Impact of Newly Measured Nuclear Reaction Rates on ²⁶Al Ejected Yields from Massive Stars

Abstract

Over the last three years, the rates of all the main nuclear reactions involving the destruction and production of ²⁶Al in stars (²⁶Al(n, p)²⁶Mg, ²⁶Al(n, α)²³Na, ²⁶Al(p, γ)²⁷Si and ²⁵Mg(p, γ)²⁶Al) have been re-evaluated thanks to new high-precision experimental measurements of their crosssections at energies of astrophysical interest, considerably reducing the uncertainties in the nuclear physics affecting their nucleosynthesis. We computed the nucleosynthetic yields ejected by the explosion of a high-mass star (20 M_⊙, Z = 0.0134) using the FRANEC stellar code, considering two explosion energies, 1.2 × 10⁵¹ erg and 3 × 10⁵¹ erg. We quantify the change in the ejected amount of ²⁶Al and other key species that is predicted when the new rate selection is adopted instead of the reaction rates from the STARLIB nuclear library. Additionally, the ratio of our ejected yields of ²⁶Al to those of 14 other short-lived radionuclides (³⁶Cl, ⁴¹Ca, ⁵³Mn, ⁶⁰Fe, ⁹²Nb, ⁹⁷Tc, ⁹⁸Tc, ¹⁰⁷Pd, ¹²⁶Sn, ¹²⁹I, ³⁶Cs, ¹⁴⁶Sm, ¹⁸²Hf, ²⁰⁵Pb) are compared to early solar system isotopic ratios, inferred from meteorite measurements. The total ejected ²⁶Al yields vary by a factor of ~3 when adopting the new rates or the STARLIB rates. Additionally, the new nuclear reaction rates also impact the predicted abundances of short-lived radionuclides in the early solar system relative to ²⁶Al. However, it is not possible to reproduce all the short-lived radionuclide isotopic ratios with our massive star model alone, unless a second stellar source could be invoked, which must have been active in polluting the pristine solar nebula at a similar time of a core-collapse supernova.

1. Introduction

The synthesis occurring in stars (and related explosions) of the radioactive nuclide ²⁶Al, characterized by a half-life of 0.72 Myr, plays a pivotal role in enhancing our comprehension of the genesis of our solar system and the evolution of stars and galaxies, and is a subject of interest in both γ-ray astrophysics and cosmochemistry [1,2]. An excess of ²⁶Mg, the daughter isotope of ²⁶Al, is found in meteoritic calcium-aluminium-rich inclusions (CAIs), the first solids to form in the protosolar nebula. This observation provides evidence for the injection of live ²⁶Al during the early stages of the Solar System [3,4]. Consequently, unraveling the origin of ²⁶Al is imperative for a comprehensive understanding of the birth of the Sun and our planetary system. Additionally, the distinctive ²⁶Al emission of the diffuse 1809 keV line within our Galaxy, as observed by γ-ray telescopes [5], serves as a direct tracer of ongoing nucleosynthesis processes enriching the interstellar medium, with its spatial distribution indicating that outflows from massive Wolf-Rayet stars (with masses exceeding 25 M_⊙ [6,7]) and core-collapse supernovae are the principal loci for ²⁶Al production [8], contributing to approximately 70% of the observed live ²⁶Al in the Milky Way [9,10].

²⁶Al is generated during three distinct phases in the life cycle of massive stars: (i) H core burning in Wolf-Rayet stars, characterized by strong mass loss leading to the expulsion of highly ²⁶Al-enriched layers within the H convective core; (ii) explosive C/Ne burning; and (iii) C/Ne convective shell burning during pre-supernova stages. In the latter phase, the fraction of ²⁶Al that survives the subsequent explosion is ejected [11,12].

Other contributors to the presence of ²⁶Al in the Galactic environment include nova explosions [13]. Novae play a role in supplying up to 30% of the live ²⁶Al observed in the Milky Way [14,15]. Further minor sources are found in Asymptotic Giant Branch (AGB) stars, representing the concluding phase in the life cycle of low- and intermediate-mass stars [16], and Type-I X-ray bursts (assuming enough mass can actually be ejected from the accreting neutron star). In this last case, one of the main channels producing ²⁶Al is the nuclear reaction chain ²²Mg($\alpha ,p$)²⁵Al($\beta$⁺)²⁵Mg($p,\gamma$)²⁶Al, whose reaction rate uncertainty can significantly impact the modeled light curve shape [17]. Across all these sources, ²⁶Al is generated through the ²⁵Mg($p,\gamma$)²⁶Al nuclear reaction, and it is mainly destroyed (apart from its $\beta$+ decay into ²⁶Mg) by ²⁶Al($p,\gamma$)²⁷Si [11] and, if an efficient neutron source is available, by the neutron-induced nuclear reactions ²⁶$\mathrm{Al}(n,p$)²⁶$\mathrm{Mg}$ and ²⁶$\mathrm{Al}(n,\alpha$)²³$\mathrm{Na}$ [18]. The rates of all these key nuclear reactions have been re-evaluated by several authors over the last two years. Both the ²⁵Mg($p,\gamma$)²⁶Al and the ²⁶Al($p,\gamma$)²⁷Si nuclear reaction rates have been recently re-computed by the authors of Ref. [2], who presented a re-evaluation based on the most recent nuclear data and included the deep underground direct measurement performed by the LUNA collaboration [19]. Additionally, the ²⁵Mg($p,\gamma$)²⁶Al nuclear reaction was also re-studied by the authors of Ref. [20], who derived a new reaction rate based on the results of a complete experimental investigation of the 92, 130, and 189 keV resonances (interms of center of mass energy) with the Jinping Underground Nuclear Astrophysics (JUNA) lab. Finally, the authors of Ref. [21] determined a new rate for both the ²⁶$\mathrm{Al}(n,p$)²⁶$\mathrm{Mg}$ and ²⁶$\mathrm{Al}(n,\alpha$)²³$\mathrm{Na}$ nuclear reactions, of which they presented re-evaluations primarily based on high-precision measurements from the n_TOF-CERN facility ([22,23,24]), supplemented by theoretical calculations and a previous experiment [25] at higher neutron energies, to cover the full range of relevant stellar temperatures.

In this work, we re-compute the ejected nucleosynthetic yields from a 20 M_⊙ massive star at solar metallicity including all the new rates of the ²⁵Mg($p,\gamma$)²⁶Al, ²⁶Al($p,\gamma$)²⁷Si, ²⁶$\mathrm{Al}(n,p$)²⁶$\mathrm{Mg}$ and ²⁶$\mathrm{Al}(n,\alpha$)²³$\mathrm{Na}$ key nuclear reactions involving ²⁶Al. The main characteristics of our stellar models are presented in Section 2.2. We apply the new rates in the calculation of full stellar and nucleosynthesis models, quantify the impact of the new nuclear reaction rates and compare the results to key observables in Section 3. Our conclusions are presented in Section 4.

2. Computational Methods

2.1. Simulation Setup

We used the 1D stellar code FRANEC (Frascati Raphson-Newton Evolutionary Code, presented in detail in [26,27]) to calculate the hydrostatic evolution of the models presented in this work. FRANEC solves the equations describing the physical and chemical evolution of a star in hydrostatic and thermal equilibrium, assuming spherical symmetry and applying a classical Raphson-Newton method. The adopted initial composition for the solar metallicity is the one provided by Asplund, 2009 [28] and corresponds to Z = 0.01345. For the simulations, we used an extended nuclear network, already adopted in Roberti et al. [29], including 525 nuclear species extending up to Bi and more than 3000 reactions, fully coupled to the solution of the equations that describe the evolution of the star. The explosion and the associated explosive nucleosyntheses have been computed by means of the HYPERION Lagrangian hydrodynamic flux limited diffusion radiation 1D code (extensively described in [30]). HYPERION solves the full system of hydrodynamic equations governing the supernova evolution by means of the fully Lagrangian scheme of the piecewise parabolic method (PPM), a higher-order extension of Godunov’s method, described by [31]. The nuclear network used for the explosion is the same as the one used for the hydrostatic evolution of the star.

2.2. Description of the Stellar Models

To evaluate the impact of the newly measured reaction rates on ²⁶Al, we calculated the evolution of a 20 $M_{\odot }$ star at solar metallicity adopting three different reaction rate selections for the ²⁵Mg($p,\gamma$)²⁶Al, ²⁶Al($p,\gamma$)²⁷Si, ²⁶$\mathrm{Al}(n,p$)²⁶$\mathrm{Mg}$ and ²⁶$\mathrm{Al}(n,\alpha$)²⁶$\mathrm{Na}$ nuclear reaction, as summarised in

Table 1. Schematic view of the references adopted for the three reaction rate selections presented in this work.

Nuclear Reaction	STANDARD	JU-LA-BA	LA-BA
25Mg($p,\gamma$)26Al	Iliadis et al., 2010 [32]	Zhang et al., 2023 [20]	Laird et al., 2023 [2]
26Al($p,\gamma$)27Si	Iliadis et al., 2010 [32]	Laird et al., 2023 [2]	Laird et al., 2023 [2]
26Al($n,p$)26Mg	Caughlan & Fowler 1988 [33]	Battino et al., 2023 [21]	Battino et al., 2023 [21]
26Al($n,\alpha$)23Na	Angulo et al., 1999 [34]	Battino et al., 2023 [21]	Battino et al., 2023 [21]

. The nuclear reaction rates adopted in the STANDARD case are the same as in [29]; in particular, the ²⁵Mg($p,\gamma$)²⁶Al and ²⁶Al($p,\gamma$)²⁷Si reaction rates are both from [32], while the ²⁶$\mathrm{Al}(n,p$)²⁶$\mathrm{Mg}$ and ²⁶$\mathrm{Al}(n,\alpha$)²³$\mathrm{Na}$ reaction rates are from [33,34], respectively. In the LA-BA case abbreviation for “LAIRD-BATTINO”), the ²⁶$\mathrm{Al}(n,p$)²⁶$\mathrm{Mg}$ and ²⁶$\mathrm{Al}(n,\alpha$)²³$\mathrm{Na}$ reaction rates are both from [21], while the ²⁵Mg($p,\gamma$)²⁶Al and ²⁶Al($p,\gamma$)²⁷Si reaction rates are both from [2]. Most importantly, we extended both rates beyond 0.7 GK (the highest temperature provided by [2]) using the RATESMC code [32], including higher energy resonances beyond 484 keV based on the information presented in [32]. Our new high-temperature rates are presented in

Table 2. ²⁵Mg($p,\gamma$)²⁶Al reaction rates for temperatures higher than 0.7 GK on the ground state of ²⁶Al in units of [cm³/mol s].

T[GK]	Lower Limit	Median Rate	Upper Limit
0.700	1.027 × 102	1.078 × 102	1.136 × 102
0.800	1.933 × 102	2.022 × 102	2.121 × 102
0.900	3.195 × 102	3.333 × 102	3.488 × 102
1.000	4.810 × 102	5.011 × 102	5.240 × 102
1.250	1.026 × 103	1.068 × 103	1.118 × 103
1.500	1.740 × 103	1.811 × 103	1.897 × 103
1.750	2.586 × 103	2.689 × 103	2.815 × 103
2.000	3.532 × 103	3.667 × 103	3.831 × 103
2.500	5.615 × 103	5.809 × 103	6.039 × 103
3.000	7.804 × 103	8.047 × 103	8.329 × 103
3.500	9.946 × 103	1.023 × 104	1.055 × 104
4.000	1.193 × 104	1.225 × 104	1.261 × 104
5.000	1.520 × 104	1.559 × 104	1.600 × 104
6.000	1.748 × 104	1.792 × 104	1.839 × 104
7.000	1.890 × 104	1.938 × 104	1.989 × 104
8.000	1.966 × 104	2.017 × 104	2.071 × 104
9.000	1.994 × 104	2.047 × 104	2.103 × 104
10.000	1.989 × 104	2.043 × 104	2.100 × 104

,

Table 3. ²⁵Mg($p,\gamma$)²⁶Al reaction rates for temperatures higher than 0.7 GK on the isomeric state of ²⁶Al in units of [cm³/mol s].

T[GK]	Lower Limit	Median Rate	Upper Limit
0.700	2.887 × 101	3.054 × 101	3.237 × 101
0.800	5.899 × 101	6.232 × 101	6.610 × 101
0.900	1.043 × 102	1.102 × 102	1.171 × 102
1.000	1.661 × 102	1.756 × 102	1.870 × 102
1.250	3.944 × 102	4.169 × 102	4.451 × 102
1.500	7.198 × 102	7.593 × 102	8.101 × 102
1.750	1.125 × 103	1.183 × 103	1.259 × 103
2.000	1.592 × 103	1.668 × 103	1.767 × 103
2.500	2.631 × 103	2.741 × 103	2.881 × 103
3.000	3.705 × 103	3.843 × 103	4.016 × 103
3.500	4.720 × 103	4.883 × 103	5.079 × 103
4.000	5.626 × 103	5.807 × 103	6.020 × 103
5.000	7.033 × 103	7.240 × 103	7.474 × 103
6.000	7.930 × 103	8.154 × 103	8.400 × 103
7.000	8.428 × 103	8.660 × 103	8.910 × 103
8.000	8.641 × 103	8.873 × 103	9.124 × 103
9.000	8.660 × 103	8.890 × 103	9.135 × 103
10.000	8.550 × 103	8.776 × 103	9.015 × 103

and

Table 4. ²⁶Al($p,\gamma$)²⁷Si reaction rates for temperatures higher than 0.7 GK on the ground state of ²⁶Al in units of [cm³/mol s].

T[GK]	Lower Limit	Median Rate	Upper Limit
0.700	4.640 × 101	5.050 × 101	5.500 × 101
0.800	8.020 × 101	8.730 × 101	9.520 × 101
0.900	1.220 × 102	1.330 × 102	1.450 × 102
1.000	1.710 × 102	1.860 × 102	2.020 × 102
1.250	3.170 × 102	3.420 × 102	3.700 × 102
1.500	4.880 × 102	5.230 × 102	5.610 × 102
1.750	6.730 × 102	7.160 × 102	7.640 × 102
2.000	8.600 × 102	9.120 × 102	9.680 × 102
2.500	1.210 × 103	1.280 × 103	1.350 × 103
3.000	1.500 × 103	1.580 × 103	1.670 × 103
3.500	1.720 × 103	1.820 × 103	1.920 × 103
4.000	1.880 × 103	1.990 × 103	2.100 × 103
5.000	2.040 × 103	2.160 × 103	2.300 × 103
6.000	2.050 × 103	2.170 × 103	2.320 × 103
7.000	1.960 × 103	2.080 × 103	2.210 × 103
8.000	1.830 × 103	1.930 × 103	2.050 × 103
9.000	1.680 × 103	1.780 × 103	1.880 × 103
10.000	1.550 × 103	1.630 × 103	1.730 × 103

. Finally, in the JU-LA-BA case (abbreviation for “JUNA-LAIRD-BATTINO”) the reaction rates adopted are the same as in the LA-BA case, with the only exception of the ²⁵Mg($p,\gamma$)²⁶Al reaction rate which is taken from [20]. Note that we do not attempt any extrapolation of the reaction rate in all the cases where the temperature grid is not extending up to 10 GK, but instead we adopt the value of the last available experimental point.

In this work, we focused on non-magnetic and non-rotating massive star models simulations, as we plan to extend our analysis to rotating models in a follow-up study. The physics inputs and assumptions are the same as used in [27,29]; in the following, we briefly recall the main relevant features. The convective zones are treated using the mixing length theory (MLT), and their borders are defined according to the Ledoux criterion in H-burning regions and to the Schwarzschild criterion elsewhere. Semiconvection is taken into account, as discussed in [35]. The MLT and semiconvection parameters are ${\alpha }_{\mathrm{MLT}}=2.1$ and ${\alpha }_{\mathrm{semi}}=0.02$, respectively. The convective boundary mixing (CBM), implemented as 0.2 $H_{\mathrm{P}}$ of overshooting, is considered only at the edge of the H convective core during the main sequence (MS). Mass loss has been included following the prescriptions of [36,37] for the Blue Super Giant (BSG) phase, [38,39] for the Red Super Giant (RSG) phase, and [40] for the Wolf-Rayet phase (WR). We also included the dynamical mass loss caused by the approach of the luminosity of the star to the Eddington limit by removing all the zones where the Eddington factor $\Gamma =L_{\mathrm{rad}}/L_{\mathrm{EDD}}>1$. The explosion is triggered through a thermal bomb, i.e., by depositing a certain amount of thermal energy at the edge of the Fe core in the pre-supernova model. The injected energy is such that to achieve a final kinetic energy of the ejecta (measured at infinity) of $1.2\times {10}^{51}$ erg, the typical kinetic energy assumed for SN 1987A and widely used in the literature [41,42,43,44] is employed. The final mass-cut (that defines the remnant mass) is determined by the total amount of fallback ∼${10}^7$ s after the core-collapse and corresponds to 2.58 $M_{\odot }$, in agreement with other 20 $M_{\odot }$ CCSN models available in literature [12,45,46]. We calculate an additional CCSN model, imposing a final kinetic energy of $3\times {10}^{51}$ erg, to study the effect of a more energetic explosion on the production of ²⁶Al. In this case, the explosion energy is high enough to completely eject all the layers above the Fe core. The mass-cut is therefore chosen so that the ejecta contain 0.07 $M_{\odot }$ of ⁵⁶Ni, to match the amount of ⁵⁶Ni estimated from SN 1987A, and it is 1.86 $M_{\odot }$. Finally, we did not include any “$mixing$ $and$ $fall$ $back$” prescriptions in any of our CCSN simulations (see, e.g., [47,48]). A summary of the main properties of the reference model is reported in

Table 5. Summary of the key characteristics of the presupernova (PSN) and explosive models presented in this work.

Mass-loss scheme	Vink [36,37], de Jaeger [38], Nugis [40], van Loon [39]
Convection criteria	Ledoux in H burning zones, Schwarzschild elsewhere
${\alpha }_{\mathrm{MLT}}$	2.1
${\alpha }_{\mathrm{semi}}$	0.02
CBM	0.2 $H_{\mathrm{P}}$ of overshooting (H burning)
Initial mass	20 $M_{\odot }$
Initial metallicity	$1.345\times {10}^{-2}$ (Asplund 2009 [28])
Initial H mass fraction	0.721
Initial 4He mass fraction	0.265
12C mass fraction (He exhaustion)	0.286
CO core (He exhaustion)	4.91 $M_{\odot }$
Lifetime (PSN)	9.87 Myr
Total Mass (PSN)	7.46 $M_{\odot }$
$log{T}_{\mathrm{eff}}$ (PSN)	4.30
$logL/L_{\odot }$ (PSN)	5.31
Explosion scheme	Thermal bomb [30]
Thermal energy injected	$2.61\times {10}^{51}$ erg, $4.41\times {10}^{51}$ erg
Explosion energy (at infinity)	$1.20\times {10}^{51}$ erg, $3.00\times {10}^{51}$ erg
Remnant mass	2.58 $M_{\odot }$, 1.86 $M_{\odot }$

. The final structure is instead shown in Figure 1. We remind the reader that the explosive nucleosynthesis depends only on the temperature and density in the post-shock zones, and we use the following definition (e.g., [49,50,51]):

• Complete Si burning (${\mathrm{Si}}_{\mathrm{c}}$): $T_{\mathrm{shock}}>5$ GK;
• Incomplete Si burning (${\mathrm{Si}}_{\mathrm{i}}$): $5\mathrm{GK}>T_{\mathrm{shock}}>4\mathrm{GK}$;
• Explosive O burning (O): $4\mathrm{GK}>T_{\mathrm{shock}}>3.3\mathrm{GK}$;
• Explosive Ne burning (Ne): $3.3\mathrm{GK}>T_{\mathrm{shock}}>2.1\mathrm{GK}$;
• Explosive C burning (C): $2.1\mathrm{GK}>T_{\mathrm{shock}}>1.9\mathrm{GK}$.

²⁶Al in CCSNe is usually produced between explosive Ne and C zones. In our models, ²⁶Al is synthesized at temperatures between 2.60 and 1.74 GK. It is clearly visible in Figure 2, which shows the ²⁵Mg(p, $\gamma$)²⁶Al nuclear reaction rate ratio between [2,20], how, in this temperature range, the rate from [2] is very close to the rate from [20], while it is between two and three factors lower at the lowest temperatures. This is mainly due to the shifted resonance energy computed when taking the atomic binding into account, as described in [2], while this effect was not included by [20].

Below $T_{\mathrm{shock}}=1.9$ GK, the temperature is too low on the typical explosion timescale (∼$1\mathrm{s}$) to activate other major explosive burning stages, and therefore, the matter is ejected without any further reprocessing. However, some secondary processes may still occur, such as the neutron burst in the He shell, required to explain Mo and Zr anomalies in pre-solar silicon carbide grains [52].

3. Results

3.1. Impact of New Nuclear Reaction Rates on the Ejected Yields

We re-computed the complete nucleosynthesis fully coupled to the structural evolution of our massive star model, using the three network settings described in Section 2.2. The resulting ejected yields for Ne, Na, Mg, Al and Si stable isotopes and ²⁶Al and ⁶⁰Fe (i.e., the species most affected by the newly measured reaction rates we test in this work) are presented in

Table 6. Total explosive ejected yields (in M_⊙) of ²⁶Al and other key species of our models for different selections of nuclear reaction rates (see main text for more details).

Species	STANDARD	JU-LA-BA	LA-BA
	1.2 × ${\mathbf{10}}^{\mathbf{51}}$ erg
20Ne	5.60 × 10−1	5.60 × 10−1	5.60 × 10−1
23Na	1.07 × 10−2	1.07 × 10−2	1.07 × 10−2
24Mg	9.32 × 10−2	9.32 × 10−2	9.33 × 10−2
25Mg	1.66 × 10−2	1.66 × 10−2	1.66 × 10−2
26Mg	1.65 × 10−2	1.65 × 10−2	1.65 × 10−2
26Al	7.02 × 10−5	2.75 × 10−5	2.65 × 10−5
27Al	1.14 × 10−2	1.13 × 10−2	1.14 × 10−2
28Si	2.13 × 10−2	2.14 × 10−2	2.13 × 10−2
29Si	3.17 × 10−3	3.17 × 10−3	3.18 × 10−3
30Si	1.96 × 10−3	1.94 × 10−3	1.95 × 10−3
60Fe	1.19 × 10−5	1.19 × 10−5	1.17 × 10−5
	3 × ${\mathbf{10}}^{\mathbf{51}}$ erg
20Ne	4.79 × 10−1	4.79 × 10−1	4.79 × 10−1
23Na	8.80 × 10−3	8.79 × 10−3	8.79 × 10−3
24Mg	9.81 × 10−2	9.78 × 10−2	9.82 × 10−2
25Mg	1.44 × 10−2	1.44 × 10−2	1.44 × 10−2
26Mg	1.44 × 10−2	1.44 × 10−2	1.44 × 10−2
26Al	9.68 × 10−5	3.36 × 10−5	3.26 × 10−5
27Al	1.20 × 10−2	1.19 × 10−2	1.20 × 10−2
28Si	2.82 × 10−1	2.82 × 10−1	2.82 × 10−1
29Si	4.99 × 10−3	5.03 × 10−3	5.03 × 10−3
30Si	6.94 × 10−3	6.88 × 10−3	6.89 × 10−3
60Fe	1.14 × 10−5	1.14 × 10−5	1.12 × 10−5

. Only minor differences are present between the LA-BA and JU-LA-BA cases. Indeed, the total rate of the ²⁵Mg($p,\gamma$)²⁶Al between 2.60 and 1.74 GK from Laird et al., 2023 [2] is very close to the rate from JUNA [20], differing on average by about 4%. Since the only difference between the JU-LA-BA and LA-BA cases is the different ²⁵Mg($p,\gamma$)²⁶Al reaction rate, this causes the small difference of the same order between the two cases in the final ²⁶Al abundance, being 4% lower in LA-BA compared to JU-LA-BA. Larger differences are instead present between the STANDARD case and both the JU-LA-BA and LA-BA cases, due to the different ²⁶Al($n,\alpha$)²³Na and ²⁶Al($n,p$)²⁶Mg reaction rates adopted. In particular, the ²⁶Al ejected yields change by a factor between 2 and 3 depending on the explosion energy. This is consistent with the results of [21], who noticed how the final abundance of ²⁶Al produced in their supernova simulations was about a factor of 2 lower when their ²⁶Al($n,\alpha$)²³Na and ²⁶Al($n,p$)²⁶Mg reaction rates were adopted (as in JU-LA-BA and LA-BA) instead of those from [33,34] (as in STANDARD).

3.2. Comparison to Ess Ratios

The early Solar System (ESS) was enriched in short-lived radionuclides (SLRs, with half-lives ranging between 0.1 and 100 Myr) at the time of its formation 4.6 Gyrs ago [4]. This evidence is gleaned from the examination of meteorite compositions, wherein excesses of daughter isotopes originating from radioactive nuclei are detected. While the majority of SLRs possess half-lives long enough (exceeding approximately 2 Myr) to be adequately explained by their production in the Galaxy [53], isotopes with shorter half-lives (such as ²⁶Al, ³⁶Cl and ⁴¹Ca) defy explanation through galactic chemical evolution (GCE) processes alone. Consequently, the presence and abundance of these isotopes necessitate one or more additional local stellar sources, potentially originating from the same molecular cloud as the Sun. These local sources would have directly enriched the protosolar nebula from which the Sun formed or the pre-existing proto-planetary disk (for comprehensive reviews of various scenarios, see, e.g., [4]).

We compared the yields of each of our stellar model to ESS abundances, as outlined in [4] and Lawson et al. (in prep). In order to probe the formation of the Solar System, we consider isotopes that have half-lives comparable to the time frame of the formation of the Solar System from its molecular cloud (approx. 15 Myrs, [54]), and compare their abundances with the ejected yields from our models. The aim is to test the hypothesis that a core-collapse supernova event could have produced all the SLRs in the proportion needed to reproduce their ESS abundances measured in calcium-aluminium-rich inclusions (CAIs, the first solids to have formed in the protosolar nebula), hence shaping the environment for the birth of the Sun. SLR daughter and reference signatures can be detected in meteoritic samples and are further used to estimate the abundances of SLRs in the ESS [4,55]. A list of these SLRs, daughter isotopes, reference isotopes, half-lives, mean lifetime and ESS abundance ratios can be found in

Table 7. The full list of SLRs considered in this work, listed with reference isotopes, T_1/2, $\tau$, and ESS ratios. Data gathered from [4]. If no error is given, the value is an upper limit.

SLR	Daughter	Reference	T1/2 (yr)	$\mathit{\tau }$ (yr)	ESS Ratio
26Al	26Mg	27Al	7.170 × 105	1.034 × 106	(5.23 ± 0.13) × 10−5
36Cl	36S	35Cl	3.010 × 105	4.343 × 105	(2.44 ± 0.65) × 10−5
41Ca	41K	40Ca	9.940 × 104	1.434 × 105	(4.6 ± 1.9) × 10−9
53Mn	53Cr	55Mn	3.740 × 106	5.396 × 106	(7 ± 1) × 10−6
60Fe	60Ni	56Fe	2.620 × 106	3.780 × 106	(1.01 ± 0.27) × 10−8
92Nb	92Zr	92Mo	3.470 × 107	5.006 × 107	(3.2 ± 0.3) × 10−5
97Tc	97Mo	98Ru	4.210 × 106	6.074 × 106	<1.1 × 10−5
98Tc	98Ru	98Ru	4.200 × 106	6.059 × 106	<6 × 10−5
107Pd	107Ag	108Pd	6.500 × 106	9.378 × 106	(6.6 ± 0.4) × 10−5
126Sn	126Te	124Sn	2.300 × 105	3.318 × 105	<3 × 10−6
129I	129Xe	127I	1.570 × 107	2.265 × 107	(1.28 ± 0.03) × 10−4
135Cs	135Ba	133Cs	2.300 × 106	3.318 × 106	<2.8 × 10−6
146Sm	142Nd	144Sm	6.800 × 107	9.810 × 107	(8.28 ± 0.44) × 10−3
182Hf	182W	180Hf	8.900 × 106	1.284 × 107	(1.018 ± 0.043) × 10−4
205Pb	205Tl	204Pb	1.730 × 107	2.496 × 107	(1.8 ± 1.2) × 10−3

. The nucleosynthesis of the SLRs in both massive star and CCSN are detailed in [12].

In order to calculate the contribution of ejecta from a model to the ESS, we must take into account the effect of dilution and radioactive decay. We can calculate a dilution factor (f) for each model, defined by

$$f={\displaystyle \frac{M_{SLR}^{ESS}}{M_{SLR}^{*}}}$$

(1)

Here, $M_{SLR}^{ESS}$ is the observed abundance of a reference SLR and $M_{SLR}^{*}$ is the total ejected mass of the SLR from the model.

We use ²⁶Al to define the f for each model ejecta, as the ²⁶Al/²⁷Al ratio is very well established [56,57]. A time delay is then applied to the ejecta to account for the time between the ejection of the SLRs by the stellar source and its incorporation into the CAIs. The decrease in the abundance of the radioactive species is then calculated using the radioactive decay equation

$$M_{SLR}\left(t\right)=M_{SLR}\left(t_0\right)e^{-\frac{ln\left(2\right)\Delta t}{T_{1/2}}}$$

(2)

In this equation, the initial mass of the isotope ($M_{SLR}\left(t_0\right)$) is decayed by the half-life of the SLR ($T_{1/2}$). This provides the final decayed abundance of the ejecta ($M_{SLR}\left(t\right)$) after a time delay.

The SLR yields of each model are decayed using a time delay of 2.5 Myr. This is a good representative value [4], determined by examining the decay profiles of each SLR (with a dilution f calculated using ²⁶Al, Lawson et al. in prep).

In Figure 3, we present the ESS abundance predicted by our 1.2 × ${10}^{51}$ erg explosive model for each SLR once decayed using Equation (2) and diluted using ²⁶Al to calculate f. In order to fully replicate the ESS abundances, each isotope should lie on (or potentially below) the dashed threshold line. Any SLR above this threshold are overproduced and require an additional ²⁶Al source. SLRs below the threshold may be polluted by an additional event or through galactic chemical evolution (GCE) contributions. We separate the top and bottom panels of Figure 3 into isotopes with lower/equal and greater half-lives than ⁶⁰Fe, respectively. Isotopes in the top panel have the shortest half-lives, and therefore, their ratio to ²⁶Al should ideally lie very close to the ESS value, since there would not be enough time for GCE to fill the gap in case of very low values (which can instead play a role for the longer-lived isotopes in the bottom panel). The models presented in this work are plotted as single points (a blue diamond for the standard model, blue hexagon for JU-LA-BA and a red star for LA-BA) and are compared to the 23 models with initial mass 20$M_{\odot }$ of [12] (covering an explosion range of 0.53–8.86 × ${10}^{51}$ erg and remnant mass range of 1.74–3.40$M_{\odot }$), which are plotted as box-plots and represent the impact of different stellar physics inputs. As the LA-BA and JU-LA-BA models have less ²⁶Al, they have an increased ${\mathrm{M}}_{\mathrm{model}}$/${\mathrm{M}}_{\mathrm{ESS}}$ across all isotopes.

Starting with the isotopes with half-lives less than 2.62 Myr (the T^1/2 of ⁶⁰Fe, top panel of Figure 3), ³⁶Cl is underproduced in the stellar models presented in this work. However, considering the range of ³⁶Cl/²⁶Al covered by [12] models, it might be in agreement with the observed ESS values if different explosion energies are considered. ⁴¹Ca and ¹²⁶Sn are both underproduced, even considering different stellar physics inputs. A similar case is represented by ⁶⁰Fe and ¹³⁵Cs, which are overproduced in both our models and in those by [12]. Considering the isotopes with half-lives longer than 2.62 Myr (bottom panel of Figure 3), ⁵³Mn, and ⁹⁸Tc are in agreement within nuclear and stellar uncertainties with their ESS values, while ⁹²Nb and ¹⁴⁶Sm are underproduced, but could be brought into agreement with ESS values by GCE contributions. On the other hand, ⁹⁷Tc, ¹⁰⁷Pd, ¹²⁹I, ¹⁸²Hf and ²⁰⁵Pb are all overproduced.

Therefore, we notice how, even considering the most recent and up-to-date nuclear physics input, only 5 out of the 14 SLRs considered here are consistent with their observed ESS values. Two potential solutions could be represented by either a different astrophysical scenario that is able to perform better against observations, or an additional pollution event producing more ²⁶Al and less of the overproduced SLRs (such as ⁶⁰Fe) that happened close in time (within about 2.5 Myr) and space to a core-collapse supernova.

4. Conclusions

In this work, we computed the evolution of a high-mass star (20 M_⊙, Z = 0.01345) and the nucleosynthetic yields ejected by its explosion at two different energies, $1.2\times {10}^{51}$ erg and $3\times {10}^{51}$ erg. We included all the updated rates of the relevant nuclear reactions for ²⁶Al nucleosynthesis, i.e., ²⁶Al($n,p$)²⁶Mg, ²⁶Al($n,\alpha$)²³Na, ²⁶Al($p,\gamma$)²⁷Si and ²⁵Mg($p,\gamma$)²⁶Al.

We found a substantial decrease in the ejected amount of ²⁶Al in the JU-LA-BA and LA-BA cases, between a factor of two and three, compared to the STANDARD case. This is consistent with the impact of the newest ²⁶Al($n,\alpha$)²³Na and ²⁶Al($n,p$)²⁶Mg reaction rates found by [21].

We then compared the ejected yields of our models for 14 SLRs (³⁶Cl, ⁴¹Ca, ⁵³Mn, ⁶⁰Fe, ⁹²Nb, ⁹⁷Tc, ⁹⁸Tc, ¹⁰⁷Pd, ¹²⁶Sn, ¹²⁹I, ³⁶Cs, ¹⁴⁶Sm, ¹⁸²Hf, ²⁰⁵Pb) with their abundances in the ESS, in order to check if a core-collapse supernova could represent a realistic scenario for the circumstances and the environment of the birth of the Sun. Only 5 out of the 14 SLRs are consistent with their observed ESS abundances. At least two potential solutions should be considered: (1) A different astrophysical scenario that is able to perform better against observations. One possibility may be represented by rotating Wolf-Rayet stars, where the rotation-enhanced mass loss may limit the amount of hydrogen to be converted into ¹⁴N during the core H-burning phase and hence the amount of neutrons released by the ²²Ne($\alpha ,n$)²⁵Mg during the He-burning phase (since ²²Ne is produced by a sequence of $\alpha$-captures on ¹⁴N). This may therefore limit the production of n-capture SLRs like ⁶⁰Fe and ¹³⁵Cs. (2) An additional pollution event producing more ²⁶Al and less of the overproduced SLRs (such as ⁶⁰Fe) that happened close in time (within about 2.5 Myr) and space to a core-collapse supernova.