Constraining nucleosynthesis in neutrino-driven winds: observations, simulations and nuclear physics

A promising astrophysical site to produce the lighter heavy elements of the first $r$-process peak ($Z = 38-47$) is the moderately neutron rich ($0.4<Y_e<0.5$) neutrino-driven ejecta of explosive environments, such as core-collapse supernovae and neutron star mergers, where the weak $r$-process operates. This nucleosynthesis exhibits uncertainties from the absence of experimental data from $(\alpha,xn)$ reactions on neutron-rich nuclei, which are currently based on statistical model estimates. In this work, we report on a new study of the nuclear reaction impact using a Monte Carlo approach and improved $(\alpha,xn)$ rates based on the Atomki-V2 $\alpha$ Optical Model Potential ($\alpha$OMP). We compare our results with observations from an up-to-date list of metal-poor stars with [Fe/H] $<$ -1.5 to find conditions of the neutrino-driven wind where the lighter heavy elements can be synthesized. We identified a list of $(\alpha,xn)$ reaction rates that affect key elemental ratios in different astrophysical conditions. Our study aims on motivating more nuclear physics experiments on $(\alpha, xn)$ reactions using current and the new generation of radioactive beam facilities and also more observational studies of metal-poor stars.


INTRODUCTION
Solving the mystery of the origin of the heavy elements (Z > 26) in the universe has been a long-standing effort in nuclear astrophysics. Roughly half of them are produced via the rapid neutron capture process (rprocess) (Horowitz et al. 2019;Cowan et al. 2021, for two recent reviews), although its astrophysical site or sites are still under discussion. The recent detection of a binary neutron star merger (NSM) via both gravitational waves (GW170817) (Abbott et al. 2017) and the electromagnetic follow-up transient (AT2017gfo) (Drout et al. 2017) has reignited the interest for the origin of psaltis@theorie.ikp.physik.tu-darmstadt.de the r -process. Watson et al. (2019) identified strontium (Z = 38) in the merger ejecta, supporting the NSM as a site for r -process nucleosynthesis. However, Galactic Chemical Evolution (GCE) modelling suggests that this cannot be the sole site (Côté et al. 2019) and other similarly exotic environments, such as Magneto-rotational Supernovae explosions (MR-SNe) can contribute to the Galactic r -process abundance budget (Winteler et al. 2012;Nishimura et al. 2017;Reichert et al. 2021).
Despite the ongoing discussions about its origin, the r -process shows a unique robustness in its abundance pattern, with a couple of exceptions; the lighter region of the first peak, namely at Z = 38 − 47 (Sneden et al. 2008) and also the actinides, where "actinide-boost" stars with enhanced thorium and uranium abundances compared to the main r-process have been recently ob-arXiv:2204.07136v1 [astro-ph.HE] 14 Apr 2022 served Holmbeck et al. 2019). In addition to the s-process and r-process, the lighter heavy elements are also produced by an additional process (e.g., a light element primary process (LEPP) Travaglio et al. (2004) and the weak r-process Montes et al. (2007)).
In recent years, there has been an intense observational effort to identify the elemental composition of metal-poor (old) stars, which are thought to be polluted by few or even a single r-process event. Honda et al. (2004); Roederer et al. (2010); Hansen et al. (2012); Roederer et al. (2014); Niu et al. (2015); Aoki et al. (2017), to name a few, have focused on stars that show an enhancement in their lighter heavy elements compared to the Z = 55-75 region of the solar r-process abundance pattern (Sneden et al. 2008, Figure 11). Such stars are called limited-r or "Honda-like" stars, due to the seminal work of Honda et al. (2004) in the giant HD 122563. They are identified according to the standard classification, [Eu/Fe] 1 < 0.3, [Sr/Ba] > 0.5, and [Sr/Eu] > 0.0 (Frebel 2018;Hansen et al. 2018). Metalpoor stars that show a robust r-process pattern and have [Eu/Fe] > +1.0 and [Ba/Eu] < 0.0 are called r-II or "Sneden-like" stars, from the work of Sneden et al. (2003) in CS 22892-052 (also known as Sneden's star). This observational effort has offered valuable data to compare our nucleosynthesis theories to.
One of the proposed sites to produce the lighter heavy elements (Z = 38 − 47) in a primary process is the slightly neutron-rich (0.4< Y e <0.5) neutrino-driven ejecta of core-collapse supernovae explosions (CCSNe) or NSMs. In such conditions, a weak r -process (also referred to as α-process) can operate (Woosley & Hoffman 1992;Witti et al. 1994;Qian & Woosley 1996;Hoffman et al. 1997;Wanajo et al. 2001;Arcones & Montes 2011;Hansen et al. 2014;Bliss et al. 2017Bliss et al. , 2018. The nucleosynthesis starts from Nuclear Statistical Equilibrium (NSE), since the ejected material is at high temperature and density, and when the temperature falls to T 9 ≈ 5where T 9 is the temperature in units of 10 9 K -an α-rich freeze-out occurs. After that, the nucleosynthesis proceeds mainly through α-and proton-induced reactions on neutron-rich nuclei, until the temperature drops to T 9 ≈ 2. Pereira & Montes (2016) and Bliss et al. (2017) have shown that the nucleosynthesis flow strongly depends on (α, n) reactions on neutron-rich nuclei, which help to synthesize nuclei with larger atomic number Z. 1 In the bracket notation [X/Y] = log where the symbols , represent the stellar and solar values, respectively.
Due to scarce experimental data, current weak rprocess nucleosynthesis calculations employ (α, n) reaction rates based on the statistical Hauser-Feshbach formalism (Hauser & Feshbach 1952). Unfortunately, such reaction rates can be uncertain as much as two orders of magnitude in the relevant temperature region (T 9 = 2 − 5) (Pereira & Montes 2016). The main contribution in the aforementioned uncertainties originates from the α-nucleus potential (αOMP), which has been identified as the prime nuclear physics uncertainty for this scenario (Pereira & Montes 2016;Mohr 2016).
Recently, Bliss et al. (2020) performed a nuclear reaction sensitivity study to explore the impact of the (α, n) reactions for the weak r -process nucleosynthesis using reaction rates calculated from the Hauser-Feshbach code TALYSv1.6 (Koning et al. 2007). In that study the authors used the TALYS Global α-Optical model Potential (GAOP) which is based on a spherical potential approach by Watanabe (1958). Bliss et al. (2020) used a Monte Carlo technique to vary all the (α, n) reaction rates in the network by random factors, sampled from a log-normal distribution with µ = 0 and σ = 2.4, which corresponds to factors between 0.1 and 10 in the 68.3% coverage (log(10) = 2.4). They identified a list of 45 (α, n) reactions that impact the elemental abundances of the lighter heavy elements.
That work motivated numerous experimental studies, and new measurements of few (α, n) reaction cross sections for nuclei close to stability have been recently reported Szegedi et al. 2021). More experiments are currently proposed or are analyzed in nuclear physics facilities around the world.
In the present work, we build on the technique of Bliss et al. (2020), using new, constrained rates of (α, xn) reactions, based on the Atomki-V2 αOMP (Mohr et al. 2020). In addition, we compare for the first time our nuclear reaction impact study results with abundance observations of metal-poor stars with [Fe/H]<-1.5 that show an enhancement in their lighter heavy element distribution and are considered candidates for the weak rprocess. For the astrophysical conditions that can reproduce the abundance observations, we identify a list of 21 (α, n) reactions that their rates need to be constrained experimentally.
This paper is structured as follows: in Section 2 we present the different astrophysical conditions that we selected for our study. In Section 3, we present our up-todate compilation of elemental abundances of metal-poor stars with an enhanced production of the first r-process elements. In Section 4, we provide an introduction to the Atomki-V2 potential and its advantages. In Sections 5 and 6 we present the impact study and its results, along with the comparison to observational data. Finally, in Section 7 we conclude and discuss our results.

SELECTION OF THE ASTROPHYSICAL CONDITIONS
The conditions of the neutrino-driven wind ejecta are uncertain but also critical for the weak r -process. Bliss et al. (2018) explored the relevant phase space (Y e , entropy per baryon and expansion timescale) using a steady-state model. In the subsequent impact study of Bliss et al. (2020), 36 representative trajectories from the CPR2 group -the conditions that produce lighter heavy elements -were chosen to test the importance of individual (α, n) reaction rates (Bliss 2020). In the present work, we study more than half of the aforementioned trajectories (20), which we summarize in Table 1. In Figure 1 we map them in the neutron-toseed (Y n /Y seed )α-to-seed (Y α /Y seed ) space at T 9 = 3, where Y seed is the sum of the abundances of all nuclei heavier than helium. Note that each loci of thermodynamic trajectories roughly corresponds to a different Y e of the wind ejecta.  Bliss et al. (2018) in the Yn/Y seed -Yα/Y seed phase space at T9 = 3. The black stars represent the tracers studied in Bliss et al. (2020). The subset of white stars were selected for this study. The arrows indicate the loci of tracers with the same Ye, which decreases with decreasing Yα/Y seed .

COMPILATION OF ABUNDANCE OBSERVATIONS FROM METAL-POOR STARS
In Table 2 we present an up-to-date list of elemental abundances from thirteen metal-poor stars with [Fe/H] < -1.5 that are identified as "Honda-like" stars and also four "Sneden-like" stars. Typically metal-poor stars are those with [Fe/H] < -1.0. However, the abundance cut we chose can eliminate stars that could have been polluted by Type Ia supernovae explosions. Most of the stars in our compilation have observed elemental abundances in the mass region of Z= 38-46, between strontium and palladium. Unfortunately, despite the wealth of observational data of metal-poor stars in the literature, one can find only a small sample of objects where more than few elements other than the Sr-Y-Zr triplet in the Z = 38-46 range are observed and reported (Abohalima & Frebel 2018). Studies that identify "Hondalike" stars, Hansen et al. (2018) for example, only report measurements for select heavy elements, such as strontium, barium and europium. Although these elements are very important, since they are used for the usual classification of r-process stars, observations of niobium, molybdenum, ruthenium, palladium and silver are also crucial to help us identify the astrophysical conditions that produce the first r-process peak elements, as we discuss in Section 6.2. · · · · · · · · · · · · Barklem et al. (2005) HD 4306 -0.08(9) -0.99(18) -0.22(9) · · · · · · · · · · · · Honda et al.

Atomki-v2: A NEW αOMP
As we mentioned in Section 1, (α, n) reactions on intermediate and heavy mass nuclei, due to their high nuclear level density, are calculated within the statistical (Hauser-Feshbach) model. Within this framework, an (α, n) reaction cross section in the laboratory is defined as: where T i are the transmission coefficients for the i th channel and b n = T n / i T i is the branching ratio for neutron decay. Above the neutron threshold, the neutron transmission T n is dominating over the other transmissions T i and for this reason b n is close to unity. The compound nucleus formation cross section depends solely on T α,0 and thus only on the chosen αOMP. It has been shown that different αOMPs predict cross sections that disagree up to two orders of magnitude at the astrophysically interesting energies (Pereira & Montes 2016;Mohr 2016). This sensitivity mainly results from the tail of the imaginary part of the αOMP at radii outside the colliding nuclei (Mohr et al. 2020).
A different approach to calculate the compound nucleus formation is by employing a transmission through a real potential (pure barrier transmission model or PBTM). This approach avoids the above mentioned complications with the tail of the imaginary potential. The real part of the αOMP is relatively well constrained; e.g., in the case of the Atomki-V1 potential, the real part of the αOMP is calculated from a double-folding approach, and its parameters are finetuned to experimental data of low-energy elastic scattering (Mohr et al. 2017(Mohr et al. , 2013. It was shown in Mohr et al. (2020) that this approach is able to reproduce compound formation cross sections at very low sub-Coulomb energies with deviations below a factor of two over a wide range of masses. Recent experimental results Szegedi et al. 2021) on stable isotope elements show a very good agreement with the predicted cross sections and thus confirm the predictive power of this approach.
The new Atomki-V2 αOMP (Mohr et al. 2020) combines the real part of the Atomki-V1 potential with a narrow, deep, and sharp-edged imaginary potential of Woods-Saxon type. The benefits of this combination are twofold. First, this parameterization ensures that the compound formation cross section is practically identical to the successful PBTM approach; thus, the excellent reproduction of the experimental compound formation cross sections persists. Second, the Atomki-V2 potential can be implemented in standard codes for nuclear reaction cross sections in the statistical model; thus, the branching ratios b i towards the different exit channels in Equation (1) can be calculated in the usual way without further effort.
A compilation of α-induced reaction rates for nuclei between iron and bismuth (26 ≤ Z ≤ 83) using the Atomki-V2 αOMP was recently published by Mohr et al. (2021). The calculations were performed using a modified version of the TALYS code (Koning et al. 2007). In general, the calculation of astrophysical reaction rates at high temperatures has to take into account that lowlying excited states in the target nuclei may be thermally populated. This compilation also includes the effect of thermal excitation and not just transmission from the ground state of the compound nucleus. For our impact study, we will use a conservative estimated uncertainty factor of three for the astrophysical reaction rates which is larger than a factor of two found between predicted and experimental cross sections of stable nuclei. This factor accounts for any increase in reaction rate uncertainties for nuclei away for stability, along the weak rprocess path.

IMPACT STUDY ON (α, n) REACTION RATES
The impact study was performed using the nuclear reaction network WinNet (Winteler 2011;Winteler et al. 2012) using 4053 nuclei up to hafnium (Z = 72), which are connected with ≈ 54,000 reactions. Theoretical weak reactions are taken from Langanke & Martínez-Pinedo (2001), and neutrino reactions from (Langanke & Kolbe 2001, see also Fröhlich et al. 2006 for details about the neutrino reactions). All other reaction rates, except for the (α, n) reactions, are adopted from the JINA REA-CLIB compilation (Cyburt et al. 2010). For the calculations of the neutrino-driven trajectories from the study of Bliss et al. (2020), the electron fraction Y e evolves in nuclear statistical equilibrium (NSE) at a temperature of T 9 = 10 and assume that NSE stands down to T 9 = 7.
For each selected trajectory, we performed 10 4 network calculations varying simultaneously all ∼ 4300 (α, xn) reactions between iron (Z=26) and hafnium (Z= 72). Specifically, each (α, xn) reaction was multiplied by a randomly sampled variation factor p. The sampling was performed from a log-normal distribution with µ = 0 and σ = 1.10, which corresponds to factors between 0.33 and 3 in the 68.3% coverage. We selected these limits, due to the success of the Atomki-V2 αOMP to agree with experimental data to within a factor of 3. Log-normal distributions are ideal for such studies because they are defined only for p ≥ 0 (Parikh et al. 2008;Longland et al. 2010;Rauscher et al. 2016;Nishimura et al. 2019;Bliss et al. 2020). The log-normal distribu-tion is a versatile tool for statistical analysis that is used in a wide variety of disciplines, such as finance (Black & Karasinski 1991) and epidemiology (Linton et al. 2020). Since the forward and reverse reactions are connected by detailed balance in weak r-process conditions, we used the same variation factor for the latter. One might argue that multiplying all the relevant reaction rates by a constant factor for the whole temperature range produces unrealistic results for the final abundances, since it omits any temperature dependence they might have. Longland (2012) showed that this approach is in fact a good approximation when using theoretical reaction rates. In particular, the temperature dependence of a rate has a minimal effect in nucleosynthesis yields, when many rates are sampled and varied simultaneously.

RESULTS
In this section we discuss the main results of our impact study and the comparison to observational data. Figure 2 shows a probability histogram of the log 10 (Sr/Y) ratio, which is also one of the main observables in metal-poor stars. The strontium-yttriumzirconium triplet is of extreme importance in nuclear astrophysics, since it can be produced by a variety of processes (Travaglio et al. 2004). To select the number and widths of the bins in Figure 2, we use the Freedman-Diaconis rule (Freedman & Diaconis 1981) which takes into account both the sample size, and its spread. We can create similar graphs to compare our impact study results with observations of metal-poor stars in a statistically meaningful manner, as we shall present in Section 6.1. We choose the representative trajectory MC06 (Y e = 0.40, s = 56 k B /nucleon, τ = 63.6 ms) as a benchmark to compare our results with the work of Bliss et al. (2020). Figure 3 shows the elemental abundance distribution at time t = 1 Gy, when all nuclei in the network have decayed to stability. The two calculations agree well, and in addition, we can note that the overall uncertainty of our calculations is smaller, which can be attributed to the fact that we are using a factor of 3 lower (α, xn) reaction rate uncertainties.  In Figure 4 we compare the kernel density estimates (KDEs) (Izenman 1991) for a select list of elemental abundance ratio distributions between our study and Bliss et al. (2020). Once again, the distributions using the constrained (α, xn) reaction rate uncertainties based on the Atomki-v2 αOMP are much narrower compared to the GAOP ones, and in most cases the 1σ uncertainty is comparable to observational errors (Honda et al. 2004, for example) (δX ∼ 0.2 dex). This can be further illustrated in Figure 5, where we show the bivariate KDE of log 10 (Y/Zr) versus log 10 (Sr/Zr) from the representative trajectory MC06. Note that the 1 and 2σ contours cover the 39.3% and 86.5% of the total volume, respectively.

Comparison to elemental abundance ratios of metal-poor stars
The novelty of the present impact study is that we focus on elemental abundance ratios instead of single elemental abundances. In Figure 6 we compare bivariate probability density KDEs for all the Monte Carlo trajectories to the observations of Table 2 for different com-binations of elemental ratio pairs, namely Sr/Zr, Y/Zr, Nb/Zr, Mo/Zr, Ru/Zr and Pd/Zr. Note that in some cases, for example the Pd/Zr vs Mo/Zr ratios not all MC trajectories are shown, since their production of Z > 42 elements is below our abundance threshold of Y min = 10 −10 . It should be noted that for all the ratios shown in Figure 6, no overproduction of other abundances is found.
We chose the above elemental ratio pairs to investigate the nuclear physics impact via the (α, n) reaction rates to the production of elements with the most observational statistics, that is the Sr-Y-Zr triplet, the Z = 38-42 and the Z = 42-46 regions, respectively. In addition to the Table 2 compilation, we included additional data from the works of Peterson (2013) and Hansen et al. (2012), to provide a larger sample of metal-poor stars, that are not necessarily identified as "Honda-like" or "Sneden-like".
As Figure 6 demonstrates, using elemental abundance ratios instead of elemental abundances is a powerful tool to constrain the astrophysical conditions of the weak r-process. By reducing the associated nuclear physics uncertainties -experimentally determining the relevant (α, xn) reaction rates (Section 6.3) to reduce the size of the contours -and more precise observations (smaller error bars) we would be able to identify the conditions where the lighter heavy elements can be produced in the neutrino-driven wind ejecta.

Which astrophysical conditions can produce the lighter heavy elements?
According to our analysis, we can identify individual conditions of the neutrino-driven ejecta where the lighter heavy elements can be produced. This is demonstrated in Figure 6, where bivariate KDE contours are compared with the elemental abundance ratios from the metal-poor stars compiled in Section 3. Figure 6 depicts the interconnection between three different aspects of nuclear astrophysics: (i) elemental abundance observations in metal-poor stars (data points), (ii) astrophysical modeling (location of the different contour lines) and (iii) the nuclear physics impact (size are covered area of the contours).
In Figure 7 we map the thermodynamic trajectories that reproduce the observed abundance ratios of  reproduce the observations. As it has been discussed in detail in the literature (Woosley & Hoffman 1992;Qian & Woosley 1996, for example), the entropy per baryon in neutrino-driven winds is related to temperature and density, s ∝ T 3 /ρ. High entropy per baryon leads to more free nucleons and less seed nuclei, which results in a larger neutron-to-seed ratio and thus an increased production of lighter heavy nuclei (Arcones & Bliss 2014). Note that in the case of the Nb/Zr ratio ( in Figure 7), given the scarcity of observational data (only four "Honda-like" stars in Table 2) and the fact that most of the stars in our compilation have only observational upper limits, there are many conditions that can reproduce it within uncertainties.
In the lower panel of Figure 7 we map the thermodynamic trajectories of our study in the entropy per baryon vs. expansion timescale space. The expansion timescale exhibits similar effects to the entropy, with fast winds (low expansion timescale τ ) leading to lower seed nuclei available for nucleosynthesis, compared to a slower wind (higher expansion timescale τ ).
Trying to identify the astrophysical conditions that produce specific regions of the Z= 38-46 region, we can find connections in the way that the entropy per baryon, Y e and expansion timescale affect the weak r-process. Specifically, for the Sr/Zr and Y/Zr ratios, for which we have the largest observational dataset, Figure 7 shows that conditions with relatively low Y e (0.40 Y e 0.46) and entropy per baryon (55 s 85 k B /nucleon) are more favorable to match these observations. We should stress that the neutrino-driven ejected material, such as in a core-collapse supernova explosion (Witt et al. 2021), follows a distribution of Y e , entropy per baryon, and expansion timescale, meaning that there is not a single astrophysical condition in a given explosion. Rather, it is the combination of many different conditions that provide the final abundance pattern.
Interestingly enough, there are only a handful of tracers that can reproduce the heavier elemental ratios, Ru/Zr and Pd/Zr (10, 29 and 31). These three conditions have entropy per baryon 55 s 75 k B /nucleon and 0.40 Y e 0.43.
The works of Hansen et al. (2014) and Arcones & Bliss (2014) have explored a wide range of conditions of the neutrino-driven wind ejecta and focused on the production of Sr, Y, Zr, and Ag both in neutron-and protonrich winds. One of their main results, which we can confirm in our study, is that small variations in the conditions, produce large effects in the final abundances of the lighter heavy elements.
Another interesting result, shown in Figure 6, is that even though we use the same uncertainty for the (α, n) reaction rates for all the Monte Carlo trajectories, the size of the contours for different astrophysical conditions are different. This can be attributed to the sensitivity of each condition to changes in the main nuclear reaction channel, (α, n) on neutron-rich nuclei, which moves the nucleosynthesis flow to heavier species.
It is worth adding that proton-rich conditions of the neutrino-driven wind can also produce the elements between strontium and silver, via the νp-process (Fröhlich et al. 2006). This scenario operates in the neutrondeficient side of the chart of nuclides and the nucleosynthesis is flowing via sequences of (p, γ) and (n, p) reactions. Arcones & Bliss (2014) have showed that in the νp-process, the abundance pattern of Z= 38-47 elements is homogeneous and changes smoothly when varying the wind parameters (entropy, expansion timescale and Y e ), in contrast to the neutron-rich conditions.

The most important (α, n) reactions
We have identified the most important reactions for each astrophysical condition of the neutrino-driven ejecta by the correlation between reaction rate variation and elemental abundance ratio. For this, we employ the Spearman's correlation coefficient r corr (Spearman 1904), which is defined as: (2)  where n = 10 4 is the number of calculations we performed for each thermodynamic trajectory, R denotes the ranks of a rate variation {p 1 , p 2 , · · · , p n } and the final abundance ratio {Y r1 , Y r2 , · · · Y rn }. R(p) and R(Y r ) are the average ranks for rate variation and elemental abundance ratio. r corr lies in the [−1, 1] space, and (-)1 shows a perfect monotonic (anti-)correlation of the two quantities. The Spearman's correlation is suitable for such studies, due to the non-linearity between reaction rate variations and resulted abundances. In previous nuclear reaction sensitivity studies (Nishimura et al. 2019;Rauscher et al. 2016, for example) the Pearson correlation coefficient (Pearson 1895) has been employed, which instead evaluates the linear relationship between the two aforementioned quantities.
Using the above analysis, we investigated for the first time which (α, xn) reaction rates affect elemental abundance ratios of the first r-process peak elements that are observed in metal-poor stars. We identified (α, xn) reactions as important when their |r corr | > 0.20 for a specific ratio. We investigated all the thermodynamic trajectories of Table 1, despite the fact that some of them do not match the observations in Figure 6. As we discussed in Section 6.2, a combination of different astrophysical conditions could also explain the observations in metal-poor stars. In the Appendix we also report our results for (α, n) reactions that affect elemental abundances for all the MC trajectories of Table 1 and compare our results with the work of Bliss et al. (2020).
To guide the experimental nuclear physics community for future measurements, we have grouped these 35 (α, xn) reactions rates, that are also listed in Table 3, according to how many ratios affect and in how many conditions of the neutrino-driven wind.

CONCLUSIONS AND DISCUSSION
We performed a new study about the impact of the (α, n) reaction rates for the production of the lighter heavy elements via the weak r-process, using the Atomki-V2 αOMP, and thermodynamical trajectories from Bliss et al. (2018), which cover a broad range of astrophysical conditions. The (α, xn) reactions rates based on the successful Atomki-V2 αOMP lead to a smaller uncertainty in the production of the lighter heavy elements (Z = 38-47). However there are still uncertain reactions that need to be studied experimentally. In particular, as we move away from stability to more neutron-rich species, these uncertainties may be larger than the predicted from current theoretical models.
It is clear from our discussion in Section 3 that elemental observations in the first r-process peak of metal-poor stars are scarce. We expect that the next generation of Earth-and space-based telescopes, such as the 4-metre multi-object spectroscopic telescope (4MOST) (De Jong et al. 2012), WEAVE at the William Herschel Telescope (Dalton et al. 2016) and VLT's CUBES (Genoni et al. 2022;Hansen 2022), along with dedicated surveys, will provide more observations of metal-poor stars that will help constrain our current models and provide better predictions of the weak r-process. Data from more elements between strontium and silver of metal-poor stars that have already been observed in the past are also valuable.
There is a strong dependence of weak r-process nucleosynthesis to the entropy per baryon, expansion timescale and electron fraction of the wind ejecta. In Section 6.2 we discussed which conditions can reproduce the observed elemental ratios. Our results suggest that there seems to be a relationship between the conditions of the wind and the production of Z= 38-46 elements, but more observational data are crucial to compare our models to. We should also note that even though there are Monte Carlo trajectories that failed to reproduce elemental abundance ratios in Figure 6, they should not be completely dismissed from further analysis. In explosive astrophysical environments, the neutrino-driven ejected material is described by a distribution of Y e , entropy, and expansion timescale, and thus a combination of multiple such conditions could be able to reproduce the observed abundance ratios.
In this work, we identified 35 (α, n) reactions that impact elemental abundance ratios of Z = 38-47 elements, for different astrophysical conditions. Experimentally determining all the above cross sections and reaction rates will provide better constrains in the model parameters of the weak r -process and also help nuclear theorists to better understand the peculiarities of the αOMP in the intermediate nuclear mass regime.
It is evident from Figure 8 that most of the reactions we identified as important in our study are located beyond or at the N = 50 shell closure nuclei, something that has been also noted in Bliss et al. (2020). As in the main r-process, the (n, γ) − (γ, n) equilibrium leads to an accumulation of material in the neutron shell closure nuclei which act as waiting points. (α, n) reactions on these nuclei help matter to move to heavier masses, and thus they are important for the weak r-process. Additionally, Table 3 shows that the isotopic chains of krypton, rubidium, strontium and zirconium provide the majority of the identified (α, n) reactions.
It is important to emphasize that the (α, n) reactions we highlight in this study include nuclear species that are readily available at sufficient intensities in the cur- Table 3. (α, xn) reaction rates for which the Spearman's coefficient is |rcorr| ≥ 0.20 in neutrino-driven wind trajectories for the given elemental ratios of Figure 6. Normal text entries indicate the ones that match the abundance ratios within 1σ in observational errors and 2σ confidence intervals based on the variation of (α, xn) reaction rates. The entries in italics do not match the observed abundance ratios, but are included for completeness. See the text for details.

Reaction
Affected  2,4,7,8,10,17,20,25,26,31 87 Kr(α,n) Sr/Zr, Y/Zr, Nb/Zr 0. 20-0.39, 0.21-0.30 4, 7, 8, 12, 16, 20, 25, 30 88 Kr(α, n) Sr/Zr, Y/Zr, Nb/Zr, Mo/Zr, Ru/Zr, Nb/Zr, Pd/Zr 0. 21-0.74, 0.32-0.40 2, 4, 6, 7, 8, 10, 17, 20, 22 25, 26, 28, 29, 30 rent and the next generation of radioactive beam facilities, such as the Facility for Rare Isotope Beams (FRIB) -see Figure 8, FAIR at GSI, the Californium Rare Isotope Breeder Upgrade (CARIBU) at ATLAS and ISAC/ARIEL at TRIUMF. We believe that it is a great opportunity for the experimental nuclear astrophysics community to directly measure the cross sections of the (α, n) reactions we identified in the present study and constrain the production of the lighter heavy elements of the first r-process peak. Our work shows the potential of combining astronomical observations, hydrodynamic simulations and both theoretical and experimental nuclear physics to understand the origin of the heavy elements in the cosmos.    Table 4 we present the list of 37 (α, n) reactions which affect elemental abundances by more than a factor of 2 in the 2σ distribution and also have a Spearman's coefficient |r corr | ≥ 0.20. The relevant discussion can be found in Section 6.3. Note that in the work of Bliss et al. (2020), an elemental variation of a factor of 5 was chosen as the threshold to identify a (α, n) reaction as important. Given that we varied all the (α, xn) reaction rates by a factor of 3, compared to a factor of 10 of Bliss et al. (2020), we assume that a lower threshold is justified. We should also point out that even when we were increasing this lower limit to factors of 4 or 5, the list of important reactions was not significantly affected.
Comparing this list of reactions with the one of Bliss et al. (2020), we observe a very good agreement in target nuclei with 36 < Z < 41. Our calculations were not impacted significantly by lighter mass reactions, such as 59,68 Fe or 74,76 Ni. There are only 6 (α, n) reactions out of the 37 that were not included in the work of Bliss et al. (2020), namely 79 Ga, 86 Br, 91 Rb, 96 Y, 100,102 Mo. In both works, the (α, n) reactions along the isotopic lines of krypton (Z= 36), strontium (Z= 38) and zirconium (Z= 40) are very important since the respective β − decays feed the first r-process peak elements. Table 4. Element (Z) and wind trajectories for which the Spearman's coefficient is |rcorr| ≥ 0.20 and the elemental abundance varies by more that a factor of 2 within 2σ of the abundance distribution (for elemental abundances Y> 10 −10 ).