Isospin asymmetric nuclear matter and properties of axisymmetric neutron stars Partha Roy Chowdhury1,∗ Abhijit Bhattacharyya1,† and D. N. Basu2‡ 1Department of Physics, University of Calcutta, 92 A.P.C. Road, Kolkata-700 009, India 2Variable Energy Cyclotron Centre, 1/AF Bidhan Nagar, Kolkata-700 064, India (Dated: January 7, 2011) Pure hadronic compact stars, above a limiting value (≈1.6 M⊙) of their gravitational masses, to whichpredictionsofmost ofotherequationsofstate(EoSs)arerestricted,canbereachedfrom the equation of state (EoS) obtained using DDM3Y effective interaction. This effective interaction is foundtobequitesuccessfulinprovidingunifieddescriptionofelasticandinelasticscattering,various radioactivitiesandnuclearmatterproperties. Wepresentasystematicstudyofthepropertiesofpure hadroniccompact stars. Theβ-equilibratedneutronstarmatterusingthisEoS withathincrustis able to describe highly-massive compact stars, such as PSR B1516+02B with a mass M=1.94+0.17 −0.19 M⊙ and PSR J0751+1807 with a mass M=2.1±0.2 M⊙ to a 1σ confidencelevel. 1 1 PACSnumbers: 21.65.Cd,21.65.Ef,26.60.Kp 0 2 n The theoretical study of the nuclear equation of state Weinvestigatetheimpactsofthecompressionmodulus a isafieldofresearchwhichties togetherdifferentareasof andsymmetryenergyofnuclearmatteronthemaximum J physics. Nuclear EoS is of great interest as its features mass of NSs in view of the recent constraints from the 6 control the stability of neutron star (NS), the evolution isospin diffusion in heavy-ion collisions at intermediate of the universe, supernova explosion, nucleosynthesis as energies [6, 7]. In the present work, the density depen- ] h wellascentralcollisionsofheavynuclei. Extensivestud- dent M3Y effective interaction (DDM3Y) [8] which pro- t ies in the past two decades of nuclear matter created videsaunified descriptionofthe elasticandthe inelastic - l at subnormal or supernormal density in heavy ion col- scattering [9, 10], various radioactivites [11–15] and nu- c lisions have resulted in experimental constraints on the clear matter properties [16–18], is employed to obtain u nuclear EoS of symmetric matter. Recent astrophysi- EoS of the β equilibrated NS matter. This EoS is used n [ cal observations of massive neutron stars and heavy-ion to carry out a systematic study of the static as well as data are confronted with our present understanding of rotating NSs in view of the recent observations of the 2 the EoS of dense hadronic matter. We argue that the massive compact stars. v data from massive neutron stars and pulsars provide an As mentioned above, the nuclear matter EoS is calcu- 2 0 importantcross-checkbetweenhigh-density astrophysics latedusingtheisoscalarandtheisovectorcomponentsof 9 and heavy-ion physics. The density dependence of nu- M3Yinteractionalongwithdensitydependence. Theen- 1 clear symmetry energy (NSE) obtained by using nuclear ergy per nucleon for isospin asymmetric nuclear matter . EoS plays an important role for modelling the structure is given by 2 0 of the neutron stars (NSs) and the dynamics of super- 0 nova explosions since a series of observables (e.g. slope 3h¯2k2 ρJ C 1 L of NSE, the value of NSE at nuclear density etc.) can ǫ(ρ,X)=[ F]F(X)+( v )(1−βρn) (1) v: be determined from the knowledge of symmetry energy. 10m 2 i Thestiffnessofthehigh-densitymattercontrolsthemax- where X=ρn−ρp is the isospin asymmetry with ρ , ρ X imum mass of compact stars. New measurements of the ρn+ρp n p and ρ=ρ +ρ being the neutron, proton and nucleon n p r propertiesofpulsarspointtowardslargemassesandcor- a number densities respectively, kF is Fermi momentum respondingly to a rather stiff EoS [1] characterized by in case of SNM, F(X)=[(1+X)5/3+(1−X)5/3] and J rep- symmetric nuclear matter (SNM) incompressibility 250- 2 v resents the volume integrals of the isoscalar and the 270 MeV or more. In a recent analysis of x-ray burster EXO 0748-676 (M= 2.10±0.28 M ) it is even claimed isovector parts of the M3Y interaction. The details of ⊙ the present methodology may be obtained in Ref.[16]. that soft nuclear EoS are ruled out [2]. In addition, it is However, for solving the Einstein’s equations for stellar arguedinRef.[2]thatcondensatesanddeconfinedquarks structure, we need to consider the total energy density [3] may not exist in the cores of NSs. Recently, new ob- including mass (also called, the mass-energy density) ε servations for the mass and the radius of compact stars which is related to the ǫ and baryon number density ρ have been obtained which provide stringent constraints as ε = ρ(ǫ+m) where m (∼ 938.919 MeV) is the aver- on the EoS of strongly interacting matter at high densi- age of the neutron and proton masses in MeV unit. As ties [4, 5]. the saturationenergy per nucleon ǫ =−15.26MeV and 0 the saturation density ρ =0.1533 fm−3 are used in this 0 work, the corresponding total energy density at satura- ∗ [email protected] tionisε0=141.597MeV/fm3=2.524×1014gm/cm3. Ob- † [email protected] viously, as these two parameters ǫ and ρ are extracted 0 0 ‡ [email protected] on the basis of information on finite nuclei, they put the 2 400 DD-F DD-F 200 KVOR 300 KVOR V) DBHF V) DBHF e This Work e M M This Work 100 200 (m A ( y Es E/ 100 0 -50 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 ρ -3 ρ -3 (fm ) (fm ) FIG. 1. (Color online) (a) (Left) Density dependent behavior of symmetry energy (Esym vs. ρ) determined by different approaches. (b) (Right) The EoS determined by different models (E/A vs. ρ) are shown for beta equilibrated charge neutral neutron star matter. two constraints to the high density behaviour of any nu- DBHFandourEoSare0.1469,0.1600,0.1810and0.1533 clear matter EoS. The pressure-density relationship for fm−3 respectively. SotheDBHFusesconsiderablylarger the present EoS is consistent with the experimental flow density whereas the ρ used by other two models are 0 data[19]confirmingits highdensitybehaviour. The val- slightlydifferentfromtheexperimentallyextractedvalue ues of other important quantities, L, K , K and of 0.1533 fm−3. The values of NSE at saturation den- sym asy K , defined and calculated in Refs.[17, 18], also agree sitycalculatedbyDD-F,KVOR,DBHFandourEoSare τ extremely well with those extracted from experiments. 31.6, 32.9, 34.4 and 30.71 MeV respectively. It is clear thatDBHFslightlyoverestimatesthevalueofNSEatρ The NSE given by Esym(ρ) = ǫ(ρ,1)− ǫ(ρ,0) has a which is roughly around 30 MeV. 0 value of 30.71±0.26 MeV at the saturation density ob- AnegativeNSEathighdensitiesimpliesthatthepure tained from this calculation which satisfies one of the neutron matter becomes the most stable state. Conse- constraints on the high density EoS. At higher densi- quently, pure neutron matter exists near the core [20] of ties the NSE (see Fig.1(a)) using DDM3Y interaction peaks at ρ ≈ 1.95ρ and becomes negative at ρ ≈ 4.7ρ . the NSs. Although the present EoS is ‘stiff’ since the 0 0 SNM incompressibility K = 274.7±7.4 MeV, but the In Fig.1 the symmetry energy and energy per nucleon ∞ NSE is ‘super-soft’because it increasesinitially with nu- (E/A) are plotted as a functions of the baryon density. cleonicdensityuptoabouttwotimesthenormalnuclear Theequationofstatefortheβ-stablechargeneutralneu- density and then decreases monotonically (hence ‘soft’) tron star matter (see Fig.1(b)) is calculated numerically andbecomes negative (hence ‘super-soft’)athigher den- using Eq.(1) with β-equilibrated proton fraction deter- sities[16,17]. Thisisconsistentwiththerecentevidence mined from the NSE. These plots compare the symme- fora soft NSE atsuprasaturationdensities [21] andwith try energy functions and the EoSs determined by the thefactthatthesuper-soft[22]nuclearsymmetryenergy present calculation and several relativistic models (e.g. is preferred by the FOPI/GSI experimental data on the DD-F, KVOR, DBHF) [4] for the neutron star matter. π+/π− ratio in relativistic heavy-ion reactions for the InFig.1(a),theNSEcalculatedbythephenomenological stability of NSs. RMF models using density dependent masses and cou- pling constants (e.g. DD-F, KVOR) goes on increasing Theβ equilibriumprotonfractionx [23]ofaNScon- β withdensityandneverbecomenegative. Intherelativis- sisting of neutrons (n), protons (p) and electrons (e) is ticDirac-Brueckner-Hartree-Fock(DBHF)approach,the completely governed by the density dependent behavior NSE increases more rapidly with density indicating very of NSE. Contrary to the relativistic models like DD-F, large proton fraction at higher density. This shows an KVOR, DBHF etc. this work does not support the fast opposite trendto the NSE function determinedfromour cooling via direct nucleon URCA process as the maxi- EoS. The saturation density (ρ ) used in DD-F, KVOR, mum x is 4.4% only. Recently it has been concluded 0 β 3 2 ) 2 s ) s s a s m a m r Static a r ol 1 Rotating a Static l S o 1 ( DD-F S Rotating M KVOR ( DD-F M DBHF KVOR 0 DBHF 0 0.5 1 1.5 0 ρ -3 0 4 8 12 16 20 (fm ) c R (km) FIG. 2. (Color online) (a) (Left) The variation of NS masses (M) is shown with central density (ρc). The results of this calculation are denoted as ‘Static’ and ’Rotating’ for non-rotating and rotating NSs at the Keplerian limit. (b) (Right) Mass- Radius relationship is given for static and rotating stars at Keplerian frequency using this EoS. The other three plots (DD-F, KVOR,DBHF) present thesame relationship for static star only. theoretically that an acceptable EoS of asymmetric nu- rotatingstellarstructurearesolvedbyusingourEoSfol- clear matter shall not allow the direct URCA process lowingtheprocedureadoptedbyKomatsu,Eriguchiand to occur in NSs with masses below 1.5 solar masses [4]. Hachisu [27, 30]. We choose the ‘rns’ code written by N. However, the possibility of fast cooling [24, 25] via di- Stergioulas [31] in calculating rotating as well as static rect hyperon URCA or any other processes enhancing NS properties. neutrino emissivities like π− and K− condensates may In Fig.2(a), we have shown the mass of the stars as a not be completely ruled out. Also a recent experimen- function of central baryon density (ρ ). Our results are c tal observation suggests [26] high heat conductivity and plottedforthestaticandKeplerianlimit. Thisisobvious enhanced core cooling process indicating the enhanced fromthe Fig. 2(a)thatfor the samemasscomparatively level of neutrino emission, which may be due to Cooper less central density appears for the rotating stars due pairing. Further theoretical studies and sufficient obser- to centrifugal action. It may be noted that as angular vationaldataareneededtoshedsomelightonthecooling frequency (Ω) becomes greater, the structure of NS [32] phenomenon of NS. gets changed not only because of centrifugal flattening, Letus nowexplorethe variouspropertiesofstatic and butalsobecauseitistakingplaceagainstthebackground rotating NSs using the proposed EoS. To study the ro- of a radially dependent frame dragging frequency. For tating stars the following metric is used: comparisonwehavealsoplottedtheresultsofthreeother EoSs as mentioned earlier in the text. The maximum ds2 =−e(γ+ρ)dt2+e2α(dr2+r2dθ2) (2) mass for the static case is about 1.92 M⊙ with radius ∼ 9.7 km and for the rotating case it is about 2.27 M +e(γ−ρ)r2sin2θ(dφ−ωdt)2 with radius ∼ 13.1 km. So a mass higher than 1.92 M⊙ ⊙ wouldruleoutastaticstarasfarasthisEoSisconcerned. wherethe gravitationalpotentialsγ,ρ,αandω arefunc- The phenomenological RMF models DD-F and KVOR tions of polar coordinates r and θ only. The Einstein’s predictmaximummass aroundtwice solarmass for non- field equations for the three potentials γ,ρ,α have been rotating star. The relativistic DBHF model calculates solved using Green’s function technique [27–29] and the themaximummass∼2.33M andtherefore,theDBHF fourth potential ω has been determined from other po- ⊙ predictsmassiveNSevenforstaticcase. Thisisalsoclear tentials. All the physical quantities may then be deter- fromtheFig.2(b)wherethemass-radiusrelationshipsfor mined from these potentials [30]. The matter inside the all the above EoSs are shown. NS is approximated as a perfect fluid. Solution of the potentials and hence the calculation of physical quan- To summarise, we have presented a nuclear EoS at tities can be done numerically. The field equations for supersaturation densities which satisfies both the con- 4 straints from NS and heavy ion collision phenomenol- to disfavour too stiff behavior of the EoS. The data ogy. Ourresultsshowthatwiththestellarconfiguration, from massive NSs and pulsars may provide an impor- whichcontainalargefractionofβequilibratedNSmatter tant cross-check between high-density astrophysics and with a thin crust is able to describe highly-massivecom- heavy-ionphysics. Thevariationofpressurewithdensity pact stars, such as the one associated to the millisecond for the present EoS is consistent with the experimental pulsars PSR B1516+02B with a mass M=1.94+0.17 M flow dataconfirming its highdensity behaviour. We find −0.19 ⊙ (1σ)[33]andPSRJ0751+1807,witha massM=2.1±0.2 that the large values of gravitational masses (≃2.0 M ) ⊙ M to a 1σ confidence level (and 2.1+0.4 M to a 2σ for the NSs are possible with the present EoS with the ⊙ −0.5 ⊙ confidence level) [34]. In the case of PSR J1748-2021B, SNM incompressibility K =274.7±7.4 MeV, which is ∞ a millisecond pulsar in the Globular Cluster NGC 6440, rather ‘stiff’ enough at high densities to allow compact the measured mass is M=2.74+0.41 M (2σ) [35]. There stars with large values of gravitational masses ∼ 2 M −0.51 ⊙ ⊙ are few other EoSs which can explain such a high mass while the corresponding symmetry energy is ‘super-soft’ forstaticcase,however,they failtoexplaintheexpected as preferred by FOPI/GSI experimental data. Thus the behavior of the NSE. We would like to mention at this DDM3Y effective interaction which is found to provide stage thata starmay not rotateas fast asKeplerianfre- unifieddescriptionofelasticandinelasticscattering,var- quency due to r-mode instability [36]. There have been ious radioactivites and nuclear matter properties, also suggestions that r-mode may limit the time period to provides excellent description of β equilibrated NS mat- 1.5 ms [37]. However, pulsar rotating faster (e.g. PSR ter to allow the recent observations of the massive com- J17482446ad)than this limit has already been observed pact stars. [38]. Furtherobservationsandabetterr-modemodelling The research work of P. Roy Chowdhury is sponsored may shed more light on this issue. by the UGC (No.F.4-2/2006(BSR)/13-224/2008(BSR)) Modern constraints from the mass and mass-radius- underDr. D.S.KothariPostdoctoralFellowshipScheme. relation measurements require stiff EoS at high densi- The work of A. 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