Equation of state of the beta - Stable nuclear matter in neutron stars and proto - neutron stars

baryon densities. Such an enhancement of the proton fraction in PNS 96 Chapter 5. HF study of the -stable PNS matter 0 1 2 3 4 S /A ν-free T=10 MeV 20 40 80 CDM3Y6 CDM3Y6s ν-trapped T=10 MeV 20 40 80 0 40 80 120 160 0.2 0.4 0.6 0.8 1 T (M eV ) nb (fm-3) ν-free S/A =4 2 1 CDM3Y6 CDM 3Y6s 0.2 0.4 0.6 0.8 1 S/A=4 2 1 ν-trapped Figure 5.3: Entropy per baryon (upper panel) and temperature (lower panel) as function of baryon number density nb of the -stable PNS matter given by the CDM3Y6 interaction [13, 9] (thick lines) and its soft CDM3Y6s version [6] (thin lines) in the -free (left panel) and -trapped (right panel) cases. matter by neutrino trapping was also found in the BHF calculation by Vida~na et al. [94] that includes hyperons. The suppression of the impact of the nuclear symmetry energy in the presence of trapped neutrinos is mainly explained by the fact that the PNS matter is more symmetric and weak processes, like p+ e $ n+ e, can proceed in both directions. It is clearly seen in Fig. 5.2 that the more symmetric matter the less impact of the symmetry energy. This eect is also illustrated in the density proles of entropy and temperature of the -stable PNS matter given by the CDM3Y6 and CDM3Y6s interactions shown in Fig. 5.3. One can see on the left panel of Fig. 5.3 that in the -free case, the dierence between the sti and soft symmetry-energy scenarios is strongest at high baryon densities so that 5.2. EOS of PNS matter 97 0.01 0.1 1 x k CDM3Y6 S/A=1 n p e µ S/A=2 CDM3Y6 n p e µ CDM3Y6 S/A=4 n p e µ 0.01 0.1 0.2 0.4 0.6 0.8 1 CDM3Y6s S/A=1 n p e µ 0.2 0.4 0.6 0.8 1 nb (fm-3) CDM3Y6s S/A=2 n p e µ 0.2 0.4 0.6 0.8 1 CDM3Y6s S/A=4 n p e µ Figure 5.4: Particle fractions as function of baryon number density nb in the -free and -stable PNS matter at entropy per baryon S=A = 1; 2 and 4, given by the CDM3Y6 interaction [9] (upper panel) and its soft CDM3Y6s version [6] (lower panel). the center of PNS becomes much hotter when the symmetry energy is soft. Such an impact to the density proles of entropy and temperature by the symmetry energy is, however, suppressed in the presence of trapped neutrinos (see right panel of Fig. 5.3). In both cases, the eect of the symmetry energy is clearly demon- strated: at high baryon densities, the asy-sti interactions always give higher electron and proton fractions compared with those given by the asy-soft interactions. Such an impact of the symmetry energy is more pro- nounced in the -free case, where the electron fraction xe predicted by the asy-soft models reduces signicantly to below 0.1 at high baryon densi- ties. As discussed in Ref. [6], the dierence in the electron fraction caused 98 Chapter 5. HF study of the -stable PNS matter 0.01 0.1 1 x k CDM3Y6S/A=1 n p,e νe CDM3Y6S/A=2 n p,e νe CDM3Y6S/A=4 n p e µ νe 0.01 0.1 0.2 0.4 0.6 0.8 1 CDM3Y6sS/A=1 n p,e νe 0.2 0.4 0.6 0.8 1 nb (fm-3) CDM3Y6sS/A=2 n p,e νe 0.2 0.4 0.6 0.8 1 CDM3Y6sS/A=4 n p e µ νe Figure 5.5: The same as Fig. 5.4 but for the -trapped, -stable matter of the PNS. by dierent slopes of the symmetry energy at high baryon densities has a drastic eect on the -emission process during the cooling of neutron star. Namely, in the soft symmetry-energy scenario, the direct Urca process is strongly quenched because of Ye < 11% [6, 56], while it is well allowed in the sti symmetry-energy scenario. The dierence in the electron frac- tion caused by dierent slopes of the symmetry energy is less pronounced in very hot PNS matter at S=A = 4. Such state of hot PNS matter has been shown to occur at the onset of collapse of a very massive (40 M) progenitor to black hole [76, 4]. Based on dierent EOS's given by dierent density dependent NN in- teractions, the composition of the -stable PNS in terms of the constituent- particle fractions at entropy per baryon S=A = 1; 2 and 4 is quite dierent (see Figs. 5.4-5.7). The results shown in Figs. 5.4 and 5.6 were obtained 5.2. EOS of PNS matter 99 0.01 0.1 1 x k SLy4 S/A=1 n p e µ S/A=2 SLy4 n p e µ SLy4 S/A=4 n p e µ 0.01 0.1 0.2 0.4 0.6 0.8 1 M3Y-P7 S/A=1 n p e µ 0.2 0.4 0.6 0.8 1 nb (fm-3) M3Y-P7 S/A=2 n p e µ 0.2 0.4 0.6 0.8 1 M3Y-P7 S/A=4 n p e µ Figure 5.6: The same as Fig. 5.4, but given by the SLy4 version [19] of Skyrme interac- tion (upper panel) and M3Y-P7 interaction parametrized by Nakada [16] (lower panel). for the -free PNS matter that corresponds the late stage of the evolu- tion of PNS when most of neutrinos have escaped from the core. In this case the dierence between the sti- and soft symmetry-energy scenarios is quite signicant as discussed above for the neutron-proton asymmetry. A comparison with the BHF results obtained for the -stable PNS matter at entropy S=A = 1; 2 (see upper panels of Figs. 5.4 and 5.6, and Fig. 4 of Ref. [27]) shows that the asy-sti interactions (CDM3Y3, CDM3Y6, and SLy4) give the results rather close to the BHF results, while the results given by the asy-soft interactions (CDM3Y3s, CDM3Y6s, and M3Y-P7) are very dierent from the BHF results. The results shown in Figs. 5.5 and 5.7 were obtained for the -trapped PNS matter with the electron lepton fraction Ye  0:4, as expected during the rst stage of the core- 100 Chapter 5. HF study of the -stable PNS matter 0.01 0.1 1 x k SLy4S/A=1 n p,e νe SLy4S/A=2 n p e νe µ SLy4S/A=4 n p e νe µ 0.01 0.1 0.2 0.4 0.6 0.8 1 M3Y-P7S/A=1 n p,e νe 0.2 0.4 0.6 0.8 1 nb (fm-3) M3Y-P7S/A=2 n p e νe µ 0.2 0.4 0.6 0.8 1 M3Y-P7S/A=4 n p e νe µ Figure 5.7: The same as Fig. 5.6 but for the -trapped, -stable PNS matter. collapse supernovae [3, 27]. In such a scenario, the electron conversion to muon is unlikely at low temperatures or entropy, and the muon fraction was found much suppressed at S=A = 1 and 2. At high temperatures of the PNS matter at S=A = 4, the electron chemical potential e increases and exceeds the muon threshold, and muons appear again. It is interesting to note the dierence between results obtained in the -trapped and -free cases. Although -equilibrium implies the weak pro- cesses l + p $ n + l, in the -free PNS matter most of neutrinos have escaped and cannot contribute to these processes, and they proceed prefer- ably in one direction (from left to right). As a result, the PNS matter undergoes the neutronization process [29, 95]. Therefore, the -free PNS matter is always more neutron-rich than the -trapped matter as shown above in Fig. 5.2. 5.2. EOS of PNS matter 101 We discuss also the eect of increasing temperature (or equivalently increasing entropy) on the particle fractions shown in Figs. 5.4-5.7. In general, the increase of temperature tends to reduce the dierence between the neutron and proton fractions (nn decreases while np increases with T ). In a similar way, the dierence between e and  becomes also less pronounced with the increasing temperature (ne decreases and n increases with T ). Such a phenomenon is expected to occur between particles of similar nature, and the hot matter tends to quench the dierence in their masses. 5.2.2 Impact of the in-medium nucleon eective mass Beside the symmetry energy, the impact of nucleon eective mass to the thermal properties of NM was found quite signicant in the previous section, and it is of interest to explore the eect of m in the -stable PNS matter at dierent temperatures and entropy. The density proles of neutron and proton eective masses in the -free and -stable PNS matter at entropy per baryon S=A = 1; 2 and 4, obtained with dierent density dependent NN interactions are shown in Fig. 5.8. One can see that the density dependence of nucleon eective mass in the hot -stable PNS matter is quite similar to that at zero temperature shown in Figs. 4.8 and 4.9. While nucleon eective mass given by the CDM3Y6 interaction saturates at nb  0:2 fm3 and rises up to above unity at high baryon densities (in agreement with the BHF calculation [71] that includes 3-body forces), the m values given by other interactions steadily decrease with the increasing baryon density. The fall of nucleon eective mass given by the M3Y-P7 and Sly4 interactions is very drastic at high densities and it should result, therefore, on very high temperature in the center of PNS. The density proles of temperature in the -free and - 102 Chapter 5. HF study of the -stable PNS matter 0.3 0.5 0.7 0.9 1.1 m n * /m CDM3Y6 S/A=1 S/A=2 S/A=4 D1N 0.3 0.5 0.7 0.9 0.2 0.4 0.6 0.8 nb (fm-3) M3Y-P7 0.2 0.4 0.6 0.8 SLy4 0.3 0.5 0.7 0.9 1.1 m p * / m CDM3Y6 S/A=1 S/A=2 S/A=4 D1N 0.3 0.5 0.7 0.9 0.2 0.4 0.6 0.8 nb (fm-3) M3Y-P7 0.2 0.4 0.6 0.8 SLy4 Figure 5.8: Density prole of neutron and proton eective mass in the -free and - stable PNS matter at entropy per baryon S=A = 1; 2 and 4, given by the HF calculation using the CDM3Y6 [9] and M3Y-P7 [16] interactions (left panel), the D1N version of Gogny interaction [18] (right panel) and SLy4 version [19] of Skyrme interaction (right panel). 5.3. Proto-neutron star in the hydrodynamical equilibrium 103 0 50 100 150 200 T (M eV ) CDM3Y6 S/A=1 S/A=2 S/A=4 D1N 0 50 100 150 0.2 0.4 0.6 0.8 1 nb (fm-3) M3Y-P7 0.2 0.4 0.6 0.8 1 SLy4 Figure 5.9: Density prole of temperature in the -free and -stable PNS matter at entropy per baryon S=A = 1; 2 and 4, deduced from the HF results obtained with the same density dependent NN interactions as those considered in Fig. 5.8. stable PNS matter at entropy per baryon S=A = 1; 2 and 4, given by the same density dependent NN interactions are shown in Fig. 5.9. It can be seen that the rise of temperature with the increasing baryon density given by the M3Y-P7 and Sly4 interactions is indeed very sti, to T above 200 MeV at high densities. We will see below that such dierent behaviors of nucleon eective mass also in uence strongly the hydrostatic conguration of hot PNS. 5.3 Proto-neutron star in the hydrodynam- ical equilibrium We used the total internal energy E and pressure P inside PNS at en- tropy per baryon S=A = 0; 1; 2 and 4 given by the dierent EOS's discussed 104 Chapter 5. HF study of the -stable PNS matter 0 0.5 1 1.5 2 2.5 M m a x /M Ο. CDM3Y6ν-free CDM3Y6ν-trapped 0 0.5 1 1.5 2 12 18 24 30 R (km) CDM3Y6sν-free 12 18 24 30 CDM3Y6sν-trapped S/A=1 S/A=2 S/A=4 Figure 5.10: Gravitational mass (in unit of solar mass M) of the -stable, -free (left panel) and -trapped (right panel) 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 [9] (upper panel) and its soft CDM3Y6s version [6] (lower panel). The circle at the end of each curve indicates the last stable conguration. above in previous section as inputs for the Tolman-Oppenheimer-Volkov (TOV) equations [38]. The -stable hydrostatic conguration of hot PNS in dierent scenarios given by the solutions of the TOV equations are pre- sented in Tables 5.1 and 5.2, and Figs. 5.10-5.12. In order to explore explicitly the eects caused by the nuclear sym- metry energy to the stable conguration of PNS, we have considered solu- tions of the TOV equations given by the EOS's based on the CDM3Y3 and CDM3Y6 interactions [9] and their soft versions CDM3Y3s and CDM3Y6s [6]. The only dierence between these two groups of the CDM3Yn in- teraction is the modeling of the IV density dependence that gives dier- 5.3. Proto-neutron star in the hydrodynamical equilibrium 105 Table 5.1: Properties of the -free and -trapped, -stable PNS at entropy per baryon S=A = 0; 1; 2 and 4, given by the solutions of the TOV equations using the EOS's based on the CDM3Y3, CDM3Y6 interactions [9] and their soft CDM3Y3s, CDM3Y6s versions [6]. Mmax and Rmax are the maximum gravitational mass and radius; nc; c; Pc, and Tc are the baryon number density, mass density, total pressure, and temperature in the center of PNS. Ts is temperature of the outer core of PNS, at baryon density s  0:63  1015 g/cm3. Results at S=A = 0 represent the stable conguration of cold (-free) NS [6]. EOS S=A Mmax Rmax nc c Pc Tc Ts (M) (km) (fm3) (*) (**) (MeV) (MeV) CDM3Y3 0 1.59 9.70 1.44 3.16 494.4 0.0 (-free) 1 1.64 9.60 1.53 3.16 538.9 19.7 17.6 2 1.65 10.14 1.40 2.90 446.8 49.4 37.9 4 1.94 14.67 0.78 1.58 180.5 113.6 84.6 CDM3Y3 1 1.67 10.59 1.44 2.90 470.8 18.7 14.3 (-trapped) 2 1.69 11.40 1.29 2.59 373.9 44.7 30.1 4 1.99 15.67 0.75 1.45 163.1 91.4 66.9 CDM3Y3s 0 1.13 9.36 1.61 3.26 261.1 0.0 (-free) 1 1.18 9.53 1.52 3.07 253.0 37.4 18.4 2 1.33 10.47 1.30 2.66 239.0 71.4 39.0 4 1.93 15.28 0.67 1.33 130.6 120.3 86.1 CDM3Y3s 1 1.55 10.84 1.46 2.90 390.3 18.0 14.2 (-trapped) 2 1.58 11.31 1.30 2.59 319.0 46.5 30.1 4 1.93 14.91 0.77 1.50 161.8 91.5 69.0 CDM3Y6 0 1.95 10.23 1.20 2.74 627.3 0.0 (-free) 1 1.97 10.25 1.24 2.73 626.5 10.1 17.4 2 1.93 10.68 1.18 2.59 524.1 33.7 37.5 4 2.12 14.37 0.80 1.63 235.5 114.1 86.8 CDM3Y6 1 2.03 11.11 1.18 2.51 553.9 9.7 18.3 (-trapped) 2 2.00 11.46 1.13 2.37 467.7 24.8 29.6 4 2.11 14.29 0.82 1.68 246.6 89.7 66.6 CDM3Y6s 0 1.42 9.74 1.46 3.06 340.4 0.0 (-free) 1 1.65 9.20 1.50 3.45 652.7 17.1 18.3 2 1.37 11.13 1.16 2.44 212.9 65.4 38.8 4 1.96 14.24 0.80 1.67 206.9 121.5 84.5 CDM3Y6s 1 2.04 11.67 1.02 2.02 314.7 12.3 13.9 (-trapped) 2 1.88 11.68 1.12 2.30 353.9 34.6 29.8 4 2.06 14.24 0.84 1.68 229.4 91.2 66.7 *1015g/cm3 **MeV fm3 106 Chapter 5. HF study of the -stable PNS matter 0 0.5 1 1.5 2 2.5 M m a x /M Ο. CDM3Y3ν-free CDM3Y3ν-trapped 0 0.5 1 1.5 2 12 18 24 30 R (km) CDM3Y3sν-free 12 18 24 30 ν-trapped CDM3Y3s S/A=1 S/A=2 S/A=4 Figure 5.11: The same as Fig. 5.10 but given by the CDM3Y3 interaction [9] (upper panel) and its soft CDM3Y3s version [6] (lower panel). ent slopes of the free symmetry energy at high NM densities and dier- ent temperatures as shown in Fig. 4.5. One can see in Tables 5.1 and Figs. 5.10 and 5.11 that in the -free case, the impact of the symmetry energy is quite strong at entropy per baryon S=A = 0; 1 and 2. For exam- ple, the PNS maximum gravitational mass obtained with the CDM3Y3s interaction at S=A = 1 is Mmax(CDM3Y3s)  1:18 M, and it increases to Mmax(CDM3Y3)  1:64 M when the density dependence of the sym- metry energy is changed from the soft to the sti behavior. Similarly with the CDM3Y6 interaction, we foundMmax(CDM3Y6s)  1:65M and Mmax(CDM3Y6)  1:97 M at S=A = 1 (see Fig. 5.10). Thus, the dif- ference in the slope of the symmetry energy at high NM densities could lead to a dierence of 0:3 0:5 M in the predicted Mmax value. We note further that Mmax obtained with the CDM3Y6 interaction at S=A = 0; 1 5.3. Proto-neutron star in the hydrodynamical equilibrium 107 Table 5.2: The same as Table 5.1 but obtained with the EOS's based on the SLy4 version [19] of Skyrme interaction, M3Y-P7 interaction parametrized by Nakada [16], and D1N version [18] of Gogny interaction. EOS S=A Mmax Rmax nc c Pc Tc Ts (M) (km) (fm3) (*) (**) (MeV) (MeV) SLy4 0 2.05 9.96 1.21 2.86 860.4 0.0 (-free) 1 2.06 10.07 1.18 2.74 776.3 96.2 30.4 2 2.11 10.66 1.08 2.51 666.3 162.9 61.9 4 2.37 14.60 0.70 1.54 279.8 214.0 122.6 SLy4 1 2.16 11.00 1.13 2.51 717.7 56.2 19.5 (-trapped) 2 2.16 11.16 1.10 2.44 661.9 92.1 41.1 4 2.14 12.07 1.03 2.30 564.9 148.9 81.9 M3Y-P7 0 2.07 10.05 1.17 2.82 869.9 0.0 (-free) 1 2.14 10.23 1.13 2.66 827.6 60.9 23.5 2 2.23 11.09 1.00 2.00 645.8 104.9 48.09 4 2.40 13.86 0.75 1.73 365.3 174.8 109.5 M3Y-P7 1 2.15 10.49 1.14 2.73 999.7 42.7 16.1 (-trapped) 2 2.22 11.09 1.07 2.44 738.9 75.7 38.0 4 2.27 11.68 1.05 2.30 693.0 138.8 77.2 D1N 0 1.23 7.75 2.36 5.24 819.9 0.0 (-free) 1 1.32 8.46 1.88 4.10 581.3 97.9 22.8 2 1.61 10.31 1.29 2.82 389.3 139.9 43.1 4 2.22 14.28 0.77 1.67 282.9 148.9 84.7 D1N 1 1.82 12.20 1.07 2.05 249.5 32.8 15.1 (-trapped) 2 1.96 12.22 1.06 2.11 314.1 64.0 31.0 4 2.28 14.28 0.85 1.77 352.6 103.0 66.5 is quite close to the neutron star mass of 1:97 M observed recently by Demores et al. [78]. Because the isospin dependences of the CDM3Y3 and CDM3Y6 interactions are nearly the same, the dierence of about 0:3 M in the Mmax values at S=A = 0; 1 obtained with these two interactions in the -free case should be due to dierent nuclear incompressibilities K0 (see Table 2.6). Concerning other NN interactions under study, only the 108 Chapter 5. HF study of the -stable PNS matter 0 0.5 1 1.5 2 2.5 M m a x /M Ο. SLy4ν-free ν-trapped SLy4 0 0.5 1 1.5 2 12 18 24 30 R (km) ν-free M3Y-P7 12 18 24 30 ν-trapped M3Y-P7 S/A=1 S/A=2 S/A=4 Figure 5.12: The same as Fig. 5.10 but given by the SLy4 version of Skyrme interaction [19] (upper panel) and M3Y-P7 interaction parametrized by Nakada [16] (lower panel). Sly4 and M3Y-P7 interactions give the Mmax values at S=A = 0; 1 close to the observed neutron star mass of 1:97 M [78]. The D1N interaction gives a too low Mmax  1:23 and 1.32 M at S=A = 0 and 1, respectively, while the M3Y-P5 interaction gives unstable results for the PNS congu- ration at high temperatures and baryon densities, and are not included in the present discussion. We note that the instability of the results given by the M3Y-P5 interaction is a purely numerical one that is likely due to the parametrization of this version of M3Y-Pn interaction. In the presence of trapped neutrinos, the eect caused by dierent slopes of the symmetry energy is diminished already at entropy S=A = 1 and 2, and stable values of the maximum mass and radius of PNS are rather close, independent of the behavior of the symmetry energy (see Table 5.1, Figs. 5.10 and 5.11). We note that the microscopic BHF calculation of the 5.3. Proto-neutron star in the hydrodynamical equilibrium 109 -trapped, isentropic PNS [27] gives the maximum mass Mmax  1:95 M at entropy per baryon S=A = 1 and 2, and the corresponding Rmax  10:3 and 10.8 km. Thus, only the PNS conguration obtained with the CDM3Y6 interaction has the maximum mass close to the BHF result, while the maximum radius is slightly larger than that predicted by the BHF calculation. At much higher temperatures of the PNS matter at S=A = 4, the impact of the symmetry energy becomes weaker and similar -stable con- gurations were found with both the asy-sti and asy-soft interactions in both the -free and -trapped scenarios. However, the eect of the in- creasing entropy on the maximum gravitational mass and radius of the -stable PNS is very signicant, and we found a strong increase of Mmax and Rmax at entropy per baryon S=A = 4 compared to those obtained at S=A = 0 (see Tables 5.1 and 5.2, and Figs. 5.10-5.12). The hot PNS at S=A = 4 is substantially expanded in its mass (
Mmax  0:3 0:5 M) and size (
Rmax  4 5 km), while the pressure in the core is decreased by a factor up to three. The case of the D1N interaction is extreme with the dierences
Mmax  1 M and
Rmax  6:5 km found between the cold neutron star and hot PNS at S=A = 4 (both being in -equilibrium without trapped neutrinos). With a minor impact of the symmetry energy at entropy S=A = 4, a very substantial dierence in temperature of hot PNS matter given by dierent density dependent interactions (shown in the last two columns of Tables 5.1 and 5.2) is directly related to the dierence in nucleon eective mass illustrated in Figs. 5.8-5.9. While the maximum gravitational masses obtained with the D1N, M3Y-P7 and Sly4 interactions are only slightly larger than that obtained with the CDM3Y6 interaction by
Mmax  0:1 M, the dierence in temperature of PNS at entropy S=A = 4 is very large, up to about 100 MeV. The recent hydrodynamic simulation of the 110 Chapter 5. HF study of the -stable PNS matter gravitational collapse of a massive 40M protoneutron progenitor to black hole [76] has shown that the -stable matter (with very small neutrino fraction) in the outer core of PNS, is reaching S=A  4 at baryon density s  0:63 1015 g/cm3 and temperature Ts  80 90 MeV, depending on the inputs of the EOS of -stable PNS matter (see Fig. 16 of Ref. [76]). For the purpose of comparison, we have also deduced temperature of the PNS layer of baryon density of around 0:63 1015 g/cm3 from out mean- eld results (see the last column of Tables 5.1 and 5.2), and found that only the CDM3Yn and D1N interactions give temperature Ts in the -free scenario comparable with that obtained in the hydrodynamic simulation. The temperature Tc in the center of -stable hydrostatic PNS given by the CDM3Yn interaction is about 90 and 110 MeV in the -trapped and -free scenarios, respectively. The Tc values given by the D1N, M3Y-P7 and Sly4 interactions are much too high (see Tables 5.2), and this eect is clearly due to the unrealistic behavior of nucleon eective mass predicted by these interactions at high baryon densities (see Fig. 5.8). From the above discussion, only the asy-sti CDM3Yn interaction seems to reproduce the conguration of hot PNS comparable with that given by the hydrodynamic simulation. We discuss our mean-eld results for the -free and -stable PNS at S=A = 4 in more details. Because the simulated density prole of temper- ature reaches its maximum at the total entropy per baryon S=A  4, the authors of Refs. [76, 4] have compared the stable hydrostatic congurations obtained with dierent EOS's for the -free, -stable PNS at S=A = 4 with the results of their hydrodynamic simulation. What they found is quite remarkable: the gravitational mass MG enclosed inside the shock at the onset of collapse of the 40M protoneutron progenitor to black hole (open squares in Fig. 5.13) given by the simulation is rather close to the maxi- mum gravitational mass Mmax given by the same EOS of hot PNS matter 5.3. Proto-neutron star in the hydrodynamical equilibrium 111 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 0.0 0.2 0.4 0.6 0.8 1.0 t B H ( s ) CDM3Y6 D1N M3Y-P7 SLy4 M/M Solar Figure 5.13: Delay time tBH from the onset of the collapse of a 40 M progenitor until the black hole formation as function of the enclosed gravitational massMG (open squares) given by the hydrodynamic simulation [76, 4], and Mmax values given by the solution of the TOV equations using the same EOS for the -free and -stable PNS at S=A = 4 (open circles). The Mmax values given by the present mean-eld calculation of the -free and -stable PNS at S=A = 4 using dierent density dependent NN interactions are shown on the correlation line interpolated from the results of simulation. (open circles in Fig. 5.13). As a result, almost a linear correlation was found between the Mmax value and the delay time tBH from the onset of collapse until black hole formation predicted by the simulation [76] (see the solid line in Fig. 5.13). We have combined, therefore, the results of this interesting study from Refs. [76, 4], and presented the correlation between tBH, MG and Mmax values in Fig. 5.13. Although the tBH and MG values that might be obtained from the hydrodynamic simulation using the EOS's considered in the present work are still uncertain, we can use the corre- 112 Chapter 5. HF study of the -stable PNS matter lation found between tBH and Mmax in Refs. [76, 4] to roughly estimate tBH based on Mmax values given by the density dependent NN interactions under study. Focusing on the CDM3Y6, Sly4, and M3Y-P7 interactions which give the Mmax value at S=A = 0 close to the observed neutron star mass of around 1.97 M [78], we nd that tBH  0:5 s might be obtained with the EOS based on the CDM3Y6 interaction, and tBH  0:6 0:7 s with the EOS's based on the Sly4 and M3Y-P7 interactions. These values seem to satisfy the realistic boundary condition tBH & 0:5 s discussed in Ref. [76]. Conclusion A consistent Hartree-Fock calculation of cold and hot asymmetric NM has been performed to obtain the EOS of the npe matter of the neu- tron star and proto-neutron star in  equilibrium, using several realistic versions of the in-medium density dependent NN interaction that give two dierent behaviors of the NM symmetry energy at supranuclear densities (the soft and sti scenarios). In general, the EOS given by the soft-type interactions tend to give the pressure lower than the empirical values at high densities. Dierent EOS's of the NS core supplemented by the EOS of the NS crust given by the compressible liquid drop model have been used for the input of the TOV equations to study how dierent behaviors of the symmetry energy aect the model prediction of the NS properties. The EOS's obtained with the sti-type interactions were found to give a consistently reasonable description of the empirical data for the NS mass and radius, and to comply well with the causality condition. In compari- son with the same empirical NS data, the soft-type interactions were found less successful, especially, the two versions of the Gogny interaction cer- tainly seem to need an appropriate modication before they can be used in the TOV equations to study the structure of the neutron star. The vital role of the NM symmetry energy was demonstrated in our specic test of the CDM3Yn interactions, where we found a signicant reduction of the maximum gravitational mass MG and radius RG away from the empirical boundaries when the slope of the NM symmetry energy was changed from 113 114 Chapter 5. HF study of the -stable PNS matter the sti behavior to the soft one. It is natural to expect that if hyperons (and other possible constituents like kaons or quark matter) are included at high baryon densities, the MG and RG values given by the soft-type in- teractions can be driven to a region lying well below all existing empirical estimates. The impact of nuclear symmetry energy and nucleon eective mass to thermal properties of asymmetric NM has been studied in details. We found that the two groups of the interactions (the so-called asy-sti and asy-soft) that give the sti and soft behaviors of the symmetry energy of NM at zero temperature also give the same behaviors of the free symmetry energy of hot NM. The free symmetry energy and nucleon eective mass obtained with the asy-sti CDM3Y3 and CDM3Y6 interactions are in a good agreement with the prediction of the microscopic BHF calculations [27, 71] using the Argonne NN interaction, while those results given by the asy-soft interactions dier signicantly from the BHF results at high NM densities. We have tested the quadratic approximation for the dependence of the free symmetry energy on the neutron-proton asymmetry  by comparing the results of the HF calculation at dierent asymmetries . While this ap- proximation remains reasonable for the internal symmetry energy Esym=A at dierent temperatures, it becomes much poorer for the free symmetry energy Fsym=A with the increasing temperature. Such a breakdown of the quadratic approximation is due to the contribution of entropy. The HF calculation of hot NM has been done in a ne grid of the baryon number densities, neutron-proton asymmetries, temperatures and entropies to in- vestigate the density proles of temperature and entropy of hot NM. A very signicant impact of the nucleon eective mass m to the thermodynamic properties of hot NM was found, which is directly related to the momen- tum dependence of the nucleon mean-eld potential. At the given baryon Chapter 5. HF study of the -stable PNS matter 115 density, the smaller the nucleon eective mass the larger the temperature of NM. The density dependent NN interactions were further used to generate the EOS of baryon matter in the uniform PNS core, which was smoothly connected to the EOS of the inhomogeneous PNS crust by Shen et al. [5] for the mean-eld study of -stable PNS matter at nite temperature. Our study considered two dierent scenarios: the -trapped matter with the electron lepton fraction Ye  0:4 and the total entropy per baryon S=A = 1; 2 and 4, which mimics the initial stage of PNS; the -free matter at S=A = 1; 2 and 4, which is close to the late stage of PNS when most neutrinos have escaped. The impact of nuclear symmetry energy was found signicant in the -free PNS matter where the dynamic neutron-proton asymmetry, baryon and lepton compositions obtained with the asy-sti and asy-soft interactions are quite dierent. The high-density behaviors of the density proles of temperature and entropy of the -free, -stable PNS matter are strongly aected not only by the symmetry energy but also by the nucleon eective mass. Although the impact of the symmetry energy was found less signicant in the presence of trapped neutrinos, we found that the symmetry energy still aects strongly the neutrino fraction in the -trapped PNS matter, with x . 0:1 given by the asy-sti interactions at high baryon densities, and x & 0:1 given by the asy-soft interactions. Using the inputs of the TOV equation based on dierent EOS's, we ob- tained the -stable hydrostatic conguration of PNS at the total entropy per baryon S=A = 1; 2 and 4 in both the -free and -trapped scenarios. In the absence of trapped neutrinos, dierent slopes of the symmetry energy at high baryon densities were shown to give a dierence of 0.3 to 0.5 M in the maximum gravitational mass Mmax predicted with the CDM3Yn interactions. For the -trapped, -stable PNS, the eect of the symmetry energy is diminished already at the entropy S=A = 1 and 2, and the stable 116 Chapter 5. HF study of the -stable PNS matter values of the maximum gravitational mass and radius of the PNS are rather close, independent of the behavior of the symmetry energy. However, the impact of the nucleon eective mass remains drastic in both -trapped and -free cases, with temperature of the PNS matter being inversely propor- tional to m. In particular, the dierence in temperature in both the outer core and center of PNS given by dierent density dependent NN interac- tions at S=A = 4 is mainly due to the dierence in m because the eect of the symmetry energy is diminished at this high entropy. A special attention was given to the conguration of the -free PNS at entropy S=A = 4, which was shown by Hempel et al. [76] to occur at the onset of the collapse of a massive (40 M) protoneutron progenitor to black hole. We found that at very high temperatures of PNS matter at S=A = 4, the impact of the symmetry energy becomes weaker and the similar -stable congurations of PNS were obtained with both the asy-sti and asy-soft interactions. The Mmax and Rmax values were found strongly increased at the entropy per baryon S=A = 4, with the dierence
Mmax  0:3 0:5 M and
Rmax  4 5 km compared to the results obtained at S=A = 0. Thus, the hot PNS at S=A = 4 is substantially expanded in size, with the decreased central pressure and density. In the outer core of hot PNS being at entropy S=A = 4, only temperature given by the asy-sti CDM3Yn interaction is comparable to that predicted by the hydrodynamic simulation. The maximum gravitational masses Mmax obtained for the -stable and -free PNS at S=A = 4 using dierent EOS's of hot NM were used to estimate the time tBH of collapse of the 40 M progenitor to black hole, based on a correlation found between tBH and Mmax from the hydrodynamic simulation [76, 4]. From a more general viewpoint, the present mean-eld study illustrates the large impact of the nuclear symmetry energy and nucleon eective mass in dense and hot baryon matter. This eect becomes more complicated Chapter 5. HF study of the -stable PNS matter 117 after the inclusion of trapped neutrinos which weakens the correlation of the thermodynamic properties of PNS with the symmetry energy. The densities and temperatures reached in the core of PNS at entropy of S=A  4 or higher might imply a phase transition to new degrees of freedom as discussed, e.g., in Refs. [95]. Thus, PNS is the most extreme compact object which requires the most advanced knowledge in the nuclear and QCD physics. It is, however, very short-lived because it lasts only a minute or so before being collapsed to black hole or cooled down to neutron star. Thus, the observation of the next supernova explosion in our galaxy will certainly provide the new and fascinating information about the hot PNS matter that is extremely dicult to obtain in the terrestrial nuclear physics laboratories. The present mean-eld results were obtained in the absence of an ex- ternal electromagnetic eld. This is, however, only an ideal condition. 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