⇦Chinese Physics Letters, 2021, Vol. 38, No. 5, Article code 052101 From Finite Nuclei to Neutron Stars: The Essential Role of High-Order Density Dependence in Effective Forces Chong-Ji Jiang (蒋崇基), Yu Qiang (强雨), Da-Wei Guan (管大为), Qing-Zhen Chai (柴清祯), Chun-Yuan Qiao (乔春源), and Jun-Chen Pei (裴俊琛)* Affiliations State Key Laboratory of Nuclear Physics and Technology, School of Physics, Peking University, Beijing 100871, China Received 28 December 2020; accepted 10 March 2021; published online 2 May 2021 Supported by the National Key R&D Program of China (Grant No. 2018YFA0404403), the National Natural Science Foundation of China (Grant Nos. 11975032, 11835001, 11790325, and 11961141003).
*Corresponding author. Email: peij@pku.edu.cn Citation Text: Jiang C J, Qiang Y, Guan D W, Chai Q Z, and Qiao C Y et al. 2021 Chin. Phys. Lett. 38 052101    Abstract A unified description of finite nuclei and equation of state of neutron stars presents both a major challenge and also opportunities for understanding nuclear interactions. Inspired by the Lee–Huang–Yang formula of hard-sphere gases, we develop effective nuclear interactions with an additional high-order density dependent term. While the original Skyrme force SLy4 is widely used in studies of neutron stars, there are not satisfactory global descriptions of finite nuclei. The refitted SLy4${'}$ force can improve descriptions of finite nuclei but slightly reduces the radius of neutron star of 1.4$M_{\odot}$ with $M_{\odot}$ being the solar mass. We find that the extended SLy4 force with a higher-order density dependence can properly describe properties of both finite nuclei and GW170817 binary neutron stars, including the mass-radius relation and the tidal deformability. This demonstrates the essential role of high-order density dependence at ultrahigh densities. Our work provides a unified and predictive model for neutron stars, as well as new insights for the future development of effective interactions. DOI:10.1088/0256-307X/38/5/052101 © 2021 Chinese Physics Society Article Text Neutron stars (NSs), which are compact objects born in a supernova explosion,[1] attract nuclear physicists' interest because they are a unique laboratory for studying the properties of ultradense nuclear matter.[2,3] The astrophysical observables of an NS mass, radius and the mass-radius relation can provide constraints on nuclear theories extrapolated from properties of finite nuclei.[4] On 17 August 2017, the gravitational waves emitted from the merging binary NS GW170817[5] were detected by the Advanced LIGO and the Advanced Virgo, marking a new era of multi-messenger astronomy. The GW170817 event not only extends our knowledge of masses and radii of NSs, but also provides a new astronomical observable, the quadrupolar tidal deformability,[6] as a further constraint.[7–9] The theoretical descriptions of an NS mass, radius and tidal deformability depend solely on the equation of state (EOS) of nuclear matter.[10–13] Therefore, seeking a unified description of properties of finite nuclei and EOS of neutron stars is a major challenge. Because quantum chromodynamics (QCD) is non-perturbative at low energies, it is in principle possible to obtain realistic model-independent nuclear forces from lattice QCD calculations.[14] However, this kind of precise calculation is so formidable that it can only be accomplished by today's supercomputers. The chiral effective field theory based on QCD can derive accurate realistic forces for light nuclei but can propagate large uncertainties towards heavy nuclei and dense nuclear matter, due to the less constrained many-body forces.[15] Therefore, various effective nuclear interactions have been extensively explored for the EOS of neutron stars.[16,17] The effective interactions are developed by taking into account the medium effects; i.e., dependent on surrounding nuclear densities, in contrast to realistic forces in free spaces. The effective interactions are rather well constrained around the saturation density with experimental studies of finite nuclei. However, the extrapolation of effective interactions is very risky to neutron stars, which could be several times denser than the saturation density. Experiments have attempted to study the correlation between finite nuclei and neutron stars by measurements of neutron skins of neutron-rich nuclei, combined with model-dependent analyses.[18,19] It is expected that improved measurements of symmetry energies via heavy ion collisions can directly study EOS at high densities.[20] In this Letter, we focus on the applicability of the widely used Skyrme forces for neutron stars. The Skyrme force is a low-momentum effective force that has been very successfully used in descriptions of structures and dynamics of the whole nuclear chart.[21,22] Among the many Skyrme forces, the SLy4 parameterization[23,24] is the best choice for descriptions of neutron stars and has been widely used as a reference in astrophysical calculations.[6] The development of SLy4 has taken into account ab initio UV14+UVII results of EOS of pure neutron matter at high densities.[23] Many effective forces with a soft EOS have been excluded by observations of neutron stars of around 2$M_{\odot}$ with $M_{\odot}$ being the solar mass.[25] However, the inclusion of hyperons in NS core is questionable due to its soft EOS.[16,17] For example, the best descriptions of binding energies of finite nuclei are obtained by the UNEDF0 parameterization with a global RMSE of 1.5 MeV, but has failed to properly describe neutron stars.[26] Meanwhile, the SLy4 force has a global RMSE of 4.37 MeV, which is too large.[27] In Skyrme forces, it is crucial to include a repulsive density dependent term to describe the nuclear saturation properties and pressure.[28] The density dependent term also simulates the effects of 3-body and many-body forces beyond the low-momentum two-body interactions. It is understood that the induced 3-body and many-body forces are particularly significant in low-momentum (or soft) interactions, which would be less significant in realistic (or hard) interactions with a high momentum cutoff.[29] In this case, the density dependent term in soft Skyrme forces should be carefully treated. We speculate that it must be insufficient for Skyrme forces with a simple density dependent term to describe a variety of nuclear systems, from dilute nuclear halos to dense neutron stars. For systems of hard spheres, its exact energy density functional was derived by Lee, Huang, Yang in a series of publications.[30,31] The energy density functional form is written as[32] $$ \begin{alignat}{1} \varepsilon ={}&\frac{3\hbar^{2}}{10m}(3\pi^{2})^{2/3}\rho^{5/3}+\frac{\hbar^{2}\pi a}{m}\rho^2 \\ &+\frac{2\hbar^{2}a^{2}3^{4/3}\pi^{2/3}}{35m}(11-2\ln2)\rho^{7/3}\\ &+0.78\frac{\hbar^{2}a^{3}3^{5/3}\pi^{10/3}}{2m}\rho^{8/3} ,~~ \tag {1} \end{alignat} $$ in which $a$ denotes a finite radius of particles. The higher order density dependent terms appear naturally as a result of a finite radius of particles. The higher-order correction certainty impacts the EOS, which has been proven precisely by experiments of cold atomic gases.[33] The finite size effects of nucleons are related to repulsive hard cores in realistic forces but have never been explicitly considered by effective nuclear forces. The aim of this work is to study the influences of high-order density dependences in extended Skyrme forces on observables of neutron stars. Extended Skyrme Forces. The standard Skyrme force is a low momentum effective interaction for Skyrme Hartree–Fock calculations and time-dependent dynamic calculations. It includes two-body terms with the momentum expansion to the second order and a density dependent term $\rho^{\gamma}({\boldsymbol r}_1-{\boldsymbol r}_2)$. The density dependent term simulates the three-body and many body correlations. However, the power of density dependence $\gamma$ is largely uncertain. Inspired by the Lee–Huang–Yang formula, we extended the Skyrme force with an additional higher-order density dependent term. The energy per nucleon $\frac{E}{A}(\rho)$ in nuclear matter in terms of density $\rho$ is written as[27] $$\begin{alignat}{1} \frac{E}{A}(\rho)={}& \frac{3\hbar^2}{10m}(3\pi^2)^{2/3}\rho^{2/3}+\frac{1}{4}t_0(1-x_0)\rho \\ &+\frac{3}{40}\big[t_1(1-x_1)+3t_2(1+x_2)\big](3\pi^2)^{2/3}\rho^{5/3} \\ &+\frac{1}{24}\sum\limits_{i}t_{3i}(1-x_{3i})\rho^{\gamma_i+1}.~~ \tag {2} \end{alignat} $$ Note that Eq. (2) is for the pure neutron matter. The energy density functional of combined protons and neutrons is given in Ref. [23]. In the extended Skyrme forces, there are two density dependent terms associated with $t_{3i}(i=1,2)$ and $\gamma_i$. We take $\gamma_1=1/6$ and $\gamma_2=1/2$ based on SLy4 forces.[27] This can be seen as a reasonable and perturbative improvement and it is different from the series of expansions of $\rho^{n/3}$ terms as in Ref. [34]. The pressure is given by $ P(\rho)=\rho^2\frac{\partial\frac{E}{A}(\rho)}{\partial\rho}$. The symmetry energy is given by the energy difference between pure neuron matter and symmetric nuclear matter. We only refit the momentum-independent parameters $t_0, t_{3i},x_0, x_{3i}$ for simplicity. These momentum-independent parameters are directly correlated in the s-wave channel. Other parameters are kept the same as SLy4 forces. We obtained different parameter sets by fitting binding energies and charge radii of groups of light nuclei, heavy nuclei, and global nuclei. We also refitted the SLy4 force with the group of global nuclei. The details of the optimizations and lists of parameterizations were given in Ref. [27]. Equilibrium of $npe\mu$ Matter. The nuclear matter in a neutron star is expected to be a mixture of baryons (protons and neutrons) and leptons in equilibrium with respect to weak interactions.[13,23] The relative fractions of $npe\mu$ components are crucial for the rate of neutrino cooling of neutron stars. The total energy density is a function of particle densities $\rho_n, \rho_p, \rho_{e}, \rho_{\mu}$, and is written as $$\begin{alignat}{1} \epsilon(r)={}&\epsilon_b(\rho_n, \rho_p)+\rho_n m_nc^2+\rho_p m_pc^2+\epsilon_{e}(\rho_{e})\\ & +\epsilon_{\mu}(\rho_{\mu})+\rho_{e}m_{e}c^2+\rho_{\mu}m_{\mu}c^2.~~ \tag {3} \end{alignat} $$ Note that the charge neutrality is kept as $\rho_p=\rho_{e}+\rho_{\mu}$. The equilibrium of the chemical potentials (defined as $\mu_{j}=\partial \epsilon/\partial \rho_j$) is written as $$ \mu_n=\mu_p+\mu_{e},\quad\mu_{\mu}=\mu_{e}.~~ \tag {4} $$ Based on the equilibrium equation, we can calculate the densities of various components at a given baryon density and thus the EOS at $npe\mu$ equilibrium is obtained. The Tolman–Oppenheimer–Volkov Equation. The TOV equation describes the static density distributions of spherical neutron star at gravitational equilibrium, which is derived from general relativity.[35,36] The TOV differential equation in terms of pressure $P(r)$, energy density $\epsilon(r)$, and mass $M(r)$ is written as $$\begin{alignat}{1} \frac{{d}P}{{d}r}={}&-\frac{GM\epsilon}{r^2c^2} \Big(1+ \frac{P}{\epsilon}\Big)\\ &\cdot\Big(1+ \frac{4\pi r^3 P}{ Mc^2}\Big)\Big(1- \frac{2GM}{rc^2}\Big)^{-1}.~~ \tag {5} \end{alignat} $$ The mass $M(r)$ is the total mass contained inside radius $r$, while $M(0)=0$. With calculated EOS $P(\rho)$, the TOV equation can be numerically solved from the center with an initial density to the surface where the pressure becomes zero. We obtain the density distribution and total mass, and the associated radius under certain boundary conditions. It can be seen that EOS is crucial to determine the mass-radius relation of neutron stars. Tidal Deformability. The dimensionless tidal deformability $\varLambda$ is a functional of dimensionless compactness $\mathcal{C}=GM/Rc^2$.[9] $\varLambda$ becomes an important additional constraint on theoretical EOS because it was first inferred from gravitational waves of GW170817.[6] The tidal deformability $\varLambda$ is obtained by solving equations as follows.[37,38] First, a differential equation is solved to obtain $y(r)$. Here $y_R$ is the value of $y(R)$ when $R$ is the NS radius. $$ \frac{{d}y(r)}{{d}r}=-\frac{1}{r}[{y(r)}^2+y(r)F(r)+r^2Q(r)].~~ \tag {6} $$ The above functions $F(r)$ and $Q(r)$, related to the total mass $M(r)$ and pressure $P(r)$, energy density $\epsilon(r)$ at radius $r$, can be written as[9] $$ F(r)=\Big\{1-4\pi r^2G\frac{\epsilon-P}{c^4}\Big\}\Big(1-\dfrac{2GM}{rc^2}\Big)^{-1},~~ \tag {7} $$ $$\begin{alignat}{1} Q(r)={}&\Big\{\frac{4\pi G}{c^4}\Big[5\epsilon+9P+\frac{\epsilon+P}{\partial P/\partial \epsilon}\Big]-\frac{6}{r^2}\Big\}\Big(1-\frac{2GM}{rc^2}\Big)^{-1}\\ & -\frac{4G^2M^2}{r^4c^4}\Big(1+\frac{4\pi r^3P}{Mc^2}\Big)^2\Big(1-\frac{2GM}{rc^2}\Big)^{-2}.~~ \tag {8} \end{alignat} $$ After obtaining $y_R$, the tidal deformability can be calculated as[37,38] $$\begin{alignat}{1} \varLambda={}&\frac{16}{15}(1-2\mathcal{C})^2[2+2\mathcal{C}(y_R-1)-y_R]\{2\mathcal{C}[6-3y_R\\ &+3\mathcal{C}(5y_R-8)]+4\mathcal{C}^3[13-11y_R+\mathcal{C}(3y_R-2)\\ &+2\mathcal{C}^2(1+y_R)]+3(1-2\mathcal{C})^2[2-y_R\\ &+2\mathcal{C}(y_R-1)]\ln{(1-2\mathcal{C})}\}^{-1}.~~ \tag {9} \end{alignat} $$ Results. We first performed global Skyrme–Hartree–Fock + BCS calculations of 603 even-even nuclei. The obtained binding energies of 603 nuclei[39] and charge radii of 339 nuclei[40] are compared with experimental data, as shown in Table 1. For SLy4 force, the overall rms of deviations in binding energies is about 4.38 MeV, which is very large. In particular, SLy4 seriously underestimates the binding energies of heavy and superheavy nuclei remarkably.[27] This undermines its descriptions of even larger systems, such as neutron stars. We also see that overall rms of the refitted SLy4${'}$ is about 2.86 MeV, which is improved significantly. We also performed calculations using extended SLy4 forces with an additional high-order density dependent term. The SLy4E$_{\rm light}$ is excellent for light nuclei but its precision reduces towards heavy nuclei. SLy4E$_{\rm heavy}$ is acceptable for both light and heavy nuclei. SLy4E$_{\rm global}$ is further improved compared to SLy4E$_{\rm heavy}$ and can be used for global calculations of finite nuclei with an rms of 2.57 MeV. The charge radii are also generally improved by extended SLy4 forces. The SLy4E$_{\rm light}$ was adopted for studies of collective excitations of weakly bound nuclei.[41] The extended Skyrme forces based on SkM$^{*}$ and UNEDF0 are very successful for global descriptions of finite nuclei and nuclear driplines,[42,43] but are not successful for descriptions of neutron stars.
Table 1. Global calculations of binding energies of 603 even-even nuclei and charge radii of 339 nuclei are compared with experimental data.[39,40] Calculations are performed with the original SLy4, the refitted SLy$4{'}$, and three extended SLy4E parameter sets. The standard root-mean-square (rms) deviations of binding energies $\sigma_{_{\scriptstyle E}}$ (in MeV) and charge radii $\sigma_r$ (in fm) are listed.
Parameter SLy4 SLy$4{'}$ Global Light Heavy
$\sigma_{_{\scriptstyle E}}(Z\leq28)$ 2.22 2.832 2.302 1.426 2.281
$\sigma_{_{\scriptstyle E}}(Z\leq82)$ 3.18 2.582 2.002 2.381 2.373
$\sigma_{_{\scriptstyle E}}({\rm all})$ 4.372 2.860 2.303 3.013 2.576
$\sigma_{r}({\rm all})$ 0.028 0.0287 0.0244 0.0203 0.0244
cpl-38-5-052101-fig1.png
Fig. 1. (a) Pressure of pure neutron matter and (b) symmetry energy as a function of nuclear density, corresponding to different Skyrme parameter sets, see text for details.
Next, we study the influence of the high-order density dependent term on the nuclear EOS. The pressures of neutron matter by different extended SLy4 forces are shown in Fig. 1(a). For various modified SLy4 forces at the saturation point ($\rho_0=0.16$ fm$^{-3}$), the pressure ranges from 2.2 to 2.5 MeV. For densities lower than the saturation density, the pressures are almost the same. In the high density region, we can see that SLy4$'$ gives the lowest pressure, while SLy4 and SLy4E$_{\rm global}$ give the highest pressure. The pressure is also related to the sound of speed in dense nuclear matter.[23] It is obvious that the high-order density dependent term can increase the pressure at high densities. This is understandable given that the repulsive high-order term increases the pressure at high densities, which is particular important for descriptions of neutron stars. The symmetry energies is related to EOS and has been extensively studied by experiments of heavy ion collisions.[20] The symmetry energy at the saturation density is around 32 MeV for SLy4 forces, while it has large uncertainties at high densities. In our cases, it can be seen that SLy4$'$, SLy4E$_{\rm light}$, SLy4E$_{\rm heavy}$ give very soft symmetry energies at high densities. The symmetry energies of SLy4E$_{\rm global}$ are slightly smaller than that of SLy4 results.
cpl-38-5-052101-fig2.png
Fig. 2. Mass-radius relation obtained from EOS of different Skyrme parameter sets. The dashed regions correspond to constrained mass and radius of the two NSs in GW170817 at $90\%$ confidence level.[6]
We now solve the TOV equation to study the properties of neutron stars with various Skyrme forces. Note that we consider the neutron star matter at the $npe\mu$ equilibrium, which leads to a slightly soft EOS compared to the pure neutron matter. Figure 2 shows the calculated mass-radius relation. The crucial criteria for testing effective forces is to reproduce the observed maximum mass of neutron stars, which is about 2$M_{\odot}$.[24] The soft EOS cannot support such a large mass. It can be seen that the maximum masses of all forces are larger than 2$M_{\odot}$. The secret of the series of successful SLy4 forces is that the $x_2$ parameter is $-1.0$.[23] SLy4$'$ results in a slightly smaller maximum mass than that of SLy4. The $M$–$R$ relation from SLy4 and SLy4E$_{\rm global}$ are almost the same. We can see that SLy4$'$ results in the smallest radii of 11.42 km for an NS with 1.4$M_{\odot}$. Meanwhile, SLy4 and SLy4E$_{\rm global}$ correspond to the largest radii of 11.67 and 11.70 km, respectively. It can be understood that the radii of 1.4 $M_{\odot}$ and the maximum NS mass are exactly correlated with the pressure at high densities, rather than the symmetry energy. In contrast to precise measurements of NS masses, there are large uncertainties in measurements of NS radii. The Bayesian analysis of GW170817 binary NSs results in a radius of $11.9\pm1.4$ km for two NSs.[6] We find that our results agree well with the $M$–$R$ relation of GW170817 binary neutron stars. Another Bayesian analysis of GW170817 provides a more stringent constraint of radii of 12.36$^{+0.52}_{-0.38}$ km and 12.32$^{+0.66}_{-0.43}$ km, respectively.[7] The GW170817 measurement excluded very stiff EOS.
cpl-38-5-052101-fig3.png
Fig. 3. Calculated tidal deformability $\varLambda$ as a function of NS's mass, with different Skyrme parameter sets. The inferred tidal deformability with uncertainties from Bayesian analysis of GW180817 binary NSs is shown. The inferred tidal deformability of 1.4$M_{\odot}$ is also shown.
Finally, the calculated tidal deformability $\varLambda$ with the five SLy4 parameter sets are shown in Fig. 3 as a function of NS's mass. We see that, generally, the tidal deformability is very large for NS with small masses. SLy4 and SLy4E$_{\rm global}$ result in the largest tidal deformability, which is related to the largest $R_{1.4}$. Meanwhile, SLy4$'$ results in smallest tidal deformability, corresponding to the smallest $R_{1.4}$. The differences in tidal deformability from various Skyrme forces are correlated with the $M$–$R$ relation. The Bayesian analysis of the GW170817 event suggests that at $90\%$ confidence level, the tidal deformabilities $\varLambda_{1,2}$ of the binary stars are $\varLambda_1=255^{+416}_{-171}$ and $\varLambda_2=661^{+858}_{-375}$, respectively.[7] The tidal deformability of 1.4$M_{\odot}$ is estimated to be $\varLambda_{1.4}=190^{+390}_{-120}$ at $90\%$ level.[6,7] We see that SLy4 and SLy4E$_{\rm global}$ results agree better with the observed tidal deformability of GW170817. In summary, the observations of GW170817 binary neutron stars, including for the first time the tidal deformability from gravitational waves, have provided a great opportunity for constraining the EOS of dense nuclear matter. It is known that the EOS of dense nuclear matter has large uncertainties which are related to the large uncertainties of nuclear interactions at high densities.[10,44–46] There are also extensive studies to constrain EOS or symmetry energies at high densities based on correlations with other nuclear observables.[10,44–46] It is crucial to develop a unified and consistent description of finite nuclei and neutron stars. We have developed a series of extended SLy4 forces with an additional high-order density dependent term by fitting the properties of finite nuclei, as inspired by the Lee–Huang–Yang formula. The original SLy4 force is widely used to study neutron stars with the maximum mass above 2.0$M_{\odot}$ but is not precise for finite nuclei. The refitted SLy4$'$ without high-order density dependence can improve the descriptions of finite nuclei but its EOS becomes soft. The extended SLy4E$_{\rm global}$ is satisfactory for finite nuclei and neutron stars simultaneously. Our results agree well with the GW170817 observations. We have demonstrated the essential role of the high-order density dependent term in effective nuclear interactions at ultrahigh densities. Our unified SLy4E$_{\rm global}$ descriptions provide a predictive model for studying neutron stars, as well as new understandings of effective interactions. Realistic studies of neutron stars (including superfluidity and non-uniform crusts) will be of interest in the future. We expect that more precise observations of gravitational waves in the future will provide more stringent constraints on nuclear interactions in ultradense nuclear matter. References Cosmic Rays from Super-NovaeThe equation of state of hot, dense matter and neutron starsEquations of state for supernovae and compact starsNuclear matter and neutron-star properties calculated with the Skyrme interactionGW170817: Observation of Gravitational Waves from a Binary Neutron Star InspiralGW170817: Measurements of Neutron Star Radii and Equation of StateConstraining the Neutron Star Equation of State Using Multiband Independent Measurements of Radii and Tidal DeformabilitiesProperties of the Binary Neutron Star Merger GW170817Tidal Love Numbers of Neutron StarsThe Equation of State of Nuclear Matter: From Finite Nuclei to Neutron StarsA unified equation of state of dense matter and neutron star structureNeutron Star Structure and the Equation of StateNuclear matter and neutron star properties calculated with a separable monopole interactionNuclear Force from Lattice QCDNew ideas in constraining nuclear forcesSkyrme interaction and nuclear matter constraintsNeutron star equation of state: Quark mean-field (QMF) modeling and applicationsComplete Electric Dipole Response and the Neutron Skin in Pb 208 Measurement of the Neutron Radius of Pb 208 through Parity Violation in Electron ScatteringConstraints on the Density Dependence of the Symmetry EnergyDetermination of the Equation of State of Dense MatterRecent progress and new challenges in isospin physics with heavy-ion reactionsCircumstantial Evidence for a Soft Nuclear Symmetry Energy at Suprasaturation DensitiesCVII. The nuclear surfaceHartree-Fock Calculations with Skyrme's Interaction. I. Spherical NucleiA Skyrme parametrization from subnuclear to neutron star densitiesA Skyrme parametrization from subnuclear to neutron star densities Part II. Nuclei far from stabilitiesA Massive Pulsar in a Compact Relativistic BinaryEnergy density functional for nuclei and neutron starsExtension and parametrization of high-order density dependence in Skyrme forcesThe mean-field theory of nuclear structure and dynamicsConsistent three-nucleon forces in the nuclear many-body problemQuantum-Mechanical Many-Body Problem with Hard-Sphere InteractionMany-Body Problem in Quantum Mechanics and Quantum Statistical MechanicsSaturation of Nuclear ForcesPrecision Measurements of Collective Oscillations in the BEC-BCS CrossoverExploring the extended density-dependent Skyrme effective forces for normal and isospin-rich nuclei to neutron starsStatic Solutions of Einstein's Field Equations for Spheres of FluidOn Massive Neutron CoresEffects of symmetry energy on the radius and tidal deformability of neutron stars in the relativistic mean-field modelTidal deformability of neutron stars with realistic equations of state and their gravitational wave signatures in binary inspiralThe AME2016 atomic mass evaluation (II). Tables, graphs and referencesTable of experimental nuclear ground state charge radii: An updateProbing surface quantum flows in deformed pygmy dipole modesGlobal analysis of Skyrme forces with higher-order density dependenciesConstraints on the neutron drip line with the newly observed Na 39 Tides in merging neutron stars: Consistency of the GW170817 event with experimental data on finite nucleiInsights on Skyrme parameters from GW170817Constraints on the symmetry energy and its associated parameters from nuclei to neutron stars
[1] Baade W and Zwicky F 1934 Proc. Natl. Acad. Sci. USA 20 259
[2] James M L and Madappa P 2016 Phys. Rep. 621 127
[3] Oertel M, Hempel M, Klähn T, and Typel S 2017 Rev. Mod. Phys. 89 015007
[4] Rikovska S J, Miller J C, Koncewicz R, Stevenson P D, and Strayer M R 2003 Phys. Rev. C 68 034324
[5] Abbott B P et al. 2017 Phys. Rev. Lett. 119 161101
[6] Abbott B P et al. 2018 Phys. Rev. Lett. 121 161101
[7] Fasano M, Abdelsalhin T, Maselli A, and Ferrari V 2019 Phys. Rev. Lett. 123 141101
[8] Abbott B P et al. 2019 Phys. Rev. X 9 011001
[9] Hinderer T 2008 Astrophys. J. 677 1216
[10] Burgio G F and Vidaña I 2018 Universe 6 119
[11] Douchin F and Haensel P 2001 Astron. & Astrophys. 380 151
[12] James M L and Madappa P 2001 Astrophys. J. 550 426
[13] Stone J R, Stevenson P D, Miller J C, and Strayer M R 2002 Phys. Rev. C 65 064312
[14] Ishii N, Aoki S, and Hatsuda T 2007 Phys. Rev. Lett. 99 022001
[15] Tews I, Davoudi Z, Ekstrom A, Holt J D, and Lynn J E 2020 J. Phys. G 47 103001
[16] Dutra M, Lourenço O, Sá Martins J S, Delfino A, Stone J R, and Stevenson P D 2012 Phys. Rev. C 85 035201
[17] Li A, Zhu Z Y, Zhou E P, Dong J M, Hu J N, and Xia C J 2020 J. High Energy Astrophys. 28 19
[18] Tamii A et al. 2011 Phys. Rev. Lett. 107 062502
[19] Abrahamyan S et al. (PREX Collaboration) 2012 Phys. Rev. Lett. 108 112502
[20] Tsang M B, Zhang Y X, Danielewicz P, Famiano M, Li Z X, Lynch W G, and Steiner A W 2009 Phys. Rev. Lett. 102 122701
Danielewicz P, Lacey R, and Lynch W G 2002 Science 298 1592
Li B A, Chen L W, and Che M K 2008 Phys. Rep. 464 113
Xiao Z G, Li B A, Chen L W, Yong G C, and Zhang M 2009 Phys. Rev. Lett. 102 062502
[21] Skyrme T H R 1956 Philos. Mag. 1 1043
[22] Vautherin D and Brink D M 1972 Phys. Rev. C 5 626
[23] Chabanat E, Bonche P, Haensel P, Meyer J, and Schaeffer R 1997 Nucl. Phys. A 627 710
[24] Chabanat E, Bonche P, Haensel P, Meyer J, and Schaeffer R 1998 Nucl. Phys. A 635 231
[25] Antoniadis J, Freire P C C, Wex N et al. 2013 Science 340 1233232
[26] Erler J, Horowitz C J, Nazarewicz W, Rafalski M, and Reinhard P G 2013 Phys. Rev. C 87 044320
[27] Xiong X Y, Pei J C, and Chen W J 2016 Phys. Rev. C 93 024311
[28] Negele J W 1982 Rev. Mod. Phys. 54 913
[29] Grange P, Lejeune A, Martzolff M, and Mathiot J F 1989 Phys. Rev. C 40 1040
[30] Huang K and Yang C N 1957 Phys. Rev. 105 767
[31] Lee T D and Yang C N 1957 Phys. Rev. 105 1119
[32] DeDominicis C and Martin P C 1957 Phys. Rev. 105 1419
[33] Altmeyer A, Riedl S, Kohstall C, Wright M J, Geursen R, Bartenstein M, Chin C, Denschlag J H, and Grimm R 2017 Phys. Rev. Lett. 98 040401
[34] Agrawal B K, Dhiman S K, and Kumar R 2006 Phys. Rev. C 73 034319
[35] Tolman R C 1939 Phys. Rev. 55 364
[36] Oppenheimer J R and Volkoff G M 1939 Phys. Rev. 55 374
[37] Hu J N, Bao S S, Zhang Y, Nakazato K I, Sumiyoshi K, and Shen H 2020 Prog. Theor. Exp. Phys. 2020 043D01
[38] Hinderer T, Lackey B D, Lang R N, and Read J S 2010 Phys. Rev. D 81 123016
[39] Wang M, Audi G, Kondev G G, Huang W J, Naimi S, and Xu X 2017 Chin. Phys. C 41 030003
[40] Angeli I and Marinova K P 2013 At. Data Nucl. Data Tables 99 69
[41] Wang K, Kortelainen M, and Pei J C 2017 Phys. Rev. C 96 031301
[42] Zuo Z W, Pei J C, Xiong X Y, and Zhu Y 2018 Chin. Phys. C 42 064106
[43] Chai Q Z, Pei J C, Fei N, and Guan D W 2020 Phys. Rev. C 102 014312
[44] Malik T, Agrawal B K, De J N, Samaddar S K, Providência C, Mondal C, and Jha T K 2019 Phys. Rev. C 99 052801
[45] Tsang C Y, Tsang M B, Danielewicz P, Fattoyev F J, Lynch W G 2019 Phys. Lett. B 796 1
[46] Zhang Y X, Liu M, Xia C J, Li Z X, and Biswal S K 2020 Phys. Rev. C 101 034303