Thermal evolution of neutron stars containing a quark core

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U  A

MSc Astronomy & Astrophysics

Master Thesis

Thermal evolution of neutron stars

containing a quark core

by

Stefanos Tsiopelas

10850015

February 2018

September 2015 - February 2018

54 ECTS

Supervisors:

Nanda Rea, Dr.

Justin Elfritz, Dr.

Examiners:

A. Watts, A.Pr.

R. Wijnands, Dr.

Anton Pannekoek

Institute

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Μαζί μου κουβαλάω της Ελλάδας το γκρίζο, κι αν για λίγο το αφήσω να ’ναι εδώ όταν γυρίσω.

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1 Introduction

Neutron stars are compact remnants of massive stars which form in supernova explosions. The violent birth process, renders these objects extremely dense, rapidly spinning, hot and highly magnetized. Understanding their structure and evolution requires a multilateral approach including high-energy, nuclear and plasma physics in parameter regimes impossible to probe terrestrially. They are equally interesting from an observational point of view as well, since they are detected to be emitting in diﬀerent parts of the spectrum, such as the

radio, x-ray and γ−ray bands.

In order to describe the properties of the highly-condensed matter in their interior, statistical, particle and nuclear physics have to be summoned for the establishment of an appropriate Equation of State (EoS). Neutron stars are widely thought to consist mainly of neutrons, protons and electrons.

Nev-ertheless, as Quantum Chromodynamics (QCD) suggest a deconﬁnement of

quarks inside their supranuclear cores, this should not always be the case. Theoretical models that have been implemented thus far to explain the emission proﬁle, indicate that the main power source is not the same for all neutron stars. As a result, the few thousands of observed ones, are divided into accretion-powered, pulsars, magnetars, central compact objects and iso-lated thermally-emitting. Despite the signiﬁcant level of understanding re-garding the latter’s thermal evolution, their cooling scenario remains strongly dependent on the assumed EoS.

The purpose of this work, is to investigate the structure and thermal evo-lution of thermally-emitting neutron stars consisting of a quark core and a hadronic outer region, imposing a mixed phase to mediate the undergoing phase transition. The thesis can be divided in two parts. In the ﬁrst one, a brief presentation of neutron stars (NSs) and hybrid stars (HSs) in chapter

2, is followed by the development of a suitable EoS, employing the MIT-bag

model along with a unified hadronic EoS for describing each phase and even-tually the whole star (Ch.3). The subsequent chapter 4 reviews the obtained mass-radius curves of the derived configurations as well as the distribution of hadronic and quark populations in the interior of the constructed stars, offering the background needed to proceed into studying their cooling mech-anisms.

The second part begins in chapter 5 with outlining the thermal evolution of compact stars and the way it is altered by the presence of a magnetic ﬁeld. In

chapter 6, it continues with the presentation of all the possible νe− ¯νe emission

mechanisms due to hadronic and quark matter that contribute to the cool-ing of hybrid stars. Finally, this study is concluded (Ch.7) by presentcool-ing the

cooling pattern of the stars obeying the constructed EoS and a comparison to

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2. Compact Objects 4

2 Compact Objects

After the exhaustion of its nuclear inventory, a star reaches the last evolution-ary stage of its life. Before entering though, it expels most of its mass either in a violent way, setting oﬀ a supernova explosion, or in a less violent one, creating a planetary nebula. The stellar remnant is expected to undergo total gravita-tional collapse and transform into a black hole. This is indeed what happens if the remnant surpasses the Tolman-Oppenheimer-Volkoﬀ mass limit.

Therefore, stellar death does not necessarily mean that the star stops ex-isting. In fact, the majority of them, continue their lives as the gravitational collapse halts due to an ostensibly unexpected event: as the star keeps shrink-ing, the constituent gas starts exhibiting its fermionic aspect. This leads to gravity being balanced by the pressure of the degenerate fermions. If those are the electrons, the new stable configuration is called a white dwarf, while in the case that degenerate neutrons stop the collapse, it is called a N S. The supernova explosion that creates the second ones together with conservation of angular momentum and magnetic flux, causes them to be extremely dense, hot, rapidly spinning and hosting strong magnetic fields, rendering them ob-jects of high astrophysical interest.

To the latter class of compact objects, QCD imply that a subclass should be added: hybrid stars. The principal idea behind this suggestion, is that for densities relevant to the deep interior of neutron stars, the existence of quark

matter becomes favorable. As a result, the three lightest quarks (up, down and

strange) should appear in their core at a suﬃciently high baryon density.

2.1 Neutron stars

2.1.1 Prediction and discovery

The possible existence of NS has been studied by the astrophysicists since the mid-1930s. The origin and the history of this research ﬁeld has a few particularly intriguing features that are worth noting. Firstly, their existence was proposed by W.Baade and F.Zwicky [11]. Based on the principles of quantum mechanics and radioactivity, they formulated that such astrophysical objects are physically plausible. However, it was not until the 1960s that their existence was conﬁrmed by radioastronomers. Namely, by A.Hewish and

S.Okoye, who reported ’an unusual source of high radio brightness temperature in

the Crab Nebula’ [40]. Soon thereafter it was proven that this type of radiation

comes from a neutron star. Moreover, as the ’Crab Nebula’ turned out to be

in reality a supernova remnant, the detection provided an indication of their formation mechanism.

The work of S.Chandrasekhar places the lower limit on the mass of a

neutron star at ∼ 1.4 M_⊙ [21]. On the other hand, the upper limit, also

known as Tolman-Oppenheimer-Volkoﬀ limit (TOV), is still undeﬁned. So far,

the heaviest neutron star to have ever been observed has a mass of∼ 2 M_⊙ [10].

The radius of an object that heavy is constrained to being just a few kilometers, with observational evidence placing the range of the radii from 6 to 16 km.

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2.1 Neutron stars 5

Therefore, plausible theoretical models along with astronomical observations,

suggest that densities in the central regions can reach the order of 1015 g/cm3.

By comparison, the density of an atomic nucleus is approximately equal to 3· 1014 _g/cm3 _{(saturation density).}

2.1.2 Internal properties

The interior of a neutron star, can be divided in two main regions: the core and the crust with each one of them further divided into an inner and an outer part. The core is by far more extended than the crust, comprising almost all

of the star’s total mass (∼ 99%).

The outer crust occupies an extremely thin layer, starting from the surface of the star and reaching up to a few hundred meters deep at most. Its con-stituents are ions of atomic nuclei and ultra-relativistic electrons. The latter

being degenerate as well, e− are capable of triggering the process of inverse

beta decay (p + e− → n + νe). Towards the bottom of the outer crust, as the

density rises, neutronization becomes more intense, lowering the concentra-tions of protons inside the ions. This subregion stops at a density of about

4· 1011 g/cm3 (neutron drip point), where neutrons start escaping the nuclei.

In the inner crust that follows, the remaining neutron-rich ions form a lat-tice, among which the relativistic electrons and the degenerate neutrons move. This comprises the largest part of the kilometer-thick crust, that ends as the density approaches the nuclear saturation point. Towards the inner edge of this region, the free neutrons increase, with their pressure eventually

surpass-ing that of electrons ∼ (4 · 1012 _g/cm3_{). However, the latter ones still play a}

role, converting the nuclei from spherical into having other geometrical shapes.

This so-called pasta phase is the prelude of the core.

Deeper into the star, nuclei are completely dissolved, with the outer core consisting of neutrons, protons, electrons and muons (npeµ matter). From

this region and on, charge neutrality and β−equilibrium conditions determine

the distribution of each particle. As a result, neutrons comprise the largest fraction, hence the term neutron star. The uncertainty about the behavior of highly-condensed matter makes it hard to deﬁne the end of the outer core. Nonetheless, it is assumed to lie at a density of about twice the nuclear satu-ration density.

Unlike to the crust, modeling the innermost part of the neutron star is not

an easy task. Depending on the Equation of state (EoS) that is implemented

for the description, the composition varies (see Fig.1). In general, modern EoS claim that hadronic reactions generate massive baryons and mesons, while oth-ers suggest that npeµ matter exists till the very center [2, 31, 66]. Furthermore, the maximum possible mass of a neutron star is strongly coupled to the choice of EoS [25].

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2.1 Neutron stars 6

Figure 1: Schematic proﬁle of the interior of a neutron star indicating the matter

composition and the approximate spatial extensions of each region along with the transition densities between them [56].

2.1.3 Superﬂuidity & hadrons

The attractive force between the hadrons inside the neutron star can induce

Cooper pairing at densities relevant for neutron star interiors. The

phe-nomenon of pairing applies to all the particles that are fermions, as long

as their temperature drops below a critical value Tc that diﬀers from particle

to particle. In this particular case, both neutrons and protons are expected to enter a paired state, with the former being more susceptible to this phase transition.

In fact, pairing of neutrons is expected to be starting from the inner crust and extend until the very center of the star. The pattern, though, switches

from 1_S

0 at the onset region to 3P2 in the core [63]. In the superﬂuid phase,

neutrons respond to the NS rotation by forming a set of vortices aligned with its spin axis [13]. On the other hand, protons manifest supercritical proper-ties only inside the core of the star. Their non-negative charge leads them to being superconducting and the electric field to deviate from being purely Ohmic [28]. In addition, since protons at this state are considered to form a type II superconductor, it is expected that the magnetic field in the core splits into an array of fluxoids [68].

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2.2 Hybrid stars 7

Despite not aﬀecting the macroscopic properties of a neutron star, this col-lection of attributes has grave consequences on its thermal evolution. Namely, it lowers the speciﬁc heat and thermal conductivity of the particles, suppresses

their neutrino emissivity and therefore slowing the cooling of the star(Minimal

cooling scenario, see 5.1). Regarding the created vortices, they are thought to be linked with the glitches frequently observed on pulsars.

2.2 Hybrid stars

2.2.1 Prediction by QCD

The rapid development of perturbative QCD during the last decades, has in-spired astrophysicists to investigate the possibility of deconfined quark matter occurring in the interior of compact objects. Their cores, reaching baryon densities as high as a few times the nuclear saturation density, are optimal candidates for being host to free quarks. This assumption implies that as the central region of the star is approached, a first order phase transition from confined (hadronic) to deconfined (quark) matter happens. In the QCD phase diagram (Fig.2), such a conversion is represented by the line separating liquid nuclear matter from cold quark matter.

As it is also illustrated in the ﬁgure, ordinary neutron stars lie in the regime of low temperatures and of average quark chemical potentials (µ). They oc-cupy the space to the left of the transition boundary, while hybrid stars (HS), in principle contain a core that extends along the dashed line. In total, the outer part of the latter can be fully described by a typical nuclear equation of state, and the inner, converted one, by an appropriate EoS derived from QCD in the weak-coupling limit.

2.2.2 Internal structure

The deconfinement of quarks invokes the main difference between the struc-ture of a NS and a hybrid star. It is obvious that any distinction between these two classes emerges from the undergoing phase transition. In particular, a HS core is expected to have a completely altered composition, consisting of 3-flavor quark matter: up (u), down (d) and strange (s) quarks, as heavier quark production is not favored for the relevant range of chemical potentials. The crust on the other hand, remaining unaffected, is expected to hold the same properties with that of an ordinary neutron star.

In order to describe the quark matter of HS in an astrophysical context, the general consensus is to start from considering massless u,d and a mas-sive strange quark. This population, is to be supplemented with electrons, so

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Figure 2: Schematic phase diagram of matter at extreme density and temperature.

The vertical axis represents temperature. The horizontal axis is the quark chemical potential µ. As µ increases at low temperature, the system passes from vacuum to nuclear matter at µ∼ 310 MeV . At an unknown higher chemical potential there is a transition to quark matter, which, at suﬃciently high density, is expected to be in the color-ﬂavor-locked (CFL) phase. Neutron star matter is in the lower part of the phase diagram, ranging up to an unknown maximum density at the center of the star [6].

that the constraints of charge neutrality and β-equlibirum are satisﬁed. Due to the imprecise knowledge about the details of strong interaction, though, many uncertainties are introduced in the thermodynamic description of quark matter. Although the MIT-bag model developed by Chodos et. al [23], has been widely used in literature to simulate the behavior of quarks under the conditions found in the core of HSs, it contains parameters of still unknown exact values [3, 18, 22]. The strange quark’s mass, the value of the strong in-teraction coupling constant and the Bag constant, are the main ones that aﬀect the construction of an appropriate EoS.

Another issue of discrepancy in studying the interior of such stars, is the way the phase transition is performed. The key question to be asked is whether the two phases are divided by a sharp interface or do they create a mixed phase where both coexist [8, 32].

Nuclear and quark matter, can be simultaneously present only when their pressures are equal. Considering the case of both being separately neutral, this condition is satisﬁed only in a certain value of the baryon chemical potential

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µb, that is mainly dependent on value chosen for the Bag constant. This fact,

ultimately results in an abrupt density jump in the interior of the star that sharply separates hadrons from quarks. In that case, the structure of the hy-brid star is simple: a quark core surrounded by a hadronic mantle.

Relaxing the charge neutrality condition, demanding instead that the mix-ture of nucleons and quarks is neutral, the possibility of a mixed phase becomes plausible. Under this constraint, hadronic and quark pressures can be equal

over a range of µb and not for just a single value of baryon chemical potential

as before. This translates to a coexistence of hadrons and quarks over a broad region inside the star. As a result, the HS consists of a quark core, a mixed phase and hadrons in its outer part. In any case, the deciding factor about which one of the two possible structures is favored, is the Coulomb energy and surface tension that contribute in the energetics of the two phases [8].

Regardless of what the exact internal structure might be, the simple pres-ence of quarks, can leave a hint on the astrophysical properties of a compact object. Given that including them will lead to a softer EoS than purely hadronic ones, the maximum mass of a hybrid star is expected to be lower than the

ob-servational limit of 2 M_⊙. Furthermore, since interactions between quarks oﬀer

new emission mechanisms, hybrid stars are expected to have a distinct thermal evolution to that of ordinary neutron stars [13, 61].

2.2.3 Superconductivity & quarks

Like hadrons, quarks can undergo Bardeen–Cooper–Schrieﬀer (BCS) pairing,

and thus step into a superconducting phase when they reach a critical

temper-ature Tc,q [12]. The color charge associated with the strong interaction permits

a diversity in pairing patterns, several of which have been investigated by dif-ferent authors [5, 9, 15]. To name a few important ones, two quarks of a particular combination of ﬂavor and color are thought to pair, while the third

one remains unpaired, in a state called 2SC phase. In a temperature further

below Tc,q, the motif shifts, as forming colorless triplets is favored. The system

is described to be in a color-ﬂavor-locked phase(CFL), where quarks carry, as

the name suggests, a particular combination of color and ﬂavor. For example,

one such triad may consist of a red up (ru), a blue down (bd) and a green

strange (gs) quark.

Although the limits between the unpaired quark matter, 2SC and CFL phases are still a subject of investigation, it has been suggested that detection of superconducting quarks can in principle be sought in observable quanti-ties of compact objects. In fact, a major eﬀect of the two supercritical states is expected to be caused on the thermal evolution of a compact object, in a level adequate enough to render neutron stars and hybrid stars

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indistinguish-2.2 Hybrid stars 10

able [65]. Nevertheless, there still exists a theoretical uncertainty on the exact size and temperature dependence of the pairing gaps in both phases.

Under these circumstances, any prediction about the cooling of a star con-taining superconducting quark matter is sensitive to the imposed parameters and inevitably with large uncertainties. As it is beyond the scope of this work to investigate the discrepancy that covers the phenomenon of color supercon-ductivity, only the case of unpaired quark matter will be treated.

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3. Equations of State used 11

3 Equations of State used

This chapter is dedicated to the presentation of the equations of state imple-mented in order to construct hybrid stars and the method used to model the emergence of pure quark matter. For the hadronic part, the EoS employed was the one developed by Douchin and Haensel, originally applied for describ-ing ordinary neutron stars [25, 37]. For the quark counterpart, the MIT-Bag model was used [23, 29]. In total, three diﬀerent hybrid EoS were created by

keeping the strong interaction coupling constant as and the mass of the strange

quark ms ﬁxed, while tuning the value of the bag constant B. The transition

from the confined to the deconfined phase was considered to proceed through a mixed phase, following the Gibbs equilibrium condition. In this way, it is ensured that pressure and energy jumps are absent at the critical densities and the final EoS is smooth over the whole range of chemical potentials.

3.1 Douchin-Haensel hadronic EoS

Douchin and Haensel suggested a uniﬁed equation of state for describing the whole of the neutron star interior, covering an energy density range spanning

from 106 _{to 10}15_g/cm3_{. The basic assumptions behind the} _{D-H EoS are the}

zero temperature approximation and a Skyrme-type eﬀective nucleon-nucleon interaction between the constituents [25]. The nuclei belonging to the crustal

part were modeled using the compressible liquid drop model (CLDM) that,

being classical, treats the nucleon and atomic numbers Z and A as continuous variables. After the weak ﬁrst-order phase transition between the crust and

the core at 0.076 f m−3, the hadronic matter is considered as an homogeneous

npeµ gas interacting through the SLy4 forces, ﬁrst proposed by Chabanat et

al [20]. This model of interacting forces is by construction compatible with that of Wiringa, rendering it highly applicable for neutron rich matter [66].

In the crust, the CLDM allows for the total energy density of the system

to be expressed as the sum of the bulk, the surface tension and the Coulomb contributions, to which the electron energy density is added, reading in total

Etot = EN,bulk+ EN,surf + ECoul+ Ee . (1)

The two terms in the middle are dependent on the shape of the nuclei, so that their expressions vary towards the bottom of the crust. In particular, beyond the neutron drip point, where nuclei coexist with the pure neutron gas, the authors have considered ﬁve diﬀerent types of nuclear structures which, start-ing from the top, are spheres, rods, slabs, tubes and bubbles. As the purpose

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3.1 Douchin-Haensel hadronic EoS 12

of the present work is the investigation of hybrid stars, the analysis of the D-H EoS focuses on the core part, where quark matter is expected to appear. Further details about the method used to develop the EoS of the crust can be found in Douchin & Haensel 2000 and Douchin et al.2000 [24, 26].

Once inside the core, the net energy density can be calculated by adding each particle species’ energy, including the rest mass of the hadrons and the terms referring to their interaction. The leptons on the other hand, are con-sidered as non-interacting Fermi gases. As a result, one reaches

Etot = EN(nn, np) + nnmn+ npmp+ Ee(ne) + Eµ(nµ), (2)

where ni represents the number density of the species i. At this part of the

EoS, the independent variable is the total baryon density.

10-6 ₁₀-5 ₁₀-4 ₁₀-3 ₁₀-2 ₁₀-1 ₁₀0 nb [fm−3] 109 1010 1011 1012 1013 1014 1015 En er gy [ g/ cm 3]

Figure 3: Plot of energy against baryon density using the hadronic EoS of Douchin

& Haensel 2001 [25]. For the sake of clarity, the diagram is restricted to covering the energy range from 109 to 3· 1015g/cm3. The dotted vertical line marks the neutron drip point and the dashed one the crust-core interface.

In this context, for a given nb the population of each kind of particle is

calculated imposing the conditions of β−equilibrium and charge neutrality.

The former ones are expressed through the notion of chemical potential µi =

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µn= µp+ µe, (3)

µe = µµ, (4)

while the latter one through that of particle fraction xi = ni/nb as

xp = xe+ xµ. (5)

Using the expression for the energy of asymmetric nuclear matter provided by Wiringa et al. [66] , it is possible to numerically solve the above equations and

determine every ni. Eq.(3) can then be rewritten in the form

µn− µp = µe⇒ (3π2_n b) 2/3 2mn [ (1− xp)2/3− x2/3p ] + 4 (1− 2xp) V2(nb) = (3π2xe)1/3, (6)

where the term V2(nb) represents the interaction energy of symmetric nuclear

matter, and it is set by interpolating tabulated data of nuclear and neutron matter calculations [66]. The resulting distributions of particles are shown in Fig.4. 10-3 ₁₀-2 ₁₀-1 ₁₀0 n_b [fm−3_] 10-3 10-2 10-1 100 xi [ fm − 3]

n

p

e

−

_µ

−

Figure 4: Relative abundances of particles in equilibrium throughout the outer crust,

the inner crust and the core for the hadronic EoS of Douchin & Haensel 2001 [25]. The brown dash-dotted line shows the distribution of neutrons inside nuclei, while the normal one that of free neutrons. Red dashed and red dotted lines indicate the baryon density at the beginning of the inner crust and that of the core respectively.

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Neutrons start dripping out of nuclei at a baryon density of 2·10−4f m−3,

af-ter which the region of the inner crust begins. The density jump in the neutron

star population (full brown line) at 0.076 f m−3, is characteristic of the

ﬁrst-order transition from the crust to the core. Muons appear after nb ≃ 0.12 fm−3,

where the electric chemical potential surpasses the value of mµ = 105.66 M eV.

Finally, the pressure may be calculated when the number densities are

known for the whole range of nb. Starting from the expression for the total

energy, it is possible to yield a one-parameter EoS by performing the differ-entiation P (nb) = n2b d dnb ( E(nb) nb ) (7) The overview of the pressure as a function of the total energy density is illustrated in Fig.5. It is noticeable, that after reaching the neutron drip point, it softens considerably and stiffens as E keeps rising. On the other side of the crust-core interface, the EoS becomes stiffer. It should be noted, though, that it is only safe to assume that dense matter follows the particular model up to nb ≈ 3 no. Any extension to higher densities is based on extrapolation [25].

109 ₁₀10 ₁₀11 ₁₀12 ₁₀13 ₁₀14 ₁₀15 Energy [g/cm3_] 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 Pr es su re [ er g/ cm 3]

Figure 5: Plot of the D-H hadronic EoS covering the outer crust, the inner crust

and the core of a neutron star, using the tabulated data from Douchin & Haensel 2001 [25]. For the sake of clarity, it is restricted to covering the energy range from 109 to 3· 1015g/cm3.

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3.2 The MIT-bag model 15

3.2 The MIT-bag model

The MIT-bag model is a phenomenological equation of state, according to which quarks constitute a degenerate Fermi gas. It was ﬁrst developed for the description of non-interacting strange quark matter [23]. Later works man-aged to incorporate the strong interaction force to the basic model while others stressed the importance of including the strange quark mass in the calculation of neutrino emissivities from a quark gas [4,29] [17,27,44]. Given the purpose of the present thesis, both of them will be taken into account.

The parameters as and mq together with the fundamental parameter B1/4,

that represents the conﬁnement of quarks into hadrons, fully characterize the quark gas. Based on estimations used by other authors, in the subsequent analysis of this work, the former ones will be assigned the indicative values of

αs = 0.4, mu = md = 0 and ms = 175 M eV, while the bag constant will be

ranging in order to investigate its eﬀect on the transition density [3, 5, 7, 34]. The description of the quark gas begins by deﬁning the thermodynamic potential of each kind of particles as a function of their respective chemical

potential. Restricting αs in the leading term and including the massive terms

up to O(m4 s), they read [29] Ωu,d =− ( 1− 2αs π ) µ4_u,d 4π2 , (8) Ωs =− ( 1−2αs π ) µ4_s 4π2 + 1 4π2 [ 3µ2_sm2_s− 3 2m 4 sln ( 2µs ms )] , (9) Ωe =− µ4 e 12π2, (10) Ωtot = ∑ i Ωi+ B . (11)

In this context, the total pressure is the opposite of Ωtot and the number

densities are extracted by diﬀerentiation with respect to the appropriate µi,

P =−Ωtot =− ∑ i Ωi− B = ∑ i Pi− B, (12)

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3.2 The MIT-bag model 16

ni =−

∂Ωi

∂µi

. (13)

The latter quantities can be calculated through solving the equations set by

the charge neutrality condition and β−equilibrium for neutrino-free matter,

Q = 2 3nu− 1 3nd− 1 3ns− ne = 0 (14) µd= µu+ µe (15) µs= µd (16)

The above system of (13)− (15) becomes simpliﬁed and solvable by

repa-rameterizing the expression for the chemical potentials of each kind in terms

of the baryon chemical potential 3µ and the electron chemical potential µe,

according to the standard law

µi = biµ− qiµe, (17)

where biis the baryon number and qi the electric charge of each particle species.

In this way, the weak equilibrium conditions are automatically satisﬁed and

the problem transforms into yielding a µe for a given µ, by taking just the

requirement of neutrality into account, so that

Q(µ, µe) = 0 . (18)

Moreover, the total quark pressure can be described using one single inde-pendent variable, that of µ. It is evident that choosing between different values of the bag constant does not have any effect on the particle distributions, as it does not get involved in calculating the appropriate chemical potentials. On the other hand, quark pressure and ergo the chemical potential at which quarks come into existence are very sensitive to B. A larger one delays their appearance since it implies that a higher value of pressure must be reached in order to overcome confinement. For large enough chemical potentials though, the massive terms become negligible and pressures converge, indicating the importance of considering them when studying transitions that proceed with a mixed phase.

1/4₌₁₅₈

_MeV

B

1/4₌₁₆₀

_MeV

Figure 6: Plot of pressure against quark chemical potential for EoS developed using

αs = 0.4, mu = md = 0, ms = 175 M eV and a set of diﬀerent bag constants. A

higher µ corresponds to a higher baryon density at the beginning of the quark phase. For the noted values of B1/4, nb ranges between 0.26 and 0.38 f m−3.

What does cause a difference on the quark and electron populations is the exclusion of the mass or the interaction constant. In the first case, electrons are entirely absent. Quark matter is charge neutral by itself, so leptons are not needed and number densities are equal between all flavors. In the second case, the distribution of each quark species keeps increasing as the total baryon density is raised. The effects raised by switching on and off each term, are shown in Fig.7 and 8.

0.0 0.5 1.0 1.5 2.0 baryon density [fm−3_] 10-5 10-4 10-3 10-2 10-1 100 nu m be r d en sit y [ fm − 3]

u

d

s

e

−

Figure 7: Splitting of quark number densities and appearance of electrons (dashed

lines) when considering a massive strange quark of ms = 175 M eV for developing

the quark EoS. Normal lines correspond to those of a massless, non-interacting quark gas, in which nu = nd = ns.

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3.2 The MIT-bag model 18 0.0 0.5 1.0 1.5 2.0 2.5 3.0 baryon density [fm−3_] 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 ∆ ni [ fm − 3]

Figure 8: Diﬀerence in the number densities of massless quarks between the

non-interacting and the non-interacting case for EoS developed using the MIT-bag model. ∆n = nnon_q −int− nint_q .

Finally, the remaining quantity for describing completely the thermody-namic properties of the quark gas is energy. Once the total pressure and the number densities are known, it can be computed through the formula

ϵ =−P +∑

i

1/4₌₁₅₈

_MeV

B

1/4₌₁₆₀

_MeV

Figure 9: Pressure against energy for diﬀerent bag constants, denoting the softening

of the quark matter EoS as B1/4rises. The EoS have been constructed using αs = 0.4, mu = md = 0, ms = 175 M eV and the set of the indicated bag constants.

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3.3 Mixed phase 19

3.3 Mixed phase

The scenario of transition to quark matter through a mixed phase was ﬁrst proposed by Glendenning [32]. In his work, it is shown that the Gibbs con-struction for modeling the conversion is only possible when global, rather than local, charge neutrality is assumed, so that pressure can vary along the transition region [34]. As a consequence, mixed phase is found to consist of opposite charged phases, with the rarer being immersed in the dominant one. Furthermore the emerging geometric structures are thought to be similar to the ones encountered in the transition performed in the inner crust. However, in his, as well as in this, analysis any phenomena arising from this feature, such as Coulomb interactions and surface tensions were neglected, in absence of precise understanding of their properties in the QCD-scale. In this picture of the mixed phase, the fraction of volume occupied from quark matter is denoted by χ, and the leptons are considered as part of the hadronic phase.

In order to locate where the mixed phase starts, one needs ﬁrst to ﬁnd the

combinations of µ and µe corresponding to the charge neutral hadronic phase

before using them in the expression for the pressure of the quark phase [34].

The baryon chemical potential at which Pq intersects with the hadronic

pres-sure, denotes the beginning of the mixed phase. The upper limit, above which lies the regime of pure quark matter, is deﬁned by following the complemen-tary process.

B1/4(M eV ) nb,in(f m−3) nb,f(f m−3)

150 0.28 0.83

153 0.37 0.86

155 0.43 0.89

Table 1: Range of the mixed phase with respect to total baryon densities for values

of the bag constant that give plausible transition densities.

Knowing the range of the mixed phase, the next step comprises of determin-ing the value of electron chemical potential that satisﬁes the Gibbs equilibrium condition. According to it, the pressures of hadronic and quark matter must be equal so that the coexistence of both phases is ensured. In practice, it is calculated by solving Eq.(20) for a given baryon chemical potential belonging to the mixed phase.

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3.3 Mixed phase 20 320 340 360 380 400 420 440 460 480 500 µ[MeV] 1033 1034 1035 1036 Pr es su re [ er g/ cm 3]

Figure 10: Plot of the method used to determine the range of the mixed phase for

the case of B1/4 _{= 153 M eV} _{and α}

s = 0.4, mu = md = 0, ms = 175 M eV. The

crossing points between the hadron pressure (blue lines) and the quark pressure (red lines) mark its edges. Dashed and normal lines represent charged and neutral matter respectively.

Having calculated the values of those variables, it is possible to compute the number density for all particles (Fig.11) by taking into account the relations

required for β−stable matter described earlier in this chapter. These

quanti-ties end up yielding a positively charged hadron phase and a negative quark phase. In simple terms, the amount of electrons and muons needed to render the two pressures equal, is not able to set the net charge of each phase to zero. The global charge neutrality condition,

χQq+ (1− χ)Qh = 0⇒ χ =

Qh

Qh− Qq

, (21)

determines then the fraction of volume occupied by quark phase for any point of the mixed region. Using this, the total baryon number density and the total energy density are deﬁned respectively as

nb = χnq+ (1− χ)nH, (22)

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3.3 Mixed phase 21 320 340 360 380 400 420 440 460 480 500 µ[MeV] 10-5 10-4 10-3 10-2 10-1 100 ni [ fm − 3]

p

n

µ

−

e

−

u

d

s

Figure 11: Population of particles for the hybrid EoS constructed using B1/4 = 155 M eV. Gray dashed lines indicate the edges of the mixed phase.

1011 ₁₀12 ₁₀13 ₁₀14 ₁₀15 Energy [g/cm3_] 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 Pr es su re [ er g/ cm 3]

Hadronic phase

Mixed phase

Quark phase

Figure 12: The constructed hybrid EoS for B1/4 _{= 150 M eV}_{. Red dotted line}

rep-resents the outer edge of the inner crust and red dashed line the crust-core interface. The continuity of pressure as the constitution changes from hadrons to quarks is characteristic of a transition through a mixed phase.

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3.3 Mixed phase 22

Combining theD-H EoS and the MIT-bag model with the parameters

pre-sented in 3.2, resulted that a mixed phase is realistic for bag constants between

150 and 155 M eV . Values below and over the accepted range, were found

leading to transition densities either lower or by far higher than the nuclear saturation density, and where thus discarded. The density threshold for the appearance of quarks is largely dependent on B. In any case though, the

region of coexistence of quark and hadronic matter is approximately 0.5 f m−3

wide and becomes more restricted as the bag constant is raised. An example of the physically accepted generated equations of state to be implemented for constructing hybrid stars and studying their thermal evolution is displayed in Fig.12 .

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4. Structure of hybrid stars 23

4 Structure of hybrid stars

4.1 TOV equations

4.1.1 Derivation

The first step towards deriving the equations of structure comprises defining an appropriate metric to describe the spacetime inside the hybrid star. As in the case of neutron stars, it would be more helpful to use spherical coordinates, as it simplifies all the mathematical derivations. Thus, the simplest metric that describes the interior of a compact object is

ds2 =−gttdt2+ grrdr2+ r2dϑ2 + r2sin2ϑdϕ2. (24)

The non-diagonal terms of the metric tensor are equal to zero for

diﬀer-ent reasons. Either because the coordinate system is orthonormal (grϑ= grϕ=

gϑϕ = gtϑ = gtϕ = 0), or because time reversal symmetry is required(−gtr = gtr).

Furthermore, for a physically accepted spacetime, the respective requirement for the metric dictates that all of its terms should be positive. Since the latter two are already fulﬁlling this criterion, it is suﬃcient to express the former two ones as exponential functions, so that

gtt = e2Φ, (25)

grr = e2Λ. (26)

The explicit formulas for both gtt and grr above can be derived by solving the

Field Equations, providing analytic expressions for both the lapse function eΦ

and the spatial curvature factor eΛ. In order to proceed further into

extract-ing the equations of structure of the star, it is necessary to make use of the expression for the Stress-Energy tensor, which reads

Tµν = (ϵ + P ) uµuν + P gµν, (27)

where ϵ is the energy density of the ﬂuid, P its pressure and u its four-velocity. The star being non-rotating means that only the temporal component of u is non-zero. Furthermore, the normalization condition of the velocity four-vector

sets the equalities at ut=-eΦ and ut=e−Φ. Therefore, the stress-energy tensor is

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4.1 TOV equations 24

Ttt = ϵ e2Φ, (28)

Trr= P e2Λ, (29)

Tϑϑ = P r2, (30)

Tϕϕ = P r2sin2ϑ. (31)

Using the metric deﬁned above, one can construct the Einstein tensor, whose components read [48] Gtt = 1 r2e 2Φ d dr [ r(1− e−2Λ)], (32) Grr =− 1 r2 e 2Λ(₁_{− e}−2Λ)₊2 rΦ ′_, ₍₃₃₎ Gϑϑ = r2e−2Λ [ Φ′′+ (Φ′)2+Φ ′ r − Φ ′_Λ′₋ Λ′ r ] , (34) Gϕϕ =sin2ϑ Gϑϑ. (35)

Finally, one can apply these expressions together with Eqs.(28)-(31) in order to solve the Field Equations and ﬁnd explicit expressions for Φ(r), Λ(r). Since the metric (Eq.24) represents a static spacetime, only the ﬁrst two are neces-sary for it:

Gtt = 8πTtt ⇒ 1 r2e 2Φ d dr [ r(1− e−2Λ)]= 8πϵe2Φ ⇒ d dr [ r(1− e−2Λ)]= 8πϵr2, (36) This differential equation is simplified after defining the auxiliary function, that should be considered as a ’mass function’, reading [60]

r(1− e−2Λ)= 2m(r), (37)

So that,

dm

dr = 4πϵr

2_. ₍₃₈₎

Applying the second Field Equation, one can reach to a diﬀerential equation for Φ(r): Grr = 8πTrr ⇒ − 1 r2e 2Λ(₁_{− e}−2Λ)₊ 2 rΦ ′ _{= 8πP e}2Λ _⇒

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⇒ dΦ

dr =

m(r) + 4πP r3

r [r− 2m(r)] . (39)

In both Field Equations, the pressure and the energy density of the fluid are functions of the distance from the center of the star. Therefore, the Tolman-Oppenheimer-Volkoff (TOV) version of this equation, according to which, the radius of the star is defined to be at the distance at which the pressure drops to zero, is more convenient [54]. Using the energy conservation for the radial coordinate in order to extract¹ a relation between Φ and P ,

T_;νrν = ∂T rν ∂xν +T κν Γ_κνr +TrκΓ_κνν = ∂T rr ∂r +T tt Γ_ttr+TrrΓ_rrr+TϑϑΓ_ϑϑr +TϕϕΓ_ϕϕr +TrrΓ_rνν = 0 ⇒ P ′ ( 1− 2m r ) + Φ′(P + ϵ) ( 1− 2m r ) = 0⇒ dΦ dr =− 1 ϵ + P dP dr, (40)

Eq.(39) ﬁnally simpliﬁes to dP

dr =−(ϵ + P )

m(r) + 4πP r3

r2_[1− 2m(r)/r]. (41)

4.1.2 Solution

In practice, Eq.(41) is the relativistic version of the hydrostatic equilibrium, ap-propriate for studying compact objects such as neutron or hybrid stars. The ef-fective corrections to the Newtonian analogue, are the substitution of the terms

describing the energy density and the mass with ϵ + P and m(r) + 4πP r3_,

re-spectively, as well as the introduction of the term referring to the curved space, [1− 2m(r)/r]. Along with the equation for the mass function (Eq.38), they form a system which deﬁnes the internal structure of the stars in question.

This system can be solved only by supplementing it with a relation connect-ing pressure and energy density. Thus, the EoS is the way through which the microscopic properties of matter enter the stellar structure equations. When it comes to the dense matter lying in the interior of compact objects, the un-certainty about the exact microphysics in that domain, permits various sug-gestions for describing those properties. In fact, given that the solution of the coupled diﬀerential equations, is particularly sensitive to the assumption, dif-ferent models lead to diﬀerent results (see Fig.13).

¹Christoﬀel symbols are computed out of gµν through Γρσµ = 12g

µν_[g

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The procedure for extracting the total mass and radius of the star begins

by choosing an arbitrary, albeit appropriate, central density ρc and setting the

boundary condition m(0) = 0. Both equations are then solved numerically up to the point at which the pressure nulliﬁes (P (R) = 0). The total gravitational mass of the star is given by the value of the ’mass function’ at this radius (M = m(R)). In the natural unit system² being used here, the mass carries units of cm. The respective value in solar masses can be calculated using the

conversion equation 1.48· 105_{cm = 1 M}

⊙.

By repeating the above-mentioned steps for a set of diﬀerent central den-sities, and therefore central pressures, one is provided with a set of mass and radius combinations. The net result is a sequence of stars parameterized by

central density. These M (ρc), R(ρc) relations are unique for each equation

of state and are commonly described in the form of mass-radius diagrams. Typical diagrams of this kind, for various examples of dense matter EoS, are shown in the ﬁgure below.

Figure 13: Mass-radius curves for compact stars following diﬀerent EoS models [67]. The

solid lines correspond to regular neutron star solutions with different EoS with normal matter composition (i.e. n, p, e, µ). The dashed lines correspond to solutions for EoS also containing exotic matter such as hyperons, kaon condensates and quarks. The dotted lines correspond to strange quark solutions. It is worth noting how the M-R curve is affected by adopting different equations of state. Dot-dashed lines refer to observed pulsars J1614− 2230 (green),

J 0348 + 0432(red) and Vela X-1 (blue).

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4.2 Results for hybrid stars 27

4.2 Results for hybrid stars

For the purpose of this work, the equation of state used is the one derived in Ch.3, where the transition from hadronic to quark matter proceeds through a mixed phase. The conﬁned phase is described by the uniﬁed EoS by Douchin

and Haensel(D-H), while the deconﬁned was considered to follow the

MIT-bag model, taking into account the interactions between quarks and the mass

of the strange quark [25, 37] [23, 29]. In total, three diﬀerent values of

the bag constant B1/4 _{were used to generate three families of hybrid stars;}

150, 153 and 155 M eV . The range of central densities for each value was

se-lected so that the constructed stellar conﬁgurations cover both the possibility of a quark core and a mixed phase in the very center of the star. The resulting mass-radius curves are shown in Fig.14.

One can spot that the hybrid stars are less massive than the neutron stars

following the D-H equation of state. This is due to the softening of the EoS

once quarks are included, as they can withstand gravity to a lesser degree, leading to lower mass conﬁgurations. Furthermore, quark-containing stars are more compact than the ones consisting entirely of npeµ matter. An ad-ditional feature worth noting is the subtle downward move of the MR curves as the bag constant is raised. The explanation for that trend is similar; as the

B1/4 _{becomes higher, the pressure for a given energy range drops, rendering}

the EoS softer and the respective stars lighter.

As regards the stability of the stars, it can be examined through plotting total mass against central energy density. In particular, the part of the curves

for which dM /dρcis negative represents conﬁgurations that are unstable when

subject to radial oscillations. For the hybrid stars in question, this branch is shown with dashed lines in Fig.15 . Below the ﬁlled dot, which marks the most massive star of each stellar sequence, all stars are stable and invulnerable

to radial perturbations [38]. It is also evident that ρc,max is aﬀected by the

value of B. As the latter drops, the value of the former drops as well. In any case, in consultation to the examples shown in Fig.13 too, it is well established that if quarks do exist in the interior of compact stars, the maximum possible mass becomes considerably lower.

B1/4_{(M eV )} _ρ

c,max(g/cm3) Mmax(M⊙) R (km)

150 24.50· 1014 1.71 10.72

153 25.00· 1014 _1.68 _10.58

155 25.50· 1014 _1.66 _10.54

Table 2: Central densities, masses and radii of the most massive stable conﬁgurations

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4.2 Results for hybrid stars 28 9.0 9.5 10.0 10.5 11.0 11.5 12.0 Radius [km] 1.2 1.4 1.6 1.8 2.0 Ma ss [ M¯ ]

B

1/4₌₁₅₀

_MeV

B

1/4₌₁₅₃

_MeV

B

1/4₌₁₅₅

_MeV

D

−

H

Figure 14: Mass-radius curves of the constructed hybrid stars with a mixed phase,

compared to the one of neutron stars following the uniﬁed hadronic EoS of Douchin-Haensel [25]. 1015 ₁₀16 Central density [g/cm3_] 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Ma ss [ M¯ ]

B1/4₌₁₅₀_MeV _B1/4₌₁₅₃_MeV _B1/4₌₁₅₅_MeV

Figure 15: Mass versus central density of the stars constructed with the hybrid EoS

developed using the MIT-bag model and the D-H hadronic equation of state. Normal lines represent stable conﬁgurations, ﬁlled circles the ones with the maximum possible mass and dashed lines stars under unstable equilibrium.

n

_b,tot

Figure 16: Populations of particles for the most massive hybrid star of each constructed

stellar family. The gray shaded area represents the crust and the gray dashed-dotted lines denote the limits of the mixed phase for every case.

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The distributions of the particles as a function of the radial distance from the center of the star are shown in Fig.16. For reasons of clarity, the most massive configurations of each sequence were chosen, details of which can be found in Table 2. Based on the composition, a hybrid star is divided in four regions: the quark core, the mixed phase, a mantle consisting of neutron star matter and the crust. In fact, the first two cover most of the stellar interior irrespective of the choice for the bag constant. What does depend upon the selection of B, though, is the range of the mixed phase. Namely, the lower the value of the bag constant, the lower the baryon density at which decon-finement sets off and the larger the region of coexistence between hadrons and quarks.

Moving inwards from the crust, above 1.3 · 1014_g/cm3 _{and before reaching}

the baryon densities at which the phase transition happens, the constituent particles are free neutrons, protons, electrons and muons. Beyond the critical

nb, the number density of protons increases, ensuring that the hadron phase

remains positive throughout the mixed phase. Electrons and muons on the other hand, decrease as a result of satisfying the Gibbs condition of equilibrium between the pressures of the two phases. The amount of neutrons is decided

by the requirement of β equilibrium (µn = µp+ µe). As the total baryon

num-ber density rises, further towards the center of the star, the population of each quark species rises. The net quark charge is negative in the mixed phase, so that global charge neutrality in this shell is maintained. Once the number density of strange quarks surpasses that of protons, the latter start decreas-ing until they become depleted, along with neutrons, markdecreas-ing the start of the quark core.

At that region the phase transition has been completed and down quarks are the most abundant species, with up and strange following in this order. Given that the last ones have a non negligible mass for chemical potentials rel-evant to the interior of hybrid stars, a small quantity of electrons is needed to keep the phase charge neutral. According to the model, the extent of the core depends on the value of the central density, with less dense stars possessing

a smaller one. Below a suﬃciently low ρc though, quarks would exist only

as a component of a mixed phase and for an even lower one, they would be absent.

The two subsequent plots, illustrate the energy density and pressure pro-ﬁles, excluding the crust part, of three stars belonging to diﬀerent stellar se-quences. Starting from the higher bag constant, their central densities are 23,

21.30 and 17.80 · 1014 _g/cm3 _{respectively. The form of the curves is typical of}

a hybrid star containing a mixed phase. The energy is continuous throughout the star and the pressure falls smoothly from the center to the outer part [33].

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The alternative model, where the phase transition from pure hadronic matter to quark matter occurs abruptly at a particular value of the pressure, would result in a jump at the corresponding radial distance.

0 2 4 6 8 10 12 r [km] 1015 En er gy [ g/ cm 3]

B1/4₌₁₅₀_MeV _B1/4₌₁₅₃_MeV _B1/4₌₁₅₅_MeV

Figure 17: Indicative density proﬁles of hybrid stars constructed with diﬀerent

val-ues of B1/4. Their mass-radius combinations are 1.69 M⊙, 11.20 km (dotted line), 1.67 M_⊙, 10.82 km(dashed line) and 1.66 M⊙, 10.70 km(dash-dotted line).

0 2 4 6 8 10 12 r [km] 1033 1034 1035 Pr es su re [ er g/ cm 3]

B1/4₌₁₅₀_MeV _B1/4₌₁₅₃_MeV _B1/4₌₁₅₅_MeV

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5. Thermal evolution 32

5 Thermal evolution

Born out of supernova explosions, compact stars are assumed to start their life having very high internal temperatures and cool down through a com-bination of thermal radiation from their surface and neutrino emission from their interior. From the ﬁrst day of their lives, when the temperature in their

interior has already dropped from ∼1011 _{K to 10}9_{− 10}10 _{K, until the age of}

several thousand years, thermal energy is carried away mainly in the form of

νe radiation [55, 57].

At such temperatures, the interior of compact objects is transparent to neu-trinos, so they escape eﬃciently. In both hybrid and ordinary neutron stars, neutrinos are generated through a plethora of emission mechanisms, sum-marized in Table 3 at the end of Ch.6. Neutrino emissivities are strongly

dependent on temperature, so that when Tcore becomes suﬃciently low,

pho-ton emission from the surface overtakes. Beyond this point, theneutrino cooling

era is over and the photon cooling era begins.

Measurements of the surface temperature and the luminosity of compact objects can yield significant information about the nature of their contained matter and underlying magnetic field, because, in principle, different composi-tions leave different footprints. Prediccomposi-tions on thermal evolution are therefore sensitive to the used equation of state, the magnetic field strength, the as-sumption regarding the chemical abundances of the envelope and the possible pairing of the constituent particles. The variety of these factors results to a wide range of cooling histories when their numberless combinations are con-sidered. The final outcome of those theoretical calculations are a set of curves describing detectable features, such as luminosity, as a function of time, which are testable through observations in the X-ray part of the spectrum. One of the most promising conclusions that can be made out of an appropriate analysis of a source’s cooling pattern, is the verification of quark matter’s presence inside compact objects.

However, not all observed neutron stars are suitable for a study of that pur-pose. A substantial amount of them are binary x-ray sources, which means that the accretion luminosity prevails over surface thermal emission, thereby discarding any possibility of deriving useful inferences. This fact renders iso-lated compact objects as ideal candidates. Among this population, though, lie young pulsars associated with supernova remnants, whose surface luminos-ity is contaminated by the non-thermal emission caused by the eﬀect of the magnetic ﬁeld on high energy particles of the magnetosphere. The best oppor-tunities for studying thermal emission from neutron or hybrid stars therefore range in the ensemble of isolated middle-aged sources.

36 L [ er g/ se c ]

RPP

HB

XINS

MAG

Figure 19: Observational data of isolated neutron stars in quiescence showing a clear thermal component. Each source is classiﬁed as rotation-powered pul-sar (RPP), high- ⃗B radio pulsar (HB), X-ray isolated neutron star (XINS) or

mag-netar (MAG). The sample has been obtained from www.neutronstarcooling.info and www.atnf.csiro.au/people/pulsar/psrcat [52], where all details about every source are

given.

5.1 Cooling equations & thermal evolution models

The mathematical description of the thermal evolution requires a general

rel-ativistic treatment. Using the metric adopted in Ch.4, the energy balance

equation for the star reads cv,toteΦ ∂ T ∂ t + ⃗∇ · ( e2ΦF⃗ ) = e2 Φ(H − Q) , (42) ⃗ F = −e−Φˆκ· ⃗∇(eΦT), (43)

with cv being the total speciﬁc heat per unit volume, ⃗F the thermal ﬂux in the

diﬀusion limit, ˆκ the thermal conductivity tensor, H the heating power per

unit volume andQ the sum of neutrino and photon emissivities that represent

the energy losses. For isolated neutron stars, the heating power is provided by all the mechanisms converting magnetic or rotational energy into heat. The basic prerequisite for solving this equation is an EoS which describes the ther-modynamic properties of the compact star. The microphysics of the selected

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model is what becomes translated into inputs for the deﬁnition of the param-eters mentioned above.

The speciﬁc heat of the star is regulated by the core since this is the part containing the bulk of the stellar mass. Depending on whether the free

parti-cles are relativistic or not, the value of cv,i for each population equals

cv,i = π2nikB T TF,i = π2nikB T pF,i ∼(π2ni )2/3 kBT erg cm3· K , (44) c_v,i = 1 2π 2_n ikB T TF,i = 1 2π 2_n ikB2 T EF,i = π2nik2Bm ⋆ i T p2 F,i ∼(π2ni )1/3 k_B2m⋆T erg cm3· K (45) respectively. Compared to ordinary neutron stars, hybrid stars are expected to have a larger heat capacity due to up and down quarks belonging in the ﬁrst class. In the case of pairing in hadronic or quark matter, the speciﬁc heat of paired particles becomes exponentially suppressed and has no essential

con-tribution in the total cv [9, 50]. Regarding the crustal part, that is common in

both kinds of stars, the speciﬁc heat is lower in total and originates from the ionic lattice and the ultra-relativistic electrons, except from the layers where neutrons are not superﬂuid [57].

Neutrinos and antineutrinos are not the only particles carrying away the heat, as electrons, neutrons, quarks and even phonons are also able to transport it throughout the stellar interior of either neutron or hybrid stars. Thermal conductivity is in practice determined by the mean free path of the particles and the associated scatterings between them. In the interior of neutron and hybrid stars, the mean energy exchanged during each collision is much smaller than the mean thermal energy of each particle. Therefore, it is safe to follow the relaxation time approximation and express the tensors of thermal conduc-tivity of each degenerate species as [39, 63]

ˆ κi = π2_k2 BniT 3EF,i ˆ τth i . (46)

The relaxation tensor ˆτth

i is the sum of the inverse of all the collision

frequen-cies of the processes relevant to heat transport. In the crust, this task is mainly performed be electrons, given that the lattice restricts the mean free path of neutrons while that of phonons is smaller than that of electrons [63]. Unless

superﬂuid though, neutrons are expected to dominate κcrust in temperatures

lower than 108 _{K [57]. In the core, the highly conducting electrons (quarks)}

dominate the thermal conductivity of the neutron (hybrid) star during the initial stages of cooling, rendering it rapidly isothermal [15, 63]. Overall, as it

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holds for speciﬁc heat, κcrust is lower than κcore, causing the crust to cool faster

in young stars.

In weakly magnetized compact objects, when no heating mechanisms are assumed, the commonly proposed cooling scenarios during the neutrino cool-ing era, are the enhanced and the minimal coolcool-ing [13, 68]. The ﬁrst class refers to the case where the energy losses occur via direct Urca channels asso-ciated with hadron and/or quark populations. The characteristic of this kind

of fast processes is the T6 _{dependence of the ν}

e/ ¯νe emission rate, regardless of

the core composition. As a result, the cooling curve of a star that follows the

enhanced cooling scenario, forms a characteristic sudden drop in Tsurf and L

as early as in the ﬁrst few centuries (see Fig.20). For lighter stars the neu-trino production is lower; the kinematic conditions are hardly favorable and even when they are, the volume in which fast processes are allowed is smaller. Therefore, they are predicted to have a higher surface temperature and lumi-nosity than heavier stars. A drop below a critical temperature though, can invoke pairing on the particles at an early stage, thus lowering its cooling rate by suppressing the total neutrino emissivity.

As already mentioned in Ch.2, pairing is thought to appear in protons, neutrons and quarks inducing suppression on their microscopic properties. Although a complete, precise description of the phenomenon for densities relevant to neutron star matter is still pending, it is widely applied for investi-gating its eﬀects on thermal evolution. Most studies agree that once it switches

on for a certain species, their speciﬁc heat reduces by a factor R ∼ e−∆/T,

where ∆ is the gap parameter, linked to the respective critical temperature

via the standard BCS pairing relation κBTcr ≈ 0.57∆ [12,13,55,59]. A further

drop in temperature, causes a further reduction of heat capacity to the point of being equal to that of leptons [57]. Neutrino emissivity out of paired baryons becomes suppressed as well, since particles in such a supercritical state have to overcome the energy gap, in order to interact with another particle. In

the range of temperatures T ≪ Tcr, this behavior is illustrated numerically

using a yet another set of control functions Rχ(T /Tc), which diﬀers between

each particle species χ [68]. Nevertheless, this does not imply that the total neutrino emission switches oﬀ, as a new mechanism arises (Fig.20); that of

Cooper Pair breaking-formation (CPBF), which is possible for both hadrons

and quarks. In a way, pairing compensates for the delay in the cooling of the star by providing an eﬀective channel for carrying the heat away.

CPBF is deemed the main energy loss mechanism in the second class of thermal evolution scenarios, minimal cooling, that refers to ordinary neutron stars inside which any presence of exotic matter, such as pions, kaons or quarks, is excluded. According to it, direct Urca reactions are not

Thermal evolution of neutron stars containing a quark core

U  A

MSc Astronomy & Astrophysics

Master Thesis

Thermal evolution of neutron stars

containing a quark core

by

Stefanos Tsiopelas

10850015

February 2018

September 2015 - February 2018

Supervisors:

Nanda Rea, Dr.

Justin Elfritz, Dr.

Examiners:

A. Watts, A.Pr.

R. Wijnands, Dr.

Anton Pannekoek

Institute

Contents

1

Introduction

2

Compact Objects

2.1

Neutron stars

2.2

Hybrid stars

3

Equations of State used

3.1

Douchin-Haensel hadronic EoS

n

p

e

µ

3.2

The MIT-bag model

B

MeV

B

MeV

B

MeV

B

MeV

B

MeV

B

MeV

B

MeV

u

d

s

e

B

MeV

B

MeV

B

MeV

B

MeV

B

MeV

B

MeV

B

MeV

3.3

Mixed phase

p

n

µ

e

u

d

s

Hadronic phase

_µ

_MeV

_MeV

_MeV

_MeV

_MeV

_MeV

_MeV

_MeV

_MeV

_MeV

_MeV

_MeV

_MeV

_MeV

_MeV

_MeV

_MeV

_MeV

_MeV

_MeV

₁₀

₁₀

₁₀

₁₀

₁₀

₁₀