Impact of the symmetry energy on nuclear pasta phases and crust-core transition in neutron stars

We study the impact of the symmetry energy on properties of nuclear pasta phases and crust-core transition in neutron stars. We perform a self-consistent Thomas-Fermi calculation employing the relativistic mean-field model. The properties of pasta phases presented in the inner crust of neutron stars are investigated and the crust-core transition is examined. It is found that the slope of the symmetry energy plays an important role in determining the pasta phase structure and the crust-core transition. The correlation between the symmetry energy slope and the crust-core transition density obtained in the Thomas-Fermi approximation is consistent with that predicted by the liquid-drop model.

Figure 1

Symmetry energy $E_{\text{sym}}$ as a function of the baryon density $n_b$ for modified versions of TM1 (upper panel) and IUFSU (lower panel) with several values of $L$ at saturation density. The symmetry energy is fixed at a density of $0.11{\mathrm{fm}}^{-3}$.

Figure 2

Phase diagrams for the two sets of models generated from IUFSU (left panel) and TM1 (right panel). Different colors represent droplet, rod, slab, tube, bubble, and homogeneous phases as indicated in the legend.

Figure 3

Energy per nucleon of the pasta phase relative to that of homogeneous matter, $\Delta E$, as a function of $n_b$ obtained from the TF (solid lines) and CP (dashed lines) calculations using the original IUFSU model. Different colors represent different pasta phases as shown in Fig. 2.

Figure 4

Size of the Wigner-Seitz cell, $r_{\text{ws}}$ (thick lines), and that of the inner part, $r_d$ (thin lines), as a function of $n_b$ obtained in the TF approximation. The results calculated with the smallest $L$ in TM1 ($L=40$ MeV) and IUFSU ($L=47.2$ MeV) are shown by solid lines, where the pasta phase structure changes from droplet (green) to rod (red), slab (blue), tube (violet), and bubble (cyan) as the density increases. For comparison, the results with the largest $L$ in TM1 ($L=110.8$ MeV) and IUFSU ($L=110$ MeV) are shown by green-dashed lines, where only the droplet configuration appears before the crust-core transition.

Figure 5

Comparison of equilibrium properties of the inner crust between the TF (solid lines) and CP (dashed lines) methods using the original IUFSU model. The size of the Wigner-Seitz cell, $r_{\mathrm{ws}}$ (thick lines), and that of the inner part, $r_d$ (thin lines) (a), the electron chemical potential ${\mu }_e$ (b), and the average proton fraction $Y_p$ (c) are plotted as a function of $n_b$. The different colors represent different pasta phases.

Figure 6

Equilibrium properties of the Wigner-Seitz cell obtained in the TF approximation for modified versions of TM1 (upper panel) and IUFSU (lower panel) with several values of $L$. The average proton fraction $Y_p$ (a), the neutron densities of the liquid phase and the gas phase, $n_{n,L}$ and $n_{n,G}$, at the center or boundary of the cell (b), and the proton density of the liquid phase, $n_{p,L}$, at the center or boundary of the cell (c) are plotted as a function of $n_b$. Small jumps are observed at the transition between different pasta shapes.

Figure 7

Same as Fig. 6, but for chemical potentials of electrons ${\mu }_e$ (a), neutrons ${\mu }_n$ (b), and protons ${\mu }_p$ (c).

Figure 8

Density distributions of neutrons (solid lines) and protons (dashed lines) in the Winger-Seitz cell at $n_b=0.04,0.06,0.07,0.08,$ and $0.09{\text{fm}}^{-3}$ (top to bottom) obtained in the TF approximation. The results with the original IUFSU model ($L=47.2$ MeV) are shown by thick black lines. For comparison, the results with the largest $L$ in the set of IUFSU ($L=110$ MeV) are shown by thin red lines in the top panel. The cell boundary is indicated by the hatching.

Figure 9

Crust-core transition density $n_{b,t}$ (a), proton fraction at the transition point, $Y_{p,t}$ (b), and pressure at the transition point, $P_t$ (c), as a function of $L$ obtained in the TF approximation using the two sets of models generated from TM1 (red-solid line with squares) and IUFSU (blue-dashed line with circles). The results obtained with the original TM1 and IUFSU models are indicated by the filled square and circle, respectively.