Comparing treatments of weak reactions with nuclei in simulations of core-collapse supernovae
Hiroki Nagakura, Shun Furusawa, Hajime Togashi, Sherwood Richers, Kohsuke Sumiyoshi, Shoichi Yamada
aa r X i v : . [ a s t r o - ph . H E ] D ec Draft version December 27, 2018
Typeset using L A TEX twocolumn style in AASTeX61
COMPARING TREATMENTS OF WEAK REACTIONS WITH NUCLEI IN SIMULATIONS OFCORE-COLLAPSE SUPERNOVAE
Hiroki Nagakura,
1, 2
Shun Furusawa, Hajime Togashi, Sherwood Richers, Kohsuke Sumiyoshi, andShoichi Yamada
7, 8 Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544 TAPIR, Walter Burke Institute for Theoretical Physics, Mailcode 350-17, California Institute of Technology, Pasadena, CA 91125, USA Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS), RIKEN 2-1 Hirosawa, Wako, Saitama 351-0198, Japan Nishina Center for Accelerator-based Science, RIKEN 2-1 Hirosawa, Wako, Saitama 351-0198, Japan North Carolina State University, Raleigh, NC 27607 Numazu College of Technology, Ooka 3600, Numazu, Shizuoka 410-8501, Japan Advanced Research Institute for Science & Engineering, Waseda University, 3-4-1 Okubo, Shinjuku, Tokyo 169-8555, Japan Department of Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku, Tokyo 169-8555, Japan
ABSTRACTWe perform an extensive study of the influence of nuclear weak interactions on core-collapse supernovae (CCSNe),paying particular attention to consistency between nuclear abundances in the equation of state (EOS) and nuclear weakinteractions. We compute properties of uniform matter based on the variational method. For inhomogeneous nuclearmatter, we take a full ensemble of nuclei into account with various finite-density and thermal effects and directly usethe nuclear abundances to compute nuclear weak interaction rates. To quantify the impact of a consistent treatmentof nuclear abundances on CCSN dynamics, we carry out spherically symmetric CCSN simulations with full Boltzmannneutrino transport, systematically changing the treatment of weak interactions, EOSs, and progenitor models. We findthat the inconsistent treatment of nuclear abundances between the EOS and weak interaction rates weakens the EOSdependence of both the dynamics and neutrino signals. We also test the validity of two artificial prescriptions for weakinteractions of light nuclei and find that both prescriptions affect the dynamics. Furthermore, there are differences inneutrino luminosities by ∼
10% and in average neutrino energies by 0 . − Keywords: supernovae: general—neutrinos—hydrodynamics
Corresponding author: Hiroki [email protected]
Nagakura et al. INTRODUCTIONA star more massive than about 10 M ⊙ will build upan iron core massive enough to collapse under its owngravity. During the few tenths of a second of collapse,the stellar material of ever-increasing density emits alarge number of electron neutrinos, making the mattervery neutron-rich. Once the core reaches a few times nu-clear saturation density, the strong nuclear force quicklystiffens the equation of state (EOS), halting the collapseand driving a shock wave out through the rest of thestar. If enough energy is absorbed below the shock, theshock is driven through the rest of the star, resultingin a core-collapse supernova (CCSN, see Janka (2017)for a review). However, the mechanism by which theexplosion is driven is not well understood.In the absence of direct probes of the internal dynam-ics, simulations are used in an effort to understand theimportant physics. CCSN dynamics depend sensitivelyon the flow of energy, momentum, and lepton numberdetermined by the EOS and weak interaction rates, butboth the EOS and weak interactions are very challeng-ing to treat accurately. Properties of matter at super-nuclear densities are poorly understood at the momentdue to theoretical difficulties and weak constraints fromexperiments and observations of neutron stars (see a re-cent review by Oertel et al. (2017)). One major chal-lenge facing a theoretical description of nuclear abun-dances relevant to CCSNe is the presence of extremelyheavy nuclei at densities and electron fractions where noexperimental data are available. The uncertainty of thestate of these nuclei obscures the effects of neutrino-nucleus interactions, which can significantly influenceCCSNe.The first theoretical nuclear EOS applicable to CC-SNe simulations was developed by (Hillebrandt & Wolff1985). Subsequently, Lattimer-Swesty (Lattimer & Douglas Swesty1991) and Shen (Shen et al. 1998a,b) were developedand used extensively in the CCSNe community fordecades. The Lattimer-Swesty EOS uses Skyrme-typeinteractions with multi-body terms for uniform mat-ter and a compressible liquid drop model for non-uniform matter. On the other hand, the Shen EOSwas developed based on the relativistic mean field(RMF) approach with TM1 parameter set for uni-form matter and the Thomas-Fermi model for non-uniform matter. During the last few years, extensivestudies have been conducted on development of bet-ter EOS (see e.g., Fischer et al. (2014); Oertel et al.(2017)), and new EOS tables are available online (see e.g., stellarcollapse.org, CompOSE EOS database , andNCT ). Some are constructed using the common RMFapproach but with different nuclear parameter sets(see e.g., Hempel et al. (2012)), while others employa liquid-drop model with Skyrme type interactions(da Silva Schneider et al. 2017). The Skyrme DFTmethod proposed by Pocahontas Olson et al. (2016)is commonly used to construct EOS, but this methodis based on an approximate Hartree-Fock computa-tion that relies on Skyrme-type effective interaction.The variational method (VM) is an alternative to theSkyrme DFT method that is capable of producing anEOS with a much more realistic treatment of nuclearforces (see Sec. 2 for more details).During the collapse and post-bounce phases of CC-SNe, the matter temperature of the inner core exceeds0.5 MeV. In such a high temperature state, nuclearstatistical equilibrium (NSE) is achieved and the nu-clear abundances are determined entirely by the den-sity, temperature, and electron fraction. The outer coreat large radii is at a lower density and temperature,and a full reaction network would be needed to obtainthe most realistic nuclear abundances in this region. Inboth the Lattimer-Swesty and Shen formulations, theabundances of heavy nuclei are computed under thesingle nucleus approximation (SNA), in which the en-semble of heavy nuclei is accounted for using a singlerepresentative nucleus. However, the average mass andcharge number of heavy nuclei in the SNA are signif-icantly different from those computed using a full en-semble of nuclei in some thermodynamical states rele-vant to CCSNe (Souza et al. 2009; Burrows & Lattimer1984). In addition, the full ensemble of nuclear data ismandatory to evaluate accurate electron capture rates,which is one of the most important input physics for thegravitational collapse of an iron core (Langanke et al.2003; Hix et al. 2003). Indeed, Sullivan et al. (2016)and Furusawa et al. (2017c) showed that electron cap-ture rates are dominated by ∼
100 nuclei during thecollapse phase, which should be included in CCSN sim-ulations to accurately model deleptonization of the innercore.Weak interactions other than electron capture also de-pend on the nuclear species, which means imposing theSNA in simulations of CCSNe carries the risk of makingneutrino-matter interaction rates incorrect. Motivatedby the need for a realistic EOS, multi-nuclear EOS weredeveloped to account for an ensemble of atomic nuclei https://compose.obspm.fr http://user.numazu-ct.ac.jp/~sumi/eos ASTEX Comparing treatments of weak reactions with nuclei in simulations of core-collapse supernovae M NS ∼ M ⊙ (Demorest et al. 2010;Antoniadis et al. 2013) provide a stringent constrainton lower limit of the maximum mass of a neutronstar. More recently, a joint analysis of the gravitationalwaves and their electromagnetic counterparts from theneutron star merger event GW170817 (Abbott et al.2017a,b,c) suggests that the maximum neutron starmass is M NS . . M ⊙ (Margalit & Metzger 2017;Shibata et al. 2017). In addition, according to an analy-sis of the tidal deformabilities of neutron stars obtainedfrom GW170817 by Shibata et al. (2017), the neutronstar radius for M NS ∼ . M ⊙ should be less than 13km.This is consistent with other constraints by X-ray ob-servations (see e.g., Steiner et al. (2013)). Althoughambiguity remains, all of these constraints drasticallynarrow down possible candidates of the nuclear EOSfor uniform matter. The allowable parameter space isso narrow, in fact, that both the Lattimer-Swesty andShen EOS do not satisfy all constraints. The Shen EOSpredicts a cold neutron star radius for M NS = 1 . M ⊙ of around 14 . . The weak reaction rate table was computed byold nuclear-statistical-equilibrium (NSE) prescriptions(Furusawa et al. 2011, 2013b). In this study we improveon this comparison by performing simulations that useboth EOS with weak interaction rates consistent withnuclear abundances in each EOS. Sullivan et al. (2016);Titus et al. (2018) also recently investigated the sen-sitivity of CCSNe to electron-captures by heavy nucleiunder the consistent treatment of nuclear abundance be-tween EOS and electron capture of heavy nuclei. Theyfocus mainly on the impact of uncertainties of reactionrates but did not discuss the influence of inconsistencyof nuclear abundance between EOS and weak reactionrates, which we will address in this paper. In addition,we also study the influence of electron and positroncaptures by light nuclei on CCSNe, which were alsoneglected in these previous papers.This paper is organized as follows. We first introducethe essence of constructing multi-nuclear VM EOS inSec. 2 and new electron and positron capture rates ofnuclei in Sec. 3. In Sec. 4, we summarize numericalset up for CCSN simulations. We present the resultsof CCSN simulations in Sec. 5, which is divided intothree subsections. In Sec. 5.1, we apply our new VMand FYSS EOS to spherically symmetric CCSN simula-tions of a 11.2 M ⊙ progenitor in Woosley et al. (2002),compare the difference of CCSN dynamics between twoEOS, and discuss the differences between this and ourprevious study (Furusawa et al. 2017c). In Sec. 5.2, weinvestigate the influence of light nuclei by performingtwo more simulations with artificial prescriptions of lightnuclei and quantify the errors. In Sec. 5.3 we study theprogenitor dependence of these results by carrying outsimulations for 15, 27 and 40 M ⊙ progenitors and discussthe ”Mazurek’s law”, mass-radius relation of protoneu-tron star (PNS), and neutrino signals. Finally we wrapup this paper with summary in Sec. 6. MULTI-NUCLEAR VARIATIONAL EOSIn this work, we use a recently-developed EOS basedon the variational method (VM) (Togashi & Takano2013; Togashi et al. 2017), in which the two-body nu-clear potential is expressed with the Argonne v18 po-tential (Wiringa et al. 1995) and three-body forces areincluded to satisfy the experimental constraints of thesaturation properties. This EOS was recently extendedto include an ensemble of nuclei Furusawa et al. (2017c), See also other previous works e.g., Couch (2013); Suwa et al.(2013); Pan et al. (2018) for the study of EOS dependence of CC-SNe simulations.
Nagakura et al. employing the same framework as in Furusawa et al.(2017b). In addition, this new multi-nuclear VMEOS takes into account washout of shell effects inthe nuclear-statistical-equilibrium (NSE) computations,which brings a large change to weak interaction ratesat high temperatures ( ∼ http://user.numazu-ct.ac.jp/~sumi/eos/. In this section, we describe the essence of physicsin multi-nuclear VM EOS (Togashi & Takano 2013;Togashi et al. 2017; Furusawa et al. 2017c). The freeenergy density is calculated with variational many-bodytheory. The Hamiltonian consists of the kinetic, theAV18 two-body potential (Wiringa et al. 1995) andthe UIX three-body potential (Carlson et al. 1983;Pudliner et al. 1995) terms. This formulation is ex-tended to non-uniform nuclear matter at sub-nucleardensities to compute the free energy density f n,p of un-bound nucleons that drip from heavy nuclei, as well asthe bulk energies E bulkAZ of heavy nuclei. The free energydensity of non-uniform nuclear matter is given by f = ηf n,p + X AZ n AZ × (cid:26) κT (cid:20) ln (cid:18) n AZ g AZ ( M AZ T / π ~ ) / (cid:19) − (cid:21) + M AZ (cid:27) , (1)where n AZ , M AZ , and g AZ are the number density,mass, and internal degrees of freedom for nucleus withthe mass number A and atomic number Z . The quan-tities f n,p are defined as the local free energy densityfor dripped nucleons in the vapor volume V ′ . The lat-ter is calculated as V ′ = V − V N where V is the totalvolume and V N is the volume occupied by nuclei. Theexcluded volume effects for nucleons and nuclei are ac-counted for in η = V ′ /V = 1 − P AZ ( n AZ A/n sAZ ) and κ = 1 − n B /n , respectively, where n is the nuclearsaturation density of symmetric matter. We also takeinto account the dependence of the saturation densitiesof heavy nuclei n sAZ on the iso-spin of each nucleus, theproton fraction Z/A , and temperature.The mass of a heavy nucleus with atomic number 6 ≤ Z ≤ A ≤ M AZ = E bulkAZ + E surfAZ + E CoulAZ + E shellAZ . (2)The surface energy E surfAZ and Coulomb energy E CoulAZ depend on number densities of uniformly-distributeddense electrons and dripped nucleons, and on shapechanges of heavy nuclei from normal droplets to bubblesjust below nuclear normal density. The shell energies of heavy nuclei E shellAZ , which represent quantum effectssuch as neutron- and proton-magic numbers and pair-ing, are taken from experimental and theoretical massdata (Audi et al. 2014; Koura et al. 2005). The tem-perature dependencies of E shellAZ and g AZ are also phe-nomenologically taken into account. The mass modelof the light nuclei with Z < M AZ = M dataAZ + ∆ E CoulAZ + ∆ E P auliAZ + ∆ E selfAZ where M dataAZ isthe experimental mass, ∆ E CoulAZ is Coulomb energy shift,and a quantum approach is incorporated to evaluatePauli-energy shift ∆ E P auliAZ and self-energy shift ∆ E selfAZ (R¨opke 2009; Typel et al. 2010).We also use an EOS computed with relativisticmean field theory (RMF) using the TM1 parameter set(Furusawa et al. 2017b), which we call the FYSS EOS,as a point of comparison for the VM EOS describedabove. ELECTRON AND POSITRON CAPTURESWe calculate weak interaction rates of nuclei consis-tently with the nuclear abundances provided by theEOS. The electron capture rates for heavy nuclei areevaluated in the same way as in Furusawa et al. (2017a).For some nuclei, we use theoretical reaction data inLanganke & Mart´ınez-Pinedo (2000), Langanke et al.(2003), Oda et al. (1994) and Fuller et al. (1982), whichare based on the shell model or its extension. It shouldbe noted, however, that these theoretical computa-tions do not cover the all of nuclei that appear inCCSNe. We adopt an analytical function of the Q -value (Langanke et al. 2003) for electron capture ratesonto neutron-rich and/or heavy nuclei where data isunavailable : λ AZ = (ln2) BK (cid:18) Tm e c (cid:19) × (cid:2) F ( η AZ ) − χ AZ F ( η AZ ) + χ AZ F ( η AZ ) (cid:3) , (3)where K = 6146 sec., χ AZ = ( Q AZ − ∆ E ) /T , η AZ =( µ e + Q AZ − ∆ E ) /T , µ e is the electron chemical poten-tial , and F k is the relativistic Fermi integral of order k .The Q -value is defined as Q AZ = M AZ − M A,Z − . Itshould be noted that our treatment of nuclear masses isdifferent from that used by Langanke et al. (2003) whouse data available for isolated nuclei to set the nuclearmasses in dense matter. By contrast, we utilize the orig-inal mass formula to take into consideration various in-medium effects such as the surface tension reduction and We refer the reader to Fig. 5 in Furusawa et al. (2017a) whichdisplays the corresponding data or formula for each nucleus in(N,Z) plane.
ASTEX Comparing treatments of weak reactions with nuclei in simulations of core-collapse supernovae A & ν e + H ←→ e − + p + p, (4)(ponn) : ¯ ν e + H ←→ e + + n + n, (5)(el2h) : ν e + n + n ←→ e − + H , (6)(po2h) : ¯ ν e + p + p ←→ e + + H , (7)(el3he) : ν e + H ←→ e − + He , (8)(po3h) : ¯ ν e + He ←→ e + + H . (9)For neutrino absorptions on deuterons (Eqs. 4 and 5), weuse vacuum cross-section data (Nakamura et al. 2001).To account for the medium modifications of deuteronmass (i.e., ∆ E CoulAZ , ∆ E P auliAZ , and ∆ E selfAZ ), we intro-duce the shifted neutrino injection energy E ∗ ν = E ν + m ∗ H − m H , where the deuteron mass is given by m H invacuum and m ∗ H in medium. The in-medium deuteronmass is evaluated by the same mass model in the EOS.The neutrino absorption rate of Eq. (4) is then1 /λ ( E ν ) = n H Z d p e (cid:20) d σ ν H d p e ( E ∗ ν ) (cid:21) (1 − f e ( E e )] , (10)where n H is deuteron number density and f e denotesthe Fermi-Dirac distribution of electrons, and similarlyfor Eq. (5). We evaluate the rates of the electron andpositron capture on two nucleons forming a deuteron(leftward reactions of Eqs. (4) and (5)) through the de-tailed balance with the absorption rate. We also ignoreother minor reactions involving deuterons, such as pairprocesses and neutral-current breakup reactions, since they are less dominant than the charged current in-teractions of deuterons as described in Eqs. (4) to (7)(Furusawa et al. 2013a; Nasu et al. 2015).We evaluate the rates of electron and positron cap-ture on deuterons (Eqs. (6) and (7)) under the assump-tion that the matrix elements of electron and positroncaptures are equivalent to those of neutrino absorptions(Eqs. (4) and (5)). This assumption is reasonable forCCSNe conditions, in which the energy deposited to therelative motion between two nucleons is negligible. Theresult is that d σ el2h d p ν ≈
12 d σ ν H d p e , (11)where the factor of 2 comes from the difference in spindegrees of freedom between neutrinos and electrons.The three-nucleon nuclei H and He interact withneutrinos via breakup or charge exchange, the latter ofwhich is the dominant neutrino opacity source. There-fore, we treat only the charge exchange reaction as de-scribed in Eqs. (8) and (9). We calculate those ratesas1 /λ ( E ν ) = n H (cid:20) G F V ud π ( ~ c ) (cid:21) p e E e [1 − f e ( E e )] B ( GT ) , (12)where n H is triton number density, B ( GT ) = 5 . V ud = 0 .
967 as in Fischer et al. (2016). We do nottake in-medium effects on nuclear masses into accountin these reactions. We do not expect this assumption tohave a large effect, but further work would be requiredto determine the quantitative effects.We neglect inelastic scatterings between alpha-particles and neutrinos in this study for simplicity andbecause, similar to the recoil treatment in nucleon-neutrino interactions, the energy exchange by scat-terings between neutrinos and alpha-particles is quitesmall. However, inelastic scattering with alpha parti-cles may still play a non-negligible role to the shockrevival if the shock wave has reached close to the revival(Ohnishi et al. 2007), though the precise treatment ofsuch a small energy exchange is technically challengingfor mesh-based methods. NUMERICAL SET UP
Table 1 . Summary of modelsModel EOS ECPH EPCPL Progenitor Mass( M ⊙ )V112 VM new new 11.2 Table 1 continued
Nagakura et al.
Table 1 (continued)
Model EOS ECPH EPCPL Progenitor Mass( M ⊙ )F112 FYSS new new 11.2OV112 VM old no 11.2OF112 FYSS old no 11.2NV112 VM new no 11.2DV112 VM new nucleons 11.2V15 VM new new 15V27 VM new new 27V40 VM new new 40 Note —The ECPH column denotes whether rates of electron cap-ture on heavy nuclei are inconsistent (old) or consistent (new)with nuclear abundances in the EOS. The EPCPL column de-notes whether electron and positron captures on light nuclei areincluded (new), neglected (no), or treated as weak interactionson free nucleons (nucleons). OV112 and OF112 are the samemodels as in Furusawa et al. (2017c).
All simulations presented in this paper are per-formed by our multi-dimensional (multi-D) neutrinoradiation hydrodynamics code (Nagakura et al. 2014,2017). We solve the special relativistic Boltzmannequations for neutrino transport. We employ the cen-tral scheme (Kurganov & Tadmor 2000) for numericallysolving the Newtonian hydrodynamics equations (seealso Nagakura et al. 2011). We solve the Poisson equa-tion for Newtonian gravity using the mass integrationmethod. Details of the formulation and numerical meth-ods in our code were presented in a series of our previ-ous papers (Sumiyoshi & Yamada 2012; Nagakura et al.2014; Shibata et al. 2014; Nagakura et al. 2017) andthe reliability of our code was established by a de-tailed comparison with Monte-Carlo transport methodin Richers et al. (2017a). For all simulations, we em-ploy the same weak interaction rates as those used in(Nagakura et al. 2018), except for electron and positroncaptures on nuclei. In our simulations, we cover thespatial domain of 0 ≤ r ≤ ≤ ǫ ≤ ◦ ≤ ¯ θ ≤ ◦ ). The momentum-space resolution is almost the same as that in otherspherically symmetric Boltzmann simulations (see e.g.,Sumiyoshi et al. (2008); Lentz et al. (2012b,a)). Asshown in Richers et al. (2017a), however, the limitedresolution yields results in errors at some level. Forinstance, neutrino luminosities at our current resolu- tion are likely underestimated by several percent due tonumerical diffusion. Though fully resolved calculationsare desirable, we do not expect that qualitative trendsin the weak dependence of CCSN dynamics on weakinteractions vary strongly with resolution.Tab. 1 lists a summary of the models simulated in thiswork. V112 is our fiducial model which employs 11.2 M ⊙ progenitor in (Woosley et al. 2002), VM EOS andthe improved electron and positron captures on nuclei asdescribed in Sec. 3. F112 is identical to V112, except inthat it employs the FYSS EOS. Note that the weak re-action rates in F112 are consistently computed with thenuclear abundances provided by the FYSS EOS. ModelsOV112 (VM EOS) and OF112 (FYSS EOS) are from ourprevious work (Furusawa et al. 2017c). These modelsemploy an old FD+RPA prescription (Juodagalvis et al.2010) for electron captures on heavy nuclei and neglectelectron and positron captures on light nuclei. In modelNV112, all input physics are the same as V112 exceptthat weak interactions with light-nuclei are neglected.In model DV112, weak interactions with light nuclei ex-cept for alpha particles are artificially replaced by weakreactions with their constituent free nucleons, which issimilar to the treatment of light nuclei in previous stud-ies that employ the SNA. Models V15, V27 and V40use the same physics as V112, but use 15 M ⊙ , 27 M ⊙ ,and 40 M ⊙ progenitors, respectively. We run simula-tions up to 400ms after the bounce for V112, and 500msfor other progenitors. We stop the simulation of V112earlier than those of other progenitors since the outer-most mass shell of the progenitor in our computationaldomain would reach the shock wave between 400ms and500ms. RESULTS5.1.
VM EOS vs. RMF EOS
Figure 1 shows the time evolution of central densityduring the collapsing phase for V112, F112, OV112 andOF112. The contraction of inner core in V112 and F112evolves more slowly than in previous models (OV112 andOF112) , which is due to the difference in deleptoniza-tion rates. This is apparent in Fig. 2, which shows thatboth the rates of cooling and deleptonization are consid-erably slower in the new models. The stable nuclei with N = 28 (e.g., Fe) are very susceptible to electron cap-ture and generate high deleptonization rates (see alsoFig. 9 in Furusawa et al. (2017a)). Our new NSE modeltakes the washout of the shell effect into account, which Hereafter, we sometimes collectively refer V112 and F112 as”new models”, and OV112 and OF112 as ”previous models.”
ASTEX Comparing treatments of weak reactions with nuclei in simulations of core-collapse supernovae
0 20 40 60 80 100 120 140 160 180 ρ [ g / c m ] Time [ms]V112F112OV112OF112
Figure 1.
Time evolution of central density during the col-lapsing phase for V112, F112, OV112 and OF112. reduces the nuclear abundance of A ∼
50 ( N ∼ ρ c = 10 g cm − in Fig. 3. The top two panels show that both the delep-tonization and cooling rates in the inner core are higherin models using the VM EOS than in models using theFYSS EOS. As discussed in Furusawa et al. (2017c), thisis mainly due to the temperature difference resultingfrom the different EOSs where the temperature tendsto be higher in VM EOS than in FYSS EOS (see e.g.,the right middle panel in Fig. 3). The VM EOS has asmaller symmetry energy of nuclei than the FYSS EOS,which yields smaller mass fraction of heavy nuclei. Asa result, the entropy per baryon and the adiabatic in-dex tend to be higher in the VM EOS than in the FYSSEOS, which facilitate the increase of temperature duringadiabatic contraction.As displayed in the middle left panel of Fig. 3, the dif-ference of electron fraction at the center between mod-els V112 and F112 is larger than the difference betweenmodels OV112 and OF112. This is attributed to the factthat the previous models artificially suppress the differ-ence between EOS by sharing between them a singlecommon table of electron capture rates on heavy nu-clei. Both deleptonization and cooling rates (upper twopanels of Fig. 3) for r . cm in the new models arehigher than those of the previous models, though the op-posite is true outside of this region just as in the initial conditions (see Fig. 2). The difference of deleptonizationbetween new and old models also come from another rea-son which is the different treatment of electron captureon A &
100 heavy nuclei. Because of the more rapiddeleptonization and neutrino cooling in the new mod-els, the contraction of inner core proceeds faster thanthose in the previous models at this phase. The modelsusing the VM EOS (V112 and OV112) cool and delep-tonize faster than models using the FYSS EOS (F112and OF112). Note that at this early time, though thedeleptonization rate in model OF112 is lower than thatin model F112, the central electron fraction in modelF112 is larger than in model OF112. This is just a ves-tige of the opposite ordering of deleptonization rates atthe onset of collapse (Fig. 2).We also find that models using the VM EOS show abigger difference of neutrino reactions between new andprevious models than models using the FYSS EOS (seeFig. 2 and upper panels in Fig. 3). As we have alreadymentioned, the models with the VM EOS have higherinner core temperature than do models with the FYSSEOS. Since higher temperature facilitates deleptoniza-tion by more electron capture, the difference betweennew and previous models is more prominent when usingthe VM EOS.The dynamics deviate remarkably between V112 andF112 with increasing the central density. In Figs. 4 and5, we display the same quantities as in Fig. 3, but atlater times during the collapse phase when the centraldensity ρ c reaches 10 and 10 g cm − , respectively.The electron fraction remains lower and the temperaturehigher in model V112 than in model F112 at these latertimes. Since both deleptonization and neutrino coolingare suppressed by neutrino absorption and scattering inthe high density region, the matter profile becomes lesssensitive to the difference of electron capture rates andthe scattering opacities for neutrinos. As a result, thedifference of electron fraction and temperature betweentwo models remains locked in even in the late phases ofcollapse. Indeed, the difference in deleptonization ratebetween V112 and F112 is subtle at the center (see e.g.,top left panel in Fig. 5) despite the fact that the scat-tering opacity in V112 is remarkably lower than F112.This difference in opacity is apparent in the bottom leftpanel of Fig. 5, noting that opacity of coherent scatter-ing of neutrinos by heavy nuclei is proportional to ¯ A .The difference of ¯ A is mainly attributed to the differ-ence of temperature. In fact, the ordering of ¯ A followsthe opposite ordering of temperature (see middle rightpanel of Fig. 5).Fig. 6 shows the enclosed mass as a function of radiusat the time of core bounce. The outer core in models Nagakura et al. -0.16-0.12-0.08-0.04 01e+06 1e+07 1e+08 1e+09 Q N [ / s ] Radius [cm] V112F112OV112OF112 -8-6-4-2 01e+06 1e+07 1e+08 1e+09 Q e [ e r g / g / s ] Radius [cm]
Figure 2.
Radial profiles of deleptonization rate ( Q N , left) and neutrino cooling rate ( Q e , right) during the collapse of a 11 . M ⊙ progenitor when the central density is ρ c = 6 × g cm − which corresponds to the time of beginning of our simulation. V112and F112 employ the VM and FYSS EOS, respectively. OV112 and OF112 are similar, but use an old prescription for weakinteraction rates on heavy nuclei that is inconsistent with the nuclear abundances in the EOS. Consistency between weakinteractions and nuclear abundances in the EOS results in significantly slower cooling and deleptonization. using the VM EOS (V112 and OV112) are less compactthan that those in models using the FYSS EOS (F112and OF112) for r . × cm. Above this radius, thecompactness ( M/r ) in models using the same neutrinophysics converges. However, the new models (V112 andF112) have slightly larger compactness than the previ-ous (OV112 and OF112) models. This is also due to thepreviously mentioned detail that the new models col-lapse more slowly. Though this makes the inner core(left panel) less dense and increases the length of timebetween the onset of collapse and core bounce. Thedifferent neutrino interactions between the old and newmodels have little effect on the matter at large radii(right panel), so an increased collapse time in the newmodels allows the matter at large radii more time to fallin before core bounce, resulting in higher compactness.We display radial profiles of several important quanti-ties at the time of core bounce (defined as when the post-shock entropy per baryon first reaches 3 k B ) in Fig. 7as a function of mass coordinate. As was apparent inFig. 6, we find that the new models (V112 and F112)have systematically lower central density and electronfraction but higher temperature and entropy than thosein the old models (OV112 and OF112). The systematicdifferences can be interpreted as follows. Except at theonset of collapse, the deleptonization rate is larger inthe new models, which yields smaller electron and pro- ton fractions in iron core. The smaller proton fractionmakes nuclear matter stiffer since the repulsive force isenhanced by the larger asymmetry between proton andneutron mass fractions. In addition, the smaller elec-tron fraction leads to the higher temperature becauseof the reduction of electron degeneracy pressure duringcollapse. Thus, the thermal contribution to the totalpressure in the new models is larger than in the previousmodels. Because of these two effects, the core bouncein the new models takes place at a lower central densitythan the previous models. The lepton fraction is directlyrelated to the inner core mass at bounce, so the orderingof the mass coordinate of the shock at bounce (entropyplot in Fig. 7) follows the ordering of the lepton fractionin the inner core.As shown in the third panel on the right in Fig. 7,there is a substantial amount of light nuclei just underthe shock wave. This tells us that the prompt shockwave is not powerful enough to completely photodisin-tegrate heavy nuclei into nucleons (see e.g., Figs. 10). Aswe will discuss later, various light nuclei are generatedin the almost entire region of post-shock flows in thepost-bounce phase. The role of light nuclei to CCSNdynamics and neutrino signals will be analyzed in thenext subsection.Despite the fact that there are many differences be-tween V112 and F112 during the collapse and bounce ASTEX Comparing treatments of weak reactions with nuclei in simulations of core-collapse supernovae -2-1.5-1-0.5 0 Q N [ s - ] V112F112OV112OF112 -9-6-3 0 Q e [ e r g g - s - ] Y e T [ M e V ]
50 60
70 80 A - Radius [cm] 0.9 0.92 0.94 X h Radius [cm]
Figure 3.
Comparison between different EOS and weak interaction assumptions during the collapse phase when the centraldensity is ρ c = 10 g / cm . Q N is the net leptonization rate, Q e is heating rate, Y e is electron fraction, T is the temperature, ¯ A is average number of nucleons in heavy nuclei, and X h is the mass fraction of heavy nuclei. V112 and F112 employ the VM andFYSS EOS, respectively. OV112 and OF112 are similar, but use an old prescription for weak interaction rates on heavy nucleithat is inconsistent with the nuclear abundances in the EOS. Nagakura et al. -12-9-6-3 0 Q N [ s - ] V112F112OV112OF112 -9-6-3 0 Q e [ e r g g - s - ] Y e T [ M e V ]
50 60 70
80 90 100 A - Radius [cm] 0.8 0.84 0.88 X h Radius [cm]
Figure 4.
Comparison between different EOS and weak interaction assumptions during the collapse phase when the centraldensity is ρ c = 10 g / cm . Quantities and models are the same as in Fig. 3. ASTEX Comparing treatments of weak reactions with nuclei in simulations of core-collapse supernovae -30-20-10 0 Q N [ s - ] V112F112OV112OF112 -6-4-2 0 Q e [ e r g g - s - ] Y e T [ M e V ]
60 80 100
120 140 A - Radius [cm] 0.8 0.84 X h Radius [cm]
Figure 5.
Comparison between different EOS and weak interaction assumptions during the collapse phase when the centraldensity is ρ c = 10 g / cm . Quantities and models are the same as in Fig. 3. Nagakura et al. M [ M ⊙ ] Radius [10 cm]V112F112OV112OF112 0.8 0.9 1 1.1 1.2 1.3 1 2 3 4 5 6 M [ M ⊙ ] Radius [10 cm] Figure 6.
Enclosed mass versus radius at the time of core bounce. The left panel shows the region where 10 cm ≤ r ≤ cmand the right panel shows the region where 10 cm ≤ r ≤ × cm. ASTEX Comparing treatments of weak reactions with nuclei in simulations of core-collapse supernovae ρ [ g / c m ] V112F112OV112OF112 0.2 0.4 0.6 0.8 1 X h T [ M e V ] X a s [ k B ]
20 40 60 A | Y e Mass coordinate [M ⊙ ] | Y l Mass coordinate [M ⊙ ] | Figure 7.
Comparison between different EOS and weak interaction assumptions at core bounce. From top to bottom, we plotbaryon mass density ( ρ ), temperature ( T ), entropy per baryon ( s ) and electron fraction ( Y e ) in the left column, while we plotthe mass fraction of heavy nuclei ( X h ), the mass fraction of light nuclei ( X a ), the average mass number ( ¯ A ) and the leptonfraction ( Y l ) in the right column. All quantities are displayed as a function of mass coordinate in this figure. Nagakura et al. R s [ k m ] T b [ms] V112F112OV112OF112 100 150 200 25030 60 90 120 150 R s [ k m ] T b [ms] Figure 8.
Evolution of the shock radius as a function of time after core bounce. The right panel magnifies the peak region.Models V112 and OV112 use the VM EOS, while models F112 and OF112 use the FYSS EOS. Models V112 and F112 use anew prescription for weak interactions on nuclei, while models OV112 and OF112 use an old prescription (see Secs. 2 to 3).
ASTEX Comparing treatments of weak reactions with nuclei in simulations of core-collapse supernovae ∼ b > R s < ∼ ∼ ∼ ν e ) luminosity, which can be clearly seen in the inseton the top left panel of the figure. Note that we mea-sure the neutrino luminosity at r = 500km, and it takes ∼ . r ∼ L p ) for allmodels are in the range 4 < L p / (10 erg s − ) <
5. Aswe can see in this figure, model V112 has a smaller L p than those in F112. This may be attributed to the factthat model F112 has a slightly stronger bounce shockwave than model V112. As discussed already, the delep-tonization of the iron core during the collapsing phasein F112 is less active than V112, which forms a largerinner core and then generates a stronger shock wave inF112.After the neutronization burst, the time evolution of ν e luminosity is almost identical between V112 and F112until T b ∼ ν e in model V112 is higherthan in model F112. The difference can be interpreted ina similar way as the EOS dependence of the shock tra-jectory. In the late post-bounce phase, the weakeningcontribution to the post-shock flow from thermal pres-sure makes the EOS dependence of the PNS structuremore prominent. This makes the post-shock structurein model V112 is more compact than in model F112.As a result, the neutrinosphere in model V112 is at asmaller radius, which results in less luminous neutrinoemission but with higher mean energy. We also finda systematic EOS dependence of the peak of ¯ ν e lumi-nosity (50 . T b . Nagakura et al. L ν e [ e r g / s ] V112F112OV112OF112 0 10 20 30 40 50 0 5 10 15 20 9 10 11 12 13 14 E ν e [ M e V ] L ν e [ e r g / s ] | E ν e [ M e V ] | L ν x [ e r g / s ] T b [ms] |
13 14 15 16 0 100 200 300 400 E ν x [ M e V ] T b [ms] | Figure 9.
Time evolution of the neutrino luminosity (left) and the mean energy (right) for ν e (top), ¯ ν e (middle) and ν x (bottom). Models V112 and OV112 use the VM EOS, while models F112 and OF112 use the FYSS EOS. Models V112 andF112 use a new, consistent prescription for weak interactions on nuclei, while models OV112 and OF112 use and old prescription.We measure these quantities at r = 500km. ASTEX Comparing treatments of weak reactions with nuclei in simulations of core-collapse supernovae ν e is stronger than inthe FYSS EOS models.We find a systematic difference in the time evolutionof ν e luminosity between new and previous models. Asshown in the top left panel of Fig. 9, the time evolu-tion is a bit faster for the new models. This is againdue to the difference of the structure of the outer core.However, the systematic difference between the new andprevious models is not so remarkable for electron-typeanti-neutrinos (¯ ν e ) in the early post bounce phase (seethe middle left panel). Since the ¯ ν e neutrinosphere islocated at a smaller radius than the ν e neutrinosphere,it is more shielded from the accretion of the outer coreand the light curve of ¯ ν e in the early post-bounce phaseis less affected by the difference of the outer core struc-ture. This systematic difference for ¯ ν e between new andprevious models arises from ∼ . M ⊙ mass co-ordinate reaches the shock. The systematic differencesin the accretion rate are due to slightly faster collapsein the new models (see the discussion in reference toFig. 6). Interestingly, there are no such systematic dif-ferences of light curve in heavy leptonic neutrinos ( ν x )between new and previous models (see the bottom leftpanel in Fig. 9). This may be attributed to the fact thatthe characteristics of ν x signals are determined deeperinside of PNS than other species, which is less sensitiveto the structure of outer core, at least up to ∼ Influence of light nuclei
We turn our attention to the impact of light nucleion CCSNe. In this study, we analyze three models withdifferent treatments of weak reactions with light nuclei.Our fiducial model V112 takes into account electron andpositron captures on light nuclei self-consistently withVM EOS. Model NV112 artificially neglects all interac-tions with light nuclei. Weak reactions with light nucleiin model DV112 are artificially treated as if the lightnuclei were dissociated into free nucleons. See Sec. 4 formodel details.In the collapse phase, no remarkable differences arefound in the time evolution of fluid dynamics and theneutrino signals among three models, even though theabundance of light nuclei is not small. Indeed, thereare some regions with X a >
10% (see e.g., Fig. 11in Furusawa et al. (2017c)). Such an insensitiveness tolight nuclei in the collapse phase is attributed to thefact that the large weak reaction rates relevant to heavynuclei such as the electron capture and coherent scat-tering make the effect of light nuclei imperceptible. The effect of light nuclei becomes non-negligible in the post-bounce phase, when heavy nuclei become less dominantas a result of photodissociation of heavy nuclei in thepost-shock region.Therefore, we only focus on the post-bounce phasebelow. Before comparing among three models in detail,we first analyze some important characteristics of lightnuclei based on the results of V112, and then quantifyhow these prescriptions change both CCSN dynamicsand neutrino signals.5.2.1.
Characteristics of light nuclei in CCSNe
The radial distributions of mass fractions of light andheavy nuclei in the early post bounce phase (T b ≤ X a rises withthe rapid drop of X h . We first look at the propertiesof X a in the post-shock region. As mentioned alreadyin Sec. 5.1, the energy of the shock wave is not enoughto completely decompose heavy nuclei into nucleons; in-stead some light-nuclei are formed. In the post-shockflows, the mass fraction of light nuclei decreases inwardfrom the shock, and then again increases around themass shell of ∼ . M ⊙ . Such an excess of light nuclei issustained for a long time during the post-bounce phase.The generation and durability of the excess of light nu-clei can be understood as follows. At the core bounce,the shock wave is generated around the mass shell of ∼ . M ⊙ in model V112. In the very early phase, theabundance of light nuclei ( X a ) behind the shock waveincreases with the shock propagation. Once the shockwave is strong enough to destroy heavy nuclei into manynucleons, X a decreases. As a result, the nascent shockdraws the spike profile of radial distributions of light nu-clei. Because of the large neutrino opacity, the matterevolves almost adiabatically with time. Although thedensity increases with time by the contraction of PNS,the constant entropy and proton fraction (right panelsin Fig. 10) work to sustain the same X a . Note that theentropy and proton fraction change in the neutrino dif-fusion time scale, and so the characteristic time scale ofchange of the abundance of light nuclei is also dictatedby the neutrino diffusion.Interestingly, the mass fraction of light nuclei in thepre-shock region (in particularly close to the shock wave)drastically change with time in the early post-bouncephase as shown in the top left panel in Fig. 10. Thisis due to the fact that many neutrinos are absorbed inthe pre-shock matter. At the shock front, neutrinos areabsorbed mostly by free protons, but ∼
10% and ∼ Nagakura et al. X a bounce3ms5ms10ms 0 0.2 0.4 0.6 0.8 0.4 0.6 0.8 1 1.2 X h Mass coordinate [M ⊙ ] 0.1 0.2 0.3 0.4 0.5 Y e s [ k B ] Mass coordinate [M ⊙ ] Figure 10.
Radial profiles of the mass fraction of light nuclei (top left), heavy nuclei (bottom left), electron fraction (top right),and entropy per baryon (bottom right) in model V112. Red, blue, green and black colors denote 0ms, 3ms, 5ms and 10ms afterthe bounce, respectively.
Fig. 11 shows that significant amounts of light nucleicontinue to exist in the post-shock region in the midand late post-bounce phases. Almost all nuclei are pho-todissociated into light nuclei (mainly alpha-particles)right behind the shock wave, which is clearly visible inthe snapshot of 100ms after bounce in the left panel ofFig. 11. This phase roughly corresponds to the timewhen the shock wave reaches the maximum radius inV112 (see e.g., Fig. 8). As shown in the right panel ofFig. 11, the mass fraction of light nuclei is more thantwice that of nucleons. On the other hand, the abun-dance of light nuclei decreases as the stalled shock waverecedes, which is displayed in two lines for 200ms and400ms after bounce in the left panel. This is becausethe kinetic energy per baryons in the pre-shock accre-tion flows is larger for the smaller shock radius. As a result, the post-shock temperature tends to be high, andthen the photodissociation of heavy nuclei become morecomplete. Therefore, the effect of light nuclei on thedynamics of CCSNe declines with time. It should benoted, however, that the influence of light nuclei wouldbe enhanced in multi-D cases since the shock radius isgenerally larger than in spherically symmetric simula-tions.Next, let us take a look at the abundance of individ-ual components of light nuclei. Their radial distribu-tions are displayed in Fig. 12 at 5ms (left) and 100ms(right) after bounce. As can be seen in both panels,deuterons are the dominant light nucleus around thesurface of PNS, but the mass fraction declines withincreasing radius. Instead, alpha particles dominateclose to the shock wave. It should be noted, however,
ASTEX Comparing treatments of weak reactions with nuclei in simulations of core-collapse supernovae X a Radius [km] X a X p X n X h T b =100ms Figure 11.
Left: Radial distribution of mass fraction of light nuclei in the post-bounce phase for model V112. Color distinguishesthe different post-bounce times of 50ms (red), 100ms (blue), 200ms (green) and 400ms (black) after bounce, respectively. Right:radial distributions of mass fractions of light nuclei (red), free protons (blue), free neutrons (green) and heavy nuclei (black) atthe 100ms after bounce in model V112. -3 -2 -1 X D X T X H X α X O T b =5ms -6 -5 -4 -3 -2 -1 T b =5ms T b =100ms Figure 12.
Radial profiles of mass fraction of each light nucleus. Different colors denote mass fractions of deuteron ( X D , red),triton ( X T , blue), heliton ( X H , green), alpha ( X α , purple) and the sum of other light nuclei ( X O , black), respectively. The leftand right panels display T b = 5ms and T b = 100ms, respectively. Nagakura et al. that the energy transfer to neutrinos by reactions withdeuterons is comparable to that with nucleons and isroughly ten times more efficient than that with alphaparticles (Sumiyoshi & R¨opke 2008; Arcones et al. 2008;Hempel et al. 2012; Furusawa et al. 2013a). This makesdeuterons an effective source of opacity even where themass fraction is small, giving them a primary role forboth cooling and heating.In Fig. 13, we display radial profiles of frequency-integrated emissivities of neutrinos by weak interac-tions with light nuclei (Eqs. (4) to (9)), along withemission from electron capture, positron capture,electron-positron pair annihilation, and nucleon-nucleonbremsstrahlung radiation. As displayed in these pan-els, contributions from light nuclei are subdominant toelectron and positron capture in the entire post-bouncephase, even though the mass fraction of light nucleioverwhelms that of nucleons in some regions (see alsoFischer et al. (2016)). This is attributed to the factthat both electron and positron captures by light nu-clei require more energy than those in nucleons sincethey need to break up or excite from the nuclear boundstate, which results in reducing the reaction rates aswell as decreasing the average energy of emitted neu-trinos. We find that the next-dominant emissivity atT b = 5 ms comes from electron capture on deuterons(el2h), which contributes ∼
10% of total ν e emissivity at r ∼ b = 100ms electron neutrino absorptionon deuterons (elpp) overwhelms the emissivity of ”el2h”in the region r > < ν e emissivities of light nuclei in the cooling andheating regions, respectively. However, they are < ν e , which means their impacton CCSNe is weak.Weak reactions with light nuclei should predominantlyaffect the low-energy end of the neutrino energy spec-trum. Indeed, as shown in upper panels of Fig. 14,”el2h” overwhelms the emissivity of ”ecp” at r . . r ∼ ∼ . ν e the emissivity frompositron captures by light nuclei never overwhelm ”pcp”,which is simply because the mass fraction of neutrons isremarkably larger than others.5.2.2. Impact of artificial prescriptions of light nuclei
In Sec. 5.2.1, we discuss characteristics of light nu-clei and find some evidence that the impacts of electronand positron captures of light nuclei on CCSNe are sub-dominant. As we will see below, however, artificial pre-scriptions for weak reactions with light nuclei in modelsNV112 and DV112 (see Sec. 4) change the dynamics ofCCSNe and neutrino signals.We first discuss the shock trajectories in models us-ing different prescriptions for light nuclei displayed inFig. 15. All three models are almost identical frombounce to T b ∼ Q e >
0, which lies at 100km . r < R s inthis snapshot) for model NV112 (DV112) is always more(less) efficient than in model V112, which naturally ex-plains the trend in the shock radius.In Fig. 17, we display radial profiles of properties ofthe neutrino radiation field and neutrino heating rates inthe gain region. The left and right panels are for ν e and¯ ν e , respectively. In the bottom panels, we also show theenergy deposition rate of the reaction ( e − + p ↔ ν e + n )for ν e and ( e + + n ↔ ¯ ν e + p ) for ¯ ν e , which are denotedas Q n and Q p , respectively. These reactions are themain processes of neutrino heating in the gain region.Note that we show the difference from model V112 inthis figure, and red lines mark zero on y-axis. We firstfocus on the difference between NV112 and V112. Theenergy density ( E ), energy flux ( F ) and mean energy( E m ) for both ν e and ¯ ν e are consistently larger thanthose in V112 because ignoring all interactions with lightnuclei reduces the opacity of neutrinos. This facilitatesneutrino diffusion from the cooling region, which resultsin a large neutrino energy density and flux in the gainregion. The reduction of neutrino opacity also causesthe neutrinosphere to be located farther inside the PNS,which results in an increased ν e mean energy in the gainregion. As a result, ν e energy deposition in the gainregion is larger, which can be seen in the bottom panelsin Fig. 17. The mean energy of ¯ ν e in model NV112 is notdifferent from model V112 since the opacity from weakinteractions of light nuclei is much smaller than that ofnucleon scattering at the PNS surface, which is reflectedin a smaller change in heating from ¯ ν e than from ν e .Model DV112 (black lines in Fig. 17) has the oppo-site trend as NV112. Recall that light nuclei are treatedas if decomposed into free nucleons for the purposes ofweak interactions in model DV112. We first look intothe trend for ¯ ν e . As clearly seen in the right panels, all ASTEX Comparing treatments of weak reactions with nuclei in simulations of core-collapse supernovae -8 -6 -4 -2 R [ / s ] ecppairnbrel2helppel3he T b =5ms ν e -10 -8 -6 -4 -2
10 20 30 40 50 60 70 R [ / s ] Radius [km]pcppairnbrpo2hponnpo3h T b =5ms ν e ν e - -8 -6 -4 -2 R [ / s ] T b =5ms T b =100ms ν e ν e - ν e -8 -6 -4 -2 R [ / s ] Radius [km]T b =5ms T b =100ms ν e ν e - ν e - Figure 13.
Radial profiles of frequency-integrated neutrino emissivities by various interactions. Upper and lower panels arefor ν e and ¯ ν e , respectively. The relevant interactions are electron capture by free protons (ecp), positron capture by freeneutrons (pcp), electron-positron pair annihilation (pair), nucleon bremsstralung radiation (nbr), electron neutrino and anti-neutrino absorption on deuterons (elpp and ponn, respectively), electron and positron capture on deuterons (el2h and po2h,respectively), electron capture on He (el3he), and positron capture on H (po3h). See Eqs. (4) to (9) for reaction details. Theleft and right columns correspond to times T b = 5ms and T b = 100ms. E , F , E m and Q p are smaller than those in V112. This isattributed to the fact that the scattering with nucleonsdominates the ¯ ν e opacity. Since the abundances of nucle-ons is artificially increased in model DV112, the scatter-ing opacity is larger and the neutrino sphere shifts out-ward. This decreases the mean neutrino energy, slowsthe escape of neutrinos, and decreases neutrino energyflux and density. However, the energy density and fluxfor low energy ¯ ν e ( . ν e reaction rates deepinside the PNS, the increased mass fraction of nucle-ons significantly enhances the ¯ ν e emissivity. Since the neutrinospheres for low energy neutrinos is located ata smaller radius than those for higher energy neutrinos,low energy neutrinos in the gain region are more suscep-tible to the change in bremsstrahlung emissivity in thePNS. The increase of low energy neutrinos also reducesthe mean energy of neutrinos as displayed in the rightand third row of Fig. 17.On the other hand, the response due to the prescrip-tion for light nuclei in model DV112 for ν e is less sen-sitive than ¯ ν e . Indeed, as displayed in the the first andsecond panels on the left side of Fig. 17, the differencesof energy density and flux between models DV112 andV112 are much smaller than those for ¯ ν e , though thedifference is in the same direction as NV112. Since elec-2 Nagakura et al. -3 -2 -1 R a t i o t o ec p T b =5ms ν e E=5.0MeV -3 -2 -1
10 20 30 40 50 60 70 R a t i o t o p c p Radius [km]T b =5ms ν e ν e - E=5.0MeV -3 -2 -1 R a t i o t o ec p el2helppel3heT b =5ms T b =100ms ν e ν e - ν eE=5.0MeV -3 -2 -1 R a t i o t o p c p Radius [km]po2hponnpo3hT b =5ms T b =100ms ν e ν e - ν e - E=5.0MeV
Figure 14.
Radial profiles of the ratio of emissivities from interactions with light nuclei relative to electron capture (top) orpositron capture (bottom) for 5Mev electron neutrinos (top) and electron anti-neutrinos (bottom). The relevant interactionsare electron neutrino and anti-neutrino absorption on deuterons (elpp and ponn, respectively), electron and positron capture ondeuterons (el2h and po2h, respectively), electron capture on He (el3he), and positron capture on H (po3h) as in Eqs. (4) to (9). tron capture on free protons dominates the emissivity,the increased mass fraction of free protons enhances the”ecp” process, which causes the larger neutrino energyflux and density in the gain region. By the same to-ken, the increased absorption opacity from the inverseprocess increases the radius of neutrino sphere, whichreduces E m .Note that the difference of E m between two modelsdepends on radius as shown in Fig. 17. E m in NV112 is smaller than that in V112 for r . . r . r & E m between the two models. Indeed, the difference inenergy deposition is more sensitive to the differences inmatter profiles than to differences in the treatment of ASTEX Comparing treatments of weak reactions with nuclei in simulations of core-collapse supernovae R s [ k m ] T b [ms] V112NV112DV112 100 150 200 25030 60 90 120 150 R s [ k m ] T b [ms] Figure 15.
Evolution of the shock radius as a function of time after core bounce to see the impact of weak interactions of lightnuclei. The right panel magnifies the peak region. Model V112 (red) is the fiducial model and includes weak interactions withlight nuclei. Model NV112 (blue) neglects weak interactions with light nuclei. Model DV112 (black) treats nucleons in lightnuclei as if they were unbound and free for the purposes of weak interaction rates (see Sec. 4). -15-10-5 0 5 10 50 100 150 200 250 300 Q e [ e r g g - s - ] Radius [km] V112NV112DV112
Figure 16.
Heating rates at T b = 100ms after the bouncefor several models. Model V112 (red) is the fiducial modeland includes weak interactions with light nuclei. ModelNV112 (blue) neglects weak interactions with light nuclei.Model DV112 (black) treats nucleons in light nuclei as if theywere unbound and free for the purposes of weak interactionrates (see Sec. 4). light nuclei (see below). These two effects compete witheach other in terms of neutrino heating in the gain re-gion, though reduction in E m ends up winning out and the ν e energy deposition is smaller than in model V112(left panel of Fig. 17).To see the effects of light nuclei in more detail, wecalculate the steady-state neutrino radiation field byletting the neutrino field relax on a fixed fluid back-ground. This approach has been frequently used toanalyze the qualitative trends of neutrino-matter in-teractions in CCSNe (see e.g., Sumiyoshi et al. (2015);Richers et al. (2017a)). Given fluid distributions at100ms after the bounce in model V112, we computethe steady state neutrino radiation fields by the sameprescriptions as V112, NV112 and DV112. The radialdistributions of the differences in net gain between thedynamical and steady-state radiation fields using thesame weak interaction treatments are displayed as dot-ted lines in Fig. 18. As can be clearly seen in this figure,the differences between different prescriptions of lightnuclei in steady state models are much smaller thanthose in dynamical models. This tells us that the ar-tificial prescription mostly influences CCSN dynamicsthrough feedback to matter. The effect would be moreserious in multi-D cases, since the fluid dynamics aremore sensitive to small changes in neutrino energy de-position (Burrows et al. 2016).Finally, we discuss the influence of the two artificialprescriptions of light nuclei on the neutrino signal. Wedisplay the time evolution of neutrino luminosities andmean energies in Fig. 19. As shown in the upper pan-4 Nagakura et al. ∆ E / E [ × - ] NV112-V112DV112-V112 ν e -4-3-2-1 0 1 2 ∆ E / E [ × - ] ν e ν e - ∆ F / F [ × - ] ν e ν e - -4-3-2-1 0 1 ∆ F / F [ × - ] ν e ν e - -2-1 0 1 2 3 ∆ E m / E m [ × - ] ν e ν e - -3-2-1 0 ∆ E m / E m [ × - ] ν e ν e - -1 0 1120 140 160 180 200 ∆ Q n / Q n [ × - ] Radius [km] ν e ν e - -1 0 1120 140 160 180 200 ∆ Q p / Q p [ × - ] Radius [km] ν e ν e - Figure 17.
Radial profiles of various quantities relevant to neutrino heating in the gain region at T b = 100ms relative to modelV112. All quantities are normalized by the values in the fiducial model V112. Model V112 (red lines mark zero on y-axis) is thefiducial model. The left and right columns are for ν e and ¯ ν e , respectively. From the top, we show the neutrino energy density( E ), energy flux ( F ) and mean energy ( E m ), all measured in the fluid rest frame. The bottom panels show the net gain fromprocesses involving ν e (denoted Q n ) and ¯ ν e (denoted Q p ) from neutrino emission and absorption by free nucleons. Model V112(red) is the fiducial model and includes weak interactions with light nuclei. Model NV112 (blue) neglects weak interactions withlight nuclei. Model DV112 (black) treats nucleons in light nuclei as if they were unbound and free for the purposes of weakinteraction rates (see Sec. 4). ASTEX Comparing treatments of weak reactions with nuclei in simulations of core-collapse supernovae -6-4-2 0 2 4 6 120 140 160 180 200 ∆ Q e [ e r g g - s - ] Radius [km]NV112-V112DV112-V112SteadySteady
Figure 18.
Radial profiles of the difference of the net gain byneutrino emission and absorption. Solid lines show the dif-ference in net gain at T b = 100ms between a model with anartificial treatment of light nuclei (NV112 or DV112) and thefiducial model V112. Dashed lines show the difference in netgain between the steady-state radiation fields on the V112matter background using different prescriptions for light nu-clei. Model NV112 (blue) neglects weak interactions withlight nuclei. Model DV112 (black) treats nucleons in lightnuclei as if they were unbound and free for the purposes ofweak interaction rates (see Sec. 4). Note that for dynamicalmodels (solid lines), we measure the difference from modelV112. On the other hand, for steady state model (dashedlines), we measure the difference from the steady state modelwith the same prescription of the weak interactions with lightnuclei as model V112. els, both neutrino luminosity and mean energy for ν e in model NV112 are overestimated compared to modelV112. Meanwhile the same quantities in model DV112are not so different from those in model V112. Thesetrends are in consistent with those discussed above inreference to the left column of Fig. 17 due to a higheropacity and emissivity from electron capture reactions.It should be noted, however, that the peak neutrinoluminosity at the neutrino burst in model DV112 isroughly ∼
5% smaller than in model V112.For ¯ ν e and ν x , the artificial prescription in DV112changes the time evolution of both the luminosity andmean energy. Again, the difference between DV112 andV112 can be understood in the same way as in previousdiscussions (see right column in Fig. 17). That is, the ar-tificial increase in the number of free nucleons increasesthe opacity from nucleon scattering and the emissivityfrom nucleon-nucleon bremsstrahlung. Fig 20 shows thedistribution function f for outgoing (¯ θ = 0 ◦ ) neutrinosat r = 500km and T b = 100ms in model V112 (top panel), along with the differences from this spectrum inmodels NV112 and DV112 (bottom panel). f is a func-tion of the spacetime coordinates x µ and the neutrinofour momentum p i . Since the latter satisfies the condi-tion p µ p µ = − m ν , where m ν denotes the mass of theneutrino (which is assumed to be zero in this paper),only three of the four momentum components are in-dependent. The differences between model DV112 andV112 shows similar trends for ¯ ν e and ν x . The neutrinoexcess in the low energy side in model DV112 is mostlydue to the artificial increase of emissivity of nucleon-nucleon bremsstrahlung. On the other hand, the neu-trino number depletion in the high-energy side for modelDV112 is due to the artificial increase of opacity fromnucleon scattering.Here is a short summary of the impact of artificialprescriptions of light nuclei to CCSNe. As discussed inSec. 5.2.1, electron and positron captures by light nu-clei are subdominant weak interaction processes in CC-SNe. However, the artificial prescriptions quantitativelychange the dynamics (as measured by shock trajectory)of CCSNe. Ignoring light nuclei (as in model NV112)tends to produce an artificially larger shock radius dueto the overestimation of neutrino energy deposition inthe gain region (see Figs. 16 and 18). On the contrary,another artificial prescription which the abundances oflight nuclei are decomposed into nucleons and countedas free nucleons (as in model DV112) underestimatesboth the shock radius and neutrino heating. The pri-mary cause of the difference of neutrino heating is thatignoring weak interactions in light nuclei enhances es-caping neutrinos from the cooling region by reducingthe opacity, which results in enhanced neutrino absorp-tion in the gain region (for model NV112). On thecontrary, the artificial increase of the mass fraction offree nucleons in model DV112 artificially increases thenucleon scattering and nucleon-nucleon bremsstrahlungopacities. They change the neutrino flux and the spec-trum, which reduces the neutrino absorption in the gainregion. The impact on neutrino heating rate in the gainregion is ∼ Progenitor dependence
The so-called ”Mazurek’s law” during the collapsephase (see e.g., Bruenn (1985); Liebend¨orfer et al.6
Nagakura et al. L ν e [ e r g / s ] V112NV112DV112 0 10 20 30 40 50 0 5 10 15 20 9 10 11 12 13 14 E ν e [ M e V ] L ν e [ e r g / s ] | E ν e [ M e V ] | L ν x [ e r g / s ] T b [ms] |
13 14 15 16 0 100 200 300 400 E ν x [ M e V ] T b [ms] | Figure 19.
Time evolution of the neutrino luminosity (left) and the mean energy (right) for ν e (top), ¯ ν e (middle) and ν x (bottom) for three models with different treatments of weak reactions with light nuclei. We measure these quantities at r = 500km. Model V112 (red) is the fiducial model and includes weak interactions with light nuclei. Model NV112 (blue)neglects weak interactions with light nuclei. Model DV112 (black) treats nucleons in light nuclei as if they were unbound andfree for the purposes of weak interaction rates (see Sec. 4). ASTEX Comparing treatments of weak reactions with nuclei in simulations of core-collapse supernovae -16 -13 -10 -7 -4 -1 f V112 ν e - -0.2-0.100.10.2 0 20 40 60 80 100 ∆ f / f Energy [MeV]NV112-V112DV112-V112 ν e - -12 -10 -8 -6 -4 -2 f ν x - -0.2-0.100.1 0 20 40 60 80 100 ∆ f / f Energy [MeV] ν x - Figure 20.
Upper: the distribution function of outgoing (¯ θ = 0 ◦ ) ¯ ν e (left) and ν x (right) as a function of energy at fluid restframe for V112 model. The spectra are measured in r = 500km at T b = 100ms. The bottom panels display those in NV112(blue) and DV112 (thick) but subtracting the V112. We also normalized the difference by V112. The red line marks at zero ony-axis. Nagakura et al. (2003)) states that in strongly interacting systems, suchas the inner core (i.e., the part of the core that is soni-cally connected), changes in the model inputs are com-pensated for and lead to small differences in the output.This, combined with similar properties of the stellarcore at the onset of collapse due to the Chandrasekharcriterion, leads to universal inner core properties in thecollapse of different progenitors. This is because boththe distributions of electron fraction and entropy shouldbe self-regulated by electron captures during collapse.A higher electron fraction enhances the rates of electroncapture onto free protons due to the larger number offree protons and electrons, which facilitates the delep-tonization and then results in reducing the electronfraction. Similarly, a high entropy state enhances elec-tron capture rates, which facilitates neutrino coolingand reduces the excess of entropy. In previous stud-ies, we have observed such a self-regulation mechanismwhen electron captures of free protons dictate the timeevolution of both deleptonization and neutrino cool-ing (see e.g., Kachelrieß et al. (2005); O’Connor & Ott(2013)). However, if the rates of electron captures ontoheavy nuclei overwhelm those of captures onto freeprotons, it is not so clear that collapse of differentprogenitors should show the same universal structure(see e.g., Liebend¨orfer et al. (2003)). This is becausethe deleptonization and cooling rates are no longer asimple function of electron fraction and temperature,but rather sensitively depend on the abundances ofheavy nuclei. As a matter of fact electron captures ontoheavy nuclei are always dominant during the entire col-lapse phase regardless of the progenitor (Langanke et al.2003; Juodagalvis et al. 2010; Sullivan et al. 2016) (andsee also Fig. 23).In this light, there are many previous works thatdiscuss the progenitor dependence of dynamics andneutrino signals in CCSNe (see e.g., Mayle et al.(1987); Sumiyoshi et al. (2008); Nakazato et al. (2013);Liebend¨orfer et al. (2003); Kachelrieß et al. (2005);O’Connor & Ott (2013); Summa et al. (2016); Bruenn et al.(2016); Richers et al. (2017b); Radice et al. (2017);Ott et al. (2017)). This study, however, is a fist at-tempt to understand the progenitor dependence of theCCSNe in the context of up-to-date rates of electroncapture on heavy and light nuclei consistently with amulti-nuclear EOS with a realistic nuclear force. To dothis, we compare four progenitor models with ZAMSmasses of 11 . M ⊙ (V112), 15 M ⊙ (V15), 27 M ⊙ (V27),and 40 M ⊙ (V40), whose matter profiles at the initial
10 100 1000 ρ [ g / c m ] Radius [km] V112V15V27V40
Figure 21.
Density profiles of progenitors at the initialcollapse phase ( ρ c = 1 . × g / cm ) for models V112 (red),V15 (blue), V27 (green), and V50 (black). collapse phase are shown in Figs. 21 and 22 . As shownin these figures, the matter distributions in the differentprogenitors are quite different from each other. Amongour selected progenitors, higher-mass progenitors showa higher core electron fraction and higher core entropyper baryon. This means that more massive progenitorsare expected to have more efficient deleptonization andneutrino cooling in the inner core, allowing the differ-ent models to be driven closer to the same inner corestructure as collapse proceeds.We first take a look whether electron captures by freeprotons or heavy nuclei are dominant for deleptoniza-tion and cooling of the core during the collapse phase.Figure 23 shows that in the inner core (quasi-horizontalsegments of the curves on the left sides of the plots),electron capture by heavy nuclei dominates that by freeprotons at all times during the collapse phase. However,this is not the case outside of the inner core. In general,the electron chemical potential should be larger than themass difference ∆ np between neutrons and protons forelectron capture by free protons, and larger than the nu-clear Q value for capture by heavy nuclei. The nuclear Qvalue includes not only ∆ np but also the biding energydifference between parent and daughter nuclei, whichmeans that Q value is larger than ∆ np . In the outercore, although the electron chemical potential exceeds∆ np , i.e., meeting the requirement of electron captures The time snapshot in these figures corresponds to the begin-ning of our simulations for 11 . M ⊙ but 160ms, 172ms and 197msfor 15 M ⊙ , 27 M ⊙ and 40 M ⊙ , respectively. ASTEX Comparing treatments of weak reactions with nuclei in simulations of core-collapse supernovae Y e Radius [km]V112V15V27V40 1 2 3 4 5 6 10 100 1000 s [ k B ] Radius [km]
Figure 22.
Radial profiles of electron fraction (left) and entropy per baryon (right) at the initial collapse phase ( ρ c =1 . × g / cm ) for models V112 (red), V15 (blue), V27 (green), and V50 (black). by free protons, it is not large enough compared to thenuclear Q value due to the low temperature and density.As a result, electron capture by heavy nuclei is stronglysuppressed. It is also important to note that the elec-tron fraction in the outer core is larger than in the innercore, which creates a supports a larger amount of freeprotons and results in a higher electron capture rate byfree protons than by heavy nuclei. At later times duringcollapse (i.e., lower panels), the ratio of electron captureson heavy nuclei to protons approach the same value be-tween different simulations in the inner core. Thesetrends are true for both the rate of change of leptonnumber (left plots) and for the rate of change of inter-nal energy (right plots). Even in the late collapse phase( ρ c > g / cm ), the electron capture of heavy nu-clei remains dominant in the inner core, which supportsthe need for a consistent treatment of nuclear abun-dances in electron capture rates Langanke et al. (2003);Juodagalvis et al. (2010); Sullivan et al. (2016).Larger mass progenitors are much more dominated bycaptures onto free protons than lower mass progenitors.This is due to the fact that there is a higher entropyper baryon in the calculations starting from high-massprogenitors (visible as a higher temperature in Figs. 24to 26). Because of the temperature difference, the aver-age mass (bottom left panels) and abundance of heavynuclei (bottom right panels) is higher in models usinglower-mass progenitors. Interestingly, despite the dif-ferences in cooling rates, deleptonization rates, temper-atures, and nuclear abundances, the difference of elec-tron fraction in the inner core among models almostdisappears by the time when the central density reaches10 g / cm (see the middle left panel in Fig. 24). Even in the late collapse phase, the time evolution of the centralelectron fraction is almost identical between the mod-els starting from different progenitors, just as in modelsusing more approximate treatments of electron capture(e.g., Liebendoerfer (2005)). Deleptonization and neu-trino cooling rates become insensitive to these differ-ences after the time at ρ c ∼ g / cm (see top panels),since neutrinos become trapped in the inner core.At the time of core bounce, the shock wave is gen-erated for all progenitors at the mass shell between0 . M ⊙ and 0 . M ⊙ (see Fig. 27). The minor progen-itor dependence arises due to the difference in the ther-mal component of the pressure. As discussed above,model V112 has the lowest temperature in the inner coreamong the models using different progenitors, whichmeans that the thermal pressure is the weakest and thetotal mass of inner core becomes the smallest. On theother hand, the weaker thermal pressure support allowsa higher central density (top left panel), so the innercore is the most compact. As during the collapse phase,the electron fraction and lepton fraction (bottom pan-els) are consistent among the models. At bounce, thedensity is large enough in the inner core that there areno light or heavy nuclei, though the abundances of nu-clei outside the inner core follow the same ordering asduring collapse (i.e., larger progenitors have more lightnuclei and fewer heavy nuclei).Contrary to that of the inner core, the compactness ofouter core increases with progenitor mass (see Fig. 28).The trend simply reflects the density profiles in the dif-ferent progenitors at r & Nagakura et al. -2 -1 Q N ( p ) / Q N ( A ) V112V15V27V40Initial 10 -2 -1 Q E ( p ) / Q E ( A ) Initial10 -2 -1 Q N ( p ) / Q N ( A ) Initial ρ c = 10 g/cm -2 -1 Q E ( p ) / Q E ( A ) Initial ρ c = 10 g/cm -3 -2 -1 Q N ( p ) / Q N ( A ) Initial ρ c = 10 g/cm ρ c = 10 g/cm -3 -2 -1 Q E ( p ) / Q E ( A ) Initial ρ c = 10 g/cm ρ c = 10 g/cm -3 -2 -1 Q N ( p ) / Q N ( A ) Radius [cm]Initial ρ c = 10 g/cm ρ c = 10 g/cm ρ c = 10 g/cm -3 -2 -1 Q E ( p ) / Q E ( A ) Radius [cm]Initial ρ c = 10 g/cm ρ c = 10 g/cm ρ c = 10 g/cm Figure 23.
Radial profiles of the ratio of the rates of electron capture onto free protons to those onto heavy nuclei. Left panelsshow deleptonization rates and right panels show cooling rates. From top to bottom, we display the initial profile, followed byprofiles at times when the central density is ρ c = 10 g / cm , ρ c = 10 g / cm , and ρ c = 10 g / cm . ASTEX Comparing treatments of weak reactions with nuclei in simulations of core-collapse supernovae -4-3-2-1 0 Q N [ s - ] V112V15V27V40 -3-2-1 0 Q e [ e r g g - s - ] Y e T [ M e V ]
50 60 70 801e+06 1e+07 1e+08 1e+09 A - Radius [cm] 0.7 0.75 0.8 0.85 0.9 0.95 11e+06 1e+07 1e+08 1e+09 X h Radius [cm]
Figure 24.
Comparison between different progenitors during the collapse phase when the central density is ρ c = 10 g / cm . Q N is the net leptonization rate, Q e is heating rate, Y e is electron fraction, T is the temperature, ¯ A is average number ofnucleons in heavy nuclei, and X h is the mass fraction of heavy nuclei. Results are shown for models V112 (red), V15 (blue),V27 (green), and V50 (black). Nagakura et al. -15-12-9-6-3 0 Q N [ s - ] V112V15V27V40 -12-9-6-3 0 Q e [ e r g g - s - ] Y e T [ M e V ]
50 60 70 80 90 1001e+06 1e+07 1e+08 1e+09 A - Radius [cm] 0.65 0.7 0.75 0.8 0.85 0.9 0.95 11e+06 1e+07 1e+08 1e+09 X h Radius [cm]
Figure 25.
Same as Fig. 24 but when ρ c = 10 g / cm . the post-bounce phase (see in Fig. 29). Changes in theslopes in Fig. 28 result in sudden change in the accre-tion rate when that part of the star passes through theshock, reducing the ram pressure, but also reducing theneutrino luminosity.The trajectory of shock wave in each model is shownin Fig. 30. For all models, the shock wave reachesthe maximum radius at similar time after the bounce(T b ∼ ∼ b ∼ r ∼ ASTEX Comparing treatments of weak reactions with nuclei in simulations of core-collapse supernovae -30-20-10 0 Q N [ s - ] V112V15V27V40 -4-2 0 Q e [ e r g g - s - ] Y e T [ M e V ]
40 50 60 70 80 1e+06 1e+07 1e+08 1e+09 A - Radius [cm] 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1e+06 1e+07 1e+08 1e+09 X h Radius [cm]
Figure 26.
Same as Fig. 24 but when ρ c = 10 g / cm . interface passes through the shock wave by the end ofour simulations, which also correlates with the suddendecrease of the mass accretion rate in Fig. 29. The shockwave in V27 and V40 suspends the recession for a whileonce the Si/Si-O interface hits the shock front, whichare shown in the left panel of Fig. 30.The driving force of shock propagation during theearly shock expansion phase is the core bounce, whichis insensitive to the differences in the outer core struc-ture between different models. Interestingly, except formodel V112, the shock trajectory at later times also de-pends weakly on the progenitor structure. For instance, the difference in maximum shock radius between modelsV15, V27 and V40 is less than 6%. In the late phase (be-yond ∼
150 ms after core bounce), the shock trajectoryis roughly identical for all four progenitor models. Sucha weak dependence is mainly attributed to the competi-tion between mass accretion rates and neutrino heating.The large mass accretion rate in model V40, for example,pushes the shock wave back with a large ram pressure.On the other hand, the shock wave is pushed out bythe enhancement of neutrino heating in the gain region,which is due to the increase of both accretion compo-nents of neutrino luminosity and the baryon mass in4
Nagakura et al. ρ [ g / c m ] V112V15V27V40 0.2 0.4 0.6 0.8 1 X h T [ M e V ] X a s [ k B ]
20 40 60 A | Y e Mass coordinate [M ⊙ ] | Y l Mass coordinate [M ⊙ ] | Figure 27.
Comparison at core bounce between models starting from different progenitors. From top to bottom, we plotbaryon mass density ( ρ ), temperature ( T ), entropy per baryon ( s ) and electron fraction ( Y e ) in the left column, while we plotthe mass fraction of heavy nuclei ( X h ), the mass fraction of light nuclei ( X a ), the average mass number ( ¯ A ) and the leptonfraction ( Y l ) in the right column. All quantities are displayed as a function of mass coordinate in this figure. Models are V112(11 . M ⊙ , red), V15 (15 M ⊙ , blue), V27 (27 M ⊙ , green), and V40 (40 M ⊙ , black). ASTEX Comparing treatments of weak reactions with nuclei in simulations of core-collapse supernovae M [ M ⊙ ] Radius [10 cm]V112V15V27V40 0.8 1 1.2 1.4 1.6 1.8 2 2.2 5 10 15 20 25 30 35 40 45 50 M [ M ⊙ ] Radius [10 cm] Figure 28.
Enclosed mass versus radius at the time of core bounce for models V112 (red), V15 (blue), V27 (green) and V40(black). The left panel shows the region where 10 cm ≤ r ≤ cm and the right panel shows the region where 10 cm ≤ r ≤ × cm. Larger progenitors lead to more compact density profiles at bounce. Nagakura et al. M [ M ⊙ s - ] T b [ms]V112V15V27V40 . Figure 29.
Mass accretion rates measured at r = 500km asa function of time for models using different progenitors. Themore compact density profiles of larger progenitors generallycause higher accretion rates at early times. the gain region (see left panels in Fig. 31 for the timeevolution of neutrino luminosity and the left column inFig. 32 for relevant quantities to neutrino heating).The sudden decrease of mass accretion rate when theSi/Si-O interface accretes through the shock simply re-duces the ram pressure, which results in rapid expansionof the shock wave. This is visible as a bump in the shocktrajectory in all models in Fig. 30, but is exaggerated inmodel V112 because the drop in ram pressure occurswhile the prompt shock is still expanding. On the otherhand, after the shock stagnates, the time evolution ofshock wave is dictated by the balance between the rampressure from mass accretion and neutrino heating fromhot accreted mass, as described above. This negativefeedback softens the impact of the accretion rate dropin models V15, V27, and V40, and returns the shockin model V112 to a location similar to the other threemodels by 150 ms after bounce.The time evolution of the luminosity and the meanenergy are displayed in Fig. 31. The peak ν e luminos-ity at the neutronization burst varies by ∼
20% amongthe models, with higher luminosities coming from highermass progenitors (top left panel). Among our mod-els, large mass progenitors lead to more compact outercores (see also Fig. 28), which means that the post-shock flow also becomes also compact. As a result, the Note that the trend between mass and compactness would notbe monotonic outside of this mass range in reality. See below inmore details post-shock flow is more opaque for neutrinos models us-ing more massive progenitors, which causes the neutri-nos to decouple with matter at the larger radius. Suchan increase of radius of neutrino sphere causes the en-hancement of neutrino luminosity.As shown in the left column of Fig. 31, the larger massof our models tend to have a higher neutrino luminosityfor all species of neutrinos due to the larger accretionrate. Importantly, the information of density structureof outer core is imprinted in the neutrino signal. Forinstance, the sudden decrease of luminosities which areseen in ν e and ¯ ν e for V15 and V27 at T b ∼ . M ⊙ . M . M ⊙ or M & M ⊙ . Hence, neutrinosignals cannot be a direct tool to determine the progeni-tor mass in real observations. However, the neutrino sig-nals could be still useful for narrowing down the range ofprogenitor candidates in particular for low mass progeni-tors ( M . M ⊙ ). Indeed, according to Sukhbold et al.(2016), these progenitors roughly monotonically corre-late with the compactness with their mass.One of the standard diagnostics for assessing the ef-fectiveness of the neutrino heating mechanism is the ra-tio of the advection to the heating time scales (see e.g.,(Murphy & Burrows 2008; Janka et al. 2016)). We de-fine the advection time scale T adv = M g / ˙ M where M g and ˙ M denote the mass of gain region and mass accre-tion rate at r = 500km, respectively. The heating timescale is T heat = | E tot | / ˙ Q , where E tot and ˙ Q denote thetotal energy of matter (the sum of gravitational, kineticand thermal energy) and neutrino heating rate in thegain region, respectively. Naturally, we find that thisratio is correlated to the shock trajectory. As shown inthe bottom right panel in Fig. 32, the ratio in modelV112 is remarkably larger than in other models, espe-cially at T b < ASTEX Comparing treatments of weak reactions with nuclei in simulations of core-collapse supernovae R s [ k m ] T b [ms] V112V15V27V40 100 150 200 25030 60 90 120 150 R s [ k m ] T b [ms] Figure 30.
Same as Fig. 8 but for V112 (red), V15 (blue), V27 (green) and V40 (black). advection time scale in model V112, which is more thanthree times as large as other models at the same time(see the top right panel in Fig. 32). Though none ofthe models explode, this diagnostic suggests that modelV112 was closest to explosion, followed by models V27,V40, and V15.To see the progenitor dependence of PNS structure, weplot the radius of and mass enclosed by several densitycontours in Fig. 33. Plotting mass-radius relations inthis way allow us to analyze the EOS dependence of CC-SNe in a way that naturally includes thermal effects thatare generally not negligible in PNSs. The mass-radiusrelation for the iso-density surface of ρ = 10 g / cm (topleft panel) is almost identical among all progenitors be-cause this contour is at all times deep within the PNS,the structure of which is rather universal (see Fig. 27).The matter distribution at this contour evolves quasi-steadily and adiabatically. We also see the universalityin the mass-radius relation at the lower density contoursat early times, though the universality does not hold atlater times (i.e., the curves recede in radius and deviatefrom each other). Note also that the timing of the ap-pearance of the turnover depends on the model. ModelV112 deviates from the rest at the lowest value of en-closed mass (e.g., a remarkable deviation can be seenaround the point of M ∼ . M ⊙ and r ∼ ρ = 10 g / cm contour).To understand the cause of the turn-over and the pro-genitor dependence thereof, we compare the matter pro-file among models at the time when the enclosed massfor the iso-density surface of ρ = 10 g / cm reaches M = 1 . M ⊙ , which are displayed in Fig. 34. A largemass accretion rate prevents the PNS from cooling (i.e.,the PNS evolves adiabatically). When the accretion ratedrops, the PNS is able to cool and condense on a diffu-sion timescale. Since less massive progenitors experiencea drop in accretion rate sooner, they break off the uni-versal adiabatic curve first. Indeed, both the entropy(bottom left panel) and electron fraction (bottom rightpanel) at r ∼ r > ρ = 10 g / cm . This is simply due to the factthat neutrino diffusion timescale is shorter for the lessopaque layer. CONCLUSIONS AND DISCUSSIONWe present spherically symmetric CCSNe simula-tions with full Boltzmann neutrino transport under self-consistent treatment of nuclear abundances betweena multi-nuclear EOS and weak interactions. In mostof our simulations, we employ the newly developed8
Nagakura et al. L ν e [ e r g / s ] E ν e [ M e V ] V112V15V27V40 1 2 3 4 5 6 7 L ν e [ e r g / s ] | E ν e [ M e V ] | L ν x [ e r g / s ] T b [ms] |
13 14 15 16 0 100 200 300 400 500 E ν x [ M e V ] T b [ms] | Figure 31.
Time evolution of the neutrino luminosity (left) and the mean energy (right) for ν e (top), ¯ ν e (middle) and ν x (bottom). The progenitors used for model V112, V15, V27, and V40 have masses of 11 . M ⊙ , 15 M ⊙ , 27 M ⊙ , and 40 M ⊙ ,respectively. ASTEX Comparing treatments of weak reactions with nuclei in simulations of core-collapse supernovae M g [ M ⊙ ] V112V15V27V40 135103050100 T a d v [ m s ] E [ e r g / s ] . T h ea t [ m s ] .
246 100 200 300 400 500 η [ % ] T b [ms] . T a d v / T h ea t T b [ms] . Figure 32.
Time evolution of several diagnostic quantities for CCSNe simulations. In the left column, the mass contained inthe gain region (top), the energy deposition rate (middle) and the heating efficiency (bottom). In the right panel, the advectiontime scale in the post-shock flow (top), the heating time scale (middle) and ratio of the former to the latter (bottom). Colorsare the same as Fig. 30. Nagakura et al. M [ M ⊙ ] Radius [km]V112V15V27V40 ρ = 10 [g/cm ] 0.4 0.8 1.2 1.6 2 15 18 21 24 27 30 M [ M ⊙ ] Radius [km] ρ = 10 [g/cm ] 0.4 0.8 1.2 1.6 220 24 28 32 36 40 44 48 M [ M ⊙ ] Radius [km] ρ = 10 [g/cm ] 0.4 0.8 1.2 1.6 230 40 50 60 70 80 90 M [ M ⊙ ] Radius [km] ρ = 10 [g/cm ] Figure 33.
Radius of and mass enclosed by density contours during the post-bounce evolution in models V112 (red), V15(blue), V27 (green) and V40 (black). In general, time increases from the bottom of the curves to the top. Each panel shows adifferent density contour, as labeled in the plot.
VM EOS, which is one of the most up-to-date nuclearEOS, originally developed by Togashi & Takano (2013);Togashi et al. (2017) and further extended to multi-nuclear treatments by Furusawa et al. (2017c). Givenbaryon density, temperature and proton (electron) frac-tion, our new EOS table provides us not only otherthermodynamical quantities, but also a full distributionof nuclear abundances in nuclear statistical equilibrium.We use these abundances to construct a new weak reac-tion rate table including electron and positron capturesby light nuclei that is consistent with the nuclear abun-dances in the EOS.We then carry out CCSN simulations with these de-tailed physics inputs and study the EOS dependence(in Sec. 5.1), influence of light nuclei (in Sec. 5.2) and progenitor dependence (in Sec. 5.3) of CCSN dynamicsand neutrino signals. The key findings in this study aresummarized as follows.1. Inconsistent treatments of electron capture ratewith nuclear abundances in EOS weaken the EOSdependence of deleptonization and neutrino cool-ing in the early part of the collapse phase (seeradial profiles of Q N , Q e and Y e in Fig. 3).2. As pointed out by previous studies (Langanke et al.2003; Hix et al. 2003), the appropriate treatmentof electron capture by heavy nuclei is importantto determine the structure of the supernova coreduring the collapse. We find that our incomplete-treatments of electron captures of heavy nucleichange the shock radius by up to a few percent ASTEX Comparing treatments of weak reactions with nuclei in simulations of core-collapse supernovae ρ [ g / c m ] V112V15V27V40 51015202530 T [ M e V ] s [ k B ] Radius [km] 0.10.20.30.4 0 10 20 30 40 50 Y e Radius [km]
Figure 34.
Radial profiles of density (top left), temperature (top right), entropy per baryon (bottom left) and electronfraction (bottom right) in the post-bounce phase of V112, V15, V27 and V40 at times T b = 335ms, 186ms, 171ms, and 154ms,respectively. For all models, the enclosed mass at the radius of ρ = 10 g / cm is 1 . M ⊙ . The brown doted line in the top leftpanel guides the eye to ρ = 10 g / cm . (see Fig. 8). Neutrino luminosity and average en-ergy for both ν e and ¯ ν e also changed by ∼ ∼ ∼ ∼ . − Nagakura et al. ferences from artificial prescriptions of light nucleiare largely due to the fact that they cause dif-ferences in the evolution of the matter, while thesteady-state calculations use the same matter pro-file as the dynamical ones.5. Different progenitors evolve to a common innercore density and electron fraction structure duringthe collapse phase, despite the fact that the reac-tion rate is dominated by the electron capture ofheavy nuclei and different progenitors have differ-ent abundances of heavy nuclei in the inner core.It should be noted, however, that the temperaturedifference continues to remain up to core bounce,which causes a slight difference in the location forthe formation of the shock at core bounce (seeFig. 27). This universality persists beyond corebounce until deleptonization and neutrino coolinginfluence on the structure of PNS. Departures fromthe universal structure appear earlier at largerradii and in models with less compact outer cores(see Sec. 5.3).6. The time evolution of the shock wave in the11 . M ⊙ model (V112) between 50 and 150msafter bounce is very different from other moremassive progenitor models. This is attributed tothe fact that the Si/Si-O interface in model V112hits the shock wave while the prompt shock isstill expanding. On the other hand, for othermassive progenitor models, the interface passesthrough the shock wave after the shock stagnates.At this phase, the time evolution of shock waveis dictated by the balance between the ram pres-sure from mass accretions and neutrino heating.The sudden drop of mass accretion rate due tothe arrival of the Si/Si-O interface at shock wavecauses not only weakening the ramp pressure bymass accretions but giving the negative feedbackto neutrino luminosity, which results in less im-pact to the shock dynamics than that in V112.It should be noted that the shock wave in V112quickly returns to match the shock radius of theother progenitors at ∼ ν e luminosity in the neutronizationburst is comparable to or even overwhelms the differ-ence in ν e luminosity between models using the VMand FYSS EOSs (see Sec.5.1). It is important to em-phasize that such a quantitative argument for the neu-trino signals is only possible by using simulations witha consistent treatment of the EOS and nuclear weak in-teractions. Indeed, the smaller progenitor dependenceof the neutonization burst in previous studies (see e.g.,Kachelrieß et al. (2005)) may be due to the artificiallycommon weak reaction treatments. Thus observationsof the neutronization burst alone, even if the effects ofneutrino oscillations can be disentangled from the super-nova dynamics, are unlikely to shed light on propertiesof the nuclear EOS.It should be noted, however, that the neutrino sig-nals shown in Fig. 31 signals could potentially be usefulfor extracting the matter profile of progenitors, sincethe luminosities and average energies trace the accre-tion history. For instance, model V112 shows a re-markably lower neutrino luminosity and average neu-trino energy during the middle and late post-bouncephase. According to the recent multi-D simulations (seee.g., O’Connor & Couch (2018); Radice et al. (2017)),this dependence of neutrino signals on progenitor prop-erties is retained at least up to the time of shock re-vival. Therefore, observations of the luminosity and av-erage energy could help pin down the mass and com-pactness of the progenitor, but more work is required toassess whether this trend is robust and well-junderstoodenough to interpret an isolated CCSN signal.However, care must be taken to disentangle neu-trino signal changes due to the Si/Si-O interface passingthrough the shock from those due to shock revival, sincethey have similar characteristics (see also Summa et al.(2016); Seadrow et al. (2018)). To understand the de-generacy, we need accurate multi-dimensional modelsthat exhibit successful shock revival, along with an un-derstanding of the effects neutrino oscillations have onthe neutrino signal at Earth and the characteristics ofneutrino detectors. We are currently running axisym-metric CCSNe simulations with our up-to-date inputphysics and address these issues. The results of thisstudy will lay the foundation of analyzing these compli-cated multi-dimensional modeling of CCSNe.Last but not least, a consistent treatment of the EOSand nuclear weak interactions is important for carry-ing out accurate nucleosynthesis computations in par-ticular for the ν -process which occurs during PNS cool-ing phase. This process causes spallations of nucleonsfrom heavy nuclei including r-process elements, which ASTEX Comparing treatments of weak reactions with nuclei in simulations of core-collapse supernovae
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