Neutron star mass formula with nuclear saturation parameters for asymmetric nuclear matter

Low-mass neutron stars are directly associated with the nuclear saturation parameters because their central density is definitely low. We have already found a suitable combination of nuclear saturation parameters for expressing the neutron star mass and gravitational redshift, i.e., $\eta \equiv (K_0{L}^2{)}^{1/3}$ with the incompressibility for symmetric nuclear matter, $K_0$, and the density-dependent nuclear symmetry energy, $L$. In this study, we newly find another suitable combination given by ${\eta }_{\tau }\equiv (-K_{\tau }{L}^5{)}^{1/6}$ with the isospin dependence of incompressibility for asymmetric nuclear matter, $K_{\tau }$, and derive the empirical relations for the neutron star mass and gravitational redshift as a function of ${\eta }_{\tau }$ and the normalized central number density. With these empirical relations, one can evaluate the mass and gravitational redshift of the neutron star, whose central number density is less than threefold the saturation density, within $\sim 10\%$ accuracy, and the radius within a few percent accuracy. In addition, we discuss the neutron star mass and radius constraints from the terrestrial experiments, using the empirical relations, together with those from the astronomical observations. Furthermore, we find a tight correlation between ${\eta }_{\tau }$ and $\eta$. With this correlation, we derive the constraint on $K_{\tau }$ as $-348\le K_{\tau }\le -237\mathrm{MeV}$, assuming that $L=60\pm 20$ and $K_0=240\pm 20\mathrm{MeV}$.

Figure 1

Experimental and theoretical constraints on $K_{\tau }$ obtained so far. The constraint of $-348\le K_{\tau }\le -237\mathrm{MeV}$ ($-718\le K_{\tau }\le -245\mathrm{MeV}$) is obtained from the empirical relation given by Eq. (12), assuming that $L=60\pm 20\mathrm{MeV}$ [30, 31] and $K_0=240\pm 20\mathrm{MeV}$ [45] ($L=20–145\mathrm{MeV}$ [2] and $K_0=200–315\mathrm{MeV}$ [46, 47]).

Figure 2

The left panel is the relation between $L$ and $K_{\tau }$ for the OI-EOSs (open circles) and the Skyrme-type EOS models (filled circles). The right panel is the relation between $\eta$ and ${\eta }_{\tau }$. The thick solid line denotes the fitting formula given by Eq. (12). With this empirical relation together with the fiducial value of $\eta$, i.e., $70.6\le \eta \le 118.5\mathrm{MeV}$ (shaded region), the expected value of ${\eta }_{\tau }$ is in the range of $53.8\le {\eta }_{\tau }\le 102.2\mathrm{MeV}$ (shaded region). In a similar way, the value of ${\eta }_{\tau }$ is expected in the range of $30.4\le {\eta }_{\tau }\le 189.3\mathrm{MeV}$ (the region between two horizontal dashed lines) with the conservative value of $\eta$, i.e., $43.1\le \eta \le 187.8\mathrm{MeV}$ (the region between two vertical dashed lines).

Figure 3

The mass (top panel) and gravitational redshift (bottom panel) for the neutron star models with $n_c/n_0=1$, 2, and 3, constructed with various EOSs, are plotted as a function of ${\eta }_{\tau }$. The thick solid lines are fitting lines given by Eqs. (13) and (14).

Figure 4

The coefficients in the fitting formula (13) and (14) are shown as a function of $n_c/n_0$, where the top and bottom panels correspond to the results for $a_i^m$ and $a_i^z$, respectively. In both panels, the solid lines denote the fitting of $a_i^m$ and $a_i^z$ given by Eqs. (15) and (16).

Figure 5

The relative deviation of neutron star mass (top panel) and gravitational redshift (middle panel) estimated with the empirical formulas from those determined with the specific EOSs is shown as a function of $n_c/n_0$. The bottom panel is the relative deviation of the neutron star radius estimated with the empirical formulas for the stellar mass and gravitational redshift from that of the TOV solution.

Figure 6

The expected region in the relation between neutron star mass and radius is shown with the shaded region on the right-bottom side of the figure by adopting the constraint on $K_{\tau }$ from experiments at RCNP, i.e., $K_{\tau }=-550\pm 100\mathrm{MeV}$ [51], together with $L=60\pm 20\mathrm{MeV}$ as a fiducial value of $L$, which corresponds to ${\eta }_{\tau }=59.9–113\mathrm{MeV}$. We also show the allowed region with the empirical relation as a function of $\eta$, assuming the fiducial values of $L=60\pm 20\mathrm{MeV}$ and $K_0=240\pm 20\mathrm{MeV}$. For reference, we show the constraints obtained from the astronomical observations, i.e., PSR $\mathrm{J}0030+0451$ and MSP $\mathrm{J}0740+6620$ with NICER, the shaded region with the observation of x-ray bursts, and $1.4M_{\odot }$ neutron star radius from GW170817 (see text for the details), together with the theoretical mass-radius curves constructed with some EOSs. The shaded region on the left-top side of the figure denotes the excluded region from the causality.

Figure 7

The expected region of the neutron star mass and radius, using the empirical relations as a function of ${\eta }_{\tau }$. The shaded region corresponds to the constraint on neutron star mass and radius with $L=60\pm 20$ and $K_{\tau }=-550\pm 100\mathrm{MeV}$, which is the same as the shaded region shown in Fig. 6. In the left panel, the expected regions are additionally shown for $L=60\pm 30$, $L=60\pm 10$, $L=60\pm 5$, and $L=60\mathrm{MeV}$ together with $K\tau =-550\pm 100\mathrm{MeV}$. In the right panel, the region enclosed with the dashed, solid, and dotted lines correspond to the region expected with $L=60\pm 20$ and $K_{\tau }=-550\mathrm{MeV}$, $L=60$ and $K_{\tau }=-550\pm 100\mathrm{MeV}$, and $L=60$ and $-348\le K_{\tau }\le -237\mathrm{MeV}$ respectively.