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Nucleation of Quark Matter in Neutron Star Cores T

Nucleation of Quark Matter in Neutron Star Cores T

The Astrophysical Journal, 608:945–956, 2004 June 20 # 2004. The American Astronomical Society. All rights reserved. Printed in U.S.A.

NUCLEATION OF IN CORES T. Harko, K. S. Cheng, and P. S. Tang Department of Physics, University of Hong Kong, Pokfulam Road, Hong Kong, China; [email protected], [email protected], [email protected] Received 2004 January 14; accepted 2004 March 8

ABSTRACT We consider the general conditions of quark droplet formation in high-density neutron matter. The growth of the quark bubble (assumed to contain a sufficiently large number of particles) can be described by means of a Fokker-Planck equation. The dynamics of the nucleation essentially depends on the physical properties of the medium in which it takes place. The conditions for quark bubble formation are analyzed within the framework of both dissipative and nondissipative (with zero bulk and shear viscosity coefficients) approaches. The conversion time of the to a quark star is obtained as a function of the of the neutron matter and of the microscopic parameters of the quark nuclei. As an application of the formalism obtained, we analyze the first-order transition from neutron matter to quark matter in rapidly rotating neutron star cores, triggered by the gravitational energy released during the spinning down of the neutron star. The endothermic conversion process, via gravitational energy absorption, could take place in a very short time interval, of the order of a few tens of seconds, in a class of dense compact objects with very high magnetic fields, called . Subject headings: dense matter — : general — : interiors — stars: neutron

1. INTRODUCTION collapse, and surface radiation, have also been studied (Cheng & Harko 2000, 2003; Harko & Cheng 2000, 2002; Ng et al. Since Witten (1984), following early proposals by Itoh 2003). (1970) and Bodmer (1971), suggested that The physical mechanisms of the transition from neutron matter, consisting of u-, d-, and s-, is energetically the matter to quark matter in an astrophysical background have most favorable , the problem of the existence of been studied in several models. The first is that of Olinto strange quark stars has been intensively investigated in the (1987), who used a nonrelativistic diffusion model. As such, physical and astrophysical literature. The possibility that some this is a slow combustion model, with the burning front compact objects could be strange stars remains an interesting propagating at a speed of approximately 10 m s1.Thisis and intriguing, but still open, question. Witten (1984) also determined primarily by the rate at which one of the down proposed two possible formation scenarios for : quarks inside the is converted, through a weak decay, the quark- phase transition in the early universe, and to a strange quark: d þ u ! s þ u. The second method of conversion of neutron stars into strange ones at ultrahigh describing the conversion process was first suggested by densities. In the theories of , quark bag Horvath & Benvenuto (1988) and analyzed in detail by models suppose that breaking of the physical vacuum takes Lugones et al. (1994) and Lugones & Benvenuto (1995), who place inside . As a result, vacuum energy densities modeled the conversion as a detonation. In this case the inside and outside a hadron become essentially different, and conversion rate is several orders of faster than that the vacuum pressure on the bag wall equilibrates the pressure predicted by the slow combustion model. This model is based of quarks, thus stabilizing the system. If the hypothesis of on the relativistic shock waves and combustion theory. But quark matter is true, then some neutron stars could actually be regardless of the way in which the transformation occurs, an strange stars, made entirely of strange matter (Alcock et al. initial seed of quark matter is needed to start the process. 1986; Haensel et al. 1986). However, there are general argu- Real neutron stars have two conserved charges: electric and ments against the existence of strange stars (Caldwell & baryonic. Therefore, a neutron star has more than one inde- Friedman 1991). For an extensive review of pendent component, and in this sense is a ‘‘complex’’ system properties, see Cheng et al. (1998a). (Glendenning 1992). The characteristics of a first-order phase Several mechanisms have been proposed for the formation transition, such as the deconfinement transition, are very dif- of quark stars. Quark stars are expected to form during the ferent in the two cases. In the complex system the conserved collapse of the core of a massive star after a ex- charges can be shared by the two phases in equilibrium in plosion, as a result of a first- or second-order phase transition, different concentrations in each phase. The mixed phase, resulting in deconfined quark matter (Dai et al. 1995). The formed from hadrons and quarks, cannot exist in a simple proto–neutron star core, or the neutron star core, is a favorable body in the presence of because the pressure in that environment for the conversion of ordinary matter to strange phase is a constant (Glendenning 2000). This causes a dis- quark matter (Cheng et al. 1998b). Another possibility is that continuity in the density distribution in the star occurring at some neutron stars in low-mass X-ray binaries can accrete the radius where Gibbs criteria are satisfied. The isospin sufficient mass to undergo a phase transition to become symmetry energy in neutron-rich matter will exploit the de- strange stars (Cheng & Dai 1996). This mechanism has also gree of freedom of readjusting the charges between hadronic been proposed as a source of radiation emission for cosmo- and quark phases in equilibrium so as to reduce the symmetry logical gamma-ray bursts (Cheng & Dai 1998). Some basic energy to an extent consistent with charge conservation. properties of strange stars, such as mass, radius, cooling, Regions of hadronic matter will have a net positive charge 945 946 HARKO, CHENG, & TANG Vol. 608 neutralized by a net negative charge on the quark matter also been discussed, with reference to formation in regions (Glendenning 2000). Coulomb repulsion will prevent ultrarelativistic heavy-ion collisions. the regions of like charge from growing too large, and the A different approach to the problem of thermal nucleation surface energy will act in the opposite sense in preventing was taken by Olesen & Madsen (1994). They used the as- them from becoming too small. Consequently, the mixed sumption (standard in the theory of bubble nucleation in first- phase will form a Coulomb lattice so as to minimize the sum order phase transitions) that bubbles form at a rate given by 4 of Coulomb and surface interface energy at each proportion R T exp (Wc=T), where Wc is the minimum work re- of phases. As the quark phase becomes more abundant, the quired to form a bubble with radius rc. As a main result it droplets merge to form rods, rods merge to form slabs, etc. follows that if the bag constant lies in the interval where three- (Glendenning 2000). Hence, the actual geometric phase of flavor but not two-flavor quark matter is stable at zero pressure quark nuggets in a neutron star evolves as a function of the and temperature (145 MeV B1=4 163 MeV), then all or pressure and of the surface energy between dense nuclear parts of a neutron star will be converted into strange matter, matter and quark matter. during the first second of its existence. For the bag constant In order to describe the process of the formation and evo- above the stability interval, a partial transformation is still lution of quark nuclei in neutron stars, one must use nucle- possible (Olesen & Madsen 1994). ation theory. The goal of nucleation theory is to compute the Heiselberg (1995) calculated the rate of formation of quark probability that a bubble or droplet of the A phase appears in a matter droplets in neutron stars from a combination of bub- system initially in the B phase near the critical temperature ble formation rates in cold degenerate and high-temperature (Landau & Lifshitz 1980). Homogeneous nucleation theory matter, taking into account nuclear matter calculations of applies when the system is pure. Nucleation theory is appli- the viscosity and thermal conductivity. The droplet formation cable for first-order phase transitions when the matter is not rate is that of Langer & Turski (1973),pffiffiffi given by I ¼ (k=2) 3=2 4 dramatically supercooled or superheated. If substantial super- 0 exp (Wc=T ), where 0 ¼ (2=3 3)(=T) (rc=q) is cooling or superheating is present, or if the phase transition is the statistical prefactor, which measures the phase-space vol- second order, then the relevant dynamics is spinodal decom- ume of the saddle point around Rc that the droplet has to pass position (Shukla & Mohanty 2001). on its way to the lower energy state. Here q is the quark The nucleation of strange quark matter inside hot, dense correlation length. The dynamical prefactor determines the 2 3 nuclear matter was investigated, using Zel’dovich’s kinetic droplet growth rate and is given by k ¼ (2=w r c )½kT þ theory of nucleation, by Horvath et al. (1992). By assuming 2(4=3 þ ),wherew is the enthalpy difference and k, , that the newly formed strange quark matter bubble can be and are the thermal conductivity and the shear and bulk described by a simple bag model containing Nq ultrarelativistic viscosities, respectively. The droplet formation rate can be quarks, and by assuming that the time evolution of the radius of expressed as (Heiselberg 1995) the bubble is described by an equation of the form dr=dt ¼ 7=2 2 3 (r R )= ,whereR is the critical radius of the bubble and 50 c w c I ¼ 20 400 exp 185 134 20 s1 km3; w is the weak interaction timescale, the nucleation rate is 2 3=2 2 P10w T P10T10 given by 10 10 ð3Þ 2 1=2 3=4 1 1=2 ¼ 2:2 10 ðÞT= Ncq w exp 3:1Ncq =T ; ð1Þ where 400 is the quark chemical potential in units of 400 MeV. The total number N of droplets formed by nucleation is the where Ncq 300 is the critical quark number, is the surface integrated rate over the volume of the neutron star and the tension of the bubble, and nn and nq are the particle number timeR after the nucleationR process starts at the moment t0, N ¼ R 4r2 dr 1 IðÞprðÞ; TtðÞ dt,whereR is the radius densities in the neutron and quark phase, respectively (Horvath 0 t0 et al. 1992) . In this way a lower bound for the temperature of the neutron star. The pressure and temperature depend sen- of the nucleation, T 2:1 MeV, can be obtained. The effects sitively on the equation of state (EOS) of the nuclear and quark of the curvature energy term on thermal strange quark matter matter. To convert the core of the neutron star into quark nucleation have been considered by Horvath (1994), who matter, at least one droplet must be formed, i.e., N > 1. Then, derived, within the same approach, the following expression by equation (3), the condition for the formation of at least 2 2=3 1=3 for the nucleation rate: one droplet requires 24 MeV fm (Pc;10Tc;10), where Pc and Tc are the core values of the pressure and temperature, r2n n T 1=2 4R2 respectively (Heiselberg 1995). The pressure and enthalpy ¼ c q n exp c : ð2Þ 4w 3T difference depends strongly on the EOS of nuclear and quark matter, and hence the droplet formation rate cannot be reliably All the effects of the curvature enter only through the value estimated. of Rc. As a general conclusion, we find that even though the The effect of subcritical hadron bubbles on an inhomoge- curvature term acts against strange quark matter nucleation, neous first-order quark-hadron phase transition was studied the physical temperature of a just-born proto–neutron star is, in Shukla et al. (2000). The transition from nuclear matter in any model, more than enough to drive an efficient boiling of (consisting of neutrons, protons, and electrons) to matter the neutron material (the observations of the neutrino flux containing strangeness was considered, within a mean field from SN 1987A are consistent with an of type model and using Langer nucleation theory, by Norsen T 4 MeV). The possibility of stable strange quark matter (2002). An estimate of the time it takes for the new phase to (both bulk and quasi-bulk) at finite temperature, and some of appear at various densities and times in the cooling history of its properties, have been investigated within the framework of a proto–neutron star was also given. the dynamical density-dependent quark mass model of con- On the other hand, as a result of an increase of the cen- finement by Chakrabarty (1993). The behavior of the surface tral density of the star, due, for example, to or spin- tension and the stability of quark droplets at T 6¼ 0have ning down, a metastable, supercompressed neutron phase can No. 2, 2004 NUCLEATION OF QUARK MATTER IN NEUTRON STAR CORES 947 appear, with a star consisting of the new phase surrounded by viscosity coefficients. In the nucleation theory of Csernai & normal neutron matter (Grassi 1998). This situation is similar Kapusta (1992a), the transition from the neutron to quark state to a gas undergoing a phase transition to its liquid state if is possible only for normal matter having viscous properties. compressed to a volume Vc at fixed T.Ifitisslowlycom- For zero bulk and shear viscosities the transition is impossible. pressed, it may stay in the vapor state even for V < Vc (similar This assumption is, however, too restrictive, and the transition examples are liquid water freed of dissolved air, which can be can also take place for a perfect (no viscosity or heat con- heated above boiling temperature without the formation of duction) neutron matter fluid. We also derive the expression of vapor and cooled below freezing temperature without solidi- the rate formation for this case. fication). The transition from the supercompressed neutron An alternative point of view of the transformation of neu- phase to the quark phase can take place over a very long time, tron matter into strange matter can be developed if one which in some cases can be longer than the estimated life of assumes that the conversion process is endothermic. In this the neutron star (Grassi 1998). Therefore, an increase in the case the strange quark matter formation inside a neutron star is density of the neutron star does not automatically lead to the triggered by an external source of energy (accretion from the transition to quark matter inside the star. companion star or spin-down). Pulsars are born with an The effect of the magnetic field on the quark star structure enormous angular momentum and rotational energy, which and on the nucleation process of the quark bubbles has been they radiate over a long period of time via electromagnetic studied in Chakrabarty (1995, 1996) and Chakrabarty & Sahu radiation and electron-positron pair emission. When rotating (1996). In the presence of strong magnetic fields the EOS of rapidly, a is centrifugally flattened. With decreasing strange quark matter changes significantly. The strange stars angular velocity the central density of the star is increasing become more compact, the magnetic field reducing the mass and may attain the critical density necessary for a phase and radius of the star. The surface energy and the curvature transition. First at the center and then in an expanding region, term of the quark phase both diverge, with the surface tension the neutron matter will be converted to a highly compressible diverging logarithmically, while the curvature term diverges quark matter phase (Glendenning et al. 1997). The conversion much faster. Therefore, the thermal nucleation of quark bub- of neutron matter to quark matter alters the moment of inertia bles in a compact metastable state of neutron matter is com- of the star, and the epoch over which conversion takes place pletely forbidden in the presence of a strong magnetic field. will be signaled in the spin-down characteristics of the pulsar. These results for the formation of quark bubbles in neutron A measurable quantity, the braking index ¨ =˙ 2,where is matter have been obtained by using the Csernai & Kapusta the angular velocity of the star, could be an observational theory of nucleation (Csernai & Kapusta 1992a, 1992b) and indicator of the slow transition from neutron to quark phase its extension including the thermal conductivity of dense (Glendenning et al. 1997). By using the formalism developed matter (Venugopalan & Vischer 1994). In this theory both the for the nucleation of quark matter, we reconsider the idea of hadron and quark materials are considered as substances with phase transition from neutron matter to quark matter in ro- zero baryonic number and are treated by using a relativistic tating compact stellar objects, by assuming that the nucleation formalism. The principal result of this approach is the sug- and the growth of the quark droplets and the formation of a gestion that the prefactor k is proportional to the transport quark core in a neutron star are a result of the transfer of the coefficients (viscosity and thermal conductivity) of the neutron gravitational energy to the quark droplets, as a result of the matter; thus, when these coefficients vanish, a quark bubble spinning down of the rapidly rotating neutron star. necessarily does not form. The present paper is organized as follows. The nucleation However, Ruggeri & Friedman (1996) argued that the en- kinetics of strange matter bubbles is considered, within the ergy flow does not vanish in the absence of any heat con- framework of the Zel’dovich nucleation theory, in x 2. The duction or viscous damping. Since the change of energy expressions for the rate of formation of quark matter are de- density e in time is given, in the low-velocity limit, by the rived, in both the perfect and dissipative neutron fluid case, in conservation equation @e=@t ¼:(wv) (Csernai & Kapusta x 3. The transition of a neutron star to a quark star, due to 1992a), where w is the enthalpy and v is the velocity, this increase of the central density as a result of the spinning down, implies that the energy flow wv is always present. Therefore, is considered in x 4. In x 5 we discuss our results and give an expression for the prefactor can be derived that does not conclusions. vanish in the absence of viscosity. The viscous effects cause 2. NUCLEATION KINETICS OF STRANGE MATTER only a small perturbation to the prefactor. The differences IN NEUTRON STARS between the Csernai-Kapusta (CK; Csernai & Kapusta 1992a) and Ruggieri-Friedman (RF; Ruggeri & Friedman 1996) re- We consider that the change from the metastable neutron sults are due to the technical differences in the treatment of phase to the stable quark phase occurs as the result of fluc- the pressure gradients. tuations in a homogeneous medium, formed of neutrons, in A generalized approach, following the CK formalism and which small quantities of the quark phase (called bubbles or leading to the prefactor in both viscous and nonviscous nuclei) are randomly generated. Since the process of creation regimes, was developed by Shukla et al. (2001). Unlike in of an interface is energetically unfavorable, it follows that Csernai & Kapusta (1992a), the linearized relativistic hydro- when a quark nucleus is below a certain size, it is unstable and dynamics equations have been solved in all regions. disappears again. Surface effects disfavor the survival of small It is the purpose of the present paper to extend the bubbles below the radius Rc (called critical size; the nuclei of Zel’dovich nucleation theory for the formation of quark this size are called critical nuclei or bubbles), which is nothing droplets in neutron matter, by taking into account the effects of but the value that extremizes the thermodynamical work W both the energy flow and thermal conductivity and shear and necessary to create the bubbles. Only nuclei whose size r is bulk viscosities. As a result, a more general expression for the above the value Rc are stable and can survive (Landau & quark matter droplet rate formation can be obtained, which is Lifshitz 1980). The nuclei are assumed to be macroscopic also valid in the limiting case of vanishing conductivity and objects containing a large number of particles (quarks). 948 HARKO, CHENG, & TANG Vol. 608

For strongly degenerate neutron matter, the thermodynamic by substituting for r in the right-hand side of equation (6) the work W necessary to create a quark bubble is (Alcock & zeroth-order approximation given by equation (7). Then, in Olinto 1989; Mardor & Svetitsky 1991; Madsen & Olesen first order, the critical radius is given by 1991; Olesen & Madsen 1994; Horvath 1994) pffiffiffiffiffiffiffiffiffiffiffi R ¼ 1 þ 1 þ b ; ð10Þ Â ÀÁÀÁÃ4 c C W ¼ n P P r3 þ 4r2 þ 8r þ E ; q q n q n 3 c where ð4Þ  ÀÁ C 4 2 4 b ¼ 2 þ q n R : ð11Þ where Pq and Pn are the pressures of the quark matter and 2 3 c0 neutron matter, respectively (exterior to the bubble, assumed to be spherical), is the surface tension, nq is the particle For values of C ¼ 20 MeV fm3, ¼ 18 MeV fm1, in number density of quark matter, and are the chemical po- 3 3 q n the range (75–100 MeV) , q 0:4e fm , and negligible tentials of each phase, is the curvature coefficient, and Ec ¼ the critical radius is of the order of R ¼ 2 7fm.The 3 2 2 n c 5 Z e =r is the Coulomb energy of the droplet (Heiselberg Coulomb correction has the effect of increasing the critical et al. 1993). Here Ze V is the excess charge of ¼ð q nÞ d radius. The value r ¼ Rc corresponds to the limit beyond the droplet with volume Vd compared with the surrounding which large quantities of the quark phase begin to be formed. medium, and q and n are the charge densities of the quark In fact, it is more appropriate to refer not to a limit point and nuclear matter, respectively. Therefore, the Coulomb en- r ¼ R , but to a critical range of values of r near that point, 2 2 5 c ergy of the droplet is given by Ec ¼ 16 (q n) r =15. with width r (T=4)1=2 (Landau & Lifshitz 1980). The Assuming that the quark matter is immersed in a uniform fluctuational development of nuclei in this range can still, with background of electrons and that it is in equilibrium, with high probability, throw them back into the subcritical range, d ¼ s ¼ u þ e, it follows that the total electric charge but nuclei beyond the critical range will inevitably develop 2 in the quark phase can be expressed as q eq(ms =2 into the new quark phase. 2 2eq)= ,whereu d q (Heiselberg et al. 1993). The minimum critical work required to form a stable quark 3 Generally, q n. bubble is therefore given by The appropriate form of the curvature energy can be  3 ÀÁ2 obtained from (Madsen 1993; Heiselberg et al. 1993) 4 2 W ¼ FðbÞ 1 þ 4 Gðb; b Þ ; ð12Þ Z hi c 3C2 q n C3 0 gr 1 1 E ¼ dk k 1 þ exp k curv 3 T 0"# where we denote g2r T 2 ¼ 1 þ O ; ð5Þ F b 2 21 b 3=2 3b 13 6 ð Þ¼ þ ðÞþ þ ð Þ and where g is the statistical weight and is the chemical potential. "# 2 2 ÀÁpffiffiffiffiffiffiffiffiffiffiffi 5 ÀÁpffiffiffiffiffiffiffiffiffiffiffiffiffi Hence, the curvature coefficient is given by ¼ 3 =8 1 1 b 1 1 b 4 1 2 þ þ 1 þ þ 0 18 MeV fm (=300 MeV) (Heiselberg et al. 1993). Gðb; b0Þ¼ ÀÁffiffiffiffiffiffiffiffiffiffiffi : 3=2 5 p 4 Requiring W to be an extreme, @W=@r ¼ 0, gives the fol- 2 þ 21ðÞþ b þ3b 1 þ 1 þ b lowing equation for the value of the critical radius r ¼ R : c ð14Þ 2 4½nqð ÞðPq PnÞr þ 8r þ 8 q n Within a purely thermodynamic approach, one can pose 2 ÀÁ 16 2 ¼ r4: ð6Þ only the problem of calculating the probability of occurrence 3 q n in a medium of fluctuating nuclei of various sizes, the medium being regarded as in equilibrium. Instead of the thermody- By neglecting the Coulomb energy, we obtain for the crit- namic probability of nucleation, it is more convenient to use ical radius Rc0 of the bubble the expression the equilibrium distribution function for nuclei of various pffiffiffiffiffiffiffiffiffiffiffiffiffi sizes existing in the medium, denoted f0(r). Here f0 dr is the Rc0 ¼ 1 þ 1 þ b0 ; ð7Þ number of nuclei per unit volume of the medium with sizes in C the range dr. According to the thermodynamic theory of where fluctuations (Landau & Lifshitz 1980), ÀÁ C ¼nqðq nÞþ Pq Pn ¼nq þ P > 0 ð8Þ f0ðÞr exp½WrðÞ=T : ð15Þ and Near r ¼ Rc the thermodynamic work can be expressed as  2 3 ÀÁ2 b0 ¼ 2Cjj = > 0: ð9Þ 4 24 2 WrðÞ¼ 2 FðbÞ 1 þ q n 3 GðbÞ 3C 5 C pffiffiffiffiffiffiffiffiffiffiffi ÀÁ The parameter b0 can also be represented as b0 3 2 2 6(C=20 MeV fm3)(=18 MeV fm1)½=(7 MeV)32 (Horvath 12C 1 þ b 32 q n 1994).   pffiffiffiffiffiffiffiffiffiffiffi 3 ÀÁ 3 3 1 2 To find the value of the critical radius for a nonnegligible 1 þ 1 þ b 3C ðÞr Rc : ð16Þ electrostatic energy of the bubble, we use an iterative method, No. 2, 2004 NUCLEATION OF QUARK MATTER IN NEUTRON STAR CORES 949

Therefore, for the equilibrium distribution function we find formed in stationary conditions per unit time and per unit the expression volume the expression ( sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffiffiffiffiffiffiffiffiffiffiffi ÀÁÀÁffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi ÀÁ p 2 p 3 3 2 2 12C3 1 b 32 3 1 1 b f0ðrÞ¼f0ðÞRc exp 12C 1 þ b 32 q n þ q n þ þ j ¼ 3 ) ! 3C T  ffiffiffiffiffiffiffiffiffiffiffi 3 ÀÁ BR f R : 22 3 p 3 1 2 ðÞc 0ðÞc ð Þ 1 þ 1 þ b 3C T ðÞr Rc ; ð17Þ Above the critical range, the distribution function is constant: having reached that point, the nucleus becomes steadily larger, where f0(Rc) ¼ C0 exp (Wc=T), C0 ¼ const. with practically no change in the reverse direction. Accord- In order to estimate the coefficient C0 of the exponential in ingly, we can neglect the term containing the derivative @f =@r f0(Rc), we follow Landau & Lifshitz (1980) and assume 2 in the flux, leading to j ¼ Af . From the significance of the flux it C0 ¼ nqnnRc ,wherenn and nq are the particle number densi- follows that the coefficient A acts as a velocity in size space, ties in the neutron and quark phases, respectively. Thus, we A ¼ (dr=dt)macro (Landau & Lifshitz 1980). Therefore, we find obtain  T dr 3 B ¼ 2 4 0 f ðÞ¼R n n R exp FðbÞ W ðrÞ dt macro 0 c q n c 3C2T   ÀÁ pffiffiffiffiffiffiffiffiffiffiffi ÀÁ ÀÁ2 3 3 2 2 2 ¼ 3C T 2 12C 1 þ b 32 q n 1 þ 4q n Gðb; b0Þ : ð18Þ C3 !   pffiffiffiffiffiffiffiffiffiffiffi 3 1 3 dr Let f (t; r) be the kinetic size distribution function of the 1 þ 1 þ b ðÞr Rc : ð23Þ dt nuclei. The elementary process that changes the size of a macro nucleus is the attachment to it, or the loss by it, of a quark The rate of growth of the bubble radius near the critical droplet, and this is to be regarded as a small change, since the radius is given by (dr=dt)macro ¼ k(r Rc), where k is the nuclei are considered to be macroscopic objects. Therefore, dynamical prefactor (Langer & Turski 1973; Csernai & the growth of the nuclei is described by a Fokker-Planck Kapusta 1992a). equation (Landau & Lifshitz 1980) The prefactor k has been evaluated, by solving the equations @ftðÞ; r @j of relativistic fluid dynamics in all regions, in Shukla et al. ¼ ; ð19Þ (2001). The result (also taking into account heat conduction) is @t @r sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  where j ¼B(@f =@r) þ Af is the flux in the size space. Here 2 wn 1 4 k ¼ þ knT þ 2 n þ n ; B is the nuclear size diffusion coefficient and A is connected R3 ðÞw 2 c2 R3ðÞw 2 3 with B by a relationship that follows from the fact that for c s c ð24Þ an equilibrium distribution j ¼ 0. Therefore, we find A ¼ 0 BW (r)=T. where cs is a constant (the velocity of the sound in the medium In the case of a continuous stationary phase transition around the saddle configuration) and kn, n,andn are the process, we have j ¼ const. The constant flux is just the thermal conductivity and shear and bulk viscosity coefficients number of nuclei passing through the critical range per unit of the neutron matter, respectively. The first term in the above time per unit volume of the medium; i.e., it defines the rate of equation is the same as obtained by Ruggeri & Friedman the process. With the condition of constant flux we obtain (1996), corresponding to the case of nonviscous matter. The Bf0(@=@r)( f =f0) ¼ j, giving second term is similar to the results obtained by Csernai & Z Kapusta (1992a) and Venugopalan & Vischer (1994), but with f dr 2 ¼j þ const: ð20Þ a minor difference; i.e., instead of 4 there is a factor cs in the f0 Bf0 numerator. The constant in this equation and j are found from the 3. NEUTRON MATTER TO QUARK MATTER boundary conditions for small and large r. The fluctuation TRANSITION RATES probability increases rapidly with decreasing size, and small nuclei have a high probability of occurrence. This is expressed From equation (24) for the prefactor it follows that the rate by the boundary condition f =f0 ! 1asr ! 0. The boundary of formation of quark bubbles in neutron matter is given by vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi condition for large r can be established by noting that above u the critical range the function f increases without limit, 2 u 3 0 nqnnRc t 3C T j ¼ pffiffiffiffiffiffiffiffiffiffiffi ÀÁÀÁpffiffiffiffiffiffiffiffiffiffiffi whereas the true distribution function f (r) remains finite. This 2 3 2 3 3 12C 1 þ b 32q n 1 þ 1 þ b situation is expressed by imposing the boundary condition sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! f =f ¼ 0forr !1. The solution that satisfies the above 0 2 w fgk T þ 24½ðÞ=3 þ conditions is (Landau & Lifshitz 1980) n þ n n n 3 2 2 3 2 Z Z Rc ðÞw c R ðÞw f 1 dr 1 1 dr  s c ¼ j ; ¼ : ð21Þ 43 f0 r Bf0 j 0 Bf0 exp f ðbÞ 3C2  In these equations the integrand has a sharp maximum at ÀÁ2 2 1 r ¼ Rc. By extending the integration with respect to r from 1 þ 4q n GbðÞ; b0 T : ð25Þ 1 to +1, one obtains for the number of viable quark nuclei C3 950 HARKO, CHENG, & TANG Vol. 608

To obtain the number of the net strange quark matter where pF is the Fermi momentum and ˜1; ˜2 aresomequan- bubbles, the rate j must be multiplied by the time interval tities related to particle-particle cross sections, estimated in available for prompt nucleation, t, and by the volume V0,at Danielewitz (1984). The contributions of the bulk viscosity which the nucleation can take place in the dense core (Horvath and thermal conductivity are negligible, and the main contri- et al. 1992). We impose the condition that at least one quark bution is from the shear viscosity, which is typically of the bubble appears (which would suffice to convert the whole order of 50 MeV fm2 (Heiselberg 1995). But in the case neutron star). Therefore, using equation (25), one obtains the of an ideal neutron gas, with zero viscosity, there will be no following general condition for a quark bubble formation in a bubble growth, and in this case the transition from neutron neutron star core: matter to quark matter in astrophysical objects cannot take place. ¼ jtV0 1: ð26Þ Assuming that the energy flow does not vanish in the ab- The quark bubble formation rate essentially depends on the sence of any heat conduction or viscous damping, and con- enthalpy difference in the two phases. In the density range of sidering that the viscous effects are small and can be interest, all the neutron matter EOS can be very well param- neglected, the condition of the formation of a quark bubble inside the dense core of a neutron star becomes eterized as polytropes, Pn ¼ Kn (Lugones et al. 1994). The B vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi corresponding energy density in the neutron phase is "n ¼ u u 3 nBmn þ½1=( 1)KnB, leading to wn ¼ "n þ Pn ¼ nBmn þ 1 nqnn t 3C RcTwn pffiffiffi pffiffiffiffiffiffiffiffiffiffiffi ÀÁÀÁpffiffiffiffiffiffiffiffiffiffiffi ½=( 1)KnB. We also assume that the formed quark bubble w 3 2 2 3 2 12C 1 þ b 32q n 1 þ 1 þ b consists of u-andd-quarks in the ratio 1: 2; only later weak  3 ÀÁ2 interactions may change the composition to an energetically 4 2 more favorable state. The quarks’ chemical potentials are re- exp 2 FðbÞ 1 þ 4q n 3 GbðÞ; b0 1=3 3C T C lated by d ¼ 2 u, and, assuming chemical equilibrium tV 1: 30 across the phase boundary, we also have n ¼ u þ 2d ¼ 0 ð Þ 4=3 (1 þ 2 )u (Olesen & Madsen 1994). Then the pressure in the quark phase is (assuming a simple bag model) Pq ¼ For r > Rc the time growth of the radius of the quark 4 4 2 ½(u þ d)=4 B and "q ¼ 3Pq þ B (Cheng et al. 1998a), droplets can be described by the equation (Csernai & Kapusta 4 4 2 giving wq ¼½(u þ d)= 3B. Hence, the enthalpy dif- 1992a; Shukla et al. 2001) ference in the two phases can be approximated by dr r Rc 4 þ 4 kR2 ; ð31Þ w ¼ u d 3B n m Kn : ð27Þ dt c r2 2 B n 1 B Assuming that the energy flow is provided by the viscous with the general solution effects only, one obtains for the prefactor the expression "# 2 (Venugopalan & Vischer 1994) 1 1 r r r Rc t þ þ ln 4 ; ð32Þ  k 2 R R R 2 4 c c c k ¼ knT þ 2 n þ n : ð28Þ 2 3 3 ðÞw Rc where we have used the initial condition r(0) ¼ 2Rc.There- In the limit of zero baryon number, kn ! 0 and we obtain fore, for a neutron star with radius R, the time tconv required for the result of Csernai & Kapusta (1992a). If the matter is the conversion of the whole star to a quark star can be obtained baryon-rich, but viscous damping is negligible, n;n ! 0, from and we obtain the results of Langer & Turski (1973) and "# 2 Kawasaki (1975). 1 1 R R R Rc Therefore, the condition of the formation of a quark bubble tconv þ þ ln 4 : ð33Þ k 2 Rc Rc Rc is given by vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u Since R 3 R , with a very good approximation we find u 3 c 1 nqnn t 3C T pffiffiffiffiffiffiffiffiffiffiffi ÀÁÀÁpffiffiffiffiffiffiffiffiffiffiffi 2 ðÞw 2R 3 2 2 3 1 R c 12C 1 þ b 32q n 1 þ 1 þ b t : ð34Þ  conv 2k R 4 43 c k T þ 2 þ exp FðbÞ n 3 n n 3C2T  Neglecting the viscous effects, and by assuming again that ÀÁ2 the neutron matter is described by a polytropic EOS, while the 2 1 þ 4q n GbðÞ; b0 tV0 1: ð29Þ quark phase obeys the simple bag model EOS, one obtains for C3 the conversion time

The transport properties of dense matter have been inten- 2 sively investigated in both high-energy physics and astro- ðÞw 2 tconv pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R physical frameworks (Flowers & Itoh 1976, 1979, 1981; 8R w ÈÉÂÃÀÁc n Hakim & Mornas 1993). The shear viscosity and thermal 4 4 2 2 þ = 3B nBmn ½=ðÞ 1 Kn conductivity k for neutron matter have been derived by ¼ u qd ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiÈÉB R2: Danielewitz (1984) from the Uhlenbeck-Uehling equation 8Rc nBmn þ ½=ðÞ 1 KnB (see also Sawyer 1989; Cutler et al. 1990). They are given 2 1 5 2 2 3 2 1 ð35Þ by ¼ (16 ) ( pF=mn˜1)T and k ¼ 5( pF=mn˜2)T =96, No. 2, 2004 NUCLEATION OF QUARK MATTER IN NEUTRON STAR CORES 951

The volume V(t) of the neutron matter converted to quark where matter after a time interval t < t is given, in this simple conv 1=2 model, by ðÞ1 e2 ÀÁ 31ðÞ e2 f ðeÞ¼ 3 2e2 arcsin e : ð41Þ e3 e2 217=4ðÞw R 3=4 Vt n c t3=2: 36 ðÞ 3=2 ð Þ Assuming that the total mass of the star is a constant, one 3ðÞw obtains the following relation between the variation a, , If the transition from neutron to quark phase is driven by ,ande of the equatorial radius, angular velocity, density, viscous processes only, then and eccentricity, respectively: 3 a 1 e e tconv þ ¼ : ð42Þ ÈÉÀÁ a 1 e2 4 4 2 2 u þ d = 3B nBmn ½=ðÞ 1 KnB 2 Rc R ; The variation of the eccentricity with respect to the angular 2 k T 24=3 fgn þ ½ðÞn þ n velocity is given by ð37Þ e Gf ðeÞð=Þ ¼ ; ð43Þ and GgðeÞ  where 5 3=2 3=2 2 4 3=2 VðtÞ knT þ 2 n þ n t : ð38Þ f ðeÞ df ðeÞ 3ðÞw 3R3=2 3 gðÞ¼e c e de pffiffiffiffiffiffiffiffiffiffiffiffiffi 29ðÞ 2e2 1 e2 þ ðÞ8e2 9 arcsin e 4. ENDOTHERMIC NUCLEATION OF QUARK ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi : ð44Þ 4 2 MATTER BUBBLES IN THE CORE OF e 1 e ROTATING NEUTRON STARS With the use of the above equations the variation of the In the previous sections we have considered the process of gravitational energy of the neutron star, corresponding to a nucleation of quark bubbles in neutron star cores, and we have simultaneous change in equatorial radius, angular velocity, derived the conditions for quark matter formation in an as- and density, is given by  trophysical context. In the present section we apply the results 3 GM 2 1 previously obtained to analyze the possibility of the transition EgrðÞ¼a;; AeðÞ BeðÞ ; ð45Þ from neutron to quark phase during the spinning down of a 5 a G rapidly rotating neutron star. In this case the rotational energy where we denote of the star can be used to trigger the phase transition. The minimum energy density required to create a quark 1 e ffiffiffiffiffiffiffiffiffiffiffiffiffi1 arcsin e bubble is hðeÞ¼ 2 p þ 2 ; ð46Þ 3 1 e e 1 e2 e 3W u ¼ c hðeÞ hðeÞ f ðeÞ 1 q 4R3 AðeÞ¼ ; BðeÞ¼ þ : ð47Þ c  gðeÞ gðeÞ 3 C 24 ÀÁ2 ÀÁffiffiffiffiffiffiffiffiffiffiffi 2 The gravitational energy density released as the result of the ¼ p 3 FðbÞ 1 þ q n 3 GðbÞ : 1 þ 1 þ b 5 C slowing down of the neutron star can trigger the endothermic ð39Þ phase transition at the center of the star. Inside a sphere of radius Rq the gravitational energy density is An estimation of equation (39) for C 20 MeV fm3,  2 1 3 3 9 GM 1 18 MeV fm , (75 MeV) , q 0:4e fm ,and u a;; R3 Ae Be : 34 3 grðÞ¼ q ðÞ ðÞ n 0givesuq 3:76 10 ergs cm . 20 a G We consider that the evolution of the rotating neutron star ð48Þ can be approximated by a sequence of MacLaurin spheroids (Chandrasekhar 1986; Cheng et al. 1992). The mass of the This energy should be greater than or equal to the mini- star is denoted by M andthemajorandminoraxisofthestar mum energy necessary for the formation of a quark bubble, by a and c (equatorial and radius). Then the eccentricity given by equation (39). Therefore, the gravitational energy 1=2 of the star is defined according to e ¼½1 (c=a)2 .Let converts to the quark phase the neutron matter inside a sphere ¼ const denote the density of the star, corresponding to a of radius given value of the angular velocity and of the equatorial radius a. M 2=3 a 1=3 R a;; 2:88 Hence, the basic equations describing the mass, hydrostatic qðÞ¼ 6 M 10 equilibrium, and gravitational energy Egr of the rotating star  1 1=3 are (Chandrasekhar 1986) AeðÞ BeðÞ G ÀÁ 4 1=2 ÀÁpffiffiffiffiffiffiffiffiffiffiffi M ¼ a3 1 e2 ; 2 ¼ 2Gf ðeÞ; 1 þ 1 þ b C1=3 3 hi 1=3 106 cm: 1=3 ðÞ 3 GM 2 arcsin e 2 þ 21ðÞþ b 3=2 þ 3b E ¼ ; ð40Þ gr 5 a e ð49Þ 952 HARKO, CHENG, & TANG Vol. 608

In order for the whole neutron star to be converted to a TABLE 1 quark star, the condition R ðÞ¼; aðÞ; must hold. On Variation of the Central Density of the Rapidly Rotating Neutron for Different Equations of State the other hand, for typical neutron star densities of the order of 1014–1015 gcm3,thetermA(e)/G is much smaller than Mstat Rstat B(e)(/)/, 1 EOS (M) (km) (s ) c /c 1 AeðÞ TBeðÞ : ð50Þ A...... 1.6551 8.368 10011 1.26 G AU...... 2.1335 9.411 10587 0.97 FPS...... 1.7995 9.281 8874.9 0.91 For small eccentricities e the function B(e) can be approxi- L ...... 2.7002 13.7 6482.9 1.244 1 mated by 3 . M...... 1.8045 11.6 4437 4.189 Therefore, we obtain the following condition for the relative change in the central density, necessary to convert a neutron Note.—Variation of the central density of the rapidly rotating neutron star star to a strange star, due to the endothermic gravitational for different EOSs (Cook et al. 1994): A (Pandharipande 1971), AU (Wiringa energy transfer in the whole volume of the star: et al. 1988), FPS (Lorentz et al. 1993), L (Pandharipande et al. 1976), and hi M (Pandharipande & Smith 1975). 3=2 C 2 þ 21ðÞþ b þ3b M 2=3 0:125 ÀÁffiffiffiffiffiffiffiffiffiffiffi : ð51Þ canonical pulsars (with low magnetic fields of the order of p 3 11 13 1 þ 1 þ b M B 10 10 G) in the sense that they spin down much more rapidly. Foraneutronstarofaroundtwosolarmasses,M 2 M,and If we assume that the spin-down of the is com- by assuming C 20 MeVand b 6, we obtain = 1:85. pletely determined by the torque of its relativistic wind Of course, this result is strongly dependent on the precise emission, generated via the magnetic dipole radiation, then the numerical values of C and b, which are generally poorly time variation of the angular velocity of the star is given by known; C is also a neutron matter EOS-dependent parameter. Shapiro & Teukolsky (1983), For example, assuming for C a value of 10 MeV will lead to = 0:92. An increase in the mass M of the star will also 2 23 I˙ ¼ ; ð52Þ decrease the value of the relative change in the density of the 3 c3 star necessary to convert the neutron star. At present, we have very little observational knowledge where I is the moment of inertia of the star and ¼ R3B is the regarding how fast newborn neutron stars can rotate. The most magnetic dipole moment. When the star is born with a spin well-known young pulsar is the Crab pulsar, which was born period much shorter than the observed one, the age of the star with a rotational period of about 20 ms (Manchester & Taylor is the spin-down age sd ¼ P=P˙ ,whereP and P˙ are the present 1977). If so, during its lifetime the pulsar’s central density only spin period and its time derivative, respectively. If the spin- increases by about c=c 0:001 (Ma & Xie 1996). Cur- down is entirely due to the magnetic dipole radiation, we rently, the fastest rotating pulsar known is PSR 1937+214, obtain whichhasaperiodof1.55msor 4000 s1, but it has 3 Ic3 weak magnetic field, and it has been suggested that it is spun sd ¼ ¼ : ð53Þ up by accretion (Alpar et al. 1982). However, it has been ˙ 2 B2R62 suggested that some rapidly spinning millisecond pulsars are By adopting for the moment of inertia and the radius of the be born by accretion-induced collapse from white dwarfs magnetar the typical values I ¼ 1045 gcm2 and R ¼ 106 cm, (Arons 1983). Therefore, it cannot be ruled out that some andbyassumingthatthestarwasbornwithamagnetic pulsars may be born with millisecond periods and strong field of the order of B 1015 G and with an angular velocity magnetic field. 1 of 0 ¼ 6000 rad s , the spin-down age is given by sd The theoretical investigation of rotating general relativistic 1125 s. objects performed by Cook et al. (1994) shows that for some From equation (52) it follows that the decay law of the realistic EOSs of neutron matter, this variation of the central angular velocity of the star is of the form density can be achieved during the complete spin-down of the star (see Table 1). qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0 The central density increase due to the spin-down can be ¼ ÀÁ; ð54Þ 2 2 6 3 easily realized in a short time interval for a special class of ðÞ4=3 B 0R =c I t þ 1 stellar-type objects, called magnetars. Magnetars are compact objects with superstrong magnetic fields of the order B where 0 is the initial angular velocity of the star. 1015 G or even higher (Duncan & Thompson 1992; Paczynski Hence, in a time interval of around t 106 sandfora 1992; Thompson & Duncan 1995, 1996). It is now believed magnetic field B ¼ 1015 G the angular velocity of the star 1 1 that soft gamma repeaters (SGRs)—a small class (four con- decreases from 0 ¼ 6000 rad s to ¼ 142 rad s .Foran firmed and one candidate) of high-energy transients discovered initial angular velocity of the order of ¼ 8000 rad s1 and 16 through their emission of bright X-ray/gamma-ray bursts, for a magnetic field of the order of B 10 G, sd 5s.Ina which repeat on timescales of seconds to years—are magne- time interval of around t 600 s the angular velocity of the tars. There is evidence that the giant SGR flares involve the star decreases to 579 rad s1. cooling of a confined e -photon plasma in an ultrastrong Therefore, in a magnetar density changes, due to the spin- magnetic field (Lyubarsky et al. 2002). Some authors (Cheng down, take place in a short interval of time. Hence, they can & Dai 1998; Usov 2001), motivated in part by the super- provide enough energy to trigger spontaneous nucleation to Eddington of the SGRs’ giant flares, have sug- the quark phase, once the central density of the star goes gested that they are strange stars. The magnetars differ from the above the critical density. No. 2, 2004 NUCLEATION OF QUARK MATTER IN NEUTRON STAR CORES 953

The nucleation of quark droplets in the presence of strong due to dissipation is slow, causing the temperature to vary magnetic fields essentially depends on the strength of the slowly across the bubble wall. We also considered that the magnetic field. The energy of a quantum particle changes phase transition is strongly first order, releasing considerable significantly if the magnetic field is of the order or exceeds the amounts of heat. 2 3 critical value Bc ¼ m c =e f,wherem is the mass of the par- The growth of the critical size quark bubbles nucleated in ðeÞ ticle. For electrons the value of the critical field is Bc ¼ 4:4 the first-order transition in the neutron star core is governed 1013 G. In order to calculate the value of the critical magnetic by the dynamical prefactor k. There are several expressions for field for quarks, we have to use the estimations for the quark the prefactor obtained in the physical literature. In the Csernai- masses. However, there are many uncertainties in the numer- Kapusta approach (Csernai & Kapusta 1992a), the prefactor ical values of the quark masses, which play an essential role in essentially depends on the dissipative properties (shear, bulk the calculation of the critical magnetic field and are generally viscosity, and thermal conductivity) of neutron matter. poorly known. In the case of free quarks, the current quark Therefore, in a nonviscous medium with n ¼ n ¼ kn ¼ 0, mass is considered to be in the range of mq 5–10 MeV. the growth of the quark bubbles cannot take place. The main ðqÞ Then the corresponding critical magnetic field is Bc physical assumption is that the energy flow is provided by the 2 ðeÞ 15 (1 2)10 Bc (4:4 8:8)10 G. For the current quark viscous effects only. mass, this is the typical strength of the magnetic field at which On the contrary, one can argue that the energy flow does not the cyclotron lines begin to occur. In this limit the cyclotron vanish in the absence of heat conduction or viscous damping quantum is of the order of or greater than its rest energy. This is (Ruggeri & Friedman 1996; Shukla et al. 2001) . In this case, equivalent to the requirement that the de Broglie wavelength is and in the limit of zero viscosity, the prefactor depends only of the order of or greater than the Larmor radius of the particle on two scale parameters, the correlation length and the in the magnetic field (Chakrabarty 1995). However, from the critical radius of the quark bubble Rc. For a viscous medium physical point of view a better description of the quark mass the prefactor can simply be written as the sum of the viscous can be obtained by using, instead of the current quark mass, the and nonviscous terms, with the viscous term not affecting the effective or constitute quark mass, which takes into account the growth process significantly. The previous (viscous) results effects of the strong interactions, and which is of the order can be reobtained by using some assumptions for the velocity mq 100–300 MeV (Cheng et al. 1998a). The critical of sound in the medium around the saddle configuration. magnetic field corresponding to the effective quark mass could The condition for the formation of at least a quark bubble in ðqÞ 17 18 be as high as Bc 10 10 G. the neutron matter as a result of the thermal fluctuation is given The study of the behavior of the surface and curvature by equation (29). The condition includes the cases of both energy terms in strong magnetic fields has shown that both viscous and ideal neutron matter. It is extremely sensitive with ðqÞ quantities diverge for B > Bc , with the curvature term di- respect to the numerical values of the microscopic parameters verging much faster. Consequently, in the presence of strong characterizing the quark bubble, such as the surface tension , magnetic fields the rate of stable quark droplet formation per the bag constant B, or the curvature coefficient . In discussing unit volume I T 4 exp (W=T) tends to zero, I ! 0, and the astrophysical implications of equation (29), we consider therefore there cannot be any thermal nucleation of quark separately the two limiting cases, corresponding to viscous droplets in neutron star cores (Chakrabarty 1995, 1996). and nonviscous neutron matter, respectively. The value of B inside a magnetar is not known, and it is not In the case of ideal neutron matter, the condition of the certain if the interior magnetic field is much stronger than the formation of a quark bubble is given by equation (30). This surface field. If in the interior of the magnetar the value of the equation determines at which temperature the phase transition ðqÞ 15 magnetic field can exceed the critical value Bc 5 10 G, takes place. For example, in the case of the Walecka mean which represents the most conservative estimation of the field equation of state (Walecka 1975), with K ¼ 15758 and critical magnetic field, corresponding to the free quark mass ¼ 4:95 (Lugones et al. 1994), the temperature Tc necessary for m ¼ 5 MeV, then the rate of droplet formation can be con- the formation of a stable quark bubble in a second (t ¼ 1s), 3 siderably reduced and the transition from neutron matter to in a volume V0 with radius R0 ¼ 1km(V0 ¼ 4=3km ), quark matter could not take place. and at the center of the neutron star is around Tc ¼ 12 On the other hand, by taking into account that the physical 13 MeV, for ¼ (85 MeV)3. Here we have also assumed the mass of the quark is the effective mass, the corresponding standard values for the quark matter chemical potential and for ðqÞ 17 18 3 critical magnetic field could be of the order Bc 10 10 the bag constant, u ¼ d ¼ 280 MeV and B ¼ 60 MeV fm G, a value that could be higher than the magnitude of the (Cheng et al. 1998a). Since the temperature of a newborn magnetic field inside the magnetars. For magnetic fields neutron star is around 10 MeV (Olesen & Madsen 1994), this ðqÞ smaller than Bc , the endothermic nucleation process can take value of the surface tension seems to rule out, for the given place in a short time interval during the spin-down of the equation of state, the possibility of quark nuclei forming in the magnetar. neutron matter. However, a small variation in the value of , ¼ (75 MeV)3, reduces the temperature for quark nuclei for- 5. DISCUSSIONS AND FINAL REMARKS mation to Tc ¼ 4 6 MeV, a value that does not exclude the In this paper we have considered a kinetic theory of strange possibility that quark nuclei formation can be initiated in the matter nucleation in neutron stars. We derived general nec- early stages of neutron star evolution. Smaller values of ,of essary conditions for neutron to quark matter conversion. the order of ¼ (50 MeV)3, can lower the phase transition There have been several assumptions made in the derivation temperature even more, making it possible even for cold neu- of the results, mostly related to the form of the growth speed tron stars. The transition temperature is relatively insensitive to of the bubble. The result is valid only if nonlinear effects can the details of the equation of state of the dense neutron matter. be ignored, and the linearized hydrodynamic equations are If the temperature at which the phase transition is initiated is applicable. Furthermore, we have assumed that the radii of the strongly dependent on the microscopic model adopted for bubbles are larger than the correlation length and that heating describing the neutron matter properties, the total transition 954 HARKO, CHENG, & TANG Vol. 608 time of the neutron star to a quark star, given by equation (35), (Danielewitz 1984). Consequently, by neglecting the heat is much less dependent on the details of the equation of state conduction and the bulk viscosity, the conversion time or the exact numerical values of the surface tension or cur- becomes vature of the bubble. Generally, for a neutron star with radius 2 2 R ¼ 12 15 km, the conversion time is very long, of the order 3 ðÞw 3 ðÞw t R R2 R T 2R2: ð56Þ 6 3 3 conv c c 2 of 10 yr for (75 MeV) .For (50 MeV) or even 32 n 32 0ðÞn=n0 lower, the conversion time could have values of the order of 107 yr, which are still of the same order of magnitude as the In small temperatures, the shear viscosity becomes very lifetime of a neutron star (107 yr). Hence, in the present ap- large. Hence, for a shear viscosity–driven phase transition the proach, which is limited to the consideration of linear effects conversion time could be very small. only in the hydrodynamic description of the first-order phase In Figure 1 we have represented tconv given by equation (56) transitions for an ideal neutron fluid, the growth of the quark as a function of temperature, for different equations of state nuclei is very slow. Therefore, the results of the present of neutron matter. We have considered four equations of analysis suggest that even though quark nuclei could appear state, namely, the equation of state (EOS) of the free neutron inside the neutron star at a very early stage, the transition to a gas (Shapiro & Teukolsky 1983), the Bethe-Johnson EOS pure quark phase ends only in the final stage of its evolution. (Shapiro & Teukolsky 1983), the Lattimer-Ravenhall EOS If the neutron matter–quark matter phase transition is (Lattimer & Ravenhall 1978), and the Walecka EOS (Walecka mainly driven by viscous processes, as in the CK scenario, 1975). In all cases we calculated the conversion time for a then the time necessary for a neutron star to convert to a quark neutron star with radius R ¼ 10 km. star follows from equations (28) and (34). In order to nu- As one can see from the figure, in this case the conversion merically estimate the conversion time, we need an estimation time is much shorter as compared to the ideal neutron matter of viscosity and conduction coefficients of nuclear matter. We case or thermal conduction–driven phase transition. Time tconv also consider two limiting cases for the phase transition. Be- is rapidly decreasing with the temperature. For a neutron star 14 3 low the critical density ¼ c 10 gcm , cold neutron with rapidly cooling processes (e.g., direct , kaon matter consists of two distinct phases. At low density, neutron condensation), the stellar temperature can decrease to less than star matter consists of nuclei that form a solid lattice for 102 MeV in a timescale of 104 yr (Tsuruta 1998). According temperatures below the melting temperature and a sea of rel- to Figure 1, the conversion can take place on a timescale less ativistic electrons. When the density reaches 1011 gcm3, than 103 yr if the spin-down timescale is less than this neutrons begin to ‘‘drip’’ from the neutron-rich nuclei and timescale. This can be the case if this neutron star is a mag- then, in addition to the relativistic electrons, there is a sea of netar. We should point out that the interiors of magnetars are nonrelativistic neutrons. At the critical density the neutron- not quantum liquid, i.e., superfluid or superconducting, be- rich nuclei dissolve, leaving seas of degenerate neutrons, cause the strong magnetic field can destroy the cooper protons, and electrons. The neutrons in cold neutron star pairs (Chau et al. 1992). matter are quite likely superfluid, and therefore the transport On the other hand, if the neutrons and protons are superfluid, coefficients are dominated by electrons, with the motion of then the transport properties are completely determined by electrons determined by electron-electron and electron-proton electron-electron scattering. In fact, in this case the viscosity of 1 4 2 5 scattering (Flowers & Itoh 1979). As a result, the viscosity is the nuclear fluid is given by ¼ (15 =2pe )(2pe=kTF) 2 4 2 much lower, but the thermal conductivity is only slightly ½5=2 þ 3(me=pe) þ (me=pe) T (Flowers & Itoh 1976), where 1=2 smaller as compared to the case in which both neutrons and is the fine-structure constant, kTF ¼ (4pee=) is the protons form normal fluids (Flowers & Itoh 1979). Hence, we Thomas-Fermi wavevector, and me , pe ,ande are the electron can neglect the viscosity coefficients with respect to the mass, momentum, and energy, respectively. thermal conductivity and approximate kn by kn k014=T8, We assume that the electron gas inside the neutron star is 23 with k0 10 (Flowers & Itoh 1979). A slightly different extreme relativistic and strongly degenerate. Then it follows 1=3 expression for the leading term in thermal conductivity has that pe pF ¼ (3ne=8) (Shapiro & Teukolsky 1983), with 1=4 been proposed in Danielewitz (1984), knT k0(n=n0) , pF the Fermi momentum and ne the electron number density. 3 3 with k0 ¼ 0:15 fm and n0 ¼ 0:145 fm . Hence, by using Since ne can be related to the total particle number n by means this last expression, we obtain for tconv of the relation ne ¼ Yen, with Ye the mean number of electrons per baryon, it follows that p (3Y n =8)1=3(n=n )1=3. 2 e e 0 0 ðÞw Hence, we obtain for the shear viscosity coefficient of the t R R2: ð55Þ conv c 1=4 nuclear matter 4k0ðÞn=n0 ( "# 2 The transition time is independent of the temperature of the 15 85=337=6 3=2 5 m ðÞ8 1=3 1 e neutron matter. It depends only on the microscopic properties 5=3 5=3 þ 3 1=3 1=3 2 ðÞ3Yen0 ðÞn=n0 2 ðÞ3Yen0 ðÞn=n0 of the quark bubble (surface tension and curvature coefficient), "#) density, and the equation of state of the matter inside the star. 1=3 4 3 meðÞ8 2 For u ¼ d ¼ 280 MeV, B ¼ 60 MeV fm ,and þ T : ð57Þ 3 1=3 1=3 (75 MeV) equation (55) gives, for a neutron star with radius ðÞ3Yen0 ðÞn=n0 R ¼ 12 km and for several equations of state, values of the 11 conversion time of the order tconv 10 yr. The variation of the conversion time, as a function of In the case of young and hot neutron stars, the main dis- temperature, is represented, for a superfluid neutron and pro- sipative mechanism that could drive the phase transition is the ton core of the star with Ye ¼ 0:1, in Figure 2. shear viscosity of the neutron matter. The temperature de- In this case, since the viscosity of the superfluid nuclear pendence of the shear viscosity coefficient can be approxi- matter is much smaller, the conversion time is longer than for 2 2 3 2 mated by n (0=T )(n=n0) , with 0 ¼ 1700 MeV fm a nuclear fluid dominated by the neutron viscosity. However, No. 2, 2004 NUCLEATION OF QUARK MATTER IN NEUTRON STAR CORES 955

Fig. Fig. 1.—Variation of the conversion time tconv for a neutron star with radius 2.—Variation of the conversion time tconv for a neutron star with radius R ¼ 10 km, for different EOSs of the neutron matter: free neutron EOS (solid R ¼ 10 km to a quark star, in the case of the presence of a superfluid core, for curve), Bethe-Johnson EOS (dotted curve), Lattimer-Ravenhall EOS (short- different EOSs of the neutron matter: free neutron EOS (solid curve), Bethe- dashed curve), and Walecka EOS (long-dashed curve). Johnson EOS (dotted curve), Lattimer-Ravenhall EOS (short-dashed curve), and Walecka EOS (long-dashed curve). in this case the phase transition from neutron matter to quark matter can also take place at very low temperatures. Conse- neutron stars quark matter and baryonic matter may be con- quently, a very short tconv , of the order of hours or a few days, tinuously connected. Then the growth of the interface between is also allowed in the framework of this model. the neutron and CFL quark phase could be the result of ex- It is widely accepted that at asymptotic densities the ground terior energy absorption, via an endothermic mechanism. state of QCD with ms ¼ 0 is the color-flavor locked (CFL) In summary, by using the kinetic nucleation theory, we have phase (Alford et al. 1998; Rajagopal & Wilczek 2001). In this considered the possibility that the neutron matter–quark matter phase, color gauge symmetry is completely broken, as are phase transition could be triggered by the gravitational energy both chiral symmetry and baryon number. The effective cou- released during the spin-down of a pulsar. This process could pling is weak and the low-energy properties can be determined take place in a very short time interval, of the order of a few by using methods from the theory of superconductivity. For tens of seconds, in a class of dense compact objects, with very large gaps ( 100 MeV), the CFL phase is rigorously high magnetic fields, called magnetars. However, the per- electrically neutral, despite the unequal quark masses and even centage of neutron star matter that can be converted into quark in the presence of electron chemical potential. The transition matter depends on the initial period, the neutron star mass, and from nuclear matter to quark matter via bubble nucleation can the equation of state of the neutron and quark phases. An im- be greatly simplified if the transition occurs directly to quark portant and open issue is how we could determine if such a matter in the CFL phase (Rajagopal & Wilczek 2001). For a process is taking place or has already taken place in some given baryonic chemical potential ¯, electrically neutral nu- pulsars or magnetars, as well as what the observational sig- clear matter and electrically neutral quark matter have differ- natures of the endothermic neutron matter–quark matter tran- ent values of the electronic chemical potential e.Sincee sition are. In particular, it would be interesting to know if such must be continuous across any interface, a mixed phase region a process could be a possible gamma-ray burst mechanism, as is formed, within which positively charged nuclear matter and suggested by Berezhiani et al. (2003). All of these issues will negatively charged quark matter with the same e coexist at be considered in a subsequent paper (P. S. Tang et al. 2004, in any given radius. The growth of the quark droplets is at the preparation). expense of the nuclear matter. However, if the quark matter is in the CFL phase, an interface between bulk nuclear matter with nonzero electron number Ne 6¼ 0andCFLquarkmatter with Ne ¼ 0 may bep stable,ffiffiffi as long as e satisfies the condition This work is supported by an RGC grant of the Hong Kong 2 jms =4¯ j < = 2,where is the variation of the government of the SAR of China. The authors would like to chemical potential (Rajagopal & Wilczek 2001). Therefore, thank the anonymous referee for comments that helped to the existence of the CFL phase supports the idea that in improve the manuscript.

REFERENCES Alcock, C., Farhi, E., & Olinto, A. 1986, ApJ, 310, 261 Chakrabarty, S. 1996, Phys. Rev. D, 54, 1306 Alcock, C., & Olinto, A. 1989, Phys. Rev. D, 39, 1233 Chakrabarty, S., & Sahu, P. K. 1996, Phys. Rev. D, 53, 4687 Alford, M., Rajagopal, K., & Wilczek, F. 1998, Phys. Lett. B, 422, 247 Chandrasekhar, S. 1986, Ellipsoidal Figures of Equilibrium (New York: Dover) Alpar, M. A., Cheng, A. F., Ruderman, M. A., & Shaham, J. 1982, Nature, Chau, H. F., Cheng, K. S., & Ding, K. Y. 1992, ApJ, 399, 213 300, 728 Cheng, K. S., Chau, W. Y., Zhang, J. L., & Chau, H. F. 1992, ApJ, 396, 135 Arons, J. 1983, Nature, 302, 301 Cheng, K. S., & Dai, Z. G. 1996, Phys. Rev. Lett., 77, 1210 Berezhiani, Z., Bombaci, I., Drago, A., Frontera, F., & Lavagno, A. 2003, ApJ, ———. 1998, Phys. Rev. Lett., 80, 18 586, 1250 Cheng, K. S., Dai, Z. G., & Lu, T. 1998a, Int. J. Mod. Phys. D, 7, 139 Bodmer, A. R. 1971, Phys. Rev. D, 4, 1601 Cheng, K. S., Dai, Z. G., Wei, D. M., & Lu, T. 1998b, Science, 280, 407 Caldwell, R. R., & Friedman, J. R. 1991, Phys. Lett. B, 264, 143 Cheng, K. S., & Harko, T. 2000, Phys. Rev. D, 30, 083001 Chakrabarty, S. 1993, Phys. Rev. D, 48, 1409 ———. 2003, ApJ, 596, 451 ———. 1995, Phys. Rev. D, 51, 4591 Cook, G. B., Shapiro, S. L., & Teukolsky, S. A. 1994, ApJ, 424, 823 956 HARKO, CHENG, & TANG

Csernai, L. P., & Kapusta, J. I. 1992a, Phys. Rev. D, 46, 1379 Lugones, G., Benvenuto, O. G., & Vucetich, H. 1994, Phys. Rev. D, 50, 6100 ———. 1992b, Phys. Rev. Lett., 69, 737 Lyubarsky, Y., Eichler, D., & Thompson, C. 2002, ApJ, 580, L69 Cutler, C., Lindblom, L., & Splinter, R. J. 1990, ApJ, 363, 603 Ma, F., & Xie, B. 1996, ApJ, 462, L63 Dai, Z. G., Peng, Q. H., & Lu, T. 1995, ApJ, 440, 815 Madsen, J. 1993, Phys. Rev. Lett., 70, 391 Danielewitz, P. 1984, Phys. Lett. B, 146, 168 Madsen, J., & Olesen, M. L. 1991, Phys. Rev. D, 43, 1069 Duncan, R. C., & Thompson, C. 1992, ApJ, 392, L9 Manchester, R. N., & Taylor, J. H. 1977, Pulsars (San Francisco: Freeman) Flowers, E., & Itoh, N. 1976, ApJ, 206, 218 Mardor, I., & Svetitsky, B. 1991, Phys. Rev. D, 44, 878 ———. 1979, ApJ, 230, 847 Ng, C. Y., Cheng, K. S., & Chu, M. C. 2003, Astropart. Phys., 19, 171 ———. 1981, ApJ, 250, 750 Norsen, T. 2002, Phys. Rev. C, 65, 045805 Glendenning, N. K. 1992, Phys. Rev. D, 46, 1274 Olesen, M. L., & Madsen, J. 1994, Phys. Rev. D, 49, 2698 ———. 2000, Compact Stars, Nuclear Physics, Particle Physics and General Olinto, A. V. 1987, Phys. Lett. B, 192, 71 Relativity (New York: Springer) Paczynski, B. 1992, Acta Astron., 42, 145 Glendenning, N. K., Pei, S., & Weber, F. 1997, Phys. Rev. Lett., 79, 1603 Pandharipande, V. R. 1971, Nucl. Phys. A, 174, 641 Grassi, F. 1998, ApJ, 492, 263 Pandharipande, V. R., Pines, D., & Smith, R. A. 1976, ApJ, 208, 550 Haensel, P., Zdunik, J. L., & SchaeAer, R. 1986, A&A, 160, 121 Pandharipande, V. R., & Smith, R. A. 1975, Nucl. Phys. A, 237, 507 Hakim, R., & Mornas, L. 1993, Phys. Rev. C, 47, 2846 Rajagopal, K., & Wilczek, F. 2001, Phys. Rev. Lett., 86, 3492 Harko, T., & Cheng, K. S. 2000, Phys. Lett. A, 266, 249 Ruggeri, F., & Friedman, W. A. 1996, Phys. Rev. D, 53, 6543 ———. 2002, A&A, 385, 947 Sawyer, R. S. 1989, Phys. Lett. B, 233, 412 Heiselberg, H. 1995, in Quark Matter Droplet Formation in Neutron Stars, Shapiro, S. L., & Teukolsky, S. A. 1983, Black Holes, White Dwarfs and International Symposium on Strangeness and Quark Matter, ed. S. Kumar & Neutron Stars (New York: Wiley) J. Madsen (Singapore: World Scientific), 338–342 Shukla, P., & Mohanty, A. K. 2001, Phys. Rev. C, 64, 054910 Heiselberg, H., Pethick, C. J., & Staubo, E. F. 1993, Phys. Rev. Lett., 70, 1355 Shukla, P., Mohanty, A. K., & Gupta, S. K. 2001, Phys. Rev. D, 63, 014012 Horvath, J. E. 1994, Phys. Rev. D, 49, 5590 Shukla, P., Mohanty, A. K., Gupta, S. K., & Gleiser, M. 2000, Phys. Rev. C, Horvath, J. E., & Benvenuto, O. G. 1988, Phys. Lett. B, 213, 516 62, 054904 Horvath, J. E., Benvenuto, O. G., & Vucetich, H. 1992, Phys. Rev. D, 45, 3865 Thompson, C., & Duncan, R. C. 1995, MNRAS, 275, 255 Itoh, N. 1970, Prog. Theor. Phys., 44, 291 ———. 1996, ApJ, 473, 322 Kawasaki, K. 1975, J. Stat. Phys., 12, 365 Tsuruta, S. 1998, Phys. Rep., 292, 1 Landau, L. D., & Lifshitz, E. M. 1980, Statistical Physics (Oxford: Pergamon) Usov, V. V. 2001, Phys. Rev. Lett., 787, 21101 Langer, J. S., & Turski, L. A. 1973, Phys. Rev. A, 8, 3230 Venugopalan, R., & Vischer, A. P. 1994, Phys. Rev. E, 49, 5849 Lattimer, J. M., & Ravenhall, D. G. 1978, ApJ, 223, 314 Walecka, J. D. 1975, Phys. Lett. B, 59, 109 Lorentz, C. P., Ravenhall, D. G., & Pethick, C. J. 1993, Phys. Rev. Lett., 70, 379 Wiringa, R. B., Fiks, V., & Fabrocini, A. 1988, Phys. Rev. C, 38, 1010 Lugones, G., & Benvenuto, O. G. 1995, Phys. Rev. D, 52, 1276 Witten, E. 1984, Phys. Rev. D, 30, 272

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