Nucleosynthesis in the Innermost Ejecta of Neutrino-driven Supernova Explosions in Two Dimensions

All the SN hydrodynamical trajectories have been computed from 2D general relativistic simulations with energy-dependent ray-by-ray-plus neutrino transport based on a variable Eddington factor technique as implemented in the SN code Vertex (Rampp & Janka 2002; Buras et al. 2006; Müller et al. 2010). Except for the pseudorelativistic ECSN model of Wanajo et al. (2011b), general relativity is treated in the extended conformal flatness approximation (Cordero-Carrión et al. 2009). A modern set of neutrino interaction rates (the "full rates" set of Müller et al. 2012b) has been used for the simulations. The explosions were obtained self-consistently and thus the models contained no free parameters.

The initial pre-SN models were adopted from Nomoto (1987; an $8.8\,{M}_{\odot }$ star with an O–Ne–Mg core with solar metallicity), A. Heger (unpublished;⁸ $9.6\,{M}_{\odot }$ and $8.1\,{M}_{\odot }$ models⁹ with metallicities of Z = 0 and $Z={10}^{-4}\,{Z}_{\odot }$, respectively), Woosley et al. (2002; an $11.2\,{M}_{\odot }$ solar metallicity model, s11.2 in their paper), Woosley & Weaver (1995; a $15\,{M}_{\odot }$ solar metallicity model, s15s7b2 in their paper), and Woosley et al. (2002; a $27\,{M}_{\odot }$ solar metallicity model, s27.0 in their paper). The corresponding explosion models are labeled as e8.8, z9.6, u8.1, s11, s15, and s27 hereafter. The relevant model parameters are summarized in Table 1. For more details on the individual explosion models, we refer the reader to the original publications on e8.8 (Wanajo et al. 2011b), z9.6 (Janka et al. 2012), u8.1 (Müller et al. 2012a), s11 and s15 (Müller et al. 2012b), and s27 (Müller et al. 2012a).

Table 1.  Parameters of SN Explosion Models

Model	Mproga	Zprogb	MCOc	MHed	${\rm{\Delta }}\theta$ e	texplf	tfing	mfinh	Ediagi
	(${M}_{\odot }$)	(${Z}_{\odot }$)	(${M}_{\odot }$)	(${M}_{\odot }$)	(deg)	(ms)	(ms)	(${M}_{\odot }$)	(${10}^{50}\,\mathrm{erg}$)
e8.8	8.8	1	1.377	1.377	1.4	92	362	>1.377j	0.9
z9.6	9.6	0	1.371	1.69	1.4	125	1420	>1.373k	0.6
u8.1	8.1	10−4	1.385	1.92	1.4	177	335	1.373	0.4
s11l	11.2	1	1.89	2.84	2.8	213	922	1.39	0.4
s15m	15	1	2.5	4.5	2.8	569	779	1.45	1.3
s27n	27	1	7.2	9.2	1.4	209	790	1.74	1.9

Notes.

^aProgenitor mass at the zero-age main sequence. ^bMetallicity at the zero-age main sequence. ^cCO core mass at collapse. ^dHe core mass at collapse. ^eAngular resolution. ^fPost-bounce time of explosion, defined as the point in time when the average shock radius $\langle {r}_{\mathrm{sh}}\rangle$ reaches $400\ \mathrm{km}$. ^gFinal post-bounce time reached in the simulation. ^hMass coordinate corresponding to the average shock radius at the end of the simulation. ⁱDiagnostic explosion energy at the end of simulation. ^jNote that an artificial envelope with $\rho \propto {r}^{-3}$ was used in Wanajo et al. (2011b) outside the He core. It is therefore not appropriate to give a more precise mass coordinate for the final position of the shock in the hydrogen shell. ^kBy the end of the simulation, the shock has crossed the outer grid boundary at a mass coordinate of $1.373\,{M}_{\odot }$ in model z9.6. ^lSame as model s11.2 in Woosley et al. (2002) and Müller et al. (2012b). ^mSame as model s15s7b2 in Woosley & Weaver (1995) and Müller et al. (2012b). ⁿSame as model s27.0 in Woosley et al. (2002) and Müller et al. (2012a).

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Note that although we include two models of subsolar metallicity, the limited number of models does not permit us to discuss the dependency on metallicity in this paper.

Snapshots of the electron fraction, Y_e, and the entropy per nucleon, S, in these simulations at late times are shown in Figure 1. One can see roughly spherical structures for e8.8 and z9.6 (top panels) and more strongly asymmetric features with a dipolar or quadrupolar geometry for the other models (middle and bottom panels). The different explosion dynamics reflect the core structures of pre-SN stars with steeper to shallower density gradients in the order of e8.8, z9.6, u8.1, s11, s15, and s27 (see Figure 8 in Janka et al. 2012). The ECSN progenitor for e8.8 is a super-asymptotic giant branch (SAGB) star with an O–Ne–Mg core surrounded by a very dilute H–He envelope, while progenitors of CCSNe have Fe cores embedded by dense O–Si shells. For e8.8, therefore, the explosion sets in very early at a postbounce time of ${t}_{\mathrm{pb}}\sim 80$ ms before vigorous convection can develop. Overturn driven by the Rayleigh–Taylor instability only occurs when the explosion is underway, but the plumes do not have sufficient time to merge into large structures so no global asymmetry emerges. For the CCSN cases (except for z9.6), by contrast, the explosions gradually start after the multidimensional effects, i.e., convection or the standing-accretion-shock instability (SASI; Blondin et al. 2003), have reached the nonlinear regime (the onset of the explosion occurs at ${t}_{\mathrm{pb}}\sim 500$ ms for s15 and ${t}_{\mathrm{pb}}\sim 150\ldots 200$ ms for the others; see Table 1 and Figure 14 in Janka et al. 2012). Moreover, due to the presence of a relatively dense and massive O shell, the shock expansion is slow enough for a dominant unipolar or bipolar asymmetry to emerge after shock revival.

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Models z9.6 and u8.1 stand apart from the more massive CCSN models s11, s15, and s27 since stars close to the Fe-core formation limit exhibit evolutionary and structural similarities to ECSN progenitors (Woosley & Heger 2010; Jones et al. 2013, 2014). Specifically, these progenitors have very thin O and C shells between the core and the low-density He and H envelope. Because of these peculiarities, model z9.6 (and to some degree u8.1) is a case on the borderline of ECSN-like explosion behavior (Müller 2016). Its progenitor exhibits the steepest core-density gradient near the core-envelope interface among the CCSN cases, resulting in a similarly rapid expansion of the neutrino-heated ejecta and hence in Y_e and S structures rather similar to those of e8.8. The progenitor of u8.1 has a slightly shallower core-density gradient than that of z9.6, so the propagation of the shock is slightly slower than for z9.6 but is still faster than for typical Fe-core progenitors. Since the width of the ECSN channel is subject to considerable uncertainties (see Poelarends et al. 2008; Jones et al. 2013, 2014; Doherty et al. 2015; Jones et al. 2016, and references therein), CCSNe from this mass range are particularly interesting as a possible alternative source for "ECSN-like" nucleosynthesis (i.e., the characteristic nucleosynthesis in neutron-rich neutrino-heated ejecta with moderate entropy as discussed in Wanajo et al. 2011b, 2013a, 2013b).

As in Wanajo et al. (2011b) and similar to Wongwathanarat et al. (2017), we compute the trajectories for our nucleosynthesis calculations from the 2D data files (with a time spacing of $0.25\,\mathrm{ms}$) instead of co-evolving tracer particles during the simulation. In order not to follow the tracers during multiple convective overturns with the risk of accumulating discretization errors, we integrate the tracer trajectories backward in time from an appropriate point during the simulation (i.e., starting either from the end of the simulation or from a time when the bulk of the ejecta have cooled sufficiently as in the case of z9.6). This procedure greatly reduces the number of required trajectories to adequately sample the ejecta: selecting the final locations of the tracer particles only within the region of interest reduces the total mass of ejecta that must be covered (less than $0.03\,{M}_{\odot }$) and makes it easy to use higher-mass resolution where it is most needed, i.e., in the neutrino-processed ejecta. For a more general overview of the problems in extracting tracer trajectories from multidimensional simulations, we refer the reader to Harris et al. (2017).

Some nucleosynthesis-relevant properties for all SN models are summarized in Table 2. It is important to note that we only calculate the nucleosynthesis for the "early ejecta"—i.e., we consider only the material that has been ejected in neutrino-driven outflows or undergone explosive burning in the shock by the end of the simulations. The contributions of the outer shells (i.e., the H–He envelopes of e8.8, z9.6, and u8.1 and material outside the middle of the O-burning shell in s11, s15, and s27) and later neutrino-driven ejecta to the total yields are neglected. For the massive CCSNe (s11, s15, s27), the nucleosynthetic contribution for the outer part of the O shell therefore remains uncertain, and long-term simulations to several seconds (Müller 2015; Müller et al. 2017) will be necessary to determine it. As the shock has only traversed part of the O shell in these models we also miss part of the contribution of hydrostatic and explosive burning to heavy element production from O/C/He shells, so our yields can only be understood as lower limits for some species (among them ${}^{26}\mathrm{Al}$ and ${}^{60}\mathrm{Fe}$). We also disregard the possibility of late-time fallback of the initial ejecta, but this may not be a major uncertainty since the most advanced parameterized 1D models of neutrino-driven explosions suggest that this plays a minor role (Ertl et al. 2016) for the progenitor mass range considered here.

Table 2.  Properties of Innermost SN Ejecta

Model	Type	MPNSa	Mejb	${M}_{\mathrm{ej},{\rm{n}}}$ c	${Y}_{{\rm{e}},\min }$ d	${Y}_{{\rm{e}},\max }$ e	Sminf	Smaxg
		(${M}_{\odot }$)	(${10}^{-3}\,{M}_{\odot }$)	(${10}^{-3}\,{M}_{\odot }$)			(${k}_{{\rm{B}}}\,{\mathrm{nuc}}^{-1}$)	(${k}_{{\rm{B}}}\,{\mathrm{nuc}}^{-1}$)
e8.8	ECSN	1.36	11.4	5.83	0.398	0.555	9.80	383
z9.6	CCSN	1.36	12.4	4.94	0.373	0.603	12.6	27.8
u8.1	CCSN	1.36	7.69	3.24	0.399	0.612	9.83	29.9
s11	CCSN	1.36	14.1	0.133	0.474	0.551	6.64	34.7
s15	CCSN	1.58	15.9	0.592	0.464	0.598	6.78	36.7
s27	CCSN	1.65	27.3	0.759	0.387	0.601	5.19	44.0

Notes.

^aBaryonic mass of the proto-neutron star at the end of simulation. ^bTotal mass in the innermost ejecta. ^cEjecta mass with ${Y}_{{\rm{e}}}\lt 0.4975$. ^dMinimal Y_e evaluated at ${T}_{9}=10$. ^eMaximal Y_e evaluated at ${T}_{9}=10$. ^fMinimal asymptotic entropy (at the end of the simulation). ^gMaximal asymptotic entropy (at the end of the simulation).

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For models e8.8, z9.6, and u8.1, explosive burning is already complete at the end of the simulations: the outer shell (composed of H and He) will not add significant yields of heavy elements and no more explosive nucleosynthesis is taking place as the shock propagates to the stellar envelope after the end of the simulations. However, we still miss small amounts of ejected material from the neutrino-driven wind (Duncan et al. 1986) in these progenitors. This material may undergo weak r- or $\nu p$-process nucleosynthesis, though the most recent calculations of wind nucleosynthesis for e8.8 suggest only a relatively unspectacular production of Fe-group elements and some $\nu p$-process nuclides with small production factors (Pllumbi et al. 2015), which turn out to be insignificant compared to the contribution from the early ejecta that we investigate here. For e8.8, z9.6, and u8.1, the yields presented here thus cover essentially the complete nucleosynthesis of heavy elements in these progenitors.

The case is different for s11, s15, and s27, where the shock has progressed only to the middle of the O/Si shell. For these models, substantial amounts of ashes from O, Ne, C, and He burning thus remain to be ejected and will contribute significantly to the total production factors. The postshock temperatures at the end of our simulations also remain sufficiently high for some additional explosive O burning to take place. Moreover, strong accretion downflows still persist in these models. This keeps the neutrino luminosities high and allows the ejection of neutrino-heated matter to continue well beyond $1\,{\rm{s}}$ after the onset of the explosion (Müller 2015; Bruenn et al. 2016). Our nucleosynthesis calculations therefore place only a lower bound on the production of Fe-group and trans-Fe group elements in the SN core of these models. It is noteworthy that this may partly contribute to the unexpectedly small mass of ⁵⁶Ni obtained for these models, which remains significantly smaller than expected for "ordinary" SNe (e.g., $0.07\,{M}_{\odot }$ for SN 1987A; Bouchet et al. 1991).

Aside from the limitation of our analysis to the nucleosynthesis during the first few hundred milliseconds after shock revival, the nucleosynthesis in our models remains subject to other uncertainties: the simulations were conducted assuming axisymmetry (2D), and the explosion energies at the end of simulations are only ($0.3\mbox{--}1.5)\times {10}^{50}$ erg (see Figure 15 in Janka et al. 2012), which are either less energetic than typical observed events (several 10⁵⁰ erg; e.g., Kasen & Woosley 2009; Nomoto et al. 2013) or still in a phase of steep rise, i.e., not converged to their final values. We discuss the possible repercussions of this in Sections 4.8 and 5.

Figure 2 shows the Y_e distributions in the ejecta for all models evaluated at ${T}_{9}=10$, where T₉ is the temperature in units of 10⁹ K. For a trajectory with a maximum temperature of ${T}_{9,\max }\lt 10$, we show Y_e at ${T}_{9}={T}_{9,\max }$ instead. ${\rm{\Delta }}{M}_{\mathrm{ej}}$ is the ejecta mass in each Y_e bin with an interval of ${\rm{\Delta }}{Y}_{{\rm{e}}}=0.005$. The minimum and maximum values of Y_e for all models are given in Table 2 (8th and 9th columns, respectively). The Y_e distributions shown here are similar to those inferred from the hydro data at late times except for small (up to a few %) shifts toward ${Y}_{{\rm{e}}}\approx 0.5$ as a result of some numerical diffusion and mixing (clipping of extrema) that suppresses the tails of the Y_e distribution in the hydro at late times.¹⁰ We find that the low-mass models (e8.8, z9.6, and u8.1) have appreciable amounts of neutron-rich ejecta (40%–50%; 7th column in Table 2). This is due to the faster growth of the shock radii for these low-mass cases: as a result of the higher ejecta speed, less time is available to increase the Y_e by neutrino processing as material is ejected from the neutron-rich environment near the gain radius, where conditions close to equilibrium between neutrino absorption and electron (and to some extent positron) captures are maintained and lead to a low Y_e. By contrast, the bulk of the ejecta are proton-rich (96%–99%; Table 2) in the massive models (s11, s15, and s27). Here the ejecta expand more slowly, allowing neutrino absorption to reset the Y_e to an equilibrium value determined by the competition of electron neutrino and antineutrino absorption (Qian & Woosley 1996; Fröhlich et al. 2006). With similar electron neutrino and antineutrino mean energies, the equilibrium value tends to lie above ${Y}_{{\rm{e}}}=0.5$ because of the proton-neutron mass difference.

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Figure 3 illustrates the distributions of ejecta masses as functions of Y_e and the asymptotic entropy per nucleon, S (in units of Boltzmann's constant k_B). All SN models show spikes of ${Y}_{{\rm{e}}}\approx 0.50$ components as leftovers of the initial composition of shocked material that never undergoes strong neutrino processing, and they show a positive correlation of S with Y_e in the neutrino-processed ejecta. The latter fact is reasonable because neutrino heating raises both Y_e and S. We find, however, a larger scatter of the entropies at a given Y_e for the massive SN models (s11, s15, and s27). This reflects the more vigorous motions arising from convective instability and SASI for more massive cases as well as stronger temporal and spatial variations in the neutrino irradiation.

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Note that the very high entropies at ${Y}_{{\rm{e}}}\approx 0.5$ for e8.8 (up to ${S}_{\max }=383{k}_{{\rm{B}}};$ last column in Table 2) stem from the shock heating of the outgoing material colliding with the dilute SAGB envelope. However, a sizable postshock entropy, $S\gt 100{k}_{{\rm{B}}}$, is reached only in shells that never reach nuclear equilibrium (${T}_{9}\lt 3$; Janka et al. 2008; Kuroda et al. 2008), so these shells do not contribute to the production of heavy elements (such as an r-process; suggested by Ning et al. 2007). Except for this component of shock-heated ejecta with low temperatures, e8.8 has a maximum entropy of nucleosynthesis-relevant material of $\approx 25{k}_{{\rm{B}}}$, and more massive models have larger values (S_max; last column in Table 2).

The core-collapse simulations were stopped at $t={t}_{\mathrm{fin}}$ for our models, where ${t}_{\mathrm{fin}}=0.423\,{\rm{s}}$, 0.605 s, 0.474 s, 0.767 s, 0.947 s, and 1.13 s after core bounce for e8.8, z9.6, u8.1, s11, s15, and s27, respectively. At these times, the temperatures in the ejecta were still high, in particular for massive models, so nucleosynthesis would be still active. The temperature (T) and density (ρ) thus need to be extrapolated for nucleosynthesis calculations. It is known that the late-time evolution of the density is well approximated by $\rho \propto {t}^{-2}$ (e.g., Arcones et al. 2007). We thus assume

$$\begin{eqnarray}&&\rho {(t)={c}_{1}(t-{t}_{1})}^{-2}\quad (t\gt {t}_{\mathrm{fin}}),\end{eqnarray} \tag{ 1 }$$

where the constants c₁ and t₁ are determined to get a smooth connection of the density at $t={t}_{\mathrm{fin}}$. The expansion of the ejecta is almost adiabatic at this stage (i.e., ${T}^{3}/\rho \approx \mathrm{constant}$),¹¹ and thus the temperature is extrapolated as

$$\begin{eqnarray}&&T{(t)={c}_{2}(t-{t}_{1})}^{-2/3}\quad (t\gt {t}_{\mathrm{fin}})\end{eqnarray} \tag{ 2 }$$

with t₁ from Equation (1) and c₂ determined so that the temperature matches the value at $t={t}_{\mathrm{fin}}$. The radius r for $t\gt {t}_{\mathrm{fin}}$, which is needed to calculate the rates of neutrino interactions, is obtained from Equation (1) with the assumption of steady-state conditions, i.e., ${r}^{2}\rho {v}_{r}=\mathrm{constant}$ (Panov & Janka 2009; Wanajo et al. 2011a), where v_r is the radial component of velocity at $t={t}_{\mathrm{fin}}$ and constant afterward. Selected temporal evolutions of density and temperature are shown in Figures 4 and 5, respectively, where the extrapolations are indicated by red curves. The evolutions for e8.8 and z9.6 appear to be nearly self-similar as expected from their almost spherical expansions (Figure 1). For more massive models, in particular s11, s15, and s27, a variety of evolutions can be seen because of highly asymmetric vigorous convective, and SASI motions.

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We first analyze the nucleosynthesis in detail for z9.6, since this model covers the widest range in Y_e (Y_e = 0.373–0.603; Figure 2). Figure 6 shows the neutron-, proton-, and α-to-seed abundance ratios (${Y}_{{\rm{n}}}/{Y}_{{\rm{h}}}$, ${Y}_{{\rm{p}}}/{Y}_{{\rm{h}}}$, and ${Y}_{\alpha }/{Y}_{{\rm{h}}}$) at a point when the temperatures decrease to ${T}_{9}=3$ for all the trajectories of model z9.6. We consider this temperature as it approximately corresponds to the end of nuclear equilibrium: this temperature is close to the conventional critical temperature for the termination of quasi-nuclear equilibrium (QSE; ${T}_{9}=4$; Meyer et al. 1998) to the beginning of a $\nu p$-process (${T}_{9}=3$; Fröhlich et al. 2006), and the beginning of an r-process (${T}_{9}=2.5$; Woosley et al. 1994). Here the seed abundance Y_h is defined as the total abundance of all nuclei heavier than He. The other SN models show similar dependencies of ${Y}_{{\rm{n}}}/{Y}_{{\rm{h}}}$, ${Y}_{{\rm{p}}}/{Y}_{{\rm{h}}}$, and ${Y}_{\alpha }/{Y}_{{\rm{h}}}$ on Y_e (not shown here), indicating that Y_e (rather than entropy and the expansion timescale) is most crucial for the nucleosynthesis in our case. For this reason, we describe our nucleosynthetic results for model z9.6 in terms of Y_e. According to Figure 6, we can distinguish the following types of nucleosynthesis regimes in our SN models: among the (strongly) neutrino-processed ejecta, we find an NSE regime for ${Y}_{{\rm{e}}}\lt 0.43$, a QSE regime for $0.43\leqslant {Y}_{{\rm{e}}}\lt 0.5$, and nucleosynthesis by charged-particle capture processes for ${Y}_{{\rm{e}}}\geqslant 0.5$. No r-process is expected in our models because of neutron-to-seed ratios far below unity over the entire range of Y_e (Figure 6). We note that in all of these regimes (and different from "classical" explosive nucleosynthesis), the neutrino-processed material is initially in NSE and the QSE/charged-particle capture regime is reached after an NSE phase as the temperatures decrease during the expansion of the ejecta.

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In addition to these three regimes, there is a fourth regime of material with ${Y}_{{\rm{e}}}\approx 0.5$ and lower entropy; this comprises the ejecta that immediately expand after being shocked and undergo classical explosive nucleosynthesis. The nucleosynthesis in this regime is not the primary subject of this paper; for models e8.8, z9.6, and u8.1 it is unimportant in terms of yields and for s11, s15, and s27, we merely cover the explosive burning of a part of the O shell, so there is no basis for an extensive analysis of this regime. Moreover, the physical principles of classical explosive nucleosynthesis are well known in the literature (Arnett 1996).

It is worth noting that although the regimes of NSE/QSE are also encountered in classical explosive nucleosynthesis, the conditions for the realization of these regimes are very different: for classical explosive burning, the nucleosynthesis is determined by the peak temperature (which depends on the explosion energy and the mass–radius profile of the progenitor) and in the NSE/QSE regime it is determined on the Y_e in the progenitor. In the case of the neutrino-processed ejecta, the initial temperature is always high enough for NSE to obtain, and the interplay of multidimensional fluid flow and weak interactions of neutrinos, electrons, and positrons determines the conditions at freeze-out from NSE/QSE, i.e., Y_e, entropy, and the expansion timescale.

3.1.1. NSE

For ${Y}_{{\rm{e}}}\lt 0.43$, all the ratios ${Y}_{{\rm{n}}}/{Y}_{{\rm{h}}}$, ${Y}_{{\rm{p}}}/{Y}_{{\rm{h}}}$, and ${Y}_{\alpha }/{Y}_{{\rm{h}}}$ are considerably smaller than unity ($\lt 0.01$) at ${T}_{9}=3$ (Figure 6). The global distribution of final nucleosynthetic abundances is thus mostly determined in NSE at a high temperature (${T}_{9}\gt 5$).¹⁴ The subsequent QSE does not substantially change the abundance distribution because of the small amounts of free nucleons and α particles. This is due to relatively small entropies ($S\sim 14\,{k}_{{\rm{B}}}/\mathrm{nuc};$ Figure 3) for these neutron-rich ejecta and the neutron-richness itself. Under such conditions the three-body process $\alpha {(\alpha n,\gamma )}^{9}$ Be (followed by ⁹Be ${(\alpha ,\gamma )}^{12}$ C), rather than triple-α, is fast enough to form the NSE cluster by assembling free nucleons and α particles. Such neutron-rich conditions also disfavor α emission (i.e., to avoid being more neutron-rich). In NSE, the resulting abundance distribution is independent of specific reactions. What determines the abundance distribution are the binding energies per nucleon ($B/A;$ shown in Figure 7, left). As the Y_e of nuclides (indicated by white lines) decreases from ∼0.50 to ∼0.40, the nuclides with the maximal B/A shift from ⁵⁶Ni (Z = N = 28) to ⁴⁸Ca (Z = 20 and N = 28) and ⁸⁴Se (Z = 34 and N = 50; see Hartmann et al. 1985).

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We find such NSE-like nucleosynthesis features in Figure 8, which shows the abundance distribution in selected trajectories of model z9.6 when the temperatures decrease to ${T}_{9}=5$, 4, and 3 as well as the abundances at the end of the calculations. The top three panels correspond to the trajectories with NSE-like conditions of ${Y}_{{\rm{e}}}\lt 0.43$ and relatively low entropies ($\sim 14\,{k}_{{\rm{B}}}/\mathrm{nuc}$). Major abundance peaks are already formed at ${T}_{9}=5$ (red lines). The abundance patterns are almost frozen when the temperature decreases to ${T}_{9}=4$ (defined as the end of NSE) and do not change significantly during the subsequent evolution.

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3.1.2. QSE

3.1.3. Charged-particle Capture Process

For ${Y}_{{\rm{e}}}\geqslant 0.50$, ⁵⁶Ni dominates in the heavy QSE cluster, and the α-rich freeze-out from QSE (${T}_{9}\sim 4$) leads to an α-process (Woosley & Hoffman 1992). This greatly enhances the abundances of α-elements with $A\sim 12$–40 (with multiples of four) as can be seen in the right-middle panel of Figure 8 (for ${Y}_{{\rm{e}}}=0.500$). For ${Y}_{{\rm{e}}}\gt 0.51$, both ${Y}_{\alpha }/{Y}_{{\rm{h}}}$ and ${Y}_{{\rm{p}}}/{Y}_{{\rm{h}}}$ are greater than unity at ${T}_{9}=3$ (Figure 6). The freeze-out is thus followed by α-capture and proton-capture processes. We find in the bottom three panels of Figure 8 (${Y}_{{\rm{e}}}=0.525$, 0.550, and 0.603) that nuclei in a wide range of $A\sim 10$–70, including odd-Z elements, are substantially enhanced by these charged-particle capture processes after the temperature drops below ${T}_{9}=3$.

Figure 9 compares the final abundances with (blue) and without (cyan) neutrino reactions for the trajectory that has the highest ${Y}_{{\rm{e}}}=0.603$ in model z9.6 (same as that in the right-bottom panel of Figure 8). This indicates that the enhancement of nuclei with A = 60–70 is due to a $\nu p$-process in which the faster $(n,p)$ and $(n,\gamma )$ reactions with the free neutrons supplied by ${\bar{\nu }}_{e}$ capture on free protons replace the slower ${\beta }^{+}$-decays (Fröhlich et al. 2006; Pruet et al. 2006; Wanajo 2006). Note that the $\nu p$-process in our result is very weak and only the nuclei up to $A\sim 70$ are produced despite its substantial proton-richness (up to ${Y}_{{\rm{e}}}=0.603$). By contrast, Wanajo et al. (2011a) have shown that the $\nu p$-process in neutrino-driven wind with similar proton-richness can produce nuclei of up to $A\sim 120$ (see also Pruet et al. 2006; Arcones et al. 2012). This discrepancy is due to the lower entropies and longer expansion timescales in the early dynamical ejecta, which reduce ${Y}_{{\rm{p}}}/{Y}_{{\rm{h}}}$ at the beginning of a $\nu p$-process (${T}_{9}\sim 3$), than those in the late-time neutrino-driven wind.

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In Section 3.1 we found that the iron-group (from Ca to Cu) and light trans-iron (from Zn to Mo) species can be produced in the innermost ejecta of z9.6, the model with the widest range in Y_e in the ejecta. No heavier elements are produced in any of our six models. In Figures 10 and 11, the final mass fractions of stable isotopes from K to Mo are presented as functions of Y_e for all the trajectories of model z9.6. The mass fractions of selected radioactive isotopes (before decay) are also shown in the right-bottom panel of Figure 11. We find from these figures that few isotopes exhibit maximum abundances near ${Y}_{{\rm{e}}}=0.5$, in particular the light trans-iron species (from Zn to Mo). Overall, Ni and light trans-iron species are predominantly formed in neutron-rich ejecta, although the weak $\nu p$-process in proton-rich ejecta plays a subdominant role for those up to Ge. Several isotopes such as ⁴⁸Ca, ⁵⁰Ti, ⁵⁴Cr, and the radioactive nuclide ⁶⁰Fe are made only in very neutron-rich ejecta with ${Y}_{{\rm{e}}}\sim 0.40\mbox{--}0.43$ (Wanajo et al. 2013a, 2013b). This sensitivity clearly demonstrates the importance of nucleosynthesis studies based on multidimensional SN simulations with detailed multigroup neutrino transport, which are indispensable for accurately determining the Y_e distribution in the ejecta.

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For all SN models, the mass-integrated yields are compared to the solar abundance (Lodders 2003). Figure 12 shows the elemental mass fractions in the total ejecta with respect to their solar values (not to the initial compositions of progenitors), i.e., "production factors," for these models. The total ejecta mass from each SN is taken to be the sum of the ejected mass from the core and the outer envelope—that is, ${M}_{\mathrm{prog}}-{M}_{\mathrm{PNS}}$ in Tables 1 and 2. Here the outer envelope is assumed to be metal free (i.e., H and He only), which is reasonable for the models near the low-mass end of the SN range, e8.8, z9.6, and u8.1: the part of the H and He shells that were not included in our nucleosynthesis calculations would not contribute substantially to the yields for the range of nuclei considered here ($Z\geqslant 19$) regardless of progenitor metallicity. For the more massive models s11, s15, and s27, one should bear in mind that there is an important additional contribution to heavy elements (those heavier than He) from the outer envelope (see Woosley et al. 2002; Woosley & Heger 2007 and references therein).

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In each panel of Figure 12, we show a normalization band in yellow that covers the range within $1\,\mathrm{dex}$ of the maximum production factor and one tenth of that. The elements that reside in this band can (at least in part) originate from SNe represented by each model, provided that their production factors are greater than ∼10 (e.g., Woosley & Heger 2007). As an order-of-magnitude estimate, the production factors of ∼40 for light trans-iron elements from Zn to Zr in our representative model z9.6 suggest that such low-mass CCSNe can be the major sources of these elements if these events account for a few 10% of all CCSNe and would still contribute sizable amounts if they made up ∼10% of all CCSNe. This suggests that SNe at the low-mass end of the progenitor spectrum as represented by models e8.8, z9.6, and u8.1 could be an important source of light trans-iron elements (Section 4.6).

Note that there is a remarkable agreement of the nucleosynthesis of z9.6 with that of e8.8, which is a consequence of the similarity of their pre-SN core structures with a steep density gradient outside the core and a very dilute outer envelope. In both cases, the resulting fast explosion leads to the ejection of appreciable amounts of neutron-rich material. However, model u8.1, with a core structure very similar to (but with a slightly shallower density gradient than) that of z9.6, results in a different nucleosynthetic trend with deficiencies of several elements between Zn and Zr. This indicates that only a slight difference of pre-SN core-density structures can lead to substantially different nucleosynthesis outcomes.

For more massive models, the production factors are smaller than ∼10 for all elements, despite their greater ejecta masses compared to low-mass models (M_ej in Table 2). The maximum production factor of ∼1 for s11 provides no element responsible for the Galactic chemical evolution. For s15 and s27, only Zr with the production factor ∼10 is the element that can be originated from these types of CCSNe. However, these SNe are possible contributors of some isotopes as discussed in Section 4.2. Moreover, the non-neutrino processed "outer ejecta" from these stars, which are not fully included in our study, contribute significantly to the chemical enrichment of the Galaxy as discussed in Section 4.1.

Figure 13 compares the isotopic abundances with the solar values for all SN models. The maximum isotopic production factor for each model is generally greater than that of elements (Section 4.1) and therefore places tighter constraints on the contribution of relevant SNe to the Galaxy. In Table 3, we list the five largest production factors for each model. Note that most of the isotopes listed here are made in nuclear equilibrium, so uncertainties in individual nuclear reaction rates are irrelevant. Models e8.8 and z9.6 exhibit appreciable production factors of 354 (⁸⁶Kr) and 168 (⁸²Se), respectively. This implies that such low-mass SNe account for ∼10% of all CCSN events (according to the reason described in Section 4.1), supposing that these trans-iron elements originate solely from this class of events. A similar contribution of u8.1-like events can be expected with its largest production factor of 149 (⁷⁴Se). The maximum production factor of 1.48 for s11 indicates that no species are dominantly produced in the innermost ejecta of such SNe. For s15 and s27, the largest production factors are 24.8 and 43.1 (⁷⁴Se), respectively, which are sizably greater than those of elements (∼10; Figure 13). This indicates that such intermediate-mass and massive CCSNe can be important sources of several species listed in Table 3.

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To be more quantitative, we consider ⁸²Se, with the largest production factor for model z9.6, as representative. By assuming ${f}_{{\rm{z}}9.6}$ to be the fraction of z9.6-like events to all CCSNe, we have (Wanajo et al. 2011b)

$$\begin{eqnarray}&&\displaystyle \frac{{f}_{{\rm{z}}9.6}}{1-{f}_{{\rm{z}}9.6}}=\displaystyle \frac{{X}_{\odot }{(}^{82}\mathrm{Se})/{X}_{\odot }{(}^{16}{\rm{O}})}{{M}_{{\rm{z}}9.6}{(}^{82}\mathrm{Se})/\langle M{(}^{16}{\rm{O}})\rangle }=0.164,\end{eqnarray} \tag{ 3 }$$

where ${X}_{\odot }{(}^{82}\mathrm{Se})=1.38\times {10}^{-8}$ and ${X}_{\odot }{(}^{16}{\rm{O}})=6.60\times {10}^{-3}$ are the mass fractions in the solar system (Lodders 2003), ${M}_{{\rm{z}}9.6}{(}^{82}\mathrm{Se})=1.91\times {10}^{-5}\,{M}_{\odot }$ is the ejecta mass of ⁸²Se for z9.6, and $\langle M{(}^{16}{\rm{O}})\rangle =1.5\,{M}_{\odot }$ is the production of ¹⁶O by massive CCSNe averaged over the stellar initial mass function (IMF) between $13\,{M}_{\odot }$ and $40\,{M}_{\odot }$ (see Wanajo et al. 2009). Equation (3) gives ${f}_{{\rm{z}}9.6}=0.14$, which corresponds to a mass window of ${\rm{\Delta }}{M}_{\mathrm{prog}}\sim 1\,{M}_{\odot }$ near the low-mass end of the SN progenitor spectrum, ${M}_{\mathrm{prog}}\sim 9\,{M}_{\odot }$. It is currently uncertain whether this mass window for z9.6-like progenitors is reasonable or not, as only a slight difference of core-density structure leads to a very different nucleosynthetic result of u8.1. If we apply Equation (3) to model e8.8 by replacing ⁸²Se with ⁸⁶Kr (Table 3), then we get ${f}_{{\rm{e}}8.8}=0.085$. This corresponds to a mass window of ${\rm{\Delta }}{M}_{\mathrm{prog}}\sim 0.5\,{M}_{\odot }$, which is in reasonable agreement with the prediction from synthetic SAGB models at solar metallicity (Poelarends et al. 2008). If other production channels were to contribute significantly to these elements, then the allowed rate of ECSNe or a z9.6-like explosion would, of course, be lower.

It is thus conceivable that a combination (or either) of ECSNe and low-mass CCSNe accounts for the production of light trans-iron species in the Galaxy. For the more massive models s11, s15, and s27, a lack of self-consistent nucleosynthesis yields in the outer envelopes precludes a quantitative estimate of their contributions.

Our results described in Sections 4.1 and 4.2 shows that SNe near the low-mass end produce appreciable amounts of trans-iron species despite their small ejecta masses (compared to those of massive models; M_ej in Table 2). The reason can be found in Figure 14, which shows the fraction of the ejecta that originates in neutron-rich matter (${Y}_{{\rm{e}}}\lt 0.4975$, ${M}_{\mathrm{ej},{\rm{n}}}$ in Table 2) for a given species. For low-mass models e8.8, z9.6, and u8.1, the light trans-iron isotopes of A = 64–90 are almost exclusively produced in neutron-rich ejecta, whereas proton-rich matter plays a minor role (see also Figures 10 and 11). The subdominant roles of SNe from massive progenitors (represented by s11, s15, and s27) to the production of these species can be understood as a result of the small mass of neutron-rich ejecta for these stars (${M}_{\mathrm{ej},{\rm{n}}}$ in Table 2 and Figure 2).

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Among the massive models, s27 has a relatively larger amount of neutron-rich ejecta compared to the two others, with the Y_e down to ∼0.4. This is a consequence of the fact that model s27 exhibits an earlier explosion with a rapidly increasing shock radius because of prominent SASI activity that is absent in other models. These neutron-rich ejecta lead to a relatively flat trend of production factors in this model, similar to what we found for e8.8 and z9.6 (Figures 12 and 13). However, the bottom panel of Figure 13 indicates that about half of the species between A = 64 and 90 originate from neutron-rich and proton-rich ejecta. In fact, 43% of ⁶⁴Zn in s27 comes from the proton-rich ejecta. As described in Section 3.1.3, a weak $\nu p$-process is responsible for the production of these isotopes in the proton-rich ejecta. Greater entropies of the ejecta (Figure 3), slower expansion compared to that for low-mass models, and higher neutrino luminosities and mean energies also enhance the efficiency of the $\nu p$-process.

Models e8.8 and z9.6 exhibit appreciable production factors of ⁴⁸Ca (18.9 and 18.6, respectively), which is made in neutron-rich NSE (or α-deficient QSE; Meyer et al. 1996; Wanajo et al. 2013a) as can be seen in Figures 8 and 10 (middle-top panels, ${Y}_{{\rm{e}}}\sim 0.4$). The amounts are still not large enough to regard these low-mass SNe as the main contributors of ⁴⁸Ca, although a reduction of the entropies by about 30% would lift the values to a satisfactory level (Wanajo et al. 2013a). A rare class of high-density SNe Ia, in which similar physical conditions are expected, has also been suggested as a possible source of ⁴⁸Ca (Woosley 1997).

One of the outstanding features of our nucleosynthesis result is the production of all the stable isotopes of Zn (A = 64, 66, 67, 68, and 70), an element whose origin remains a mystery. Among our explored models, the low-mass models e8.8 and z9.6 exhibit nearly flat production factors over A = 64–70. This fact implies that the element Zn, or all its stable isotopes, could originate from ECSNe or low-mass CCSNe as represented by e8.8 and z9.6, respectively. As can be seen in the right-bottom panel of Figure 10, Zn isotopes are predominantly made in neutron-rich ejecta with ${Y}_{{\rm{e}}}\sim 0.4\mbox{--}0.5$ and models e8.8 and z9.6 produce a sufficient amount of ejecta in this range to contribute most of the solar inventory of Zn. There is still a discrepancy of a factor of 10 between the largest (⁶⁶Zn) and smallest (⁶⁷Zn) production factors, which might be cured if the Y_e distributions for these models were slightly modified. Alternatively, some of the neutron-rich Zn isotopes may originate from hydrostatic neutron-capture processes in massive stars (Woosley & Weaver 1995; Nomoto et al. 2013). CCSNe represented by u8.1 cannot be the single source of all Zn isotopes because of the descending trend of production factors (Figure 15). Model s27, which shows a flat trend of production factors, cannot be representative of Zn contributors either because of the small production factors (1.2–5.2).

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To date, only hypernova models (Umeda & Nomoto 2002; Tominaga et al. 2007; Nomoto et al. 2013) have been proposed as sources of ⁶⁴Zn (except for an ECSN model in Wanajo et al. 2011b), the dominant isotope of Zn in the solar system. In their hypernova models high entropies of ejecta lead to a strong α-rich freeze-out from nuclear equilibrium, resulting in an appreciable production of ⁶⁴Zn. However, this mechanism does not coproduce the other isotopes and requires additional contributors such as hydrostatic neutron captures in massive stars. As described in Section 3.1.3, the $\nu p$-process also produces ⁶⁴Zn only and therefore cannot be the main mechanism responsible for making Zn.

The nearly flat production factors over the wide range of A = 64–90 (except for Ga and As; Figure 13) in e8.8 and z9.6 also suggest that such low-mass SNe are the dominant sources of light trans-iron elements from Zn to Zr as suggested by Wanajo et al. (2011b) if their mass window is ${\rm{\Delta }}{M}_{\mathrm{prog}}\sim 0.5\mbox{--}1$. This is a consequence of the fact that the abundant neutron-rich ejecta for these models (Figure 2) cover the range of Y_e (down to ∼0.40) where the production of these species becomes maximal (Figures 10 and 11). By contrast, the results for model u8.1 (with a pre-SN core structure similar to that of z9.6) imply that such SNe can only be a source of the proton-rich isotopes of these trans-iron elements. This is due to the small amount of neutron-rich ejecta with ${Y}_{{\rm{e}}}\sim 0.40\mbox{--}0.45$.

In previous studies, the production of such light trans-iron species has often been attributed to the weak s-process (e.g., Käppeler et al. 2011). Woosley & Heger (2007) showed that the s-process in massive stars produces appreciable amounts of light trans-iron isotopes but only between A = 65–85 with a descending trend of production factors. It also has been claimed that significant production of these elements can occur in stars above $25\,{M}_{\odot }$ via the weak s-process (Pignatari et al. 2010). However, Sukhbold et al. (2016) pointed out that such light trans-iron species (in particular those above As) were sizably underproduced in their work because a large number of stars formed black holes instead of exploding and therefore did not contribute to weak s-process nuclides. A major contribution of the weak s-process from stars above $25\,{M}_{\odot }$ is also unlikely considering observational evidence against the successful explosions of stars in this mass range (Smartt 2015). Note that Chieffi & Limongi (2013) find that sufficient rotation-induced mixing in massive stars appreciably increases the amount of weak s-process elements. We speculate that the weak s-process in massive stars may still be responsible for several trans-iron elements, in particular such as Ga and As, which are underproduced in our models e8.8 and z9.6 (Figures 12 and 13).

It is interesting to note that model s27 gives abundant ⁹²Mo (with the third largest production factor, 30.7, in Table 3), an important p-nuclide whose astrophysical origin has been a long-lasting problem. In our case ⁹²Mo is made in slightly neutron-rich ejecta (${Y}_{{\rm{e}}}\sim 0.47$; see middle panel of Figure 8 and middle-bottom panel of Figure 11) during nuclear equilibrium (as suggested by Hoffman et al. 1996; Wanajo 2006). With a production factor of ∼30, such a type of CCSNe—i.e., relatively early explosions with dense outer envelopes—s27-like explosions would need to account for ∼30% of all CCSNe to explain the solar abundance of ⁹²Mo. A potential problem is that the other p-isotope ⁹⁴Mo cannot be made in neutron-rich nuclear equilibrium (Hoffman et al. 1996; Wanajo 2006). However, the $\nu p$-process in the neutrino-driven wind may add these isotopes with a high ⁹⁴Mo/⁹²Mo ratio (Pruet et al. 2006; Wanajo 2006; Wanajo et al. 2011a). Note that we do not find large production factors for the p-isotopes of Ru and Pd as in Pruet et al. (2006). This is because these isotopes were made by the $\nu p$-process (Section 3.1.3) in their late-time (≳1 s) wind ejecta with high entropies ($S\sim 70\,{k}_{{\rm{B}}}$), while the hydrodynamical simulations of models s15 and s27 stopped at ∼0.8 s after core bounce with entropies still below $S=50\,{k}_{{\rm{B}}}$.

The masses of ⁵⁶Ni ejected from all models are listed in Table 4 (7th column), along with those of ⁵⁷Ni (last column). These values, in the range $\sim 0.002\mbox{--}0.006\,{M}_{\odot }$, are about one order of magnitude smaller than typical observed values (e.g., $0.07\,{M}_{\odot }$ for SN 1987A; Bouchet et al. 1991). However, the ⁵⁶Ni masses in our study should be taken as lower limits for the massive models s11, s15, and s27, for which we omit the outer envelopes including large parts of the Si/O layer. We therefore likely underestimate the amount of ⁵⁶Ni produced by explosive nucleosynthesis in the SN shock. Moreover, 2D models face a generic difficulty in determining the amount of ⁵⁶Ni made by explosive burning in the shock, since the shocked material is funneled around the neutrino-driven outflows onto the proto-neutron star with little mixing by the Kelvin–Helmholtz instability into the neutrino-driven ejecta (Müller 2015). The production of ⁵⁶Ni by explosive burning also depends on the postshock temperatures and could be related to the slow rise of the explosion energy to only ($0.3\mbox{--}1.5)\times {10}^{50}$ erg at the end of the simulations.

In addition, the core-collapse simulations of s15 and s27 stopped at a time when accretion was still ongoing and the mass ejection rate in the neutrino-driven outflows was still high so that we may miss some late-time contribution to ⁵⁶Ni in the neutrino-driven ejecta (see, e.g., Wongwathanarat et al. 2017). In fact, the late-time ejecta of s15 and s27 are mostly proton rich and are dominated by ⁵⁶Ni and α particles (Figure 6 and the bottom right panel of Figure 11).

For the low-mass models e8.8, z9.6, and u8.1, the listed values can be regarded as the final ones; the ${}^{56}\mathrm{Ni}$ mass is definitely very small (∼0.002–$0.003\,{M}_{\odot }$) for these cases. It has already been argued before for e8.8 (Kitaura et al. 2006; Wanajo et al. 2011b; Wanajo 2013) that the small ${}^{56}\mathrm{Ni}$ may be consistent with the value estimated for some observed low-luminosity SNe (e.g., Hendry et al. 2005; Pastorello et al. 2007) with small ejecta masses. The ${}^{56}\mathrm{Ni}$ mass and the other explosion properties of the three low-mass models are also consistent with the remnant composition and reconstructed light curve of the Crab supernova SN 1054 (Smith 2013; Tominaga et al. 2013; Moriya et al. 2014). From the nucleosynthetic point of view, low-mass iron-core SNe thus appear to be an equally viable explanation for these events compared to ECSNe.

The low-mass models e8.8 and z9.6 produce appreciable amounts of ⁶⁰Fe, $3.61\times {10}^{-5}\,{M}_{\odot }$, and $3.14\times {10}^{-5}\,{M}_{\odot }$, respectively (6th column in Table 4), which is comparable to the IMF-averaged ejection mass from CCSNe, ($2.70\mbox{--}3.20)\times {10}^{-5}\,{M}_{\odot }$ in Sukhbold et al. (2016). Note that in massive stars, ⁶⁰Fe is produced by successive neutron captures from Fe isotopes (Timmes et al. 1995; Limongi & Chieffi 2006) in the outer envelope, which is not included in our analysis of the massive models s11, s15, and s27. In e8.8 and z9.6, ⁶⁰Fe forms in neutron-rich NSE (${Y}_{{\rm{e}}}\sim 0.42\mbox{--}0.43$; right-bottom panel of Figure 11) as suggested by Wanajo et al. (2013b). This indicates that ECSNe or low-mass CCSNe account for at least ∼10% (see ${f}_{{\rm{e}}8.8}$ and ${f}_{{\rm{z}}9.6}$ in Section 4.2) of live ⁶⁰Fe in the Galaxy. Models other than e8.8 and z9.6 produce little ⁶⁰Fe in their innermost ejecta because of their small masses of neutron-rich ejecta.

Table 4 also lists the masses of other important radioactive isotopes, ²⁶Al, ⁴¹Ca, ⁴⁴Ti, and ⁵³Mn, which are, however, negligibly small compared to the contribution from the outer envelopes of massive stars (and from neutrino-driven wind for ⁴⁴Ti; Wongwathanarat et al. 2017). Recent work by Sukhbold et al. (2016) suggests the IMF-averaged amount of ²⁶Al from massive stars to be $(2.80\mbox{--}3.63)\times {10}^{-5}\,{M}_{\odot }$. Their resultant mass ratio of ⁶⁰Fe to ²⁶Al, ∼1 (although a factor of two smaller than previous estimates, e.g., Woosley & Heger 2007), conflicts with the mass ratio ∼0.34 inferred from gamma-ray observation (flux ratio 0.148 ± 0.06; Wang et al. 2007). Taking the observational value as a constraint, this would suggest that Sukhbold et al. (2016) overestimated the ⁶⁰Fe mass by about a factor of three. This can be attributed to uncertainties in the relevant reaction rates (Woosley & Heger 2007; Tur et al. 2010) while for our cases ⁶⁰Fe forms in nuclear equilibrium and thus individual reactions are irrelevant. If this is the case, then ECSNe or low-mass SNe can contribute to live ⁶⁰Fe up to ∼30% of the amount in the Galaxy.