Neutron star mass formula with nuclear saturation parameters

A neutron star is produced as a compact remnant through a supernova explosion, which occurs at the last moment of a massive star’s life. The neutron stars are in extreme states, which is hard to be realized in terrestrial laboratories. In particular, due to the nature of the nuclear saturation properties, it is quite difficult to obtain the nuclear information in a higher density region through the terrestrial experiments. This is a reason why the equation of state (EOS) for neutron star matter has not been fixed yet. Namely, the structure of the neutron star and its maximum mass are not exactly determined. Thus, the observations of the neutron stars and/or the phenomena associated with the neutron stars are quite important for understanding the physics in such extreme states.

For example, the discovery of the $2M_{\odot}$ neutron stars D10 ; A13 ; C20 ; F21 has ruled out some of the soft EOSs as the EOS for neutron star matter. That is, the EOS, with which the maximum mass does not reach the observed mass, can be ruled out. In addition, the light bending induced by the strong gravitational field, which is one of the important relativistic effects, modifys the pulsar light curve, which principally tells us the stellar compactness, i.e., the ratio of the stellar mass to its radius (e.g., PFC83 ; LL95 ; PG03 ; PO14 ; SM18 ; Sotani20a ). In practice, through the observations with the Neutron star Interior Composition Explorer (NICER) operating on an International Space Station, the mass and radius of PSR J0030+0451 Riley19 ; Miller19 and PSR J0740+6620 Riley21 ; Miller21 are constrained. Owing to the gravitational wave observations in the event of GW170817 gw170817 , the tidal deformability of the neutron star just before the merger of the binary neutron stars is also constrained, which tells us that the $1.4M_{\odot}$ neutron star radius should be less than $13.6$ km Annala18 . Furthermore, it is proposed that the neutron star mass and radius may be determined with the technique of asteroseismology thorough the future gravitational wave observations (e.g., AK1996 ; AK1998 ; STM2001 ; SH2003 ; SYMT2011 ; PA2012 ; DGKK2013 ; Sotani2020 ; SD2021 ). These astronomical constraints on the neutron star mass and radius indirectly constrain the EOS for neutron star matter especially for a higher density region.

On the other hand, terrestrial experiments are obviously important for extracting the nuclear information, which also constrains the EOS for neutron star matter, even though the resultant constraint may be mainly around the nuclear saturation density. Up to now a lot of experiments worldwide have been done to fix the nuclear saturation parameters. Owing to these attempts, some of the saturation parameters have been constrained well, but many parameters, especially for higher order terms, still remain uncertain (see Sec. II for more detail). For instance, even the constraint on the density-dependence of the nuclear symmetry energy, which is recently reported from two large facilities in Japan and the USA, still has large uncertainties SPIRIT ; PREXII . Additionally, since the EOS for neutron star matter can be characterized by the nuclear saturation parameters, the neutron star properties may be also associated with the saturation parameters. Thus, to improve our understanding of the nuclear properties, the constraint on the neutron star mass and radius from the astronomical observations are quite important as well as constraints on the nuclear properties from the terrestrial experiments.

Nevertheless, even if one would accurately observe the neutron star mass and/or radius, it is difficult to directly discuss the nuclear saturation parameters. This is because the neutron star properties are associated with the EOSs, which can be characterized by the nuclear saturation parameters, but direct connection between the neutron star properties and nuclear saturation parameters is still unclear. To partially solve this difficulty, we have already found a suitable combination of the nuclear saturation parameters, with which the low-mass neutron star models can be expressed well SIOO14 . In this study, we extend the previous work and try to derive the empirical formulas for the mass and gravitational redshift of neutron star models constructed with the central density up to threefold nuclear saturation density, which helps us to directly discuss the association between the neutron star properties and nuclear saturation parameters.

This manuscript is organized as follows. In Sec. II, we briefly mention the EOSs considered in this study. In Sec. III, we systematically examine the neutron star models and derive the empirical formulas for the neutron star mass and its gravitational redshift as a function of the nuclear saturation parameters. Finally, in Sec. IV, we conclude this study. Unless otherwise mentioned, we adopt geometric units in the following, $c=G=1$, where $c$ and $G$ denote the speed of light and the gravitational constant, respectively.

To construct the neutron star models by solving the Tolman-Oppenheimer-Volkoff (TOV) equation, one has to assume an EOS for neutron star matter. In this study, we mainly adopt the phenomenological nuclear EOS models, focusing only on the unified EOS, i.e., the neutron star crust EOS is constructed with the same nuclear model as in the neutron star core EOS. As a phenomenological macroscopic model, we adopt the EOSs proposed by Oyamatsu and Iida (hereafter referred to as the OI-EOS) OI03 ; OI07 . The OI-EOSs are constructed with the Padé-type potential energies in such a way as to reproduce empirical masses and radii of stable nuclei, using a simplified version of the extended Thomas-Fermi theory. On the other hand, as a phenomenological Skyrme-type model, we adopt KDE0v, KDE0v1 KDE0v , SLy2, SLy4, SLy9 SLy4 ; SLy9 , SKa SKa , SkI3 SkI3 , and SkMp SkMp . In addition, we also adopt the Shen EOS Shen , which is based on the relativistic mean field theory, and the Togashi EOS Togashi17 , which is derived by the variational many-body calculation with AV18 two-body and UIX three-body potentials. In Fig. 1, we show the mass and radius relation for the neutron star models constructed with the EOSs adopted in this study, where the stellar models with $n_{c}/n_{0}=1$, 2, and 3 are shown with the marks. We note that some of the stellar models with $n_{c}/n_{0}=1$ are out of the panel due to the large radius. One can observe that some of EOSs are obviously ruled out from the $2M_{\odot}$ observations D10 ; A13 ; C20 ; F21 or the radius constraint from the GW170817 Annala18 , but in order to examine with the wide parameter space, we adopt even such EOSs in this study.

In any case, the bulk energy per nucleon for the uniform nuclear matter at zero temperature can generally be expressed as a function of the baryon number density, $n_{\rm b}=n_{n}+n_{p}$, and an asymmetry parameter, $\alpha=(n_{n}-n_{p})/n_{\rm b}$, with the neutron number density, $n_{n}$, and the proton number density, $n_{p}$;

where $w_{s}$ corresponds to the energy per nucleon of symmetric nuclear matter, while $S$ denotes the density-dependent symmetry energy. Additionally, $w_{s}$ and $S$ can be expanded around the saturation density, $n_{0}$, of the symmetric nuclear matter as a function of $u=(n_{\rm b}-n_{0})/(3n_{0})$;

The coefficients in these expressions are the nuclear saturation parameters, with which each EOS is characterized. The parameters for the adopted EOSs are concretely listed in Table 1, where $\eta$, $\xi$, $\eta_{sy}$, and $\xi_{sy}$ are the specific combination of the nuclear saturation parameters (see the following sections for details), defined by

Among the nuclear saturation parameters, $n_{0}$, $w_{0}$, and $S_{0}$ are well constrained as $n_{0}\approx 0.15-0.16$ fm^-3, $w_{0}\approx-15.8$ MeV OHKT17 , and $S_{0}\approx 31.6\pm 2.7$ MeV Li19 . Meanwhile, $K_{0}$ and $L$ are more difficult to be determined from the terrestrial experiments, because these parameters are the density derivative at the saturation point, i.e., one needs to know the information not only at the saturation point but also in wider range around the saturation point. The constraints on these parameters are gradually improved and the current fiducial values are $K_{0}=230\pm 40$ MeV KM13 and $L=58.9\pm 16$ MeV Li19 , even though the constraints on $L$ recently reported from two large facilities still have a large uncertainty, i.e., $42\leq L\leq 117$ MeV with S$\pi$RIT by the Radioactive Isotope Beam Factory at RIKEN in Japan SPIRIT and $L=106\pm 37$ MeV with PREX-II by the Thomas Jefferson National Accelerator Facility in Newport News, the United States PREXII . Moreover, the saturation parameters in higher order terms, such as $Q$, $K_{sym}$, $Q_{sym}$, are almost unconstrained from the experiments, but they are theoretically predicted as $-800\leq Q\leq 400$ MeV, $-400\leq K_{sym}\leq 100$ MeV, and $-200\leq Q_{sym}\leq 800$ MeV Li19 .

It is known that $S_{0}$ is strongly associated with $L$ as $S_{0}\approx 28+0.075L$ OI03 ; LL13 . In a similar way, we find that $K_{0}+K_{sym}$ is also strongly associated with $L$, adopting 118 models for the Skyrme-type EOSs listed in Ref. Danielewicz09 and 304 models for OI-EOSs. We plot $K_{0}+K_{sym}$ as a function of $L$ in Fig. 2, where the thick solid line denotes the fitting formula given by

The similar correlation has been reported in Ref. Tews17 , which is shown in Fig. 2 with the dotted line. This type of correlation may be very useful for constraining the value of $K_{sym}$ with using the constraints on $K_{0}$ and $L$, because the uncertainty in $K_{sym}$ is still very large. In practice, by assigning the fiducial values of $K_{0}$ and $L$ mentioned above in Eq. (8), one can find that $-186\leq K_{sym}\leq 13$ MeV.

The neutron star structure is determined by solving the TOV equations together with the appropriate EOS. The neutron star may sometimes be considered as a huge nucleus, but its structure is quite dense, compared to atomic nuclei. Nevertheless, since the density inside low-mass neutron stars is definitely low, their mass seems to be strongly associated with the nuclear saturation properties. In practice, it has been found that the mass, $M$, and gravitational redshift, $z\equiv(1-2GM/Rc^{2})^{-1/2}-1$, for the low-mass neutron stars, whose central density is less than twice the nuclear saturation density, are well expressed as a function of $\eta$ defined by Eq. (4) and $u_{c}=\rho_{c}/\rho_{0}$, where $\rho_{c}$ and $\rho_{0}$ are the central energy density and the energy density corresponding to the nuclear saturation density, i.e., $\rho_{0}=2.68\times 10^{14}$ g/cm³ SIOO14 . That is, one can estimate neutron star mass and radius by combining the empirical formulas, $M=M(u_{c},\eta)$ and $z=z(u_{c},\eta)$. In practice, assuming the recent experimental constraints obtained with S$\pi$RIT and PREX-II SPIRIT ; PREXII , one can show the allowed region in the neutron star mass and radius relation SNN22 . Using this new parameter $\eta$, one can also discuss the rotational properties of the low-mass neutron stars SSB16 and the possible maximum mass of neutron stars Sotani17 ; SK17 . In this study, we try to extend this type of empirical formulas even for higher central density up to three times saturation density. This is because, as the central density becomes larger, the empirical formulas discussed in more detail below lose accuracy. This may come from the additional EOS dependence, such as higher order coefficients in Eqs. (2) and (3). In addition, for reference, we show the mass of neutron stars with $n_{c}=3n_{0}$ constructed with the EOSs considered in this study in Table 1, with which one may be able to adopt our empirical relations discussed below if the stellar mass is less than $\simeq 0.68-1.90M/M_{\odot}$ (average value is $1.34M_{\odot}$).