Metastability of hadronic compact stars
aa r X i v : . [ a s t r o - ph ] F e b Metastability of hadronic compact stars
Ignazio Bombaci, Prafulla K. Panda,
2, 3
Constan¸ca Providˆencia, and Isaac Vida˜na Dipartimento di Fisica “Enrico Fermi”, Universit`a di Pisa,and INFN Sezione di Pisa, Largo Bruno Pontecorvo 3, I-56127 Pisa, Italy Indian Association for the Cultivation of Sciences, Jadavpur, Kolkata-700 032, India Centro de F´ısica Te´orica, Department of Physics,University of Coimbra, 3004-516 Coimbra, Portugal Departament d’Estructura i Constituents de la Mat`eria. Universitatde Barcelona, Avda. Diagonal 647, E-08028 Barcelona, Spain
Pure hadronic compact stars, above a threshold value of their gravitational mass (central pressure),are metastable to the conversion to quark stars (hybrid or strange stars). In this paper, we presenta systematic study of the metastability of pure hadronic compact stars using different relativisticmodels for the equation of state (EoS). In particular, we compare results for the quark-mesoncoupling (QMC) model with those for the Glendenning–Moszkowski parametrization of the non-linear Walecka model (NLWM). For QMC model, we find large values ( M cr = 1 . . M ⊙ ) for thecritical mass of the hadronic star sequence and we find that the formation of a quark star is onlypossible with a soft quark matter EoS. For the Glendenning–Moszkowski parametrization of theNLWM, we explore the effect of different hyperon couplings on the critical mass and on the stellarconversion energy. We find that increasing the value of the hyperon coupling constants shifts thebulk transition point for quark deconfinement to higher densities, increases the stellar metastabilitythreshold mass and the value of the critical mass, and thus makes the formation of quark starsless likely. For the largest values of the hyperon couplings we find a critical mass which may be ashigh as 1.9 - 2.1 M ⊙ . These stellar configurations, which contain a large central hyperon fraction( f Y,cr ∼ M = 1 . +0 . − . M ⊙ . PACS number(s): 97.60.s, 97.60.Jd, 26.60.Dd, 26.60.Kp
I. INTRODUCTION
The nucleation of quark matter in neutron stars has been studied by many authors, due to its potential connectionwith explosive astrophysical events such as supernovae and gamma ray burst. Some of the earlier studies on quarkmatter nucleation (see e.g., [1, 2, 3] and references therein) dealt with thermal nucleation in hot and dense hadronicmatter. In these studies, it was found that the prompt formation of a critical size drop of quark matter via thermalactivation is possible above a temperature of about 2 − T = 0)neutrino free hadronic matter [8, 9, 10, 11, 12, 13, 14, 15].Quantum fluctuations could form, in principle, a drop of β -stable quark matter (hereafter the Q β phase). However,this process is strongly suppressed with respect to the formation of a non β -stable drop by a factor ∼ G N/ F ermi , where N ∼ − β -stabledrop would involve the almost simultaneous conversion of ∼ N/ u ) and down ( d ) quarks into strange ( s ) quarks.Alternatively, quantum fluctuations can form a non β -stable drop (hereafter the Q ∗ phase), in which the flavor contentof the quark phase is equal to that of the β -stable hadronic phase at the same pressure [9, 10, 12]. Since no flavorconversion is involved, there are no suppressing Fermi factors, and a Q ∗ drop can be nucleated much more easily.Once a critical-size Q ∗ drop is formed, the weak interactions will have enough time to act, changing the quark flavorfraction of the deconfined droplet to lower its energy, and a drop of the Q β phase is formed. This first seed of quarkmatter will trigger the conversion [16, 17, 18] of the pure hadronic star to a hybrid star or to a strange star (dependingon the details of the equation of state for quark matter used to model the phase transition). The stellar conversionprocess liberates a total energy of the order of 10 erg [18].When finite-size effects at the interface between the quark and hadron phases are taken into account, it is necessaryto have an overpressure ∆ P = P − P > P , to create a drop of deconfinedquark matter. As a consequence, pure hadronic stars with values of the central pressure larger than P are metastableto the decay (conversion) to hybrid stars or to strange stars [11, 12, 13, 14, 15]. The mean lifetime of the metastablestellar configuration is related to the time needed to nucleate the first drop of quark matter in the stellar center anddepends dramatically on the value of the stellar central pressure [11, 12, 13, 14, 15].The possibility of having in nature both metastable hadronic stars and stable quark stars, has led the authors ofref. [12] to extend the concept of limiting mass of a “neutron star” with respect to the classical one introduced byOppenheimer and Volkoff [19]. Since metastable HS with a “short” mean-life time are very unlikely to be observed,the extended concept of limiting mass has been introduced in view of the comparison with the values of the mass ofcompact stars deduced from direct astrophysical observation (see sect. 3.1 of ref. [12] for the definition of the limitingmass , M lim , of compact stars in the case of metastable pure hadronic stars).As it is well known, neutron star mass measurements give one of the most stringent test on the overall stiffness ofdense matter EoS. Recent measurements of Post Keplerian orbital parameters in relativistic binary stellar systems(containing millisecond pulsars) give strong evidence for the existence of highly-massive “neutron stars”. For example,the compact star associated to the millisecond pulsar PSR B1516+02B in the Globular Cluster NGC 5904 (M5) hasa mass M = 1 . +0 . − . M ⊙ (1 σ ) [20]. In the case of PSR J1748-2021B, a millisecond pulsar in the Globular ClusterNGC 6440, the measured mass is M = 2 . +0 . − . M ⊙ (2 σ ) [21]. These measurements challenge most of the existingmodels for dense matter EoS.In this work, we carry out a systematic study of the properties of metastable hadronic compact stars obtainedwithin different relativistic mean-field models for the equation of state (EoS) of hadronic matter. In particular, wecompare the predictions of the Quark-Meson Coupling (QMC) model [22, 23] with those of the non-linear Waleckamodel (NLWM) [24] parametrizations given by Glendenning–Moszkowski (GM) [25].For the quark phase we have adopted a phenomenological EOS [26] which is based on the MIT bag model forhadrons. The parameters here are: the mass m s of the strange quark, the so-called pressure of the vacuum B (bagconstant) and the QCD structure constant α s . For all the quark matter model used in the present work, we take m u = m d = 0, m s = 150 MeV and α s = 0.In the QMC model quark degrees of freedom are explicitly taken into account: baryons are described as a system ofnon-overlapping MIT bags which interact through the effective scalar and vector mean fields. The coupling constantsare defined at the quark level. An attractive aspect of the model is that different phases of hadronic matter, from verylow to very high baryon densities, can be described within the same underlying model, namely the MIT bag model:matter at low densities is a system of nucleons interacting through meson fields, with quarks and gluons confinedwithin MIT bag; at very high density one expects that baryons and mesons dissolve and the entire system of quarksand gluons becomes confined within a single, big MIT bag.In the case of the Glendenning–Moszkowski EoS [25], we have paid special attention to the role played by thehyperon-meson couplings. In fact, all previous works on metastable hadronic stars [11, 12, 13, 14, 15] have uniquelyconsidered the case of “low” values for these quantities ( x σ = 0 . σ meson tonucleon– σ meson coupling). As it is well known, larger values of the hyperon-meson couplings (constrained by theempirical binding energy of the Λ particle in nuclear matter) make the EoS stiffer and increase the value of theOppenheimer–Volkoff mass for the hadronic stellar sequence [25]. In addition, as we demonstrate in the present work,increasing the values of the hyperon-meson couplings shifts the bulk transition point for quark deconfinement tohigher densities and increments the value of the critical mass M cr (see ref.[11, 12, 13] and Section III for the explicitdefinition of this quantity) for the hadronic stellar sequence. Thus our study is relevant in connections with the recentmeasurements of highly-massive “neutron stars” mentioned above.A brief review of the NLW and QMC models is given in Section II. The quantum nucleation of a quark matterdrop inside hadronic matter is briefly reviewed in Section III. Our main results are presented in Section IV, whereasthe main conclusions are given in Section V II. THE FORMALISM
In the present section we review the models used in this work, namely the GM parametrizations [25] of the NLWMand the quark-meson coupling (QMC) model including hyperons.
A. The non-linear Walecka model
The Lagrangian density, including the baryonic octet, in terms of the scalar σ , the vector-isoscalar ω µ and thevector-isovector ~ρ µ meson fields reads (see e.g. [4, 27, 28]) L = L hadrons + L leptons (1)where the hadronic contribution is L hadrons = L baryons + L mesons (2)with L baryons = X baryons ¯ ψ [ γ µ D µ − M ∗ B ] ψ, (3)where D µ = i∂ µ − g ωB ω µ − g ρB ~t B · ~ρ µ , (4)and M ∗ B = M B − g σB σ. The quantity ~t B designates the isospin of baryon B . The mesonic contribution reads L mesons = L σ + L ω + L ρ , (5)with L σ = 12 ( ∂ µ σ∂ µ σ − m σ σ ) + 13! κσ + 14! λσ , (6) L ω = −
14 Ω µν Ω µν + 12 m ω ω µ ω µ , Ω µν = ∂ µ ω ν − ∂ ν ω µ , (7) L ρ = − ~B µν · ~B µν + 12 m ρ ~ρ µ · ~ρ µ , ~B µν = ∂ µ ~ρ ν − ∂ ν ~ρ µ − g ρ ( ~ρ µ × ~ρ ν ) (8)For the lepton contribution we take L leptons = X leptons ¯ ψ l ( iγ µ ∂ µ − m l ) ψ l , (9)where the sum is over electrons and muons. In uniform matter, we get for the baryon Fermi energy ǫ F B = g ωB ω + g ρB t B ρ + q k F B + M ∗ B , with the baryon effective mass M ∗ B = M − g σB σ. We will use the GM1 and GM3 parametrizations of NLWM [25] both fitted to the bulk properties of nuclear matter:for GM1 (GM3) the compressibility is 300 (240) MeV and the effective mass at saturation M ∗ = 0 . M ( M ∗ =0 . M ). The inclusion of hyperons involves new couplings, the hyperon-nucleon couplings: g σB = x σB g σ , g ωB = x ωB g ω , g ρB = x ρB g ρ . For nucleons we take x σB , x ωB , x ρB = 1 and for hyperons we will consider the couplingsproposed by Glendenning and Moszkowski [25]. They have considered the binding energy of the Λ in nuclear matter, B Λ , (cid:18) B Λ A (cid:19) = −
28 MeV = x ω g ω ω − x σ g σ σ (10)to establish a relation between x σ and x ω . Moreover, known neutron star masses restrict x σ to the range 0 . − . x ρ = x σ and will consider x σ = 0 . , . , . B. The quark-meson coupling model for hadronic matter
In the QMC model, the nucleon in nuclear medium is assumed to be a static spherical MIT bag in which quarksinteract with the scalar and vector fields, σ , ω and ρ and these fields are treated as classical fields in the mean fieldapproximation [22, 23]. The quark field, ψ q ( x ), inside the bag then satisfies the equation of motion: (cid:20) i / ∂ − ( m q − g qσ σ ) − g qω ω γ + 12 g qρ τ z ρ (cid:21) ψ q ( x ) = 0 , q = u, d, s, (11)where m q is the current quark mass, and g qσ , g qω and g qρ and denote the quark-meson coupling constants. Thenormalized ground state for a quark in the bag is given by ψ q ( r , t ) = N q exp ( − iǫ q t/R B ) (cid:18) j ( x q r/R B ) iβ q ~σ · ˆ rj ( x q r/R B ) (cid:19) χ q √ π , (12)where ǫ q = Ω q + R B (cid:18) g qω ω + 12 g qρ τ z ρ (cid:19) ; β q = s Ω q − R B m ∗ q Ω q + R B m ∗ q , (13)with the normalization factor given by N − q = 2 R B j ( x q ) (cid:2) Ω q (Ω q −
1) + R B m ∗ q / (cid:3) . x q , (14)where Ω q ≡ q x q + ( R B m ∗ q ) , m ∗ q = m q − g qσ σ , R B is the bag radius of the baryon, and χ q is the quark spinor. Thequantities ψ q , ǫ q , β q , N q , Ω q , m ∗ q all depend on the baryon considered. The bag eigenvalue, x q , is determined by theboundary condition at the bag surface j ( x q ) = β q j ( x q ) . (15)The energy of a static bag describing baryon B consisting of three ground state quarks can be expressed as E bag B = X q n q Ω q R B − Z B R B + 43 π R B B B , (16)where Z B is a parameter which accounts for zero-point motion and B B is the bag constant. The effective mass of anucleon bag at rest is taken to be M ∗ B = E bag B . The equilibrium condition for the bag is obtained by minimizing theeffective mass, M ∗ B with respect to the bag radius d M ∗ B d R ∗ B = 0 . (17)For the QMC model, the equations of motion for the meson fields in uniform static matter are given by m σ σ = X B g σB C B ( σ ) 2 J B + 12 π Z k B M ∗ B ( σ )[ k + M ∗ B ( σ )] / k dk , (18) m ω ω = X B g ωB (2 J B + 1) k B (cid:14) (6 π ) , (19) m ρ ρ = X B g ρB I B (2 J B + 1) k B (cid:14) (6 π ) . (20)In the above equations J B , I B and k B are respectively the spin, isospin projection and the Fermi momentum of thebaryon species B . For the hyperon couplings we take x ω = 0 .
78 and x ρ = 0 .
7. The coupling x σ is an output of themodel and is approximately equal to 0.7. Note that the s -quark is unaffected by the σ and ω mesons i.e. g sσ = g sω = 0 . In Eq. (18) we have g σB C B ( σ ) = − ∂M ∗ B ( σ ) ∂σ = − ∂E bag B ∂σ = X q = u,d n q g qσ S B ( σ ) (21)where S B ( σ ) = Z bag d r ψ q ψ q = Ω q / R B m ∗ q (Ω q − q (Ω q −
1) + R B m ∗ q / q ≡ ( u, d ) . (22)The total energy density and the pressure including the leptons can be obtained from the grand canonical potentialand they read ε = 12 m σ σ + 12 m ω ω + 12 m ρ ρ + X B J B + 12 π Z k B k dk (cid:2) k + M ∗ B ( σ ) (cid:3) / + X l π Z k l k dk (cid:2) k + m l (cid:3) / , (23) P = − m σ σ + 12 m ω ω + 12 m ρ ρ + 13 X B J B + 12 π Z k B k dk [ k + M ∗ B ( σ )] / + 13 X l π Z k l k dk [ k + m l ] / . (24)For the bag radius we take R N = 0 . Z N and B N for nucleons are obtained by fitting thenucleon mass M = 939 MeV and enforcing the stability condition for the bag at free space. The values obtained are Z N = 3 . B / N = 211 .
303 MeV for m u = m d = 0 MeV and Z N = 4 . B / N = 210 .
854 MeV for m u = m d = 5 . B B , for all baryons and the parameter Z B and R B of the otherbaryons are obtained by reproducing their physical masses in free space and again enforcing the stability conditionfor their bags. Note that for a fixed bag value, the equilibrium condition in free space results in an increase of thebag radius and a decrease of the parameters Z B for the heavier baryons. The set of parameters used in the presentwork is given in Ref. [29].Next we fit the quark-meson coupling constants g qσ , g ω = 3 g qω and g ρ = g qρ for the nucleon to obtain the correctsaturation properties of the nuclear matter, E B ≡ ǫ/ρ − M = − . ρ = ρ = 0 .
15 fm − , a sym = 32 . K = 257 MeV and M ∗ = 0 . M . We have g qσ = 5 . g ωN = 8 .
981 and g ρN = 8 . m σ = 550 MeV, m ω = 783 MeV m ρ = 770 MeV. III. QUANTUM NUCLEATION OF QUARK MATTER IN HADRON STARS
Let us consider a pure hadronic star whose central pressure (density) is increasing due to spin-down or due to massaccretion (from a companion or from the interstellar medium). As the central pressure approaches the deconfinementthreshold pressure P (see Fig. 2), a drop of non β -stable quark matter ( Q ∗ ), but with flavor content equal to thatof the β -stable hadronic phase, can be formed in the central region of the star. The process of drop formation isregulated by its quantum fluctuations in the potential well created from the difference in the Gibbs free energies ofthe hadron and quark phases [9, 10, 11] U ( R ) = 43 πn b,Q ∗ ( µ Q ∗ − µ H ) R + 4 πσ R (25)where R is the radius of the Q ∗ droplet (supposed to be spherical), n b,Q ∗ is the quark baryon number density, µ Q ∗ and µ H are the quark and hadron chemical potentials at a fixed pressure P and σ is the surface tension for thesurface separating the hadron from the Q ∗ phase. Notice that µ is the same as the bulk Gibbs energy per baryon g = ( P + ǫ ) /n B = ( P i µ i n i ) /n B . Notice also that we have neglected the term associated with the curvature energy,and also the terms connected with the electrostatic energy, since they are known to introduce small corrections[10, 12]. The value of the surface tension σ for the interface separating the quark and hadron phase is poorly known,and typically values used in the literature range within 10 −
50 MeV fm − [10, 32].The time needed to form the first drop (nucleation time) can be straightforwardly evaluated within a semi-classicalapproach [9, 10]. First one computes, in the Wentzel–Kramers–Brillouin (WKB) approximation, the ground stateenergy E and the oscillation frequency ν of the drop in the potential well U ( R ). Then, the probability of tunnelingis given by p = exp (cid:20) − A ( E )¯ h (cid:21) (26)where A is the action under the potential barrier which in a relativistic framework reads A ( E ) = 2 c Z R + R − p [2 M ( R ) c + E − U ( R )][ U ( R ) − E ] , (27)being R ± the classical turning points and M ( R ) = 4 π ρ H (cid:18) − n b,Q ∗ n H (cid:19) R (28)the droplet effective mass, with ρ H and n H the hadron energy density and the hadron baryon number density,respectively. The nucleation time is then equal to τ = ( ν p N c ) − , (29)where N c is the number of virtual centers of droplet formation in the star. A simple estimation gives N c ∼ [9, 10]. The uncertainty in the value of N c is expected to be within one or two orders of magnitude. In any case, allthe qualitative features of our scenario will not be affected by this uncertainty. As a consequence of the surface effectsit is necessary to have an overpressure ∆ P = P − P > P to create a dropof deconfinement quark matter in the hadronic environment. The higher the overpressure, the easier to nucleate thefirst drop of Q ∗ matter. In other words, the higher the mass of the metastable pure hadronic star, the shorter thetime to nucleate a quark matter drop at the center of the star.In order to explore the astrophysical implications of quark matter nucleation, following ref. [11, 12], we introducethe concept of critical mass for the hadronic star sequence. The critical mass M cr is the value of the gravitational massof a metastable hadronic star for which the nucleation time is equal to one year: M cr = M HS ( τ = 1 yr ). Therefore,pure hadronic stars with M HS > M cr are very unlikely to be observed, while pure hadronic stars with M HS < M cr are safe with respect to a sudden transition to quark matter. Then M cr plays the role of an effective maximum massfor the hadronic branch of compact stars (see discussion in Ref. [12]). While the Oppenheimer–Volkov maximum massis determined by the overall stiffness of the equation of state for hadronic matter, the value of M cr will depend inaddition on the properties of the intermediate non β -stable Q ∗ phase. IV. RESULTS AND DISCUSSION
In this section we present and discuss our results for stellar configurations obtained using the equation of state (EoS)models described in section II. In particular, we determine the region of the pure hadronic star sequence where thesecompact stars are metastable, the value of the corresponding critical mass M cr , and the final fate of this configurationafter quark matter nucleation, i.e. whether it will evolve to a quark star or to a black hole.In Fig 1 the EoS for the models discussed are plotted for the range of densities of relevance for the discussion thatfollows. For GM1 and GM3 we have considered three different hyperon-meson coupling as discussed above. TheQMC EoS corresponds approximately to x σ = 0 .
7. A higher value of the hyperon couplings x i corresponds to stifferEoSs: at high densities we have vector dominance defined by the magnitude of x ω , x ρ . It is clear from Fig. 1 thatthe onset of hyperons (represented by the change of slope in the EoS curves) occurs for the smaller x σ values at lowerdensities. The nucleonic EoS for QMC is very soft and therefore the onset of hyperons occurs at quite high densities, ε = 373 .
87 MeV/fm . As a consequence although QMC is softer than GM1 EoS at lower densities, it becomes, athigher densities, stiffer than GM1( x σ = 0 .
6) and very close to GM1( x σ = 0 . Q ∗ phaseusing the various EoS models (couple of continuous and dashed curves with the same color) considered in the presentwork. It is clearly seen that in the case of the GM1 or GM3 EoS models, the lower the value of the hyperon coupling x σ , the softer the EoS (see also Fig 1) and the lower the pressure P at the crossing between the hadronic and the Q ∗ phase. This will give rise to lower critical masses for the smaller x σ values (see Tabl. I-III below). The Q ∗ phase -3 ]0255075100 P r e ss u r e [ M e V f m - ] GM1, x s =0.6GM1, x s =0.7GM1, x s =0.8GM3, x s =0.6GM3, x s =0.7GM3, x s =0.8QMC FIG. 1: Hadronic Eos for QMC and for GM1 and GM3 with the different hyperon-meson couplings discussed in the text. is very sensitive to the particle content and it is due to this fact that, although in Fig. 2 the EoS for QMC is softerthan the EoS for GM1 with x σ = 0 . Q ∗ phase occurs at higher pressures. A similarobservation occurs in the figure with the GM3 results. This behavior will reflect itself on values of the critical masses M cr .In Fig. 3, we show the mass-radius (MR) curve for pure HS within the QMC model for the EoS of the hadronicphase, and that for hybrid stars or strange stars for different values of the bag constant B . The configuration markedwith an asterisk on the hadronic MR curves represents the hadronic star for which the central pressure is equal tothe threshold value P and the quark matter nucleation time is τ = ∞ . The full circle on the hadronic star sequencerepresents the critical mass configuration, in the case σ = 30 MeV/fm . The full circle on the HyS (SS) mass-radiuscurve represents the hybrid (strange) star which is formed from the conversion of the hadronic star with M HS = M cr .We assume [18] that during the stellar conversion process the total number of baryons in the star (or in other wordsthe stellar baryonic mass) is conserved. Thus the total energy liberated in the stellar conversion is given by thedifference between the gravitational mass of the initial hadronic star ( M in ≡ M cr ) and that of the final hybrid orstrange stellar configuration with the same baryonic mass ( M fin ≡ M QS ( M bcr ) ): E conv = ( M in − M fin ) c . (30)As we can see from Fig. 3, for the case of the QMC model, the region of metastability of pure hadronic stars (thepart of the MR curve between the asterisk and the full circle) is very narrow. For this hadronic EoS, the quark starsequence can be populated only in the case of “small” values of the bag constant ( B ≤
80 MeV/fm , in this case thefinal star is a strange star). In all the other cases the critical mass hadronic star will form a black hole.For comparison we plot the MR curve obtained with the GM1 parametrization for the same surface tension ( σ =30MeV/fm ) and two values for bag constant ( B = 75 and 100 MeV/fm ). We consider the two extreme values ofthe hyperon couplings studied in this work. The dots and stars have the same meaning as in Fig. 3. We see thatfor the cases plotted the only configuration that does not end in a black hole has the smallest bag constant andhyperon coupling considered. In the present model, however, the configuration with the central pressure P and the M cr configuration are quite separated, contrary to what was observed with QMC, Fig. 3.The larger mass difference between the star with the central pressure P and the one with the M cr occurs whenthese stars have small masses. A small change in the central energy density corresponds to a large change in themass. If instead of plotting the MR graph we would have plotted the corresponding mass–central pressure (MP)graph a larger difference between these two configurations would be expected. This is seen in Figs. 5 and 6 wherethe mass-pressure curves for the family of stars obtained respectively within QMC and GM1 are plotted for two bag
50 60 70 80 90 100 110 120Pressure [MeV fm -3 ]11001150120012501300 G i bb s e n e r gy p e r p a r ti c l e [ M e V ] GM1, x σ =0.6GM1, x σ =0.7GM1, x σ =0.8QMCP =60.073 P =70.759 P =87.162B=100 MeV fm -3 P =100.6150 60 70 80 90 100 110 120 130 140 150 160 170Pressure [MeV fm -3 ]110011501200125013001350 G i bb s e n e r gy p e r p a r ti c l e [ M e V ] GM3, x σ =0.6GM3, x σ =0.7GM3, x σ =0.8QMC P =87.598 P =110.62 P =154.29B=100 MeV fm -3 P =100.61 FIG. 2: The Gibbs‘ energy per particle for the β -stable hadronic phase (continuous curves) and for the respective Q ∗ phase(dashed curves). The upper panel refers to the GM1 and the lower panel to the GM3 EoS. The results for the QMC model areplotted in both panels. constants and two values of the surface tension ( σ = 10 and 30 MeV/fm ). We conclude that when the M cr star isalmost on top of the P star in the MR curves, these stars lie on or close to the plateau that contains the maximummass configuration. A large separation between these two configurations corresponds to a phase transition whichoccurs during the rise of the MR curve before the plateau. Due to the softness of the QMC EOS, hyperons set on atquite large energy densities and the star with the central P pressure only occurs at high densities. We also concludethat a smaller surface tension hastens the transition and the critical mass is closer to the P mass.In Tables I, II, and III we give the gravitational ( M cr ) and baryonic ( M bcr ) critical mass values for the hadronic starsequence, together with the central hyperon fraction ( f Y,cr = n Y /n B , i.e. the ratio between the total hyperon numberdensity and the total baryon number density at the center of the critical mass star). We also report the value of thegravitational mass ( M fin ) of the final quark star configuration and the total energy [18] E conv = ( M cr − M fin ) c released in the stellar conversion process, assuming baryon mass conservation ( i.e. no matter ejection) [18]. Thegravitational ( M QS,max ) and the baryonic ( M bQS,max ) mass of the maximum mass configuration for the quark (hybridor strange) star sequence are also included. The value of the latter quantity is relevant to establish whether the M / M s un M / M s un B=150 MeV fm -3 B=100 MeV fm -3 B=85 MeV fm -3 B=75 MeV fm -3 go to BH go to BHgo to BH FIG. 3: Mass-radius relation for a pure HS described within the QMC model and that of the HyS or SS configurations forseveral values of the bag constant and m s = 150 MeV and α s = 0. The configuration marked with an asterisk represents in allcases the HS for which the central pressure is equal to P . The conversion process of the HS, with a gravitational mass equalto M cr , into a final HyS or SS is denoted by the full circles connected by an arrow. In all the panels σ is taken equal to 30MeV/fm . critical mass hadronic star will evolve to a quark star ( M bcr < M bQS,max ) or will form a black hole ( M bcr > M bQS,max ).The entries in Tables I, II [49], and III are relative respectively to the GM1, GM3 and QMC equation of state for thehadronic phase. For the quark phase we consider four different values of bag constants, 75, 85, 100 and 150 MeV/fm ,and two different values for quark-hadron surface tension, 10 and 30 MeV. Notice that for the quark matter parameterset adopted in the present work (see Section I), strange quark matter is absolutely stable [33, 34] only for B = 75MeV/fm .Some comments are in order: the critical masses increase with the increase of the hyperon couplings. This increasecan be as large as 0.3 - 0.4 M ⊙ when x σ changes from 0.6 to 0.8; the critical mass is also dependent on the particlecontent, namely of the strangeness content, and this explains the different relative positions for the different bagpressures of the QMC result which essentially corresponds to x σ = 0 .
7. Due to the fact that the EoS for QMC isvery soft, the hyperon onset occurs at quite high densities and therefore the critical mass is always quite high forthis model. The critical mass increases with the bag constant because a larger bag constant corresponds to a stifferquark EoS and therefore the phase transition to the quark phase will occur at larger densities. When the critical masshadronic star is converted to a black hole, this is indicated in Tables I, II, and III with a entry BH, in the columnsfor M fin and E conv (no energy will be radiated as soon as the star pass the event horizon). Notice that, in the caseof the GM3 model with x σ = 0 . B = 150 MeV/fm there is no entry for the critical mass value (and for M bcr , M fin and E conv ) since in this case the nucleation time of the maximum mass hadronic star ( M HS,max ) is much largerthan one year ( i.e. the star is metastable with a life-time comparable or much higher than the age of the universe).We observe from the results in tables I, II, that increasing the value of the hyperon coupling constants (for fixed B and σ ) reduces the central hyperon fraction ( f Y,cr ) of the critical mass star, and increases the energy released duringthe conversion into a quark or hybrid star (for those configurations which will not form a black hole).In Fig. 7, we show the internal composition for the hadronic star with a gravitational mass M = 2 . M ⊙ andradius R = 12 . x σ = 0 .
8. This star corresponds to the criticalmass configuration when we consider B = 150 MeV/fm and σ = 30 MeV/fm (see table I). As we see, this star has aconsiderable central hyperon fraction ( f Y,cr = 0 . R Y ∼ . R Y ≤ r ≤ R crust ) with a thickness of about 3.4 km. The0 M / M s un M / M s un B=100 MeV fm -3 B=100 MeV fm -3 B=75 MeV fm -3 B=75 MeV fm -3 x σ =0.6x σ =0.6 x σ =0.8x σ =0.8go to BHgo to BH go to BH FIG. 4: Mass-radius relation for a pure HS described within the GM1 parametrization and that of the HyS or SS configurationsfor two values of the bag constant ( B = 75 and 100 MeV/fm ) and two values of the hyperon-meson coupling ( x σ = 0 . , and0.8) and m s = 150 MeV and α s = 0. The configuration marked with an asterisk represents in all cases the HS for which thecentral pressure is equal to P . The conversion process of the HS, with a gravitational mass equal to M cr , into a final HyS orSS is denoted by the full circles connected by an arrow. In all the panels σ is taken equal to 30 MeV/fm . stellar crust extends from R crust up to R .It has been argued by several authors [35, 36, 37, 38] that if strange quark matter (SQM) is absolutely stable[33, 34], then all compact stars are likely to be strange stars. The argument in favor of this thesis is the following:if the interstellar medium is sufficiently contaminated by quark nuggets ( i.e. lumps of SQM), then the presence of asingle quark nugget in the interior of a “normal” neutron star (hadronic star) is sufficient to trigger the conversionof the star to a strange star [34, 35]. Likely the quark nugget contamination of the interstellar medium is the resultof the merging of strange stars in binary systems [36, 39]. Under these conditions, compact star progenitors couldcapture a quark nugget during their lives ( i.e. during the various nuclear burning stages of the stellar evolution).Thus, according to this argument, a strange quark seed will be present in all new born compact stars, and thus theconversion to a strange star will happen immediately, without a metastable hadronic star being formed first. Thisis a plausible scenario, however it would be relevant only for a few of the stellar models considered in our work, i.e. those relative to the value B = 75 MeV/fm for the bag constant (see Tables I, II, and III) for which SQM isabsolutely stable, and will not have any effect upon the existence of metastable hadronic compact stars in all theother cases considered in the present work. The magnitude of the flux of quark nuggets in the interstellar medium(which is a crucial quantity for the validity of the scenario of ref.s [35, 36, 37, 38]) has been estimated [36] making theassumption that all pulsars exhibiting glitches must be “normal” neutron stars (hadronic stars), not strange stars.This assumption is based primarily on the nearly total lack of models for the glitch phenomenon with strange stars(see anyhow ref. [40, 41]), while such models have been quite successfully developed in the case of hadronic stars (see e.g. ref. [42, 43, 44, 45]). However, recent studies have established the possibility of an inhomogeneous crystallinecolor superconducting phase (LOFF phase) in the interior of strange stars (see [46, 47] and references therein quoted),or the likely existence of a SQM crystalline crust in strange stars [48]. These late theoretical developments rise thepossibility to explain pulsar glitches with strange star based models, and thus require as useful a recalculation of theastrophysical limits of the flux of quark nuggets. The scenario discussed in the present work is an alternative to thescenario [35, 36, 37, 38] according to which all compact stars are strange stars, which requires that SQM is absolutelystable.1 M / M s un c [MeV fm -3 ]00.511.52 M / M s un c [MeV fm -3 ] 00.511.52B=150 MeV fm -3 B=150 MeV fm -3 B=75 MeV fm -3 B=75 MeV fm -3 σ =10 MeV fm -2 σ =10 MeV fm -2 σ =30 MeV fm -2 σ =30 MeV fm -2 go to BHgo to BH FIG. 5: Mass-pressure relation for a pure HS described within the QMC model and that of the HyS or SS configurationstwo values of the Bag constant(75 and 100 MeV/fm ) and m s = 150 MeV and α s = 0. The configuration marked with anasterisk represents in all cases the HS for which the central pressure is equal to P . The conversion process of the HS, with agravitational mass equal to M cr , into a final HyS or SS is denoted by the full circles connected by an arrow. Two values of thesurface energy σ were considered 10 MeV/fm (left) and 30 MeV/fm (right). V. CONCLUSIONS
It has been recently shown [11, 12, 13, 14, 15] that pure hadronic compact stars, above a threshold value of theirgravitational mass, are metastable to the conversion to quark stars. In this work we have done a systematic study ofthe metastability of pure hadronic compact stars using different relativistic hadronic models for the equation of stateof hadronic dense matter. In particular, we have used and compared the quark-meson coupling (QMC) model withthose for the Glendenning–Moszkowski parametrization of the non-linear Walecka model (NLWM). In the case of theQMC model, we have obtained that the region of metastability of pure hadronic stars is very narrow. For the GMmodel, we have investigated the effect of the hyperon couplings on the critical mass of the hadronic star sequence andon the stellar conversion energy. We have found that increasing the value of the hyperon coupling constants shiftsthe bulk transition point for quark deconfinement to higher densities, increasing the value of the critical mass for thehadronic stellar sequence, and thus makes the formation of quark stars less likely. The nucleonic EoS for QMC is verysoft and therefore the onset of hyperons occurs at quite high densities, which gives rise to large critical masses. Theconversion to a quark star will occur only for a small value of the bag constant. Finally we point out that both QMCand GM1 with the largest values of the hyperon-meson couplings predict limiting masses [12] which may be as highas 1.9 - 2.1 M ⊙ .These values would be able to describe highly-massive compact stars, such as the one associated to the millisecondpulsars PSR B1516+02B [20], and nearly the one in PSR J1748-2021B [21]. Acknowledgments
This work was partially supported by FEDER/FCT (Portugal) under the projects POCI/FP/63918/2005 andPTDC/FIS/64707/2006 and by the Ministero dell’Universit`a e della Ricerca (Italy) under the PRIN 2005 project2 M / M s un c [MeV fm -3 ]00.511.52 M / M s un c [MeV fm -3 ] 00.511.52B=100 MeV fm -3 B=100 MeV fm -3 B=75 MeV fm -3 B=75 MeV fm -3 σ =10 MeV fm -2 σ =10 MeV fm -2 σ =30 MeV fm -2 σ =30 MeV fm -2 go to BH go to BH FIG. 6: The same as Fig 5 for the GM1 parametrization with the x σ = 0 . Theory of Nuclear Structure and Nuclear Matter . [1] J. E. Horvath, O. G. Benvenuto and H. Vucetich, Phys. Rev. D (1992) 3865.[2] J. E. Horvath, Phys. Rev. D (1994) 5590.[3] M. L. Olesen and J. Madsen, Phys. Rev. D (1994) 2698.[4] M. Prakash, I. Bombaci, M. Prakash, P. J. Ellis, J. M. Lattimer and R. Knorren, Phys. Rep. (1997) 1.[5] G. Lugones and O. G. Benvenuto, Phys. Rev. D (1998) 083001.[6] O. G. Benvenuto and G. Lugones, Mon. Not. R. A. S. (1999) L25.[7] I. Vida˜na, I. Bombaci and I. Parenti, J. Phys. G (2005) S1165.[8] F. Grassi, ApJ (1998) 263.[9] K. Iida and K. Sato, (1997) Prog. Theor. Phys. 98, 277.[10] K. Iida and K. Sato, Phys. Rev. C 58 (1998) 2538.[11] Z. Berezhiani, I. Bombaci, A. Drago, F. Frontera and A. Lavagno, Astrophys. J. (2003) 1250.[12] I. Bombaci, I. Parenti and I. Vida˜na,
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