Sensitivity of ⁴⁴Ti and ⁵⁶Ni Production in Core-collapse Supernova Shock-driven Nucleosynthesis to Nuclear Reaction Rate Variations

Stars with initial mass ≳8 M_⊙ undergo a core-collapse supernova (CCSN) explosion after core fuel exhaustion (Woosley & Janka 2005). The CCSN enriches the interstellar medium by releasing the isotopes synthesized throughout its life cycle and during CCSN nucleosynthesis. Among the ejected isotopes, ⁴⁴Ti and ⁵⁶Ni are produced near the boundary of the protoneutron star remnant and the ejecta, and are thus studied to gain insight into details of the CCSN mechanism (Young et al. 2006; Young & Fryer 2007; Fryer et al. 2012, 2018).

Observations of ⁴⁴Ti in CCSN remnants are possible due to characteristic γ-rays at 67.9, 78.3, and 1157.0 keV emitted in the decay sequence ⁴⁴Ti (t_1/2 = 59.1 yr) $\to \,{}^{44}\mathrm{Sc}$ (t_1/2 = 3.97 hr) $\to \,{}^{44}\mathrm{Ca}$ (stable) (Chen et al. 2011). ⁵⁶Ni, on the other hand, follows the much briefer decay sequence ⁵⁶Ni (t_1/2 = 6.08 day) $\to \,{}^{56}\mathrm{Co}$ (t_1/2 = 77.2 day) $\to \,{}^{56}\mathrm{Fe}$ (Junde et al. 2011). As such, CCSN ⁵⁶Ni production is generally inferred from observations of the remnant iron, e.g., for the Cassiopeia A (CasA) supernova of 1671 AD (Eriksen et al. 2009).

Comparing such observational constraints to yields from CCSN model calculations offers the opportunity to constrain properties of the CCSN explosion, such as the explosion energy and duration, as well as the remnant mass  (Aufderheide et al. 1991; Young et al. 2006; Young & Fryer 2007; Müller et al. 2017; Fryer et al. 2018; Sawada & Maeda 2019). However, model calculation results for ⁴⁴Ti and ⁵⁶Ni yields have been shown to have significant sensitivities to variations in nuclear reaction rates (The et al. 1998; Hoffman et al. 1999; Magkotsios et al. 2010). This is particularly problematic as many of the relevant reaction rates have poor experimental constraints. Nuclear physics measurements can be prioritized with the aid of sensitivity studies, whereby nuclear reaction rates are varied within an uncertainty factor and the impact on the model calculation results is assessed.

Large-scale investigations of nuclear reaction rate uncertainties impacting shock-driven nucleosynthesis in CCSN were previously performed by The et al. (1998) and Magkotsios et al. (2010). These pioneering works used analytic temperature–density trajectories in nucleosynthesis postprocessing calculations, varying nuclear reaction rates by factors of /100 and ×100. They identified several nuclear reaction rates that can impact ⁴⁴Ti and ⁵⁶Ni nucleosynthesis for temperature and density trajectories approximating material heated by the expanding shock following a supernova core bounce. We build on prior work in the following ways. (1) We perform stellar evolution and CCSN calculations with one-dimensional models in an effort to focus on astrophysical conditions that are most relevant to comparisons with astronomical observations. (2) We vary nuclear reaction rates by realistic rate variation factors based on existing nuclear physics constraints in order to avoid focusing on cases which are already sufficiently constrained. (3) We cross-check whether the prioritized reaction rate lists resulting from prior studies are comprehensive. We note that similar work has been done by Hoffman et al. (2010) and Tur et al. (2010), but for a much smaller set of reaction rate variations.

Our paper is organized as follows. In Section 2 we discuss the model details for massive star evolution and subsequent CCSN. In Section 3 we detail temperature and density evolution during CCSN for comparison with prior postprocessing work. In Section 4 we present the explosion details, nucleosynthetic yields of ⁴⁴Ti and ⁵⁶Ni from our baseline CCSN calculations, and compare to yields determined by observations and previous modeling efforts. In Section 5 we present our list of varied reaction rates and adopted reaction rate variation factors. In Section 6 we present nuclear physics sensitivities, explaining the rate impacts in terms of the nuclear reaction network. In Section 7 we compare our results with the postprocessing calculations of The et al. (1998) and Magkotsios et al. (2010). We conclude in Section 8, with brief suggestions for future work involving astrophysics model calculations and nuclear physics experiments.

We performed one-dimensional, multizone model calculations using Modules for Experiments in Stellar Astrophysics² (MESA), which is an open-source code for modeling stellar evolution and stellar explosions (Paxton et al. 2011, 2015, 2018, 2019). In order to perform CCSN, we ran our simulations in two different stages using MESA version 7624, consisting of evolution to core collapse followed by the supernova explosion.

In stage one, we evolved nonrotating, solar metallicity (Z = 0.02) M_prog = 15 M_⊙, 18 M_⊙, 22 M_⊙, and $25\,{M}_{\odot }$ progenitors from zero age main sequence (ZAMS) to the onset of core-collapse, as this is within the suspected ZAMS mass range for progenitors of CCSN remnants with observed ⁴⁴Ti (Pérez-Rendón et al. 2002; Young et al. 2006; Wongwathanarat et al. 2017). Our stage one calculations are based on Farmer et al. (2016), using the 204 isotope network mesa_204.net,³ the JINA ReacLib reaction rate library version V2.0 2013-04-02 (Cyburt et al. 2010), and the Dutch wind scheme that is based on Nieuwenhuijzen & de Jager (1990), Nugis & Lamers (2000), Vink et al. (2001), and Glebbeek et al. (2009) with an efficiency scale factor η = 0.8 (Maeder & Meynet 2001). The spatial resolution resulted in the number of zones varying between ∼450 and 1200 zones, with the number of zones tending to increase in later stages of stellar evolution. The onset of core-collapse was defined as the time when any mass zone at the interior of the star exceeded an infall velocity of 1000 km s⁻¹ (Farmer et al. 2016).

Stage two took the final stellar model of stage one and carried this into the CCSN phase, following the procedure described by Paxton et al. (2015), which is divided into four distinct steps:

• 1.  
  Run MESA using inlist_adjust, where we adjust several control parameters for initiating CCSN simulations. The astrophysical parameters were chosen following Farmer et al. (2016). Notable among them are absence of any prescription to mimic rotation and the choice of the Dutch wind scheme with η = 0.8.
• 2.  
  Run MESA using inlist_remove_core, where we define and remove the stellar core. The outer mass coordinate of the core, i.e., the deepest zone of the model, is defined by the inner mass boundary I_b, located at the mass coordinate just outside the Fe core (as specified in Table 1).
• 3.  
  Run MESA using inlist_edep, which employs the "thermal bomb" mechanism (e.g., Young & Fryer 2007; Paxton et al. 2015) to inject energy E_inj ∼  10⁵¹ erg (Fryer et al. 2012) into a thin mass shell ${\rm{\Delta }}{M}_{\mathrm{shell}}\,=0.05\,{M}_{\odot }$ above I_b (the innermost ∼20 zones) over a time period t_inj = 20 ms. As specified in Table 1, we inject E_inj = 0.5 foe⁴ in 15 M_⊙ and 18 M_⊙ models; 1.32 foe in 15 M_⊙, 18 M_⊙, and 22 M_⊙; and 3.5 foe in 22 M_⊙ and 25 M_⊙ ZAMS stars. These explosion energies were roughly based on the findings of Vartanyan et al. (2019), Fryer et al. (2012), Paxton et al. (2015), and Nomoto (2014), where the specific energies were somewhat arbitrarily chosen in order to span a plausible range.We limited ourselves to a single t_inj to keep the overall number of calculations manageable. Our choice of t_inj was motivated in part by the finding of Fryer et al. (2012) that most supernovae have explosion energies E_exp > 10⁵¹ erg, which can be reached if the explosion occurs less than 250 ms after core bounce. Additionally, the analysis of Sawada & Maeda (2019) suggests short explosion timescales are more consistent with observations. The specific choice of t_inj = 20 ms corresponds to the fast-explosion timescale for thermal bomb models explored in Young & Fryer (2007). To explore the impact of this choice, we performed CCSN calculations for the 18 M_⊙ model using E_inj = 1.32 foe and t_inj = 3, 20, and 200 ms, comparing the yield of ⁴⁴Ti (see Section 4). The 20 ms calculations were in closer agreement with the ⁴⁴Ti yields of 3D models and observations, further motivating our choice. Similarly, we explored using ΔM_shell = 0.02 M_⊙, finding little impact on the ⁴⁴Ti and ⁵⁶Ni yields.
• 4.  
  Run MESA using inlist_explosion, which follows the thermodynamic and nucleosynthetic evolution of the stellar envelope due to the shock wave propagating outward from the thermal bomb energy deposition region. Nuclear reaction rates were varied in this step of the calculations.

Table 1.  Key Model Properties and the Resultant M(⁴⁴Ti) and M(⁵⁶Ni) with ΔM_shell = 0.05 M_⊙ for the Baseline Calculations Performed in This Work

${M}_{\mathrm{prog}}$	Einj	Eexp	Fecore	Ib	Mcut	T0	Log10(ρ0)	Ye	M(44 Ti)	M(56 Ni)	$\tfrac{M{(}^{44}\mathrm{Ti})}{M{(}^{56}\mathrm{Ni})}$
(M⊙)	(foe)	(foe)	(M⊙)	(M⊙)	(M⊙)	(GK)	ρ0(gm cm−3)		×10−5 (M⊙)	×10−2 (M⊙)	×10−5
	0.50	0.24	1.40	1.45	1.450				1.22	9.0	13.9
	0.50	0.24	1.40	1.45	1.500				0.43	5.0	8.14
15	0.50	0.24	1.40	1.45	1.600	4.848–2.483	7.120–5.962	0.497–0.499	0.08	0.3	23.9
	1.32	1.13	1.40	1.45	1.450				1.19	16.0	7.17
	1.32	1.13	1.40	1.45	1.500				0.24	14.0	1.78
	1.32	1.13	1.40	1.45	1.600	5.522–2.763	7.106–5.967	0.497–0.499	0.21	6.0	3.71
	0.50	0.07	1.46	1.45	1.450				1.10	14.0	8.02
	0.50	0.07	1.46	1.45	1.500				0.30	11.0	2.75
18	0.50	0.07	1.46	1.45	1.622	5.252–1.877	7.111–5.573	0.497–0.499	0.17	1.0	10.9
	1.32	0.93	1.46	1.45	1.450				1.09	20.0	5.42
	1.32	0.93	1.46	1.45	1.500				0.32	17.0	1.85
	1.32	0.93	1.46	1.45	1.622	6.176–2.609	7.278–5.915	0.497–0.499	0.27	8.0	3.49
	1.32	0.70	1.58	1.60	1.600				1.14	22.0	5.25
	1.32	0.70	1.58	1.60	1.650				0.48	20.0	2.35
22	1.32	0.70	1.58	1.60	1.820	5.704–2.651	7.199–5.957	0.498–0.499	0.46	9.0	4.85
	3.50	2.99	1.58	1.60	1.600				2.35	32.0	7.29
	3.50	2.99	1.58	1.60	1.650				1.14	32.0	3.60
	3.50	2.99	1.58	1.60	1.820	6.950–2.979	7.440–5.940	0.498–0.499	1.00	20.0	4.90
	3.50	2.79	1.61	1.60	1.600				10.6	34.0	31.0
25	3.50	2.79	1.61	1.60	1.650				9.99	34.0	29.4
	3.50	2.79	1.61	1.60	1.814	6.675–1.250	7.360–4.813	0.498–0.499	9.88	25.0	39.0

Note. Fe_core is the mass of the iron core. T₀, Log₁₀(ρ₀), and Y_e represent corresponding values from M₄ to the mass coordinate with no oxygen-burning. All other properties are described in the text.

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As the shock wave powered by the thermal bomb propagates toward the stellar surface, it changes the thermodynamic conditions in the regions of the star that it sweeps through. We quantify the impact on thermodynamic conditions by comparing to analytic temperature–density trajectories, namely exponential and power law, often used in postprocessing studies (e.g., The et al. 1998; Magkotsios et al. 2010). Both trajectories are characterized by a peak temperature T₀ and peak density ρ₀, followed by expansion and cooling with constant T³ρ⁻¹ (Magkotsios et al. 2010).

The exponential trajectory is based on Fowler & Hoyle (1964), where material heated to T₀ and compressed to ρ₀ undergoes adiabatic expansion, and the expansion timescale τ is equal to the freefall timescale:

$$\begin{eqnarray}&&\tau ={\left(24\pi G{\rho }_{0}\right)}^{-1/2}\approx 446{\left({\rho }_{0}\right)}^{-1/2}\,{\rm{s}},\end{eqnarray} \tag{ 1 }$$

where G is Newton's gravitation constant. The associated temporal evolution of temperature and density are described by Magkotsios et al. (2010), Harris et al. (2017), and Fryer et al. (2018)

$$\begin{eqnarray}&&T(t)={T}_{0}\exp \left(-\displaystyle \frac{t}{3\tau }\right)\qquad \rho (t)={\rho }_{0}\exp \left(-\displaystyle \frac{t}{\tau }\right).\end{eqnarray} \tag{ 2 }$$

The power-law profile is based on a constant-velocity (homologous) expansion (see e.g., Magkotsios et al. 2010; Fryer et al. 2018), where the temporal evolution of shock-heated material is described as

$$\begin{eqnarray}&&T(t)=\displaystyle \frac{{T}_{0}}{2t+1}\qquad \rho (t)=\displaystyle \frac{{\rho }_{0}}{{\left(2t+1\right)}^{3}}.\end{eqnarray} \tag{ 3 }$$

To inform comparisons with prior postprocessing studies using these analytic trajectories, we separately fit the temperature and density evolution of each zone of our models using Equations (2) and (3). We found that the power-law profile provides a superior reproduction of our data. This is consistent with Fryer et al. (2018), who found the power-law describes their data particularly well once the shock-heated material drops out of nuclear statistical equilibrium (NSE).

Magkotsios et al. (2010) studied the trends for ⁴⁴Ti and ⁵⁶Ni production in the T₀–ρ₀ plane for both exponential and power-law profiles. Figure 1 shows a color map of their ⁴⁴Ti production across the T₀–ρ₀ phase space for the power-law profile (Equation (3)) with electron-fraction Y_e = 0.498, which is close to the average Y_e = 0.498–0.499 range that we observe across each of our progenitor stars. For each M_prog and E_inj used in our CCSN calculations, we determined T₀ and ρ₀ for each mass zone, including these as data points in Figure 1. The figure includes labels for the four regions identified by Magkotsios et al. (2010) that are characterized by specific nuclear burning patterns and govern the yield of ⁴⁴Ti. Region A corresponds to incomplete silicon burning. Region B corresponds to normal freeze-out from NSE, where the abundance is largely determined from the Q-values of the reactions (Woosley et al. 1973; Meyer 1994; The et al. 1998; Hix & Thielemann 1999). Region C corresponds to the chasm region, where nucleosynthesis is characterized by the flow of material from a low-A quasistatic equilibrium (QSE) cluster containing ⁴⁴Ti to a high-A cluster containing ⁵⁶Ni. Region D corresponds to the α-rich freeze-out (Woosley et al. 1973).

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Magkotsios et al. (2010) demonstrated that the yield of ⁴⁴Ti in shock-driven nucleosynthesis depends sensitively on the location in the T₀–ρ₀ plane and the expansion profile followed by the shock-heated material. The reaction rate sensitivities reported by that work were therefore specified by phase space region and expansion profile, including the impact on the topology of the phase space (e.g., chasm widening). Figure 1 shows that our M_prog = 15 M_⊙ E_inj = 0.5 foe model is limited to lower values of T₀ and does not cross the chasm region (Region C), while any other combination of E_inj with progenitor mass M_prog extends to higher T₀ and ρ₀, into the α-rich freeze-out (Region D). The increase in maximum T₀ and ρ₀ with increasing M_prog and E_inj is also apparent.

Prior to discussing the impact of varied nuclear reaction rates on ⁴⁴Ti and ⁵⁶Ni production, we first present results from our calculations using the baseline reaction rate library. The yields presented in Table 1 were determined by integrating the total mass of ⁴⁴Ti or ⁵⁶Ni outward from a lower-bound in mass known as the mass cut M_cut (Diehl & Timmes 1998). To help assess the impact of this arbitrary boundary on our results, we explored three choices of M_cut: I_b, I_b + ΔM_shell, and M₄. The latter is defined as the mass zone where entropy per nucleon s = 4, beyond which higher-dimensional models have found most material avoids fallback onto the protoneutron star (Ertl et al. 2016). In order to compare with postprocessing calculations, we determine X(⁴⁴Ti) and X(⁵⁶Ni) over the region between M_cut and the mass coordinate where there is no longer oxygen-burning, determining the mass-fraction per zone and taking an average weighted by the mass of each zone.

Figure 2 shows selected mass fractions following shock-driven nucleosynthesis for example calculations, with one value of M_cut shown for context. Clearly the choice of M_cut and E_inj will significantly affect yields of ⁴⁴Ti and ⁵⁶Ni, highlighting the need to explore nuclear sensitivities for multiple scenarios.

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Figure 3 compares yields of ⁴⁴Ti and ⁵⁶Ni from our baseline calculations using various combinations of M_prog, E_inj, and M_cut to yields from model calculations of earlier studies, as well as inferred yields from astronomical observations. The boundary for observations of 1987A is based on the results of Boggs et al. (2015) ($M{(}^{44}\mathrm{Ti})=(1.5\pm 0.3)\times {10}^{-4}\,{M}_{\odot }$), Jerkstrand et al. (2011) ($M{(}^{44}\mathrm{Ti})=(1.5\pm 0.5)\times {10}^{-4}\,{M}_{\odot }$), Grebenev et al. (2012) ($M{(}^{44}\mathrm{Ti})=(3.1\pm 0.8)\times {10}^{-4}\,{M}_{\odot }$), and Seitenzahl et al. (2014) ($M{(}^{44}\mathrm{Ti})=(0.55\pm 0.17)\times {10}^{-4}\,{M}_{\odot }$, $M{(}^{56}\mathrm{Ni})=(7.1\,\pm 0.3)\times {10}^{-2}\,{M}_{\odot }$). The boundary for observations of CasA is from Wang & Li (2016) ($M{(}^{44}\mathrm{Ti})=(1.3\pm 0.4)\times {10}^{-4}\,{M}_{\odot }$), Siegert et al. (2015) ($M{(}^{44}\mathrm{Ti})=(1.37\pm 0.19)\times {10}^{-4}\,{M}_{\odot }$), Grefenstette et al. (2014) ($M{(}^{44}\mathrm{Ti})=(1.25\pm 0.3)\times {10}^{-4}\,{M}_{\odot }$), and Eriksen et al. (2009) ($M{(}^{56}\mathrm{Ni})=5.8\mbox{--}16.0\times {10}^{-2}\,{M}_{\odot }$). The boundary labeled W15-2-cw-llb is from the 3D model calculation of Wongwathanarat et al. (2017) that resembled CasA. We note that comparisons to CasA may not be particularly relevant for this work, as this supernova is suspected to have originated from a progenitor that lost much of its hydrogen envelope (Koo et al. 2020). Results from 1D model calculations of Rauscher et al. (2002), Limongi & Chieffi (2003), and Chieffi & Limongi (2013) are also shown for M_prog = 15–25 M_⊙. While our ⁵⁶Ni yields are similar to these previous 1D calculations, our ⁴⁴Ti yields are generally lower, especially for our 18 and 22 M_⊙ progenitors. This is likely due to the different explosion mechanism (the other works do not employ a thermal bomb) and our choice of E_inj and t_inj (Young & Fryer 2007; Fryer et al. 2012, 2018), but a more comprehensive cross-model comparison, which is beyond the scope of the present work, would be necessary to comment further.

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To provide further context for comparisons to earlier 1D modeling work, we calculate the explosion energy E_exp as described by Aufderheide et al. (1991) and restated in Equations (4) and (5):

$$\begin{eqnarray}&&{E}_{\exp }={E}_{\mathrm{inj}}+{E}_{{\rm{b}}}+{\rm{\Delta }}{E}_{{\rm{n}}},\end{eqnarray} \tag{ 4 }$$

where

$$\begin{eqnarray}&&{E}_{{\rm{b}}}={E}_{{\rm{k}}}-{E}_{\mathrm{Gv}}+{E}_{\mathrm{int}}.\end{eqnarray} \tag{ 5 }$$

Here E_k is the total kinetic energy, E_Gv is the total gravitational energy, E_int is the total internal energy, E_b is the total binding energy, and ΔE_n is the nuclear binding energy released by burning. All the energy values are calculated from I_b to the surface of a star. Compared to previous works (e.g., Aufderheide et al. 1991; Rauscher et al. 2002; Limongi & Chieffi 2003), our calculations with the lower of two E_inj for a given M_prog result in relatively low E_exp, though the high E_inj M_prog = 22 M_⊙ and 25 M_⊙ cases result in relatively high E_exp.

For each combination of M_prog and E_inj listed in Table 1, we performed reaction rate variations and assessed the change in ⁴⁴Ti and ⁵⁶Ni yields relative to the baseline calculation. Given the computational expense, we chose to vary a single reaction rate at a time, as opposed to varying several rates at once in a Monte Carlo fashion (e.g., Bliss et al. 2020). We limited our investigation to the 49 reaction rates identified as high impact by Magkotsios et al. (2010), excluding the weak rates identified in that work, along with two additional reaction rates chosen to test the completeness of the 49-rate list. The reaction list, given in Table 2, includes the additional two reactions ${}^{27}\mathrm{Al}{(\alpha ,n)}^{30}{\rm{P}}$ and ${}^{39}{\rm{K}}{(p,\alpha )}^{36}\mathrm{Ar}$ based on suspected impact due to the reaction network flow (see Figure 7).

Table 2.  Reaction Rate Variation Factors (RR)

	(α, n), (n, α)			(α, p), (p, α)			(p, γ)
Reactions	RR $(\uparrow )(\downarrow )$	Reac #	Reactions	RR $(\uparrow )(\downarrow )$	Reac #	Reactions	RR $(\uparrow )(\downarrow )$	Reac #
${}^{10}{\rm{B}}{(\alpha ,n)}^{13}{\rm{N}}$	20% (1)	1	${}^{44}\mathrm{Ti}{(\alpha ,p)}^{47}{\rm{V}}$	44% (10)a	7	${}^{45}{\rm{V}}{(p,\gamma )}^{46}\mathrm{Cr}$	10	33
${}^{11}{\rm{B}}{(\alpha ,n)}^{14}{\rm{N}}$	10% (2)	2	${}^{40}\mathrm{Ca}{(\alpha ,p)}^{43}\mathrm{Sc}$	15% (11)	8	${}^{41}\mathrm{Sc}{(p,\gamma )}^{42}\mathrm{Ti}$	100	34
${}^{23}\mathrm{Mg}{(n,\alpha )}^{20}\mathrm{Ne}$	100	3	${}^{17}{\rm{F}}{(\alpha ,p)}^{20}\mathrm{Ne}$	100	9	${}^{43}\mathrm{Sc}{(p,\gamma )}^{44}\mathrm{Ti}$	10	35
${}^{9}\mathrm{Be}{(\alpha ,n)}^{12}{\rm{C}}$	10% (3)	4	${}^{21}\mathrm{Na}{(\alpha ,p)}^{24}\mathrm{Mg}$	100	10	${}^{44}\mathrm{Ti}{(p,\gamma )}^{45}{\rm{V}}$	100	36
${}^{42}\mathrm{Ca}{(\alpha ,n)}^{45}\mathrm{Ti}$	21% (4)	5	${}^{27}\mathrm{Al}{(\alpha ,p)}^{30}\mathrm{Si}$	100	11	${}^{57}\mathrm{Ni}{(p,\gamma )}^{58}\mathrm{Cu}$	10	37
${}^{34}{\rm{S}}{(\alpha ,n)}^{37}\mathrm{Ar}$	16% (5)	6	${}^{55}\mathrm{Co}{(\alpha ,p)}^{58}\mathrm{Ni}$	10	12	${}^{40}\mathrm{Ca}{(p,\gamma )}^{41}\mathrm{Sc}$	100	38
${}^{27}\mathrm{Al}{(\alpha ,n)}^{30}{\rm{P}}$	10	50	${}^{48}\mathrm{Cr}{(\alpha ,p)}^{51}\mathrm{Mn}$	100	13	${}^{44}{\rm{V}}{(p,\gamma )}^{45}\mathrm{Cr}$	100	39
	$(\alpha ,\gamma )$		${}^{52}\mathrm{Fe}{(\alpha ,p)}^{55}\mathrm{Co}$	100	14	${}^{43}\mathrm{Ti}{(p,\gamma )}^{44}{\rm{V}}$	10	40
Reactions	RR $(\uparrow )(\downarrow )$	Reac #	${}^{54}\mathrm{Fe}{(\alpha ,p)}^{57}\mathrm{Co}$	8% (12)	15	${}^{42}\mathrm{Sc}{(p,\gamma )}^{43}\mathrm{Ti}$	10	41
${}^{40}\mathrm{Ca}{(\alpha ,\gamma )}^{44}\mathrm{Ti}$	25% (6)	24	${}^{56}\mathrm{Ni}{(\alpha ,p)}^{59}\mathrm{Cu}$	10	16	${}^{57}\mathrm{Cu}{(p,\gamma )}^{58}\mathrm{Zn}$	100	42
${}^{12}{\rm{C}}{(\alpha ,\gamma )}^{16}{\rm{O}}$	20% (7)(8)	25	${}^{6}\mathrm{Li}{(\alpha ,p)}^{9}\mathrm{Be}$	100	17	${}^{20}\mathrm{Ne}{(p,\gamma )}^{21}\mathrm{Na}$	12% (15)b	43
${}^{7}\mathrm{Be}{(\alpha ,\gamma )}^{11}{\rm{C}}$	100	26	${}^{13}{\rm{N}}{(\alpha ,p)}^{16}{\rm{O}}$	100	18	${}^{47}{\rm{V}}{(p,\gamma )}^{48}\mathrm{Cr}$	10	44
${}^{24}\mathrm{Mg}{(\alpha ,\gamma )}^{28}\mathrm{Si}$	10c	27	${}^{42}\mathrm{Ca}{(\alpha ,p)}^{45}\mathrm{Sc}$	21% (13)	19	${}^{42}\mathrm{Ca}{(p,\gamma )}^{43}\mathrm{Sc}$	21% (16)	45
${}^{42}\mathrm{Ca}{(\alpha ,\gamma )}^{46}\mathrm{Ti}$	15% (9)	28	${}^{43}\mathrm{Sc}{(\alpha ,p)}^{46}\mathrm{Ti}$	10	20	${}^{39}{\rm{K}}{(p,\gamma )}^{40}\mathrm{Ca}$	100d	46
	$(p,n)$, $(n,p)$		${}^{58}\mathrm{Ni}{(\alpha ,p)}^{61}\mathrm{Cu}$	25% (14)	21	${}^{57}\mathrm{Co}{(p,\gamma )}^{58}\mathrm{Ni}$	10	47
Reactions	RR $(\uparrow )(\downarrow )$	Reac #	${}^{38}\mathrm{Ca}{(\alpha ,p)}^{41}\mathrm{Sc}$	100	22	${}^{54}\mathrm{Fe}{(p,\gamma )}^{55}\mathrm{Co}$	12% (17)	48
${}^{57}\mathrm{Ni}{(n,p)}^{57}\mathrm{Co}$	100	29	${}^{34}\mathrm{Ar}{(\alpha ,p)}^{37}{\rm{K}}$	100	23	${}^{52}\mathrm{Mn}{(p,\gamma )}^{53}\mathrm{Fe}$	10	49
${}^{56}\mathrm{Co}{(p,n)}^{56}\mathrm{Ni}$	100	30	${}^{39}{\rm{K}}{(p,\alpha )}^{36}\mathrm{Ar}$	100	51
${}^{27}\mathrm{Si}{(n,p)}^{27}\mathrm{Al}$	100	31
${}^{11}{\rm{C}}$(n,p)${}^{11}{\rm{B}}$	100	32

Notes. The rate numbers are used as identifiers in Figures 4–6.

^aSubsequent evaluations (Hoffman et al. 2010; K. Chipps et al. 2020, in preparation) suggest that Sonzogni et al. (2000) underestimate the rate uncertainty and that a factor of 3 or more would be more appropriate. ^bLyons et al. (2018) have since updated the rate, but the rate and uncertainty are similar in our temperature range of interest. ^cThe evaluation of Adsley et al. (2020) suggests a much smaller factor may be appropriate; however, they focused on lower temperatures than are of interest for this work. ^dOur calculations were performed prior to the rate evaluation of Longland et al. (2018), who suggest ∼ ×10 is likely a more appropriate factor for the temperature range of interest. However, we note that experimental coverage is incomplete for the relevant center of mass energies.

References. (1) Gibbons & Macklin (1959), (2) Wang et al. (1991), (3) Wrean et al. (1994), (4) Cheng & King (1979), (5) Scott et al. (1993), (6) Robertson et al. (2012), (7) Plag et al. (2012), (8) deBoer et al. (2017), (9) Mitchell et al. (1985), (10) Sonzogni et al. (2000), (11) Howard et al. (1974), (12) Tims et al. (1991), (13) Buckby & King (1983), (14) Vlieks et al. (1974), (15) Rolfs et al. (1975), (16) Vlieks et al. (1978), (17) Kennett et al. (1981).

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Similar past studies have often adopted reaction rate variation factors ×100 and /100 (e.g., The et al. 1998; Magkotsios et al. 2010), since these studies aimed to find all plausible sensitivities. However, concerted efforts from the nuclear physics community have reduced the uncertainties for several reaction rates well below two orders of magnitude uncertainty. Furthermore, decades of comparison between measured data and Hauser–Feshbach calculations have shown that nuclear reactions proceeding through compound nuclei at relatively high nuclear level densities have theoretical predictions generally within an order of magnitude of measurements (see, e.g., Mohr 2015). As such, we aimed to adopt more realistic rate variation factors, taking into account existing experimental and theoretical data.

Reaction rate variation factors used here are given in Table 2. When experimental data were available over the astrophysically relevant energies or a reaction rate evaluation existed, the uncertainty from that work was used as the rate variation factor. In the absence of such information, the rate was either assigned an uncertainty factor of 10 or 100, depending on whether or not the Hauser–Feshbach formalism was thought to be valid, following the approach of Cyburt et al. (2016).

For Hauser–Feshbach validity, we adopt the heuristic that more than 10 nuclear levels must exist within an MeV of the excitation energy populated in the compound nucleus for a particular reaction (Wagoner 1969; Rauscher et al. 1997). To determine the relevant excitation energy, we use the Gamow window approximation for the center of mass reaction energy of astrophysical interest (Rolfs & Rodney 1988):

$$\begin{eqnarray}&&{E}_{{\rm{G}}}=0.1220\ {\left({Z}_{1}^{2}{Z}_{2}^{2}\displaystyle \frac{{A}_{1}{A}_{2}}{({A}_{1}+{A}_{2})}{T}_{9}^{2}\right)}^{1/3}\,\mathrm{MeV},\end{eqnarray} \tag{ 6 }$$

where Z_i are the proton numbers of the two reactants, A_i are their mass numbers, T₉ is the environment temperature in gigakelvin, and k_B is the Boltzmann constant. The 1/e width of the window about E_G is

$$\begin{eqnarray}&&{{\rm{\Delta }}}_{G}=4\sqrt{\left(\displaystyle \frac{{E}_{{\rm{G}}}\ {k}_{{\rm{B}}}\ {T}_{9}}{3}\right)}\,\mathrm{MeV}.\end{eqnarray} \tag{ 7 }$$

While more accurate methods exist to determine the excitation energy window of astrophysical interest (Newton et al. 2007; Rauscher 2010), we chose the more approximate Gamow window method given the uncertainty of nuclear level structure for the nuclides involved and the choice of a single temperature for which to calculate the energies of astrophysical interest.

We calculated the Gamow window at T₉ = 4 as this is where the calculations of Magkotsios et al. (2010) fall out of QSE. We determined whether at the bottom of the window (E_G − Δ_G/2) more than 10 nuclear levels per MeV of excitation were present, based on the microscopic nuclear level densities calculated by Goriely et al. (2008). For high level-density cases, we consider a reaction to be in the statistical regime and assign an uncertainty factor of 10, based on the typical spread in predictions from Hauser–Feshbach calculations (e.g., Pereira & Montes 2016). For low level-density cases, we consider a reaction to be in the resonant regime and assign an uncertainty factor of 100, based on the significant rate modification that can occur due to addition/removal of a particular resonance or change in a key resonance's properties (e.g., Cavanna et al. 2015).

Figures 4–6 show the sensitivity study results for ⁴⁴Ti, ⁵⁶Ni, and ⁴⁴Ti/⁵⁶Ni, respectively, for all combinations of M_prog, E_inj, and M_cut modeled here, where the reaction numbers correspond to the designations in Table 2 and the reported ratios are to the baseline results of Table 1. In each plot the upper panel corresponds to a reaction rate increase, while the bottom panel corresponds to a reaction rate decrease. We show the sensitivities for various combinations of E_inj and M_cut to give a sense of the robustness of our results to changes in model assumptions. However, for the ensuing discussion we limit ourselves to the results for M_cut = M₄.

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In the following, we separately discuss reactions impacting ⁴⁴Ti, ⁵⁶Ni, and their ratio. Motivated by typical observational uncertainties (discussed in Section 4), we only discuss cases (listed in Table 3) that affect these yields by a factor of two or more for at least one set of model conditions, as we deem this to be significant for model-observation comparisons. Of course as observations improve, it is likely that even smaller sensitivities will be of interest in the future. Note that in general we see fewer reaction rate variations having a significant impact for larger M_prog calculations. This is because, for those cases, much of the ⁴⁴Ti and ⁵⁶Ni beyond M_cut was synthesized during stellar evolution at radii not significantly impacted by the outgoing shock (Chieffi & Limongi 2017).

Table 3.  Ratios to the ${M}_{\mathrm{cut}}={M}_{4}$ Baseline Calculation Results of Table 1 for Reaction Rates Significantly Impacting $M{(}^{44}\mathrm{Ti})$, $M{(}^{56}\mathrm{Ni})$, and/or $M{(}^{44}\mathrm{Ti})/M{(}^{56}\mathrm{Ni})$

		$M{(}^{44}\mathrm{Ti})$/$M{(}^{44}\mathrm{Ti}){| }_{\mathrm{baseline}}$							$M{(}^{56}\mathrm{Ni})$/$M{(}^{56}\mathrm{Ni}){| }_{\mathrm{baseline}}$							$[M{(}^{44}\mathrm{Ti})$/$M{(}^{56}\mathrm{Ni})$]/[$M{(}^{44}\mathrm{Ti})$/$M{(}^{56}\mathrm{Ni})]{| }_{\mathrm{baseline}}$
Reactions	RR	15 M${}_{\odot }$		18 M${}_{\odot }$		22 M${}_{\odot }$		25 M${}_{\odot }$	15 M${}_{\odot }$		18 M${}_{\odot }$		22 M${}_{\odot }$		25 M${}_{\odot }$	15 M${}_{\odot }$		18 M${}_{\odot }$		22 M${}_{\odot }$		25 M${}_{\odot }$
		0.5	1.32	0.5	1.32	1.32	3.5	3.5	0.5	1.32	0.5	1.32	1.32	3.5	3.5	0.5	1.32	0.5	1.32	1.32	3.5	3.5
${}^{17}{\rm{F}}{(\alpha ,p)}^{20}\mathrm{Ne}$	×100	0.8	1.3	0.9	1.1	1.1	0.8	0.9	0.8	0.6	0.3	0.5	0.7	0.8	0.7	0.9	2.1	2.8	2.1	1.5	1.0	1.3
	×0.01	1.2	1.0	1.1	1.3	1.2	1.3	1.0	1.4	1.1	1.1	1.0	1.0	1.0	1.0	0.8	1.0	1.0	1.2	1.2	1.3	1
${}^{27}\mathrm{Al}{(\alpha ,p)}^{30}\mathrm{Si}$	×100	0.8	1.0	0.7	1.0	0.8	0.8	0.9	0.9	0.8	0.4	0.8	0.8	0.9	0.9	1.0	1.2	1.9	1.2	0.9	0.9	1.0
	×0.01	1.2	1.0	1.0	1.2	1.0	1.1	1.0	1.6	1.0	1.3	1.1	1.1	1.0	1.0	0.8	0.9	0.9	1.1	0.9	1.0	1.0
${}^{55}\mathrm{Co}{(\alpha ,p)}^{58}\mathrm{Ni}$	×10	0.9	1.6	0.7	2.0	1.7	1.8	2.1	1.6	1.1	1.4	1.1	1.1	1.0	1.1	0.6	1.4	0.5	1.8	1.5	1.7	2.0
	×0.1	1.3	1.0	1.2	1.1	1.0	0.9	1.0	1.3	1.0	1.1	1.0	1.0	1.0	1.0	1.0	1.0	1.1	1.1	1.0	0.9	1.0
${}^{48}\mathrm{Cr}{(\alpha ,p)}^{51}\mathrm{Mn}$	×100	0.6	0.8	0.4	0.7	0.7	0.9	1.0	1.6	1.2	1.5	1.2	1.1	1.1	1.1	0.3	0.7	0.2	0.6	0.6	0.8	0.9
	×0.01	1.2	1.1	1.1	1.0	1.1	1.0	1.0	1.4	1.0	0.9	1.0	1.0	1.0	1.0	0.9	1.1	1.2	1.0	1.2	1.0	1.0
${}^{52}\mathrm{Fe}{(\alpha ,p)}^{55}\mathrm{Co}$	×100	0.7	2.8	0.5	1.1	1.0	1.3	1.0	2.1	1.1	1.6	1.2	1.1	1.1	1.1	0.3	2.4	0.3	1.0	0.9	1.2	0.9
	×0.01	1.2	1.0	1.0	1.0	0.9	1.1	1.0	1.3	1.0	1.3	1.0	1.0	1.0	1.0	0.9	1.0	0.8	1.0	0.9	1.1	1.0
${}^{54}\mathrm{Fe}{(\alpha ,p)}^{57}\mathrm{Co}$	×1.08	1.1	1.0	0.8	1.0	1.0	1.1	1.0	1.3	1.0	0.8	1.0	1.0	1.0	1.0	0.8	1.0	1.0	0.9	1.0	1.1	1.0
	×0.92	1.1	1.0	0.8	1.0	0.9	0.9	3.6	1.2	1.0	0.7	1.0	1.0	1.0	1.0	0.9	1.0	1.2	1.0	1.0	1.0	3.6
${}^{56}\mathrm{Ni}{(\alpha ,p)}^{59}\mathrm{Cu}$	×10	0.8	0.5	0.6	0.8	0.8	1.0	1.0	2.2	1.0	1.4	1.0	1.1	1.0	1.0	0.4	0.5	0.4	0.8	0.7	1.0	1.0
	×0.1	1.2	1.1	1.3	1.2	1.2	1.1	1.0	1.5	1.0	1.4	1.0	1.0	1.0	1.0	0.8	1.1	0.9	1.2	1.3	1.1	1.0
${}^{13}{\rm{N}}{(\alpha ,p)}^{16}{\rm{O}}$	×100	0.8	1.1	0.5	1.0	0.8	0.6	0.9	0.7	0.4	0.1	0.5	0.5	0.6	0.7	1.0	2.6	3.6	1.8	1.6	1.0	1.3
	×0.01	1.3	1.9	0.9	1.4	1.4	1.6	1.1	1.7	1.2	1.3	1.0	1.1	1.0	1.0	0.8	1.6	0.7	1.4	1.3	1.6	1.0
${}^{43}\mathrm{Sc}{(\alpha ,p)}^{46}\mathrm{Ti}$	×10	1.2	2.1	1.0	1.1	1.1	1.2	1.0	1.5	1.1	1.1	1.0	1.0	1.0	1.0	0.8	1.9	0.9	1.1	1.1	1.2	1.0
	×0.1	1.1	1.0	0.9	1.2	1.0	1.0	1.7	1.4	1.0	1.0	1.0	1.0	1.0	1.0	0.8	1.0	0.9	1.2	1.0	1.0	1.6
${}^{57}\mathrm{Ni}{(n,p)}^{57}\mathrm{Co}$	×100	4.0	4.2	4.2	4.8	3.1	1.8	1.1	0.3	0.8	0.4	0.8	0.7	0.9	0.8	13.4	5.1	10.0	5.7	4.3	2.1	1.4
	×0.01	0.7	0.6	0.5	1.1	0.8	1.0	1.0	1.8	1.1	1.6	1.1	1.0	1.0	1.0	0.4	0.5	0.3	1.1	0.8	1.0	1.0
${}^{56}\mathrm{Co}{(p,n)}^{56}\mathrm{Ni}$	×100	0.5	1.0	0.3	1.0	0.8	0.9	1.0	2.4	1.2	1.8	1.3	1.1	1.1	1.1	0.2	0.8	0.2	0.8	0.7	0.8	0.9
	×0.01	1.6	1.8	1.6	2.0	1.6	1.2	1.0	1.2	0.9	1.0	0.9	0.9	0.9	0.9	1.4	2.0	1.7	2.3	1.8	1.3	1.1
${}^{47}{\rm{V}}{(p,\gamma )}^{48}\mathrm{Cr}$	×10	0.7	0.5	0.5	0.5	0.7	0.9	1.0	2.1	1.1	1.4	1.1	1.1	1.0	1.1	0.3	0.4	0.4	0.5	0.6	0.9	0.9
	×0.1	2.0	2.4	2.0	2.2	2.0	1.5	1.1	1.0	1.0	0.7	1.0	0.9	1.0	0.9	2.2	2.5	2.7	2.3	2.2	1.6	1.1
${}^{42}\mathrm{Ca}{(p,\gamma )}^{43}\mathrm{Sc}$	×1.21	1.1	0.9	1.0	1.0	1.0	0.9	1.0	1.4	1.1	1.2	1.0	1.0	1.0	1.0	0.8	0.8	0.9	1.0	1.0	0.9	1.0
	×0.79	1.1	1.0	1.0	1.1	0.9	2.9	1.0	1.2	1.0	1.1	1.0	1.0	1.0	1.0	0.9	1.0	0.9	1.2	0.9	2.9	1.0
${}^{39}{\rm{K}}{(p,\gamma )}^{40}\mathrm{Ca}$	×100	5.2	3.5	4.0	3.5	3.4	2.4	1.2	0.6	1.0	1.1	1.0	1.0	1.0	1.0	9.2	3.5	3.5	3.4	3.3	2.4	1.2
	×0.01	0.1	0.2	0.1	0.2	0.2	0.5	0.9	1.1	1.0	0.8	0.9	0.9	1.0	1.0	0.1	0.2	0.2	0.2	0.2	0.5	0.9
${}^{57}\mathrm{Co}{(p,\gamma )}^{58}\mathrm{Ni}$	×10	2.1	2.3	1.9	2.0	1.5	1.1	1.1	1.2	0.9	0.9	0.9	0.9	1.0	0.9	1.8	2.5	2.2	2.2	1.7	1.2	1.2
	×0.1	1.2	0.6	0.9	0.8	0.8	0.8	1.0	1.9	1.1	1.1	1.0	1.0	1.0	1.0	0.6	0.6	0.8	0.7	0.8	0.8	0.9
${}^{52}\mathrm{Mn}{(p,\gamma )}^{53}\mathrm{Fe}$	×10	0.7	0.6	0.5	0.9	0.9	1.1	1.0	1.8	1.1	1.2	1.0	1.1	1.0	1.1	0.4	0.5	0.5	0.9	0.8	1.0	0.9
	×0.1	1.5	1.4	1.6	1.4	1.3	1.3	1.0	0.9	0.9	0.9	0.9	0.9	0.9	0.9	1.6	1.5	1.9	1.5	1.4	1.4	1.1
${}^{27}\mathrm{Al}{(\alpha ,n)}^{30}{\rm{P}}$	×10	1.0	1.0	1.0	1.0	2.9	0.9	1.3	1.2	1.0	0.9	1.0	1.0	1.0	1.0	0.9	1.0	1.1	1.0	3.0	0.9	1.4
	×0.1	1.2	1.1	1.1	0.9	1.1	1.0	1.0	1.2	1.0	1.4	1.0	1.0	1.0	1.0	1.0	1.1	0.8	0.9	1.1	1.0	1.0
${}^{39}{\rm{K}}{(p,\alpha )}^{36}\mathrm{Ar}$	×100	0.1	0.2	0.1	0.3	0.3	0.6	0.7	0.9	0.8	0.6	0.9	0.8	0.9	0.9	0.1	0.3	0.2	0.3	0.3	0.6	0.8
	×0.01	3.3	2.6	2.4	2.6	2.4	2.0	1.2	1.0	1.0	1.2	1.0	1.0	1.0	1.0	3.4	2.6	2.1	2.6	2.4	2.0	1.2

Note. Ratios greater than a factor of two increase or reduction are bolded for readability. The RR column lists the reaction rate multiplication factor. Subcolumns under each ${M}_{\mathrm{prog}}$ indicate ${E}_{\mathrm{inj}}$ in foe.

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The explanations we provide for the sensitivities we observe are based on analyzing the flow in the nuclear reaction network (see Figure 7), along with guidance from the discussions in The et al. (1998) and Magkotsios et al. (2010). The flow for converting isotope i to isotope j over a timestep is ${{ \mathcal F }}_{{ij}}=\int \left({\dot{Y}}_{i\to j}-{\dot{Y}}_{j\to i}\right){dt}$, where ${\dot{Y}}_{i\to j}$ is the rate that the abundance of isotope i is depleted by reactions converting i to j. As shown in Figure 1, the relevant conditions for this work are primarily those known as α-rich and silicon-rich freeze-out. For silicon-rich conditions (Region A), nucleosynthesis yields are mostly characterized by nuclear masses and the initial abundances before the explosion, with some flow between many small QSE clusters. For α-rich freeze-out (Region D), much larger QSE clusters are formed initially and a significant excess of α-particles are available. In that case nucleosynthesis is characterized by the flow between these large QSE clusters and the fusion of α-particles into heavier nuclides with Z = 2n, A = 4n, where n is an integer. Therefore, our reaction rate sensitivities fall into three categories: those influencing the helium burning, those near boundaries of the large QSE clusters that exist in α-rich freeze-out at early times, and those near ⁴⁴Ti that can influence nucleosynthesis as that QSE cluster dissolves into many smaller QSE clusters.

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6.1. Reactions Impacting ⁴⁴Ti

6.2. Reactions Impacting ⁵⁶Ni

6.3. Reactions Impacting ⁴⁴Ti/⁵⁶Ni

It is interesting to compare the results of the present work to previous studies exploring nuclear reaction rate sensitivities of shock-driven nucleosynthesis. In particular, we focus our discussion on the postprocessing studies of The et al. (1998) and Magkotsios et al. (2010), as these studies also explored the impact of individual reaction rate variations for a large number of rates. The pioneering work of The et al. (1998) used an exponential temperature and density trajectory (see Equation (2)) with T₀ = 5.5 GK, ρ₀ = 10⁷ g cm⁻³, Y_e = 0.497, and 0.499, and reaction rate variation factors of 100. Magkotsios et al. (2010) expanded on this work, exploring reaction rate sensitivities with rate variation factors of 100 for exponential and power-law trajectories (see Equation (3)), a large phase space of T₀–ρ₀ (see Figure 1), and Y_e = 0.48–0.52. In comparison, our model calculations have average Y_e = 0.498–0.499, thermodynamic trajectories more closely resembling the power-law forms, with mass zones covering the T₀–ρ₀ phase-space as shown in Figure 1, and focused mostly on the strong reactions identified as important by Magkotsios et al. (2010) using rate variation factors informed by the literature.

While our list of influential reactions shares several similarities with previous works, there are some conspicuous differences. In particular, we do not highlight many of the reactions that both prior studies identified as high-impact,⁵ namely, ⁴⁰Ca(α, γ)⁴⁴Ti, ⁴⁴Ti(α, p)⁴⁷V, ${}^{45}{\rm{V}}{(p,\gamma )}^{46}\mathrm{Cr}$, and ⁵⁷Ni(p, γ)⁵⁸Cu. For each of these reactions, the likely explanation is that we used far smaller rate variation factors (see Table 2). For the first two in this set, this is based on the significant efforts in the nuclear physics community to reduce these reaction rate uncertainties. For the latter two, our reduced rate variation factors (10 as opposed to 100) are based on the presumed accuracy of Hauser–Feshbach reaction rate predictions given the high nuclear level densities involved.

Our relative insensitivity to variations in the ${}^{40}\mathrm{Ca}{(\alpha ,\gamma )}^{44}\mathrm{Ti}$ and ${}^{44}\mathrm{Ti}{(\alpha ,p)}^{47}{\rm{V}}$ rates when using experimentally constrained uncertainties⁶ is in agreement with Hoffman et al. (2010) who sampled a slightly lower ρ₀ region of the thermodynamic phase space with models of CasA from Magkotsios et al. (2008). They found that remaining uncertainties in ${}^{40}\mathrm{Ca}{(\alpha ,\gamma )}^{44}\mathrm{Ti}$ contribute a ∼20% variation to X(⁴⁴Ti), while ${}^{44}\mathrm{Ti}{(\alpha ,p)}^{47}{\rm{V}}$ uncertainties (estimated there as a factor of 3) lead to a 70% variation. Similarly, Tur et al. (2010) found the 3α → ¹²C and ${}^{12}{\rm{C}}{(\alpha ,\gamma )}^{16}{\rm{O}}$ reactions are sufficiently constrained from the standpoint of ⁴⁴Ti CCSN nucleosynthesis.

For the remaining high-impact cases of Magkotsios et al. (2010), the sensitivity identified in their work is for a region of phase space that our models did not populate. For ${}^{17}{\rm{F}}{(\alpha ,p)}^{20}\mathrm{Ne}$, ${}^{21}\mathrm{Na}{(\alpha ,p)}^{24}\mathrm{Mg}$, and ${}^{40}\mathrm{Ca}{(\alpha ,p)}^{43}\mathrm{Sc}$, Magkotsios et al. (2010) only find a significant impact for relatively proton-rich initial conditions. For ${}^{41}\mathrm{Sc}{(p,\gamma )}^{42}\mathrm{Ti}$ and ${}^{44}\mathrm{Ti}{(p,\gamma )}^{45}{\rm{V}}$, Magkotsios et al. (2010) observe significant variations of X(⁴⁴Ti) only for quite high-T₀ low-ρ₀ conditions.

Indeed the significant reaction rate sensitivities that we identify in Table 2 are all highlighted as of secondary importance by Magkotsios et al. (2010). However, since their "secondary" designation is a general term for less than a factor of 10 variation in X(⁴⁴Ti) for any region within their full T₀–ρ₀ phase space, a quantitative comparison is not possible. Qualitatively, we find a significant sensitivity in M(⁴⁴Ti) and/or $M{(}^{44}\mathrm{Ti})/M{(}^{56}\mathrm{Ni})$ for the majority of (α, p) and (p, γ) reactions identified by Magkotsios et al. (2010) as impacting the chasm depth, and a subset of the (p, n) reactions found to impact the chasm width. The connection of our results to the chasm are not surprising, as all of our model calculations have mass zones very near to or directly over this region.

Perhaps more interesting are the reaction rate sensitivities that we find in our work that were not identified by Magkotsios et al. (2010). We included ${}^{39}{\rm{K}}{(p,\alpha )}^{36}\mathrm{Ar}$ and ${}^{27}\mathrm{Al}{(\alpha ,n)}^{30}{\rm{P}}$ as a test of the completeness of the previous survey, finding significant sensitivities for some choices of model conditions. This is not entirely unexpected, as The et al. (1998) identified ${}^{27}\mathrm{Al}{(\alpha ,n)}^{30}{\rm{P}}$ and ${}^{36}\mathrm{Ar}{(\alpha ,p)}^{39}{\rm{K}}$, the reverse of ${}^{39}{\rm{K}}{(p,\alpha )}^{36}\mathrm{Ar}$, as influential for some choices of Y_e. Nonetheless, this highlights the need for larger-scale reaction rate sensitivity studies for shock-driven nucleosynthesis using one-dimensional models.

We performed nuclear reaction rate sensitivity studies for ⁴⁴Ti and ⁵⁶Ni production in CCSN shock-driven nucleosynthesis using the code MESA. We evolved a range of M_prog stars to core collapse and induced an artificial explosion with a range of E_inj, analyzing nucleosynthesis yields for a range of M_cut, in order to gauge the robustness of our results to model calculation assumptions. For each set of model assumptions, we varied the strong reactions previously identified by Magkotsios et al. (2010) as influential for ⁴⁴Ti production, including two additional rates to test the completeness of the influential reaction rate set, using reaction rate variation factors based on uncertainty constraints in the literature.

We find a significant impact on ⁴⁴Ti and/or ⁵⁶Ni nucleosynthesis for only a subset of the influential reaction rates of Magkotsios et al. (2010), though this is likely attributable to the smaller reaction rate variation factors and different astrophysical conditions used in our work. Additionally, we find nucleosynthesis sensitivities for reaction rates not identified by Magkotsios et al. (2010). This indicates that larger-scale reaction rate sensitivity studies exploring an expanded set of reaction rates is desirable.

From the present work we conclude that additional effort is required from the nuclear physics community to reduced the reaction rate uncertainties for ${}^{13}{\rm{N}}{(\alpha ,p)}^{16}{\rm{O}}$, ${}^{17}{\rm{F}}{(\alpha ,p)}^{20}\mathrm{Ne}$, ${}^{27}\mathrm{Al}{(\alpha ,p)}^{30}\mathrm{Si}$, ${}^{43}\mathrm{Sc}{(\alpha ,p)}^{46}\mathrm{Ti}$, ${}^{48}\mathrm{Cr}{(\alpha ,p)}^{51}\mathrm{Mn}$, ${}^{52}\mathrm{Fe}{(\alpha ,p)}^{55}\mathrm{Co}$, ${}^{54}\mathrm{Fe}{(\alpha ,p)}^{57}\mathrm{Co}$, ${}^{55}\mathrm{Co}{(\alpha ,p)}^{58}\mathrm{Ni}$, ${}^{56}\mathrm{Ni}{(\alpha ,p)}^{59}\mathrm{Cu}$, ${}^{57}\mathrm{Ni}{(n,p)}^{57}\mathrm{Co}$, ${}^{56}\mathrm{Co}{(p,n)}^{56}\mathrm{Ni}$, ${}^{39}{\rm{K}}{(p,\gamma )}^{40}\mathrm{Ca}$, ${}^{42}\mathrm{Ca}{(p,\gamma )}^{43}\mathrm{Sc}$, ${}^{47}{\rm{V}}{(p,\gamma )}^{48}\mathrm{Cr}$, ${}^{52}\mathrm{Mn}{(p,\gamma )}^{53}\mathrm{Fe}$, ${}^{57}\mathrm{Co}{(p,\gamma )}^{58}\mathrm{Ni}$, ${}^{27}\mathrm{Al}{(\alpha ,n)}^{30}{\rm{P}}$, and ${}^{39}{\rm{K}}{(p,\alpha )}^{36}\mathrm{Ar}$.

We thank the Ohio Supercomputer Center for providing computational resources (Ohio Supercomputer Center 1987), R. Farmer for making his pre-supernova evolution inputs for MESA available and for assistance in their implementation, and F. X. Timmes for sharing the data from Magkotsios et al. (2010). This work was supported by the U.S. Department of Energy under grants DE-FG02-88ER40387 and DE-SC0019042, the National Nuclear Security Administration under grant DE-NA0003909, and has benefited from support by the National Science Foundation under grant PHY-1430152 (Joint Institute for Nuclear Astrophysics–Center for the Evolution of the Elements).

Software: Modules for Experiments in Stellar Astrophysics (MESA)(Paxton et al. 2011, 2013, 2015, 2018).