Impacts of the 12 C ( 𝛼, 𝛾 ) 16 O reaction rate on 56 Ni nucleosynthesis in pair-instability supernovae

Nuclear reactions are key to our understanding of stellar evolution, particularly the 12 C ( 𝛼, 𝛾 ) 16 O rate, which is known to significantly influence the lower and upper ends of the black hole (BH) mass distribution due to pair-instability supernovae (PISNe). However, these reaction rates have not been sufficiently determined. We use the MESA stellar evolution code to explore the impact of uncertainty in the 12 C ( 𝛼, 𝛾 ) 16 O rate on PISN explosions, focusing on nucleosynthesis and explosion energy by considering the high resolution of the initial mass. Our findings show that the mass of synthesized radioactive nickel ( 56 Ni ) and the explosion energy increase with 12 C ( 𝛼, 𝛾 ) 16 O rate for the same initial mass, except in the high-mass edge region. With a high (about twice the STARLIB standard value) rate, the maximum amount of nickel produced falls below 70 𝑀 ⊙ , while with a low rate (about half of the standard value) it increases up to 83.9 𝑀 ⊙ . These results highlight that carbon "preheating" plays a crucial role in PISNe by determining core concentration when a star initiates expansion. Our results also suggest that the onset of the expansion, which means the end of compression, competes with collapse caused by helium photodisintegration, and the maximum mass that can lead to an explosion depends on the 12 C ( 𝛼, 𝛾 ) 16 O reaction rate.


INTRODUCTION
Pair Instability Supernovae (PISNe) are the explosive deaths of very massive stars, which have been theoretically predicted (e.g., Barkat et al. 1967;Fryer et al. 2001;Heger et al. 2003) and a good candidate has recently been discovered (Schulze et al. 2024).In very massive stars that form massive helium cores ( He ≳ 45 ⊙ ; Heger & Woosley 2002), the electron-positron creation reactions take place in the core soften the equation of state, and reduce the adiabatic index  below 4/3 (Fraley 1968).To be specific, thermal energy is converted into the rest mass of the electron-positron pairs, decreasing the pressure (Rakavy & Shaviv 1967).The instability induced by this pressure reduction causes the core to collapse, leading to explosive oxygen and silicon burning (Rakavy et al. 1967).If the explosive oxygen burning provides enough energy, its thermonuclear energy can reverse the collapse, leading the entire star to explode with no remnant behind it.It is also predicted from stellar evolutionary theory that when massive progenitors become PISNe, we can observe ★ E-mail: h-kawashimo@g.ecc.u-tokyo.ac.jp the luminous transients (10 44 erg s −1 or brighter at peak) for several months (e.g., Heger & Woosley 2002;Scannapieco et al. 2005;Kasen et al. 2011;Dessart et al. 2013).
Since a PISN completely destroys stars and leaves no compact objects behind, it has been thought that there is a pair-instability mass gap in the black hole mass distribution at 50−130 ⊙ , corresponding to the progenitors of the mass region where PISN occurs (Heger & Woosley 2002;Woosley et al. 2007;Belczynski et al. 2016;Woosley 2017Woosley , 2019;;Spera & Mapelli 2017).Hence, the upper limit of the mass gap is considered to be determined by the mass range of PISNe and the lower limit by the transition between PISNe and pulsational pair-instability supernovae (PPISN) (cf.Farmer et al. 2020).However, this conjecture is now challenged by GW190521 which has two black holes with masses of 66 +17 −18  ⊙ and 85 +21 −14  ⊙ (Abbott et al. 2020a,b;Estellés et al. 2022), and the PISN condition is required to be reconsidered (cf.Nitz & Capano 2021;Abbott et al. 2024;Kinugawa et al. 2021;Moreno Méndez et al. 2023).
The 12 C(, ) 16 O reaction rate is one of the most influential nuclear reactions in the evolution of stars (Tur et al. 2009(Tur et al. , 2010)), and this is also true for PISNe (Takahashi 2018).However, the 12 C(, ) 16 O reaction rate is difficult to determine experimentally with the current measurement sensitivity and remains highly uncertain (deBoer et al. 2017).Therefore, it is important to perform astrophysical simulations that take this uncertainty into account (e.g., Weaver & Woosley 1993;Kikuchi et al. 2015;Mehta et al. 2022;Farag et al. 2022).
Recently, the uncertainty in the 12 C(, ) 16 O reaction rate was found to affect the range of PI mass gaps (Farmer et al. 2019(Farmer et al. , 2020;;Costa et al. 2021) (cf.Mehta et al. 2022).It suggested that black holes can be generated in mass regions previously thought to be PI mass gaps, and has attracted attention in explaining GW190521 1 .From there, when considering stellar mass distribution, it is expected that the 12 C(, ) 16 O reaction rate also affects the event rate of PISNe (Tanikawa et al. 2023).Thus, the effect of the 12 C(, ) 16 O reaction rate on PISNe is a noteworthy issue from the standpoint of optical observations.However, it is not clear how the uncertainties of the 12 C(, ) 16 O reaction rate affect the brightness of individual PISNe.
The amount of radioactive nickel 56 Ni that determines the brightness of an SN is important as information is directly related to observations.It will be helpful to predict the detectability of PISNe by upcoming observatories (Moriya et al. 2019;Regős et al. 2020;Moriya et al. 2022a,b;Tanikawa et al. 2023;Aguado et al. 2023).In addition, nickel synthesis is also an important topic from galactic chemical evolution since nickel is eventually turned into iron and supplied to space.In this study, we have used stellar evolution calculations to consider PISNe that occur under various 12 C(, ) 16 O rates and calculate the amount of 56 Ni produced and the explosion energy.
This paper is structured as follows.In section 2, we explain the investigation methods.In section 3, we show our results and discuss our findings.We conclude the paper in Section 4.

Setup
We utilize version 15140 of the stellar evolution code MESA (Paxton et al. 2011(Paxton et al. , 2013(Paxton et al. , 2015(Paxton et al. , 2018(Paxton et al. , 2019;;Jermyn et al. 2023) to simulate the evolutionary process of helium cores.These cores either collapse to form black holes or undergo explosive events known as Pair-Instability Supernovae (PISNe).The input parameter configuration is based on the default model choices outlined by Marchant et al. (2019), specifically referred to as the ppisn setup within MESA-r15140 2 .Note that we determined the success or failure of PISN using the same criteria as in Marchant et al. (2019).We suppose that a PISN succeeds when all parts of the star exceed the escape velocity, and the calculation is terminated at that time.We also determine failure based on the central density exceeding 10 12 g cm −3 and the maximum infall velocity of the central Fe core exceeding 8 × 10 8 cm s −1 .
In our simulations, we initiate the process by employing a nonrotating model of hydrogen-free helium stars with a metallicity of  = 10 −5 .Given our specific focus on understanding the 56 Ni amount and explosion energy in the PISN explosions and resolving the transition 1 Note that there are many suggestions to fill the PI mass gaps without changing 12 C( , ) 16 O reaction rate (e.g.Rodriguez et al. 2019;Di Carlo et al. 2020;Fishbach & Holz 2020;Umeda et al. 2020;González et al. 2021;De Luca et al. 2021;Cruz-Osorio et al. 2021;Tanikawa et al. 2021;Ziegler & Freese 2021;Rizzuto et al. 2022;Costa et al. 2022;Siegel et al. 2022;Ziegler & Freese 2022;Moreno Méndez et al. 2023;Volpato et al. 2023). 2 We note that one alteration from the original ppisn setup involves omitting inlist switching based on helium depletion to avoid potential failures during the handoff between inlists.
between successful PISN and CC models, we conducted calculations using various initial mass ranges.We initially explored a broad range of initial masses, spanning from 40 to 180  ⊙ , with increments of 5  ⊙ .Within this range, the occurrence of PISN explosions was confirmed through calculations performed in increments of 1  ⊙ .Furthermore, we conducted simulations with finer resolution, using increments of 0.1  ⊙ near the upper boundary of the mass range and subsequently employing increments of 0.01  ⊙ in the immediate vicinity of the uppermost edge (see Appendix D).Our investigated mass range covers between 70 and 150  ⊙ near the region of the PISN BH mass gap, as revealed by previous studies (Marchant et al. 2019).
The evolution of helium stars serves as a valuable laboratory for investigating the evolution of massive stars experiencing pairinstability.This is because a majority of massive stars are believed to have shed their outer hydrogen layers, thereby exposing their helium cores.Furthermore, the properties of these stars in their final phase are strongly influenced by the mass of their helium cores (Woosley 2017;Marchant et al. 2019).It is important to note that progenitors of merging binary black holes also undergo the loss of their hydrogen envelopes as a result of binary interactions unless their metallicity is nearly zero or convective overshoot is ineffective (e.g., Tanikawa et al. 2022).
We utilize the approx21_plus_co56.netnuclear reactions network integrated into the MESA framework.This network has been proven to be efficient and accurate in estimating explosion energy and the quantity of synthesized 56 Ni during explosive nucleosynthesis (Longland et al. 2010;Sallaska et al. 2013;Iliadis et al. 2015Iliadis et al. , 2016;;Farmer et al. 2019).For nuclear reaction rates, we adopt the default rates provided by MESA in this version, which are based on NACRE (Angulo et al. 1999) and JINA REACLIB (Cyburt et al. 2010).However, there is one exception, namely the 12 C(, ) 16 O rate, which is discussed in detail in Section 2.2.
For hydrodynamics, the setup uses the HLLC method, which is useful for modeling shock waves (Toro et al. 1994).The simulation is switched from hydrostatic to dynamical when the stellar global stability index falls below its critical value, 4/3.This index is calculated using the local pressure  and the local density , as represented by the equation below: where Γ 1 is the local first adiabatic exponent.This corresponds to the time when neutrino cooling is progressing rapidly (cf.Marchant et al. 2019;Farmer et al. 2019).

2.2
The treatment of the 12 C(, ) 16 O rate The treatment of the 12 C(, ) 16 O rate is the most important part of this paper, and it is essentially based on the previous studies by Farmer et al. (2019Farmer et al. ( , 2020)).We utilize STARLIB reaction rate library, which provides the median nuclear reaction rate, ⟨ c.s. ⟩ med , and the associated uncertainty factor, f.u., at temperatures ranging from  = 10 6 to 10 10 K (Sallaska et al. 2013).Following the approach of Longland et al. (2010), we assume that all reaction rates provided by STARLIB follow a log-normal probability distribution.The lognormal distribution is characterized by the position parameter  and spread parameter , respectively. .
(2) These parameters can be obtained using the median rate ⟨ c.s. ⟩ med and the factor uncertainty f.u.represented in STARLIB as follows.
In a lognormal distribution, the natural logarithm of the random variable ( = ln ) follows a normal distribution.= ⟨ c.s. ⟩ med • (f.u.)  . (5) Figure 1 shows the 12 C(, ) 16 O rate as a function of temperature, normalized to the median STARLIB rate ⟨ c.s. ⟩ ±•  /⟨ c.s. ⟩ med .⟨ c.s. ⟩ med and its uncertainty are from Kunz et al. (2002).Hereafter, when referring to the reaction rate ⟨ c.s. ⟩ ±•  , we simply denote it a ± • .To examine the effects of 12 C(, ) 16 O burning rate, we simulate stellar models using calculated 12 C(, ) 16 O rates ranging from −2 to +2 in increments of 1.It is important to note that we refer to the 0 series -representing the most probable values -as the standard series.

Overviews for PISN
In this section, we begin by discussing the typical characteristics of PISNe and the reliability of our explosion model using the standard 12 C(, ) 16 O rate. Figure 2  In Section 3.2, we provide the findings regarding the correlation between the 12 C(, ) 16 O rate and the properties of the PISN explosion, specifically the explosion energy, as final total energy (Section 3.2.1)and the synthesis of nickel (Section 3.2.2).Subsequently, we explore the underlying physics behind these correlations in Section 3.2.3.All results are presented in tabular form in Appendix D. We note that the total energy is the sum of kinetic, gravitational, and internal energy.In the final phase, the stars are sufficiently expanded, and no gravitational binding so that the explosion energy is approximately equal to the kinetic energy.

Explosion energy
Figure 3 illustrates the relationship between the explosion energy  expl and the initial He core mass  init,He for each 12 C(, ) 16 O reaction rate.Each color corresponds to a different reaction rate.When we fix the initial He core mass, we observe that models with higher 12 C(, ) 16 O rates exhibit higher explosion energies.Furthermore, within each series of the same 12 C(, ) 16 O rate, we observe a consistent pattern: the explosion energy gradually increases on the low-mass side and then sharply decreases in the high-mass region (for more discussion, see Appendix C).This behavior is observed across all models.In the increasing trend region, we also observe that the maximum explosion energy increases as the 12 C(, ) 16 O rate decreases.

56 Ni synthesis
Figure 4 displays the synthesized nickel mass at the final step as a function of the initial helium star mass. 4Notably, within the models sharing the same initial mass, a higher 12 C(, ) 16 O rate results in increased nickel synthesis.Similar to the explosion energy, we observe that the amount of synthesized nickel in the most massive progenitors is greater at lower 12 C(, ) 16 O reaction rates.However, we do not observe a point where the trend abruptly changes within each series.

Carbon "preheating"
Our findings reveal that within the same progenitor mass, a higher 12 C(, ) 16 O rate leads to increased total energy and the synthesis of radioactive nickel.This observation aligns with previous studies (Takahashi 2018;Farmer et al. 2020), which suggest that these trends with the 12 C(, ) 16 O rate stem from the carbon-burning process preceding the explosive oxygen burning that triggers PISNe.We describe the "preheating" process by observing energy gaining just before oxygen burning.Figure 5 presents the time trajectories of the total carbon mass and total energy for the initial He core mass  init,He = 100 ⊙ , in comparison to the standard 12 C(, ) 16 O rate and ±2 models.The time  = 0 corresponds to when the central temperature  c reaches log  c (K) = 9.5 in each model, marking the onset of explosive oxygen burning (Truran & Arnett 1970;Woosley et al. 1973).At high 12 C(, ) 16 O reaction rates (+2), the carbon is already depleted at the end of helium burning ( ≈ −80s).Consequently, limited carbon burning occurs, and the energy remains stagnant until the onset of explosive oxygen burning.In contrast, at low 12 C(, ) 16 O rates, a substantial amount of carbon persists, leading to carbon preheating that boosts the total energy prior to explosive oxygen burning.As a result, the star becomes unbound without awaiting explosive oxygen burning, leading to a gradual growth in total energy.Note that this preheating process is considered to occur within the CO core.This discussion is consistent with the known fact that PISNe are driven by explosive oxygen burning initiated in the CO core.

The maximum mass limit of the explosion
In this section, we elaborate on the fact that heavier stars become explodable in low 12 C(, ) 16 O rate environments.The upper limit of explodable initial mass, which represents the upper boundary of the PI mass gap, is primarily determined by photodisintegration (Takahashi et al. 2016;Takahashi 2018).We anticipate that nickel production and decomposition will transpire concurrently within hightemperature environments.The abundance pattern of a star that undergoes a failed PISN and is just prior to collapse reveals a decrease in nickel around its center, with helium constituting the majority of the components.The two panels in Figure 6   This condition is given by log (( He )) = 11.7974 where and  He is the residual number fraction of 4 He (see Appendix B for the derivation of Eq. ( 6) and ( 7)).Based on Figure 6(a), it is evident that all series of 12 C(, ) 16 O rates exhibit nearly identical characteristics.Furthermore, Figure 6(b) demonstrates that the transition from explosion to implosion occurs at  He ≈ 0.96 across all series.This finding suggest that the upper limit of PISNe is determined by the initiation of 4 He → 2n + 2p photodisintegration.
Figure 7 shows the evolution of central density and temperature for progenitors with  He = 115, 120, 125, 130 ⊙ .The square points indicate the onset of expansion (see Figure 2), and the trajectories of the unexploded model are depicted as dashed lines in corresponding colors.This figure also demonstrates that the central evolution of progenitors with the same initial He mass follows a consistent trajectory regardless of the reaction rate.The position of the onset of expansion varies.Moreover, as explained in Figure 2, it is evident that, for a given 12 C(, ) 16 O reaction rate, the position of the onset of explosion shifts to higher temperatures and pressures as the mass increases.By combining these results with the discussion in Figure 6, it can be inferred that endpoints associated with higher 12 C(, ) 16 O rates surpass the photodisintegration condition at relatively lower masses.Conversely, the endpoints for lower 12 C(, ) 16 O reaction rates occur at lower temperatures and pressures, limiting only higher-mass stars to cross the photodisintegration condition.For instance, focusing on the case of 120 ⊙ in Figure 7, higher reaction rates undergo more intense contraction, nearing the He photodisintegration border.When considering 125 ⊙ , any progenitors experience higher temperatures and pressures compared to the case of 120 ⊙ , resulting in a shift in the endpoint.As a consequence, the +2 model exceeds the He photodisintegration border and fails to explode as PISN.Conversely, the −2 model, farthest from the He photodisintegration border, remains sufficiently distant even for a 130 ⊙ case, indicating that it would not exceed the He photodisintegration border without becoming even more massive.

SUMMARY
We conducted stellar evolution calculations to investigate the impact of 12 C(, ) 16 O rates on 56 Ni nucleosynthesis in pair-instability supernovae (PISNe).Our findings indicate that lower 12 C(, ) 16 O reaction rates result in a greater amount of synthesized nickel in the heaviest explodable progenitor stars.For instance, the upper-mass limit of the synthesized nickel mass changes from 67 ⊙ (+2) to 83 ⊙ (−2), corresponding to 125 ⊙ (+2) and 160 ⊙ (−2) for the maximum mass of exploding progenitors.The shift of those mass ranges has already found in previous studies for lower-mass side as PPISN-PISN transition line, and our findings are consistent with the same trends for these insights (Regős et al. 2020;Costa et al. 2021;Woosley & Heger 2021).The novelty of this study lies in the systematic calculations of the synthesized nickel mass, which has not been investigated in the previous works.The change in the synthesized nickel mass may be attributed to the carbon preheating process.Additionally, we demonstrated that distinct 12 C(, ) 16 O reaction rates give rise to varying ranges of explodable masses due to the interplay between He photodisintegration and the preheating effect.
Note that these results will be affected by the size of the nuclear reaction network.We probably overestimate the amount of nickel produced and the rate of energy absorption by photodisintegration due to the current small network (see Renzo et al. 2020;Farmer et al. 2016, also Appendix C.).However, this overestimation is small enough compared to the amount of change from 12 C(, ) 16 O reaction varying.Our main results, therefore, are reliable even after taking into account the uncertainty of the size of the network.On the other hand, it should be noted that some previous studies have shown that larger networks synthesize more nickel (Marchant et al. 2019;Renzo et al. 2020).
Our findings have implications for estimating the detectability of PISNe, particularly regarding their dependence on the 12 C(, ) 16 O rate (e.g., Pan et al. 2012;Moriya et al. 2019Moriya et al. , 2022a,b;,b;Wong et al. 2019;Regős et al. 2020).Tanikawa et al. (2023) conducted population synthesis calculations to investigate the impact of 12 C(, ) 16 O rates on PISN discoveries using the Euclid space tele-scope (Laureĳs et al. 2011).They found that PISNe would be more frequently detected in the standard 12 C(, ) 16 O case compared to the −3 12 C(, ) 16 O case due to a higher intrinsic PISN event rate in the former case.However, their assumptions about identical light curves for PISNe with different 12 C(, ) 16 O rates raised concerns about the validity of their results.Finally, our results can address these concerns.Figure 4 indicates that PISNe in the low 12 C(, ) 16 O rate case tends to be fainter than those in the standard 12 C(, ) 16 O case when the initial He star masses are fixed.Although the maximum luminosity of PISNe gradually increases as 12 C(, ) 16 O rates decrease, it will not significantly impact PISN detectability.This is because PISNe with higher He star masses are already rare due to initial stellar mass functions in which the number of stars decreases with their masses increasing (Salpeter 1955;Schneider et al. 2018).
In the future, we will further investigate this argument by combining binary population synthesis calculations with PISN light curves, particularly for the low 12 C(, ) 16 O case.
Then, the abundances in nuclear equilibrium are given by the Saha's equation, where  i is the number density,  i is the spin degree of freedom, and  i is the mass for i particle, respectively.The number density is expressed by where  i is the number fraction and  is the density.From reaction (B1), we note that  = 28.3MeV.Also and By assuming we get Combining these equations, we obtain As a result, LHS of Eq. ( B2) is rewritten as Thus Eq. (B2) reads which is Eq. ( 6).

APPENDIX C: COMPARISON WITH THE PREVIOUS WORK
In this appendix, we validate the reliability of our calculations by comparing them to previous studies that employed alternative calculation methods.
Figure C1 illustrates the relationship between the initial He core mass  init,He and (a) the final energy of the explosion  expl , and (b) the synthesized 56 Ni mass.These plots represent models with a standard 12 C(, ) 16 O rate.We have included the results by Heger & Woosley (2002) for comparison.We confirm a positive correlation between the amount of synthesized 56 Ni and the explosion energy for the initial He core mass.Importantly, our results obtained without magnification of the 12 C(, ) 16 O rate show reasonable consistency with previous studies.
Note that in Figure C1(a), also in Figure 3, we observe a drop in the explosion energy at the heavier end of the initial He core mass, which has not been reported in previous studies (e.g., Heger & Woosley 2002).It is possible that the explodable upper mass limit of PISNe, primarily governed by He photodisintegration, leads to the "freeze-out" of photodisintegrated elements from iron, resulting in a portion of the explosion energy being captured as rest mass energy.However, it is important to note that this is speculative, and we have not identified the exact physical cause of this trend.

Figure 2 .
Figure2. c - c trajectories for models with initial He core masses of 40, 95, 110, 130, and 160 ⊙ , using the standard 12 C( , ) 16 O rate.Each color corresponds to the initial mass of the progenitor, except for the black line, which represents  = 4/3, the border of gravitational instability.The 40 and 160  ⊙ models undergo core collapse, while the other models result in PISN explosions.Square points indicate the maximum temperature experienced in exploding models, marking the beginning of the expansion.Beyond these points, the trajectories of explodable models turn back adiabatically.

Figure 3 .Figure 4 .
Figure 3.The relationship between the explosion energy  expl and the initial He core mass  init,He for different 12 C( , ) 16 O reaction rates.Each color in the plot corresponds to a specific reaction rate as described in Figure 1.
depict the maximum central temperature and the corresponding central density experienced by each model, represented by the square points in Figure 2. Dashed lines in the figure represent the condition for photodisintegration of 4 He → 2n + 2p, which uses helium produced from 56 Ni → 14 4 He.

Figure 5 .
Figure5.The time evolution of total carbon mass (top panel) and the total energy (bottom panel) for the initial He core mass  He = 100 ⊙ , comparing the standard and ±2 12 C( , ) 16 O rate models.The time origin  = 0 is defined as the moment when the central temperature  c reaches log  c (K) = 9.5 in each model, marking the onset of explosive oxygen burning that triggers the PISN.Squares represent the residual carbon mass at log  c (K) = 9.5, while triangles correspond to log  c (K) = 9.3, which is the beginning of neon burning, it consumes residue of carbon burning.The left panel displays the difference between them.In the bottom panel, the dashed horizontal line indicates the transition between negative and positive total energy.

Figure 6 .Figure 7 .Figure A1 .
Figure 6.The maximum central temperature and corresponding central density reached by each model.The colors represent different reaction rates, following the same convention as Figure 1.The top panel (a) displays all exploding models, while the bottom panel (b) zooms in on the region highlighted in purple in panel (a).The grey dashed lines indicate the threshold for 4 He → 2n + 2p photodisintegration with various  He .

Figure A2 .Figure C1 .
Figure A2.The time evolution of the central temperature (top panel) and central density (bottom panel) for models with an initial helium star mass of 100 ⊙ .The dashed vertical lines represent the instances when log  c (K) = 9.3, indicating the initiation of carbon preheating.