MNRAS 000, 1–11 (2020) Preprint 23 April 2020 Compiled using MNRAS LATEX style file v3.0

The accuracy of post-processed nucleosynthesis

Eduardo Bravo?E.T.S. Arquitectura del Vallès, Universitat Politècnica de Catalunya, Carrer Pere Serra 1-15, 08173 Sant Cugat del Vallès, Spain

Accepted 2020 March 27. Received 2020 March 26; in original form 2020 January 14

ABSTRACTThe computational requirements posed by multi-dimensional simulations of type Ia su-pernovae make it difficult to incorporate complex nuclear networks to follow the releaseof nuclear energy along with the propagation of the flame. Instead, these codes usuallymodel the flame and use simplified nuclear kinetics, with the goal of determining asufficiently accurate rate of nuclear energy generation and, afterwards, post-processingthe thermodynamic trajectories with a large nuclear network to obtain more reliablenuclear yields. In this work, I study the performance of simplified nuclear networkswith respect to reproduction of the nuclear yields obtained with a one-dimensionalsupernova code equipped with a large nuclear network. I start by defining a strategyto follow the properties of matter in nuclear statistical equilibrium (NSE). I proposeto use published tables of NSE properties, together with a careful interpolation rou-tine. Short networks (iso7 and 13α) are able to give an accurate yield of 56Ni, afterpost-processing, but can fail by order of magnitude in predicting the ejected mass ofeven mildly abundant species (> 10−3 M�). A network of 21 species reproduces thenucleosynthesis of the Chandrasekhar and sub-Chandrasekhar explosions studied herewith average errors better than 20% for the whole set of stable elements and isotopesfollowed in the models.

Key words: nuclear reactions, nucleosynthesis, abundances – supernovae: general –white dwarfs

1 INTRODUCTION

Type Ia supernovae (SNIa) are the result of the thermonu-clear disruption of carbon-oxygen white dwarf (WD) stars,due to their destabilization by a companion star in a bi-nary or a ternary system (Hillebrandt & Niemeyer 2000;Howell 2011; Katz & Dong 2012; Maoz et al. 2014). Thespectra and light curves of SNIa can be recorded startingshortly after the explosion (Zheng et al. 2013; Hosseinzadehet al. 2017; Miller et al. 2018) and continuing until a fewyears later (Graur et al. 2018; Jacobson-Galán et al. 2018;Maguire et al. 2018), if the luminosity and the distance tothe event allow it. Besides providing valuable insights intothe systematics of SNIa (Branch et al. 1993; Filippenko 1997;Poznanski et al. 2002; Howell et al. 2005; James et al. 2006;Arsenijevic et al. 2008; Branch et al. 2009, to cite only afew), these data allow researchers to constrain the propertiesof the explosion, from fundamental parameters, such as theejected mass, explosion energy, and synthesized mass of 56Ni(Stritzinger et al. 2006; Scalzo et al. 2014), to second-orderdetails, such as the ejected amounts of other radioactive iso-topes, notably 57Ni and 55Fe (Graur et al. 2016; Dimitriadiset al. 2017; Shappee et al. 2017; Yang et al. 2018; Li et al.

? E-mail: eduardo.bravo@upc.edu

2019). Whereas observational data are usually reported to-gether with their error bars, there is scarce knowledge of theeffects of the diverse sources of uncertainty related to su-pernova models. Indeed, Bravo & Mart́ınez-Pinedo (2012);Parikh et al. (2013); Bravo (2019) proved that the nucle-osynthesis of one-dimensional SNIa models is robust withrespect to variations in individual reaction rates, at leastduring the explosion phase.

Currently, two competing models may account for thebulk of SNIa. In one model, the exploding WD is close to theChandrasekhar-mass limit and the disruption is total; there-fore, the mass of the ejecta is fixed a priori. The most ac-cepted explosion mechanism of massive WDs is delayed det-onation (DDT, Khokhlov 1991; Poludnenko et al. 2019), inwhich the thermonuclear burning wave propagates subsoni-cally at first and turns into a detonation later, typically oneor two seconds after thermal runaway. In the other model,the mass of the exploding WD is substantially less than theChandrasekhar limit, the burning wave propagates as a det-onation from the very beginning, and the instability leadingto the detonation may be a consequence of the burning of athin helium layer, accumulated on top of the WD after accre-tion from a secondary star, or it may be due to the mergingor collision with another degenerate star (e.g. a second WD;

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Woosley & Weaver 1994; Woosley & Kasen 2010; Sim et al.2010; Kushnir et al. 2013; Shen et al. 2018).

Although there is not consensus about the level of asym-metry involved in SNIa explosions, most polarization mea-surements are indicative of small deviations from sphericalsymmetry (Wang et al. 1996; Howell et al. 2001; Maundet al. 2010; Maeda et al. 2010a; Maund et al. 2013). One-dimensional models are able to account for the properties ofSNIa in the visible (Höflich & Khokhlov 1996; Nugent et al.1997; Tanaka et al. 2011; Blondin et al. 2013; Hoeflich et al.2017) and gamma bands (Churazov et al. 2014, 2015, seeDiehl et al. 2014; Isern et al. 2016 for a different view), andthose of their remnants in the X-ray and radio bands (e.g.Badenes et al. 2006; Lopez et al. 2011; Mart́ınez-Rodŕıguezet al. 2018). However, to understand SNIa it is necessaryto simulate the explosion in three dimensions, either to ac-count for hydrodynamical instabilities and turbulence (e.g.Plewa et al. 2004; Röpke et al. 2006; Bravo & Garćıa-Senz2006; Kasen & Woosley 2007), or to address the inherentasymmetrical configuration of colliding or merging WDs.

The computational requirements posed by three-dimensional hydrodynamics make it difficult to incorporatecomplex nuclear networks to follow the release of nuclearenergy along with the propagation of the flame. Usually,multi-dimensional supernova codes need to model the flameby making use of simplified nuclear kinetics, with the goalsof giving an accurate rate of nuclear energy generation andcomputing the explosion in a reasonable time. Afterwards,the thermodynamic trajectories of the integration nodes canbe post-processed with the aid of a large nuclear networkto obtain more reliable values of the supernova yields (e.g.Thielemann et al. 1986; Bravo et al. 2010; Townsley et al.2016; Leung & Nomoto 2017). Although, recently, severalgroups have designed algorithms to incorporate large nu-clear networks into multi-dimensional models of SNIa (Pa-patheodore 2015; Kushnir & Katz 2019), most research stillfollows the post-processing approach.

In this work, I address the question of the performanceof simplified nuclear networks with respect to the reproduc-tion of the correct nuclear yields1. After a brief explanationof the methodology used (Sect. 2), the first point I treatis the definition of a strategy to follow the properties ofmatter in a state of nuclear statistical equilibrium (NSE,Sect. 3). In the following section, I test several simplifiednuclear networks for the accuracy of the nucleosynthesis ob-tained through post-processing (Sect. 4). Two more sectionsaddress the performance of simplified nuclear networks forhigh-metallicity WD progenitors and the effect of differentcriteria for switching on and off NSE routines. A final sectionis dedicated to the conclusions of the present work.

2 METHODOLOGY

The simulations presented in this work are performed withthe same supernova code described in Bravo et al. (2019),where extensive details of the method of computation canbe found. The code integrates the hydrodynamic evolution

1 In the present work, by correct nuclear yields I mean those ob-tained with the same supernova code without the simplifications

described in the text.

using a large nuclear network, solves the Saha equationsfor NSE, when applicable, and calculates the neutroniza-tion rate and associated neutrino energy losses at each timestep, computing the weak interaction rates using the NSEcomposition. The hydrodynamics is followed in one dimen-sion, assuming spherical symmetry, but the nucleosyntheticprocesses through which matter passes are qualitatively2 thesame as in any three-dimensional SNIa simulation, hence itis appropriate to assess the accuracy of the computed nucle-osynthesis. Furthermore, the supernova models generated bythis code have been compared successfully with observed op-tical spectra (S. Blondin, private communication), gamma-ray emission from SN2014J (Churazov et al. 2014, 2015; Is-ern et al. 2016) and the X-ray spectra of supernova remnants(Badenes et al. 2006).

The default nuclear network used here is the same asin Bravo & Mart́ınez-Pinedo (2012) and can be found incolumn BM-P in Table 1. This network includes all stableisotopes up to molybdenum, but the code allows to definedifferent nuclear networks by simply listing the species tobe followed. The nuclear reactions linking these species areincluded automatically in the network, together with a basicset formed by the fusion reactions triple-α, 12C+12 C, 16O+16

O, and 12C+16 O. All reaction rates are taken from the JINAREACLIB3 compilation (Cyburt et al. 2010), in the versionof November 6, 2008.

As references, I have selected two explosion models suit-able for normal-luminosity SNIa, characterized by an ejectedmass of 56Ni about M(56Ni) ∼ 0.5−0.7 M�. The first model,sub-MCh, is a central detonation of a sub-ChandrasekharWD of mass MWD = 1.06 M� (model 1p06 Z9e-3 std in Bravoet al. 2019). The second one, MCh, is a delayed detonationof a massive WD with central density ρc = 3× 109 g cm−3and a deflagration-to-detonation transition density ρDDT =2.4 × 107 g cm−3 (model ddt2p4 Z9e-3 std in Bravo et al.2019). Both models assume an initial composition made ofequal masses of 12C and 16O, contaminated with 22Ne as ap-propriate for progenitor metallicity 0.009 and other metalsfrom sodium to indium in solar proportions with respect to22Ne.

As a first test of convergence of the supernova codewith respect to the size of the nuclear network, I have re-computed models MCh and sub-MCh with two different nu-clear networks, both of them sufficiently large to give ac-curate nuclear energy generation rates. The first network(netAKh) is that employed in the SNIa models computed byAlexei Khokhlov and reported in Blondin et al. (2013, 2017).It uses 144 isotopes, including all stable isotopes betweenneon and copper, with the exception of 48Ca. The secondnetwork (nse7) includes 260 isotopes and was introduced byKushnir (2019) and used in their study of the structure ofdetonation waves in SNIa. It includes all stable isotopes upto arsenic, with the exception of 76Ge. Table 1 shows bothnetworks.

To assess the accuracy of the nucleosynthetic yields Iuse a set of fourteen indicators, all of them expressed asper cent relative differences between the quantities obtained

2 The precise thermodynamic histories of mass shells may be a

function of the dimensionality of the model.3 http://groups.nscl.msu.edu/jina/reaclib/db/.

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Post-processing accuracy 3

Table 1. Nuclear networks for the convergence study.

BM-P netAKh nse7Z Amin −Amax Amin −Amax Amin −Amaxn 1 - 1 1 - 1 1 - 1H 1 - 4 1 - 1 1 - 3

He 3 - 9 4 - 4 3 - 6Li 4 - 11 - 6 - 7

Be 6 - 14 - 7 - 10

B 7 - 17 - 10 - 11C 8 - 20 12 - 13 11 - 14

N 10 - 21 13 - 13 13 - 15

O 12 - 23 16 - 16 15 - 18F 14 - 25 - 17 - 19

Ne 16 - 27 20 - 22 19 - 23

Na 18 - 34 23 - 23 21 - 25Mg 20 - 35 23 - 26 23 - 28

Al 22 - 36 27 - 27 25 - 30

Si 24 - 38 27 - 32 27 - 33P 26 - 40 30 - 33 29 - 35

S 28 - 42 31 - 36 30 - 37Cl 30 - 44 35 - 37 32 - 39

Ar 32 - 46 36 - 41 34 - 42

K 34 - 49 39 - 43 37 - 45Ca 36 - 51 40 - 46 38 - 48

Sc 38 - 52 41 - 47 41 - 51

Ti 40 - 54 43 - 50 43 - 53V 42 - 56 45 - 52 45 - 55

Cr 44 - 58 47 - 56 47 - 57

Mn 46 - 60 49 - 60 49 - 59Fe 49 - 63 51 - 62 50 - 62

Co 51 - 65 53 - 61 52 - 64

Ni 53 - 69 56 - 64 54 - 66Cu 55 - 71 57 - 65 56 - 68

Zn 57 - 78 59 - 66 58 - 70

Ga 61 - 81 - 61 - 72Ge 63 - 83 - 64 - 74

As 65 - 85 - 69 - 75Se 67 - 87 - 75 - 75

Br 69 - 90 - -

Kr 71 - 93 - -Rb 73 - 99 - -

Sr 77 - 100 - -

Y 79 - 101 - -Zr 81 - 101 - -

Nb 85 - 101 - -

Mo 87 - 101 - -Tc 89 - 101 - -Ru 91 - 101 - -

Rh 93 - 101 - -Pd 95 - 101 - -

Ag 97 - 101 - -Cd 99 - 101 - -

In 101 - 101 - -

in a test model with respect to the results in the refer-ence hydrodynamic model that uses the default reactionnetwork. Therefore, I compare sub-Chandrasekhar modelswith model 1p06 Z9e-3 std, and Chandrasekhar-mass mod-els with ddt2p4 Z9e-3 std. The indicators are as follows:

• the discrepancy of the final kinetic energy, a1 = ∆K/K,• the discrepancy of the ejected mass of 56Ni,

a2 = ∆M(56Ni)/M(56Ni),• the discrepancy of the ejected mass of 57Ni ,

b1 = ∆M(57Ni)/M(57Ni),

• the discrepancy of the ejected mass of 55Fe,b2 = ∆M(55Fe)/M(55Fe),• a measure of the discrepancy, c1, based on the average of

the squared deviations of the logarithm of the ejected massof the elements,

σlog,ele =

√1

Nele

∑log2[

M′(Z)M(Z)

], (1)

c1 = 10σlog,ele −1 , (2)

• a measure of the discrepancy, c2, based on the average ofthe squared deviations of the logarithm of the ejected massof the isotopes,

σlog,iso =

√1

Niso

∑log2[

M′(AZ)M(AZ)

], (3)

c2 = 10σlog,iso −1 , (4)

• a measure of the discrepancy, c3, based on a weightedaverage of the squared deviations of the logarithm of theejected mass of the elements,

σwm,ele =

√∑ω(Z) log2

[M′(Z)M(Z)

], (5)

c3 = 10σwm,ele −1 , (6)

• a measure of the discrepancy, c4, based on a weightedaverage of the squared deviations of the logarithm of theejected mass of the isotopes,

σwm,iso =

√∑ω(AZ) log2

[M′(AZ)M(AZ)

], (7)

c4 = 10σwm,iso −1 , (8)

• the maximum relative discrepancy, d1, in the mass ofthe elements with final yield M(Z) > 10−3 M�,• the maximum relative discrepancy, d2, in the mass of the

elements with final yield in the range 10−3 > M(Z)> 10−6 M�,• the maximum relative discrepancy, d3, in the mass of the

elements with final yield in the range 10−6 > M(Z)> 10−12 M�,• the maximum relative discrepancy, e1, in the mass of

the isotopes with final yield M(AZ) > 10−3 M�,• the maximum relative discrepancy, e2, in the mass of the

isotopes with final yield in the range 10−3 > M(AZ)> 10−6 M�,and• the maximum relative discrepancy, e3, in the mass of

the isotopes with final yield in the range 10−6 > M(AZ) >10−12 M�.

The quantities M(Z) and M(AZ) are the masses, in M�, ofelement Z and isotope AZ in the reference model, M′ standsfor the same quantities for the test model, Nele and Niso are,respectively, the number of different elements and isotopesejected and the weighting functions are defined as:

ω(Z) =1[

log2 M(Z)]×∑[1/ log2 M(Z)] , (9)

ω(AZ) =1[

log2 M(AZ)]×∑[1/ log2 M(AZ)] . (10)

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4 Bravo

Table 2. Nucleosynthetic indicators.

a1 a2 b1 b2 c1 c2 c3 c4 d1 d2 d3 e1 e2 e3(%) (%) (%) (%) (%) (%) (%) (%) (%) (%) (%) (%) (%) (%)

Convergence study

Network ModelnetAKh MCh 0.3 0.2 0.4 0.0 1.5 1.5 0.3 0.4 1.5 6.1 1.3 1.5 3.9 6.1nse7 MCh 0.0 0.0 0.0 0.0 0.3 0.3 0.0 0.0 0.2 0.7 1.4 0.2 0.5 1.4netAKh sub-MCh 0.3 0.0 0.3 0.3 0.7 0.8 0.2 0.2 1.2 1.8 0.8 1.2 1.8 1.6nse7 sub-MCh 0.0 0.0 0.0 0.0 0.3 0.3 0.1 0.1 0.0 1.4 0.3 0.0 1.3 1.4

NSE table interpolationInterpolator Model

linlog MCh 0.2 0.6 11 7.6 4.3 23 0.9 4.4 7.4 12 0.8 22 100 360linlin MCh 0.0 0.0 0.1 0.0 1.0 8.3 0.2 1.5 0.9 4.7 0.3 8.2 32 82loglog MCh 0.2 0.6 13 7.1 4.3 23 0.9 4.4 7.4 12 0.8 22 101 355poly MCh 0.1 0.5 13 7.4 4.8 28 0.9 5.0 7.9 13 1.9 24 120 490spline MCh 0.0 0.0 0.1 0.0 0.1 0.2 0.0 0.1 0.1 0.2 0.4 0.6 0.7 0.8peνn MCh 0.0 0.1 0.3 0.1 0.2 0.6 0.1 0.1 0.3 0.3 1.4 0.8 1.1 4.2den×10 MCh 0.0 0.1 0.4 0.0 0.2 0.5 0.1 0.1 0.3 0.3 0.6 0.8 1.0 3.1Simplified networks

Network Model

iso7 MCh 0.5 9.4 86 29 58 120 12 23 64 620 14 310 1500 590013α MCh 3.7 4.3 70 27 43 93 7.4 17 49 380 8.8 220 910 3100net21 MCh 4.3 3.2 6.4 6.5 8.9 8.4 3.5 4.1 16 17 12 17 18 14iso7 sub-MCh 0.8 6.1 26 31 33 43 11 15 52 140 9.8 92 170 32013α sub-MCh 1.2 3.9 21 27 23 30 7.5 11 40 93 7.4 66 110 180net21 sub-MCh 1.4 1.1 0.0 2.4 3.6 3.7 1.4 1.6 5.0 9.4 4.0 5.0 6.1 9.5

High metallicity progenitorNetwork Model

net21 MCh 3.8 4.5 12 4.4 11 12 3.7 7.4 19 19 17 21 20 31net23 MCh 6.0 2.5 2.8 9.0 15 18 4.2 7.6 28 29 22 28 69 26net21 sub-MCh 2.4 5.9 9.4 6.2 16 18 6.8 9.8 20 29 22 25 30 32net23 sub-MCh 2.9 0.1 1.0 4.5 5.5 6.5 1.7 3.1 7.6 11 7.0 10 13 10

Transition NSE � net21Condition Model

TNSE = 5.5×109 K MCh 4.3 3.2 6.4 6.5 8.9 8.4 3.5 4.1 16 17 12 17 18 14TNSE = 5.0×109 K MCh 3.5 1.6 10 7.0 5.2 5.3 1.9 2.4 8.4 7.9 8.3 9.6 14 8.3Tout = 4×109 K MCh 3.3 0.5 14 8.0 4.3 5.1 1.2 1.9 9.7 9.7 4.8 12 20 16Tout = 3×109 K MCh 3.3 1.1 1.4 1.6 19 40 2.9 8.9 21 100 5.1 70 260 530ρNSE0 = 108 g cm−3 MCh 4.4 3.3 4.1 4.6 8.8 8.3 3.6 4.2 17 17 12 17 18 15ρNSE0 = 4×107 g cm−3 MCh 4.3 3.2 6.7 6.3 8.9 8.4 3.5 4.1 17 17 12 17 19 14

With these definitions, indicators c3 and c4 provide a mea-sure of the mean deviation of the most abundant species,while indicators c1 and c2 give a measure of the deviation ofthe yields of all species. Indicators a2, b1, and b2 are eval-uated from the ejected masses of the isotopes 100 s afterthermal runaway. All the indicators from c1 to e3 refer tothe elemental or isotopic yields of isotopes between carbonand krypton after radioactive decays.

For reference, the kinetic energy and masses of radioac-tive isotopes ejected in the two reference models are thefollowing: in model ddt2p4 Z9e-3 std, K = 1.42 × 1051 erg,M(56Ni) = 0.685 M�, M(57Ni) = 7.15×10−3 M� and M(55Fe) =9.65×10−3 M�; in model 1p06 Z9e-3 std, K = 1.32×1051 erg,M(56Ni) = 0.664 M�, M(57Ni) = 1.01×10−2 M� and M(55Fe) =3.21×10−3 M�.

The thermodynamic trajectories obtained with the ne-tAKh and nse7 networks have been fed to a post-processingnuclear code that uses the same network and reaction rates

as in the BM-P network4. Table 2 show the results in therows under the header “Convergence study“. The agreementwith the results of the hydrodynamic calculation using thedefault network is very satisfactory. The direct measure ofthe nuclear energy released, that is the final kinetic energy, isreproduced in all four calculations to better than 0.3%. Theyields of the radioactive isotopes are reproduced to within0.4% with both networks and, in particular, the yield of 56Nito better than 0.2%.

The nucleosynthesis of stable isotopes and elements alsoconverges, where both netAKh and nse7 obtain similar rat-ings. The mean deviation of the most abundant elements andisotopes is 6 0.4%, while that representative of all ejectedspecies lies in the range 0.3-1.5%. Finally, the maximum de-viation of elements and isotopes whose yield is larger than10−3 M� is 6 1.5%, while that of the remaining elements andisotopes with yields > 10−12 M� is less than ∼ 6%.

4 The same strategy has been applied to all the calculations pre-

sented in the following sections.

MNRAS 000, 1–11 (2020)

Post-processing accuracy 5

3 NUCLEAR STATISTICAL EQUILIBRIUM

One of the key ingredients of simulations of SNIa explosionsis the treatment of NSE in matter burnt at high density. Thecomposition of matter in NSE can be calculated by solvinga set of Saha equilibrium equations linking the abundancesof all isotopes to two arbitrarily chosen abundances (or com-binations thereof), which play the role of independent vari-ables, plus two closure relationships that account for theconservation of baryon number and the electrical neutralityof matter. The procedure is usually iterative, which makesit inefficient for a multi-dimensional hydrodynamic compu-tation of a supernova explosion. Therefore, in this sort ofsimulation, it is usual to rely on interpolation of a table ofNSE states, pre-computed on a net of density, ρ, tempera-ture, T , and electron mole number, Ye, nodes, the denser thebetter. Usually, the table gives the main properties of mat-ter in NSE, including nuclear binding energy, mean molarnumber, neutronization rate and neutrino energy loss rate.

One example of this kind of table of NSE properties isgiven in Seitenzahl et al. (2009). Recently, Bravo et al. (2019)reported that the final yields computed using their NSE ta-ble might disagree by order of magnitude for some isotopeswith respect to those obtained computing the NSE stateproperties on the fly in the hydrodynamical calculation. Thediscrepancy did not affect significantly either the total en-ergy release, i.e. the final kinetic energy, or the ejected massof 56Ni, and was attributed to the interpolation procedureapplied to obtain the NSE properties out of the ρ, T , and Yetable nodes.

Here, I argue that the culprit for the discrepancy justmentioned is relying on the interpolation at the densitynodes. To illustrate the situation, Fig. 1 shows the neutrinoenergy loss-rate, εν in a sample of the Seitenzahl et al. (2009)table nodes of ρ, T , and Ye for typical values during a super-nova explosion. While the energy-loss rate changes smoothlybetween consecutive temperature and electron mole num-ber nodes, the dependence on density is more complex andthe values of εν change by four orders of magnitude be-tween ρ node numbers 12 and 20 (ρ = 2× 108 g cm−3 andρ = 2 × 1010 g cm−3). This huge change makes the resultsof the interpolation in density sensitive to the interpolationprocedure. Indeed, the plot of εν versus density in betweennodes 12 and 20 in Fig. 1 is suggestive of a linear dependencebetween logεν and logρ (the table nodes are equispaced inlogρ).

To test the impact of the NSE table interpolationscheme I have chosen different interpolants and computedthe difference in NSE properties with respect to those ob-tained by solving the NSE Saha equilibrium equations. Theinterpolants used and the designations given are the follow-ing:

• linear interpolation of NSE properties, for example ενor the neutronization rate Ẏe, with respect to logρ (linlog);• linear interpolation with respect to ρ (linlin);• linear interpolation of log Ẏe and logεν with respect to

logρ (loglog);• third order polynomial of the NSE properties, for ex-

ample Ẏe, with respect to logρ (poly);• cubic spline fitting the NSE properties with respect to

ρ (spline);• a physically motivated interpolation function (peνn).

1x1016

1x1017

1x1018

1x1019

1x1020

1x1021

0 5 10 15 20 25 30

εν (

erg

g-1

s-1

)

interpolation node number

εν vs ρ

εν vs T

εν vs Ye

Figure 1. Sample of variation of the neutrino energy loss-rate as

function of density in the range 2×105 −2×1010 g cm−3 (at fixedT = 9×109 K and Ye = 0.5 mol g−1, red dots), temperature in therange 3× 109 − 1.05× 1010 K (at fixed ρ = 2× 109 g cm−3 and Ye =0.5 mol g−1, green downward triangles) and electron mole numberin the range 0.4400−0.5025 mol g−1 (at fixed ρ= 2×109 g cm−3 andT = 9×109 K, blue upward triangles) in the interpolation nodes ofthe NSE table.

The last interpolator is motivated by the dominant role ofprotons in the neutronization rate of NSE matter in SNIamodels (Fuller et al. 1985; Brachwitz et al. 2000; Bravo2019). Hence it seems natural to interpolate using the samefunction that describes the dependence of the p(e−,ν)n rateor the associated neutrino energy emission rate on den-sity (Fuller et al. 1985). The effective log( f t)-values char-acterizing electron captures by protons are almost constant,whereas the rate dependence on ρ is given by the so-calledmodified phase space factor (Eqs. 3 and 6 in Fuller et al.1985), Ie. The neutrino energy emission rate depends on ρthrough the appropriate phase-space factor (Eq. 7 in Fulleret al. 1985), Jνe . Instead of computing the relativistic Fermiintegrals that appear in the definition of these space factors,I calculate approximate values taking advantage of Eqs. 15in Fuller et al. (1985).

Figures 2 and 3 show the relative error between Ẏe ob-tained using the different interpolants and the exact neu-tronization rate obtained solving the NSE Saha equilibriumequations. The error goes to zero at the table nodes, butcan reach up to 1, i.e. 100% error, at low densities and hightemperatures. The dashed lines show the most relevant com-bination of density and temperature for the SNIa MCh modelin two conditions: when electron captures start in NSE andthe neutronization rate is maximum (left columns) and whena value of Ye = 0.47 mol g−1 is attained, a condition that isonly reached in the innermost ∼ 0.15 M� of the WD (rightcolumns). The WD mass shells go through ρ-T conditionsfor which the relative error in Ẏe is as high as 5− 20 % forthe most simple interpolators: linlog, linlin, loglog and poly.On the other hand, the upper bound on the maximum errorin the same conditions is ∼ 3−5 % when either the spline orthe peνn interpolator is used.

Table 2 shows the impact of the different interpolatorson the final yields of the MCh model, under the heading ”NSEtable interpolation“. The nucleosynthetic results confirm the

MNRAS 000, 1–11 (2020)

6 Bravo

Ye = 0.50 Ye = 0.47

5×109

6×109

7×109

8×109

9×109

0.001 0.01 0.1 1relative error 0.001 0.01 0.1 1relative error

5×109

6×109

7×109

8×109

9×109

T (

K)

0.001 0.01 0.1 1relative error 0.001 0.01 0.1 1relative error

7.8 8 8.2 8.4 8.6 8.8 9 9.2 9.4 9.6

5×109

6×109

7×109

8×109

9×109

0.001 0.01 0.1 1relative error

log ρ (g cm-3

)

7.8 8 8.2 8.4 8.6 8.8 9 9.2 9.4 9.6

0.001 0.01 0.1 1relative error

Figure 2. Relative error between the exact neutronization rate, Ẏe, and that computed by interpolation on a NSE table, for two valuesof the electron mole number, Ye = 0.50 mol g−1 (left column) and Ye = 0.47 mol g−1 (right column). The results are shown for differentinterpolation schemes, from top to bottom: linear interpolation of Ẏe versus logρ, linear interpolation of Ẏe versus ρ and linear interpolationof log Ẏe versus logρ. The relative error is colour coded according to the colour bar at the top of the plot. The dashed lines show thedensity and temperature at which the mass shells of model MCh start experiencing electron captures in NSE (left column) and at thetime they reach an electron mole number Ye = 0.47 mol g−1 (right column).

intuition gained with Figs. 2 and 3: the most accurate inter-polators are the cubic spline and peνn, which perform almostequally well. All interpolants lead to accurate values of thefinal kinetic energy and just negligible errors in the mass of56Ni synthesized. The relevant differences appear when onelooks into the nucleosynthesis of less abundant species. Forinstance, using linlog, loglog and poly leads to errors in theejected mass of 57Ni and 55Fe of about 10%, average errorsin the isotopic yields of 20−30% (indicator c2), and order-of-magnitude errors in the yields of some isotopes with yieldssmaller than 10−3 M� (indicators e2 and e3). On the otherhand, the maximum isotopic error obtained with the cubicspline interpolator is 0.8 % and that obtained with peνn is4.2 %. The behaviour of interpolator linlin is intermediatebetween the two groups above.

Table 2 also shows the result of increasing the resolu-tion of the NSE table by up to ten times more density nodes(”den×10“), with ∆ logρ = 0.025. With a table of this size,even using a linlog interpolant gives very good results, com-parable to both the spline and the peνn interpolators withthe original table with ∆ logρ = 0.25.

In the calculations reported in the following sections, Iwork with simplified nuclear networks and use a table to ob-tain the properties of NSE matter, with the aim of testing asfaithfully as possible the strategies commonly used in manySNIa explosion models. In these tests, it is important thatthe treatment of NSE matter does not introduce additionalerrors, beside those attributable to simplified networks. Forthis purpose, I use the cubic spline interpolator and the NSEtable with ∆ logρ = 0.25.

4 SIMPLIFIED NUCLEAR NETWORKS

Because of the huge difference between the size of a WDand the width of thermonuclear burning waves, multi-dimensional simulations of SNIa need to allocate most mem-ory and CPU resources to solve the hydrodynamic equationsover as large a range of length-scales as possible (e.g. Gamezoet al. 2003). In turn, the nuclear kinetics must be solvedwith a reduced nuclear network, with the goal that the nu-clear energy must be released as faithfully as possible, as if

MNRAS 000, 1–11 (2020)

Post-processing accuracy 7

Ye = 0.50 Ye = 0.47

5×109

6×109

7×109

8×109

9×109

0.001 0.01 0.1 1relative error 0.001 0.01 0.1 1relative error

5×109

6×109

7×109

8×109

9×109

T (

K)

0.001 0.01 0.1 1relative error 0.001 0.01 0.1 1relative error

7.8 8 8.2 8.4 8.6 8.8 9 9.2 9.4 9.6

5×109

6×109

7×109

8×109

9×109

0.001 0.01 0.1 1relative error

log ρ (g cm-3

)

7.8 8 8.2 8.4 8.6 8.8 9 9.2 9.4 9.6

0.001 0.01 0.1 1relative error

Figure 3. Same as Fig. 2 but for different interpolation schemes, from top to bottom: third order polynomial of Ẏe versus logρ, cubicspline of Ẏe versus ρ and peνn.

Table 3. Simplified nuclear networks.

iso7 13α net21 net23Z A A A A

n - - 1 1p - - 1 1

He 4 4 4 4

C 12 12 12 12O 16 16 16 16

Ne 20 20 20 20,22Mg 24 24 24 24,25Si 28 28 28 28

S - 32 32 32

Ar - 36 36 36Ca - 40 40 40

Ti - 44 44 44Cr - 48 48 48Fe - 52 52,53,54, 52,53,54

55,56 55,56Co - - 55,56 55,56

Ni 56 56 56 56

a complete nuclear network were used. Other compositionalproperties that affect the equation of state, such as the elec-tron mole number and the mean molar weight, must also be

reproduced accurately in order not to change the explosiondevelopment. In this section, I show the impact of the useof reduced nuclear networks on the nucleosynthesis of boththe MCh and the sub-MCh models in one dimension.

First, I test two small networks, designed iso7 and 13α,that are widely used in multi-dimensional simulations ofSNIa, plus a slightly larger network designed to improvethe nucleosynthetic results, named net21. Table 3 shows thecomposition of each network. In these calculations, all nu-clear reactions linking species present in the network directlyare accounted for and the rates are taken from the REACLIBcompilation (see Sect. 2) unless otherwise stated here.

The iso7 network, introduced by Timmes et al. (2000)as a simplification of the nine-isotope reaction network de-scribed in detail in Table 1 of Woosley (1986), has beenused in 2D simulations of DDT models of exploding WDs(e.g. Leung & Nomoto 2018). This network is crafted forefficient computation of nuclear energy generation in multi-dimensional calculations of explosive burning stages fromcarbon-burning onwards. It assumes two quasi-equilibriumgroups of isotopes, the silicon group and the iron group, andhard-wires the nucleosynthetic flows between both groupsin a single step that link the abundances of 28Si and 56Ni. This step is computed making use of Eqs. 6-8 of Timmes

MNRAS 000, 1–11 (2020)

8 Bravo

Figure 4. Accuracy of the nucleosynthesis obtained with the net21 network as compared to model ddt2p4 Z6p75e-2 std. Top: final

ejected mass of stable isotopes in the reference model. Bottom: percentage error of the mass yield of each isotope when the simplified

network is used, with respect to the reference model.

et al. (2000). Timmes et al. (2000) warned that the iso7and 13α networks (and, in general, any α-network) mightgive energy generation rates wrong by order of magnitude ifYe . 0.49 mol g−1.

The 13α network, introduced by Mueller (1986) andlater by Livne & Arnett (1995) as a simplification of a largernetwork described in Weaver et al. (1978), has been used inSPH simulations of merging and colliding WDs (Raskin et al.2014; Dan et al. 2015). The version of the 13α network usedin the present work follows Timmes (1999) and Timmes et al.(2000), where it is applied a special treatment to the linksbetween α-nuclei from magnesium onwards: above a temper-ature of 2.5×109 K, the flows from (α,p) reactions, followedby (p,γ), are added to the flows from (α,γ). It is importantto note that, in this version of the 13α network5, the linkfrom an α-nucleus, 2ZZ, to the next one, 2Z+4(Z+2), through(α,p) reactions takes into account the possibility that (p,α)follows instead of (p,γ):

2ZZ�2Z+3 (Z + 1)�2Z+4 (Z + 2) . (11)

For instance, in the conversion of 28Si into 32S through 31P,the four reactions that follow have to be considered besides28Si(α,γ)32S,

28Si�31 P�32 S . (12)

Assuming that the abundance of the intermediate nucleus,31P in this example, is established by the equilibrium of thedirect and reverse reactions in Eq. 11, the overall rate ofchange of the molar fraction of 2Z+4(Z+2), Y

[2Z+4(Z + 2)

],

5 See also http://cococubed.asu.edu/code pages/burn helium.shtml

from 2ZZ is given by

Reff(α,γ)Y[2ZZ]Yα −Reff(γ,α)Y

[2Z+4(Z + 2)

], (13)

where the effective rates, Reff(α,γ) and Reff(γ,α), are

Reff(α,γ) = R(α,γ) + R(α,p)R(p,γ)

R(p,α) + R(p,γ)(14)

and

Reff(γ,α) = R(γ,α) + R(γ,p)R(p,α)

R(p,α) + R(p,γ), (15)

and R(α,p), R(p,α), R(γ,p), and R(p,γ) are the true ratesof the reactions 2ZZ→2Z+3(Z+1), 2Z+3(Z+1)→2ZZ,2Z+4(Z+2)→2Z+3(Z+1), and 2Z+3(Z+1)→2Z+4(Z+2), re-spectively. As before, all true rates have been computed asin the REACLIB compilation.

In both networks, iso7 and 13α, the electron mole num-ber in the initial model is Ye = 0.5 mol g−1. The electronmole number is allowed to change during NSE, due to elec-tron captures, but is kept fixed when the composition iscomputed by integration of the nuclear network, because itdoes not include weak interactions.

Table 2 shows the accuracy of the nucleosynthesis ob-tained by post-processing the thermodynamic trajectoriesbelonging to these two networks, under the heading ”Simpli-fied networks“. Generally, the performance of the simplifiednetworks in the Mch models is worse than in the sub-MChmodels. The kinetic energy is reproduced reasonably withthe simplified networks, slightly better with iso7 than with13α, and the error in the yield of 56Ni is within 6-10% wheniso7 is used but ∼ 4% with 13α.

The errors in the yields of radioisotopes 55Fe and 57Ni,

MNRAS 000, 1–11 (2020)

Post-processing accuracy 9

which are often constrained observationally by the late-timelight curves of SNIa, are about 30%, with the exception of57Ni in MCh, the abundance of which is wrong by nearlyan order of magnitude. These errors are comparable withthe maximum deviation obtained in the yields of the mostabundant elements, d1 ∼ 40−60%, while the maximum errorin the abundance of the isotopes (indicators e1 to e3) maybe up to several orders of magnitude. Among those withthe largest yields, the isotopes that present the maximumdeviation are mostly part of the iron group: 40Ca, 52,53Cr,55Mn, 57Fe, and 60,62Ni.

The net21 network is an extension of the 13α networkthat includes additional isotopes of the iron group plus freeprotons and neutrons (its full composition can be seen inTable 3), in order to obtain a more reliable representationof the nucleosynthesis of the most deficient nuclides fromthe results of the iso7 and 13α networks. The net21 networkmakes use of the same hard-wiring of rates as in Eqs. 14 and15, with the exception of the chain

52Fe�55 Co�56 Ni . (16)

Since 55Co is included explicitly in the network, the effectiverates between 52Fe and 56Ni given by Eqs. 14 and 15 aresubstituted by the corresponding true rates, R(α,γ) and R(γ,α).

When using the net21 network in the hydrocode, theerrors in the kinetic energy and the yield of 56Ni after post-processing are about 1− 5%. On the other hand, the errorsin the yields of 55Fe and 57Ni are less than 7% and the max-imum error in the predicted abundance of any isotope orelement is less than 20% (indicators d1 to e3).

The errors in the post-processed nucleosynthesis afterusing the net21 network in the hydrocode are comparableto, although slightly larger than, those obtained with thelarge networks used in the convergence study, netAkh andnse7. Therefore, the performance of net21 would barely beimproved with other simplified networks based on no morethan a few tenths of nuclides. I have experimented withlarger networks, including a group of CNO isotopes (13,14C,14N, 17O) and the intermediate species in Eq. 11, from 27Alto 51Mn, with no significant improvement in the performanceover that of network net21. I have also probed reducing thenumber of iron-group isotopes in the network, but the accu-racy of the nucleosynthesis was worse than with net21.

5 NUCLEAR POST-PROCESSING FOR HIGHMETALLICITY PROGENITORS

In this section, I test the accuracy of the post-processednucleosynthesis with respect to the initial metallicity of theprogenitor star. Network net21, as well as iso7 and 13α, isnot capable to describe an initial composition of carbon-oxygen material with an excess of neutrons over protons.This is because the initial metallicity of the progenitor staris encoded, at the time of formation of a carbon-oxygen WD,in the abundance of 22Ne (Timmes et al. 2003), which isnot a part of any of the simplified networks discussed inthe previous section. Hence one may wonder whether theaccuracy of the net21 network degrades for high metallicityprogenitors.

Here, I introduce a new network, net23, that comple-ments the net21 network with the inclusion of 22Ne and

25Mg (Table 3). As just explained, the presence of 22Neserves the purpose of building initial models with non-zeroneutron-excess, as it is done in the hydrodynamic modelsthat use the full network (see Sect. 2). The isotope 25Mgprovides a simple route for the burning of 22Ne through22Ne(α,n)25Mg(α,n)28Si.

For this test, I have selected an initial metallicity ofthe WD as high as Z = 0.0675. Table 2 shows the resultsof the post-processed nucleosynthesis using networks net21and net23, for both the MCh and the sub-MCh models, underthe heading ”High metallicity progenitor“. Network net21performs slightly better in the MCh model, whereas net23does better in the sub-MCh model, but the overall accuracyof both networks is similar. When the nucleosynthesis errorsin the high metallicity calculations are compared with thosein the Z = 0.009 models, using the same networks, the resultsare slightly better at low metallicity, but not significantlydifferent.

Figure 4 shows the percentage error in the prediction ofthe abundances of the stable isotopes between carbon andkrypton after post-processing the thermodynamic trajecto-ries obtained with the net21 network, compared with thoseusing the default network in the hydrocode. The largest er-ror, 31%, belongs to 48Ca, the abundance of which is below10−6 M�, while all other isotopes are predicted with errorssmaller than ∼ 25%.

6 TRANSITION FROM THE NUCLEARNETWORK TO NSE AND VICE VERSA

The simultaneous inclusion in a simulation of a simplifiednetwork and an NSE routine, as described in Sect. 3, raisesthe question of the criteria for the transition between bothtreatments. The transition between the nuclear network andNSE is usually defined in terms of the temperature, wheredifferent values are used for assuming NSE, TNSE, and leav-ing it, Tout. In the hydrocode used in this work, a third pa-rameter, ρNSE0, allows acceleration of the burning of shellshit by a deflagration front.

Silicon exhaustion is a milestone for achieving NSE. Itis reached at a temperature somewhere in between ∼ 5 ×109 K and ∼ 6× 109 K with a slight dependence on density(e.g., Fig. 20 in Woosley et al. 1973). In the calculationspresented in this section, I have adopted a unified value ofTNSE, independent of matter density6, as detailed in Table 2.

The freezing-out of nuclear reactions when NSE mattercools is more complex, because at low densities the abun-dance of free particles (especially α particles) may be suf-ficiently large to affect the composition significantly. Harriset al. (2017) studied the impact of the value of Tout in thecontext of core-collapse supernovae. Usual values of the tem-perature at which matter is assumed to leave NSE lie in therange from ∼ 2× 109 K to ∼ 5× 109 K. Again, in the calcu-lations presented in this section, there is a single value ofTout, independent of density, at variance with the methodadopted in the hydrocode using the default network (Bravoet al. 2019).

6 In the hydrocode using the default network, there are two dif-ferent values of the minimum temperature to achieve NSE, de-

pending on density; see Bravo et al. (2019) for further details

MNRAS 000, 1–11 (2020)

10 Bravo

Following the passage of a deflagrative front, shells areassumed to achieve NSE if their density is larger than thethird parameter, ρNSE0. By default, I adopt conservative val-ues for the three parameters: TNSE = 6×109 K, Tout = 5×109 Kand ρNSE0 = 8×107 g cm−3.

As can be seen in Table 2, the precise value of TNSE doesnot affect the error in the post-processed nucleosynthesis ofthe MCh model significantly, as long as it is in the rangefrom ∼ 5× 109 K to ∼ 6× 109 K7. On the other hand, whenthe threshold for leaving NSE, Tout, takes on a value between4×109 K and 5×109 K the accuracy of the nucleosynthesis issatisfactory, but when this parameter goes down to 3×109 Kthe results worsen: for instance, the maximum error in thepredicted isotopic yields increases by orders of magnitude.Finally, the accuracy of the nucleosynthesis is not affectedby the value of ρNSE0, at least within the range explored inthis work and presented in Table 2, ρNSE0 = 4× 107 g cm−3to 108 g cm−3.

As explained in Appendix B4 of Bravo et al. (2019),detonated matter is not assumed to be in NSE if its den-sity is below ρNSE0, irrespective of its temperature. In prac-tice, it implies that, in the present models, freeze-out fromNSE occurs at low entropy. Therefore, the effect of the NSEparameters just described limits to the so-called normal orparticle-poor freeze-out. On the other hand, it is remarkablethat alpha-rich freeze-out is managed by integration of thenet21 network with very good accuracy, in spite of its smallsize.

7 CONCLUSIONS

Three-dimensional hydrodynamical simulations are neces-sary to predict the outcome of several explosion scenariosand compare them with SNIa observational data. Whereasgreat efforts have been made to obtain reliable nucleosyn-thetic yields and explore their dependence on a numberof simulation parameters (e.g. Maeda et al. 2010b; Seiten-zahl et al. 2013), until now very few works have been pub-lished addressing the accuracy of the resulting nucleosyn-thesis with respect to the use of simplified nuclear networks(Papatheodore 2015)8. In the present work, I use a super-nova code, capable of integrating the nuclear kinetic equa-tions using a large nuclear network and the hydrodynamicalequations simultaneously, as a benchmark for testing the re-sults of several simplifying assumptions related to nuclearkinetics. These simplifications are related to the use of asmall nuclear network and the use of tabulated properties ofmatter in nuclear statistical equilibrium. I define a set of 14indicators related to the accuracy of the nucleosynthesis.

The method used in this work does not allow testing ofall the strategies currently used in multi-dimensional sim-ulations of SNIa to follow the nuclear energy generation-rate accurately during a thermonuclear explosion. For in-stance, several studies (e.g. Travaglio et al. 2004; Dubeyet al. 2012; Leung & Nomoto 2017) use Lagrangian tracerparticles smartly distributed through the simulated space

7 Recall that the errors reported in this subsection have to becompared with the reference model, that is the net21 MCh modelunder the heading ”Simplified networks“.8 https://trace.tennessee.edu/utk graddiss/3454/

to advect the thermodynamic properties of the underlyingEulerian cells. Nucleosynthesis is then obtained following apost-processing step on the tracer particles, the number ofwhich is much less than the original Eulerian nodes of thesimulation. Other studies (e.g. Calder et al. 2007; Fink et al.2010; Townsley et al. 2016) adopt a nuclear energy genera-tion rate linked to the nature of the burning wave (whethera detonation or a deflagration) and the density of fuel.

I propose to use published tables of NSE properties(neutronization rate, mean molar number, mean nuclearbinding energy and neutrino energy loss-rate, as functionsof density, temperature and electron molar number) butputting great care into interpolation between their valuesat the tabulated density points. The interpolation in tem-perature and electron mole number is not so critical andcan simply be linear. A cubic spline interpolation in den-sity gives the most precise results, the accuracy of which isbetter than 1% in all 14 nucleosynthetic indicators. Alter-natively, a simple linear interpolation on an NSE table withhigh resolution in density, ∆ logρ = 0.025, may give almostas accurate results as the cubic spline interpolation of thestandard NSE table with ∆ logρ = 0.25.

I have tested several simplified nuclear networks forthe accuracy of the nucleosynthesis obtained after post-processing, compared with the nucleosynthesis resulting di-rectly from the supernova code when the default, large nu-clear network is used. Short networks (iso7 and 13α) are ableto give an accurate yield of 56Ni after post-processing, butcan fail by an order of magnitude in predicting the ejectedmass of even mildly abundant species (> 10−3 M�). I findthat a network of 21 species, net21, reproduces the nucle-osynthesis of Chandrasekhar and sub-Chandrasekhar explo-sions nicely, their average errors being better than 10% forthe most abundant elements and isotopes (yields larger than10−3 M�) and better than 20% for the whole set of stableelements and isotopes followed in the model.

In these explosion models, the NSE state is switched onand off according to three criteria based on local tempera-ture and density. The temperature at which it can be safelyassumed that matter will achieve NSE can adopt any valuebetween 5×109 K and 6×109 K without affecting the accu-racy of the post-processed nucleosynthesis significantly. Fornormal freeze-out of NSE, equilibrium abundances can be as-sumed for temperatures in excess of ∼ 4×109 K. However, ifthe NSE state is kept until a temperature as low as 3×109 K,the resulting post-processed nucleosynthesis can be almostan order of magnitude wrong even for isotopes with yield> 10−3 M�. On the other hand, the alpha-rich freeze-out ofNSE is managed satisfactorily by the net21 network, in spiteof its small size.

ACKNOWLEDGEMENTS

This work has been supported by the MINECO-FEDERgrants AYA2015-63588-P and PGC2018-095317-B-C21.

REFERENCES

Arsenijevic V., Fabbro S., Mourão A. M., Rica da Silva A. J.,2008, A&A, 492, 535

MNRAS 000, 1–11 (2020)

http://dx.doi.org/10.1051/0004-6361:200810675https://ui.adsabs.harvard.edu/abs/2008A&A...492..535A

Post-processing accuracy 11

Badenes C., Borkowski K. J., Hughes J. P., Hwang U., Bravo E.,

2006, ApJ, 645, 1373

Blondin S., Dessart L., Hillier D. J., Khokhlov A. M., 2013, MN-RAS, 429, 2127

Blondin S., Dessart L., Hillier D. J., Khokhlov A. M., 2017, MN-RAS, 470, 157

Brachwitz F., et al., 2000, ApJ, 536, 934

Branch D., Fisher A., Nugent P., 1993, AJ, 106, 2383

Branch D., Chau Dang L., Baron E., 2009, PASP, 121, 238

Bravo E., 2019, A&A, 624, A139

Bravo E., Garćıa-Senz D., 2006, ApJ, 642, L157

Bravo E., Mart́ınez-Pinedo G., 2012, Phys. Rev. C, 85, 055805

Bravo E., Domı́nguez I., Badenes C., Piersanti L., Straniero O.,

2010, ApJ, 711, L66

Bravo E., Badenes C., Mart́ınez-Rodŕıguez H., 2019, MNRAS,

482, 4346

Calder A. C., et al., 2007, ApJ, 656, 313

Churazov E., et al., 2014, Nature, 512, 406

Churazov E., et al., 2015, ApJ, 812, 62

Cyburt R. H., et al., 2010, ApJS, 189, 240

Dan M., Guillochon J., Brüggen M., Ramirez-Ruiz E., Rosswog

S., 2015, MNRAS, 454, 4411

Diehl R., et al., 2014, Science, 345, 1162

Dimitriadis G., et al., 2017, MNRAS, 468, 3798

Dubey A., Daley C., ZuHone J., Ricker P. M., Weide K., GrazianiC., 2012, ApJS, 201, 27

Filippenko A. V., 1997, ARA&A, 35, 309

Fink M., Röpke F. K., Hillebrandt W., Seitenzahl I. R., Sim S. A.,

Kromer M., 2010, A&A, 514, A53+

Fuller G. M., Fowler W. A., Newman M. J., 1985, ApJ, 293, 1

Gamezo V. N., Khokhlov A. M., Oran E. S., Chtchelkanova A. Y.,

Rosenberg R. O., 2003, Science, 299, 77

Graur O., Zurek D., Shara M. M., Riess A. G., Seitenzahl I. R.,Rest A., 2016, ApJ, 819, 31

Graur O., et al., 2018, ApJ, 859, 79

Harris J. A., Hix W. R., Chertkow M. A., Lee C. T., Lentz E. J.,Messer O. E. B., 2017, ApJ, 843, 2

Hillebrandt W., Niemeyer J. C., 2000, Annual Review of Astron-omy and Astrophysics, 38, 191

Hoeflich P., et al., 2017, preprint, (arXiv:1707.05350)

Höflich P., Khokhlov A., 1996, ApJ, 457, 500

Hosseinzadeh G., et al., 2017, ApJ, 845, L11

Howell D. A., 2011, Nature Communications, 2, 350

Howell D. A., Höflich P., Wang L., Wheeler J. C., 2001, ApJ, 556,302

Howell D. A., et al., 2005, ApJ, 634, 1190

Isern J., et al., 2016, A&A, 588, A67

Jacobson-Galán W. V., Dimitriadis G., Foley R. J., Kilpatrick

C. D., 2018, ApJ, 857, 88

James J. B., Davis T. M., Schmidt B. P., Kim A. G., 2006, MN-

RAS, 370, 933

Kasen D., Woosley S. E., 2007, ApJ, 656, 661

Katz B., Dong S., 2012, arXiv e-prints, p. arXiv:1211.4584

Khokhlov A. M., 1991, A&A, 245, 114

Kushnir D., 2019, MNRAS, 483, 425

Kushnir D., Katz B., 2019, arXiv e-prints, p. arXiv:1912.06151

Kushnir D., Katz B., Dong S., Livne E., Fernández R., 2013, ApJ,

778, L37

Leung S.-C., Nomoto K., 2017, preprint, (arXiv:1710.04254)

Leung S.-C., Nomoto K., 2018, ApJ, 861, 143

Li W., et al., 2019, arXiv e-prints, p. arXiv:1906.07321

Livne E., Arnett D., 1995, ApJ, 452, 62

Lopez L. A., Ramirez-Ruiz E., Huppenkothen D., Badenes C.,Pooley D. A., 2011, ApJ, 732, 114

Maeda K., et al., 2010a, Nature, 466, 82

Maeda K., Röpke F. K., Fink M., Hillebrandt W., Travaglio C.,Thielemann F., 2010b, ApJ, 712, 624

Maguire K., et al., 2018, MNRAS, 477, 3567

Maoz D., Mannucci F., Nelemans G., 2014, Annual Review of

Astronomy and Astrophysics, 52, 107

Mart́ınez-Rodŕıguez H., et al., 2018, ApJ, 865, 151Maund J. R., et al., 2010, ApJ, 725, L167

Maund J. R., et al., 2013, MNRAS, 433, L20Miller A. A., et al., 2018, ApJ, 852, 100

Mueller E., 1986, A&A, 162, 103

Nugent P., Baron E., Branch D., Fisher A., Hauschildt P. H.,1997, ApJ, 485, 812

Papatheodore T. L., 2015, PhD thesis, University of Tennessee,

KnoxvilleParikh A., José J., Seitenzahl I. R., Röpke F. K., 2013, A&A, 557,

A3

Plewa T., Calder A. C., Lamb D. Q., 2004, ApJ, 612, L37Poludnenko A. Y., Chambers J., Ahmed K., Gamezo V. N., Taylor

B. D., 2019, Science, 366, aau7365

Poznanski D., Gal-Yam A., Maoz D., Filippenko A. V., LeonardD. C., Matheson T., 2002, PASP, 114, 833

Raskin C., Kasen D., Moll R., Schwab J., Woosley S., 2014, ApJ,788, 75

Röpke F. K., Hillebrandt W., Niemeyer J. C., Woosley S. E., 2006,

A&A, 448, 1Scalzo R., et al., 2014, MNRAS, 440, 1498

Seitenzahl I. R., Townsley D. M., Peng F., Truran J. W., 2009,

Atomic Data and Nuclear Data Tables, 95, 96Seitenzahl I. R., et al., 2013, MNRAS, 429, 1156

Shappee B. J., Stanek K. Z., Kochanek C. S., Garnavich P. M.,

2017, ApJ, 841, 48Shen K. J., Kasen D., Miles B. J., Townsley D. M., 2018, ApJ,

854, 52

Sim S. A., Röpke F. K., Hillebrandt W., Kromer M., Pakmor R.,Fink M., Ruiter A. J., Seitenzahl I. R., 2010, ApJ, 714, L52

Stritzinger M., Mazzali P. A., Sollerman J., Benetti S., 2006,

A&A, 460, 793Tanaka M., Mazzali P. A., Stanishev V., Maurer I., Kerzendorf

W. E., Nomoto K., 2011, MNRAS, 410, 1725Thielemann F.-K., Nomoto K., Yokoi K., 1986, A&A, 158, 17

Timmes F. X., 1999, ApJS, 124, 241

Timmes F. X., Hoffman R. D., Woosley S. E., 2000, ApJS, 129,377

Timmes F. X., Brown E. F., Truran J. W., 2003, ApJ, 590, L83

Townsley D. M., Miles B. J., Timmes F. X., Calder A. C., BrownE. F., 2016, ApJS, 225, 3

Travaglio C., Hillebrandt W., Reinecke M., Thielemann F. K.,

2004, A&A, 425, 1029Wang L., Wheeler J. C., Li Z., Clocchiatti A., 1996, ApJ, 467,

435

Weaver T. A., Zimmerman G. B., Woosley S. E., 1978, ApJ, 225,1021

Woosley S. E., 1986, in Audouze J., Chiosi C., Woosley S. E., eds,Saas-Fee Advanced Course 16: Nucleosynthesis and ChemicalEvolution. p. 1

Woosley S. E., Kasen D., 2010, preprint, (arXiv:1010.5292)Woosley S. E., Weaver T. A., 1994, ApJ, 423, 371

Woosley S. E., Arnett W. D., Clayton D. D., 1973, ApJS, 26, 231Yang Y., et al., 2018, ApJ, 852, 89Zheng W., et al., 2013, ApJ, 778, L15

This paper has been typeset from a TEX/LATEX file prepared by

the author.

MNRAS 000, 1–11 (2020)

http://dx.doi.org/10.1086/504399http://cdsads.u-strasbg.fr/abs/2006ApJ...645.1373Bhttp://dx.doi.org/10.1093/mnras/sts484http://dx.doi.org/10.1093/mnras/sts484http://cdsads.u-strasbg.fr/abs/2013MNRAS.429.2127Bhttp://dx.doi.org/10.1093/mnras/stw2492http://dx.doi.org/10.1093/mnras/stw2492http://cdsads.u-strasbg.fr/abs/2017MNRAS.470..157Bhttp://dx.doi.org/10.1086/308968http://cdsads.u-strasbg.fr/abs/2000ApJ...536..934Bhttp://dx.doi.org/10.1086/116810https://ui.adsabs.harvard.edu/abs/1993AJ....106.2383Bhttp://dx.doi.org/10.1086/597788https://ui.adsabs.harvard.edu/abs/2009PASP..121..238Bhttp://dx.doi.org/10.1051/0004-6361/201935095https://ui.adsabs.harvard.edu/abs/2019A%26A...624A.139Bhttp://dx.doi.org/10.1086/504713http://cdsads.u-strasbg.fr/abs/2006ApJ...642L.157Bhttp://dx.doi.org/10.1103/PhysRevC.85.055805http://cdsads.u-strasbg.fr/abs/2012PhRvC..85e5805Bhttp://dx.doi.org/10.1088/2041-8205/711/2/L66http://cdsads.u-strasbg.fr/abs/2010ApJ...711L..66Bhttp://dx.doi.org/10.1093/mnras/sty2951http://cdsads.u-strasbg.fr/abs/2019MNRAS.482.4346Bhttp://dx.doi.org/10.1086/510709http://cdsads.u-strasbg.fr/abs/2007ApJ...656..313Chttp://dx.doi.org/10.1038/nature13672http://cdsads.u-strasbg.fr/abs/2014Natur.512..406Chttp://dx.doi.org/10.1088/0004-637X/812/1/62http://cdsads.u-strasbg.fr/abs/2015ApJ...812...62Chttp://dx.doi.org/10.1088/0067-0049/189/1/240http://cdsads.u-strasbg.fr/abs/2010ApJS..189..240Chttp://dx.doi.org/10.1093/mnras/stv2289https://ui.adsabs.harvard.edu/abs/2015MNRAS.454.4411Dhttp://dx.doi.org/10.1126/science.1254738http://cdsads.u-strasbg.fr/abs/2014Sci...345.1162Dhttp://dx.doi.org/10.1093/mnras/stx683http://cdsads.u-strasbg.fr/abs/2017MNRAS.468.3798Dhttp://dx.doi.org/10.1088/0067-0049/201/2/27https://ui.adsabs.harvard.edu/abs/2012ApJS..201...27Dhttp://dx.doi.org/10.1146/annurev.astro.35.1.309https://ui.adsabs.harvard.edu/abs/1997ARA&A..35..309Fhttp://dx.doi.org/10.1051/0004-6361/200913892http://cdsads.u-strasbg.fr/abs/2010A%26A...514A..53Fhttp://dx.doi.org/10.1086/163208http://adsabs.harvard.edu/abs/1985ApJ...293....1Fhttp://dx.doi.org/10.1126/science.1078129http://cdsads.u-strasbg.fr/abs/2003Sci...299...77Ghttp://dx.doi.org/10.3847/0004-637X/819/1/31https://ui.adsabs.harvard.edu/abs/2016ApJ...819...31Ghttp://dx.doi.org/10.3847/1538-4357/aabe25http://cdsads.u-strasbg.fr/abs/2018ApJ...859...79Ghttp://dx.doi.org/10.3847/1538-4357/aa76dehttps://ui.adsabs.harvard.edu/abs/2017ApJ...843....2Hhttp://dx.doi.org/10.1146/annurev.astro.38.1.191http://dx.doi.org/10.1146/annurev.astro.38.1.191http://cdsads.u-strasbg.fr/abs/2000ARA%26A..38..191Hhttp://arxiv.org/abs/1707.05350http://dx.doi.org/10.1086/176748http://cdsads.u-strasbg.fr/abs/1996ApJ...457..500Hhttp://dx.doi.org/10.3847/2041-8213/aa8402https://ui.adsabs.harvard.edu/abs/2017ApJ...845L..11Hhttp://dx.doi.org/10.1038/ncomms1344http://cdsads.u-strasbg.fr/abs/2011NatCo...2E.350Hhttp://dx.doi.org/10.1086/321584http://adsabs.harvard.edu/abs/2001ApJ...556..302Hhttp://adsabs.harvard.edu/abs/2001ApJ...556..302Hhttp://dx.doi.org/10.1086/497119https://ui.adsabs.harvard.edu/abs/2005ApJ...634.1190Hhttp://dx.doi.org/10.1051/0004-6361/201526941http://cdsads.u-strasbg.fr/abs/2016A%26A...588A..67Ihttp://dx.doi.org/10.3847/1538-4357/aab716http://cdsads.u-strasbg.fr/abs/2018ApJ...857...88Jhttp://dx.doi.org/10.1111/j.1365-2966.2006.10508.xhttp://dx.doi.org/10.1111/j.1365-2966.2006.10508.xhttps://ui.adsabs.harvard.edu/abs/2006MNRAS.370..933Jhttp://dx.doi.org/10.1086/510375http://cdsads.u-strasbg.fr/abs/2007ApJ...656..661Khttps://ui.adsabs.harvard.edu/abs/2012arXiv1211.4584Khttp://cdsads.u-strasbg.fr/abs/1991A%26A...245..114Khttp://dx.doi.org/10.1093/mnras/sty3121https://ui.adsabs.harvard.edu/abs/2019MNRAS.483..425Khttps://ui.adsabs.harvard.edu/abs/2019arXiv191206151Khttp://dx.doi.org/10.1088/2041-8205/778/2/L37https://ui.adsabs.harvard.edu/abs/2013ApJ...778L..37Khttp://arxiv.org/abs/1710.04254http://dx.doi.org/10.3847/1538-4357/aac2dfhttps://ui.adsabs.harvard.edu/abs/2018ApJ...861..143Lhttps://ui.adsabs.harvard.edu/abs/2019arXiv190607321Lhttp://dx.doi.org/10.1086/176279https://ui.adsabs.harvard.edu/abs/1995ApJ...452...62Lhttp://dx.doi.org/10.1088/0004-637X/732/2/114http://cdsads.u-strasbg.fr/abs/2011ApJ...732..114Lhttp://dx.doi.org/10.1038/nature09122http://cdsads.u-strasbg.fr/abs/2010Natur.466...82Mhttp://dx.doi.org/10.1088/0004-637X/712/1/624http://cdsads.u-strasbg.fr/abs/2010ApJ...712..624Mhttp://dx.doi.org/10.1093/mnras/sty820http://cdsads.u-strasbg.fr/abs/2018MNRAS.477.3567Mhttp://dx.doi.org/10.1146/annurev-astro-082812-141031http://dx.doi.org/10.1146/annurev-astro-082812-141031http://dx.doi.org/10.3847/1538-4357/aadaechttp://cdsads.u-strasbg.fr/abs/2018ApJ...865..151Mhttp://dx.doi.org/10.1088/2041-8205/725/2/L167https://ui.adsabs.harvard.edu/abs/2010ApJ...725L.167Mhttp://dx.doi.org/10.1093/mnrasl/slt050https://ui.adsabs.harvard.edu/abs/2013MNRAS.433L..20Mhttp://dx.doi.org/10.3847/1538-4357/aaa01fhttps://ui.adsabs.harvard.edu/abs/2018ApJ...852..100Mhttps://ui.adsabs.harvard.edu/abs/1986A&A...162..103Mhttp://dx.doi.org/10.1086/304459http://cdsads.u-strasbg.fr/abs/1997ApJ...485..812Nhttp://dx.doi.org/10.1051/0004-6361/201321518http://cdsads.u-strasbg.fr/abs/2013A%26A...557A...3Phttp://cdsads.u-strasbg.fr/abs/2013A%26A...557A...3Phttp://dx.doi.org/10.1086/424036http://cdsads.u-strasbg.fr/abs/2004ApJ...612L..37Phttp://dx.doi.org/10.1126/science.aau7365https://ui.adsabs.harvard.edu/abs/2019Sci...366.7365Phttp://dx.doi.org/10.1086/341741https://ui.adsabs.harvard.edu/abs/2002PASP..114..833Phttp://dx.doi.org/10.1088/0004-637X/788/1/75http://cdsads.u-strasbg.fr/abs/2014ApJ...788...75Rhttp://dx.doi.org/10.1051/0004-6361:20053926http://cdsads.u-strasbg.fr/abs/2006A%26A...448....1Rhttp://dx.doi.org/10.1093/mnras/stu350http://cdsads.u-strasbg.fr/abs/2014MNRAS.440.1498Shttp://dx.doi.org/10.1016/j.adt.2008.08.001http://cdsads.u-strasbg.fr/abs/2009ADNDT..95...96Shttp://dx.doi.org/10.1093/mnras/sts402https://ui.adsabs.harvard.edu/abs/2013MNRAS.429.1156Shttp://dx.doi.org/10.3847/1538-4357/aa6eabhttps://ui.adsabs.harvard.edu/abs/2017ApJ...841...48Shttp://dx.doi.org/10.3847/1538-4357/aaa8dehttp://cdsads.u-strasbg.fr/abs/2018ApJ...854...52Shttp://dx.doi.org/10.1088/2041-8205/714/1/L52http://adsabs.harvard.edu/abs/2010ApJ...714L..52Shttp://dx.doi.org/10.1051/0004-6361:20065514http://cdsads.u-strasbg.fr/abs/2006A%26A...460..793Shttp://dx.doi.org/10.1111/j.1365-2966.2010.17556.xhttp://cdsads.u-strasbg.fr/abs/2011MNRAS.410.1725Thttp://cdsads.u-strasbg.fr/abs/1986A%26A...158...17Thttp://dx.doi.org/10.1086/313257https://ui.adsabs.harvard.edu/abs/1999ApJS..124..241Thttp://dx.doi.org/10.1086/313407http://adsabs.harvard.edu/abs/2000ApJS..129..377Thttp://adsabs.harvard.edu/abs/2000ApJS..129..377Thttp://dx.doi.org/10.1086/376721http://cdsads.u-strasbg.fr/abs/2003ApJ...590L..83Thttp://dx.doi.org/10.3847/0067-0049/225/1/3http://cdsads.u-strasbg.fr/abs/2016ApJS..225....3Thttp://dx.doi.org/10.1051/0004-6361:20041108https://ui.adsabs.harvard.edu/abs/2004A&A...425.1029Thttp://dx.doi.org/10.1086/177617https://ui.adsabs.harvard.edu/abs/1996ApJ...467..435Whttps://ui.adsabs.harvard.edu/abs/1996ApJ...467..435Whttp://dx.doi.org/10.1086/156569https://ui.adsabs.harvard.edu/abs/1978ApJ...225.1021Whttps://ui.adsabs.harvard.edu/abs/1978ApJ...225.1021Whttp://arxiv.org/abs/1010.5292http://dx.doi.org/10.1086/173813http://cdsads.u-strasbg.fr/abs/1994ApJ...423..371Whttp://dx.doi.org/10.1086/190282http://cdsads.u-strasbg.fr/abs/1973ApJS...26..231Whttp://dx.doi.org/10.3847/1538-4357/aa9e4chttps://ui.adsabs.harvard.edu/abs/2018ApJ...852...89Yhttp://dx.doi.org/10.1088/2041-8205/778/1/L15https://ui.adsabs.harvard.edu/abs/2013ApJ...778L..15Z

1 Introduction2 Methodology3 Nuclear statistical equilibrium4 Simplified nuclear networks5 Nuclear post-processing for high metallicity progenitors6 Transition from the nuclear network to NSE and vice versa7 Conclusions