Nucleosynthesis of r-Process Elements by Jittering Jets in Core-Collapse Supernovae
aa r X i v : . [ a s t r o - ph . H E ] N ov NUCLEOSYNTHESIS OF R-PROCESS ELEMENTS BYJITTERING JETS IN CORE-COLLAPSE SUPERNOVAE
Oded Papish and Noam Soker ABSTRACT
We calculate the nucleosynthesis inside the hot bubble formed in the jittering-jets model for core collapse supernovae (CCSNe) explosions, and find the forma-tion of several × − M ⊙ of r-process elements. In the jittering-jets model fastjets launched from a stochastic accretion disk around the newly formed neutronstar are shocked at several thousands km, and form hot high-pressure bubbles.These bubbles merge to form a large bubble that explode the star. In the cur-rent study we assume a spherically symmetric homogenous bubble, and follow itsevolution for about one second during which nuclear reactions take place. Thejets last for about one second, their velocity is v j = 0 . c , and their total energyis ∼ erg. We use jets’ neutron enrichment independent on time, and fol-low the nuclear reactions to the formation of the seed nuclei up to Z ≤
50, onwhich more neutrons will be absorbed to form the r-process elements. Based onthe mass of the seed nuclei we find the r-process element mass in our idealizedmodel to be several × − M ⊙ , which is slightly larger than the value deducedfrom observations. More realistic calculations that relax the assumptions of ahomogenous bubble and constant jets composition might lead to agreement withobservations.
1. INTRODUCTION
The mechanism for the explosion of core-collapse (CC) supernovae (SNe) is still un-known. Most popular are models based on explosion driven by neutrinos. Less popularare models based on jet-driven explosions. (e.g. LeBlanc & Wilson 1970; Meier et al. 1976;Bisnovatyi-Kogan et al. 1976; Khokhlov et al. 1999; MacFadyen et al. 2001; H¨oflich et al.2001; Fargion 2003; Woosley & Janka 2005; Couch et al. 2009, 2011; Lazzati et al. 2011). Re-cent observations (e.g., Wang et al. 2001; Leonard et al. 2001, 2006; Elmhamdi et al. 2003; Department of Physics, Technion – Israel Institute of Technology, Haifa 32000, Israel; [email protected]; [email protected] ∼
300 km, and revive the shock. Namely, they eject the material in that region. Insome theoretical studied the jets were injected at large distances beyond the stalled-shockradius (e.g., Khokhlov et al. 1999; H¨oflich et al. 2001; Maeda & Nomoto 2003; Couch et al.2009, 2011). H¨oflich et al. 2001 injected slow and fast jets at a radius of 1200 km in a heliumstar for about two seconds. They show the possibility of exploding the star with slow jets.Their results show that the polarization in SN199em is consistent with slow jets. In modelm2r1hot of Couch et al. 2011 a jet was injected near the speed of sound, leading to theformation of a hot bubble. In others, like MacFadyen et al. (2001), the jets were injectedmuch closer to the neutron star (NS), at 50 km. In the simulations of MacFadyen et al.(2001) the jets were injected at a much later time in the explosion, and are less relevant toour goal of exploding a star with jets. Kohri et al. (2005) propose that disk-wind energy isable to revive a stalled shock and help to produce a successful supernova explosion.Our jittering-jet model for explosion (Soker 2010; Papish & Soker 2011, hereafter Paper1 ) is based on the following points, that differ in several ingredients from the models citedabove (for more detail see Paper 1). (1) We don’t try to revive the stalled shock. Tothe contrary. Our model requires the material near the stalled-shock to fall inward andform an accretion disk around the newly born NS or black hole (BH). (2) We conjecturethat due to stochastic processes and the stationary accretion shock instability (SASI; e.g.Blondin & Mezzacappa 2007) segments of the post-shock accreted gas (inward to the stalledshock wave) possess local angular momentum. When they accreted they form and accretiondisk with rapidly varying axis direction. (3) We assume that the accretion disk launchestwo opposite jets. Due to the rapid change in the disk’s axis, the jets can be intermittentand their direction rapidly varying. These are termed jittering jets. (4) We show in Paper1 that the jets penetrate the infalling gas up to a distance of few × × penetrating jet feedback mechanism. (5) The jets deposit their energy inside thestar via shock waves, and form two hot bubbles, that eventually merge and accelerate therest of the star and lead to the explosion. In section 2 below we use self similar calculationsto further explore this process. (6) The jets are launched only in the last phase of accretion 3 –onto the NS. For the required energy the jets must be launched from the very inner regionof the accretion disk.Nucleosynthesis can occurs in the expanding jets (or disk winds) and in the postshockregion. Previous studies include Cameron (2001) who discussed the nucleosynthesis insidejets launched at a velocity of 0 . c from an accretion disk around a rapidly rotating proto NS.He suggested the possibility of creating r-process elements inside the jets. Nishimura et al.(2006) simulated the r-process nucleosynthesis during a jet powered explosion. In theirsimulation a rotating star with a magnetic field induces a jetlike outflow during the collapsewhich explode the star. Neutrinos play no role in the simulation. Unlike their model, ourmodel does not have a large scale rotation of the star, and the jets penetrate farther awaycreating hot bubbles.In our jittering-jets explosion model the jets are launched close to the NS where thegas is neutron-rich (e.g., Kohri et al. 2005). In section 3 we examine the implications of thison the nuclear reactions (nucleosynthesis) in the inflated hot bubble. The properties of thebubbles are as derived in section 2. Our discussion and summary are in section 4.
2. DYNAMICAL EVOLUTION OF THE INFLATED BUBBLES
In this section we describe an approximate model for the inflated bubbles and theirdynamical evolution. The analysis here is similar to that conducted by Volk & Kwok (1985)to study the evolution of the spherical hot bubble in planetary nebulae. During the activephase of the jets we derive a self-similar analytical solution to the gas-dynamical equations.At later times the solution is numerical.The jittering jets form wide bubbles that occupy most of the volume up to the distancethey have reached; eventually the bubbles merge (see Paper 1). Our basic assumption istherefore that the two inflated bubbles merge to form one large bubble. The low densityhigh energy volume inside R s is termed hereafter the ‘spherical bubble’. If we equate thevolume of the assumed spherical bubble with the total volume of the wide inflated bubbles,the radius of the spherical bubble R s is only slightly smaller that the distance the inflatedbubbles have reached. This assumption leads to a spherically symmetric flow that allows aself similar solution for constant power jets. We also assume that the gas inside the bubbleis homogeneous, i.e., the composition, density, and pressure are constant inside the bubble.The energy of the jets injected into the bubble with a power of ˙ E j forces the bubble toexpand. The expanding spherical bubble pushes the dense surrounding gas supersonicallyoutward, forming a forward shock ahead of this dense shell. The boundary of the dense 4 –shell and the bubble is a contact discontinuity, across which the pressure is constant but notthe density or the composition. This flow structure is drawn schematically in Fig. 1. Themass M s in the shell is the swept-up ambient gas. As the dense shell is thin, we take theradius of the forward shock to be equal to the radius of the contact discontinuity (which isthe radius of the spherical bubble) R s . The pre-shock (up-stream) ambient density profileat radius r > R s , is taken to be a power law, with the scaling from Wilson et al. (1986) andMikami et al. (2008) (see Paper 1) ρ s ( r ) = Ar β = 1 . × (cid:16) r
100 km (cid:17) − . g cm − , . r . km . (1)The spherical flow obeys the following conservation equations (Volk & Kwok 1985) dM s dt = 4 πR s ρ ( R s ) ˙ R s , (2) ddt (cid:16) M s ˙ R s (cid:17) = 4 πR s P ˙ R s , (3) ddt (cid:0) πR s P (cid:1) = ˙ E j − πR s P ˙ R s , (4)where P is the pressure inside the bubble. Equations (2) - (4) describe the conservationof mass, momentum, and energy respectively. The energy inside the bubble includes the Stellar core
Forward shockContact discontinuityJetsJets termination shock
Sphericalbubble
NS Rs
Stellar core
Fig. 1.—: Schematic drawing of the inflated spherical bubble. The spherical bubble ispowered by jittering jets, i.e., they change their direction at a high rate, launched from anaccretion disk around the newly formed neutron star. This spherical bubble explodes thestar according to our model. R-process elements are fused inside the bubble. The typicalradius during the jets’ active phase is 3000-10000 km. 5 –thermal energy of the gas and the radiation energy. We neglect losses by neutrinos (seePaper 1) and energy production and sink from nuclear reactions.During the active time period of the jets the solution to equations (2) - (4) is a self-similar solution, which we take in the form R s ( t ) = R t α . (5)Using equations (2) - (4) we get the following parameters α = 3 β + 5 , R β +50 = ( β + 3)( β + 5) πA (2 β + 7)( β + 8) ˙ E j . (6)A short time of ∼ . r ∼
15 km for a time period of t s = 1 − ,
000 km s − (Cameron 2001). This velocity islarger than the velocity used in Paper 1 as it better fits the escape velocity from the surfaceof the newly formed NS. The velocity is chosen to be the escape velocity from the neutronstar, as we are interested in the jets emerging from the neutron star vicinity. Some studiesinject the jets at distances of > km withinseveral seconds. The total mass carried by the two jets is either 0 . M ⊙ or 0 . M ⊙ ,corresponding to a total injected energy of E = 1 . × erg or E = 1 . × ergrespectively. For the parameters of ˙ E = 1 . × erg s − , and active phase time of t s = 1 s,for example, the solution during the jets’ active phase is R s ( t ) = 6 . × t . cm , < t < t s = 1 s . (7)For later times we numerically integrate equations (2) - (4) with ˙ E j = 0, and using thethe results of the self similar solution at t = t s as initial conditions. For a check, we alsonumerically integrate the equations for the full time of the solution. The numerical solutioncoincides with the self-similar solution after a very short time. The full numerical solutionfor three cases are shown in Fig. 2. The plot shows the radius R s , temperature T , density ρ , and entropy s of the bubble as a function of time. The parameters for the three cases aregiven in the figure caption. The jets’ power was taken to be a constant for 0 < t < t s , and˙ E = 0 for later times. This gives the little bump at t = t s in the graph.The two cases differ by jets’ power and having active time of t s = 1 s, are similar intheir general behavior. As well, changing the active phase duration from t s = 1 s to t s = 2 s 6 –does not make large differences in the dynamical properties (the extra density line in theleft panel of fig. 2.). All cases lead to explosion. Later we will show that these cases aredifferent in nucleosynthesis outcomes. l og ( T K ) l og ( ρ g c m − ) l og ( K B / nu c l eon ) l og ( R s c m ) ρρ l sTR s ρ sTR s Fig. 2.—: Left: Radius R s , Temperature T , density ρ , and entropy s of the spherical bubbleas a function of time for the model with jets’ power of ˙ E = 1 . × erg s − and a jets’active phase lasting t s = 1 s. ρ l is the density for a case with ˙ E j = 1 × erg s − for thesame duration of T s = 1 s. Right: the same but for a model with ˙ E = 0 . × erg s − and an active phase duration of t s = 2 sFor the typical parameters expected in the model the main results of this section areas follows. (1) The spherical bubble reaches a typical radius of ∼ km at the end of thejets’ active phase. (2) The temperature relevant for nucleosynthesis ( T ∼ × K) occursat about one seconds from the beginning of the jet injection. (3) The density in the bubbleof ∼ g cm − at that time implies that nuclear reactions will be of a high enough rateto be significant. For that, in the next section we study the nucleosynthesis inside the bubble. 7 –
3. NUCLEOSYNTHESIS INSIDE THE BUBBLE
In the previous section we found the temperature inside the bubble to start at ∼ K,and to decrease due to adiabatic cooling to 2 . × K in ∼ T ≃ × K andstop at T ≃ . × K. During this time fresh material is injected into the bubble fromthe jets. As the jets are launched from very close to the neutron star, they are composedof highly enriched neutron material (Kohri et al. 2005). During the expansion of the jetsthey adiabatically cool, and nucleons might fuse to give α particles and heavier nuclei (e.g.,Cameron 2001; Maeda & Nomoto 2003; Fujimoto et al. 2008). However, the jets are even-tually shocked with a post shock temperature of > K. At that temperature all nucleirapidly disintegrate, and a gas composed of free nucleons is formed. Our nucleosynthesiscalculations start from the free-nucleons post shock jets’ gas. At t ≃ . T ≃ × K, the free nucleons fusionrate overcomes the disintegration rate and α particles start to be accumulated. During thetime up to ∼ α particle fuse to form heavier nucleiuntil α freeze-out is reached.The nuclear reaction network is similar to that given in Woosley & Hoffman (1992).The reaction rates are taken from the JINA Reaclib Database (Cyburt et al. 2010), andinclude reactions with 1,2, and 3 body interactions and beta decays. The reaction networkis integrated assuming a uniform composition in the bubble and a continues injection ofprotons and neutrons from the jets until the jets terminated. For the electron fraction Y e weuse values for the accretion disk around a neutron star as calculated by Kohri et al. (2005).For parameters relevant to our model we find from the calculations of Kohri et al. (2005)that the neutron to proton ratio is in the range n/p ≃ −
10, namely, Y e ≃ . − .
17. Herewe integrate the reaction network for 3 different electron fractions Y e = 0 . , . , .
25. Thiscorrespond to a neutron to proton ratio of n/p = 10 , , and 3. We study nucleosynthesis forjets’ active phase of 1 s and 2 s. The network is solved independently of the hydrodynamicssolution of section 2 (Hix & Thielemann 1999).The nucleosynthesis results are summarized in Fig. 3. The plots show the mass fractionof neutrons , α particles, and of total seed elements during the evolution of the bubble. Byseed elements we refer to nuclei on which further neutron capture will occur to synthesis ther-process nuclei (Woosley & Hoffman 1992; Witti et al. 1994). The inclusion of all nuclearprocesses beyond the seed nuclei is beyond the scope of the present paper. They will bestudied in a forthcoming paper where a full multi-dimensional gasdynamical code will beused to study the interaction of the jets with the core material.We assume that the post-shock freshly injected jets’ material is fully mixed inside the 8 –bubble. This is based on the expected formation of vortices inside the bubble by the jitteringjets. The post-shock velocity for γ = 4 / ∼ ,
000 km s − , and for a bubble’s radius of R s ≃ ∼ . >
100 K B / nucleon as is required forr-process elements production (Hoffman et al. 1997). Our flow structure differs from calcu-lations where there is no mixing and the entropy is almost constant during nucleosynthesis(e.g. Witti et al. 1994; Woosley et al. 1994; Arcones et al. 2007; Kuroda et al. 2008).As well, the continuous injection of nucleons as nucleosynthesis takes place reduces thefinal mass of seed nuclei. This is seen by comparing the mass fraction of seed nuclei for thetwo simulated cases, of 1 s and 2 s jets active phase duration (Table 1 and Fig. 3). Thetable shows also the sensitivity of the nucleosynthesis production to the value Y e assumedat the base of the jet.Model I Model II Model IIITotal mass 0 .
60 0 .
60 0 .
60 0 .
36 0 .
36 0 .
36 0 .
60 0 .
60 0 . . . . . . . .
85 0 .
85 0 . Y e .
09 0 .
17 0 .
25 0 .
09 0 .
17 0 .
25 0 .
09 0 .
17 0 .
25n 79% 60% 42% 79% 60% 42% 81% 63% 46% α
8% 14% 21% 8% 14% 20% 14% 24% 33%Seed 13% 25% 36% 13% 25% 37% 4% 10% 17%Table 1:: Mass fraction of neutrons, α particles and seed elements for different electronfractions Y e . Models I and II are for jets with a duration of t s = 1 s. Model III is for jetswith a duration of t s = 2 s. Total mass is in units of 10 − M ⊙ . Power is in units of 10 erg.The most likely case is for jets’ active phase of 1 − Y e = 0 . − .
17, corresponding to a neutron toproton ratio of n/p = 5 −
10 (Kohri et al. 2005). Also, the total energy might be slightlyless that 1 . × erg used here. From those values we find the seed nuclei mass fractionto be 0 . − .
2, and the corresponding total seed nuclei mass to be 10 − − − M ⊙ , with 9 –Fig. 3.—: Evolution of the mass fraction of neutrons, α particles, and seed nuclei for ther-process. Left: Model I with t s = 1 s and power of ˙ E = 1 . × erg. Right: Model IIIwith t s = 2 s and power of ˙ E = 0 . × erg. In each panel the initial ratio of electron tonucleon Y e is given. The total mass is 0 . M ⊙ for both cases.more likely values in the range 10 − − × − M ⊙ . After neutrons are absorbed (beyondthe scope of this paper) and form the r-process elements, the total mass of the r-processelements is 2 − ∼ several × − M ⊙ .¿From the solar abundance Mathews & Cowan (1990) deduced that the average massof r-process material ejected in a CCSN is ≈ − M ⊙ . Our simple and idealized modeloverproduces r-process elements, but not by much. In future 3D numerical simulations threeassumptions that have been used here will be relaxed. These might reduce the production ofr-process elements by a factor of ∼ −
5. (1) Simulating precessing jets will cause deviationfrom sphericity. We expect that in some regions the production of r-process elements willbe less efficient. (2) Adding the mass at the termination shocks of the jets will result in aninhomogeneous bubble. Basically, the flow will differ by having regions where the mattercools adiabatically with no addition of entropy (low entropy regions), and regions of highentropy where gas is added. This might lead to less efficient production of r-process elementsin some regions. (3) We will change the the neutron enrichment (or Y e ) of the jets with time.It is quite possible that at early times the jets are less neutron enriched than at later timewhen the accretion disk is depleted and more mass comes from closer to the NS. 10 –
4. SUMMARY
Our main goal was to examine the nucleosynthesis inside the bubbles formed by thejets in the the jittering-jets model for core collapse SN explosion (Paper 1). The two jetsare launched from an unrelaxed accretion disk around the newly formed NS. Because ofstochastic accretion of mass and angular momentum the disk’s axis is rapidly changing andthe disk might even be intermittent. The jets penetrate to a distance of few × penetrating jetfeedback mechanism. To facilitate a solution in the scope of the present paper we assumed that the bubblesformed by the two jittering jets merge to form one large spherical bubble, as shown schemat-ically in Fig. 1. The spherical solution under these assumptions is composed of two phases.In the first one, the jets’ active phase, energy and mass are injected into the bubble at con-stant rates. The second phase starts when the jets’ cease, and the bubble starts to expandadiabatically. The gasdynamical equations in spherical symmetry were solved analyticallyusing a self-similar solution for the jets’ active phase, and numerically thereafter (section 2).Beyond the assumptions of jittering jets that have the power to explode the star andthe formation of a spherical bubble, all quantitative parameters have been used before bysome studies. We did not adjust or played with any quantitative parameter in the solutionspresented here. (1) The jets’ total energy of 1 − . × erg is taken from the energy requiredto explode CCSNe. (2) The jets’ velocity of v j = 0 . c comes from the escape velocity nearthe NS surface. This value for jets’ velocity from NS has been used before, e.g., Cameron(2001). (3) The energy and velocity determine the total mass carried by the jets. (4) The ∼ − Y e ) of the jets’ material is taken from the calculation of Kohri et al. (2005).The assumption of a spherical bubble that has the energy to explode the star and thequantitative parameters listed above determine the properties and evolution of the bubble.From these we calculated the nucleosynthesis inside the bubble. In the limited scope of thepresent paper we numerically integrated a reaction network that follows the fusion up to theseed nuclei with Z ≤
50. The results of the nucleosynthesis calculations for the three studiedcases are presented in Table 1 and Fig. 3. We note that Ye = 0:25 is above the expectedvalue during the main phase of the jets (Kohri et al. 2005), but might be applicable at early 11 –time before the NS is fully relaxed by neutrino cooling.During the integration of the network an α freeze-out is reached, i.e., when the numberof α particles does not change anymore. We take the heavy nuclei from the α freeze-outto be the seed elements for r-process elements. From the mass of seed elements for thetypical parameters expected in this study, Y e = 0 .
1, energy of 10 erg, and active jets’duration of t s ≃ − × − M ⊙ . This is a few times larger than ≈ − M ⊙ , the average mass of r-processelements per CCSN deduced from observations (Mathews & Cowan 1990). We note that inmany observed cases the production of r-process elements is much below the average (e.g.,Sneden et al. 2010 and references therein). It is possible, therefore, that the conditions usedhere are met only in a fraction of CCSNe. For example, in some cases the value of Y e at thebase of the jets is larger than used here, namely Y e & .
3, where the number of neutrons isnot sufficient to produce r-process elements.One strong assumption of the present work is that the bubble is homogeneous in tem-perature and composition. This assumption will be relaxed in a future study when the gas-dynamical equations will be solved with a multi-dimensional numerical code. The accuratetreatment of the inflation process of the bubble will justify the inclusion of a more extendednuclear reaction network. Nevertheless, our results here strongly suggest that within thecontext of the jittering-jets model for core collapse SN explosions, the nucleosynthesis ofthe r-process elements is a a likely possibility. This outcome strengthen the possibility thatCCSNe are driven by jets.We thank an anonymous referee for helpful comments. This research was supported bythe Asher Fund for Space Research at the Technion and the Israel Science foundation.
REFERENCES
Arcones, A., Janka, H.-T., & Scheck, L. 2007, A&A, 467, 1227Arnould, M., Goriely, S., & Takahashi, K. 2007, Phys. Rep., 450, 97Bethe, H. A. 1990, Reviews of Modern Physics, 62, 801Bisnovatyi-Kogan, G. S., Popov, I. P., & Samokhin, A. A. 1976, Ap&SS, 41, 287Blondin, J. M., & Mezzacappa, A. 2007, Nature, 445, 58Brandt, T. D., Burrows, A., Ott, C. D., & Livne, E. 2011, ApJ, 728, 8 12 –Cameron, A. G. W. 2001, ApJ, 562, 456Chugai, N. N., Fabrika, S. N., Sholukhova, O. N., et al. 2005, Astronomy Letters, 31, 792Couch, S. M., Wheeler, J. C., & Milosavljevi´c, M. 2009, ApJ, 696, 953Couch, S. M., Pooley, D., Wheeler, J. C., & Milosavljevi´c, M. 2011, ApJ, 727, 104Cyburt, R. H., et al. 2010, ApJS, 189, 240Elmhamdi, A., Danziger, I. J., Chugai, N., et al. 2003, MNRAS, 338, 939Fargion, D. 2003, Chinese Journal of Astronomy and Astrophysics Supplement, 3, 472Fujimoto, S.-i., Nishimura, N., & Hashimoto, M.-a. 2008, ApJ, 680, 1350Hanke, F., Marek, A., Mueller, B., & Janka, H.-T. 2011, arXiv:1108.4355Hix, W. R., & Thielemann, F.-K. 1999, Journal of Computational and Applied Mathematics,109, 321Hoffman, R. D., Woosley, S. E., & Qian, Y.-Z. 1997, ApJ, 482, 951H¨oflich, P., Khokhlov, A., & Wang, L. 2001, 20th Texas Symposium on relativistic astro-physics, 586, 459Khokhlov, A. M., H¨oflich, P. A., Oran, E. S., et al. 1999, ApJ, 524, L107Kohri, K., Narayan, R., & Piran, T. 2005, ApJ, 629, 341Kuroda, T., Wanajo, S., & Nomoto, K. 2008, ApJ, 672, 1068Lazzati, D., Morsony, B. J., Blackwell, C. H., & Begelman, M. C. 2011, arXiv:1111.0970LeBlanc, J. M., & Wilson, J. R. 1970, ApJ, 161, 541Leonard, D. C., Filippenko, A. V., Ardila, D. R., & Brotherton, M. S. 2001, ApJ, 553, 861Leonard, D. C., Filippenko, A. V., Ganeshalingam, M., et al. 2006, Nature, 440, 505MacFadyen, A. I., Woosley, S. E., & Heger, A. 2001, ApJ, 550, 410Maeda, K., & Nomoto, K. 2003, ApJ, 598, 1163Mathews, G. J., & Cowan, J. J. 1990, Nature, 345, 491Meier, D. L., Epstein, R. I., Arnett, W. D., & Schramm, D. N. 1976, ApJ, 204, 869 13 –Mikami, H., Sato, Y., Matsumoto, T., & Hanawa, T. 2008, ApJ, 683, 357Nishimura, S., Kotake, K., Hashimoto, M.-a., Yamada, S., Nishimura, N., Fujimoto, S., &Sato, K. 2006, ApJ, 642, 410Nordhaus, J., Burrows, A., Almgren, A., & Bell, J. 2010, ApJ, 720, 694Papish, O., & Soker, N. 2011, MNRAS, 1321Sneden, C., Roederer, I., Cowan, J., & Lawler, J. E. 2010, Nuclei in the Cosmos.,Smith, N., Cenko, S. B., Butler, N., et al. 2011, arXiv:1108.2868Soker, N. 2010, MNRAS, 401, 2793Volk, K., & Kwok, S. 1985, A&A, 153, 79Wang, L., Howell, D. A., H¨oflich, P., & Wheeler, J. C. 2001, ApJ, 550, 1030Wilson, J. R., Mayle, R., Woosley, S. E., & Weaver, T. 1986, Annals of the New YorkAcademy of Sciences , 470, 267Witti, J., Janka, H.-T., & Takahashi, K. 1994, A&A, 286, 841Woosley, S. E., & Hoffman, R. D. 1992, ApJ, 395, 202Woosley, S. E., Wilson, J. R., Mathews, G. J., Hoffman, R. D., & Meyer, B. S. 1994, ApJ,433, 229Woosley, S., & Janka, T. 2005, Nature Physics, 1, 147