Effect of the Nuclear Equation of State on Relativistic Turbulence-induced Core-collapse Supernovae

To study the effects of the EOS on the explosion properties, we simulated the explosion of four representative progenitors with initial main-sequence masses of 9, 15, 20, and 25 M_⊙ from Sukhbold et al. (2016) using seven different EOSs summarized in Table 1. We employ some of the most common EOSs used for CCSN simulations to be consistent with previous results presented in the literature (e.g., Lattimer & Swesty 1991; O'Connor & Ott 2011; Janka 2012; Hempel et al. 2012; Couch 2013; Steiner et al. 2013; Suwa et al. 2013; Fischer et al. 2014; Char et al. 2015; Olson et al. 2016; Furusawa et al. 2017; Richers et al. 2017; Burrows et al. 2018; Morozova et al. 2018; Nagakura et al. 2018; Schneider et al. 2019; Harada et al. 2020; Yasin et al. 2020), despite the fact that some of these EOSs are inconsistent with recent observational and/or experimental constraints.

Table 1. Parameters Characterizing the Equations of State Utilized in This Study

EOS	Type	n0	J	L	K0	${m}_{n}^{* }$/mn	${m}_{p}^{* }$/mp	${{\rm{M}}}_{\max }$
		(fm−3)	(MeV baryon−1)	(MeV baryon−1)	(MeV baryon−1)	⋯	⋯	(M⊙)
LS220	SRO	0.1549	28.61	73.81	219.85	1.0000	1.0000	2.04
APR	SRO	0.1600	32.59	58.47	266.00	0.6980	0.6987	2.19
KDE0v1	SRO	0.1646	34.58	54.70	227.53	0.7440	0.7443	1.97
SLy4	SRO	0.1595	32.00	45.96	229.90	0.6950	0.6953	2.05
SFHo	RMF	0.1583	31.57	47.10	245.40	0.7609	0.7606	2.05
DD2	RMF	0.1491	32.73	57.94	242.70	0.5628	0.5622	2.14
HShen	RMF	0.1455	36.95	110.99	281.00	0.6340	⋯	2.21

Note. The first four EOSs are calculated using the SRO code from Schneider et al. (2017), while the last three are calculated using RMF theory. The parameter n₀ is the nuclear saturation density, J is the symmetry energy at saturation density, L is the slope of the symmetry energy, K₀ is the nuclear incompressibility, ${m}_{n}^{* }$ is the nucleon effective mass, and m_n is the nucleon mass.

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The nuclear properties that can most significantly impact the dynamics of CCSNe and determine the properties of the PNS formed in the collapse (Schneider et al. 2019; Yasin et al. 2020) are (i) the zero-temperature saturation density of symmetric nuclear matter n₀, (ii) the effective nucleon mass in symmetric matter at saturation density ${m}_{n}^{* }$, (iii) the nuclear incompressibility K₀, and (iv) the density-dependent symmetry energy _sym(n). The latter is defined as the isovector component of the energy per baryon, obtained by expanding the energy per baryon around its value for symmetric matter:

$$\begin{eqnarray}&&{\epsilon }_{B}{(n,y)={\epsilon }_{B}(n,y=1/2)+{\epsilon }_{\mathrm{sym}}\delta (y)}^{2},\end{eqnarray} \tag{ 1 }$$

where y is the proton fraction and δ(y) = 1–2y. Hence, the symmetry energy is defined as

$$\begin{eqnarray}&&{\epsilon }_{\mathrm{sym}}={\epsilon }_{B}(n,y=1/2)-{\epsilon }_{B}(n,y=0).\end{eqnarray} \tag{ 2 }$$

It is also common to employ an alternative definition of the symmetry energy, which can be derived by expanding _B (n, y) around its value for symmetric matter (i.e., _B (n, y = 1/2)), yielding

$$\begin{eqnarray}&&{\epsilon }_{\mathrm{sym}}={\epsilon }_{B}{(n,y=1/2)+{\epsilon }_{\mathrm{sym}}\delta (y)}^{2}+{ \mathcal O }[\delta {(y)}^{4}].\end{eqnarray} \tag{ 3 }$$

The two definitions agree up to quartic terms.

A brief summary of these EOSs is as follows:

• 1.  
  The APR EOS (Akmal et al. 1998; Schneider et al. 2019) is based on a thorough variational calculation, including realistic two- and three-body nuclear forces to derive the ground-state energies of pure neutron matter and symmetric nuclear matter. It utilizes a Hamiltonian that includes the two-body Argonne V18 potential extracted by fitting two-nucleon experimental scattering data, as well as the more empirical three-body Urbana IX potential, which is fit in part to light nuclei. This is the only EOS considered here that includes a phase transition to a neutral pion condensate. It is a generalization of the original EOS at zero temperature from Akmal et al. (1998), where the nuclear potentials are extended to the finite-temperature part. At low densities it is connected to a network of ∼3300 nuclei in nuclear statistical equilibrium (NSE) using the Schneider–Roberts–Ott (SRO) code (Schneider et al. 2017).
• 2.  
  The LS220 (Lattimer & Swesty 1991), KDE0v1 (Agrawal et al. 2005), and SLy4 (Chabanat et al. 1997) parameterizations are taken from Schneider et al. (2017). For these EOSs nuclear matter is described using Skyrme models for nuclear interactions. In the transition to nuclear saturation density nuclei are described using the compressible liquid-drop model in the Single Nucleus Approximation (SNA). The version of LS220 utilized here is in excellent agreement with the original work by Lattimer & Swesty (1991). At low densities these EOSs are coupled to an NSE network of ∼3300 nuclei using the SRO code (Schneider et al. 2017).
• 3.  
  The DD2 and SFHo EOSs are calculated using the RMF approach of Hempel & Schaffner-Bielich (2010) with the parameterization of the interactions between nucleons from Typel et al. (2010) and Steiner et al. (2013), respectively. Nuclear matter is treated using an NSE approach for the transition from nonuniform to uniform nuclear matter. These two EOSs differ from the EOSs described above in that they do not employ the SNA in the vicinity of the saturation density. Rather, several thousand nuclei are taken into account, whose masses and binding energies, when available, are taken from experiment.
• 4.  
  The HShen EOS from Shen et al. (2011) is also based on RMF theory. This EOS differs from the SFHo and DD2 in that it uses an SNA based on a Thomas–Fermi model to describe matter in the approach to nuclear matter density.

The first four EOSs were generated using the SROEOS code from Schneider et al. (2017). Therefore, they share the same treatment of inhomogeneous phases at subnuclear densities. This consists of a finite-temperature compressible liquid-drop model in which heavy nuclei are described using a single-nucleus approximation. All of the adopted EOSs in Table 1 can thus be grouped into two categories: SRO type (the first four), and the RMF type (the last three). The gravitational mass versus radius relationships for cold neutron stars are shown in Figure 1, alongside some observational constraints on the mass and radius of neutron stars. As can be seen, not all these EOSs are consistent with the current observational constraints. Nevertheless, these formulations should represent a reasonable gamut of possible EOSs.

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All of the simulations presented in this paper were run using a modified version of the open-source code GR1D (O'Connor & Ott 2010; O'Connor 2015). The original code is based on general relativistic, spherically symmetric hydrodynamics and radiation transport. The Boltzmann equation that describes the propagation of neutrinos is solved with a two-moment scheme, the so-called M1 transport (Shibata et al. 2011; Cardall et al. 2013), while neutrino opacities were generated using the open-source code NuLib (O'Connor 2015).

We use the same spatial grid for all the simulations, i.e., 700 zones with a constant resolution of 0.3 km in the inner 20 km and then logarithmically spaced outer zones extending to a radius of 15,000 km. The only exception are the progenitors with very steep density gradients (i.e., the ones with masses smaller than 12 M_⊙), for which we place the outer boundary at the radius where the density reaches 6 × 10⁴ g cm⁻³, which can vary between 5000 and 10,000 km.

For the neutrinos, we use 18 energy groups logarithmically spaced from 1 to 280 MeV. We adopted the opacities described in Bruenn (1985), with weak-magnetism and recoil corrections from Horowitz (2002) and a virial correction to neutrino−nucleon scattering from Horowitz et al. (2017). We include inelastic neutrino−electron scattering and velocity-dependent terms in the transport equation according to O'Connor (2015).

Neutrino-heated turbulence plays a crucial role in the shock revival and explodability, as many previous studies have noted (Couch & Ott 2015; Radice et al. 2016, 2018; Mabanta & Murphy 2018). In essence, material behind the expanding shock is heated by neutrinos escaping the PNS. This increases the entropy and leads to a negative entropy gradient that triggers turbulent convection. The rising turbulent fluid elements can then transfer energy to the shock. This makes spherical explosions triggered by neutrino-heated turbulent convection physically well motivated.

To analyze the impact of each EOS on the explosion, we used a modified version of GR1D described in Boccioli et al. (2021) based on the STIR model of Couch et al. (2020). With this version, one can include the effects of turbulent convection in our spherically symmetric simulations by using a time-dependent mixing-length theory (MLT) approach and therefore trigger an explosion.

The difference between our model and the original STIR is that we explicitly take GR into account. Despite yielding small differences in the shock dynamics, GR can have a significant effect on the explodability of SNe as a function of progenitor mass (Boccioli et al. 2021). Here we summarize the main features of this model, but more details can be found in Mabanta & Murphy (2018), Couch et al. (2020), and Boccioli et al. (2021). The metric adopted in GR1D is the following:

$$\begin{eqnarray}&&\begin{array}{l}{{ds}}^{2}={g}_{\mu \nu }{x}^{\mu }{x}^{\nu }\\ \,=\,-\alpha {(r,t)}^{2}{{dt}}^{2}+X{(r,t)}^{2}{{dr}}^{2}+{r}^{2}d{{\rm{\Omega }}}^{2},\end{array}\end{eqnarray} \tag{ 4 }$$

where the t − t component α is the lapse function and the r − r component X is a function of the gravitational mass of the system at radius r.

In addition to the standard hydrodynamic equations solved in GR1D (O'Connor & Ott 2010), we have added an equation to account for the time evolution of the turbulent energy:

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{\partial {{Dv}}_{\mathrm{turb}}^{2}}{\partial t}+\displaystyle \frac{1}{{r}^{2}}\displaystyle \frac{\partial }{\partial r}\left[\displaystyle \frac{\alpha {r}^{2}}{X}D({v}_{\mathrm{turb}}^{2}v-{D}_{{\rm{K}}}{\rm{\nabla }}{v}_{\mathrm{turb}}^{2})\right]\\ \quad =\,-\alpha X\left(\rho {v}_{\mathrm{turb}}^{2}\displaystyle \frac{\partial v}{\partial r}+\rho {v}_{\mathrm{turb}}{\omega }_{\mathrm{BV}}^{2}{{\rm{\Lambda }}}_{\mathrm{mix}}-\rho \displaystyle \frac{{v}_{\mathrm{turb}}^{3}}{{{\rm{\Lambda }}}_{\mathrm{mix}}}\right),\end{array}\end{eqnarray} \tag{ 5 }$$

where v_turb is the turbulent velocity and D = X ρ W is the conserved density, with $W={(1-{v}^{2})}^{-1/2}$ being the Lorentz factor. The proper physical velocity is defined as v = Xv^r , where v_r is the coordinate velocity and D_K = α_K v_turbΛ_mix is a diffusion coefficient. The two main quantities characterizing the model are the mixing length

$$\begin{eqnarray}&&{{\rm{\Lambda }}}_{\mathrm{mix}}={\alpha }_{\mathrm{MLT}}\frac{P}{\rho \,d\phi /{\rm{d}}r}\end{eqnarray} \tag{ 6 }$$

and the Brunt−Väisälä frequency

$$\begin{eqnarray}&&{\omega }_{\mathrm{BV}}^{2}=\frac{{\alpha }^{2}}{\rho {{hX}}^{2}}\left(\frac{d\phi }{dr}-v\frac{\partial v}{\partial r}\right)\left(\frac{\partial \rho (1+\epsilon )}{\partial r}-\frac{1}{{c}_{s}^{2}}\frac{\partial P}{\partial r}\right),\end{eqnarray} \tag{ 7 }$$

where $\phi =\mathrm{ln}\alpha$ reduces to the Newtonian potential in the nonrelativistic limit. The quantity h = 1 + + P/ρ is the relativistic enthalpy, while is the internal energy, P is the pressure, and c_s is the sound speed.

The above equations depend on two parameters: α_MLT and α_K. Additional diffusion coefficients analogous to D_K are added to the evolution equations for internal energy, electron fraction, and neutrino energy density, increasing the number of parameters in the model to 5. However, we fix α_K and the other mixing-length parameters to 1/6, consistent with the choice of Müller et al. (2016) and Couch et al. (2020), since the model is not very sensitive to their value. The only parameter left is α_MLT, which determines the magnitude of the mixing length and therefore the scale of convection in the gain region.

Two of the main quantities of interest for an exploding SN are the trajectory of the shock as a function of time and the explosion energy. We display these in Figure 6 for the four progenitors described in Section 1. Since the simulations we performed extend to less than 1 s of the post-bounce phase, a calculation of the final total explosion energy is not possible. Instead, one can define a diagnostic explosion energy (Buras et al. 2006; Marek & Janka 2009; Müller et al. 2012; Bruenn et al. 2016) as the integral of the binding energy E_bind in regions where E_bind > 0.

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In the GR case, following Müller et al. (2012), one can define E_bind in terms of the lapse function α, the density ρ, the specific internal thermal energy _th, the pressure P, and the Lorentz factor W:

$$\begin{eqnarray}&&{E}_{\mathrm{bind}}=\alpha \left[(\rho +\rho {\epsilon }_{\mathrm{th}}+P){W}^{2}-P\right]-\rho W.\end{eqnarray} \tag{ 9 }$$

Here it is important to distinguish between the traditional internal energy adopted in classical thermodynamics and the internal energy used in the context of CCSNe (see Appendix A of Bruenn et al. 2016). The latter definition incorporates the binding energy of nuclei in the expression of the internal energy. This allows negative values of , hence making the definition of the total energy of the fluid problematic. A way around this is to define _th as the difference between the internal energy —taken from the EOS at a given density, temperature, and electron fraction—and the internal energy ₀, defined as the internal energy at zero temperature and at the same density and electron fraction. ⁴

From this definition, it is clear that ₀ represents the binding energy of the nuclei, which at low temperature is the only significant contribution to the internal energy. Finally, the diagnostic explosion energy can be defined as

$$\begin{eqnarray}&&{E}_{\mathrm{diag}}={\int }_{{E}_{\mathrm{bind}}\gt 0}{E}_{\mathrm{bind}}dV,\end{eqnarray} \tag{ 10 }$$

where d V is the proper volume.

As can be seen from the shock radius and the diagnostic energy, the LS220 EOS gives the earliest and strongest explosion, followed by the APR, KDE0v1, SLy4, SFHo, DD2, and HShen. This hierarchy is maintained across all progenitors, with the only exception being the 9 M_⊙ progenitor, for which the hierarchy of the explosion energies is different. Specifically, the SFHo and the DD2 generate comparatively stronger explosions in the 9 M_⊙ progenitor than in the more massive progenitors, as can be seen from the diagnostic energy for this progenitor.

In addition to this, it should be noted that the 9 M_⊙ progenitor leads to very early and faint explosions (the diagnostic energy in Figure 6 has been multiplied by 8 so that it could more clearly be represented on the plot), and the decrease around 450 ms post-bounce is a numerical artifact due to the shock leaving the outer boundary of the simulation. This trend for the 9 M_⊙ progenitor relates to the fact that these stars explode as electron-capture SNe (Isern et al. 1991). A more detailed analysis of the differences between this progenitor and the more massive ones is given in the Appendix. For now we will focus on the 15, 20, and 25 M_⊙ progenitors.

Since our convection model is designed to reproduce turbulence in the gain region, one can analyze how the explosion is affected by this mechanism. To make the comparison among different EOSs, which have different gain and shock radii, we define a dimensionless quantity

$$\begin{eqnarray}&&{\xi }_{\mathrm{gain}}=\displaystyle \frac{r-{r}_{\mathrm{gain}}}{{r}_{\mathrm{shk}}-{r}_{\mathrm{gain}}},\end{eqnarray} \tag{ 11 }$$

such that ξ_gain = 0 when r = r_gain and ξ_gain = 1 when r = r_shock. The profiles of the turbulent velocity in the gain region are plotted in Figure 7 for two different time snapshots.

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At ∼50 ms post-bounce the gain region forms and the turbulent velocity starts building up. After that, as can be seen in the snapshot at ∼150 ms, the profiles become smoother and the peak velocity steadily grows. Overall, the turbulent velocity profiles do not significantly change with progenitors and EOSs.

Nevertheless, some global trends can be identified, since, for example, the LS220 and the SFHo consistently yield the largest peak turbulent velocities, while the smallest corresponds to the SLy4. By looking at Figure 6, however, it is clear that the strongest explosions are given by the LS220 and the APR, which yields a smaller peak turbulent velocity than the SFHo. Hence, there does not seem to be any direct correlation between the strength of convection in the gain region and the strength of the explosion as a function of EOS.

It is interesting to note that if one only considers SRO-type EOSs or only RMF-type EOSs, the hierarchy of turbulent velocities follows the hierarchy of explosion strengths. Namely, for SRO-type EOSs, from largest to lowest peak turbulent velocity, we find LS220, APR, KDE0v1, SLy4, which is the same as the hierarchy for explosion strengths. For RMF-type EOSs the hierarchy of peak turbulent velocities reads SFHo, DD2, HShen, which is also the same as the hierarchy of explosion strength.

Putting SRO and RMF types together, however, disrupts the correlation between explosion strength and convective velocity, as can be seen from the zoomed-in panels of Figure 7. There, one can see that SFHo, DD2, and HShen yield a quite large turbulent velocity compared to the SRO-type EOSs, despite having low explosion energies, or not exploding at all. We will come back to this when we discuss neutrino luminosities in Section 4.4. We suggest that this difference in the turbulent velocity profiles (and neutrino luminosities, as discussed later) between SRO-type and RMF-type EOSs lies most likely in the different treatment of nuclear matter as it approaches saturation density.

The PNS is a very high density environment, and therefore its interior can be greatly impacted by the EOS. In Figure 8 we show the central density, temperature, and entropy per baryon as a function of time for different progenitors and EOSs. Note that the central entropy in the first 50 ms after bounce correlates extremely well with the strength of the explosion. Indeed, by comparing it to the explosion energy and shock trajectory, one can clearly see that the EOS with the highest central entropy (LS220) yields the earliest and strongest explosion, while the EOS with the lowest central entropy (HShen) does not explode at all, and after the initial stalling of the shock at ∼150 km is the EOS with the quickest recollapse. In general, the hierarchy of central entropies is the same as the hierarchy of explosion strength for all progenitors, with the possible exception of the SLy4 and KDE0v1 EOSs, which we will discuss at the end of this section.

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As a historical note, this correlation between explodability and early-time central entropy was suggested by Hans Bethe (Bethe 1990), the explanation being the fact that entropy in the core is generated by the onset of nonequilibrium processes due to deleptonization during collapse and energy exchange between matter and the nonthermal neutrinos. The amount of entropy generated then affects the radius at which the shock forms and hence the explodability. For example, the lower central Y_e and temperature for the LS220 EOS are consistent with stronger deleptonization and larger entropy.

Another way to illustrate the same concept, from the point of view of MLT, is that the minimum value of α_MLT needed to trigger an explosion is very low (typically around 1.2/1.3, depending on the progenitor) for the EOS with the highest central entropy (LS220), while it is very large (typically around 1.7/1.8, depending on the progenitor) for the EOS with the lowest central entropy (HShen). On the other hand, the central temperature and density do not correlate with the strength of explosion.

A similar relationship was found in the studies of Schneider et al. (2019) and Yasin et al. (2020), who focused only on one type of EOS, i.e., a liquid-drop model that uses Skyrme interactions between nucleons at high densities. With their setup they were able to vary different parameters (such as the effective mass of nucleons at saturation density, the symmetry energy, the incompressibility, etc.) one by one, while keeping all the other parameters fixed. Therefore, they were able to pinpoint which property of the EOS has the largest impact on the explosion.

Their analysis revealed that the effective mass of nuclei at saturation density m*_n is the quantity that most impacts the structure of the PNS, and therefore the explosion, since it determines how rapidly it contracts (Yasin et al. 2020) and changes the temperature of the neutrinospheres for all flavors (Schneider et al. 2019). They also pointed out that the effective mass affects the central entropy of the PNS, in that a larger effective mass would lead to a larger central entropy.

Our approach is somewhat different compared to these previous studies. We include not only EOSs based on Skyrme interactions, like the LS220, KDE0v1, and SLy4, but also the APR EOS and EOSs based on RMF theory. The APR EOS shares the same properties as the Skyrme-type EOSs near saturation density; therefore, we group them together under the label "SRO-type." However, the nuclear potentials for the APR EOS are derived from nucleon–nucleon scattering, rather than from the energy density of nuclear matter. Alongside the EOS developed by Togashi et al. (2017), it is one of the few EOSs available for SN simulations developed starting from realistic nuclear potentials. The EOSs calculated using RMF theory use either an NSE (SFHo and DD2) or a Thomas–Fermi (HShen) treatment of nuclei near saturation density. By considering this variety of EOS formulations, we can investigate the subtle differences among the most common frameworks used to calculate the nuclear EOS.

When we compare EOSs calculated using Skyrme interactions (i.e., LS220, KDE0v1, and SLy4), we find that the effective mass indeed correlates with the strength of explosion. However, this is not true when we compare all of the RMF-type and SRO-type EOSs, since, for example, the SFHo has a larger effective mass than the SLy4 but generates weaker and somewhat delayed explosions. Also, despite the APR sharing the same properties as the Skyrme-type EOSs near saturation density, it does not fit into the correlation between the effective mass and the strength of explosion. For example, it has a smaller effective mass than KDE0v1 but yields stronger explosions. This is not completely unexpected since the density dependence of the effective mass is more complicated in the APR model (Schneider et al. 2019). It also shows that the effective mass is not the only parameter that can predict the strength of explosion. Instead, the only quantity that seems to correlate with the strength of the explosion is the central entropy, regardless of the framework used to calculate the nuclear EOS.

A slight deviation from the correlation between explosion and central entropy is represented by the case of KDE0v1 and SLy4. These two EOSs yield shock trajectories and explosion energies that are very similar across progenitors, with KDE0v1 always giving a slightly stronger explosion. If one looks at the central entropy right after bounce, however, KDE0v1 gives a slightly smaller value than SLy4. This shows that, despite m*_n being the parameter that impacts the PNS structure the most, not surprisingly the other nuclear parameters can also change the structure of the PNS. In particular, assuming a Fermi liquid theory, one expects the central entropy to be roughly given by ${S}_{c}\sim m{* }_{n}{T}_{c}/{\rho }_{c}^{2/3}$ (Baym & Pethick 1991), where S_c , T_c , and ρ_c are the central entropy, temperature, and density, respectively. KDE0v1 has a larger nucleon effective mass m*_n , which increases the central entropy, but it also has a larger saturation density, which has the effect of increasing the central density, hence lowering the central entropy. These competing effects can slightly disrupt the hierarchy of central entropy, but since in our analysis the EOSs differ for several nuclear parameters, we do not expect a perfect correlation. Nonetheless, the correlation between central entropy right after bounce and the strength of the explosion remains quite robust.

It is not completely unexpected that the correlation between strength of explosion and nucleon effective mass breaks down when comparing SRO-type and RMF-type EOSs. The definition of effective mass and entropy is different in the two frameworks (and within SRO type, it varies between the APR and the Skyrme-type EOSs). This is especially true since relativistic effects, which are taken into account by RMF models but not by the SRO models, can be significant. As can be seen in Figure 9, however, for densities between 10¹⁴ and 10¹⁵ g cm⁻³ the hierarchy of effective masses is the same as the hierarchy of entropies. At first, this seems to be in contrast with the fact that there is no clear correlation between effective nucleon mass at saturation density and central entropy in our simulations. However, from Figure 8 one can see that different EOSs yield very different thermodynamic conditions inside the PNS. Temperatures, densities, and electron fractions span ranges of 10–20 MeV, (3–4) × 10¹⁴ g cm⁻³, and 0.27–0.3, respectively. Therefore, comparing different EOSs at a fixed temperature and electron fraction can be misleading if one wants to analyze the impact of the EOS on the explosion of CCSNe. The bottom panel of Figure 8 confirms this. Here one can see that the hierarchy of the nucleon effective masses at the center of the newly formed PNS is different from the hierarchy of entropies and therefore does not correlate with the strength of explosion.

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Convection within the PNS has been shown to be very important since it can change the neutrino signal (Roberts et al. 2012; Mirizzi et al. 2016; Müller 2020) and the gravitational wave signal (Marek et al. 2009; Yakunin et al. 2010; Müller et al. 2013; Andresen et al. 2017; Morozova et al. 2018) and trigger the so-called lepton-number emission self-sustained asymmetry (Tamborra et al. 2014). However, its impact on the explosion is still unclear, and more high-resolution 3D simulations are needed to clarify its role.

In a recent paper, Nagakura et al. (2020, hereafter NBRV20) analyzed the convection inside the PNS for 14 progenitors from Sukhbold et al. (2016), with masses ranging from 9 to 60 M_⊙, using the SFHo EOS. Their conclusion was that there does not seem to be any direct influence on the shock revival from the PNS convection at early times (and vice versa). The most significant correlation they found is between more massive PNSs and more vigorous convection.

The quantity chosen to represent the "vigor" of convection is the total turbulent energy in the PNS, defined as

$$\begin{eqnarray}&&{E}_{\mathrm{turb}}^{\mathrm{pns}}={\int }_{0}^{{R}_{\mathrm{pns}}}\rho {v}_{\mathrm{turb}}^{2}{dV}.\end{eqnarray} \tag{ 12 }$$

The top panels in Figure 10 show the evolution of ${E}_{\mathrm{turb}}^{\mathrm{pns}}$ for the various progenitors and EOSs.

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In Figure 11 we show the time evolution of baryonic mass of the PNS for the various EOSs and progenitors considered in this paper. This figure, together with Figure 10, confirms the trend found by NBRV20, i.e., that more massive PNSs show more vigorous convection. This is true for all EOSs; hence, we confirm that the trend noted in the simulations of NBRV20 holds true regardless of the EOS used.

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It is also interesting to analyze each progenitor separately and study how ${E}_{\mathrm{turb}}^{\mathrm{pns}}$ correlates with the strength of the explosion. For each progenitor, as can be seen in Figure 10, the hierarchy of ${E}_{\mathrm{turb}}^{\mathrm{pns}}$ is the same as the hierarchy of the explosion energy, i.e., earlier explosions with larger explosion energies have a more vigorous PNS convection. From the analysis carried out in Section 4.2, one could then conclude that a larger central entropy leads to stronger convection, since both are correlated with a stronger explosion.

Another detail that emerges from our analysis is that the SRO-type EOSs seem to generate overall stronger PNS convection and form larger convective layers. By looking at the top two panels of Figure 10, it is evident how there is a clear separation between the curves that represent SRO-type EOSs and the curves that represent RMF-type EOSs.

We postpone a more thorough analysis of PNS convection and this effect in particular to future work. The reason for this is that the turbulent convection model that we used is designed to reproduce convection in the gain region, and therefore it is not accurate inside the PNS. In fact, by looking at the turbulent velocity near the PNS in Figure 2, it is clear that there is a discrepancy with the 3D results. Overall, the turbulent velocities we find are roughly a factor of 10–20 smaller than in 3D. Hence, the turbulent energy in the PNS ${E}_{\mathrm{turb}}^{\mathrm{pns}}$ is also about two orders of magnitude smaller than the expectation from 3D simulations.

As pointed out by Couch et al. (2020), the convection inside the PNS is extremely efficient in STIR, and therefore it quickly erases the unstable thermodynamic gradients. This shuts off further growth of the turbulent velocity, which explains why in STIR it is so small. This is confirmed by the fact that, counterintuitively, smaller values of α_MLT exhibit larger turbulent velocities inside the PNS. This indicates that, for large values of α_MLT, convection is too efficient to allow the turbulent velocity to build up.

Another reason why STIR is inadequate to predict the PNS convection is that it does not take lepton gradients into account. The neutrinos trapped inside the PNS alter the full lepton gradient. This is not merely equal to the electron fraction gradient as it is in the gain region and as it is assumed in our STIR model.

Calculating the full lepton gradient, however, is not trivial (Roberts et al. 2012) since it involves complicated thermodynamic derivatives. Hence, we leave a more detailed treatment of PNS convection to a future work.

Despite the inadequacy of STIR in predicting the magnitude of the turbulent velocity in the PNS, we find that it correctly predicts the width and the mass of the PNS convective layer, when compared to the results of NBRV20, and we find the same correlation between the mass of the PNS and the strength of convection.

Since the pioneering work of Bethe & Wilson (1985), it has been clear that delayed neutrino heating is a primary mechanism enabling the explosion. The EOS changes the thermodynamic conditions of the PNS, and therefore it indirectly changes the neutrino–matter interaction. This in turn changes the neutrino observable properties, such as the luminosity evolution and the energy spectrum. The bulk of emission of neutrinos happens inside the PNS, where neutrinos decouple from matter. This is usually identified as the region in the vicinity of the neutrinosphere, defined as the layer inside the PNS at which the neutrino opacity is equal to 2/3. Therefore, changes in temperature and density inside the PNS will affect the temperature and density profiles where the neutrinos decouple, thereby modifying the emission properties of neutrinos.

As discussed in Schneider et al. (2019), a larger value of the effective mass at saturation changes the observable properties of the emergent neutrinos, since it causes a faster contraction of the PNS. This shifts the location of the different neutrinospheres to smaller radii, larger temperatures, and lower densities. The only exception are the heavy lepton neutrinos. In this case the density of the neutrinosphere increases as the effective mass increases. It should also be noted that neutrinos of different energy decouple at different radii, since the opacity depends on the energy-dependent neutrino interaction cross section, but for simplicity one usually refers to a single neutrino-averaged neutrinosphere, defined using an energy-averaged opacity.

To analyze the impact of the EOS on the neutrino properties, we focus on the 20 M_⊙ progenitor, although a similar behavior can be seen in the other progenitors as well. The location and thermodynamic properties of the neutrinosphere for different flavors are shown in Figure 12, and the luminosities and average energies are shown in Figure 13.

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If we limit the analysis to the SRO-type EOSs, the trends found by Schneider et al. (2019) are confirmed in our simulations. That is, the SRO-type EOSs with a larger effective mass yield a hotter neutrinosphere located at smaller radii. Similarly, larger effective masses lead to larger neutrino luminosities and energies for all flavors. If we include the RMF-type EOSs in this analysis, however, these trends are disrupted, and, for example, the SFHo, with a smaller effective mass at saturation than the LS220, has larger neutrinosphere temperatures, neutrino energies, and luminosities for all flavors.

In their paper, Schneider et al. (2019) show the neutrino properties for nonexploding models. In the present work, however, some of the EOSs lead to explosions (due to the turbulent convection), and therefore some neutrino properties will be different. In particular, as the explosion sets in, the accretion onto the PNS stops, and the neutrino emission leads to fast cooling of the inner regions. This accelerates the contraction of the PNS and moves the neutrinospheres to regions of higher density. In addition to this, the temperature stops increasing and becomes roughly constant. As a consequence, the average neutrino energy also tends to a constant value, while the luminosity sharply decreases, since without accretion the neutrino production rapidly diminishes. Hence, in these simulations, the EOS is responsible for differences in the neutrino signal only in the first 150–200 ms, when turbulent convection has not yet triggered an explosion. After that, the different explosion dynamics will dramatically change the neutrino signal.

As pointed out in previous studies (Hempel et al. 2012; Steiner et al. 2013), the EOS can impact the neutrino signal in several ways. For example, the presence of light nuclei near the neutrinospheres could in principle modify the neutrino emission and absorption rates and therefore affect the neutrino signal. However, since we use the standard Bruenn (1985) neutrino opacities, which do not include interactions with light nuclei, we do not see this effect in our simulations. The main quantities that can affect the neutrino luminosities and average energy are mass accretion and the location of the neutrinosphere. Larger mass accretion results in larger luminosities, whereas neutrinospheres located at smaller radii (and therefore higher temperatures) yield larger neutrino energies. A more detailed analysis of these effects can be found in Hempel et al. (2012).

It is also worth mentioning that, given the different treatment of nuclei near saturation density (namely, SNA or NSE), the fraction of heavy nuclei in inhomogeneous phases will be different in each EOS. This can also alter the neutrino signal, since a different treatment of heavy nuclei at intermediate densities will change the electron fraction (and therefore the deleptonization) of the core. Smaller electron fractions lead to more compact configurations and therefore larger luminosities and energies (for a more detailed description, see Hempel et al. 2012). The contraction of the PNS can also be influenced by the presence of heavy nuclei below saturation density, as well as by the inclusion of general relativistic effects (taken into account only by RMF-type EOSs). Fast contractions will quickly increase the temperatures of the neutrinosphere and therefore increase the average energies of neutrinos. A more detailed discussion can be found in Hempel et al. (2012) and Steiner et al. (2013).

Overall, our simulations do not show a clear correlation between neutrino properties and strength of explosion. However, if one analyzes the neutrino luminosity during the accretion phase in models with no turbulence (i.e., simple spherically symmetric simulations without the inclusion of STIR), one can see the correlation between effective mass and luminosity for SRO-type models (see Figure 14). On the other hand, the RMF-type models show comparatively larger luminosities, although the hierarchy (SFHo, DD2, HShen) is the same as the hierarchy of explosion strength. Hence, we see that within SRO-type or RMF-type frameworks the correlation between luminosity and explosion strength holds. However, this is disrupted when analyzing EOSs across both frameworks, as already pointed out in Section 4.1 for the case of the turbulent velocity in the gain region.

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One can conclude that, compared to SRO-type EOSs, RMF-type EOSs have larger luminosities and turbulent velocities in the gain region but generate weaker explosions. The reason behind this most likely lies in the different treatment of nuclei near saturation density. A similar difference can be found in the 9 M_⊙ progenitor, discussed in the Appendix, where a different treatment of nuclear matter near saturation density leads to larger central densities, which then modify the explosion energies. Hence, not only the nuclear properties but also the treatment of nuclear matter near saturation density can affect the explosion of CCSNe.