Importance of ^{56}Ni production on diagnosing explosion mechanism of core-collapse supernova
aa r X i v : . [ a s t r o - ph . H E ] A p r MNRAS , 1– ?? (2017) Preprint 18 September 2018 Compiled using MNRAS L A TEX style file v3.0
Importance of Ni production on diagnosing explosionmechanism of core-collapse supernova
Yudai Suwa ⋆ , Nozomu Tominaga , , and Keiichi Maeda , Center for Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto, 606-8502, Japan Department of Physics, Faculty of Science, and Engineering, Konan University, 8-9-1 Okamoto, Kobe, Hyogo 658-8501, Japan Kavli Institute for the Physics, and Mathematics of the Universe (WPI), University of Tokyo, Kashiwa, Chiba 277-8583, Japan Department of Astronomy, Kyoto University, Oiwake-cho, Kitashirakawa, Sakyo-ku, Kyoto 606-8502, Japan
Accepted. Received.
ABSTRACT Ni is an important indicator of the supernova explosions, which characterizes lightcurves. Nevertheless, rather than Ni, the explosion energy has often been paid at-tention from the explosion mechanism community, since it is easier to estimate fromnumerical data than the amount of Ni. The final explosion energy, however, is diffi-cult to estimate by detailed numerical simulations because current simulations cannotreach typical timescale of saturation of explosion energy. Instead, the amount of Niconverges within a short timescale so that it would be a better probe of the explosionmechanism. We investigated the amount of Ni synthesized by explosive nucleosyn-thesis in supernova ejecta by means of numerical simulations and an analytic model.For numerical simulations, we employ Lagrangian hydrodynamics code in which neu-trino heating and cooling terms are taken into account by light-bulb approximation.Initial conditions are taken from Woosley & Heger (2007), which have 12, 15, 20, and25 M ⊙ in zero age main sequence. We additionally develop an analytic model, whichgives a reasonable estimate of the amount of Ni. We found that, in order to pro-duce enough amount of Ni, O (1) Bethe s − of growth rate of the explosion energy isneeded, which is much larger than that found in recent exploding simulations, typically O (0 .
1) Bethe s − . Key words:
The important product of supernova nucleosynthesis is Ni,which drives supernova brightness. A typical amount of Niby canonical supernovae is estimated as O (0 . M ⊙ (Hamuy2003; Smartt 2009), which can be measured by exponen-tial tail from the late light curve with low ambiguity. Incontrast, the explosion energy, which has been used as anindicator of the explosion simulations, needs two observables(light curve and spectrum) to be estimated, since it is in-terfered by a product of ejecta mass and velocity. It im- ⋆ E-mail: [email protected] A typical amount of Ni of nearby supernovae (1987A, 1993J,and 1994I) is ≈ M ⊙ (e.g., Arnett et al. 1989; Woosley et al.1994; Iwamoto et al. 1994). More precisely, from light curve we can estimate the geomet-rical mean of diffusion timescale of photons and hydrodynami-cal timescale, p t diff t hyd ∼ p M ej κ/v ej , where M ej is the ejectamass, κ is opacity, and v ej is the typical velocity of the ejecta(Arnett 1982). The velocity can be independently measured bythe spectrum. By assuming the opacity with a reasonable value plies that the amount of Ni has a smaller systematic errorcompared to the explosion energy. Indeed, for SN 1998bwas an example, the estimated explosion energy ranges from2 to 25 Bethe (1 Bethe ≡ erg) (H¨oflich et al. 1999;Nakamura et al. 2001; Maeda et al. 2006), depending on de-tails of radiation transfer simulations and the ejecta struc-ture assumed in such models, and methods to derive thephysical quantities from observables. On the other hand,the estimated amount of Ni is converged between 0.2 and0.4 M ⊙ . In addition, production of Ni has been suggestedto be sensitive to the explosion mechanism, that is, the en-ergy deposition rate rather than the total explosion energyitself (see, e.g. Maeda & Tominaga 2009; Suwa & Tominaga2015).The mechanism of supernova explosions is still undera thick veil, even though it has been already more than80 years from the original idea by Baade & Zwicky (1934), κ ≈ . g − , we can resolve the degeneracy between massand velocity.c (cid:13) Suwa, Tominaga, and Maeda more than 50 years from the first numerical simulation(Colgate & White 1966), and more than 30 years from thefirst simulation of delayed explosion (Bethe & Wilson 1985),which is the current standard scenario of supernova explo-sion mechanism.After a few decades of unsuccessful explosionera (Rampp & Janka 2000; Liebend¨orfer et al. 2001;Thompson et al. 2003; Sumiyoshi et al. 2005), wehave some exploding simulations since Buras et al.(2006) (e.g. Marek & Janka 2009; Suwa et al. 2010;Takiwaki et al. 2012; M¨uller et al. 2012; Bruenn et al.2013; Nakamura et al. 2015; Lentz et al. 2015; M¨uller 2015;Pan et al. 2016; O’Connor & Couch 2015; Burrows et al.2016), in which multidimensional hydrodynamics equationsare solved simultaneously with spectral neutrino transport.However, most of simulations have been performed intwo-dimension (with axial symmetry). Three-dimensionalsimulations without any spacial symmetry employed haveshown worse results than two dimensional ones (Hanke et al.2012; Couch 2013; Takiwaki et al. 2014; Lentz et al. 2015),since three dimensional turbulence leads to an energy cas-cade from large scale to small scale (normal cascade), whiletwo dimensional one makes it opposite (inverse cascade).It is known that a large scale, i.e. global, turbulence aidsthe explosion, so that some results from two-dimensionalsimulations might reflect a numerical artifact and thesesimulations might well overestimate the explosion energy.The state-of-the-art simulations have shown slow in-crease of the explosion energy. As summarized in Table 1,the growing rate of the explosion energy is typically O (0 . − , especially for 3D simulations. Therefore, it canbe argued that, by neutrino heating mechanism, these sim-ulations require at least a few second to get a canonicalexplosion energy, i.e. 1 Behte. It should be noted that theexplosion energy estimated in explosion simulations is not adirect observable, since there is bound (totally negative en-ergy) material above the shock and it reduces the explosionenergy when it is swept by the shock.The explosion energy is related to the Ni syn-thesis, since to synthesize Ni the temperature needsto be T ∼ > × K. The postshock temperatureis scaled by the explosion energy as T = 1 . × K( r shock / − / ( E exp / / , where r shock is the shock radius (Woosley et al. 2002). Therefore with E exp = 1 Bethe, Ni can be generated for r shock ∼ < v s is roughly 10 km s − after theonset of the explosion, it takes only a few hundred millisec-onds to reach this radius. If the growth rate of the explosionenergy is small and it takes a few second to achieve 1 Bethe,it is not trivial whether Ni is synthesized by explosive nu-cleosynthesis in the ejecta.In this paper, we investigate Ni production as an in-dicator of the explosion mechanism. First we perform nu-merical simulations of supernova explosion in Section 2. Bycalibrating with numerical simulation data about shock andtemperature evolution, we construct an analytic model that These simulations are all starting from stellar evolutionary re-sults. By changing initial condition, the growth rate of the ex-plosion energy can be ≈ − even in spherical symmetry(Suwa & M¨uller 2016). Table 1.
Properties of recent explosion simulationsAuthor(s) ZAMS mass a ˙ E exp b ( M ⊙ ) (Bethe s − )2D (axisymmetric)Bruenn et al. (2016) 12, 15, 20, 25 1.5 – 3Suwa et al. (2016) 12 – 100 0.5 – 0.7Pan et al. (2016) 11, 15, 20, 21, 27 1 – 5O’Connor & Couch (2015) 12, 15, 20, 25 0.5 – 1Nakamura et al. (2016) 17 0.4Summa et al. (2016) 11.2 – 28 1Burrows et al. (2016) 12, 15, 20, 25 1 – 33DLentz et al. (2015) 15 0.2Melson et al. (2015) 9.6 0.6M¨uller (2015) 11.2 0.4Takiwaki et al. (2016) 11.2, 27 0.4 – 2 a Not only the mass, evolution codes are also different. b Note that these numbers are quite rough estimates in the earlyphase ( ∼
100 ms after the onset of explosion) based on figures inthe literature. describes shock and temperature evolution, which are im-portant ingredients of Ni production, and give constrainton the growth rate of the explosion energy to synthesizeenough Ni in Section 3. This analytic model is useful toinvestigate Ni production for a broader parameter spaceof both the explosion properties and progenitor structure.We summarize our results and discuss their implications inSection 4.
We employ blcode , which is a prototype code of
SNEC (Morozova et al. 2015) and a pure hydrodynamics code based on Mezzacappa & Bruenn (1993), as a base. It solvesNewtonian hydrodynamics in Lagrange coordinate. Basicequations are given by ∂r∂M = 14 πr ρ , (1) DvDt = − GMr − πr ∂P∂M , (2) DǫDt = − P DDt (cid:18) ρ (cid:19) + H − C , (3)where r is radius, M is mass coordinate, ρ is density, v is radial velocity, t is time, G is the gravitational con-stant, P is pressure, and ǫ is specific internal energy. D/Dt means Lagrange derivative. Artificial viscosity byvon Neumann & Richtmyer (1950) is employed to capturea shock. Neutrino heating and cooling are newly addedin this work by a method used in the literature (e.g.Murphy & Burrows 2008), in which neutrino cooling is given Both codes are available from https://stellarcollapse.org.MNRAS , 1– ?? (2017)6 Ni production of CCSN as a function of temperature and neutrino heating is a func-tion of radius with a parametric neutrino luminosity. Heat-ing term, H , and cooling term, C , are given as H =1 . × erg g − s − × (cid:18) L ν e erg s − (cid:19) (cid:16) r (cid:17) − (cid:18) T ν e (cid:19) , (4) C =1 . × erg g − s − (cid:18) T (cid:19) . (5)Here, we fixed neutrino temperature as T ν e = 4MeV. In ad-dition, we take into account these terms only in postshockregime. We do not take into account optical depth terms(see Nordhaus et al. 2010; Hanke et al. 2012) for simplic-ity. We modify inner boundary conditions so that the inner-most mass element does not shrink within 50 km from thecenter to mimic the existence of a protoneutron star. TheHelmholtz equation of state by Timmes & Arnett (1999) isused. Initial composition is used for equation of state.The initial conditions are the 12, 15, 20, and 25 M ⊙ models from Woosley & Heger (2007). Properties of the pro-genitor models are given in Table 2. In this table, we showmass coordinate, radius, and density at a mass coordinatewhich has s = 4 k B baryon − , since the current understand-ing of shock launch is that it is realized when a mass ele-ment with s = 4 k B baryon − is accreting onto the shock.In the fifth column, we show the “compactness parameter”(O’Connor & Ott 2011), which is defined as ξ M = M/M ⊙ R ( M ) / , (6)where R ( M ) is the radius of the sphere whose mass coor-dinate is M . According to O’Connor & Ott (2011), smallervalues of ξ M are better for explosions, but note that theyused ξ . , which is different from our values. The sixth col-umn gives µ M (Ertl et al. 2016), which is defined as µ M = dMdr (cid:12)(cid:12)(cid:12)(cid:12) r = R ( M ) = 4 πρR ( M ) , (7)in units of M ⊙ / dM/dr by computing the numerical deriva-tive at the mass shell where s = 4 k B baryon − with a massinterval of 0 . M ⊙ . Here we instead simply use the secondequality in equation (7) to compute dM/dr analytically.They showed that for a given value of M s =4 , a smaller µ M isbetter for an explosion. From seventh to tenth columns givethe same quantities as ones from third to sixth columns, butdifferent mass coordinate M s =4 + 0 . M ⊙ .The mass cut is determined by M s =4 − . M ⊙ . We em-ploy 1000 grids with mass resolution of 10 − M ⊙ so that1 M ⊙ is included in numerical regime. To check the impactof this choice, we additionally perform a simulation with amass cut of M s =4 − . M ⊙ with 1100 grid points and findno significant difference from standard grid model. We alsoperformed a simulation with 1500 grids points and with thesame total mass (i.e. 33% better rezolution) and found nosignificant differences. Therefore, the numerical results thatwill be shown below are insensitive to numerical setup.In the following, we use the so-called diagnostic explo-sion energy, which is defined as the integral of the sum ofspecific internal, kinetic, and gravitational energies over allzones, in which it is positive, as an approximate estimate of the explosion energy. Note that this energy is not directobservables, since there is bound (totally negative energy)material above the shock and it reduces the explosion energywhen it is swept by the shock. The results are summarized in Table 3. Model names aredenoted as WH07sAALB, where the two digits AA indicatethe progenitor mass, and a digit B indicates the neutrinoluminosity (see the second and third columns in the sametable). WH07 means Woosley & Heger (2007). t exp is the ex-plosion onset time (time at the diagnostic explosion energybecoming positive) measured from protoneutron star (PNS)formation time, which is determined by the innermost masselement reaching r = 50 km. t T =5 presents post-explosiontime when the temperature just after the shock becomes T = 5, where T = T / K, and the next column givesexplosion energy at the same time. ˙ E exp , T =5 is the growthrate of the explosion energy during t T =5 . E exp , is explo-sion energy at 1 s after the explosion onset. M PNS is finalPNS mass which is estimated by the locally bound materialbelow shock wave. The last column gives the mass of Niwhich is calculated as the mass of the material whose max-imum temperature is over 5 × K. The range implies theuncertainty in the simulation. Since the PNS mass (i.e. so-called mass cut in canonical nucleosynthesis studies) evolvesin time, we give minimum and maximum mass with maxi-mum temperature being beyond 5 × K above PNS. Themaximum value includes a component ejected as a neutrino-driven wind. Whether this component synthesizes Ni ornot depends on the evolution of electron fraction Y e , whichis altered by neutrino irradiation from PNS. Since it is be-yond the scope of this study, we do not discuss it below.Figure 1 presents time evolution of radial velocity, den-sity, and temperature as a function of mass coordinate formodel WH07s20L4. It is clearly shown that a stalled shock isformed at first and then once the Si/O layer ( ≈ . M ⊙ ) ac-cretes onto the shock, the shock eventually begins to propa-gate outward (indicated by the positive post-shock velocity)because of the rapid decrease of the ram pressure.Figure 2 gives maximum temperature distribution as afunction of mass coordinate found in model WH07s20L4 (redline) and analytic estimate based on the explosion energy(blue line). The analytic estimate is given by solving thefollowing equation: E exp = 4 π r s aT f ( T ) , (8)where a = 7 . × − erg cm − K − is the radiation con-stant and r s is the shock radius. A temperature-dependentfunction f ( T ) = 1+(7 / T / ( T +5 .
3) (Freiburghaus et al.1999; Tominaga 2009), which takes into account both radia-tion and non-degenerate electron and positron pairs, is usedhere. Since the temperature range is not large, f ( T = 5) =2 .
44 also gives a rather good agreement with numerical re-sult. This factor makes the temperature smaller by 20% thanone without the correction. In this estimate, the postshocktemperature in the ejecta is assumed to be a constant inspace, which is indeed realized in the simulation (see Figure1). In the analytic estimate shown in the figure, we take the
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MNRAS , 1– ???? (2017) Suwa, Tominaga, and Maeda
Table 2.
Precollapse properties of the SN progenitors from Woosley & Heger (2007)Name M s =4 a R M s =4 b ρ M s =4 c ξ M s =4 d µ M s =4 e R M s =4 +0 . M ⊙ f ρ M s =4 +0 . M ⊙ g ξ M s =4 +0 . M ⊙ h µ M s =4 +0 . M ⊙ i ( M ⊙ ) (1000 km) (10 g cm − ) (1000 km) (10 g cm − )WH07s12 1.530 2.813 0.168 0.544 0.084 4.655 0.035 0.350 0.048WH07s15 1.818 3.770 0.129 0.482 0.116 4.924 0.051 0.390 0.079WH07s20 1.824 2.654 0.268 0.687 0.119 3.646 0.133 0.528 0.112WH07s25 1.901 2.803 0.317 0.678 0.157 3.771 0.131 0.531 0.118 a Mass with s = 4 k B baryon − . b Radius with s = 4 k B baryon − . c Density with s = 4 k B baryon − . d Compactness parameter of M s =4 , see Eq. (6). e µ parameter determined by Eq. (7) in units of M ⊙ / f Radius with M s =4 + 0 . M ⊙ . g Density with M s =4 + 0 . M ⊙ . h Compactness parameter of M s =4 + 0 . M ⊙ . i µ parameter of M s =4 + 0 . M ⊙ in units of M ⊙ / Table 3.
Summary of simulationsName progenitor L ν e , a t exp b t T =5 c E exp ,T =5 d ˙ E exp , T =5 e E exp , f M PNS g M Nih (10 erg s − ) (ms) (ms) (Bethe) (Bethe s − ) (Bethe) ( M ⊙ ) ( M ⊙ )WH07s12L1 WH07s12 1 — — — — — — —WH07s12L2 WH07s12 2 553 97 0.093 0.950 0.147 1.527 0.023 – 0.047WH07s12L3 WH07s12 3 361 130 0.230 1.769 0.478 1.456 0.068 – 0.098WH07s12L4 WH07s12 4 233 149 0.366 2.447 0.981 1.315 0.097 – 0.226WH07s15L2 WH07s15 2 — — — — — — —WH07s15L3 WH07s15 3 580 135 0.166 1.230 0.164 1.820 0.060 – 0.079WH07s15L4 WH07s15 4 409 151 0.358 2.362 0.502 1.737 0.086 – 0.135WH07s15L5 WH07s15 5 267 160 0.481 3.007 1.057 1.648 0.107 – 0.196WH07s20L2 WH07s20 2 — — — — — —WH07s20L3 WH07s20 3 307 197 0.344 1.752 0.575 1.806 0.118 – 0.151WH07s20L4 WH07s20 4 249 175 0.392 2.238 0.791 1.769 0.110 – 0.166WH07s20L5 WH07s20 5 236 169 0.458 2.709 1.042 1.736 0.107 – 0.196WH07s25L2 WH07s25 2 — — — — — — —WH07s25L3 WH07s25 3 427 210 0.379 1.801 0.591 1.943 0.125 – 0.149WH07s25L4 WH07s25 4 238 183 0.431 2.354 0.981 1.852 0.113 – 0.172WH07s25L5 WH07s25 5 226 171 0.492 2.874 1.220 1.822 0.111 – 0.197 a Neutrino luminosity. b Time between NS formation and explosion onset. c Time between explosion onset and postshock temperature being T = 5 × K. d Explosion energy at a time when postshock temperature is T = 5 × K. e E exp ,T =5 /t T =5 . f Explosion energy at 1 s after explosion onset. g PNS mass at the last time of simulation. h Ni mass. explosion energy and shock radius from the correspondingnumerical simulation.Figure 3 presents time evolution of mass accretion rate( ˙ M ) of non-exploding models. Mass accretion rates are mea-sured at r = 500 km. Since there is a correlation betweenmass accretion rate and explosion criteria via critical neu-trino luminosity (Burrows & Goshy 1993), the mass accre-tion rate is a good measure to discuss explodability. As isknown, the mass accretion rate becomes almost constantwhen Si/O layer is accreting (see, e.g. Suwa et al. 2016),which is seen in these simulation as well, especially in mod- els WH07s20 and WH07s25. The constant values of accre-tion rate are dependent on the initial progenitor structure.From table 3, one sees that models with high L ν e exhibitsimilar explosion onset time (4th column) for WH07s20and WH07s25, which have rapid transient in mass accre-tion rate (green and purple lines). Meanwhile, WH07s12 andWH07s15 do not show such a clear transition, i.e. the explo-sion onsets earlier for higher L ν e , since these progenitor donot have a drastic change of mass accretion rate (red andblue lines). MNRAS , 1– ?? (2017)6 Ni production of CCSN -6-5-4-3-2-1 0 1 2 3 1.6 1.7 1.8 1.9 2 2.1 2.2 V e l o c it y ( k m / s ) Mass (M ⊙ ) t pb =5mst pb =200mst pb =300mst pb =400mst pb =700ms10 D e n s it y ( g / c m ) Mass (M ⊙ ) t pb =5mst pb =200mst pb =300mst pb =400mst pb =700ms10 T e m p e r a t u r e ( K ) Mass (M ⊙ ) t pb =5mst pb =200mst pb =300mst pb =400mst pb =700ms Figure 1.
Time evolution of the radial velocity (top) density(middle) and temperature (bottom) as a function of mass coor-dinate for model WH07s20L4. Each line indicates different timefrom 5 ms to 700 ms after the bounce (i.e., postbounce time). Theshock begins expansion at t pb ∼
300 ms.
In this section, we derive the temperature evolution basedon a simple analytic model and justify it with numericalresults explained in the previous section.
As known in star formation field, there is a self-similar so-lution of stellar collapse, so-called “expansion-wave collapsesolution” (Shu 1977). This solution implies that the densitystructure inside rarefaction wave becomes ρ ( r ) ∝ r − / and T e m p e r a t u r e ( K ) Mass (M ⊙ )WH07s20L4Analytic Figure 2.
Maximum temperature distributions of a numericalsimulation (red line) and analytic expression (blue line). Numer-ical model employs s20 model of Woosley & Heger (2007) andneutrino luminosity L ν = 4 × erg s − and the consequentgrowth rate of the explosion energy is ≈ . × erg s − . M a ss acc r e ti on r a t e ( M ⊙ s - ) Time (s) WH07s12WH07s15WH07s20WH07s25
Figure 3.
Mass accretion rate as a function of time after NSformation, measured at r = 500 km. All models shown here arenon-exploding models with a small neutrino luminosity. r − outside for isothermal gas. Suto & Silk (1988) extendedthis solution for adiabatic flow with arbitrary adiabatic in-dex and showed that ρ ∝ r − / profile inside rarefactionwave is obtained irrespective of adiabatic index.From modern supernova simulations, typical progeni-tors lead to a constant mass accretion rates when Si/O layeris accreting (see Appendix A of Suwa et al. 2016). With thisfact and continuity equation, ∂ t ρ + r − ∂ r ( r ρv ) = 0, where ∂ t = ∂/∂t and ∂ r = ∂/∂r , one recognizes that the densitystructure does not evolve, i.e. ∂ t ρ = 0, since r ρv = ˙ M/ π becomes constant.The current understanding of explosion onset is the fol-lowing. A rapid density decrease between Si/O layers leadsto decrease of the ram pressure above the shock due to de-creasing mass accretion rate. It results in a shock expansion,since the thermal pressure changes more slowly and over- MNRAS , 1– ????
Mass accretion rate as a function of time after NSformation, measured at r = 500 km. All models shown here arenon-exploding models with a small neutrino luminosity. r − outside for isothermal gas. Suto & Silk (1988) extendedthis solution for adiabatic flow with arbitrary adiabatic in-dex and showed that ρ ∝ r − / profile inside rarefactionwave is obtained irrespective of adiabatic index.From modern supernova simulations, typical progeni-tors lead to a constant mass accretion rates when Si/O layeris accreting (see Appendix A of Suwa et al. 2016). With thisfact and continuity equation, ∂ t ρ + r − ∂ r ( r ρv ) = 0, where ∂ t = ∂/∂t and ∂ r = ∂/∂r , one recognizes that the densitystructure does not evolve, i.e. ∂ t ρ = 0, since r ρv = ˙ M/ π becomes constant.The current understanding of explosion onset is the fol-lowing. A rapid density decrease between Si/O layers leadsto decrease of the ram pressure above the shock due to de-creasing mass accretion rate. It results in a shock expansion,since the thermal pressure changes more slowly and over- MNRAS , 1– ???? (2017) Suwa, Tominaga, and Maeda whelms the ram pressure. Therefore, when the base of theoxygen layer arrives at the shock, the shock expands and,simultaneously, the density structure above shock becomesquasi-stationary. In the following we neglect time evolutionof density structure above a shock wave.
The shock velocity is given by Eq. (19) of Matzner & McKee(1999) as v s = 0 . (cid:18) E exp M ej (cid:19) / (cid:18) M ej ρ ( r s ) r s (cid:19) . , (9)where E exp , M ej , and r s are explosion energy, ejecta mass,and shock radius, respectively. The ejecta mass is given by M ej ( t, r s ) = ˙ Mt + Z r s r mc πr ρ ( r ) dr, (10)where r mc is the radius of mass cut, i.e. the initial positionof the shock. We assume the density profile as (see previoussubsection) ρ ( r ) = ρ R (cid:16) rR (cid:17) − / , (11)where ρ R and R are constants. Adopting the mass accre-tion rate as ˙ M = 4 πr s ρ ( r s ) v acc ( r s ) = 2 πρ R R / √ GM ( v acc = v ff / p GM/ r s , where v ff = p GM/r is free-fallvelocity, is used), we get M ej ( t, r s ) = 2 πρ R R / (cid:20) √ GMt + 43 (cid:16) r / s − r / (cid:17)(cid:21) . (12)Here, we also assume a constant mass accretion rate.By assuming r s = v s t with a constant shock velocity v s , one finds that at the early time ( t ∼ < GM/v s =0 .
19 ( M/ . M ⊙ ) ( v s / km s − ) − s), a contribution frommass accretion (the first term in square bracket of Eq. 12)dominates the ejecta mass, and at the late time the sweptmass contribution (the second term in bracket) dominates.In the following we evaluate shock evolutions in two extremecases: i) ejecta mass is dominated by accreted mass and ii)ejecta mass is dominated by swept mass. Let us assume that M ej = ˙ Mt by neglecting swept masscontribution (second term in the square brackets of Eq. 12).We also assume a constant growth rate of the explosion en-ergy, ˙ E exp , which gives E exp = ˙ E exp t , for simplicity. Since v s = dr s /dt , by introducing Eq. (12) to Eq. (9), we obtainthe following time evolution of the shock: r s ( t ) = .
86 ˙ E / ˙ M . ρ . R R . / t . + r . / ! / . . (13)Here we use an initial condition that r s ( t = 0) = r mc . Theorigin of time (i.e. t = 0) is determined by the shock transi-tion from a steady-accretion shock to an expanding shock,i.e. the onset time of the explosion. Here, we leave ˙ M as afree parameter because v acc is not always half free-fall veloc-ity. This is because a fluid element starts to fall down afterthe rarefaction waves passes it and before that it stays inhydrostatic configuration with no bulk velocity. Accretion rate based on progenitor structure will be given in Section3.5. Here we take into account swept mass contribution alone inEq. (12), which is M ej ( t, r s ) = 8 π ρ R R / ( r / s − r / ) . (14)Assuming r s ≫ r mc and taking the leading order term of( r mc /r s ), we can integrate Eq. (9) as " − . (cid:18) r mc r s (cid:19) / r / s − . r / = 0 . ρ − / R R − / ˙ E / t / , (15)where we imposed an initial condition of r = r mc for t = 0.We can get shock evolution by solving this algebraic equa-tion numerically.By comparing Eq. (13) and solution of Eq. (15) witha direct integrated solution of Eq. (9), we find that, in theparameter regime we are interested in, shock evolution iswell captured by these approximate solutions (i.e. Eqs. 13and 15). For instance, with ρ R = 10 g cm − , R = 1000km, ˙ M = 0 . M ⊙ s − , and ˙ E exp = 1 Bethe s − , differencesbetween these three solutions keep within ∼
20% for r s ∼ < ,
000 km. Therefore, in Section 3.5 we use Eq. (13) toestimate temperature evolution, since this solution can bewritten in simply analytic manner.
In the above estimates, we assumed that all shocked mate-rials which accrete or are swept are included in ejecta mass.This assumption is not always correct, since part of themaccrete onto a neutron star when postshock velocity is notoutgoing. From Rankine-Hugoniot condition, we have fol-lowing relation at shock frame: ρ pre v pre = ρ post v post , (16)where quantities with “pre” mean pre-shock states and“post” mean post-shock states. By going to rest frame, wehave ρ pre ( v pre − v s ) = ρ post ( v post − v s ) . (17)The preshock and postshock densities are related by ρ post = βρ pre , where β ≈ In order tomake postshock velocity positive (i.e. v post > v s > − v pre / ( β − ≈ − v pre /
3. Note that preshock velocity isnegative ( v pre < This value is different from a strong shock limit, ρ post /ρ pre = 7,for γ = 4 /
3, where γ is adiabatic index. This is because the Machnumber of preshocked accretion flow is M ≈ ρ post /ρ pre = 4 . , 1– ?? (2017)6 Ni production of CCSN ejecta mass excluding infalling material at the onset of theexplosion as follows. M ej ( t ≈ , r s ) = h . GM ) − / E / ( ρ R R / ) − . r . s i / . (18)= 0 . M ⊙ M − . . E . , ρ − . R, R − . r . s, , (19)where M . = M/ . M ⊙ , E exp , = E exp / erg, ρ R, = ρ R / g cm − , R = R/ cm, and r s, = r s / cm. Herewe assume | v pre | = p GM/ r s and ρ ( r s ) = ρ R ( R/r s ) / .This equation implies that ejecta mass at the very begin-ning of the explosion ( E exp = 10 erg in this estimate) isnegligible. For a case with a slow growth of the explosionenergy, i.e. a small ˙ E exp , ejecta mass should keep small andmost of mass, which accretes onto the shock or swept by theshock, must go through the ejecta and accrete onto a centralobject (a neutron star or a black hole).Note that for large ˙ E exp cases, a shock is rapidly ac-celerated and accreting and swept materials are followingthe shock as ejecta. Therefore, assumption employed in theprevious subsection is validated. In this subsection, we derive a critical value of the heatingrate to produce the explosion, based on discussion of a crit-ical neutrino luminosity in the literature. Below this criticalvalue, the shock cannot be launched.It is well known that there is a critical neutrino lu-minosity to produce an explosion driven by neutrino heat-ing mechanism. Burrows & Goshy (1993) indicated a criticalneutrino luminosity as a function of mass accretion rate as L ν e ∝ ˙ M / . , in which neutrino average energy was as-sumed to be a constant. More recently, subsequent stud-ies updated the expression of critical neutrino luminosityby taking into account other physical parameters, e.g. neu-trino average energy, PNS radius, etc. Here, we utilize Janka(2012), which gives L ν,c ( ˙ M ) ∝ ˙ M / M / . In the current setup, we found that the critical neutrino luminosity for s20 is L ν e ,c ≈ . × erg s − , with M NS ≈ . M ⊙ (determinedby M s =4 ) and ˙ M ≈ . M ⊙ s − . By changing parametersto M NS ≈ . M ⊙ and ˙ M ≈ . M ⊙ s − , which are rele-vant for s12, we get L ν e ,c ≈ . × erg s − . For s15, i.e. M NS ≈ . M ⊙ and ˙ M ≈ . M ⊙ s − , L ν e ,c ≈ . × ergs − . These values are roughly consistent with our numer-ical results. Since the mass accretion rate evolution is notconstant in s12 and s15, the direct comparison is not verymeaningful. We do not try to derive more precise estimate,because it is not the main focus of this study.The heating rate by neutrino can be estimated withEqs. (83) and (86) of Janka (2001) as˙ E exp = H − C = 3 . × erg s − (2 L ν e , ) ρ s, r s, ( r s /r g ) , (20)where r g is the gain radius, which is ≈
100 km, and ρ s, is density behind the shock in units of 10 g cm − .With ˙ M = 4 πr ρv and compression ratio β = 4, ρ s, =0 .
14 ˙ M . M − / , . ( r s, / − / , where ˙ M . = ˙ M / . M ⊙ s − , M NS , . = M NS / . M ⊙ , which are relevant for WH07s20. v = p GM NS / r s is again used. For L ν e , = 4, r s, = 2, and r s /r g = 2, we get ˙ E exp = 2 . × erg s − , whichroughly agrees with model WH07s20L4 (see Table 2). Bycombining critical neutrino luminosity, we can derive a crit-ical heating rate to produce an explosion as˙ E exp ,c = 1 . × erg s − ˙ M / . M / . ( r s, / / ( r s / r g ) . (21)This is slightly larger than a consequent value of modelWH07s20L3 ( ˙ E exp = 1 . × erg s − ). The inconsistencyis originated from the simplification of the analytic model,which we do not discuss further. The temperature can be estimated by4 π r s aT ζ = E int + ˙ E exp t, (22)where E int is the initial internal energy and ζ = 2 .
44 (seeSection 2.2). Here we assume that E int is compensating forgravitational binding energy at onset of the explosion (i.e.the explosion energy becomes positive) so that it does notappear in the expression of the explosion energy. From thisequation, the temperature is written as T = (cid:18) E int + ˙ E exp t )4 πr s aζ (cid:19) / (23)= 6 . × K( E int , + ˙ E exp , t ) / r − / s, , (24)where E int , = E int / erg and ˙ E exp , = ˙ E exp / ergs − . By combining Eqs. (13) and (24), we get T =6 . × K( E int , + ˙ E exp , t ) / ×
320 ˙ E / , ˙ M . ρ . R, R . / t . + r . / , ! − / . , (25)where ˙ M = ˙ M/M ⊙ s − .Next, we derive E int that dominates the temperatureevolution in the early phase, from stellar structure. Since astanding accretion shock turns to a runaway phase when thethermal pressure in postshock regime becomes larger thanthe ram pressure in preshock regime, the time evolution ofram pressure is crucial. The preshock ram pressure can beevaluated by the free-fall model as P ram = ρv = ˙ M s πr s v acc , (26)where M s is a total mass enclosed by the shock and ˙ M s is mass accretion rate at the shock. Here we assume that M s + ˙ M s δt ≈ M s , where δt is the timescale we are interestedin. The mass accretion rate is (Woosley & Heger 2012)˙ M s = dM s dt ff = 2 M s t ff ρ ¯ ρ − ρ , (27)where ρ is the density at t = 0 and ¯ ρ = 3 M s / (4 πr )is the mean density inside r (initial radius of the massshell). t ff is the free-fall time, which is t ff = p π/ (32 G ¯ ρ ) = p π r / (8 GM s ) (Kippenhahn & Weigert 1990). By combin-ing them and using ¯ ρ ≫ ρ , we get P ram = 43 π GM s r ρ (cid:18) r r s (cid:19) / . (28) MNRAS , 1– ????
320 ˙ E / , ˙ M . ρ . R, R . / t . + r . / , ! − / . , (25)where ˙ M = ˙ M/M ⊙ s − .Next, we derive E int that dominates the temperatureevolution in the early phase, from stellar structure. Since astanding accretion shock turns to a runaway phase when thethermal pressure in postshock regime becomes larger thanthe ram pressure in preshock regime, the time evolution ofram pressure is crucial. The preshock ram pressure can beevaluated by the free-fall model as P ram = ρv = ˙ M s πr s v acc , (26)where M s is a total mass enclosed by the shock and ˙ M s is mass accretion rate at the shock. Here we assume that M s + ˙ M s δt ≈ M s , where δt is the timescale we are interestedin. The mass accretion rate is (Woosley & Heger 2012)˙ M s = dM s dt ff = 2 M s t ff ρ ¯ ρ − ρ , (27)where ρ is the density at t = 0 and ¯ ρ = 3 M s / (4 πr )is the mean density inside r (initial radius of the massshell). t ff is the free-fall time, which is t ff = p π/ (32 G ¯ ρ ) = p π r / (8 GM s ) (Kippenhahn & Weigert 1990). By combin-ing them and using ¯ ρ ≫ ρ , we get P ram = 43 π GM s r ρ (cid:18) r r s (cid:19) / . (28) MNRAS , 1– ???? (2017) Suwa, Tominaga, and Maeda -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 T e m p e r a t u r e ( K ) Mass (M ⊙ )WH07s12L2WH07s15L3WH07s20L4WH07s25L4Analytic Figure 4.
Maximum temperature distributions of four numericalsimulations (colored solid lines) and analytic expression (blackdashed lines). For analytic models, we use ˙ E exp taken from Table3, ρ and R of M s =4 +0 . M ⊙ which are taken from Table 2, originof mass coordinate set to M s =4 , and r mc , = 2. ˙ M in analyticmodels are 0.15 (WH07s12), 0.2 (WH07s15), 0.3 (WH07s20), and0.3 M ⊙ s − (WH07s25), respectively, which are taken from Figure3. Numerical results are horizontally sifted by 0 . M ⊙ (WH07s12,WH07s20, and WH07s25) and 0 . M ⊙ (WH07s15) leftward fordirect comparison with analytic lines. Therefore, the internal energy of postshock regime is givenby e int = 3 P ram = 4 π GM s r ρ (cid:18) r r s (cid:19) / . (29)Here we assume that the pressure is dominated by radiationcomponent, i.e. γ = 4 / E int can be estimated as E int = 4 πr s × P rad (30)= 163 GM s ρ r / r / s (31)= 3 . × (cid:18) M s . M ⊙ (cid:19) (cid:18) ρ g cm − (cid:19) × (cid:16) r s
100 km (cid:17) / (cid:16) r (cid:17) / erg . (32)Note that this value is not an actual total internal energyincluded by the shock, but is a rough estimate of an initialinternal energy of the ejecta which consists of a thin shellthat is promptly exploding.In Figure 4, we show a comparison between numericalresults and analytic solutions for the maximum temperaturedistribution as a function of mass coordinate. We pick upWH07s12L2, WH07s15L3, WH07s20L4, and WH07s25L4,for typical models, since these models start exploding whenthe mass accretion rate is (almost) constant (see Table 3 andFigure 3). For analytic models, we solve shock evolution byEq. (13), which only includes accreted mass in the ejectamass, but we also add swept mass by using Eq. (11) andvalues ( ρ and R ) at M = M s =4 + 0 . M ⊙ from Table 2. Thisapproximation works well, since the shock evolution by Eq.(13) is not largely different from a direct numerical integra-
0 0.05 0.1 0.15 0.2 T e m p e r a t u r e ( K ) Mass (M ⊙ )1D ( E · exp,51 =2)1D ( E · exp,51 =4)3D ( E · exp,51 =2)3D ( E · exp,51 =4) Figure 5.
The same plot as Figure 4, but only analytic solutionsshown for the model WH07s20. Solid and dashed lines indicateone-dimensional (1D) evolution and three-dimensional (3D) ones,respectively. Red and blue lines indicate different growth rates ofthe explosion energy, ˙ E exp , respectively. A critical temperaturefor Ni production ( T = 5 × K) is also presented by greyhorizontal line. tion of Eq. (9) (see Section 3.2.2). In addition, we use ˙ E exp taken from Table 3, origin of mass coordinate set to M s =4 ,and r mc , = 2. ˙ M in analytic models are 0.15 (WH07s12),0.2 (WH07s15), 0.3 (WH07s20), and 0.3 (WH07s25), respec-tively, which are taken from Figure 3. Numerical resultsare horizontally shifted by 0 . M ⊙ (WH07s12, WH07s20,and WH07s25) and 0 . M ⊙ (WH07s15) leftward for directcomparison with analytic lines in Figure 4. These shifts areshowing systematic error in analytic models, but these er-ror is small enough to discuss conventional amount of Ni,i.e. 0.07 M ⊙ (for SN 1987A, 1993J, and 1994I). Numericaland analytic models of WH07s20L4 and WH07s25L4 agreerather well for most regime, since these models have consid-erably constant mass accretion rate (see Figure 3). On theother hand, WH07s12L2 and WH07s15L3 show deviationbetween numerical and analytic models, especially in thelate time (i.e. large mass coordinate), because these modelshave evolving mass accretion rates that break our assump-tion. Nevertheless, temperature profile where we are inter-ested in, i.e. T >
5, are well reproduced by the analyticmodels.
Next, let us introduce multidimensional (multi-D) effectsin the analytic model. It turns out from recent neutrino-radiation hydrodynamics simulations that postshock pres-sure is not determined by thermal pressure alone, but tur-bulent pressure (i.e. Reynolds stress) is also contributing.Roughly speaking, the turbulent pressure becomes compa-rable to the thermal pressure (e.g. Couch & Ott 2015). Inaddition, at the propagating phase the kinetic energy be-comes comparable to the internal energy in the ejecta (see,e.g. Figure 14 in Bruenn et al. 2016). Therefore, it is natu-ral to introduce a factor ( ≈ .
5) for internal energy amountin Eqs. (25) and (32), to take into account multi-D effects,
MNRAS , 1– ?? (2017)6 Ni production of CCSN i.e. E int + ˙ E exp t → (cid:16) E int + ˙ E exp t (cid:17) . (33)Figure 5 shows the impact of multi-D effect on the tem-perature evolution. As is shown, the temperature of multi-D model decreases compared to one-dimensional model. Wealso represent the dependence of ˙ E exp in this figure. Roughlyspeaking, multi-D models produce half amount of Ni ofone-dimensional models, which is consistent with conse-quence of Yamamoto et al. (2013), in which they performedhydrodynamics simulations as well as nucleosynthesis calcu-lations of 1D and 2D (axial symmetry).Even below the critical heating rate derived for the 1Dcases, successful explosions were observed in multi-D simula-tions. Multi-D effect is not only reducing the internal energyas explained above, but also reducing critical neutrino lu-minosity (e.g. Murphy & Burrows 2008; Hanke et al. 2012;Couch 2013). Previous works typically showed that multi-D simulations imply a smaller critical neutrino luminosityfor the explosion than 1D ones by ∼ E exp is pro-portional to L ν e , the critical heating rate would be also re-duced by ∼
20% in multi-D simulations. In addition, multi-D simulations would produce partial explosions. In partic-ular, it is often seen in two-dimensional simulations that apart of material explodes (polar direction) and other partforms a downflow accreting onto a PNS. These structure re-duces both diagnostic explosion energy and ejecta mass, andleads to smaller amount of Ni. We employ the followingexpression to take into account partial explosion effect onthe amount of Ni; M Ni = M Ni ,c ˙ E exp ˙ E exp , c , (34)where M Ni ,c is the amount of Ni corresponding to criticalheating rate in multi-D model. It is worthy to note thatspherical symmetric explosion maximizes the amount of Ni(Maeda & Tominaga 2009; Suwa & Tominaga 2015). Ni mass
In this subsection, we explain the amount of Ni dependingon the explosion energy growth rate and progenitor models.Figure 6 presents the amount of Ni as a function of ˙ E exp in 1D cases. All parameters other than ˙ E exp are the sameas Figure 4. Thick lines give analytic estimate and coloredregion show uncertainty of models. For instance, neutrino-driven wind increases the amount of Ni, definitely depen-dent on Y e profile of the wind, and fallback of ejecta con-versely decreases Ni. Since the impact of these effects islargely uncertain, we here roughly present error region with ± . M ⊙ as a guideline. It should be noted that this fig-ure implies discrepancy between our numerical models andanalytic model, especially for WH07s12 and WH07s15 witha rather larger ˙ E exp than critical value, since these modelsshow time-evolving mass accretion rate, which breaks theassumption employed in the analytic model. The numeri-cal models, however, employ a constant neutrino luminosity,which means feedback effects of mass accretion rate evolu-tion are neglected. A natural expectation of the feedback ef-fect is that the neutrino luminosity decreases as the mass ac-cretion rate decreases. Then, shock launch is obtained once M N i ( M ⊙ ) Explosion energy growth rate (10 erg)WH07s12/1DWH07s15/1DWH07s20/1D Figure 6.
The amount of Ni as a function of the growth rate ofthe explosion energy, ˙ E exp . Horizontal grey line indicates a canon-ical value of Ni, 0 . M ⊙ . Thick lines give analytic estimate withthe same parameter sets as Figure 4 but different ˙ E exp . Coloredregions present possible error with ± . M ⊙ , which is caused by,for instance, neutrino-driven wind upwards or fallback downward.The left endpoints correspond to the critical ˙ E exp , which are esti-mated by Eq. (21). Since WH07s25 indicate rather similar resultas WH07s20 (see Figure 4), it is not shown in this figure. M N i ( M ⊙ ) Explosion energy growth rate (10 erg)WH07s12/mDWH07s15/mDWH07s20/mD Figure 7.
The same plot as Figure 6, but for multi-dimensionalcases, in which reduction of thermal energy (Eq. 33), reduction ofthe critical heating rate (by 20% from Figure 6), and reductionof ejecta mass (Eq. 34) are taken into account. The reduction ofejecta mass is only taken into account below the critical heatingrate, which makes bend of lines around ˙ E exp , ≈ the mass accretion rate reaches a stationary state with aconstant mass accretion rate, which exists for WH07s12 andWH07s15 as well, but rather late time (see Figure 3). There-fore, our analytic model works well.In Figure 7, we show the amount of Ni by multi-D cases, in which reduction of thermal energy (Eq. 33),reduction of critical heating rate (by 20% from 1D) andreduction of ejecta mass (Eq. 34) are all taken into ac-count. As is shown, to achieve enough Ni synthesis, we
MNRAS , 1– ????
MNRAS , 1– ???? (2017) Suwa, Tominaga, and Maeda need rather large growth rate of the explosion energy,larger than ≈ − for WH07s20 and even larger forWH07s12 and WH07s15. Note that in this estimate, we donot include contribution from neutrino-drive wind which islargely uncertain in this study. Bruenn et al. (2016) indi-cated the amount of ejected Ni, in which both explosivenucleosynthesis component and neutrino-driven wind com-ponent are included, as 0.035 (WH07s12), 0.077 (WH07s15),0.065 (WH07s20), and 0.074 (WH07s20) M ⊙ , respectively.The growth rate of the explosion energy is roughly, ∼ . ∼ ∼ . ∼ − , respectively. Therefore, by takingcontributions of explosive nuclear burning from our ana-lytic model, we find that neutrino-driven wind contributesfor ∼ .
02 (WH07s12), ∼ .
04 (WH07s15), ∼ . M ⊙ (WH07s20 and WH07s25), respectively. It is worthy to notethat their simulations in 2D exceptionally succeeded to pro-duce enough Ni, but their 3D model (Lentz et al. 2015)exhibited a much smaller ˙ E exp than 2D (see Table 1), whichimplies difficulty of Ni synthesis in their 3D simulation. Ni is an important indicator of the supernova explo-sion, which characterizes light curves, particularly late decayphase. In principle, the amount of Ni can be directly mea-sured by light curve alone, while ejecta mass and explosionenergy are estimated by combining light curve and spectrumproperties. Nevertheless, the explosion energy has often beenpaid attention from explosion mechanism community, sinceit is easier to estimate from numerical data than the amountof Ni. The final explosion energy, however, is difficult toestimate by detailed numerical simulations, which solve hy-drodynamics equations as well as neutrino-radiation transferequation. This is because current simulations can reach only O (1) s, but the explosion energy can grow even after. On theother hand, Ni should be generated within short timescaleafter the onset of the explosion, i.e. O (0 .
1) s, because inorder to synthesize Ni high temperature ( > × K)is necessary and temperature decreases rather fast as theshock propagates. Therefore, the amount of Ni is betterindicator for the explosion condition.In this paper, we investigated the amount of Ni syn-thesized by explosive nucleosynthesis in supernova ejecta bymeans of numerical simulations and an analytic model. Fornumerical simulations, we employ Lagrangian hydrodynam-ics code in which neutrino heating and cooling terms aretaken into account by light-bulb approximation. Initial con-ditions are taken from Woosley & Heger (2007), which have12, 15, 20, and 25 M ⊙ in zero age main sequence. We ad-ditionally developed the analytic model, which gives a rea-sonable estimate of the amount of Ni. We found that toproduce enough amount of Ni (0.07 M ⊙ ), we need O (1)Bethe s − of growth rate of the explosion energy, which ismuch larger than canonical exploding simulations, typically O (0 .
1) Bethe s − .It should be noted that a recent model fitting studysuggested that the distribution of M ( Ni) in normal type-II supernovae is rather broad, i.e. from 0.005 to 0.28 M ⊙ (M¨uller et al. 2017). Our model implies that these diversitycan be mainly produced by different progenitor masses, i.e. lighter progenitor models would produce less Ni than moremassive progenitors. However, it should be also noted thatestimates of local supernovae are concentrating around 0.07 M ⊙ (e.g., Arnett et al. 1989). With precise measurements of M ( Ni) and the ejecta mass (related to progenitor mass),it is able to give stringent constraint on the explosion mech-anism of core-collapse supernovae. The current study alsoimplies that in order to produce enough amount of Ni,progenitor models which have a large value of compactnessparameter are preferred. This is reasonable because a pro-genitor model, which has a small compactness parameter,is extended and temperature of important mass coordinate( ∼ . M ⊙ above shock launching point) cannot be highenough to synthesize Ni. This trend is opposite to theexplodability, which prefers a small value of compactnessto produce the successful explosion. These two observationsmay indicate that there is a limited parameter space of pro-genitors, which can explain both the explodability and Niproduction simultaneously.
ACKNOWLEDGEMENTS
This study was supported in part by the Grant-in-Aid forScientific Research (Nos. 26800100, 15H02075, 15H05440,16H00869, 16H02158, 16H02168, 16K17665, and 17H02864).YS was supported by MEXT as “Priority Issue on Post-Kcomputer” (Elucidation of the Fundamental Laws and Evo-lution of the Universe) and JICFuS. TN and KM were sup-ported by the World Premier International Research CenterInitiative (WPI Initiative), MEXT, Japan. Discussions dur-ing the YITP workshop YITP-T-16-05 on “Transient Uni-verse in the Big Survey Era: Understanding the Nature ofAstrophysical Explosive Phenomena” were useful to com-plete this work.
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