Equations of State for Hadronic Matter and Mass-Radius Relations of Neutron Stars with Strong Magnetic Fields

: Neutron star is an important object for us to verify the equation of state of hadronic matter. For a speciﬁc choice of equations of state, mass and radius of a neutron star are determined, for which there are constraints from observations. According to some previous studies, since the strong magnetic ﬁeld acts as a repulsive force, there is a possibility that neutron stars with strong magnetic ﬁelds may have relatively heavier masses than other non-magnetized neutron stars. In this paper, the structure of a neutron star with a strong internal magnetic ﬁeld is investigated by changing its internal functional form to see how much the neutron star can be massive and also how radius of a neutron star can be within a certain range.


Introduction
To study a neutron star (NS) is an interesting and important subject in nuclear physics as it is a unique object of superdense hadronic matter, where its central density might be several times larger than the nuclear saturation density.Equation of state (EoS) for the nuclear matter that depends on the energy density functional is often utilized to discuss various properties of NSs, for instance, the mass-radius (MR) relation.
Historically, Oppenheimer and Volkoff (independently, Tolman) were the first who estimated its maximum mass of M max ∼ 0.71 solar mass (M ) by employing a noninteracting strongly degenerate relativistic gas of neutrons in 1939 [1,2].Discoveries of NSs with masses around twice the solar mass have made a strong impact on the nuclear physics community who had once believed that the maximum mass of an NS should not exceed 1.56 M due to the occurrence of kaon condensation [3,4].Indeed, several NSs with around 2 M have been confirmed since 2010.The NS PSR J1614-2230 has the observed mass of 1.97 ± 0.04 M [5] and the NS PSR J0348+0432 has mass of 2.01 ± 0.04 M [6].In 2021, the mass and equatorial radius of the millisecond-pulsar (MSP) J0740+6620 was constrained to be M = 2.072 +0.067 −0.066 M and R 2.072 M = 12.39 +1.30 −0.98 km, respectively [7][8][9][10].If NS matter consists of only nucleons with strong two-or three-body repulsive forces, massive NSs more than 2 M can be easily described.However, hyperons are energetically favorable when the baryon chemical potential is large enough in the inner core of an NS.This leads to a reduction in the Fermi pressure to soften the EoS and to a reduction in the predicted maximum mass less than 2 M .This is called the hyperon puzzle.Since discoveries of massive NSs, many people have searched for the optimal EoSs that can achieve masses over 2 M .To obtain NS masses more than 2 M is one of the goals to be achieved for the optimal EoSs.
There are some characteristic NSs: millisecond-pulsar, which has a short rotational period of less than 10 ms, and magnetar, which has a powerful magnetic field on its surface.As a part of external conditions to support over 2 M , the extremely rapid rotation and powerful magnetic field should be considered in the EoS study.In our recent studies [11,12], EoSs for a rapidly rotating NS with some deformation and a fixed magnetic field have been investigated.Although an unrealistically rapid rotation can modify the NS mass from 1.89 M to 2 M , the rotational effects on the mass increase were found to be not so significant for the realistic frequency range of the observed millisecond-pulsars.
A magnetar is a type of the NS that has a strong magnetic field on the surface.Recently, up to 30 magnetars have been observed [13,14].Maximumly, about 10 15 G of the magnetic field on the surface has been found by the observation [13], but we do not yet know the generation mechanism of the strong magnetic fields, where and how they come from.
Another constraint on the EoS comes from the NS radius.From the observation of the GW170817 gravitational wave event, the upper limit radius is 13.6 km [15] at the mass of M = 1.4 M .Moreover, another reports the upper limit radius 13.76 km [16] at M = 1.4 M .The lower limit radius was reported as 10.68 +0. 15  −0.04 km [17] at M = 1.6 M .The upper limit of radius by NICER observations of PSR J0030+0451 is 13.02 +1. 24 −1.06 km [18] at M = 1.44 +0.15 −0.14 M .In this paper, the internal structure and MR relation of NS, including hyperon populations, are investigated by changing the functional form of a strong magnetic field.The strength of the magnetic field is fixed constant as 10 12 G on the surface and over 10 18 G near the center.The shape of the magnetic field is parametrized by two parameters α and γ.We first formulate the relativistic mean field (RMF) theory, including hyperons.Here, we introduce 12 different EoSs [19,20] depending on the different coupling constants between baryons and mesons.These EoSs have similar saturation properties at nuclear saturation density.This RMF includes a strong magnetic field, which has a functional form of baryon number density ρ.In this work, we solve the Tolman-Oppenheimer-Volkov (TOV) equations to obtain masses and radii for various EoSs by changing α and γ parameters freely.Here, the MR relations are verified with respect to the constraints that maximum masses are within the range of observed masses and the radii at 1.4 solar mass are less than 13.76 km.
This paper is organized as follows.The formulation of the calculation is given in Section 2. The results are given in Section 3. Finally, a summary is given in Section 4.
For seven kinds of EoSs employed in the present study, nuclear properties at saturation number density ρ 0 are given in Table 1.Here, TM1 and TM2 are different with respect to the slope parameter L, where L is closely related to the radius of an NS.The NL3 parametrization is fitted to the ground-state properties of both stable and unstable nuclei.This parametrization predicts very large, purely nucleonic NS maximum masses, but a symmetry energy slope parameter L is too large.Thus, we also consider the parametrization NL3ωρ with a softer density dependence of the symmetry energy due to the inclusion of the nonlinear ωρ term.

Magnetic Fields
The neutron stars can be massive because of the pressure from magnetic fields.In this work, we investigate an NS with a strong magnetic field inside the NS.From observations, only the strength of the magnetic field on the surface is known and we need to assume the strength and the functional form of the magnetic field inside the NS.
In the following, a density-dependent magnetic field strength is adopted in this study [26,30], where B 0 indicates the strength in a denser region than that of the saturation number density ρ 0 (0.153 fm −3 ) and B s indicates the strength of magnetic fields on the surface.Here, we adopt B s = 10 12 G.Since Equation ( 13) does not satisfy div B = 0 in general, this form of the magnetic fields must be used with caution as a simple way to implement magnetic fields in an NS.With this problem in mind, however, it should be emphasized that for particles in a very small region of an NS, they feel almost a constant magnetic field, for which div B = 0 is practically satisfied.For more realistic magnetic fields, different equations might be used [31].
Even if one assumes the form of Equation ( 13), one has free parameters α, γ, and B 0 .Up to now, there is no work of changing α and γ parameters freely.In this work, we arbitrarily change α and γ to investigate their effects on the radii and masses of NSs.

EoS of Hadronic Matter with Magnetic Fields
The energy density and the pressure of hadronic matter in the presence of the magnetic fields are given as respectively, where ε m and p m are the energy density and the pressure of hadrons where the contribution B 2 /2 from the magnetic fields are neglected.However, it should be noted that, without B 2 /2, the effects of the magnetic fields to ε m and p m are included through the interaction between the magnetic field and each particle's magnetic moment.All other details are given in Appendix A of Reference [12].

Comparison of the Case without Magnetic Fields
Figure 1 shows the MR relations of 12 EoSs without assuming any magnetic fields.The orange and light green dashed areas indicate pulsars PSR J0740+6620 and PSR J0030+ 0451, respectively.The thick area shows the one within 68% credibility and the thinner area shows the one within 90% credibility.Each EoS curve must go through these areas if they satisfy the observational constraints.The arrow indicates the upper limit of radius for 1.4 M from the gravitational wave event GW170817.
Only three EoSs (NL3a, NL3ωρ-a, and DDME2-a) barely satisfy the MR constraints (within 90% credibility for the maximum mass), but among them, only NL3ωρ-a (K = 271.8MeV) satisfies the radius constraint at 1.4 M .It should be noted that DDME2 (K = 250.9MeV) is a relatively soft EoS at nuclear saturation density, but it becomes harder in a higher density region.Table 3 summarizes the maximum masses, radius at 1.4 M , and radius at 2.072 M for 12 EoSs without magnetic fields.Only three kinds of EoSs (NL3, NL3ωρ, and DDME2) give masses of twice the solar mass.As general features, the maximum mass becomes smaller when hyperons are included, but the maximum mass increases when the magnetic field is implemented.The average radius increases when a strong magnetic field is implemented.

Changing α and γ Parameters of the Magnetic Field Function
In this section, the magnetic field B(ρ) in Equation ( 13) is also implemented, where α and γ parameters are changed as free parameters.
The B(ρ) as a function of baryon number density ρ for various α parameters is shown in Figure 2.Each line indicates a magnetic field with B 0 = 2.5 × 10 18 G and γ = 2.In Figure 2, as α becomes larger, the strength of the magnetic field B changes more abruptly.Even at baryon number density ρ = 0.01 fm −3 and α = 0.01 (black solid line), the strength of magnetic field is 1 × 10 14 G.This indicates that parameter γ strongly affects the strength of the magnetic field near the surface of an NS.Furthermore, the radius at 1.4 M becomes larger when α becomes larger.
Next, the magnetic field B(ρ) as a function of baryon number density ρ for various γ parameters is shown in Figure 3.Each line indicates a magnetic field with B 0 = 2.5 × 10 18 G and α = 0.05.For ρ > 0.3 fm −3 , B(ρ) is almost constant B 0 = 2.5 × 10 18 G.Here, magnetic field strengths at saturation density ρ 0 are given as B(ρ 0 ) = 1.23 × 10 17 G for any value of γ.Magnetic field strengths B(ρ 0 ) at saturation density ρ 0 for various α parameters are shown in Table 4.
For γ = 4, α =0.01.  Figure 5 shows the MR relations of GM3 EoS with magnetic fields.The following EoSs go through 68% credibility with respect to the maximum mass and the radius at 1.4 M .
For γ = 5, α = 0.01. Figure 6 shows the MR relations of NL3ωρ-a EoS with magnetic fields.The following EoSs go through 68% credibility with respect to the maximum mass and the radius at 1.4 M .
Figure 7 shows the MR relations of NL3ωρ-b EoS with magnetic fields.In this case, no EoSs go through 68% credibility with respect to the maximum mass and the radius at 1.4 M .Table 5. Maximum mass (M max ) in unit of M and radius (km) at M = 1.4 M (R 1.4M ) and radius (km) at M = 2.072 M (R 2.072 M ) for GM1 EoS with magnetic fields (B s = 10 12 G, B 0 = 2.5 × 10 18 G).Table 7. Maximum mass (M max ) in unit of M and radius (km) at M = 1.4 M (R 1.4 M ) and radius (km) at M = 2.072 M (R 2.072 M ) for NL3ωρ-a EoS with magnetic fields (B s = 10 12 G, B 0 = 2.5 × 10 18 G).Figure 8 shows the MR relations of TM2ωρ-a EoS with magnetic fields.The following EoSs go through 68% credibility with respect to the maximum mass and the radius 1.4 M .
For γ = 4, α =0.01. Figure 9 shows the MR relations of TM2ωρ-b EoS with magnetic fields.The following EoSs go through 68% credibility with respect to the maximum mass and the radius at 1.4 M .
For γ = 4, α =0.01.     Figure 10 shows the MR relations of DDME2-a (left) and DDME2-b (right) EoSs with magnetic fields.The following EoSs for DDME2-a go through 68% credibility with respect to the maximum mass and the radius at 1.4 M .

Summary
In this work, the MR relations for magnetized neutron stars (NSs) have been investigated using twelve EoSs (seven kinds of EoSs).The NSs consist of nucleons and hyperons.They are implemented with magnetic fields, whose forms are changed by the shape parameters α and γ.The α and γ are the damping factor and the exponent of the density dependence of a magnetic field, respectively.
First, we have found that, without any magnetic fields, only NL3ωρ-a EoS satisfies the constraint for mass and radius from observations.Second, we have found that many varieties of EoSs with the implement of strong magnetic fields give maximum masses over 2 M and satisfy the constraints from the observations at the same time.
To summarize, using different shapes of the magnetic field, in this work, we have demonstrated that many kinds of EoSs with strong magnetic fields can satisfy the constraints on the maximum mass and the radius within the range by observations.Other EoSs with lower incompressibility (K < 240 MeV) than those EoSs examined in this work will be studied in future work.

Figure 4 .
Figure 4. MR relations of GM1 EoS with magnetic fields.The strength of surface magnetic field strength B s is 10 12 G, and the central magnetic field strength B 0 is 2.5 × 10 18 G.The arrow and colored hatched areas are the same as in Figure 1.

Figure 5 .
Figure 5. MR relations of GM3 EoS with magnetic fields.The strength of surface magnetic field strength B s is 10 12 G, and the central magnetic field strength B 0 is 2.5 × 10 18 G.The arrow and colored hatched areas are the same as in Figure 1.

Figure 6 .
Figure 6.MR relations of NL3ωρ-a EoS with magnetic fields.The strength of surface magnetic field strength B s is 10 12 G, and the central magnetic field strength B 0 is 2.5 × 10 18 G.The arrow and colored hatched areas are the same as in Figure 1.

Figure 7 .
Figure 7. MR relations of NL3ωρ-b EoS with magnetic fields.The strength of surface magnetic field strength B s is 10 12 G, and the central magnetic field strength B 0 is 2.5 × 10 18 G.The arrow and colored hatched areas are the same as in Figure 1.

6 Figure 8 .
Figure 8. MR relations of TM2ωρ-a EoS with magnetic fields.The strength of surface magnetic field strength B s is 10 12 G, and the central magnetic field strength B 0 is 2.5 × 10 18 G.The arrow and colored hatched areas are the same as in Figure 1.

Figure 9 .
Figure 9. MR relations of TM2ωρ-b EoS with magnetic fields.The strength of surface magnetic field strength B s is 10 12 G, and the central magnetic field strength B 0 is 2.5 × 10 18 G.The arrow and colored hatched areas are the same as in Figure 1.

Figure 10 .
Figure 10.MR relations of DDME2-a (left) and DDME2-b (right) EoSs with magnetic fields.The strength of surface magnetic field strength B s is 10 12 G, and the central magnetic field strength B 0 is 2.5 × 10 18 G.The arrow and colored hatched areas are the same as in Figure 1.

Table 3 .
Maximum masses, radius at M = 1.4 M , and radius at M = 2.072 M for 12 EoSs without magnetic fields.
are shown in Tables 5-12 for each EoSs.The MR relations by changing α and γ are given in Figures 4-10 for those EoSs with various incompressibilities, which satisfy the observational constraints.

Table 9 .
Maximum mass (M max ) in unit of M and radius (km) at M = 1.4 M (R 1.4 M ) and radius (km) at M = 2.072 M (R 2.072 M ) for TM2ωρ-a EoS with magnetic fields (B s = 10 12 G, B 0 = 2.5 × 10 18 G).