EXPLOSIVE NUCLEOSYNTHESIS IN THE NEUTRINO-DRIVEN ASPHERICAL SUPERNOVA EXPLOSION OF A NON-ROTATING 15 M_☉ STAR WITH SOLAR METALLICITY

To calculate the structure and evolution of the collapsing star, we solve the Newtonian hydrodynamic equations

$$\begin{equation} \frac{D{\rho }}{D{t}}+\rho \nabla \cdot \mathbf {v}=0, \end{equation} \tag{ 1 }$$

$$\begin{equation} \rho \frac{D{\mathbf {v}}}{D{t}}=-\nabla P -\rho \nabla (\Phi +\Phi _{c}), \end{equation} \tag{ 2 }$$

$$\begin{equation} \rho \frac{d}{dt}\displaystyle {\Bigl (\frac{e}{\rho } \Bigr)} = - P \nabla \cdot \mathbf {v} + Q_{\rm E}, \end{equation} \tag{ 3 }$$

$$\begin{equation} \frac{D{Y_e}}{D{t}} = Q_{\rm N}, \end{equation} \tag{ 4 }$$

where $\rho,P,\mathbf {v},e$, and Y_e are the mass density, the pressure, the fluid velocity, the internal energy density, and the electron fraction, respectively. We denote the Lagrange derivative as D/Dt. The gravitational potential of fluid and the central object with a mass of M_in, Φ and Φ_c, are evaluated with

$$\begin{equation} \bigtriangleup {\Phi } = 4\pi G \rho \end{equation} \tag{ 5 }$$

and

$$\begin{equation} \Phi _c = - \frac{{\it GM}_{\rm in}}{r}, \end{equation} \tag{ 6 }$$

where G is the gravitational constant. We note that M_in continuously increases due to mass accretion through the inner boundary.

Q_E and Q_N are the source terms that describe the rate of change per unit volume in Equations (3) and (4), respectively, and will be summarized in Appendices A and B. In the present study, we take into account the absorption of electron and anti-electron neutrinos as well as neutrino emission through electron and positron captures, electron–positron pair annihilation, nucleon–nucleon bremsstrahlung, and plasmon decays. We assume that the fluid is axisymmetric and that neutrinos are isotropically emitted from the neutrino spheres with given luminosities and with the Fermi–Dirac distribution of given temperatures (Ohnishi et al. 2006). Rates for the absorption of neutrinos and neutrino emission through electron and positron captures are taken from Scheck et al. (2006, Appendix D). Geometrical factor f_ν is set to be

$$\begin{equation} f_\nu \, = \, {\frac{1}{2}} \big [ 1+\sqrt{1-(R_\nu /r)^2} \; \big ], \end{equation} \tag{ 7 }$$

as in Scheck et al. (2006). Here, R_ν is the radius of the neutrino sphere and is simply estimated with the relation, $L_{\nu } = \frac{7}{16}\sigma T_{\nu }^{4} \cdot 4\pi R_{\nu }^{2}$ for a given set of the luminosity L_ν and temperature T_ν (Ohnishi et al. 2006), where σ is the Stefan–Boltzmann constant. We adopt rates for the emission of neutrinos ($\nu _{\rm e},{\bar{\nu }}_{\rm e},\nu _x,\bar{\nu }_x$) through pair annihilation, bremsstrahlung, and plasmon decays as in Ruffert et al. (1996, Appendix B). Moreover, we include the heating term in Q_E due to the absorption of neutrinos on ⁴He and the inelastic scatterings on ⁴He via neutral currents (Haxton 1988; Ohnishi et al. 2007).

The numerical code for the hydrodynamic calculations employed in this paper is based on the ZEUS-2D code (Stone & Norman 1992; Ohnishi et al. 2006). We use a realistic equation of state (EOS) based on the relativistic mean field theory (Shen et al. 1998). For a lower density regime (ρ < 10⁵ g cm⁻³), where no data is available in the EOS table with the Shen EOS, we use another EOS, which includes contributions from an ideal gas of nuclei, radiation, and electrons and positrons with arbitrary degrees of degeneracy (Blinnikov et al. 1996). We carefully connect two EOS at ρ = 10⁵ g cm⁻³ for physical quantities to vary continuously in density at a given temperature (Fujimoto et al. 2006).

First, we perform a spherical symmetric hydrodynamic simulation of the core collapse of a 15 M_☉ non-rotating star with the solar metallicity (Woosley & Weaver 1995) using a hydrodynamic code (Kotake et al. 2004) for about 10 ms after core-bounce, when the bounce shock turns into a standing accretion shock and the protoneutron star (PNS) grows to ∼1.2 M_☉. Then, we map distributions of densities, temperatures, radial velocities, and electron fractions of the spherical symmetric simulation to initial distribution for 2D hydrodynamic simulations. After the remap, the central region inside 50 km in radius is excised to follow a long-term postbounce evolution (e.g., Scheck et al. 2006; Kifonidis et al. 2006). We impose velocity perturbations on the unperturbed radial velocity in a dipolar manner and follow the postbounce evolution. The spherical coordinates are used in our simulations, and the computational domain is extended over $50 {\, \rm km}\le r \le \rm 50{,}000 {\, \rm km}$ and 0 ⩽ θ ⩽ π, or from the Fe core to inner O-rich layers, which are covered with 500(r) × 128(θ) meshes. The mass is 3.17 M_☉ in the computational domain. We note that convective motion occurs at the onset of the 2D simulation with the above meshes, while the motion does not appear in the case of coarser mesh points of 500(r) × 60(θ). Evolution of the explosion energy and of the mass ejection rate are very similar to those for high-resolution simulations with 300(r) × 196(θ) meshes ($50 {\, \rm km}\le r \le \rm 3000 {\, \rm km}$ and 0 ⩽ θ ⩽ π) for about 350 ms after the core bounce. Therefore, the resolution of the simulations with 500(r) × 128(θ) meshes seems to be appropriate for the present study. However, the resolution may be too low to follow later time evolution of the explosion toward homologous expansion (Gawryszczak et al. 2010). For the high-resolution simulation, the minimum grid size in the radial- and lateral(θ)-directions, δr and δθ, is 1 km and π/196, respectively, while δr = 1 km and δθ = π/128 for our fiducial set (i.e., 500(r) × 128(θ) mesh points).

We have performed simulations for models with the electron–neutrino luminosities, $L_{\nu _e} =$3.7, 3.9, 4.0, 4.2, 4.5, 4.7, and 5.0 × 10⁵² erg s⁻¹ for 1–2 s after the core bounce, when a shock front has reached a layer with $r = \rm 10{,}000{\, \rm km}$ in almost all directions. We take the input neutrino luminosities as above because the revival of the stalled bounce shock occurs only for models with $L_{\nu _e} \ge$ 3.9 × 10⁵² erg s⁻¹, and also because for models with $L_{\nu _e} > 5.0 \times 10^{52} \rm \, erg \,s^{-1}$, the star explodes too early for the SASI to grow, as will be discussed later. We set $L_{\bar{\nu _e}} = L_{\nu _e}$ and $L_{\nu _x} = 0.5 L_{\nu _e}$, where $L_{\bar{\nu _e}}$ and $L_{\nu _x}$ are the luminosities of anti-electron neutrino and other types (μ, τ, anti-μ, and anti-τ), respectively. We consider models with neutrino temperatures, $T_{\nu _e}$, $T_{\bar{\nu _e}}$, and $T_{\nu _x}$ as 4 MeV, 5 MeV, and 10 MeV, respectively (Ohnishi et al. 2006). The adopted neutrino luminosities are comparable to those with a more accurate transport scheme, but the temperatures are slightly higher (Marek et al. 2009; Marek & Janka 2009). We present hydrodynamic and nucleosynthetic results for cases with lower neutrino temperatures in Section 4.1.

We confirm that the explosion is highly aspherical and l = 1 and l = 2 modes are dominant as shown in Kifonidis et al. (2006), Ohnishi et al. (2006), and Scheck et al. (2006), although the shape of the explosion strongly depends on numerical details, such as mesh resolution and boundary conditions (Kifonidis et al. 2006; Scheck et al. 2006). The entropy contour of the hydrodynamic simulation is shown in Figure 1 for the case with $L_{\nu _e} = 4.5 \times 10^{52} \rm \, erg \, s^{-1}$. Most of the SN ejecta have entropy less than 20k_B, where k_B is the Boltzmann constant. Entropy reaches 70k_B for small amounts of the ejecta.

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We find that for models with $L_{\nu _e} \ge 3.9 \times 10^{52} \rm \, erg \, s^{-1}$, the star explodes aspherically via the neutrino heating aided by SASI. Figure 2(a) shows explosion energies as a function of $L_{\nu _e}$ for all the exploded models. The energies are estimated at an epoch of 500 ms after the explosion and slightly increase after the epoch. Kinetic and thermal energies of the explosion are also shown in Figure 2(a). The thermal energies dominate the kinetic ones.

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Higher $L_{\nu _{\rm e}}$ makes the onset of explosion earlier and the mass of the PNS (M_pNS) smaller. Note that we estimate M_pNS at t = t_exp + 500 ms, because M_pNS only slightly increases later than 500 ms after t_exp. Here, t_exp indicates the timescale when the explosion sets in, which can typically be estimated when the mass ejection rate at 100 km reaches 0.1M_☉ s⁻¹ in our 2D simulations. Figure 2(b) shows the explosion time t_exp and M_pNS as a function of $L_{\nu _{\rm e}}$. Except for the lowest luminosity model ($L_{\nu _e} \le 3.9 \times 10^{52} \rm \, erg \, s^{-1}$), M_pNS is in the range of 1.54–1.70 M_☉. These values are much larger than the so-called mass cuts in the spherical models of 15 M_☉ progenitors by Hashimoto (1995) and Rauscher et al. (2002), which are 1.30 M_☉ and 1.32 M_☉, respectively. The mass of the PNS, however, becomes larger (1.68 M_☉) due to the fallback of ejecta (Rauscher et al. 2002).

In order to calculate the chemical composition of the SN ejecta, we need the Lagrangian evolution of physical quantities, such as density, temperature, and velocity of the material. We adopt a tracer particle method (Nagataki et al. 1997; Seitenzahl et al. 2010) to calculate the Lagrangian evolution of the physical quantities from the Eulerian evolution obtained from our simulations. The Lagrangian evolution is followed during the 2D aspherical simulation as well as during the spherical collapsing phase. To obtain information on mass elements, 6000 tracer particles are placed in the regions from 300 extending to 10,000 km (the O-rich layer). We have confirmed that the estimated energies and masses of the ejecta with 6000 particles are equal to the ones with 3000 particles within ∼1% accuracy, and the obtained abundance profiles are also very similar.

Initial abundances of the particles are set to be those of the star just before the core collapse (Rauscher et al. 2002), in which 1400 nuclei are taken into account. We note that a pre-SN model in Rauscher et al. (2002) has smaller helium, carbon–oxygen, and oxygen–neon core masses, compared with those in Woosley & Weaver (1995), due to coupled effects through the inclusion of mass loss and the revisions of opacity and nuclear inputs (Rauscher et al. 2002). The mass of a particle in a layer is weighted to the mass of the layer. We note that the minimum mass of the particles is ∼10⁻⁴ M_☉. We find that more than one fifth of the particles are ejected due to the aspherical explosion.

Next, we calculate abundances and masses of the SN ejecta. Ejecta that are located on the inner region of the star ($r_{\rm ej,cc} \le \rm 10{,}000{\, \rm km}$) before the core collapse have maximum temperatures high enough for elements heavier than C to burn explosively. Here, r_{ej, cc} is the radius of the ejecta at the core collapse. We therefore follow abundance evolution of the ejecta from the inner region using a nuclear reaction network, which includes 463 nuclide from neutron, proton to Kr (Fujimoto et al. 2004). We will discuss effects of neutrino interactions on heavy nuclei and uncertainty in nuclear reaction rates on nucleosynthetic results in Sections 4.2 and 4.3, respectively. The abundances of ejecta from the outer region ($r_{\rm ej,cc} > \rm 10{,}000{\, \rm km}$) are set to be those before the core collapse (Rauscher et al. 2002). We note that the mass of the outer region, or $r_{\rm ej,cc} > \rm 10{,}000{\, \rm km}$, is 10.4 M_☉. Moreover, when temperatures of the ejecta are greater than 9 × 10⁹ K, we set chemical composition of the ejecta to be that in nuclear statistical equilibrium, whose abundances are expressed with simple analytical expressions, specified by the density, temperature, and electron fraction.

Electron fractions of the ejecta are re-evaluated during SN explosion coupled with the nuclear reaction network. The change in Y_e is taken into account through electron and positron captures on heavy nuclei, in addition to electron and positron captures on neutrons and protons, as well as absorption of ν_e and ${\bar{\nu }}_{\rm e}$ on neutrons and protons. The captures and absorptions on neutrons and protons are also taken into account in the hydrodynamic simulations. The rates for the captures and the absorptions are adopted from Fuller et al. (1980, 1982) and Scheck et al. (2006), respectively.

It should be emphasized that post-processing electron fractions are slightly different (up to 10%) from those estimated with hydrodynamic simulations, in which the evolution of electron fractions is followed. This is because abundances of neutrons and protons in the network calculations are slightly different from those estimated with EOS in the hydrodynamic simulations. We note that 463 nuclei are taken into account in the network calculations, while only neutrons, protons, ⁴He, and a representative heavier nuclide are evaluated with EOS.

In the neutrino-heating-dominated region, the abundances of neutrons and protons in the network calculations are larger than those evaluated with EOS. Hence, if we perform hydrodynamic simulations, in which abundances of nucleons are reliably evaluated with the reaction network, the neutrino heating rates in the simulations could increase compared to those in the current study, since the neutrino heating through the absorption of ν_e and ${\bar{\nu }}_{\rm e}$ is dominant over the other heating reactions, and the heating rates via the absorption are proportional to the abundances of the nucleons.

The explosion energies also might increase. We emphasize that abundances of SN ejecta chiefly depend on the explosion energy and the mass of SN ejecta from an inner region, not on $L_{\nu _{\rm e}}$, as will be shown later.

Maximum densities, ρ_max, and maximum temperatures, T_max, are good indicators for the composition of SN ejecta (Thielemann et al. 1996). Figure 3(a) shows ρ_max as a function of T_max of the ejecta for $L_{\nu _e} = 4.5 \times 10^{52} \rm \, erg \, s^{-1}$. Most of the ejecta with relatively low densities (<10⁸ g cm⁻³) have ρ_max and T_max similar to those of ejecta in the spherical model of core-collapse SNe (Thielemann et al. 1998). For some particles that have very high densities along with high temperatures (ρ_max ⩾ 10⁹ g cm⁻³ and T_max ⩾ 10¹⁰ K), electron captures operate to some extent, so that these particles become slightly neutron-rich, $Y_e\rm (\rm 10{,}000{\, \rm km}) < 0.48$. Here, $Y_e\rm (\rm 10{,}000{\, \rm km})$ represents the electron fraction for tracer particles evaluated when the particles reach $r=\rm 10{,}000{\, \rm km}$. This may be a useful quantity for measuring Y_e of the ejecta, since Y_e closely freezes out at $r > \rm 10{,}000{\, \rm km}$ (except through β-decays at a later epoch). All ejecta with T_max > 10¹⁰ K fall down to the heating region ⩽200–300 km to be heated via neutrinos (Figure 3(b)). For these neutrino-heated ejecta, ρ_max ranges from 10⁸ to 2 × 10¹⁰ g cm⁻³.

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For ejecta with higher ρ_max, electron captures on protons proceed more efficiently to make Y_e smaller. Time evolution of the physical quantities of such an ejecta is shown during the infall of the ejecta near the cooling region (r < 100 km) in Figure 4(a). As the density and temperature rise to more than 10⁹ g cm⁻³ and 10¹⁰ K, respectively, the electron fraction decreases due to the electron captures. When the ejecta start to be released via neutrino heating at t = 0.42 s, the electron fraction increases through the absorption of ν_e by neutrons. Finally, $Y_e \rm (\rm 10{,}000{\, \rm km})$ becomes 0.461 for the ejecta.

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On the other hand, the electron captures on protons are not efficient for a proton-rich ejecta with $Y_e\rm (\rm 10{,}000{\, \rm km}) = 0.559$, as shown in Figure 4(b). This is because the densities of the inner region (r ⩽ 200 km) are relatively low (⩽10⁸ g cm⁻³) due to the mass ejection during an earlier phase (⩾t_exp). The electron fraction therefore remains constant and rises from 0.5 to 0.559 via the ν_e absorption in an inner region r ⩽ 200 km. The proton richness in the ejecta is caused by the small energy difference between ν_e and ${\bar{\nu }}_{\rm e}$. For $T_{\nu _{\rm e}}$ and $T_{{\bar{\nu }}_{\rm e}}$ adopted in our simulations, the relation, $4(m_n -m_p) > \epsilon _{{\bar{\nu }}_{\rm e}} -\epsilon _{\nu _{\rm e}}$, holds, which leads to Y_e > 0.5 (Fröhlich et al. 2006a), where m_n and m_p are masses of neutrons and protons, and $\epsilon _{{\bar{\nu }}_{\rm e}}$ and $\epsilon _{\nu _{\rm e}}$ are energies of anti-electron and electron neutrinos, respectively. We note that $\epsilon _{{\bar{\nu }}_{\rm e}} = 15.8\, {\rm MeV}$ for $T_{{\bar{\nu }}_{\rm e}} = 5\, {\rm MeV}$ and $\epsilon _{\nu _{\rm e}}= 12.6\, {\rm MeV}$ for $T_{\nu _{\rm e}} = 4\, {\rm MeV}$.

Figure 5 shows masses as a function of $Y_e \rm (\rm 10{,}000\,km)$ of ejecta from the inner region $r_{\rm ej,cc} \le \rm 10{,}000{\, \rm km}$. We find that most of the ejecta (98.8%) have electron fractions of 0.49–0.5. Small fractions of the ejecta, 0.9% and 0.3% in mass, are slightly neutron-rich (0.46 < Y_e < 0.49) and proton-rich (0.5 < Y_e < 0.56), respectively. Masses of the slightly neutron- and proton-rich ejecta are larger for models with larger L_ν, while the mass fractions of these ejecta are comparable for all the models.

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Electron fraction of the ejecta with the minimum radial position r_min ⩽ 200–300 km changes due to high neutrino flux and/or efficient e^± capture (Figure 4(a)). Figure 6 shows $Y_e \rm (\rm 10{,}000\,km)$ as a function of r_min of the ejecta for a model with $L_{\nu _e} = 4.5 \times 10^{52} \rm \, erg \, s^{-1}$. Hereafter, we refer to the ejecta with r_min ⩽ 200 km as the primary ejecta, which have high maximum densities ⩾10⁸ g cm⁻³ and temperatures ⩾10¹⁰ K (Figure 3). The ejecta are heated through the neutrino heating. On the other hand, the others are refereed as the secondary ejecta, heated chiefly via the shock wave driven by the primary ejecta.

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It is true that Y_e of the primary ejecta can change if we adopt a more accurate neutrino-transfer scheme instead of the simplified light-bulb transfer scheme. Abundances of the primary ejecta are therefore highly uncertain because of the uncertainty of their Y_e. On the other hand, for the secondary ejecta, Y_e changes chiefly through the neutrino absorptions, but the changes in Y_e are found to be less than 1% for almost all the secondary ejecta. In addition, for our typical 2D models that produce energetic explosions ($L_{\nu _e} \ge 4.5 \times 10^{52} \rm \, erg \, s^{-1}$), the masses of the primary ejecta occupy only about 2% (8.7 × 10⁻³ M_☉) of the ejecta from the inner region (0.41 M_☉ for $r_{\rm ej,cc} \le \rm 10{,}000{\, \rm km}$). Therefore, Y_e of the secondary ejecta, whose mass is much larger than that of the primary ejecta, is unlikely to be largely changed even if we use a more accurate neutrino-transfer scheme. We conclude that masses of abundant nuclei, such as ¹⁶O, ²⁸Si, and ⁵⁶Ni, do not largely change in the SN ejecta. We will discuss this point in Section 4.4.

The value of Y_e of the primary ejecta depends on the epoch of the ejection. The primary ejecta that eject in an early phase (before an epoch of 200–300 ms after the explosion [t_ej = 230 ms]) are mainly neutron-rich, while the primary ejecta are proton-rich in the later phase. The proton-rich primary ejecta corresponds to neutrino-driven winds that are possibly not neutron-rich but proton-rich (Fischer et al. 2010; Hüdepohl et al. 2010). Figure 7 shows $Y_e \rm (\rm 10{,}000\,km)$ as a function of t(T_max) of ejecta for a model with $L_{\nu _e} = 4.5 \times 10^{52} \rm \, erg \, s^{-1}$, where t(T_max) is defined as the time when the temperature of ejecta attains its maximum value, T_max. We note that the gas starts to be ejected just after t = t(T_max), as shown in Figure 4. We note that the dependence of Y_e on the epoch of the ejection also appears in a spherical simulation of SN explosion of a star with an ONeMg core using an elaborated code taking into account an accurate neutrino transfer scheme (Kitaura et al. 2006; Wanajo et al. 2009).

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Masses of nuclei, such as ⁵⁶Ni, ⁵⁷Ni, ⁵⁸Ni, and ⁴⁴Ti, have been estimated in some SN remnants. For SN1987A, masses of ⁵⁶Ni and ⁴⁴Ti are deduced to be ≃ 0.07 M_☉ (Shigeyama et al. 1988; Woosley 1988) and (1–2) × 10⁻⁴ M_☉ (Nagataki 2000 and references therein), respectively. The estimated mass of ⁴⁴Ti is comparable to that in Cas A (1.6^+0.6_{− 0.3} × 10⁻⁴ M_☉; Renaud et al. 2006) and greater than that in the youngest Galactic SN remnant G1.9+0.3 ((1–7) × 10⁻⁵ M_☉; Borkowski et al. 2010), which may originate from a Type Ia event. Figure 8 shows masses of SN ejecta ejected from the inner region ($r_{\rm ej,cc} \le \rm 10{,}000{\, \rm km}$), M_{ej, in}, and masses of ⁵⁶Ni and ⁴⁴Ti. Masses of the ejecta from the inner region of $r_{\rm ej,cc} \le \rm 10{,}000{\, \rm km}$ (solid line with filled squares) and masses of ⁵⁶Ni (dashed line with filled circles) and ⁴⁴Ti (dotted line with filled triangles) are shown with a value multiplied by a factor of 1, 10, and 10⁴, respectively. We find that masses of the ejecta and ⁵⁶Ni roughly correlate with the neutrino luminosities. The masses of ⁵⁶Ni and ⁴⁴Ti are less than and comparable to those in the spherical model (Rauscher et al. 2002), or 0.11 M_☉ and 1.4 × 10⁻⁵ M_☉, respectively. Note that the explosion energy is 1.2 × 10⁵¹ erg in the model. We find that ⁴⁴Ti relative to ⁵⁶Ni is much smaller than those in the solar system, in SN1987A, and in Cas A, but comparable to that in the SN remnant G1.9+0.3. ⁵⁷Ni relative to ⁵⁶Ni is comparable to that in the solar system and in SN1987A, while ⁵⁸Ni relative to ⁵⁶Ni is overproduced compared with that in the solar system and in SN1987A, because it is abundantly produced in the slightly neutron-rich ejecta (Hashimoto 1995; Nagataki et al. 1997).

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It should be emphasized that ⁴⁴Ti is underproduced in our simulations of the SN explosion, contrary to the overproduction in the previous 2D results (Nagataki et al. 1997), in which the explosion energy is aspherically and artificially added and the remnant mass is set to a value in order that the ejected mass of ⁵⁶Ni reproduces the mass observed in SN1987A in Nagataki et al. (1997). The underproduction is possibly caused by a larger remnant mass in our models. This is because the explosion energy and the ratio of the explosion energy on the polar axis to that on the equatorial plane are comparable to those evaluated in our simulation, and masses of ⁴⁴Ti as well as ⁵⁶Ni have been shown to strongly depend on the value of the remnant mass in 2D calculations (Young et al. 2006).

In order to compare estimated abundances with those of the solar system (Anders & Grevesse 1989), we have integrated masses of nuclei over all the ejecta to evaluate the abundances of the SN ejecta. Figure 9(a) shows overproduction factors after decays as a function of the mass number, A for $L_{\nu _e} = 4.5 \times 10^{52} \rm \, erg \, s^{-1}$, in which the explosion energy is 1.3 × 10⁵¹ erg s⁻¹ (Figure 2(a)) and M_{ej, in} is 0.41 M_☉ (Figure 8). We find that the abundance pattern of the SN ejecta is similar to that of the solar system. We note that ¹⁷O, which is underproduced in the ejecta, can be abundantly synthesized in Type Ia SNe, and that ¹⁵N and ¹⁹F are comparably produced if neutrino effects are taken into account (Woosley & Weaver 1995), as shown later in Figure 14. We point out that ⁶⁴Zn, which is underproduced in the spherical case (Rauscher et al. 2002), is abundantly produced in slightly neutron-rich ejecta (0.46 ⩽ Y_e ⩽ 0.49).

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In the ejecta, however, the neutron-rich Ni, ⁶²Ni, is overproduced. The overproduction of the neutron-rich Ni isotopes has also appeared in the spherical models (Hashimoto 1995; Rauscher et al. 2002) as well as in the asymmetric model (Nagataki et al. 1997). The overproduction has been suppressed if the electron fraction of slightly neutron-rich material (Y_e = 0.495 just before the core collapse) near the mass cut is artificially modified to 0.499 (Hashimoto 1995; Nagataki et al. 1997). Hence, it has been remarked that the overabundance of the neutron-rich Ni isotopes may inherit the uncertainty in the progenitor model, in particular in the neutronization due to the combined effects of convective mixing, electron captures, and positron decays during their Si burring stage (Hashimoto 1995). In our aspherical models, the overabundances of Ni in the primary neutron-rich ejecta are not large, other than ⁶⁰Ni, as we will discuss later (Section 4.4). The overproduction of Ni mainly takes place in slightly neutron-rich secondary ejecta (0.49 < Y_e < 0.5). We find that the electron fractions of the secondary ejecta are 0.4985 just before the core collapse and decrease by up to 1% through the neutrino absorptions. Therefore, the overabundance of ⁶²Ni possibly inherits the uncertainty not only in the progenitor model but also in the change for Y_e during the SN explosion. Moreover, the overproduction of Ni isotopes is also shown to be reduced in the ejecta from the spherical SN explosion induced via artificially enhanced neutrino heating (Fröhlich et al. 2006a).

On the other hand, the integrated abundances are largely different from those in the solar system for $L_{\nu _e} = 3.9 \times 10^{52} \rm \, erg \, s^{-1}$, in which E_exp = 0.45 × 10⁵¹ erg s⁻¹ and M_{ej, in} = 0.20 M_☉. Nuclei with A ⩾ 28 are deficient in the model (Figure 9(b)), compared with O, which is mainly ejected from outer layers. The overproduction factors of nuclei lighter than ²⁷Al are very similar for both models, because these nuclei are mainly synthesized during the hydrostatic burning. Nuclei heavier than Si are mainly produced during the SN explosion through the explosive burning in ejecta from the inner region ($r_{\rm ej,cc} \le \rm 10{,}000{\, \rm km}$).

The abundance patterns for models with M_{ej, in} ∼ (0.4–0.5) M_☉, or those with $L_{\nu _e} =$ (4.5, 4.7, and 5.0) × 10⁵² erg s⁻¹, are similar to those in the solar system. M_{ej, in}, which corresponds to masses of a neutron star, estimated as (1.54–1.62) M_☉ for these models (Figure 2(b)), are comparable to the baryonic mass (around 1.5 M_☉) of a neutron star for observed neutron-star binaries, in which the progenitor of the neutron star seems to be a star with an Fe core (Schwab et al. 2010). Moreover, E_exp/M_ej = (0.92–1.2) × 10⁵⁰ erg M⁻¹_☉ evaluated for the models are comparable to the values E_exp/M_ej = 0.76 × 10⁵⁰ erg M⁻¹_☉ estimated in SN 1987A (Shigeyama & Nomoto 1990). The time from the core collapse to the explosion, t_exp, is ∼0.2 s (Figure 2(b)), which is enough to grow a low-mode SASI. The growth is appropriate for explaining the distributions and velocities of nuclei observed in SN1987A, as shown in Kifonidis et al. (2006).

The value of M_{ej, in} depends on E_exp as well as aspherical matter infall to the neutron star, while the masses of ⁵⁶Ni correlate with E_exp (Figures 2(a) and 8). Figure 10 shows the position of SN ejecta before the core collapse for a model with $L_{\nu _e} = 4.5 \times 10^{52} \rm \, erg \, s^{-1}$. Filled squares, filled circles, filled triangles, and open triangles indicate ejecta that correspond to the complete Si burning (T_{max, 9} ⩾ 5 where T_{max, 9} = T_max/10⁹ K), the incomplete Si burning (4 ⩽ T_{max, 9} ⩽ 5), the O burning (3.3 ⩽ T_{max, 9} ⩽ 4), and the C/Ne burning (1 ⩽ T_{max, 9} ⩽ 3.3), respectively. We find that whole of the iron core (∼2000 km) collapses to the PNS and that the infall of material to the PNS is aspherical; larger amounts of the gas infall from the upper hemisphere compared with from the lower half. Larger amounts of material aspherically infall to the neutron star for a model with $L_{\nu _e} = 4.5 \times 10^{52} \rm \, erg \, s^{-1}$ compared with the model with $L_{\nu _e} = 5.0 \times 10^{52} \rm \, erg \, s^{-1}$. Though E_exp is comparable, M_{ej, in} for $L_{\nu _e} = 4.5 \times 10^{52} \rm \, erg \, s^{-1}$ is smaller than that for $L_{\nu _e} = 5.0 \times 10^{52} \rm \, erg \, s^{-1}$.

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Distributions of the energy and abundances of SN ejecta are highly aspherical, as one can expect from the entropy distribution (Figure 1). The distributions are shown for $L_{\nu _e} = 4.5 \times 10^{52} \, \rm erg \, s^{-1}$ at t = 1.5 s in Figure 11. The energy and abundances are presented in units of 10⁴⁷ erg and normalized by the solar abundances, respectively, for logarithm scale. It should be emphasized that the distributions change during the later expansion phase (Kifonidis et al. 2006; Gawryszczak et al. 2010). We note that the secondary ejecta located in outer layers ($r_{\rm ej,cc} > \rm 10{,}000{\, \rm km}$) are not shown in the figures. The high-energy ejecta concentrates on the shock front, in particular in polar regions (θ < π/4, θ > 3π/4). Both Si and O burning proceed to produce ⁵⁶Ni and ²⁸Si abundantly in the region. We emphasize that a deformed shell structure forms with Fe, Si, and O layers from inside to outside in the secondary ejecta heated via the shock wave. Composition of the secondary ejecta depends chiefly on T_max, which is determined via the shock heating. The shell structure is therefore formed in the secondary ejecta, because T_max gradually decreases from inner to outer layers, where the structure is highly aspherical and deformed. We note that Ca is co-produced with ²⁸Si, as in the spherical models.

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The primary ejecta, however, do not form such a structure. This is because physical properties of the ejecta, which are mainly heated via neutrinos to be >10¹⁰ K (Figure 4), are independent from their positions; thus, their compositions are also independent. A fraction of the primary ejecta that are mainly composed of Fe and Ni is found to be mixed into the deformed layers, while a small amount of the secondary ejecta falls toward the neutron star near the equatorial plane. Consequently, the ejecta with abundant O and/or Si appear in the central region (r < 2000 km). Due to such mixings and infall, some of the Si- and O-rich ejecta can penetrate more deeply into the Fe-rich ejecta, which is in sharp contrast to spherical models.

We have averaged energy and masses of nuclei of the inner ejecta over a radial direction. We note that the energy and masses of the outer ejecta ($r_{\rm ej,cc} > \rm 10{,}000{\, \rm km}$) are not included in an averaging procedure. If we include ejecta from the outer layers, the asymmetry of ¹⁶O becomes small because of the existence of spherically distributed ¹⁶O in the layers. Figure 12(a) shows the averaged energy and masses of ¹⁶O, ²⁸Si, and ⁵⁶Ni. Asphericity of the energy is prominent; the ratio of the energy at θ = π/2 to that at θ = 0 is ∼4, and the ratio of the maximum to the minimum energy is ∼8. We note that the ratios are comparable to those in 2D models (Nagataki et al. 1997, models A2 and A3). Only a small amount of ⁵⁶Ni exists near the equatorial plane (θ ∼ (0.8–1.6) rad.), where the energy is lower compared with that near the polar axis. The asymmetry of ²⁸Si mass is similar to that of the energy and smaller than that of ⁵⁶Ni. Mass distribution of ⁴⁴Ti is very similar to that of ⁵⁶Ni, as in spherical models, because both nuclei are produced in the secondary ejecta through α-rich freezeout. The peaks of ²⁸Si mass around the polar directions are misaligned with those of ⁵⁶Ni around θ ∼ π/4 and 3π/4, although the misalignment could disappear due to the lateral motion of the ejecta during the later explosion phase (⩽300 s) as pointed out in Gawryszczak et al. (2010). We note that the lateral motion does not follow during the later phase in the present study.

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In addition to the mass fractions of nuclei, number ratios of nuclei relative to O (¹⁶O) are important quantities, because the ratios are indicative of the production and destruction mechanism of nuclei. Angular distributions of the number ratios are shown for the inner ejecta ($r_{\rm ej,cc} \le \rm 10{,}000{\, \rm km}$) averaged over a radial direction for $L_{\nu _e} = 4.5 \times 10^{52} \rm \, erg \, s^{-1}$ in Figure 12(b). The ratio of Ne (²⁰Ne, solid line) is 0.1–0.2 for a direction where the O/Ne burning does not effectively operate, while it becomes much lower in a region where the O/Ne burning effectively operates to synthesize abundant Si and/or Fe relative to O. Distribution of the number ratio N(Mg)/N(O) (²⁴Mg, dash-dotted line) is similar to that in the spherical case, because Mg is produced during both the stellar evolution and the explosion through the same process, or the O/Ne burning, with which N(Mg)/N(O) becomes ∼0.05 for a 15 M_☉ progenitor. Here, N(X) is the number of nuclei, X. The ratio of Si (²⁸Si, dashed line) anti-correlates with that of Fe (the sum of ⁵⁶Ni, ⁵⁶Fe, and ⁵⁴Fe; dotted line). This is because Fe is produced via the burning of Si.

We have assumed the neutrino spheres with given luminosities and with the Fermi–Dirac distribution of neutrino temperatures, and considered models with neutrino temperatures, $T_{\nu _e}$, $T_{\bar{\nu _e}}$, and $T_{\nu _x}$ as 4 MeV, 5 MeV, and 10 MeV, respectively. The temperatures are, however, slightly lower and change with time, as shown in simulations using a more elaborate numerical code (Marek et al. 2009; Marek & Janka 2009). In order to evaluate the dependence of hydrodynamic and nucleosynthetic results on neutrino temperatures, we have performed simulations for three models with lower neutrino temperatures: $T_{\nu _e} = 3.6 \, \rm MeV$, $T_{\bar{\nu _e}}= 4.5 \, \rm MeV$, and $T_{\nu _x} = 9 \, \rm MeV$, and $L_{\nu _e} =$ (4.0, 4.5, and 5.0) × 10⁵² erg s⁻¹. We find that, for a given $L_{\nu _e}$, the explosions are much weaker for a lower T_ν model, in spite of only 10% changes in neutrino temperatures. Table 1 summarizes hydrodynamic and nucleosynthetic properties for the nine models that produce explosions, in which T_νs are changed in the three ways mentioned above. The explosion energies are (0.35, 0.35, and 0.45) × 10⁵¹ erg for models with $L_{\nu _e} =$ (4.0, 4.5, and 5.0) × 10⁵² erg s⁻¹, respectively. Mass of the ejecta from the inner region ($r_{\rm ej,cc} \le \rm 10{,}000{\, \rm km}$), M_{ej, in}, and ⁵⁶Ni mass are 0.11–0.21 M_☉ and 0.011–0.017 M_☉, respectively. Figure 13 shows overproduction factors for the low T_ν model with $L_{\nu _e} = 5.0 \times 10^{52} \rm \, erg \, s^{-1}$. We emphasize that the overproduction factors are very similar to those for the high T_ν model with $L_{\nu _e} = 3.9 \times 10^{52} \rm \, erg \, s^{-1}$, in which the explosion energy and the ejecta mass from the inner region are comparable to the above model, or 0.45 × 10⁵¹ erg and 0.020 M_☉, respectively. We conclude that the abundances of the ejecta do not directly depend on T_ν but mainly depend on the explosion energy and the ejecta mass.

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Table 1. Model Parameters and Properties of the Explosion for Nine Exploded Models

$T_{\nu _{\rm e}}$	$T_{{\bar{\nu }}_{\rm e}}$	$L_{\nu _{\rm e}}$	Eexp	texp	MpNS	Mej, in	M(56Ni)	M(44Ti)
4.0	5.0	3.9	0.45	1.20	1.84	0.20	1.85e-2	5.29e-6
4.0	5.0	4.0	0.80	0.31	1.70	0.34	2.72e-2	1.31e-5
4.0	5.0	4.2	1.02	0.31	1.65	0.39	4.51e-2	8.04e-6
4.0	5.0	4.5	1.30	0.23	1.62	0.41	5.49e-2	1.52e-5
4.0	5.0	4.7	1.03	0.21	1.54	0.49	5.10e-2	1.14e-5
4.0	5.0	5.0	1.30	0.18	1.56	0.48	5.44e-2	1.05e-5
3.6	4.5	4.0	0.35	1.41	1.93	0.11	1.10e-2	6.95e-6
3.6	4.5	4.5	0.35	1.19	1.87	0.17	1.40e-2	5.53e-6
3.6	4.5	5.0	0.45	0.62	1.83	0.21	1.73e-2	6.38e-6

Notes. Each column shows $T_{\nu _{\rm e}}$, $T_{{\bar{\nu }}_{\rm e}}$, $L_{\nu _{\rm e}}$, E_exp, t_exp, and masses (M_pNS, M_{ej, in}, M(⁵⁶Ni), and M(⁴⁴Ti)), in units of MeV, MeV, 10⁵² erg s⁻¹, 10⁵¹ erg, s, and M_☉, respectively.

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Effects of neutrino interactions on SN ejecta have previously been investigated for spherical SN explosion (Woosley et al. 1990; Woosley & Weaver 1995). We have calculated abundances of the ejecta taking into account ν interactions on heavy nuclei (Goriely et al. 2001) for $L_{\nu _e} = 4.5 \times 10^{52} \rm \, erg \, s^{-1}$. Cross sections for the neutrino reactions are taken from Woosley et al. (1990), in which rates are included for neutral and charged current reactions on He and from C to Kr. Figure 14 shows ratios of the abundances of the ejecta with ν interactions to those without the interactions. We find that the increase of the abundances due to the interactions of ν with heavy nuclei is up to 50%, except for ¹⁹F whose ratio is 2.1. ¹⁹F is enhanced via the neutral and charged current interactions of ²⁰Ne, or ²⁰Ne(νν', p)¹⁹F and ²⁰Ne$({\bar{\nu }}_{\rm e},e^+n)$¹⁹F, respectively. ¹⁵N and ⁵⁰V are synthesized through ¹⁶O(νν', p)¹⁵N and ⁵⁰Cr$({\bar{\nu }}_{\rm e},e^+)$⁵⁰V, respectively. We note that the enhancement of light elements, such as Li, Be, and B, via the ν interactions is not fully taken into account in our calculation, because the composition of ejecta from outer layers ($r_{\rm ej,cc} >\rm 10{,}000{\, \rm km}$) is fixed to the pre-SN composition, although the production of these elements is efficient in the outer layers through neutrino interactions (Woosley et al. 1990; Yoshida et al. 2008). Moreover, neutrino absorptions on D may be important for nucleosynthesis as well as the dynamics of the SN explosion (Nakamura et al. 2009).

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We have adopted reaction rates mainly taken from the REACLIB database in our nuclear reaction network presented in Section 3.1. The database was recently updated and is continuously maintained by the Joint Institute for Nuclear Astrophysics (JINA) REACLIB project (Cyburt et al. 2010). We have calculated abundances of the ejecta for $L_{\nu _e} = 4.5 \times 10^{52} \rm \, erg \, s^{-1}$, adopting reaction rates taken from the JINA REACLIB V1.0 database. Figure 15 shows ratios of the abundances of the ejecta, adopting reaction rates in the JINA REACLIB V1.0 database to those in REACLIB database. We find that the differences are small, up to 30%, between abundances with the JINA REACLIB V1.0 and REACLIB databases. If we use newly evaluated α-capture rates on ⁴⁰Ca and ⁴⁴Ti, yields of ⁴⁴Ti are likely to be lower (Hoffman et al. 2010).

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As shown in Section 3.3, the ejecta consist of the primary ejecta, whose Y_e and thus abundances are highly uncertain, and the secondary ejecta, which have a relatively definite composition and are much heavier than the primary ejecta. In order to clarify the uncertainty of the estimate of abundances of the ejecta, we have evaluated abundances of the ejecta integrated over ejecta without the primary ejecta and have compared the abundances with those summed over all the ejecta (Figure 9[a]). Figure 16 shows ratios of the abundances of all the ejecta to those without the primary ejecta for $L_{\nu _e} = 4.5 \times 10^{52} \rm \, erg \, s^{-1}$. We find that abundances of most of the nuclei with A ⩽ 70 do not greatly change within a factor of two. The overproduction factors are therefore very similar to those in Figure 9(a). The ratios are greater than 1.5 for ⁴³Ca, ⁴⁵Sc, ⁴⁷Ti, ⁵⁰Ti, ⁵⁴Cr, ⁶⁰Ni, ⁶⁴Zn, ⁶⁶Zn, and ⁷⁰Ge, in particular, 9.0, 2.8, and 3.1 for ⁶⁴Zn, ⁶⁶Zn, and ⁷⁰Ge, respectively. These nuclei are therefore chiefly synthesized in the primary ejecta, not in the secondary ejecta. Moreover, ⁵⁰Ti, ⁵⁴Cr, ⁶⁰Ni, ⁶⁴Zn, ⁶⁶Zn, and ⁷⁰Ge are synthesized in neutron-rich primary ejecta, while ⁴³Ca, ⁴⁵Sc, ⁴⁷Ti, and ⁶⁰Ni are abundantly produced in proton-rich primary ejecta through νp-processes (Fröhlich et al. 2006a, 2006b; Pruet et al. 2006; Wanajo 2006).

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In short, abundances of ⁶⁴Zn, ⁶⁶Zn, and ⁷⁰Ge are highly uncertain, but those of the other nuclei are relatively definite. If a fraction of the primary ejecta could become much larger, abundances might be highly uncertain, in particular for the nuclei that are produced in the primary ejecta. The fraction is, however, unlikely to be much larger, because the ejection of the secondary ejecta is driven by the shock wave caused by the primary ejecta. It should be noted that the mass fractions of the primary ejecta to the secondary ejecta are comparable for all the models, although the explosion energies and ejected masses from the inner region are diverse among the models.

Moreover, matter near the PNS is blown off via neutrino-driven winds during later evolution of the star (>2 s). The ejecta could be proton-rich rather than neutron-rich (Y_e ∼ 0.5–0.6) and have high entropy (>50k_B; Fischer et al. 2010; Hüdepohl et al. 2010). The νp-process could operate in the ejecta to synthesize light p-nuclei (Fröhlich et al. 2006a, 2006b; Pruet et al. 2006; Wanajo 2006). However, the process may not make large contributions to the ejected masses of abundant nuclei, since the mass ejection rate through the winds is small at the later epoch (Hüdepohl et al. 2010).

Recently, Kimura et al. (2009) and Uchida et al. (2009) analyzed the metal distribution of the Cygnus loop using the data obtained by the Suzaku and XMM-Newton observations. The progenitor of the Cygnus loop is a core-collapse SN explosion whose progenitor mass ranges from ∼12–15 M_☉. The ejecta distributions are asymmetric to the geometric center; the ejecta of O and Ne are distributed more in the northwest rim, while the ejecta of Si and Fe are distributed more in the southwest of the Cygnus loop. Since the material in this middle-aged SN remnant has not yet been completely mixed, the observed asymmetry is considered to still remain a trace of inhomogeneity produced at the moment of explosion. This evidence may allow us to speculate that the asymmetry of the heavy element observed in the Cygnus loop may come from globally asymmetric explosions explored in this work. We try to seek relevance in the following.

Figure 17 shows number ratios of Ne, Mg, Si, and Fe relative to O observed in the Cygnus loop and those of ejecta for $L_{\nu _e} = 4.5 \times 10^{52} \rm \, erg \, s^{-1}$. We find that the ratios observed in the Cygnus loop (solid line) are comparable to those averaged over ejecta from the inner lower region ($r_{\rm ej,cc} \le \rm 10{,}000{\, \rm km}$ and θ ⩾ 2π/3) (dotted line). The ratios averaged over all the ejecta (dash-dotted line) are the same as those averaged over ejecta from the inner equatorial region ($r_{\rm ej,cc} \le \rm 10{,}000{\, \rm km}$ and π/3 < θ < 2π/3) (dashed line). We note that the ratios averaged over the ejecta from the inner upper region ($r_{\rm ej,cc} \le \rm 10{,}000{\, \rm km}$ and θ ⩽ π/3) are comparable to but slightly lower than those averaged over the ejecta from the inner lower region (dotted line).

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N(Ne)/N(O) and N(Mg)/N(O) are independent from an averaging region and are comparable to all our aspherical models. These ratios are therefore concluded to be determined during the hydrostatic evolution of the progenitor and are thus a good indicator for the progenitor mass. On the other hand, ratios of Si and Fe relative to O are much higher in the inner region, in particular in the inner lower region, compared with the outer region (${\ge} \rm 10{,}000{\, \rm km}$). N(Si)/N(O) and N(Fe)/N(O) averaged over all the ejecta are much smaller than those observed in the Cygnus loop. Therefore, incomplete radial mixing of Fe and Si with O is required to explain the high Fe and Si ratios in the Cygnus loop. In fact, the averaged abundances of the Cygnus loop have a correlation with the Si- and Fe-rich regions (Kimura et al. 2009; Uchida et al. 2009). Our results suggest that the abundance ratios for Ne and Mg as well as for Si and Fe observed in the Cygnus loop could be well reproduced with the SN ejecta from the inner region of a 15 M_☉ progenitor.

In order to draw a robust conclusion to the findings obtained in the current 2D simulations, it is indispensable to move on to 3D simulations (e.g., Blondin & Mezzacappa 2007; Iwakami et al. 2008, 2009; Nordhaus et al. 2010; Wongwathanarat et al. 2010). In 2D, the growth of SASI and the large-scale convection tend to develop along the coordinate symmetry axis preferentially, thus suppressing the anisotropies in explosions as well as in the resulting explosive nucleosynthesis. An encouraging piece of news to us is that the 3D simulations cited above are, at least, in favor of the SASI-aided low-mode explosions, in which the major axis of the expanding shock has a certain direction at the moment of explosion (albeit free from the coordinate symmetry axis). By incorporating the present scheme into the 3D simulations of Iwakami et al. (2008, 2009), we plan to clarify these issues as a sequel to this study.

In Appendices A and B, we summarize our treatment of the rate of change in Y_e and energy per unit volume, Q_N and Q_E, respectively. We take into account the absorption of electron and anti-electron neutrinos as well as neutrino emission through electron and positron captures, electron–positron pair annihilation, nucleon–nucleon bremsstrahlung, and plasmon decays. Moreover, we include the heating term in Q_E due to the absorption of neutrinos on ⁴He and the inelastic scatterings on ⁴He via neutral currents (Haxton 1988; Ohnishi et al. 2007). We chiefly follow the treatments in Appendix D of Scheck et al. (2006) and in Appendix B of Ruffert et al. (1996 and references therein).

The heating rate per unit volume through the absorption of ν_e on neutrons is described as

$$\begin{equation} {\cal Q}^{\rm a}_{\nu _{\rm e}}= \sigma c \frac{L_{\nu _{\rm e}} \, n_{\rm n}}{4\pi r^2 c f_{\nu _{\rm e}}} \, \frac{ \big\langle \epsilon _{\nu _{\rm e}}^3\big\rangle +2\Delta \big\langle \epsilon _{\nu _{\rm e}}^2\big\rangle +\Delta ^2 \langle \epsilon _{\nu _{\rm e}}\rangle }{\langle \epsilon _{\nu _{\rm e}}\rangle } \, \Theta (\langle \epsilon _{\nu _{\rm e}}\rangle), \end{equation} \tag{ A1 }$$

where Δ = (m_n − m_p)c², n_n is the number density of neutrons and $\sigma = 4 G_{\rm F}^2 m_{\rm e}^2 \hbar ^2 / \pi c^2= \frac{1}{4}(3\alpha _{\rm w}^2+1) \sigma _0 / (m_{\rm e}c^2)^2$, with the Fermi coupling constant G_F, the reduced Planck constant ℏ, the electron mass m_e, α_w = 1.254, and σ₀ = 1.76 × 10⁻⁴⁴ cm². Here, the nth energy moment of ν_e, $\langle \epsilon _{\nu _i}^n\rangle$, is given by

$$\begin{equation} \langle \epsilon _{\nu _i}^n\rangle = (k_{\rm B} T_{\nu _i})^n \, \frac{ {\cal F}_{2+n}(\eta _{\nu _i}) }{ {\cal F}_2(\eta _{\nu _i}) }, \end{equation} \tag{ A2 }$$

and a factor for the Pauli blocking, $\Theta (\langle \epsilon _{\nu _i}\rangle)$, is approximated as

$$\begin{equation} \Theta (\langle \epsilon _{\nu _i}\rangle) = 1-f_{\rm FD}\left(\frac{\langle \epsilon _{\nu _i}\rangle +\Delta }{k_{\rm B}T},\eta _{\rm e^-} \right)\!{,} \end{equation} \tag{ A3 }$$

where $\eta _{\nu _i}$ is the chemical potential of ν_i in units of k_BT and is set to be 0, $\eta _{\rm e^-}$ is the chemical potential of electrons in units of k_BT, k_B is the Boltzmann constant, and ${\cal F}_n(\eta)$ is defined as

$$\begin{equation} {\cal F}_n(\eta) \equiv \int _0^{\infty } {dx} \; x^n \, f_{\rm FD}(x,\eta) \end{equation} \tag{ A4 }$$

using the Fermi–Dirac distribution function,

$$\begin{equation} f_{\rm FD}(x,\eta) = \frac{1}{1 \; + \; \exp (x - \eta)}. \end{equation} \tag{ A5 }$$

The heating rate via the absorption of ${\bar{\nu }}_{\rm e}$ on protons is given by

$$\begin{eqnarray} &&{\cal Q}^{\rm a}_{{\bar{\nu }}_{\rm e}}= \sigma c \frac{L_{{\bar{\nu }}_{\rm e}} \, n_{p}}{4\pi r^2 c f_{{\bar{\nu }}_{\rm e}}} \frac{ \big\langle \epsilon _{{\bar{\nu }}_{\rm e}}^3\big\rangle ^{\!\star } +3\Delta \big\langle \epsilon _{{\bar{\nu }}_{\rm e}}^2\big\rangle ^{\!\star } +3\Delta ^2 \langle \epsilon _{{\bar{\nu }}_{\rm e}}\rangle ^{\!\star }+\Delta ^3 \big\langle \epsilon _{{\bar{\nu }}_{\rm e}}^0\big\rangle ^{\!\star } }{\langle \epsilon _{{\bar{\nu }}_{\rm e}}\rangle },\nonumber\\ \end{eqnarray} \tag{ A6 }$$

where the nth energy moment of ${\bar{\nu }}_{\rm e}$, $\langle \epsilon _{\bar{\nu }_i}^n\rangle ^{\star }$, is given by

$$\begin{equation} \langle \epsilon _{\bar{\nu }_i}^n\rangle ^{\star } = (k_{\rm B} T_{\bar{\nu }_i})^n \, \frac{ {\cal F}_{2+n}(\eta _{\bar{\nu }_i} - \Delta /k_{\rm B}T_{\bar{\nu }_i}) }{ {\cal F}_2(\eta _{\bar{\nu }_i}) }. \end{equation} \tag{ A7 }$$

The energy emission rate of ν_e per unit volume via electron capture on protons is given by

$$\begin{equation} {\cal Q}^{\rm e}_{\nu _{\rm e}}= \frac{\sigma c}{2} \; n_{\rm p} \, n_{{\rm e}^-} \big[ \big\langle \epsilon _{{\rm e}^-}^3\big\rangle ^{\!\star } +2\Delta \big\langle \epsilon _{{\rm e}^-}^2\big\rangle ^{\!\star } +\Delta ^2 \langle \epsilon _{{\rm e}^-}\rangle ^{\!\star }\big], \end{equation} \tag{ A8 }$$

and that of ${\bar{\nu }}_{\rm e}$ via positron capture on neutrons by

$$\begin{equation} {\cal Q}^{\rm e}_{{\bar{\nu }}_{\rm e}}= \frac{\sigma c}{2} \; n_{\rm n} \, n_{{\rm e}^+} \big[ \big\langle \epsilon _{{\rm e}^+}^3\big\rangle +3\Delta \big\langle \epsilon _{{\rm e}^+}^2\big\rangle +3\Delta ^2 \langle \epsilon _{{\rm e}^+}\rangle +\Delta ^3 \big]. \end{equation} \tag{ A9 }$$

Here, the number densities of electrons and positrons are described by

$$\begin{equation} n_{\mathrm{e^\mp }} = {8\pi \over (hc)^3}\,(k_{\mathrm{B}} T)^3 {\cal F}_2(\pm \eta _{\mathrm{e}^-}), \end{equation} \tag{ A10 }$$

where h is the Planck constant and the nth energy moments of electrons and positrons, 〈ⁿ_e〉 and 〈ⁿ_e〉^⋆, are given by

$$\begin{equation} \langle \epsilon _{\rm e}^n\rangle = (k_{\rm B}T)^n \, \frac{{\cal F}_{2+n}(\eta _{\mathrm{e}})}{{\cal F}_2(\eta _{\mathrm{e}})} \end{equation} \tag{ A11 }$$

and

$$\begin{equation} \langle \epsilon _{\rm e}^n\rangle ^{\star } = (k_{\rm B}T)^n \, \frac{{\cal F}_{2+n}(\eta _{\mathrm{e}}-\Delta /k_{\rm B}T)}{{\cal F}_2(\eta _{\mathrm{e}})}. \end{equation} \tag{ A12 }$$

The energy emission rates of ν_e and ${\bar{\nu }_e}$ per unit volume via electron–positron pair annihilations are given by

$$\begin{eqnarray} {\cal Q}^{\rm pa}_{\nu _{\rm e}} &=& Q^{\rm pa}_{{\bar{\nu }}_{\rm e}} = \frac{C_V^2+ C_A^2}{36} \left(\frac{8\pi }{(h c)^3}\right)^2 \frac{\sigma _0 c}{(m_e c^2)^2} (k_{\rm B}T)^9\nonumber\\ &&\times\, [ {\cal F}_{4}(\eta _e) {\cal F}_{3}(-\eta _e) + {\cal F}_{3}(\eta _e) {\cal F}_{4}(-\eta _e) ] P_{{\rm pair},\nu _{\rm e}} P_{{\rm pair},{\bar{\nu }_e}},\nonumber\\ \end{eqnarray} \tag{ A13 }$$

where $C_A = \frac{1}{2}$, $C_V = \frac{1}{2} + 2 \sin ^2 \theta _{\rm W}$, and sin ²θ_W = 0.23. Here, $P_{{\rm pair},\nu _{\rm e}}$ and $P_{{\rm pair},{\bar{\nu }_e}}$ are factors for the phase space blocking of ν_e and ${\bar{\nu }_e}$ for the pair processes, respectively, and the factor for ν_i is approximately expressed as

$$\begin{equation} P_{{\rm pair},\nu _i} \simeq \left\lbrace 1 + \exp \left[-\left(\frac{1}{2}\frac{{\cal F}_4(\eta _{e})}{{\cal F}_3(\eta _e)} +\frac{1}{2}\frac{{\cal F}_4(-\eta _{e})}{{\cal F}_3(-\eta _e)} -\eta _{\nu _i}\right) \right]\right\rbrace ^{-1}. \end{equation} \tag{ A14 }$$

The energy emission rate of ν_x per unit volume via the pair annihilations is given by

$$\begin{eqnarray} {\cal Q}^{\rm pa}_{\nu _x} &=& \frac{(C_V -C_A)^2 +(C_V +C_A -2)^2}{18} \left(\frac{8\pi }{(h c)^3}\right)^2\nonumber\\ &&\times\, \frac{\sigma _0 c}{(m_e c^2)^2} (k_{\rm B}T)^9 [ {\cal F}_{4}(\eta _e) {\cal F}_{3}(-\eta _e)\nonumber\\ && +\, {\cal F}_{3}(\eta _e) {\cal F}_{4}(-\eta _e) ] (P_{{\rm pair},\nu _x})^2. \end{eqnarray} \tag{ A15 }$$

The energy emission rate per unit volume through nucleon–nucleon bremsstrahlung of a single neutrino pair is well approximated with

$$\begin{eqnarray} {\cal Q}^{\rm br}_{\nu \bar{\nu }} &=& 1.04 \times 10^{30} \zeta \left(X_n^2 +X_p^2 +\frac{28}{3}X_n X_p \right) \left(\frac{\rho }{10^{14}{\, \rm g\,cm^{-3}}} \right)^2\nonumber\\ &&\times\, \left(\frac{T}{1 \, {\rm MeV} } \right)^{5.5} \, \, {\rm erg} \, {\rm cm} ^{-3} \, \, {\rm s} ^{-1}, \end{eqnarray} \tag{ A16 }$$

where ζ is set to be 0.5 (Burrows et al. 2000), and X_n and X_p are the mass fraction of neutrons and protons, respectively.

The emission rates of ν_e and ${\bar{\nu }_e}$ per unit volume via plasmon decay are given by

$$\begin{eqnarray} R^{\rm pl}_{\nu _{\rm e}} &=& R^{\rm pl}_{{\bar{\nu }_e}} = C_V^2 \frac{\pi ^3}{3 \alpha _* (hc)^6} \frac{\sigma _0 c}{(m_e c^2)^2} (k_{\rm B}T)^8 \gamma ^6 e^{-\gamma }\nonumber\\ &&\times\,(1+\gamma) P_{{\rm plas},\nu _{\rm e}} P_{{\rm plas},{\bar{\nu }_e}}, \end{eqnarray} \tag{ A17 }$$

where α_* = 1/137.036 is the fine-structure constant, $\gamma = \gamma _0 \sqrt{\eta _e^2 +\pi ^2/3}$ with $\gamma _0 = 2\sqrt{\alpha _*/3\pi } = 5.565 \times 10^{-2}$, and $P_{{\rm plas},\nu _{\rm e}}$ and $P_{{\rm plas},{\bar{\nu }_e}}$ are factors for the phase space blocking of ν_e and ${\bar{\nu }_e}$ for plasmon decay processes, respectively. The factor for ν_i is approximately expressed as

$$\begin{equation} P_{{\rm plas},\nu _i} \simeq \left\lbrace 1 + \exp \left[-\left(1+ \frac{1}{2}\frac{\gamma ^2}{1+\gamma } -\eta _{\nu _i}\right) \right]\right\rbrace ^{-1}. \end{equation} \tag{ A18 }$$

The emission rate of ν_x per unit volume via plasmon decay is given by

$$\begin{equation} R^{\rm pl}_{\nu _x} = (C_V-1)^2 \frac{4\pi ^3}{3 \alpha _* (hc)^6} \frac{\sigma _0 c}{(m_e c^2)^2} (k_{\rm B}T)^8 \gamma ^6 e^{-\gamma }(1+\gamma) (P_{{\rm plas},\nu _x})^2. \end{equation} \tag{ A19 }$$

The energy emission rates of ν_i per unit volume via plasmon decay are given by

$$\begin{equation} {\cal Q}^{\rm pl}_{\nu _i} = R^{\rm pl}_{\nu _i} \cdot k_{\rm B} T \left(1 + \frac{1}{2}\frac{\gamma ^2}{1+\gamma } \right). \end{equation} \tag{ A20 }$$

In addition to the heating processes through the neutrino absorption on nucleons, the heating processes due to neutrino–helium interactions are taken into account, as in Ohnishi et al. (2007). Through the absorption of ν_e and ${\bar{\nu }}_{\rm e}$ and the neutrino–helium inelastic scatterings on nuclei via neutral currents, ν + (A, Z) → ν + (A, Z)*, the heating rate per unit volume, ${\cal Q}^{\alpha }$, is evaluated as

$$\begin{eqnarray} {\cal Q}^{\alpha } &=& \frac{\rho X_{A}}{m_{\rm B}} \frac{31.6\, {\rm MeV} }{(r/10^{7}\,{\rm cm})^{2}}\nonumber\\ &&\times\,\left[ \frac{L_{\nu _{{\rm e}}}}{10^{52}\,{\rm erg \,s}^{-1}} \left(\frac{5\, {\rm MeV} }{T_{\nu _{{\rm e}}}}\right) \frac{A^{-1}\big\langle \sigma _{\nu _{{\rm e}}}^{+}E_{\nu _{{\rm e}}} +\sigma _{\nu _{{\rm e}}}^{0}E_{{\rm ex}}^{{\rm A}} \big\rangle _{T_{\nu _{{\rm e}}}}}{10^{-40}\,{\rm cm}^2\, {\rm MeV} } \right. \nonumber \\ &&+\, \frac{L_{\bar{\nu }_{{\rm e}}}}{10^{52}\,{\rm erg \,s}^{-1}} \left(\frac{5\, {\rm MeV} }{T_{\bar{\nu }_{{\rm e}}}}\right) \frac{A^{-1}\big\langle \sigma _{\bar{\nu }_{{\rm e}}}^{-}E_{\nu _{{\rm e}}} +\sigma _{\bar{\nu }_{{\rm e}}}^{0}E_{{\rm ex}}^{{\rm A}} \big\rangle _{T_{\bar{\nu }_{{\rm e}}}}}{10^{-40}\,{\rm cm}^2\, {\rm MeV} } \nonumber \\ &&+\left.\,\frac{L_{\nu _{x}}}{10^{52}\,{\rm erg \,s}^{-1}} \left(\frac{10\, {\rm MeV} }{T_{\nu _{x}}}\right) \frac{A^{-1}\big\langle \sigma _{\nu _{x}}^{0}E_{{\rm ex}}^{{\rm A}} +\sigma _{\bar{\nu }_{x}}^{0}E_{{\rm ex}}^{{\rm A}} \big\rangle _{T_{\nu _{x}}}}{10^{-40}\,{\rm cm}^2\, {\rm MeV} } \right],\nonumber\\ \end{eqnarray} \tag{ A21 }$$

where X_A is the mass fraction of the nucleus and m_B is the atomic mass unit (Haxton 1988). The last term denotes the sum of the contributions from μ and τ neutrinos. The cross section for each neutral current is evaluated by the following fitting formula:

$$\begin{equation} A^{-1}\big\langle \sigma _{\nu }^{0}E_{{\rm ex}}^{{\rm A}} +\sigma _{\bar{\nu }}^{0}E_{{\rm ex}}^{{\rm A}} \big\rangle _{T_{\nu }} = \alpha \left[ \frac{T_{\nu } - T_{0}}{10\, {\rm MeV} } \right]^{\beta }, \end{equation} \tag{ A22 }$$

where α, β, and T₀ are given in Table I of Haxton (1988), and are chosen to be α = 1.24 × 10⁻⁴⁰ MeV cm², β = 3.82, and T₀ = 2.54 MeV (Ohnishi et al. 2007). In the first and second terms on the right-hand side of Equation (A21), the contributions from the charged current reactions, σ⁺_ν and σ⁻_ν, are also taken into account according to Table II of Haxton (1988); that is, $A^{-1}\langle \sigma _{\nu _{{\rm e}}}^{+}E_{\nu _{{\rm e}}} \rangle _{T_{\nu _{{\rm e}}}} = 0.30 \times 10^{-42} \, {\rm MeV} \,{\rm cm} ^2$ for $T_{\nu _{\rm e}} = 4\, {\rm MeV}$ and $A^{-1}\langle \sigma _{\bar{\nu }_{{\rm e}}}^{-}E_{\bar{\nu }_{{\rm e}}} \rangle _{T_{\bar{\nu }_{{\rm e}}}} = 1.20 \times 10^{-42} \, {\rm MeV} \,{\rm cm} ^2$ for $T_{{\bar{\nu }}_{\rm e}} = 5\, {\rm MeV}$.

The rate of change in energy in unit volume, Q_E, is evaluated as

$$\begin{eqnarray} Q_{\rm E} &=& \left({\cal Q}^{\rm a}_{\nu _{\rm e}}+{\cal Q}^{\rm a}_{{\bar{\nu }}_{\rm e}}\right) +{\cal Q}^{\alpha } -\left({\cal Q}^{\rm e}_{\nu _{\rm e}}+{\cal Q}^{\rm e}_{{\bar{\nu }}_{\rm e}}\right) -3{\cal Q}^{\rm br}_{\nu \bar{\nu }}\nonumber\\ && -\,\left({\cal Q}^{\rm pa}_{\nu _{\rm e}} +{\cal Q}^{\rm pa}_{{\bar{\nu }}_{\rm e}} +4{\cal Q}^{\rm pa}_{\nu _x}\right) -\left({\cal Q}^{\rm pl}_{\nu _{\rm e}} +{\cal Q}^{\rm pl}_{{\bar{\nu }}_{\rm e}} +4{\cal Q}^{\rm pl}_{\nu _x}\right),\nonumber\\ \end{eqnarray} \tag{ A23 }$$

where a factor of three in $3{\cal Q}^{\rm br}_{\nu \bar{\nu }}$ corresponds to three types of neutrino pairs $\nu \bar{\nu }$.