Empirical neutron star mass formula based on experimental observables

We derive the empirical formulas expressing the mass and gravitational redshift of a neutron star, whose central density is less than threefold the nuclear saturation density, as a function of the neutron-skin thickness or the dipole polarizability of ${}^{208}\mathrm{Pb}$ or ${}^{132}\mathrm{Sn}$, especially focusing on the eight Skyrme-type effective interactions. The neutron star mass and its gravitational redshift can be estimated within $\approx 10\%$ errors with our formulas, while the neutron star radius is also expected within a few $\%$ errors by combining the derived formulas. Owing to the resultant empirical formulas, we find that the neutron star mass and radius are more sensitive to the neutron-skin thickness of ${}^{208}\mathrm{Pb}$ than the dipole polarizability of ${}^{208}\mathrm{Pb}$ or ${}^{132}\mathrm{Sn}$.

Figure 1

Neutron-skin thickness, $\mathrm{\Delta }{R}_n$, expected with each EOS model is shown as a function of the corresponding value of $L$. The top and bottom panels correspond to the result for ${}^{208}\mathrm{Pb}$ and ${}^{132}\mathrm{Sn}$. The fitting lines given by Eqs. (4a) and (4b) are shown with the solid lines, while the fitting line derived in Ref. [50] is also shown with the dotted line in the top panel.

Figure 2

Dipole polarizability multiplied with $S_0$ for each EOS model is shown as a function of $L$, where the top and bottom panels correspond to the result for ${}^{208}\mathrm{Pb}$ and ${}^{132}\mathrm{Sn}$. The fitting lines given by Eqs. (4c) and (4d) are shown with the solid lines, while the fitting line derived in Ref. [51] is also shown with the dotted line in the top panel. For reference, the experimental value of ${\alpha }_{\mathrm{D}}^{\mathrm{Pb}}{S}_0$ is shown with the shaded region in the top panel, using the experimental value ${\alpha }_{\text{D}}^{\text{Pb}}=20.1\left(6\right){\mathrm{fm}}^3$ [62] together with $S_0\approx 31.6\pm 2.7\mathrm{MeV}$ [23] as the fiducial value of $S_0$.

Figure 3

The mass of the neutron stars constructed with each EOS model is shown as a function of $\mathrm{\Delta }{R}_n^{\mathrm{Pb}}$ in the left panel and $\mathrm{\Delta }{R}_n^{\mathrm{Sn}}$ in the right panel, where the neutron star mass for $u_{\mathrm{c}}=1$, 2, and 3 are shown. The fitting lines are given by Eqs. (5a) and (5b).

Figure 4

The coefficients in Eqs. (5a), (5b), (7a), and (7b) are shown as a function of $u_{\mathrm{c}}$, where the top and bottom panels correspond to the coefficients in the formulas with the data of ${}^{208}\mathrm{Pb}$ and ${}^{132}\mathrm{Sn}$, respectively, while the solid lines denote the fitting lines given by Eqs. (6a), (6b), (8a), and (8b).

Figure 5

The gravitational redshift of the neutron stars constructed with each EOS model is shown as a function of $\mathrm{\Delta }{R}_n^{\mathrm{Pb}}$ in the left panel and $\mathrm{\Delta }{R}_n^{\mathrm{Sn}}$ in the right panel for the stellar models with $u_{\mathrm{c}}=1$, 2, and 3. The fitting lines are given by Eqs. (7a) and (7b).

Figure 6

Relative deviation of the stellar mass ($M$) and gravitational redshift ($z$) estimated with the empirical relations [Eqs. (5a)–(8b)] from those determined as the TOV solutions is shown as a function of $u_{\mathrm{c}}$. The bottom panels correspond to the relative deviation of the stellar radius ($R$) predicted with the empirical formulas for $M$ and $z$ from that determined as the TOV solutions. The left and right panels correspond to the results obtained from the empirical formulas as a function of $\mathrm{\Delta }{R}_n^{\mathrm{Pb}}$ and $\mathrm{\Delta }{R}_n^{\mathrm{Sn}}$, respectively.

Figure 7

The mass of the neutron stars constructed with each EOS model is shown as a function of ${\alpha }_{\mathrm{D}}^{\mathrm{Pb}}{S}_0$ in the left panel and ${\alpha }_{\mathrm{D}}^{\mathrm{Sn}}{S}_0$ in the right panel, where we show the results for $u_{\mathrm{c}}=1$, 2, and 3. The fitting lines are given by Eqs. (9a) and (9b).

Figure 8

The coefficients in Eqs. (9a), (9b), (11a), and (11b) are shown as a function of $u_{\mathrm{c}}$, where in the top and bottom panels correspond to the coefficients in the formulas with the data of ${}^{208}\mathrm{Pb}$ and ${}^{132}\mathrm{Sn}$, respectively, while the solid lines denote the fitting lines given by Eqs. (10a), (10b), (12a), and (12b).

Figure 9

The gravitational redshift of the neutron stars constructed with each EOS model is shown as a function of ${\alpha }_{\mathrm{D}}^{\mathrm{Pb}}{S}_0$ in the left panel and ${\alpha }_{\mathrm{D}}^{\mathrm{Sn}}{S}_0$ in the right panel, where we show the results for $u_{\mathrm{c}}=1$, 2, and 3. The fitting lines are given by Eqs. (11a) and (11b).

Figure 10

Same as in Fig. 6, but with the empirical formulas using ${\alpha }_{\mathrm{D}}^{\mathrm{Pb}}$ in the left panel and ${\alpha }_{\mathrm{D}}^{\mathrm{Sn}}$ in the right panel.

Figure 11

The neutron star mass and radius predicted with the empirical formulas with $\mathrm{\Delta }{R}_n^{\mathrm{Pb}}$ (top left), $\mathrm{\Delta }{R}_n^{\mathrm{Sn}}$ (top right), ${\alpha }_{\mathrm{D}}^{\mathrm{Pb}}{S}_0$ (bottom left), and ${\alpha }_{\mathrm{D}}^{\mathrm{Sn}}{S}_0$ (top left), where we assume that $\mathrm{\Delta }{R}_n^{\mathrm{Pb}}/\left(0.2\mathrm{fm}\right)=0.9, \mathrm{\Delta }{R}_n^{\mathrm{Sn}}/\left(0.2\mathrm{fm}\right)=1.2, {\alpha }_{\mathrm{D}}^{\mathrm{Pb}}{S}_0/\left(600\mathrm{MeV}{\mathrm{fm}}^3\right)=1.13, {\alpha }_{\mathrm{D}}^{\mathrm{Sn}}{S}_0/\left(600\mathrm{MeV}{\mathrm{fm}}^3\right)=0.56$ as their test values. The dotted, dashed, and solid lines denote the region of the neutron star mass and radius with $\pm 5\%, \pm 10\%$, and $\pm 15\%$ deviation from the test values. In the bottom left panel, we also show the shaded region as the predicted region from the experimental value of ${\alpha }_{\text{D}}^{\text{Pb}}=20.1\left(6\right){\mathrm{fm}}^3$ together with the fiducial value of $S_0$ (see text for details).

Figure 12

The neutron star mass and radius predicted with our empirical formula given by Eqs. (5a), (6a), (7a), and (8a), assuming that $\mathrm{\Delta }{R}_n^{\mathrm{Pb}}=(0.9\pm 0.09)\times 0.2\mathrm{fm}$. In the figure, we also plot the constraints from the astronomical observations in GW170817, by NICER (PSR J0030+451 and MSP J0740+6620), via x-ray bursts, and with the magnetar QPOs (GRB 200415A), together with the mass and radius region predicted with the mass formulas with the nuclear saturation parameters, assuming $L=60\pm 20$ and $K_0=240\pm 20\mathrm{MeV}$. The top left region is excluded from the causality. For reference, the mass and radius for the neutron stars constructed with the SKa, SkMp, and SLy4 EOSs are shown with the dotted lines. See the text for the details.