Neutron star mass formula with nuclear saturation parameters for asymmetric nuclear matter Hajime Sotani1 2and Shinsuke Ota3 1Astrophysical Big Bang Laboratory RIKEN Saitama 351-0198 Japan

2025-05-02 0 0 902.81KB 11 页 10玖币

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

Hajime Sotani1, 2, ∗and Shinsuke Ota3

1Astrophysical Big Bang Laboratory, RIKEN, Saitama 351-0198, Japan

2Interdisciplinary Theoretical & Mathematical Science Program (iTHEMS), RIKEN, Saitama 351-0198, Japan

3Research Center for Nuclear Physics (RCNP), Osaka University, Ibaragi, Osaka 567-0047, Japan

(Dated: October 24, 2022)

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., η≡(K0L2)1/3with the incompressibility for

symmetric nuclear matter, K0, and the density-dependent nuclear symmetry energy, L. In this study, we newly

ﬁnd another suitable combination given by ητ≡(−KτL5)1/6with the isospin dependence of incompressibility

for asymmetric nuclear matter, Kτ, and derive the empirical relations for the neutron star mass and gravitational

redshift as a function of ητand 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 ∼10% accuracy, and the radius within a few % 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 ητand η. With this correlation, we derive the constraint on Kτas −348 ≤Kτ≤ −237 MeV,

assuming that L= 60 ±20 and K0= 240 ±20 MeV.

PACS numbers: 04.40.Dg, 26.60.+c, 21.65.Ef

I. INTRODUCTION

Neutron star is a massive remnant left after a supernova explosion, which happens at the last moment of the massive star’s life.

The density inside the star becomes much higher than the standard nuclear density, ρ0= 2.7×1014 g/cm3, and the gravitational

and magnetic ﬁelds inside/around the star are signiﬁcantly stronger than those observed in our solar system [1]. The neutron

star mass and radius strongly depend on the equation of state (EOS) for neutron star matter under the β-equilibrium. The mass

of a neutron star model with a higher central density generally becomes larger, even though the EOS is not ﬁxed yet. So, the

astronomical observations of neutron stars or their activities tell us the information about the EOS for a relatively higher density

region, while the terrestrial experiments tell us that for a lower density region (e.g., Fig. 2 in Ref. [2]).

In practice, the discovery of 2Mneutron stars [3–6] has excluded some soft EOSs, with which the expected maximum mass

is less than the observed mass. This discovery simultaneously reveals the problem that most of the EOSs with hyperon are too

soft to construct the 2Mneutron star, i.e., the so-called hyperon puzzle. Meanwhile, the gravitational wave event, GW170817

[7], tells us the information on the tidal deformability of the neutron star, which leads to the constraint that a 1.4Mneutron

star radius should be less than 13.6 km [8]. We note that the constraint on neutron star radius may become more stringent in

view of the existing multi-messenger observational data [9, 10]. The light bending due to the strong gravitational ﬁeld induced

by the neutron star is also one of the important phenomena to see the neutron star properties. That is, the pulsar light curve

from the rotating neutron star would be modiﬁed due to this relativistic effect and one could get the neutron star properties

by carefully observing it (e.g., [11–16]). Based on this idea, the Neutron star Interior Composition Explorer (NICER) is now

operating on an International Space Station (ISS) and it has already announced the constraint on two neutron stars properties,

i.e., PSR J0030+0451 [17, 18] and PSR J0740+6620 [19, 20]. Furthermore, the direct detection of the gravitational waves from

the neutron star in the future may enable us to extract the neutron star properties (e.g., [21–29]).

On the other hand, the EOS in a lower density region is also gradually constrained through terrestrial nuclear experiments,

but still, there are large uncertainties in EOS parameters (or in neutron star properties) constrained from terrestrial experiments.

For instance, the ﬁducial value of the density-dependent nuclear symmetry energy Lis L'60 ±20 MeV [30, 31], while the

constraints of Lobtained recently seem to be signiﬁcantly larger than the ﬁducial value [32, 33]. This is because one has to

usually transform the experimental constraint to the EOS parameters, even if the information determined via experiments is

associated with some aspects of nuclear EOS. Then, one can eventually discuss the neutron star mass and radius as a solution of

the Tolman-Oppenheimer-Volkoff (TOV) equation. Anyway, the terrestrial experiments are deﬁnitely crucial for understanding

the neutron star EOS as well as the astronomical observation of neutron stars.

∗sotani@yukawa.kyoto-u.ac.jp

arXiv:2210.11651v1 [nucl-th] 21 Oct 2022

TABLE I. EOS parameters adopted in this study, K0,n0,L,Q,Ksym,Qsym,Kτ, and Qτare listed, while ηand ητare speciﬁc combinations

with them given by η=K0L21/3and ητ=−KτL51/6.

EOS K0n0L Q Ksym Qsym KτQτη ητ

(MeV) (fm−3) (MeV) (MeV) (MeV) (MeV) (MeV) (MeV) (MeV) (MeV)

OI-EOSs 200 0.165 35.6 -759 -142 801 -221 2017 63.3 48.2

0.165 67.8 -761 -27.6 589 -176 2909 97.2 79.5

220 0.161 40.2 -720 -144 731 -254 1915 70.9 54.7

0.161 77.6 -722 -9.83 486 -221 2779 110 92.4

240 0.159 45.0 -663 -146 642 -291 1760 78.6 61.4

0.158 88.2 -664 10.5 363 -274 2559 123 107

260 0.156 49.8 -589 -146 535 -333 1551 86.4 68.4

0.155 99.2 -590 32.6 219 -338 2246 137 122

280 0.154 54.9 -496 -146 410 -378 1285 94.5 75.7

0.153 111 -498 57.4 54.4 -412 1834 151 138

300 0.152 60.0 -386 -146 266 -429 962 103 83.3

0.151 124 -387 86.1 -133 -499 1310 167 157

KDE0v 229 0.161 45.2 -373 -145 523 -342 1187 77.6 63.4

KDE0v1 228 0.165 54.7 -385 -127 484 -363 1317 88.0 75.0

SLy2 230 0.161 47.5 -364 -115 507 -325 1183 80.3 65.4

SLy4 230 0.160 45.9 -363 -120 522 -323 1175 78.7 63.7

SLy9 230 0.151 54.9 -350 -81.4 462 -327 1215 88.4 73.9

SKa 263 0.155 74.6 -300 -78.5 175 -441 940 114 100

SkI3 258 0.158 101 -304 73.0 212 -412 1276 138 127

SkMp 231 0.157 70.3 -338 -49.8 159 -369 1086 105 92.7

In order to discuss the neutron star mass and radius directly with the EOS parameters constrained somehow from the ex-

periments, it may be useful to derive a kind of empirical formula expressing the neutron star properties as a function of EOS

parameters if it exists. In practice, we have already found the empirical relations expressing the mass (M) and gravitational

redshift (z) of a low-mass neutron star as a function of the suitable combination of the nuclear saturation parameters, η, and

the central density of neutron star [34], with which the neutron star mass and radius expected from the terrestrial experiments

can be discussed as in Ref. [2]. On the other hand, in this study, we consider deriving another type of empirical relations for

the low-mass neutron stars, focusing on the nuclear saturation parameters for asymmetric nuclear matter, and then discuss the

impact of the uncertainties in the nuclear saturation parameters on the neutron star mass and radius.

This manuscript is organized as follows. In Sec. II, we brieﬂy mention the EOSs considered in this study and the saturation

parameters in nuclear matter. In Sec. III, we examine the neutron star models and derive the empirical formulas for the neutron

star mass and its gravitational redshift as a function of a suitable combination of the nuclear saturation parameters we derived in

this study. Using the newly derived empirical formulas, in Sec. IV, we discuss the neutron star mass and radius together with the

constraints from the astronomical and experimental observations. Finally, in Sec. V, we conclude this study. Unless otherwise

mentioned, we adopt geometric units in the following, c=G= 1, where cand Gdenote the speed of light and the gravitational

constant, respectively.

II. EOS FOR NEUTRON STAR MATTER

To construct the neutron star models theoretically, one has to prepare the EOS for neutron star matter. We note that, in order

to discuss the neutron star properties with the EOS parameters, one has to adopt the uniﬁed EOS. That is, the EOS describing

the neutron star core is constructed with the same nuclear model as in the EOS for the neutron star crust. We note that, if one

constructs the neutron star model with the EOS (unlike the uniﬁed EOS), which is assembled by connecting the EOS for the core

region to the different EOS for the crust region at an appropriate transition density, the radius of a neutron star whose central

density is around the transition density strongly depends on the selection of the transition density (e.g., Ref. [35]). That is,

unless the uniﬁed EOSs are adopted, one can not discuss the empirical relations, expressing the neutron star models, as we will

derive in Sec. III. In this study, to systematically see the EOS dependence of the neutron star properties, we particularly adopt

the phenomenological EOS proposed by Oyamatsu and Iida [36, 37] (hereafter referred to as OI-EOS) and the EOSs with the

Skyrme-type interaction listed in Table I.

OI-EOSs are based on the Pad´

e-type potential energies and constructed in such a way that the nucleus models with a simpliﬁed

version of the extended Thomas-Fermi theory should become consistent with the empirical masses and radii of stable nuclei.

We note that most of the OI-EOSs adopted here may not fulﬁll either constraint of 2Mobservation or the 1.4Mneutron star

radius constrained from the GW170817, because the OI-EOSs are constructed by focusing on the behavior around the saturation

point. Nevertheless, since we discuss the neutron star properties constructed with the central density, which is not so high, and

since the OI-EOSs are deﬁnitely suitable for systematical study, we adopt the OI-EOSs in this study. On the other hand, we

also consider the EOSs with various Skyrme-type interactions, such as KDE0v, KDE0v1 [38], SLy2, SLy4, SLy9 [39, 40], SKa

[41], SkI3 [42], and SkMp [43]. We note that the non-relativistic EOSs may break the causality in a high density region, where

the resultant neutron star models are not realistic. But, in this study, we focus only on the density region less than threefold the

saturation density, where all the EOSs adopted in this study do not break the causality.

Even though the EOS generally depends on the nuclear model, compositions, and interaction, the bulk energy per nucleon for

the uniform nuclear matter at zero temperature can anyhow be expressed as a function of the baryon number density, nb, and an

asymmetry parameter, α, such as

A=ws(nb) + α2S(nb) + O(α3),(1)

where nband αare given by nb=nn+npand α= (nn−np)/nbwith the neutron number density, nn, and the proton number

density, np;wscorresponds to the energy per nucleon of symmetric nuclear matter (α= 0); and Sdenotes the density-dependent

symmetry energy. Additionally, wsand Scan be expanded around the saturation density, n0, of the symmetric nuclear matter as

a function of u= (nb−n0)/(3n0);

ws(nb) = w0+K0

2u2+Q

6u3+O(u4),(2)

S(nb) = S0+Lu +Ksym

2u2+Qsym

6u3+O(u4).(3)

The coefﬁcients in this expansion correspond to the nuclear saturation parameters, and each EOS has its own set of nuclear

saturation parameters. Among the nuclear saturation parameters, n0,w0, and S0are especially well-constrained from the

terrestrial experiments, i.e., n0≈0.15 −0.16 fm−3,w0≈ −15.8MeV [44], and S0≈31.6±2.7MeV [31]. The constraint on

K0and Lis relatively more difﬁcult. This is because one needs to obtain the nuclear data in a wide density range to constrain

K0and L, which are a kind of density derivative. The current ﬁducial values of K0and Lare K0= 240 ±20 MeV [45] and

L= 60 ±20 MeV [30, 31]. Meanwhile, since the determination of the values of K0and Lare strongly model-dependent, one

may consider K0= 200 −315 MeV [46, 47] and L= 20 −145 MeV [2] as their conservative values, which cover almost all

predictions from various experiments. The experimental constraints on the saturation parameters for higher order terms, such as

Q,Ksym, and Qsym, are almost disorganized, but they are theoretically evaluated as −800 ≤Q≤400,−400 ≤Ksym ≤100,

and −200 ≤Qsym ≤800 MeV [31].

On the other hand, considering E/A as the energy per particle of inﬁnite asymmetric nuclear matter, one can also expand it

around an isospin dependent saturation density, ˜n0(α)'n01−3(L/K0)α2[48], such as

A= ˜w0(˜n0) + ˜

K0(˜n0)

2˜u2+˜

Q(˜n0)

6˜u3+O(˜u4),(4)

where ˜uis deﬁned as ˜u= (nb−˜n0)/(3˜n0)and the coefﬁcients in this expansion are related to the saturation parameters in Eqs.

(2) and (3) through

˜w0(˜n0) = w0+S0α2+O(α3),(5)

K0(˜n0) = K0+Kτα2+O(α3),(6)

Q0(˜n0) = Q+Qτα2+O(α3).(7)

Here, Kτand Qτare respectively the isospin dependence of incompressibility and skewness coefﬁcient at the saturation density,

˜n0(α), given by

Kτ=Ksym −6L−Q

L, (8)

Qτ=Qsym −9Q

L. (9)

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NeutronstarmassformulawithnuclearsaturationparametersforasymmetricnuclearmatterHajimeSotani1,2,andShinsukeOta31AstrophysicalBigBangLaboratory,RIKEN,Saitama351-0198,Japan2InterdisciplinaryTheoretical&MathematicalScienceProgram(iTHEMS),RIKEN,Saitama351-0198,Japan3ResearchCenterforNuclearPhysics(RCNP)...

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Neutron star mass formula with nuclear saturation parameters for asymmetric nuclear matter Hajime Sotani1 2and Shinsuke Ota3 1Astrophysical Big Bang Laboratory RIKEN Saitama 351-0198 Japan

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