Investigation of the inner edge of neutron star crusts: Temperature dependence and related effects

The mutual correlation between the nuclear equation of state (EOS) and the bulk properties of neutron stars (NS) is crucial in probing both of them. Here, we use EOSs of hot $npe$ ($npe\mu$) nuclear matter, based on the density-dependent CDM3Y-Paris nucleon-nucleon interaction in the nonrelativistic Hartree-Fock scheme, to investigate the temperature dependence of the core-crust transition properties under $\beta$ equilibrium, at the inner edge of NS. We use four EOSs that provide symmetric nuclear matter saturation incompressibility of 218 and 252 MeV, with two parametrizations of the density dependence of the isovector part of the M3Y force. We found that the softer EOS estimates larger proton fraction in the NS matter and indicates a wider range for direct Urca (DU) cooling process within the core center of NSs. Increasing the temperature decreases the density corresponding to the threshold proton fraction for DU process, increasing the region for the DU process inside NSs. The muons decrease the isospin asymmetry of the $npe\mu$ NS matter at its core center, its thermal pressure, and the DU threshold density. The muon fraction slightly changes with temperature. A value of about half the proton fraction is inferred for the $\beta$-stable muon fraction of hot $npe\mu$ matter, around the core center. Based on the four considered EOSs, the liquid core-solid crust transition density, pressure, and proton fraction are estimated to increase from $(0.54\pm 0.02){\rho }_0, 0.36\pm 0.12\mathrm{MeV}{\mathrm{fm}}^{-3}$, and 0.03, respectively at $T=0\mathrm{MeV}$, to $(0.85\pm 0.04){\rho }_0, 7.36\pm 0.52\mathrm{MeV}{\mathrm{fm}}^{-3}$, and 0.14, respectively at $T=50\mathrm{MeV}$.

Figure 1

Density dependence of the proton fraction ($x_p$) in $\beta$ stable cold (a) $npe$ and (b) $npe\mu$ matters, based on the M3Y-Paris $\mathit{NN}$ interaction using two EOSs CDM3Y3 ($K_0=218\mathrm{MeV}$) and CDM3Y6 ($K_0=252\mathrm{MeV}$), with the density dependence of the isovector part of the $\mathit{NN}$ force that derived in terms of the corresponding isoscalar density dependence (DIV) [70] and that derived independently (IIV) [56].

Figure 2

Density dependence of the proton fraction in $\beta$ stable $npe$ matter based on the density-dependent (a) CDM3Y3-Paris-218 (IIV) and (b) CDM3Y6-Paris-252 (IIV) $\mathit{NN}$ interaction at temperatures $T=0$, 10, 20, 30, 40, and 50 MeV represented. Panels (c) and (d) show the corresponding electron chemical potentials for the two EOSs, respectively. Panels (e) and (f) show the same as panels (a) and (b), respectively, but for $npe\mu$ matter.

Figure 3

Same as Figs. 2 and 2 but for the muon fraction ($x_{\mu }$).

Figure 4

(a)–(d) Same as Figs. 2, 2, 2, and 2, respectively, but for the free energy of baryons [Eq. (15)] over the density range $\rho \le 1{\mathrm{fm}}^{-3}$. (e) Same as Fig. 4 but for the entropy per nucleon.

Figure 5

Same as Fig. 2 but for the thermal pressure, $P_{th}=P\left(T\right)-P\left(0\right)$, inside NSs, over the density range $\rho \le 1{\mathrm{fm}}^{-3}$.

Figure 6

Temperature dependence of the core-crust (a) transition density ${\rho }_t$ (${\mathrm{fm}}^{-3}$) [Eq. (23)], (b) transition proton fraction $x_{pt}$ [Eqs. (17) and (18)], and (c) transition pressure $P_t$ (MeV ${\mathrm{fm}}^{-3}$) [Eqs. (21) and (22)] in NSs based on the CDM3Y3-218 (DIV), CDM3Y6-252 (DIV), CDM3Y3-218 (IIV), and CDM3Y6-252 (IIV) EOSs.