Equation of state of neutron star matter, and the nuclear symmetry energy

The nuclear mean-field potentials obtained in the Hartree-Fock method with different choices of the in-medium nucleon-nucleon (NN) interaction have been used to study the equation of state (EOS) of the neutron star (NS) matter. The EOS of the uniform NS core has been calculated for the $\mathit{npe}\mu$ composition in the $\beta$ equilibrium at zero temperature, using version Sly4 of the Skyrme interaction as well as two density-dependent versions of the finite-range M3Y interaction (CDM3Y$n$ and M3Y-P$n$), and versions D1S and D1N of the Gogny interaction. Although the considered effective NN interactions were proven to be quite realistic in numerous nuclear structure and/or reaction studies, they give quite different behaviors of the symmetry energy of nuclear matter at supranuclear densities that lead to the soft and stiff scenarios discussed recently in the literature. Different EOS’s of the NS core and the EOS of the NS crust given by the compressible liquid drop model have been used as input of the Tolman-Oppenheimer-Volkov equations to study how the nuclear symmetry energy affects the model prediction of different NS properties, like the cooling process as well as the gravitational mass, radius, and moment of inertia.

Figure 1

HF results for the NM symmetry energies $S(n_b)$ given by the density-dependent NN interactions under study. The shaded (magenta) region marks the empirical boundaries deduced from the analysis of the isospin diffusion data and double ratio of neutron and proton spectra data of HI collisions [34, 36]. The square and triangle are the constraints deduced from the consistent structure studies of the GDR [47] and neutron skin [49], respectively. The circles and crosses are results of the ab initio calculation by Akmal, Pandharipande, and Ravenhall (APR) [50] and microscopic Monte Carlo (MMC) calculation by Gandolfi et al. [51], respectively.

Figure 2

Energy of the symmetric NM and pure neutron matter calculated in the HF approximation (1) and (2) using the in-medium NN interactions that give a stiff behavior of $S(n_b)$ as shown in the upper panel of Fig. 1. The circles and crosses are results of the ab initio calculation by APR [50] and MMC calculation by Gandolfi et al. [51], respectively.

Figure 3

The same as Fig. 2 but obtained with the in-medium NN interactions that give a soft behavior of $S(n_b)$ as shown in the lower panel of Fig. 1.

Figure 4

The fractions $x_j=n_j/n_b$ of constituent particles of the NS matter obtained from the solutions of Eqs. (6) and (8) using the mean-field potentials given by the M3Y-P5 and D1N interactions.

Figure 5

The same as Fig. 4 but using the mean-field potentials given by the CDM3Y4 and Sly4 interactions. The circles are $n_j$ values calculated at the maximum central densities $n_c$ given by the solution of the TOV equations.

Figure 6

The proton fraction $x_p$ of the $\beta$-stable NS matter obtained from the solutions of Eqs. (6) and (8) using the mean-field potentials given by the stiff-type CDM3Y$n$ and Sly4 interactions. The circles are $n_p$ values calculated at the maximum central densities $n_c$ given by the TOV equations. The thin lines are the corresponding DU thresholds (10).

Figure 7

The pressure inside the NS matter obtained with the in-medium NN interactions that give stiff (upper panel) and soft (lower panel) behaviors of $S(n_b)$, in comparison with the empirical data points deduced from the astronomical observation of neutron stars [58]. The shaded band shows the uncertainties associated with the data determination. The circles are $P$ values calculated at the corresponding maximum central densities given by the TOV equations and the vertical arrows indicate the baryon densities above which the NS matter predicted by the M3Y-P3 and M3Y-P4 interactions becomes superluminal (see Fig. 13).

Figure 8

The NS gravitational mass versus its radius obtained with the EOS’s given by the stiff-type (upper panel) and soft-type (lower panel) in-medium NN interactions, in comparison with the empirical data (shaded contours) deduced by Özel et al. [58] from recent astronomical observations of neutron stars. The circles are values calculated at the maximum central densities. The thick solid (red) line is the limit allowed by the general relativity [63].

Figure 9

The same as Fig. 8, but in comparison with the empirical data (shaded contours) deduced by Steiner et al. [59] from the observation of the x-ray burster 4U 1608-52.

Figure 10

The NS moment of inertia versus its gravitational mass obtained with the EOS’s given by the stiff-type (upper panel) and soft-type (lower panel) in-medium NN interactions, in comparison with the empirical data (shaded contours) deduced from the mass-radius data by Özel et al. [58] using Eq. (13). The circles are values calculated at the maximum central densities.

Figure 11

The same as Fig. 10, but in comparison with the empirical data (shaded contours) deduced from the mass-radius data by Steiner et al. [59] for the x-ray burster 4U 1608-52 using Eq. (13).

Figure 12

The gravitational mass $M$ given by different EOS’s of the NS matter plotted versus the corresponding total baryon mass $M_b$. The shaded rectangle is the empirical value inferred from observations of the double pulsar PSR J0737-3059 by Podsiadlowski et al. [66].

Figure 13

The adiabatic sound velocity versus baryon density obtained with the EOS’s given by the stiff-type (upper panel) and soft-type (lower panel) in-medium NN interactions. The thick solid (red) lines are the subluminal limit ($v_s⩽c$), and the vertical arrows indicate the baryon densities above which the NS matter predicted by the M3Y-P3 and M3Y-P4 interactions becomes superluminal (see details in the text).

Figure 14

The same as in Fig. 1 but $S(n_b)$ curves in the lower panel were obtained with the soft CDM3Y$n$s interactions.

Figure 15

The same as Fig. 4 but using the mean-field potentials given by the CDM3Y6 and CDM3Y6s interactions.

Figure 16

The same as in Fig. 7 but $P(n_b)$ curves in the lower panel were obtained with the soft CDM3Y$n$s interactions.

Figure 17

The same as Fig. 8 but mass-radius curves in the lower panel were obtained with the soft CDM3Y$n$s interactions.

Figure 18

The same as Fig. 9 but mass-radius curves in the lower panel were obtained with the soft CDM3Y$n$s interactions.