2.1.1 Classical electron-capture supernova models

The steep density gradient outside the core in ECSN-like progenitors is immediately relevant for the dynamics of the ensuing supernova because it implies a rapid decline of the mass accretion rate $\dot{M}$ as the edge of the core reaches the stalled accretion shock. A rapid drop in $\dot{M}$ implies a decreasing ram pressure ahead of the shock and a continuously increasing shock radius (though the shock remains a stationary accretion shock for at least $\mathord {\sim } 50\, {\rm ms}$ after bounce and longer for some ECSN-like progenitor models). Under these conditions, neutrino heating can easily pump sufficient energy into the gain region to make the accreted material unbound and power runaway shock expansion. As a result, the neutrino-driven mechanism works for ECSN-like progenitors even under the assumption of spherical symmetry. Using modern multi-group neutrino transport, this was demonstrated by Kitaura et al. (Reference Kitaura, Janka and Hillebrandt2006) for the progenitor of Nomoto (Reference Nomoto1984, Reference Nomoto1987) and confirmed in subsequent simulations by different groups (Janka et al. Reference Janka, Müller, Kitaura and Buras2008; Burrows et al. Reference Burrows, Dessart, Livne, Immler, Weiler and McCray2007b; Fischer et al. Reference Fischer, Whitehouse, Mezzacappa, Thielemann and Liebendörfer2010). The explosions are characterised by a small explosion energy of $\mathord {\sim } 10^{50} \, {\rm erg}$ (Kitaura et al. Reference Kitaura, Janka and Hillebrandt2006; Janka et al. Reference Janka, Müller, Kitaura and Buras2008) and a small nickel mass of a few $ 10^{-3}\,\text{M}_\odot$ (Wanajo et al. Reference Wanajo, Nomoto, Janka, Kitaura and Müller2009).

Even though multi-dimensional effects are not crucial for shock revival in these models, they are not completely negligible. Higher entropies at the bottom of the gain layer lead to convective overturn driven by Rayleigh–Taylor instability shortly after the explosion is initiated (Wanajo, Janka, & Müller Reference Wanajo, Janka and Müller2011). Simulations in axisymmetry (2D) showed that this leads to a modest increase of the explosion energy in Janka et al. (Reference Janka, Müller, Kitaura and Buras2008); an effect which is somewhat larger in more recent models (von Groote et al., in preparation). The effect of Rayleigh–Taylor overturn on the ejecta composition is, however, much more prominent (see Section 2.2).

2.1.2 Conditions for ECSN-like explosion dynamics

Not all of the newly available supernova progenitor models at the low-mass end (Jones et al. Reference Jones2013, Reference Jones, Hirschi and Nomoto2014; Woosley & Heger Reference Woosley and Heger2015b) exhibit a similarly extreme density profile as the model of Nomoto (Reference Nomoto1984, Reference Nomoto1987); in some of them, the density gradient is considerably more shallow (Figure 1). This prompts the questions: How steep a density gradient is required outside the core to obtain an explosion that is triggered by a rapid drop of the accretion rate and works with no or little help from multi-D effects? In reality, there will obviously be a continuum between ECSN-like events and neutrino-driven explosions of more massive stars, in which multi-D effects are crucial for achieving shock revival. Nonetheless, a rough distinction between the two different regimes is still useful, and can be based on the concept of the critical neutrino luminosity of Burrows & Goshy (Reference Burrows and Goshy1993).

Burrows & Goshy (Reference Burrows and Goshy1993) showed that stationary accretion flow onto a proto-neutron star in spherical symmetry is no longer possible if the neutrino luminosity L _ν (which determines the amount of heating) exceeds a critical value $L_{\rm crit}(\dot{M})$ that is well approximated by a power law in $\dot{M}$ with a small exponent, or, equivalently, if $\dot{M}$ drops below a threshold value for a given luminosity. This concept has recently been generalised (Janka Reference Janka2012; Müller & Janka Reference Müller and Janka2015; Summa et al. Reference Summa, Hanke, Janka, Melson, Marek and Müller2016; Janka, Melson, & Summa Reference Janka, Melson and Summa2016) to a critical relation for the (electron-flavour) neutrino luminosity L _ν and neutrino mean energy E _ν as a function of mass accretion rate $\dot{M}$ 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 the core, M, L _ν, and E _ν do not depend dramatically on the stellar structure outside the core during the early post-bounce: The proto-neutron star mass is inevitably $M\approx 1.4\,\text{M}_\odot$ , and since the neutrino emission is dominated by the diffusive neutrino flux from the core, the neutrino emission properties are bound to be similar to the progenitor of Nomoto (Reference Nomoto1984), i.e., one has L _ν ~ 5 × 10⁵² erg s⁻¹ and E _ν ≈ 11 MeV (Hüdepohl et al. Reference Hüdepohl, Müller, Janka, Marek and Raffelt2009), with a steady decrease of the luminosity towards later times. Using calibrated relations for the ‘heating functional’Footnote ² L _ν E ² _ν (Janka et al. Reference Janka, Melson and Summa2016), this translates into a critical mass accretion rate of $\dot{M}_{\rm crit} \approx 0.07\,\text{M}_\odot \, {\rm s}^{-1}$ for ECSN-like progenitors.

To obtain similarly rapid shock expansion as for the $8.8\,\text{M}_\odot$ model of Nomoto (Reference Nomoto1984), $\dot{M}$ must rapidly plummet well below this value. This can be translated into a condition for the density profile outside the core using analytic expressions for the infall time t _infall and accretion rate $\dot{M}$ for mass shell m, which are roughly given by (Woosley & Heger Reference Woosley and Heger2012, Reference Woosley and Heger2015a; Müller et al. Reference Müller, Heger, Liptai and Cameron2016b),

(1) $$\begin{equation} t_{\rm infall}=\sqrt{\frac{\pi }{4 G \bar{\rho }}} =\sqrt{\frac{\pi ^2 r^3}{3 G m}}, \end{equation}$$

and

(2) $$\begin{equation} \dot{M}=\frac{2 m }{t_{\rm infall}} \frac{\rho }{\bar{\rho } - \rho }, \end{equation}$$

where $\bar{\rho }$ is the average density inside the mass shell. For progenitors with little mass outside the core, we have

(3) $$\begin{equation} \dot{M}\approx \frac{2 m }{t_{\rm infall}} \frac{\rho }{\bar{\rho }} =\frac{8\rho }{3}\sqrt{3 G m r^3}. \end{equation}$$

Using $m=1.4\,\text{M}_\odot$ and assuming that $\dot{M}$ needs to drop at least to $M_{\rm crit}=0.05\,\text{M}_\odot \, {\rm s}^{-1}$ within 0.5 s after the onset of collapse to obtain ECSN-like explosion dynamics, one finds that the density needs to drop to

(4) $$\begin{equation} \rho \lesssim \frac{1}{8}\sqrt{\frac{3}{G m}}\dot{M}_{\rm crit} r^{-3/2} \end{equation}$$

for a radius r < 2 230 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 (Reference Nomoto1984) to fulfil this criterion. ECSN-like explosion dynamics is expected alike for the modern $8.8\,\text{M}_\odot$ ECSN progenitor of Jones et al. (Reference Jones2013) and low-mass iron cores (A. Heger, private communication) of $8.1\,\text{M}_\odot$ (with metallicity Z = 10⁻⁴) and $9.6\,\text{M}_\odot$ (Z = 0), though the low-mass iron-core progenitors are a somewhat marginal case.

2.1.3 Low-mass iron-core progenitors

Simulations of these two low-mass iron progenitors with $8.1\,\text{M}_\odot$ (Müller, Janka, & Heger Reference Müller, Janka and Heger2012b) and $9.6\,\text{M}_\odot$ (Janka et al. Reference Janka, Hanke, Hüdepohl, Marek, Müller and Obergaulinger2012; Müller et al. Reference Müller, Janka and Marek2013 in 2D; Melson, Janka, & Marek Reference Melson, Janka and Marek2015a in 3D) nonetheless demonstrated that the structure of these stars is sufficiently extreme to produce explosions reminiscent of ECSN models: Shock revival sets in early around 100 ms after bounce, aided by the drop of the accretion rate associated with the infall of the thin O and C/O shells, and the explosion energy remains small (5 × 10⁴⁹. . .10⁵⁰ erg).

As shown by Melson et al. (Reference Melson, Janka and Marek2015a), there are important differences to ECSNe, however: Whilst shock revival also occurs in spherical symmetry, multi-dimensional effects significantly alter the explosion dynamics. In 1D, the shock propagates very slowly through the C/O shell after shock revival, and only accelerates significantly after reaching the He shell. Without the additional boost by convective overturn, the explosion energy is lower by a factor of $\mathord {\sim } 5$ compared to the multi-D case. Different from ECSNe, somewhat slower shock expansion provides time for the small-scale convective plumes to merge into large structures as shown for the $9.6\,\text{M}_\odot$ model of Janka et al. (Reference Janka, Hanke, Hüdepohl, Marek, Müller and Obergaulinger2012) in Figure 2.

Figure 2.

Entropy s (left half of plot) and electron fraction $Y_\text{e}$ (right half) in the $9.6\,\text{M}_\odot$ explosion model of Janka et al. (Reference Janka, Hanke, Hüdepohl, Marek, Müller and Obergaulinger2012) and Müller et al. (Reference Müller, Janka and Marek2013) 280 ms after bounce. Large convective plumes push neutron-rich material from close to the gain region out at high velocities.

Both for the $8.8\,\text{M}_\odot$ model of Wanajo et al. (Reference Wanajo, Janka and Müller2011) and the low-mass iron-core explosion models, the dynamics of the Rayleigh–Taylor plumes developing after shock revival is nonetheless quite similar. The entropy of the rising plumes is roughly $\mathord {\sim } 15 \ldots 20 k_{\rm b}/{\rm nucleon}$ compared to $\mathord {\sim }10 k_{\rm b}/{\rm nucleon}$ in the ambient medium. For such an entropy contrast, balance between buoyancy and drag forces applies a limiting velocity of the order of the speed of sound. This limit appears to be reached relatively quickly in the simulations. Apart from the very early growth phase, the plume velocities should therefore not depend strongly on the initial seed perturbations; they are rather set by bulk parameters of the system, namely the post-shock entropy at a few hundred kilometres and the entropy close to the gain radius, which together determine the entropy contrast of the plumes. This will become relevant later in our discussion of the nucleosynthesis of ECSN-like explosions.

2.2.1 1D electron-capture supernovae models—early ejecta

Nucleosynthesis calculations based on modern, spherically symmetric ECSN models were first performed by Hoffman, Müller, & Janka (Reference Hoffman, Müller and Janka2008) and Wanajo et al. (Reference Wanajo, Nomoto, Janka, Kitaura and Müller2009). The results of these calculations appeared to point to a severe conflict with observational constraints, showing a strong overproduction of N = 50 nuclei, in particular ⁹⁰Zr, due to the ejection of slightly neutron-rich material (electron fraction $Y_\text{e}\gtrsim 0.46$ ) with relatively low entropy (s ≈ 18k_b /nucleon) immediately after shock revival. Hoffman et al. (Reference Hoffman, Müller and Janka2008) inferred that such nucleosynthesis yields would only be compatible with chemogalactic evolution if ECSNe were rare events occurring at a rate no larger than once per 3 000 yr.

The low $Y_\text{e}$ -values in the early ejecta stem from the ejection of matter at relatively high velocities in the wake of the fast-expanding shock. In slow outflows, neutrino absorption on neutrons and protons drives $Y_\text{e}$ to an equilibrium value that is set by the electron neutrino and anti-neutrino luminosities $L_{\nu _\text{e}}$ and $L_{\bar{\nu }_\text{e}}$ , the ‘effective’ mean energiesFootnote ³ $\varepsilon _{\nu _\text{e}}$ and $\varepsilon _{{\bar{\nu }}_\text{e}}$ , and the proton–neutron mass difference Δ = 1.293 MeV as follows (Qian & Woosley Reference Qian and Woosley1996):

(5) $$\begin{equation} Y_\text{e} \approx \left[1+\frac{L_{\bar{\nu }_\text{e}} (\varepsilon _{\bar{\nu }_\text{e}}-2 \Delta )}{L_{{\nu }_\text{e}} (\varepsilon _{{\nu }_\text{e}}+2 \Delta )}\right]^{-1}. \end{equation}$$

For the relatively similar electron neutrino and anti-neutrino luminosities and a small difference in the mean energies of 2. . .3 MeV in modern simulations, one typically finds an asymptotic value of $Y_\text{e}>0.5$ , i.e., proton-rich conditions. To obtain low $Y_\text{e}<0.5$ in the ejecta, neutrino absorption reactions need to freeze out at a high density (small radius) when the equilibrium between the reactions $n(\nu _\text{e},\text{e}^-)p$ and $p(\nu _\text{e},\text{e}^+)n$ is still skewed towards low $Y_\text{e}$ due to electron captures $p (\text{e}^-,\nu _\text{e}) n$ on protons. Neglecting the difference between arithmetic, quadratic, and cubic neutrino mean energies and assuming a roughly equal contribution of $n(\nu _\text{e},\text{e}^-)p$ and $p(\bar{\nu }_\text{e},\text{e}^+)n$ to the neutrino heating, one can estimate that freeze-out roughly occurs when (cp. Equation (81) in Qian & Woosley Reference Qian and Woosley1996),

(6) $$\begin{equation} \frac{v_\text{r}}{r} \approx \frac{2 m_\text{N} \dot{q}_\nu }{E_{\nu _\text{e}}+E_{\bar{\nu }_\text{e}}}, \end{equation}$$

where $m_\text{N}$ is the nucleon mass, $\dot{q}_\nu$ is the mass-specific neutrino heating rate, r is the radius, and $v_\text{r}$ is the radial velocity. Since $\dot{q}_\nu \propto r^{-2}$ , freeze-out will occur at smaller r, higher density, and smaller $Y_\text{e}$ for higher ejection velocity.

2.2.2 Multi-D effects and the composition of the early ejecta

Since high ejection velocities translate into lower $Y_\text{e}$ , the Rayleigh–Taylor plumes in 2D simulations of ECSNe (Figure 2 in Wanajo et al. Reference Wanajo, Janka and Müller2011) and explosions of low-mass iron cores (Figure 2) contain material with even lower $Y_\text{e}$ than found in 1D ECSN models. Values of $Y_\text{e}$ as low as 0.404 are found in Wanajo et al. (Reference Wanajo, Janka and Müller2011).

Surprisingly, Wanajo et al. (Reference Wanajo, Janka and Müller2011) found that the neutron-rich plumes did not aggravate the problematic overproduction of N = 50 nuclei in their 2D ECSN model. This is due to the fact that the entropy in the neutron-rich lumps is actually smaller than in 1DFootnote ⁴ (but higher than in the ambient medium), which changes the character of the nucleosynthesis by reducing the α-fraction at freeze-out from nuclear statistical equilibrium (NSE). The result is an interesting production of trans-iron elements between Zn and Zr for the progenitor of Nomoto (Reference Nomoto1984, Reference Nomoto1987); the production factors are consistent with current rate estimates for ECSNe of about 4% of all supernovae (Poelarends et al. Reference Poelarends, Herwig, Langer and Heger2008). Subsequent studies showed that neutron-rich lumps in the early ejecta of ECSNe could contribute a sizeable fraction to the live ⁶⁰Fe in the Galaxy (Wanajo, Janka, & Müller Reference Wanajo, Janka and Müller2013b), and might be production sites for some other rare isotopes of obscure origin, such as ⁴⁸Ca (Wanajo, Janka, & Müller Reference Wanajo, Janka and Müller2013a). Due to the similar explosion dynamics, low-mass iron-core progenitors exhibit rather similar nucleosynthesis (Wanajo et al., in preparation; Harris et al., in preparation). The results of these nucleosynthesis calculations 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 element primary process; Travaglio et al. Reference Travaglio, Gallino, Arnone, Cowan, Jordan and Sneden2004; Wanajo & Ishimaru Reference Wanajo and Ishimaru2006; Qian & Wasserburg Reference Qian and Wasserburg2008; Arcones & Montes Reference Arcones and Montes2011; Hansen et al. Reference Hansen2012; Ting et al. Reference Ting, Freeman, Kobayashi, De Silva and Bland-Hawthorn2012).

Since $Y_\text{e}$ in the early ejecta of ECSNe and ECSN-like explosion is sensitive to the neutrino luminosities and mean energies and to the ejection velocity of the convective plumes (which may be different in 3D compared to 2D, or exhibit stochastic variations), Wanajo et al. (Reference Wanajo, Janka and Müller2011) also explored the effect of potential uncertainties in the minimum $Y_\text{e}$ in the ejecta on the nucleosynthesis. They found that a somewhat lower $Y_\text{e}$ of ~ 0.3 in the plumes might make ECSNe a site for a ‘weak r-process’ that could explain the enhanced abundances of lighter r-process elements up to Ag and Pd in some metal-poor halo stars (Wanajo & Ishimaru Reference Wanajo and Ishimaru2006; Honda et al. Reference Honda, Aoki, Ishimaru, Wanajo and Ryan2006).

Whether the neutron-rich conditions required for a weak r-process can be achieved in ECSNe or low-mass iron-core supernovae remains to be determined. Figure 3 provides a tentative glimpse on the effects of stochasticity and dimensionality on the $Y_\text{e}$ in neutron-rich plumes based on several 2D and 3D explosion models of a $9.6\,\text{M}_\odot$ low-mass iron-core progenitor (A. Heger, private communication) conducted using the FMT transport scheme of Müller & Janka (Reference Müller and Janka2015).Footnote ⁵ Stochastic variations in 2D models due to different (random) initial perturbations shift the minimum $Y_\text{e}$ in the ejecta at most by 0.02. This is due to the fact that the Rayleigh–Taylor plumes rapidly transition from the initial growth phase to a stage where buoyancy and drag balance each other and determine the velocity (Alon et al. Reference Alon, Hecht, Ofer and Shvarts1995). 3D effects do not change the distribution of $Y_\text{e}$ tremendously either, at best they tend to shift it to slightly higher values compared to 2D, which is consistent with a somewhat stronger braking of expanding bubbles in 3D as a result of the forward turbulent cascade (Melson et al. Reference Melson, Janka and Marek2015a). It thus appears unlikely that the dynamics of convective overturn is a major source of uncertainty for the nucleosynthesis in ECSN-like explosions, though confirmation with better neutrino transport is still needed.

Figure 3.

Binned distribution of the electron fraction $Y_\text{e}$ in the early ejecta for different explosion models of a $9.6\,\text{M}_\odot$ star 270 ms after bounce. The plots show the relative contribution ΔM _ej/M _ej to the total mass of (shocked) ejecta in bins with $\Delta Y_\text{e}=0.01$ . The upper panel shows the $Y_\text{e}$ -distribution for the 2D model of Janka et al. (Reference Janka, Hanke, Hüdepohl, Marek, Müller and Obergaulinger2012) computed using the vertex-coconut code (Müller et al. Reference Müller, Janka and Dimmelmeier2010). The bottom panel illustrates the effect of stochastic variations and dimensionality using several 2D models (thin lines) and a 3D model computed with the coconut-fmt code of Müller & Janka (Reference Müller and Janka2015) (thick lines). Note that the dispersion in $Y_\text{e}$ in the early ejecta is similar for both codes, though the average $Y_\text{e}$ in the early ejecta is spuriously low when less accurate neutrino transport is used (fmt instead of Vertex). The bottom panel is therefore only intended to show differential effects between different models, and is not a prediction of the absolute value of $Y_\text{e}$ . It suggests that (i) stochastic variations do not strongly affect the distribution of $Y_\text{e}$ in the ejecta, and that (ii) the resulting distribution of $Y_\text{e}$ in 2D and 3D is relatively similar.

If these events are indeed sites of a weak r-process, the missing ingredient is likely to be found elsewhere. Improvements in the neutrino opacities, such as the proper inclusion of nucleon potentials in the charged-current interaction rates (Martínez-Pinedo et al. Reference Martínez-Pinedo, Fischer, Lohs and Huther2012; Roberts, Reddy, & Shen Reference Roberts, Reddy and Shen2012), or flavour oscillations involving sterile neutrinos (Wu et al. Reference Wu, Fischer, Huther, Martínez-Pinedo and Qian2014) could lower $Y_\text{e}$ somewhat. Wu et al. (Reference Wu, Fischer, Huther, Martínez-Pinedo and Qian2014) found a significant reduction of $Y_\text{e}$ by up to 0.15 in some of the ejecta, but these results may depend sensitively on the assumption that collective flavour oscillations are still suppressed during the phase in question. Moreover, Wu et al. (Reference Wu, Fischer, Huther, Martínez-Pinedo and Qian2014) pointed out that a reduction of $Y_\text{e}$ with the help of active-sterile flavour conversion might require delicate fine-tuning to avoid shutting off neutrino heating before the onset of the explosion due to the disappearance of $\nu _\text{e}$ ’s (which could be fatal to the explosion mechanism).

Moreover, whether ECSNe necessarily need to co-produce Ag and Pd with Sr, Y, and Zr is by no means clear. Whilst observed abundance trends may suggest such a co-production, the abundance patterns of elements between Sr and Ag in metal-poor stars appear less robust (Hansen, Montes, & Arcones Reference Hansen, Montes and Arcones2014); and the failure of unaltered models to produce Ag and Pd may not be indicative of a severe tension with observations.

2.2.3 Other nucleosynthesis scenarios for electron-capture supernovae

There are at least two other potentially interesting sites for nucleosynthesis in ECSN-like supernovae. For ‘classical’ ECSN-progenitors with more extreme density profiles, it has been proposed that the rapid acceleration of the shock in the steep density gradient outside the core can lead to sufficiently high post-shock entropies (s ~ 100 k_b /nucleon) and short expansion time-scales (τ_exp ~ 10⁻⁴ s) to allow r-process nucleosynthesis in the thin shells outside the core (Ning, Qian, & Meyer Reference Ning, Qian and Meyer2007). This has not been borne out by numerical simulations, however (Janka et al. Reference Janka, Müller, Kitaura and Buras2008; Hoffman et al. Reference Hoffman, Müller and Janka2008). When the requisite high entropy is reached, the post-shock temperature has already dropped far too low to dissociate nuclei, and the expansion timescale does not become sufficiently short for the scenario of Ning et al. (Reference Ning, Qian and Meyer2007) to work. The proposed r-process in the rapidly expanding shocked shells would require significantly different explosion dynamics, e.g., a much higher explosion energy.

The neutrino-driven wind that is launched after accretion onto the proto-neutron star has been completely subsided has long been discussed as a potential site of r-process nucleosynthesis in supernovae (Woosley et al. Reference Woosley, Wilson, Mathews and Hoffman1994; Takahashi, Witti, & Janka Reference Takahashi, Witti and Janka1994; Qian & Woosley Reference Qian and Woosley1996; Cardall & Fuller Reference Cardall and Fuller1997; Thompson, Burrows, & Meyer Reference Thompson, Burrows and Meyer2001; Arcones, Janka, & Scheck Reference Arcones, Janka and Scheck2007; Arcones & Thielemann Reference Arcones and Thielemann2013). ECSN-like explosions are in many respects the least favourable site for an r-process in the neutrino-driven wind since they produce low-mass neutron stars, which implies low wind entropies and long expansion timescales (Qian & Woosley Reference Qian and Woosley1996), i.e., conditions that are detrimental to r-process nucleosynthesis. However, ECSNe are unique in as much as the neutrino-driven wind can be calculated self-consistently with Boltzmann neutrino transport (Hüdepohl et al. Reference Hüdepohl, Müller, Janka, Marek and Raffelt2009; Fischer et al. Reference Fischer, Whitehouse, Mezzacappa, Thielemann and Liebendörfer2010) without the need to trigger an explosion artificially. These simulations revealed a neutrino-driven wind that is not only of moderate entropy (s ≲ 140k_b /nucleon even at late times), but also becomes increasingly proton-rich with time, in which case the νp-process (Fröhlich et al. Reference Fröhlich, Martínez-Pinedo, Liebendörfer, Thielemann, Bravo, Hix, Langanke and Zinner2006) could potentially operate. The most rigorous nucleosynthesis calculations for the neutrino-driven wind in ECSNe so far (Pllumbi et al. Reference Pllumbi, Tamborra, Wanajo and Janka2015) are based on simulations that properly account for nucleon interaction potentials in the neutrino opacities (Martínez-Pinedo et al. Reference Martínez-Pinedo, Fischer, Lohs and Huther2012; Roberts et al. Reference Roberts, Reddy and Shen2012) and have also explored the effects of collective flavour oscillations, active-sterile flavour conversion. Pllumbi et al. (Reference Pllumbi, Tamborra, Wanajo and Janka2015) suggest that wind nucleosynthesis in ECSNe is rather mundane: Neither does the νp-process operate nor can neutron-rich conditions be restored to obtain conditions even for a weak r-process. Instead, they find that wind nucleosynthesis mainly produces nuclei between Sc and Zn, but the production factors are low, implying that the role of neutrino-driven winds in ECSNe is negligible for this mass range for the purpose of chemogalactic evolution.

3.3.1 Conditions for runaway shock expansion

Even without the complications of multi-D fluid flow, the physics of shock revival is subtle. In spherical symmetry, one can show that for a given mass accretion rate $\dot{M}$ , there is a maximum (critical) electron-flavour luminosity L _ν at the neutrinosphere above which stationary accretion flow onto the proto-neutron star is no longer possible (Burrows & Goshy Reference Burrows and Goshy1993; cp. Section 2). This also holds true if the contribution of the accretion luminosity due to cooling outside the neutrinosphere is taken into account (Pejcha & Thompson Reference Pejcha and Thompson2012). The limit for the existence of stationary solutions does not perfectly coincide with the onset of runaway shock expansion, however. Using 1D light-bulb simulations (i.e., neglecting the contribution of the accretion luminosity), Fernández (Reference Fernández2012) and Gabay, Balberg, & Keshet (Reference Gabay, Balberg and Keshet2015) showed that the accretion flow becomes unstable to oscillatory and non-oscillatory instability slightly below the limit of Burrows & Goshy (Reference Burrows and Goshy1993). Moreover, it is unclear whether the negative feedback of shock expansion on the accretion luminosity and hence on the neutrino heating could push models into a limit cycle (cp. Figure 28 of Buras et al. Reference Buras, Rampp, Janka and Kifonidis2006a) even above the threshold for non-stationarity.

Since an a priori prediction of the critical luminosity, $L_\nu (\dot{M})$ is not feasible, heuristic criteria have been developed (Janka & Keil Reference Janka, Keil, Labhardt, Binggeli and Buser1998; Janka, Kifonidis, & Rampp Reference Janka, Kifonidis, Rampp, Blaschke, Glendenning and Sedrakian2001; Thompson Reference Thompson2000; Thompson, Quataert, & Burrows Reference Thompson, Quataert and Burrows2005; Buras et al. Reference Buras, Janka, Rampp and Kifonidis2006b; Murphy & Burrows Reference Murphy and Burrows2008; Pejcha & Thompson Reference Pejcha and Thompson2012; Fernández Reference Fernández2012; Gabay et al. Reference Gabay, Balberg and Keshet2015; Murphy & Dolence Reference Murphy and Dolence2015) to gauge the proximity of numerical supernova models to an explosive runaway (rather than for pinpointing the formal onset of the runaway after the fact, which is of less interest). The most commonly used criticality parameters are based on the ratio of two relevant timescales for the gain region (Janka & Keil Reference Janka, Keil, Labhardt, Binggeli and Buser1998; Janka et al. Reference Janka, Kifonidis, Rampp, Blaschke, Glendenning and Sedrakian2001; Thompson Reference Thompson2000; Thompson et al. Reference Thompson, Quataert and Burrows2005; Buras et al. Reference Buras, Janka, Rampp and Kifonidis2006b; Murphy & Burrows Reference Murphy and Burrows2008), namely the advection or dwell time τ_adv that accreted material spends in the gain region, and the heating timescale τ_heat over which neutrino energy deposition changes the total or internal energy of the gain region appreciably. If τ_adv > τ_heat, neutrino heating can equalise the net binding energy of the accreted material before it is lost from the gain region, and one expects that the shock must expand significantly due to the concomitant increase in pressure. Since this expansion further increases τ_adv, an explosive runaway is likely to ensue.

The timescale criterion τ_adv/τ_heat > 1 has the virtue of being easy to evaluate since the two timescales can be defined in terms of global quantities such as the total energy E _{tot, g} in the gain region, the volume-integrated neutrino heating rate $\dot{Q}_\nu$ , and the mass M _g in the gain region (which can be used to define $\tau _{\rm adv}=M_{\rm gain}/\dot{M}$ under steady-state conditions). The significance of these global quantities for the problem of shock revival is immediately intuitive, though care must be taken to define the heating timescale properly. Thompson (Reference Thompson2000), Thompson et al. (Reference Thompson, Quataert and Burrows2005), Murphy & Burrows (Reference Murphy and Burrows2008), and Pejcha & Thompson (Reference Pejcha and Thompson2012) define τ_heat as the timescale for changes in the internal energy E _int in the gain region:

(7) $$\begin{equation} \tau _{\rm heat} =\frac{E_{\rm int}}{\dot{Q}_\nu } \end{equation}$$

based on the premise that shock expansion is regulated by the increase in pressure (and hence in internal energy). This definition yields unsatisfactory results, however. The criticality parameter can be spuriously low at shock revival if this definition is used (τ_adv/τ_heat < 0.4).

By defining τ_heat in terms of the total (internal+kinetic+potential) energyFootnote ⁶ of the gain region (Buras et al. Reference Buras, Janka, Rampp and Kifonidis2006b):

(8) $$\begin{equation} \tau _{\rm heat} =\frac{E_{\rm tot,g}}{\dot{Q}_\nu }, \end{equation}$$

the criterion τ_adv/τ_heat > 1 becomes a very accurate predictor for non-oscillatory instability (Fernández Reference Fernández2012; Gabay et al. Reference Gabay, Balberg and Keshet2015). This indicates that the relevant energy scale to which the quasi-hydrostatic stratification of the post-shock region is the total energy (or perhaps the total or stagnation enthalpy) of the gain region, and not the internal energy. This is consistent with the observation that runaway shock expansion occurs roughly once the total energy or the Bernoulli integral (Fernández Reference Fernández2012; Burrows et al. Reference Burrows, Hayes and Fryxell1995) reach positive values somewhere (not everywhere) in the post-shock region, which is essentially what the timescale criterion estimates. What is crucial is that the density and pressure gradients between the gain radius and the shock (and hence the shock position) depends sensitively on the ratio of enthalpy h (or the internal energy) and the gravitational potential, rather than on enthalpy alone. Under the (justified) assumption that quadratic terms in $v_\text{r}^2$ in the momentum and energy equation are sufficiently small to be neglected in the post-shock region, one can show (see Appendix A) that the logarithmic derivative of the density ρ in the gain region is constrained by

(9) $$\begin{equation} \frac{\partial \ln \rho }{\partial \ln r} >-\frac{3GM}{rh}, \end{equation}$$

where M is the proto-neutron star mass. Once h > GM/r or even e _int > GM/r (where e _int is the internal energy per unit mass), significant shock expansion must ensue due to the flattening of pressure and density gradients.

Janka (Reference Janka2012), Müller & Janka (Reference Müller and Janka2015), and Summa et al. (Reference Summa, Hanke, Janka, Melson, Marek and Müller2016) have also pointed out that the timescale criterion can be converted into a scaling law for the critical electron-flavour luminosity L _ν and mean energy E _ν in terms of the proto-neutron star mass M, the accretion rate $\dot{M}$ , and the gain radius r _g,

(10) $$\begin{equation} (L_\nu E_\nu ^2)_{\rm crit} \propto (\dot{M} M)^{3/5} r_{\rm g}^{-2/5}. \end{equation}$$

The concept of the critical luminosity, the timescale criterion, and the condition of positive total energy or a positive Bernoulli parameter at the gain radius are thus intimately related and appear virtually interchangeable considering that they remain approximate criteria for runaway shock expansion anyway. This is also true for some other explosion criteria that have been proposed, e.g., the antesonic condition of Pejcha & Thompson (Reference Pejcha and Thompson2012), which states that the sound speed c _s must exceed a certain fraction of the escape velocity v _esc for runaway shock expansion somewhere in the accretion flow:

(11) $$\begin{equation} c_{\rm s}^2>3/16 v_{\rm esc}^2. \end{equation}$$

Approximating the equation of state as a radiation-dominated gas with an adiabatic index γ = 4/3 and a pressure of P = ρe _int/3 = ρh/4, one finds that the antesonic condition roughly translates to

(12) $$\begin{equation} \frac{c_{\rm s}^2}{3/16 v_{\rm esc}^2} =\frac{4/3 P/\rho }{3/8 GM/r} =\frac{32 e_{\rm int}}{27 GM/r} =\frac{8h}{9GM/r}>1, \end{equation}$$

i.e., the internal energy and the enthalpy must be close to the gravitational binding energy (even if the precise critical values for e _int and h may shift a bit for a realistic equation of state).Footnote ⁷

3.3.2 Impact of multi-D effects on the heating conditions

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

The canonical interpretation has long been that the runaway condition τ_adv > τ_heat remains the decisive criterion in multi-D, and that multi-D effects facilitate shock revival mainly by increasing the advection time-scale τ_adv (Buras et al. Reference Buras, Janka, Rampp and Kifonidis2006b; Murphy & Burrows Reference Murphy and Burrows2008). Especially close to criticality, τ_heat is also shortened due to feedback processes—better heating conditions imply that the net binding energy in the gain region and hence τ_heat must decrease.

Whilst simulations clearly show increased advection timescales in multi-D compared to 1D (Buras et al. Reference Buras, Janka, Rampp and Kifonidis2006b; Murphy & Burrows Reference Murphy and Burrows2008; Hanke et al. Reference Hanke, Marek, Müller and Janka2012) as a result of larger shock radii, the underlying cause for larger accretion shock radii in multi-D is more difficult to pinpoint. Ever since the first 2D simulations, both the transport of neutrino-heated high-entropy material from the gain radius out to the shock (Herant et al. Reference Herant, Benz, Hix, Fryer and Colgate1994; Janka & Müller Reference Janka and Müller1996) as well as the ‘turbulent pressure’ of convective bubbles colliding with the shock (Burrows et al. Reference Burrows, Hayes and Fryxell1995) have been invoked to explain larger shock radii in multi-D. Both effects are plausible since they change the components P (thermal pressure) and ρv⊗v (where v is the velocity) of the momentum stress tensor that must balance the ram pressure upstream of the shock during stationary accretion.

That the turbulent pressure plays an important role follows already from the high turbulent Mach number $\mathord {{\sim }0.5}$ in the post-shock region (Burrows et al. Reference Burrows, Hayes and Fryxell1995; Müller et al. Reference Müller, Janka and Heger2012b) before the onset of shock revival, and has been demonstrated quantitatively by Murphy, Dolence, & Burrows (Reference Murphy, Dolence and Burrows2013) and Couch & Ott (Reference Couch and Ott2015) using spherical Reynolds decomposition to analyse parameterised 2D and 3D simulations. Using a simple estimate for the shock expansion due to turbulent pressure, Müller & Janka (Reference Müller and Janka2015) were even able to derive the reduction of the critical heating functional in multi-D compared to 1D in terms of the average squared turbulent Mach number ⟨Ma²⟩ in the gain region,

(13) $$\begin{eqnarray} (L_\nu E_\nu ^2)_{\rm crit,2D} &\approx & (L_\nu E_\nu ^2)_{\rm crit,1D} \left(1+\frac{4\langle {\rm Ma}^2 \rangle }{3}\right)^{-3/5}\\ \nonumber &\propto & (\dot{M} M)^{3/5} r_{\rm g}^{-2/5} \left(1+\frac{4\langle {\rm Ma}^2 \rangle }{3}\right)^{-3/5}, \end{eqnarray}$$

and then obtained (L _ν E ² _ν)_{crit, 2D} ≈ 0.75(L _ν E ² _ν)_{crit, 1D} in rough agreement with simulations using a model for the saturation of non-radial fluid motions (see Section 3.3.3).

Nonetheless, there is likely no monocausal explanation for better heating conditions in multi-D. Yamasaki & Yamada (Reference Yamasaki and Yamada2006) found, for example, that convective energy transport from the gain radius to the shock also reduces the critical luminosity (although they somewhat overestimated the effect by assuming constant entropy in the entire gain region). Convective energy transport reduces the slope of the pressure gradient between the gain radius (where the pressure is set by the neutrino luminosity and mean energy) and the shock, and thus pushes the shock out by increasing the thermal post-shock pressure. That this effect also plays a role alongside the turbulent pressure can be substantiated by an analysis of neutrino hydrodynamics simulations (Bollig et al. in preparation).

Only a detailed analysis of the properties of turbulence in the gain region (Murphy & Meakin Reference Murphy and Meakin2011) combined with a model for the interaction of turbulence with a non-spherical accretion shock will reveal the precise combination of multi-D effects that conspire to increase the shock radius compared to 1D. This is no prerequisite for understanding the impact of multi-D effects on the runaway condition as encapsulated by a phenomenological correction factor in Equation (13), since effects like turbulent energy transport, turbulent bulk viscosity, etc. will also scale with the square of the turbulent Mach number in the post-shock region just like the turbulent pressure. They are effectively lumped together in the correction factor (1 + 4/3⟨Ma²⟩)^−3/5. The turbulent Mach number in the post-shock region is thus the crucial parameter for the reduction of the critical luminosity in multi-D, although the coefficient of ⟨Ma²⟩ still needs to be calibrated against multi-D simulations (and maybe different in 2D and 3D).

This does not imply, however, that the energetic requirements for runaway shock expansion in multi-D are fundamentally different from 1D: Runaway still occurs roughly once some material in the gain region first acquires positive total (internal+kinetic+potential) energy e _tot; and the required energy input for this ultimately stems from neutrino heating.Footnote ⁸

3.3.3 Saturation of instabilities

What complicates the role of multi-D effects in the neutrino-driven mechanism is that the turbulent Mach number in the gain region itself depends on the heating conditions, which modify the growth rates and saturation properties of convection and the SASI. Considerable progress has been made in recent years in understanding this feedback mechanism and the saturation properties of these two instabilities.

The linear phases of convection and the SASI are now rather well understood. The growth rates for buoyancy-driven convective instability are expected to be of order of the Brunt–Väisälä frequency ω_BV, which can be expressed in terms of P, ρ, c _s, and the local gravitational acceleration g asFootnote ⁹

(14) $$\begin{equation} \omega _{\rm BV}^2=g \left(\frac{1}{\rho } \frac{\partial \rho }{\partial r}- \frac{1}{\rho c_{\rm s}^2}\frac{\partial P}{\partial r}\right), \end{equation}$$

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

(15) $$\begin{equation} \omega _{\rm BV}^2 \sim \frac{G M \dot{Q}_\nu }{4\dot{M} r_{\rm g} ^2 c_{\rm s}^2 \left(r_{\rm sh}-r_{\rm g}\right)} \sim \frac{3\dot{Q}_\nu }{4\dot{M} r_{\rm g} \left(r_{\rm sh}-r_{\rm g}\right)}, \end{equation}$$

using c ² _s ≈ GM/(3r _g) at the gain radius (cp. Müller & Janka Reference Müller and Janka2015). An important subtlety is that advection can stabilise the flow so that ω² _BV > 0 is no longer sufficient for instability unless large seed perturbations in density are already present. Instability instead depends on the more restrictive criterion for the parameter χ (Foglizzo et al. Reference Foglizzo, Scheck and Janka2006):

(16) $$\begin{equation} \chi = \int \limits _{r_{\rm g}}^{r_{\rm sh}}\frac{\omega _{\rm BV}}{|v_r|}{\rm d}r, \end{equation}$$

with χ ≳ 3 indicating convective instability.

The scaling of the linear growth rate ω_SASI of SASI modes is more complicated, since it involves both the duration τ_cyc of the underlying advective-acoustic cycle as well as a quality factor $\mathcal {Q}$ for the conversion of vorticity and entropy perturbations into acoustic perturbation in the deceleration region below the gain region and the reverse process at the shock (Foglizzo et al. Reference Foglizzo, Scheck and Janka2006, Reference Foglizzo, Galletti, Scheck and Janka2007):

(17) $$\begin{equation} \omega _{\rm SASI} \sim \frac{ \ln |\mathcal {Q}|}{\tau _{\rm cyc}}. \end{equation}$$

For realistic models with strong SASI, one finds $\ln |\mathcal {Q}| \sim 2$ (Scheck et al. Reference Scheck, Janka, Foglizzo and Kifonidis2008; Müller et al. Reference Müller, Janka and Heger2012b). SASI growth appears to be suppressed for χ ≳ 3 probably because convection destroys the coherence of the waves involved in the advective-acoustic cycle (Guilet et al. Reference Guilet, Sato and Foglizzo2010). Interestingly, the demarcation line χ = 3 between the SASI- and convection-dominated regimes is also valid in the non-linear regime if χ is computed from the angle- and time-averaged mean flow (Fernández Reference Fernández2012); and both the SASI and convection appear to drive χ close to this critical value (Fernández Reference Fernández2012).

Both in the SASI-dominated regime and the convection-dominated regime, large growth rates are observed in simulations. It only takes a few tens of milliseconds until the instabilities 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 sensitive to the saturation mechanism than to initial conditions, so that the onset of shock revival is only subject to modest stochastic variations (Summa et al. Reference Summa, Hanke, Janka, Melson, Marek and Müller2016). Exceptions apply when the heating conditions vary rapidly, e.g., due to the infall of a shell interface or extreme variations in shock radius (as in the light-bulb models of Cardall & Budiardja Reference Cardall and Budiardja2015), and the runaway condition is only narrowly met or missed (Melson et al. Reference Melson, Janka and Marek2015a; Roberts et al. Reference Roberts, Ott, Haas, O’Connor, Diener and Schnetter2016).

The saturation properties of convection were clarified by Murphy et al. (Reference Murphy, Dolence and Burrows2013), who determined that the volume-integrated neutrino heating rate $\dot{Q}_\nu$ and the convective luminosity L _conv in the gain region roughly balance each other. This can be understood as the result of a self-adjustment process of the accretion flow, whereby a marginally stable, quasi-stationary stratification with χ ≈ 3 is established (Fernández Reference Fernández2012). Müller & Janka (Reference Müller and Janka2015) showed that this can be translated into a scaling law that relates the average mass-specific neutrino heating rate $\dot{q}_\nu$ in the gain region to the root mean square average δv of non-radial velocity fluctuations:

(18) $$\begin{equation} \delta v \sim \left[ \dot{q}_\nu (r_{\rm sh}-r_{\rm g}) \right]^{1/3}. \end{equation}$$

That a similar scaling should apply in the SASI-dominated regime is not immediately intuitive. Müller & Janka (Reference Müller and Janka2015) in fact tested Equation (18) using a SASI-dominated 2D model and argued that self-adjustment of the flow to χ ≈ 3 will result in the same scaling law as for convection-dominated models. However, models suggest that a different mechanism may be at play in the SASI-dominated regime. Simulations are at least equally compatible with the mechanism proposed by Guilet et al. (Reference Guilet, Sato and Foglizzo2010), who suggested that saturation of the SASI is mediated by parasitic instabilities and occurs once the growth rate of the parasite equals the growth rate of the SASI: Assuming that the Kelvin–Helmholtz instability is the dominant parasite, a simple order-of-magnitude estimate for saturation can be obtained by equating ω_SASI and the average shear rate:

(19) $$\begin{equation} \omega _{\rm SASI} \sim \frac{\delta v}{\Lambda }, \end{equation}$$

where Λ is the effective width of the shear layer. Kazeroni, Guilet, & Foglizzo (Reference Kazeroni, Guilet and Foglizzo2016) find that the Kelvin–Helmholtz instability operates primarily in directions where the shock radius is larger, which suggests Λ = r _{sh, max} − r _g. This results in a scaling law that relates the velocity fluctuations to the average radial velocity $\langle v_\text{r} \rangle$ in the gain region:

(20) $$\begin{equation} \delta v \sim \omega _{\rm SASI} \Lambda \sim \frac{\ln |\mathcal {Q}| (r_{\rm sh,max}-r_{\rm g})}{\tau _{\rm adv}} \sim \ln |\mathcal {Q}| \, |\langle v_\text{r} \rangle |, \end{equation}$$

where we assumed τ_cyc ≈ τ_adv. The quality factor $\mathcal {Q}$ can in principle change significantly with time and between different models. Nonetheless, together with the assumption of a roughly constant quality factor, Equation (20) appears to capture the dynamics of the SASI in 3D quite well for a simulation of an $18\,\text{M}_\odot$ progenitor with the coconut-fmt code (Müller & Janka Reference Müller and Janka2015) as illustrated in Figure 4.

Figure 4.

Comparison of the root-mean-square average δv of non-radial velocity component in the gain region (black) with two phenomenological models for the saturation of non-radial instabilities in a SASI-dominated 3D model of an $18\,\text{M}_\odot$ star using the coconut-fmt code. The red curve shows an estimate based on Equation (18), which rests on the assumption of a balance between buoyant driving and turbulent dissipation (Murphy et al. Reference Murphy, Dolence and Burrows2013; Müller & Janka Reference Müller and Janka2015). The blue curve shows the prediction of Equation (20), which assumes that saturation is regulated by a balance between the growth rate of the SASI and parasitic Kelvin–Helmholtz instabilities (Guilet, Sato, & Foglizzo Reference Guilet, Sato and Foglizzo2010). Even though Equation (20) assumes a constant quality factor $|\mathcal {Q}|$ to estimate the SASI growth rate, it appears to provide a good estimate for the dynamics of the model. Interestingly, the saturation models for the SASI- and convection dominated regimes give similar results during later phases even though the mechanism behind the driving instability is completely different.

Equation (18) for the convection-dominated regime and Equation (20) apparently predict turbulent Mach numbers in the same ballpark. This can be understood by expressing $\dot{q}_\nu$ in terms of the accretion efficiency $\eta _{\rm acc} =L_\nu /(GM \dot{M}/r_{\rm g})$ and the heating efficiency $\eta _{\rm heat} =\dot{Q}_\nu /L_\nu$ :

(21) $$\begin{eqnarray} \dot{q}_\nu &=&\frac{\dot{Q}_\nu }{M_{\rm g}}= \eta _{\rm heat} \eta _{\rm acc} \frac{GM \dot{M}}{r_{\rm g} M_{\rm g}}= \eta _{\rm heat} \eta _{\rm acc} \frac{GM}{r_{\rm g} \tau _{\rm adv}} \\ \nonumber &=& \eta _{\rm heat} \eta _{\rm acc} \frac{GM}{r_{\rm sh} \tau _{\rm adv}}\frac{r_{\rm sh}}{r_{\rm gain}}. \end{eqnarray}$$

If we neglect the ratio r _sh/r _g and approximate the average post-shock velocity as $| \langle v_{\rm r} \rangle | \approx \beta ^{-1} \sqrt{GM/r_{\rm sh}}$ (where β is the compression ratio in the shock), we obtain

(22) $$\begin{equation} \dot{q}_\nu \sim \eta _{\rm heat} \eta _{\rm acc} \frac{\beta ^2 |\langle v_\text{r} \rangle |^2 }{\tau _{\rm adv}}, \end{equation}$$

and hence

(23) $$\begin{equation} \delta v \sim (\eta _{\rm heat} \eta _{\rm acc} \beta ^2)^{1/3} |\langle v_\text{r} \rangle |. \end{equation}$$

For plausible values (e.g., η_heat = 0.05η_acc = 2, β = 10), one finds $\delta v \sim 2 |\langle v_\text{r} \rangle |$ , i.e., the turbulent Mach number at saturation is of the same order of magnitude in the convection- and SASI-dominated regimes (where at least $\ln |\mathcal {Q}| \mathord {\sim } 2$ can be reached).

Equations (18) and (20) remain order-of-magnitude estimates; and either of the instabilities may be more efficient at pumping energy into non-radial turbulent motions in the gain region, as suggested by the light-bulb models of Fernández (Reference Fernández2015) and Cardall & Budiardja (Reference Cardall and Budiardja2015). These authors find that the SASI can lower the critical luminosity in 3D considerably further than convection. Fernández (Reference Fernández2010) attributes this to the emergence of the spiral mode of the SASI (Blondin & Mezzacappa Reference Blondin and Mezzacappa2007; Fernández Reference Fernández2010) in 3D, which can store more non-radial kinetic energy than the SASI sloshing mode in 2D, but this has yet to be borne out by self-consistent neutrino hydrodynamics simulations (see Section 3.4 for further discussion).

3.3.4 Why do models explode more easily in 2D than in 3D?

How can one explain the different behaviour of 2D and 3D models in the light of our current understanding of the interplay between neutrino heating, convection, and the SASI? It seems fair to say that we can presently only offer a heuristic interpretation for the more pessimistic evolution of 3D models.

The most glaring difference between 2D and 3D models (especially in the convection-dominated regime) prior to shock revival lies in the typical scale of the turbulent structures, which are smaller in 3D (Hanke et al. Reference Hanke, Marek, Müller and Janka2012; Couch Reference Couch2013b; Couch & Ott Reference Couch and Ott2015), whereas the inverse turbulent cascade in 2D (Kraichnan Reference Kraichnan1967) artificially channels turbulent kinetic energy to large scales. This implies that the effective dissipation length (or also the effective mixing length for energy transport) are smaller in 3D, so that smaller dimensionless coefficients C appear in relations like Equation (18),

(24) $$\begin{equation} \delta v =C \left[ \dot{q}_\nu (r_{\rm sh}-r_{\rm g}) \right]^{1/3}, \end{equation}$$

and the turbulent Mach number will be smaller for a given neutrino heating rate. Indeed, for the $18\,\text{M}_\odot$ model shown in Figure 4, we find

(25) $$\begin{equation} \delta v =0.7 \left[ \dot{q}_\nu (r_{\rm sh}-r_{\rm g}) \right]^{1/3} \end{equation}$$

in 3D rather than what Müller & Janka (Reference Müller and Janka2015) inferred from 2D models (admittedly using a different progenitor),

(26) $$\begin{equation} \delta v =\left[ \dot{q}_\nu (r_{\rm sh}-r_{\rm g}) \right]^{1/3}. \end{equation}$$

Following the arguments of Müller & Janka (Reference Müller and Janka2015) to infer the correction factor $\left(1+\frac{4\langle {\rm Ma}^2 \rangle }{3}\right)^{-3/5}$ for multi-D effects in Equation (13), one would then expect a considerably larger critical luminosity in 3D, i.e., (L _ν E ² _ν)_{crit, 3D} ≈ 0.85(L _ν E ² _ν)_{crit, 1D} instead of (L _ν E ² _ν)_{crit, 2D} ≈ 0.75(L _ν E ² _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 that the critical luminosities in 2D and 3D are still very close to each other. This already indicates that more subtle effects may be at play in 3D that almost compensate the stronger effective dissipation of turbulent motions. The fact that simulations typically show transient phases of stronger shock expansion and more optimistic heating conditions in 3D than in 2D (Hanke et al. Reference Hanke, Marek, Müller and Janka2012; Melson et al. Reference Melson, Janka, Bollig, Hanke, Marek and Müller2015b) also points in this direction.

Furthermore, light-bulb models (Handy, Plewa, & Odrzywołek Reference Handy, Plewa and Odrzywołek2014) and multi-group neutrino hydrodynamics simulations (Melson et al. Reference Melson, Janka and Marek2015a; Müller Reference Müller2015) have demonstrated that favourable 3D effects come into play after shock revival. These works showed that 3D effects can lead to a faster, more robust growth of the explosion energy provided that shock revival can be achieved in the first place.

The favourable 3D effects that are responsible for this may already counterbalance the adverse effect of stronger dissipation in the pre-explosion phase to some extent: Energy leakage from the gain region by the excitation of g-modes is suppressed in 3D because the forward turbulent cascade (Melson et al. Reference Melson, Janka and Marek2015a) and (at high Mach number) the more efficient growth of the Kelvin–Helmholtz instability (Müller Reference Müller2015) brake the downflows before they penetrate the convectively stable cooling layer. Moreover, the non-linear growth of the Rayleigh–Taylor instability is faster for three-dimensional plume-like structures than for 2D structures with planar (Yabe, Hoshino, & Tsuchiya Reference Yabe, Hoshino and Tsuchiya1991; Hecht et al. Reference Hecht, Ofer, Alon, Shvarts, Orszag and McCrory1995; Marinak et al. Reference Marinak1995) or toroidal geometry (as in the context of Rayleigh–Taylor mixing in the stellar envelope during the explosion phase; Kane et al. Reference Kane, Arnett, Remington, Glendinning, Müller, Fryxell and Teyssier2000; Hammer, Janka, & Müller Reference Hammer, Janka and Müller2010), which might explain why 3D models initially respond more strongly to sudden drops in the accretion rate at shell interfaces and exhibit better heating conditions than their 2D counterparts for brief periods. Finally, the difference in the effective dissipation length in 3D and 2D that is reflected by Equations (25) and (26) may not be universal and depend, e.g., on the heating conditions or the χ-parameter; the results of Fernández (Reference Fernández2015) in fact demonstrate that under appropriate circumstances more energy can be stored in non-radial motions in 3D than in 2D in the SASI-dominated regime.

3.4.1 Rotation and beyond

Nakamura et al. (Reference Nakamura, Kuroda, Takiwaki and Kotake2014) and Janka et al. (Reference Janka, Melson and Summa2016) pointed out that rapid progenitor rotation can facilitate explosions in 3D. Janka et al. (Reference Janka, Melson and Summa2016) ascribed this partly to the reduction of the pre-shock infall velocity due to centrifugal forces, which decreases the ram pressure ahead of the shock. Even more importantly, rotational support also decreases the net binding energy |e _tot| per unit mass in the gain region in their models. They derived an analytic correction factor for the critical luminosity in terms of the average-specific angular momentum j in the infalling shells:

(27) $$\begin{equation} (L_\nu E_\nu ^2)_{\rm crit,rot} \approx (L_\nu E_\nu ^2)_{\rm crit} \times \left(1-\frac{j^2}{2 G M r_{\rm sh}}\right)^{3/5}. \end{equation}$$

Assuming rapid rotation with j ≫ 10¹⁶ cm² s⁻¹, one can obtain a significant reduction of the critical luminosity by several 10% as Janka et al. (Reference Janka, Melson and Summa2016) tested in a simulation with a modified rotation profile.Footnote ¹⁰ For very rapid rotation, other explosion mechanisms also become feasible, such as the magnetorotational mechanism (Akiyama et al. Reference Akiyama, Wheeler, Meier, Lichtenstadt, Meier and Lichtenstadt2003; Burrows et al. Reference Burrows, Dessart, Livne, Immler, Weiler and McCray2007b; Winteler et al. Reference Winteler, Käppeli, Perego, Arcones, Vasset, Nishimura, Liebendörfer and Thielemann2012; Mösta et al. Reference Mösta2014), or explosions driven by the low-T/W spiral instability (Takiwaki, Kotake, & Suwa Reference Takiwaki, Kotake and Suwa2016).

However, current stellar evolution models do not predict the required rapid rotation rates for these scenarios for the generic progenitors of type IIP supernovae. The typical specific angular momentum at a mass coordinate of $m=1.5\,\text{M}_\odot$ is only of the order of j ~ 10¹⁵ cm² s⁻¹ in models (Heger, Woosley, & Spruit Reference Heger, Woosley and Spruit2005) that include angular momentum transport by magnetic fields generated by the Tayler–Spruit dynamo (Spruit Reference Spruit2002), and asteroseismic measurements of core rotation in evolved low-mass stars suggest that the spin-down of the cores may be even more efficient (Cantiello et al. Reference Cantiello, Mankovich, Bildsten, Christensen-Dalsgaard and Paxton2014). For such slow rotation, centrifugal forces are negligible; Equation (27) suggests a change of the critical luminosity on the per-mil level. Neither is rotation expected to affect the character of neutrino-driven convection appreciably because the angular velocity Ω in the gain region is too small. The Rossby number is well above unity:

(28) $$\begin{equation} {\rm Ro}\sim \frac{|v_r|}{ (r_{\rm sh}-r_{\rm g}) \Omega } \sim \frac{r_{\rm s}^2}{\tau _{\rm adv} j} \sim 10, \end{equation}$$

assuming typical values of τ_adv ~ 10 ms and r _sh ~ 100 km.

Magnetic field amplification by a small-scale dynamo or the SASI (Endeve, Cardall, & Budiardja Mezzacappa Reference Endeve, Cardall and Mezzacappa2010; Endeve et al. Reference Endeve, Cardall, Budiardja, Beck, Toedte, Mezzacappa and Blondin2012) could also help to facilitate shock revival with magnetic fields acting as a subsidiary to neutrino heating but without directly powering the explosion as in the magnetorototational mechanism. The 2D simulations of Obergaulinger et al. (Reference Obergaulinger, Janka and Aloy2014) demonstrated that magnetic fields can help organise the flow into large-scale modes and thereby allow earlier explosions, though the required initial field strengths for this are higher ( $\mathord {\sim } 10^{12} \, {\rm G}$ ) than the typical values predicted by stellar evolution models.

3.4.2 Higher neutrino luminosities and mean energies?

Another possible solution for the problem of missing or delayed explosions in 3D lies in increasing the electron flavour luminosity and mean energy. This is intuitive from Equation (13), where a mere change of $\mathord {\sim }5\%$ in both L _ν and E _ν results in a net effect of 16%, which is almost on par with multi-D effects.

The neutrino luminosity is directly sensitive to the neutrino opacities, which necessitates precision modelling in order to capture shock propagation and heating correctly (Lentz et al. Reference Lentz, Mezzacappa, Bronson Messer, Hix and Bruenn2012a, Reference Lentz, Mezzacappa, Bronson Messer, Hix and Bruenn2012b; Müller et al. Reference Müller, Janka and Marek2012a; see also Section 4), as well as to other physical ingredients of the CCSN problem that influence the contraction of the proto-neutron star, such as general relativity and the nuclear equation of state (Janka Reference Janka2012; Müller et al. Reference Müller, Janka and Marek2012a; Couch Reference Couch2013a; Suwa et al. Reference Suwa, Takiwaki, Kotake, Fischer, Liebendörfer and Sato2013; O’Connor & Couch Reference O’Connor and Couch2015). Often such changes to the neutrino emission come with counterbalancing side effects (Mazurek’s law); e.g., stronger neutron star contraction will result in higher neutrino luminosities and mean energies, but will also result in a more tightly bound gain region, which necessitates stronger heating to achieve shock revival.

That the lingering uncertainties in the microphysics may nonetheless hold the key to more robust explosions has long been recognised in the case of the equation of state. Melson et al. (Reference Melson, Janka, Bollig, Hanke, Marek and Müller2015b) pointed out that missing physics in our treatment of neutrino-matter interactions may equally well be an important part of the solution of the problem shock revival. Exploring corrections to neutral-current scattering cross-section due to the ‘strangeness’ of the nucleon, they found that changes in the neutrino cross-section on the level of a few 10% were sufficient to tilt the balance in favour of explosion for a $20\,\text{M}_\odot$ progenitor. Whilst Melson et al. (Reference Melson, Janka, Bollig, Hanke, Marek and Müller2015b) deliberately assumed a larger value for the contribution of strange quarks to the axial form factor of the nucleon than currently measured (Airapetian et al. Reference Airapetian, Akopov, Akopov, Andrus, Aschenauer, Augustyniak and Avakian2007), the deeper significance of their result is that Mazurek’s law can sometimes be circumvented so that modest changes in the neutrino opacities still exert an appreciable effect on supernova dynamics. A re-investigation of the rates currently employed in the best supernova models for the (more uncertain) neutrino interaction processes that depend strongly on in-medium effects (charged-current absorption/emission, neutral current scattering, Bremsstrahlung; Burrows & Sawyer Reference Burrows and Sawyer1998, Reference Burrows and Sawyer1999; Reddy et al. Reference Reddy, Prakash, Lattimer and Pons1999; Hannestad & Raffelt Reference Hannestad and Raffelt1998) may thus be worthwhile (see Bartl, Pethick, & Schwenk Reference Bartl, Pethick and Schwenk2014; Rrapaj et al. Reference Rrapaj, Holt, Bartl, Reddy and Schwenk2015; Shen & Reddy Reference Shen and Reddy2014 for some recent efforts).