The Status of Multi-Dimensional Core-Collapse Supernova Models · 2012;Kiziltan et...

Publications of the Astronomical Society of Australia (PASA)c© Astronomical Society of Australia 2016; published by Cambridge University Press.doi: 10.1017/pas.2016.xxx.

The Status of Multi-Dimensional Core-Collapse SupernovaModels

B. Muller1,2,31Astrophysics Research Centre, School of Mathematics and Physics, Queen’s University Belfast, Belfast, BT7 1NN, United [email protected] Centre for Astrophysics, School of Physics and Astronomy, Monash University, VIC 3800, Australia3Joint Institute for Nuclear Astrophysics, University of Notre Dame, IN 46556, USA

AbstractNumerical models of core-collapse supernova explosions powered by the neutrino-driven mechanism have matured con-siderable in recent years. Explosions at the low-mass end of the progenitor spectrum can routinely be simulated in 1D,2D, and 3D today and already allow us to study supernova nucleosynthesis based on first-principle models. Results ofnucleosynthesis calculations indicate that supernovae of the lowest masses could be important as contributors of somelighter neutron-rich elements beyond iron. The explosion mechanism of more massive stars is still under investigation,although first 3D models of neutrino-driven explosions employing multi-group neutrino transport have recently becomeavailable. Together with earlier 2D models and more simplified 3D simulations, these have elucidated the interplay be-tween neutrino heating and multi-dimensional hydrodynamic instabilities in the post-shock region that is essential forshock revival. However, some physical ingredients may still need to be added or improved before simulations can ro-bustly explain supernova explosions over a wide mass range. We explore possible issues that may affect the accuracy ofnumerical supernova simulations, and review some of the ideas that have recently been explored as avenues to robustexplosions, including uncertainties in the neutrino rates, rapid rotation, and an external forcing of non-radial fluid mo-tions by strong seed perturbations from convective shell burning. The “perturbation-aided” neutrino-driven mechanismand the implications of recent 3D simulations of shell burning in supernova progenitors are discussed in detail. The effi-cacy of the perturbation-aided mechanism in some progenitors is illustrated by the first successful multi-group neutrinohydrodynamics simulation of an 18M progenitor with 3D initial conditions. We conclude with a few speculations aboutthe potential impact of 3D effects on the structure of massive stars through convective boundary mixing.

Keywords: supernovae: general – hydrodynamics – instabilities – neutrinos – stars: massive – stars: evolution

1 INTRODUCTION

The explosions of massive stars as core-collapse supernovae(CCSNe) constitute one of the most outstanding problemsin modern astrophysics. This is in no small measure due tothe critical role of supernova explosions in the history ofthe Universe. Core-collapse supernovae figure prominentlyin the chemical evolution of galaxies as the dominant pro-ducers, e.g., of elements between oxygen and the iron group(Arnett 1996; Woosley et al. 2002), and supernova feedbackis a key ingredient in the modern theory of star formation(Krumholz 2014). The properties of neutron stars and stellar-mass black holes (masses, spins, kicks; Ozel et al. 2010,2012; Kiziltan et al. 2013; Antoniadis et al. 2016; Arzou-manian et al. 2002; Hobbs et al. 2005) cannot be understoodwithout addressing the origin of these compact objects instellar explosions.

Why (some) massive stars explode is, however, a dauntingproblem in its own right regardless of the wider implications

of supernova explosions: The connection of supernovae ofmassive stars with the gravitational collapse to a neutron starhas been postulated more than eighty years ago (Baade &Zwicky 1934), and the best-explored mechanism for power-ing the explosion, the neutrino-driven mechanism, has gonethrough several stages of “moulting” in the fifty years afterits conception by Colgate & White (1966). Yet the problemof the supernova explosion mechanism still awaits a defini-tive solution. The rugged path towards an understanding ofthe explosion mechanism merely reflects that core-collapsesupernovae are the epitome of a “multi-physics” problemthat combines aspects of stellar structure and evolution, nu-clear and neutrino physics, fluid dynamics, kinetic theory,and general relativity. We cannot recapitulate the history ofthe field here and instead refer the reader to the classical andmodern reviews of Bethe (1990), Arnett (1996), Mezzacappa(2005), Kotake et al. (2006), Janka et al. (2007), Burrowset al. (2007a), Janka (2012), and Burrows (2013) as startingpoints.

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The longevity of the supernova problem should not bemisinterpreted: Despite the occasional detour, supernovatheory has made steady progress, particularly so duringthe last few years, which have seen the emergence of ma-ture – and increasingly successful – multi-dimensional first-principle simulations of the collapse and explosion of mas-sive stars as well as conceptual advances in our understand-ing of the neutrino-driven explosion mechanism and its in-terplay with multi-dimensional hydrodynamic instabilities.

1.1 The Neutrino-driven Explosion Mechanism in itsModern Flavour

Before we review these recent advances, it is apposite tobriefly recapitulate the basic idea of the neutrino-driven su-pernova mechanism in its modern guise. Stars with zero-age main sequence masses above & 8M and with a heliumcore mass . 65M (the lower limit for non-pulsational pair-instability supernovae; Heger & Woosley 2002; Heger et al.2003) develop iron cores that eventually become subjectto gravitational instability and undergo collapse on a free-fall time-scale. For low-mass supernova progenitors withhighly degenerate iron cores, collapse is triggered by thereduction of the electron degeneracy pressure due to elec-tron captures; for more massive stars with higher core en-tropy and a strong contribution of radiation pressure, photo-disintegration of heavy nuclei also contributes to gravita-tional instability. Aside from these “iron core supernovae”,there may also be a route towards core collapse from super-AGB stars with O-Ne-Mg cores (Nomoto 1984, 1987; Poe-larends et al. 2008; Jones et al. 2013, 2014; Doherty et al.2015), where rapid core contraction is triggered by electroncaptures on 20Ne and 24Mg;1 hence this sub-class is desig-nated as “electron-capture supernovae” (ECSNe).

According to modern shell-model calculations (Langanke& Martınez-Pinedo 2000; Langanke et al. 2003), the elec-tron capture rate on heavy nuclei remains high even dur-ing the advanced stages of collapse (Langanke et al. 2003)when the composition of the core is dominated by increas-ingly neutron-rich and massive nuclei. Further deleptonisa-tion during collapse thus reduces the lepton fraction Ylep toabout 0.3 according to modern simulations (Marek et al.2005; Sullivan et al. 2016) until neutrino trapping occursat a density of ∼1012 g cm−3. As a result, the homologouslycollapsing inner core shrinks (Yahil 1983), and the shockforms at a small enclosed mass of ∼0.5M (Langanke et al.2003; Hix et al. 2003; Marek et al. 2005) after the corereaches supranuclear densities and rebounds (“bounces”).Due to photodisintegration of heavy nuclei in the infallingshells into free nucleons as well as rapid deleptonisation inthe post-shock region once the shock breaks out of the neu-trinosphere, the shock stalls a few milliseconds after bounce,

1Whether the core continues to collapse to a neutron star depends criticallyon the details of the subsequent initiation and propagation of the oxygendeflagration during the incipient collapse (Isern et al. 1991; Canal et al.1992; Timmes & Woosley 1992; Schwab et al. 2015; Jones et al. 2016a).

i.e. it turns into an accretion shock with negative radial ve-locity downstream of the shock. Aided by a continuous re-duction of the mass accretion rate onto the young proto-neutron star, the stalled accretion shock still propagates out-ward for ∼70 ms, however, and reaches a typical peak radiusof ∼150 km before it starts to recede again.

The point of maximum shock expansion is roughly co-incident with several other important changes in the post-shock region: Photons and electron-positron pairs becomethe dominant source of pressure in the immediate post-shockregion, deleptonisation behind the shock occurs more gradu-ally, and the electron neutrino and antineutrino luminositiesbecome similar. Most notably, a region of net neutrino heat-ing (“gain region”) emerges behind the shock. In the “de-layed neutrino-driven mechanism” as conceived by Bethe &Wilson (1985) and Wilson (1985), the neutrino heating even-tually leads to a sufficient increase of the post-shock pressureto “revive” the shock and make it re-expand, although thepost-shock velocity initially remains negative. Since shockexpansion increases the mass of the dissociated material ex-posed to strong neutrino heating, this is thought to be a self-sustaining runaway process that eventually pumps sufficientenergy into the post-shock region to allow for the develop-ment of positive post-shock velocities and, further down theroad, the expulsion of the stellar envelope.

Modern simulations of core-collapse supernovae that in-clude energy-dependent neutrino transport, state-of-the artmicrophysics, and (to various degrees) general relativisticeffects have demonstrated that the neutrino-driven mech-anism is not viable in spherical symmetry (Rampp &Janka 2000, 2002; Liebendorfer et al. 2001, 2004, 2005;Sumiyoshi et al. 2005; Buras et al. 2006a,b; Muller et al.2010; Fischer et al. 2010; Lentz et al. 2012a,b), except forsupernova progenitors of the lowest masses (Kitaura et al.2006; Janka et al. 2008; Burrows et al. 2007b; Fischer et al.2010), which will be discussed in Section 2.

In its modern guise, the paradigm of neutrino-driven ex-plosions therefore relies on the joint action of neutrino heat-ing and various hydrodynamic instabilities to achieve shockrevival. As demonstrated by the first generation of multi-dimensional supernova models in the 1990s (Herant et al.1994; Burrows et al. 1995; Janka & Muller 1995, 1996),the gain region is subject to convective instability due tothe negative entropy gradient established by neutrino heat-ing. Convection can be suppressed if the accreted mate-rial is quickly advected from the shock to the gain radius(Foglizzo et al. 2006). Under these conditions, the stand-ing accretion shock instability (SASI; Blondin et al. 2003;Blondin & Mezzacappa 2006; Foglizzo et al. 2007; Laming2007; Yamasaki & Yamada 2007; Fernandez & Thompson2009a,b) can still grow, which is mediated by an advective-acoustic cycle (Foglizzo 2002; Foglizzo et al. 2007; Guilet &Foglizzo 2012) and manifests itself in the form of large-scalesloshing and spiral motions of the shock. The precise mech-anism whereby these instabilities aid shock revival requirescareful discussion (see Section 3.3), but their net effect can

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The Status of Multi-Dimensional Core-Collapse Supernova Models 3

be quantified using the concept of the “critical luminosity”(Burrows & Goshy 1993) for the transition from a steady-state accretion flow to runaway shock expansion: In effect,convection and/or the SASI reduce the critical luminosity inmulti-D by 20 . . . 30% (Murphy & Burrows 2008; Nordhauset al. 2010; Hanke et al. 2012; Fernandez 2015) compared tothe case of spherical symmetry (1D).

1.2 Current Questions and Structure of This Review

We cannot hope to comprehensively review all aspects of thecore-collapse supernova explosion problem, even if we limitourselves to the neutrino-driven paradigm. Instead we shallfocus on the following topics that immediately connect tothe above overview of the neutrino-driven mechanism:

• The neutrino-driven explosion mechanism demonstra-bly works at the low-mass end of supernova progen-itors. In Section 2, we shall discuss the specific ex-plosion dynamics in the region around the mass limitfor iron core formation, i.e. for ECSN progenitors andstructurally similar iron core progenitors. We shall alsoconsider the nucleosynthesis in these explosions; sincethey are robust, occur early after bounce, and can easilybe simulated until the explosion energy has saturated,explosions of ECSN and ECSN-like progenitors cur-rently offer the best opportunity to study core-collapsesupernova nucleosynthesis based on first-principle ex-plosion models.

• For more massive progenitors, it has yet to be demon-strated that the neutrino-driven mechanism can pro-duce robust explosions in 3D with explosion proper-ties (e.g. explosion energy, nickel mass, remnant mass)that are compatible with observations. In Section 3, weshall review the current status of 3D supernova simu-lations, highlighting the successes and problems of thecurrent generation of models and detailing the recentprogress towards a quantitative understanding of theinterplay of neutrino heating and multi-dimensionalfluid flow.

• In the wake of a rapid expansion of the field ofcore-collapse supernova modelling, a wide variety ofmethods have been employed to investigate the super-nova problem with a continuum from a rigorous first-principle approach to parameterised models of lim-ited applicability that are only suitable for attackingwell-circumscribed problems. In Section 4, we presentan overview of the different numerical approaches tosimulations of neutrino-driven explosions and providesome guidance for assessing and comparing simulationresults.

• The problem of shock revival by the neutrino-drivenmechanism has not been conclusively solved. In Sec-tion 5, we shall review one of the promising ideas thatcould help explain supernova explosions over a widerange of progenitors, viz. the suggestion that shock re-

vival may be facilitated by strong seed perturbationsfrom prior convective shell burning in the infallingO or Si shells (Arnett & Meakin 2011; Couch & Ott2013; Muller & Janka 2015; Couch et al. 2015; Mulleret al. 2016a); and we shall also discuss some otherperspectives opened up by current and future three-dimensional simulations of late burning stages in su-pernova progenitors.

Potential observational probes for multi-dimensional fluidflow in the supernova core during the first ∼1 s exist in theform of the neutrino and gravitational wave signals, but weshall not touch these in any depth and instead point thereader to topical reviews (Ott 2009; Kotake 2013 for grav-itational wave emission; Mirizzi et al. 2016 for the neutrinosignal) as well as some of the major publications of recentyears (gravitational waves: Muller et al. 2013; Yakunin et al.2015; Nakamura et al. 2016; neutrinos: Tamborra et al. 2013,2014a; Muller & Janka 2014) Neither do we address alterna-tive explosion scenarios here. and refer the reader to Janka(2012) for a broader discussion that covers, e.g. the mag-netorotational mechanism as the most likely explanation forhypernovae with explosion energies of up to ∼1052 erg.

2 THE LOW-MASS END ELECTRON-CAPTURESUPERNOVAE AND THEIR COUSINS

Stars with zero-age main sequence (ZAMS) masses in therange ∼8 . . . 10M exhibit structural peculiarities duringtheir evolution that considerably affect the supernova explo-sion dynamics if they undergo core collapse. The classicalpath towards electron-capture supernovae (ECSNe; Nomoto1984, 1987), where electron captures on 24Mg and 20Ne ina degenerate O-Ne-Mg core of ∼ 1.37M drive the core to-wards collapse, best exemplifies these peculiarities: Only asmall C/O layer is present on top of the core, and the Helayer has been effectively whittled down by dredge-up. Theconsequence is an extremely steep density gradient betweenthe core and the high-entropy hydrogen envelope (Figure 1).While this particular scenario is beset with many uncer-tainties (Siess 2007; Poelarends et al. 2008; Jones et al.2013, 2014, 2016a; Doherty et al. 2015; Schwab et al. 2015;Woosley & Heger 2015b), recent studies of stellar evolutionin the mass range around 9M have demonstrated that thereis a variety of paths towards core-collapse that result in asimilar progenitor structure (Jones et al. 2013; Woosley &Heger 2015b), though there is some variation, e.g. in themass of the remaining He shell due to a different history ofdredge-up events. From the perspective of supernova explo-sion dynamics, the crucial features in the mass range around9M are the small mass of the remaining C/O shell and therapid drop of the density outside the core; both are sharedby ECSN progenitors and the lowest iron-core progenitors.This is illustrated in Figure 1 (see also Figure 7 in Jones et al.2013 and Figure 4 in Woosley & Heger 2015b).

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101 102 103 104 105

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100101102103104105106107108109

10101011

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3] N8.8

J8.8z9.6u8.1s10.09s15s11.2

Figure 1. Density profiles of several low-mass supernova progenitors il-lustrating the conditions for ECSN-like explosions. Profiles are shown forthe 8.8M ECSN-progenitor of Nomoto (1984, 1987) (N8.8, black), the 8.8M “failed massive star” of Jones et al. (2013) (J8.8, purple), low-massiron core progenitors (A. Heger, private communication) of 9.6M (z9.6,with Z=0, red) and 8.1M (u8.1, with Z = 10−4, blue), and iron progenitorswith 10.09M and 15M (s10.09 and s15, from Muller et al. 2016b, yellowand cyan) and 11.2M (s11.2 from Woosley et al. 2002, green). The thickdashed vertical line roughly denotes the location of the shell that reaches theshock 0.5 s after the onset of collapse. Slanted dashed lines roughly demar-cate the regime where the accretion rate onto the shock reaches 0.05M s−1

(thick dashed line), 5 × 10−3 M s−1 (thin), and 5 × 10−4 M s−1 (thin) (seeSection 2.1.2 for details and underlying assumptions). ECSN-like explosiondynamics is expected if the density profile intersects the grey region.

2.1 Explosion Dynamics in ECSN-like Progenitors

2.1.1 Classical Electron-Capture Supernova ModelsThe steep density gradient outside the core in ECSN-likeprogenitors is immediately relevant for the dynamics of theensuing supernova because it implies a rapid decline of themass accretion rate M as the edge of the core reaches thestalled accretion shock. A rapid drop in M implies a decreas-ing ram pressure ahead of the shock and a continuously in-creasing shock radius (though the shock remains a stationaryaccretion shock for at least ∼50 ms after bounce and longerfor some ECSN-like progenitor models). Under these condi-tions, neutrino heating can easily pump sufficient energy intothe gain region to make the accreted material unbound andpower runaway shock expansion. As a result, the neutrino-driven mechanism works for ECSN-like progenitors evenunder the assumption of spherical symmetry. Using modernmulti-group neutrino transport, this was demonstrated by Ki-taura et al. (2006) for the progenitor of Nomoto (1984, 1987)and confirmed in subsequent simulations by different groups(Janka et al. 2008; Burrows et al. 2007b; Fischer et al. 2010).The explosions are characterised by a small explosion en-ergy of ∼1050 erg (Kitaura et al. 2006; Janka et al. 2008) anda small nickel mass of a few 10−3M (Wanajo et al. 2009).

Even though multi-dimensional effects are not crucial forshock revival in these models, they are not completely neg-ligible. Higher entropies at the bottom of the gain layer leadto convective overturn driven by Rayleigh-Taylor instability

shortly after the explosion is initiated (Wanajo et al. 2011).Simulations in axisymmetry (2D) showed that this leads toa modest increase of the explosion energy in Janka et al.(2008); an effect which is somewhat larger in more recentmodels (von Groote et al., in preparation). The effect ofRayleigh-Taylor overturn on the ejecta composition is, how-ever, much more prominent (see Section 2.2).

2.1.2 Conditions for ECSN-like Explosion DynamicsNot all of the newly available supernova progenitor mod-els at the low-mass end (Jones et al. 2013, 2014; Woosley& Heger 2015b) exhibit a similarly extreme density profileas the model of Nomoto (1984, 1987); in some of them thedensity gradient is considerably more shallow (Figure 1).This prompts the questions: How steep a density gradientis required outside the core to obtain an explosion that istriggered by a rapid drop of the accretion rate and workswith no or little help from multi-D effects? In reality, therewill obviously be a continuum between ECSN-like eventsand neutrino-driven explosions of more massive stars, inwhich multi-D effects are crucial for achieving shock revival.Nonetheless, a rough distinction between the two differentregimes is still useful, and can be based on the concept ofthe critical neutrino luminosity of Burrows & Goshy (1993).

Burrows & Goshy (1993) showed that stationary accre-tion flow onto a proto-neutron star in spherical symmetryis no longer possible if the neutrino luminosity Lν (whichdetermines the amount of heating) exceeds a critical valueLcrit(M) that is well approximated by a power law in M witha small exponent, or, equivalently, if M drops below a thresh-old value for a given luminosity. This concept has recentlybeen generalised (Janka 2012; Muller & Janka 2015; Summaet al. 2016; Janka et al. 2016) to a critical relation for the(electron-flavour) neutrino luminosity Lν and neutrino meanenergy Eν as a function of mass accretion rate M and proto-neutron star mass M as well as additional correction factors,e.g., for shock expansion due to non-radial instabilities.

For low-mass progenitors with tenuous shells outside thecore, M, Lν, and Eν do not depend dramatically on the stel-lar structure outside the core during the early post-bounce:The proto-neutron star mass is inevitably M ≈ 1.4M, andsince the neutrino emission is dominated by the diffusiveneutrino flux from the core, the neutrino emission proper-ties are bound to be similar to the progenitor of Nomoto(1984), i.e. one has Lν ∼ 5 × 1052 erg s−1 and Eν ≈ 11 MeV(Hudepohl et al. 2009), with a steady decrease of the lumi-nosity towards later times. Using calibrated relations for the“heating functional”2 LνE2

ν (Janka et al. 2016), this translatesinto a critical mass accretion rate of Mcrit ≈ 0.07M s−1 forECSN-like progenitors.

To obtain similarly rapid shock expansion as for the8.8M model of Nomoto (1984), M must rapidly plummetwell below this value. This can be translated into a condi-

2This compact designation for LνE2ν has been suggested to me by H.-

Th. Janka.

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tion for the density profile outside the core using analyticexpressions for the infall time tinfall and accretion rate M formass shell m, which are roughly given by (Woosley & Heger2012, 2015a; Muller et al. 2016b),

tinfall =

√π

4Gρ=

√π2r3

3Gm, (1)

and

M =2m

tinfall

ρ

ρ − ρ, (2)

where ρ is the average density inside the mass shell. For pro-genitors with little mass outside the core, we have

M ≈2m

tinfall

ρ

ρ=

8ρ3

√3Gmr3. (3)

Using m = 1.4M and assuming that M needs to drop at leastto Mcrit = 0.05M s−1 within 0.5 s after the onset of collapseto obtain ECSN-like explosion dynamics, one finds that thedensity needs to drop to

ρ .18

√3

GmMcritr−3/2 (4)

for a radius r < 2230 km.Figure 1 illustrates that the density gradient at the edge of

the core can be far less extreme than in the model of Nomoto(1984) to fulfil this criterion. ECSN-like explosion dynamicsis expected alike for the modern 8.8M ECSN progenitor ofJones et al. (2013) and low-mass iron cores (A. Heger, pri-vate communication) of 8.1M (with metallicity Z = 10−4)and 9.6M (Z=0), though the low-mass iron core progeni-tors are a somewhat marginal case.

2.1.3 Low-mass Iron Core ProgenitorsSimulations of these two low-mass iron progenitors with8.1M (Muller et al. 2012b) and 9.6M (Janka et al. 2012;Muller et al. 2013 in 2D; Melson et al. 2015a in 3D) nonethe-less demonstrated that the structure of these stars is suffi-ciently extreme to produce explosions reminiscent of ECSNmodels: Shock revival sets in early around 100 ms afterbounce, aided by the drop of the accretion rate associatedwith the infall of the thin O and C/O shells, and the explo-sion energy remains small (5 × 1049 . . . 1050 erg).

As shown by Melson et al. (2015a), there are importantdifferences to ECSNe, however: While shock revival alsooccurs in spherical symmetry, multi-dimensional effects sig-nificantly alter the explosion dynamics. In 1D, the shockpropagates very slowly through the C/O shell after shock re-vival, and only accelerates significantly after reaching the Heshell. Without the additional boost by convective overturn,the explosion energy is lower by a factor of ∼5 compared tothe multi-D case. Different from ECSNe, somewhat slowershock expansion provides time for the small-scale convec-tive plumes to merge into large structures as shown for the9.6M model of Janka et al. (2012) in Figure 2.

Figure 2. Entropy s (left half of plot) and electron fraction Ye (right half) inthe 9.6M explosion model of Janka et al. (2012) and Muller et al. (2013)280 ms after bounce. Large convective plumes push neutron-rich materialfrom close to the gain region out at high velocities.

Both for the 8.8M model of Wanajo et al. (2011) andthe low-mass iron-core explosion models, the dynamics ofthe Rayleigh-Taylor plumes developing after shock revival isnonetheless quite similar. The entropy of the rising plumes isroughly ∼15 . . . 20kb/nucleon compared to ∼10kb/nucleonin the ambient medium. For such an entropy contrast, bal-ance between buoyancy and drag forces applies a limitingvelocity of the order of the speed of sound. This limit ap-pears to be reached relatively quickly in the simulations.Apart from the very early growth phase, the plume veloci-ties should therefore not depend strongly on the initial seedperturbations; they are rather set by bulk parameters of thesystem, namely the post-shock entropy at a few hundredkilometres and the entropy close to the gain radius, whichtogether determine the entropy contrast of the plumes. Thiswill become relevant later in our discussion of the nucle-osynthesis of ECSN-like explosions.

2.2 Nucleosynthesis

2.2.1 1D Electron-Capture Supernovae Models – EarlyEjecta

Nucleosynthesis calculations based on modern, sphericallysymmetric ECSN models were first performed by Hoff-man et al. (2008) and Wanajo et al. (2009). The resultsof these calculations appeared to point to a severe con-flict with observational constraints, showing a strong over-production of N = 50 nuclei, in particular 90Zr, due to theejection of slightly neutron-rich material (electron fractionYe & 0.46) with relatively low entropy (s ≈ 18kb/nucleon)immediately after shock revival. Hoffman et al. (2008) in-ferred that such nucleosynthesis yields would only be com-

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6 B. Muller

0.40 0.45 0.50 0.55Ye

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ej

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0.00

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∆M

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Figure 3. Binned distribution of the electron fraction Ye in the early ejectafor different explosion models of a 9.6M star 270 ms after bounce. Theplots show the relative contribution ∆Mej/Mej to the total mass of (shocked)ejecta in bins with ∆Ye = 0.01. The upper panel shows the Ye-distributionfor the 2D model of Janka et al. (2012) computed using the Vertex-CoCoNuT code (Muller et al. 2010). The bottom panel illustrates the effectof stochastic variations and dimensionality using several 2D models (thinlines) and a 3D model computed with the CoCoNuT-FMT code Muller &Janka (2015) (thick lines). Note that the dispersion in Ye in the early ejectais similar for both codes, though the average Ye in the early ejecta is spu-riously low when less accurate neutrino transport is used (FMT instead ofVertex). The bottom panel is therefore only intended to show differentialeffects between different models, and is not a prediction of the absolute valueof Ye. It suggests that i) stochastic variations do not strongly affect the dis-tribution of Ye in the ejecta, and that ii) the resulting distribution of Ye in2D and 3D is relatively similar.

patible with chemogalactic evolution if ECSNe were rareevents occurring at a rate no larger than once per 3,000 years.

The low Ye-values in the early ejecta stem from the ejec-tion of matter at relatively high velocities in the wake of thefast-expanding shock. In slow outflows, neutrino absorptionon neutrons and protons drives Ye to an equilibrium valuethat is set by the electron neutrino and antineutrino lumi-nosities Lνe and Lνe , the “effective” mean energies3 ενe andενe , and the proton-neutron mass difference ∆ = 1.293 MeV

3ε is given in terms of the mean-square 〈E2〉 and the mean energy 〈E〉, asε = 〈E2〉/〈E〉. Tamborra et al. (2012) can be consulted for the ratio of thedifferent energy moments during various evolutionary phases.

as follows (Qian & Woosley 1996),

Ye ≈

[1 +

Lνe (ενe − 2∆)Lνe (ενe + 2∆)

]−1

. (5)

For the relatively similar electron neutrino and antineutrinoluminosities and a small difference in the mean energies of2 . . . 3 MeV in modern simulations, one typically finds anasymptotic value of Ye > 0.5, i.e. proton-rich conditions. Toobtain low Ye < 0.5 in the ejecta, neutrino absorption re-actions need to freeze out at a high density (small radius)when the equilibrium between the reactions n(νe, e−)p andp(νe, e+)n is still skewed towards low Ye due to electroncaptures p(e−, νe)n on protons. Neglecting the difference be-tween arithmetic, quadratic, and cubic neutrino mean ener-gies and assuming a roughly equal contribution of n(νe, e−)pand p(νe, e+)n to the neutrino heating, one can estimatethat freeze-out roughly occurs when (cp. Eq. 81 in Qian &Woosley 1996),

vr

r≈

2mN qνEνe + Eνe

, (6)

where mN is the nucleon mass, qν is the mass-specific neu-trino heating rate, r is the radius and vr is the radial veloc-ity. Since qν ∝ r−2, freeze-out will occur at smaller r, higherdensity, and smaller Ye for higher ejection velocity.

2.2.2 Multi-D Effects and the Composition of the EarlyEjecta

Since high ejection velocities translate into lower Ye, theRayleigh-Taylor plumes in 2D simulations of ECSNe (Fig-ure 2 in Wanajo et al. 2011) and explosions of low-mass ironcores (Figure 2) contain material with even lower Ye thanfound in 1D ECSN models. Values of Ye as low as 0.404 arefound in Wanajo et al. (2011).

Surprisingly, Wanajo et al. (2011) found that the neutron-rich plumes did not aggravate the problematic overproduc-tion of N = 50 nuclei in their 2D ECSN model. This is due tothe fact that the entropy in the neutron-rich lumps is actuallysmaller than in 1D4 (but higher than in the ambient medium),which changes the character of the nucleosynthesis by re-ducing the α-fraction at freeze-out from nuclear statisticalequilibrium (NSE). The result is an interesting productionof trans-iron elements between Zn and Zr for the progenitorof Nomoto (1984, 1987); the production factors are consis-tent with current rate estimates for ECSNe of about 4% ofall supernovae (Poelarends et al. 2008). Subsequent studiesshowed that neutron-rich lumps in the early ejecta of ECSNecould contribute a sizeable fraction to the live 60Fe in theGalaxy (Wanajo et al. 2013b), and might be production sitesfor some other rare isotopes of obscure origin, such as 48Ca(Wanajo et al. 2013a). Due to the similar explosion dynam-ics, low-mass iron-core progenitors exhibit rather similar nu-

4The dynamical reasons for this difference between 1D and multi-D mod-els have yet to be investigated. Conceivably shorter exposure to neutrinoheating in 2D due to faster expansion (which is responsible for the lowerYe) also decreases the final entropy of the ejecta.

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cleosynthesis (Wanajo et al., in preparation; Harris et al., inpreparation). The results of these nucleosynthesis calcula-tions tallies with the observed abundance trends in metal-poor stars that suggest a separate origin of elements like Sr,Y, and Zr from the heavy r-process elements (light elementprimary process; Travaglio et al. 2004; Wanajo & Ishimaru2006; Qian & Wasserburg 2008; Arcones & Montes 2011;Hansen et al. 2012; Ting et al. 2012).

Since Ye in the early ejecta of ECSNe and ECSN-like ex-plosion is sensitive to the neutrino luminosities and mean en-ergies and to the ejection velocity of the convective plumes(which may be different in 3D compared to 2D, or exhibitstochastic variations), Wanajo et al. (2011) also explored theeffect of potential uncertainties in the minimum Ye in theejecta on the nucleosynthesis. They found that a somewhatlower Ye of ∼ 0.3 in the plumes might make ECSNe a sitefor a “weak r-process” that could explain the enhanced abun-dances of lighter r-process elements up to Ag and Pd in somemetal-poor halo stars (Wanajo & Ishimaru 2006; Honda et al.2006).

Whether the neutron-rich conditions required for a weakr-process can be achieved in ECSNe or low-mass iron-coresupernovae remains to be determined. Figure 3 provides atentative glimpse on the effects of stochasticity and dimen-sionality on the Ye in neutron-rich plumes based on several2D and 3D explosion models of a 9.6M low-mass iron coreprogenitor (A. Heger, private communication) conducted us-ing the FMT transport scheme of Muller & Janka (2015).5

Stochastic variations in 2D models due to different (random)initial perturbations shift the minimum Ye in the ejecta atmost by 0.02. This is due to the fact that the Rayleigh-Taylorplumes rapidly transition from the initial growth phase to astage where buoyancy and drag balance each other and de-termine the velocity (Alon et al. 1995). 3D effects do notchange the distribution of Ye tremendously either, at bestthey tend to shift it to slightly higher values compared to2D, which is consistent with a somewhat stronger brakingof expanding bubbles in 3D as a result of the forward turbu-lent cascade (Melson et al. 2015a). It thus appears unlikelythat the dynamics of convective overturn is a major sourceof uncertainty for the nucleosynthesis in ECSN-like explo-sions, though confirmation with better neutrino transport isstill needed.

If these events are indeed sites of a weak r-process, themissing ingredient is likely to be found elsewhere. Improve-ments in the neutrino opacities, such as the proper inclusionof nucleon potentials in the charged-current interaction rates(Martınez-Pinedo et al. 2012; Roberts et al. 2012), or flavouroscillations involving sterile neutrinos (Wu et al. 2014) couldlower Ye somewhat. Wu et al. (2014) found a significant re-duction of Ye by up to 0.15 in some of the ejecta, but these

5The FMT neutrino transport scheme cannot be relied upon for precise pre-dictions of the value of Ye, but should be sufficiently accurate for exploringdifferential effects such as differences between plume expansion in 2D and3D.

results may depend sensitively on the assumption that collec-tive flavour oscillations are still suppressed during the phasein question. Moreover, Wu et al. (2014) pointed out that areduction of Ye with the help of active-sterile flavour conver-sion might require delicate fine-tuning to avoid shutting off

neutrino heating before the onset of the explosion due to thedisappearance of νe’s (which could be fatal to the explosionmechanism).

Moreover, whether ECSNe necessarily need to co-produce Ag and Pd with Sr, Y, and Zr is by no means clear.While observed abundance trends may suggest such a co-production, the abundance patterns of elements between Srand Ag in metal-poor stars appear less robust (Hansen et al.2014); and the failure of unaltered models to produce Ag andPd may not be indicative of a severe tension with observa-tions.

2.2.3 Other Nucleosynthesis Scenarios forElectron-Capture Supernovae

There are at least two other potentially interesting sites fornucleosynthesis in ECSN-like supernovae. For “classical”ECSN-progenitors with more extreme density profiles, ithas been proposed that the rapid acceleration of the shockin the steep density gradient outside the core can lead tosufficiently high post-shock entropies (s ∼ 100 kb/nucleon)and short expansion time-scales (τexp ∼ 10−4 s) to allow r-process nucleosynthesis in the thin shells outside the core(Ning et al. 2007). This has not been borne out by numeri-cal simulations, however (Janka et al. 2008; Hoffman et al.2008). When the requisite high entropy is reached, the post-shock temperature has already dropped far too low to dis-sociate nuclei, and the expansion time-scale does not be-come sufficiently short for the scenario of Ning et al. (2007)to work. The proposed r-process in the rapidly expandingshocked shells would require significantly different explo-sion dynamics, e.g. a much higher explosion energy.

The neutrino-driven wind that is launched after accre-tion onto the proto-neutron star has been completely sub-sided has long been discussed as a potential site of r-processnucleosynthesis in supernovae (Woosley et al. 1994; Taka-hashi et al. 1994; Qian & Woosley 1996; Cardall & Fuller1997; Thompson et al. 2001; Arcones et al. 2007; Arcones& Thielemann 2013). ECSN-like explosions are in manyrespects the least favourable site for an r-process in theneutrino driven wind since they produce low-mass neutronstars, which implies low wind entropies and long expan-sion time-scales (Qian & Woosley 1996), i.e. conditions thatare detrimental to r-process nucleosynthesis. However, EC-SNe are unique inasmuch as the neutrino-driven wind can becalculated self-consistently with Boltzmann neutrino trans-port (Hudepohl et al. 2009; Fischer et al. 2010) without theneed to trigger an explosion artificially. These simulationsrevealed a neutrino-driven wind that is not only of moder-ate entropy (s . 140kb/nucleon even at late times), but alsobecomes increasingly proton-rich with time, in which casethe νp-process (Frohlich et al. 2006) could potentially oper-

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8 B. Muller

ate. The most rigorous nucleosynthesis calculations for theneutrino-driven wind in ECSNe so far (Pllumbi et al. 2015)are based on simulations that properly account for nucleoninteraction potentials in the neutrino opacities (Martınez-Pinedo et al. 2012; Roberts et al. 2012) and have also ex-plored the effects of collective flavour oscillations, active-sterile flavour conversion. Pllumbi et al. (2015) suggest thatwind nucleosynthesis in ECSNe is rather mundane: Nei-ther does the νp-process operate nor can neutron-rich con-ditions be restored to obtain conditions even for a weak r-process. Instead, they find that wind nucleosynthesis mainlyproduces nuclei between Sc and Zn, but the production fac-tors are low, implying that the role of neutrino-driven windsin ECSNe is negligible for this mass range for the purposeof chemogalactic evolution.

2.3 Electron-Capture Supernovae – Transients andRemnants

Although the explosion mechanism of ECSNe is in manyrespects best understood among all core-collapse supernovatypes from the viewpoint of explosion mechanism, unam-biguously identifying transients as ECSNe has proved moredifficult. It has long been proposed that SN 1054 was anECSN (Nomoto et al. 1982) based on the properties of itsremnant, the Crab nebula: The total mass of ejecta in the neb-ula is small (. 5M; Davidson & Fesen 1985; MacAlpine &Uomoto 1991; Fesen et al. 1997), as is the oxygen abundance(Davidson et al. 1982; Henry & MacAlpine 1982; Henry1986), which is in line with the thin O-rich shells in ECSNprogenitors. Moreover, the kinetic energy of the ejecta isonly about . 1050 erg (Fesen et al. 1997; Hester 2008) as ex-pected for an ECSN-like event. Whether the Crab originatesfrom a classical ECSN or from something slightly differentlike a “failed massive star” of Jones et al. (2013) continues tobe debated; MacAlpine & Satterfield (2008) have argued, forexample, against the former interpretation based on a highabundance ratio of C vs. N and the detection of some ashesof oxygen burning (S, Ar) in the nebula.

It has been recognised in recent years that the (recon-structed) light curve of SN 1054 – a type IIP supernova witha relatively bright plateau – is also compatible with the lowexplosion energy of . 1050 erg predicted by recent numeri-cal simulations. Smith (2013) interpreted the bright plateau,which made SN 1054 visible by daytime for ∼3 weeks, asthe result of interaction with circumstellar medium (CSM).The scenario of Smith (2013) requires significant mass loss(0.1M for about 30 years) shortly before the supernova,which may be difficult to achieve, although some channelstowards ECSN-like explosions could involve dramatic massloss events (Woosley & Heger 2015b). Subsequent numeri-cal calculations of ECSN light curves (Tominaga et al. 2013;Moriya et al. 2014) demonstrated, however, that less extremeassumptions for the mass loss are required to explain the op-tical signal of SN 1054; indeed a very extended hydrogenenvelope may be sufficient to explain the bright plateau, and

CSM interaction with the progenitor wind may only be re-quired to prevent the SN from fading too rapidly.

Several other transients have also been interpreted as EC-SNe, e.g. faint type IIP supernovae such as SN 2008S (Bot-ticella et al. 2009). Smith (2013) posits that ECSNe are ob-served type IIn-P supernovae with circumstellar interactionlike SN 1994W with a bright plateau and a relatively sharpdrop to a faint nickel-powered tail, but again the requiredamount of CSM is not easy to explain. All of these candidateevents share low kinetic energies and small nickel masses asa common feature and are thus prima facie compatible withECSN-like explosion dynamics. Variations in the envelopestructure of ECSN-progenitors (e.g. envelope stripping in bi-naries) may account for the very different optical signatures(Moriya et al. 2014).

The peculiar nucleosynthesis in ECSNe-like explosionsmay also leave observable fingerprints in the electromag-netic signatures. The slightly neutron-rich character of theearly ejecta results in a strongly supersolar abundance ra-tio of Ni to Fe after β-decays are completed (Wanajo et al.2011). Such high Ni/Fe ratios are seen in the nebular spec-tra of some supernovae (Jerkstrand et al. 2015a,b). ECSNecan only explain some of these events, however; many ofthem exhibit explosion energies and Nickel masses that areincompatible with an ECSN.

3 3D SUPERNOVA MODELS OF MASSIVEPROGENITORS

In more massive progenitors with extended Si and O shells,the mass accretion rate onto the shock does not drop asrapidly as in ECSN-like explosions. Typically, one finds arelatively stable accretion rate of a few 0.1M s−1 during theinfall of the O shell, which implies a high ram pressure aheadof the shock. Under these conditions, it is no longer trivialto demonstrate that neutrino heating can pump a sufficientamount of energy into the post-shock region to power run-away shock expansion. 1D simulations of the post-bouncephase using Boltzmann solvers for the neutrino transportconvincingly demonstrated that neutrino-driven explosionscannot be obtained under such conditions in spherical sym-metry (Liebendorfer et al. 2001; Rampp & Janka 2000; Bur-rows et al. 2000a). Much of the work of recent years hastherefore focused on better understanding and accuratelymodelling how multi-dimensional effects in supernovae fa-cilitate neutrino-driven explosions – an undertaking first be-gun in the 1990s with axisymmetric (2D) simulations em-ploying various approximations for neutrino heating andcooling (Herant et al. 1992; Yamada et al. 1993; Herant et al.1994; Burrows et al. 1995; Janka & Muller 1995, 1996). 2Dsimulations have by now matured to the point that multi-group neutrino transport and the neutrino-matter interactionscan be modelled with the same rigour as in spherical symme-try (Livne et al. 2004; Buras et al. 2006a; Muller et al. 2010;Bruenn et al. 2013; Just et al. 2015; Skinner et al. 2015), orwith, still with acceptable accuracy for many purposes (see

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Section 4 for a more careful discussion), by using some ap-proximations either in the transport treatment or the neutrinomicrophysics (Suwa et al. 2010; Muller & Janka 2015; Panet al. 2016; O’Connor & Couch 2015; Roberts et al. 2016).

3.1 Prelude – First-principle 2D Models

The current generation of 2D supernova simulations withmulti-group neutrino transport has gone a long way towardsdemonstrating that neutrino heating can bring about explo-sion in conjunction with convection or the SASI. Thanks tosteadily growing computational resources, the range of suc-cessful neutrino-driven explosion models has grown fromabout a handful in mid-2012 (Buras et al. 2006b; Marek& Janka 2009; Suwa et al. 2010; Muller et al. 2012a) to ahuge sample of explosion models with ZAMS masses be-tween 10M and 75M, different metallicities, and differ-ent choices for the supranuclear equation of state (Mulleret al. 2012b; Janka et al. 2012; Suwa et al. 2013; Bruennet al. 2013; Obergaulinger et al. 2014; Nakamura et al. 2015;Muller 2015; Bruenn et al. 2016; O’Connor & Couch 2015;Summa et al. 2016; Pan et al. 2016).

Many of the findings from these simulations remain im-portant and valid after the advent of 3D modelling: The 2Dmodels have established, among other things, the existenceof distinct SASI- and convection-dominated regimes in theaccretion phase, both of which can lead to successful explo-sion (Muller et al. 2012b) in agreement with tunable, pa-rameterised models (Scheck et al. 2008; Fernandez et al.2014). They have shown that “softer” nuclear equations ofstate that result in more compact neutron stars are gener-ally favourable for shock revival (Janka 2012; Suwa et al.2013; Couch 2013a). The inclusion of general relativistic ef-fects, whether by means of the conformally-flat approxima-tion (CFC) or, less rigorously, an effective pseudo-relativisticpotential for Newtonian hydrodynamics, was found to havea similarly beneficial effect (CFC: Muller et al. 2012a;pseudo-Newtonian: O’Connor & Couch 2015). Moreover,there are signs that the 2D models of some groups convergewith each other; simulations of four different stellar models(12, 15, 20, 25M) of Woosley & Heger (2007) by Summaet al. (2016) and O’Connor & Couch (2015) have yieldedquantitatively similar results.

Despite these successes, 2D models have, by and large,struggled to reproduce the typical explosion properties ofsupernovae. They are often characterised by a slow and un-steady growth of the explosion energy after shock revival.Usually the growth of the explosion energy cannot be fol-lowed beyond 2 . . . 4 × 1050 erg after simulating up to ∼1 sof physical time (Janka et al. 2012; Nakamura et al. 2015;O’Connor & Couch 2015), i.e. below typical observed val-ues of 5 . . . 9 × 1050 erg (Kasen & Woosley 2009; Pejcha &Prieto 2015). Only the models of Bruenn et al. (2016) reachsignificantly higher explosion energies. While the explosionenergy often has not levelled out yet at the end of the sim-ulations and may still grow significantly for several seconds

(Muller 2015), its continuing growth comes at the expense oflong-lasting accretion onto the proto-neutron star. This mayresult in inordinately high remnant masses. Thus, while 2Dmodels appeared to have solved the problem of shock re-vival, they faced an energy problem instead.

3.2 Status of 3D Core-Collapse Supernova Models

Before 3D modelling began in earnest (leaving aside tenta-tive sallies into 3D by Fryer & Warren 2002), it was hopedthat 3D effects might facilitate shock revival even at ear-lier times than in 2D, and that this might then also pro-vide a solution to the energy problem, since more energycan be pumped into the neutrino-heated ejecta at early timeswhen the mass in the gain region is larger. These hopeswere already disappointed once several groups investigatedthe role of 3D effects in the explosion mechanism using asimple “light-bulb” approach, where the neutrino luminosityand mean energy during the accretion phase are prescribedand very simple approximations for the neutrino heatingand cooling terms are employed. Although Nordhaus et al.(2010) initially claimed a significant reduction of the criti-cal neutrino luminosity for shock revival in 3D compared to2D based on such an approach, these results were affected bythe gravity treatment (Burrows et al. 2012) and have not beenconfirmed by subsequent studies. Similar parameterised sim-ulations have shown that the critical luminosity in 3D isroughly equal to 2D (Hanke et al. 2012; Couch 2013b; Bur-rows et al. 2012; Dolence et al. 2013) and about 20% lowerthan in 1D, though the results differ about the hierarchy be-tween 2D and 3D.

Subsequent supernova models based on multi-group neu-trino transport yielded even more unambiguous results:Shock revival in 3D was either not achieved for progeni-tors that explode in 2D (Hanke et al. 2013; Tamborra et al.2014b), or was delayed significantly (Takiwaki et al. 2014;Melson et al. 2015b; Lentz et al. 2015). These first disap-pointing results need to be interpreted carefully, however:A detailed analysis of the heating conditions in the non-exploding 3D models of 11.2M, 20M, and 27M progen-itors simulated by the Garching supernova group revealedthat these are very close to shock revival (Hanke et al. 2013;Hanke 2014; Melson et al. 2015b). Moreover, the 3D modelsof the Garching group are characterised by more optimisticheating conditions, larger average shock radii, and higher ki-netic energies in non-spherical motions compared to 2D forextended periods of time; the same is true for the delayed(compared to 2D) 3D explosion of Lentz et al. (2015) of a15M progenitor. It is merely when is comes to sustainingshock expansion that the 3D models prove less resilient thantheir 2D counterparts, which transition into an explosive run-away more robustly.

The conclusion that 3D models are only slightly less proneto explosion is reinforced by the emergence of the firstsuccessful simulations of shock revival in progenitors with20M (Melson et al. 2015b) and 15M (Lentz et al. 2015)

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10 B. Muller

using rigorous multi-group neutrino transport and the bestavailable neutrino interaction rates. There is also a num-ber of 3D explosion models based on more simplified ap-proaches to multi-group neutrino transport (Takiwaki et al.2012, 2014; Muller 2015; Roberts et al. 2016).

3.3 How Do Multi-D Effects Facilitate ShockRevival?

Despite these encouraging developments, several questionsnow need to be addressed to make further progress: Whatis the key to robust 3D explosion models across the entireprogenitor mass range for which we observe explosions (i.e.at least up to 15 . . . 18M; see Smartt et al. 2009 and Smartt2015)? This question is tightly connected to another, morefundamental one, namely: What are the conditions for an ex-plosive runaway, and how do multi-dimensional effects mod-ify them?

3.3.1 Conditions for Runaway Shock ExpansionEven without the complications of multi-D fluid flow, thephysics of shock revival is subtle. In spherical symmetry,one can show that for a given mass accretion rate M, thereis a maximum (“critical”) electron-flavour luminosity Lν atthe neutrinosphere above which stationary accretion flowonto the proto-neutron star is no longer possible (Burrows &Goshy 1993; cp. Section 2). This also holds true if the con-tribution of the accretion luminosity due to cooling outsidethe neutrinosphere is taken into account (Pejcha & Thomp-son 2012). The limit for the existence of stationary solu-tions does not perfectly coincide with the onset of runawayshock expansion, however. Using 1D light-bulb simulations(i.e. neglecting the contribution of the accretion luminos-ity), Fernandez (2012) and Gabay et al. (2015) showed thatthe accretion flow becomes unstable to oscillatory and non-oscillatory instability slightly below the limit of Burrows &Goshy (1993). Moreover, it is unclear whether the negativefeedback of shock expansion on the accretion luminosity andhence on the neutrino heating could push models into a limitcycle (cp. Figure 28 of Buras et al. 2006a) even above thethreshold for non-stationarity.

Since an a priori prediction of the critical luminosity,Lν(M) is not feasible, heuristic criteria have been devel-oped (Janka & Keil 1998; Janka et al. 2001; Thompson2000; Thompson et al. 2005; Buras et al. 2006b; Murphy &Burrows 2008; Pejcha & Thompson 2012; Fernandez 2012;Gabay et al. 2015; Murphy & Dolence 2015) to gauge theproximity of numerical supernova models to an explosiverunaway (rather than for pinpointing the formal onset of therunaway after the fact, which is of less interest). The mostcommonly used criticality parameters are based on the ra-tio of two relevant time-scales for the gain region (Janka &Keil 1998; Janka et al. 2001; Thompson 2000; Thompsonet al. 2005; Buras et al. 2006b; Murphy & Burrows 2008),namely the advection or dwell time τadv that accreted mate-rial spends in the gain region, and the heating time-scale τheat

over which neutrino energy deposition changes the total orinternal energy of the gain region appreciably. If τadv > τheat,neutrino heating can equalise the net binding energy of theaccreted material before it is lost from the gain region, andone expects that the shock must expand significantly due tothe concomitant increase in pressure. Since this expansionfurther increases τadv, an explosive runaway is likely to en-sue.

The time-scale criterion τadv/τheat > 1 has the virtue of be-ing easy to evaluate since the two time-scales can be definedin terms of global quantities such as the total energy Etot,g inthe gain region, the volume-integrated neutrino heating rateQν, and the mass Mg in the gain region (which can be usedto define τadv = Mgain/M under steady-state conditions). Thesignificance of these global quantities for the problem ofshock revival is immediately intuitive, though care must betaken to define the heating time-scale properly. Thompson(2000), Thompson et al. (2005), Murphy & Burrows (2008),and Pejcha & Thompson (2012) define τheat as the time-scalefor changes in the internal energy Eint in the gain region,

τheat =Eint

Qν

, (7)

based on the premise that shock expansion is regulated bythe increase in pressure (and hence in internal energy). Thisdefinition yields unsatisfactory results, however. The criti-cality parameter can be spuriously low at shock revival ifthis definition is used (τadv/τheat < 0.4).

By defining τheat in terms of the total (inter-nal+kinetic+potential) energy6 of the gain region (Buraset al. 2006b),

τheat =Etot,g

Qν

, (8)

the criterion τadv/τheat > 1 becomes a very accurate predic-tor for non-oscillatory instability (Fernandez 2012; Gabayet al. 2015). This indicates that the relevant energy scale towhich the quasi-hydrostatic stratification of the post-shockregion is the total energy (or perhaps the total or stagnationenthalpy) of the gain region, and not the internal energy. Thisis consistent with the observation that runaway shock expan-sion occurs roughly once the total energy or the Bernoulliintegral (Fernandez 2012; Burrows et al. 1995) reach pos-itive values somewhere (not everywhere) in the post-shockregion, which is essentially what the time-scale criterion es-timates. What is crucial is that the density and pressure gra-dients between the gain radius and the shock (and hencethe shock position) depends sensitively on the ratio of en-thalpy h (or the internal energy) and the gravitational po-tential, rather than on enthalpy alone. Under the (justified)assumption that quadratic terms in v2

r in the momentum andenergy equation are sufficiently small to be neglected in thepost-shock region, one can show (see Appendix A) that the

6Note that rest-mass contributions to the internal energy are excluded in thisdefinition.

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logarithmic derivative of the density ρ in the gain region isconstrained by

∂ ln ρ∂ ln r

> −3GM

rh, (9)

where M is the proto-neutron star mass. Once h > GM/r oreven eint > GM/r (where eint is the internal energy per unitmass), significant shock expansion must ensue due to theflattening of pressure and density gradients.

Janka (2012), Muller & Janka (2015) and Summa et al.(2016) have also pointed out that the time-scale criterioncan be converted into a scaling law for the critical electron-flavour luminosity Lν and mean energy Eν in terms of theproto-neutron star mass M, the accretion rate M, and the gainradius rg,

(LνE2ν )crit ∝ (MM)3/5r−2/5

g . (10)

The concept of the critical luminosity, the time-scale crite-rion, and the condition of positive total energy or a positiveBernoulli parameter at the gain radius are thus intimately re-lated and appear virtually interchangeable considering thatthey remain approximate criteria for runaway shock expan-sion anyway. This is also true for some other explosion cri-teria that have been proposed, e.g. the antesonic conditionof Pejcha & Thompson (2012), which states that the soundspeed cs must exceed a certain fraction of the escape ve-locity vesc for runaway shock expansion somewhere in theaccretion flow,

c2s > 3/16v2

esc. (11)

Approximating the equation of state as a radiation-dominated gas with an adiabatic index γ = 4/3 and a pres-sure of P = ρeint/3 = ρh/4, one finds that the antesonic con-dition roughly translates to

c2s

3/16v2esc

=4/3P/ρ

3/8GM/r=

32eint

27GM/r=

8h9GM/r

> 1, (12)

i.e. the internal energy and the enthalpy must be close tothe gravitational binding energy (even if the precise criticalvalues for eint and h may shift a bit for a realistic equation ofstate).7

3.3.2 Impact of Multi-D Effects on the HeatingConditions

Why do multi-D effects bring models closer to shock revival,and how is this reflected in the aforementioned explosioncriteria? Do these explosion criteria even remain applicablein multi-D in the first place?

7This argument holds only for stationary 1D flow, however. In multi-D, theantesonic condition becomes sensitive to fluctuations in the sound speed,which limits its usefulness as diagnostic for the proximity to explosion.The fluctuations will be of order δcs/cs ∼ δρ/ρ, i.e. of the order of thesquare of the turbulent Mach number. This explains why high values ofc2

s /v2esc are encountered in multi-D even in non-exploding models (Muller

et al. 2012a). A similar problem occurs if the shock starts to oscillatestrongly in 1D close to the runaway threshold.

The canonical interpretation has long been that the run-away condition τadv > τheat remains the decisive criterionin multi-D, and that multi-D effects facilitate shock revivalmainly by increasing the advection time-scale τadv (Buraset al. 2006b; Murphy & Burrows 2008). Especially close tocriticality, τheat is also shortened due to feedback processes– better heating conditions imply that the net binding energyin the gain region and hence τheat must decrease.

While simulations clearly show increased advection time-scales in multi-D compared to 1D (Buras et al. 2006b; Mur-phy & Burrows 2008; Hanke et al. 2012) as a result of largershock radii, the underlying cause for larger accretion shockradii in multi-D is more difficult to pinpoint. Ever since thefirst 2D simulations, both the transport of neutrino-heatedhigh-entropy material from the gain radius out to the shock(Herant et al. 1994; Janka & Muller 1996) as well as the“turbulent pressure” of convective bubbles colliding with theshock (Burrows et al. 1995) have been invoked to explainlarger shock radii in multi-D. Both effects are plausible sincethey change the components P (thermal pressure) and ρv ⊗ v(where v is the velocity) of the momentum stress tensor thatmust balance the ram pressure upstream of the shock duringstationary accretion.

That the turbulent pressure plays an important role followsalready from the high turbulent Mach number ∼ 0.5 in thepost-shock region (Burrows et al. 1995; Muller et al. 2012b)before the onset of shock revival, and has been demonstratedquantitatively by Murphy et al. (2013) and Couch & Ott(2015) using spherical Reynolds decomposition to analyseparameterised 2D and 3D simulations. Using a simple es-timate for the shock expansion due to turbulent pressure,Muller & Janka (2015) were even able to derive the reduc-tion of the critical heating functional in multi-D compared to1D in terms of the average squared turbulent Mach number〈Ma2〉 in the gain region,

(LνE2ν )crit,2D ≈ (LνE2

ν )crit,1D

(1 +

4〈Ma2〉

3

)−3/5

(13)

∝ (MM)3/5r−2/5g

(1 +

4〈Ma2〉

3

)−3/5

,

and then obtained (LνE2ν )crit,2D ≈ 0.75(LνE2

ν )crit,1D in roughagreement with simulations using a model for the saturationof non-radial fluid motions (see Section 3.3.3).

Nonetheless, there is likely no monocausal explanation forbetter heating conditions in multi-D. Yamasaki & Yamada(2006) found, for example, that convective energy transportfrom the gain radius to the shock also reduces the criticalluminosity (although they somewhat overestimated the ef-fect by assuming constant entropy in the entire gain region).Convective energy transport reduces the slope of the pres-sure gradient between the gain radius (where the pressureis set by the neutrino luminosity and mean energy) and theshock, and thus pushes the shock out by increasing the ther-mal post-shock pressure. That this effect also plays a rolealongside the turbulent pressure can be substantiated by an

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12 B. Muller

analysis of neutrino hydrodynamics simulations (Bollig etal. in preparation).

Only a detailed analysis of the properties of turbulence inthe gain region (Murphy & Meakin 2011) combined with amodel for the interaction of turbulence with a non-sphericalaccretion shock will reveal the precise combination of multi-D effects that conspire to increase the shock radius comparedto 1D. This is no prerequisite for understanding the impactof multi-D effects on the runaway condition as encapsulatedby a phenomenological correction factor in Equation (13),since effects like turbulent energy transport, turbulent bulkviscosity, etc. will also scale with the square of the turbulentMach number in the post-shock region just like the turbulentpressure. They are effectively lumped together in the correc-tion factor (1 + 4/3〈Ma2〉)−3/5. The turbulent Mach numberin the post-shock region is thus the crucial parameter forthe reduction of the critical luminosity in multi-D, althoughthe coefficient of 〈Ma2〉 still needs to be calibrated againstmulti-D simulations (and may be different in 2D and 3D).

This does not imply, however, that the energetic require-ments for runaway shock expansion in multi-D are funda-mentally different from 1D: Runaway still occurs roughlyonce some material in the gain region first acquires posi-tive total (internal+kinetic+potential) energy etot; and the re-quired energy input for this ultimately stems from neutrinoheating.8

3.3.3 Saturation of InstabilitiesWhat complicates the role of multi-D effects in the neutrino-driven mechanism is that the turbulent Mach number in thegain region itself depends on the heating conditions, whichmodify the growth rates and saturation properties of convec-tion and the SASI. Considerable progress has been made inrecent years in understanding this feedback mechanism andthe saturation properties of these two instabilities.

The linear phases of convection and the SASI are nowrather well understood. The growth rates for buoyancy-driven convective instability are expected to be of order ofthe Brunt-Vaisala frequency ωBV, which can be expressedin terms of P, ρ, cs, and the local gravitational acceleration gas9

ω2BV = g

(1ρ

∂ρ

∂r−

1ρc2

s

∂P∂r

), (14)

8This is not at odds with the findings of Murphy & Burrows (2008) andCouch & Ott (2015), who noticed that the neutrino heating rate in light-bulb and leakage-based multi-D simulations at runaway is smaller than in1D. Due to a considerably different pressure and density stratification (cf.Figure 3 in Couch & Ott 2015, which shows a very steep pressure gradientbehind the shock in the critical 1D model), the gain region needs to becomemuch more massive in 1D than in multi-D before the runaway conditionτadv/τheat > 1 is met. Therefore both the neutrino heating rate Qν and thebinding energy Etot of the gain region are higher around shock revival in1D (as both scale with Mgain).

9Note that different sign conventions for ωBV are used in the literature; hereω2

BV > 0 corresponds to instability.

0.0 0.1 0.2 0.3 0.4 0.5 0.6

time after bounce [s]

0.0

0.5

1.0

1.5

2.0

δv[c

ms−

1]

×109

δv

0.7× [qν(rsh,min − rg)]1/3

(rsh,max − rg)/τadv

Figure 4. Comparison of the root-mean-square average δv of non-radialvelocity component in the gain region (black) with two phenomenologicalmodels for the saturation of non-radial instabilities in a SASI-dominated3D model of an 18M star using the CoCoNuT-FMT code. The red curveshows an estimate based on Equation (18), which rests on the assumption ofa balance between buoyant driving and turbulent dissipation (Murphy et al.2013; Muller & Janka 2015). The blue curve shows the prediction of Equa-tion (20), which assumes that saturation is regulated by a balance betweenthe growth rate of the SASI and parasitic Kelvin-Helmholtz instabilities(Guilet et al. 2010). Even though Equation (20) assumes a constant qualityfactor |Q| to estimate the SASI growth rate, it appears to provide a goodestimate for the dynamics of the model. Interestingly, the saturation modelsfor the SASI- and convection dominated regimes give similar results dur-ing later phases even though the mechanism behind the driving instabilityis completely different.

which becomes positive in the gain region due to neutrinoheating. A first-order estimate yields,

ω2BV ∼

GMQν

4Mr2gc2

s

(rsh − rg

) ∼ 3Qν

4Mrg

(rsh − rg

) , (15)

using c2s ≈ GM/(3rg) at the gain radius (cp. Muller & Janka

2015). An important subtlety is that advection can stabilisethe flow so that ω2

BV > 0 is no longer sufficient for insta-bility unless large seed perturbations in density are alreadypresent. Instability instead depends on the more restrictivecriterion for the parameter χ (Foglizzo et al. 2006),

χ =

rsh∫rg

ωBV

|vr |dr, (16)

with χ & 3 indicating convective instability.The scaling of the linear growth rate ωSASI of SASI modes

is more complicated, since it involves both the duration τcycof the underlying advective-acoustic cycle as well as a qual-ity factor Q for the conversion of vorticity and entropy per-turbations into acoustic perturbation in the deceleration re-gion below the gain region and the reverse process at theshock (Foglizzo et al. 2006, 2007),

ωSASI ∼ln |Q|τcyc

. (17)

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For realistic models with strong SASI, one finds ln |Q| ∼ 2(Scheck et al. 2008; Muller et al. 2012b). SASI growthappears to be suppressed for χ & 3 probably because con-vection destroys the coherence of the waves involved inthe advective-acoustic cycle (Guilet et al. 2010). Interest-ingly, the demarcation line χ = 3 between the SASI- andconvection-dominated regimes is also valid in the non-linearregime if χ is computed from the angle- and time-averagedmean flow (Fernandez 2012); and both the SASI and convec-tion appear to drive χ close to this critical value (Fernandez2012).

Both in the SASI-dominated regime and the convection-dominated regime, large growth rates are observed in simu-lations. It only takes a few tens of milliseconds until the in-stabilities reach their saturation amplitudes. For this reason,the turbulent Mach number and the beneficial effect of multi-D effects on the heating conditions are typically more sen-sitive to the saturation mechanism than to initial conditions,so that the onset of shock revival is only subject to modeststochastic variations (Summa et al. 2016). Exceptions applywhen the heating conditions vary rapidly, e.g. due to the in-fall of a shell interface or extreme variations in shock radius(as in the light-bulb models of Cardall & Budiardja 2015),and the runaway condition is only narrowly met or missed(Melson et al. 2015a; Roberts et al. 2016).

The saturation properties of convection were clarified byMurphy et al. (2013), who determined that the volume-integrated neutrino heating rate Qν and the convective lumi-nosity Lconv in the gain region roughly balance each other.This can be understood as the result of a self-adjustmentprocess of the accretion flow, whereby a marginally sta-ble, quasi-stationary stratification with χ ≈ 3 is established(Fernandez 2012). Muller & Janka (2015) showed that thiscan be translated into a scaling law that relates the averagemass-specific neutrino heating rate qν in the gain region tothe root mean square average δv of non-radial velocity fluc-tuations,

δv ∼[qν(rsh − rg)

]1/3. (18)

That a similar scaling should apply in the SASI-dominated regime is not immediately intuitive. Muller &Janka (2015) in fact tested Equation (18) using a SASI-dominated 2D model and argued that self-adjustment ofthe flow to χ ≈ 3 will result in the same scaling law asfor convection-dominated models. However, models suggestthat a different mechanism may be at play in the SASI-dominated regime. Simulations are at least equally compat-ible with the mechanism proposed by Guilet et al. (2010),who suggested that saturation of the SASI is mediated byparasitic instabilities and occurs once the growth rate of theparasite equals the growth rate of the SASI: Assuming thatthe Kelvin-Helmholtz instability is the dominant parasite, asimple order-of-magnitude estimate for saturation can be ob-

tained by equating ωSASI and the average shear rate,

ωSASI ∼δvΛ

(19)

where Λ is the effective width of the shear layer. Kazeroniet al. (2016) find that the Kelvin-Helmholtz instability oper-ates primarily in directions where the shock radius is larger,which suggests Λ = rsh,max − rg. This results in a scaling lawthat relates the velocity fluctuations to the average radial ve-locity 〈vr〉 in the gain region,

δv ∼ ωSASIΛ ∼ln |Q|(rsh,max − rg)

τadv∼ ln |Q| |〈vr〉|, (20)

where we assumed τcyc ≈ τadv. The quality factor Q can inprinciple change significantly with time and between differ-ent models. Nonetheless, together with the assumption ofa roughly constant quality factor, Equation (20) appears tocapture the dynamics of the SASI in 3D quite well for a sim-ulation of an 18M progenitor with the CoCoNuT-FMT code(Muller & Janka 2015) as illustrated in Figure 4.

Equation (18) for the convection-dominated regime andEquation (20) apparently predict turbulent Mach numbers inthe same ballpark. This can be understood by expressing qνin terms of the accretion efficiency ηacc = Lν/(GMM/rg) andthe heating efficiency ηheat = Qν/Lν,

qν =Qν

Mg= ηheatηacc

GMMrgMg

= ηheatηaccGM

rgτadv(21)

= ηheatηaccGM

rshτadv

rsh

rgain.

If we neglect the ratio rsh/rg and approximate the averagepost-shock velocity as |〈vr〉| ≈ β

−1 √GM/rsh (where β is thecompression ratio in the shock), we obtain

qν ∼ ηheatηaccβ2|〈vr〉|

2

τadv, (22)

and hence

δv ∼ (ηheatηaccβ2)1/3|〈vr〉|. (23)

For plausible values (e.g. ηheat = 0.05 ηacc = 2, β = 10), onefinds δv ∼ 2|〈vr〉|, i.e. the turbulent Mach number at satura-tion is of the same order of magnitude in the convection-and SASI-dominated regimes (where at least ln |Q|∼2 can bereached).

Equations (18) and (20) remain order-of-magnitude es-timates; and either of the instabilities may be more effi-cient at pumping energy into non-radial turbulent motionsin the gain region, as suggested by the light-bulb models ofFernandez (2015) and Cardall & Budiardja (2015). Theseauthors find that the SASI can lower the critical luminos-ity in 3D considerably further than convection. Fernandez(2010) attributes this to the emergence of the spiral mode ofthe SASI (Blondin & Mezzacappa 2007; Fernandez 2010) in3D, which can store more non-radial kinetic energy than theSASI sloshing mode in 2D, but this has yet to be borne out

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14 B. Muller

by self-consistent neutrino hydrodynamics simulations (seeSection 3.4 for further discussion).

3.3.4 Why Do Models Explode More Easily in 2D Thanin 3D?

How can one explain the different behaviour of 2D and 3Dmodels in the light of our current understanding of the inter-play between neutrino heating, convection, and the SASI? Itseems fair to say that we can presently only offer a heuris-tic interpretation for the more pessimistic evolution of 3Dmodels.

The most glaring difference between 2D and 3D mod-els (especially in the convection-dominated regime) prior toshock revival lies in the typical scale of the turbulent struc-tures, which are smaller in 3D (Hanke et al. 2012; Couch2013b; Couch & Ott 2015), whereas the inverse turbulentcascade in 2D (Kraichnan 1967) artificially channels turbu-lent kinetic energy to large scales. This implies that the effec-tive dissipation length (or also the effective mixing length forenergy transport) are smaller in 3D, so that smaller dimen-sionless coefficients C appear in relations like Equation (18),

δv = C[qν(rsh − rg)

]1/3, (24)

and the turbulent Mach number will be smaller for a givenneutrino heating rate. Indeed, for the 18M model shown inFigure (4), we find

δv = 0.7[qν(rsh − rg)

]1/3(25)

in 3D rather than what Muller & Janka (2015) inferred from2D models (admittedly using a different progenitor),

δv =[qν(rsh − rg)

]1/3. (26)

Following the arguments of Muller & Janka (2015) to infer

the correction factor(1 +

4〈Ma2〉

3

)−3/5for multi-D effects in

Equation (13), one would then expect a considerably largercritical luminosity in 3D, i.e. (LνE2

ν )crit,3D ≈ 0.85(LνE2ν )crit,1D

instead of (LνE2ν )crit,2D ≈ 0.75(LνE2

ν )crit,1D in 2D.Such a large difference in the critical luminosity does not

tally with the findings of light-bulb models that show thatthe critical luminosities in 2D and 3D are still very close toeach other. This already indicates that more subtle effectsmay be at play in 3D that almost compensate the strongereffective dissipation of turbulent motions. The fact that sim-ulations typically show transient phases of stronger shockexpansion and more optimistic heating conditions in 3D thanin 2D (Hanke et al. 2012; Melson et al. 2015b) also points inthis direction.

Furthermore, light-bulb models (Handy et al. 2014) andmulti-group neutrino hydrodynamics simulations (Melsonet al. 2015a; Muller 2015) have demonstrated that favourable3D effects come into play after shock revival. These worksshowed that 3D effects can lead to a faster, more robustgrowth of the explosion energy provided that shock revivalcan be achieved in the first place.

The favourable 3D effects that are responsible for thismay already counterbalance the adverse effect of strongerdissipation in the pre-explosion phase to some extent: En-ergy leakage from the gain region by the excitation of g-modes is suppressed in 3D because the forward turbulentcascade (Melson et al. 2015a) and (at high Mach number)the more efficient growth of the Kelvin-Helmholtz instabil-ity (Muller 2015) brake the downflows before they penetratethe convectively stable cooling layer. Moreover, the non-linear growth of the Rayleigh-Taylor instability is faster forthree-dimensional plume-like structures than for 2D struc-tures with planar (Yabe et al. 1991; Hecht et al. 1995; Mari-nak et al. 1995) or toroidal geometry (as in the context ofRayleigh-Taylor mixing in the stellar envelope during theexplosion phase; Kane et al. 2000; Hammer et al. 2010),which might explain why 3D models initially respond morestrongly to sudden drops in the accretion rate at shell in-terfaces and exhibit better heating conditions than their 2Dcounterparts for brief periods. Finally, the difference in theeffective dissipation length in 3D and 2D that is reflected byEquations (25) and (26) may not be universal and depend,e.g., on the heating conditions or the χ-parameter; the resultsof Fernandez (2015) in fact demonstrate that under appropri-ate circumstances more energy can be stored in non-radialmotions in 3D than in 2D in the SASI-dominated regime.

3.4 Outlook: Classical Ideas for More RobustExplosions

The existence of several competing – favourable and un-favourable – effects in 3D first-principle models does notchange the fundamental fact that they remain more reluc-tant to explode than their 2D counterparts. This suggests thatsome important physical ingredient are still lacking in cur-rent simulations. Several avenues towards more robust ex-plosion models have recently been explored. Some of theproposed solutions have a longer pedigree and revisit ideas(rapid rotation in supernova cores, enhanced neutrino lumi-nosities) that have been investigated on and off in super-nova theory already before the advent of 3D simulations.The more “radical” solution of invoking strong seed per-turbations from convective shell burning to boost non-radialinstabilities in the post-shock region will be discussed sepa-rately in Section 5.

3.4.1 Rotation and BeyondNakamura et al. (2014) and Janka et al. (2016) pointed outthat rapid progenitor rotation can facilitate explosions in 3D.Janka et al. (2016) ascribed this partly to the reduction of thepre-shock infall velocity due to centrifugal forces, which de-creases the ram pressure ahead of the shock. Even more im-portantly, rotational support also decreases the net bindingenergy |etot| per unit mass in the gain region in their models.They derived an analytic correction factor for the critical lu-minosity in terms of the average specific angular momentum

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j in the infalling shells,

(LνE2ν )crit,rot ≈ (LνE2

ν )crit ×

(1 −

j2

2GMrsh

)3/5

. (27)

Assuming rapid rotation with j 1016 cm2 s−1, one can ob-tain a significant reduction of the critical luminosity by sev-eral 10% as Janka et al. (2016) tested in a simulation with amodified rotation profile.10 For very rapid rotation, other ex-plosion mechanisms also become feasible, such as the mag-netorotational mechanism (Akiyama et al. 2003; Burrowset al. 2007b; Winteler et al. 2012; Mosta et al. 2014), or ex-plosions driven by the low-T/W spiral instability (Takiwakiet al. 2016).

However, current stellar evolution models do not pre-dict the required rapid rotation rates for these scenarios forthe generic progenitors of type IIP supernovae. The typ-ical specific angular momentum at a mass coordinate ofm = 1.5M is only of the order of j ∼ 1015 cm2 s−1 in models(Heger et al. 2005) that include angular momentum transportby magnetic fields generated by the Tayler-Spruit dynamo(Spruit 2002), and asteroseismic measurements of core rota-tion in evolved low-mass stars suggest that the spin-downof the cores may be even more efficient (Cantiello et al.2014). For such slow rotation, centrifugal forces are negli-gible; Equation (27) suggests a change of the critical lumi-nosity on the per-mil level. Neither is rotation expected toaffect the character of neutrino-driven convection apprecia-bly because the angular velocity Ω in the gain region is toosmall. The Rossby number is well above unity,

Ro ∼|vr |

(rsh − rg)Ω∼

r2s

τadv j∼ 10, (28)

assuming typical values of τadv ∼ 10 ms and rsh ∼ 100 km.Magnetic field amplification by a small-scale dynamo or

the SASI (Endeve et al. 2010, 2012) could also help to fa-cilitate shock revival with magnetic fields acting as a sub-sidiary to neutrino heating but without directly powering theexplosion as in the magnetorototational mechanism. The 2Dsimulations of Obergaulinger et al. (2014) demonstrated thatmagnetic fields can help organise the flow into large-scalemodes and thereby allow earlier explosions, though the re-quired initial field strengths for this are higher (∼1012 G)than the typical values predicted by stellar evolution mod-els.

3.4.2 Higher Neutrino Luminosities and MeanEnergies?

Another possible solution for the problem of missing or de-layed explosions in 3D lies in increasing the electron flavourluminosity and mean energy. This is intuitive from Equa-tion (13), where a mere change of ∼5% in both Lν and Eν

10One should bear in mind, though, that rotation also decreases the neutrinoluminosity and mean neutrino energy because it leads to larger neutronstar radii (Marek & Janka 2009).

results in a net effect of 16%, which is almost on par withmulti-D effects.

The neutrino luminosity is directly sensitive to the neu-trino opacities, which necessitates precision modelling inorder to capture shock propagation and heating correctly(Lentz et al. 2012a,b; Muller et al. 2012a; see also Sec-tion 4), as well as to other physical ingredients of the core-collapse supernova problem that influence the contractionof the proto-neutron star, such as general relativity and thenuclear equation of state (Janka 2012; Muller et al. 2012a;Couch 2013a; Suwa et al. 2013; O’Connor & Couch 2015).Often such changes to the neutrino emission come withcounterbalancing side effects (Mazurek’s law); e.g. strongerneutron star contraction will result in higher neutrino lumi-nosities and mean energies, but will also result in a moretightly bound gain region, which necessitates stronger heat-ing to achieve shock revival.

That the lingering uncertainties in the microphysics maynonetheless hold the key to more robust explosions has longbeen recognised in the case of the equation of state. Mel-son et al. (2015b) pointed out that missing physics in ourtreatment of neutrino-matter interactions may equally wellbe an important part of the solution of the problem shockrevival. Exploring corrections to neutral-current scatteringcross section due to the “strangeness” of the nucleon, theyfound that changes in the neutrino cross section on the levelof a few ten percent were sufficient to tilt the balance infavour of explosion for a 20M progenitor. While Melsonet al. (2015b) deliberately assumed a larger value for thecontribution of strange quarks to the axial form factor of thenucleon than currently measured (Airapetian et al. 2007), thedeeper significance of their result is that Mazurek’s law cansometimes be circumvented so that modest changes in theneutrino opacities still exert an appreciable effect on super-nova dynamics. A re-investigation of the rates currently em-ployed in the best supernova models for the (more uncertain)neutrino interaction processes that depend strongly on in-medium effects (charged-current absorption/emission, neu-tral current scattering, Bremsstrahlung; Burrows & Sawyer1998, 1999; Reddy et al. 1999; Hannestad & Raffelt 1998)may thus be worthwhile (see Bartl et al. 2014; Rrapaj et al.2015; Shen & Reddy 2014 for some recent efforts).

4 ASSESSMENT OF SIMULATIONMETHODOLOGY

Considering what has been pointed out in Section 3 – thecrucial role of hydrodynamic instabilities and the delicatesensitivity of shock revival to the neutrino luminosities andmean energies – it is natural to ask: What are the require-ments for modelling the interplay of the different ingredientsof the neutrino-driven mechanism accurately? This questionis even more pertinent considering that the enormous expan-sion of the field during the recent years has sometimes pro-duced contradictory results, debates about the relative im-

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16 B. Muller

portance of physical effects, and controversies about the ap-propriateness of certain simulation methodologies.

Ultimately, only the continuous evolution of the simu-lation codes, the inclusion of similar physics by differentgroups, and carefully designed cross-comparisons will even-tually produce a “concordance model” of the neutrino-drivenmechanism and confirm that simulation results are robustagainst uncertainties. For 1D neutrino hydrodynamics sim-ulations, this has largely been achieved in the wake of thepioneering comparison paper of Liebendorfer et al. (2005),which has served as reference for subsequent method pa-pers and sensitivity studies in 1D (Muller et al. 2010; Lentzet al. 2012a,b; O’Connor 2015; Just et al. 2015; Summa et al.2016). Similar results of the Garching-QUB collaboration(Summa et al. 2016) and O’Connor & Couch (2015) withmulti-group neutrino transport indicate a trend to a similarconvergence in 2D, and more detailed comparisons are un-derway (see, e.g., https://www.authorea.com/users/1943/articles/97450/_show_article for efforts coor-dinated by E. O’Connor). Along the road to convergence,it appears useful to provide a preliminary review of someissues concerning the accuracy and reliability of supernovasimulations.

4.1 Hydrodynamics

Recently, the discussion of the fidelity of the simulations hasstrongly focused on the the hydrodynamic side of the prob-lem. As detailed in Section 3, multi-D effects play a crucialrole in the explosion mechanism, and are regulated by a bal-ance of driving (by neutrino heating through buoyancy, or byan inherent instability of the flow like the SASI) and dissi-pation.

4.1.1 Turbulence in Supernova SimulationsThis balance needs to be modelled with sufficient physicaland numerical accuracy. On the numerical side, the chal-lenge consists in the turbulent high-Reynolds number flow,and the question arises to what extent simulations with rel-atively coarse resolution can capture this turbulent flow ac-curately. Various authors (Handy et al. 2014; Abdikamalovet al. 2015; Radice et al. 2015; Roberts et al. 2016) havestressed that the regime of fully developed turbulence can-not be reached with the limited resolution affordable to coverthe gain region (∼100 zones, or even less) in typical models,and Handy et al. (2014) thus prefer to speak of “perturbedlaminar flow” in simulations. Attempts to quantify the effec-tive Reynolds number of the flow using velocity structurefunctions and spectral properties of the post-shock turbu-lence (Handy et al. 2014; Abdikamalov et al. 2015; Radiceet al. 2015) put it at a few hundred at best, and sometimeseven below 100.

This is in line with rule-of-thumb estimates based onthe numerical diffusivity for the highest-wavenumber (odd-even) modes in Godunov-based schemes as used in manysupernova codes. This diffusivity can be calculated analyti-

cally (Appendix D of Muller 2009; see also Arnett & Meakin2016 for a simpler estimate). For Riemann solvers that takeall the wave families into account (e.g. Colella & Glaz 1985;Toro et al. 1994; Mignone & Bodo 2005; Donat & Marquina1996), the numerical kinematic viscosity νnum in the sub-sonic regime is roughly given in terms of the typical velocityjump per cell δvgs and the cell width δl as νnum ∼ δl δvgs. Re-lating δvgs to the turbulent velocity v and scale l of the largesteddy as δvgs ∼ v(δl/l)1/3 (i.e. assuming Kolmogorov scaling)yields a numerical Reynolds number of

Re =vlνnum

∼

(lδl

)4/3

= N4/3, (29)

where N is the number of zones covering the largest eddyscale. For more diffusive solvers like HLLE (Einfeldt 1988),one obtains νnum ∼ δl cs ∼ δl v Ma−1 instead and

Re ∼ (l/δl)Ma ∼ N Ma, (30)

i.e. such solvers are strongly inferior for subsonic flow withlow Mach number Ma.

Such coarse estimates are to be taken with caution,however. The numerical dissipation is non-linear and self-regulated as typical of implicit large-eddy simulations(ILES, Boris et al. 1992; Grinstein et al. 2007). In fact,the estimates already demonstrate that simply comparingthe resolution in codes with different solvers and grid ge-ometries can be misleading. Codes with three-wave solverslike Vertex-Prometheus (Rampp & Janka 2002; Buras et al.2006a) and CoCoNuT-FMT (Muller & Janka 2015) of theMPA-QUB collaboration, Flash (Fryxell et al. 2000) asused in Couch (2013a) and subsequent work by S. Couchand E. O’Connor, and the VH-1 hydro module (Blondinet al. 1991) in the Chimera code of the Oak Ridge-FloridaAtlantic-NC State collaboration, have less stringent resolu-tion requirements than HLLE-based codes (Ott et al. 2012;Kuroda et al. 2012). The reconstruction method, specialtweaks for hydrostatic equilibrium (or an the lack of sucha treatment), as well as the grid geometry and grid-inducedperturbations (Janka et al. 2016; Roberts et al. 2016) alsoaffect the behaviour and resolution-dependence of the simu-lated turbulence.

4.1.2 Resolution Requirements – A Critical AssessmentRegardless of the employed numerical schemes, the fact re-mains that the achievable numerical Reynolds number in su-pernova simulations is limited, and that the regime of fullydeveloped turbulence (Re 1000) will not be achieved inthe near future, as it would require & 512 radial zones in thegain region alone. The question for supernova models, how-ever, is not whether all the facets of turbulence in inviscidflow can be reproduced, but whether the flow properties thatmatter for the neutrino-driven mechanism are computed withsufficient accuracy. In fact, one cannot even hope that sim-ply cranking up the numerical resolution with ILES methodswould give the correct solution: In reality, non-ideal effectssuch as neutrino viscosity and drag (van den Horn & van

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Weert 1984; Burrows 1988; Jedamzik et al. 1998; Guiletet al. 2015) come into play, and deviations of the turbu-lent Prandtl number from unity as well as MHD effects likea small-scale dynamo (see Section 3.4) can complicate thepicture even for non-rotating, weakly magnetised supernovacores. These effects will likely not grossly alter the dynamicsof convection and the SASI, but the physical reality may beslightly different from the limit of infinite resolution if theseeffects are not accounted for and inviscid flow is assumedinstead.

At the end of the day, these additional complications andthe finite resolution probably have a limited effect on super-nova dynamics, since they only affect a correction term tothe critical luminosity such as (1 + 4/3〈Ma2〉)−3/5 in Equa-tion (13) through the effective dissipation length that deter-mines the non-dimensional coefficient in Equation (18). Ifwe repeat the analytic estimate for Lcrit of Muller & Janka(2015), but assume stronger dissipation and decrease theircritical Mach number at shock revival Ma2

crit = 0.4649 by10%, then Equation (13) suggests an increase of the criti-cal luminosity from 74.9% of the 1D value to of 76.6% ofthe 1D value, which is a minute change. Modelling turbulentdissipation within 10% uncertainty thus seems wholly suffi-cient given that one can hardly hope to achieve 1% accuracyfor the neutrino luminosities and mean energies.

The turbulent dissipation does not change without boundswith increasing resolution, but eventually reaches an asymp-totic limit at high Reynolds numbers. Although most super-nova simulation may not fully reach this asymptotic regime,they do not fall far short of it: The works of Handy et al.(2014) and Radice et al. (2015, 2016) suggest that this levelof accuracy in the turbulent dissipation can be reached evenwith moderate resolution (< 100 grid points per direction,∼ 2 resolution in angle in spherical polar coordinates) in thegain region with higher-order reconstruction methods andaccurate Riemann solvers. Problems due to stringent reso-lution requirements may still lurk elsewhere, though, e.g.concerning SASI growth rates as already pointed out tenyears ago by Sato et al. (2009). Resolution studies and cross-comparisons thus remain useful, though cross-comparisonsare of course hampered by the different physical assump-tions used in different codes and the feedback processesin the supernova core. For this reason a direct comparisonof, e.g., turbulent kinetic energies and Mach numbers be-tween different models is not necessarily meaningful. Thedimensionless coefficients governing the dynamics of non-radial instabilities such the proportionality constant ηconv =

vturb/[qν(rsh − rg)] in Equation (18) or the quality factor Q inEquation (17) may be more useful metrics of comparison.

4.2 Neutrino Transport

The requirements on the treatment of neutrino heating andcooling are highly problem-dependent. The physical princi-ples behind convection and the SASI can be studied withsimple heating and cooling functions in a light-bulb ap-

proach, and such an approach is indeed often advantageousas it removes some of the feedback processes that com-plicate the analysis of full-scale supernova simulations. Tomodel the fate and explosion properties of concrete progeni-tors in a predictive manner, some form of neutrino transportis required, and depending on the targeted level of accuracy,the requirements become more stringent; e.g. higher stan-dards apply when it comes to predicting supernova nucle-osynthesis. There is no perfect method for neutrino transportin supernovae as yet. Efforts toward a solution of the full 6-dimensional Boltzmann equation are underway (e.g. Cardallet al. 2013; Peres et al. 2014; Radice et al. 2013; Nagakuraet al. 2014), but not yet ripe for real supernova simulations.

Neutrino transport algorithms (beyond fully parame-terised light-bulb models) currently in use for 1D and multi-D models include:

• leakage schemes as, e.g., in O’Connor & Ott (2010),O’Connor & Ott (2011), Ott et al. (2013) and Couch &O’Connor (2014)

• the isotropic diffusion source approximation (IDSA) ofLiebendorfer et al. (2009),

• one-moment closure schemes employing prescribedflux factors (Scheck et al. 2006), flux-limited diffusionas in the Vulcan code (Livne et al. 2004; Walder et al.2005), the Chimera code (Bruenn 1985; Bruenn et al.2013) and the Castro code (Zhang et al. 2013; Dolenceet al. 2015), or a dynamic closure as in the CoCoNuT-FMT code,

• two-moment methods employing algebraic closures in1D (O’Connor 2015) and multi-D (Obergaulinger &Janka 2011; Kuroda et al. 2012; Just et al. 2015; Skin-ner et al. 2015; O’Connor & Couch 2015; Robertset al. 2016; Kuroda et al. 2016) or variable Edding-ton factors from a model Boltzmann equation (Bur-rows et al. 2000b; Rampp & Janka 2002; Buras et al.2006a; Muller et al. 2010),

• discrete ordinate methods for the Boltzmann equation,mostly in 1D (Mezzacappa & Bruenn 1993; Yamadaet al. 1999; Liebendorfer et al. 2004) or, at the expenseof other simplifications, in multi-D (Livne et al. 2004;Ott et al. 2008; Nagakura et al. 2016; only for staticconfigurations: Sumiyoshi et al. 2015).

This list should not be taken as a hierarchy of accuracy;it mere reflects crudely the rigour in treating one aspect ofthe neutrino transport problem, i.e. the angle-dependence ofthe radiation field in phase space. When assessing neutrinotransport methodologies, there are other, equally importantfactors that need to be taken into account when comparingdifferent modelling approaches.

Most importantly, the sophistication of the microphysicsvaries drastically. On the level of one-moment and two-moment closure models, it is rather the neutrino micro-physics that decides about the quantitative accuracy. The3D models of the MPA-QUB group (Melson et al. 2015a,b;

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18 B. Muller

Janka et al. 2016) and the Chimera team (Lentz et al. 2015)currently represent the state-of-the-art in this respect; thoughother codes (O’Connor 2015; Just et al. 2015; Skinner et al.2015; Kuroda et al. 2016) come close.

Often, the neutrino physics is simplified considerably,however. Some simulations disregard heavy flavour neutri-nos altogether (e.g Suwa et al. 2010; Takiwaki et al. 2012),or only treat them by means of a leakage scheme (Takiwakiet al. 2014; Pan et al. 2016). This affects the contraction ofthe proto-neutron star and thus indirectly alters the emissionof electron flavour neutrinos and the effective inner bound-ary for the gain region as well.

Among multi-D codes, energy transfer due to inelasticneutrino-electron scattering (NES) is routinely taken into ac-count only in the Vertex code (Rampp & Janka 2002; Buraset al. 2006a; Muller et al. 2010) of the MPA-QUB collabo-ration, the Alcar code (Just et al. 2015), the Chimera codeof the Chimera team (Bruenn 1985; Bruenn et al. 2013), andthe Fornax code of the Princeton group (Skinner et al. 2015).Without NES (Bruenn 1985) and modern electron capturerates (Langanke et al. 2003), the core mass at bounce islarger and the shock propagates faster at early times (Lentzet al. 2012a,b). In multi-D, this can lead to unduly strongprompt convection. Because of this problem, a closer lookat the bounce dynamics is in order whenever explosions oc-cur suspiciously early (< 100 ms after bounce). Parameteris-ing deleptonisation during collapse (Liebendorfer 2005) pro-vides a workaround to some extent.

The recoil energy transfer in neutrino-nucleon scatteringeffectively reshuffles heavy flavour neutrino luminosity toelectron flavour luminosity in the cooling region (Mulleret al. 2012a) and hence critically influences the heating con-ditions in the gain region. Among multi-D codes, only Ver-tex and Chimera currently take this into account, and thecode CoCoNuT-FMT (Muller & Janka 2015) uses an effec-tive absorption opacity for heavy flavour neutrinos to mimicthis phenomenon.

Vertex and Chimera are also the only multi-D codes toinclude the effect of nucleon-nucleon correlations (Burrows& Sawyer 1998, 1999; Reddy et al. 1999) on absorption andscattering opacities. Nucleon correlations have a huge im-pact during the cooling phase, which they shorten by a fac-tor of several (Hudepohl et al. 2009). Their role during thefirst second after bounce is not well explored. Consideringthat the explosion energetics are determined on a time-scaleof seconds (Muller 2015; Bruenn et al. 2016), it is plausiblethat the increased diffusion luminosity from the neutron stardue to in-medium corrections to the opacities may influencethe explosion energy to some extent.

Gray schemes (Fryer & Warren 2002; Scheck et al. 2006;Kuroda et al. 2012) cannot model neutrino heating and cool-ing accurately; an energy-dependent treatment is neededbecause of the emerging neutrino spectra are highly non-thermal with a pinched high-energy tail (Janka & Hillebrandt1989; Keil et al. 2003).

Some multi-D codes use the ray-by-ray-plus approxima-tion (Buras et al. 2006a), which exaggerates angular vari-ations in the radiation field, and has been claimed to leadto spuriously early explosions in some cases in conjunctionwith artificially strong sloshing motions in 2D (Skinner et al.2015). Whether this is a serious problem is unclear in thelight of similar results of Summa et al. (2016) for ray-by-ray-plus models and O’Connor & Couch (2015) for fully two-dimensional two-moment transport. On the other hand, fullymulti-dimensional flux limited diffusion approaches smearout angular variations in the radiation field too strongly (Ottet al. 2008).

Neglecting all or part of the velocity-dependent terms inthe transport equations potentially has serious repercussions.Neglecting only observer correction (Doppler shift, com-pression work, etc.) as, e.g. in Livne et al. (2004) can al-ready have an appreciable impact on the dynamics (Buraset al. 2006a; Lentz et al. 2012a). Disregarding even theco-advection of neutrinos with the fluid (O’Connor 2015;Roberts et al. 2016) formally violates the diffusion limit andeffectively results in an extra source term in the opticallythick regime due to the equilibration of matter with laggingneutrinos,

qν ≈ ρ−1v · ∇Eeq (31)

where Eeq is the equilibrium neutrino energy density. Judg-ing from the results of O’Connor & Couch (2015) andRoberts et al. (2016), which are well in line with re-sults obtained with other codes, the effect may not be tooserious in practice, though. It should also be noted that(semi-)stationary approximations of the transport equation(Liebendorfer et al. 2009; Muller & Janka 2015) avoid thisproblem even if advection terms are not explicitly included.

Leakage-based schemes as used, e.g., in Ott et al. (2012),Couch & Ott (2015), Abdikamalov et al. (2015), and Couchet al. (2015) also manifestly fail to reproduce the diffusionlimit. Here, however, the violation of the diffusion limit isunmistakable and can severely affect the stratification of thegain region and, in particular, the cooling region. Togetherwith ad hoc choices for the flux factor for calculating theheating rate, this can result in inordinately high heating effi-ciencies immediately after bounce and a completely invertedhierarchy of neutrino mean energies. It compromises the dy-namics of leakage models to an extent that they can only beused for very qualitative studies of the multi-D flow in thesupernova core.

There is in fact no easy lesson to be learned from thepitfalls and complications that we have outlined. In manycontexts approximations for the neutrino transport are per-fectly justified for a well-circumscribed problem, and feed-back processes sometimes mitigate the effects of simplifyingassumptions. It it crucial, though, to be aware of the impactthat such approximations can potentially have, and our (in-complete) enumeration is meant to provide some guidancein this respect.

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0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

time after bounce [s]

101

102

103

104

rad

ius

[km

]

3D progenitor1D progenitor

Figure 5. Impact of pre-collapse asphericities on shock revival in 3D multi-group neutrino hydrodynamics simulations of an 18M progenitor. The plotshows the minimum, maximum (solid lines) and average (dashed) shockradii for a model using 3D initial conditions (black) from the O shell burn-ing simulation of Muller et al. (2016a) and a spherically averaged versionof the same progenitor (red). The gain radius (dash-dotted) and the proto-neutron star radius (dotted, defined by a fiducial density of 1011 g cm−3) areshown only for the model starting from 3D initial conditions; they are vir-tually identical for both models. A neutrino-driven explosion is triggeredroughly 0.25 s after bounce aided by the infall of the convectively perturbedoxygen shell in the model using 3D initial conditions. The simulation start-ing from the 1D progenitor model exhibits steady and strong SASI oscilla-tions after 0.25 s, but does not explode at least for another 0.3 s.

5 FUTURE DIRECTIONS: MULTI-D EFFECTS INSUPERNOVA PROGENITORS

Given the sophisticated simulation methodology employedin the best currently available supernova codes, one may betempted to ask whether another missing ingredient for robustneutrino-driven explosion is to be sought elsewhere. One re-cent idea, first proposed by Couch & Ott (2013), focuses onthe progenitor models used in supernova simulations. Thetwist consists in an extra “forcing” of the non-radial motionsin the gain region by large seed perturbations in the infallingshells. Such seed perturbations will arise naturally in activeconvective burning shells (O burning, and perhaps also Siburning) that reach the shock during the first few hundredmilliseconds after bounce.

5.1 Role of Pre-Collapse Perturbations in theNeutrino-Driven Mechanism

In default of multi-D progenitor models, this new varia-tion of the neutrino-driven mechanism was initially studiedby imposing large initial perturbations by hand in leakage-based simulations (Couch & Ott 2013, 2015) and multi-group neutrino hydrodynamics simulations (Muller & Janka2015); the earlier light-bulb based models of Fernandez(2012) also touched parts of the problem. The results ofthese investigations were mixed, even though some of thesecalculations employed perturbations far in excess of whatestimates based on mixing-length theory (Biermann 1932;

Bohm-Vitense 1958) suggest: For example, Couch & Ott(2013) used transverse velocity perturbations with a peakMach number of Ma = 0.2 in their 3D models, and found asmall beneficial effect on shock revival, which, however, wastantamount to a change of the critical neutrino luminosity byonly ∼2%. The more extensive 2D parameter study of differ-ent solenoidal and compressive velocity perturbations anddensity perturbations by Muller & Janka (2015) establishedthat both significant perturbation velocities (Ma & 0.1) aswell as large-scale angular structures (angular wavenumber` . 4) need to be present in active convective shell in orderto reduce the critical luminosity appreciably, i.e. by & 10%.

These parametric studies already elucidated the physicalmechanism whereby pre-collapse perturbations can facilitateshock revival. Muller & Janka (2015) highlighted the impor-tance both of the infall phase as well as the interaction ofthe perturbations with the shock. Linear perturbation theoryshows that the initial perturbations are amplified during col-lapse (Lai & Goldreich 2000; Takahashi & Yamada 2014).This not only involves a strong growth of transverse veloc-ity perturbations as δvt ∝ r−1, but even more importantly aconversion of the initially dominating solenoidal velocityperturbations with Mach number Maconv into density pertur-bations δρ/ρ ≈ Ma (Muller & Janka 2015) during collapse,i.e. the relative density perturbations are much larger aheadof the shock than during quasi-stationary convection, whereδρ/ρ ≈ Ma2.11

Large density perturbations ahead of the shock imply apronounced asymmetry in the pre-shock ram pressure anddeform the shock, creating fast lateral flows as well as post-shock density and entropy perturbations that buoyancy thenconverts into turbulent kinetic energy. The direct injectionof kinetic energy due to infalling turbulent motions may alsoplay a role (Abdikamalov et al. 2016), though it appears tobe subdominant (Muller & Janka 2015; Muller et al. 2016a).A very crude estimate for the generation of additional turbu-lent kinetic energy due to the different processes as well asturbulent damping in the post-shock region has been used byMuller et al. (2016a) to estimate the reduction of the criticalluminosity as,

(LνE2ν )crit,pert ≈ (LνE2

ν )crit,3D

(1 − 0.47

Maconv

`ηaccηheat

), (32)

in terms of the pre-collapse Mach number Maconv of eddiesfrom shell burning, their typical angular wavenumber `, andthe accretion efficiency ηacc = Lν/(GMMrgain) and heatingefficiency ηheat during the pre-explosion phase.

A more rigorous understanding of the interaction betweeninfalling perturbations, the shock, and non-radial motions inthe post-shock region is currently emerging: Abdikamalovet al. (2016) studied the effect of upstream perturbations on

11I am indebted to T. Foglizzo for pointing out that this conversion ofvelocity perturbations into density perturbations is another instance ofadvective-acoustic coupling (Foglizzo 2001, 2002), so that there is adeep, though not immediately obvious, connection with the physics ofthe SASI.

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20 B. Muller

Figure 6. Top row: Radial velocity in units of cm s−1 (top left) and mass fraction of Si (top right) at the onset of collapse in the 3D progenitor modelof an 18M star of Muller et al. (2016a). Bottom row: Entropy in units of kb/nucleon (bottom left) and mass fraction of Si (bottom right) in the ensuingneutrino-driven explosion 1.43 s after bounce from. All plots show equatorial slices from the 3D simulation. It can be seen that the geometry of the initialconditions is still imprinted on the explosion to some extent with stronger shock expansion in the direction of updrafts of Si rich ashes in the O burning shell.This is a consequence of the forced deformation of the shock around the onset of the explosion.

the shock using the linear interaction approximation of Rib-ner (1953) and argue, in line with Muller et al. (2016a), thata reduction of the critical luminosity by > 10% is plausi-ble. Their estimate may, however, be even too pessimistic asthey neglect acoustic perturbations upstream of the shock.Different from Abdikamalov et al. (2016), the recent anal-ysis of Takahashi et al. (2016) also takes into account thatinstabilities or stabilisation mechanisms operate in the post-shock flow, and studied the (linear) response of convectiveand SASI eigenmodes to forcing by infalling perturbations.A rigorous treatment along these lines that explains the sat-uration of convective and SASI modes as forced oscillatorswith non-linear damping remains desirable.

5.2 The Advent of 3D Supernova Progenitor Models

The parametric studies of Couch & Ott (2013, 2015) andMuller & Janka (2015) still hinged on uncertain assump-tions about the magnitude and scale of the seed perturba-tions left by O and Si shell burning. Various pioneeringstudies of advanced shell burning stages (O, Si, C burning)(Arnett 1994; Bazan & Arnett 1994, 1998; Asida & Arnett2000; Kuhlen et al. 2003; Meakin & Arnett 2006, 2007b,a;Arnett & Meakin 2011; Viallet et al. 2013; Chatzopouloset al. 2014) merely indicated that convective Mach num-bers of a few 10−2 and the formation of large-scale eddiesare plausible, but did not permit a clear-cut judgement aboutwhether pre-collapse perturbations play a dynamical role inthe neutrino-driven mechanism.

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Figure 7. Radial velocity in units of cm s−1 (shown in 90-wedges in the left half of each plot) and mass fraction XO of oxygen during the last minutesof shell burning in an 12.5M progenitor. Snapshots at at 175 s (top left), 66 s (top right), 24 s (bottom left) before collapse, and at the onset of collapse(bottom right) are shown. The residual oxygen in a thin, almost O-depleted shell (red) starts to burn vigorously due to the contraction of the core (top right).As the entropy of this shell increases and matches that of an almost unprocessed, O-rich shell (blue) and the active Ne shell (cyan), it expands outwardsby “encroachment” (bottom left), but there is insufficient time for the shells to merge completely before collapse (bottom right). Note that this is not aqualitatively new phenomenon in 3D; similar events occur in 1D stellar evolution models.

The situation has changed recently with the advent ofmodels of convective shell burning that have been evolvedup to collapse. The idea here is to calculate the last fewminutes prior to collapse to obtain multi-dimensional ini-tial conditions, while ignoring potential long-term effects in3D such as convective boundary mixing (which we discussin Section 5.3). Couch et al. (2015) performed a 3D sim-ulation of the last minutes of Si shell burning in a 15Mstar. The simulation was limited to an octant, and nuclearquasi-equilibrium during Si burning was only treated witha small network. More importantly, the evolution towards

collapse was artificially accelerated by artificially increas-ing electron capture rates in the iron core. As pointed outby Muller et al. (2016a), this can alter the shell evolutionand the convective velocities considerably. Since the shellconfiguration and structure at collapse varies considerablyin 1D models, such an exploratory approach is nonethelessstill justified (see below).

Muller et al. (2016a) explored the more generic casewhere Si shell burning is extinguished before collapse andthe O shell is the innermost active convective region. In their3D simulation of the last five minutes of O shell burning

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22 B. Muller

in an 18M progenitor, they circumvented the aforemen-tioned problems by excising the non-convective Fe and Sicore and contracting it in accordance with a 1D stellar evo-lution model. Moreover, Muller et al. (2016a) simulated theentire sphere using an overset Yin-Yang grid (Kageyama &Sato 2004; Wongwathanarat et al. 2010) as implemented(with some improvements) in the Prometheus supernovacode (Melson 2013; Melson et al. 2015a).

The implications of these simulations for supernova mod-elling are mixed. The typical convective Mach number inCouch et al. (2015) was only ∼0.02, and while they foundlarge-scale motions, the scale of the pre-collapse perturba-tions was still limited by the restriction to octant symmetry.Perturbations of such a magnitude are unlikely to reduce thecritical luminosity considerably (Section 5.1). Consequently,supernova simulations starting from 1D and 3D initial con-ditions using a leakage scheme performed by Couch et al.(2015) did not show a qualitative difference; both 1D and3D initial conditions result in explosions, though the shockexpands slightly faster in the latter case. The use of a leakagescheme and possible effects of stochasticity preclude definiteconclusions from these first results.

The typical convective Mach number in the 18M modelof Muller et al. (2016a) is considerably larger (∼0.1), andtheir simulation also showed the emergence of a bipolar(` = 2) flow structure, which lead them to predict a relativelylarge reduction of the critical luminosity by 12 . . . 24%,which would accord a decisive role to 3D initial conditionsin the neutrino-driven mechanism at least in some progen-itors. A first 3D multi-group neutrino hydrodynamics sim-ulation of their 18M progenitor using the CoCoNuT-FMTcode appears to bear this out (Muller et al. 2016, in prepa-ration): Figure 5 shows the shock radius both for two sim-ulations using 3D and 1D initial conditions, respectively:In the former case, shock revival occurs around 250 ms af-ter bounce thanks to the infall of the convectively perturbedoxygen shell, whereas no explosion develops in the refer-ence simulation by the end of the run more than 600 ms af-ter bounce. An analysis of the heating conditions indicatesthat the non-exploding reference model is clearly not a nearmiss at 250 ms. The effect of 3D initial conditions is thusunambiguously large and sufficient to change the evolutionqualitatively. Moreover, the model indicates that realistic su-pernova explosion energies are within reach in 3D as well:The diagnostic explosion energy reaches 5 × 1050 erg andstill continues to mount by the end of the simulation 1.43 safter bounce. It is also interesting to note that the initialasymmetries are clearly reflected in the explosion geometry(Figure 6) as speculated by Arnett & Meakin (2011). Inci-dentally, the model also shows that the accretion of convec-tive regions does not lead to the formation of the “accretionbelts” proposed by Gilkis & Soker (2014) as an ingredientfor their jittering-jet mechanism.

Whether 3D initial conditions generally play an importantrole in the neutrino-driven mechanism cannot be answeredby studying just two progenitors, aside from the fact that

the models of Couch et al. (2015) and Muller et al. (2016a)still suffer from limitations. The properties (width, nuclearenergy generation rate) and the configuration of convectiveburning shells at collapse varies tremendously across differ-ent progenitors in 1D stellar evolution models as, e.g., theKippenhahn diagrams in the literature indicate (Heger et al.2000; Chieffi & Limongi 2013; Sukhbold & Woosley 2014;Cristini et al. 2016) indicate. The interplay of convectiveburning, neutrino cooling, and the contraction/re-expansionof the core and the shells sometimes leave inversions in thetemperature stratification and a complicating layering of ma-terial at different nuclear processing stages. For this reason,1D stellar evolution models sometimes show a highly dy-namic behaviour immediately prior to collapse with shells ofincompletely burnt material flaring up below the innermostactive shell. This is illustrated by follow-up work to Mulleret al. (2016a) shown in Figure 7, where a partially processedlayer with unburnt O becomes convective shortly before col-lapse due to violent burning and is about to merge with theoverlying O/Ne shell before collapse intervenes.

The diverse shell configurations in supernova progenitorsneed to be thoroughly explored in 3D before a general ver-dict on the efficacy of convective seed perturbations in aidingshock revival can be given. Since the bulk properties of theflow (typical velocity, eddy scales) in the interior of the con-vective shells are apparently well captured by mixing-lengththeory (Arnett et al. 2009; Muller et al. 2016a), the convec-tive Mach numbers and eddy scales predicted from 1D stellarevolution models can provide guidance for exploring inter-esting spots in parameter space.

5.3 Convective Boundary Mixing – How Uncertain isthe Structure of Supernova Progenitors?

In what we discussed so far, we have considered multi-D ef-fects in advanced convective burning stages merely becauseof their role in determining the initial conditions for stel-lar collapse. They could also have an important effect onthe secular evolution of massive stars long before the super-nova explosion, and thereby change critical structural prop-erties of the progenitors, such as the compactness parameter(O’Connor & Ott 2011). While mixing-length theory (Bier-mann 1932; Bohm-Vitense 1958) may adequately describethe mixing in the interior of convective zones,12 the mix-ing across convective boundaries is less well understood, andmay play an important role in determining the pre-collapsestructure of massive stars along with other non-convectiveprocesses (e.g. Heger et al. 2000; Maeder & Meynet 2004;Heger et al. 2005; Young et al. 2005; Talon & Charbonnel2005; Cantiello et al. 2014) for mixing and angular momen-tum transport. That some mixing beyond the formally un-stable regions needs to be included has long been known

12The story may be different for angular momentum transport in convectivezones, which deserves to revisited (see Chatzopoulos et al. 2016 for acurrent study in the context of Si and O shell burning).

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(Kippenhahn & Weigert 1990). Phenomenological recipesfor this include extending the mixed region by a fraction ofthe local pressure scale height, or adding diffusive mixingin the formally stable regions with a calibrated functionaldependence on the distance to the boundary (Freytag et al.1996; Herwig et al. 1997).

The dominant mechanism for convective boundary mix-ing during advanced burning stages is entrainment (Fer-nando 1991; Meakin & Arnett 2007b; Viallet et al. 2015) dueto the growth of the Kelvin-Helmholtz or Holmboe instabil-ity at the shell interfaces. For interfaces with a discontinuousdensity jump as often encountered in the interiors of evolvedmassive stars, the relevant dimensionless number for suchshear-driven instabilities is the bulk Richardson number RiB.For entrainment driven by turbulent convection, one has

RiB =gl δρ/ρ

v2conv

, (33)

in terms of the local gravitational acceleration g, the densitycontrast δρ/ρ at the interface, the typical convective velocityvconv in the convective region, and the integral scale l of theconvective eddies. Equating l with the pressure scale heightl = P/ρg allows us to re-express RiB in terms of the convec-tive Mach number Maconv and the adiabatic exponent γ,

RiB =δρ

ρ

glv2

conv=δρ

ρ

Pρv2

conv=δρ

ρ

1γMa2

conv. (34)

Deep in the stellar core, Maconv is typically small duringmost evolutionary phases, and RiB is large so that the con-vective boundaries are usually very “stiff” (Cristini et al.2016).

Various power laws for the entrainment rate have beenproposed in the general fluid dynamics literature (Fernando1991; Strang & Fernando 2001) and astrophysical studies(Meakin & Arnett 2007b) of interfacial mixing driven byturbulent convection on one side of the interface. In the as-trophysical context, it is convenient to translate these into apower law for the mass flux Mentr of entrained material intothe convective region,

Mentr = 4πr2ρvconvA Ri−nB , (35)

with a proportionality constant A and a power-law exponentn. Here ρ is the density on the convective side of the inter-face.

A number of laboratory studies (Fernando 1991; Strang& Fernando 2001) and astrophysical simulations (Meakin& Arnett 2007b; Muller et al. 2016a) suggest values ofA ∼ 0.1 and n = 1. This can be understood heuristically byassuming that layer of width δl ∼ Av2

conv/(g δρ/ρ) always re-mains well mixed,13 and that a fraction δl/l of the massflux Mdown = 2πr2ρvconv in the convective downdrafts comesfrom this mixed layer.

This estimate is essentially equivalent to another one pro-posed in a slightly different context (ingestion of unburnt He

13The width of this region will be determined by the criterion that the gra-dient Richardson number is about 1/4.

during core-He burning; Constantino et al. 2015) by Spruit(2015), who related the ingestion (or entrainment) rate into aconvective zone to the convective luminosity Lconv. Spruit’sargument can be interpreted as one based on energy conser-vation; work is needed to pull material with positive buoy-ancy from an outer shell down into a deeper one, and theenergy that is tapped for this purpose comes from convec-tive motions. Since Lconv ∼ 4πr2ρv3

conv, we can write Equa-tion (35) as

Mentr = A ×4πr2ρv3

conv

gl δρ/ρ≈ A ×

Lconv

gl δρ/ρ, (36)

which directly relates the entrainment rate to the ratio ofLconv and the potential energy of material with positive buoy-ancy after downward mixing over an eddy scale l. The en-trainment law (35), the argument of Spruit (2015), and theproportionality of the entrainment rate with Lconv found inthe recent work of Jones et al. (2016b) on entrainment inhighly-resolved idealised 3D simulation of O shell burningappear to be different sides of the same coin.

5.4 Long-Term Effects of Entrainment on the ShellStructure?

How much will entrainment affect the shell structure of mas-sive stars in the long term? First numerical experimentsbased on the entrainment law of Meakin & Arnett (2007b)were performed by Staritsin (2013) for massive stars on themain sequence 14 and did not reveal dramatic differences inthe size of the convective cores compared to more familiar,calibrated recipes for core overshooting.

Taking Equation (36) at face value allows some interest-ing speculations about the situation during advanced burningstages. Since the convective motions ultimately feed on theenergy generated by nuclear burning Eburn, we can formulatea time-integrated version of Equation (36) for the entrainedmass ∆Mentr over the life time of a convective shell,

GMr

δρ

ρ∆Mentr . AEburn, (37)

GMr

δρ

ρ∆Mentr . AMshell∆Q, (38)

where Mshell is the (final) mass of the shell, and ∆Q is thenuclear energy release per unit mass. With GM/r ∼ 2eint instellar interiors, we can estimate ∆Mentr in terms ∆Q and theinternal energy eint at which the burning occurs,15

∆Mentr . AMshell

(δρ

ρ

)−1∆Q2eint

. (39)

14It is doubtful whether entrainment operates efficiently for core H burning,though. Here diffusivity effects are not negligible for convective boundarymixing, which is thus likely to take on a different character (Viallet et al.2015).

15 eint at the shell boundary may be the more relevant scale, but the convec-tive luminosity typically decreases even more steeply with r than eint, soour estimate is on the safe side for formulating an upper limit.

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24 B. Muller

For O burning at ∼ 2 × 109 K and with ∆Q ≈0.5 MeV/nucleon, the factor ∆Q/(2eint) is of orderunity. Typically, the density contrast δρ/ρ between adjacentshells is also not too far below unity. Since A ≈ 0.1, thissuggests that the shell growth due to entrainment comesup to at most a few tens of percent during O shell burningunless δρ/ρ is rather small to begin with. Thus, a result ofentrainment might be that convective zones may swallowthin, unburnt shells with a small density contrast beforebounce, whereas the large entropy jumps between the majorshells are maintained and even enhanced as a result of thiscannibalisation.

For C burning, the long-term effect of entrainment couldbe somewhat larger than for O burning due to the lowertemperature threshold and the higher ratio ∆Q/2eint; for Siburning, the effect should be smaller. During earlier phasesour estimates break down because the convective flux car-ries only a small fraction of the energy generation by nuclearburning. If this is taken into account, the additional growthof convective regions due to entrainment is again of a modestscale (Spruit 2015).

5.5 Caveats

The estimates for the long-term effect of entrainment onthe growth of convective regions in Section 5.4 are to betaken with caution, however. They are not only crude, time-integrated zeroth-order estimates; the entrainment law (36)is by no means set in stone. Current astrophysical 3D simula-tions only probe a limited range in the critical parameter RiB,and tend to suffer from insufficient resolution for high RiB,as shear instabilities develop on smaller and smaller scales.

As a result, it cannot be excluded that the entrainmentlaw (35) transitions to a steeper slope in the astrophysi-cally relevant regime of high RiB. Experiments also com-pete with the difficulties of a limited dynamic range inReynolds, Prandtl, and Peclet number, and remain incon-clusive about the regime of high RiB that obtains in stel-lar interiors. Power-law exponents larger than n = 1 (up ton = 7/4) have also been reported in this regime as alterna-tives to n = 1 (Fernando 1991; Strang & Fernando 2001; Fe-dorovich et al. 2004). A power-law exponent n > 1 wouldimply a strong suppression of entrainment in stellar interi-ors under most circumstances, and the long-term effect ofentrainment would be negligible. Moreover, magnetic fieldswill affect the shear-driven instabilities responsible for con-vective boundary mixing (Bruggen & Hillebrandt 2001).

Finally, most of the current 3D simulations of convec-tive boundary mixing suffer from another potential problem;the balance between nuclear energy generation and neutrinocooling that obtains during quasi-stationary shell burningstages is typically violated, or neutrino cooling is not mod-elled at all. Jones et al. (2016b) pointed out that this may beproblematic if neutrino cooling decelerates the buoyant con-vective plumes and reduces the shear velocity at the interfa-

cial boundary. Only sufficiently long simulations will be ableclarify whether the strong entrainment seen in some numer-ical simulations is robust or (partly) specific to a transientadjustment phase.

Thus, it remains to be seen whether convective boundarymixing has significant effects on the structure of supernovaprogenitors. Even if it does, it is not clear whether it willqualitatively affect the landscape of supernova progenitors.The general picture of the evolution of massive stars maystay well within the bounds of the variations that have beenexplored already, albeit in a more parametric way (see, e.g.,Sukhbold & Woosley 2014).

6 CONCLUSIONS

It is evident that our understanding of the supernova explo-sion mechanism has progressed considerably over the lastfew years. While simulations of core-collapse supernovaehave yet to demonstrate that they can correctly reproduceand explain the whole range explosions that is observedin nature, there are plenty of ideas for solving the remain-ing problems. Some important milestones from the last fewyears have been discussed in this paper, and can be sum-marised as follows:

• ECSN-like explosions of supernova progenitors withthe lowest masses (8 . . . 10M) can be modelled suc-cessfully both in 2D and in 3D. Regardless of theprecise evolutionary channel from which they origi-nate, supernovae from the transition region betweenthe super-AGB star channel and classical iron-corecollapse supernovae share similar characteristics, i.e.low explosion energies of ∼1050 erg and small nickelmasses of a few 10−3M. Due to the ejection of slightlyneutron-rich material in the early ejecta, they are an in-teresting source site for the production of the lighterneutron-rich trans-iron elements (Sr, Y, Zr), and arepotentially even a site for a weak r-process up to Agand Pd (Wanajo et al. 2011). An unambiguous identifi-cation of ECSN-like explosions among observed tran-sients is still pending, however, although there are var-ious candidate events.

• Though it has yet to be demonstrated that the neutrino-driven explosion mechanism can robustly account forthe explosions of more massive progenitors, first suc-cessful 3D models employing multi-group neutrinotransport have recently become available. The reluc-tance of the first 3D models to develop explosions dueto the different nature of turbulence in 3D proves to beno insurmountable setback; and even the unsuccessful3D models computed so far appear to be close to ex-plosion.

• Some of the recent 2D models produced by differentgroups (Summa et al. 2016; O’Connor & Couch 2015)show similar results, which inspires some confidencethat the simulations are now at a stage where mod-

PASA (2016)doi:10.1017/pas.2016.xxx


elling uncertainties due to different numerical method-ologies are under reasonable control, though they havenot been completely eliminated yet. We have addressedsome of the sensitivities to the modelling assumptionin this paper, including possible effects of numericalresolution as well as various aspects of the neutrinotransport treatment.

• Recent studies have helped to unravel how the inter-play between neutrino heating and hydrodynamic in-stabilities works quantitatively, and they have clarifiedwhy neutrino-driven mechanism can be obtained witha considerably smaller driving luminosity in multi-D.

• There is a number of ideas about missing physics thatcould make the neutrino-driven mechanism robust fora wider range of progenitors. These include rapid rota-tion (Nakamura et al. 2014; Janka et al. 2016; thoughstellar evolution makes this unlikely as a generic ex-planation), changes in the neutrino opacities (Melsonet al. 2015b), and a stronger forcing of non-radial insta-bilities due to seed perturbations from convective shellburning (Couch & Ott 2013; Couch et al. 2015; Muller& Janka 2015; Muller et al. 2016a).

• 3D initial conditions for supernova simulations havenow become available (Couch et al. 2015; Muller et al.2016a), and promise to play a significant and benefi-cial role in the explosion mechanism. A first 3D multi-group simulation starting from a 3D initial model ofan 18M progenitor has been presented in this review.The model has already reached an explosion energy of5 × 1050 erg, and suggests that the observed range ofexplosion energies may be within reach of 3D simula-tions.

• Nonetheless, the study of 3D effects in supernova pro-genitors is yet in its infancy. A thorough explorationof the parameter space is required in order to judgewhether they are generically important for our un-derstanding of supernova explosions. This is not onlytrue with regard to the 3D pre-collapse perturbationsfrom shell burning that are crucial to the “perturbation-aided” neutrino-driven mechanism. The role of con-vective boundary mixing on the structure of supernovaprogenitors also deserves to be explored.

Many of these developments are encouraging, though thereare also hints of new uncertainties that may plague super-nova theory in the future. Whether the new ideas of recentyears will prove sufficient to explain shock revival in core-collapse supernovae remains to be seen. The perspectives arecertainly good, but obviously a lot more remains to be donebefore simulations and theory can fully explain the diversityof core-collapse events in nature. There is no need to feara shortage of fruitful scientific problems concerning the ex-plosions of massive stars.

7 ACKNOWLEDGEMENTS

The author acknowledges fruitful discussions with R. Bollig,A. Burrows, S. Couch, E. Lentz, Th. Foglizzo, A. Heger, F. Herwig,W. R. Hix, H.-Th. Janka, S. Jones, T. Melson, R. Kotak, J. Mur-phy, K. Nomoto, E. O’Connor, L. Roberts, S. Smartt, H. Spruit,and M. Viallet. Particular thanks go to A. Heger, S. Jones, andK. Nomoto for providing density profiles of ECSN-like progenitorsfor Figure 1, to H.-Th. Janka for critical reading, and to T. Mel-son and M. Viallet for long-term assistance with the developmentof the Prometheus code. Part of this work has been supported bythe Australian Research Council through a Discovery Early CareerResearcher Award (grant DE150101145). This research was under-taken with the assistance of resources from the National Compu-tational Infrastructure (NCI), which is supported by the AustralianGovernment. This work was also supported by resources providedby the Pawsey Supercomputing Centre with funding from the Aus-tralian Government and the Government of Western Australia, andby the National Science Foundation under Grant No. PHY-1430152(JINA Center for the Evolution of the Elements). Computationswere performed on the systems raijin (NCI) and Magnus (Pawsey),and also on the IBM iDataPlex system hydra at the Rechenzentrumof the Max-Planck Society (RZG) and at the Minnesota Supercom-puting Institute.

A The Density Gradient in the Post-Shock Region

Neglecting quadratic terms in the velocity and neglecting the self-gravity of the material in the gain region, one can write the mo-mentum and energy equation for quasi-stationary accretion onto theproto-neutron star in the post-shock region as,

1ρ

∂P∂r

= −GMr2 , (A1)

∂

∂r

(h −

GMr

)=

qνvr, (A2)

in terms of the pressure P, the density ρ, the proto-neutron starmass M, the enthalpy h, the mass-specific net neutrino heating rateqν, and the radial velocity vr. For a radiation-dominated gas, onehas h ≈ 4P/ρ, which implies,

14∂h∂r

+h4∂ ln ρ∂r

= −GMr2 , (A3)

and by taking ∂h/∂r from Equation (A2),

qν4vr

+h4∂ ln ρ∂r

= −3GM4r2 . (A4)

Solving for the local power-law slope α = ∂ ln ρ/∂ ln r of the den-sity yields,

α = −3GM

rh−

rqνvrh

. (A5)

Since qν > 0 and vr < 0 in the gain region before shock revival, thisimplies a power-law slope α that is no steeper than,

α ≥ −3GM

rh. (A6)

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