NEUTRINO-DRIVEN SUPERNOVA OF A LOW-MASS IRON-CORE PROGENITOR BOOSTED BY THREE-DIMENSIONAL TURBULENT CONVECTION

In 3D convective overturn in the neutrino-heated postshock layer develops at ${{t}_{{\rm pb}}}\;\gtrsim$ 70 ms, showing the well-known Rayleigh–Taylor mushrooms (Figure 1, left) and increasing nonradial velocities (colors in Figure 2). The postshock convection reaches its maximum activity between about 100 ms and 200 ms after bounce with a prominent maximum at ∼120 ms, at which time the shock starts its accelerated expansion (cf. Figures 2 and 3) and the postshock flow becomes highly turbulent (Figure 1, middle). At ${{t}_{{\rm pb}}}\;\gtrsim$ 200 ms the shock and postshock matter expand with ∼25,000 km s⁻¹, and the ejecta attain a nearly self-similar structure (Figure 1, right). This is the time when convection and turbulence behind the shock become weaker again (less intense red in Figure 2) and mass shells leaving the PNS surface indicate that the compact remnant begins to lose material in the low-density, high-entropy baryonic wind driven by neutrino heating above the neutrinosphere.

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In our multi-dimensional simulations the shock stagnates only for ∼120–130 ms and then expands to $\left\langle {{R}_{{\rm shock}}} \right\rangle \approx 6000$ km at 400 ms postbounce (p.b.) compared to only ∼2000 km in the 1D case (Figure 3). Angle-averaged quantities X(r) are computed according to

$$\begin{eqnarray}&&\langle X(r)\rangle \equiv \frac{\int X(r)d{\Omega }}{\int d{\Omega }}.\end{eqnarray} \tag{ 3 }$$

Positive total energies in the postshock layer develop only shortly after the onset of outward shock acceleration. Interestingly, in 3D the shock expands faster and the explosion energy increases more steeply and to a higher value than in 2D. At ${{t}_{{\rm pb}}}=400$ ms the diagnostic energy reaches

$$\begin{eqnarray}&&E_{{\rm expl}}^{3{\rm D}}(400\;{\rm ms})\sim (0.77\;\ldots \;1.05)\times {{10}^{50}}\;{\rm erg}\end{eqnarray} \tag{ 4 }$$

in 3D with an increase at a rate of ∼(0.75 ... 1.25) × 10⁵⁰ erg s⁻¹ due to the PNS wind for the cases of minimal and maximal energy, respectively, whereas it is only ∼(0.67 ... 0.92) × 10⁵⁰ erg in 2D with a similar growth rate by the PNS-wind power as in 3D.⁴

While the onset of the explosion in multi-dimensions is aided by buoyant convection, the steeper rise of the explosion energy in 3D can be understood by a smaller net energy-loss rate, ${{\dot{Q}}_{{\rm cool}}}$, in the neutrino-cooling layer during 100 ms $\lesssim \;{{t}_{{\rm pb}}}\lesssim 200$ ms (Figure 3). ${{\dot{Q}}_{{\rm cool}}}$ is evaluated between the mean gain radius, R_gain, and an inner radius ${{R}_{0}}\equiv r({{\tau }_{{{\nu }_{e}}}}=3)$, which is (somewhat arbitrarily) defined at an optical depth of 3 for ${{\nu }_{e}}$ (all quantities are evaluated on angle-averaged profiles). In contrast, the integrated net neutrino-heating rate in the gain layer, ${{\dot{Q}}_{{\rm gain}}}$, and the heating efficiency, $\eta _{{\rm heat}}^{{\rm gain}}={{\dot{Q}}_{{\rm gain}}}{{({{\dot{E}}_{{{\nu }_{e}}}}+{{\dot{E}}_{{{{\bar{\nu }}}_{e}}}})}^{-1}}$ (with ${{\dot{E}}_{{{\nu }_{i}}}}$ being the direction-averaged luminosities of ${{\nu }_{e}}$ and ${{\bar{\nu }}_{e}}$), do not exhibit any significant 3D–2D differences (Figure 3).

The reduced energy-loss rate in the cooling layer for 100 ms $\lesssim {{t}_{{\rm pb}}}\lesssim 200$ ms is associated with a systematically lower specific internal energy, $\bar{e}_{\operatorname{int}}^{{\rm cool}}\equiv \left( \int _{{{R}_{0}}}^{{{R}_{{\rm gain}}}}dV\rho e \right)M_{{\rm cool}}^{-1}$, averaged over the cooling-layer mass ${{M}_{{\rm cool}}}=\int _{{{R}_{0}}}^{{{R}_{{\rm gain}}}}dV\rho$ (Figure 3). This suggests lower temperatures in the cooling layer, less efficient neutrino emission, and therefore an inward shift of the gain radius in 3D compared to 2D: $R_{{\rm gain}}^{3D}\lt R_{{\rm gain}}^{2D}$ (Figure 3).

Consequently, the mass of the gain layer, ${{M}_{{\rm gain}}}=\int _{{{R}_{{\rm gain}}}}^{\left\langle {{R}_{{\rm shock}}} \right\rangle }dV\rho$, is larger and, correspondingly, the mass in the cooling layer, ${{M}_{{\rm cool}}}$, smaller in 3D. While the 3D–2D difference of M_cool varies with time because the definition of R₀ depends on model-specific details, the gain-mass difference grows to ${\Delta }{{M}_{{\rm gain}}}\approx 1.2\times {{10}^{-3}}\;{{M}_{\odot }}$ between 100 and 200 ms and remains essentially constant afterwards. The nuclear recombination energy of this mass difference,

$$\begin{eqnarray}&&{\Delta }{{M}_{{\rm gain}}}\times {{e}_{{\rm recomb}}}\sim (1.6\;\ldots \;2.0)\times {{10}^{49}}\;{\rm erg}\gtrsim {\Delta }E_{{\rm expl}}^{3{\rm D}-2{\rm D}}\end{eqnarray} \tag{ 5 }$$

for ${{e}_{{\rm recomb}}}\sim (7\;\ldots \;8.8)\;{\rm MeV}/{\rm nucleon}\approx (6.76\;\ldots \;8.49)$ $\times \;{{10}^{18}}$erg g⁻¹ (depending on whether free nucleons recombine to α-particles or iron-group elements) can easily account for the 3D–2D differences of the explosion energy in Figure 3.

Turbulent pressure in the gain layer is understood to be supportive for the onset of neutrino-driven explosions (Murphy et al. 2013; Couch & Ott 2014; Müller & Janka 2014b). However, for our low-mass SN progenitor this effect cannot be crucial for the observed 2D–3D difference. Despite the faster and more energetic explosion, the 3D model exhibits a lower turbulent kinetic energy in the gain layer during the crucial period 100 ms $\lesssim \;{{t}_{{\rm pb}}}\lesssim \;200$ ms, visible by the nonradial contribution $E_{{\rm kin},\theta ,\phi }^{{\rm gain}}=\int _{{{R}_{{\rm gain}}}}^{\left\langle {{R}_{{\rm shock}}} \right\rangle }dV\;\frac{1}{2}\rho (v_{\theta }^{2}+v_{\phi }^{2})$ in Figure 3. Taking into account the larger mass in the gain layer, it means that the nonradial fluid motions are much less vigorous in the 3D case.

In contrast, the nonradial kinetic energy in the cooling layer, $E_{{\rm kin},\theta ,\phi }^{{\rm cool}}=\int _{{{R}_{0}}}^{{{R}_{{\rm gain}}}}dV\;\frac{1}{2}\rho (v_{\theta }^{2}+v_{\phi }^{2})$, is smaller in the 2D simulation despite the higher mass of this layer in 2D. This suggests that nonradial motions are much more efficiently damped by dissipation in stronger deceleration shocks and due to symmetry constraints (reflective axis boundaries) in the cooling layer of the 2D model.

The reason for more favorable explosion conditions in 3D becomes clearer through radial profiles of the turbulent accretion flow in the gain and cooling layers (Figures 4 and 5). Here, angular averages of quantities X(r) over infalling matter, i.e., over regions with radial velocity v_r < 0, are computed according to

$$\begin{eqnarray}&&{{\langle X(r)\rangle }_{{{v}_{r}}\lt 0}}\equiv \frac{\int X(r){\Theta }(-{{v}_{r}})d{\Omega }}{\int {\Theta }(-{{v}_{r}})d{\Omega }},\end{eqnarray} \tag{ 6 }$$

where Θ(x) is the heaviside function with Θ(x) = 1 for $x\geqslant 0$ and Θ(x) = 0 otherwise.

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Radial velocity profiles of v_r, angle-averaged over convective downdrafts in the postshock accretion layer, exhibit stronger local variations with higher extrema and pronounced intermittency-like behavior in 2D, whereas the angle-averaged infall velocities in the 3D model appear smoother and show less extreme fluctuations (Figure 4, left). This points to a larger number of smaller-scale convective structures, thus reducing the statistical variance, and suggests less vigorous radial mass motions in the 3D model, which is compatible with the lower nonradial kinetic energy in the gain layer between 100 and 200 ms p.b. (cf. Figure 3).

This conclusion is supported by cross-sectional cuts (Figure 5), where narrow convective downflows in 2D possess higher velocities, in agreement with radial profiles of v_r averaged over inflows and time-averaged over 100 ms $\leqslant {{t}_{{\rm pb}}}\leqslant 200$ ms (Figure 4). The higher infall velocities in 2D in the gain layer and a fair part of the cooling layer are associated with a greater mass-accretion rate around the gain radius (Figure 4). Moreover, in 2D a larger fraction of the sphere in the gain layer is subtended by downflows, which is expressed by the "downflow filling factor,"

$$\begin{eqnarray}&&{{\alpha }_{{{v}_{r}}\lt 0}}\equiv \frac{\int {\Theta }(-{{v}_{r}})d{\Omega }}{\int d{\Omega }},\end{eqnarray} \tag{ 7 }$$

in Figure 4. In contrast, in the cooling layer $\alpha _{{{v}_{r}}\lt 0}^{2{\rm D}}\lt \alpha _{{{v}_{r}}\lt 0}^{3{\rm D}}$. All these findings are in line with higher kinetic energies of radial and nonradial flows in the gain layer and slightly lower nonradial kinetic energy in the cooling layer for the 2D model (Figure 3).

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At first glance, the higher mass-accretion rate around R_gain in 2D seems to contradict the 2D–3D differences of the flow geometry visible in Figure 5: in 2D one counts one or two prominent convective downdrafts, consistent with a convective cell pattern characterized by low spherical-harmonics modes, i.e., large-scale structures with few downflows. In contrast, many more downflow funnels exist in 3D, although with smaller infall velocities. Nevertheless, the 2D model channels more mass per time through the gain radius. Obviously, the total volume of toroidal "sheets" formed by axi-symmetric (2D) downflows is bigger than the volume encompassed by 3D accretion funnels. Exactly this is expressed by the larger ${{\alpha }_{{{v}_{r}}\lt 0}}$ in the gain layer, which, together with bigger infall velocities, accounts for the higher mass-accretion rate in 2D.

The reduced rate and less powerful infall of mass through the gain radius is therefore the fundamental reason for more favorable explosion conditions in 3D. Higher downflow velocities in 2D lead to more violent deceleration of the accretion downdrafts in shocks as they penetrate into the cooling layer (see Figure 5 and the more dramatic decrease of $|{{\left\langle {{v}_{r}} \right\rangle }_{{{v}_{r}}\lt 0}}|$ in 2D in Figure 4). On the one hand this causes stronger gravity-wave activity below R_gain, which can be recognized by larger angular variations of v_r for the 2D model in this region in Figure 5. On the other hand the impact of downflows dissipates kinetic energy, for which reason the 2D simulation shows a lower kinetic energy of nonradial mass motions in the cooling layer despite its higher nonradial kinetic energy in the heating layer (Figure 3). More important, however, is the enhanced shock heating of the cooling layer, which leads to higher temperatures in this region in 2D (Figure 4). Estimates confirm that this difference of the angle-averaged temperature of several 10⁹ K (or several percent of the local angle-averaged temperature) is responsible for the larger net cooling rate of the 2D model just below R_gain (Figure 4), because local temperature fluctuations are considerably bigger than the relative differences of the angular averages and neutrino emission rates by e^±-captures and e⁺e⁻-pair annihilation scale with T ⁶ and T ⁹, respectively. Consequently, one finds $R_{{\rm gain}}^{2{\rm D}}\gt R_{{\rm gain}}^{3{\rm D}}$ during 170 ms $\lesssim \;{{t}_{{\rm pb}}}\lesssim$ 220 ms (Figure 4, bottom left). Since the temperature in the gain layer is only marginally higher in 3D, the net heating rates between R_gain and shock are hardly distinguishable for 2D and 3D (Figure 4, bottom right).

The 2D–3D differences in the convective postshock layer, in particular higher downflow velocities and larger mass-infall rate in 2D, can be attributed to well-known differences of 2D and 3D turbulence discussed previously (e.g., Burrows et al. 2012; Hanke et al. 2012; Couch 2013; Dolence et al. 2013; Abdikamalov et al. 2014; Couch & O'Connor 2014): turbulent energy cascading leads to fragmentation of large-scale flows to smaller-scale vortices in 3D, whereas the inverse energy cascade in 2D tends to enhance kinetic energy on the largest possible scales. The turbulent fragmentation (e.g., through Kelvin–Helmholtz instability) of the postshock accretion flow affects the flow characteristics most strongly near the gain radius, for which reason 2D–3D differences of ${{\dot{M}}_{{{v}_{r}}\lt 0}}$ increase as the infalling gas approaches R_gain (Figure 4). Because of enhanced turbulent energy in small-scale vortices, the 3D flow becomes less coherent, and turbulent small-scale structures keep mass in the gain layer and reduce ${{\dot{M}}_{{{v}_{r}}\lt 0}}$, while in 2D the mass-infall rate continues to grow all the way toward R_gain.