. Under the most extreme conditions in models with and we find . At higher metallicities is lower. Because of the reduced CBM efficiency in massive AGB and S-AGB models (see Sect. 2.1.4) the size of pocket decreases substantially with increasing initial mass. Additionally, the pressure scale height at the core-envelope interface decreases with increasing initial mass which leads to a further decrease of the CBM in the parametrized model. This leads to pockets in S-AGB models below at .
Dredge-out is found in the most massive AGB models during second DUP when the convective He-burning shell grows in mass and merges with the convective envelope. This leads to the enrichment of the surface with products of He-shell burning (Ritossa et al., 1999). H can be entrained into the He-burning convection zone and ignite as a flash. This is another H-ingestion event (Gil-Pons & Doherty, 2010; Jones et al., 2016). We find dredge-out in S-AGB models with at and . The flash at produces a peak in luminosity of up to . The maximum H-burning luminosities agree well with Jones et al. (2016). The initial masses of our stellar models with dredge-out are below the lower initial mass limit of dredge-out of as reported by Gil-Pons & Doherty (2010), presumably due to difference in the core overshooting prescription.
3.2.4 Carbon flame quenching in S-AGB stars
In the , S-AGB model the propagation of the C flame toward the center is quenched (Denissenkov et al., 2013). The C-flame quenching depends sensitively on the assumption of CBM, which is essentially unconstrained. If CBM at the bottom of the C-burning shell is efficient enough to quench the flame, then the result is a hybrid core. It consists of a inner C-O core of surrounded by thicker layers of O, Ne and Mg. For stellar models with the first C-burning flash occurs at closer to the center than at higher metallicity. The C burning moves outward in mass through a series of convective C-shell burning episodes. The location of the first C ignition is further outwards for models of higher metallicity due to the higher degeneracy of the lower core masses (García-Berro et al., 1997; Siess, 2007). The onset of C burning coincides with the beginning of the second DUP for the , model. At higher metallicity the C burning starts earlier than at lower metallicity. The difference in metallicity has a qualitatively similar effect on convective C burning as the difference in initial mass between and shown in Fig. 3 in Farmer et al. (2015). Possible implications of hybrid WDs are discussed in Denissenkov et al. (2017).
|: Initial stellar mass.|
|: H-free core mass at the first TP.|
|: Approximated mean stellar luminosity.|
|: Approximated mean stellar radius.|
|: Number of TPs.|
|: Number of TPs with TDUP.|
|: Time at first TP.|
|: Maximum dredged-up mass after a single TP.|
|: Total dredged-up mass of all TPs.|
|: Average interpulse duration of TPs.|
|: Total mass lost during the TP-AGB phase.|
|: Maximum temperature during the TP-AGB phase.|
|: Maximum size of PDCZ.|
|: Maximum He luminosity during TP-AGB phase.|
|: Maximum total luminosity during TP-AGB phase.|
|TP: TP number.|
|: Time since the first TP.|
|: Largest temperature at the bottom of the PDCZ.|
|: Temperature in the He-burning shell during deepest extend of TDUP.|
|: Temperature at the bottom of the convective envelope during deepest extend of TDUP.|
|: Minimum mass coordinate of the bottom of the He-flash convective zone.|
|: Mass coordinate of the H-free core at the time of the TP.|
|: Stellar mass at the TP.|
|Z = 0.02|
|this work||M02||P16||this work||M02||P16|