Neutrino emissivity in the quarkhadron mixed phase of neutron stars
Abstract
Numerous theoretical studies using various equation of state models have shown that quark matter may exist at the extreme densities in the cores of highmass neutron stars. It has also been shown that a phase transition from hadronic matter to quark matter would result in an extended mixed phase region that would segregate phases by net charge to minimize the total energy of the phase, leading to the formation of a crystalline lattice. The existence of quark matter in the core of a neutron star may have significant consequences for its thermal evolution, which for thousands of years is facilitated primarily by neutrino emission. In this work we investigate the effect a crystalline quarkhadron mixed phase can have on the neutrino emissivity from the core. To this end we calculate the equation of state using the relativistic meanfield approximation to model hadronic matter and a nonlocal extension of the threeflavor NambuJonaLasinio model for quark matter. Next we determine the extent of the quarkhadron mixed phase and its crystalline structure using the Glendenning construction, allowing for the formation of spherical blob, rod, and slab rare phase geometries. Finally we calculate the neutrino emissivity due to electronlattice interactions utilizing the formalism developed for the analogous process in neutron star crusts. We find that the contribution to the neutrino emissivity due to the presence of a crystalline quarkhadron mixed phase is substantial compared to other mechanisms at fairly low temperatures ( K) and quark fractions (), and that contributions due to lattice vibrations are insignificant compared to staticlattice contributions.
pacs:
21.65.Qrquark matter and 26.60.Kpequations of state and 97.60.Jdneutron stars1 Introduction
Seconds after the formation of a neutron star the mean free path of neutrinos grows beyond the star’s radius, and neutrinos from the core escape easily cooling the new star rapidly. Neutrino emission continues to be the dominant energy loss mechanism of a neutron star for thousands of years until a temperature of about K is reached NS1 (). It has been shown that the presence of quark matter in the core of a neutron star can have a significant impact on the neutrino emissivity, and suggested that this impact could have an observable effect on the star’s thermal evolution weberPPNP (); glendenning2001 (). In this work we investigate the effect that a mixed phase containing quark matter may have on the neutrino emissivity of a neutron star.
If electric charge neutrality in a neutron star is to be treated globally as proposed by Glendenning glendenning1992 (), then the first order phase transition from hadronic matter to quark matter in the core will result in a mixed phase in which both phases of matter coexist. To minimize the total isospin asymmetry energy the two phases will segregate themselves, resulting in positively charged regions of hadronic matter and negatively charged regions of quark matter, with the rare phase occupying sites on a Coulomb lattice. Further, the competition between the Coulomb and surface energy densities will cause the matter to arrange itself into energy minimizing geometric configurations glendenning2001 ().
The presence of the Coulomb lattice and the nature of the geometric configurations of matter in the quarkhadron mixed phase may have a significant effect on the neutrino emissivity from the core. More specifically, the presence of electrons in the mixed phase will lead to an additional neutrino emissivity mechanism due to interactions with the lattice. This process is analogous to neutrinopair bremsstrahlung of electrons in the neutron star crust, where ions exist on a lattice immersed in an electron gas, and for which there exists a large body of work (see, for example flowers (); itoh84a (); itoh84b (); itoh84c (); itoh84d (); pethick1997 (); kaminker1999 ()). The situation is somewhat more complicated in the quarkhadron mixed phase, but the operative interaction is still the Coulomb interaction. Thus, to estimate the neutrinopair bremsstrahlung of electrons in the quarkhadron mixed phase we rely heavily on this body of work (particularly kaminker1999 ()).
Neutrino emissivity due to the interaction of electrons with a crystalline quarkhadron mixed phase has been previously studied in this manner by Na et al. Na (). In the present work we replace the MIT Bag Model used by Na () to describe quark matter with a threeflavor nonlocal variant of the NambuJonaLasinio model. Next, we extend the range of possible geometric structures in the mixed phase beyond spherical blobs to include rods and slabs, and calculate the associated static lattice contributions to the neutrino emissivity. Phonon contributions to the emissivity for rod and slab geometries are not considered, though a comparison of the phonon and static lattice contributions for spherical blobs is given and indicates that phonon contributions may not be significant. Finally, the extent of the conversion to quark matter in the core is determined for the chosen parameterizations, and this allows for a targeted comparison between emissivity contributions from standard neutrino emission mechanisms (modified Urca, nucleonnucleon and quarkquark (NN+QQ) bremsstrahlung) and contributions from electronlattice interactions. In this work the minimal cooling paradigm is assumed, as the mechanism under investigation is not expected to compete with the direct Urca process, but may serve to enhance the cooling of neutron stars in its absence.
This paper is structured as follows. In Sec. 2 we discuss the zero temperature equation of state of a neutron star containing hadronic matter, quark matter, and a quarkhadron mixed phase. The crystalline structure of the mixed phase is described in Sec. 3. The neutrino emissivity due to interactions between electrons and the crystalline lattice in the quarkhadron mixed phase is described in Sec. 4. Our results, including neutron star properties and neutrino emissivity calculations, are presented in Sec. 5. Finally, we present our conclusions in Sec. 6.
2 Equation of State
2.1 Neutron Star Crust
The neutron star outer and inner nuclear crust exists at densities between g cm g cm weberbook (). Matter in the inner crust consists mostly of nuclei in a Coulomb lattice that is immersed in a gas of electrons and, above neutron drip ( g cm), free neutrons. In this work we use a combination of the BaymPethickSutherland and BaymBethePethick equations of state for the nuclear crust BPS (); BBP (). Compared to the neutron star core the crust has less affect on the neutron star properties that are to be studied in this work.
2.2 Confined Hadronic Phase
The hadronic phase of neutron star matter exists at densities above that of the crust and is populated by baryons () and leptons (). To model the hadronic phase we use the relativistic meanfield approximation (RMF), in which the interactions between baryons are described by the exchange of scalar (), vector (), and isovector () mesons walecka (). The meanfield Lagrangian is given by glen85:b (); weberbook (); glendenningbook (); selfinteractions (); boguta77:a (); boguta83:a ()
(1)  
The and mesons are responsible for nuclear binding while the meson is required to obtain the correct value for the empirical symmetry energy. In contrast to and mesons, which are isoscalars, the meson is an isovector field that manifests itself in the occurrence of the Pauli matrix () in Eq. (1). The cubic and quartic terms in Eq. (1) are necessary (at the relativistic meanfield level) to obtain the empirical incompressibility of nuclear matter selfinteractions (); boguta77:a (). The field tensors and are defined as and .
The mesonbaryon coupling constants (, , , , ) of the Lagrangian are set so that the properties of nuclear matter at saturation density are reproduced for the appropriate parameterization (Table 1). In this work we employ the GM1 and NL3 parameterizations as in Ref. NJL2014 (). To fix the mesonhyperon coupling constants we follow the method presented in Ref. Miyatsu2013 (). The scalar mesonhyperon coupling constants are fit to the following hypernuclear potentials at saturation density: MeV, MeV, and MeV. The vector mesonhyperon coupling constants are fixed in SU(3) flavor symmetry by the mixing angle and coupling ratio taken from the Nijmegen extendedsoftcore (ESC08) model ESC08 (). The isovector mesonhyperon coupling constants are given by the usual relations, and .
Nuclear saturation properties  GM1  NL3 

(fm)  0.153  0.148 
(MeV)  16.3  16.3 
(MeV)  300  272 
0.78  0.60  
(MeV)  32.5  37.4 
The field equations for the baryon fields follow from Eq. (1) as follows weberPPNP (); glen85:b (); weberbook (); glendenningbook (),
(1)  
The meson fields in (1) are solutions of the following field equations weberPPNP (); glen85:b (); weberbook (); glendenningbook (),
(2)  
(3)  
(4) 
In the meanfield limit, the meson field equations (2) through (4) are given by weberPPNP (); glen85:b (); weberbook (); glendenningbook ()
(5)  
(6) 
(7) 
where the effective baryon mass .
To determine the equation of state we solve a nonlinear system consisting of the meson meanfield equations and the charge conservation conditions (baryonic, electric) given by glen85:b (); weberbook (); glendenningbook ()
(8) 
(9) 
where is the total baryonic density and and are the electric charges of baryons and leptons, respectively. Particles in the hadronic phase are subject to the chemical equilibrium condition,
(10) 
where is the chemical potential and is the baryon number of particle . New baryon or lepton states are populated when the right side of equation (10) is greater than the states’ chemical potential. The baryonic and leptonic number densities (, ) are both given by
(11) 
The free parameters of the system are the meson meanfields (, , ), and the neutron and electron fermi momenta (, ). Finally, the energy density and pressure of the hadronic phase are given by glen85:b (); weberbook (); glendenningbook ()
(12)  
(13)  
2.3 Deconfined Quark Phase
If the dense interior of a neutron star contains deconfined quark matter, it will be made of up (), down (), and strange () quarks in chemical equilibrium with a small number of electrons and muons. To model the quark phase we use a nonlocal extension of the NambuJonaLasinio model (n3NJL) as described in Ref. NJL2014 (). The effective action of this model is given by
(14)  
where denotes quark flavor (), is a chiral U(3) vector that includes the light quark fields, , is the current quark mass matrix, with denote the generators of SU(3), and . The coupling constants and , the strange quark mass , and the threemomentum ultraviolet cutoff parameter , are all model parameters. Their values are taken from Ref. rehberg (), i.e., MeV, MeV, MeV, and . The vector coupling constant is treated as a free parameter.
For the meanfield approximation, the thermodynamic potential associated with of Eq. (14) is given by
(15)  
where , , and are the quark scalar, vector, and auxiliary mean fields, respectively. Moreover, we have , , and are the momentum dependent quark masses. The quantity is the form factor which introduces nonlocality into the quark interactions NJL2014 (). The auxiliary mean fields are given by
(16) 
Due to the inclusion of the vector interaction the quark chemical potentials are shifted as follows,
(17) 
(18) 
The scalar and vector mean fields are obtained by minimizing the grand thermodynamic potential,
(19) 
The quark number densities are given by
(20) 
To determine the equation of state one must solve a nonlinear system of equations for the fields and , and the neutron and electron chemical potentials and . This system of equations consists of the mean field equations,
(21) 
with cyclic permutations over the quark flavors,
(22) 
and the charge conservation equations,
(23) 
(24) 
Finally, the pressure and energy density are given by
(25) 
(26) 
where is the grand thermodynamic potential calculated for .
2.4 QuarkHadron Mixed Phase
When the pressure in the hadronic phase grows to a level equal to that of the quark phase at the same baryonic density a first order phase transition from hadronic matter to quark matter may begin. Since a theory that can treat both the hadronic and quark phases simultaneously is currently unavailable, we construct the mixed phase by blending RMF and n3NJL. Each phase is solved for separately, and then the two are blended together under the Gibbs condition, . The pressure () and energy density () in the mixed phase are given by glendenning2001 (); glendenning1992 ()
(27) 
and
(28) 
where is the quark fraction of the mixed phase. Other properties such as the particle number densities can be handled in a similar fashion.
3 Crystalline Structure of the QuarkHadron Mixed Phase
A mixed phase of hadronic and quark matter will arrange itself so as to minimize the total energy of the phase. Under the condition of global charge neutrality this is the same as minimizing the contributions to the total energy due to phase segregation, which includes the surface and Coulomb energy contributions. Expressions for the Coulomb () and surface () energy densities can be written as glendenning2001 ()
(29)  
(30) 
where () is the hadronic (quark) phase charge density, and is the radius of the rare phase structure. The quantities and in Eq. (29) are defined as
(31) 
and
(32) 
where is the dimensionality of the lattice. The quantity in Eq. (30) denotes the surface tension.
The phase rearrangement process will result in the formation of geometrical structures of the rare phase distributed in a crystalline lattice that is immersed in the dominant phase (Figure 2). The rare phase structures are approximated for convenience as spherical blobs, rods, and slabs glendenning2001 (). The spherical blobs occupy sites in a three dimensional () body centered cubic (BCC) lattice, the rods in a two dimensional () triangular lattice, and the slabs in a simple one dimensional () lattice kaminker1999 (). At both hadronic and quark matter exist as slabs in the same proportion, and at the hadronic phase becomes the rare phase with its geometry evolving in reverse order (from slabs to rods to blobs).
3.1 Surface Tension of the QuarkHadron Interface
Direct determination of the surface tension of the quarkhadron interface is problematic because of difficulties in constructing a single theory that can accurately describe both hadronic matter and quark matter. Therefore, we employ an approximation proposed by Gibbs where the surface tension is taken to be proportional to the difference in the energy densities of the interacting phases glendenning2001 (),
(33) 
where is proportional to the surface thickness which should be on the order of the strong interaction (1 fm), and is a proportionality constant. In this work we maintain the energy density proportionality but set the parameter so that the surface tension falls below 50 MeV fm, a value consistent with those suggested for in recent literature yasutake2014 (); surfacetension1 (); surfacetension2 (); surfacetension3 ().
3.2 Rare Phase Structure Size, Charge, and Number Density
The size of the rare phase structures is given by the radius () and is determined by minimizing the sum of the Coulomb and surface energies, , and solving for glendenning2001 (),
(34) 
The primitive cell of the lattice is taken to be the WignerSeitz cell, though it is simplified to have the same geometry as the rare phase structure. The WignerSeitz cell radius is set so that the cell is charge neutral.
The density of electrons in the mixed phase is taken to be uniform throughout. Charge densities in both the rare and dominant phases are also taken to be uniform, an approximation supported by a recent study by Yasutake et al. yasutake2014 (). The uniformity of charge in the rare phase also justifies the use of the nuclear form factor () presented in Section 4. The total charge number per unit volume () of the rare phase structures is given in Figure 4.
The number density of rare phase blobs will be important for calculating the phonon contribution to the emissivity. Since there is one rare phase blob per WignerSeitz cell, the number density of rare phase blobs () is simply the reciprocal of the WignerSeitz cell volume,
(35) 
4 Neutrino Emissivity in the QuarkHadron Mixed Phase
Modeling the complex interactions of electrons with a background of neutrons, protons, hyperons, muons, and quarks is an exceptionally complicated problem. However, to make a determination of the neutrino emissivity that is due to electronlattice interactions in the quarkhadron mixed phase we need only consider the Coulomb interaction between them. This simplifies the problem greatly, as a significant body of work exists for the analogous process of electronion scattering that takes place in the crusts of neutron stars.
4.1 ElectronLattice Interaction
To determine the state of the lattice in the quarkhadron mixed phase we use the dimensionless ion coupling parameter given by NS1 ()
(36) 
Below the lattice behaves as a Coulomb liquid, and above as a Coulomb crystal NS1 (). It was shown in Na et al. Na () that the emissivity due to electronblob interactions in the mixed phase was insignificant compared to other contributions
at temperatures above K. Therefore, in this work we consider temperatures in the range . At these temperatures the value of the ion coupling parameter is generally well above , and so the lattice in the quarkhadron mixed phase is taken to be a Coulomb crystal.
To account for the fact that the elasticity of scattering events is temperature dependent we need to compute the DebyeWaller factor, which is known for spherical blobs only and requires the plasma frequency and temperature given by
(37) 
(38) 
where is the mass of a spherical blob kaminker1999 (). The DebyeWaller factor is then given by
(39) 
where is a phonon or scattering wave vector, , and kaminker1999 (); baiko1995 (). In order to smooth out the charge distribution over the radial extent of the rare phase structure we adopt the nuclear form factor given in kaminker1999 (),
(40) 
Screening of the Coulomb potential by electrons is taken into account by the static dielectric factor , given in Ref. itoh1983_1 (). However, the charge number of the rare phase structures is high and the electron number density is low, so setting this factor to unity has no noticeable effect on the calculated neutrino emissivity. Finally, the effective interaction is given by kaminker1999 ()
(41) 
4.2 Neutrino Emissivity
General expressions for the neutrino emissivity due to electronlattice interactions were derived by Haensel et al. haensel1996 () for spherical blobs and by Pethick et al. pethick1997 () for rods and slabs,
(42) 
(43) 
where and are dimensionless quantities that scale the emissivities. Both and contain a contribution due to the static lattice (Bragg scattering), but we consider the additional contribution from lattice vibrations (phonons) for spherical blobs, so .
4.3 Phonon Contribution to Neutrino Emissivity
The expressions for determining the neutrino emissivity due to interactions between electrons and lattice vibrations (phonons) in a Coulomb crystal, with proper treatment of multiphonon processes, were obtained by Baiko et al. baiko1998 () and simplified by Kaminker et al. kaminker1999 (). The phonon contribution to the emissivity is primarily due to Umklapp processes in which a phonon is created (or absorbed) by an electron that is simultaneously Bragg reflected, resulting in a scattering vector that lies outside the first Brillouin zone, ziman (); raikh1983 (), where is given by Eq. (35).
The contribution to the neutrino emissivity due to phonons is contained in and given by Eq. (21) in Ref. kaminker1999 (),
(44) 
where , and the lower integration limit excludes momentum transfers inside the first Brillouin zone. The structure factor is given by Eqs. (24) and (25) in Ref. kaminker1999 ()),
(45)  
(46) 
where and denotes averaging over phonon frequencies and modes,
(47) 
It is assumed that there are three phonon modes , two linear transverse and one longitudinal. The frequencies of the transverse
modes are given by , where , , and . The frequency of the longitudinal mode is determined by Kohn’s sum rule, mochkovitch1979 ().
Umklapp processes proceed as long as the temperature , below which electrons can no longer be treated in the free electron approximation raikh1983 (). This limits the phonon contribution to the neutrino emissivity to only a very small range in temperature for a crystalline quarkhadron mixed phase (see Figure 5), and renders it negligible compared to the static lattice contribution as will be shown in the next section.
4.4 Static Lattice Contribution to Neutrino Emissivity
Pethick and Thorsson pethick1997 () found that with proper handling of electron bandstructure effects the static lattice contribution to the neutrino emissivity in a Coulomb crystal was significantly reduced compared to calculations performed in the free electron approximation. Kaminker et al. kaminker1999 () presented simplified expressions for calculating the static lattice contribution () using the formalism developed in Ref. pethick1997 (). The dimensionless quantities and that scale the neutrino emissivities for spherical blobs and rods/slabs, respectively, are given by
(48) 
and
(49) 
where is a scattering vector and restricted to linear combinations of reciprocal lattice vectors, , , and is given by Eq. (39) in Ref. kaminker1999 (). The sum over in Eqs. (48) and (49) terminates when , prohibiting scattering vectors that lie outside the electron Fermi surface.
5 Results
The neutron star equation of state has been calculated using the relativistic mean field approximation to describe the hadronic phase and the threeflavor nonlocal NambuJonaLasinio model for the quark phase, with the Gibbs condition governing the combination of the two in the mixed phase (Figure 1). Using the equation of state we solve the TolmanOppenheimerVolkoff equation tov (); tov2 () and find the massradius relationships given in Figure 6. The maximum masses of the neutron stars obtained for the given parameter sets are able to account for the recently discovered high mass pulsars PSR J3048+0432 and PSR J16142230 psrj1614 (); psrj3048 (); psrj3048b (), excluding GM1 with no vector coupling. It is evident from Figure 6 that increasing the vector coupling constant increases the maximum mass for the particular parameterization.
Figures 7 and 8 show the relative particle densities for the NL3 and GM1 parameterizations and three different values of the quark vector coupling constant. Hyperonization does not occur at all in the NL3 parameterization, as it is preceded by the low density onset of the quarkhadron phase transition at times nuclear density. The same is true of the GM1 parameterization except in the case that . Here the onset of the quarkhadron phase transition occurs at a much higher density due to the presence of the and hyperons which soften the equation of state considerably, an effect that can be seen in the right panel of Figure 1. The low density onset of the quarkhadron phase transition is due in part to the choice of mesonhyperon coupling constants, which have been shown to postpone the onset of hyperonization, stiffening the low density equation of state Miyatsu2013 (). Figure 6 shows that neutron stars within about of their maximum mass contain a quarkhadron mixed phase in their core, with most possessing a maximum quark fraction of around 30% (see Table 2).
GM1  NL3  
0  0.05  0.10  0  0.05  0.10  
1.89  2.05  2.20  2.04  2.24  2.43  
0.32  0.31  0.15  0.31  0.30  0.32  
[1/fm]  0.75  0.76  0.80  0.61  0.61  0.62  
[MeV/fm]  851  883  969  687  696  726 
Figure 10 shows the neutrino emissivity that is due to the crystalline structure of the quarkhadron mixed phase for all parameterizations and temperatures between K, as well as the modified Urca and bremsstrahlung (NN+QQ) emissivities for comparison.
Electronphonon interactions contribute to the neutrino emissivity when the mixed phase consists of spherical blobs ( and ) and only at (Figure 5). Figure 9 shows that the staticlattice contribution to the emissivity dominates the phonon contribution rendering it negligible, particularly at quark fractions relevant to the neutron stars of this work ().
Equations (48) and (49) indicate that the staticlattice contribution is calculated as a sum over scattering vectors that satisfy . At the onset of the mixed phase the electron Fermi momentum is at a maximum, which is particularly large in magnitude due to the lack of hyperons that would typically aid in the charge neutralization process. However, as the quarkhadron phase transition proceeds the negatively charged down and strange quarks take over the process of charge neutralization, resulting in a rapidly decreasing electron number density (). This and the exponentially decreasing size of the Wigner Seitz cell in the spherical blob phase () lead to the steep decline in (Figure 11), which accounts for the rapid decrease of the neutrino emissivity in the mixed phase.
The geometrical structure of the quarkhadron mixed phase terminates with the rod phase at for nearly all the chosen parameterizations. Up to this point the neutrino emissivity due to the structure of the mixed phase is either larger or comparable to the modified Urca and bremsstrahlung (NN+QQ) emissivities for K. The emissivities in the NL3 and GM1 parameterizations are comparable, though the effect of the mixed phase structure appears more substantial for NL3 due to lower modified Urca and bremsstrahlung (NN+QQ) emissivities. The emissivity at very low quark fraction () may be overestimated due to the finite blob radius at that results from the fact that . Finally, beyond the rodslab structure transition at the electronlattice contribution to the overall neutrino emissivity is negligible, though this is beyond the extent of the mixed phase of the neutron stars in this work.
6 Summary and Conclusions
Exploring the properties of compressed baryonic matter, or, more generally, strongly interacting matter at high densities and/or temperatures, has become a forefront area of modern physics braun_munzinger09:a (). Experimentally, such matter is being created in relativistic particle colliders such as the Relativistic Heavy Ion Collider RHIC at Brookhaven and the Large Hadron Collider (LHC) at Cern, and great advances in our understanding of such matter are expected from the next generation of collision experiments at FAIR (Facility for Antiproton and Ion Research at GSI) and NICA (Nuclotonbased Ion Collider fAcility at JINR) CBMbook11:a (); NICA ().
Complementary to these experiments, astrophysics provides a natural laboratory in which to explore the physics of compressed baryonic matter too (see becker09:a (); iau12:a (); emmi14:a () and references therein). The Hubble Space Telescope and Xray satellites such as Chandra and XMMNewton in particular have proven especially valuable. New astrophysical instruments such as the Five hundred meter Aperture Spherical Telescope (FAST), the square kilometer Array (skA), Fermi Gammaray Space Telescope (formerly GLAST), Astrosat, ATHENA (Advanced Telescope for High ENergy Astrophysics), and the Neutron Star Interior Composition Explorer (NICER) promise the discovery of tens of thousands of new neutron stars. Of particular interest will be the proposed NICER mission, which is dedicated to the study of the extraordinary gravitational, electromagnetic, and nuclearphysics environments embodied by neutron stars. NICER will explore the exotic states of matter in the core regions of neutron stars, confronting nuclear theory with unique observational constraints.
With that in mind, we focus in this paper on quark deconfinement in the cores of neutron stars. The neutron star equation of state for cold catalyzed matter ( MeV) has been determined using the relativistic mean field (RMF) approximation to model the hadronic phase and the nonlocal threeflavor NambuJonaLasinio model (n3NJL) for the quark phase. The massradius results indicate that a neutron star containing quark matter in the core can account for the high mass of the recently discovered pulsars PSR J3048+0432 and PSR J16142230, and that a maximum mass neutron star can be expected to contain approximately 30% quark matter at the center. If the surface tension between hadronic and quark matter is low as suggested in the recent literature, a phase transition that results in a mixed phase will occur in the core of a neutron star. The relaxed condition of global charge neutrality will lead to charge segregation in the mixed phase resulting in the formation of a crystalline lattice of quark matter immersed in a hadronic matter background. Expanding on Na et al. Na (), we considered the presence of two additional geometrical structures in the mixed phase in addition to spherical blobs: rods, and slabs (Figure 2).
Using the formalism developed for analogous neutrinopair bremsstrahlung processes in the neutron star crust we have estimated the neutrino emissivity due to electronlattice interactions in the quarkhadron mixed phase. The emissivity is highly dependent on the electron number density, which has been shown to decrease considerably in the presence of negatively charged hyperons and quarks (Figures 7 and 8). However, we have shown that at temperatures between K and K and quark fractions less than around 30% the neutrino emissivity due to electronlattice interactions is significant when compared to the standard baryon and quark modified Urca and bremsstrahlung (NN+QQ) processes (Figure 10). Further, we have also shown that the emissivity due to electronphonon interactions is insignificant compared to contributions from Bragg diffraction at temperatures above which Umklapp processes are frozen out (Figure 9).
Before we can determine the effect the presence of quark matter and the crystalline structure of the quarkhadron mixed phase has on the thermal evolution of a neutron star the following steps need to be taken. First, RMF should be replaced with a model for hadronic matter that softens the equation of state and produces results for neutron star radii that are more compatible with observations and recent statistical studies (see for example steiner2010 (); lattimer2012 ()). To this end, we are currently working on combining an RMF model that accounts for density dependence in the values of the mesonbaryon coupling constants with the threeflavor n3NJL model (see for example Fuchs1995 (); Typel1999 (); Hofmann2001 (); Ryu2011 (); Colucci2013 (); vanDalen2014 (); Oertel2015 (); Benic2015 ()). Next, the thermal conductivity and specific heat should be calculated for the quarkhadron mixed phase as outlined in Na et al. Na () using the updated equation of state for quark matter (n3NJL) and accounting for additional rare phase geometries (rods, slabs). Finally, these results would be incorporated into a neutron star cooling simulation capable of properly accounting for the complexity of the crystalline quarkhadron mixed phase.
Acknowledgments
This work is supported through the National Science Foundation under grants PHY1411708 and DUE1259951. Additional computing resources are provided by the Computational Science Research Center and the Department of Physics at San Diego State University. GAC and MGO acknowledge financial support by CONICET and UNLP (Project identification code 11/G119), Argentina.
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