Sensitivity study of explosive nucleosynthesis in Type Ia supernovae: I. Modification of individual thermonuclear reaction rates

Sensitivity study of explosive nucleosynthesis in Type Ia supernovae:
I. Modification of individual thermonuclear reaction rates

Eduardo Bravo eduardo.bravo@upc.edu Dept. Física i Enginyeria Nuclear, Univ. Politècnica de Catalunya, Carrer Pere Serra 1-15, 08173 Sant Cugat del Vallès, Spain    Gabriel Martínez-Pinedo g.martinez@gsi.de Technische Universität Darmstadt, Institut für Kernphysik, Schlossgartenstr. 2, 64289 Darmstadt, Germany GSI Helmholtzzentrum für Schwerioneneforschung, Planckstr. 1, 64291 Darmstadt, Germany
July 29, 2019
Abstract
Background

Type Ia supernovae contribute significantly to the nucleosynthesis of many Fe-group and intermediate-mass elements. However, the robustness of nucleosynthesis obtained via models of this class of explosions has not been studied in depth until now.

Purpose

We explore the sensitivity of the nucleosynthesis resulting from thermonuclear explosions of massive white dwarfs with respect to uncertainties in nuclear reaction rates. We lay particular emphasis on indentifying the individual reactions rates that most strongly affect the isotopic products of these supernovae.

Method

We have adopted a standard one-dimensional delayed detonation model of the explosion of a Chandrasekhar-mass white dwarf, and have post-processed the thermodynamic trajectories of every mass-shell with a nucleosynthetic code in order to obtain the chemical composition of the ejected matter. We have considered increases (decreases) by a factor of ten on the rates of 1196 nuclear reactions (simultaneously with their inverse reactions) repeating the nucleosynthesis calculations after modification of each reaction rate pair. We have computed as well hydrodynamic models for different rates of the fusion reactions of C and of O. From the calculations we have selected the reactions that have the largest impact on the supernova yields, and we have computed again the nucleosynthesis using two or three alternative prescriptions for their rates, taken from the JINA REACLIB database. For the three reactions with the largest sensitivity we have analyzed as well the temperature ranges where a modifications of their rates has the strongest effect on nucleosynthesis.

Results

The nucleosynthesis resulting from the Type Ia supernova models is quite robust with respect to variations of nuclear reaction rates, with the exception of the reaction of fusion of two C nuclei. The energy of the explosion changes by less than when the rates of the reactions or are multiplied by a factor of or . The changes in the nucleosynthesis due to the modification of the rates of these fusion reactions are as well quite modest, for instance no species with a mass fraction larger than 0.02 experiences a variation of its yield larger than a factor of two. We provide the sensitivity of the yields of the most abundant species with respect to the rates of the most intense reactions with protons, neutrons, and alphas. In general, the yields of Fe-group nuclei are more robust than the yields of intermediate-mass elements. Among the species with yields larger than  M, S has the largest sensitivity to the nuclear reaction rates. It is remarkable that the reactions involving elements with have a tiny influence on the supernova nucleosynthesis. Among the charged particle reactions, the most influential on supernova nucleosynthesis are , , and . The temperatures at which a modification of their rate has a larger impact are in the range  GK.

Conclusions

The explosion model (i.e., the assumed conditions and propagation of the flame) chiefly determines the element production of Type Ia supernovae, and derived quantities like their luminosity, while the nuclear reaction rates used in the simulations have a small influence on the kinetic energy and final chemical composition of the ejecta. Our results show that the uncertainty in individual thermonuclear reaction rates cannot account for discrepancies of a factor of two between isotopic ratios in Type Ia supernovae and those in the solar system, especially within the Fe-group

pacs:
26.30.Ef, 26.30.-k, 26.50.+x, 97.60.Bw

I Introduction

Thanks to their high luminosity, Type Ia supernovae (SNIa) are used routinely as standard candles to measure cosmological distances. They are instrumental to our current understanding of the Universe, providing evidence for its accelerated expansion Riess et al. (1998); Perlmutter et al. (1998); Schmidt et al. (1998); Perlmutter et al. (1999); Riess et al. (2001); Tonry (2005). Type Ia supernovae play also an important role in the chemical evolution of galaxies, being responsible for most of the Fe-group elements and smaller amounts of Silicon, Sulfur, Argon, and Calcium (see e.g. Timmes et al. (1995); Thielemann et al. (2003); Matteucci et al. (2009)). The elemental composition is evident in optical and infrared spectra recorded from days to months after the explosion (see e.g. Stehle et al. (2005); Mazzali et al. (2007)) and in X-ray spectra of their remnants visible for hundreds of years (for a review see Badenes (2010)). Finally, SNIa are one of the key targets for -ray astronomy, as a source of a variety of radioactive isotopes (see e.g. Isern et al. (2006)).

This is the first paper of a series in which we will study the sensitivity of the nucleosynthesis produced in SNIa with respect to uncertainties in nuclear data. In this paper, we study the sensitivity to variations in rates of thermonuclear reactions (fusion reactions, radiative captures, and transfer reactions). In forthcoming publications, we will study the sensitivity to uncertainties in nuclear masses and in weak interaction rates. Studies of the effect of nuclear data uncertainties in different astrophysical scenarios have been published from time to time during the past few decades, e.g. Bahcall et al. (1982); Krauss and Romanelli (1990); Smith et al. (1993); Iliadis et al. (2002); Izzard et al. (2007); The et al. (1998); Hix et al. (2003); Arcones and Martínez-Pinedo (2011) to cite only a few, although none of them has dealt with SNIa. These works followed different methodologies to test the impact of nuclear reaction rates. For instance, ref. The et al. (1998) varied the rate of individual nuclear reactions relevant for Ti nucleosynthesis in order to determine which reactions were a prime target for the experimental measurement of their cross sections. On the other hand ref. Hix et al. (2003) designed a numerical experiment to measure the uncertainty of the nucleosynthesis of nova explosions. To this end, they followed a Monte Carlo approach in which they varied simultaneously by random factors all the reaction rates in their network. The focus of this second approach was on the final nova nucleosynthesis rather than in determining the individual reactions that are most influential. Finally, ref. Arcones and Martínez-Pinedo (2011) used theoretical nuclear reaction rates based on four different nuclear mass models to determine their impact on the r-process abundances. In this case, the emphasis was on testing different nuclear models. In the present work, we wish to determine the individual nuclear reactions most influential on the nucleosynthesis of SNIa, hence we will follow the same strategy as ref. The et al. (1998).

At present, the favored model of SNIa is the thermonuclear explosion of a carbon-oxygen white dwarf (WD) near the Chandrasekhar mass that accretes matter from a companion star in a close binary system Hillebrandt and Niemeyer (2000). Other models, such as the sub-Chandrasekhar models or the double degenerate scenario, although not completely ruled out, either have difficulties in explaining the gross features of the spectrum and light curve of normal SNIa, or face severe theoretical objections, see e.g. Nugent et al. (1997); Woosley and Kasen (2010); Bravo et al. (2011); Napiwotzki et al. (2002); Saio and Nomoto (1998); Segretain et al. (1997). Super-Chandrasekhar models have been proposed to explain a few overluminous SNIa Jeffery et al. (2006); Howell et al. (2006); Scalzo et al. (2010); Tanaka et al. (2010); Silverman et al. (2011) but, given the scarcity of observations despite of their high intrinsic luminosity they are thought to represent at most a few percent of all SNIa explosions. Moreover, the properties of the progenitors of super-Chandrasekhar SNIa and the explosions themselves are not well understood. Thus, we will concentrate our efforts on the study of a reference SNIa Chandrasekhar-mass model Badenes et al. (2005a).

Even though the hydrostatic evolution of SNIa progenitors lasts for several Gyrs while the thermonuclear explosion lasts for a few seconds at most, the outcome is nearly independent of the history of the white dwarf prior to its explosive ignition. This fact is commonly denoted as ’stellar amnesia’ Höflich et al. (2003). The only link between the white dwarf at ignition time and its previous evolution comes through its chemical composition (C, O, Ne, and other trace species) and the distribution of hot spots that are the seeds of the emerging thermonuclear flame. The influence of uncertain reaction rates, specifically that of the reaction , on the chemical composition of massive white dwarfs has been studied by ref. Straniero et al. (2003), who found that the central C/O ratio might vary by a factor of at ignition time. On the other hand, the effect of different C/O ratios on supernova luminosity and nucleosynthesis was studied in ref. Hoeflich et al. (1998), therefore accounting implicitly for a variation on the rate of capture on C. They found that the C/O ratio can have a sizeable impact on the ejecta composition. On the contrary, ref. Röpke et al. (2006) reached the opposite conclusion after analyzing the same problem with their three-dimensional deflagration models of SNIa.

Prior to the SNIa explosion there is a phase of carbon simmering that lasts  yrs and involves temperatures below  K. During this phase the neutron excess of matter can be raised due to electron captures on N and Na. The leading thermonuclear reactions during carbon simmering are, aside from C+C, reactions that participate in the transmutation of C into O: , , and . However, the timescale of neutronization is controlled by the C fusion reaction and the rate of electron captures Piro and Bildsten (2008); Chamulak et al. (2008). Thus, we do not expect that a modification of the rates of radiative captures and transfer reactions can affect appreciably the neutronization of the white dwarf and, hence, the final supernova composition. In this work, we will consider only modifications of the thermonuclear reaction rates during the explosive phase of the supernova.

The temperature range relevant for explosive nucleosynthesis in SNIa is approximately  K to  K. However, at densities and temperatures in excess of  g cm and  K nuclei attain a nuclear statistical equilibrium state (NSE) in which the chemical composition, for given temperature, density and electron mole fraction, is determined by nuclear bulk properties (masses and partition functions), i.e. it does not depend on the reaction rates. In these conditions, NSE erases any imprint of the previous thermodynamic evolution of matter, and reaction rates do not play any role until matter leaves NSE (freeze-out process). The minimum temperature relevant for nucleosynthesis in SNIa depends on the type of combustion front. For a detonation, a shock heats the fuel to temperatures  K, the precise value depending mainly on density, before nuclear reactions start modifying the chemical composition. On the other hand, the process of combustion within a subsonic flame presents two different phases. Below a critical temperature,  K, the matter temperature is set by heat diffusion from the hot ashes, while above the nuclear energy released by combustion dominates over heat diffusion. Thus, we do not expect modifying the thermonuclear reaction rates below  K to have an impact on the final chemical composition.

The plan of the paper is as follows. In the next Section, we detail the methodology used to achieve our goals. We describe the post-processing code used to integrate the nuclear evolutionary equations, the characteristics of our reference SNIa model, the selection of the nuclear reactions to test for variations in their rates, and the ways in which we have modified these rates. In Section III, we present the results of the sensitivity study with respect to the fusion reactions of C, O, and the reaction, which are the reactions that rule the initial steps of thermonuclear combustion in SNIa. We test modifications of the first two reaction rates for effects on the propagation of the flame during a SNIa explosion. In Section IV, we present the results of the sensitivity study with respect to thermonuclear reaction rates involving protons, neutrons, and particles. We have followed different strategies in modifying these reaction rates, using either a fixed enhancement factor or a temperature dependent one. We have tested as well the use of different prescriptions for the most influential reaction rates, taken from recent literature. For a few reactions we have explored the temperature range where a modification of their rates have a stronger impact on the supernova yields. Finally, in Section V, we summarize and give our conclusions.

Ii Methodology

ii.1 Integration of the nuclear evolutionary equations

n 1 1 Al 22 36 Fe 49 63 Y 79 101
H 1 4 Si 24 38 Co 51 65 Zr 81 101
He 3 9 P 26 40 Ni 53 69 Nb 85 101
Li 4 11 S 28 42 Cu 55 71 Mo 87 101
Be 6 14 Cl 30 44 Zn 57 78 Tc 89 101
B 7 17 Ar 32 46 Ga 61 81 Ru 91 101
C 8 20 K 34 49 Ge 63 83 Rh 93 101
N 10 21 Ca 36 51 As 65 85 Pd 95 101
O 12 23 Sc 38 52 Se 67 87 Ag 97 101
F 14 25 Ti 40 54 Br 69 90 Cd 99 101
Ne 16 27 V 42 56 Kr 71 93 In 101 101
Na 18 34 Cr 44 58 Rb 73 99
Mg 20 35 Mn 46 60 Sr 77 100
Table 1: Nuclear network

We have computed the chemical composition of a reference SNIa model with the nucleosynthetic code CRANK (Code for the Resolution of an Adaptive Nuclear networK). CRANK is a post-processing code that integrates the temporal evolution of a nuclear network for given thermal and structural (density) time profile, and initial composition. We have selected the nuclear reactions that contribute most to the synthesis of abundant species. Then, we have recomputed the nucleosynthesis modifying the rate of each one of the selected reactions.

The inputs to CRANK are the nuclear data and the thermodynamic trajectories, as a function of time, of each mass shell of the supernova model. The evolutionary equations for the nuclear composition follow the time evolution of the molar fraction, , or abundance of each species until the temperature falls below  K, after which time the chemical composition is no longer substantially modified. The nuclear network is integrated with an implicit, iterative method with adaptive time steps. The iterative procedure ends when the molar abundances of all species with  mol g have converged to better than a relative variation of .

The nuclear species present in the network are dynamically determined during the calculation. Initially, the network is defined by those species with an appreciable abundance ( mol g) plus n, p, and alphas and the nuclei that can be reached from any of the abundant species by any one of the reactions included in the network. A reaction rate is included in the network only if the predicted change of a molar abundance in the next time step, , is larger than a threshold:

(1)

A similar method of integration of the nuclear evolutionary equations using an adaptive network has been described in ref. Rauscher et al. (2002).

Our nuclear network consists of a maximum of 722 nuclei, from free nucleons up to In, linked by three fusion reactions: , C+C, and O+O, electron and positron captures, and decays, and 12 reactions per each nucleus with : , , , , , , , , , , , and . We show the nuclear network in Table 1. From the whole set of reactions that might be included in the calculations, only 3138 enter effectively into the reaction network equations during the integration of the thermodynamic trajectories in our SNIa model.

The thermonuclear reaction rates, nuclear masses and partition functions are taken from the REACLIB compilation Cyburt et al. (2010). Both theoretical and experimental thermonuclear reaction rates are fitted in the JINA 111http://groups.nscl.msu.edu/jina/reaclib/db/ REACLIB library by an analytic function with seven parameters. The fits are usually better than 5% although deviations up to 30% are possible. The authors estimate an additional uncertainty typically of order 30% in the original reaction rates. Electron screening to thermonuclear reactions in the strong, intermediate and weak regimes was taken into account Itoh et al. (1979); Salpeter and van Horn (1969). In general, in the conditions achieved during thermonuclear supernova explosions the electron screening factors are small Khokhlov et al. (1997). Weak interaction rates were taken from Fuller et al. (1982); Martínez-Pinedo et al. (2000).

ii.2 Type Ia supernova model

Our reference SNIa model is the one-dimensional delayed-detonation model DDTc in Badenes et al. (2005b), characterized by its deflagration-to-detonation (DDT) transition density,  g cm. The supernova progenitor is a Chandrasekhar mass white dwarf of central density  g cm and uniform composition: 49.5% C, 49.5% O, and 1% Ne by mass. In this model the flame begins as a subsonic deflagration flame near the center of the star. As the flame propagates through the star, the pressure rises and the star expands. When the flame reaches a zone with a low enough density, , there is a transition to a supersonic detonation that burns most of the remaining fuel. Finally, the nuclear energy released is enough to unbind the whole star and eject its matter into the interstellar medium. In Fig. 1 we show the profiles of the most relevant physico-chemical quantities affecting the nucleosynthesis.

Figure 1: Profiles of physico-chemical properties accross the reference model, as a function of the Lagrangian mass coordinate (zero at the center). Left: Peak temperature and density achieved at each mass shell during the supernova explosion (thick solid line). Star marks have been located every 0.1 M, with the center of the white dwarf at the top right end of the solid line, and the surface at its bottom left end. The plane has been divided according to approximate locations of different explosive nucleosynthetic processes (dashed and dotted lines). We indicate as well the Lagrangian mass coordinate at which the solid line crosses the dashed and dotted lines. Right: Maximum molar fractions of neutrons, protons, and alphas achieved at a given mass coordinate at any time during the explosion, and final electron mole number (dot-dashed line). Note that the neutron molar fraction has been scaled up by a factor of for presentation purposes.

This kind of SNIa model generates a layered structure (see Fig. 2) in which the inner several tenths of a solar mass achieve maximum temperatures high enough ( K) to process matter into NSE, undergoing copious electron captures. When matter expands the composition is relaxed out of NSE and consists mainly of iron group elements with isotopic fractions determined by the electron mole number resulting from the electron captures phase. As can be seen in Fig. 1, in our reference model the electron captures modify the progenitor electron mole number only in the central  M. The zone were the transition from deflagration to detonation takes place, at a mass coordinate of , can be identified by the trough in the , , and profiles. The central reach NSE, from which roughly experience a moderately -rich freeze-out. Shortly after the detonation forms, it propagates fast through the white dwarf, which has no time to relax its structure before the combustion front burns most of the remaining fuel (this condition can be identified in Fig. 1 by the crowding of the star symbols between and ). Between Lagrangian mass coordinates of and the peak temperatures and densities are high enough to experience Si-burning and achieve quasi-statistical equilibrium (QSE) of the Fe-group, although this group does not achieve equilibrium with the Si-group. Farther out from the center, a tinier amount of mass is subject to explosive oxygen and neon burning, and only a few thousandths of a solar mass experience only explosive carbon burning. The mass of unburned carbon ejected by the supernova explosion is on the same order, in agreement with the upper limits deduced by Folatelli et al. (2011). We note that all the nucleosynthetic processes deemed relevant in SNIa feature in our reference model.

For reference, we give in Table 2 the nucleosynthesis obtained for this supernova model. The composition given in this and forthcoming tables corresponds to a time of one day after beginning of the explosion, hence there appear radioactive as well as stable nuclides. We have included in this table all nuclides whose ejected mass is  M, with the exception of Al, that has been included because it is an interesting radionuclide. The ejected mass of Ni is 0.675 M, and the kinetic energy of the ejecta is  erg, both values deemed typical for normal bright SNIa. The resulting chemical composition (Fig. 2) compares well with the abundance stratification induced from observations of normal SNIa as, for instance, SN2003du (e.g., Fig. 8 in Tanaka et al. (2011b), who estimated that the ejected mass of Ni was 0.65 M). Model DDTc also provides an excellent match to the X-ray spectrum of the remnant of SN1572 (Tycho), a prototype of SNIa (see Fig. 7 in Badenes et al. (2006)).

Figure 2: Chemical composition of the reference model as a function of the final velocity. The curves labelled as Fe and Ni include only stable isotopes. The thick curve is the mass fraction of Ni.
Nucleus Ejected mass range111Range of maximum temperatures achieved in the shells in which 90% of each nuclide is produced. Nucleus Ejected mass range111Range of maximum temperatures achieved in the shells in which 90% of each nuclide is produced.
(M) (GK) (M) (GK)
C destroyed K 2.6–4.0
O destroyed Ca 4.0–5.2
Ne 2.0–2.8 Ti 3.8–5.6
Na 2.0–3.2 V 4.2–5.2
Mg 2.4–3.4 V 4.2–5.2
Mg 2.0–3.4 Cr 4.0–5.2 and
Mg 2.0–3.2 Cr 3.8–5.6
Al 2.0–3.0 Mn 4.2–5.2
Al 2.2–3.4 Mn 4.4–5.2 and
Si 2.8–5.0 Fe 4.2–5.2 and
Si 2.2–3.6 Fe 4.2–5.2 and
Si 2.4–3.6 Fe
P 2.4–3.8 Co
S 3.2–5.0 Ni
S 2.6–4.0 Ni
S 2.6–3.8 Ni
Cl 2.4–4.0 Ni
Ar 3.6–5.0 Ni
Ar 2.6–4.2 Ni
Ar 3.2–4.0
Table 2: Nucleosynthesis of the reference Type Ia supernova model

The maximum abundances of free protons, neutrons, and -particles attained during the explosion are shown in the right panel of Fig. 1, as a function of the Lagrangian mass coordinate within the exploding white dwarf. These profiles can be used to gain insight into the expected sensitivities of the nucleosynthesis with respect to different types of nuclear reactions, to be discussed in the next chapters. Neutrons are always the less abundant nucleons by orders of magnitude, thus we expect that the nucleosynthesis will not be too sensitive to reactions with neutrons except, perhaps, in the outer  M. Note that neutrons are relatively abundant in the very center of the white dwarf, because of the lower that results from efficient electron captures in NSE matter at high density, but nucleosynthesis in these layers is not expected to be sensitive to the rate of any particular reaction with neutrons because the chemical composition there is controlled by the Saha equation until matter cools to low temperatures. Protons and -particles have similar abundances within the inner  M, although their maximum molar fractions decrease steadily outwards within the detonated matter ( M). Beyond  M, the maximum abundance achieved by protons is much lower than that of -particles. The maximum temperatures attained in these layers stay below  K, implying that the thermonuclear combustion hardly goes beyond O-burning. Thus, we expect that the products of O-burning will be mostly sensitive to reactions with -particles.

The above analysis can be complemented with an examination of the molar fluxes due to different reaction types, e.g. , etc. In Fig. 3 we show the evolution of the net molar fluxes in two representative mass shells of our SNIa model, grouped by reaction type. The net molar fluxes of a given reaction type in a mass shell are accumulated in time according to:

(2)

where is density, is Avogadro’s number, is the molar fraction of species , and the time integral extends from thermal runaway until the temperature goes below  K. The summation extends to all reactions of the given type, from which their inverse reactions are subtracted, e.g. in the computation of the net molar fluxes of the type reactions all the reactions are considered inverse reactions and their contributions are deducted from those of the direct, , reactions.

In the left panel of Fig. 3 we show the net molar fluxes in a mass shell located at a Lagrangian mass coordinate of  M. This layer was hit by the detonation wave  s after central thermal runaway, when its density was  g cm, and heated to  K by the shock front associated with the detonation. Above this temperature, it is the energy release by nuclear reactions which controls the evolution of temperature. The temperature rises very fast at the beginning due to rapid burning of carbon and oxygen, mainly to produce silicon and sulfur. About 1 ms after being shocked, a maximum temperature of  K is achieved. Later, matter expands and cools with a longer timescale (it takes 0.1 s to cool by  K) while most of the nuclear reactions are nearly in equilibrium with their inverse reactions. During the heating phase, it is the C reaction which dominates the nuclear fluxes, followed by radiative captures of protons and -particles once the temperature exceeds  K. Compared to the plethora of reactions with light particles unleashed by the carbon fusion reaction, the contribution of the O reaction is quite modest until the temperature exceeds  K. Above  K there is a sharp increase in the cumulative molar fluxes belonging to , , and reactions, which reach similar levels. On the other hand, during the cooling phase there is little additional contribution to the net molar fluxes, and and reactions attain a level similar to that of O, while reactions are the ones that process the smallest mass. Note that, in this mass shell, the final cumulative molar flux due to the O fusion reaction is about a factor four smaller than that due to the C fusion reaction. Taking this mass shell as representative of layers that experience incomplete Si-burning, we expect that the products of this nucleosynthetic process will be most sensitive to , , and reactions, and their inverses. Note that in shells that achieve a temperature high enough to reach NSE all the molar fluxes established prior to NSE are irrelevant, because NSE erases all memory of previous nuclear processes with the exception of weak interactions.

The right panel of Fig. 3 shows the net molar fluxes in a mass shell located at a Lagrangian mass coordinate of  M. In this case the maximum temperature achieved was  K, because the density at the time of detonation impact (at  s) was only  g cm. Due to the small value of the maximum temperature, the C reaction dominates the molar fluxes at all times. Oxygen burning is incomplete, the final molar flux due to the O fusion reaction being about 20 times smaller than that due to the C fusion reaction. Even reactions process more matter than the O fusion reaction. Taking this mass shell as representative of layers that do not go beyond carbon burning, we expect that the products of this nucleosynthetic process will be most sensitive to the rate of the C reaction and, to a lesser extent, to and reactions.

Figure 3: (Color online) Cumulative net molar fluxes of direct reactions minus inverse reactions, grouped by reaction type, compared to the C-fusion and O-fusion reaction fluxes, as functions of temperature. Each reaction type is identified by a different line type and a label, in each label an is drawn to recall that the fluxes take into account direct and inverse reactions. Thin (black) curves represent those cases in which the (accumulated) molar flux of the direct reaction is larger than that of the inverse reaction, while thick (red) curves belong to the opposite case. Vertical lines are drawn to separate the heating phase from the cooling phase. Left: Evolution at Lagrangian mass coordinate of  M. Right: Evolution at Lagrangian mass coordinate of  M.

ii.3 Selection of the nuclear reactions

As explained before, only 3138 nuclear reactions exceed the threshold of Eq. 1 and are actually included in the nucleosynthesis calculation. However, most of these reactions contribute negligibly to the determination of the final chemical composition of the supernova ejecta. In order to determine the most relevant reactions, we define the total mass processed by a nuclear reaction, between particle and nucleus , in all the mass shells of the supernova model :

(3)

where is the mass of shell of the supernova model, and is the baryon number of species . In the computation of the integral we have not taken into account reactions above  K, because at such temperatures the direct and inverse reactions are in equilibrium, causing the nuclear abundances to be determined by properties of the nuclei involved (mass, partition function) instead of the reaction rates. For mass shells that went through NSE, the computation of the integral in Eq. 3 starts when the temperature drops below  K, since their chemical composition is insensitive to the nuclear history prior to the NSE state (with the exception of weak interactions, whose effect is not addressed in the present work).

The reactions we have selected for careful study are the three fusion reactions plus those for which  M. This warrants that we test all the reactions able to contribute significantly to the synthesis of every species whose yield is larger than the chosen  M. Each time we integrate the nuclear evolutionary equations we modify by the same factor the direct and inverse reactions. Following this procedure, we find that the nucleosynthesis at this chosen level could be sensitive to 1096 (pairs of) reactions in addition to the above mentioned three fusion reactions.

Table 3 gives the masses processed by the three fusion reactions and the top ten radiative captures and transfer reactions, where the masses processed by the inverse reactions have been subtracted from those of the direct reactions. The quoted values of give a quite generous upper limit of the impact these reactions might have on the resulting nucleosynthesis of the supernova, as the subsequent nuclear reactions destroy the products of earlier reactions. As we will see in the following, the top ten reactions listed in Table 3 are not in fact the most influential reactions.

Reaction Reaction
C 0.524 0.70
O 0.198 0.68
0.67
0.93 0.65
0.84 0.64
0.83 0.63
0.77
Table 3: Masses processed by the fusion reactions and the top ten radiative captures and transfer reactions

ii.4 Modification of the reaction rates

As a first approach to study the sensitivity to the different reaction rates, we modify them, one by one, by a fixed factor, either equal to or , repeating the nucleosynthesis calculation for each variation. As mentioned previously, each time we modify the rate of a reaction we modify as well by the same factor the rate of the inverse reaction, in order to maintain detailed balance.

The Gamow energies in the reactions that play a significant role in the nucleosynthesis of Type Ia supernovae go from a few tenths of a MeV (for instance,  MeV for the reaction at  K) to nearly ten MeV (e.g.,  MeV for the reaction at  K). It is expected, both from theoretical and experimental arguments, that the uncertainties in the rates at low temperatures are larger than at high temperatures Hoffman et al. (1999). Most of the theoretical reaction rates we have used are based on an statistical model of nuclei, which assumes formation of a compound nucleus with a high level density, a condition generally satisfied at high temperatures. Furthermore, experimental measurements of nuclear cross sections involving high-Z nuclei are generally difficult to perform at energies below the Coulomb barrier. Consequently, we use a second approach in which the reaction rates are modified by applying a factor that is a monotonic decreasing (exponential) function of the temperature. We have applied the following temperature dependent factor to each reaction rate:

(4)

where or , is the fixed factor applied in the first approach. Of course, we are not trying to convey that Eq. 4 is representative of the uncertainty of all the reactions studied here (see Sections IV.3 and IV.4), but it provides a convenient way to invetigate the effects of a temperature dependent rate error.

Iii Sensitivity to the rate of fusion of carbon and of oxygen, and the triple–alpha reaction.

We have checked the effect of varying each fusion reaction rate by the factors given above, either taking them fixed or as function of the temperature. Because the fusion reactions are relevant for the nuclear energy generation in the supernova explosion, we have recomputed the hydrodynamics with the modified reaction rates and give the results in Section III.1. When we kept unchanged the thermodynamic trajectories of the reference model, but the reaction rates were modified in the nucleosynthetic code, we obtained the results shown in Section III.2.

iii.1 Rate modified in the hydrodynamic explosion model

The nuclear energy release of the supernova is more sensitive to the rate of the O fusion reaction than to that of C. The final kinetic energy of the ejecta varies by less than 1% when the C reaction rate is varied by a factor of or , either fixed or as a function of the temperature given by Eq. (4). In contrast, the same relative variation in the rate of the O reaction produces a change of kinetic energy of up to . We ascribe this lack of sensitivity to the relatively small amount of mass that does not experience complete carbon or oxygen burning. Figure 4 shows the final chemical profiles in the outermost  M of ejecta, where the changes in the C and the O reaction rates are most influential. The three panels show the profiles belonging to our reference model and the models in which either the carbon or the oxygen fusion rates are increased by a factor of ten, both in the hydrodynamic as well as in the nucleosynthetic codes. As can be seen, increasing the C rate by a factor of ten barely affects the limits of the region undergoing carbon burning, which move outwards  M. On the other hand, when the O reaction rate is enhanced by the same factor the limits of the oxygen burning region move outwards  M. We conclude that the impact of the rates uncertainties on the energy of the supernova is negligible.

Figure 4: Final chemical profile within the outer layers of the SNIa ejecta for three of the computed models: our reference model (top), the model with the reaction increased by a constant factor of ten (middle), and the model with the reaction increased by a constant factor of ten (bottom). In this plot, the mass coordinate is zero at the white dwarf surface and increases inwards.
Figure 5: (Color online) Ratio of mass ejected for each nuclide with a modified C+C reaction rate with respect to the mass ejected in the reference model, as a function of the mass fraction in the reference model (most abundant species are located to the right of each figure). Note that the species included in Table 3 are those with an ejected mass larger than  M. Vertical lines link the results obtained for the same nuclide when the rate is either increased or decreased. The reaction rate was modified both in the hydrodynamics calculation as well as in the nucleosynthetic code. Top: Rate multiplied by a fixed factor, either (green empty circles) or (red filled circles). Bottom: Rate multiplied by a factor function of temperature given by Eq. (4), with either (green empty circles) or (red filled circles).

Figures 5 to 8 and Table 3 show the impact of the changes in the fusion rates of C and of O on the nucleosynthesis of the Type Ia supernova, when we modified the rates in the full supernova simulation. Figures 5 and 6 show the results sorted by final mass fraction of the product species. The mass fractions of the most abundant species are insensitive to the rate of fusion of C. As one goes to smaller abundances, the scatter of the yield ratio is larger. Among the species with mass fraction greater than there is only one nuclide that is significantly affected by the modification of the rate of C: not surprisingly it is Mg. When the factor that modifies the C fusion rate is a function of temperature, Eq. (4), the effect on the yields of all species is dramatically reduced (bottom frame in Fig. 5): no species experiences an increase larger than a factor of two in its abundance, and only a few species with quite small mass fractions () experience a reduction of more than a factor of two in their yields when the C fusion reaction rate is multiplied by a factor of ten.

Figure 6: (Color online) Ratio of mass ejected for each nuclide with a modified O+O reaction rate with respect to the mass ejected in the reference model, as a function of the mass fraction in the reference model (most abundant species are located to the right of each figure). Note that the species included in Table 3 are those with an ejected mass larger than  M. Vertical lines link the results obtained for the same nuclide when the rate is either increased or decreased. We modified the reaction rate both in the hydrodynamics calculation as well as in the nucleosynthetic code. Top: Rate multiplied by a fixed factor, either (green empty circles) or (red filled circles). Bottom: Rate multiplied by a factor function of temperature given by Eq. (4), with either (green empty circles) or (red filled circles).

When the O rate is modified (Fig. 6) the impact is in general smaller than when the C fusion rate was modified. However, many of the most abundant species are more sensitive to the O fusion rate than to the C rate because the products of C-burning (mainy O, Ne, and Mg) are in general less abundant than the products of O-burning (mainly Si, S, Ar, and Ca).

Figure 7 presents the same results as Fig. 5 from another perspective: the impact of the modification of the C reaction rate is shown against the element atomic number. The trend that can be observed in this figure is that increasing the C fusion rate (green empty circles) decreases the abundances both of CNO nuclei and of IMEs between Phosphorus and Titanium, and increases the abundances of Magnesium, Aluminum, and Silicon, while elements beyond Vanadium are scarcely affected at all. If the rate of C fusion is decreased (red filled circles) the trend is inverted, but the yields are in general more sensitive to a decrease in this rate than to an increase by the same factor.

Figure 7: (Color online) Same as Fig. 5 but plotted as a function of the atomic number of the product nucleus. Note that not all the isotops shown here appear in Table 3. Top: Rate multiplied by a fixed factor, either (green empty circles) or (red filled circles). Bottom: Rate multiplied by a factor function of temperature given by Eq. (4), with either (green empty circles) or (red filled circles).

As can be seen in Fig. 8, an increase in the O reaction rate results in a small decrease in the production of elements up to Magnesium and an increase in elements from Chlorine to Chromium. The effect on the mass fractions is much smaller than that due to variations in the C fusion rate.

Figure 8: (Color online) Same as Fig. 6 but plotted as a function of the atomic number of the product nucleus. Note that not all the isotops shown here appear in Table 3. Top: Rate multiplied by a fixed factor, either (green empty circles) or (red filled circles). Bottom: Rate multiplied by a factor function of temperature given by Eq. (4), with either (green empty circles) or (red filled circles).

We give in Table 3 the sensitivity of the yield of each one of the species included in Table 2 to the rate of fusion reactions. There, is the logarithmic derivative of the mass ejected of species with respect to the enhancement factor of each fusion reaction, (note that when using Eq. 4, represents the maximum enhancement factor, attained at low temperatures),

(5)

where is the mass ejected of species for , and is the corresponding mass when . According to this definition, a value of means that the abundance of species approximately doubles for a constant enhancement factor of in the corresponding fusion reaction rate. Similarly, a relative change in the abundance of a species by would correspond to , and a change by would derive from .

Most notable is the robustness of the production of most Fe-group isotopes, notably of Ni. When the enhancement factor is computed from Eq. 4, there is no species with , neither with respect to the rate of C nor with respect to the O rate, with the exceptions of C and K, respectively.

Nucleus 222Enhancement factor function of temperature according to Eq. 4. 222Enhancement factor function of temperature according to Eq. 4.
C 4.8E-1 1.7E-1 3.7E-2 1.6E-2
O 1.9E-2 6.0E-3 1.3E-1 5.2E-2
Ne 6.0E-2 2.8E-2 5.8E-2 2.6E-2
Na 3.1E-1 7.9E-2 6.4E-2 2.6E-2
Mg 3.7E-1 9.2E-2 1.1E-1 4.6E-2
Mg 2.8E-1 4.2E-2 3.0E-2 5.0E-3
Mg 2.8E-1 5.4E-2 8.2E-2 3.7E-2
Al 4.1E-2 4.0E-2 1.3E-2 6.0E-3
Al 2.5E-1 7.4E-2 6.5E-2 2.7E-2
Si 1.1E-2 6.0E-3 7.2E-2 2.9E-2
Si 1.3E-1 1.6E-2 2.5E-2 1.0E-3
Si 2.8E-1 7.0E-2 1.6E-1 7.1E-2
P 3.1E-2 1.0E-3 1.3E-2 3.0E-3
S 1.4E-2 0. 5.7E-2 2.2E-2
S 1.9E-2 9.0E-3 1.7E-2 8.0E-3
S 6.7E-2 5.0E-2 7.2E-2 3.0E-2
Cl 2.5E-1 6.7E-2 1.2E-1 6.0E-2
Ar 1.9E-2 2.0E-3 1.6E-2 5.0E-3
Ar 1.5E-1 3.0E-2 1.8E-1 7.8E-2
Ar 1.9E-1 4.2E-2 1.2E-1 4.8E-2
K 2.2E-1 3.6E-2 2.3E-1 1.0E-1
Ca 2.3E-2 6.0E-3 2.1E-2 1.1E-2
Ti 3.1E-2 9.0E-3 5.6E-2 2.8E-2
V 2.4E-2 1.0E-2 4.1E-2 1.6E-2
V 1.9E-2 5.0E-3 2.2E-2 1.1E-2
Cr 4.0E-3 6.0E-3 8.0E-2 3.2E-2
Cr 1.2E-2 3.0E-3 9.0E-3 4.0E-3
Mn 2.0E-2 1.2E-2 4.0E-2 1.5E-2
Mn 1.4E-2 7.0E-3 2.8E-2 1.2E-2
Fe 0. 3.0E-3 1.3E-2 5.0E-3
Fe 0. 0. 0. 0.
Fe 0. 2.0E-3 5.0E-3 3.0E-3
Co 5.0E-3 4.0E-3 1.3E-2 5.0E-3
Ni 9.0E-3 9.0E-3 2.8E-2 1.2E-2
Ni 2.0E-3 0. 0. 0.
Ni 1.0E-3 0. 0. 0.
Ni 2.0E-3 1.0E-3 1.0E-3 0.
Ni 4.0E-3 2.0E-3 2.0E-3 2.0E-3
Ni 6.0E-3 2.0E-3 0. 2.0E-3
Table 4: Sensitivity of the nucleosynthesis to the rate of fusion reactions: Rate modified in the hydrodynamic and nucleosynthetic codes (see also Figs. 5 to 8)333Values of less than 1.0E-3 have been put to 0..

iii.2 Rate modified only in the nucleosynthetic code

Figures 9 and 10 show the impact of the changes in the fusion rates of C and of O on the nucleosynthesis of Type Ia supernovae when the rates are modified only in the nucleosynthetic code. They can be compared with Figs. 7 and 8, respectively, to evaluate the relevance of incorporating the modified rates into the hydrodynamic code. The trends visible in these figures are qualitatively similar, irrespectively if the reaction rate has been modified in the hydrodynamic calculations or not.

Figure 9: (Color online) Ratio of mass ejected for each nuclide with a modified C+C reaction rate with respect to the mass ejected in the reference model, as a function of the atomic number of the product nucleus. We modified the reaction rate only in the nucleosynthetic code. Top: Rate multiplied by a fixed factor, either (green empty circles) or (red filled circles). Bottom: Rate multiplied by a factor function of temperature given by Eq. (4), with either (green empty circles) or (red filled circles).
Figure 10: (Color online) Ratio of mass ejected for each nuclide with a modified O+O reaction rate with respect to the mass ejected in the reference model, as a function of the atomic number of the product nucleus. We modified the reaction rate only in the nucleosynthetic code. Top: Rate multiplied by a fixed factor, either (green empty circles) or (red filled circles). Bottom: Rate multiplied by a factor function of temperature given by Eq. (4), with either (green empty circles) or (red filled circles).

In Fig. 11 we show the yield ratios belonging to modified rate of the reaction. The influence of the rate of this reaction focuses on a few elements: Nitrogen, Nickel, Copper, and Zinc (specially the isotopes of Nickel and Zinc produced during alpha-rich freeze-out of NSE) inversely correlate with the factor of enhancement of the reaction, while Titanium and, to a lesser extent, Scandium, Manganese, and Iron (specially the isotopes produced during explosive Si-burning) are more abundant when the reaction is faster. These results can be explained by the fact that for a faster rate an alpha-rich freeze-out occurs at lower temperature and density. As a result, the dotted-line in Fig. 1 (left) shifts down when the reaction is faster, increasing the yield of species made in normal freeze-out at the expense of alpha-rich freeze-out products.

Figure 11: (Color online) Ratio of mass ejected for each nuclide when the reaction rate is modified to the mass ejected in the reference model, as a function of the atomic number of the product nucleus. We modified the reaction rate only in the nucleosynthetic code. Top: Rate multiplied by a fixed factor, either (green empty circles) or (red filled circles). Bottom: Rate multiplied by a factor function of temperature given by Eq. (4), with either (green empty circles) or (red filled circles).

We give in Table 5 the sensitivity of the yield of each one of the species included in Table 2 to the rate of fusion reactions, when they are modified only in the nucleosynthetic code. This table can be compared to Table 3 to avaluate the importance of running a hydrodynamic code with the reaction rates modified or take the thermodynamic profiles of a reference model and modifying the rates only in a post-processing code. One finds that the sensitivities shown in both tables are qualitatively similar. Although the precise values of for given species are not equal, the rating of the species that are most sensitive to any fusion reaction rate is the same in both tables. Given the pre-eminence of the fusion reaction rates with respect to the release of nuclear energy, we conclude that for SNIa this kind of study can be safely carried out with a post-processing code, using a set of thermodynamic trajectories obtained with a supernova hydrodynamics code where the reaction rates remain unchanged.

Nucleus 222Enhancement factor function of temperature according to Eq. 4. 222Enhancement factor function of temperature according to Eq. 4. 222Enhancement factor function of temperature according to Eq. 4.
C 3.8E-1 1.9E-1 0. 0. 0. 0.
O 1.3E-2 6.0E-3 7.2E-2 3.1E-2 1.0E-3 0.
Ne 1.0E-1 5.2E-2 0. 0. 0. 0.
Na 1.7E-1 6.5E-2 1.3E-2 5.0E-3 1.0E-3 0.
Mg 3.9E-1 1.1E-1 6.8E-2 3.2E-2 1.0E-3 0.
Mg 6.4E-2 2.0E-2 3.0E-2 1.7E-2 0. 0.
Mg 1.1E-1 4.3E-2 4.3E-2 2.1E-2 1.0E-3 0.
Al 2.4E-2 5.0E-3 7.5E-2 4.3E-2 0. 0.
Al 2.5E-1 7.4E-2 3.0E-2 1.4E-2 1.0E-3 0.
Si 5.0E-3 0. 7.4E-2 3.4E-2 2.4E-2 7.0E-3
Si 1.4E-1 2.7E-2 7.9E-2 4.3E-2 2.0E-3 1.0E-3
Si 2.8E-1 8.5E-2 1.2E-1 6.4E-2 2.0E-3 1.0E-3
P 1.3E-2 1.6E-2 6.0E-2 3.2E-2 2.0E-3 0.
S 2.3E-2 7.0E-3 3.6E-2 1.5E-2 1.7E-2 5.0E-3
S 9.6E-2 4.1E-2 9.0E-2 4.3E-2 1.0E-3 0.
S 9.6E-2 7.0E-2 1.1E-2 7.0E-3 2.0E-3 1.0E-3
Cl 3.2E-1 9.5E-2 1.7E-1 8.7E-2 1.0E-3 3.0E-3
Ar 2.3E-2 7.0E-3 2.0E-2 9.0E-3 7.0E-3 2.0E-3
Ar 2.1E-1 5.5E-2 1.7E-1 7.0E-2 2.0E-3 0.
Ar 2.4E-1 7.4E-2 1.4E-1 6.9E-2 2.0E-3 0.
K 2.7E-1 6.0E-2 1.7E-1 7.6E-2 1.7E-2 1.0E-2
Ca 2.6E-2 9.0E-3 7.2E-2 3.3E-2 6.0E-3 2.0E-3
Ti 2.4E-2 9.0E-3 9.2E-2 4.2E-2 1.4E-1 9.6E-2
V 7.0E-3 2.0E-3 6.8E-2 3.0E-2 2.9E-2 1.5E-2
V 8.0E-3 1.0E-3 4.1E-2 1.7E-2 5.0E-3 0.
Cr 6.0E-3 3.0E-3 5.6E-2 2.4E-2 1.4E-2 5.0E-3
Cr 0. 3.0E-3 1.8E-2 7.0E-3 1.7E-2 5.0E-3
Mn 2.0E-3 1.0E-3 7.3E-2 3.1E-2 3.5E-2 1.3E-2
Mn 2.0E-3 3.0E-3 4.4E-2 1.8E-2 3.2E-2 1.0E-2
Fe 2.0E-3 3.0E-3 3.0E-3 2.0E-3 3.5E-2 1.3E-2
Fe 2.0E-3 3.0E-3 2.6E-2 1.2E-2 4.8E-2 1.7E-2
Fe 0. 0. 0. 0. 0. 0.
Co 2.0E-3 0. 6.0E-3 2.0E-3 1.1E-2 4.0E-3
Ni 3.0E-3 0. 1.6E-2 8.0E-3 1.2E-2 5.0E-3
Ni 2.0E-3 0. 1.0E-3 0. 4.7E-2 1.7E-2
Ni 0. 0. 0. 0. 5.6E-2 2.2E-2
Ni 2.0E-3 0. 0. 0. 1.1E-1 4.5E-2
Ni 2.0E-3 0. 0. 0. 1.4E-1 5.8E-2
Ni 2.0E-3 0. 0. 0. 1.6E-1 6.4E-2
Table 5: Sensitivity of the nucleosynthesis to the rate of fusion reactions: Rate modified only in the nucleosynthetic code (see also Figs. 9 to 11)555Values of less than 1.0E-3 have been put to 0..

Iv Sensitivity to the rate of radiative captures and transfer reactions

We will discuss in this section the sensitivity of the nucleosynthesis to changes in the rate of radiative captures and transfer reactions. We measure the sensitivity in a similar way as with respect to the rate of the fusion reactions, by defining as in Eq. 5. The meaning of is now the logarithmic derivative of the mass ejected of species with respect to the enhancement factor, , of a reaction between particle and nucleus , while is the mass ejected of species when an enhancement factor (either fixed or function of temperature) is applied to the rate of the reactions and , and is the corresponding mass when .

We start by analyzing the results obtained with a fixed enhancement factor (our first approach). In Section IV.2 we present the results obtained when we compute the enhancement factor with a decreasing uncertainty (see Eq. 4, our second approach). Then, we select the reactions to which the nucleosynthesis is most sensitive and analyze in Section IV.3 the results achieved by adopting different prescriptions for their reaction rates, chosen among the most recent literature. Finally, in Section IV.4 we analyze the temperature ranges in which a modification of a given reaction rate affects most the chemical composition of the supernova ejecta.

iv.1 Fixed rate enhancement factor

Tables 7 to 17 give, for each reaction pair that has a significant impact on the nucleosynthesis, the nuclei for which (more than twofold increase or decrease in the yield when the rate in enhanced or decreased by a factor of ten), and those for which (relative increase or decrease of the yield between and a factor of two). Although we only list in the tables the direct reactions, the inverse reactions contributed as well to the changes in the nucleosynthetic yield. There is only one reaction pair for which , it is the reaction and the species whose abundance is mostly affected is S. Each table shows the reactions belonging to a given type, e.g. , sorted according to the total mass they processed in our reference model, (see Table 3).

Parent nuclide Nuclei with Nuclei with
Si S Ne, Mg, Al, S
Fe Mn, Fe, Co
S S Mg, Al, Si, P, S
Ar Cl, Ar
Ti Sc
Mg Al, S O, Ne, Mg, P, S
Mg Ne O, Mg, Al
Fe Fe
O O Mg
Ti Ti
Si Si
Ne Ne
S P
Cl Cl
C Ne, Mg, Sc
P P
Table 6: Sensitivity of the nucleosynthesis to the rate of reactions with the parent nuclide given in the first column.777The reactions listed are those that processed more than  M in the reference model (see Table 3) and with any .

The species most sensitive to changes in the rates of reactions (Table 7) are O, Al, Ne, and S. The yields of all these nuclides are small, of order  M. Apart from the neutron captures on iron isotopes, all the reactions listed in Table 7 involve IMEs or CNO elements as the parent nuclides. Among the species with , the most abundant are Si and S, both with yields on the order of a few times  M, suggesting that the temperature range where reaction rates most affect the final nucleosynthesis is approximately  K, in agreement with our analysis in Section II.2.

Parent nuclide Nuclei with Nuclei with Nuclei with
Si Ne, Mg, Al, S, Ca, Ti
Co Mn, Fe
Ni Cu
S S
Mn Cr
Cl S, Cl, Ar
Si S P Ne, Na, Mg, Al, Si,
P, S, Cl, Ar, Ca,
Ti
Al Al, S, Ca Na, Mg, Al, P, S, Cl,
Ti
Co Cr, Fe, Co
K K, Ca
Cu Ni, Cu, Zn
Co Co
V Ti
S Ca, Ti
P Ca
Mn Cr
V Ti
Ti Ti
Al Al
Mn Mn
Sc Ti
Co Fe, Co
Mn Mn
Mg Mg Na, Mg, Al, Si, S , Ca,
Ti
Cl Cl
Mg Al Ne, Mg, S
Sc Ca
Na Sc
Fe Fe
Sc Ca, Sc, Ti
Zn Cu
Cu Cu, Zn
Sc Ca
Cl Cl
Ne Ne
F N
C Ne
Ga Cu, Zn
Ga Zn
Table 7: Sensitivity of the nucleosynthesis to the rate of reactions with parent nuclide given in the first column. 999The reactions listed are those that processed more than  M in the reference model (see Table 3) and with any .

Radiative captures of protons are by far the group of reactions whose rate most strongly determine the final abundances of the supernova explosion, as can be deduced from Table 9. The reaction with the largest of the whole network is , for which as many as 20 product species have . The species most affected by changes in the rates of the proton capture group of reactions are N, Mg, Al, Al, P, S, and Ca. Among these, Al is the species with the largest yield,  M. Within the species with there are important products of the supernova explosion such as Mg, Mg, Si, Si, P, S, S, Cl, Ar, Cr, Cr, and Fe. The parent nuclei involved in these reactions cover a wide range from C to Ga.

Parent nuclide Nuclei with Nuclei with
Mn Mn
Co Co, Ni, Zn
Si Mg, Al, Si, P, S
P P
Fe Mn, Fe
Co Co
V V
Ti Ti
Al Al
Mn Mn
Sc Ti
Mn Mn
Sc Sc
Cl Cl
Cu Cu, Zn
Mg Al
Ne N
S S
Table 8: Sensitivity of the nucleosynthesis to the rate of reactions with parent nuclide given in the first column. 111111The reactions listed are those that processed more than  M in the reference model (see Table 3) and with any .

The rates of