Effects of the symmetry energy on properties of neutron star crusts near the neutron drip density

We study the effects of the symmetry energy on the neutron drip density and properties of nuclei in neutron star crusts. The nonuniform matter around the neutron drip point is calculated by using the Thomas-Fermi approximation with the relativistic mean-field model. The neutron drip density and the composition of the crust are found to be correlated with the symmetry energy and its slope. We compare the self-consistent Thomas-Fermi approximation with other treatments of surface and Coulomb energies and find that these finite-size effects play an essential role in determining the equilibrium state at low density.

Figure 1

Binding energy per nucleon of ${}^{208}\mathrm{Pb}$ vs the symmetry energy slope $L$ with different choices of $n_{\mathrm{fix}}$ based on the TM1 and IUFSU parametrizations.

Figure 2

Energy per nucleon, $E$, as a function of the average baryon density $n_b$ obtained using the TF, CLD, and CP methods with different RMF parametrizations (NL3, TM1, FSU, and IUFSU). For comparison, the results of homogeneous matter (Hom.) are also plotted.

Figure 3

Comparison of the bulk energy per nucleon, $E_{\mathrm{bulk}}$ (top panel), the electron kinetic energy per nucleon, $E_e$ (middle panel), and the Coulomb energy per nucleon, $E_{\mathrm{Coul}}$ (bottom panel) among the CP, CLD, and TF methods using the TM1 parametrization.

Figure 4

Electron fraction $Y_e$ as a function of $n_b$ obtained using the CP, CLD, and TF methods.

Figure 5

Wigner-Seitz cell radius $r_{\mathrm{ws}}$ and droplet radius $r_d$ as a function of $n_b$ obtained using the TF, CLD, and CP methods. $r_d$ in the CP and CLD methods is given in Eq. (21), while it is defined by $r_d=\sqrt{\frac{5}{3}}{\langle r_p^2\rangle }^{1/2}$ in the TF approximation.

Figure 6

Proton number $Z$ of the droplet as a function of $n_b$ obtained using the TF, CLD, and CP methods.

Figure 7

Neutron drip density $n_{\mathrm{drip}}$ as a function of $L$ 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 filled circle, respectively. The results of NL3 and FSU are represented by the up and down triangles.

Figure 8

Same as Fig. 7, but for the cell radius $r_{\mathrm{ws}}$ at the neutron drip density.

Figure 9

Properties of the nucleus at the neutron drip density vs $L$ obtained in the TF calculation. The nucleon number $A$ and the proton number $Z$ (a), the proton fraction $Z/A$ (b), the rms radius of the neutron $R_n$ and that of the proton $R_p$ (c), and the neutron skin thickness $\Delta r_{np}=R_n-R_p$ (d) are plotted.

Figure 10

Nucleon density distributions in the Wigner-Seitz cell at the neutron drip density obtained with $L=110.8$ MeV (black solid lines) and $L=40$ MeV (red dashed lines) in the set of TM1.

Figure 11

Properties of the nucleus in neutron star crusts, such as the proton number $Z$ (a), the nucleon number $A_d$ (b), and the proton fraction $Z/A_d$ (c), as a function of $n_b$ obtained in the TF approximation using the two sets of generated models.

Figure 12

Equilibrium properties of the Wigner-Seitz cell as a function of $n_b$ obtained in the TF approximation using the two sets of generated models. The cell radius $r_{\mathrm{ws}}$ and the proton rms radius $R_p$ (a), the neutron density at the center, $n_n\left(0\right)$, and that at the boundary, $n_n\left(r_{\mathrm{ws}}\right)$ (b), and the proton density at the center, $n_p\left(0\right)$ (c), are plotted.

Figure 13

Chemical potentials of electrons, ${\mu }_e$ (a), neutrons, ${\mu }_n$ (b), and protons, ${\mu }_p$ (c), as a function of $n_b$ obtained in the TF approximation with the smallest and largest values of $L$ in the two sets of generated models.

Figure 14

Density distributions of neutrons (upper curves) and protons (lower curves) in the Wigner-Seitz cell at average baryon densities $n_b=0.01,0.03,$ and $0.05{\mathrm{fm}}^{-3}$ (top to bottom) obtained with $L=110.8$ MeV (black solid lines) and $L=40$ MeV (red dashed lines) in the set of TM1. The cell radius $r_{\mathrm{ws}}$ is indicated by the hatching.