Mean-field study of hot $\beta$-stable protoneutron star matter: Impact of the symmetry energy and nucleon effective mass

A consistent Hartree-Fock study of the equation of state (EOS) of asymmetric nuclear matter at finite temperature has been performed using realistic choices of the effective, density-dependent nucleon-nucleon $\left(\mathit{NN}\right)$ interaction, which were successfully used in different nuclear structure and reaction studies. Given the importance of the nuclear symmetry energy in the neutron star formation, EOSs associated with different behaviors of the symmetry energy were used to study hot asymmetric nuclear matter. The slope of the symmetry energy and nucleon effective mass with increasing baryon density was found to affect the thermal properties of nuclear matter significantly. Different density-dependent $\mathit{NN}$ interactions were further used to study the EOS of hot protoneutron star (PNS) matter of the $npe\mu \nu$ composition in $\beta$ equilibrium. The hydrostatic configurations of PNS in terms of the maximal gravitational mass $M_{\max }$ and radius, central density, pressure, and temperature at the total entropy per baryon $S/A=1,2$, and 4 have been determined in both the neutrino-free and neutrino-trapped scenarios. The obtained results show consistently a strong impact of the symmetry energy and nucleon effective mass on thermal properties and composition of hot PNS matter. $M_{\max }$ values obtained for the (neutrino-free) $\beta$-stable PNS at $S/A=4$ were used to assess time $t_{\mathrm{BH}}$ of the collapse of a $40M_{\odot }$ protoneutron progenitor to a black hole, based on a correlation between $t_{\mathrm{BH}}$ and $M_{\max }$ found from the hydrodynamic simulation by Hempel et al. [Astrophys. J. 748, 70 (2012)].

Figure 1

Free energy per particle $F/A$ of symmetric nuclear matter (SNM) and pure neutron matter (PNM) at different temperatures given by the HF calculation (1)–(13) using the CDM3Y3 (right column) and CDM3Y6 (left column) interactions [39] (lines), in comparison with the BHF results (symbols) by Burgio and Schulze [12].

Figure 2

The same as Fig. 1 but for the HF results obtained with the M3Y-P5 (right column) and M3Y-P7 (left column) interactions parametrized by Nakada [27, 28].

Figure 3

The same as Fig. 1 but for the HF results obtained with the D1N version [30] of Gogny interaction (left column) and SLy4 version [31] of Skyrme interaction (right column).

Figure 4

Free symmetry energy per particle $F_{\mathrm{sym}}/A$ of pure neutron matter $\left(\delta =1\right)$ at different temperatures given by the HF calculations (15) using the CDM3Y3 and CDM3Y6 interactions [39] and their soft versions CDM3Y3s and CDM3Y6s [8] (lines), in comparison with the BHF results (symbols) by Burgio and Schulze [12].

Figure 5

The same as Fig. 4 but for the HF results given by the M3Y-P5 and M3Y-P7 interactions parametrized by Nakada [27, 28] and the D1N version [30] of Gogny interaction and SLy4 version [31] of Skyrme interaction.

Figure 6

Free symmetry energy (15) (bottom row) and internal symmetry energy (17) (top row) at different temperatures, given by the HF calculation using the CDM3Y6 interaction [39]. $(F_{\mathrm{sym}}/A)/{\delta }^2$ curves must be very close if the quadratic approximation (16) is valid.

Figure 7

Density profile of the neutron- (top) and proton effective mass (bottom) at different neutron-proton asymmetries $\delta$ given by the HF calculation using the CDM3Y6 and CDM3Y3 interactions [39], and the D1N version of Gogny interaction [30], in comparison with the BHF results (symbols) by Baldo et al. [52].

Figure 8

The same as Fig. 7 but for the HF results obtained with the M3Y-P7 and M3Y-P5 interactions [27, 28] and the SLy4 version [31] of Skyrme interaction.

Figure 9

Density profile of temperature in the isentropic and symmetric NM given by the HF calculation using different density-dependent $\mathit{NN}$ interactions, in comparison with that given by the approximation (23) for the fully degenerate Fermi (DF) system at $T\ll T_F$.

Figure 10

Density profile of entropy per particle $S/A$ of symmetric nuclear matter (SNM) and pure neutron matter (PNM) at different temperatures, deduced from the HF results (lines) obtained with the CDM3Y6 and CDM3Y3 interactions [39], in comparison with the BHF results by Burgio and Schulze (symbols) [12].

Figure 11

The same as Fig. 10 but for the HF results obtained with the M3Y-P7 and M3Y-P5 interactions [27, 28].

Figure 12

The same as Fig. 10 but for the HF results obtained with the D1N version [30] of Gogny interaction and SLy4 version [31] of Skyrme interaction.

Figure 13

Symmetry part of the entropy per particle (19) of pure neutron matter at different temperatures given by the CDM3Y3 and CDM3Y6 interactions [39] and their soft CDM3Y3s and CDM3Y6s versions [8]. $S_{\mathrm{sym}}/A$ is scaled by the corresponding temperature to have the curves well distinguishable at different $T$.

Figure 14

Pressure (33) of the isentropic $\nu$-free (left column) and $\nu$-trapped (right column) $\beta$-stable PNS matter at different baryon densities $n_b$ and entropy per baryon $S/A=1,2$, and 4. The EOS of the PNS crust is given by the RMF calculation by Shen et al. [7, 55], and the EOS of the uniform PNS core is given by the HF calculation using the CDM3Y6 interaction [39] (top) and its soft CDM3Y6s version [8] (bottom). The transition region matching the PNS crust with the uniform core is shown as the dotted lines.

Figure 15

Neutron-proton asymmetry $\delta$ of the $\nu$-free (left column) and $\nu$-trapped (right column) $\beta$-stable PNS matter at different baryon number densities $n_b$ and entropy per baryon $S/A=1,2$, and 4. The EOS of the homogeneous PNS core is given by the HF calculation using the CDM3Y6 interaction [39] and its soft CDM3Y6s version [8].

Figure 16

Entropy per baryon (top row) and temperature (bottom row) as function of baryon number density $n_b$ of the $\beta$-stable PNS matter given by the CDM3Y6 interaction [23, 39] (thick lines) and its soft CDM3Y6s version [8] (thin lines) in the $\nu$-free (left column) and $\nu$-trapped (right column) cases.

Figure 17

Density profile of neutron effective mass in the $\nu$-free and $\beta$-stable PNS matter at entropy per baryon $S/A=1,2$, and 4, given by the HF calculation using the CDM3Y6 [39] and M3Y-P7 [28] interactions (left column), the D1N version of Gogny interaction [30] (right column) and SLy4 version [31] of Skyrme interaction (right column).

Figure 18

The same as Fig. 17, but for proton effective mass.

Figure 19

Density profile of temperature in the $\nu$-free and $\beta$-stable PNS matter at entropy per baryon $S/A=1,2$, and 4, deduced from the HF results obtained with the same density-dependent $\mathit{NN}$ interactions as those considered in Fig. 17.

Figure 20

Particle fractions as function of baryon number density $n_b$ in the $\nu$-free and $\beta$-stable PNS matter at entropy per baryon $S/A=1,2$, and 4, given by the CDM3Y6 interaction [39] (top row) and its soft CDM3Y6s version [8] (bottom row).

Figure 21

The same as Fig. 20, but for the $\nu$-trapped, $\beta$-stable matter of the PNS.

Figure 22

The same as Fig. 20, but given by the SLy4 version [31] of Skyrme interaction (top row) and M3Y-P7 interaction parametrized by Nakada [28] (bottom row).

Figure 23

The same as Fig. 22, but for the $\nu$-trapped, $\beta$-stable PNS matter.

Figure 24

Gravitational mass (in unit of solar mass $M_{\odot }$) of the $\beta$-stable, $\nu$-free (left column) and $\nu$-trapped (right column) PNS at entropy $S/A=1,2$ and 4 as function of the radius (in km), based on the EOS of the homogeneous PNS core given by the CDM3Y6 interaction [39] (top row) and its soft CDM3Y6s version [8] (bottom row). The circle at the end of each curve indicates the last stable configuration.

Figure 25

The same as Fig. 24 but given by the CDM3Y3 interaction [39] (top row) and its soft CDM3Y3s version [8] (bottom row).

Figure 26

The same as Fig. 24 but given by the SLy4 version of Skyrme interaction [31] (top row) and M3Y-P7 interaction parametrized by Nakada [28] (bottom row).

Figure 27

Delay time $t_{\mathrm{BH}}$ from the onset of the collapse of a $40M_{\odot }$ progenitor until the black hole formation as a function of the enclosed gravitational mass $M_{\mathrm{G}}$ (open squares) given by the hydrodynamic simulation [9, 10] and $M_{\max }$ values given by the solution of the TOV equations using the same EOS for the $\nu$-free and $\beta$-stable PNS at $S/A=4$ (open circles). The $M_{\max }$ values given by the present mean-field calculation of the $\nu$-free and $\beta$-stable PNS at $S/A=4$ using different density-dependent $\mathit{NN}$ interactions are shown on the correlation line interpolated from the results of simulation.