Is Strong SASI Activity the Key to Successful Neutrino-Driven Supernova Explosions?
Florian Hanke, Andreas Marek, Bernhard Mueller, Hans-Thomas Janka
aa r X i v : . [ a s t r o - ph . S R ] J un D raft version J uly
25, 2018
Preprint typeset using L A TEX style emulateapj v. 5 / / IS STRONG SASI ACTIVITY THE KEY TO SUCCESSFUL NEUTRINO-DRIVEN SUPERNOVA EXPLOSIONS? F lorian H anke , A ndreas M arek , B ernhard M¨ uller , and H ans -T homas J anka Max-Planck-Institut f¨ur Astrophysik, Karl-Schwarzschild-Str. 1, D-85748 Garching, Germany; [email protected], [email protected]
Draft version July 25, 2018
ABSTRACTFollowing a simulation approach of recent publications we explore the viability of the neutrino-heating explo-sion mechanism in dependence on the spatial dimension. Our results disagree with previous findings. While wealso observe that two-dimensional (2D) models explode for lower driving neutrino luminosity than sphericallysymmetric (1D) models, we do not find that explosions in 3D occur easier and earlier than in 2D. Moreover, wefind that the average entropy of matter in the gain layer hardly depends on the dimension and thus is no gooddiagnostic quantity for the readiness to explode. Instead, mass, integrated entropy, total neutrino-heating rate,and nonradial kinetic energy in the gain layer are higher when models are closer to explosion. Coherent, large-scale mass motions as typically associated with the standing accretion-shock instability (SASI) are observed tobe supportive for explosions because they drive strong shock expansion and thus enlarge the gain layer. While2D models with better angular resolution explode clearly more easily, the opposite trend is seen in 3D. Weinterpret this as a consequence of the turbulent energy cascade, which transports energy from small to largespatial scales in 2D, thus fostering SASI activity. In contrast, the energy flow in 3D is in the opposite direction,feeding fragmentation and vortex motions on smaller scales and thus making the 3D evolution with finer gridresolution more similar to 1D. More favorable conditions for explosions in 3D may therefore be tightly linkedto e ffi cient growth of low-order SASI modes including nonaxisymmetric ones. Subject headings: supernovae: general — hydrodynamics — stars: interiors — neutrino INTRODUCTION
Recent simulations in two dimensions (2D) with sophisti-cated neutrino transport have demonstrated that the neutrino-driven mechanism, supported by hydrodynamic instabilitiesin the postshock layer, can yield supernova explosions atleast for some progenitor stars (11.2 and 15 M ⊙ ones inMarek & Janka 2009; M¨uller et al. 2012). The explosions oc-cur relatively late after bounce and tend to have fairly lowenergy, being “marginal” or only slightly above the “criti-cal threshold” in this sense. Suwa et al. (2010) obtained asimilar explosion for a 13 M ⊙ progenitor in their axisym-metric simulations. However, the Oak Ridge group hasfound stronger and earlier explosions for a wider range ofprogenitors (Bruenn et al. 2009), while in purely Newtoniansimulations with multi-dimensional neutrino di ff usion (in-cluding energy dependence but without energy-bin coupling)Burrows et al. (2006, 2007) could not see any success of thedelayed neutrino-driven mechanism.While the reason for the discrepant results of these sim-ulations cannot be satisfactorily understood on the basis ofpublished results, the marginality of the 2D explosions of theGarching group and the lack of neutrino-driven explosionsin the simulations by Burrows et al. (2006, 2007) raises theimportant question about the influence of the third spatialdimension on the post-bounce evolution of collapsing stel-lar cores. Three-dimensional (3D) fluid dynamics with theirinverse turbulent energy cascade compared to the 2D caseare likely to change the flow pattern on large scales as wellas small scales. They could have an influence on the ex-istence and the growth rate of nonradial hydrodynamic in-stabilities in the supernova core even in the absence of stel-lar rotation (see, e.g., Iwakami et al. 2008) but in particularwith a moderate amount of angular momentum in the progen-itor star (e.g., Blondin & Mezzacappa 2006a; Iwakami et al.2009; Fern´andez 2010), and thus could lead to di ff erences in the hydrodynamic and thermodynamic conditions for theoperation of the neutrino-heating mechanism. In particular,3D flows might cause important changes of the dwell time ofpostshock matter in the layer where neutrinos deposit energy,which is a crucial aspect deciding about the viability and ef-ficiency of the neutrino-driven supernova mechanism (someaspects of this were discussed by Murphy & Burrows 2008and Marek & Janka 2009).Indeed, employing a simplified treatment of neutrino e ff ectsby including local neutrino-cooling and heating terms for achosen value of the neutrino luminosity and spectral temper-ature instead of solving the computationally intense neutrinotransport, Nordhaus et al. (2010) found considerably easierand earlier explosions in 3D than in 2D. In the context ofthe concept of a critical value of the neutrino luminosity that(for a given mass accretion rate onto the stalled supernovashock) has to be exceeded to obtain neutrino-driven explo-sions (Burrows & Goshy 1993; Janka & M¨uller 1996; Janka2001; Yamasaki & Yamada 2005; Murphy & Burrows 2008;Pejcha & Thompson 2011; Fern´andez 2012), they quantifiedthe improvement of 3D relative to 2D by a 15–25% reduc-tion of the critical luminosity value. In particular, they ob-served that 3D postshock convection leads to higher averageentropies in the neutrino-heating layer, thus improving theconditions for shock revival due to a significant stretching ofthe residence time of matter in the layer where it gains energyfrom neutrinos. Very recently, Takiwaki et al. (2011) reportedenhanced maximum dwell times of a small fraction of the ma-terial in the gain region in a 3D simulation compared to the 2Dcase of an 11.2 M ⊙ star, but could not unambiguously link thise ff ect to an easier explosion of the 3D model. In particular,their 3D simulation showed a shock expansion that was moredelayed than in the 2D run, and the 3D conditions did not ap-pear more favorable for an explosion with respect to a varietyof quantities like the net heating rate, the heating timescale orthe profiles of maximum and minimum entropies. Hanke, Marek, M¨uller, & JankaIn this paper we present a comparative investigation for11.2 M ⊙ and 15 M ⊙ progenitors in one, two, and three di-mensions along the lines of the study by Nordhaus et al.(2010), varying the driving neutrino luminosities used in time-dependent collapse simulations of the two stars. While ourresults for spherically symmetric (1D) and 2D models basi-cally confirm the dimension-dependent di ff erences found byMurphy & Burrows (2008) and Nordhaus et al. (2010), ourcalculations do neither exhibit a strict 1D-2D-3D hierarchy ofthe average entropy in the gain layer, nor do they show anyclear signs that 3D e ff ects facilitate the development of theexplosion better than nonradial motions in 2D. Attempting tounderstand the reason for this puzzling finding we vary theresolution of the spherical coordinate grid used for our 2Dand 3D simulations. The outcome of these studies reflectsthe action of the energy flow within the turbulent energy cas-cade. The latter transports the driving energy provided byneutrino heating and gravitational energy release in the ac-cretion flow from small to large scales in 2D and oppositein 3D. Models in 2D show growing large-scale asymmetryand quasi-periodic time variability and explode clearly easierwith higher resolution, whereas in 3D better resolved mod-els are observed to become more similar to the 1D case andthus to be farther away from an explosion. This suggests thatthe success of the neutrino-driven mechanism could be tightlylinked to the initiation of strong non-radial mass motions inthe neutrino-heated postshock layer on the largest possiblescales, implying that the easier explosions of our 2D mod-els with higher resolution are a consequence of more violentactivity due to the standing accretion-shock instability (SASI;Blondin et al. 2003), whereas the better resolved 3D modelsfor the employed artificial setup of supernova-core conditionstend to reveal considerably reduced amplitudes of low-orderspherical-harmonics modes of nonradial deformation and thusbehave more similar to the 1D case.The paper is structured as follows. In Sect. 2 we brieflydescribe our numerics and implementation of neutrino sourceterms. In Sect. 3 we give an overview of the simulations pre-sented in this paper. Our investigations of the dependence ofthe critical luminosity on the spatial dimension will be pre-sented in Sect. 4 and results of resolution studies in Sect. 5.An interpretation of our findings will follow in Sect. 6. Sec-tion 7 contains the summary and conclusions. In App. A wepresent 1D simulations that document our e ff orts to repro-duce the results of previous, similar studies in the literature bystraightforwardly applying the neutrino treatment described inthese works. NUMERICAL SETUP
We solve the equations of hydrodynamics reflecting theconservation of mass, momentum, and energy, ∂ρ∂ t + ∇ · ( ρ v ) = , (1) ∂ρ v ∂ t + ∇ · ( ρ v ⊗ v ) + ∇ P = − ρ ∇ Φ , (2) ∂ e ∂ t + ∇ · [( e + P ) v ] = − ρ v · ∇ Φ + ρ (cid:0) Q + ν − Q − ν (cid:1) , (3)where ρ is the mass density, v the fluid velocity, Φ the grav-itational potential, P the pressure, and e the total (inter-nal + kinetic) fluid energy density. These equations are in-tegrated in a conservative form (for which reason the en- ergy equation is solved for the total energy density) us-ing the explicit, finite-volume, higher-order Eulerian, multi-fluid hydrodynamics code P rometheus (Fryxell et al. 1991;M¨uller et al. 1991a,b). It is a direct implementation of thePiecewise Parabolic Method (PPM) of Colella & Woodward(1984) using the Riemann solver for real gases developed byColella & Glaz (1985) and the directional splitting approachof Strang (1968) to treat the multi-dimensional problem. Inorder to prevent odd-even coupling (Quirk 1994) we switchfrom the original PPM method to the HLLE solver of Einfeldt(1988) in the vicinity of strong shocks. The advection of nu-clear species is treated by the Consistent Multi-fluid Advec-tion (CMA) scheme of Plewa & M¨uller (1999).To facilitate comparison with Nordhaus et al. (2010) wealso employ the high-density equation of state (EoS) ofShen et al. (1998) and do not include general relativistic cor-rections. We use the monopole approximation of the Poissonequation to treat Newtonian self-gravity.To make our extensive parameter study possible, we use thelocal source terms applied by Murphy & Burrows (2008) andNordhaus et al. (2010) instead of detailed neutrino transport(see Janka 2001 for a derivation of these source terms). Inthis approach the neutrino heating and cooling rates Q + ν and Q − ν are given by Q + ν = . · L ν e erg s − ! T ν e ! (4)
100 km r ! (cid:16) Y n + Y p (cid:17) e − τ e ff " ergg s , Q − ν = . · (cid:18) T (cid:19) (cid:16) Y n + Y p (cid:17) e − τ e ff " ergg s . (5)These approximations depend on local quantities, namely thedensity ρ , the temperature T , the distance from the center ofthe star r , and the neutron and proton number fractions Y n and Y p , respectively. In Eq. (4) the electron-neutrino luminosity L ν e is a parameter and is assumed to be equal to the electronantineutrino luminosity L ¯ ν e = L ν e . The neutrino temperature T ν e is set to 4 MeV.The employed source terms, Eqs. (4) and (5), without thefactors e − τ e ff are valid for optically thin regions only. In or-der to model the transition to neutrino trapping at high op-tical depths, we follow Nordhaus et al. (2010) and multiplythe heating and cooling terms by e − τ e ff to suppress them in theinner opaque core of the proto-neutron star. Here, the opticaldepth for electron neutrinos and antineutrinos is defined as τ e ff ( r ) = Z ∞ r κ e ff ( r ′ ) d r ′ . (6)The e ff ective opacity κ e ff was derived by Janka (2001) andgiven in Murphy et al. (2009): κ e ff ≈ . · − · X n , p ρ g cm − ! T ν e ! cm − , (7)where X n , p = (cid:16) Y n + Y p (cid:17) accounts for composition averag-ing. In multi-dimensional simulations we evaluate the radialintegration for the optical depth τ e ff independently for eachlatitude θ in 2D and each pair of latitudinal and azimuthalangles ( θ , φ ) in 3D. Note that in Murphy & Burrows (2008)the exponential suppression factor is absent in the heating andcooling terms (or was not mentioned), which otherwise agreeASI activity as key to successful neutrino-driven SN explosions? 3with ours, while no definition of the e ff ective optical depth τ e ff is given in Nordhaus et al. (2010). The factor (cid:16) Y n + Y p (cid:17) in Eqs. (4) and (5) is included in Nordhaus et al. (2010), butnot in Murphy & Burrows (2008) and Murphy et al. (2009).The time period from the onset of the collapse until 15 msafter bounce was tracked with the P rometheus -V ertex code(Rampp & Janka 2002) in 1D including its detailed, multi-group neutrino transport, all relevant neutrino reactions and a“flashing treatment” for an approximative description of nu-clear burning during infall. This means that until 15 ms afterbounce we describe neutrino e ff ects including the evolution ofthe electron fraction Y e with high sophistication. At 15 ms af-ter core bounce we switch to the simple neutrino heating andcooling terms and upon mapping from 1D to 2D also impose(on the whole computational grid) random zone-to-zone seedperturbations with an amplitude of 1% of the density to breakspherical symmetry.Although during the subsequent evolution we apply theheating and cooling expressions of Eqs. (4) and (5) follow-ing Nordhaus et al. (2010) and Murphy & Burrows (2008),we refrain from adopting their treatment of changes of theelectron fraction Y e . Following a suggestion by Liebend¨orfer(2005), they prescribed Y e simply as a function of density, Y e ( ρ ), instead of solving a rate equation with source termsfor electron neutrino and antineutrino production and destruc-tion consistently with the expressions employed for neutrinoheating and cooling. Liebend¨orfer (2005) found that sucha parametrization, supplemented by a corresponding entropysource term (and a neutrino pressure term in the equation ofmotion), yields results in good agreement with 1D simula-tions including neutrino transport during the collapse phaseuntil the moment of bounce-shock formation. A universal Y e - ρ -relation, which serves as the basis of this approximation andcan be inferred for the infalling matter during the homolo-gous collapse phase, however, applies neither for the evolu-tion of the shocked accretion flow in the post-bounce phasenor for expanding neutrino-heated gas (see, e.g., Fig. 4.9 inThielemann et al. 2011). For example, a comparison with su-pernova models with detailed neutrino transport shows thatthe Y e - ρ -relation fitted to the homologous phase overestimatesthe deleptonization of the gas in most of the gain layer but un-derestimates the lepton loss of matter in the cooling layer andneutrinospheric region. Moreover, the question arises how thelepton evolution shall be treated in matter that reexpands andthus moves from high to low density? Even more, the entropysource term introduced in Eq. (5) of Liebend¨orfer (2005) isdesigned to specifically account for gas-entropy changes dueto neutrino production by electron captures and subsequentenergy transfers in (neutrino-electron) scatterings. The cor-responding energy loss or gain rate of the medium throughthe escape or capture of electron neutrinos with mean energy E ν e , which is given by δ Q ν e /δ t = E ν e δ Y e /δ t , is not includedin the heating and cooling terms Q + ν and Q − ν of Eqs. (4,5) ofthe present work. Adding it as an extra term would imply par-tial double-counting of the energy exchange via electron neu-trinos, and omitting it means to overestimate the entropy in-crease in infalling, deleptonizing matter and to underestimateentropy gains of decompressed gas with growing Y e . Becauseof the long list of such inconsistencies, whose implicationsare hard to judge or control, we decided to ignore Y e changesof the stellar medium in our post-bounce simulations (exceptfor the models discussed in App. A).This choice can be justified, but it is certainly not a perfect approach because it may also exclude e ff ects of importancein real supernova cores, whose physical processes require theinclusion of neutrino transport for an accurate description ofthe energy and lepton-number evolution. One of the unde-sired consequences of keeping Y e fixed in the accretion flowafter core bounce is an overestimation of the electron pressurein the gas settling onto the forming neutron star. In order toenforce more compression and thus to ensure close similar-ity of our results to the 1D models studied by Nordhaus et al.(2010) and Murphy & Burrows (2008), we have to enhancethe net cooling of the accreted matter by reducing the e ff ec-tive opacity κ e ff by a factor of 2.7 compared to the value givenin Murphy et al. (2009) and in Eq. (7). This reduction factor ischosen such that our simulations reproduce the minimum val-ues of the critical luminosity found to be necessary for trigger-ing explosions of the 15 M ⊙ progenitor (2 . · erg s − ) andfor the 11.2 M ⊙ star (1 . · erg s − ) in the 1D simulationsof Murphy & Burrows (2008). Without the reduction factor of κ e ff , our models turn out to explode too easily because of weakcooling (see the results in App. A). We stress that any expo-nential suppression factor of the heating and cooling rates inEqs. (4,5) is a pragmatic and ad hoc procedure to bridge thetransition from the optically thin to the optically thick regime,where neutrino transport is most complicated. From transporttheory neither the exponential factor nor the exact definitionof the optical depth of the exponent can be rigorously derived.In our reference set of standard simulations, we employ 400non-equidistant radial zones, which are distributed from thecenter to an outer boundary at 9000 km. The latter is suf-ficiently far out to ensure that the gas there remains at restduring the simulated evolution periods. The radial zones arechosen such that the resolution ∆ r / r is typically better than2%. In the multi-dimensional models with standard resolu-tion, we employ a polar grid with an angular bin size of 3 ◦ (60 θ - and 120 φ -zones). For the high-resolution 1D and multi-dimensional models of Sects. 4 and 5 we compute with up to800 radial zones and in 2D with an angular resolution down to0.5 ◦ , in 3D down to 1.5 ◦ . The additional radial grid zones aredistributed such that the region between 20 and 400 km, i.e.,not only the cooling layer around the neutrinosphere but alsothe gain layer between gain radius and shock, are resolvedsignificantly better. While full convergence of the 1D resultsrequires 600 radial zones or more, most of the plots compar-ing models for di ff erent dimensions show simulations withour standard resolution of only 400 radial cells (unless statedotherwise), because this is the limit for which we could per-form a larger set of 3D runs in acceptable time. To avoid anextremely restrictive Courant-Friedrich-Lewy (CFL) timestepin our 3D calculations, we simulate the inner core above adensity of ρ = g / cm in spherical symmetry. INVESTIGATED PROGENITORS AND MODELS
Our models are based on the 15 M ⊙ progenitor star s15s7b2of Woosley & Weaver (1995) and an 11.2 M ⊙ progenitor ofWoosley et al. (2002). The calculations for these progenitorswith our standard angular resolution of 3 ◦ are summarized inTable 1. This table is arranged such that horizontal rows havethe same driving luminosities for simulations performed indi ff erent dimensions. Varying the prescribed driving luminos-ity L ν e from run to run we present for each of the 11.2 M ⊙ and15 M ⊙ progenitors several 1D, 2D, and 3D simulations. Allof the 1D and 2D simulations cover at least 1s after bounce.The nonexploding 3D simulations with standard angular res-olution of 3 degrees were not stopped until at least 600 ms af- Hanke, Marek, M¨uller, & Janka PSfrag replacements ˙ M [ M ⊙ / s ] @ m t pb [s] F ig . 1.— Time evolution of the mass accretion rate, ˙ M ( r ) = π r ρ ( r ) | v ( r ) | ,evaluated at 500 km for the 11.2 M ⊙ and the 15 M ⊙ progenitors in nonexplod-ing models. TABLE 111.2 M ⊙ and M ⊙ results with standard grid of radialzones and degrees angular resolution .1D 2D 3D L ν e a t expb ˙ M expc t exp ˙ M exp t exp ˙ M exp (10 erg / s) (ms) ( M ⊙ / s) (ms) ( M ⊙ / s) (ms) ( M ⊙ / s)11.2 M ⊙ − − − − − − − − − −
731 0.0851.0 − −
563 0.082 537 0.0861.1 − −
461 0.0911.2 − −
357 0.104 319 0.1121.3 819 0.082 307 0.1141.4 499 0.088 241 0.126 232 0.1301.5 380 0.100 232 0.1301.6 345 0.106 203 0.13715 M ⊙ − − − − − − − − − − − − − −
876 0.197 612 0.2262.3 − −
428 0.261 426 0.2612.4 − −
442 0.2552.5 − −
283 0.313 281 0.3142.6 710 0.215 285 0.3122.7 489 0.247 262 0.3162.8 390 0.271 242 0.322 236 0.3242.9 281 0.314 235 0.3253.0 258 0.318 236 0.3243.1 248 0.320 220 0.327 a Electron-neutrino luminosity. b Time after bounce of onset of explosion. A “ − ” symbol indicates that themodel does not explode during the simulated period of evolution. c Mass accretion rate at onset of explosion. ter bounce. For models with higher resolution the simulationtimes are given in Table 2.In Table 1 the time of the onset of an explosion, t exp , and themass accretion rate at that time, ˙ M exp , are listed as characteris-tic quantities of the models. The beginning of the explosion isdefined as the moment of time t exp when the shock reachesan average radius of 400 km (and does not return lateron),while nonexploding models are denoted by a “ − ” symbol. Inmultidimensional simulations the corresponding shock posi-tion is defined as the surface average over all angular direc-tions, h R S i ≡ π H d Ω R S ( ~ Ω ). The lowest driving luminosity yielding an explosion for a given value of the mass accretionrate is termed the critical luminosity (Burrows & Goshy 1993;Murphy & Burrows 2008). We determine the mass accretionrate ˙ M ( r ) = π r ρ ( r ) | v ( r ) | at the time of the onset of the ex-plosion just exterior to the shock, i.e. at a radius of 500 km,where the infalling envelope is spherical (except for the smallseed perturbations imposed on the multi-dimensional mod-els). In Fig. 1 the mass accretion rates of the 11.2 M ⊙ and15 M ⊙ progenitors are depicted for the nonexploding runswith the lowest driving neutrino luminosities. Because theshock can be largely deformed in multi-dimensional simula-tions and its outermost parts can extend beyond 500 km (andthus impede a clean determination of the mass accretion rate)when its average radius just begins to exceed 400 km, we referto the functions plotted in Fig. 1 for defining the mass accre-tion rate at the time when the explosion sets in. CRITICAL LUMINOSITY AS FUNCTION OF DIMENSION
Based on steady-state solutions of neutrino-heated and-cooled accretion flows between the stalled shock and theproto-neutron star surface, Burrows & Goshy (1993) identi-fied a critical condition that can be considered to separate ex-ploding from nonexploding models. They found that for agiven value of the mass infall rate ˙ M onto the accretion shocksteady-state solutions cannot exist when the neutrino-heatingrate in the gain layer is su ffi ciently large, i.e., for neutrinoluminosities above a threshold value L ν . This result can becoined in terms of a critical condition L ν ( ˙ M ) expressing thefact that either the driving luminosity has to be su ffi cientlyhigh or the damping mass accretion rate enough low for anexplosion to become possible. The critical ˙ M - L ν -curve wasinterpreted by Burrows & Goshy (1993) as a separating linebetween the region above, where due to the non-existenceof steady-state accretion solutions explosions are expected totake place, from the region below, where neutrino energy in-put behind the shock is insu ffi cient to accelerate the stalledshock outwards and thus to trigger an explosion.This interpretation of the steady-state results was consis-tent with hydrodynamical simulations of collapsing and ex-ploding stars in 1D and 2D by Janka & M¨uller (1996). Per-forming a more extended parameter study than the latter work,Murphy & Burrows (2008) explored the concept of a criticalcondition systematically with time-dependent hydrodynami-cal models. They showed that a critical luminosity indeedseparates explosion from accretion and confirmed that thisvalue is lowered by ∼
30% when going from spherical sym-metry to two dimensions, at least when a fixed driving lumi-nosity is adopted and feedback e ff ects of the hydrodynamicson the neutrino emission are ignored. Some 2D e ff ects includ-ing possible consequences of rotation were discussed beforeon grounds of steady-state models by Yamasaki & Yamada(2005, 2006), while Janka (2001) tried to include time-dependent aspects of the shock-revival problem and tookinto account an accretion component of the neutrino lumi-nosity in addition to the fixed core component. The influ-ence of such an accretion contribution was more recentlyalso estimated by Pejcha & Thompson (2011), who solvedthe one-dimensional steady-state accretion problem betweenthe neutron star and the accretion shock along the lines ofBurrows & Goshy (1993), but attempted to obtain a deeperunderstanding of the critical condition by comprehensivelyanalysing the structure of the accretion layer and of the lim-iting steady-state solution in dependence of the stellar condi-tions. They found that the critical value for the neutrino lumi-ASI activity as key to successful neutrino-driven SN explosions? 5 o o o o o PSfrag replacements ˙M [M ⊙ / s] L ν e [ e r g / s ] o o o o o PSfrag replacements L ν e [ e r g / s ] t exp [ms] F ig . 2.— Critical curves for the electron-neutrino luminosity ( L ν e ) versus mass accretion rate ( ˙ M ) (left plot) and versus explosion time t exp (right plot) forsimulations in 1D (black), 2D (blue), and 3D (red) with standard resolution. The accretion rate is measured just outside of the shock at the time t exp when theexplosion sets in. For the 15 M ⊙ progenitor 1D results are displayed for 400, 600, and 800 radial zones. Higher radial resolution compared to the standard 1Druns makes explosions slightly more di ffi cult; convergence is achieved for ≥
600 radial bins. Multi-dimensional results are shown for di ff erent angular resolutions,where available, but always computed with 400 radial zones. Note that 2D simulations with improved angular zoning explode more easily, whereas in 3D onlyone case was computed (the 11.2 M ⊙ simulation for L ν e = . · erg s − ) that developed an explosion also with better angular resolution. PSfrag replacements h R S i [ k m ] h R S i [ k m ] t pb [s] 11.2 M ⊙ ⊙ F ig . 3.— Time evolution of the average shock radius as function of the post-bounce time t pb for simulations in one (thin dashed lines), two (thin solidlines), and three dimensions (thick lines). The shock position is defined asthe surface average over all angular directions. The top panel shows resultsfor the 11.2 M ⊙ progenitor and the bottom panel for the 15 M ⊙ progenitor, allobtained with our standard resolution. Di ff erent electron-neutrino luminosi-ties (labelled in the plots in units of 10 erg s − ) are displayed by di ff erentcolors. nosity is linked to an “antesonic condition” in which the ratioof the adiabatic sound speed to the local escape velocity in thepostshock layer reaches a critical value above which steady-state solutions of neutrino-heated accretion flows cannot beobtained. By performing high-resolution hydrodynamic sim- ulations Fern´andez (2012) found that radial instability is a suf-ficient condition for runaway expansion of an initially stalledcore-collapse supernova shock if the neutrinospheric param-eters do not vary with time and if heating by the accretionluminosity is neglected. However, the threshold neutrino lu-minosities for the transition to runaway instability are in gen-eral di ff erent from the limiting values for steady-state solu-tions of the kind discussed by Burrows & Goshy (1993) andPejcha & Thompson (2011). Nordhaus et al. (2010) general-ized the hydrodynamic investigations of Murphy & Burrows(2008) to include 3D models and found another reduction ofthe threshold luminosity for explosion by 15–25% comparedto the 2D case.Despite the basic agreement of the outcome of these investi-gations it should be kept in mind that it is not ultimately clearwhether the simple concept of a critical threshold conditionseparating explosions from failures (and the dependences ofthis threshold on dimension and rotation for example) holdsbeyond the highly idealized setups considered in the men-tioned works. None of the mentioned systematic studies bysteady-state or hydrodynamic models was able to include ad-equately the complexity of the feedback between hydrody-namics and neutrino transport physics. In particular, none ofthese studies could yield the proof that the non-existence ofa steady-state accretion solution for a given combination ofmass accretion rate and neutrino luminosity is equivalent tothe onset of an explosion. The latter requires the persistenceof su ffi ciently strong energy input by neutrino heating for asu ffi ently long period of time. This is especially importantbecause Pejcha & Thompson (2011) showed that the total en-ergy in the gain layer is still negative even in the case of thelimiting accretion solution that corresponds to the critical lu-minosity . Within the framework of simplified modeling se-tups, however, the question cannot be answered whether sucha persistent energy input can be maintained in the environ-ment of the supernova core.Following the previous investigations byMurphy & Burrows (2008) and Nordhaus et al. (2010)we performed hydrodynamical simulations that track the Note that Fern´andez (2012) demonstrated that transition to runaway oc-curs when the fluid in the gain region reaches positive specific energy.
Hanke, Marek, M¨uller, & Jankapost-bounce evolution of collapsing stars for di ff erent, fixedvalues of the driving neutrino luminosity. Since the massaccretion rate decreases with time according to the densityprofile that is characteristic of the initial structure of theprogenitor core (see Fig. 1 for the 11.2 and 15 M ⊙ starsconsidered in this work), each model run probes the criticalvalue of ˙ M exp at which the explosion becomes possible forthe chosen value of L ν = L ν e = L ¯ ν e . The collection ofvalue pairs ( ˙ M exp , L ν e ) defines a critical curve L ν ( ˙ M ). Theseare shown for our 1D, 2D, and 3D studies with standardresolution for both progenitor stars in the left panel of Fig. 2and in the case of the 15 M ⊙ star can be directly comparedwith Fig. 1 of Nordhaus et al. (2010). Table 1 lists, as afunction of the chosen L ν e , the corresponding times t exp whenthe onset of the explosion takes place and the mass accretionrate has the value of ˙ M exp . The post-bounce evolution of acollapsing star proceeds from high to low mass accretion rate(Fig. 1), i.e., from right to left on the horizontal axis of theleft panel of Fig. 2. When ˙ M reaches the critical value forthe given L ν e , the model develops an explosion. The rightpanel of Fig. 2 visualizes the functional relations between theneutrino luminosities L ν e and the explosion times t exp for bothprogenitors and for the simulations with di ff erent dimensions.At first glance Fig. 2 reproduces basic trends that are vis-ible in Figs. 17 of Murphy & Burrows (2008) and in Fig. 1of Nordhaus et al. (2010). For example, the critical luminos-ity increases for higher mass accretion rate and the values forspherically symmetric models are clearly higher than thosefor 2D simulations. However, a closer inspection reveals in-teresting di ff erences compared to the previous works. • In general the slopes of our critical L ν e ( ˙ M )-relationsappear to be considerably steeper and in the case ofthe 15 M ⊙ star they exhibit a very steep rise above˙ M ≈ . M ⊙ s − . This means that explosions forhigher mass accretion rates are much harder to obtainand therefore the tested driving luminosities in our sim-ulations do not lead to explosions earlier than about200 ms after bounce, independent of whether the mod-eling is performed in 1D, 2D or 3D. In contrast, thecritical curves given by Nordhaus et al. (2010) show amoderately steep increase over the whole range of plot-ted mass accretion rates between about 0.1 M ⊙ s − andmore than 0.5 M ⊙ s − . • Nonradial flows in the 2D case, by which the residencetime of accreted matter in the gain layer could be ex-tended or more matter could be kept in the gain re-gion, reduce the critical luminosities by at most ∼ M ⊙ star and .
25% forthe 11.2 M ⊙ model, which is a somewhat smaller dif-ference than found by Murphy & Burrows (2008) andNordhaus et al. (2010). • Most important, however, is the fact that we cannot con-firm the observation by Nordhaus et al. (2010) that 3Dprovides considerably more favorable conditions for anexplosion than 2D. Our critical curves for the 2D and3D cases nearly lie on top of each other. There areminor improvements of the explosion conditions in 3Dvisible in both panels of Fig. 2 and the numbers of Ta-ble 1, e.g., a 10% reduction of the smallest value of L ν for which an explosion can be obtained for the 11.2 M ⊙ star, a ∼
260 ms earlier explosion for the lowest lumi-nosity driving the 15 M ⊙ explosion (2 . · erg s − ), o o o o o o o o o o o o PSfrag replacements < s > [ k B / b a r yon ] < s > [ k B / b a r yon ] < s > [ k B / b a r yon ] t pb [s] F ig . 4.— Time evolution of the mass-weighted average entropy in the gainregion for one-dimensional (thin dotted lines), two-dimensional (thin solidlines), and three-dimensional (thick lines) simulations with di ff erent angularresolutions (corresponding to di ff erent colors). The top panel displays the11.2 M ⊙ results for an electron-neutrino luminosity of L ν e = . · erg s − ,the middle panel shows the 15 M ⊙ runs for an electron-neutrino luminosityof L ν e = . · erg s − , and the bottom panel the 15 M ⊙ models for L ν e = . · erg s − . The strong decrease of the average entropy that terminatesa phase of continuous, slow increase signals the onset of the explosion whena growing mass of cooler (low-entropy) gas is added into the gain layer afterbeing swept up by the expanding and accelerating shock wave. and a tendency of slightly faster 3D explosions for alltested luminosities (see Fig. 3 and Table 1). All of thesemore optimistic 3D features, however, will disappearfor simulations with higher resolution as we will see inSect. 5.Before we discuss the origin of the di ff erences betweenour results and those of Murphy & Burrows (2008) andNordhaus et al. (2010) we would like to remark that the kinksASI activity as key to successful neutrino-driven SN explosions? 7 F ig . 5.— Scatter-plots of the entropy structure as function of radius for simulations of the 11.2 M ⊙ progenitor with an electron-neutrino luminosity of L ν e = . · erg s − at 400 ms (left) and 600 ms (right) after core bounce. The red dots correspond to the 2D results, black ones to 3D, the light blue line is theentropy profile of the 1D simulation, and the dark-blue and green curves are mass-weighted angular averages of the 2D and 3D models, respectively. Bothmulti-dimensional simulations were performed with an angular resolution of two degrees and both yield explosions (at ∼
530 ms p.b. in 2D and ∼
570 ms p.b. in3D; see Table 2). Note that di ff erent from Fig. 4, unshocked material at a given radius is included when computing angular averages. The dispersion of entropyvalues in the unshocked flow of 2D and 3D simulations is a consequence of the imposed density-seed perturbations (cf. Sect. 2), which grow in the supersonicalinfall regime (see Buras et al. 2006b).F ig . 6.— Scatter-plots of the entropy structure as function of radius for simulations of the 15 M ⊙ progenitor with an electron-neutrino luminosity of L ν e = . · erg s − at 350 ms (left) and 700 ms (right) after core bounce. The red dots correspond to the 2D results, black ones to 3D, the light blue line is theentropy profile of the 1D simulation, and the dark-blue and green curves are mass-weighted angular averages of the 2D and 3D models, respectively. Bothmulti-dimensional simulations were performed with an angular resolution of 1.5 degrees. While the 2D model develops an explosion setting in ∼
720 ms afterbounce, the 3D model does not produce an explosion (Table 2). Note that di ff erent from Fig. 4, unshocked material at a given radius is included when computingangular averages. The dispersion of entropy values in the unshocked flow of 2D and 3D simulations is a consequence of the imposed density-seed perturbations(cf. Sect. 2), which grow in the supersonical infall regime (see Buras et al. 2006b). Hanke, Marek, M¨uller, & Jankaand even nonmonotonic parts of the curves shown in Fig. 2in particular for the multi-dimensional cases are connectedto our definition of the explosion time as being the momentwhen the mean shock radius exceeds 400 km. Especially incases where the shock deformation is large (which is an issuemainly in some of the 2D simulations) this definition is asso-ciated with significant uncertainty in the determination of theexact explosion time t exp and therefore also of the correspond-ing mass accretion rate ˙ M exp .In addition to the results for our standard resolution, Fig. 2presents 1D models with higher radial resolution (600 and800 radial zones). The critical curves with better zoning ex-hibit the same overall slopes as those of the standard runs,but there is a slight shift towards higher values of the criticalluminosity (or, equivalently, a small shift to lower ˙ M exp andlater t exp ). This is caused by somewhat larger neutrino energylosses from the cooling layer with better radial resolution, ane ff ect which makes explosions more di ffi cult. We stress thatthis resolution-dependent cooling e ff ect is a consequence ofthe simplified neutrino-loss treatment. The employed cool-ing rate is not able to reproduce real transport behavior andleads to the development of a pronounced gaussian-like den-sity peak (with density excess of a few compared to its sur-roundings) in the neutrino-cooling layer. This local densitymaximum has a strong influence on the integrated energy lossby neutrino emission. With better zoning the peak becomesbetter resolved and even grows in size. Convergence of 1Dresults seems to be achieved for ≥
600 radial zones, but in 2Dand 3D the artificial density peak prevents numerical conver-gence for all employed radial zonings (cf. Sect. 5). Figure 2also displays some data points for multi-dimensional modelswith better angular resolution (all of them, however, com-puted with 400 radial mesh points). These will be discussedin Sect. 5.A more detailed analysis, which we will report on belowand in App. A, reveals that the exact values of the critical lu-minosities as well as the detailed slope of the critical curvesseem to depend strongly on the employed description of neu-trino e ff ects, whose implementation is subject to a significantdegree of arbitrariness if detailed neutrino transport is not in-cluded in the modeling (cf. the discussion in Sect. 2). Thefact that Murphy & Burrows (2008) found fairly good over-all agreement between their critical relations L ν ( ˙ M ) and thoseobtained by Burrows & Goshy (1993) is likely to be linked toa basically similar treatment of the neutrino e ff ects.Before ∼ M ⊙ progenitor changes much more rapidly thanduring the subsequent evolution (Fig. 1). For the correspond-ing ˙ M values in excess of ∼ M ⊙ s − , the accretion shockis therefore not as close to steady-state conditions as later on.We see a steep rise of our critical curves at ˙ M & . M ⊙ s − ( t exp . .
25 s), which is a very prominent di ff erence com-pared to the results of Burrows & Goshy (1993), who as-sumed steady-state accretion, but in particular also comparedto the hydrodynamic results of Murphy & Burrows (2008),and Nordhaus et al. (2010) even in the 1D case. In order toexplore possible reasons for this di ff erence, we performed 1Dsimulations in which the neutrino treatment is copied fromNordhaus et al. (2010) as closely as possible (i.e., the re-duction factor of 2.7 in the exponent of e − τ e ff is not appliedand deleptonization is taken into account by using a Y e ( ρ ) re-lation, but not the corresponding entropy changes proposedby Liebend¨orfer 2005; for more details on these results, see App. A). These runs reveal that the steep rise of our L ν ( ˙ M )-curves is caused by a strong increase of the neutrino-coolingrate with higher values of ˙ M , in particular when we applyour neutrino treatment. The corresponding energy losses in-hibit explosions for low values of the driving luminosity. Thestronger cooling is linked mainly to our reduction of the ef-fective optical depth τ e ff , which we had to apply in order toreconcile the mass accretion rates and explosion times withthe lowest driving luminosities for which Murphy & Burrows(2008) had obtained explosions for the 11.2 and 15 M ⊙ stars(cf. Sect. 2). For example, in the case of the 15 M ⊙ progen-itor a driving luminosity of L ν e = . · erg s − triggersan explosion at t exp ≈
250 ms p.b. and ˙ M exp ≈ . M ⊙ s − (Table 1 and Fig. 2), whereas with an implementation of neu-trino e ff ects closer to that of Nordhaus et al. (2010) the explo-sion sets in at t exp ≈
120 ms p.b. and ˙ M exp ≈ . M ⊙ s − (seeFig. 19 in App. A). Shortly before this moment (at 75 ms afterbounce) the total energy loss by neutrino cooling is about 10times lower with the scheme of Nordhaus et al. (2010) thanwith our neutrino implementation. The latter yields an inte-grated energy-loss rate of ∼ · erg s − and significant cool-ing even at densities between 10 and 10 g cm − , where theNordhaus et al. (2010) treatment shows essentially no cool-ing. Neither the magnitude of the total neutrino-energy lossrate nor the region of energy extraction with our modelingapproach are implausible and in disagreement with detailedtransport simulations during a stage when the mass accretionrate still exceeds 1 M ⊙ s − (cf., e.g., Fig. 20 in Buras et al.2006a). In contrast, the Nordhaus et al. (2010) treatment ap-pears to massively underestimate the neutrino energy extrac-tion from the accretion flow during this time.These findings demonstrate that the results of the critical L ν e ( ˙ M )-relation in 1D can be quantitatively as well as qualita-tively di ff erent with di ff erent approximations of neutrino heat-ing and in particular of neutrino cooling. Moreover, this givesreason for concern that the di ff erences of the critical explosionconditions for 2D and 3D simulations seen by Nordhaus et al.(2010) might have been connected to their treatment of theneutrino physics, in particular also because the decrease ofthe critical luminosity from 2D to 3D they found was only15–25%, which is a relatively modest change (much smallerthan the 1D-2D di ff erence) and thus could easily be overruledby other e ff ects. Our results for 2D and 3D simulations witha di ff erent implementation of neutrino sources confirm thisconcern.In the Nordhaus et al. (2010) paper the average entropy ofthe matter in the gain region, h s ( t ) i , was considered to be asuitable diagnostic quantity that reflects the crucial di ff erenesof 1D, 2D, and 3D simulations concerning their relevance forthe supernova dynamics. While in the spherically symmet-ric case accreted matter moves through the gain region on theshortest, radial paths, nonradial motions can increase the timethat shock-accreted plasma can stay in the gain layer and ab-sorb energy from neutrinos. This can raise the mean entropy,internal energy, and pressure in the postshock region and thussupport the revival of the stalled supernova shock. This seemsto happen in the simulations by Nordhaus et al. (2010), whofound that turbulent mass motions in 3D can even improve theconditions for the neutrino-heating mechanism compared tothe 2D case. A crude interpretation of this di ff erence can begiven by means of random-walk arguments, considering themass motions in convective and turbulent structures as di ff u-sive process in the gain layer (Murphy & Burrows 2008). TheASI activity as key to successful neutrino-driven SN explosions? 9question, however, is whether this e ff ect is a robust 2D-3Ddi ff erence and whether it is the crucial key to successful ex-plosions by the neutrino-heating mechanism.Our results at least raise doubts. Figure 4 displays the timeevolution of the mean entropy in the gain layer for 11.2 anda 15 M ⊙ models computed with driving luminosities near theminimum value for which we obtained explosions. While the11.2 M ⊙ model explodes with a luminosity of 1 . · erg s − for all tested resolutions in more than one dimension despiteminimal di ff erences between the values of h s ( t ) i compared tothe 1D counterpart, the 15 M ⊙ progenitor develops an explo-sion for the chosen luminosity of 2 . · erg s − only in thecase of higher-resolution 2D runs (this will be further dis-cussed in Sect. 5). These successful cases, however, do notstick out by especially high values of h s ( t ) i . On the contrary,they even have lower mean entropies than the unsuccessful3D models! It is obvious that Fig. 4 does not exhibit the clear1D-2D-3D hierarchy visible in Fig. 5 of the Nordhaus et al.(2010) paper, which was found there to closely correlate withthe explosion behavior of their models. Instead, the di ff er-ences between simulations in the di ff erent dimensions arefairly small, and even two-dimensional flows, which unques-tionably allow for explosions also when none happen in 1D,do not appear more promising than the 1D case in terms ofthe average entropy of the matter in the gain layer . Similarly,3D models possess slightly (insignificantly?) higher values of h s ( t ) i but do not show a clear tendency of easier explosions,in particular not the better resolved simulations (see Sect. 5).The radial entropy structures of 1D, 2D, and 3D runs forboth progenitors, shown in Figs. 5 and 6 once before an explo-sion begins and another time around the onset of an explosionin at least one of the runs, demonstrate that low-entropy accre-tion downdrafts and high-entropy rising plumes of neutrino-heated plasma lead to large local variations of the entropy perbaryon of the matter in the gain layer (scatter regions in theplots). However, the mass-weighted angular averages of theentropies reveal much smaller di ff erences between the 1D and2D cases than visible in Fig. 4 of the Nordhaus et al. (2010)paper and in Fig. 13 of Murphy & Burrows (2008), and ex-hibit no obvious signs of more advantageous explosion con-ditions in the 3D cases compared to 2D. The noticeable dif-ferences in the radial profiles seem to be insu ffi cient to causemajor di ff erences in the mean entropies computed by addi-tional radial averaging (see Fig. 4).How can this discrepancy compared to Nordhaus et al.(2010) and Murphy & Burrows (2008) be explained, and howcan one understand the fact that 2D e ff ects play a supportiverole for neutrino-driven explosions? We will return to thesequestions in Sect. 6, but before that we shall present our re-sults of multi-dimensional simulations with varied resolutionin the following section. MODELS WITH HIGHER RESOLUTION
In order to test whether our results for the multi-dimensional models depend on the agreeably moderate 3 ◦ an-gular resolution used in the standard runs, we performed alarge set of simulations with finer grid spacing especially inthe angular directions, but also in radial direction. For this We stress that our basic findings are independent of the exact way how thegain radius of the multi-dimensional models is determined, i.e., whether theevaluation is performed with an angularly averaged gain radius or a direction-dependent gain radius. The outer boundary of the integration volume is de-fined by the shock position, which usually forms a non-spherical surface inthe multi-dimensional case. TABLE 2M ultidimensional models with different resolution .Mass a Dim b L ν e c Ang. d N r e t expf ˙ M expg t simh ( M ⊙ ) (10 Res. (ms) (ms)erg / s)11.2 2D 0.8 3 ◦ − − ◦ − − ◦ − − ◦ − − ◦ − − ◦ − − ◦
400 731 0.085 95411.2 3D 0.9 2 ◦ − − ◦
400 563 0.082 105311.2 2D 1.0 2 ◦
400 527 0.086 1053 ◦
400 537 0.086 68411.2 3D 1.0 2 ◦
400 572 0.082 761
15 2D 2.0 3 ◦ − − ◦ − − ◦ − − ◦ − −
15 3D 2.0 3 ◦ − − ◦ − −
15 2D 2.1 3 ◦ − − ◦ − − ◦
400 719 0.210 101615 2D 2.1 1 ◦
400 575 0.232 101615 2D 2.1 0.5 ◦
400 657 0.220 1016
15 3D 2.1 3 ◦ − − ◦ − − ◦ − −
15 2D 2.2 3 ◦
400 876 0.197 101615 2D 2.2 2 ◦
400 557 0.238 82515 2D 2.2 1.5 ◦
400 556 0.239 101615 2D 2.2 1 ◦
400 424 0.262 101615 2D 2.2 0.5 ◦
400 365 0.288 1016
15 3D 2.2 3 ◦
400 612 0.226 93215 3D 2.2 2 ◦ − −
15 2D 2.1 3 ◦ − − ◦ − − ◦ − − ◦ − − ◦ − − ◦ − − ◦
600 780 0.203 101615 2D 2.1 1 ◦ − − ◦
600 749 0.211 101615 2D 2.1 0.5 ◦
800 886 0.198 1016
15 3D 2.1 3 ◦ − − ◦ − −
15 2D 2.2 3 ◦ − − ◦ − − ◦
600 683 0.218 101615 2D 2.2 2 ◦ − − ◦
600 713 0.215 101615 2D 2.2 1.5 ◦
800 863 0.196 101615 2D 2.2 1 ◦
600 761 0.214 101615 2D 2.2 1 ◦
800 961 0.194 101615 2D 2.2 0.5 ◦
600 588 0.232 101615 2D 2.2 0.5 ◦
800 643 0.224 1016
15 3D 2.2 3 ◦
600 803 0.202 97515 3D 2.2 3 ◦
800 857 0.195 97715 3D 2.2 2 ◦ − − ◦ − − a Progenitor model. b Dimensionality. c Electron-neutrino luminosity. d Angular Resolution. e Number of radial zones. f Time of onset of explosion. g Mass accretion rate at onset of explosion. h Simulation time. o o o o o o o o o o o o PSfrag replacements h R S i [ k m ] h R S i [ k m ] h R S i [ k m ] t pb [s] F ig . 7.— Evolution of the average shock radius as a function of time (in sec-onds after bounce) for one-dimensional (dashed lines), two-dimensional (thinsolid lines), and three-dimensional (thick solid lines) simulations employ-ing di ff erent angular resolutions (color coding). The top panel displays the11.2 M ⊙ model for an electron-neutrino luminosity of L ν e = . · erg s − ,the middle panel shows the 15 M ⊙ star for an electron-neutrino luminos-ity of L ν e = . · erg s − , and the bottom panel the 15 M ⊙ results for L ν e = . · erg s − . purpose we concentrated on cases around the minimum val-ues of the driving luminosity that triggered explosions of bothprogenitors in our standard runs. The results are listed in Ta-ble 2. They indicate a very interesting trend: 2D models withfiner angular zoning tend to explode more readily, whereasbetter angular resolution in 3D simulations turns out to havethe opposite e ff ect.In the case of the 11.2 M ⊙ progenitor, for example, the 3Dexplosion found to set in at about 730 ms p.b. with an angularzone size of 3 ◦ for L ν e = . · erg s − cannot be repro-duced with an angle binning of 2 ◦ . Moreover, a luminosity of L ν e = . · erg s − leads to an explosion of the 3D model o o o o o o o o o o o o PSfrag replacements σ S [ k m ] σ S [ k m ] σ S [ k m ] t pb [s] F ig . 8.— Evolution of the standard deviation for the shock asphericity asa function of post-bounce time for two-dimensional (thin solid lines) andthree-dimensional (thick solid lines) simulations employing di ff erent angularresolutions (color coding). As in Fig. 7 the top panel displays the exploding11.2 M ⊙ models for an electron-neutrino luminosity of L ν e = . · erg s − .The middle panel contains the results for the 15 M ⊙ star with an electron-neutrino luminosity of L ν e = . · erg s − , where 2D runs with higherresolution lead to explosions while 3D runs do not. The bottom panel showsthe 15 M ⊙ case for L ν e = . · erg s − . It is remarkable that the 3D runwith 2 ◦ angular resolution does not explode whereas the one with angle binsof 3 ◦ explodes earlier than its 2D counterpart and develops a very large shockdeformation at the time the explosion sets in. at ∼
540 ms after bounce with a 3 ◦ -grid, but ∼
35 ms later when2 ◦ are used. The corresponding 2D models show the inversetrend as visible in the top panel of Fig. 7.Note that the average shock radii plotted in Fig. 7 as wellas Fig. 3 exhibit alternating periods of increase and decreasein particular in 2D simulations. Such features are a conse-quence of the strong sloshing motions of the accretion shockand of the associated time-dependent, large global shock de-ASI activity as key to successful neutrino-driven SN explosions? 11
0 100 200
300 ms pb r [km] r [km] s [ k B / b a r yon ] D s [ k B / b a r yon ] D
0 100 200
350 ms pb r [km] r [km] s [ k B / b a r yon ] D s [ k B / b a r yon ] D
0 100 200
400 ms pb r [km] r [km] s [ k B / b a r yon ] D s [ k B / b a r yon ] D
0 300 600
600 ms pb r [km] r [km] s [ k B / b a r yon ] D s [ k B / b a r yon ] D F ig . 9.— Snapshots of the evolution of the 11.2 M ⊙ model with an electron-neutrino luminosity of L ν e = . · erg s − and 2 ◦ angular resolution at post-bounce times of t pb = formations, which are typical of violent activity by the SASI.In 3D the corresponding wiggles and local maxima of the av-erage shock trajectory are much less pronounced. A measureof the degree of shock asphericity, irrespective of the rela-tive weights of di ff erent spherical harmonics components, isthe standard deviation of the shock radius defined by σ S ≡ q π H d Ω [ R S ( ~ Ω ) − h R S i ] . The standard deviations corre-sponding to the average shock radii of Fig. 7 are plotted inFig. 8, which confirms the mentioned di ff erence between 2Dand 3D runs.In spite of this 2D-3D di ff erence of the shock aspheric-ity, an inspection of cross-sectional snapshots of our best re-solved simulations of the 11.2 M ⊙ progenitor with an electron-neutrino luminosity of L ν e = . · erg s − reveals thatthe sizes of the convective plumes and the structure of theneutrino-heated postshock layer are fairly similar in the 2Dand 3D cases before explosion (which in both models de-velops shortly after 500 ms): In Fig. 9 it is di ffi cult to judgeby eye inspection whether the displayed simulation was con-ducted in 2D (left half-panels) or 3D (right half-panels). Even after the explosions have taken o ff the global deformation ofthe shock in both cases is not fundamentally di ff erent in thesense that low-order spherical harmonics modes (dipolar andquadrupolar components) determine the global asymmetry ofthe shock surface and in particular of the distribution of down-drafts and expanding bubbles in the gain region (see Fig. 9,lower right panel, and Fig. 10). At a closer inspection one cannotice some secondary di ff erences in the morphology of theconvective and downflow features. Despite the same angu-lar resolution the images of Figs. 9 and 10 reveal more smallstructures in the 3D case compared to the 2D data, which ap-pear more coherent, smoother, and less fragmented into finersubstructures and filaments. We will refer to this observationin Sect. 6.Our 15 M ⊙ runs with varied resolution confirm the trendsseen for the 11.2 M ⊙ progenitor. For a neutrino luminosityof L ν e = . · erg s − , for which neither 2D nor 3D sim-ulations with standard resolution produce an explosion, wefind that 2D models with angular binning of 1.5 ◦ or better doexplode, whereas explosions in 3D cannot be obtained withangular zones in the range from 1.5 ◦ to 3 ◦ (Fig. 7, middle2 Hanke, Marek, M¨uller, & Janka F ig . 10.— Upper row:
Quasi-three-dimensional visualization of the 11.2 M ⊙ simulations in 2D (upper left panel) and 3D (upper right panel) with an electron-neutrino luminosity of L ν e = . · erg s − and an angular resolution of 2 ◦ , comparing the structure at 700 ms p.b., roughly 150 ms after the onset of theexplosions. Since the explosion started slightly earlier in the 2D model (see the upper panel of Fig. 7 and Table 2) the shock is more extended in the left image.While in this case the shock possesses a much stronger dipolar deformation component than in 3D (cf. Fig. 9, lower right panel), the distribution of accretionfunnels and plumes of neutrino-heated matter exhibits a hemispheric asymmetry in both cases. Because of the axisymmetry of the 2D geometry this concerns thehemispheres above and below the x - y -plane in the upper left plot, whereas the virtual equator lies in the plane connecting the upper left and lower right corners ofthe top right image and the lower left and upper right corners of the bottom right picture. Note that the jet-like axis feature in the upper left figure is a consequenceof the symmetry constraints of the 2D setup, which redirect flows moving towards the polar grid axis. Such artifacts do not occur in the 3D simulation despite theuse of a polar coordinate grid there, too. Lower row:
Ray-tracing and volume-rendering images of the three-dimensional explosion of the 11.2 M ⊙ progenitor forthe same simulation and time displayed in the upper right image. The left lower panel visualizes the outer boundaries of the buoyant bubbles of neutrino-heatedgas and the outward driven shock, which can be recognized as a nearly transparent, enveloping surface. The visualization uses the fact that both are entropydiscontinuities in the flow. The infalling matter in the preshock region appears as di ff use, nebular cloud. The right lower panel displays the interior structure bythe entropy per nucleon of the plasma (red, yellow, green, light blue, dark blue correspond to decreasing values) within the volume formed by the high-entropybubbles, whose surface is cut open by removing a wide cone facing the observer. Note the clear dipolar anisotropy with stronger explosion towards the north-westdirection and more accretion at the south-east side of the structure. panel; Table 2). The 2D simulations exhibit violent SASIsloshing motions and the quasi-periodic appearance of largeshock asymmetries (Fig. 8, middle panel), and the 2D modelwith 1.5 ◦ angular zoning explodes with a huge prolate defor-mation (Fig. 11). A similar behavior is seen for the 15 M ⊙ runs with L ν e = . · erg s − : While all 2D models com-puted with angular zone sizes between 0.5 ◦ and 3 ◦ explode,we observe an explosion for the 3D calculation with 3 ◦ butnone for the case with 2 ◦ angular binning (Table 2 and Fig. 7, bottom panel). It is highly interesting that the 3 ◦ case, wherethe explosion occurs more readily in 3D than in 2D, is as-sociated with a large asphericity of the supernova shock atthe time the 3D run begins to develop the successful blast(Fig. 8, bottom panel). Note again that the structures of thehigher-resolved 3D model in Fig. 11 reveal finer details andfragmentation into smaller filaments than the corresponding2D simulation, despite both having the same zone sizes in theangular directions.ASI activity as key to successful neutrino-driven SN explosions? 13
0 100 200
250 ms pb r [km] r [km] s [ k B / b a r yon ] D s [ k B / b a r yon ] D
0 100 200
350 ms pb r [km] r [km] s [ k B / b a r yon ] D s [ k B / b a r yon ] D
0 300
550 ms pb r [km] r [km] s [ k B / b a r yon ] D s [ k B / b a r yon ] D
0 300 600
700 ms pb r [km] r [km] s [ k B / b a r yon ] D s [ k B / b a r yon ] D F ig . 11.— Snapshots of the post-bounce evolution of the 15 M ⊙ model with an electron-neutrino luminosity of L ν e = . · erg s − and angular resolutionof 1.5 ◦ at t pb = The data listed in Table 2 contain the clear message that2D models with better angular resolution usually develop ex-plosions earlier in contrast to 3D runs, which explode lateror not at all when the angular zoning is finer. There can be2D exceptions to the general trend (e.g., the 15 M ⊙ cases with L ν e = . · erg s − and 0.5 ◦ and 1 ◦ resolution for 400 ra-dial zones; see also Fig. 7, middle panel), which are eithera ff ected by the di ffi culty to exactly determine the onset of theblast in cases with a highly deformed shock, or which can bestochastic outliers associated with the chaotic processes lead-ing to the explosion. Note in this context that at late times˙ M ( t ) is very flat and therefore di ff erences in t exp correspondto only small di ff erences in ˙ M exp . For this reason the begin-ning of the explosion can be shifted by minor perturbations,e.g. connected to stochastic fluctuations. It is also possiblethat for special circumstances the symmetry axis of the 2Dgeometry has an influence on such a non-standard behaviorbecause of its e ff ect to redirect converging flows outwards orinwards and thus to have a positive feedback on the violenceof the SASI activity.Improved radial resolution for fixed angular grid turns outto have a negative influence on the possibility of an explosionalso in multi-dimensional simulations. 2D runs with betterradial zoning (600 or even 800 instead of 400 radial zones) fail to develop explosions or explode significantly later thantheir less well resolved counterparts (see the 15 M ⊙ results for L ν e = . · erg s − and 2 . · erg s − in Table 2). In gen-eral, in 1D, 2D, and 3D improved radial resolution shifts theonset of the explosion to later times monotonically. The onlysuccessful 3D model in the set that is useful for the presentdiscussion, a 15 M ⊙ run with L ν e = . · erg s − and 3 ◦ angular resolution, supports our findings in 2D. It shows anincreasingly delayed explosion for better radial resolution, al-though the results with 600 and 800 radial zones appear to benearly converged. All non-exploding 3D models do not yieldsuccesses also with higher radial resolution.The conclusion that good radial resolution is very impor-tant for reliable results, in particular when the explosionis “marginal”, would not be a surprise, because Sato et al.(2009) have pointed out the importance of the radial zoningclose to the neutron star and around the supernova shock in or-der to accurately capture the entropy and vorticity productionat the shock and to determine growth times and oscillation fre-quencies of the SASI. The latter is unquestionably an essentialingredient for the success of the neutrino-driven mechanismin our 2D runs and it may as well be a crucial componentfor the mechanism to work in 3D. As mentioned in Sect. 4,however, the sensitive influence of the radial zoning in the4 Hanke, Marek, M¨uller, & Jankadiscussed model set is mainly a consequence of an artificialdensity peak developing in the neutrino-loss region because ofthe use of the simplified neutrino-cooling treatment. The nar-row shape of this numerical artifact in the density structure,which enhances the energy emission by neutrinos, can evencause a dependence of the results on the particular choice ofthe grid-cell locations. Di ff erent from the 1D runs the resultsof multi-dimensional simulations with more than ∼
600 radialzones do not seem to converge. Since the local density max-imum lies between 10 g cm − and 10 g cm − in the corethat is treated spherically symmetrically in our simulations,we interpret this phenomenon as the consequence of a sub-tle feedback between higher zoning and cooling strength onthe one hand and multi-dimensional processes in the accre-tion layer on the other hand. Because of the artificial natureof the underlying density feature, however, we did not furtherexplore this finding.Finally, we remark that prior to our present work Scheck(2007) has already performed resolution studies with a largeset of 2D simulations, in which he varied the lateral zonewidth between 0.5 ◦ and 4 ◦ . In addition, he conducted three3D simulations with angular bin sizes of 2–4 ◦ . However, in-stead of the highly simplified heating and cooling descrip-tion used by us he employed the much more sophisticatedapproximation for grey neutrino transport described in detailin Scheck et al. (2006). This approximation included, e.g.,the feedback of accretion on the neutrino emission propertiesand on the corresponding energy and lepton number trans-port by neutrinos and antineutrinos of all flavors, as well as amore elaborate description of neutrino-matter interactions indetailed dependence on the thermodynamical state of the stel-lar plasma. Scheck (2007) was not interested in a systematicexploration of the critical explosion condition, but his projectwas focussed on investigating the possibility of hydrodynamicpulsar kicks by successful asymmetric explosions. Despitethe grave di ff erences of the neutrino treatments and numer-ical setups, the results obtained by Scheck (2007) are com-patible with our present findings: 2D simulations with higherresolution turned out to yield explosions significantly earlierand thus also more energetically than the low-resolution runs.Within the tested range of angular resolutions Scheck (2007)did not observe any significant di ff erences between his 3Dmodels. This, however, may just be a consequence of the factthat the models were clearly above the threshold conditionsfor an explosion and did not linger along the borderline be-tween blast and failure. INTERPRETATION AND DISCUSSION
In this section we discuss the meaning of our results in com-parison to previous studies and present an interpretation thatcould explain the main trends found in our multi-dimensionalsimulations with varied resolution.
Variation with dimension
In Sects. 4 and 5 we have reported that our simulations donot support the central finding by Nordhaus et al. (2010) thatthe tendency to explode is a monotonically increasing func-tion of dimension. While we confirm more favorable explo-sion conditions in 2D than in 1D, we do not observe that in 3Dconsiderably lower driving luminosities are needed for a suc-cess of the neutrino-driven mechanism than in 2D. Moreover,we cannot confirm the finding by Nordhaus et al. (2010) thatthe mass-weighted average of the entropy per nucleon in thegain region, h s ( t ) i , is a quantity that is suitable as an indicator o o o o o o o o o o o o PSfrag replacements m a ss i ng a i n l a y e r[ M ⊙ ] m a ss i ng a i n l a y e r[ M ⊙ ] m a ss i ng a i n l a y e r[ M ⊙ ] t pb [s]11.2-1.015.0-2.1 F ig . 12.— Time evolution of the mass in the gain region (in seconds afterbounce) for simulations in 1D (thin dotted line), 2D (thin solid lines), and 3D(thick lines). The multi-dimensional models are displayed for all employedangular resolutions depicted by di ff erent colors. The top panel shows theresults for the 11.2 M ⊙ star with an electron-neutrino luminosity of L ν e = . · erg s − , the middle panel the results for the 15 M ⊙ runs with L ν e = . · erg s − , and the bottom panel the 15 M ⊙ models for L ν e = . · erg s − . The di ff erent cases are the same as in Figs. 7 and 8. of the proximity of models to an explosion and thus can serveas an explanation of di ff erences between 1D, 2D, and 3D sim-ulations. In particular, our 3D models turned out to haveslightly higher mean entropies than corresponding 2D cases(Fig. 4) without developing better explosion conditions. Thisraises the question why our models, and multi-dimensionalsimulations in general, have produced successful explosionsby the neutrino-heating mechanism when corresponding 1Dmodels fail?It is by no means obvious that h s ( t ) i should increase in thegain layer in the multi-dimensional case. While neutrino en-ergy deposition naturally leads to a rise of the entropy of theheated gas, the averaging process over the volume of the gainASI activity as key to successful neutrino-driven SN explosions? 15 o o o o o o o o o o o o PSfrag replacements h ea ti ng r a t e [ e r g / s ] h ea ti ng r a t e [ e r g / s ] h ea ti ng r a t e [ e r g / s ] t pb [s]11.2-1.015.0-2.1 F ig . 13.— Analogous to Fig. 12, but for the time evolution of the total netrate of neutrino heating in the gain region. layer also encompasses the downdrafts carrying cool matterfrom the postshock region towards the gain radius and theneutron star. These downdrafts are much denser, they arehardly heated by neutrinos because of their extremely rapidinfall, and they can contain more mass than the surrounding,dilute bubbles that are inflated by the expanding, neutrino-heated plasma. It is therefore not clear that the spatial (mass-weighted) average h s ( t ) i grows in multi-dimensions comparedto 1D runs.Moreover, it is not even clear that convective overturn in thegain layer must lead to an average entropy of the neutrino-heated gas itself that is higher than in 1D simulations. Dif-ferent from the 1D case high-entropy matter becomes buoy-ant and begins to float in the multi-dimensional environment.Thus the heated gas is quickly carried away from the vicinityof the gain radius, where neutrino-energy deposition is maxi-mal, to larger radii. Such dynamics of the gas can well limit o o o o o o o o o o o o PSfrag replacements a dv ec ti on / h ea ti ng ti m e s ca l ea dv ec ti on / h ea ti ng ti m e s ca l ea dv ec ti on / h ea ti ng ti m e s ca l e t pb [s]11.2-1.015.0-2.1 F ig . 14.— Analogous to Fig. 12, but for the time evolution of the ratio ofadvection timescale to heating timescale in the gain layer. the amount of energy and entropy that is stored in individualchunks of matter. Little, if any of the gas is subject to mul-tiple overturn cycles bringing the gas close to the gain radiusmore than once as suggested by the “convective engine” pic-ture of Herant et al. (1994) but questioned by Burrows et al.(1995). Instead, the majority of the heated gas either expandsto larger distances, pushing shock expansion, or, in the disad-vantageous case, is swept back below the gain radius (e.g. bylarge-amplitude sloshing motions of the shock), where it losesits energy again by e ffi ciently reradiating neutrinos .Our results imply that the dominant e ff ect that makes the The real multi-dimensional situation is even more complicated. Thementioned violent sloshing motions of the shock can cause strong shock-heating of the postshock matter as discussed in detail by Scheck et al. (2008),Blondin et al. (2003), and Blondin & Mezzacappa (2006b), thus not onlymassively a ff ecting the postshock flow but also providing an additional en-tropy source besides neutrino-energy deposition. o o o o o o o o o o o o PSfrag replacements E θ,ϕ kinE θ,ϕ kinE θ,ϕ kinE θ kinE θ kinE θ kinE ϕ kinE ϕ kinE ϕ kinE θ kin,2DE θ kin,2DE θ kin,2D e n e r gy [ e r g ] e n e r gy [ e r g ] e n e r gy [ e r g ] t pb [s]11.2-1.015.0-2.1 F ig . 15.— Kinetic energies of angular mass motions in the gain layer asfunctions of time after bounce for the 11.2 M ⊙ runs with an electron-neutrinoluminosity of L ν e = . · erg s − (top panel) and the 15 M ⊙ runs with L ν e = . · erg s − (middle panel) and L ν e = . · erg s − (bottompanel). Thin solid lines correspond to the lateral kinetic energy of 2D mod-els, while for 3D simulations (thick lines) the lateral, azimuthal, and totalkinetic energies are represented by dotted, dashed, and solid line styles, re-spectively. Both angular directions contribute essentially equally to the totalkinetic energy of nonradial motions in the 3D case. As in Figs. 7, 8, 12,and 13, di ff erent colors depict di ff erent angular resolutions. It is visible thatfor models closer to a success of the neutrino-driven mechanism the angu-lar kinetic energy exhibits larger temporal variations and an overall trend ofincrease as the onset of the explosion is approached. multi-dimensional case more favorable for an explosion thanspherical symmetry is associated with an inflation of theshock radius and postshock layer, driven by the buoyant riseand expansion of the plumes of neutrino-heated plasma. Incourse of the postshock volume becoming more extended, theintegrated mass M gain in the heating layer increases comparedto the 1D case. This can be seen in Fig. 12, which displaysthe mass in the gain layer as function of post-bounce time for the 11.2 and 15 M ⊙ runs with the di ff erent neutrino luminosi-ties and resolutions already shown in Figs. 7 and 8. While themass-averaged entropy h s i in the gain region hardly changes,the integral value of the entropy, M gain h s i , clearly increaseswith models coming closer to explosion. This dependence isparticularly well visible when 2D and 3D models with di ff er-ent resolutions are compared with each other.The longer dwell times of matter in the gain layer of2D simulations observed by Murphy & Burrows (2008),which correspond to the advection times τ adv evaluated byBuras et al. (2006b) and Marek & Janka (2009) (though dif-ferent ways of calculation have been considered, in particu-lar for the multi-dimensional case), are a manifestation thata growing mass accumulates in the postshock region to getenergy-loaded by neutrino absorption and to finally drive thesuccessful supernova blast. In near-steady-state conditionsthe mass accretion rate through the gain layer is equal tothe mass infall rate ˙ M ahead of the shock, where it is de-termined by the core structure of the progenitor star. Since τ adv ≈ M gain / ˙ M (cf. Marek & Janka 2009) a larger value of τ adv correlates with a higher mass M gain . Accordingly, thetotal net heating rate Q gain and thus the heating e ffi ciency ǫ ≡ Q gain / ( L ν e + L ¯ ν e ) = Q gain / (2 L ν ) of the gas residing in thegain layer is also higher for models that develop an explosion(see Fig. 13 and Murphy & Burrows 2008). Figure 14 showsthe ratio of the advection (dwell) timescale to the neutrino-heating timescale in the gain layer (our evaluation employsthe Newtonian analog of the formulas given in M¨uller et al.2012) for the models also displayed in Figs. 12 and 13. Thesame trends as in the previous images can be seen. Modelscloser to an explosion exhibit higher values of the timescaleratio. The ratio approaches unity, indicating the proximity to arunaway instability, roughly around the time when we definethe onset of an explosion (i.e., when the average shock ra-dius of Fig. 7 passes 400 km). The quality of the coincidenceof these moments, however, di ff ers from model to model anddepends on the exact definition used for the timescales and theaccuracy at which the relevant quantities can be evaluated inhighly perturbed flows. Nevertheless, the evolutionary trendof the timescale ratio (increasing or decreasing) is suggestivefor whether a simulation run leads to a successful explosionor failure.A larger mass in the gain layer and higher total net energydeposition rate are therefore better indicators of the proxim-ity of our models to explosion than the mean entropy of thegas in this region, which does not exhibit the 1D-2D-3D hi-erarchy with dimension found previously by Nordhaus et al.(2010). As discussed in Sect. 4 the main reason for thisdiscrepancy are most probably the di ff erent treatments ofneutrino lepton number losses and our consequential recal-ibration of the energy source terms. This leads to signifi-cantly higher energy drain from the cooling layer in our sim-ulations. While this hypothesis is supported by tests thatwe conducted in 1D, we cannot be absolutely certain thatno other e ff ects play a role for the discrepancies betweenour results and those of Nordhaus et al. (2010), because de-tailed cross-comparisons are not available and our knowledgeof the details of the implementation of neutrino e ff ects byNordhaus et al. (2010) may be incomplete. Other potentialreasons for di ff erences may be connected to the hydrodynam-ics scheme (P rometheus with a higher-order Godunov solverand directional splitting vs. C astro with unsplit methodology,Almgren et al. 2010), the employed grid (polar coordinatesvs. structured grid with adaptive refinement by a nested hi-ASI activity as key to successful neutrino-driven SN explosions? 17erarchy of rectangular grids), potentially —though not verylikely— the use of a 1D core above 10 g cm − in our simula-tions, or di ff erences in the exact structure and properties of theinfall region upstream of the stalled shock as a consequenceof di ff erent treatments of the collapse phase until 15 ms af-ter core bounce (due to full neutrino transport plus a nuclear-burning approximation in the P rometheus -V ertex code vs.the simple deleptonization scheme of Liebend¨orfer 2005) orof di ff erent seeding of nonradial hydrodynamic instabilities(in our case by imposed, small random seed perturbations ofthe density), or linked to di ff erences of the low-density EoSoutside of the application regime of the Shen et al. (1998)EoS.Despite these uncertainties about the exact cause of thedi ff erences, whose ultimate elimination will require system-atic and time-consuming studies, our results, as they are,send a clear message: The outcome of the 1D-2D-3D com-parison and the e ff ects of the third dimension advertised byNordhaus et al. (2010) “as a key to the neutrino mechanismof core-collapse supernova explosions” are not at all robustresults. Instead, the exact slope of the critical explosion con-dition L ν ( ˙ M ), its location, and its shift with dimension, aswell as the existence of a 1D-2D-3D hierarchy of the mass-averaged entropy in the gain layer seem to depend sensitivelyon subtle details of the neutrino treatment or other numericalaspects of the simulations. Resolution dependence
Let us now turn to the second, highly interesting questionin connection with our set of models, namely to the resolu-tion dependence of our results. Our set of simulations per-formed with di ff erent angular binnings reveals that quantitiesthat turned out to diagnose healthy conditions for an explo-sion, i.e. the growth of the average shock radius, the degree ofshock deformation, or the mass and total heating in the gainlayer (but not the mass-averaged entropy of the matter in thegain region), show a clear dependence on the angular zoning(see Figs. 7, 8, 12, and 13 in contrast to Fig. 4). In particular,2D models with better angular resolution exhibit more favor-able conditions and explode more readily (in agreement withresults obtained by Scheck 2007 with a more sophisticatedtreatment of neutrino transport than the simple heating andcooling source terms applied in our investigation), whereas3D models obey the opposite behavior. What does the reso-lution dependence of our simulations tell us about the mech-anism leading to explosions in our models? And how can weunderstand the puzzling finding that 2D and 3D runs followopposite trends when the angular resolution is refined?We interpret this as a manifestation of two aspects or facts:(1) The success of our models, at least in the neighborhoodof the explosion threshold, is fostered mainly by large-scale mass flows as associated with strong SASI activ-ity, but not by enhanced fragmentation of structures andvortex motions on small spatial scales.(2) Our resolution study reflects the consequences of theturbulent energy cascade, which redistributes energyfed into the flow by external sources in opposite direc-tions in 2D and 3D: While in 3D the turbulent energyflow goes from large to small scales, it pumps energyfrom small to large spatial scales in 2D.Point (1) is supported by the kinetic energies of nonradialmass motions in the gain layer of the 2D and 3D models plot- ted in Fig. 15. From this picture it is obvious that in the caseof successful models the angular kinetic energy is higher andshows an overall trend of growth in time until the blast hastaken o ff . Moreover, the spiky maxima and minima of quasi-periodic variations, which are indicative of the presence oflow-order SASI modes, are significantly larger for explodingmodels. This does not only hold for 2D models, whose lateralkinetic energies exhibit variation amplitudes of several 10%and partially up to even ∼
50% of the time-averaged value. Itis also true for 3D models, although in this case the ampli-tudes are generally smaller and the nonradial kinetic energyis split essentially equally into lateral and azimuthal contribu-tions. When comparing successful runs in 2D with those in3D, our studies suggest that in both cases the shock exhibits agrowing degree of asphericity (expressed by the standard de-viation of the shock deformation plotted in Fig. 8) when theexplosion is approached, and the kinetic energy of nonradialmass motions reaches roughly the same magnitude (Fig. 15),at least for models near the explosion threshold.Actually a variety of observations can be interpreted as sup-port of the hypothesis that flows on the largest possible scalesrather than on small scales play a crucial role for the successof the neutrino-heating mechanism in our simulations: • The strength of low-mode SASI activity in 2D modelsas indicated by growing fluctuations of the angular ki-netic energy and of the shock deformation (Figs. 15,8) increases with higher resolution in clear correlationwith an earlier onset of the explosion (Fig. 7). StrongerSASI activity obviously facilitates explosions, which isvisible by a growing average shock radius as well aslarger mass and higher total heating rate in the gainlayer. • Exploding models in 2D as well as in 3D exhibit largeshock deformation at the time of explosion (althoughthe relative asphericity σ S / R S of the shock surface issomewhat smaller in 3D than in 2D; see Figs. 7 and 8). • More fine structure on small spatial scales, which canbe seen in 3D models computed with higher resolutionin Figs. 9–11, does not imply improved conditions foran explosion. • Exploding 2D models are not connected with the high-est mean entropies in the gain region (Fig. 4). This fuelsdoubts in a random-walk picture where turbulent vortexmotions on small scales enlarge the residence time ofmatter in the gain layer (Murphy & Burrows 2008) andthus could allow for more energy absorption of suchmass elements from neutrinos. If this e ff ect occurred,it does not concern a major fraction of the mass in thegain region.Point (2) is the only plausible argument we can give forexplaining the opposite response to higher angular resolutionthat we discovered in our 2D and 3D simulations. The se-quence of 2D runs with gradually reduced lateral zone sizesreflects the growing violence of large-scale flows by higherfluctuation amplitudes of the kinetic energy in the gain layer(Fig. 15) and larger temporal variations of the average shockradius and shock deformation (Figs. 7, 8). In contrast, moreenergy on small spatial scales in the 3D case manifests itselfby a progressing fragmentation of the flow, leading to a grow-ing richness of vortex structures and finer filaments in the case8 Hanke, Marek, M¨uller, & Janka l E ( l ) [ e r g / c m ] i n j ec ti on f o r w a r d e n s t r ophy ca s ca d e i nv e r s e e n e r gy ca s ca d e d i ss i p a ti on ~ l - / ~ l -
2D - 1.5°2D - 2°2D - 3°3D - 1.5° 1 10 100multipole order l E ( l ) [ e r g / c m ] i n j ec ti on f o r w a r d e n e r gy ca s ca d e d i ss i p a ti on ~ l - /
3D - 1.5°3D - 2°3D - 3°2D - 1.5° F ig . 16.— Turbulent energy spectra E ( l ) as functions of the multipole order l for di ff erent angular resolution. The spectra are based on a decomposition ofthe azimuthal velocity v θ into spherical harmonics at radius r =
150 km and 400 ms post-bounce time for 15 M ⊙ runs with an electron-neutrino luminosity of L ν e = . · erg s − . Left:
2D models with di ff erent angular resolution (black, di ff erent thickness) and, for comparison, the 3D model with the highestemployed angular resolution (grey). Right:
3D models with di ff erent angular resolution and, for comparison, the 2D model with the highest employed angularresolution (grey). The power-law dependence and direction of the energy and enstrophy cascades (see text) are indicated by red lines and labels for 2D in theleft panel and 3D in the right panel. The left vertical, dotted line roughly marks the energy-injection scale, and the right vertical, dotted line denotes the onset ofdissipation at high l for the best displayed resolution. of 3D models with smaller angle bins (Figs. 9–11). As a con-sequence, 3D models with higher angular resolution becomemore similar to the 1D case in various quantities that we con-sidered as explosion indicators, see, e.g., Figs. 7, 8, 12, and13.Both the powerful coherent mass motions of the SASI layerin 2D and the vivid activity in small vortex structures in the3D environment are fed by two external sources which sup-ply the postshock layer with an inflow of fresh energy: (i)gravitational potential energy that is released by the continu-ous stream of matter falling through the accretion shock and(ii) energy deposition by neutrinos. The energy stored in thefluid is then redistributed towards small or large scales ac-cording to the turbulent cascades characteristic of two- andthree-dimensional environments.Direct evidence for the action of di ff erent turbulent energycascades in 2D and 3D can be obtained by considering theenergy spectrum E ( k ) of turbulent motions as a function ofwavenumber k in the gain region. The spectral shape of E ( k )can already be adequately established by considering only theazimuthal velocity v θ at a given radius using a decompositioninto spherical harmonics Y lm ( θ, φ ): E ( l ) = l X m = − l (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Ω Y ∗ lm ( θ, φ ) √ ρ v θ ( r , θ, φ ) d Ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (8)Here, the velocity fluctuations have been expressed in termsof the multipole order l instead of the wave number k . A sum-mation over the energies of modes with the same l has beencarried out , and in order to obtain smoother spectra, we av-erage E ( l ) over 30 km in radius and over 10 timesteps. Oneexpects that the resulting spectrum E ( l ) directly reflects theproperties of E ( k ) such as the slopes in di ff erent regimes ofthe turbulent cascade . The computed spectra E ( l ) (Fig. 16) Note that a factor √ ρ has been introduced to ensure that the integratedenergy of all modes sums up to the total kinetic energy contained in az-imuthal motions at radius r =
150 km (modulo a normalization factor) (cp.Endeve et al. 2012). For the precise relation between Fourier and spherical harmonics powerspectra, the reader may consult Chapter 21 of Peebles (1993). For a power-law spectrum E ( k ) ∝ k α , one obtains E ( l ) ∝ (2 l + Γ ( l + α/ + / / Γ ( l − α/ + / E ( l ) ∝ l α in the limit of large l . In practice, the power-lawindices of E ( l ) and E ( k ) appear to correspond well to each other already for indeed confirm the predictions from 3D and (planar) 2D tur-bulence theory, at least for su ffi ciently high multipole order l . In 3D (right panel), a power-law spectrum with E ( l ) ∝ l − / (Landau & Lifshitz 1959) develops at intermediate wavenum-bers as the resolution is increased, reflecting the transfer ofenergy from large to small scales in a forward cascade un-til dissipation takes over at large l . At high resolution, theenergy contained in small-scale disturbances increases, as thedissipation range moves to larger l . One observes that the 5 / l ( l . l ≈
10, i.e. atscales typical for growing convective plumes.By contrast, the power-law dependence E ( l ) ∝ l − / approx-imately holds for l .
10 in 2D as a result of the reverse en-ergy cascade (Kraichnan 1967). The energy injected at l ≈ ff erent power-law index( E ( l ) ∝ l − ).This appears to be a natural explanation for the predomi-nance of large-scale and small-scale structures in 2D and 3D,respectively. Moreover, this picture suggests that as dissipa-tion a ff ects the “injection scale” at l ≈
10 less with increasingresolution, the di ff erences between 2D and 3D become morepronounced with finer grid zoning.Our analysis of the spectral properties of turbulence thusfurther strengthens our view that the trends seen in our sim-ulations strongly suggest that nonradial kinetic energy avail-able on large scales, not on small scales, assists the devel-opment of an explosion by the neutrino-heating mechanism.This explains why 2D models with higher angular resolutiontend to explode earlier and thus at higher values of the mass-accretion rate than less resolved models. On the other hand,the energy “drain” by vortex motions on ever smaller scales—with the same reservoir of pumping energy per unit massbeing available from accretion and neutrino heating— disfa-vors explosions in better resolved 3D models.We therefore conclude that the key to the mechanism ofcore-collapse supernova explosions seems intrinsically andtightly linked to the question how much kinetic energy of l &
4. Empirically, broken power laws transform in a similar manner.
ASI activity as key to successful neutrino-driven SN explosions? 19the matter in the gain region can be accumulated in nonra-dial fluid motions on the largest possible scales, i.e., in thelowest-order spherical harmonics modes of nonradial hydro-dynamic instabilities. The predominant growth of such flowsis typical of SASI activity, whose lowest-order spherical har-monics modes possess the highest growth rates (Blondin et al.2003; Blondin & Mezzacappa 2006b; Foglizzo et al. 2006,2007; Ohnishi et al. 2006). Strong SASI motions drive shockexpansion, increase the gain layer and its mass content, al-low a larger fraction of the accreted matter to stay in thegain layer and be exposed to e ffi cient neutrino heating, andthus aid the development of an explosion (Scheck et al. 2008;Marek & Janka 2009). However, our models do not show asystematic trend of higher average entropies of the matter inthe gain layer for models closer to explosion. Instead, we findthat such models have larger mass, larger nonradial kineticenergy, larger total neutrino-heating rate, and larger total en-tropy in the gain layer. SUMMARY AND CONCLUSIONS
We have performed a systematic study of the post-bounceevolution of supernova cores of 11.2 and 15 M ⊙ and their ex-plosion by the neutrino-heating mechanism in 1D, 2D, and3D, employing simple neutrino cooling and heating termswith varied values of the driving luminosity. We conceptu-ally followed previous studies by Murphy & Burrows (2008)and Nordhaus et al. (2010), but did not apply the deleptoniza-tion treatment that they adopted from Liebend¨orfer (2005),who introduced it for an approximative description of neu-trino losses during the infall phase until core bounce. We ar-gued (Sect. 2) that this approximation —with or without thesource term proposed by Liebend¨orfer (2005) to account forentropy generation in neutrino-electron scatterings— does notprovide a suitable treatment of the evolution of the electronabundance after core bounce. Therefore we did not considerchanges of the net electron fraction Y e of the stellar plasmaat times later than 15 ms after bounce, up to which the col-lapse was followed with the P rometheus -V ertex code includ-ing full neutrino transport. While ignoring Y e changes subse-quently is certainly not a good approximation, it is not neces-sarily more unrealistic than describing the lepton-number evo-lution during the accretion phase of the stalled shock by thescheme of Liebend¨orfer (2005). As a consequence, we had toreplace an exponential factor e − τ e ff , which was introduced inan ad hoc way by Murphy et al. (2009) and Nordhaus et al.(2010) to damp the neutrino source terms at high opticaldepths τ e ff , by e − τ e ff / . in order to reproduce the minimumluminosity found to yield explosions in the 1D simulations byMurphy & Burrows (2008) and Nordhaus et al. (2010). Thismodification led to enhanced neutrino losses in the coolinglayer, which were better compatible with total energy lossrates found in simulations with detailed neutrino transport,e.g., in Buras et al. (2006a), and is responsible for some ofthe findings and di ff erences discussed in Sect. 4.Our results and conclusions can be briefly summarized asfollows:1. We cannot reproduce the exact slopes and relative loca-tions of the critical curves L ν ( ˙ M ) of 1D, 2D, and 3Dsimulations found by Nordhaus et al. (2010). Whileour results confirm the well-known fact that explosionsin 2D occur for a lower driving luminosity L ν than in1D when the mass accretion rate ˙ M is fixed, we cannotdiscover any significant further reduction when we go from 2D to 3D.2. We cannot confirm that the mass-averaged entropy ofthe matter in the gain region, h s i , is a good diagnosticquantity for the proximity to an explosion. As we ar-gued in Sect. 6.1, it is neither clear nor necessary that h s i is higher for cases where explosions are obtainedmore readily. Our successful 2D models do not exhibitlarger mean entropies than the corresponding 1D cases,which fail to explode. Instead, we observed that the to-tal mass, total entropy, total neutrino-heating rate, andthe nonradial kinetic energy in the gain layer are higherin cases that develop an explosion.3. We conclude that the tendency for an explosion asa monotonically increasing function of dimension aswell as the 1D-2D-3D hierarchy of h s ( t ) i found byNordhaus et al. (2010) are not robust results. Theyseem to be sensitive to subtle di ff erences of the ap-proximations of neutrino e ff ects (and / or to other di ff er-ences in the numerical treatments of the models). Itis therefore unclear how far studies with radical sim-plifications of the neutrino physics (without detailedenergy and lepton-number source terms and transport;no feedback between accretion and neutrino proper-ties) can yield results that are finally conclusive for theexplosion-triggering processes in real supernova cores.4. Increasing the angular resolution we observed a cleartendency of 2D models to explode earlier, in agree-ment with previous results by Scheck (2007), who em-ployed a more sophisticated treatment of neutrino ef-fects based on the transport approximation describedin Scheck et al. (2006). In contrast, 3D models showthe opposite trend and in a variety of quantities andaspects become more similar to their 1D counterparts.The easier explosion of the 2D models is connected toan enhanced violence of large-scale mass motions inthe postshock region due to SASI activity, whereas 3Dmodels with better angular resolution appear to developless strength in low-order SASI modes.5. We interpret this finding as a consequence of the op-posite turbulent energy cascades in 2D and 3D. In 2Dthe energy continuously pumped into the gain layer byneutrino heating and the release of gravitational bindingenergy flows from small to large scales and thus helpsto power coherent mass motions on the largest possiblespatial scales. In contrast, in 3D this energy seems toinstigate flow vorticity and fragmentation of structureson small scales. Evidence for this interpretation is pro-vided by Fig. 16.6. We also conclude from our resolution studies that thepresence of violent mass motions connected to low-order SASI modes is favorable for an explosion (inagreement with arguments given by Marek & Janka2009 and Scheck et al. 2008). This is supported by thefact that 2D and 3D models that are closer to explosionshow signs of growing power in large-scale mass mo-tions (signalled by growing fluctuations of the kineticenergy of nonradial velocity components) and in partic-ular develop significant shock deformation and globalejecta asymmetries when the explosion sets in.0 Hanke, Marek, M¨uller, & Janka7. We found that higher radial resolution makes explo-sions more di ffi cult with the setup chosen for the inves-tigated set of models. Higher resolution turned out toprevent explosions or to let them occur later in simula-tions in 1D, 2D, and 3D. This result could be diagnosedto be a consequence of a local density maximum in theneutrino-cooling layer, which grows with higher reso-lution and enhances the energy loss by neutrino emis-sion. This density peak, however, is a numerical artifactof the employed simple neutrino-cooling treatment byan analytic source term which is exponentially dampedat high densities.The lack of very precise information on the physics in-gredients and their exact implementation, e.g., details of thetreatment of neutrino source terms, low-density EoS, and pro-genitor data when mapped into the simulation and seededwith small random perturbations, as well as a variety ofmethodical di ff erences like the hydrodynamics scheme, nu-merical grid, and the use of a 1D core at high densities ornot, prevent us from presenting a rigorous proof that couldcausally link the discrepancies between our results and thoseof Nordhaus et al. (2010) to one or more well understood rea-sons. We think that the nagging uncertainties in this contextdemand a future, involved, collaborative code-comparisonproject. This will also require considerable amounts of com-puter time for further 3D simulations, in particular with highresolution, thus needing more computer resources than avail-able to us for the described project.Despite this deficiency, however, our results suggest thatthe di ff erences of 3D compared to 2D simulations observedby Nordhaus et al. (2010) are unlikely to be a robust outcomebut seem to depend on relevant aspects of the modeling (mostprobably the neutrino physics but potentially, and not finallyexcluded, also technical aspects).We therefore conclude that the influence of 3D e ff ects onthe supernova mechanism is presently not clear. We stronglyemphasize, however, that the fact that our results do not cor-roborate improved explosion conditions in 3D compared to2D cannot be used as an argument that 3D e ff ects do not fa-cilitate the supernova explosion mechanism or are of minorimportance. We just think that in the context of the neutrino-driven mechanism the relevance and exact role of 3D fluiddynamics are not understood yet. We therefore have the opin-ion that the results obtained by Nordhaus et al. (2010) do notjustify their claims that 3D hydrodynamics o ff ers the key to afundamental understanding of the neutrino mechanism whileother physics in the supernova core, like general relativityor the properties of the nuclear EOS, are only of secondaryimportance. Though this may well be right, such statementsat the present time are premature and not supported by solidfacts and results.Our study, however, raises further important questions.How far can our understanding be developed on grounds ofmodeling approaches that employ radical simplifications ofthe neutrino physics? Which aspects of the complex inter-play between di ff erent components of the problem are linkedto the essence of how the explosion is triggered by the combi-nation of neutrino energy supply and nonradial hydrodynamicinstabilities? Examples for such mutually dependent compo-nents are the neutrino transport and hydrodynamics, the neu-tron star core evolution and fluid motions around the neutronstar, or the mass flux from the accretion shock to the deceler-ation layer (both being the coupling regions for the advective- acoustic cycle that is thought to be responsible for the SASIgrowth; e.g., Scheck et al. 2008) and the conditions in theneutrino heating and cooling layers. Much more work needsto be done to find the answers of these questions.A major shortcoming of the setup applied in previous worksand adopted also in our investigation is the neglect of neu-trino cooling and deleptonization inside the proto-neutronstar. The employed simple neutrino source terms and theirexponential suppression at high optical depths do not al-low the neutron star to evolve. Underestimating the neutronstar contraction, however, slows down the infall velocities inthe postshock layer and thus has disfavorable consequencesfor the growth of the SASI (Scheck et al. 2008), similar tothe e ff ects of reduced neutrino losses in the cooling layer(Scheck et al. 2008) or increased nuclear photodissociationbehind the stalled shock (Fern´andez & Thompson 2009). Onthe contrary, a slower postshock flow improves the conditionsfor the growth of convective instability, whose developmentis supported by a high ratio (larger than ∼ ff erences in the growth conditions for convection comparedto the SASI in collapsing stellar cores of a variety of progen-itor stars. Also the high-density equation of state and generalrelativity, influencing the contraction behavior of the nascentneutron star, can make a di ff erence (Marek & Janka 2009;M¨uller et al. 2012). Claims, based on highly simplified mod-els, that SASI is less important than convection and at mosta minor feature of the supernova dynamics (Burrows et al.2012) are therefore certainly premature.Finally, our resolution study suggests that the action of theturbulent cascade in 3D extracts energy from coherent large-scale modes of fluid motion and instead fuels fragmentationand enhanced vortex flows on small spatial scales. At leastin our 3D models with better grid zoning the appearance offiner structures in the postshock flow was connected with atendency of damping the development of explosions. Whilea finally convincing proof of such a negative feedback mayrequire much better resolved simulations than we presentlycan a ff ord to conduct (in order to minimize numerical dissipa-tion on small scales), this result implies that good resolution—considerably higher than recently used by Takiwaki et al.(2011), whose 3D simulation had only 32 azimuthal zones(corresponding to a cell size of 11.25 ◦ )— is indispensable toclarify the 3D e ff ects on the explosion mechanism. More-over, our result points to an interesting direction. It is pos-sible that the success of the neutrino-driven mechanism in3D is tightly coupled to the presence of violent SASI activ-ity, a connection that was found before —and is confirmed byour present study— to foster explosions in 2D? If so, whatis the key to instigate such violent SASI motions of the su-pernova core in three dimensions? Will they occur with abetter (more realistic) treatment of the neutrino transport andcorrespondingly altered conditions in the heating and cool-ing layers and in the contracting core of the proto-neutronASI activity as key to successful neutrino-driven SN explosions? 21 PSfrag replacements ˙ M [ M ⊙ / s ] @ m t pb [s] F ig . 17.— Mass accretion rate for the 15 M ⊙ progenitor. The red curve shows the line of Fig. 1. The black symbols represent the values extracted from oursimulations at the time t exp when the explosion sets in. Green symbols are data from Murphy & Burrows (2008) and blue symbols those from Nordhaus et al.(2010). Di ff erent symbols are used for results of 1D, 2D, and 3D simulations. star? Or are they associated with stellar rotation, which evenwith a slow rate can initiate the faster growth of spiral (non-axisymmetric) SASI modes (Blondin & Mezzacappa 2006a;Yamasaki & Foglizzo 2008; Iwakami et al. 2009; Fern´andez2010)? Or is strong SASI activity in the supernova core trig-gered by large-scale inhomogeneities in the three-dimensionalprogenitor star (Arnett & Meakin 2011), which could providea more e ffi cient seed for SASI growth than the random cell-to-cell small-amplitude perturbations employed in our simula-tions? Should the presence of large-amplitude SASI mass mo-tions indeed turn out to be the key to the neutrino mechanismin 3D, it would mean that neutrino-driven explosions are notonly a generically multi-dimensional phenomenon, but onethat is generically associated with dominant low-order modesof asymmetry and deformation from the very beginning.While this paper raises many more questions than it is ableto answer, it definitely makes clear that our understanding ofthe supernova physics in the third dimension is still in its veryinfancy. A virgin territory with distant horizons lies ahead ofus and awaits to be explored. We are grateful to Lorenz H¨udepohl for his valuable inputto di ff erent aspects of the reported project and thank ElenaErastova and Markus Rampp (Max-Planck-RechenzentrumGarching) for their help in the visualization of our 3Ddata. HTJ would like to thank Rodrigo Fern´andez, ThierryFoglizzo, Jerome Guilet, and Christian Ott for stimulat-ing and informative discussions. This work was supportedby the Deutsche Forschungsgemeinschaft through Sonder-forschungsbereich / Transregio 27 “Neutrinos and Beyond”,Sonderforschungsbereich / Transregio 7 “Gravitational-WaveAstronomy”, and the Cluster of Excellence EXC 153 “Ori-gin and Structure of the Universe”. The computations wereperformed on the Juropa cluster at the John von NeumannInstitute for Computing (NIC) in J¨ulich, partially through aDECI-6 grant of the DEISA initiative, on the IBM p690 atCineca in Italy through a DECI-5 grant of the DEISA initia-tive, and on the IBM p690 at the Rechenzentrum Garching.
APPENDIX A. SIMULATIONS WITH PARAMETRIZED DELEPTONIZATION TREATMENT FOR THE CORE-COLLAPSE PHASE
In this Appendix we briefly report on our e ff orts to reproduce the 1D results of Murphy & Burrows (2008) and Nordhaus et al.(2010) for the critical explosion conditions of the 15 M ⊙ progenitor, applying a neutrino treatment that was intended to copy theprocedure outlined in these publications as closely as possible.For this purpose we retained the exponential suppression factor e − τ e ff of Eqs. (4) and (5) without a reduction factor of 2.7 in2 Hanke, Marek, M¨uller, & Janka PSfrag replacements e l ec t r on fr ac ti on density [g / cm ]Deleptonization 1Deleptonization 2Deleptonization 3 F ig . 18.— Trajectories of the electron fraction with density deduced from di ff erent core-collapse studies (see text for details) and employed in our 1D simulationsfor parametrizing lepton losses by neutrino emission in the stellar core according to Liebend¨orfer (2005). the exponent, and the lepton evolution before and after core bounce was described by employing a predefined Y e ( ρ ) relation. Wealso aimed at reproducing the core infall of the previous works as closely as possible, because the density structure of the infallregion ahead of the stalled shock determines the mass-infall rate ˙ M ( t ) at the shock, and some di ff erences became visible whenwe compared our values with those given by Murphy & Burrows (2008) and Nordhaus et al. (2010) (Fig. 17). We thereforerecomputed the collapse phase from the onset of gravitational instability of the progenitor core through core bounce with thedeleptonization scheme of Liebend¨orfer (2005). Entropy changes were taken into account as suggested by Liebend¨orfer (2005),but were switched o ff after core bounce following Murphy & Burrows (2008) and Nordhaus et al. (2010).We tested three di ff erent cases for the functional relation Y e ( ρ ): First, we used a tabulated result for the Y e ( ρ ) evolu-tion as obtained with the P rometheus -V ertex code and state-of-the-art electron-capture rates (Langanke et al. 2003) (“Delep-tonization 1”). Second, we applied a Y e ( ρ )-table provided by Christian Ott as a co-developer of the C o C o N u T code(http: // / hydro / COCONUT / ). These data are based on collapse simulations with the V ulcan/
2D code(Livne et al. 2004) (“Deleptonization 2”). Third, we employed a fitting formula given by Liebend¨orfer (2005) for the parametersof model G15 (“Deleptonization 3”). All three Y e ( ρ ) trajectories are depicted in Fig. 18.The three sets of 1D simulations conducted for this Appendix were performed with 800 radial zones. The corresponding criti-cal luminosity curves L ν e ( ˙ M ) are displayed in Fig. 19 in comparison to those of Murphy & Burrows (2008) and Nordhaus et al.(2010) and to our results of Fig. 2 (for 400 zones, because for this resolution the calibration of the exponential suppressionfactor for best agreement with the critical curve of Murphy & Burrows (2008) was done). The overall slopes of all three curvesare similar but none of them is quantitatively or qualitatively in good agreement with those of Murphy & Burrows (2008) andNordhaus et al. (2010). Explosions in our simulations occurred significantly more readily (i.e., for lower L ν e ) than in the previousworks. This suggests less cooling in our runs, although we made all possible e ff orts to exactly follow the description of the neu-trino treatment in those papers. The steep rise and partly backward bending of our curves for ˙ M values around 0.2–0.3 M ⊙ s − canbe understood by an inspection of Fig. 20, which shows the time evolution of the shock radius for simulations with prescription“Deleptonization 1” for a selection of neutrino luminosity values. One can see that in the case of L ν e = . · erg s − theshock makes a larger excursion before it returns again. Its reexpansion, leading to an explosion, therefore happens later than inthe model with L ν e = . · erg s − , where the first shock expansion is much less strong. Correspondingly, the explosion inthe former case sets in at a later time and lower mass accretion rate than in the latter case, explaining the backward bending of L ν e ( ˙ M ) in this regime of luminosities and ˙ M . The nearly horizontal parts of the critical curves can be understood by the fact thatfor such high values of the luminosities the neutrino cooling (with the unmodified e − τ e ff suppression factor) is so weak that theASI activity as key to successful neutrino-driven SN explosions? 23 PSfrag replacements ˙M [M ⊙ / s] L ν e [ e r g / s ] HankeMurphyNordhausDeleptonization 1Deleptonization 2Deleptonization 3 F ig . 19.— Critical curves for the electron-neutrino luminosity ( L ν e ) versus mass accretion rate ( ˙ M ), representing the explosion threshold for di ff erent setsof 1D simulations of the 15 M ⊙ progenitor. The black line corresponds to our results shown as black curve in Fig. 2 (see also Table 1), red are results ofMurphy & Burrows (2008), green of Nordhaus et al. (2010), and the three additional curves (dark blue, light blue, and pink) correspond to di ff erent sets ofsimulations that we performed with the deleptonization treatment of Liebend¨orfer (2005) for the core-collapse phase and the di ff erent electron-fraction trajectoriesof Fig. 18 in our e ff ort to reproduce the 1D results of Murphy & Burrows (2008) and Nordhaus et al. (2010). explosion sets in very early (see the black line in Fig. 20) and therefore for large values of the mass accretion rate. The regionbetween ˙ M ≈ . M ⊙ s − and ˙ M ≈ . M ⊙ s − is di ffi cult to probe with a stepwise increase of L ν e , because the mass-accretion ratethere changes so rapidly that the shock shows time-dependent dynamics instead of settling into a quasi-steady state.None of the critical curves obtained with the direct implementation of the neutrino treatment described in Murphy & Burrows(2008) and Nordhaus et al. (2010) can reproduce the critical luminosity curves reported in these papers reasonably well. Wetherefore decided to proceed with the modifications described in Sect. 2.4 Hanke, Marek, M¨uller, & Janka PSfrag replacements R S [ k m ] t pb [s]11.2 M ⊙ ⊙ Deleptonization 1 F ig . 20.— Time evolution of the shock radius as a function of post-bounce time, t pb , for 1D simulations performed with the deleptonization scheme ofLiebend¨orfer (2005) for electron-fraction trajectory “Deleptonization 1” of Fig. 18. The colors correspond to di ff erent electron-neutrino luminosities, which arelabeled in the plot in units of 10 erg s − . ASI activity as key to successful neutrino-driven SN explosions? 25
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