LOCV calculation of the equation of state and properties of rapidly rotating neutron stars
aa r X i v : . [ nu c l - t h ] M a y Chinese Physics C Vol. cc, No. c (cccc) cccccc
LOCV calculation of the equation of state and properties of rapidlyrotating neutron stars
A.H. Farajian M. Bigdeli S. Belbasi
Department of Physics, University of Zanjan,Zanjan, 45371-38791, Iran
Abstract:
In this paper, we have investigated the structural properties of rotating neutron stars using the numericalRNS code and equations of state which have been calculated within the lowest order constrained variational (LOCV)approach. In order to calculate the equation of state of nuclear matter, we have used UV +TNI and AV potentials.We have computed the maximum mass of the neutron star and the corresponding equatorial radius at different angularvelocities. We have also computed the structural properties of Keplerian rotating neutron stars for the maximummass configuration, M K , R K , f K and j max . Key words:
LOCV method, neutron star matter, equation of state, rotating neutron star
PACS:
All existing studies indicate that observed neutronstars, such as millisecond pulsars (MSPs), are rotating.Recently, many MSPs have been discovered. One of themost rapidly rotating neutron stars is pulsar PSR J1748-2446ad, which has rotational frequency 716 Hz [1]. Therotational frequency f , which can be directly measured,affects the global attributes of neutron stars, specifically,maximum mass, radius, spin parameter and total mo-ment of inertia [2–6]. The maximum mass increases withrotation due to the rotational energy and there are evensuper-massive sequences [7]. So far, there have been alarge number of mass and radii measurements. The ac-curate measurement of mass for about 35 neutron starslies in the wide range of M ∼ . − . M ⊙ and the radiiof more than a dozen neutron stars lies in the range R ∼ . − . M = 1 . ± . M ⊙ [9], and PSR J0348+0432,with mass M = 2 . ± . M ⊙ [10]. These massive neu-tron stars require the equation of state (EOS) of thesystem to be rather stiff. Present radius determinationsare model dependent and subject to large uncertainties.However, some current and planned projects, such asNICER ∗ are trying to determine the radii more precisely.Theoretically, the EOSs have been applied to determineneutron star properties which should be in agreementwith the precise observations.Another important characteristic quantity for com-pact stars is the dimensionless spin parameter j ≡ cJ/GM , where J is angular momentum and M is grav- itational mass. The astrophysical estimations and impli-cations of j for different astronomical objects have beenconsidered by several authors, e.g. Refs. [11–16]. T¨or¨ok et al. have investigated the mass vs. spin parameter rela-tionship M ( j ) = M [1+ k ( j + j )] for the Z-source CircinusX-1 [15] and atoll source 4 U −
53 [16]. Kato et al. have shown that a description of the observed correla-tions of Circinus X-1 requires adopting M = 1 . − . M ⊙ as the mass of the central star in Circinus X-1 and j ∼ . j max (spin parameter of a neutron star rotatingat the Keplerian frequency), depends on the compositionof compact stars. Their results indicate that the valueof j max has an upper bound about j max ∼ . M ≥ M ⊙ [17]. Their results also indicate that there is no univer-sal upper bound for the spin parameter of quark starssimulated by the MIT bag model and it can be largerthan unity ( j max > j max of compact stars can be larger than 0 .
1) E-mail: m [email protected] ∗ cccccc-1hinese Physics C Vol. cc, No. c (cccc) cccccc also for hybrid stars, whereas the role of the crust in thetotal mass of the compact star is negligible. In this pa-per, we show that only the outer crust structure couldplay the same roles, see Section 3. Qi et al. also haveconstructed a universal formula for spin parameter ver-sus frequency, j = 0 . f /f k ) − . f /f k ) +0 . f /f k ),for different kinds of compact stars.In this study, we have investigated the structuralproperties of rapidly rotating neutron stars with andwithout outer crust structures. Here we have usedEOS for the liquid core of the neutron star which havebeen calculated within the lowest order constrained vari-ational (LOCV) method with UV +TNI [19] andAV [20] potentials. Previously, we used these EOSto determine the core-crust transition parameters andglobal attributes of core and crust for neutron stars [21]. We have employed the EOS for neutron star matterby describing the neutron star’s outer crust, inner crustand the liquid core. For the inner crust, we use the EOSwhich is calculated by Douchin and Haensel [22], and forthe outer crust, the Baym-Pethick-Sutherland EOS [23]is used. In the case of the neutron star core, we assumea charged neutral infinite system which is a mixture ofleptons and interacting nucleons. The energy density ofthis system can be obtained as follows, ε = ε N + ε l , (1)where ε N ( ε l ) is the energy density of nucleons (leptons).The energy density of leptons, which are considered as anoninteracting Fermi gas, is given by, ε lep = X l = e, µ X k ≤ k Fl ( m l c + ~ c k ) / . (2)In this equation, k Fl = (3 π ρ l ) / is the Fermi momentumof leptons. The nucleon contribution of energy density isgiven by, ε N = ρ ( E nucl + m N c ) , (3)where E nucl is the total energy per particle of asymmetricnuclear matter and ρ is the total number density, ρ = ρ p + ρ n . Here, ρ n and ρ p are number density of neutrons andprotons respectively. Table 1. Saturation density and correspondingvalues of energy per particle, incompressibilityand symmetry energy of symmetric nuclear mat-ter. Here ρ s is given in fm − and energy param-eters are in MeV. Potential ρ s E K S UV + TNI 0.17 -16.86 261 31.27AV The β -equilibrium conditions and charge neutrality ofneutron star matter impose the following coupled con-straints on our calculations, µ e = µ µ = µ n − µ p (4) ρ p = ρ e + ρ µ . (5)We find the abundance of the particles by solving thesecoupled equations and calculate the total energy and theEOS of the neutron star matter.In the following, we determine the energy per particleof asymmetric nuclear matter, E nucl , in more detail byusing the LOCV method. In our formalism, the energyper particle is written in terms of correlation function, f , and its derivatives; and approximately given up to thetwo-body term as the following form [24], E nucl ([ f ]) = 1 A h ψ | H | ψ ih ψ | ψ i = 1 A X τ = n,p X k ≤ k Fτ ~ k m τ + 12 A X ij h ij | ν (12) | ij i a , (6)where ψ = F φ is a trial many-body wave function. Here φ is the Slater determinant of wave function of A in-dependent nucleons and F = S Q i>j f ( ij ) ( S is a sym-metrizing operator) is a Jastrow form of A-body corre-lation operator. In the above equation, k Fτ = (3 π ρ τ ) / is the Fermi momentum of nucleons and ν (12) is the ef-fective potential, which is given by, ν (12) = − ~ m [ f (12) , [ ∇ , f (12)]] + f (12) V (12) f (12) . (7)Here, f (12) and V (12) are the two-body correlation andpotential, respectively. In our calculations, we used theUV +TNI and AV two-body potentials. cccccc-2hinese Physics C Vol. cc, No. c (cccc) cccccc l og P ( dyn c m - ) log e ( gr cm -3 ) UV +TNI AV Inner crust Outer crust
Fig. 1. The EOS of neutron star matter for theUV + TNI and AV potentials. The EOS ofthe outer and inner crust are also shown. In this formalism, the correlation function is consid-ered as different forms [25], and calculated by numer-ically solving of set of coupled and uncoupled Euler-Lagrange differential equations [26]. These differentialequations are a result of functional minimization of thetwo-body cluster energy with respect to the correlationfunctions variation. For more details see Refs. [26–29].A summary of our results for bulk properties of sym-metric nuclear matter for the UV +TNI and AV po-tentials are given in Table 1. In this table, we have giventhe saturation density ρ s , and the corresponding valuesof energy per particle E , incompressibility K , and nu-clear symmetry energy S . The calculated saturationproperties of symmetric nuclear matter are in excellentagreement with the experimental data [30] for the UV +TNI potential.The pressure of neutron star matter can be calculatedby the following relation, P = ρ ∂ε∂ρ − ε. (8)In Fig. 1, we have plotted the pressure of neutron starmatter at the core of the star for the mentioned poten-tials versus total energy density. In this figure we alsoshow the EOS for outer and inner crust. It is seen thatthe UV +TNI potential leads to a stiffer EOS. M ( M ) f =0.0 f =716 AV UV +TNI M ( M ) R ( km ) Fig. 2. The gravitational mass ( M ) versus circum-ferential radius ( R ) for non-rotating and rotatingneutron star with the UV +TNI and AV po-tentials. The frequency ( f ) is given in Hz. Thesolid (dashed) curve shows the result for neutronstars including (excluding) the outer crust struc-ture. f = 0 and f = 716 Hz. The solid (dashed) curveshows the result for neutron stars including (excluding)the outer crust structure. Clearly, the inclusion of theouter crust has no considerable effect on the maximummass and corresponding radius of the neutron star. How-ever, the global structure of the neutron star is sensitiveto its angular velocity, and the maximum mass increasesby increasing the rotation velocity. cccccc-3hinese Physics C Vol. cc, No. c (cccc) cccccc f k ( k H z ) M ( M ) UV +TNI AV Fig. 3. The variation of the Keplerian fre-quency ( f K ) with gravitational mass M for neu-tron stars with (solid curve) and without (dashedcurve) outer crust structure. From this figure, one can compare the results of the EOSderived using the UV +TNI and AV potentials. Ata frequency of f = 716 Hz, which corresponds to thespin period P ≈ .
39 ms, by applying the AV po-tential, we get M max /M ⊙ ≃ .
653 ( ≃ . +TNI leads to larger stellar mass and radiusin comparison with the AV potential, and we obtain M max /M ⊙ ≃ . ≃ . +TNIpotential. This is in good agreement with the resultsobtained by observations for the millisecond pulsar PSRJ0348+0432, M = 2 . ± . M ⊙ [10]. However, this pul-sar rotates with the lower frequency of ≃
25 Hz. Thisdoes not affect the good comparison, because in this fre-quency range the maximum mass has a little variancewith the rotation (see Fig. 2 and Table 2).Another crucial parameter that can be used to de-scribe rotating neutron stars is the Keplerian frequency, f k , the maximum value of frequency. We have plot-ted Keplerian frequencies versus gravitational masses inFig. 3. It is seen that f k depends on the EOS mod-els presented here. From Fig. 3, for the case of theUV +TNI potential, we find that the value of the Ke-plerian mass corresponding to our calculated frequency, f k ≃ .
93 kHz ( f k ≃ .
96 kHz) is about M k ≃ . M ⊙ ( M k ≃ . M ⊙ ) for a neutron star with (without) outercrust structure. For the case of the AV potential, wefind M k ≃ . M ⊙ ( M k ≃ . M ⊙ ) corresponding to f k ≃ .
23 kHz ( f k ≃ .
24 kHz). It is seen that the Kep-lerian mass and frequency for a neutron star with outer crust are a little lower than those of a neutron star with-out outer crust. f k ( k H z ) M ( M ) UV +TNI AV Fig. 4. The variation of the Keplerian fre-quency ( f K ) with gravitational mass M forneutron stars for precise values of Keplerianfrequency (solid curve) and those given byEq. (9) (dotted curve). We have also calculated Keplerian frequency using thefit formula proposed by Haensel et al. [6], f K = 1 . kHz (cid:18) MM ⊙ (cid:19) / (cid:18) R km (cid:19) − / , (9)where 0 . M ⊙ ≤ M ≤ . M statmax , M statmax is the maximummass of the non-rotating (static) configuration and R is the corresponding radius. The results are shown inFig. 4. As can be seen from this figure, there is a goodagreement between the precise values and those calcu-lated using the above equation, especially for the UV +TNI potential.In the following, we discuss the relation between max-imum mass and frequency in more detail. In Fig. 5, wepresent the maximum mass in units of Keplerian mass, M max ( M K ), as a function of stellar frequency in units ofKeplerian frequency, f ( f K ). This figure shows that forboth EOS employed in the present work, M max ( M K ) dis-plays a similar behavior versus f ( f K ) and, nearly, doesnot depend on the EOS. According to this behavior, wefind 0 . M K . M max ≤ . M K . In other words, the maximum mass in the Keplerian con-figuration increases about 20% compared to the maxi-mum mass of non-rotating configurations. This result is cccccc-4hinese Physics C Vol. cc, No. c (cccc) cccccc in agreement with those obtained by the universal rela-tion M k ≃ (1 . ± . M statmax proposed by Breu andRezzola [32]. M a x i m u m M a ss ( M K ) frequency ( f K ) AV UV +TNI Fig. 5. The maximum mass versus frequency fordifferent equations of state. The maximum massand frequency are given in units of Keplerian massand frequency, respectively.
Now, we focus on the treatment of the dimensionlessspin parameter j , for rotating neutron stars. Here, wewould like to consider the influence of the outer cruststructure on the spin parameter at Keplerian frequency,i.e. maximum spin parameter, j max . In order to achievethis, we shown the maximal spin parameter, j max , as afunction of gravitational mass in Fig. 6. As can be seenfrom this figure, the maximal spin parameter of the ro-tating neutron star displays different behaviors when weeither include or exclude the outer crust structure. Itis seen that j max for NSs with the outer crust structurelying in the narrow range ∼ (0.64 - 0.7) for M ≥ . M ⊙ .Therefore, we see that our result for the upper limit of j max ( ≤ .
7) is in agreement with those reported earlier[17, 18] for traditional neutron stars, while, for the neu-tron star with only inner crust structure j max is largerthan 0.7 and this value is the lower limit of j max ( ≥ . j , ofslow rotating neutron stars. In Fig. 7, we plot the spinparameter j as a function of the rotational frequencynormalized to Keplerian frequency, f /f k , for using the UV +TNI at different values of baryonic mass of neu-tron star, M b /M ⊙ = 1 , . , j m a x M ( M ) UV +TNI AV Fig. 6. The variation of the maximum spin param-eter ( j max ) with gravitational mass M for neu-tron stars with (solid curve) and without (dashedcurve) outer crust structure. It is seen that for each fixed frequency, the curves areessentially independent of mass sequence. A unified re-lationship could be fitted approximately by the formula j = 0 . f /f k ) − . f /f k ) + 0 . f /f k ), as denotedby the circles. We also show the result of the univer-sal formula j = 0 . f /f k ) − . f /f k ) + 0 . f /f k ),which has been suggested in Ref. [18], with squares, forcomparison. j f/f K M b = M b = M b = cccccc-5hinese Physics C Vol. cc, No. c (cccc) ccccccFig. 7. The dimensionless spin parameter ( j ) asa function of the rotational frequency normalizedto Keplerian frequency ( f/f K ) for the UV +TNIpotential. The circles show our fitted formula,and squares that from Ref. [18]. A summary of our results for the structural propertiesof rotating neutron stars with and without outer crustpredicted from different EOS is given in Table 2. This ta-ble also includes the maximum mass and correspondingequatorial radius for neutron stars at f = 0 , and 716 Hz,as well as the structural properties of Keplerian rotatingneutron stars for the maximum mass configuration, M K , R K , f K and j max . In this work, we have calculated the structural prop-erties of rotating neutron stars with and without outer crust structures. Here we have employed lowest or-der constrained variational approach and used the UV +TNI and AV potentials to compute the EOS of nu-clear matter. We have computed maximum mass andcorresponding equatorial radius at fixed frequency f = 0and f = 716 Hz. We have also computed the structuralproperties of Keplerian rotating neutron stars for maxi-mum mass configuration, M K , R K , f K and j max .A summary of our results for the structural propertiesof rotating neutron stars with and without outer crustpredicted from different EOS is given in Table 2. Our re-sults show that the maximal spin parameter, j max , lies inthe narrow range ∼ (0.64 - 0.7) for M ≥ . M ⊙ for theEOS considered. In the case of slow rotating neutronstars, we have suggested a unified relationship for thespin parameter j = 0 . f /f k ) − . f /f k ) +0 . f /f k )which is essentially independent of mass sequence. Fi-nally, our results in the Keplerian configuration are invery good agreement with those of other studies. Table 2. Maximum mass and corresponding equatorial radius for neutron stars at f = 0 ,
716 Hz. The structuralproperties of Keplerian rotating neutron stars for maximum mass configuration, M K , R K , f K and j max are alsogiven. The gravitational mass is given in solar masses ( M ⊙ ), R is in km and f K in kHz. The quantities inparenthesis show the results of our calculation for neutron stars without outer crust structure. Potential M f =0 R f =0 M f =716 R f =716 M K R K f K j max UV + TNI 1.99(1.99) 9.67(9.56) 2.027(2.027) 9.8(9.7) 2.36(2.40) 12.59(12.59) 1.93(1.96) 0.682(0.707)AV We wish to thank the University of Zanjan Research Councils.
References : 1901 (2006)2 J. B. Hartle, and K. S. Thorne, ApJ, : 807 (1968)3 N. Stergioulas, and J.L. Friedman, ApJ, : 306 (1995)4 M. Salgado, S. Bonazzola, E. Gourgoulhon, and P. Haensel,Astron. Astrophys, : 455 (1994)5 G. B. Cook, S. L. Shapiro, and S. A. Teukolsky, ApJ, :823 (1994)6 P. Haensel, J. L. Zdunik, M. Bejger, and J. M. Lattimer, As-tron. Astrophys, : 605 (2009)7 M. Salgado, S. Bonazzola, E. Gourgoulhon, and P. Haensel,Astron. Astrophys, : 155 (1994)8 F. ¨Ozel, and P. Freire, Annu. Rev. Astron. Astrophys, : 401(2016)9 P.B. Demorest, T. Pennucci, S.M. Ransom, M.S.E. Roberts,and J.W.T. Hessels, Nature, : 1081 (2010)10 J. Antoniadis, et al., Science, : 448 (2013)11 M. D. Duez, S. L. Shapiro, and H. J. Yo, Phys. Rev. D, :104016 (2004) 12 S. Kato, Publ. Astron. Soc. Japan, : 889 (2008)13 T. Piran, Rev. Mod. Phys, : 1143 (2005)14 M. Shibata, Phys. Rev. D, : 024033 (2003); M. Shibata,,ApJ, : 992 (2003)15 G. T¨or¨ok, P. Bakala, E. Sr´amkov´a, Z. Stuchlik, and M. Ur-banec, ApJ, : 748 (2010)16 G. T¨or¨ok, P. Bakala, E. Sr´amkov´a, Z. Stuchlik, and M. Ur-banec, and K. Goluchov´a, ApJ, : 138 (2012)17 K. W. Lo, and L. M. Lin, ApJ, : 12 (2011)18 B. Qi, N. B. Zhang, B. Y. Sun, S. Y. Wang, and J. H. Gao,RAA, , No. 4 (2016)19 I. E. Lagaris, and V. R. Pandharipande, Nucl. Phys. A, :349 (1981)20 R. B. Wiringa, V. G. J. Stoks, and R. Schiavilla, Phys. Rev.C, : 38 (1995)21 M. Bigdeli, and S. Elyasi, Eur. Phys. J. A, : 38 (2015)22 F. Douchin, and P. Haensel, Astron. Astrophys, : 151(2001)23 G. Baym, C. Pethick, and D. Sutherland, ApJ, : 299(1971) cccccc-6hinese Physics C Vol. cc, No. c (cccc) cccccc
24 J. W. Clark, and N. C. Chao, Lettere Nuovo Cimento, : 185(1968)25 J. C. Owen, R. F.Bishop, and J. M. Irvine, Nucl. Phys. A, :45 (1997)26 G. H. Bordbar,and M. Modarres, Phys. Rev. C, : 714 (1998)27 M. Modarres, and G. H.Bordbar, Phys. Rev. C, : 2781(1998) 28 M. Bigdeli, G. H. Bordbar, and A. Poostforush, Phys. Rev. C, : 034309 (2010)29 M. Bigdeli, Phys. Rev. C, : 054312 (2010)30 P. E. Haustein, At. Data Nucl. Data Tables, : 185 (1988)31 N. Stergioulas, Living Rev. Rel, : 3 (2003)32 C. Breu, and L. Rezzola, MNRAS, : 646 (2016): 646 (2016)