Effect of Inner Crust EoS on Neutron star properties
aa r X i v : . [ nu c l - t h ] D ec Effect of Inner Crust EoS on Neutron starproperties.
Ishfaq A. Rather , A. A. Usmani , S. K. Patra , Department of Physics, Aligarh Muslim University, Aligarh 202002, India Institute of Physics, Bhubaneswar 751005, India Homi Bhabha National Institute, Training School Complex, Anushakti Nagar,Mumbai 400094, IndiaE-mail: [email protected] keywords:
Unified EoS, Inner crust, Neutron star, Symmetry energy slope parameter
Abstract.
The neutron star maximum mass and the radius are investigated withinthe framework of the relativistic mean-field (RMF) model. The variation in the radiusat the canonical mass, R . , using different inner crust equation of state (EoS) withdifferent symmetry energy slope parameter is also studied. It is found that althoughthe NS maximum mass and the corresponding radius do not vary much with differentslope parameter inner crust EoS, the radius and the tidal deformability at 1.4 M ⊙ fora unified EoS is lowest among all crust EoS. For non-unified EoSs, the crust with alow symmetry energy slope parameter produces a low NS radius at the canonical mass.The non-unified EoSs follow a linear correlation between radius at 1.4 M ⊙ and the slopeparameter L sym . The properties of uniformly rotating neutron stars are also studied.The variation in the radius is seen at 1.4 M ⊙ with the slope parameter. The momentof inertia is also calculated and a little variation is seen with different crust EoS.
1. Introduction
The structure and the properties of Neutron Stars (NS) have been studied effectivelyfrom experimental as well as theoretical models. Such studies reveal the inner structureof NS and the presence of exotic phases. The results obtained from the astrophysicalobservations require several theoretical inputs for the interpretation. A coextensive ef-fort from theory and experiments have improved and provided new insights into thefield. The recently observed Gravitational Waves GW170817 [1, 2] and GW190425 [3]have constrained the NS maximum mass and radius. The presence of exotic phases likequarks inside NS has also been observed recently [4]. After the discovery of gravitationalwave GW170817, more theoretical work has been done to understand the relation be-tween EoS and NS properties through various aspects like a phase transition, symmetryenergy [5, 6, 7, 8, 9, 10, 11, 12, 13]. However, there are still numerous fundamentalproblems related to the matter under extreme conditions that are yet to be answered.The most important one is the unified theory which can describe the overall structure nner crust EoS S ( ρ ) andits derivatives ( L sym , K sym , Q sym ) affect the EoS and have been used to constrain theEoS near the saturation density. The density-dependent symmetry energy has a strongcorrelation between the pressure at saturation density and the radius of a NS [17]. It hasbeen shown [18, 19] that the slope parameter L sym is strongly correlated to the neutronskin thickness. A higher L sym value describes neutron matter with higher pressure andhence thicker neutron skin [20, 21, 22]. The curvature parameter K sym determines thecrust-core transition density and the gravitational binding energy of the NS [23]. Theskewness parameter Q sym has a large saturation in its value due to different values fromvarious models. The skewness parameter is related to the nuclear incompressibility ofthe system [24]. All these quantities affect the EoS directly or indirectly.The unified EoS describes the neutron star from its outer crust to the inner core.However, a unified EoS is generally not available. Hence the complete EoS is dividedinto three different phases: the outer crust phase, the inner crust, and the core phase.It has been shown [25, 26] that the neutron star properties like mass and radius do notdepend upon the outer crust EoS, but a particular choice of inner crust EoS and thematching of inner crust to the core EoS is critical and the variations larger than 0.5 kmhave been obtained for a 1.4 M ⊙ . For the outer crust which lies in the density range10 -10 g/cm , the Baym-Pethick-Sutherland (BPS) EoS [27], the Haensel-Pichon (HP)EoS [28] are widely used in the literature. Both these EoSs do not affect the mass andradius of a neutron star. For the matter beyond the neutron drip line ranging fromdensity 10 -10 g/cm , the inner crust EoS follows. The Baym-Bethe-Pethick (BBP)EoS [29] is usually used. However to avoid the large uncertainties in mass and radiusof a neutron star, studies have shown that for the complete unified EoS, the inner crustEoS should be either from the same model as core EoS or the symmetry energy slopeparameter should match [26, 30]. Thus a proper choice of inner crust EoS will determinethe true value of neutron star mass and radius.This paper is organized as follows. In section 2, we explain the relativistic-meanfield formalism to the study of the EoS. The Thomas-Fermi approximation within RMFframework to describe the inner crust part is also discussed. Section 3 discusses thenuclear matter properties for parameter sets used. The unified EoS by combiningthe outer crust, inner crust, and the core EoS will be discussed in section 3. TheNS properties for static as well as rotating neutron stars with different EoSs are alsodiscussed. Finally, the summary and conclusions are outlined in section 4. nner crust EoS
2. Theory and Formalism
In this section, we summarize the formulations of our EoS for nuclear matter. TheRelativistic Mean Field (RMF) theory in which the nucleons interact through the mesonexchange is adopted. The simplest relativistic Lagrangian contains the contribution from σ , ω , and ρ mesons without any cross-coupling between them [31]. The prediction oflarge nuclear matter incompressibility [32] by this model was reduced to an acceptablerange by the addition of self-coupling terms [33]. The addition of other self- and cross-couplings improved the nuclear matter properties. The Effective field theory motivatedRMF (E-RMF) is an extension to the basic RMF theory in which all possible self- andcross-couplings between the mesons are included [34, 35, 36]. The E-RMF Lagrangianused in the present work, which contains the contribution from σ -, ω - mesons upto 4thorder expansion, and ρ - and δ - mesons upto 2nd order is given by [37] L = X α = n,p ¯ ψ α (cid:26) γ µ ( i∂ µ − g ω ω µ − g ρ τ α · ρ µ ) − ( M − g σ σ ) − g δ τ α · δ (cid:27) ψ α + 12 ∂ µ σ∂ µ σ − m σ σ + ζ g ω ( ω µ ω µ ) − g σ m σ M (cid:18) k
3! + k g σ M σ (cid:19) σ + 12 m ω ω µ ω µ − F µν · F µν + 12 g σ σM (cid:18) η + η g σ M σ (cid:19) m ω ω µ ω µ + 12 η ρ m ρ M g σ σ ( ρ µ · ρ µ ) + 12 m ρ ρ µ · ρ µ − R µν · R µν − Λ ω g ω g ρ ( ω µ ω µ )( ρ µ · ρ µ ) + 12 ∂ µ δ ∂ µ δ − m δ δ , (1)where ψ is the nucleonic field and M is the nucleonic mass. m σ , m ω , m ρ , and m δ arethe masses and g σ , g ω , g ρ , and g δ are the coupling constants of σ , ω , ρ , and δ mesons re-spectively. The Euler-Lagrangian equations of motion for the meson fields are obtainedusing the relativistic mean field approximation [8] m σ σ = g σ ρ s ( r ) − m σ g σ M σ (cid:18) k k g σ σM (cid:19) + g σ M (cid:18) η + η g σ σM (cid:19) m ω ω + η ρ M g σ m ω ρ (2) m ω ω = g ω ρ ( r ) − g σ M (cid:18) η + η g σ σM (cid:19) m ω ω − ζ g ω ω − ω ( g ω g ρ ρ ) ω, (3) m ρ ρ = 12 g ρ ρ ( r ) − η ρ g σ σM m ρ ρ − ω ( g ω g ρ ω ) ρ, (4) nner crust EoS m δ δ = g δ ρ s ( r ) (5)where, ρ s ( r ) = X α = n,p < ¯ ψ α γ ψ α > = ρ sn + ρ sp = 2(2 π ) X α (cid:18) Z k α M ∗ α ( k α + M ∗ α ) / d k (cid:19) (6) ρ ( r ) = X α < ¯ ψ α ψ α > = ρ n + ρ p = 2(2 π ) (cid:18) Z k n d k + Z k p d k (cid:19) (7) ρ ( r ) = X α < ¯ ψ α τ ψ α > = ρ p − ρ n (8) ρ s ( r ) = X α < ¯ ψ α τ γ ψ α > = ρ sp − ρ sn (9)are the scalar, baryon, and isovector densities respectively. M ∗ α ( α = n, p ) is theeffective mass of nucleons given by the relation M ∗ n = M − g σ σ + g δ δ (10) M ∗ p = M − g σ σ − g δ δ (11)The expression for the energy density and pressure are obtained from the givenLagrangian using energy momentum tensor relation given by [38] T µν = X i ∂ ν φ i ∂ L ∂ ( ∂ µ φ i ) − g µν L (12)The energy density is the zeroth component and the pressure is the third componentof energy momentum tensor. Their expressions are given as [37] E = 2(2 π ) X α = n,p Z k α d kE ∗ i ( k ) + ρg ω ω + m σ σ (cid:18)
12 + k g σ σM + k g σ σ M (cid:19) − ζ g ω ω − m ω ω (cid:18) η g σ σM + η g σ σ M (cid:19) + 12 ρ g ρ ρ − (cid:18) η ρ g σ σM (cid:19) m ρ ρ − Λ ω g ρ g ω ρ ω + 12 m δ δ , (13) nner crust EoS P = 2(2 π ) X α = n,p Z k α d k k E ∗ i ( k ) − m σ σ (cid:18)
12 + k g σ σM + k g σ σ M (cid:19) + 14! ζ g ω ω + 12 m ω ω (cid:18) η g σ σM + η g σ σ M (cid:19) + 12 ρ g ρ ρ + 12 (cid:18) η ρ g σ σM (cid:19) m ρ ρ + Λ ω g ρ g ω ρ ω − m δ δ , (14)The symmetry energy usually defined as the difference in the energy per nucleonof pure neutron matter and symmetric nuclear matter plays a major role in the nuclearEoS [39]. The symmetry energy and its derivatives have a major impact on neutronstar properties like mass and radius. The symmetry energy is defined as S ( ρ ) = 12 (cid:18) ∂ E ( ρ, β ) ∂β (cid:19) β =0 (15)where, β is the asymmetric parameter which is 0 for symmetric nuclear matter(SNM) and 1 for pure neutron matter (PNM). β = ρ n − ρ p ρ n + ρ p (16)The derivatives of symmetry energy, slope parameter L sym , symmetry energycurvature K sym , and the skewness parameter Q sym are defined at the saturation density ρ as L sym = 3 ρ ∂S ( ρ ) ∂ρ (cid:12)(cid:12)(cid:12)(cid:12) ρ = ρ , (17) K sym = 9 ρ ∂ S ( ρ ) ∂ρ (cid:12)(cid:12)(cid:12)(cid:12) ρ = ρ , (18)and Q sym = 27 ρ ∂ S ( ρ ) ∂ρ (cid:12)(cid:12)(cid:12)(cid:12) ρ = ρ , (19)respectively.The inner crust which contains the inhomogenous nuclear matter is defined byapplying the Thomas-Fermi(TF) approximation [40, 41] within the framework of RMF nner crust EoS L = X α = n,p L α + X i = σ,ω,ρ,δ L i + L e + L γ + L nl , (20)where, L α = ¯ ψ α [ γ µ iD µ − M ∗ ] ψ α , (21) L σ = 12 ( ∂ µ σ∂ µ σ − m σ σ ) , (22) L ω = 12 (cid:16) −
12 Ω µν Ω µν + m ω ω µ ω µ (cid:17) , (23) L ρ = 12 (cid:16) − R µν · R µν + m ρ ρ µ · ρ µ (cid:17) , (24) L δ = 12 ( ∂ µ δ ∂ µ δ − m δ δ ) , (25) L e = ¯ ψ e [ γ µ ( i∂ µ + eA µ ) − m e ] ψ e , (26) L γ = − F µν F µν , (27) L nl = − kσ − λσ + 14 ζ Γ ω ( ω µ ω µ ) + Λ ω Γ ω Γ ρ ω ν ω ν ρ µ · ρ µ , (28)where, Γ σ , Γ ω , and Γ ρ are the density-dependent coupling constants for densit-dependentparameter sets considered. F µν and R µν are the field tensors.The Skyrme type interactions like Lattimer and Swesty EoS [46] and thecompressible liquid drop model has also been used by several authors to describe thenonuniform matter [47, 48, 49].
3. Results and Discussion
To study the effect of crust EoS on neutron star properties, we have chosen severalparameter sets to construct the core EoS. Since the outer crust EoS does not affectthe neutron star maximum mass and radius, therefore the use of several EoSs for outercrust will provide no new information regarding the NS. Hence the BPS EoS [27] hasbeen used for the outer crust part. For the inner crust part, the relativistic mean-fieldmodel with constant couplings, non-linear terms [50], and density-dependent couplings[51] have been used. These include NL3 [52] set with non-linear σ terms, TM1 [53] withnon-linear σ and ω terms, NL3 ωρ [54, 22] which includes the non-linear ωρ terms in ad-dition to the previous couplings, FSU [55] and IU-FSU [56], and the density-dependentDD-ME2 [57] and DD-ME δ [58].The nuclear matter properties at saturation density for the above considered crustEoSs are shown in table 1. The symmetry energy slope parameter L sym of the givensets lies in the range 47-118 MeV. We have considered these sets for the inner crust nner crust EoS Table 1.
Nuclear matter properties for the crust EoS at saturation density ( ρ ), energy( E ), symmetry energy ( S ), slope parameter ( L sym ), incompressibility coefficient ( K ),and skewness parameter ( Q sym ). All the values are in MeV except for the ( ρ ) whichis in fm − . Model ρ E S L sym
K Q sym
NL3 0.148 -16.24 37.3 118.3 270.7 203TM1 0.145 -16.26 36.8 110.6 280.4 -295FSU 0.148 -16.30 32.6 60.5 230.0 -523IU-FSU 0.155 -16.40 31.3 47.2 213.2 -288NL3 ωρ δ σ self-coupling terms to the models with all types of self- andcross-couplings along with the δ meson inclusion. NL3 [52], TM1 [53], IU-FSU [59],IOPB-I [8], and G3 [36] parameter sets are used to study the neutron star core. Allthe coupling constants and the meson masses for the above parameter sets are given in[53, 59, 8].The unified EoSs consisting of the outer crust, the inner crust, and the core areconstructed using the above defined parameter sets. The unified EoS follows as: BPS(for the outer crust)+ BBP, NL3, TM1, NL3 ωρ , FSU, IU-FSU, DD-ME2, and DD-ME δ (for the inner crust)+ NL3, TM1, IU-FSU, IOPB-I, and G3 (for the core). The unifiedEoS without inner crust is also constructed to see the impact of inner crust on NS prop-erties. The different unified and non-unified EoSs produced are shown in figure 1.The green dot in the inset (as well as in the main plot) of figure 1 shows thematching of outer crust with inner crust EoS while the black dot in the main plot rep-resents the crust-core transition point. The NL3 parameter set produces stiff core EoSas compared to other parameter sets. IU-FSU produces soft EoS at low density. G3 setproduces soft EoS as compared to NL3, TM1 and IOPB-I at both low and high energydensities. Among the inner crust EoS, NL3 and TM1 set produce soft EoS at very lowdensity and become stiff as the density increases. IU-FSU crust initially produces stiffEoS but becomes soft at higher energy density ( E ≈
45 MeV/fm ). For outer crust, theBBP EoS is used for all different combinations of inner crust and core EoSs as the outercrust EoS doesn’t affect the mass and radius of a NS. nner crust EoS -1 ε ( MeV/fm )10 -4 -3 -2 -1 P ( M e V / f m ) NL3TM1IU-FSUIOPB-IG30 0.4 0.8 1.2 1.6 20.0000.0010.0010.0020.0020.003 BPSBBPDD-ME2DD-ME δ FSUIU-FSUNL3NL3 ωρ TM1 . Outer Crust . Inner Crust crust-core transition .. Figure 1.
Unified EoS with different inner crust and core EoS. The inset shows thematching of outer crust with the inner crust. The green dot shows the matching pointof BPS EoS with the inner crust. The black dot represents the point of crust-coretransition.
To determine the maximum mass and the corresponding radius of a stationary andspherical neutron stars obtained using different EoSs, we use the Tolman-Oppenheimer-Volkoff (TOV) equations [60, 61]. dP ( r ) dr = − [ E ( r ) + P ( r )][ M ( r ) + 4 πr P ( r )] r (1 − M ( r ) /r ) (29)and dM ( r ) dr = 4 πr E ( r ) (30)Here M ( r ) is the gravitational mass. For the given boundary conditions P (0) = P c , M (0) = 0, with P c being the central pressure, the equations (29) and (30) are solvedto obtain the NS properties. Apart from obtaining the properties of a static NS, wealso see the impact of inner crust on uniformly rotating neutron star. The properties ofuniformly rotating neutron star are obtained using the RNS code [62, 63, 64, 65].Figure 2 shows the mass-radius relation for a static and uniformly rotating NS withNL3 core EoS. The shaded regions represent the constraints on the maximum mass of aNS by pulsars PSR J1614-2230 (1.928 ± M ⊙ [16],PSR J0348+0432(2.01 ± M ⊙ [14], and PSR J0740+6620 (2.04 +0 . − . ) M ⊙ [66], and GW190814 (2.50-2.67 M ⊙ ) [67]. Thegreen arrow represents the radius at the canonical mass of a NS which shows that themaximum value of radius should be R . ≤ .
76 km [9]. The constraints on the maxi-mum mass show that the theoretical prediction of a NS maximum mass should reach the nner crust EoS ≈ M ⊙ . But the recent observation of gravitational wave data GW190814 has asecondary component with a maximum mass in the range 2.5-2.67 M ⊙ . This secondaryobject is considered to be either light black-hole or supermassive neutron star. Fromfigure 2, it is clear that the NS maximum mass produced using different inner crustEoS varies by small margins and lies in the range 2.764-2.787 M ⊙ . The correspondingradius varies from 13.027-13.378 km. However, the radius at canonical mass is muchmore affected than the radius at maximum mass. For a NS without inner crust, theradius at the canonical mass is found out to be R . =14.987 km. With the addition ofthe inner crust, the radius decreases from 14.496-13.853 km. The neutron star with asmall radius at the canonical mass is produced by using NL3 as inner crust EoS whichsatisfies the constraints by GW170817. The NL3 set has higher value of symmetry en-ergy slope parameter L sym = 118.3 MeV,but matches completely with the core EoS andhence forms a unified EoS. Thus, we see that a unified EoS produces a neutron star withsmaller radius at the canonical mass. The other inner crust EoSs have slope parametersmaller than the NL3 set and we see that the low value slope parameter set (IU-FSU)produces a larger radius at the canonical mass as compared to the smaller value L sym set as can be seen in the figure. Thus we see that the R . has a significant relation withthe slope parameter. This is consistent with the work in references [9, 11].Figure (2) also shows the MR profiel for uniformly rotating stars (RNS). Similar tothe Static Neutron stars (SNS), the RNS maximum mass and the corresponding radiusare not much affected by the inner crust EoS, but the radius at 1.4 M ⊙ varies in a similarfashion as SNS in the range R . =19.5-21.4 km.
10 12 14 16 18 20 22 24
R (km) M ( M o ) No CrustNL3TM1FSUNL3 ωρ DD-ME δ DD-ME2IU-FSUBBP . GW190814PSR J0740+6620
PSR J0348+0432PSR J1614-2230
Figure 2.
Mass-Radius profile of SNS and RNS for NL3 core with different inner crustEoS. The solid (dashed) lines represent the rotating neutron stars (static neutron stars)The recent constraints on the mass [16, 14, 66, 67] and radius [9] of NS are also shown. nner crust EoS M ⊙ and 12.234-12.805 km, respectivelyas shown in figure. The radius at the canonical mass varies from 13.572 km for TM1crust EoS to 15.549 km produced without using the inner crust. As seen in the figure,although every EoS for TM1 core along with different inner crust satisfies the mass con-straint from recently observed gravitational wave data, no such EoS satisfies the radiusconstraint. For IU-FSU core EoS, as shown in the figure, the maximum mass and radiusvary from 1.931-1.940 M ⊙ and 11.030-11.263 km, respectively. The radius at the canon-ical mass varies from 12.295 km for IU-FSU crust EoS to 12.778 km withut inner crust,all satisfying the radius constraint. The maximum mass and the radius for RNS withTM1 and IU-FSU core are almost identical with mass in the range 2.57-2.63 M ⊙ satisfy-ing the GW190814 mass constraint and radius around 16 km. The unified EoS in TM1(L crust =110.6 MeV + L core =110.6 MeV) and IU-FSU (L crust =47.2 MeV + L core =47.2MeV) produce a NS with smaller radius for SNS as well as RNS.
10 12 14 16 18 20 22 24R (km)00.511.522.53 M ( M O ) No CrustNL3TM1FSUNL3 ωρ DD-ME δ DD-ME2IU-FSUBBP . GW190814PSR J0740+6620
PSR J0348+0432PSR J1614-2230 a) TM1
10 12 14 16 18 20 22 24R (km)00.511.522.53 M ( M O ) . GW190814PSR J0740+6620
PSR J0348+0432PSR J1614-2230 b) IU-FSU
Figure 3. a) Same as figure 2, but for a) TM1 and b) IU-FSU core EoS.
Figure 4 shows the MR profile for IOPB-I and G3 core EoS. For IOPB-I set, themaximum mass varies from 2.141-2.156 M ⊙ and the radius 11.872-12.029 km. R . variesfrom 13.118-13.508 km. Similar results follow for G3 EoS, where the NS maximum massand the corresponding radius vary slightly with different inner crust EoS. However, asusual, the radius R . varies from 12.436-14.447 km. For RNS, the radius at the canon-ical mass varies from 18.65 to 19.18 km and 17.72 to 20.86 km for IOPB-I and G3 EoS,respectively. It is to be mentioned here that both IOPB-I anf G3 set doesn’t have aunified EoS i.e, inner crust and core EoS with same symmetry energy slope parameter.However, we see that for IOPB-I core EoS, the FSU inner crust with a similar valueof slope parameter as IOPB-I, predicts smaller radius for NS at 1.4 M ⊙ than any othercrust EoS. Similar follows for G3 set ( L sym = 49.3 MeV) where IU-FSU with a slopeparameter value of 47.2 MeV gives a smaller radius neutron star. nner crust EoS
10 12 14 16 18 20 22 24R (km)00.511.522.53 M ( M O ) . GW190814PSR J0740+6620
PSR J0348+0432PSR J1614-2230 a) IOPB-I
10 12 14 16 18 20 22 24R (km)00.511.522.53 M ( M O ) GW190814 . PSR J0740+6620
PSR J0348+0432PSR J1614-2230 b) G3
Figure 4. a) Same as figure 2, but for a) IOPB-I and b) G3 core EoS.
From the above MR relations, we see that the maximum mass and the radius ofboth static and rotating neutron stars do not change by a large amount using the innercrust with different slope parameters. The unified EoSs (NL3, TM1, and IU-FSU)produce neutron stars with a small radius at the canonical mass. However, for non-unified EoSs (IOPB-I and G3), the inner crust EoS from same or different model withsmaller symmetry energy slope parameter L sym predicts a smaller radius at canonicalmass of a NS.The effect of inner crust on the static neutron star tidal deformability λ is alsostudied for all core EoSs. In addition to this, the variation in the RNS properties likeMoment of Inertia (MI) is also discussed. The tidal deformability of a NS is defined as[68, 69] λ = − Q ij E ij = 23 k R (31)where, Q ij is the induced quadrupole mass and E ij is the external field. Thedimensionless tidal deformability then follows from the λ asΛ = λM = 2 k C (32)where, k is the second love number and C = M/R is the compactness parameter. Theexpression for the love number is given as [68] k = 85 (1 − C ) [2 C ( y − n C (4( y + 1) C + (6 y − C +(26 − y ) C + 3(5 y − C − y + 6) − − C ) (2 C ( y − − y + 2) log (cid:16) − C (cid:17)o − . (33)The value of y = y ( R ) can be computed by solving the following differential equation[69, 70] r dy ( r ) dr + y ( r ) + y ( r ) F ( r ) + r Q ( r ) = 0 , (34)where, F ( r ) = r − πr [ E ( r ) − P ( r )] r − M ( r ) , (35) nner crust EoS Q ( r ) = 4 πr (cid:16) E ( r ) + 9 P ( r ) + E ( r )+ P ( r ) ∂P ( r ) /∂ E ( r ) − πr (cid:17) r − M ( r ) − (cid:20) M ( r ) + 4 πr P ( r ) r (1 − M ( r ) /r ) (cid:21) . (36)In the simplest form, the moment of inertia I is defined as the ratio of the angularmomentum to the angular velocity of a NS, I = J/ Ω. For a uniformly rotating neutronstar, the moment of inertia is defined in terms of angular frequency ω as [71, 72] I ≈ π Z R ( E + P ) e − φ ( r ) h − m ( r ) r i − ¯ ω Ω r dr, (37)where, ¯ ω is the dragging angular velocity of a uniformly rotating star, satisfying theboundary conditions¯ ω ( r = R ) = 1 − IR , d ¯ ωdr | r =0 = 0 , (38)The moment of inertia of a NS has been calculated by various people [62, 73, 74],but the variation in the value of MI with different inner crust EoS hasn’t been calculated. O )02004006008001000120014001600 Λ NL3TM1FSUIU-FSUDD-ME δ DD-ME2NL3 ωρ BBPNo crust . . Figure 5.
The relation between dimensionless tidal deformability and the mass ofa NS for NL3 core EoS with different inner crust EoS. The black dot along with thegreen arrow represents the recent constraints on the tidal deformability at 1.4 M ⊙ fromGW170817 data [2] . The grey dashed line represents the upper limit on the Λ . value[1] . Figure (5) shows the variation of the dimensionless tidal deformability with theNS mass for NL3 core EoS with different inner crust EoS. The constraints on the Λfrom the recent gravitational wave data is also shown. The grey dashed line represents nner crust EoS . =190 +390 − [1]. The NL3 unified EoS predicts the lowest value ofthe dimensionless tidal deformability at 1.4 M ⊙ , Λ . =800, due to the small radius atthe canonical mass. The other non-unified EoSs predict a value in the range 800-1400with Λ . =1400 for NL3 without the inner crust. This shows that the unified EoS isimportant in determining the NS properties that support the sontraints from recentgravitational wave data. O )02004006008001000120014001600 Λ O ) .. . . TM1 IU-FSUa) b)
Figure 6.
Same as figure (5), but for a) TM1 and b) IU-FSU core EoS.
Figure (6) shows the dimensionless tidal deformability for TM1 and IU-FSU coreEoS with different crust EoSs. For TM1 set, the unified EoS provides a low value fortidal deformability, Λ . =1060. This value increases upto to Λ . = 1587 for NS withoutinner crust. For IU-FSU, the unified EoS gives Λ . =368, and Λ . =637 without innercrust EoS. For other non-unified EoSs, the variation in the tidal deformability at 1.4 M ⊙ is very small for both TM1 and IU-FSU core EoSs.Figure (7) shows the dimensionless tidal deformability for IOPB-I and G3 core sets.For IOPB-I, the FSU crust EoS predicts the smallest value of Λ . =637, while the othercrust EoSs determine the value in the range 640-730. Similarly for G3 set, the IU-FSUgives a low value for tidal deformability, Λ . =349, while others provide a value in therange 393-450. The value increases with the increase in the value of symmetry energyslope parameter. The NS without inner crust provides a value of Λ . =914 and 620 forIOPB-I and G3 sets respectively.The variation in the moment of inertia for RNS with NL3 core EoS and differentcrust EoSs is shown in figure (8). As clear from the figure, the change in the moment nner crust EoS O )02004006008001000120014001600 Λ O ) . . . . IOPB-I G3a) b)
Figure 7.
Same as figure (5), but for a) IOPB-I and b) G3 core EoS. O )02468 I( g c m ) NL3TM1FSUNL3 ωρ DD-ME δ DD-ME2TM1BBPNo crust0.6 0.8 1 1.211.251.51.752 . PS R J - PS R J + Figure 8.
Variation of moment of inertia I with the mass of a NS for NL3 core EoSwith different crust EoS. The green arrow represents the constraints on the momentof inertia from PSR J0737-3039A [75, 76]obtained from the GW 170817 data analysis[1, 2] . of inertia with different crust EoSs is very small. For unified NL3 EoS, the value of I is calculated as I =1.53 × g cm well satisfying te constraint from PSR J0737-3039A I =1.53 +0 . − . × g cm . For the non-unified EoSs, the moment of inertia increases withthe symmetry energy slope parameter L sym as they predict a large radius.For TM1, IU-FSU, IOPB-I, and G3 core EoSs, the moment of inertia variation with nner crust EoS I =1.31 & 1.29 × g cm , respectively. With nounified EoS available for IOPB-I and G3 sets, the low symmetry energy slope parametercrust EoS provides a lower value of the moment of inertia, I =1.27 & 1.22 × g cm ,respectively. The moment of inertia doesn’t change much in all the cases, which suggestsit’s weak dependence on the slope parameter of the crust EoS. I ( g c m ) O )01234 I( g c m ) O ) . . PS R J - PS R J + TM1 IU-FSUIOPB-I G3a) b)c) d)
Figure 9.
Same as figure (8), but for a) TM1, b) IU-FSU, c) IOPB-I , and d) G3 coreEoSs.
Table 2 shows the deviation in the properties of a static NS like maximum mass,the corresponding radius, and the radius at the canonical mass for the given parametersets. It is clear that the variation in the maximum mass and the corresponding radiusare very small for EoSs, but the radius at 1.4 M ⊙ is highly impacted by the inner crustEoS. For NL3 core EoS, the variation in the radius at R . is maximum for NL3 innercrust ≈ R . , forIOPB-I and G3 core EoSs is with the FSU and IU-FSU inner crust EoS, respectively.Such large deviations in the radius, R . show that a proper choice of inner crust EoSis important to calculate the mass and radius of a NS with small uncertainities in thesevalues. The unified EoSs for NL3, TM1, and IU-FSU predict small values of radius, tidaldeformability, at the canonical mass. For RNS, the radius at 1.4 M ⊙ varies with differ-ent crust EoS. The moment of inertia doesn’t vary largely. For the core EoSs withouthaving a unified EoS, the crust with a smaller symmetry energy slope parameter L sym predicts small values of radius, the tidal deformability for SNS and radius, moment ofinertia for RNS. Thus the crust-core transition allows the construction of a stellar EoSand a precise measurement of the NS properties for both static and rotating stars.The constraints on the inner crust EoS of a NS and the proper matching of inner nner crust EoS Table 2.
Variation (in percent) in the maximum mass, corresponding radius, and theradius at 1.4 M ⊙ of a NS without inner crust and the corresponding inner crust. Model BBP IU-FSU DD-ME2 DD-ME δ NL3 ωρ FSU TM1 NL3∆ M max R max R . M max R max R . M max R max R . M max R max R . M max R max R . L sym predicts a smaller radius at 1.4 M ⊙ as compared to the one with large L sym . The same trend is followed by all parametersets with unified EoS. For non-unified EoS, the crust from same or different model butwith smaller symmetry energy slope parameter gives a low radius NS. While the innercrust does affect the radius of RNS, the moment of inertia varies only by a small factor.
4. Summay and conclusion
The NS properties like mass and radius were investigated using the relativistic mean-field (RMF) model. To study the effect of symmetry energy and its slope parameter ona neutron star, we used inner crust EoSs with different symmetry energy slope parame-ters. For the outer crust, the BPS EoS is used for all sets as the outer crust part doesn’taffect the NS maximum mass and radius. For the inner crust part, we used NL3, TM1, nner crust EoS ωρ , DD-ME δ , DD-ME2, and IU-FSU parameter sets whose slope parametervaries from 118.3-47.2 MeV. For the core part, NL3, TM1, IU-FSU, IOPB-I, and G3parameter sets are used. The unified EoS are constructed by properly matching theinner crust EoS with outer crust and core EoS. The EoSs constructed for the sphericaland symmetrical NS under charge neutral and β -equilibrium conditions ware taken asthe input into the TOV equation to obtain NS properties. It is seen that although theNS maximum mass and the corresponding radius do not change by a large amount,the radius at the canonical mass, R . are largely impacted by using inner crust EoSswith different symmetry energy slope parameter. By varying the slope parameter fromlow to high values, the radius R . also increases. Different parameter sets for core EoSare used to see if they predicted a different behavior between R . and L sym . Also, thevariation in the NS maximum mass, radius, and the radius at 1.4 M ⊙ are calculated andthe variation of about 2 km is found in the radius at the canonical mass. The prop-erties like mass, radius, and the moment of inertia of uniformly rotating stars are alsocalculated using same EoSs. It is seen that similar to SNS, the maximum mass and thecorresponding radius do not vary much, but the radius at the canonical mass is affectedby the slope parameter. The moment of inertia doesn’t vary too much with change inthe symmetry energy slope parameter L sym .There are several different aspects that need to be further studied in the currentwork. A unified EoS for the parameter sets like IOPB-I and G3 with both crust andcore part described by the same model with different slope parameter L sym will be abetter investigation to see how the radius at canonical mass behaves. The temperaturedependent variation in the NS properties like mass, radius, and the tidal deformabilitywith different symmetry energy slope parameter inner crust will provide more newinsights into this work. References [1] Abbott B P and et al (LIGO Scientific Collaboration and Virgo Collaboration) 2017
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