Crust-core interface and bulk neutron star properties
CCrust-core interface and bulk neutron star properties
Ch. Margaritis, ∗ P.S. Koliogiannis, † and Ch.C. Moustakidis ‡ Department of Theoretical Physics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece
The nuclear symmetry energy plays an important role in the description of the properties offinite nuclei as well as neutron stars. Especially, for low values of baryon density, the accuratedescription of the crust-core interface strongly depends on the symmetry energy. Usually, the wellknown parabolic approximation is employed for the definition of the symmetry energy withoutavoiding some drawbacks. In the present work, a class of nuclear models, suitable for the descriptionof the inner and outer core of neutron stars, is applied in studying the effect of higher orders ofthe expansion of the energy on the location of the crust-core transition. The thermodynamical anddynamical methods are used for the determination of the transition density n t and pressure P t .The corresponding energy density functional is applied for the study of some relevant propertiesof both non-rotating and slow rotating neutron stars. We found that the larger the value of theslope parameter L , the slower the convergence of the expansion. In addition, a universal relation ispresented between n t and L , by employing the full expression and dynamical approach. The crustalmoment of inertia is very sensitive on the location of the transition while the effects are moderatedconcerning the critical angular velocity of the r-mode instability and minimum mass configuration.The effect on the tidal deformability is less but not negligible. In any case, the use of the parabolicapproximation leads to the overestimation of n t and P t and consequently, on inaccurate predictions. PACS numbers: 26.60.+c, 97.60.Jd, 21.65.+fKeywords: Equation of state; Symmetry energy; Neutron star
I. INTRODUCTION
The equation of state (EoS) of neutron-rich nuclearmatter is the main ingredient in the study of the struc-ture and properties of neutron stars [1–5]. Moreover, theobservations of neutron stars provide useful constraintsconcerning the EoS both for low and high nuclear matterdensities. In particular, of great interest are the propertiesrelated to the interface between the crust and core, includ-ing mainly the transition density and the correspondingtransition pressure. These quantities are sensitive on thebehavior of the EoS at low densities and play importantrole on the predictions of some bulk neutron star proper-ties. To be more specific, the inner crust comprises theouter region from the density at which neutrons drip outof nuclei to the inner edge, separating the solid crust fromthe homogeneous liquid core. At the inner edge a phasetransition occurs from the high density homogeneous mat-ter to the in-homogeneous one at lower densities. It wasfound that the transition density determines the struc-ture of the inner part of the neutron star’s crust andalso is related to some finite nuclei properties includingneutron-skin, dipole polarizability, etc. [6–8].The determination of the transition density n t itselfis a very complicated problem because the inner crustmay have an intricate structure. A well established ap-proach is to find the density at which the uniform liquidbecomes unstable with respect to small-amplitude densityfluctuations, indicating the formation of nuclear clusters. ∗ [email protected] † [email protected] ‡ [email protected] This approach includes mainly the dynamical method [9–18], the thermodynamical one [19–23], the random phaseapproximation [8, 24], and the Vlasov method [25, 26].Recently, in a notable work, Carreau et al. [27] studiedthe crust-core transition within a unified meta-modelingof the nuclear EoS where the variational equations inthe crust are solved within a Compressible Liquid-Drop(CLD) approach.The motivation of the present work is twofold. Firstly,we intend to study more systematically the convergenceof the baryon energy per particle expansion around theasymmetry parameter I = ( n n − n p ) / ( n n + n p ), where n n and n p are the neutron and proton number densities, re-spectively. It is well known that keeping only a quadraticterm of I , the accuracy of the expansion is sufficient.Most of the relevant studies use this order of the expan-sion, which is well known as the parabolic approximation.In recent studies, the effects of additional terms havebeen considered, usually up to order of O ( I ) [18]. Inthe present work we extend the study up to the order of O ( I ), conjecturing that it is sufficient to gain the mainconclusion about the speed of convergence of the expan-sion. In particular, we employ the energies per baryon,originated by various nuclear models, mainly focus onthe role played by the slope of the symmetry energy L at the saturation density and its effect on the growthof the convergence speed. Moreover, we employed boththe dynamical and thermodynamical method to calculatethe transition density and pressure, corresponding to thecrust-core interface for each expansion term. We mainlyfocus on the parabolic approximation (PA) and the fullexpression (FE) (which includes all terms).Secondly, we apply our findings in order to examine theeffect of the two cases on some neutron star properties a r X i v : . [ nu c l - t h ] F e b that we expect to be sensitive on the values of n t and P t .To be more specific, we concentrate on the crustal momentof inertia both of non-rotating and slowly rotating neutronstars, the critical frequency of the r-mode instability, thetidal deformability, and the minimum mass configuration.Finally, we analyze and discuss the extent of the effects ofthe crust-core interface on each of the latter properties.The article is organized as follows: In Sec. II we presentthe energy expansion, the symmetry energy and the corre-sponding slope parameter. In Sec. III we briefly summa-rize the dynamical and the thermodynamical method forthe determination of the transition density and pressure,while in Sec. IV we present the specific neutron star prop-erties related to the crust-core interface. In Sec. V wepresent the results of the present study and discuss theirimplications. Finally, Sec. VI includes the concludingremarks. II. SYMMETRY ENERGY
The energy per baryon E b ( n, I ) of asymmetric nuclearmatter can be expanded around the asymmetry parameter I as [22, 28] E b ( n, I ) = E b ( n, I = 0) + (cid:88) k =1 E sym , k ( n ) I k . (1)The asymmetry parameter can be written as I = 1 − x , where x is the proton fraction n p /n . Moreover, inEq. (1), E b ( n, I = 0) denotes the energy per baryon ofthe symmetric nuclear matter. The coefficients of theexpansion in Eq. (1) are given by the expression E sym , k ( n ) = 1(2 k )! ∂ k E b ( n, I ) ∂I k (cid:12)(cid:12)(cid:12)(cid:12) I =0 . (2)Since the strong interaction must be symmetric under theexchange of neutrons with protons, only even powers of I appear in Eq. (1). The nuclear symmetry energy is, bydefinition, the coefficient of the quadratic term E sym , ( n ).The slope parameter L is an indicator of the stiffness ofthe EoS and is defined as L = 3 n s dE sym , ( n ) dn (cid:12)(cid:12)(cid:12)(cid:12) n = n s , (3)where n s is the nuclear saturation density.We can define also the slope parameter L k , whichcorresponds to higher order terms, using Eq. (2) (see alsoRef. [18]), according to the rule L k = 3 n s dE sym , k ( n ) dn (cid:12)(cid:12)(cid:12)(cid:12) n = n s . (4)Keeping in Eq. (1) terms only up to a quadratic one,and defining the symmetry energy as the difference be-tween the energy per baryon in pure neutron matter and symmetric nuclear matter, we lead to the parabolic ap-proximation of the symmetry energy according to thelaw E PAsym ( n ) = E b ( n, I = 1) − E b ( n, I = 0) . (5)The slope parameter L PA , that corresponds to theparabolic approximation, is found by Eq. (3) by replac-ing E sym , ( n ) with E PAsym ( n ). It is worth to mention herethat in general, the definitions of E sym , ( n ) and E PAsym ( n )do not coincide. This is the case only when the energyper baryon includes terms up to a quadratic one of theasymmetry parameter I . III. DYNAMICAL AND THERMODYNAMICALMETHOD
The crust-core interface is related to the phase tran-sition between nuclei and uniform nuclear matter. Thelatter one is nearly pure neutron matter, as the protonfraction is just a few percent, determined by the con-dition of β -equilibrium. The study of the instability of β -stable nuclear matter is based on the variation of thetotal energy density, in the framework of the Thomas-Fermi approximations [9, 10]. In the dynamical method,compared to the thermodynamical one, effects due to in-homogeneity of the density and the Coulomb interactionhave also been included. The onset of instability willoccur if the total energy, in the presence of the densityin-homogeneity, is lower than the energy of the uniformphase. The key expression for the description of theinstability reads [9, 10, 17] as U dyn ( n ) = U ( n ) + 4 (cid:112) πα (cid:126) cξ − αξ (cid:0) πx n (cid:1) / , (6)where α = e / (cid:126) c , U ( n ) = ∂µ p ∂n p − ( ∂µ p /∂n n ) ∂µ n /∂n n , (7)and ξ = 2 D nn (1 + 4 ζ + ζ ) , ζ = − ∂µ p /∂n n ∂µ n /∂n n . (8)The chemical potentials µ n and µ p are defined as µ n = (cid:18) ∂E b ∂n n (cid:19) n p , µ p = (cid:18) ∂E b ∂n p (cid:19) n n . (9)The transition density n t is determined from the condition U dyn ( n t ) = 0.The key expression of the thermodynamical method isthe following (for more details see Refs. [19–22]) C therm ( n ) = 2 n ∂E ( n, x ) ∂n + n ∂ E ( n, x ) ∂n − (cid:18) ∂ E ( n, x ) ∂n∂x n (cid:19) (cid:18) ∂ E ( n, x ) ∂x (cid:19) − , (10)and the transition density n t is now determined from thecondition C therm ( n t ) = 0.In both cases, the proton fraction x , is determined asa function of the baryon density n from the condition of β -equilibrium. In particular, in β -stable nuclear matter,the chemical equilibrium condition takes the form µ n = µ p + µ e . (11)It is easy to show that after some algebra we get [17, 22] µ n − µ p = (cid:18) − ∂E b ∂x (cid:19) n . (12)Now, since the electron chemical potential µ e is given by µ e = (cid:126) c (3 π n e ) / , (13)we finally found (cid:18) ∂E b ∂x (cid:19) n = − (cid:126) c (3 π xn ) / . (14)Eq. (14) is solved numerically and an expression for x asa function of n is found.Another important quantity is the transition pressure P t . Both baryons and leptons contribute to the totalpressure, meaning that the transition pressure is given bythe following relation [21, 22] P FEt ( n t , x t ) = n ∂E b ∂n (cid:12)(cid:12)(cid:12)(cid:12) n = n t + (cid:126) c π (cid:0) π x t n t (cid:1) / , (15)where x t is the proton fraction related to the transitiondensity. IV. APPLICATION ON NEUTRON STARPROPERTIES
The EoS of nuclear matter is the key ingredient tostudy the bulk properties of neutron stars. However,there are some specific ones which depend directly onthe location of the crust-core interface. This class ofproperties includes the fraction of moment of inertia ofthe crust with respect to the total one, the critical angularfrequency which defines the r-mode instability, the tidalpolarizability, and the minimum mass configuration. Tobe more specific, the crustal moment of inertia I crust /I and the critical angular velocity Ω c , by definition dependon the crust-core interface. On the other hand, tidaldeformability λ of a low mass neutron star is sensitive onthe specific details of the crust [29–32], while minimummass configuration mainly depends on the contributionof the crust both to M min and R min . In the latter, sincethe central density is close to the transition density, itis expected that even a slight shift to n t (and P t ) willmodify the relevant predictions. A. Crustal fraction of the moment of inertia
The crustal moment of inertia plays an important roleon the evolution of neutron stars and the exhibition ofsome specific phenomena. It is particularly interestingsince it can be inferred from observations of pulsar glitches,the occasional disruptions of the otherwise extremely reg-ular pulsations of magnetized, rotating neutron stars. Inthe case of a non-rotating (or slow rotating) neutron stara few elaborated approximations have been performedleading to analytical predictions of I crust . These approx-imations depend directly on the transition pressure P t (or/and the transition density n t ). The most used onewas provided in Ref. [33] and is written as I crust I (cid:39) πP t R M c (1 − . β − . β ) β × (cid:18) P t n t mc (1 + 5 β − β ) β (cid:19) − , (16)where β = GM/Rc is the compactness parameter.We also apply for comparison a second approxima-tion [34] given by I crust (cid:39) π R βc R P t (cid:18) − . − β β (cid:19) × (cid:20) (cid:18) R core βR − (cid:19) (cid:18) P t E t (cid:19) + · · · (cid:21) . (17)In general, the moment of inertia of rotating neutronstars exhibits dependence on the spin frequency [35] andin this case, the calculation of the crustal moment ofinertia demands special treatment. However, it is worthnoticing that approximations (16) and (17) are sufficientlyaccurate also for a slow rotating neutron star, that isrotating with angular velocity Ω (cid:28) Ω k , where Ω k is theKepler angular velocity [34]. B. Critical angular velocity for r-modes
The r -modes are oscillations of rotating stars whoserestoring force is the Coriolis force (see Refs. [36–38] andreferences therein). The gravitational radiation-driveninstability of these models has been proposed as an ex-planation for the observed relatively low spin frequenciesof young neutron stars and of accreting ones in low-massx-ray binaries (LMXBs). This instability can only occurwhen the gravitational-radiation driving time scale of the r -mode is shorter than the time scales of the various dis-sipation mechanisms that may occur in the interior of theneutron star. The instability condition reads as [36–39]1 τ GW + 1 τ ee + 1 τ nn = 0 , (18)where τ ee and τ nn are the time scales of the various dissi-pation mechanisms (due to electron-electron and neutron-neutron scattering respectively) which are considered inthe present study. The condition (18) leads to the criti-cal angular velocity Ω c which is given by (for a detailedanalysis see Refs. [17, 39, 40])Ω c Ω = (cid:18) − ˜ τ GW (˜ τ ee + ˜ τ ee )˜ τ ee ˜ τ nn (cid:19) / (cid:18) K T (cid:19) / , (19)where Ω = (cid:112) GM/ R , T is the temperature, and also τ GW = ˜ τ GW (cid:18) Ω Ω (cid:19) , τ ii = ˜ τ ii (cid:18) Ω Ω (cid:19) / (cid:18) K T (cid:19) , with ii = ee, nn . In the present work we will concentratein the case where the main damping mechanism is dueto the viscous dissipation at the boundary layer of theperfectly rigid crust and fluid core. In this case, thecorresponding critical angular velocity takes the form [17,39])Ω c = 1 . × (cid:18) R core km (cid:19) / (cid:18) E t MeV fm − (cid:19) / × (cid:32) . (cid:18) E t MeV fm − (cid:19) / (cid:33) / × I ( R core ) − / (cid:18) K T (cid:19) / . (20)From Eq. (20) is obvious the direct dependence of Ω c onthe crust-core interface via E t , as well as the indirect onevia the values of the core radius R core and the integral I ( R core ) where I ( R core ) = (cid:90) R core (cid:18) E ( r )MeV fm − (cid:19) (cid:16) r km (cid:17) d (cid:16) r km (cid:17) , (21)with E ( r ) being the energy density of neutron star matterat distance r from the center. C. Tidal deformability
Gravitational waves from the final stages of inspiralingbinary neutron stars are one of the most important sourcesfor ground-based gravitational wave detectors [41, 42].Flanagan and Hinderer [43] have pointed out that tidaleffects are also potentially measurable during the earlypart of the evolution when the waveform is relativelyclean. The tidal fields induce quadrupole moments onthe neutron stars. The response of the neutron star isdescribed by the dimensionless so-called Love number k which depends on the structure of the neutron star (bothcore and crust). The Love number is linearly related tothe tidal deformability λ according to λ = 2 R k / G . Now, k is given by [44] k = 8 β − β ) [2 − y R + ( y R − β ] × (cid:104) β (6 − y R + 3 β (5 y R − β (cid:0) − y R + β (3 y R −
2) + 2 β (1 + y R ) (cid:1) + 3 (1 − β ) [2 − y R + 2 β ( y R − − β ) (cid:105) − (22)where the quantity y R ≡ y ( R ) is determined by solving therelevant differential equation of y ( r ) simultaneously withthe Tolman-Oppenheimer-Volkoff equations [44]. It isnoted that the measure of the amplitude of the radiatedgravitational waves provides information for the tidaldeformability and consequently useful constraints on bulkneutron stars properties, including mainly the radius. Weexpect that due to the strong dependence of λ on R andalso on the Love number k , the effects of the crust-coreinterface may affect its value. In any case, this possibilityis of interest and also worth to be under consideration. D. Minimum mass configuration
The minimum neutron star mass, apart for the maxi-mum one, is also of great interest in Astrophysics [45, 46].Its knowledge is related to the case of a neutron star in aclose binary system with a more compact partner (neutronstar or black hole). In particular, the lower mass neutronstar transfers mass to the more massive object, a processwhich ultimately leads to approaching its minimum value.Finally, crossing this value, the neutron star reaches anon equilibrium configuration. The minimum mass is auniversal feature, independent of the details of the EoSand well constrained to the value M min (cid:39) . M (cid:12) . Thisis because the corresponding central densities are close tothe values of the transition densities n t . Now, since theequation of the crust is well known, all theoretical pre-dictions for M min converge. However, the correspondingradius R min is very sensitive on the details of the EoS.We expect that the location of the crust-core transitionwill affect appreciably the values of R min . In the presentstudy we investigate in which extent R min is affected bythe values of n t (and P t ). V. RESULTS AND DISCUSSION
In the present work, we compare a class of EoSs gener-ated from three different nuclear models. In particular,we employ a momentum dependent interaction model(MDI) which was presented and analyzed in previous pa-pers [47, 48]. The parametrization of the model has beenchosen in order to generate specific values for E sym , ( n s )and the slope parameter L at the saturation density n s .The second one (HLPS), is based on the microscopic TABLE I. Transition density (in units of fm − ) and pressure (in units of MeV fm − ) calculated using the full expression foreach nuclear model ( n FEt , P
FEt ), the parabolic approximation ( n PAt , P
PAt ) and approximations of each EoS based on Eq. (1) upto order 2 k ( n t , , P t , ). All calculations are performed in the framework of the thermodynamical method.Nuclear model n FEt P FEt n PAt P PAt n t , P t , n t , P t , n t , P t , n t , P t , n t , P t , MDI(65) 0.078 0.317 0.097 0.594 0.094 0.641 0.092 0.519 0.090 0.477 0.088 0.449 0.086 0.428MDI(72.5) 0.073 0.315 0.094 0.728 0.094 0.697 0.090 0.618 0.087 0.561 0.085 0.519 0.083 0.488MDI(80) 0.068 0.286 0.094 0.836 0.095 0.783 0.090 0.684 0.086 0.608 0.084 0.553 0.082 0.513MDI(95-30) 0.058 0.146 0.099 1.078 0.099 1.071 0.093 0.845 0.087 0.697 0.083 0.601 0.081 0.532MDI(95-32) 0.064 0.277 0.097 1.054 0.097 1.026 0.091 0.857 0.087 0.742 0.084 0.662 0.081 0.604MDI(100) 0.053 0.088 0.102 1.202 0.103 1.247 0.096 0.967 0.089 0.765 0.085 0.641 0.081 0.554MDI(110) 0.047 0.047 0.106 1.484 0.109 1.723 0.108 1.548 0.094 0.993 0.088 0.799 0.083 0.658HLPS(49.4) 0.089 0.551 0.097 0.694 0.098 0.675 0.095 0.651 0.094 0.630 0.093 0.614 0.092 0.602HLPS(29.5) 0.098 0.455 0.104 0.495 0.105 0.513 0.103 0.509 0.102 0.500 0.101 0.492 0.100 0.485SkI4(60.4) 0.081 0.337 0.091 0.496 0.091 0.481 0.089 0.453 0.087 0.432 0.086 0.416 0.085 0.404Ska(76.1) 0.079 0.528 0.093 0.866 0.094 0.809 0.090 0.762 0.088 0.713 0.086 0.677 0.085 0.649Sly4(46) 0.088 0.463 0.094 0.546 0.094 0.528 0.093 0.517 0.092 0.506 0.091 0.497 0.090 0.491TABLE II. Transition density (in units of fm − ) and pressure (in units of MeV fm − ) calculated using the full expression foreach nuclear model ( n FEt , P
FEt ), the parabolic approximation ( n PAt , P
PAt ) and approximations of each EoS based on Eq. (1) upto order 2 k ( n t , , P t , ). All calculations are performed in the framework of the dynamical method.Nuclear model n FEt P FEt n PAt P PAt n t , P t , n t , P t , n t , P t , n t , P t , n t , P t , MDI(65) 0.070 0.232 0.086 0.425 0.084 0.488 0.082 0.365 0.080 0.339 0.077 0.322 0.077 0.309MDI(72.5) 0.064 0.212 0.082 0.483 0.083 0.458 0.079 0.413 0.077 0.380 0.075 0.354 0.074 0.334MDI(80) 0.060 0.181 0.082 0.529 0.082 0.492 0.078 0.441 0.076 0.396 0.074 0.363 0.072 0.337MDI(95-30) 0.050 0.075 0.084 0.615 0.084 0.602 0.079 0.479 0.075 0.400 0.072 0.347 0.069 0.309MDI(95-32) 0.055 0.157 0.082 0.641 0.083 0.618 0.078 0.524 0.075 0.457 0.072 0.409 0.070 0.374MDI(100) 0.047 0.039 0.085 0.656 0.086 0.670 0.079 0.508 0.075 0.408 0.072 0.343 0.069 0.298MDI(110) 0.042 0.016 0.088 0.804 0.090 0.919 0.085 0.699 0.077 0.488 0.072 0.392 0.069 0.325HLPS(49.4) 0.079 0.415 0.087 0.525 0.087 0.509 0.085 0.493 0.084 0.478 0.083 0.466 0.082 0.457HLPS(29.5) 0.091 0.339 0.093 0.366 0.094 0.403 0.093 0.376 0.092 0.376 0.091 0.371 0.090 0.399SkI4(60.4) 0.073 0.248 0.082 0.356 0.082 0.343 0.080 0.327 0.079 0.314 0.078 0.304 0.077 0.296Ska(76.1) 0.069 0.377 0.082 0.622 0.083 0.580 0.080 0.553 0.078 0.520 0.077 0.494 0.075 0.474Sly4(46) 0.080 0.365 0.085 0.427 0.085 0.411 0.083 0.405 0.083 0.398 0.082 0.392 0.082 0.387 calculations, in the framework of chiral effective field the-ory interactions, in low densities and suitable polytropicparametrization at high nuclear densities [49]. Specifically,we employ two parametrizations of the model, the soft( L = 29 . L = 49 . L . Also, the specific cases MDI(95-30) and MDI(95-32)correspond to L = 95 MeV and E sym , ( n s ) = 30 MeVand E sym , ( n s ) = 32 MeV, respectively. The calculationsare performed using both the thermodynamical and thedynamical method. In each case, we employed the full expression of the energy per baryon of each model, theparabolic approximation (see Eq. (5)) and the correspond-ing terms of the expansion (see Eq. (1)) up to tenth order.In order to clarify further the predictions, we display theresults also in Fig. 1.In each case, the higher the order in the expansion, thelower the values of n t and P t . Even more, the higher thevalue of the slope parameter L , the larger the deviationbetween the second order predictions and the consider-ation of the full expression. In other words, accordingto our finding, the lower the value of L , the higher theaccuracy of the parabolic approximation. It is worth tomention here that the quadratic dependence of P t on n t (see Eq. (15)) is well reflected on the current predictionsand mainly on the dispersion of the results for high valuesof L . As a general rule, the thermodynamical methodleads to higher values of P t on n t compared to the dy-namical one. However, the most distinctive feature is theappearing of a universal dependence of n t on L in bothmethods concerning the full expression. We found that,independently of the employed model, there is an orderingon the mentioned dependence, where the increase of L , FEPAup to I up to I up to I up to I up to I Eq. H L L H MeV L n t H f m - L THER H a L FEPAup to I up to I up to I up to I up to I L H MeV L P t H M e V f m - L THER H b L FEPAup to I up to I up to I up to I up to I Eq. H L L H MeV L n t H f m - L DYN H c L FEPAup to I up to I up to I up to I up to I L H MeV L P t H M e V f m - L DYN H d L FIG. 1. Transition density n t and pressure P t as a function of the slope parameter L for various nuclear models. The calculationsare performed with the thermodynamical method for (a,b) and the dynamical one for (c,d). The half-filled diamonds presentthe full expression, the half-filled circles present the parabolic approximation, and the circles, the squares, the diamonds, thetriangles, and the reversed triangles present the approximation up to second, fourth, sixth, eighth, and tenth order, respectively,based on Eq. (1). The solid line in (a,c) represents the Eq. (23). leads to a decreased n t . In order to check our finding, weutilize the following expression, predicted by Steiner etal. [53] n t = S (cid:0) . − . L + 0 . L (cid:1) (fm − ) , (23)where S = E sym , ( n s ) / (30 MeV) and L = L/ (70 MeV). We found that there is an excellent agree-ment with the present results (see Fig. 1(a,c)), concerningthe dynamical method, for a large interval of L and mainlyfor different nuclear models.Moreover, we studied the effects of the crust-core in-terface on four specific properties of neutron stars. Weemployed in each case, as an example, the MDI(80) model.It has to be noted here that we found similar results forthe rest of the models (depending of course on the valueof the slope parameter L ). Crustal fraction : The effects of the symmetry energyon the crustal fraction of a non-rotating and slow rotatingwith M = 1 . M (cid:12) neutron star, are displayed in Fig. 2. Inparticular, we employed the approximations (16) and (17)for comparison. Our finding confirms previous predic-tions for a non-rotating neutron star, that is the crustal moment of inertia is very sensitive on the location of thecrust-core interface. In particular, in Fig. 2(a), we displaythe dependence of the crustal ratio I crust /I on the mass.The application of the dynamical method, using the fullexpression, leads to the lower values of I crust /I , comparedto the thermodynamical one. The two horizontal linesrepresent, each one, a possible constraint on I crust /I de-duced for the Vela pulsar (assuming a neutron star with M = 1 . M (cid:12) ). The lower limit, 0.014, was suggestedin Ref. [54], while the higher one, 0 .
07, was consideredin Refs. [55, 56] in order to explain the glitches. It isnotable, in Fig. 2(b), that the predictions of the approx-imations (16) and (17) are almost identical in the caseof the full expression (in both methods), while distinctdeviations appear in the case of the parabolic approxima-tion. We conjecture that in the case of a rapidly rotatingneutron star (close to the mass-shedding limit, that isthe Kepler angular velocity) the effects of the symmetryenergy expansion on I crust /I will be dramatic. In thiscase, one must carefully select the appropriate methodwith the full expression. Otherwise, the accuracy of thepredictions will suffer from large uncertainties. r-mode instability : In the present study we concen- * U D Y L W D W L R Q D O 0 D V V 0 , c r u s t , , crust , , crust , $ O O R Z H G ) R U E L G G H Q $ O O R Z H G ) R U E L G G H Q D 7 + ( 5 ) ( 7 + ( 5 3 $ ' <