aa r X i v : . [ a s t r o - ph ] O c t A CCEPTED FOR PUBLICATION IN T HE A STROPHYSICAL J OURNAL
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NEUTRONIZATION DURING TYPE IA SUPERNOVA SIMMERING A NTHONY
L. P
IRO AND L ARS B ILDSTEN
Kavli Institute for Theoretical Physics, Kohn Hall, University of California, Santa Barbara, CA 93106;[email protected], [email protected]
Accepted for publication in The Astrophysical Journal
ABSTRACTPrior to the incineration of a white dwarf (WD) that makes a Type Ia supernova (SN Ia), the star “simmers”for ∼ Z ) dependent neutronization through the Neabundance (as studied by Timmes, Brown, & Truran). The main consequence is that we expect a “floor” tothe level of neutronization that dominates over the metallicity contribution when Z / Z ⊙ . /
3, and it can beimportant for even larger metallicities if substantial energy is lost to neutrinos via the convective Urca process.This would mask any correlations between SN Ia properties and galactic environments at low metallicities. Inaddition, we show that recent observations of the dependences of SNe Ia on galactic environments make it clearthat metallicity alone cannot provide for the full observed diversity of events.
Subject headings: nuclear reactions, nucleosynthesis, abundances — supernovae: general — white dwarfs INTRODUCTIONThe use of Type Ia supernovae (SNe Ia) as cosmologi-cal distance indicators has intensified the need to understandwhite dwarf (WD) explosions. Of particular importance is theorigin of the Phillips relation (Phillips et al. 1999), an essen-tial luminosity calibrator. Recent models demonstrate that itcan be explained by large variations in the abundance of stableiron group elements (Woosley et al. 2007) with the dominantcause for diversity likely residing in the explosion mechanism(Mazzali et al. 2007).One additional variable is the metallicity of the WD core,which yields excess neutrons relative to protons due to theisotope Ne. This is usually expressed as Y e = X i Z i A i X i , (1)where A i and Z i are the nucleon number and charge ofspecies i with mass fraction X i . The neutronization is crit-ical for setting the production of the neutron-rich isotopes(Thielemann et al. 1986). If no weak interactions occur dur-ing the explosion, the mass fraction of Ni produced issimply X ( Ni) = 58 Y e -
28, assuming Ni and Ni are theonly burning products (Timmes et al. 2003). The neutroniza-tion also affects the explosive burning, including the laminarflame speed (Chamulak et al. 2007a). However, the metallic-ity range of progenitors is not large enough to account for thefull SNe Ia diversity (see §4), making it critical to explore allfactors that determine Y e .A potential neutronization site is the convective car-bon burning core that is active for ∼ Current address: Astronomy Department and Theoretical As-trophysics Center, University of California, Berkeley, CA 94720;[email protected] flame commences (Timmes & Woosley 1992). We show herethat protons from the C( C, p ) Na reaction during simmer-ing capture on C, and that subsequent electron captures on N and Na decrease Y e . This process continues until eitherthe explosion occurs or sufficient heavy elements have beensynthesized that they capture the protons instead.In §2, we present simmering WD core models and summa-rize the reaction chains that alter Y e . We find that one protonis converted to a neutron for each six C nuclei consumedfor burning at ρ < . × g cm - . At densities above this,an additional conversion occurs from an electron capture on Na. Hence, the Y e in the core depends on the amount ofcarbon burned during simmering and the density at which itoccurs, which we quantify in §3. We find that neutronizationduring simmering dominates for metallicities Z / Z ⊙ . / NEUTRON PRODUCTION DURING SIMMERINGThermally unstable burning begins when the energy gen-eration rate from carbon fusion, ǫ , exceeds neutrino cool-ing (Nomoto et al. 1984). The thin solid lines in Figure1 show the range of ignition curves for X ( C) = X ( O)(Yakovlev et al. 2006) with the middle line the nominalcurrent best. The carbon fuses via C( C , p ) Na and C( C , α ) Ne with branching ratios of 0 .
44 and 0 .
56, re-spectively. At “early” times the liberated protons capture onto C, while at “late” times enough heavy elements ( Na or Ne) have been produced that they capture the protons in-stead.We treat the evolution during the simmering phase as aseries of hydrostatic models consisting of an adiabatic con-vective core and an isothermal surface at 10 K. As longas the convection zone is well described as an adiabat thisis sufficient for resolving the thermal structure without theneed to explicitly solve the energy transfer equation. Theseassumptions become weaker once the central temperature is T c & × K, so that burning occurs sufficiently quicklythat there is considerable energy generation within a convec- PIRO & BILDSTENtive eddy overturn timescale (Garcia-Senz & Woosley 1995).The energy generation does come into play because it sets theheating timescale, t h ≡ c p T c /ǫ , where c p is the specific heatof the liquid ions (we use linear mixing and the Coulomb en-ergy from Chabrier & Potekhin 1998), nearly given by theclassical Dulong-Petit law c p ≈ k B /µ i m p , where µ i is the ionmean molecular weight. Since we evaluate t h using the cen-tral conditions it is a lower limit since it should include theentire heat capacity of the convective region (Piro & Chang2007; see related discussion for neutron stars in Weinberg etal. 2006). In this way, for a given thermal profile there is awell-defined heating timescale, which connects our stationarymodels to the true time evolution. The thick dashed lines inFigure 1 trace out the trajectory of the central temperature, T c ,and density, ρ c , for M = 1 . M ⊙ and M = 1 . M ⊙ (left andright, respectively), both using compositions of X ( C) = 0 . X ( O) = 0 .
48, and X ( Ne) = 0 .
02. These indicate that ρ c de-creases with increasing T c (Lesaffre et al. 2006; Piro 2007).The thick solid lines show thermal profiles near the end of thesimmering.The simmering phase ends when sub-sonic convection canno longer transport the heat outwards because the timescaleof heating is now less than the convective overturn timescale.Since the overturn timescale depends on the integrated energygeneration rate near the WD center (while we desire a localmeasure of when convection should end for simplicity), weassume that this occurs when t h ∼ t dyn ≡ ( G ρ c ) - / , the dy-namical timescale. This gives reasonable agreement to othermore careful calculations that find the simmering phase endswhen T c ≈ × K (Woosley et al. 2004). We plot t h = 10 t dyn as a dotted line in Figure 1 to indicate where simmering ends,since the strong temperature sensitivity of C fusion makesthis line rather insensitive to the choice of prefactor. If thesimmering phase ends earlier, it can be considered in the con-text of our models by just truncating our results at a slightlylower T c . 2.1. Main Reaction Cycle at Early Times
At early times, only C, O, or Ne are poten-tial proton capture nuclei. We compared these rates us-ing Caughlan & Fowler (1988), including strong screening(Salpeter & van Horn 1969). The O( p , γ ) F reaction isnegligible, whereas resonances in the Ne( p , γ ) Na reac-tion make its rate comparable to C( p , γ ) N. However, thelarger abundance of C means that it captures more pro-tons by a factor of (22 / X ( C) / X ( Ne) ≈
40. The fate ofthe synthesized N requires some discussion, as the branch-ing amongst the three relevant reactions: N( e - , ν e ) C, N( γ, p ) C, N( p , γ ) O, depends on T , ρ , and proton massfraction, X p .The production of protons is always the rate limiting step,so that each proton is almost immediately captured by C.This means that we can find X p by balancing the proton pro-duction rate from carbon fusion, λ n h σ v i + (where λ =0 .
44 is the branching ratio for the reaction C( C , p ) Naand n is the C number density) with the proton capturerate, n p n h σ v i p + , where n p is the proton number density, X p = λ X ( C)12 h σ v i + h σ v i p + . (2)This is plotted in the upper panel of Figure 2. The small F IG . 1.— Conditions in the simmering WD. Thin solid lines are the range ofcarbon ignition curves from Yakovlev et al. (2006). The arching thick dashedlines show the central trajectory for the simmering core of a 1 . M ⊙ anda 1 . M ⊙ WD (left and right, respectively). Thick solid lines are examplethermal profiles that are composed of an adiabatic convective core connectedto an isothermal exterior. Each case is taken near the end of the simmeringphase, which occurs at the dotted line labeled t h = 10 t dyn . Dashed lines arecrossing points for nuclear reactions described in the text.F IG . 2.— Mass fractions X p and X ( N) ( upper panel ) and N destructionrates ( lower panel ), versus temperature. The dotted line at T = 8 × Kdivides where N photodisintegrations becomes important. The density is3 × g cm - and X ( C) = 0 . value of X p confirms our equilibrium assumption for the pro-ton abundance, and allows us to show that the N( p , γ ) Oreaction is negligible (bottom panel of Fig. 2) in comparisonto electron captures. The bottom panel also shows the ratesfor the reactions N( e - , ν e ) C and N( γ, p ) C, making itclear that electron capture dominates for T < × K.All of the N comes from the protons synthesized by car-EUTRONIZATION DURING SN Ia SIMMERING 3
TABLE 1R
EACTION S UMMARY FOR M AIN C YCLE
Reaction Thermal Energy Release (MeV) C( C , p ) Na 2.239 C( C , α ) Ne 4.617 C( p , γ ) N 1.944 N( e - , ν e ) C 0 C( α , n ) O 2.214 C( n , γ ) C 4.947 Na( e - , ν e ) Ne a a This reaction only occurs when t h > t ec , (Fig. 1). bon fusion, therefore the N production rate is equal to thatof protons. Once again, this is the slowest step so that wecan find the N abundance by balancing n p n h σ v i p + = R ec ( N) n , where R ec ( N) is the N electron capture rate.Captures into excited states are unlikely to be dominant, al-lowing us to use the measured f t = 4 . × s from theground-state transitions. (Likewise, we use f t = 1 . × sfor Na electron captures in §2.2.) Combining with equation(2) gives X ( N) = λ (cid:2) X ( C) (cid:3) ρ N A h σ v i + R ec ( N) , (3)where N A is Avagodro’s number, which is shown in the toppanel of Figure 2. The network is completed by C( α, n ) Oand C( n , γ ) C, leading to a composition of one each of C, O, Ne, Na.There are two complications. The first is at high T ’s wherephotodisintegration of N happens faster than the electroncaptures (above the dashed curve labeled by t ec , = t ph in Fig.1). Chemical balance ( p + C ↔ N + γ ) is achieved in thislimit, fixing the proton to N ratio. The N is then slowlyremoved due to electron captures. The electron captures mustalways balance the proton production, so the N abundanceremains identical to equation (3). Hence, photodisintegrationadds steps to the reaction chain (and alters the proton density;top panel of Fig. 2) but does not modify the conclusion that allprotons released in C burning lead to N electron capture.The second complication is the reaction Na( e - , ν e ) Neat high densities ( ρ > . × g cm - ). This occurs to theright of the dashed line labeled as t h = t ec , in Figure 1, illus-trating that these electron captures only take place at certaintimes during the simmering phase, which we account for in§3. Electron captures on N would not have time to occurabove the dashed line labeled t h = t ec , in Figure 1, but this isalways after the explosion.The main reaction cycle is summarized in Table 1. Six Care consumed, producing C, O, Ne, and depending on t h ,either Ne or Na. Therefore, during each cycle either oneor two protons have been converted to neutrons.2.2.
Late Time Truncation of Neutron Production
The carbon burning ashes eventually become abundantenough to compete with C for proton captures. The rele-vant products are Ne, and either Na or Ne. The Nereactions of Ne( p , γ ) Na and Ne( p , α ) F are negligible,so we focus on Na and Ne.We plot the proton destruction rate for the most relevantreactions in Figure 3 as a function of f , the fraction of C F IG . 3.— The proton destruction rates for different processes as a functionof the fraction of C that has burned, f . The number density of Na or Ne is taken to be f n /
3. Circles denote crucial places where the protondestruction mechanism switches. that has burned. The number density of either Na or Ne istaken to be equal to the number density of C burned times2 / × λ ≈ /
3. Circles denote where the rates cross eachother, which is nearly independent of ρ . For t h < t ec , , f mustexceed ≈ .
13 before the Na( p , α ) Ne reaction becomesimportant, and ends neutronization.When t h > t ec , , Ne forms, which has a higher protoncapture cross section than Na. Burning only f = 0 .
061 isenough that Ne( p , n ) Na becomes the primary proton sink.Since this makes Na and liberates a neutron, the new Namay electron capture again to make Ne, and the reactionchain Ne( p , n ) Na( e , ν e ) Ne can repeatedly occur, makingmany free neutrons. Competing with this process are other re-action chains that burn these neutrons (for example, see Table6 of Arnett & Thielemann 1985). In this regime, it is difficultfor us to estimate all the key nuclear reactions that will takeplace. Although further neutronization is possible, we cannotfollow this without a full reaction network (Chamulak et al.2007b). Such calculations must also be coupled to a realisticmodel for the core temperature evolution (such as what wepresent here). MAXIMUM NEUTRONIZATION ESTIMATESWe set η as the number of protons that are converted toneutrons for every six C consumed, so that η = 2 ( η = 1) for t h > t ec , ( t h < t ec , ), where we approximate λ ≈ .
5. The η =2 case is an upper limit since in parts of the convection zonewhere ρ < . × g cm - the Na does not electron capture(and Ne that is mixed to lower densities by convection maydecay). The total neutronization is measured via Y e = 12 - X ( Fe)56 - X ( Ne)22 - f η X ( C)6 × , (4)which includes the initial Fe and Ne content. Neutroniza-tion halts either when the WD explodes or when freshly madeheavy elements compete for protons (Fig. 3).In the case of competition from fresh heavy elements, trun-cation at high densities occurs when f = 0 .
061 with η = 2. Themaximum change in Y e is therefore ∆ Y e , max = - . × - X ( C)0 . . (5) PIRO & BILDSTENA similar limit pertains at lower densities. One way to ex-ceed this limit in the high density case is if additional reactionchains occur (see §2.2). We show ∆ Y e , max as a dot-dashedline in Figure 4, in comparison to the ∆ Y e ’s that result from X ( Ne) = 0 .
007 and 0 .
02 ( dotted lines ). By coincidence, themaximum effect of neutronization during simmering is com-parable to that associated with a solar metallicity.The other possible limiter of neutronization is the onset ofthe explosion. The reactions in Table 1 show that Q ≈ If we let E c be the total thermal content that iswithin the convective core with respect to the initial isother-mal WD, this implies a change ∆ Y e = - η E c m p / QM c in a con-vective core of mass M c , ∆ Y e = - . × - η E c ergs M ⊙ M c , (6)For this to compete with the Ne contribution, a total energy E c = 1 . × ergs 2 η X ( Ne)0 . M c M ⊙ , (7)or 7 × ergs g - , must be released prior to the explosion.Simmering ends when dynamical burning is triggered, re-quiring T c ≈ × K (Woosley et al. 2004). If the burningoccurred within a single zone with the specific heat of §2,then reaching this T c would require ≈ . × ergs g - , inexcess of that implied by equation (7). Of course, in realitythe convective zone extends outward, so that little mass is at T c . To accurately determine the resulting neutronization, weconstruct hydrostatic WD models consisting of fully convec-tive cores as described at the beginning of §2. We considerisothermal temperatures of either 10 K or 2 × K. At anygiven moment there is a well defined M c (Lesaffre et al. 2006;Piro 2007), and we evaluate the current thermal content by in-tegrating the specific heat relative to the initially isothermalWD, E c = Z M c c p [ T ( M ) - T i ] dM , (8)where T i is the isothermal WD temperature. In this way weuse our time independent models to find the fraction of carbonthat must have burned, f , and the associated ∆ Y e as T c and M c increase with time. We assume no neutrino losses and thus all ≈
16 MeV of thermal energy contributes to heating.In Figure 4 we summarize the results of these calculations.In each case, the slope of ∆ Y e shows a break at the transi-tion from η = 2 ( t h > t ec , ) to η = 1. This break occurs laterfor more massive WDs (Fig. 1), thus these have more neu-tronization during simmering. Increasing the isothermal tem-perature decreases M c , so that it takes less burning to reach agiven T c . These fully integrated models make it clear that sub-stantial neutronization occurs prior to the explosion. In com-parison to the ∆ Y e from Ne, simmering effects dominate if X ( Ne) < .
013 or Z / Z ⊙ . /
3. This thwarts the occurrenceof high Y e SNe Ia in low metallicity progenitors. CONCLUSION AND DISCUSSION Not all of this energy will always go into heating up the core. For exam-ple, if the convective Urca process is operating, then it will take more energy(and more carbon burning) to get to sufficient temperatures. F IG . 4.— Thick dashed ( M = 1 . M ⊙ ) and solid ( M = 1 . M ⊙ ) lines showthe change of Y e as a function of the current central temperature, T c , dueto carbon burning during simmering. In each case we consider isothermaltemperatures of 10 K and 2 × K (as labeled). The dot-dashed line is themaximum change possible if proton captures on carbon are only limited bycompetition from freshly made heavy elements (eq. [5]). Dotted lines bracketthe change in Y e expected for X ( Ne) = 0 . - . We have found that significant neutronization of the WDcore occurs throughout the simmering stage of carbon burninguntil the onset of the explosion. If substantial energy is lost tothe convective Urca process (Lesaffre et al. 2005, and refer-ences therein), then the neutronization is truncated by protoncaptures onto freshly synthesized heavy elements (resultingin eq. [5]). The main consequence is a uniform “floor” to theamount of neutronization that dominates over the metallicitydependent contribution for all progenitors with Z / Z ⊙ . / η ratherthan simply setting it to 1 or 2.In closing, we highlight some important features exhib-ited by recent observations of SNe Ia. It is clear that theamount of Ni produced in SNe Ia has a dynamic range(0 . - M ⊙ ) larger than can be explained by metallicity or sim-mering neutronization. However, since an intriguing trend isthe prevalence of Ni rich events in star-forming regions it isinteresting to quantitatively explore how large the observeddiscrepancy is. Using the SNLS sample of Sullivan et al.(2006), Howell et al. (2007) found that the average stretch is s = 0 .
95 in passive galaxies (e.g. E/S0’s) and s = 1 .
05 in star-forming galaxies. Using Jha et al’s (2006) ∆ M ( B ) - s rela-tion and Mazzali et al.’s (2007) relation between ∆ M ( B ) and Ni mass we get 0 . M ⊙ ( s = 0 .
95) and 0 . M ⊙ ( s = 1 . ≈ . M ⊙ more Nithan those in large ellipticals.EUTRONIZATION DURING SN Ia SIMMERING 5Since the SN Ia rate scales with mass in ellipticals andstar formation rate in spirals (Mannucci et al. 2005; Scan-napieco & Bildsten 2005; Sullivan et al. 2006), SNe frompassive galaxies in the SNLS survey are from more massivegalaxies than the SNe in star-forming galaxies (Sullivan et al.2006). Using the mass-metallicity relation of Tremonti et al.(2004), our integration of the separate samples in Sullivan etal. (2006) yield average 12 + log(O / H) = 8 .
87 in active galax-ies and 9 . + log(O / H) = 8 . / O at high metallicities (Liang etal. 2006), the SNe in ellipticals have twice as much Necontent as those in spirals. From the relation of Timmes etal. (2003), this implies ≈
5% less Ni, whereas the ob-served decrement is > ∆ X ( Ne) ≈ .
06, or nearly3 times solar. Although we have found that simmering en-hances neutronization, the effect is not great enough ( ∆ Y e , max would give the same change in neutronization as doubling asolar metallicity), and a diverse set of core conditions wouldstill be required. A large enhancement could be present inthe core if substantial gravitational separation had occurred(Bildsten & Hall 2001; Deloye & Bildsten 2002), yet convec-tive mixing during simmering will reduce it based on the frac-tion of the star that is convective. For a convection zone thatextends out to M ⊙ , the resulting Ne enhancement would beat most ≈30%.We thank E. Brown, R. Ellis, F. Forster, A. Howell, B. Pax-ton, P. Podsiadlowski, H. Schatz and F. Timmes for discus-sions, and D. Yakovlev for sharing carbon ignition curves. Wealso thank the referee for constructive comments and ques-tions. This work was supported by the National Science Foun-dation under grants PHY 05–51164 and AST 02-05956.