Multilayer neutron stars with scalar mesons crossing term
MMultilayer neutron stars with scalar mesons crossing term
Sebastian Kubis ∗ and W(cid:32)lodzimierz W´ojcik Institute of Physics, Cracow University of Technology, Podchor¸a˙zych 1, 30-084 Krak´ow, Poland
Noemi Zabari
Henryk Niewodnicza´nski Institute of Nuclear Physics,Polish Academy of Sciences,Radzikowskiego 152, 31-342 Krak´ow, Poland
It is shown that recently proposed RMF model with σ and δ meson interaction agrees with theobservational data and presents an interesting structure with phase transition in the outer part ofneutron star core. I. INTRODUCTION
The internal structure of neutron stars (NS) is one ofthe most enigmatic subjects of nuclear astrophysics. Thelatest observation of the neutron-star merger, associatedwith gravitational waves detection, in August 2017 (Ad-vanced LIGO/Virgo), revealed details on the mass andradius of neutron star determination. The event is knownas GW170817 [1]. The estimations coming from thisevent have made the constraints on the equation of state(EOS) of nuclear matter by simultaneous determinationof mass and radius of the newly born star. Precise massmeasurements, however, without any information aboutstellar radius, come from rotating neutron star. Neutronstar of the mass 1 . ± . M (cid:12) (PSR J1614-2230) [2]and 2 . ± . M (cid:12) (PSR J0348-0432) [3] has been re-ported in a binary system with a white dwarf. In 2018,the pulsar of the mass 2 . ± . M (cid:12) (PSR J2215-5135)was identified using Fermi Large Area Telescope (FermiLAT) by spectral lines analysis [4]. A feasible way to ob-tain maximum masses of the neutron star bigger than twosolar masses is to retain befittingly stiff EOS. However,typical stiff EOS makes star less compact [5] which wouldbe in contradiction to the radius estimation coming fromGW170817. The typical soft EOS gives relatively smallerradius but might never reach the mass above 2 M (cid:12) . Inorder to reconcile those two opposite tendencies, we pro-pose a model giving the EOS, which is soft in low density(slightly above n ) and stiff at higher densities relevantfor the inner part of the NS core.In this paper, we show the EOS results for the model,which includes the σ − δ meson crossing term. Modifi-cation of this type have been recently proposed and an-alyzed, see[6, 7]. The calculations are performed in theframework of the relativistic mean field (RMF) theory.The inclusion of such types of interactions leads to su-persoft symmetry energy behavior and slope values thatare consistent with most recent studies of nuclear matterproperties. Here, we show results for the neutron starsthat utterly conform to the mass and radius constraintsmentioned above. ∗ [email protected] II. MODEL
The base of the RMF theory gives adequate La-grangian, which includes nucleons interacting throughthe exchange of four mesons. Two isoscalar mesons σ , ω and two isovector ρ , δ mesons. In this work, the proposedLagrangian is enriched by a new kind of meson-meson in-teraction e.i. the σ − δ meson crossing term [6].Here, we are limiting the demonstration of the modelonly to the most relevant parts. Due to the inclusion of σ − δ meson interaction, the corresponding part − ˜ g α σ α (cid:126)δ is added to the Lagrangian. The α exponent distin-guishes between two types of meson-meson interactions: α = 1 for linear and α = 2 for quadratic ones. This termmodifies the equation of motion and leads to an addi-tional contribution to the total energy density of matter (cid:15) σδ = ˜ g α σ α δ , (1)where δ denotes mean value of the third component ofthe (cid:126)δ field. In the framework of RMF, for the consid-ered model, there are seven free model parameters rep-resented by the following coupling constants C σ , C ω , C ρ , C δ , b , c and g α . In the model parametrization, it is moreconvenient to use g α = ˜ g α g ασ g δ instead of ˜ g α , where g σ and g δ are meson-nucleon coupling constants appearingin the standard RMF Lagrangian. For more details werefer to the work [6]. Isoscalar sector consist of C σ , C ω , b , and c coupling constants which values are attainedto reproduce the symmetric nuclear matter. In our cal-culations we use n = 0 .
16 fm − , B = −
16 MeV and K = 230 MeV. We set the value of constant C σ to be11 MeV that gives sufficiently stiff EOS which is com-patible with observations of PSR J2215-5135 and PSRJ0740-6620. In the isovector sector, C ρ , C δ and g α con-stants are chosen to obtain symmetry energy at satura-tion S ( n ) = 30 MeV and acceptable value of the sym-metry energy slope L = 3 n dS dn | n . Recent experimentaldata suggest a rather low slope, in the range between 40and 80 MeV. Such low values are obtained as a resultof the crossing term Eq.(1), if and only if ˜ g α (and thesame g α ) takes negative values. Moreover, it was foundthat in model family with quadratic coupling, the phasetransition at the low-density region emerges and leads a r X i v : . [ nu c l - t h ] J u l n [ fm - ] x μ n μ p n q > n q < x n e u t r K μ < FIG. 1. Phase coexistence diagram for the model withquadratic interaction α = 2 and C δ = 3 . . The thickline is the binodal curve with marked points being in chemicaland mechanical equilibrium. The gray area denotes unstablephases - the spinodal region. to a characteristic softening of the EOS. Whereas in themodel family with linear coupling, the phase transitiondoes not occur.For a given nuclear model, we build the EOS of nuclearmatter in β -equilibrium. The chemical potentials of eachparticle species satisfy the β -equilibrium equality µ e = µ µ = µ n − µ p , (2)where n , p , e , µ stand for neutrons, protons, electrons andmuons respectively. In order to control the slope value L , we manipulate the constant C δ while g α is fixed as itis shown in Table I - the higher C δ the lower value of theslope. A performed analysis revealed the occurrence ofphase transition for high enough C δ . Indeed, such mod-els appear to have negative values of incompressibility K µ , which signals a split into a two-phase system for thedensities where K µ < K µ = (cid:18) ∂P∂n (cid:19) µ (3)represents incompressibility under constant chemical po-tential of charge ( µ ≡ µ e = µ µ ). It is the proper quantityto determine the region where the system ceases to be sta-ble against charge fluctuations and has to split into twophases with opposite charge density [8]. The coexistenceof those two phases takes place when their pressures andchemical potentials for particles being present in both phases are equal to each other: P ( n I , x I ) = P ( n II , x II ) µ i ( n I , x I ) = µ i ( n II , x II ) , i = n, p, e. (4)These equations, called Gibbs conditions, ensure the me-chanical and chemical equilibrium between phases. Theylead to the so-called Gibbs construction, which is shownon the proton fraction versus density diagram, Fig. 1.The spinodal region, marked with a gray area, representsthe points where incompressibility is negative. Thereinno stable phase arises . The Gibbs conditions Eqs. (4) arefulfilled for pairs of points for which µ p and µ n -contoursintersect on the n − x space. These points form the curvecalled the binodal line. In the Fig.1, the line labeled x neutr represents the locally neutral phase. Above thatline all phases are positive ( n q > n q < w ( n ( I ) p − n ( I ) e − n ( I ) µ ) + (1 − w )( n ( II ) p − n ( II ) e − n ( II ) µ ) = 0 , (5)where w is the volume fraction occupied by the I-st phase, w = V ( I ) / ( V ( I ) + V ( II ) ). The Gibbs conditions Eq.(4)and global charge neutrality Eq.(5) allow the finding ofthe region of phase space where two separated phasesoccur and uniquely determine the equation of state, e.i.the pressure versus energy density relation, P ( ε ). It isworth noting that the phase separation appears for thequadratic interaction model with the most plausible valueof slope L . For presented model with C δ = 3 . and α = 2, the slope is L = 55 . C δ . The phaseseparation usually leads to the softening of the EOS incomparison to the one-phase system. This phenomenonis well visible in Fig.2 at which a set of EOSs for dif-ferent nuclear models is shown. The phase separationdramatically changes the slope of the P ( ρ ) relation forthe coupling C δ exceeding 3 . , which corresponds tothe slope L smaller than 68 MeV. The derivative of thepressure with respect to density determines the soundvelocity in the matter v s = (cid:115) ∂P∂ρ . (6)In the lower panel in Fig.2 the sound speed is shown in amoderate range of densities. The sudden decrease of v s corresponds to the phase transition region in which thespeed of sound drops to around 10 − c . III. MULTILAYER NEUTRON STARS
The set of EOSs was constructed for the quadraticmodel for the C δ coupling within 1.0 to 3.8 fm . The L [ MeV ] P [ dyn / c m ] × × × × ρ [ g / cm ] s / c FIG. 2. The EOS and speed of sound for four different C δ couplings. Corresponding symmetry energy slope L are givenin the legend. corresponding values of the symmetry energy slope L arefrom 88.5 to 45.6 MeV. The proposed nuclear RMF modeldoes not cover the region of very low densities, where theneutron star crust is formed. For the crust, the Sly4model was used [9, 10]. The joining of crustal EOS andthe EOS for the core is carried out at the point wheretheir pressure and density are equal.The structure of non-rotating NS is obtained by imple-menting given EOSs into Tolman-Oppenheimer-Volkoff(TOV) equations P (cid:48) ( r ) = − G ( (cid:15) ( r ) + P ( r )) m ( r ) + 4 πr P ( r ) r ( r − Gm ( r )) ,m (cid:48) ( r ) = 4 πr (cid:15) ( r ) , (7)where P and ε pressure and energy density, m ( r ) is thegravitational mass (in units of energy) confined insidea sphere with radial coordinate r . The TOV equationsreveal the nontrivial structure of star for the EOS withphase transition. The matter in the region of phase sep-aration resembles the structure of inner crust - clusterswith high proton fraction immersed in low proton en-vironment or pure neutron matter and has solid-state properties. Such a layer is separated from the actual starcrust by a thin layer of homogeneous liquid matter. Con-cluding, the star includes the four different layers, twosolid-like and two liquid ones. In Fig. 3 the subsequentlayers with different properties are shown. Such a mul-tilayer structure of neutron star requires further analysisin the context of rotational and vibrational properties.The solid-like internal layer should somehow manifest inthe pulsar glitching or precession.The fundamental relation between stellar parametersis the mass-radius relation. The M − R relation for various C δ couplings is presented in Fig. 4.For all considered couplings, the maximum mass is wellabove 2 M (cid:12) , which is consistent with most recent ob-servational data [2–4]. In particular, the models with Lbelow 60 MeV are in the best agreement with the mostmassive pulsar PSR J0740+6620. As the observationsof binary systems allowing for determining the mass ofthe neutron stars well enough, the precise measurementsof stellar radius of a star with a known mass are stillnot attainable. Promising results were derived from theGW signal coming from the binary NS merger [1]. Theseresults suggest a rather small radius of the star, whichindicates soft EOS. However, typical soft EOSs lead to M max lower than those for the most massive PSR. Aspecific EOS is required to find a balance between thesetwo opposite facts. It should have different behavior indifferent ranges of density. The stiffness of the EOS athigher densities controls the maximum mass of a star,whereas at lower densities (closer to the crust) refers tothe stellar radius. The results clearly show that consid-ered models, with scalar-scalar meson interactions , makethe EOS softer at moderate densities and stiffer at higherdensities. This type of EOS comes from a specific formof the symmetry energy being very soft at a lower den-sity region. This effect is most visible for the quadraticcoupling model with L around 50 MeV. It was alreadyshown by Fattoyev et al. [11] that soft symmetry energy r [ km ] ρ [ g / c m ] - ph a s e c r u s t M / M ☉ = FIG. 3. The density profiles for different stellar masses forthe model with L = 45 . is favorable regarding the results of GW170817. It wasconcluded from the analysis of the tidal deformability ofbinary components during the merging. The tidal de-formability is sensitive to the compactness ( M/R ) of astar and thus it is possible to find upper bound for theradius for a given mass. Another constraint for the stel-lar radius comes from the threshold mass for the promptcollapse to the black hole after the merging. Bauswein etal. [12] inferred that radius of a star with M = 1 . M (cid:12) cannot be smaller than 10.7 km. Both, upper and lowerboundaries for the radius are shown in the Fig. 4. As onemay see, the EOS for all L are placed within the allowableregion. However, the models with L <
60 MeV have the M max being in better agreement with the present con-straints from PSR mass measurement. The mass-radiusrelation for the EOSs with phase transition shows charac-teristic bending. For masses smaller than 0 . M (cid:12) the stel-lar radius decreases with mass. For such low masses, thecentral density is located in the two-phase region wherethe EOS is very soft. When the stellar mass is greater,the central density enters into a homogeneous matter re-gion where the EOS becomes much stiffer. Gravitationis no longer able to compress the matter in the core, andthe further increase of the star mass causes an increase ofthe star radius. In this paper, we also show the influenceof σ - δ meson interactions on the URCA process in npeµ matter. The direct URCA process plays a crucial rolein the cooling history of a of a neutron star. Applying L [MeV]88.568.155.445.6PSR J0348 + + M URCA
10 12 14 16 180.00.51.01.52.02.5 R [ km ] M / M ☉ Bauswein et al.
Fattoyev et al.
FIG. 4. Mass-radius profiles for various C δ corresponding todifferent L . The gray bands indicate the observed masses withtheir uncertainties. The horizontal segments present a regionof forbidden values of radius from GW180718 analysis. L [ MeV ] Δ R [ k m ] PSR Vela constraint0.0 0.5 1.0 1.5 2.0 2.5012345 M / M ☉ Δ I / I [ % ] FIG. 5. The crust thickness and crustal moment of inertia asa function of stellar mass for different nuclear models. dURCA threshold given by Eq.(8) (cid:18) x − (cid:16) (1 − x ) / − x / (cid:17) (cid:19) / = (cid:16) (1 − x ) / − x / (cid:17) − (cid:18) m µ (3 π n ) / (cid:19) , (8)the x urca and corresponding n urca is found. In Fig. 4the mass of a stars with the central density exceedingthe threshold n urca are indicated. A recent analysis ofthe cooling story of transient system MXB 1659-29 [13]strongly suggests the presence of rapid cooling of a neu-tron star core via the direct URCA. The direct URCA isonly allowed when the symmetry energy increasing withthe density sufficiently fast. In our model family, thesymmetry energy dependence is the opposite. More thanthat, it may decrease with the density [6]. However, thisbehavior changes at some point, and again, the symme-try energy rapidly increases, making the proton fractionsufficiently large to ensure dURCA. That is why the crit-ical stellar masses for dURCA (Fig. 4) are still in thetypical range of neutron star masses.Other parameters describing neutron star propertiesare associated with their crust. These are the crust thick-ness ∆ R and the crustal fraction of the total moment ofinertia ∆ I/I . They have particular significance due tothe possibility of their reliable observational evaluationfrom thermal relaxation of the crust and pulsar glitches[14]. A glitch phenomenon is a sudden increase in ro-tational frequencies that many pulsars exhibit. Theiranalysis based on the average rate of angular momentum
TABLE I. Basic neutron star properties for quadratic modelfamily C δ [fm ] 1.0 3.0 3.5 3.8 L [MeV] 88.5 68.1 55.4 45.6 M max /M (cid:12) R max [km] 11.23 11.51 11.61 11.71 n max c [fm − ] 1.001 0.932 0.896 0.865 n urca [fm − ] 0.315 0.358 0.364 0.360 M urca /M (cid:12) transfer between the superfluid component and the crust[15] allows for estimating the crustal moment of inertiacompared to the total moment of inertia of a star. Thus,the glitching phenomenon validates the crust-core tran-sition being of particular importance. The more recentstudy of cumulative angular momentum transfer of Velapulsar [16] points out that crust should encompass morethan 1 .
6% of the total moment of inertia. The total mo-ment of inertia I and its crustal fraction ∆ I have beencomputed using the formulas given in [17].The total moment of inertia I and its crustal frac-tion ∆ I has been computed using the formulas given in[17]. The results are shown in the Figure 5. A promis-ing method for simultaneous measurement of the massand radius of a neutron star are based on observations ofthe quiescent mode in the low mass X-ray binaries [18].Precise measurement of M and R are still out of thereach. The uncertainties for both of these quantities arequite significant. However, combined results from manyobserved LMXB shows an interesting tendency that ob-served stellar radius increases with mass [19]. This kindof relation between radius and mass is typical for EOS being soft in the low-density region and becoming stifferat higher density. Such EOS is achieved in our model forlow value of the symmetry energy slope L , see Fig. 4. IV. SUMMARY
In this work, we have shown the properties of neu-tron stars obtained in the framework of the recentlyproposed RMF model of nuclear interactions where thescalar mesons crossing term was applied [6]. Thanks tothis new type of coupling, the symmetry energy slope L may reach sufficiently low values, as terrestrial experi-ments results from heavy-ion collisions suggest. Simul-taneously, the model leads to the neutron stars whichproperties are in agreement with the most recent obser-vational data concerning the NS masses and radii. Al-though not so convincing, the cooling data suggests thatfast cooling by direct URCA cycle is present in neutronstars. Typically, the low values of symmetry energy blockthe URCA cycle by the low proton abundance in the NScore. It makes, therefore, the fast cooling questionable.However, the proposed model has interesting propertiesthat, despite low symmetry energy slope, can still achieveproton fraction sufficiently large to ensure the presenceof the direct URCA for stars with typical mass.An additional outcome of the scalar meson couplingsleads to phase separation in the outer part of the NScore. One may expect that nuclear matter with sepa-rated phases will acquire the solid-state properties. Thepresence of such secondary crust in the NS interior wouldhave interesting implications requiring further analysis. [1] B. Abbott et al. [LIGO Scientific and Virgo], Phys. Rev.Lett. ,1081 (2010).[3] J. Antoniadis et al. , Science , 6131 (2013).[4] M. Linares, T. Shahbaz, and J. Casares, Astrophys. J. , 54 (2018).[5] M. Oertel, M. Hempel, T. Kl¨ahn, and S. Typel, Rev.Mod. Phys. , 015007 (2017).[6] N. Zabari, S. Kubis, and W. W´ojcik, Phys. Rev. C , 025801 (2007).[9] E. Chabanat, P. Bonche, P. Haensel, J. Meyer andR. Schaeffer, Nucl. Phys. A , 231 (1998).[10] F. Douchin and P. Haensel, Astron. Astrophys. , 151(2001).[11] F. Fattoyev, J. Piekarewicz, and C. Horowitz, Phys. Rev. Lett. L34 (2017).[13] E. F. Brown, A. Cumming, F. J. Fattoyev, C. Horowitz,D. Page, and S. Reddy, Phys. Rev. Lett. , 109(2007).[15] B. Link, R. I. Epstein, and J. M. Lattimer, Phys. Rev.Lett. , 3362-3365 (1999).[16] N. Andersson, K. Glampedakis, W. Ho, and C. Espinoza,Phys. Rev. Lett. ,1775 (2009).[19] F. ¨Ozel and P. Freire, Ann. Rev. Astron. Astrophys.54