Properties of High-Density Matter in Neutron Stars
Fridolin Weber, Gustavo A. Contrera, Milva G. Orsaria, William Spinella, Omair Zubairi
aa r X i v : . [ a s t r o - ph . S R ] A ug PROPERTIES OF HIGH-DENSITY MATTER IN NEUTRON STARS
Fridolin Weber ∗ Department of Physics, San Diego State University,5500 Campanile Drive, San Diego, California 92182, USA andCenter for Astrophysics and Space Sciences,University of California,San Diego, La Jolla, California 92093, USA.
Gustavo A. Contrera † Department of Physics, San Diego State University,5500 Campanile Drive, San Diego, California 92182, USA,CONICET, Rivadavia 1917, 1033 Buenos Aires, Argentina,IFLP, CONICET - Dpto. de F´ısica, UNLP, La Plata, Argentina andGravitation, Astrophysics and Cosmology Group,Facultad de Ciencias Astron´omicas y Geof´ısicas, UNLP,Paseo del Bosque S/N (1900), La Plata, Argentina.
Milva G. Orsaria ‡ Department of Physics, San Diego State University,5500 Campanile Drive, San Diego, California 92182, USA,CONICET, Rivadavia 1917, 1033 Buenos Aires, Argentina andGravitation, Astrophysics and Cosmology Group,Facultad de Ciencias Astron´omicas y Geof´ısicas, UNLP,Paseo del Bosque S/N (1900), La Plata, Argentina.
William Spinella § and Omair Zubairi ¶ Department of Physics & Computational Science Researech Center,San Diego State University, 5500 Campanile Drive, San Diego, California 92182, USA. (Dated: August 4, 2014)This short review aims at giving a brief overview of the various states of matter that have beensuggested to exist in the ultra-dense centers of neutron stars. Particular emphasis is put on therole of quark deconfinement in neutron stars and on the possible existence of compact stars made ofabsolutely stable strange quark matter (strange stars). Astrophysical phenomena, which distinguishneutron stars from quark stars, are discussed and the question of whether or not quark deconfinementmay occur in neutron stars is investigated. Combined with observed astrophysical data, such studiesare invaluable to delineate the complex structure of compressed baryonic matter and to put firmconstraints on the largely unknown equation of state of such matter.
I. INTRODUCTION
Exploring the properties of compressed baryonic matter, or, more generally, strongly interacting matter at highdensities and/or temperatures, has become a forefront area of modern physics. Experimentally, the properties of suchmatter are being probed with the Relativistic Heavy Ion Collider RHIC at Brookhaven and the Large Hadron Collider(LHC at CERN). Great advances in our understanding of such matter are expected from the next generation of heavy-ion collision experiments at FAIR (Facility for Antiproton and Ion Research at GSI) and NICA (Nuclotron-bases IonCollider fAcility at JINR) [1–3].Neutron stars contain compressed baryonic matter permanently in their centers. Such stars are remnants of massivestars that blew apart in core-collapse supernova explosions. They are typically about 20 kilometers across and spinrapidly, often making many hundred rotations per second. Many neutron stars form radio pulsars, emitting radio ∗ Electronic address: [email protected] † Electronic address: contrera@fisica.unlp.edu.ar ‡ Electronic address: [email protected] § Electronic address: [email protected] ¶ Electronic address: [email protected] roperties of High-Density Matter in Neutron Stars 2waves that appear from the Earth to pulse on and off like a lighthouse beacon as the star rotates at very high
Figure 1: (Color online) Schematic structures of quark stars (left) and neutron stars (right). If the strange quark matterhypothesis should be correct, most, if not all, neutron stars should in fact be strange quark stars. speeds. Neutron stars in X-ray binaries accrete material from a companion star and flare to life with tremendousbursts of X-rays. Depending on mass and rotational frequency, gravity compresses the matter in the cores of neutronstars to densities that are several times higher than the density of ordinary atomic nuclei. At such huge densitiesatoms themselves collapse, and atomic nuclei are squeezed so tightly together that new fundamental particles maybe produced and novel states of matter be created. The most spectacular phenomena stretch from the generationof hyperons and delta particles, to the formation of meson condensates, to the formation of a plasma of deconfinedquarks (see Fig. 1). The interest in the role of quark deconfinement for astrophysics has received renewed interest bythe discovery that quark matter at low temperatures ought to be a color superconductor (see Ref. [4] and referencestherein).All these features make neutron stars superb astrophysical laboratories for a wide range of physical studies [5–12].And with observational data accumulating rapidly from both orbiting and ground based observatories spanning thespectrum from X-rays to radio wavelengths, there has never been a more exiting time than today to study neutronstars and associated catastrophic astrophysical events. The Hubble Space Telescope and X-ray satellites such asChandra and XMM-Newton in particular have proven especially valuable. New astrophysical instruments such asthe Five hundred meter Aperture Spherical Telescope (FAST), the square kilometer Array (skA), Fermi Gamma-raySpace Telescope (formerly GLAST), Astrosat, ATHENA (Advanced Telescope for High ENergy Astrophysics), andthe Neutron Star Interior Composition Explorer (NICER) promise the discovery of tens of thousands of new neutronstars. Of particular interest will be the proposed NICER mission (scheduled to launch in 2016), which is dedicatedto the study of extraordinary gravitational, electromagnetic, and nuclear-physics environments embodied by neutronstars. NICER will explore the exotic states of matter in the core regions of neutron stars, where, as mentioned justabove, density and pressure are considerably higher than in atomic nuclei, confronting nuclear theory with uniqueobservational constraints.There is also the theoretical possibility that quark matter made of up, down and strange quarks (so-called strangequark matter [13]) may be more stable than ordinary nuclear matter [14]. This so-called strange matter hypothesisconstitutes one of the most startling possibilities regarding the behavior of superdense matter, which, if true, wouldhave implications of fundamental importance for cosmology, the early universe, its evolution to the present day, andastrophysical compact objects such as neutron stars and white dwarfs (see Ref. [15] and references therein). Theproperties of compact stars made of strange quark matter, referred to as strange (quark) stars, are compared withthose of neutron stars in Table I and Fig. 1. Even after three decades of research, there is no sound scientific basis onroperties of High-Density Matter in Neutron Stars 3
Table I: Theoretical properties of strange quark stars and neutron stars compared.Strange Quark Stars Neutron StarsMade entirely of deconfined up, down, strange Nucleons, hyperons, boson condensates,quarks, and electrons deconfined quarks, electrons, and muonsAbsent Superfluid neutronsAbsent Superconducting protonsColor superconducting quarks Color superconducting quarksEnergy per baryon < ∼
930 MeV Energy per baryon >
930 MeVSelf-bound ( M ∝ R ) Bound by gravityMaximum mass ∼ M ⊙ SameNo minimum mass if bare Minimum mass ∼ . M ⊙ Radii
R < ∼ −
12 km Radii
R > ∼ −
12 kmBaryon numbers
B < ∼ Baryon numbers 10 < ∼ B < ∼ Electric surface fields ∼ to ∼ V/cm AbsentCan either be bare or enveloped in thin Always have nuclear crustsnuclear crusts (masses < ∼ − M ⊙ )Maximum density of crust set by neutron drip, i.e., Does not apply, i.e., neutron starsstrange stars posses only outer crusts posses inner and outer crustsForm two-parameter stellar sequences Form one-parameter stellar sequences which one can either confirm or reject the strange quark matter hypothesis so that it remains a serious possibility offundamental significance for various astrophysical phenomena, as discussed in the next section [16]. II. PROPERTIES OF STRANGE QUARK STARS
A bare quark star differs qualitatively from a neutron star which has a density at the surface of about 0.1 to1 g / cm . The thickness of the quark surface is just ∼ erg / s. It has been shown in Ref. [17] that this value may be exceeded bymany orders of magnitude by the luminosity of e + e − pairs produced by the Coulomb barrier at the surface of a hotstrange star. For a surface temperature of ∼ K, the luminosity in the outflowing pair plasma was calculated tobe as high as ∼ × erg / s. Such an effect may be a good observational signature of bare strange stars [17–20]. Ifthe strange star is dressed, that is, enveloped in a nuclear crust, however, the surface made of ordinary atomic matterwould be subject to the Eddington limit. Hence the photon emissivity of a dressed quark star would be the same asfor an ordinary neutron star. If quark matter at the stellar surface is in the CFL (Color-Flavor Locked) phase theprocess of e + e − pair creation at the stellar quark matter surface may be turned off. This may be different for theearly stages of a very hot CFL quark star [21].In contrast to neutron stars, the radii of self-bound quark stars decrease the lighter the stars, according to M ∝ R .The existence of nuclear crusts on quark stars changes the situation drastically [15, 22]. Since the crust is boundgravitationally, the mass-radius relationship of quark stars with crusts can be qualitatively similar to mass-radiusrelationships of neutron stars and white dwarfs [22]. In general, quark stars with or without nuclear crusts possesssmaller radii than neutron stars. This implies that quark stars have smaller mass shedding (break-up) periods thanneutron stars. Moreover, due to the smaller radii of quarks stars, the complete sequence of quark stars–and not justthose close to the mass peak, as it is the case for neutron stars–can sustain extremely rapid rotation [15, 22]. Inparticular, a strange star with a typical pulsar mass of around 1 . M ⊙ has a Kepler period in the approximate rangeof 0 . < ∼ P K / msec < ∼ . P K ∼ V/cm. If strange matter formsa color superconductor, as expected for such matter, the strength of the electric field may increase to values thatexceed 10 V/cm. The energy density associated with such huge electric fields is on the same order of magnitude asthe energy density of strange matter itself, which may alter the masses and radii of strange quark stars at the 15%roperties of High-Density Matter in Neutron Stars 4and 5% level, respectively [24].The electrons at the surface of a quark star are not necessarily in a fixed position but may rotate with respect tothe quark matter star [25]. In this event magnetic fields can be generated which, for moderate effective rotationalfrequencies between the electron layer and the stellar body, agree with the magnetic fields inferred for several CentralCompact Objects (CCOs). These objects could thus be interpreted as quark stars whose electron atmospheres rotateat frequencies that are moderately different ( ∼
10 Hz) from the rotational frequency of the quark star itself.Last but not least, we mention that the electron surface layer may be strongly affected by the magnetic field ofa quark star in such a way that the electron layer performs vortex hydrodynamical oscillations [26]. The frequencyspectrum of these oscillations has been derived in analytic form in Ref. [26]. If the thermal X-ray spectra of quarkstars are modulated by vortex hydrodynamical oscillations, the thermal spectra of compact stars, foremost centralcompact objects (CCOs) and X-ray dim isolated neutron stars (XDINSs), could be used to verify the existence ofthese vibrational modes observationally. The central compact object 1E 1207.4-5209 appears particularly interestingin this context, since its absorption features at 0.7 keV and 1.4 keV can be comfortably explained in the frameworkof the hydro-cyclotron oscillation model [26].Rotating superconducting quark stars ought to be threaded with rotational vortex lines, within which the star’sinterior magnetic field is at least partially confined. The vortices (and thus magnetic flux) would be expelled from thestar during stellar spin-down, leading to magnetic re-connection at the surface of the star and the prolific productionof thermal energy. It has been shown in Ref. [27] that this energy release can re-heat quark stars to exceptionally hightemperatures, such as observed for Soft Gamma Repeaters (SGRs), Anomalous X-Ray pulsars (AXPs), and X-raydim isolated neutron stars (XDINs), and that SGRs, AXPs, and XDINs may be linked ancestrally [27].The conversion of a neutron star to a hypothetical quark star could lead to quark novae. Such events could explaingamma ray bursts [28], the production of heavy elements such as platinum through r-process nucleosynthesis [29],and double-humped super-luminous supernovae [30].
III. NEUTRON STARSA. Non-spherical neutron stars
Usually, the structure of neutron stars is modeled with the assumption that they are perfect spheres. However, dueto to very high magnetic fields, certain classes of neutron stars, such as magnetars and neutron stars containing coresof color superconducting quark matter [31], may be deformed. The stellar structure equation of such stars is given by[32] dPdr = − ( ǫ + P ) (cid:18) r + 4 πr P − r (cid:18) − mr (cid:19) γ (cid:19) (cid:18) r (cid:18) − mr (cid:19) γ (cid:19) − . (1) −10 −5 0 5 10−10−50510−10−50510 Equatorial Radii (km) P o l a r R ad i u s ( k m ) Figure 2: Oblate spheroid ( γ = 0 . −5 0 5−505−10−8−6−4−20246810 Equatorial Radii (km) P o l a r R ad i u s ( k m ) Figure 3: Prolate spheroid ( γ = 1 . roperties of High-Density Matter in Neutron Stars 5 Equatorial Radius (km) M a ss ( M S un ) Region of Observed Neutron StarMasses γ = 1.00 (2.01 M Sun ) γ = 0.90 (2.31 M Sun ) γ = 0.80 (2.67 M Sun ) γ = 0.70 (3.11 M Sun ) Figure 4: (Color online) Mass-radius relationships of sequences of oblate strange quark stars. The mass is increasing withincreasing oblateness (decreasing values of γ ). The range of observed neutron star masses is indicated. Here, ǫ is the energy-density, P is the pressure, m is the gravitational mass, and γ is a deformation constant. Thestar’s total mass is given by M = γm . Oblate (prolate) neutron stars are obtained for γ < γ > γ = 1, equation (1) reduces to the well known Tolman-Oppenheimer-Volkoff equation,which describes the properties of spherically symmetric neutron (compact) stars [5, 6, 15].The mass-radius relationship of oblate compact stars, computed from Eq. (1) for different γ values, are shown inFig. 4.For simplicity, the equation of state of a relativistic gas of deconfined quarks described by the MIT bag model, P = ( ǫ − B ) /
3, has been used here [32]. The value of the bag constant is B = 57 MeV / fm ( B / = 145 MeV), whichmakes the quark gas absolutely stable with respect to nuclear matter. The results shown in Figs. 2 to 4 thereforeare for strange quark stars. One sees that already relatively small deviations from spherical symmetry increase themasses of strange stars substantially. B. Rotation-driven compositional changes
The change in central density of a neutron star whose frequency varies from zero to the mass shedding (Kepler)frequency can be as large as 50 to 60% [15]. This suggests that changes in the rotation rate of a neutron star maydrive phase transitions and/or lead to significant compositional changes in the star’s core [15, 33, 34]. As a case inpoint, for some rotating neutron stars the mass and initial rotational frequency may be just such that the centraldensity rises from below to above the critical density for dissolution of baryons into their quark constituents. Thismay be accompanied by a sudden shrinkage of the neutron star, effecting the star’s moment of inertia and, thus, itsspin-down behavior. As shown in Ref. [35], the spin-down of such a neutron star may be stopped or even reversedfor tens of thousands to hundreds of thousands of years [15, 35]. The observation of an isolated neutron star which isspinning-up, rather than down, could thus hint at the existence of quark matter in its core.
C. Quark deconfinement in high-mass neutron stars
Quark deconfinement in high-mass neutron stars has very recently been studied using extensions of the localand non-local 3-flavor Nambu-Jona Lasinio (NJL) model supplemented with repulsive vector interactions among thequarks (see Refs. [36, 37] and references therein). The phase transition from confined hadronic matter to deconfinedquark matter has been constructed via the Gibbs condition, which imposes global rather than local electric chargeneutrality and baryon number conservation. Depending on the strength of the quark vector repulsion, it was foundthat an extended mixed phase of confined hadronic matter and deconfined quarks can exist in neutron stars asmassive as 2 . . M ⊙ . A phase of pure quark matter inside such high-mass neutron stars, while not excluded, isonly obtained for certain parametrizations of the underlying lagrangians. The radii of all these stars are between 12and 13 km, as expected for neutron stars of that mass [38–40].roperties of High-Density Matter in Neutron Stars 6 (a) ep n Y i / M max G V = 0 M max (b) - e p n / G V = 0.09 G S - Figure 5: (Color online) Particle population of neutron star matter computed for the non-local SU(3) NJL model. The yellowareas highlight the mixed phase. The solid vertical lines indicate the central densities of the associated maximum-mass NSs.The hadronic model parametrization is NL3 and the vector repulsions are (a) G V /G S = 0 and (b) G V /G S = 0 . In Fig. 5 we show the relative particle fractions Y i ( ≡ ρ i /ρ ) of neutron star matter as a function of baryon numberdensity for the non-local NJL model and using NL3 parametrization of Ref. [41] for the hadronic model. It can beseen that by increasing the strength of the vector interaction, negatively charged particles like µ − ’s and ∆ − ’s take onthe role of electrons, whose primary duty is to make the stellar matter electrically neutral. Because of the early onsetof the ∆ population in this model, there is less need for electrons so that their number density in the mixed phase is (a) G V = 0 X = 0.72 e B C he m i c a l P o t en t i a l [ M e V ] / M max (b) e X = 0.97 B P [ M e V /f m ] / M max G V = 0.09 G S Figure 6: (Color online) Pressure P (solid lines), baryon chemical potential µ B = µ n / µ e (dotted lines) as a function of baryon number density (in units of ρ = 0 .
16 fm − ). In all the cases a non localSU(3) NJL and non linear Walecka (with parametrization NL3) models are considered for quark matter and hadronic phases,respectively. The hatched areas denote the mixed phase regions where confined hadronic matter and deconfined quark mattercoexist. Panel (a) is computed for zero vector repulsion. The impact of finite values of the vector repulsion ( G V = 0 . G S ) onthe data is shown in panel (b). reduced compared to the outcome of standard mean-field/bag model calculations. The corresponding deleptonizationdensities, i.e. the densities beyond which leptons are no longer present in quark-hybrid matter, depend on the ratio G V /G S ( ρ G V =0 = 4 . ρ and ρ G V =0 . G S = 5 . ρ ) [37].roperties of High-Density Matter in Neutron Stars 7Since we model the quark-hadron phase transition in three-space, accounting for the fact that the electric andbaryonic charge are conserved for neutron star matter, the pressure varies monotonically with the proportion of thephases in equilibrium, as shown in Fig. 6. The hatched areas shown in this figure denote the mixed phase regionswhere confined hadronic matter and deconfined quark matter coexist. The quark matter contents of the maximum-mass neutron stars computed for these equations of state are also indicated ( χ values). Pure quark matter would existfor χ = 1. Our calculations show that, in the non-local case, the inclusion of the quark vector coupling contributionshifts the onset of the phase transition to higher densities and narrows the width of the mixed quark-hadron phase,when compared to the case G V = 0. To the contrary, when the quark matter phase is represented by the local NJLmodel, the width of the mixed phase tends to be broader for finite G V /G S values [37].To account for the uncertainty in the theoretical predictions of the ratio G V /G S , we treat the vector couplingconstant as a free parameter. We observed that the non local NJL model is more sensitive to the increase of G V /G S than the local model. For G V /G S > .
09 we have a shift of the onset of the quark-hadron phase transition to higherand higher densities, preventing quark deconfinement in the cores of neutron stars. However, we can reach values of G V /G S up to 0 . D. “Backbending”–a possible signal of quark deconfinement
Whether or not quark deconfinement exists in static (non-rotating) neutron stars makes only very little differenceto their properties, such as the range of possible masses and radii, which renders the detection of quark matter insuch objects extremely complicated. This may be strikingly different for rotating neutron stars which develop quarkmatter cores in the course of spin-down. The reason being that such stars become more and more compressed as theyspin down from high to low frequencies. For some rotating neutron stars the mass and initial rotational frequencymay be just such that the central density rises from below to above the critical density for the dissolution of baryonsinto their quark constituents. This could affect the star’s moment of inertia dramatically [15, 35]. Depending onthe rate at which quark matter is produced, the moment of inertia can decrease very anomalously, and could evenintroduce an era of stellar spin-up (so-called “backbending”) lasting for ∼ years [35]. Since the dipole age ofmillisecond pulsars is about 10 years, one may estimate that roughly about 10% of the solitary millisecond pulsarspresently known could be in the quark transition epoch and thus could be signaling the ongoing process of quarkdeconfinement. Changes in the moment of inertia reflect themselves in the braking index, n , of a rotating neutronstar, as can be seen from [15, 35, 42] n (Ω) ≡ Ω ¨Ω˙Ω = 3 − I + 3 I ′ Ω + I ′′ Ω I + I ′ Ω → − I ′ Ω + I ′′ Ω I + I ′ Ω (2)where dots and primes denote derivatives with respect to time and Ω, respectively. The last relation in Eq. (2)constitutes the non-relativistic limit of the braking index. It is obvious that these expressions reduce to the canonicallimit, n = 3, if the moment of inertia is completely independent of frequency. Evidently, this is not the case for rapidlyrotating neutron stars, and it fails for stars that experience pronounced internal changes (as possibly driven by phasetransitions) which alter the moment of inertia significantly. In Ref. [43] it was shown that the changes in the momentof inertia caused by the gradual transformation of hadronic matter into quark matter may lead to n (Ω) → ±∞ atthe transition frequency where pure quark matter is produced. Such dramatic anomalies in n (Ω) are not known forconventional neutron stars (see, however, Ref. [44]), because their moments of inertia appear to vary smoothly withΩ [6]. The future astrophysical observation of a strong anomaly in the braking behavior of a pulsar may thus indicatethat quark deconfinement is occurring at the pulsar’s center. E. Quark-hadron Coulomb lattices in the cores of neutron stars
Because of the competition between the Coulomb and the surface energies associated with the positively chargedregions of nuclear matter and negatively charged regions of quark matter, the mixed phase may develop geometricalstructures (e.g., blobs, rods, slabs, as schematically illustrated in Fig. 7 [45]), similarly to what is expected of the sub-nuclear liquid-gas phase transition [46–49]. The consequences of such a Coulomb lattice for the thermal and transportproperties of neutron stars have been studied in Ref. [45]. It was found that at low temperatures of
T < ∼ K theneutrino emissivity from electron-blob Bremsstrahlung scattering is at least as important as the total contributionfrom all other Bremsstrahlung processes (such as nucleon-nucleon and quark-quark Bremsstrahlung) and modifiednucleon and quark Urca processes. It is also worth noting that the scattering of degenerate electrons off rare phaseroperties of High-Density Matter in Neutron Stars 8
Figure 7: Schematic illustration of possible geometrical structures in the quark-hadron mixed phase of neutron stars. Thestructures may form because of the competition between the Coulomb and the surface energies associated with the positivelycharged regions of nuclear matter and negatively charged regions of quark matter. blobs in the mixed phase region lowers the thermal conductivity by several orders of magnitude compared to a quark-hadron phase without geometric patterns. This may lead to significant changes in the thermal evolution of the neutronstars containing solid quark-hadron cores, which has not yet been studied.
F. Pycnonuclear reaction rates
The presence of strange quark nuggets in the crustal matter of neutron stars could be a consequence of Witten’sstrange quark matter hypothesis [14]. The impact of such nuggets on the pycnonuclear reaction rates among heavyatomic nuclei has been studied in Ref. [50]. Particular emphasis was put on the consequences of color superconductivityon the reaction rates. Depending on whether or not quark nuggets are in a color superconducting state, their electriccharge distributions differ drastically, which was found to have dramatic consequences for the pycnonuclear reactionrates in the crusts of neutron stars. Future nuclear fusion network calculations may thus have the potential to shedlight on the existence of strange quark matter nuggets and on whether they are in a color superconducting state, assuggested by QCD.
G. Rotational instabilities
The r-mode instability of a rotating neutron star dissipates the star’s rotational energy by coupling the angularmomentum of the star to gravitational waves [51–53]. This instability can be active in a newly formed isolated neutronstar as well as in old neutron stars being spun up by accretion of matter from binary stars. If the interior containsquark matter, the r-mode instability and the gravitational wave signal may carry information about quark matter[54–58].
Acknowledgments
F. W. is supported by the National Science Foundation (USA) under Grants PHY-0854699 and PHY-1411708.M. O. and G. C. acknowledge CONICET and SeCyT-UNLP (Argentina) for financial support. M. O. and G. C.are thankful for hospitality extended to them during their visits at the SDSU, supported by a NSF-CONICETInternational Cooperation Project. [1] The CBM Physics Book, B. Friman, C. H¨ohne, J. Knoll, S. Leupold, J. Randrup, R. Rapp, and P. Senger (Eds.),
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