Nuclear physics uncertainties in neutrino-driven, neutron-rich supernova ejecta
NNuclear physics uncertainties in neutrino-driven, neutron-rich supernova ejecta
J. Bliss, A. Arcones,
1, 2
F. Montes,
3, 4 and J. Pereira
3, 4 Institut f¨ur Kernphysik, Technische Universit¨at Darmstadt,Schlossgartenstr. 2, Darmstadt 64289, Germany GSI Helmholtzzentrum f¨ur Schwerionenforschung GmbH, Planckstr. 1, Darmstadt 64291, Germany National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, MI 48824, USA Background:
Neutrino-driven ejecta in core collapse supernovae (CCSNe) offer an interestingastrophysical scenario where lighter heavy elements between Sr and Ag can be synthesized. Previousstudies emphasized the important role that ( α, n ) reactions play in the production of these elements,particularly in neutron-rich and alpha-rich environments.
Purpose:
In this paper, we have investigated the sensitivity of elemental abundances to specific( α, n ) reaction-rate uncertainties under different astrophysical conditions.
Method:
The abundances of ligther heavy elements were calculated with a reaction networkunder different astrophysical conditions. ( α, n ) reaction rates were varied within their theoreticaluncertainty using a Monte Carlo approach.
Results:
The most important ( α, n ) reaction affecting the nucleosynthesis of lighter heavy nucleiwere identified for 36 representative conditions of CCSNe neutrino-drive winds.
Conclusions:
Experimental studies of these reactions will reduce the nucleosynthesis uncertain-ties and make it possible to use observations to understand the origin of lighter heavy elements andthe astrophysical conditions where they are formed.
I. INTRODUCTION
Until relatively recently, the origin of elements heav-ier than iron was thought to be limited to the s-process(weak and strong) and the r-process, with only a few iso-topes been produced by the p-process. We know nowthat several other processes may produce elements be-yond iron (e.g., ν p-process [1–3], i-process [4]). Observa-tions provide a unique window to look into the origin ofelements. By combining nucleosynthesis studies based ondifferent astrophysical conditions with observations, onecan learn about the environments where heavy elementsare synthesised [see e.g. 5].Enormous progress has been reported in our under-standing of the r-process in the last years [6]. The kilo-nova associated with the gravitational wave detection,GW170817 [7, 8] and by the first identification of a heavyelement, strontium, in its spectrum [9] are indicationsthat production of heavy elements occurs in neutron starmergers. Furthermore, since the r-process conditionsfound in such mergers may be different in different partsof the ejecta, the change of the kilonova from blue to redmay indicate that there may not be a unique r-processabundance pattern. Additional evidence for multiple r-process conditions are found in the observations of old-est stars in our galaxy and dwarf galaxies [10–12] andin recent results from galactic chemical evolution models[e.g., 13]. The scattered abundances of elements belowthe second r-process peak (from Sr to Ag), observed indifferent Eu-enriched stars, contrasts with the rather ro-bust pattern found for elements between the second andthird r-process peak. Moreover, some of the stars presenthigh abundances of lighter heavy elements (defined hereas elements from Sr to Ag) compared to elements beyondthe second r-process peak (these stars have been referred as Honda-like stars [14] or r-limited stars [15]). Already[16–18] demonstrated that at least one additional processis necessary to explain the observed solar and old-starabundances.One possible origin of these lighter heavy elements arethe neutrino-driven ejecta in core-collapse supernovae.At the end of their life, massive stars collapse, form aneutron star, and a shock wave is launched and destroysthe star. The details of the explosion have not beenfully understood but there is a consensus that standardsupernovae are driven by neutrinos and hydrodynami-cal instabilities [19–21]. A subset of supernovae may beinstead triggered by fast rotation and magnetic fields,i.e., magneto-rotational supernova (MR-SN). Neverthe-less, in both kind of explosions, part of the matter isejected by the emitted neutrinos and can become neu-tron or proton-rich. For MR-SN the magnetic pressurefacilitate the ejection of neutron-rich matter and the r-process can produce the heaviest elements ([22–24]). Incontrast, in the matter that is mostly ejected by neutri-nos (for both types of explosions), neutrino interactionsaffect the proton-neutron composition leading to condi-tions favourable to produce lighter heavy elements upto the second r-process peak or below. Current simula-tions indicate that significant part of the ejecta is protonrich [25] and that it is possible to have small amountsof fast-expanding, neutron-rich material [see, e.g. 26]. Inthis paper, we focus on the nucleosynthesis occurring inslightly neutron rich ejected material.The nucleosynthesis of this neutrino-driven, neutron-rich ejecta has been investigated by several groups [27–29]. The energy deposited by neutrinos leads to unboundmatter that expands and cools. Initially the tempera-ture of the ejected matter is high and thus the composi-tion is dominated by neutrons and protons. During the a r X i v : . [ nu c l - e x ] J a n expansion, the temperature drops and α -particles formand recombine into seed nuclei still in nuclear statisticalequilibrium (NSE). Due to the fast expansion, the tem-perature further drops and an alpha-rich NSE freeze-outoccurs at T ∼ α ,n) reactions. This is knownas alpha process, charged-particle reaction (CPR) phase,or weak r-process and lasts until the temperature dropsbelow T ∼ γ )-( γ ,n) equilibrium in every isotopic chain with the max-imum abundances at few isotopes away from stability.Furthermore, since the expansion time scale is relativelyshort (tens of mili-seconds), the beta decays are too slowcompared to ( α ,n) reactions that have dropped out ofequilibrium with their (n, α ) counterpart. Those reac-tions become the main channel to move matter towardsheavier nuclei, with a minor contribution from (p,n) and( α , γ ) reactions. In order to fully understand this process,uncertainties in both the specific astrophysical conditionsin the wind, and nuclear physics uncertainties in the re-actions involved have to be quantified and reduced.Our aim in this paper is to identify the key reactionsthat need to be measured to reduce the nuclear physicsuncertainty to be able to use observations to constrainastrophysical wind conditions. In Bliss 2017, we stud-ied the overall effect of both (astrophysical conditionsand nuclear physics uncertainties) and concluded that( α ,n) reactions are critical and still not well constrainedfrom theory or experiments [31, 32]. In a follow-up study[30], a systematic nucleosynthesis study covering all pos-sible astrophysical wind conditions was performed. Inthis paper, we explore the impact of ( α ,n) reactions us-ing a Monte Carlo study for 36 representative trajectoriescovering a broad range of astrophysical conditions. Weprovide for the first time a list of key reactions for theweak r-process in core-collapse supernovae by investigat-ing correlations, impact on the abundances, and impor-tance under different astrophysical conditions.The paper is structured as follows. We discuss theastrophysical conditions of the 36 trajectories selectedin Sect. II. The Monte Carlo method is introduced inSect. III, including the identification of key reactions. InSect. IV, we present our results including the list of keyreactions. We conclude in Sect. V. II. VARIETY IN THE ASTROPHYSICALCONDITIONS IN NEUTRINO-DRIVENSUPERNOVA EJECTA
In core-collapse supernovae, matter can be shock-heated and become unbound or it can be ejected by neu-trinos. In the neutrino-driven ejecta, neutrinos depositenough energy to unbound matter while they also changeneutrons into protons. Therefore, the properties of theneutrino-driven ejecta can vary from neutron- to proton-rich and produce elements up to around Silver [see e.g., 28, for a review]. When matter expands very fast, neutri-nos cannot act long enough and the ejected matter staysneutron rich. In slightly neutron-rich conditions (elec-tron fractions Y e between 0.4 and 0.5) the nucleosynthe-sis path runs close to stability on the neutron-rich sideor along the valley of stability.In [30], we have performed a systematic study coveringa broad range of possible astrophysical conditions. Ourstudy was based on a steady-state model for neutrino-driven winds (following [33]) using, as input parameters,the proto-neutron star masses and radii, neutrino lumi-nosities and energies, and initial electron fraction ( Y e ).This allowed us to investigate the sensitivity of the weakr-process to the wind parameters (i.e., Y e , entropy, ex-pansion timescale). The conclusion of that paper wasthat the final abundances can be divided into four dis-tinctive patterns (referred to as NSE1, NSE2, CPR1,and CPR2, see Fig. 4 in [30]) and that each of theseabundance-pattern groups are clearly correlated with theneutron-to-seed ( Y n /Y seed ) and α -to-seed ( Y α /Y seed ) ra-tios at T = 3 GK. The NSE1 and NSE2 abundance distri-butions are mainly set already during the nuclear statisti-cal equilibrium (NSE) phase. Therefore, binding energiesand partition functions of the involved nuclei, and notindividual reactions rates, determine the nucleosynthesis(see also Wanajo et al. [25] for a similar conclusion). Forwind conditions resulting in a CPR1 pattern, the finalabundances are mainly given by known Q-values of ( α, n)reactions at N = 50, as also concluded by [2, 27]. Thegroup CPR2 is linked to high Y n /Y seed and Y α /Y seed . Forthis group, individual nuclear reaction rates have a crit-ical impact on the nucleosynthesis evolution and on thefinal abundances. Moreover, the elemental abundancepatterns, which can extend up to Z = 55, are highlydependent on the specific wind conditions. Therefore, inorder to identify the most important reactions we have toexplore several astrophysical conditions that lead to dif-ferent abundances within CPR2. The conditions for thegroup CPR2 are shown in Fig. 1 with stars indicatingthe selected trajectories for our sensitivity study. Table Iprovides the wind parameters for the 36 trajectories. III. SENSITIVITY STUDY ON ( α, n) REACTIONRATE UNCERTAINTIES
For the 36 trajectories introduced before, we performeda Monte Carlo study to investigate the impact of ( α, n)reactions. A similar approach was used in other sensitiv-ity studies for Type I X-ray bursts [34], novae [35, 36],p-process [37], s-process [23, 38], r-process nucleosynthe-sis [39, 40], and ν p-process [41].We considered 909 ( α, n) reactions on stable andneutron-rich nuclei between Fe (Z=26) and Rh (Z=45).The reference (or baseline) ( α, n) reaction rates were cal-culated with TALYS 1.6 using the nuclear physics re-ferred as TALYS 1 in Pereira and Montes [31] (see theirTable II) except for the masses, which were taken from (Y /Y seed ) l o g ( Y n / Y s ee d )
12 3456 78910 11 121314 15 16171819202122 23 24 2526 27 2829 303132 3334 3536
FIG. 1: Distribution of the CPR2 tracers from Bliss et al. [30]in the Y α /Y seed - Y n /Y seed plane at 3 GK. The stars mark theastrophysical conditions for which we performed sensitivitystudies. Audi et al. [42] if available, or from the FRDM massmodel [43] otherwise.For each of the 36 trajectories considered, more than10000 network calculations were performed each with dif-ferent ( α, α, (cid:46) T (cid:46) n, γ ) and ( γ, n ) reactions makesthe creation of the heavier Z + 2 nucleus and correspond-ing neutron emission(s) independent of the particular( α, xn) production reaction channel(s). In each networkcalculation, each of the 909 ( α, p . The same factor was applied to the cor-responding forward and reverse rates. The rate varia-tion factors p was chosen to follow a log-normal distri-bution (if p is log-normally distributed, ln p is normallydistributed). Because the log-normal density function isonly defined for p ≥
0, the rate variation factors are al-ways positive. Since the average theoretically-estimateduncertainty of the ( α, n) reaction rates is about a factor10 within the temperatures of interest [31, 44] the meanvalue and the standard deviation of ln p were µ = 0 and σ = 2 .
3, respectively (corresponding to having a variationfactor between 0.1 and 10 with a probability of 68.3%).
A. Identification of key ( α, n) reactions The most influential reaction rates on the final abun-dances were identified by calculating correlation coeffi-
TABLE I: Astrophysical conditions associated with each tra-jectory Trajectory Y e Entropy Expansion time k B / nuc msMC1 0.42 129 11.7MC2 0.45 113 11.9MC3 0.45 122 10.3MC4 0.44 66 19.2MC5 0.43 66 34.3MC6 0.4 56 63.8MC7 0.47 96 11.6MC8 0.43 78 35MC9 0.40 73 28.1MC10 0.40 54 31MC11 0.44 104 13.2MC12 0.48 85 9.7MC13 0.43 64 35.9MC14 0.45 46 14.4MC15 0.48 103 20.4MC16 0.49 126 15.4MC17 0.46 132 12.4MC18 0.45 131 21.4MC19 0.41 75 9.8MC20 0.41 42 59.3MC21 0.41 31 22.2MC22 0.40 40 46.7MC23 0.41 48 37.5MC24 0.43 56 16.2MC25 0.46 96 20.9MC26 0.40 84 36.2MC27 0.42 76 10MC28 0.46 113 11.9MC29 0.41 66 41.4MC30 0.43 79 26.3MC31 0.43 71 11.4MC32 0.42 103 12.7MC33 0.49 175 14.2MC34 0.40 34 58.7MC35 0.44 48 13MC36 0.40 32 63.4 cients between the variations of the rates and the re-sulting abundance changes. The correlations were deter-mined using the Spearman’s correlation coefficient [45].The Spearman’s rank-order correlation coefficient r corr measures the strength and direction of the monotonicrelationship between two variables (i.e., rate variationfactor p and elemental abundance Y ) using their ranks.In case of a monotonic relationship the value of onevariable either increases or decreases as the other valueincreases. Previous sensitivity studies (e.g., Nishimuraet al. [23], Rauscher et al. [37]) applied the Pearson corre-lation coefficient [46] which quantifies the strength of thelinear relationship between two variables. Since nucle-osynthesis calculations frequently show non-linear rela-tions between variations of reaction rates and abundancechanges (see, e.g., Fig. 6 Iliadis et al. [47]) we rely onthe Spearman’s r corr which is better suited to deal withnon-linear behavior.The Spearman’s correlation coefficient is calculated us-ing r corr = (cid:80) ni =1 (cid:16) R ( p i ) − R ( p ) (cid:17) (cid:16) R ( Y i ) − R ( Y ) (cid:17)(cid:114)(cid:80) ni =1 (cid:16) R ( p i ) − R ( p ) (cid:17) (cid:114)(cid:80) ni =1 (cid:16) R ( Y i ) − R ( Y ) (cid:17) , (1)where n is the number of network calculations for agiven trajectory, R corresponds to the ranks of therate variation factors { p , p , ..., p n } and final abundances { Y , Y , ..., Y n } , and R ( p ) = ( (cid:80) ni =1 R ( p i )) /n and R ( Y ) =( (cid:80) ni =1 R ( Y i )) /n are the average ranks. The values of r corr range between − ≤ r corr ≤ +1. A Spearman’s cor-relation factor of +1 ( −
1) indicates a perfectly increasing(decreasing) monotonic function.Figure 2 shows several cluster-plots with elementalabundances as a function of the logarithm of the scal-ing factor p for different ( α, n ) reactions. The calcula-tions were done using the trajectory MC27 (see Table I).These cluster-plots illustrate the correlations between theabundances of Rb and Br( α, n) ( r corr = − . Sr( α, n) ( r corr = 0 . Zr( α, n) ( r corr = 0 . Zn( α, n) ( r corr = − . Kr( α, n)( r corr = − . ± σ of the average abun-dance. Figure 2 shows a very strong negative correlationbetween the variation of Br( α, n) and the Rb abun-dance, especially for log p >
0. In comparison the cor-relation between Sr( α, n) and the abundance of Ru ispositive. The absolute value of the correlation factor fora given element and reaction will be smaller the more re-actions contribute to the abundance uncertainty of thatelement (see e.g the influence of Zr( α, n) and Kr( α, n)on the abundance of Kr and Y, respectively). If r corr isclose to zero, as for Yttrium abundance and Kr( α, n),there is no significant correlation between the rate vari-ation factors and the final abundances. We have useda correlation factor value | r corr | ≥ .
20 as the thresholdto indicate a meaningful correlation between a specific( α, n ) reaction rate and an elemental abundance change.It should be noted that, for a given element, the correla-tion factor alone should not be used to rank the impor-tant reactions since the Spearman’s correlation coefficientis independent of the magnitude of the element abun-dance variation (i.e., the slope of the cluster). We there-fore identified important key reactions by 1) inspectingwhich abundances vary most in the Monte Carlo study (large slopes); 2) investigating which reaction rate varia-tions strongly correlate with absolute abundance changes(large r corr ). We focused our study on the aforemen-tioned lighter r-elements Z = 36 − IV. RESULTS
We have identified 45 ( α, n) reactions having an impacton the elemental abundances. Tables III–XIII show the( α, n) reactions which affect the elemental abundancesbetween Z = 36 −
47 and satisfy the following conditionsfor at least one of the MC1-C36 wind trajectories: • | r corr | ≥ . • An abundance variation at least a factor of 5 be-tween the maximum and minimum elemental abun-dance within 2 − σ of the average abundanceAll of the important reactions can be classified in threecategories depending on whether they involve nuclei with N < N = 50 or at N >
50. The N = 50 shell closureserves as a process bottleneck at temperatures around4-5 GK due to the ( n, γ ) − ( γ, n ) equilibrium. N = 50Isotopes with the highest abundances at those temper-atures ( Zn, Ga, Ge, As, Se, Br, Kr, Rb)determine the speed at which ( α, n) reactions move thematerial flow towards heavier nuclei. There are two waysreactions on shell closure isotopes affect the nucleosyn-thesis. In some cases, the abundance flow is stopped orslowed down at the N = 50 shell closure once the temper-atures drops below 2 GK. In these cases, the final abun-dances are established by the β -decay to stability of thoseisotopes. An example is the Kr abundance (see Tab. III)for which the largest impact is directly due the reactionson , Se. Similar cases are the abundances of Rb and Srwhich are directly affected by reactions on As, , Se, Br and Kr (for Sr). The effect of the N = 50 closureis not only due to the direct effect of bottle-necked nuclei β -decaying to stability but also due to the indirect ef-fect of hindering (or enhancing) the N >
50 abundances.Therefore reactions involving N = 50 nuclei also affectthe final abundances of Y, Zr, Nb, Mo, Ru and Rh. Forthe heavier elements, the importance of the N = 50 clo-sure is diluted by ( α, n) reactions on N >
50 nuclei. For alarge number of conditions, ( α, n) reactions on , Se di-rect the flow out of the N = 50 shell closure. Thereforethose reactions directly affect a large number of abun-dances.Since the system is in ( n, γ ) − ( γ, n ) equilibrium, reac-tions on the most abundant isotopes of a given isotopicchain can also indirectly affect final abundances. Forexample, reactions on , Kr affect the Rb abundanceeven though the heaviest stable rubidium isotope is Rb(produced by the beta decay of Kr). The , Kr ( α, n)reactions indirectly affect the amount of Kr by modi-fying the overall Kr isotopic abundances. Almost all ofthe reactions affecting elements Ru, Rh, Pd and Ag are p l og A bundan c e Y ( R b ) r corr = 0.85 Br( , n) 4 2 0 2 4log p l og A bundan c e Y ( R u ) r corr = 0.58 Sr( , n) 4 2 0 2 4log p l og A bundan c e Y ( A g ) r corr = 0.40 Zr( , n) 4 2 0 2 4log p l og A bundan c e Y ( K r) r corr = 0.20 Zn( , n) 4 2 0 2 4log p l og A bundan c e Y ( Y ) r corr = 0.09 Kr( , n) 12345678910 N u m be r o f c oun t s FIG. 2: Impact of the variation of the Br( α, n ), Sr( α, n ), Zr( α, n ), Zn( α, n ), and Kr( α, n ) reaction rates on theabundances of Rb (first panel), Ru (second panel), Ag (third panel), Kr (fourth panel) and Y (fifth panel), respectively. Thedashed lines illustrate distribution of the abundances in 95.4% of the network runs The color code denotes how often the sameabundance occurs for the same rate variation factor. There is a very strong negative correlation between Br( α, n ) and theRb abundances, a strong positive correlation between Sr( α, n ) and the Ru abundances, and no correlation between Kr andthe abundance of Y. of this type. There are a few reactions that affect finalabundances due to their effect in the neutron abundance(once the material is running out of neutrons). For ex-ample, the Sr( α, n) affects the Rb abundance due tothe change in neutron abundance at late times.Reactions on N <
50 isotopes have an impact onlyfor a limited set of similar astrophysical conditions (seeTable II and corresponding tracers in Fig. 1). An exam-ple of this are the reactions on , Ni and , , , Znaffecting the final Kr abundance. Since for some condi-tions, , Zn are the entry gateways to the N = 50 shellclosure, ( α, n) reactions on them influence Kr, Rb, Sr, Y,Zr, Nb and Mo abundances. In general, it is observedthat N <
50 ( α, n) reactions lead to smaller abundancevariation and correlation coefficients | r corr | than N = 50and N >
50 reactions. This indicates that in additionto those ( α, n) reactions other reactions contribute, to alesser degree, to the change of the final abundance.In Tab. II, we give a complete overview of all ( α, n) re-actions having an influence on the elemental abundancesbetween Z = 36 −
45 for at least one MC tracerReactionscan be grouped according to how many final abundancesare effected and for how many astrophysical conditions: • Many elemental abundances under many astro-physical conditions: , Se, − Kr, − Sr, Zr • Few elemental abundances under many astrophys-ical conditions: , Br, Y, , Zr • Many elemental abundances under few astrophys-ical conditions: Fe, Co, , Ni, , , − Zn, Ge, Ga, , , Ge • Few elemental abundances under few astrophysicalconditions: Fe, Co, , Cu, Br, − Sr, Y, , Zn, NbThe reduction of the uncertainties in the named reactionrates will contribute to better understand the formationof the lighter heavy elements and help constrain on theastrophysical conditions where they are synthesized.
V. CONCLUSIONS
Observations of the oldest stars in our galaxy and indwarf galaxies point to an extra process contributing tothe abundances of elements between Sr and Ag in addi-tion to the s-process and r-process. It is possible that thisextra process is a weak r-process that takes place in mat-ter ejected by neutrinos in core-collapse supernovae withonly slightly neutron-rich conditions (0 . < Y e < . α, n) reaction are critical to move mat-ter towards heavy elements in core-collapse supernovae[27, 44]. Even though our understanding of the weakr-process has increased in the last years, the calculatedabundances are still uncertain due to the lack of experi-mental information for ( α, n) reactions.In this paper, we have identified the most importantreactions that need to be measured to reduce the nu-clear physics uncertainty to be able to use observationsto constrain astrophysical wind conditions. We selected36 tracers from [30] representing the evolution of ejectedmatter under a broad range of astrophysical conditions.For each tracer, we have performed a Monte Carlo studyvarying over 900 ( α, n) reaction rates. In order to decidewhich reactions are most important, we used two crite-ria: an Spearman’s correlation coefficient | r corr | above0 .
20 and a significant impact on the abundance variationdue to the reaction. Among the relevant reactions one
TABLE II: Element ( Z ) and wind trajectories for which the Spearman’s coefficient satisfies | r corr | ≥ .
20 and the elementalabundance varies by at least a factor of 5 within 2 − σ of the abundance distribution. See text for details. The tracer MC1does not appear in the table because ( α, n) reactions are not important if Y n /Y seed is large and the nucleosynthesis path evolvesfurther away from stability. The MC1 tracer has only an impact on Z = 55 which is above the heaviest abundances we considerhere. Reaction Z MC tracers Fe( α, n) Ni 39 −
42, 45 34, 36 Fe( α, n) Ni 36, 37 3 Co( α, n) Cu 39 −
42, 45 20, 34, 36 Co( α, n) Cu 36, 37 3 Ni( α, n) Zn 36 −
42 2, 3, 17, 18, 32 Ni( α, n) Zn 36 −
42 2, 3, 18, 32 Cu( α, n) Ga 47 35 Cu( α, n) Ga 37 3 Zn( α, n) Ge 39 −
42 36 Zn( α, n) Ge 36, 37 −
42 2, 3, 17, 18, 32 Zn( α, n) Ge 36, 37 −
42 2, 3, 17, 18, 32 Zn( α, n) Ge 36, 37 −
42 2, 3, 18, 32 Zn( α, n) Ge 36, 37, 39 −
42 2, 3, 18, 32 Ga( α, n) As 36, 38, 39, 41 17, 32 Ge( α, n) Se 39 −
42 36 Ge( α, n) Se 36 −
39, 42 28, 33, 36 Ge( α, n) Se 36 −
39, 41 11, 17, 19, 27, 28, 33 As( α, n) Br 36, 37, 41 11, 26, 27, 28, 33 Se( α, n) Kr 36 −
42, 44, 45 2, 6, 7, 8, 9, 10, 11, 18, 19, 20, 22, 23, 24, 26, 27, 28, 29, 30, 31, 33, 34, 36 Se( α, n) Kr 36 −
42, 44, 45 2, 6, 7, 8, 9, 10, 11, 18, 19, 22, 23, 24, 26, 27, 28, 29, 30, 31 Br( α, n) Rb 37 −
39 6, 7, 8, 9, 10, 22, 23, 24, 26, 28, 29, 30, 31 Br( α, n) Rb 37, 39 6, 9, 10, 29, 31 Br( α, n) Rb 39 26 Kr( α, n) Sr 38 −
42, 44, 45, 47 4, 5, 7, 8, 13, 14, 15, 16, 20, 24, 25, 33, 34, 35 Kr( α, n) Sr 38 −
42, 45 4, 5, 7, 8, 13, 16, 20, 24, 25, 28, 30, 33, 34 Kr( α, n) Sr 37 −
42, 44, 45 2, 4, 5, 6, 7, 8, 9, 11, 13, 16, 17, 18, 19, 20, 22, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34 Kr( α, n) Sr 39, 40, 42, 44, 45 2, 6, 11, 17, 18, 19, 26, 27, 28, 29, 30, 32 Kr( α, n) Sr 37 −
42, 44 −
46 2, 3, 6, 9, 10, 11, 17, 18, 19, 22, 26, 27, 28, 29, 30, 31, 32 Rb( α, n) Y 41, 45 14, 15 Rb( α, n) Y 41, 42 5, 7, 13, 20, 34 Sr( α, n) Zr 42, 44 14, 15 Sr( α, n) Zr 42 14, 15 Sr( α, n) Zr 42, 44 −
47 4, 5, 12, 13, 14, 15, 16, 20, 35 Sr( α, n) Zr 44, 45 5, 12, 13, 16 Sr( α, n) Zr 38, 42, 44 −
47 4, 5, 6, 7, 8, 11, 12, 13, 16, 20, 21, 22, 23, 24, 25, 28, 29, 30, 31, 34 Sr( α, n) Zr 42, 44 −
47 6, 7, 9, 10, 11, 22, 26, 27, 28, 29, 30, 31 Sr( α, n) Zr 37 −
42, 44 −
47 2, 6, 7, 8, 9, 10, 11, 18, 19, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32 Y( α, n) Nb 45 4, 8, 16, 21, 23, 24, 25 Y( α, n) Nb 45, 46 8, 23, 24, 25, 30 Zr( α, n) Mo 44, 45 14, 15, 35 Zr( α, n) Mo 45 −
47 5, 12, 13, 35 Zr( α, n) Mo 44 −
47 4, 5, 6, 7, 8, 12, 13, 16, 20, 21, 22, 23, 24, 25, 29, 30, 35 Zr( α, n) Mo 44, 46, 47 4, 5, 6, 7, 8, 21, 22, 23, 24, 25, 29, 30 Zr( α, n) Mo 44, 46, 47 6, 7, 8, 22, 23, 24, 25, 29, 30 Nb( α, n) Tc 45, 46, 47 12, 13, 14, 15, 35 can distinguish three groups depending on the nuclei in-volved. Reactions of nuclei at the shell closure N = 50have a clear impact on the final abundances. Since thereis a ( n, γ ) − ( γ, n ) equilibrium matter accumulates at N = 50 leading to an enhanced importance of those nu-clei. Reaction of nuclei with N >
50 affect the abun-dances of heavier nuclei because those nuclei are reachedonce the matter overcomes the shell closure. Reactionsof nuclei with
N <
50 are less relevant. We provide a setof 45 ( α, n) reactions (Table II) that are relevant for theweak r-process in core-collapse supernovae. In additionto examining the correlation coefficient and the impacton the final abundances, we have checked the number offinal elemental abundances that are affected by one re-action and the number of tracers in which a reaction isimportant.Future experiments will reduce the uncertainties ofthese reactions and will provide improved constraints totheoretical reaction models. This is critical to be able tocombine nucleosynthesis calculations and observations tounderstand the origin of lighter heavy elements and theastrophysical conditions where they are synthesised. Acknowledgments
This work was funded by Deutsche Forschungsgemein-schaft through SFB 1245, ERC 677912 EUROPIUM,BMBF under grant No. 05P15RDFN1, and by theNational Science Foundation under Grant No. PHY-1430152 (JINA Center for the Evolution of the Ele-ments). J.B. acknowledge the MGK of the SFB 1245and the JINA Center for the Evolution of the Elementsfor support during a research stay at Michigan State Uni-versity.
Appendix A: Tables
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Heger, Astro- TABLE III: Z=36Reaction Abundance variation Correlation coefficient MC tracers Fe( α, n) 8.38 0.35 3 Co( α, n) 8.38 0.26 3 Ni( α, n) 5.37-15.66 0.2-0.29 2, 3, 17, 18, 32 Ni( α, n) 7.77-15.66 0.22-0.3 2, 3, 18, 32 Zn( α, n) 5.37-15.66 0.2-0.32 2, 17, 18, 32 Zn( α, n) 5.37-15.66 0.25-0.3 2, 17, 18, 32 Zn( α, n) 7.77-15.66 0.3-0.32 2, 18, 32 Zn( α, n) 7.77-15.66 0.21-0.25 2, 18, 32 Ga( α, n) 5.37-7.77 0.23 17, 32 Ge( α, n) 12.33 0.22 28 Ge( α, n) 5.37-36.31 0.24-0.87 11, 17, 19, 27, 28 As( α, n) 12.33-27.43 0.21-0.39 27, 28 Se( α, n) 5.46-101.0 0.54-0.81 6, 7, 8, 9, 10, 22, 23, 24, 26, 29, 30, 31 Se( α, n) 5.46-101.0 0.32-0.53 6, 7, 8, 9, 10, 22, 23, 24, 26, 29, 30, 31TABLE IV: Z=37Reaction Abundance variation Correlation coefficient MC tracers Fe( α, n) 5.54 0.36 3 Co( α, n) 5.54 0.27 3 Ni( α, n) 5.54-16.18 0.26-0.28 2, 3, 18 Ni( α, n) 5.54-16.18 0.24-0.28 2, 3, 18 Cu( α, n) 5.54 0.2 3 Zn( α, n) 15.84-16.18 0.2-0.23 2, 18 Zn( α, n) 15.84-16.18 0.31 2, 18 Zn( α, n) 15.84-16.18 0.31-0.33 2, 18 Zn( α, n) 15.84-16.18 0.23-0.26 2, 18 Ge( α, n) 6.52 0.29 33 Ge( α, n) 6.52-23.15 0.55-0.69 17, 19, 33 As( α, n) 12.13-97.21 0.24-0.42 11, 26, 27, 28 Se( α, n) 5.07-97.21 0.21-0.6 9, 10, 11, 19, 22, 26, 27, 28, 31 Se( α, n) 8.49-97.21 0.2-0.49 6, 9, 10, 11, 26, 27, 28, 31 Br( α, n) 5.07-14.13 0.69-0.9 6, 7, 8, 22, 23, 24, 29, 30 Br( α, n) 8.49-11.72 0.21-0.22 6, 29 Kr( α, n) 35.33 0.21 32 Kr( α, n) 35.33-97.21 0.22-0.37 26, 32 Sr( α, n) 35.33-52.43 0.25-0.29 10, 32phys. J. , 40 (2018).[26] S. Wanajo, H.-T. Janka, and B. M¨uller, Astrophys. J.Lett. , L15 (2011).[27] R. D. Hoffman, S. E. Woosley, G. M. Fuller, and B. S.Meyer, Astrophys. J. , 478 (1996).[28] A. Arcones and J. Bliss, J. Phys. G: Nucl. Part. Phys. , 044005 (2014).[29] A. Arcones and F.-K. Thielemann, J. Phys. G: Nucl.Part. Phys. , 013201 (2013).[30] J. Bliss, M. Witt, A. Arcones, F. Montes, and J. Pereira, Astrophys. J. , 135 (2018).[31] J. Pereira and F. Montes, Phys. Rev. C , 034611(2016).[32] P. Mohr, Phys. Rev. C , 035801 (2016).[33] K. Otsuki, H. Tagoshi, T. Kajino, and S. Wanajo, Astro-phys. J. , 424 (2000), astro-ph/9911164.[34] A. Parikh, J. Jos´e, F. Moreno, and C. Iliadis, Astrophys.J.s , 110-136 (2008).[35] M. S. Smith, W. R. Hix, S. Parete-Koon, L. Dessieux,M. W. Guidry, D. W. Bardayan, S. Starrfield, D. L. TABLE V: Z=38Reaction Abundance variation Correlation coefficient MC tracers Ni( α, n) 11.87 0.28 3 Ni( α, n) 11.87 0.28 3 Zn( α, n) 10.16-84.76 0.2-0.26 2, 3, 18, 32 Zn( α, n) 10.16-84.76 0.21-0.27 2, 3, 18, 32 Ga( α, n) 10.16 0.2 32 Ge( α, n) 8.76 0.3 33 Ge( α, n) 8.76-32.1 0.53-0.64 17, 33 Se( α, n) 5.81-297.08 0.3-0.7 6, 7, 9, 10, 11, 19, 26, 27, 28, 29, 30, 31 Se( α, n) 7.57-297.08 0.21-0.53 6, 9, 10, 11, 19, 26, 27, 28, 29, 30, 31 Br( α, n) 10.85-21.93 0.43-0.46 9, 10, 31 Kr( α, n) 6.81-10.53 0.3-0.58 4, 5, 8, 16, 24, 25 Kr( α, n) 6.81-10.53 0.34-0.39 4, 5, 8, 16, 24, 25 Kr( α, n) 5.81-32.1 0.26-0.54 4, 5, 7, 8, 16, 17, 24, 25, 30 Kr( α, n) 11.87-297.08 0.28-0.34 3, 17, 19 Sr( α, n) 5.81 0.23 7 Sr( α, n) 5.81-84.76 0.26-0.45 2, 6, 7, 18, 29, 30TABLE VI: Z=39Reaction Abundance variation Correlation coefficient MC tracers Fe( α, n) 16.14 0.22 36 Co( α, n) 16.14 0.25 36 Ni( α, n) 13.44-14.08 0.2-0.3 3, 32 Ni( α, n) 13.44-14.08 0.2-0.31 3, 32 Zn( α, n) 16.14 0.29 36 Zn( α, n) 13.44 0.21 32 Zn( α, n) 13.44-107.03 0.23-0.28 2, 3, 18, 32 Zn( α, n) 13.44-107.03 0.24-0.29 2, 3, 18, 32 Zn( α, n) 37.57 0.22 18 Ga( α, n) 13.44 0.21 32 Ge( α, n) 16.14 0.24 36 Ge( α, n) 12.42 0.23 33 Ge( α, n) 12.42-248.76 0.35-0.47 17, 33 Se( α, n) 5.71-141.15 0.22-0.6 6, 9, 10, 11, 19, 26, 27, 28, 31, 36 Se( α, n) 8.36-141.15 0.25-0.51 9, 10, 11, 19, 26, 27, 28, 31 Br( α, n) 6.42-24.93 0.21-0.64 9, 10, 26, 28, 29, 31 Br( α, n) 8.36-12.0 0.22 9, 10, 31 Br( α, n) 10.57 0.21 26 Kr( α, n) 6.62 0.3 7 Kr( α, n) 6.62-7.51 0.24-0.33 7, 30 Kr( α, n) 5.71-248.76 0.26-0.51 6, 7, 17, 29, 30, 33 Kr( α, n) 5.71-248.76 0.21-0.27 6, 17, 29, 30 Kr( α, n) 5.71-248.76 0.2-0.39 6, 17, 19, 26, 28, 29, 30 Sr( α, n) 37.57-107.03 0.32-0.4 2, 18Smith, and A. Mezzacappa, in Classical Nova Explo- sions , edited by M. Hernanz and J. Jos´e (2002), vol. 637 TABLE VII: Z=40Reaction Abundance variation Correlation coefficient MC tracers Fe( α, n) 11.26 0.22 36 Co( α, n) 6.0-11.26 0.21-0.23 34, 36 Ni( α, n) 12.76-18.57 0.25-0.29 3, 18 Ni( α, n) 12.76-18.57 0.26-0.3 3, 18 Zn( α, n) 11.26 0.27 36 Zn( α, n) 12.76 0.22 3 Zn( α, n) 12.76-50.35 0.24-0.3 2, 3, 18 Zn( α, n) 12.76-50.35 0.26-0.32 2, 3, 18 Zn( α, n) 12.76-18.57 0.22-0.25 3, 18 Ge( α, n) 11.26 0.27 36 Se( α, n) 11.26 0.2 36 Kr( α, n) 5.59-6.0 0.35-0.7 20, 34 Kr( α, n) 5.35-6.0 0.22-0.36 20, 28, 34 Kr( α, n) 5.04-6.0 0.24-0.44 11, 20, 27, 28, 34 Kr( α, n) 5.04-5.87 0.26-0.3 11, 27, 28 Kr( α, n) 5.04-5.87 0.4-0.55 11, 27, 28 Sr( α, n) 50.35 0.37 2TABLE VIII: Z=41Reaction Abundance variation Correlation coefficient MC tracers Fe( α, n) 17.74 0.23 36 Co( α, n) 9.94-17.74 0.21-0.25 34, 36 Ni( α, n) 12.45-18.5 0.25-0.29 3, 18 Ni( α, n) 12.45-18.5 0.27-0.3 3, 18 Zn( α, n) 17.74 0.29 36 Zn( α, n) 11.25-12.45 0.21-0.22 3, 32 Zn( α, n) 11.25-33.1 0.27-0.31 2, 3, 18, 32 Zn( α, n) 11.25-33.1 0.27-0.32 2, 3, 18, 32 Zn( α, n) 12.45-33.1 0.2-0.25 2, 3, 18 Ga( α, n) 11.25 0.21 32 Ge( α, n) 17.74 0.28 36 Ge( α, n) 79.77 0.44 17 As( α, n) 5.99 0.27 33 Se( α, n) 5.22-21.78 0.21-0.51 11, 19, 27, 28, 29, 33 Se( α, n) 5.22-21.78 0.27-0.54 11, 19, 27, 28 Kr( α, n) 5.11-14.26 0.2-0.8 5, 13, 14, 15, 20, 33, 34 Kr( α, n) 5.11-9.94 0.22-0.35 5, 13, 20, 33, 34 Kr( α, n) 5.11-79.77 0.21-0.34 5, 11, 13, 17, 20, 27, 28, 29, 33, 34 Kr( α, n) 5.22-79.77 0.23-0.31 11, 17, 27, 28, 29 Rb( α, n) 5.48-14.26 0.23-0.64 14, 15 Rb( α, n) 5.11-5.98 0.22-0.29 5, 13, 20 Sr( α, n) 6.78-33.1 0.25-0.44 2, 11, 27, 29of American Institute of Physics Conference Series , pp.161–166.[36] W. R. Hix, M. S. Smith, S. Starrfield, A. Mezzacappa, and D. L. Smith, Nucl. Phys. A , 620 (2003).[37] T. Rauscher, N. Nishimura, R. Hirschi, G. Cescutti,A. S. J. Murphy, and A. Heger, Mon. Not. R. Astron. TABLE IX: Z=42Reaction Abundance variation Correlation coefficient MC tracers Fe( α, n) 8.55-156.19 0.2-0.23 34, 36 Co( α, n) 8.55-156.19 0.22-0.26 34, 36 Ni( α, n) 13.63 0.25 3 Ni( α, n) 13.63 0.27 3 Zn( α, n) 156.19 0.31 36 Zn( α, n) 13.63 0.23 3 Zn( α, n) 13.63 0.31 3 Zn( α, n) 13.63 0.32 3 Zn( α, n) 13.63 0.24 3 Ge( α, n) 156.19 0.29 36 Ge( α, n) 156.19 0.2 36 Se( α, n) 5.05-104.93 0.31-0.32 18, 20 Se( α, n) 104.93 0.3 18 Kr( α, n) 5.05-8.55 0.22-0.54 7, 14, 15, 20, 34 Kr( α, n) 5.6-8.55 0.25-0.26 7, 34 Kr( α, n) 5.6-8.55 0.29-0.37 6, 7, 19, 22, 29, 30, 34 Kr( α, n) 7.02-7.42 0.2-0.3 6, 19 Kr( α, n) 5.94-8.08 0.23-0.57 6, 19, 22, 26, 29 Rb( α, n) 5.05-8.55 0.21-0.27 7, 20, 34 Sr( α, n) 5.35-8.13 0.4-0.43 14, 15 Sr( α, n) 5.35-8.13 0.25-0.3 14, 15 Sr( α, n) 5.35-8.13 0.29-0.43 14, 15 Sr( α, n) 11.02 0.2 31 Sr( α, n) 8.67-11.02 0.24-0.25 9, 10, 31 Sr( α, n) 5.6-104.93 0.24-0.6 2, 6, 7, 9, 10, 18, 22, 26, 29, 30, 31TABLE X: Z=44Reaction Abundance variation Correlation coefficient MC tracers Se( α, n) 5.61-24.77 0.22-0.31 2, 6, 18, 20, 22, 34 Se( α, n) 5.61-5.81 0.28 2, 18 Kr( α, n) 7.71-31.16 0.5-0.54 14, 15 Kr( α, n) 5.61-23.87 0.2-0.27 2, 8, 9, 18, 26, 31 Kr( α, n) 5.61-6.67 0.2-0.24 2, 18, 26 Kr( α, n) 5.61-10.05 0.22-0.47 2, 9, 10, 18, 26, 31 Sr( α, n) 7.71 0.22 15 Sr( α, n) 8.32-16.83 0.21-0.43 4, 5, 12, 13, 16, 20, 35 Sr( α, n) 8.32-9.02 0.2-0.21 5, 12, 13 Sr( α, n) 5.84-25.17 0.2-0.42 4, 5, 6, 7, 8, 12, 13, 16, 20, 21, 22, 23, 24, 25, 29, 30, 31, 34 Sr( α, n) 5.84-22.0 0.21-0.29 6, 7, 9, 10, 22, 29, 30, 31 Sr( α, n) 5.81-22.0 0.23-0.62 2, 6, 7, 9, 10, 22, 23, 26, 29, 30, 31 Zr( α, n) 7.71-31.16 0.32-0.4 14, 15 Zr( α, n) 8.32-25.17 0.22-0.4 4, 5, 8, 12, 13, 16, 21, 23, 24, 25, 35 Zr( α, n) 20.09-25.17 0.23-0.25 8, 23, 24 Zr( α, n) 20.09-25.17 0.2-0.22 8, 23, 24Soc. , 4153 (2016). [38] G. Cescutti, R. Hirschi, N. Nishimura, J. W. d. Hartogh, TABLE XI: Z=45Reaction Abundance variation Correlation coefficient MC tracers Fe( α, n) 61.36 0.25 34 Co( α, n) 50.02-61.36 0.21-0.27 20, 34 Se( α, n) 5.98-50.02 0.29-0.36 2, 20 Se( α, n) 5.44-5.98 0.22-0.3 2, 18 Kr( α, n) 10.69-31.69 0.22-0.56 4, 5, 13, 14, 15, 35 Kr( α, n) 21.89 0.25 5 Kr( α, n) 5.23-29.37 0.21-0.33 5, 6, 7, 11, 18, 19, 22, 27, 30, 32 Kr( α, n) 5.23-8.43 0.23-0.27 11, 18, 19, 27, 32 Kr( α, n) 5.23-8.43 0.4-0.53 2, 11, 18, 19, 27, 32 Rb( α, n) 31.69 0.22 14 Sr( α, n) 8.51-50.02 0.2-0.35 4, 5, 12, 13, 16, 20, 35 Sr( α, n) 8.51 0.21 16 Sr( α, n) 8.39-61.36 0.2-0.39 4, 5, 6, 7, 8, 13, 16, 21, 22, 23, 24, 25, 29, 30, 34 Sr( α, n) 21.65-29.37 0.22-0.27 6, 7, 29, 30 Sr( α, n) 5.23-29.37 0.21-0.62 2, 6, 7, 8, 11, 19, 22, 23, 24, 25, 27, 29, 30, 32 Y( α, n) 8.39-12.04 0.21-0.31 4, 8, 16, 21, 23, 24, 25 Y( α, n) 8.39-12.04 0.22-0.3 8, 23, 24, 25 Zr( α, n) 10.69-14.08 0.2-0.34 15, 35 Zr( α, n) 10.89-14.08 0.21-0.23 12, 35 Zr( α, n) 10.89-14.91 0.32-0.46 12, 13, 35 Nb( α, n) 10.69-31.69 0.23-0.53 14, 15, 35TABLE XII: Z=46Reaction Abundance variation Correlation coefficient MC tracers Kr( α, n) 6.53 0.2 27 Sr( α, n) 15.31-22.98 0.23-0.27 12, 13, 16, 20, 35 Sr( α, n) 5.9-22.15 0.2-0.27 7, 16, 28, 29, 30, 31 Sr( α, n) 5.9-16.65 0.21-0.26 6, 9, 10, 26, 27, 28, 29, 31 Sr( α, n) 5.9-16.65 0.29-0.66 6, 7, 9, 10, 22, 26, 27, 28, 29, 30, 31 Y( α, n) 9.36 0.21 30 Zr( α, n) 15.31-20.4 0.2-0.23 5, 12, 13 Zr( α, n) 8.7-42.62 0.22-0.62 4, 5, 7, 8, 12, 13, 16, 20, 21, 23, 24, 25, 30, 35 Zr( α, n) 20.4-42.62 0.2-0.3 4, 5, 8, 21, 23, 24, 25 Zr( α, n) 23.65-28.26 0.2-0.3 8, 23, 24, 25 Nb( α, n) 15.31-17.58 0.21-0.37 12, 13, 35T. 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TABLE XIII: Z=47Reaction Abundance variation Correlation coefficient MC tracers Cu( α, n) 24.85 0.23 35 Kr( α, n) 20.78 0.22 13 Sr( α, n) 17.92-27.91 0.22-0.23 12, 16 Sr( α, n) 8.2-9.1 0.21-0.26 11, 28, 31 Sr( α, n) 7.61-11.35 0.22-0.25 9, 11, 26, 27, 28, 31 Sr( α, n) 7.61-29.91 0.23-0.63 6, 9, 10, 11, 26, 27, 28, 29, 31 Zr( α, n) 17.92 0.24 12 Zr( α, n) 17.92-42.23 0.2-0.6 4, 5, 6, 7, 8, 12, 13, 16, 21, 22, 23, 24, 25, 29, 30, 35 Zr( α, n) 18.74-42.23 0.24-0.33 4, 6, 7, 8, 21, 22, 23, 24, 25, 29, 30 Zr( α, n) 18.74-42.23 0.2-0.35 6, 7, 8, 22, 23, 24, 25, 29, 30 Nb( α, n) 17.92-24.85 0.2-0.32 12, 13, 35Champagne, J. Phys. G: Nucl. Part. Phys.42