Sensitivity of Type Ia supernovae to electron capture rates
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Astronomy & Astrophysics
Sensitivity of Type Ia supernovae to electron capture rates
E. Bravo
E.T.S. Arquitectura del Vallès, Universitat Politècnica de Catalunya, Carrer Pere Serra 1-15, 08173 Sant Cugat del Vallès, Spaine-mail: [email protected]
March 21, 2019
ABSTRACT
The thermonuclear explosion of massive white dwarfs is believed to explain at least a fraction of Type Ia supernovae (SNIa). Afterthermal runaway, electron captures on the ashes left behind by the burning front determine a loss of pressure, which impacts thedynamics of the explosion and the neutron excess of matter. Indeed, overproduction of neutron-rich species such as Cr has beendeemed a problem of Chandrasekhar-mass models of SNIa for a long time. I present the results of a sensitivity study of SNIa modelsto the rates of weak interactions, which have been incorporated directly into the hydrodynamic explosion code. The weak rates havebeen scaled up / down by a factor ten, either globally for a common bibliographical source, or individually for selected isotopes. Inline with previous works, the impact of weak rates uncertainties on sub-Chandrasekhar models of SNIa is almost negligible. Theimpact on the dynamics of Chandrasekhar-mass models and on the yield of Ni is also scarce. The strongest e ff ect is found on thenucleosynthesis of neutron-rich nuclei, such as Ca, Cr, Fe, and Ni. The species with the highest influence on nucleosynthesisdo not coincide with the isotopes that contribute most to the neutronization of matter. Among the last ones, there are protons, , Fe, Co, and Ni, while the main influencers are , Mn and − Fe, in disagreement with Parikh et al (2013), who found that SNIanucleosynthesis is most sensitive to the β + -decay rates of Si, S, and Ar. An eventual increase in all weak rates on pf -shell nucleiwould a ff ect the dynamical evolution of hot bubbles, running away at the beginning of the explosion, and the yields of SNIa. Key words. nuclear reactions, nucleosynthesis, abundances – supernovae: general – white dwarfs
1. Introduction
Weak interactions on iron-group nuclei (IGN) play a key role inthe late stages of stellar evolution. In type II supernovae, firstelectron captures in the iron core reduce the pressure and startthe collapse and, later, beta-decays on neutron-rich nuclei con-tribute appreciably to the neutrino flux, help to regulate the coretemperature (Aufderheide et al. 1990), and leave an imprint onthe nucleosynthetic yield of the innermost ejected shells (Lan-ganke et al. 2011). Aufderheide et al. (1994) studied the mostrelevant electron captures in the pre-supernova evolution of mas-sive stars, and identified many iron-group nuclei that may havean influence on the conditions at supernova core collapse. Thesenuclei were subsequently targets of theoretical studies to refinethe associated weak rates. As another example, electron capturesupernovae (see Gil-Pons et al. 2018, for a recent review) arepredicted to be triggered by the transmutation of the late nucle-osinthetic products of the most massive intermediate-mass starswith low metallicity progenitors (Miyaji et al. 1980).The relevance of weak interactions for Type Ia supernovae(SNIa) depends on the progenitor system. Nowadays, there isdebate about the nature of SNIa progenitors, whether they aremore or less massive white dwarfs (WD), and whether they arepart of a single degenerate or a double degenerate binary sys-tem. While all of these scenarios have several points in favourand against (Chomiuk et al. 2012; Jacobson-Galán et al. 2018;Kilpatrick et al. 2018; Rebassa-Mansergas et al. 2019, and refer-ences therein), there are indications that SNIa may be producedby a combination of all of them (e.g. Sasdelli et al. 2017; Liuet al. 2018). In SNIa triggered by the explosion of massive WDs,first, during the pre-supernova carbon simmering phase, elec-tron captures and beta decays drive the equilibrium configura- tion of the star in response to mass accretion from a companionstar and, later, electron captures destabilize the WD and startthe dynamical phase of the explosion (e.g. Yokoi et al. 1979;Chamulak et al. 2008; Piersanti et al. 2017). Early during the ex-plosion, electron captures on IGN reduce the electron pressureand a ff ect the dynamical evolution and the nucleosynthesis ofSNIa. On the other hand, SNIa coming from the explosion ofsub-Chandrasekhar mass WDs (Woosley & Weaver 1994; Finket al. 2010; Shen et al. 2018) are not expected to be a ff ected byeither electron captures o beta-decays during the pre-explosiveor explosive phases. However, explosion of WDs more massivethan ∼ (cid:12) may drive the central regions to densities and tem-peratures high enough that the electron mole number changessignificantly.The sensitivity of SNIa nucleosynthesis to the weak ratesadopted in their modelling has been analysed in a few works,with conflicting results. Brachwitz et al. (2000) probed the con-sequences of a global change in the stellar weak rates of IGNowing to new shell model calculations of the Gamow-Teller(GT + ) strength distribution of pf -shell nuclei (Dean et al. 1998).These authors found a systematic shift in the centroid of theGT + strength distribution and lower stellar weak rates thanprior models. They explored the thermonuclear explosions ofChandrasekhar-mass WDs with central densities in the range(1 . − . × g cm − , applying approximate factors to cor-rect for the GT + centroid o ff set in those nuclei for which theshell model was not available. The new rates improve the nucle-osynthesis, reducing the historical excess production of severalneutron-rich isotopes of chromium, titanium, iron, and nickel,among others, in SNIa models. They noticed that protons dom-inate the neutronization during SNIa explosions and, since theirweak rate is not a ff ected by the uncertainties plaguing electron page 1 of 8 a r X i v : . [ a s t r o - ph . S R ] M a r & A proofs capture rates on IGN, the overall neutronization only dependsweakly on these uncertainties. They also identified odd- A andodd-odd nuclei as the largest contributors to the WD neutroniza-tion, besides protons.On the other hand, Parikh et al. (2013) analysed one three-dimensional model of SNIa and the classical one-dimensionalW7 model (Thielemann et al. 1986), and found maximal sensi-tivity of SNIa nucleosynthesis to the electron capture rates of the α elements Si, S, and Ar, whereas electron captures on IGNhad little impact on the explosion. Recently, Mori et al. (2016)have studied the impact that recent alternatives to the shell modelcalculations of Dean et al. (1998), motivated by new experimen-tal data on the electron capture rates on Ni and Co, have onSNIa nucleosynthesis. These data call for di ff erences in the GT + strength distribution, leading to reduced electron capture rates.Mori et al. (2016) found that the overall yields of the explosionare a ff ected at most by 2 −
2. Explosion models
I have computed SNIa explosion models in spherical symmetrystarting from sub-Chandrasekhar (sub- M Ch ) and Chandrasekhar-mass ( M Ch ) WDs. The hydrocode integrates simultaneously thehydrodynamics and the nuclear network, and has been describedin detail by Bravo et al. (2019). Here, I focus on the behaviourof models suitable for normal-luminosity SNIa, characterized byan ejected mass of Ni of the order of M ( Ni) ∼ . − . (cid:12) .To this end, I have selected two base models, one describ-ing the central detonation (DETO) of a sub- M Ch WD of mass M WD = .
06 M (cid:12) , and the other belonging to the delayed det-onation (DDT) of a M Ch WD with central density in the range ρ c = (2 − × g cm − , thus extending the range of ρ c exploredby Brachwitz et al. (2000). The delayed detonation model ischaracterized by two parameters, the density of transition, ρ DDT ,from a deflagration (subsonic flame propagation near the centerof the WD) to a detonation (supersonic combustion wave), andthe velocity of the flame during the deflagration phase, v def , usu-ally prescribed as a fraction of the local sound velocity, v sound .The configuration parameters are given in Table 1 together withthe main explosion properties: kinetic energy, K , and M ( Ni).Models S, S + , C3, C3_1p2, and C3_4p0 in Table 1 are thesame as models 1p06_Z9e-3_std, 1p15_Z9e-3_std, ddt2p4_Z9e-3_std, ddt1p2_Z9e-3_std, and ddt4p0_Z9e-3_std in Bravo et al.(2019), and all the details of the code and the models are thesame as in that paper unless otherwise stated in what follows.The remaining models reported in this work share the same ini-tial composition as C3 and use the same set of thermonuclearreaction rates. I have explored the impact of di ff erent weak ratesfor various parameters of the DDT and for di ff erent initial central Fig. 1.
Sources of tabulated weak interaction rates. Colours identifiesthe source of weak rates on a species: FFN82 in cyan, Oda94 in blue,MPLD00 in red, and PF03 in green. The weak rates on protons, notshown in the graph, are taken from MPLD00. The species with themost influencial individual weak rates (Sect. 4.2) are highlighted withan open black square. All of these belong to the MPLD00 tabulation. densities, but most of the simulations are variations of model C3.Its central density at thermal runaway, ρ c = × g cm − , is thatsuggested by some recent studies of the carbon simmering phase(Martínez-Rodríguez et al. 2016; Piersanti et al. 2017). A centraldensity as high as ρ c = × g cm − , or even higher, has beeninvoked to explain the composition of particular SNIa events(Dave et al. 2017), while a value as low as ρ c = × g cm − provides the best nucleosynthetic match with the Solar Systemratios of the IGN (Brachwitz et al. 2000; Mori et al. 2016).All models in Table 1 share the same set of weak reactionrates, the results of their modification are given later. Next, I ex-plain the details of the weak rates incorporated to the models. At the high densities characteristic of the central layers of ex-ploding WDs, electron captures are favoured relative to β + de-cays because the Fermi-Dirac distribution of degenerate elec-trons allows an enhancement in the electron capture rate,whereas β + decays remain insensitive to the density (Sarriguren2016). The Fermi energy of the electrons depends on density as E F (cid:39) m e (cid:104) ( ρ Y e ) / − (cid:105) with an accuracy better than 98% for ρ > , where ρ is the density in units of 10 g cm − , Y e isthe electron mole number in mol g − , and m e = .
511 MeV isthe mass of the electron in energy units. In SNIa, most of theneutronization takes place on matter that is in a nuclear statis-tical equilibrium (NSE) state, at temperatures of the order of T ∼ × K, where the Fermi energy of the electrons is E F ∼ . − + transitions, which allow a change of the nuclear angular momen-tum from parent to child nuclei by ∆ J = , ±
1. Theory and ex-periment on ground state GT distribution agree, generally, withina factor ∼ Q -value, the rates are sensitive to the detailed GT + distribution, and such distributions cannot be measured from ex-cited states. In theoretical models, the transitions starting fromexcited states are treated according to the Brink (or Brink-Axel)hypothesis: the GT + strength distribution from excited states isthe same as from the ground state, shifted by the energy of theexcited state. However, the applicability of the Brink hypothe- page 2 of 8. Bravo: Sensitivity of Type Ia supernovae to electron capture rates Table 1.
Cases studied.
Model Class Type M WD ρ c v def /v sound ρ DDT
K M ( Ni)(M (cid:12) ) (g cm − ) (g cm − ) (10 erg) (M (cid:12) )S sub- M Ch DETO 1.06 4 . × - - 1.32 0.664S + sub- M Ch DETO 1.15 9 . × - - 1.46 0.894C2 M Ch DDT 1.36 2 . × . × M Ch DDT 1.37 3 . × . × M Ch DDT 1.38 4 . × . × M Ch DDT 1.39 5 . × . × M Ch DDT 1.37 3 . × . × M Ch DDT 1.37 3 . × . × M Ch DDT 1.37 3 . × . × M Ch DDT 1.37 3 . × . × A = −
60 in the IPM approx-imation (independent-particle model), but the shell model wasapplied only to sd -shell nuclei ( A = − sd -shell nuclei including new relevantexperimental information. Dean et al. (1998) applied the shellmodel to the pf -shell nuclei with experimental data on the GT + strength distribution: V, Mn, , Fe, Co, and , , , Ni,most of these even-even nuclei, and predicted the stellar weakrates for other nuclei belonging to the iron group. Cole et al.(2012) analysed new experimental data on the GT + strength dis-tribution on pf -shell nuclei, including those already consideredby Dean et al. (1998) plus: Sc, Ti, V, and Zn. Cole et al.(2012) found that the experimental electron captures rates on , Fe were higher than the theoretical rates (Caurier et al. 1999;Langanke & Martínez-Pinedo 2000, 2001) by as much as a fac-tor two in the conditions of SNIa. Fantina et al. (2012) analysedthe same two isotopes of iron from a purely theoretical point ofview, taking into account the uncertainties associated with thedi ff erent nuclear model parameters, and concluded that the weakrates on these nuclei could change by as much as two orders ofmagnitude for the whole set of parameters explored. Sarriguren(2013, 2016) revisited the e ff ect of excited states of iron-groupnuclei and concluded that thermal excitation of nuclei in SNIacan lead to overall electron capture rates higher as well as lowerthan those accounting only for transitions from the parent groundstate. In either case, the associated uncertainty is similar to thatderived from nuclear structure.In all the models presented in this work, weak interactionrates are adopted from, in order of precedence, Martínez-Pinedoet al. (2000, hereafter, MPLD00), Oda et al. (1994, hereafter,Oda94), Pruet & Fuller (2003, hereafter, PF03), and Fuller et al.(1982b, hereafter, FFN82). For instance, if a rate appears bothin MPLD00 and in FFN82, the former is the choice. The tablesare interpolated following the procedure described in Fuller et al.(1985). Figure 1 shows the sources of each weak rate in the pro-ton number vs baryon number plane.Table 2 gives the overall change in Y e as a result of electroncaptures and β + decays, on the one hand, and of β − decays, onthe other hand, for the sub- M Ch models and for the M Ch mod-els with di ff erent initial central densities. Table 2 also lists thespecies that contribute most to the change of Y e in each model(from here on, the neutronizers ). As expected, β − decay contri-bution is negligible in all SNIa models. The two sub- M Ch modelsexperience a small change of the electron mole number, in spite Table 2.
The neutronizers : contributors to the overall ∆ Y e ∆ Y e , ec a ∆ Y e ,β − b Contributing targets(mol g − ) (mol g − ) for e.c. & β + c S 5 . × − − . × − Zn; NiS + . × − − . × − p; Zn; NiC2 1 . × − − . × − p; Co; Ni; Fe; Ni; Co; Fe; NiC3 2 . × − − . × − p; Fe; Co; Fe; Ni; Co; Ni; Co; Ni; Ni; Fe; Mn; CrC4 2 . × − − . × − p; Fe; Fe; Co; Co; Ni; Ni; Co; Fe; Ni; Ni; Mn; Cr; Fe; CoC5 3 . × − − . × − p; Fe; Fe; Co; Co; Fe; Co; Ni; Ni; Mn; Ni; Ni; Fe; Mn; Cr; Co; Ni Notes. ( a ) Global change in the electron mole number of the WD due toelectron captures and positron decays. ( b ) Global change in the electronmole number of the WD due to β − decays. ( c ) Sorted list of speciescontributing by at least 10 − mol g − . of their relatively high mass (for a sub- M Ch model). The main neutronizers in the two models are Zn and Ni, and protons aswell in the S + model.In all M Ch models, protons are the main source of neutroniza-tion, followed by several isotopes from the IGN, among whichthere are even-even, odd-odd, and odd-A nuclei. At the lowest ρ c explored, the strongest neutronization is provided, besidesprotons, by Co and Ni, while, for increasing ρ c , these twospecies are overtaken by the two iron isotopes , Fe. Amongthe IGN reported in Table 2, the only rates with direct experi-mental information about the GT + distribution are Mn, , Fe,and , Ni.
3. Three-dimensional effects
In three-dimensional models of the thermonuclear explosion ofmassive WDs, thermal runaway is usually assumed to start indiscrete volumes located slightly o ff -center (bubbles). Thesebubbles tend to float owing to the expansion caused by the re-lease of nuclear energy, and their density drops o ff sooner than page 3 of 8 & A proofs r / r time (s) Fig. 2.
Time evolution of the radial coordinate, r , of an incinerated bub-ble for a central density of the WD of ρ c = × g cm − . The radialcoordinate of the bubble is plotted normalized to its initial position, r ,for two cases: standard electron capture rates (green curve) and weakrates scaled up by a factor ten (red curve). if they remained at the center and followed the expansion ofthe whole WD. Consequently the rate of neutronization dropso ff as well. In models working with standard electron capturerates, the timescale for the bubbles to start rising o ff the center is ∼ . − . ff ect ofelectron captures into the general scheme presented in Fisher &Jumper (2015). In their appendix, Fisher & Jumper (2015) wrotea second order di ff erential equation for the evolution of the ra-dial position, r , of a hot bubble, accounting for the floatationforce and the drag on the bubble:dd t (cid:34) π R (cid:32) ρ a + ρ f (cid:33) d r d t (cid:35) = π R ( ρ f − ρ a ) g , (1)where t is time since bubble ignition, R is the bubble radius,which is assumed to increase linearly with time (at constantflame velocity), g = g ( r ) is the local acceleration of gravity, ρ f is the local density, and ρ a is the density of ashes. To incorporateelectron captures in this scheme in a simplified manner, I haveassumed that the burning is isobaric, which is valid near the cen-ter of the WD, and that the main contribution to pressure is thatof a completely degenerate gas of electrons. Then, the pressure, P is a function of the product ρ Y e and, to keep it constant, thebubble density changes with time according to˙ ρ a ρ a = − ˙ Y e Y e . (2)Equation 1 can then be integrated numerically, starting from aninitial radial coordinate of the bubble, r .Fig. 2 shows the results for a WD of ρ c = × g cm − ,both for standard electron capture rates and for weak rates in-creased by a factor ten. With standard electron capture rates,the dynamical evolution of the bubble is similar to the results of complex three-dimensional simulations. However, with in-creased weak rates the bubble remains near the center, at highdensity, for nearly a second. This shows that the weak rates havethe potential of changing the overall dynamical evolution of M Ch models.
4. Sensitivity of nucleosynthesis to the electroncapture rates
In this section, I give the results of the sensitivity of the nucle-osynthetic yields of the models to the modification of weak ratesboth globally, applying the same change to all rates coming froma given source (Sect. 4.1), and individually for selected nuclei(Sect. 4.2). In the last case, the sensitivity is measured by thelogarithmic derivative, D i , j , of the mass ejected of nucleus i withrespect to the enhancement factor of the weak rates on species j , f j , which is the factor by which these weak rates are scaled atevery density and temperature: D i , j = d log m i d log f j ≈ . (cid:32) m i , m i , . (cid:33) , (3)where m i , is the mass of nucleus i ejected for f j =
10, and m i , . is the corresponding mass when f j = .
1. Just to give a feelingof the meaning of D i , j , a value of D i , j ≈ . i doubles for a constant enhancement factor of f j =
10. For the same enhancement factor, a value of D i , j ≈ . D i , j ≈ . f j =
10 or f j = .
1. According to the dis-cussion in Sect. 2.1, it is expected that the weak rates are knownwith better accuracy, of the order of a factor two. However, asmost of the neutronizers weak rates are not tied by direct exper-imental results, I explore a slightly larger enhancement factor.
Table 3 shows the results of the models in Table 1 when all theweak rates tabulated in a given reference are scaled simultane-ously by the same factor. In general, the sensitivity to an increasein the weak rates by a factor 10 is much larger than the sensitivityto a decrease in the rates by the same factor. The only exceptionis the decrease in the rates from MPLD00, including the rates onprotons, in model C3, which causes a 10% increase in the yieldof Ni and a decrease in the yield of several neutron-rich nuclei,such as that of Cr by nearly five orders of magnitude.The sub- M Ch models are very robust against changes in theweak rates. When the MPLD00 rates are increased by a factorten, the kinetic energy changes just by 0.2% and the yield of Ni by 2%. However, it is interesting that the yield of stablenickel doubles that of the model with the standard weak rates.Modifying the rates given by Oda94 in M Ch models has prac-tically no impact on the supernova explosion dynamics and nu-cleosynthesis. The modification of the FFN82 rates has no im-pact, as well, although it should be recalled that I used their ratesonly when the isotope was not tabulated in the other sources ofweak rates.The strongest impact of the modification of weak rates isfound when all the rates in MPLD00 are increased by a factorten in model C3, leading to a 3% decrease in the final kineticenergy and a 13% decrease in the yield of Ni. In this same run,the yield of Ca increases by nearly six orders of magnitude, page 4 of 8. Bravo: Sensitivity of Type Ia supernovae to electron capture rates
Table 3.
Sensitivity to a bulk change in the rates of a given source.
Model Source a Scalingfactor a ∆ KK b ∆ M ( Ni) M ( Ni) b Relevant yield ratios c S MPLD00 ×
10 0.002 -0.021 Ni( × . Ni( × . Ni( × . Ti( × . × . Ti( × . Zn( × . Ni( × . Ni( × . + MPLD00 - p d ×
10 0.002 -0.025 Ni( × . Ni( × . Ni( × . Ti( × . ×
10 -0.003 -0.055 Ti( × . ); Ni( × . ); Cr( × . ); Ni( × . ×
10 -0.030 -0.130 Ca( × . Ni( × . ); Cr( × . Fe( × . × . Zn( × . Ni( × . Ca( × . − Cr( × . − ×
10 -0.004 -0.066 Ca( × . Ni( × . ); Cr( × . Fe( × . × . Ni( × . Zn( × . Cr( × . Ca( × . ×
10 0.000 0.000 Ca( × . Ni( × . Cr( × . Si( × . × . O( × . Ti( × . Ni( × . Ca( × . ×
10 0.000 0.000 Ca( × . Ni( × . Ti( × . S( × . × . Ne( × . Ca( × . S( × . Ne( × . ×
10 -0.006 -0.056 Se( × . ); Ca( × . ); Ni( × . Cr( × . ×
10 -0.009 -0.062 Se( × . Ca( × . ); Ni( × . Cr( × . ×
10 -0.004 -0.037 Ca( × . Ni( × . Cr( × . Fe( × . ×
10 -0.004 -0.074 Ca( × . Ni( × . ); Cr( × . Fe( × . ×
10 -0.007 -0.139 Ca( × . Ni( × . ); Cr( × . Fe( × . ×
10 -0.002 -0.047 Ca( × . Ni( × . ); Cr( × . Fe( × . Notes. ( a ) The rates from the indicated source were scaled by the factor shown. ( b ) Relative change in the final kinetic energy and the mass of Ni,with respect to the values reported for the same model in Table 1. ( c ) Ratio of the final yield of a few selected isotopes with respect to those in thesame model with no weak rates scaled. The isotopes reported here are a selection among those with non-negligible ejected mass and whose yieldis most a ff ected by the scaling. Ratios larger than 999. are given in exponential format, for example ( × × . ( d ) Sources denoted as “MPLD00 - p” mean that the rates scaled where those from MPLD00 with the exception of p (cid:28) n. that of Cr increases by a factor 8.9, and the yield of Fe de-creases by nearly 10%. However, in this model the proton weakrate was modified by the same factor as the weak rates on IGN,which seems unrealistic since the uncertainty on the proton weakrate is much smaller than those on IGN nuclei. Consequently, Iran the same model with all the rates in MPLD00 modified withthe exception of those belonging to protons, for which the stan-dard rate was applied. In this case, the impact is much smallerbut still noticeable: the yield of Ni decreases by 6.6%, that of Ca increases by more than three orders of magnitude, and thatof Cr increases by a factor 3.8, while the yield of Fe remainspractically unaltered.When the same modifications are applied to models C2, C4and C5, in order to explore the e ff ects of di ff erent ρ c , the changesin the kinetic energy and the yield of Ni are similar to those formodel C3, but the nucleosynthesis changes in a di ff erent way.The higher the initial central density, the less sensitive the yieldsof Ca and Cr to the modification of the weak rates. The yieldsof several other isotopes are especially sensitive to the weak ratesonly when ρ c is within a particular range. For instance, this is thecase of the yield of Se at ρ c ∼ (4 − × g cm − .With respect to the models with di ff erent deflagration veloc-ity, models C3_100 and C3_500, I find that the nucleosynthesisis increasingly more sensitive to the modification of the weakrates of MPLD00 with increasing v def . The reason is that, as thedeflagration velocity increases, a higher mass is burnt before thewhite dwarf expands appreciably, then the matter in NSE hasmore time to capture electrons before weak rates freeze out. Thedeflagration-to-detonation transition density, ρ DDT , does not in-fluence the sensitivity of the explosion to weak rates. The appar-ent largest sensitivity of the yield of Ni in model C3_1p2 sim-ply reflects the small amount of the isotope that is made out ofNSE in this model. In absolute terms, the total change in M ( Ni)
Fig. 3.
Final distribution of the most sensitive isotopes through theejecta in model C3. The colour represents the cumulated mass of eachisotope, starting from the center of the star, normalized to the totalejected mass of the same isotope. is very close for all three models with di ff erent ρ DDT and thesame ρ c , ∆ M ( Ni) = . − .
045 M (cid:12) . Model C3 was rerun with selected weak interaction rates modi-fied by either a factor f j =
10 or f j = .
1. The selection criteriumfor choosing the rates to be modified was that they contributed tothe global change of Y e in model C3 by at least 5 × − mol g − .The individual rates explored were the electron captures plus β + on and the β − decays to (all these were changed simultaneouslyby the same factor) the following list of species: Si, , P, S, , Cl, Ar, V, , − Cr, , , − Mn, − Fe, − Co, page 5 of 8 & A proofs Fe Fe Co Co Fe Co Ni Ni Ni Co 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16(d) pe r c en t c on t r i bu t i on t o ∆ Y e ( m o l / g ) Lagrangian mass (solar masses)0.1*p Mn Mn Fe Fe Fe Fe Fe Co Co Co Co Ni Ni Ni Fig. 4.
Contribution of individual electron captures to the neutronization close to the center as a function of the mass coordinate for aChandrasekhar-mass model with: (a) ρ c = × g cm − (model C2), (b) ρ c = × g cm − (model C3), (c) ρ c = × g cm − (modelC4), and (d) ρ c = × g cm − (model C5). In panel (c), the curves are labelled with the species name. In all panels, the contribution fromprotons has been scaled down by a factor ten. − Ni, − Cu, Zn. Although electron captures on protonsdominate the neutronization, I excluded these from the list be-cause of the smaller uncertainty of their tabulated weak rate.The kinetic energy in all these models changed by less than1% with respect to the reference model C3, and the mass ofejected Ni changed by less than 1% in all but the models withthe rates on , Co or Ni modified, in which it changed by lessthan 2%.Table 4 lists, sorted by (cid:12)(cid:12)(cid:12) D i , j (cid:12)(cid:12)(cid:12) , the species (from now on, the influencers ) whose weak rates caused a strong impact on anyspecies with a non-negligible yield. The list is leaded by iron iso-topes, then manganese isotopes and cobalt isotopes. To be pre-cise, the top influencer is Fe, whose D i , j = . Ca, which means that the yield of the last isotopewould change by a factor three if the weak rate on Fe increasedby a factor ten. , Mn and , Fe also stand out among the top influencers , and all of these impact most the production of Caand then of Ni.These findings contradict the results of Parikh et al. (2013),who identified the rates on Si, S, and Ar as the most in-fluential for the M Ch scenario, whereas the weak rates on IGNhad little impact on the nucleosynthesis. The origin of the dis-crepancy can be traced back to the treatment of weak rateswhile matter is in a NSE state. Parikh et al. (2013) had over-looked to account for the e ff ect of changes in the weak interac-tions of the NSE component (private communication). Since thetimescales for weak interactions are longer than the WD expan-sion timescale and beta-equilibrium is not reached in the NSE conditions of SNIa explosions (Brachwitz et al. 2000), quantify-ing the impact of modified individual weak reaction rates whilematter is in the NSE state requires a re-computation of the NSEcomposition at each time step, either of the post-processing cal-culation or of the hydrodynamic calculation itself, as done in thispaper. To confirm that this is the main source of the observed dif-ferences, I have rerun model C3 confining the modification of in-dividual weak rates to matter colder than 5 × K, to match theNSE criterion applied by Parikh et al. (2013). With this restric-tion, the most influential rate is the β + -decay of Ar, while theimpact of the modification of the weak rates on IGN decreasesby more than an order of magnitude with respect to the valuesreported in Table 4.Within the ten most influencer electron captures, only therates on Mn and , Fe have been determined with the aid ofexperimental information on the GT + strength distribution. Theexistence of low-lying excited states in some of the influencers contributes to make uncertain their stellar electron capture rates.For instance, the first two excited states of Fe lie at 14.4 and136.5 keV, to be compared to the thermal energy at the typicaltemperatures of NSE matter in SNIa, kT ∼
800 keV. Electroncapture from these excited states are favoured by the small dif-ference between their angular momentum and that of the groundstate of Mn, which allows a GT + transition, as opposed to theground state of Fe. A similar situation occurs for the electroncaptures on Co.Table 5 gives the species whose yield is most impacted bythe modification of a weak rate on any isotope. The most sensi- page 6 of 8. Bravo: Sensitivity of Type Ia supernovae to electron capture rates
Table 4.
The influencers : species whose weak rates modification has alarge impact on the yield of any other species. impacted species and D i , j Fe: Ca (0.50); Ni (0.22); Ti (0.09); Cr (0.06); Zn (0.05) Mn: Ca (0.44); Ni (0.19); Ti (0.07) Mn: Ca (0.42); He (0.34); Ni (0.20); Ti (0.09); Ni (0.09); Cr (0.06) Fe: Ca (0.42); Ni (0.20); Ti (0.09); Cr (0.07) Fe: He (0.40); Ca (0.34); Ni (0.17); Ni (0.12); Ti (0.09); Cr (0.07); Ni (0.06); Fe (0.06) Co: He (0.34); Ca (0.11); Ni (0.08); Ni (0.05) Co: Ca (0.20); Ni (0.10) Mn: Ca (0.18); Ni (0.07) Co: Ca (0.17); Ni (0.08) Fe: Ca (0.16); Ni (0.08) Ni: Ca (0.15); Ni (0.07) Co: Ca (0.13); Ni (0.06) Co: Ca (0.13); Ni (0.05) V: Ca (0.12); Ni (0.06) Ni: Ca (0.12); Ni (0.06) Fe: Ca (0.11) Ni: Ca (0.11); Ni (0.05) Cr: Ca (0.09) Co: Ca (0.07) Ni: He (-0.07) Cr: Ca (0.06) Ni: Ca (0.06)
Notes.
Listed are those species for which (cid:12)(cid:12)(cid:12) D i , j (cid:12)(cid:12)(cid:12) ≥ . Table 5.
The sensitive : species most impacted by the modification ofindividual weak rates. [ A Z / Fe] a influencers and D i , j He -7.2 Fe (0.40); Mn (0.34); Co (0.34); Ni (-0.07) Ca -3.7 Fe (0.50); Mn (0.44); Mn (0.42); Fe (0.42); Fe (0.34); Co (0.20); Mn (0.18); Co (0.17); Fe (0.16); Ni (0.15) Ti 0.47 Fe (0.09); Mn (0.09); Fe (0.09); Fe (0.09); Mn (0.07); Co (0.04) Cr 0.87 Fe (0.07); Fe (0.07); Mn (0.06); Fe (0.06); Mn (0.04) Fe 0.34 Fe (0.06); Fe (0.05); Mn (0.04) Ni -0.16 Ni (0.04) Ni -1.3 Fe (0.12); Mn (0.09); Co (0.08) Ni -0.09 Fe (0.06) Ni -1.1 Fe (0.22); Mn (0.20); Fe (0.20); Mn (0.19); Fe (0.17); Co (0.10); Co (0.08); Fe (0.08); Mn (0.07); Ni (0.07) Zn -2.1 Fe (0.04) Zn -2.5 Fe (0.05)
Notes.
Listed are those species for which (cid:12)(cid:12)(cid:12) D i , j (cid:12)(cid:12)(cid:12) ≥ . ( a ) [ A Z / Fe] = log[ n ( A Z ) / n (cid:12) ( A Z )] − log[ n ( Fe) / n (cid:12) ( Fe)], with n ( A Z ) the number frac-tion of isotope A Z in the supernova ejecta, and n (cid:12) ( A Z ) its abundance inthe Solar System (Lodders 2003). tive are Ca and He, but these two species are of low interestin SNIa from the nucleosynthetic point of view. For instance, inmodel C3, the ratio of the ejected mass of Ca with respect tothe main nucleosynthetic product of SNIa, Fe, normalized tothe Solar System ratio is just 2 × − . Of more interest is Ni,which is mainly sensitive to the weak rates on several isotopesof iron and manganese, with D i , j ∼ .
2. Indeed, if all the weakrates tabulated in MPLD00 minus those involving protons wererevised up by a factor 10, Ni would be overproduced with re-spect to Fe in all M Ch models (see Table 3). The species mostoverproduced in model C3 (with standard rates) are Ti, Cr,and Fe, partially because of the relatively high initial centraldensity. Their sensitivity to modification of individual weak ratesis moderate, D i , j ∼ . − . sensitive and influencer species can be under-stood as a result of the spatial coincidence between the regionsof synthesis of the former and the regions of maximum impactof the second on the final value of Y e . Figure 3 shows, for modelC3, the final distribution of the most sensitive species within theWD ejecta. Ca and Ni are synthesized in the innermost few10 − M (cid:12) of the WD, where the contribution of Fe and Mn tothe neutronization is maximal, whereas Ti, Cr, Fe, and Niare made in between mass coordinates 10 − − − M (cid:12) , wheremost of the neutronization is provided by p, − Fe and Mn.On the other hand, Ni is made in the mass range 0 . − (cid:12) ,where most electron captures occur on p, Ni and Co.Fig. 4 shows the mass coordinates within which the di ff erentspecies contribute most to the neutronization in the M Ch modelswith di ff erent initial central density. At all values of ρ c , electroncaptures on protons dominate the neutronization in the centralregions of the WD, with decreasing contribution at increasingdistance to the center, and electron captures on Ni determinethe neutronization at M (cid:38) . − . (cid:12) . The number of speciesthat contribute significantly to the neutronization increases with ρ c , but their distribution is very similar for all central densities.Therefore, it is to be expected that the sensitivities explored fora ρ c = × g cm − are representative of models with ρ c in therange (2 − × g cm − .
5. Summary
I have assessed the impact of modifications of the weak interac-tion rates on the nucleosynthesis and other explosion propertiesof SNIa models. The present work relies on the simulation ofone-dimensional models of Chandrasekhar-mass as well as sub-Chandrasekhar mass WD SNIa models through a hydrocode thatincorporates a large enough nuclear network that the nucleosyn-thesis can be obtained directly, and there is no need for nuclearpost-processing (Bravo et al. 2019).Since many of the arguments in favour of the single-degenerate scenario for SNIa progenitors rely on the detectionof neutron-rich nuclei and elements in individual SNIa and theirremnants (e.g. Höflich et al. 2004; Yamaguchi et al. 2015; Shenet al. 2018), it is of particular interest to test if the explosion ofthe heaviest sub-Chandrasekhar mass WDs is able to producethe yields required by the observations. And the result is thatit is not. The impact of modifying the electron capture rates inthe explosion of sub-Chandrasekhar WDs is small, even for pro-genitors close to the upper mass limit for WDs made of carbonand oxygen. In particular, model S + , whose progenitor mass is M WD = .
15 M (cid:12) , is unable to produce either the central hole inthe distribution of radioactive nickel (Höflich et al. 2004) or thehigh nickel to iron mass ratio detected in the supernova remnant3C397 (Yamaguchi et al. 2015). page 7 of 8 & A proofs
The impact of the explored changes of electron capture rateson the main properties of Chandrasekhar-mass models, namelythe final kinetic energy and the mass of Ni synthesized, is alsoscarce. The yield of Ni may vary by as much as ∼ . (cid:12) onlyin case of a global revision of the rates upwards by an order ofmagnitude, while the maximum variation obtained by changingindividual rates is limited to ∼ .
01 M (cid:12) . In comparison, currentobservational estimates of the amount of Ni needed to powerSNIa light curves work with uncertainties of order ∼ . (cid:12) (Scalzo et al. 2014), although there are prospects to reduce theerror budget to ∼ . (cid:12) (Childress et al. 2015).I have identified three groups of species relevant for assess-ing the impact of weak rates on SNIa: the neutronizers , whichare those that contribute most to the neutronization of mat-ter, the influencers , which are those whose eventual weak ratechange impacts most the abundance of any species with a non-negligible yield, and the sensitive , which are the isotopes whoseabundance is most impacted by a change in the weak rates. Inthe Chandrasekhar-mass models explored in this work, the neu-tronizers are, besides protons, , Fe, Co, and Ni. The in-fluencers do not match the neutronizers point by point, theyare mainly , Mn and − Fe. Finally, the sensitive are thoseneutron-rich nuclei made close to the center of the WD, andwhich are usually overproduced in Chandrasekhar-mass SNIamodels with ρ c > × g cm − : Ca, Ti, Cr, and , Ni, inagreement with Brachwitz et al. (2000) and Parikh et al. (2013).I do not support the claim by Parikh et al. (2013) that SNIanucleosynthesis is most sensitive to the modification of individ-ual β + -decay rates of Si, S, and Ar. On the other hand, thepresent results relative to the sensitivity of the nucleosynthesisof SNIa to simultaneous changes in all the weak rates grosslyagree with the findings of Parikh et al. (2013). Another e ff ect Ihave found of increasing globally the weak rates by an order ofmagnitude is a reduced floatability of hot bubbles near the center,which increases the time available for further electron capturesand a ff ects the overall dynamics of the explosion. These resultsunderline the importance of knowing the weak rates with accu-racy, at least for Chandrasekhar-mass models. Acknowledgements.
I am grateful to Gabriel Martínez-Pinedo for providing stel-lar weak rates tables on a fine grid. Thanks are due to Robert Fisher, for inter-esting discussions concerning the floatability of hot bubbles, and to Ivo Seiten-zahl and Friedrich Röpke for clarifying the origin of the di ff erences between thepresent results and those of Parikh et al. The referee, Chris Fryer, has made inter-esting suggestions to improve the presentation of this paper. This work has beensupported by the MINECO-FEDER grant AYA2015-63588-P. References