Signature of deconfinement with spin down compression

7/28/2019 Signature of deconfinement with spin down compression

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arXiv:0801

.0358v2

[astro-ph]

5Apr2009

Signature of deconfinement with spin down compression in

cooling hybrid stars

Morten Stejner

Department of Physics and Astronomy, University of Aarhus

Ny Munkegade, Bld. 1520, DK-8000 Aarhus C, Denmark.

[email protected]

and

Fridolin Weber

Department of Physics, San Diego State University5500 Campanile Dr, San Diego CA 92182-1233

and

Jes Madsen

Department of Physics and Astronomy, University of Aarhus

Ny Munkegade, Bld. 1520, DK-8000 Aarhus C, Denmark.

ABSTRACT

The thermal evolution of neutron stars is coupled to their spin down and the resultingchanges in structure and chemical composition. This coupling correlates stellar surfacetemperatures with rotational state as well as time. We report an extensive investigationof the coupling between spin down and cooling for hybrid stars which undergo a phasetransition to deconfined quark matter at the high densities present in stars at low ro-tation frequencies. The thermal balance of neutron stars is re-analyzed to incorporatephase transitions and the related latent heat self-consistently, and numerical calcula-tions are undertaken to simultaneously evolve the stellar structure and temperaturedistribution. We find that the changes in stellar structure and chemical composition

with the introduction of a pure quark matter phase in the core delay the cooling andproduce a period of increasing surface temperature for strongly superfluid stars of strongand intermediate magnetic field strength. The latent heat of deconfinement is found toreinforce this signature if quark matter is superfluid and it can dominate the thermalbalance during the formation of a pure quark matter core. At other times it is lessimportant and does not significantly change the thermal evolution.

Subject headings: Stars:neutron stars:rotation dense matter equation of state

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1. Introduction

The chemical composition of neutron stars

at densities beyond nuclear saturation re-mains uncertain with alternatives rangingfrom purely nucleonic compositions throughhyperon or meson condensates to deconfinedquark matter see e.g. Weber (2005) andPage & Reddy (2006) for recent reviews withemphasis on quark matter. A future un-derstanding of neutron star structure gainedthrough confrontation of theoretical mod-els with the now steadily growing body ofobservational facts will therefore simultane-

ously constrain fundamental elements of par-ticle and nuclear physics (Lattimer & Prakash2007). The interlinked processes of spindown and thermal cooling present intriguingprospects of gaining insight in the propertiesof matter in neutron star cores by confronta-tion with soft X-ray observations of thermalradiation from neutron star surfaces as theyboth depend sensitively on and to some extentdetermine the chemical composition.

As neutron stars spin down and contract,their structure and composition change withthe increasing density drastically if phaseboundaries are crossed so new forms of mat-ter become possible. We shall here investigatehow this influences the thermal evolution ofhybrid stars which contain large amounts ofdeconfined quark matter. The increasing den-sity and changing chemical composition fur-ther imply additional entropy production inbulk and the release of latent heat as particlescross any phase boundaries present. We there-fore re-analyze the thermal equilibrium of

compact stars to show how mixed phases maybe incorporated. We thus arrive at a naturaldescription of the latent heat of phase transi-tions in compact stars, but also find throughdirect numerical calculations that unless thestellar structure changes very rapidly the ef-fects of latent heat on the thermal evolution

are insignificant when compared to those ofthe changing chemical composition and thesurface area reduction.

Neutron stars are extremely compact ob-jects and densities in their cores reach well inexcess of the nuclear saturation density. Atsuch densities the distance between particlesis on the order of the characteristic range ofthe nuclear forces. Therefore, as was stressedrecently by Baym (2007), perturbative treat-ments in terms of few- or even many-bodyforces although highly successful in describ-ing the properties of matter below nuclear sat-

uration are no longer well defined in the coreof neutron stars. Further, the relevant degreesof freedom should include the appearance ofhyperons and possibly deconfined quarks, sotreatments in terms of nucleons alone mustalso be seen as approximative. Hyperons areexpected to appear at densities around 230,where 0 = 0.153 fm

3 is the baryon densityat nuclear saturation. The appearance of hy-perons so softens the equation of state thatpurely hadronic equations of state may not

allow stable models compatible with the accu-rately measured masses of neutron stars in bi-nary systems (Baldo et al. 2003; Schulze et al.2006). Schulze et al. (2006) further demon-strate that this conclusion is highly robustwith respect to different assumptions abouthyperon interactions and the nucleonic equa-tion of state. At best these studies are strongarguments against purely hadronic composi-tions and they certainly do show the relevanceof considering alternatives such as a transition

at high densities to deconfined quark matter.

Quark matter represents an entirely newtype of matter as opposed to just an ad-ditional degree of freedom as in the case ofhyperons in the hadronic phase and it can-not be assumed to soften the equation ofstate to the same extent (Alford et al. 2008;

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Schaffner-Bielich 2007). Hybrid stars can bebe consistent with the high masses and radiiindicated by recent observations (e.g. Ozel

(2006), Freire et al. (2008b,a); Freire (2008)),and the quark matter equation of state canfulfill the constraints imposed by heavy-ioncollision transverse flow data and K+ pro-duction (which nucleonic equations of statedo not). Further, Drago et al. (2008) foundthat only stars with a quark matter compo-nent can rotate stably without losing angularmomentum by emission of gravitational wavesthrough the r-mode instability at the 1122 Hzindicated by recent observations for the X-

ray transient XTE J1739-285 (Kaaret et al.2007). This conclusion is tentative as theobserved neutron star spin frequency awaitsconfirmation, but it again shows how the com-position of neutron stars is an open questionto be determined by a confrontation of theoryand observation, and that a deconfined quarkmatter phase remains a viable alternative.

For definiteness we shall work with theequation of state suggested by Glendenning

(1992); hereafter the G300180 equation of state.

While this equation of state is not sophisti-cated in its treatment of the quark matterphase, it is illustrative in that it allows a verylarge pure quark matter core with a transi-tion through a mixed phase at relatively lowdensities around 2-50. For our purposes it isinteresting in that it implies drastic changes incomposition with spin down the pure quarkmatter core disappears at high spin frequen-cies for instance and it therefore represents

something close to a limiting case. We shallthen be able to investigate both the effects ofsteady conversion of hadronic matter to quarkmatter and the sudden appearance of a purequark matter core in the hadronic phase.

In the so-called minimal scenario, whichexcludes exotic and very rapid neutrino pro-cesses (Page et al. 2004), the internal tem-

perature of neutron stars drops from beyond1010 K to around 109 K within a few minutesafter their birth, and neutrino cooling con-

tinues to dominate for at least the next fewthousand years until the internal temperaturehas dropped below 108 K and photon cool-ing takes over (see, e.g. Page et al. (2004);Yakovlev & Pethick (2004) for reviews). Ifthe highly efficient direct Urca neutrino emis-sion (i.e., essentially beta decays, n p +e+ e and related processes in quark matter)is active, the stars may cool very rapidly andreach very low temperatures on a timescale ofa few hundred years. As we shall see the nu-

clear direct Urca process is active in the mixedphase of hybrid stars, and may thus controlthe thermal evolution. These processes maybe suppressed by pairing of the participatingparticles however, and further the extent ofthe mixed phase depends strongly on the ro-tation frequency thus giving rise to a diverserange of possible cooling paths.

The latent heat of the phase transitionand the related release of entropy in bulkwith changing density are generally found

to be of smaller importance than the chang-ing structure and chemical composition. Itdoes not significantly delay or enhance thecooling, although it does briefly balance oreven dominate the cooling terms when thepure quark matter core is first formed for cer-tain choices of stellar parameters. The latentheat of the quark matter phase transition waspreviously considered by Miao & Xiao-Ping(2007); Miao et al. (2007) and, Xiaoping et al.(2008). These authors took a different ap-

proach to calculate the latent heat than whatis discussed below and found very significantheating terms which we do not recover.

The work of Reisenegger (1995) and latelyFernandez & Reisenegger (2005) consideredthe related process of roto-chemical heatingfor neutron stars in which weak reactions aredriven out of equilibrium by the changing den-

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sity. The resulting release of energy was foundable to maintain old stars at relatively hightemperatures determined by their rate of spin

down. This term naturally appears in ourequations for the thermal balance and shouldbe included in a full model. It is to some ex-tent complementary to the effects discussedhere, but the treatment of weak reactions be-yond equilibrium is beyond our scope in thiswork, and we shall assume chemical equilib-rium throughout. We return briefly to discussthis issue in Sect. 5.

The link between the thermal evolution of

neutron stars and their spin down has be-come a subject of some interest as an ob-servational correlation between the tempera-tures and inferred magnetic fields of neutronstars was recently discovered by Pons et al.(2007). This was interpreted as evidence formagnetic field decay and detailed work byAguilera et al. (2008,?) strongly supports thisconclusion. An alternative interpretation wasoffered recently by Niebergal et al. (2007) interms of magnetic flux expulsion from color-

flavor locked quark stars (a hypothetical classof stars consisting entirely of quark matterwhich in this case is assumed to be absolutelystable). The previously mentioned work ofDrago et al. (2008) also indicates a link be-tween the spin down and cooling of hybridstars. These correlations if they can in-deed be shown to exist complement thetraditional cooling calculations which relateonly temperature and age, and they may helpbreak the degeneracy seen between such calcu-

lations with different underlying assumptionsabout the state of matter at very high density.

In the following, we shall first revisit theequations of thermal balance for compactstars in Sect. 2 to show the effects of a time-dependent density and discuss how the pres-ence of phase transitions may be included.

In Sect. 3, we discuss the G300180 equation ofstate, the resulting stellar models and somesimple estimates for the additional terms in

the thermal balance equations in more detail.In Sect. 4 we show the results of includingthese terms in spherical isothermal coolingcalculations. We conclude with a discussionin Sect. 5.

2. Thermal Equilibrium in the Mixed

Phase

The thermal evolution of compact stars isdetermined by the equations of local energy

conservation and transport in the frameworkof general relativity. These equations balanceany energy radiated away from the star byphotons and neutrinos against changes in restmass due to nuclear reactions and changes ingravitational or internal energy as the stellarstructure evolves. In the most widely studiedscenario the stellar structure is assumed con-stant and the only energy source available toneutron stars is then their original endowmentof thermal energy (e.g., Van Riper (1991);Schaab et al. (1996); Page et al. (2004, 2006);Yakovlev & Pethick (2004) and referencestherein). Neutrino production in the core andphoton emission from the stellar surface thenensures a monotonical and sometimes veryrapid cooling of the star depending stronglyon any superfluid properties of the core. Butin addition to this a number of powerful en-ergy sources may play a role in delaying oreven reversing the cooling at various stagesin a neutron stars life see e.g. Schaab et al.(1999) for an overview of possible sources and

their effects. Here we discuss the effects ofincluding a phase transition in the energy bal-ance, which, as the stellar structure changes,must then also include redistribution of en-tropy b etween the two phases as their pro-portion changes as well as changes in surfaceand Coulomb energy for transitions througha mixed phase.

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Following Thorne (1966) or the equivalentdiscussion in Weber (1999) we consider en-ergy conservation for a spherical shell inside

of which is a baryons and which itself con-tains a baryons. The treatment given in thissection therefore assumes spherical symmetrywhich is sufficient to show the effects of aphase transition on local energy conservation we shall return later to discuss possible con-sequences of a multidimensional cooling cal-culation. The shell under consideration willchange its internal energy during a coordinatetime interval dt by

d(internal energy) =

(amount of rest mass-energy converted to

internal energy by reactions)

+ (work done on shell by gravitational

forces to change its volume during

quasi-static contraction)

(energy radiated, conducted

or convected away from the shell). (1)

It is important to note at this point that ifthe two phases are in equilibrium as shouldcertainly be expected for transitions governedby strong reactions such as that from hadronicto quark matter there is no binding en-ergy involved in the transition and thereforeno contribution to the first term on the righthand side of Eq. (1). If this was the caseand the phase transition did involve a bind-ing energy the two phases could not be inequilibrium and the star would adjust itselfaccordingly eventually becoming a strangestar in the case of the quark matter transition

if quark matter was assumed bound relative tohadronic matter at zero external pressure. Inhybrid stars this is not the assumption how-ever, and so any latent heat evolved or ab-sorbed in the phase transition follows fromthe different thermodynamical properties ofthe two phases as with any other phase tran-sition.

In the mixed phase of the G300180 equa-tion of state regions containing negativelycharged quark matter appear at densities of

about 2 times nuclear saturation and dom-inate completely at 5 times nuclear satura-tion. They allow the hadronic matter tolower its isospin asymmetry energy and be-come positively charged by including moreprotons with charge neutrality achieved glob-ally. The geometry and structure of themixed phase is determined by a balance be-tween surface tension and Coulomb repul-sion between regions of like charge. For de-tails on the phase transition we refer to e.g.

Glendenning (1992); Heiselberg et al. (1993);Glendenning (2000); Glendenning & Weber(2001); Voskresensky et al. (2003); Endo et al.(2006).

Returning to our spherical shell of baryonnumber a we will assume that its volume Vis large enough to contain a macroscopic num-ber of unit cells each containing a region filledby the rare phase whose presence may thenbe considered a microscopic property of theequation of state. The work done to change

the shells volume during contraction or expan-sion of the star must then include changes insurface and Coulomb energy as well as theusual pressure term

dW = PdV + dS + dEC (2)

where is the surface tension, S is the amountof surface area in the shell dividing the twophases and EC is the Coulomb energy con-tained in the shell. These terms can be sim-ilarly included in the first law of thermody-

namics which we write in the same notation(units with G = c = kB = 1 will be used hereand throughout this paper)

d = P+ d + T ds +

k kdYk

+ dSa + dEC

a (3)

where s is the entropy per baryon and Yk =k/ is the fraction of the baryon number den-

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sity in the form k with k running over all par-ticle species and

k Yk = 1. In the quark

matter phase each quark contributes by only13 to the baryon number density and s is thenthe entropy per three quarks. The last twoterms in Eq. (3) are the local densities of sur-face and Coulomb energy written in a formuseful for our purpose.

Inserting this in Eq. (1) and using that theamount of energy radiated, conducted or con-vected away from the shell can be written interms of the gradient of the total luminosity,Ltot, we show in Appendix A that local energybalance may be expressed as

d

da(Ltote

2) = e

T

ds

dt

a

+

k

k

dYkdt

a

(4)thus giving the contribution to Ltot from theshell in terms of the entropy production andthe difference between the chemical potentialsof particles participating in any reactions tak-ing place in the shell. e2 is the time compo-nent of a spherically symmetric metric

ds2

= e2

dt2

+ e2

dr2

+ r2

d2

+ r2

sin 2

d2

(5)found from the general relativistic structureequations for compact stars.

Ltot includes the neutrino luminosity, butsince neutrinos can be taken to immediatelyescape from the star when they are createdwithout converting into any other form of en-ergy along the way, they fulfill their own sep-arate equation of energy conservation

d

da(Le2

) =

e

2

(6)

where L is the neutrino luminosity and is the neutrino emissivity; the rate per unitvolume at which neutrino energy is created.In neutron stars convection is negligible com-pared to electron conduction and photon dif-fusion and so the remainder of Ltot can be

shown to fulfill a transport equation (Thorne1966; Weber 1999)

dda

T e

= 3

16T3

Le162r4

(7)

where is the Stefan-Boltzmann constant, is the total thermal conductivity and L =Ltot L. At the stellar center we must haveLtot(a = 0) = 0 while at the stellar surface Lmust equal the total stellar photon luminos-ity which may depend on assumptions aboutproperties of the stellar atmosphere or lackthereof.

Eqs. (4)-(7) with the appropriate bound-ary conditions can be solved to evolve thethermal structure of a stellar model. Theyhave exactly the same form as would be ex-pected in the absence of any phase transi-tions. However, the phase transition influ-ences the entropy density and this, as weshall see, gives rise to additional terms in theheat balance including latent heat and thesurface and Coulomb energies. An equiva-lent form of Eq. (4) showing the contribu-tions from surface and Coulomb terms more

explicitly can be found in Eq. (A6) of Ap-pendix A. Combining Eqs. (4) and (6), not-ing that cv = T(s/T)V and assuming con-stant structure and chemical composition wealso recover the standard cooling equation forstatic stars

d

da(Le2) = 1

e

2 + cvd(T e)

dt

(8)

Eq. (4) is useful because it allows a particu-

larly simple analysis in the presence of a phasetransition. The first thing to note is, thatparticles crossing the phase boundary wouldnot contribute to the second term on the righthand side of Eq. (4) if the two phases are inor close to equilibrium, because their chemi-cal potentials (or those of their constituents)must then be continuous across the phase

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boundary. In the following we therefore ne-glect this term, but we shall return to discussits potential importance for reactions not in

equilibrium. The first term is a different mat-ter however. The entropy per particle is afunction of density, temperature and chem-ical composition which is not required to becontinuous across the phase boundary, so par-ticles making the transition will release or ab-sorb heat accordingly. To see how this workswe write s as a sum with bulk contributionsfrom each phase according to their volume ormass fraction as well as contributions from thesurface and Coulomb energies

s =1

((1 v)S1 + vS2 + SS + SC) (9)

= (1 )s1 + s2 +1

(SS + SC) (10)

where = (1 v)1 + v2 is the average ofthe two particle densities 1,2 weighed by thevolume fraction of the dense phase, v. S1,2is the bulk entropy density in each phase andSS and SC are the surface and Coulomb con-tributions to the entropy density respectively.

We have further introduced the baryon num-ber fraction of baryons in the dense phasein a sample of matter related to the volumefraction by = v(2/), as well as the en-tropies per baryon in the respective phases,s1,2 = S1,2/1,2. At constant a and assum-ing the fraction of matter in the dense phaseand the particle densities do not depend ontemperature we then have

Tds

dt=

cv

dT

dt+ T

ds

d

d

dt

=cv

dTdt

+ Tdd

ddt

(s2 s1)

+(1 )Td

dt

ds1d

+ Td

dt

ds2d

+Td

dt

1

(SS + SC)

(11)

where the heat capacity is again a weighed

volume average in the mixed phase

cv = (1 v)cV,1 + vcV,2 . (12)

The latent heat absorbed by a particlecrossing the boundary between two phasesin equilibrium is the temperature times thedifference in entropy per particle between thetwo phases, q = T[s2s1] (Landau & Lifshitz1980); we recover this in the second term onthe right hand side of Eq. (11). If any termin Eq. (11) is negative, heat is evolved by thisterm which then heats the star and adds tothe luminosity L. In particular the latent heatis a heating term for increasing density whenthe entropy per baryon of the dense phaseis less than that of the low density phase a situation which will arise when consideringsuperfluid quark matter.

For future reference we identify the termsin Eq. (11) as follows. T

dd

ddt (s2 s1) is

identified as the latent heat, (1 )Tddt

ds1d

is identified as the hadronic bulk contribu-tion, T

ddt

ds2d is identified as the quark bulk

contribution, and the last term T(d/dt)[(SS

+SC)/] is identified as the surface and Coulombcontribution. Further we often refer to theseterms collectively as additional entropy pro-duction (or release) beyond what would beexpected at constant density.

We discuss in the following section howto calculate the bulk entropy density of thehadronic and quark matter phases in therelativistic mean field theory framework ofthe G300180 equation of state. For now let usjust remark that the entropy per particle foran ideal relativistic degenerate Fermi gas is(Landau & Lifshitz 1980)

s =(32)

23

3cT

13 = 0.02

T

MeV

fm3

13

.

(13)Taking the transition to quark matter as atransition between such gases note that the

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bag constant does not contribute to the en-tropy and remembering that each hadroncontains three quarks which further have a

color degeneracy of three, the latent heat inEq. (11) is of the order of

1033T29

9

Q

fm3

13

H

fm3

13

(14)

daQ/dt

1057/107 yrerg s1

where T9 = T /109 K, and the rate of

baryons making the transition to quark mat-ter, daQ/dt, was (arbitrarily) scaled to an

entire star being converted steadily over 10

7

years. Unless the star is very hot or changingstructure fast this is a very modest contribu-tion, and we further note that it is positivefor a contracting star and so acts to cool thestar down. However the sign is subject to thevery rough assumption that both gases maybe treated as ideal Fermi gases for the purposeof calculating their entropy, and it will changein a more detailed treatment. Specifically thequark phase may be color-superconductingwith energy gaps as high as 100 MeV and

corresponding critical temperatures of the or-der of 1011 K. Below the critical temperaturethe quark specific heat and entropy densitywould be exponentially suppressed and couldbe ignored relative to the hadronic contribu-tion in Eq. (14) which would then be a heatingterm. We shall return to both possibilities inlater sections.

We shall calculate the two bulk terms nu-merically in the following sections, so for nowwe just note that from Eq. (13) the entropy

per baryon decreases with increasing density.In a contracting star these terms will henceact as heating terms and locally be of the sameorder of magnitude as the release or absorp-tion of latent heat in Eq. (14) but of coursethey contribute throughout the star and aretherefore potentially far more important thanthe latent heat which is only significant in the

mixed phase.

Since the surface and Coulomb energiesare related by ES = 2EC in equilibrium(Glendenning 2000), their contributions tothe thermodynamic potential, and hence tothe entropy, are similarly fixed in proportion,and we need only consider one of them here.Specifically the surface part of the entropymay be found from (Landau & Lifshitz 1980)

= 0 + S = 0 + S (15)

SS = ST

= S

T, (16)

where S

is the surface contribution to thethermodynamic potential . The surface ten-sion for the quark-hadron interface, , deter-mines the geometry and extent of the mixedphase (Glendenning 1992; Heiselberg et al.1993; Voskresensky et al. 2003; Endo et al.2006). The surface tension of strangelets invacuum has been evaluated by Berger & Jaffe(1987), but the surface tension for the mixedphase remains essentially unknown. It is com-monly parameterized as being proportional tothe difference in energy density between the

two phases and the length scale L 1 fm ofthe strong interaction (Glendenning 2000)

= K [Q H] L . (17)

Assuming for simplicity that Kand L are con-stant we then get

T=

cV,Q cV,HQ H

(18)

1(SS + SC) = 3

2

S

cV,Q cV,HQ H

(19)

The corresponding term in Eq. (11) is thenof the order of the surface energy per baryontimes the ratio between thermal and total en-ergy density. It can therefore not be expectedto contribute significantly, and this expecta-tion is confirmed by the numerical results.

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3. Equation of State and Stellar Mod-

els

In our numerical work we have employedthe rotating neutron star code developed byWeber and the G300180 equation of state used byGlendenning (1992, 2000) and Glendenning & Weber(2001). Here we shall briefly describe each ofthese and the resulting stellar models.

The G300180 equation of state treats the de-confined quark matter phase in a simple ver-sion of the bag model which ignores gluon in-teractions. The confined hadronic phase is de-scribed in terms of the mean field solution to a

covariant Lagrangian that involves the baryonoctet interacting through scalar, vector andvector-isovector mesons. The precise values ofthe coupling constants for the hadronic partof the model correspond to the first set of Ta-ble 5.5 of Glendenning (2000) or Table 1 ofGlendenning & Weber (2001). For details werefer to these works where the resulting equa-tion of state as well as the underlying theoryare carefully described at zero temperature.Finite temperature expressions for pressure,energy and particle densities can be found byreinserting the Fermi distribution in the phasespace integrals of their zero temperature ex-pressions which we shall not write explicitlyhere. We refer to e.g. Glendenning (1990) forthe full temperature dependent expressions(this reference also includes gluon interactionsto first order which we ignore here).

The entropy is calculated as

s =1

T P + i

ii (20)using the finite temperature expressions out-lined above and using the highly accuratepublicly available code described in Miralles & van Riper(1996) to solve the Fermi integrals. Inkeeping with our assumption that tempera-tures remain too low to significantly influ-ence the chemical composition we neglect

contributions from thermally exited particle-antiparticle pairs.

Figs. 1 and 2 show the resulting chemicalcomposition and entropy. In the quark phasewe note that u-quarks are suppressed initiallygiving the phase a negative net charge, andthat as expected the entropy per baryon though not per quark is higher in the quarkmatter phase in the absence of any pairingphenomena. We have checked numericallythat in the absence of pairing the entropy sim-ply scales linearly with temperature withinthe temperature range we shall need.

The sum of the surface and Coulomb ener-

gies has a maximum as a function of densityaround = 0.5 0.6 fm3 with a correspond-ing minimum in the related entropy in Fig. 3.The surface and Coulomb term in Eq. (11)can therefore be either positive or negativeand either heat or cool the star accordingly.The surface and Coulomb contribution to theentropy is negative with the total entropy re-maining positive, which confirms that a struc-tured phase has lower entropy than an un-structured one.

Fig. 1. Chemical composition of the G300180equation of state as a function of total baryondensity. Individual particle densities i referto the density in regions filled with the respec-tive phase. v is the (dimensionless) volumefraction of quark matter.

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Fig. 2. Entropy per baryon at T = 1 MeVin each phase, sH and sQ, and per particle for

each particle, si, as functions of total baryondensity for the G300180 equation of state. Protonsand particles have high si because they areso rare, but contribute little to sH for the samereason.

Fig. 3. Sum of surface and Coulomb en-tropy as a function of total baryon density forthe G300180 equation of state at T = 1 MeV

We have calculated the structure of a se-quence of stars with rotation frequencies (asseen by an observer at infinity) between zeroand the limiting mass-shedding Kepler fre-quency for the G300180 equation of state. Forthis purpose we use the perturbative methodof Hartle (1967) and Hartle & Thorne (1968)as implemented in the numerical code de-

veloped by Weber which also solves self-consistently for the general relativistic Ke-pler frequency K see Weber (1999) for

the derivation of K and (Weber et al. 1991;Weber & Glendenning 1992) for further de-tails. The sequence has constant total baryonnumber A = 1.87 1057 and nonrotat-ing total gravitational mass M = 1.42 M.The Kepler frequency is then found to beK = 6168 rad s

1 corresponding to a periodof 1.02 ms.

Fig. 4. Circumferential stellar radius in the

equatorial and polar directions for rotatingstars of total nonrotating gravitational massM = 1.42 M.

Figs. 4 and 5 show a few properties ofthe models. The stars are significantly dis-torted by rotation and increase their equato-rial radius at the Kepler frequency by half thenonrotating radius (Fig. 4) while losing thepure quark matter core at = 1400 rad s1

(Fig. 5). Since the stellar photon luminosityis proportional to the surface area and must

match the energy flux emerging from the core,higher surface temperatures must also be ex-pected at low rotation frequencies for this rea-son alone. The distortion from spherical sym-metry depends on polar angle, however, andabove the frequency at which the quark mat-ter core is lost the star actually contracts inthe polar direction.

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In Fig. 5 we show the location of the phaseboundaries between pure hadronic matter, thevarious geometries of the mixed phase and

the pure quark matter phase. Here we useas the free variable the baryon number, a,contained within a surface on which the den-sity is uniform (i.e spatially but not tempo-rally constant). The Eulerian density change(d/d)r can be positive or negative depend-ing on location while the Lagrangian deriva-tive (d/d)a is always negative, and so ais a more convenient variable for some pur-poses. In Fig. 5 borders are shown at thedensities which correspond to each transition

and a is scaled to the total baryon number,A. We note that a large fraction of the staris converted to quark matter as the star spinsdown. The pure quark matter core appearsbelow = 1400 rad s1 and grows to eventu-ally comprise 30 % of the stellar gravitationalmass and 26 % of the stellar baryon number.

Fig. 5. Location of phase boundaries instars of nonrotating mass M = 1.42 M. Thelocations are given as the fractions of the totalstellar baryon number contained within a sur-face of spatially constant density correspond-ing to each transition. The inset shows theregion where the phase boundaries almost but not quite cross.

Within the framework ofHartle (1967) thechanges in pressure and density with respectto a nonrotating star are second order effects

in the rotation frequency, and as discussedby Weber & Glendenning (1992) this remainstrue even at the limiting Kepler frequency,K, where the star would shed mass from theequator. The Lagrangian density change dur-ing spindown can then be reasonably approxi-mated by the order of magnitude estimate (seealso Fernandez & Reisenegger (2005))

2

a,A

2K(21)

Fig. 6 shows how the numerically calculatedspin down compression rate compares to thisestimate in the mixed phase we have checkedit at other positions as well. For stars withhigh spin frequencies and no pure quark mat-ter core Eq. (21) is a reasonable approxima-tion although an overestimate in some regions.It is off by approximately a factor of 0.1 belowthe frequency where the pure quark mattercore forms.

Fig. 6. Frequency derivative of densityscaled to the estimate in Eq. (21) at a/A =0.33 as a function of spin frequency. The esti-mate is best at high spin frequencies but holdswithin a factor of 30 in all stars.

Eq. (21) can be used to give a rough es-timate of the effects of the additional en-

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tropy production related to density change inEq. (11)

Tdsdt = cv

dTdt + Tdsd ddt (22)

cv

dT

dt T

ds

d

2K 2 .

Taking the entropy per particle from Eq. (13)this implies that the bulk terms in Eq. (11)are of the order of

Td

dt

ds

d

2

3

2KT s

1015T2

9 B

1014 G

2

(23)

/6000

rad s1

4 fm3

13

erg s1 ,

where we have taken K = 6000 rad s1, as-

sumed a standard dipole model for the spin-down so P = (B19/3.2)

2P1 with the spinperiod, P, measured in seconds and the mag-netic field, B in units of 1019 G. As a roughestimate the total heating power from thebulk terms in Eq. (11) in the absence of pair-

ing and for a star of constant density and tem-perature with a baryon number of 1057 is thenof the order of

WBulk 1042T29

B

1014 G

2 /6000rad s1

4

fm3

13

erg s1 . (24)

This includes only the bulk terms ofEq. (11), but on general grounds one wouldexpect this to be a reasonable approximation

of the total additional entropy production.The latent heat has the same basic origin the release of entropy with changing den-sity and is therefore expected to be of thesame order of magnitude as the bulk termsin Eq. (24). In the following section we shallfind that this is justified under most, but notall, circumstances.

The estimated heating in Eq. (24) is tobe compared with the neutrino and photonluminosities which in the absence of pair-

ing phenomena can roughly be estimated as(Page et al. 2006)

Lslow 1040T89 erg s

1 (25)

Lfast 1045T69 erg s

1 (26)

L 1033T29 erg s

1 (27)

where slow refers to stars dominated by rel-atively inefficient neutrino emission processessuch as the modified Urca cycle, bremsstrahlungor the pair breaking and formation process

while fast refers to stars dominated by thehighly efficient direct Urca process.

From Eq. (24) we then see that while it maybe possible to find combinations of tempera-ture, magnetic field and rotation frequency forwhich the additional entropy production dom-inates, such configurations may be difficult torealize in nature. For instance, in order forthe heating term in Eq. (24) to dominate theneutrino luminosities the star must be a hotmillisecond magnetar. Such an ob ject would

be very short lived if it could be created in na-ture at all. For the heating term to dominatethe photon luminosity which is the more rel-evant term at late times and low temperatures extremely high magnetic fields would againbe required assuming a dipole model for thespin down. Such conditions if they were tobe realized would be very short lived andtherefore the more difficult to observe.

From these simple estimates we then ex-pect that the release of entropy related to

changing bulk density with spin down, thelatent heat of phase transitions and changesin the surface and Coulomb energy of mixedphases (smaller still) will have little impact onthe thermal evolution of compact stars and bedifficult to observe.

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4. Spherical Isothermal Cooling

Estimates cannot replace detailed calcula-

tions, and the ones discussed above further ig-nore all effects of superfluidity, the changingchemical composition and the rapid changesin structure at certain frequencies. We there-fore proceed in this section to explore theseeffects through numerical calculations takingthe simplest possible approach to solve thethermal balance in Eq. (4).

4.1. Numerical Setup

Neutron stars are commonly taken to be

isothermal after an initial thermal relaxationlasting 10-100 years in the sense that the red-shifted temperature T = T e is constantbelow a thermally insulating layer in the outercrust (Gudmundsson et al. 1983). Since weintend only to investigate the late thermalevolution of neutron stars, and as we do notexpect the heating terms discussed here tochange the isothermality by disturbing thethermal balance of any part of the star with itssurroundings, we shall work within this same

approximation. To show the full range of ef-fects, plots in the following also include youngstars not expected to be isothermal and ourresults should only be taken as indicative atsuch early times.

Assuming the star to be thermally relaxed,so the redshifted temperature T = T e isconstant below the outer crust, we then inte-grate Eq. (4) to get a simplified equation forglobal thermal balance

dT

dt =

1

C

W

L

L

(28)

with

C = da Ts

T

(29)

W =

da eT

s

(30)

L =

da

e2 (31)

L = 4R2(T

(0)S )

4F(B)e2s (32)

T(0)

S6= g

1/414 [(7)

2.25 + (/3)1.25]1/4 (33)

= T9 0.001 g1/414

7T9 (34)

where / = Q is the neutrino emissiv-ity, the surface temperature is given in units

of 106 K, TS6 = TS/106 K and T(0)S is

surface temperature at zero magnetic fieldstrength. We have used a fit appropriatefor iron envelopes to the relation betweenthe surface temperature, TS, and the innertemperature below the heat-blanketing enve-lope, T, in which g14 is the surface gravityin units of 1014 cm s2 (Potekhin et al. 1997;Potekhin & Yakovlev 2001).

The heat conductivity in the crust, whichderives mainly from the motion of electrons,becomes anisotropic in a strong magnetic field(Potekhin & Yakovlev 2001; Geppert et al.2004). It is slightly enhanced in the directionalong the magnetic field lines and stronglysuppressed in the direction orthogonal to thefield lines. The heat blanketing relation andthe surface temperature then becomes nonuni-

form, Tlocal(B,,g ,T) = T(0)

S (g, T)(B , , T ).The photon luminosity must therefore befound by integrating the locally emerging fluxand it will depend on magnetic field strength,

L(B) = L(0)F(B). For this purpose we usethe fits given by Potekhin & Yakovlev (2001)for (B , , T ) and F(B) in a dipole magneticfield. These are good for magnetic fields below1016 G, interior temperature between 107 Kand 109 K and surface temperature above105 K. F(B) reaches a minimum of 0.7around B = 1013 G and grows to about 2 at

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B = 1015 G for a star of internal tempera-ture T = 109 K. The location of the minimummoves to lower magnetic field strength for

lower internal temperature and the change inphoton luminosity can delay or accelerate thecooling at late times accordingly. The figuresin Sect. 4.2 show the effective surface tem-perature at infinity TS = T

(0)S F(B)

1/4e. Astrong magnetic field in the inner crust canalso lead to anisotropic heat flow there (seee.g. Geppert et al. (2004) and Aguilera et al.(2008)), but these effects cannot be includedhere.

For given assumptions about the neutrinoemissivity and superfluid gaps Eq. (28) issolved along with the spin down model to givethe surface temperature at infinity as a func-tion of time, rotation frequency and magneticfield strength. At each time step Eqs. (29)to (34) are solved to update the stellar struc-ture at the appropriate rotation frequency.

Our equations of thermal balance assumespherical symmetry. This effectively treats

rotation as a perturbation whose only effectsare to change the size and chemical compo-sition of the stars and to provide additionalterms in the thermal balance. This approachignores the effects of nonradial heat flowsand nonspherical perturbations of the metricwhich can be significant when the star is notspherically symmetric at very high spin fre-quencies (see Fig. 4). Two-dimensional cool-ing calculations are b eyond our scope, butthey have been performed at constant ro-

tation frequency by Schaab & Weigel (1998)and Miralles et al. (1993). For stars rotatingat a large fraction of their Kepler frequencythese authors found significant effects on thethermal evolution during the nonisothermalepoch, and polar temperatures up to 31%higher than the equatorial temperatures evenfor internally isothermal stars. The recent

work of Geppert et al. (2004); Aguilera et al.(2008,?) further includes the influence of largemagnetic fields in the crust and Joule heat-

ing by magnetic field decay in two dimensions.This results in nonradial heat flow and signif-icant heating from magnetic field decay. Wefocus on the effects of deconfinement duringspin down however and shall not pursue theseaspects here.

Unless otherwise stated we shall use theequatorial radius in Eqs. (32) to (34). Wehave performed calculations with the polarradius too and will show one such example.At high rotation frequency the surface tem-

perature is then higher and responds less tochanges in rotation frequency. We expectresults in a two-dimensional code would beintermediate between these and such a calcu-lation would be a significant improvement onwhat is presented here. With this in mind weuse the equatorial radius to show the mostpronounced effects of spin down. However,we also note that the strongest effect we havefound from spin down stems from the for-mation of a pure quark matter core. At the

rotation frequency where this happens the po-lar and equatorial radii are very similar (seeFig. 4) and a spherical approximation is rea-sonable.

Since the stellar moment of inertia I andradius also change with rotation frequencywe modify the spin down model describedin the previous section to include such ef-fects. The spin down is then determined by(Glendenning 2000)

= K

I

1 +

I

2I

1

m, I =dI

d(35)

K = (2/3)R6B2 sin2 (36)

where is the inclination angle between themagnetic axis and the rotation axis and m1is the multipolarity usually m = 3 for mag-netic dipole braking or m = 5 for gravitational

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quadrupole radiation. We take sin2 = 1 anduse the canonical value m = 3 for a dipolespin down.

As for the neutrino emissivities and super-fluid properties of the neutron star we are in-terested in the most important effects onlyand aim for transparency in the model. In thequark phase we include the direct and modi-fied Urca cycles as well as bremsstrahlung us-ing the emissivities in Blaschke et al. (2000).The us-branch of the direct Urca cycle fulfillsthe triangle inequality,

kFs < kFu + kFe (37)

where kFi is the Fermi momentum of thestrange-quark, up-quark and electron respec-tively. It is thus allowed throughout the quarkphase, but its ud-counterpart is forbidden asit cannot conserve both energy and momen-tum. This difference stems from the assumedstrange quark mass of 150 MeV.

In the hadronic phase we again include thedirect and modified Urca cycles as well asbremsstrahlung using the emissivities givenin Yakovlev et al. (2001). We ignore the ef-fects of hyperons here because of their lowabundance. In neutron stars the direct Urcareaction is often allowed only at very highdensities because it cannot fulfill both energyand momentum conservation unless the pro-ton fraction is above the value where bothcharge neutrality and the triangle inequalitycan be observed (Lattimer et al. 1991). It iscritical to note that this is not so in hybridstars with a mixed phase. The mixed phase

is possible precisely because charge neutralitydoes not have to be observed locally, and it isfavorable because the hadronic phase lowersits nuclear symmetry energy by increasing theproton fraction (Fig. 1). Hence the hadronicdirect Urca cycle is active in the mixed phase,and as it is about three orders of magnitudefaster than the us-branch of the quark direct

Urca cycle it controls the cooling of hybridstars with a mixed phase unless reduced bypairing effects.

Additional neutrino processes should beincluded in a more sophisticated treatment particularly in the crust but as stated abovewe are mainly interested in the general prop-erties of the additional entropy productionand the changing chemical composition andwe shall here leave out such terms.

If there is any attractive interaction amongparticles in a degenerate Fermi system theywill pair, and the resulting superfluidity has

important consequences for the thermal prop-erties of neutron star matter and the thermalevolution of neutron stars. Pairing in generaldelays the cooling because it suppresses mostof the neutrino emissivities and enhances theheat capacity at temperatures just below thecritical. However it also opens additional neu-trino emission processes, suppresses the heatcapacity far below the critical temperatureand enhances the thermal conductivity, andit may therefore also accelerate the cooling at

certain epochs (see e.g. Yakovlev et al. (1999)for a detailed account).

There is considerable uncertainty concern-ing the relevant superfluid phase in quarkmatter see Alford et al. (2008) for a compre-hensive review of the effects of pairing amongquarks and the possible range of phases. Forthis first investigation of consequences for thethermal evolution of a quark-hadron phasetransition during spin down we assume a sim-plified but physically transparent model for

quark matter pairing which essentially corre-sponds to pairing in the color-flavor lockedphase (Alford et al. 1999; Alford 2001). Inthis model each flavor participates equally,the gap Q,0 is independent of density andthe critical temperature is Tc = 0.72Q,0 (seeSchmitt et al. (2002)). We shall consider awide range in Q,0 partly because it is in-

15


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teresting for the present calculation, partlybecause very low gaps may be realized forhigh strange quark mass as this implies un-

equal quark Fermi momenta in the unpairedphase. The color-flavor locked phase is gener-ally understood to be so strongly suppressedin all its thermal properties as to be virtuallyinert with respect to cooling. Since we shallalso consider small gaps and since the latentheat derived from the phase transition de-pends strongly on pairing in the quark phasethis will not always apply here, but for strongquark pairing the thermal evolution is gen-erally controlled by the hadronic phase. We

further note that as shown by Jaikumar et al.(2002) (and stressed by Alford et al. (2008))Goldstone modes and their related neutrinoemissivities and heat capacities are not expo-nentially suppressed, scaling instead as T15

and T3 respectively. They will therefore dom-inate in color-flavor locked quark matter atlow temperatures, but they are exceedinglysmall and do not influence our results.

The effects of superfluidity relevant for our

purpose are that below the critical tempera-tures it suppresses the neutrino emissivitiesand entropies of the participating particleswhile also allowing additional neutrino emis-sion through Cooper pair breaking and forma-tion. The specific heat is enhanced just belowthe critical temperature and suppressed atlower temperatures. These effects are incor-porated through control functions, R(T, Tc),which multiply the relevant quantities. Theydepend only on the pairing channel, temper-

ature and gap size, and may therefore be cal-culated given expressions for the gap size mo-mentum dependence.

In the quark matter phase we use thesimplest possible control functions and sup-press the direct Urca emissivity by a fac-tor of eQ/T and the modified Urca and

bremsstrahlung emissivity by a factor ofe2Q/T, where (following Steiner et al. (2002))Q = Q,01 (T /Tc)2 is the pairing gapat the local temperature, T = Te.

The specific heat and entropy of superfluidparticles are also modified at temperatures be-low the critical temperature. In the quarkmatter phase we use the fit to the 1S0 con-trol function for the specific heat given byYakovlev et al. (1999). We further fit the re-sults of Muhlschlegel (1959) for the entropysuppression to obtain

ssf = 0.95 s0 (T/Tc)eTc/T

0.43 + 3.82(T /Tc ) 1.41 (T /Tc)2

.

(38)

Where s0 is the nonsuperfluid entropy. Thisexpression is applied to reduce the entropyof both quarks and hadrons (thus here ne-glecting the difference between different pair-ing states of neutrons). The entropy dif-ference between the superfluid and the nor-mal phase depends on temperature as s (1 T/Tc) close to the critical temperature

(Landau & Lifshitz 1980) with s0 becomingnegligible below 0.2Tc. This has the effect ofgradually turning the latent heat in Eq. (11)from a cooling term into a heating term, butit also suppresses the bulk terms in Eq. (11).

In the absence of detailed calculations ofcontrol functions for the entropy of superfluidquark matter phases in the literature we haveemployed Eq. (38) in the quark matter phasethough it is strictly relevant only for BCS su-perfluidity. Eq. (38) is, however, consistent

with the exponential suppression of entropybelow the critical temperature which wouldbe expected in any such calculation, and wedo not expect that our results are sensitive tothis choice.

Pairing further opens the important pos-sibility of neutrino emission by Cooper pairbreaking and formation in the quark phase

16


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and we include the associated emissivity asdescribed in Jaikumar & Prakash (2001) us-ing the fit to the 1S0 control function in

Yakovlev et al. (1999).

Among the nucleons pairing is predicted forneutrons in the 1S0 and

3P2 channels and forprotons in the 1S0 channel. The gaps can becalculated self-consistently but results are un-certain and vary greatly both in terms of themaximum gap size and in terms of densitydependence see e.g. Lombardo & Schulze(2001); Page et al. (2004, 2006) for collec-tions of these results. We shall here use

gaps with a phenomenological momentumdependence as suggested by Kaminker et al.(2001), Andersson et al. (2005) and recentlyAguilera et al. (2008)

(kF,N) = 0(kF,N k0)

2

(kF,N k0)2 + k21(39)

(kF,N k2)

2

(kF,N k2)2 + k23

(40)

where kF,N is the Fermi momentum of the rel-evant nucleon, N = n, p. Our choices for the

parameters 0 and ki correspond to sets a, eand h of Aguilera et al. (2008) (see that workfor references to the model calculation under-lying these fits). Eq. (40) is valid only in therange k0 < k < k2 with (kF,N) = 0 outsidethis range, and where the gaps for 1S0 and3P2 pairing of neutrons are both nonzero weuse the largest of the two gaps. We furtheruse the relations between critical tempera-ture and pairing gap listed by Yakovlev et al.(1999).

In the hadronic phase neutrons and protonsmay pair simultaneously in different chan-nels and so the resulting calculations for thecontrol functions can be quite involved. Weuse the fits to numerically calculated controlfunctions for each process and each pairingchannel compiled in Yakovlev et al. (1999) (or

Yakovlev et al. (2001) see these works fordetailed references). Where protons and neu-trons pair simultaneously we use the com-

bined factors listed in these works if availableand if not then the smallest of the two inde-pendent control functions.

The hadronic pair breaking and forma-tion process can be very powerful and maydominate the thermal evolution when pair-ing first sets in, but it was recently shownby Leinson & Perez (2006) to be stronglysuppressed in the singlet state channel byapproximately a factor of 106 relative tothe 3P2 channel (see also the recent work of

Steiner & Reddy (2009)). In the hadronicphase we therefore include pair breaking andformation only for neutrons at high densitieswhere they pair in the 3P2 channel and forthis we again use the emissivity and controlfunction given in Yakovlev et al. (1999, 2001).

4.2. Numerical Results

Figures 7 to 12 explore consequences forthe thermal history of hybrid stars of includ-ing time dependent structure and entropy pro-

duction by spin down compression in the cool-ing calculations with the approximations dis-cussed above. They show the surface temper-ature and additional entropy production at in-finity as functions of time, magnetic field andquark pairing gap energy.

In Fig. 7 we show the thermal evolution ofstars with quark pairing gap Q,0 = 10 MeV,a range of magnetic fields and initial spin pe-riod around 1 ms. The extremely rapid initialrotation is interesting because it means the

stars start out with no pure quark mattercore and undergo drastic changes in structureas a pure quark matter core develops around = 1400 rad s1. We see this in Fig. 7 asa short period of increasing temperature atthe time corresponding to this angular veloc-ity for a given magnetic field. During thisshort period the heating term from the addi-

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Fig. 7. The effect of spin down compressionon cooling curves for stars with initial spinfrequency start = 6000 rad s

1 and constant

magnetic field as labeled for each curve. Thequark core is assumed superconducting withQ,0 = 10 MeV. The increasing quark matterfraction produces slower cooling and a jump intemperature with the appearance of the purequark matter phase.

tional entropy production actually dominateswhich is shown in more detail in Fig. 10 dis-cussed below. Further, the changing neutrinoemissivity and heat capacity with the intro-duction of more quarks in the star and thereduction in stellar surface area as the purequark matter core develops all affect the ther-mal evolution. This implies somewhat slowercooling with our specific choice of param-eters and higher surface temperature. Forthe internal temperature, T, we have foundessentially a transition between two otherwisesimilar cooling tracks; one for stars with nopure quark matter core and one for stars witha fully developed quark matter core. The ef-

fective surface temperature also depends onmagnetic field strength through F(B) how-ever. For this reason such an easy interpreta-tion is difficult from Fig. 7 alone.

The magnetic field strength determines thespin down and affects the heat blanketing re-

Fig. 8. Variation of surface temperature atinfinity with magnetic field for stars at ages102, 104 and 105 years from above. Thick

lines have initial spin frequency start =6000 rad s1 while thin dashed lines startwith start = 600 rad s

1. Thin continuouslines use the polar rather than the equato-rial radius. The quark core is assumed su-perconducting with Q,0 = 10 MeV. We notea slow increase in temperature for old starsand a sudden jump at the magnetic field cor-responding to significant spin down at specificages. The suppression in temperature at lowmagnetic field and increase at high field is due

to the effects of the magnetic field of the heatblanketing relation.

lation. Fig. 8 shows how the temperaturevaries at specific ages with the magnetic fieldstrength (which is kept constant in time it-self). This is shown for initial spin frequen-cies of 6000 rad s1 and 600 rad s1 and usingthe polar radius in addition to the equatorialwhen calculating the surface temperature.

For low initial spin frequency we find no

significant effects of spin down in Fig. 8. Thechange in temperature at a specific age withincreasing magnetic field is due to the effectsof the magnetic field on the heat blanketingrelation and photon luminosity. If these ef-fects were left out the curves for low initialspin frequency would be almost constant.

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If the initial rotation frequency is suffi-ciently high to exclude the pure quark mat-ter phase it is a different matter. A strong

magnetic field initially suppresses the surfacetemperature of young stars with high rotationfrequencies, but it then gives a sharp jumpat the magnetic field strength which ensuresthat a pure quark matter core is formed at aparticular age. For older stars there is alsoa slight increase in surface temperature be-fore the formation of a pure quark matter corewhen the equatorial radius is used. If the ef-fects of the magnetic field on the heat blan-keting relation were left out the curves at high

initial spin frequency become nearly constantfor B > 1013 G and for fields too weak forspin down to set in at the particular age. Inthis case there would still be a jump in tem-perature at the magnetic field strength corre-sponding to formation of a pure quark mattercore for a specific age however.

Replacing the equatorial radius with thepolar in Eqs. (32) and (34) (thin continuouslines in Fig. 8) we find higher surface temper-atures before the introduction of pure quark

matter in the core and still a clear jump intemperature after. The gradual increase intemperature before the jump found in theother curves is absent except for old stars.This can be understood if we remember thatthe polar radius is smaller than the equato-rial and less sensitive to spin down at highrotation frequencies. We have checked that asimilar pattern may be seen in plots of tem-perature versus time using the polar radius.We expect that full two-dimensional calcula-

tions would give results which are at high ro-tation frequencies intermediate between whatwe have found using a spherical approxima-tion. The similarity between curves withpolar and equatorial radius at the jump intemperature shows that our calculations areinsensitive to the approximation at the spinfrequency where the pure quark matter core

forms and where we find the strongest effectof spin down.

Fig. 9. Numerical value of the entropy pro-duction as defined in Eq. (30) as a function oftime for a star with magnetic field B = 1011 Gand quark pairing gap Q,0 = 10 MeV. WTotis split into contributions with different phys-ical origins as explained in the text. The sur-face and Coulomb term changes sign and thenegative of this term is shown as well. Thelatent heat and bulk terms are dominant, andthe spike is caused by the appearance of a pure

quark matter phase.

In Figs. 9 and 10 we explore the importanceof the heating term in a little more detail.Fig. 9 shows the numerical value of W andits various contributions as a function of timefor a star with pairing gap Q,0 = 10 MeVand magnetic field B = 1011 G. The total en-tropy production, W, is split into contribu-tions corresponding to (integrals of) the termsin Eq. (11). WLatent corresponds to the latentheat, W

Hadronic bulkto the hadronic bulk term,

WQuark bulk to the quark bulk term and WSC

to the surface and Coulomb term of Eq. (11).Leptons do not participate in the deconfine-ment transition, so their entropy is assumedcontinuous across the phase boundary. Theyare separated from the other particles in theform of WLepton in Fig. 9.

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Fig. 10. The total entropy productionscaled to the combined neutrino and photon

luminosity for stars with quark paring gapQ,0 = 10 MeV and magnetic fields as in-dicated just above each curve. The differentline styles are meant only to help guide theeye. Note that at the peaks reaching above1 the entropy production actually dominatesthe cooling terms and the temperature in-creases. Also note that this is only p ossiblefor stars changing structure rapidly with theintroduction of a pure quark matter core.

The bulk and lepton contributions alwaysact as heating terms while the surface andCoulomb term changes sign and turns into aheating term with the onset of hadronic su-perfluidity around age 4000 years with thehadronic bulk term simultaneously fallingaway. The quark bulk term is very smallbecause for a pairing gap of 10 MeV thequark matter is strongly superconducting atall times except the very early and the quarkentropy is therefore suppressed. This also

implies, however, that hadrons making thetransition across the phase boundary essen-tially release all their entropy, and so the la-tent heat, which changes sign and becomes aheating term with the onset of quark superflu-idity, is therefore quite significant and actu-ally comes to dominate around the time whenthe hadronic bulk term falls away. The lep-

ton term and the surface and Coulomb termare relatively small but significant when thebulk terms are eliminated through the onset

of pairing. The peak in the total around age20000 years corresponds to the appearance ofa pure quark matter phase in the core andthe ensuing rapid changes in structure, andit contributes to produce the short interval ofincreasing temperature seen in Fig. 7.

Fig. 10 shows the total entropy productionscaled to the total neutrino and photon lumi-nosity, W/(L + L

), for the same set of

parameters which gave the cooling curves inFig. 7. The strong temperature dependence of

the neutrino and photon luminosities ensuresthat the scaled heating term does not increaseabove unity where heating and cooling exactlybalance except at the peaks caused by theappearance of a pure quark matter core. Wealso note that as in Fig. 7 the magnetic fieldmust be above 109 G for W to have anydiscernible effects.

Fig. 11. The total entropy productionscaled to the combined neutrino and pho-

ton luminosity for stars with quark paringgap as indicated in the legend and start =6000 rad s1. The magnetic field is constantat 1011 G. The entropy production changessign and becomes negative for low pairinggaps and thus acts as a powerful cooling termduring some epochs.

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While the latent heat for a transition toa strongly superfluid core is a heating term,the transition to a nonsuperfluid core with

higher entropy per baryon cools the star. Thisis illustrated in Fig. 11 where we plot thetotal heating term scaled to the combinedneutrino and photon luminosities at constantmagnetic field for a range of quark pairinggaps Q,0. Here we see how the entropy pro-duction changes from a heating term to a cool-ing term and may be quite dominant at cer-tain times depending on the choice of stel-lar parameters. This can also be observedin Fig. 12 where we plot surface tempera-

ture versus age with constant magnetic fieldfor a range of quark pairing gaps. The on-set of superfluidity initially delays the coolingand then accelerates it. When the pure quarkmatter core forms there is a drop in tempera-ture for very low pairing gaps and a jump forhigh pairing gaps. In fact, however, the ma-jor differences between the curves in Fig. 12are due to the differences in the thermal evo-lution of superfluid and nonsuperfluid quarkmatter rather than spin down related effects.

Old stars with a nonsuperfluid quark mattercore cool more slowly than those without aquark matter core.

A rich picture has now emerged from thecoupling between the thermal evolution andspin down compression with deconfinement.The spin down may change the thermal evo-lution of hybrid stars by heating and coolingthem at various ages and by changing thestructure and chemical composition or itmay be entirely inconsequential depending on

quark superfluidity and initial spin frequency.The effects we found turned out to dependstrongly on the inclusion of quark pairing andthe timing of the appearance of the pure quarkmatter phase in the stellar core. These aresubject to the specific assumptions made con-cerning the equation of state, stellar baryonnumber and pairing regime, so it is essential to

Fig. 12. The effect of spin down com-pression on cooling curves for stars with ini-

tial spin frequency start = 6000 rad s1

andquark pairing gap Q,0 as labelled for eachcurve. The magnetic field is constant at1011 G

stress that the results discussed in the presentsection illustrate the general point that spindown and thermal cooling may be interde-pendent, rather than providing quantitativelyreliable predictions.

Bearing this in mind we show in Fig. 13how calculations for selected pairing gaps andmagnetic field strengths measure up to obser-vational data on the thermal state of isolatedneutron stars (kindly supplied by Dany Page).This figure is shown partly as a consistencycheck for our calculations and partly to com-pare the size of the effects we have found withthe accuracy of actual measurements. Westress that the observational data are consis-tent with static cooling models (Page et al.

2004) and that we do not suggest that theeffects of spin down are necessary to explainthem. Further the effects we have found oc-cur after thermal relaxation only for mag-netic field strength below 1012 G. For thosesources in Fig. 13 for which the field is knownit is above that value (Pons et al. 2007).

Fig. 13 explores a range of quark pairing

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gaps which in our pairing scheme are den-sity independent. In nature the pairing gapwould not span a range this broad, but it

could be density dependent and variations be-tween stars in mass and density would then in-troduce variations in their thermal evolution.Similarly stars with different masses wouldhave a different phase structure and acquirepure quark matter cores at different rotationalfrequencies thus further complicating the pic-ture.

Because the hadronic direct Urca mecha-nism and the neutron pair breaking and for-mation mechanism are active in the mixed

phase our models are consistent only withthe cold sources and the effects of spin downdo not change this conclusion. That thesemechanisms are the cause of the generally lowtemperatures in our models can be seen ifthey are artificially left out. The two dot-ted lines in Fig. 13 show for illustrationonly that temperatures are much higher ifthese mechanisms are suppressed. Such a sup-pression could occur for certain quark pair-ing schemes where pairing forces the quark

phase to become electrically neutral or pos-itive rather than negative, thereby reducingthe proton content in the hadronic phase be-low the threshold for direct Urca. A self-consistent inclusion of such effects are beyondthe scope of the present investigation. Westress again that the inconsistency with hotsources seen in this figure is subject to thechoice of stellar and thermal models made forthe specific purpose of studying the effectsof deconfinement during spin down. Thus

it should not be seen as indicating a generalbreakdown in cooling theory.

The true signal of deconfinement we havefound is in the transition between coolingcurves with and without pure quark mattercores, and in the brief period of increasing orrapidly decreasing temperatures thus causedfor certain stellar parameters. As can be seen

in Fig. 13 the change in surface temperaturecaused by deconfinement is smaller than theerror bars on the observational data and it will

be difficult to test observationally on the basisof data relating only temperature and time.

Fig. 13. Comparison between observa-tional data and cooling calculations with spindown. The stars have initial spin frequencystart = 6000 rad s

1, magnetic field strengthB = 1011 G and quark pairing gaps as indi-cated. The observational data can be foundin Page et al. (2004) and sources are identifiedby numbers right above or below their temper-

ature error bars. The two dotted lines illus-trate slow cooling with artificial suppressionof neutrino emission by direct Urca and neu-tron pair breaking and formation. They haveQ,0 = 10 MeV (thick line) and 0.01 MeV(thin line). Including these mechanisms onlycold sources can be explained. The jump intemperature at the formation of a pure quarkmatter core is smaller than the error bars onthe observational data.

5. Discussion

Our intention with the present work wasto explore a possible connection between thespin down and thermal cooling of hybrid starswith a deconfined quark matter core theappearance of which might be expected to in-fluence theoretical cooling tracks. Specifically

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we were interested in the latent heat of thephase transition and the drastic changes instructure and chemical composition resulting

from the increasing density with spin down.The general formalism worked out in Sect. 2would also be relevant to models with nophase transition, however, as the bulk termsin the spin down derived entropy productionmight also be important in such cases. Whileestimates of the importance of the spin downderived entropy production did not give causeto expect strong signals of deconfinement inthe thermal evolution, numerical calculationsrevealed clear effects of the changes caused by

the appearance of a pure quark matter core.This signature is related to changes in radius,chemical composition, structure and undercertain circumstances the additional entropyproduction both in bulk and in the form oflatent heat.

Our numerical work was carried out usinga sophisticated and reliable code with respectto the stellar structure solving Hartles pertur-bative equations for the structure of rotating

compact stars self-consistently. The thermalevolution, however, was treated in a sphericalisothermal approximation with a somewhatad hoc approach to the treatment of super-fluid pairing in the quark matter phase. Stillwe believe our work is sufficiently detailedto demonstrate the general point that impor-tant signals may be derived from the inter-play between spin down and cooling of com-pact stars, which could in the long run helpanswer pressing questions about the state of

matter at high densities. The signals we havefound do not dominate the general thermalevolution of hybrid stars but they do com-plement the standard picture. By correlatingtemperature with magnetic field strength andspin frequency they may also help break thedegeneracy between models relating only tem-perature and time.

The signature of deconfinement found hereis below the present observational sensitiv-ity and not of sufficient strength to set apartthe cooling curves with temperature versustime for hybrid stars. The correlation betweentemperature and magnetic field strength pro-vides a possible alternative. To test such acorrelation it would be necessary to obtain ac-curate temperatures for a number of stars ofapproximately the same age for which the spindown could also be detected by the emission ofpulses in either radio or X-ray. It might thenbe possible to test for features in the relation

between temperature and magnetic field orequally interesting, the rotation frequency it-self which as demonstrated here could resultfrom a strong phase transition if the initial ro-tation frequency was sufficiently high to havespun out the high density phase.

Contemplating these prospects it is im-portant to consider that alternative effectsnot treated here might determine the rela-tion between spin down and thermal evolu-tion. Most pressingly the heat from magnetic

field decay was employed by Pons et al. (2007)to explain the observed correlation b etweentemperature and magnetic field and it maywell drown out any other effects althoughas noted by the authors the magnetic fielddecay itself has not yet been independentlydemonstrated. Recent work by Aguilera et al.(2008) and Aguilera et al. (2008) on the cool-ing of magnetized neutron stars in two dimen-sions also highlighted the importance of themagnetic field strength, geometry and possi-

ble decay for the thermal evolution of neutronstars. These authors find strongly anisotropicsurface temperature distributions and possi-bly an inverted temperature distribution withhot equatorial regions for middle aged stars.They further show that the effects of magneticfield decay and Joule heating can dominatethe thermal evolution at strong and interme-

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diate field strengths even to the point thatthe effects of the direct Urca process may behidden by this heating term in magnetars.

The work ofReisenegger (1995) and Fernandez & Reisenegger(2005) is also most important. These authorsconsider the second term on the right handside of Eq. (4) which is shown to give rise toso-called roto-chemical heating as the weakinteractions required to maintain chemicalequilibrium are unable to keep pace with theincreasing density. Fernandez & Reisenegger(2005) found that old millisecond pulsarsreach a quasi-equilibrium in which the photonluminosity is determined entirely by the spin

down power and remains much higher thanotherwise attainable. Their work consideredonly nucleonic equations of state but similarresults should clearly be expected for hybridstars just as part of the effects demonstratedhere should be expected to show up in nucle-onic stars.

Given the possible importance of such al-ternatives and the shortcomings of the presentstudy discussed above, a more complete treat-

ment of the interplay between the spin downand thermal evolution of neutron stars seemsdesirable before strong assertions can bemade concerning the specific shape of cool-ing tracks. As well as including all possibleenergy sources and sinks this should be ex-tended to consider a wider range of stellarmasses, equations of state, phase transitionsand pairing regimes. A two dimensional codecould further treat the influence of the mag-netic field on heat flows in the inner crust

and the effects of nonspherical heat flows forrotationally perturbed stars aspects whichare all independently well described in theliterature.

It should further be noted that the rota-tion frequency at which the pure quark mat-ter core appears depends strongly on the stel-lar baryon number and the equation of state,

and it may well be lower than discussed herethus changing the timing of spin down relatedeffects. The strongest signature of deconfine-

ment found here depends entirely on the for-mation of a pure quark matter core wherenone existed previously. If, on the other hand,a pure quark matter phase is present in thecore of stars at even the highest rotation fre-quencies as is the case for other equations ofstate we would not expect equally strong sig-nals from deconfinement with spin down com-pression.

It is possible that circumstances not con-sidered here could provide a more pronounced

link between rotation and cooling. If for in-stance the magnetic field varies over time and as discussed by Geppert (2006) this maywell be the case the timing of any strength-ening or decay of the magnetic field mayprovide entirely different spin down histories.Alternatively the temperature of stars beingspun up in accreting binaries should to someextent be determined by the changing chemi-cal composition and this could provide an al-ternative approach.

Additional energy release would also beexpected if the phase transition cannot main-tain thermodynamic or hydrostatic equilib-rium as was assumed here. As the most ex-treme example of this, we have entirely ig-nored the instabilities and corequakes whichmight accompany the appearance of a purequark matter phase in a star far from equilib-rium (Zdunik et al. 2006). Such events couldrelease large amounts of energy and signif-icantly change the thermal evolution by re-

heating the star.

While many essential issues thus remainpoorly explored we hope to have demon-strated the possible usefulness of consideringthe connection between the spin down andthermal evolution of neutron stars. Althoughthe results of such work may not be imme-

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diately applicable to observational tests, webelieve they could prove most valuable in thelong run.

We wish to acknowledge helpful conversa-tion with Sanjay Reddy on the hadronic pairbreaking and formation process and to thankDany Page for the use of his compilation ofobservational data. We are also grateful toDima Yakovlev and Oleg Gnedin for provid-ing us with an earlier version of their coolingcode, which we took advantage of when devel-oping the code used for this study.M. Stejner wishes to thank San Diego State

University for its hospitality during part ofthis work.The research of F. Weber is supported bythe National Science Foundation under GrantPHY-0457329, and by the Research Corpora-tion.J. Madsen is supported by the Danish NaturalScience Research Council.

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A. Appendix A, Derivation of the Energy Balance in Mixed Phases

To derive Eq. (4) from Eq. (1) we proceed analogously to Thorne (1966) and Weber (1999) but

work in terms of baryon number a instead of radial coordinate. We first define the nuclear energygeneration rate per baryon as the rate, q, at which rest mass is converted to internal energy asmeasured by an observer in the shell and at rest with respect to the shells reference frame

q = dmB

d(proper time)=

dmBedt

, (A1)

where mB is the average rest mass p er baryon and we work in units with. The first term on theright hand side of Eq. (1) is the amount of rest mass converted to internal energy in the entire shellwhich is then qaedt.

Next we note that by expressing the shells volume in terms of its particle density, V = (V/a)a =/a, the second term in Eq. (1) can be written

dW = Pd

1

a

+ dS + dEC (A2)

The third term represents the difference between the rates at which energy enters and leaves theshell by radiative or conductive means as measured by an observer in the shell and at rest withrespect to the shells reference frame, which may be written

Ltot(a + a)e2((a+a)(a))

Ltot(a) = Ltot(a + a)

1 + 2

d

daa

Ltot(a)

=

dLtot

da+ 2Ltot

d

da

a

= dda

Ltote2

e2a (A3)

and the energy change during a coordinate time interval, dt, is this times the proper time intervaledt. The two factors of e(a+a)(a) account for redshift and time dilation, respectively, as theenergy crosses from the inner to the outer edge of the shell, and we expanded the exponential toorder O(a) as

e2((a+a)(a)) 1 + 2[(a + a) (a)] = 1 + 2d

daa. (A4)

Using Eqs. (A1), (A2) and (A3) in Eq. (1) and expressing the change in internal energy in termsof the density of internal energy, Eint, local energy balance can then be written as

dE

int

a

= qeadt Pd

1

a

+ dS + dEC

dda

Ltote2

eadt . (A5)

Rearranging and noting that a is constant in time we then get the gradient of Ltot

d

da

Ltote

2

= e2

q+

P e

2d

dt+

e

a

dS

dt

+e

a

dECdt

ed

dt

Eint

a=constant

(A6)

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Eq. (A6) is the relation we sought giving the luminosity gradient for stars with variable structureand including the contributions from surface and Coulomb terms in the energy density. It may beconsiderably simplified however by noting that we can write the internal energy density in terms

of the total energy density as Eint/ = / mB in which case from the definition of q

q ed

dt

Eint

=

e

d

dt

d

dt

. (A7)

Inserting this in Eq. (A6) and using the first law as written in Eq. (3) finally gives the expression

d

da

Ltote

2

= e

d

dt

+ P

d

dt

a

dS

dt

1

a

dECdt

a=constant

(A8)

= e

T

ds

dt+

kk

dYkdt

a=constant

. (A9)

In this discussion we have used baryon number as the independent variable to emphasize theimportance of a Lagrangian description for stars with variable structure. This is of course notrequired for all applications and for comparison with other discussions we note that our expressionscan be converted to use radial coordinate as the independent coordinate through the standardrelation

da = 4r2

1

2m

r

12

dr (A10)

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Documents

Signature of deconfinement with spin down compression