Thermal properties and detectability of neutron stars - I cooling and heating of neutron stars

THERMAL PROPERTIES ANDDETECTABILITY OF NEUTRON

STARS - ICOOLING AND HEATING OF

NEUTRON STARS

SACHIKO TSURUTA

ResearchInstitutefor FundamentalPhysics,Kyoto University,Kybto 606, Japan,

Max-Planck-Institutfür PhysikundAstrophysik,Münche,~,FRG,

and

DepartmentofPhysics,MontanaStateUniversity,Bozeman,Monta~za59717, U.S.A.

NORTH-HOLLAND PUBLISHING COMPANY -AMSTERDAM

PHYSICSREPORTS(Review Sectionof PhysicsLetters)56, No. 5(1979)237-278.NORTH-HOLLAND PUBLISHING COMPANY

THERMAL PROPERTIES AND DETECTABILITY OF NEUTRON STARS - ICOOLING AND HEATING OF NEUTRON STARS

Sachiko TSURUTAResearch Institute for Fundamental Physics, Kyoto University, Kyoto 606, Japan,

Max-Planck-Institut für Physik und Astrophysik, München, FRG,

and Department of Physics, Montana State University,* Bozeman, Montana 59717, U.S.A.

ReceivedJanuary1979

Contents:

1. Introduction 239 3.2. Youngpulsars 2652. Theoreticalpredictions 241 3.3. Other supernovaremnants 268

2.1. Summaryof basic equations 241 3.4. Radiopulsars— Heatingproblems 2702.2. Earlier results— Summary 242 3.5. Other majorheatingmechanisms— Accretion 2712.3. Recentresults 248 3.6. Conclusion 273

3. Observationalaspects 261 Appendix 1 2743.1. Observationalmeans 261 Appendix2 275

References 275

Abstract:In this report, wefirst reviewearlierandrecentdevelopmentsin someof thermodynamicproblemsof neutronstars,especiallythoseinvolving

cooling mechanismsand theoreticalpredictionsof surface temperaturesof neutronstars.Emphasisis placed particularlyon: the effect ofequationsof stateand hencethat of nuclear and strong interactions;the effect of better treatmentof variousneutrino cooling mechanisms,especiallythoseinvolving pion condensates;andimplication of thesebetterand moredetailedtheoreticalestimateson theprospectof directlyobservingthermalradiationfrom thesurfaceof neutronstars.In connectionwith the last problem,we briefly reviewrecentdevelopmentson theobservationalside— theHEAO-B andotherprogramsalreadyexistingor expectedto beplannedfor nearfuture, which aredirectlyrelatedto theaboveproblem.In connectionwith thepossibilitiesof observingolderneutronstarswe briefly summarisevarious heatingmechanisms.

From thesestudies,we seethatexcitingpossibilitiesexist throughtheHEAO-B andsomeotherprogramswhich may berealisedin the 1980’s,that we may observeradiationdirectly from neutronstarsurfacesif theyare~(3—5)x 10

5°K.If suchradiationis detected,theobservedsurfacetemperaturesand further spectralstudiesmay give invaluable insight into various important problems,such as magneticpropertiesof densematter, equationsof state,pion condensates,and other fundamentalproblemsin nuclear, particle and high energy physics. If the surfacetemperaturesof younger membersof these stars(~l0~years)are observationallyfound to be less than (5—10) x 105°K(dependingon theindividual objects),we notethatat themomentonly pion coolingsareconsistentwith observations,andtheoutcomemay beequallyfar reaching.Among variousobservedneutronstars(pulsars)andneutronstarcandidates(e.g. supernovaremnants),theVela pulsarmay prove to be themostrewardingone.If regular pulsar-likeperiodicitiesarediscoveredin radiationsfrom anyof supernovaremnants,we can assumethepresenceofneutronstarsin theseobjects.In that case,somesupernovaremnants,suchasSN 1006, may alsoturn out to be promising.If we detectsurfaceradiationsfrom older pulsars(~l0~years),that maysupportsomeof heatingtheories.At theend, we pointout that theremay be many pointsourcesof very soft weakthermalX-raysacrossthesky (as old neutronstarsaccreteinterstellarmatter)and someof theclosestonesmay bedetectablethroughtheHEAO-B and similardevices.

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Sachiko Tsuruta, Thermal properties and detectability of neutron stars —1 239

1. Introduction

When the first few of the galactic X-ray sourceswere discoveredin the early 1960’s [1], it wasimmediatelysuggestedthat theywerecoming from neutronstars.This is becauseif theseX-rays areblackbodyradiation,the emittingbody hasto be assmall as 10 km or so, in order to emit detectableamountof radiationfrom distancestypical of astronomicalobjects,and only extremelycompactobjectssuchas neutronstarsandblackholescan do so,while isolatedblackholesarenot expectedtoemit anything*’. Indeed,theearlycrude informationfrom observationsseemedto be consistentwiththis picture. The first seriesof detailedcooling calculations[2,3] also seemedto show that neutronstarsof this rangeof temperaturecanbe consistentwith theseearlyobservationaldata.It was soondiscovered,however,through a lunar occultation experiment,that at leastthe X-ray source in theCrabNebulahasa diameterof aboutone light year [4]! It wasalsoshown that thespectrumof X-raysourcesbothin theCrabNebulaand in theConstellationScorpius(ScoX-1) werenotof theblackbodyshape.Subsequentobservationaldatamadeit clearthat atleastthedominantpart of observedX-rayswas not coming from the surfaceof neutron stars.A review of theoriesand observationsof theseearly daysis found, for instance,in ref. [3].As a possibleexplanationof this apparentdiscrepancy,Bahcall andWolf [5] pointedout, as earlyasin 1965, that if thereis considerableamountof pions inneutronstars,theyshouldcool so fastthat it would bepracticallyimpossibleto detectthem.Theydidnot claim that thereshouldbe pionsin neutronstars,but JohnBahcallsuggestedthat we shouldmakea bet on thedetectabilityof neutronstars;he, of course,would beon thenegativeside.I regretthat Ideclined this offer, becausea few years after that, a neutron starwas discovered,incredibly, as apulsar!

Sinceit becameclear thatpulsarsmustbe neutronstarsafterthediscoveriesof very shortperiodsof the Crab and the Vela pulsar (see,e.g. [6,7]), there havebeen intensive theoreticalstudiesofneutron star propertiesby various groups throughout the world. Most comprehensiveand recentreviews of thesestudiesare given, for instance,by Baym and Pethick [8j. As first suggestedbyPacini andGold, it is now generallyacceptedthatpulsarsarerotating,magneticneutronstars[6,7,9].It wasalso predictedthroughobservationsthat theassociatedmagneticfield strengthmustbe of theorder of 1012 gaussnearthe stellar surface.(See,for instance,[6,10].) The presenceof suchstrongmagneticfields is expectedto give profoundeffectson thepropertiesof neutronstars,especiallyontheir thermalproperties.This is becauseboth radiativeand conductiveopacitiesare expectedto bedrasticallyreducedalongmagneticfield lines [11,12, 13]. Observationsalso favouredthe presenceofsuperfluid nucleonsin neutron star matter, as well as solidification of the outer layers. (See,forinstance,[14,15] and referencesquoted therein.) Thermal propertiesof neutron starswere rein-vestigateddue to thesenew developments[11,13]. Specifically, the effects of magnetic fields,superfluidity and solidification were included in neutronstar cooling calculations.A reviewof theseresultswas given in an IAU symposiumin Coloradoin 1970 [11].Major conclusionsare: (1) olderradio pulsars,which are expectedto be old rotating magneticneutronstars~of 106_107 years,arevery cold, ~ 104°K,and youngerpulsarsof io~years,too, should be relatively cold, 105°K,ifnucleonsarein superfluidstates,— unlesssomeheatingmechanismsareat work, and(ii) temperaturesof young neutronstarsof 100 to 10 000 years,— the Crabpulsar, the Vela pulsar,andpossibly othersupernovaremnants,are less certain, becausecooling dependsfar more~sensitively on various

Of course,later developmentsin X-ray astronomymadeusrealisethat plasmaarounda blackhole can emitX-rays (e.g. Cyg X-1), but that

is outsidethescopeof this paper.

240 Sachiko Tsuruta. Thermal properties and detectability of neutron stars — I

differentfactorsattheseearlier years— stellarmasses,magneticfields, different assumptionsmadefornuclear interactionsand hencedifferent equationsof state,uncertaintyin superfluid theories,etc.Dependingon differentmodels,theycould be emitting thermalradiationin soft to intermediaterangeX-raysof -~ 0.3—3keV. However,sensitivitiesand resolutions(both spatialand spectral)of detectinginstruments had to be substantially increasedto distinguish betweenX-rays from neutron starsurfaces(if present)and thosefrom the surroundingnebulaeof manyof thesesupernovaremnants.Therefore,it wasdifficult to obtain further theoreticalinsight from observations.

It wasknown that neutronstarscool predominantlythroughescapingneutrinosin earlierstagesofcoolingwhile theycoolthroughphotonradiationfromthesurfacein laterstages(see,forinstance,ref. [3]).In the abovecooling calculations,all “standard” neutrinoprocesses[11]were included.Sincethen,following the discovery of neutral currents,it has been pointed out that the presenceof neutralcurr~ntswill acceleratecooling. It wasalso pointedout by Ruderman[16]that thesurfacematterof aneutronstarundersuperstrongmagneticfields might be condensedto the “magnetic metal”, andthatif so,cooling would be accelerated.On the otherhand, various mechanismsto heatthe surfacewerealso consideredby variousworkers,— thecrustcorerotationalslippage,“noise”, accretionof matterfrom various sources,etc.A reviewof thesenewerdevelopmentswasgiven in [17].A crudeestimateof theeffect of pions, aswell aspossible“magneticsolidification”, wasincluded.However,due to thelack of sensitivitiesof existing observing instruments,we could not distinguish still at that pointbetweendifferent theoreticalmodels, as long as reasonabledegreesof superfluidity and magneticfields werepresent.

During the last few years, significant progresshas been made both on the theoretical andobservationalside. Specifically, the observationalpicture hasbecomemuchbrighter.The upperlimitto the surface temperatureof the Crab neutron star, which was set at 4.7 x 106°Kby the lunaroccultationmeasurementsof the Columbiagroup [18],was furtherloweredto 3 x 106°Kby Toor andSeward[19].Thatthis is alreadyinterestingwill beshownin a subsequentsection(3.2.1).This numbermay be further lowered to —S 106°Kthrough the HEAO-B ([20], section3.2.1), which was launchedsuccessfullyin November1978. If we still detectno point sourceat suchlow temperaturelevels,theoutcomeis very exciting becausethenit is most likely that only pion condensationis consistentwithobservations,and that may stronglysupportthepresenceof pions in neutronstars.

Possibilityof thepresenceof pion condensatesin neutronstarswas first suggestedby Migdal andby Sawyer—Scalapino[21]from theoreticalgrounds. Since then, various authors investigatedthisproblem.The mostrecentaccountis found in reportsby Maxwell et al., Brown, Weise,Kiguchi, etc.[22].Otherreferencesare foundin [22].Sincetheexistenceof pions in neutronstarsand thecriticaldensity above which such pions are presentdepend on details of nuclear theoriesand stronginteractions,it is needlessto saythat the impactof suchanobservationalsupportfor thepresenceofpions in neutron starsis far reaching,and it may be of greathelp for us to understandsome offundamentalphysicsin theseareas— nuclear,particleand high energyphysics.

We point out that theseobservationaltestsneednotbe restrictedto theCrabpulsar,but they canbe extendedto other neutronstarsandneutronstarcandidates,suchas theVela pulsar,otherradiopulsars,andothersupernovaremnants.In fact, theVelapulsarmay proveto be moreexciting in thissense(seesection3.2.2). We emphasisethe realisticpossibilitiesalreadyexisting in theHEAO-B, andvariousexciting possibilitieswhich may very well bewithin ourreachin the instrumentsof the 1980’s.These observationalprospects are discussedin a subsequentsection (3), after theoretical con-siderationsaregiven in thenextsection(2).

On the theoreticalside,considerableprogresshasbeenachievedwithin the last few years,mainly

Sachiko Tsuruta, Thermal properties and detectability of neutron stars — I 241

by the NORDITA—Stony Brook group led by G. Brown, through which more realistic and betterestimateof neutronstartemperaturesbecamepossible.Using thesenewtheoreticalinformation,someof us recentlyobtainedmorerealisticestimatesof surfacetemperatures[23,24, 25]. In this report,wepresentthese new results, as well as the recent work by someother groups in this and relatedproblems.However,before that, we summarisethe basic equationsand the results shown in theearlierreviews,so thatwe cancompare.

2. Theoreticalpredictions

2.1. Summaryof basicequations

The cooling time t is obtainedfrom

= J dU/L(U). (1)

where (J0 and U~arethe energiesof the starat the time of formation of the neutronstar throughasupernovaexplosionandat the time t, respectively.L(U) is theaverageluminosity over the intervald U. (Energydecreasesby d U duringthe time intervalbetween(1—dt) and t.) The total energyU canbe expressedas

U=U(T1)=~~ (2)

where T, is the internal temperatureof the star, and U~and ~ are the thermal energiesofdegeneratefermions k (= neutrons,protons,hyperons,muons, electrons,etc.) and non-degenerateenergyof heavyions. The total energylossrateL is

L=L.>.+L~, (3a)

L~=4iroR2T~, (3b)

L~= L~+ L~+ L~’+ L~(others). (3c)

L~is photonluminosity and L~is total neutrinoluminosity. The superscriptsU, B, andp1 standforthe modified URCA, neutrinobremsstrahlung,and plasmonprocess,respectively,and L~(others)includeotherminor processes(photoneutrino,pair annihilationandneutrinosynchrotronlosses).Te

is the surfacetemperature.Since the total thermal energyand neutrinoluminosity dependon theinternal temperaturewhile the photon luminosity dependson the surfacetemperature,the stellarstructureequationshaveto be integratedfrom thesurfacetoward the interiors until the isothermalcore is reached.The generalrelativistic form of theseequationsareexpressedas

dT_ 1 3 ~p~—- -~--—~-—~L(r),

dP — — G(P/c2+ p)(4irr3P/c2+ m) 4dr r(r—2mG/c2) ‘ ( )dm 2

242 Sachiko Tsuruta, Thermal properties and detectability of neutron stars — I

alongwith theequationof state

P = P(p, T). (5)

The opacitywhich appearsin the transferequationis expressedas

1/Kl/KR+1/Kc (6)

where KR and lC~are the radiative and conductiveopacities,respectively.Under unusual circum-stances(e.g. accretionof matter from binary companions)convectionmay becomeimportant, butthat can be neglectedfor the following discussions[2].More detaileddescriptionsof basicequationsandnotationsarefoundin refs. [2,3, 11].

The aboveequationsweresolvedthrough computersfor specific neutronstarmodels(which giveadefinite setof the stellarmassM andradius R), atdifferent given surfacetemperatures.The surfacewas assumedto be the photospherewhich was determinedby standardstellar atmospheretheories(see [2,3]).

2.2. Earlier results— Summary

A sampleresult is shownin fig. 1. Here surfacetemperaturesareplotted asa functionof time fordifferent cases [11,13]. The dashed curve (I) shows the case with no magnetic fields and nosuperfluidity.The crossindicatesthepoint wherephoton cooling overtakesneutrinocooling. We notethat attheageof theCrabpulsar (—.- i03 years),thesurfacetemperatureis about4x 106°K.At a typicalageof radiopulsars~ 106 years)thestellar surfaceis still as hot asa million degrees.

Otherdashedcurvesshowtheeffect of magneticfields. A majoreffect of strongmagneticfields isthe drasticreductionof opacities.Radiativeopacitiesparallel to themagneticfield lines arefound to

10 n I I I — I I

I

LOG TIME (YEARS)

Fig. I. Surfacetemperatureasa functionoftime (cooling curves)for theneutronstarmodelV.,II atdifferentmagneticfield strengths.Curves(I),(II), (IV) and(V) standfor a uniformmagneticfield of strengths(HIHq) = 0,0.025,0.1andI, respectively,whereHq = m~c3Ihe= 4.41 x 10°gauss.In thecurve(III), thesurfacefield strengthis thesameasin thecurve(II), but the internalfield strengthis increasedby a factorof 10. Thepointswhere themajorcooling mechanismshifts from neutrinoemissionto photonemissionareindicatedby thecrosses.The solid curvesanddashedcurvesrepresenttwo extremecasesof maximumandno superfluidity,asexplainedin thetext.


be reducedas[11,12]

KR(H) = aRKR(0); with aR ~ (~/wH)2 ~ (nH)2, (7a)

whenw ~where KR(O) is the opacity with no magneticfields. w is photon frequencyand t0H is cychrotron

frequency.The conductiveopacity is similarly expressedas

= acicc(0); with ac~ 1. (7b)

The proportionality constantac is generally a complicated function of~both temperaturesanddensities, but the temperaturedependencedrops in the degenerateinteriors where conductiveopacitiesovertakeradiativeopacities.The resultof numericalintegrationsof~exactequationsfor ac isgiven in [11].The magneticfield strengthof eachcurve(from (II) through (V)) is indicatedin fig. 2 inunits of Hq = m~c3/he= 4.41 x iO’3 gauss.So, for instance,the curve (II) correspondsto H 1012

gaussand thecurve(IV) correspondsto H 5 x 1012gauss.Exceptin thecurve(III), the field strengthwasassumedto be constantover the thin skin layers wheremostof the temperaturerisestakeplace.In thecurve(III), the field strengthis increasedfrom — 1012 gaussin the raditttive transportregionstoabouta factorof 10 higherin the innerconductiveregions.At theageof th~Crabpulsar(10~years),the surfacetemperatureis increasedto nearly 10 million degrees.This is~mainly becauseanothermajor effect of magnetic fields is to suppressthe URCA process,and so the URCA neutrinoluminosity is not includedin thesecurves.Anotherreasonis that the surfacetemperatureis higherat

11 I I I

10- MODEL V711 -

9- 1/ -

E 11/LU I/U~ 8- / / -

‘IT~ /‘( / / (IV)

~7. -

y /(Ifl)

-

z / iT)LU /I-. / -

/0 /

_1 ~ - (I) (H/Hq)RC~O.O -

(U) (H/Hq)R~=O.O25

3 - if (~)(H/H~)p=0.025 -

/ (H/Hq)c=0.25

2 - (IV) (H/Hq)Rc=O.l -

(V) (H/H,~)~cl.O

1 I I I I3 4 5 6 7 8 9

LOG SURFACE TEMPERATURE (~K)

Fig. 2. Internaltemperatureasa function of surfacetemperaturefor model V,II at differentfield ~trengths~(I), (II), — — — asexplainedin fig. I.


a given internal temperaturewhich governstheearlier neutrinocooling stageswhen kT <hwy. Themagneticeffect on the differencebetweensurfacetemperaturesand internaltemperaturesis given infig. 2.

Major effectsof the presenceof superfluidity are (i) to suppressthe URCA process,and (ii) toreducespecific heatsand thus thermalenergiesof superfluidparticles,at low temperatures.Therefore,astheextremecase,weset theenergiesof baryonszeroandneglectedtheURCA process.The resultsare shown as solid curves in fig. 1. Here a major contribution to the internal energyof the star isthermal tail of degenerateelectrons.Thus the main differencebetweenthe solid curvesand dashedcurves is due to the completeabsenceof thermalenergiesof baryonsin solid curves. We caution,therefore, that the solid curvesdefinitely overestimatetheeffect of superfluidity,and the deviationsfrom reality will get moreseriousat higher temperatures.However,at sufficiently low temperaturestheyshould approachtheasymptoticlines in the lower right handcorner toward which solid curvesconverge.We may also neglectthe curve (V). This is becausethe magneticfield for this caseis—5x iO’3 gauss,which is too high according to what observationsso far seemto be suggesting.Typical field strengthsfavouredby observationsare — 10”— — 5 x 10’~gauss(seee.g. [6, 10, 26]). Thispoint will bediscussedfurther in a later section(2.3.3).We took thesefactorsinto considerationwhenwe gaveour conclusionsearlier in Introduction.In theabovesamplecalculations,we usedtheV, IIneutronstarmodel (see [11])with the following properties;M = 1.07M

0,R = 12.33 km, and pC (centraldensity)= 7.39x iO’

4 g cm3. In theabovecalculations,theURCA processwas neglectedin all exceptthedashedcurve (I). More detaileddescriptionis found in [11,13].

Due to the large differences,at given values of H, betweenthe dashedcurves and solid curvesshown above, it will be desirableto treat superfluidity more realistically. That was done in [11].Specifically, we usedinformation about the s-wave and p-wave superfluidgapswhich was the bestavailableat the time; thosegiven by the Kyoto group,TamagakiandTakatsuka[27]. (Almost sevenyears later, when Oren Maxwell reexaminedthis problem,he concludedthat they are still the bestavailable [24]and he usedthe samein his latest cooling calculations.)The reductionof energiesofsuperfluidparticleswastakeninto accountby using thefollowing equationsfor degenerateparticlesk;

U~=JC~dT.

C~= f (CO)~Y~nk41rr2dr, (8a)

D...flTkB(Xk+1)_\~(Co)k—~--~Vx~ ~

where

Xk = P~/(mkc),o c’rk / pk~

k O.Jlc I kexP~_1.44_)~for T< T~. (8b)

For thes-wavegap

kBTCk = 0.57x

For thep-wavegap

kBT~= 0.57x A(3P2)/\/2r0,with ln I’0 = 1.22.


Y~is a superfluid correction factor for particles k. Solidification of the outer heavy ion layers(p~ (2—3) X iO’~gcm3) is also included by using the following equationsfor non-degenerateheavyions;

= {f C~n10~4-n-r~dr}

where

C.., = a1k8~(O/T), (9)3

a, = ~for T> Tg, anda1 = 3 for T ~ Tm.

The intermediateregion betweenTg and Tm is interpolated.Tg is the temperatureabovewhich ionsbecomea gasand Tm is themelting temperature.The expression~( 0/T) is theDebyefunctionwhichapproacheszero as temperaturebecomesvery low (T ~ Tm). More detaileddescriptionsand othernotationsaregiven in [11].Suppressionof theURCA ratesdue to superfluiditywascalculatedby theuseof ref. [281.A typical resultis shownin fig. 3a. Here themagneticfield is about 1052gauss,and themodels(I), (II) and(III) werechosenfrom the lowestmassend,mediumandthemaximummassendof an improvedcompositneutronstarmodel constructedby theauthor,which took into accountofthebestavailableinformationat that time, of theequationof state(nuclearpotential)andcompositionof neutronstars.In themost interestingregion of —3 x 1014_1015g cm

3,the~OPEGtypepotentialoftheTamagaki’sgroupwas used,andthePandharipandehyperonequationof~statewas usedathigherdensityregions.Essentially,it is a Reidpotentialandclosestto themodelsconstructedby Baym Ct al.[29].More detaileddescription of the model (called ST model, for convenience)is given in [11].Typicalstellarparametersarelisted in table 1. We notethat fig. 3a showstheeffect of stellarmassoncooling, oncetheequationof state,magneticfield strengthandsuperfluidenergygapsare specified.

Furtherresultsare summarisedin the following figures. In fig. 3b, thecooling curve of ST model(II) obtainedwith the methoddescribedaboveis indicatedas a solid curve. It is comparedwith thetwo dashedcurveswhich were obtainedthroughthe methodsdescribedeariierfor thecasewith nosuperfluidityandmaximumpossibleeffect of superfluidity.(Seetheexplanationsof fig. 1.) The effectof stellarmasson the internaltemperature— surfacetemperaturerelationis shownin fig. 4.

I I I I I I I

COOLING OF DENSE STARS~ 8 - (H/Ilq)~~ 0025

LOG TIME (YEARS)

Fig. 3a. Coolingcurvesfor thethreehadronstarST models(I), (II) and (III) chosenin ref. [Ill, with H = 0.025H,= 1012gauss.Their propertiesaregiven in table 1. The crosseshavethe samemeaningasin fig. I.


Table 1Characteristicsof thethreemodels of hardronstarschosenin ref. [Illandused in section2.2 of this report. pC is thecentral(energy)densityincgsunits,M/M

0 is themassin solarmassunits,andR is thestellarradiusin km. Model (I) is a heavyhyperonstar,model (II) is a mediumweightneutronstar, and model (III) is a light neutronstar with neutron-rich

heavynuclei.

Model Centraldensity Mass Radiusp~(g/cm

3) M/M-~ R(km)

(I) 6.0 x10I5 1.413 7.10 Heavy

(II) 8.0 x 10I4 0.476 10.9 Medium(III) 1.1 X 1014 0.05 76.7 Light

Since theabovecomputationswere performeduntil fairly recently,the only major developmentswhich might affect the cooling problem were (i) accelerationof neutrino cooling through neutralcurrents,and(ii) further reductionof opacitiesif “magnetic solidification” takesplacein thesurfacelayers. The original suggestionof the former by Bond was soonshown to be an overestimate,andmore realistic estimateswere given subsequently[30,31]. Their inclusion, however, did not givesignificant changesto the existing results describedabove. In the latter problem, the magneticpotential was originally estimatedto be —30 keV [321.That means,surface layers will become“magnetic metal” at temperaturesbelow — l0

8°K.possibledrasticeffect of such“magnetic metal”surfaceson cooling was investigatedin [33].However,since then, it was found that the magneticpotential should more likely be a few keV, rather than —30 keV [34]. In that case “magneticsolidification” will not takeplace until temperaturesbecomelower than —.- 107°K,and this effect oncooling becomesless serious.However,it becomesimportant in anothersense,that the new lowertransitiontemperature(or magneticpotential)correspondsto theblackbodyradiationtemperatureinthe X-ray region, and thereforethe natureof the surfaceradiationwill dependsensitively on thenatureof thesurfacematter (gas,liquid, or solid). (See later discussions.)By thecurve(III) in fig. 5,we show the maximum effect of the presenceof the “magnetic metal” with the original 30 keV

I I I I I I I I I I I I

8 CASE(N)

~: CASE(S)-~T”\”\

~

LOG TIME (YEARS)

Fig. 3b. Coolingcurvesfor the ST model (II) of fig. 3a, with the sameH as in fig. 3a. The dashedcurves(S) and (N) are for the maximumsuperfluidityandno superfluidity,respectively,as definedin ref. (II]. The solid curveis thefinal bestestimatedcoolingcurve obtainedby themethoddescribedin thetext and in ref. [11].The crosseshavethe samemeaningasin fig. I.


I I I I I / J,~,(H/Hq)

50rO.025 I,’ ,/‘,/‘

10 - -

9. -

(ifi)UJ8- ~. -

I-. (II)~ 7. -

LU

-

zLUI.- 5- -z

0__J4. -

3. -

2— -

I I I I I I3 4 5 6 7 8 9

LOG SURFACE TEMPERATURE (°K)

Fig. 4. Internaltemperatureasafunctionof surfacetemperaturefor theST models(I), (II) and(III) asdescribedin fig. 3awith thesameH asinfig. 3a.

COOLINGMODEL ~

~~5~~Ifl11I

LOG TIME Iyeoral

Fig. 5. Coolingcurvesfor different casesasexplainedin thetext (section2.2). ST model (II) of fig. 3b is used.


potential, and comparethat with other effects.We caution that the effect is grossly overestimatedhere.Curve(I) correspondsto the casewith no magneticfields. In curves(II) and (III), H = 5 x 10’~gauss.Includedalsoareall theeffectsdescribedearlier(seethedescriptionsfor fig. 3a).

2.3. Recentresults

In the following, we summarisefirst our latestresults[.25],becausethesearethe latestof this kind,as far as we are awareof, and thus we could take into accountof all earlier developments,boththeoreticaland observational,and also becausethey are more quantitativethan the others we areaware of. We then summarisethe work by the others and comparethem, before we go intoobservationalproblems.

Major improvementson our earlier work which we have madehere are (i) better treatmentofneutrinoprocessesand (ii) detailedstudiesof theeffectsof equationsof state,which may proveto beimportant for our understandingof nuclear force problems.If we hope to observea neutron starthroughthermalradiationfrom thesurface,its internal temperaturegenerallyhasto be — 10a_lObo0K,correspondingto the surfacetemperatureof — l06—108°K (see fig. 2 and [3,11]). At such a hightemperaturethe stellarcooling is normally dominatedby neutrinoprOcesses.Therefore,we considerthem first.

2.3.1. NeutrinoprocessesAs neutrinoprocesseswhich can be important for our presentpurposes(thermalevolution of a

neutron star after a few monthsfollowing its formation), we have mentionedthe modified URCAprocess,theneutrinobremsstrahlungprocessand theplasmonprocess.Theseandothermoreminorprocesses(pair annihilation, photoneutnnos,etc.) were included in our previous calculationsandexplainedthere [2, 3, 11]. Drastic importanceof neutrino processesinvolving pions and possibleimportanceof the effect of neutral currentswere also pointed out. Recently, the Brown’s groupreexaminedcarefully all possibleneutrino processeswhich might give appreciableeffectson ourcurrent problem (see [22, 24, 35a, 35b1 and other referencesquoted therein). The results aresummarisedbelow. If pions are present,the beta processesinvolving pions are by far the mostimportant[22,35a], as BahcallandWolf first pointedout [5]. Theycanbe expressedas(1) The betaprocessesinvolving pions:

(a) ~+n-+n+~+i~, lOa(b) 1r+n—*n+e+i~,

and their inverseprocesses.Recently,theStony Brook groupand independentlyKiguchi of the KyQtogroup [22] found out that the final luminosity equation for pion condensatesis (qualitatively)remarkablysimilar to theonederivedfor free pionsby BahcallandWolf usingasimplerapproach[5].Other importantneutrinomechanismsarethe modified URCA process,neutrinopairbremsstrahlungfrom a neutron—neutron(nn) pair, that from a neutron—proton(np) pair, andthe ordinary neutrinobremsstrahlunginvolving electrons.Theyare:(ii) The modified URCA processes:

(a) n+n~n+p+~+i~, (lOb)(b) n+n—n+p+e+ie,

and their inverseprocesses,


(iii) The nn pair neutrinobremsstrahlungprocess:

n+n—~’n+n+v+i~ (lOc)

(iv) Thenp pair neutrinobremsstrahlungprocess:

n+p—+n+p+v+i, (lOd)

(v) The ordinaryneutrinobremsstrahlungprocess:

(Z,A)+e(Z,A)+e+v+i. (lOe)

We note that the effect of neutral currentsentersequations(lOc) and (lOd). We havere-expressedtheir emissivities in the following manner,in the form of equivalentluminosities,to suit our exactintegrations(no furtherapproximations)overthe stellar interior.For(i) (from [22]),

L= 1O~aR~6T~(ergsec~), (ha)

where a = 1 for p > p~.and

a=O.5 forp<p~.

R,,6= R~(cm)/106, where R,. is radius of thecorewithin which pions arepresent,and T9 = T~(°K)/109.

p5, is the densitywheremuonsappear—8 x io’~g cm

3.For (ii),

L~ 10~~(1+ F){ 16 (~)312r~dr6} (~)3(~) T~(ergsec’); (1lb)

F = PF(~L)/PF(e),for p > p~,and=0, forp~p~

= r(cm)/106,R

6= R(cm)/l06whereR is thestellar radius,and F is given In [5]. PN 3 x iO’4 g cm3

is thenucleardensity.For (iii),

R

61/3 * ~

L~(nn)=9x1036{J (~~-) r~dr6}(~-~) T(ergsec~). (lic)

For (iv),

L~(np) 3 x 1037{ 16 (P_)213r~dr

6} ~ (erg sec~). (lid)

For (v),

L~(C) 3 X 10~{J (P~)r~dr6} (i-) T. (ergsec’). (lie)

R~is theinnerboundaryradiusof theheavyion crust.The aboveformulae~ii)through(v) arederived


from emissivitiesgiven in [24,35]. Following Maxwell [24],we set Z2/A = 1. Note that eq. (1le) isessentially the same as the original bremsstrahlungneutrino luminosity derived by FestaandRuderman[36].

We note that pion cooling rates (I) are about io~times as efficient as the modified URCA rateswithout pions(ii), and that the luminositiesfor processes(ii), (iii) and (iv) (the modified URCA and thenn andnp pair bremsstrahlungswith neutral currents)all dependon temperaturesas T8, while thosefor (i) and (v) (pion coolingsandordinaryelectronbremsstrahlung)dependon temperaturesas ~ T6.The exactvaluesof L~aremodel dependentbecausethey generallydependon densities.In theaboveexpressions,thegreatestuncertaintiescomefrom the effective massesof neutronsandprotons,m~andm~,respectively,expressedin units of the realmassesm~andm~.

2.3.2. SuperfluidityAnother important factor is the effect of the presenceof superfluid baryonson the above

processes.Specifically, the processes(ii) and (iv), the modified URCA processesand the np pairbremsstrahlung,are greatlysuppressed(by a factorof roughly — exp(—(z~~)/kT))by the presenceofproton superfluidity,while theprocess(iii),the nn pair bremsstrahlung,is suppressedin thepresenceof neutron superfluidity(by a factorof — exp(—(~~)/kT)).(Seee.g. [281.)In theaboveexpressions,&~,and ~, are the superfluid energy gaps of protons and neutrons,respectively.In the absenceofsuperfluidbaryons,theURCA ratesaretypically 10—100 times the nn pair bremsstrahlungrates.Theyare —30 times higher than the np bremsstrahlungrates,and both (the URCA and the np) aresuppressedby protonsuperfluidity in similar ways. So Maxwell [24]neglectedthe process(iv).

A most interestingeffect of superfluiditycomesdue to the differencein compositionsof neutronsandprotons.Nearthenucleardensityprotonconcentrationis typically aroundseveralpercent[8,37].The s-wavesuperfluidgap appearsat lower densities, ~iO’~g cm3, while the p-wave gap appearsbetween— io’~and iO’~gcm3 [11,27].The sizeof a gapunfortunatelydependson theeffectivemassmentionedearlier,which is oneof themostuncertainfactorsin thepresentproblem.Variousauthorsinvestigatedthis problemandadetailedaccountis given in [25].The conclusionis thatanappropriatevalueof m~/m~,if we haveto makea choice,is —0.8 (seee.g. [27,35b, 38]), and this is thevaluebothMaxwell [24]andwe [25]usedin our calculations.It is quite possiblethat it is much smaller(see e.g.[35b,39]). So in our calculationsthat caseis also takeninto account.Effective massof protons isinvestigatedby some[40].With Maxwell [24]we mostlyusedI for m~/m~,but it canbe muchsmaller[35b,40] and that caseis also consideredin our work (seesections2.3.4 and2.3.5).

Oncetheeffectivemassesarespecified,it is easierto estimatethesuperfluidenergygaps.From thebest estimateavailable,by the Kyoto groupmentionedearlier [27],we obtainedthe valuesgiven intable 2 wherethegapsareconvertedto thecorrespondingcritical temperaturesfor convenience.(Seeeq. (8b).) Here the neutron star interior was subdividedroughly into three density regions, (I)p ~ ho’5 g cm3, (II) i0~~ p ~ iO~gcm~3,and (III) p ~ iO’~gcrn3. In order to find out the partialdensitiesof protonsin a givendensityregion,wehaveto know protoncomposition.We used[8,37] toestimateour composition.Suppressionof neutrinoluminositiesby thepresenceof superfluidnucleonswasincorporatedinto our stellar structureintegrationsby multiplying thenon-superfluidvalues(eqs.(1lb)—(h id)) by appropriatereductionfactors,which dependon exp(— T~IT)whereT~is the criticaltemperaturebelow which particlesk becomea superfluid[24,28]. This was done in eachdensityregion.(See [25]for furtherdetails.)

The netqualitativeoutcomeis as follows. The modified URCA processesand the np pair neutrino

Sachiko Tsuruta, Thermal properties anddetectability ofneutron stars — I 251

Table 2Totaldensity,protondensityandcritical superfluidtemperaturesof neutronsandprotonsin thethreedensityregionschosenfor our calculations[25].See thetext

(sections2.3.2and2.3.4) for details.

Critical temperatureTotal Proton of superfluid(°K)density density

Region (g cm3) (g cm3) Neutrons Protons

(I) p>lO’2 ~2x 10’4 0 P 2x10’(II) 1014<p< 10° 10°—Sx 1013 P 8 x 108 S 2 x 100(III) p < lO’~ <10° S lOb

processesare effectively cut off at relatively high temperatures(—2x lObo0K), while later in theevolutionwhen T

1 8 x 108°K,the nn pair bremsstrahlungwill be alsocut off (if pC < hO’5 g cm3).In

that case,amongmajorneutrinoprocessesconsideredabovein theabsenceof pions, only theprocess(v) (the ordinary bremsstrahlunginvolving electrons)will survive. In the interior where heavy ionsdisappear,the only positive ions participating in this processare protons,but sincetheyare highlydegenerate,the process(v) involving protonswill be also negligible [24].That means,amongmajorprocessesonly the bremsstrahlunginvolving heavy ions (A, Z) in the outer crust (p~ (2—3) xiO’4 g cm3) is unaffectedby the above factors.In our calculations,we retainedother more minorprocesses,too, suchastheplasmonprocessandsomeof theotherprocessesgroupedas L (others)inour earlierpapers,which arenot affectedby thesuperfluidreductionsmentionedabove.

2.3.3. Equationof stateIn order to see the effectsof equationsof state,we chosetwo extrernecases.As the hardest

equationof state(steepestdependenceof pressureP on densityp),we chosetheTI (tensor)model ofPandharipande,Pinesand Smith [41],hereaftercalledPPS-TI.This equation~of statewascriticisedbysome [42] as not fitting nuclear parametersvery well*2, but on the other hand it may explainobservationsbetter [25].A harderequationof stategenerallygives higher massesat lower densitiesandbrings the neutronstarmaximummassto highervalues(— 2M

0 for thePPS-TImodel)(also,seee.g. [43]).According to nuclearexperts,the Reid potential gives best fits to nucleardata [42].Thispotential, however,gives a very soft equationof statewhich resultsin hilher central densitiesforgiven masses,and leadsto a smallermaximummassof neutronstars,— 1.4M0 [29].Among relativelyrealisticequationsof stateconsideredfor neutron stars,we do not know of any which lies outsidethesetwo cases[8, ii, 29, 41, 43, 44]. Therefore,we chosethesetwo to stu4y theeffectsof equationsof state.Specifically, our ST model mentionedearlier and usedin [hi] 4nd our subsequentworkessentiallymadeuseof a Reid potential, and so we usedthe ST model~to representthe softestequationof state.Forthe sakeof comparisonwe set thestellarmassat M ~ 1.3M® and themagneticfield strengthat H = 5 x 1012 gauss.Therearegood reasonsfor choosingtheseparticularvaluesbothobservationallyand theoretically.On theobservationalside,severalneutronstarswhosemassescouldbe estimated(e.g. Her X-1, Sco X-1, etc.) have 1.3M® [45].From the theoreticalside, supernovatheoriespredict themassof a remnantstarto be about1.3M0[46,47].In a~ycase,from both sidesit

*!See ref. [8] for argumentsin favourof this model— (PPS-TI).


Table 3Variouspropertiesof neutronstarmodelsA and B, asdescribedin thetext. p’, M/M

0,and R areasexplainedin table I. R~,R,,, andR(I) are,respectively,thedistancefromthecenterof the starto the point where the heavy ion crust, the pion core and thedensity region(I) start,expressedin km. Thecritical densitiesat theseboundariesare,

respectively,p~= 2 x 1014 g cm3,p,~= 3 x 10°g cm3, andp(I) = 1013 g cm~3.

Model MIM0 pc(gcm_

3) R(km) R~(km) R,,(km) R(I)(km)

(A) (.317 3.5 x 1013 7.975 7.48 7.35 6.32(B) 1.320 4.2x 1014 15.974 11.20 7.50 —

seemsthat smallermassesof ~ 1M0 areunlikely [45—47].However,effectsof different stellarmasses

arediscussedlater(section2.3.5).The magneticfield strengthchosenfor our calculations,H = 5 x l012

gauss,was takenfrom observationsof a line in the spectrumof Her X-1 near53 keY which is mostlikely evena cyclotron emission(or absorption)line [26].The field strengthestimatedfrom pulsarobservationsaresomewhatlower, —10” to =~1012gaussdependingon pulsarperiods[6,10]. The latterwasderivedfrom dipole fields. Sincethesurfacemagneticfield is expectedto deviatefrom a simpledipole approximation,we should regard that the higher value,~ x 10’~gauss,which was obtaineddirectly from thesurface[26],is morereliable.

In the following we call the ST model of M = l.3M0 and H = 5 x 10’~gaussmodel A, and thePPS-TI model of the sameM and H as model B. Their characteristicsare listed in table 3. Asexpectedfrom the earlier discussions,model A is smaller and denserwith a 8 km radius andpC = 3.5 x iO’~gcm

3,while model B is largerand lighter with a — 16 km radius (abouttwice as largeas model A) andp’~ 4 x 10” gcm3.Therefore,theeffect of theheavyion crustis relativelysmall inthe former,while a largerportion of the latterconsistsof theheavyion crust.We will shortlyseethatthesedifferencesgive significanteffectson their thermalproperties,and thatby comparingour modelA and model B we canget a fairly goodideaof themaximumeffectsof equationsof state.

2.3.4. ResultsResultsare summarisedin the following. We first show the temperaturedependenceof thermal

energyandvariousluminosities*Sin fig. 6a for model A and in fig. 6b for model B. Notationsusedareas follows: E, = total thermal energy, L~,L~(C),L~(nn)and Let, are, respectively,neutrinoluminositiesdue to the modified URCA, the crust bremsstrahlung,the nn pair bremsstrahlungwithneutralcurrents,and plasmonand all otherminor processesincludedin theearlier calculations[11],andL~is photonluminosity.

At a typical internaltemperatureof abouta billion degrees,in theabsenceof superfluidity,we notein fig. 6a that the URCA neutrinoprocessdominatesover all other cooling mechanisms,the crustbremsstrahlungandthe nn pair bremsstrahlungwith neutral currentsarecomparable(both lessthantheURCA by a factorof — 100), while all other neutrinoprocesses(L~’),as well as photoncooling,are negligible. For this model (model A), this stageis reachedroughly severalmonths after theformation. This point is indicatedby the arrow marked(9) in the figure. The situationfor the Crabneutronstar is indicatedby thearrowmarked(C). Roughly thesameconditionsstill prevail asat (9),

*3 Recently,qualitatively similar resultswereobtainedindependentlyby SoyeurandBrown [35b],thoughtheyuseddifferent models.

Sachiko Tsurula, Thermal properties anddetectability of neutron stars — I 253

50 /1.1 /

MODEL A / ~/

Fig.6a. Total energyE in ergsandvariousluminosities(bothneutrinoand ~ /1photon)modelANotat,onisexpla,nedmthetext~ctionfl4 (in ~ ~

30- /~

I I I I I I I

1. 5 6 7 8 9 tO 11LOG INTERNAL TEMPERATURE~K)

MODEL B

E1~~/” ~

/ IFig. 6b. Thesameasfig. 6a, for model B. ~- 40

35 V I/I;/ ii

L~ / // (~)

,/ /C

30 / /

I I I I I I I I

4 5 6 7 8 9 10 11LOG IN1tRN41L TEMPERATUREI’K)


buthere thecrustbremsstrahlungovertakesthe nn pair bremsstrahlung,due to thesofterdependenceof theformer (on temperature)thanthe latter.The plasmonprocessbecomescomparablewith the nnbremsstrahlungonly at temperatureshigher than —ten billion degrees.In thesehigh temperatureregions the crust bremsstrahlungis negligible due to the T~dependence.We note that photonluminosity is comparablewith L~(C)andL~(nn)for theCrab.Theseconclusionsarereachedin theabsenceof superfluidities.In fig. 6b we note that the role of the crust bremsstrahlungis increasedbecauseof the lower densities associatedwith this model (B). Here the central density is 4.2 xio’~gcm~3,only slightly higherthanthe nucleardensity.The exact densitybelow which heavyionsappearis somewhatuncertain.For model B theheavy ion cruststartsat R~ 5 km if the ion criticaldensityp~= PN, thenucleardensity.It startsat 11 km if p~= 2 x lO’~gcm3. We notethat thestellarradius of model B is 16 km. Solidification of the major portion of the heavy ion regionoccurs at

one to tenbillion degrees(see,for instance,[33]).So we expectthat a large outerportion of modelB shouldbe a solid heavyion crust. We note that for this model theURCA luminosityand thecrustbremsstrahlungluminosity are comparableat 109°K,and the latter exceedsthe former in the Crab.Both asbeforeareindicatedby thearrowsmarked(9) and (C) in the figure. Significantincreaseof theL~(C)relativeto the L~’for modelB is dueto the fact that it hasa large crust (comparablewith orlarger than the centralcore). Plasmonluminosity is also increaseddue to the reduceddensities.Itbecomescomparablewith the bremsstrahlungcoolings (both L~(C)andL~(nn))at —3 X 109°K.Onthe other hand photon luminosity is lowered relative to neutrino luminosities mainly becausethetemperaturedifferencebetweenthe inner isothermalcore and the surfacebecomeslarger for thismodel. Thus it is still negligible in the Crab. We alsonotethat the ratio (L~/IL~(nn))is decreasedformodel B (to 10) becausethe central core containing protonsare smaller for this model than formodel A. Changesof the above situations in the presenceof superfluid particlesdependson theeffective massesand superfluid energy gaps which depend importantly on densities.Thesecon-siderationsaregiven belowwherewe shall summarisecooling behaviourof various cases.

Figure 7 showscooling curvesfor various possiblecases.As before model A is a 1.3M0 neutron

starof mainly a Reidpotentialtype equationof state(ST model in [11]),andmodel B is a samemassneutronstarof the PPS-TItype [41].We notethat modelA is representativeof the softestequationofstatewhile model B hasthe hardestequationof state(amongcomparatively“realistic” models),andthat for. agiven massstar,modelsof harderequationsof stategenerallycool faster.That is clearfromour presentresult (fig. 7) andour earlier studies(e.g. comparisonof theV~andthe V,, modelsin [2,3],etc.).The physicalreasonis that lower densitiesareassociatedwith the modelsof harderequationsofstate with a given mass. Therefore the highest temperaturecooling curve of model A will berepresentativeof the maximumtemperatureandthelowesttemperaturecooling curveof modelB cangive us a good estimateof the minimum temperatureof the star, at a given age. To make ourcomparisonsmore realistic,we set H = 5 x 10’~gauss,a most favouredvalue [26], and M = 1.3M0,againamostfavouredmassvalue[45—47],from theoriesandobservations.Oneexceptionis the curvemarkedBJ(H = 0), which refers to acooling behaviourof a l.3M0 modelof the Bethe—Johnsontypenuclearpotential [44]with no magneticfields. This curvewas addedfor the sakeof comparison,andalsobecausethis equationof stateis representativeof an intermediatecasebetweenthe two extremeschosenhere.

As various possibilitieswe consideredthe following cases.(Ia): No superfluidities,but we include all possibleneutrino luminosities and photon luminosity.

Both thetotal energyandneutrinoluminositiesarefor non-superfluid(normal)particles.In theenergyequationwe take account of nucleon—nucleoninteractions by multiplying eq. (8a) (= partial

Sachiko Tsuruta, Thermal properties and detectability of neutron stars — 1 255

COOLING CURVES8 M~1.3M0

(lb) (A) CV) H =5~1O12G

-~ ‘~ (1)~(Ia) (B) ~.—.

6 __

5 (Maxwell) \\ ~(8) BJ)H—0)

\~ \\ \

\ \\ \\ \ \ix ___ \“ (1(a) \ “~ ~\

(la)~ ?NO PIONS \\ ~

0 fILl1 ) ~‘

flUb) ~ \(lii) --—— PION COOLING 4

(A)MODELA2 (B) MODEL B

1 2 3 4 5 6 7 8LOG TIME (years)

Fig. 7. Variouscooling curvesfor model A and model B, taken from themost recentref. [251.Thecurves(Ia), (Ib) and (1(a) showcoolingswithno pions,while thecurves(III) showthe maximumeffect of pion coolings.The curves(Ila) includesuperflujditiesof nucleonsand all possibleneutrino and photoncoolingsexceptpion coolings.The curves(Ia) includeall cooling mechanismsexceptpion coolings,but superfluiditiesofnucleonsareneglected.In thecurve (1b), superfluiditiesandthe URCA and nucleonpair neutrinoprocessesareneglected.The arrows markedwith (C), (V) and(P) indicate,respectively,theagesof theCrab,theVela andtypical radiopulsars.Thecircles(ndicatetheupperlimit to theCrabsurfacetemperaturefrom the pastobservations.The shadedbox below the circlesindicate thepossibleCrab temperaturerangewhich can becheckedthroughtheHEAO-B andotherobservationsin near future. Thedouble headedarrow~is therangepredictedby Maxwell [24]for theCrab.Othernotationsareexplainedin thetext (sections2.3.4, 3.2.1 and 3.4).

energyof particlesk) by thecorrespondingeffectivemass(m~/mk)[24].The total luminosity

L, = L~’+ L~(nn)+ L~(np)+ L~(C)+L~t+ L~. (l2a)

Notation was explainedearlier.(Ib): No superfluiditiesas in (Ia), but we neglect the modified URCA and the nucleon pair

bremsstrahlungneutrinoprocesses.That is,

L3 = L~(C)+ L~

t+ Lv. (12b)

(ha): With superfluiditiesof both neutronsand protons, though with different transition tem-peraturesat different density regions.We usethe currently bestavailable ir~formationabout protoncomposition [8,37] and theory of superfluidity (estimatesof gaps,etc.) [27]as outlined earlier. Thevalueschosenin differentdensityregionsareasgivenearlierin table 2. Reductionof both total energyand someof luminosities as explainedearlier is takeninto accountas exactly aspossiblewith ourpresentknowledge[24,28]. Effective massesof neutronsand protons are set at (m~/m~)= 0.8 and(m /m~)= 1, respectively,following Maxwell [24]. We usethe work of TamagakiandTakatsuka[27]


to estimatethe s-waveand p-wave gapsand to convertthem to the respectivesuperfluid transitiontemperaturesT~.All neutrinoprocessesand photoncooling in eq. (12a) are included, with propersuperfluidreductionswherevertheyapply.

(hhb): With superfluidities as in (ha), but the URCA and nucleon pair bremsstrahlungsareneglected.Thatis, superfluidreductionsareincludedin the energyequations,but weusee.q. (12b) fortotal luminosity.In the abovecasespion coolingsarecompletelyneglected.

(III): The sameas (Ila), but pion coolings areadded.Equations(lla)—(lhe) are usedto get variousluminositiesas describedabove(beforeappropriate

superfluidreductionsareapplied).We note that the most importantregionfor mostof stableneutronstarsis the region (II) in table2,

with — iO’4—iO’5 g cm3,wheresuperfluid transition temperaturesareestimatedto be T~= 2x lObo0Kand T~= 8x 108°Kfor protonsand neutrons,respectively.Then whenneutronstarsurfaceradiationis in the X-ray region the internal temperatureof the star is definitely far below the criticaltemperatureof protons,but the effect of neutronsuperfluidity is doubtful. The latter critically dependson the X-ray energies.For instance,if the surfacetemperatureis abouttwo million degreesneutronsarealreadyin a superfluid state,while it is not if Te = 7X iO6°K.

Forsoft X-ray emissionwe cansafely assumethat both the URCA andthe np pair bremsstrahlungprocessesarenegligible as comparedwith someother processeswhich arenot affectedby the protonsuperfluidity,mostly L~(C)andL~(nn).For very soft X-raysor for very low densityneutronstars(with largecrusts)we expectthat L~(C)~‘ L~(nn).Also the neutrino luminositiesL~’,L~(nn)andL~(np)(the modified URCA and the nn and np pair bremsstrahlungluminosities with neutralcurrents) depend •on effective masses of neutrons and protons in the general form of~ where i and j areintegersfrom ito 3, andi+j=4 (seeeqs.(llb)—(ild)). In theabovecalculationsweused0.8and I for the neutronandprotonrelativeeffectivemasses.However,itis quite possiblethat theyare muchlower. At the lower limit of effectivemasses,all threeratesabovewill be wipedout, ascomparedwith L~(C)which is independentof the effectivemasses.So as a caseof minimum luminosity (highesttemperatureat a given age),we set L

1 = L~(C)+ L~’+ Lv (eq. 12b).(We note that the minor processesgrouped as L~t,mostly plasmon neutrinoluminosity, are alsounaffectedby superfluiditiesandeffective masses.)Sinceneglectof superfluidity meansno reductionof total energyas well, it is clearthat the case(hb) will give the maximum possibletemperaturefor agiven model at a given age. Thus the case(Ib) gives the upperboundaryof the cooling curve. Thelower boundaryis given by the case(Ia) whereall possibleneutrinoprocessesareincludedwithoutsuperfluid or magneticeffects.We maypredictthat the case(ha) will give a lower limit thanthe case(ha). We can easilyseethat that shouldnot bethe caseif wenote that the cutoffs of someof neutrinoluminosities(especiallythe URCA) by superfluidprotonsare far moreeffective thanreductionof totalenergydue to superfluidity of neutrons,becauseT~°~ T~.In fig. 7 thesemaximumand minimumtemperaturesare shown by the dashedcurves(Ib) and (ha), respectively,which makeboundariesofthe shadedregionswheretheoreticallypredictedsurfacetemperatureslie.

The curves(ha) for modelA andmodel B areindicatedby the solid curves(A) and(B). They canbe consideredas representativeof best estimatesof cooling curves. We can also regard that thecombined shadedregions of model A and model B give the acceptableregions for surfacetem-peraturesat given ages,whereeffects of equationsof state,H, superfluiditiesand uncertaintiesineffective massesare all takeninto account.As beforethe point wherephoton luminosity overtakesneutrino luminositiesare indicatedby crosses.The stars indicatethe points below which neutrons


becomesuperfluidin thedensity region(II) (seetable2). The upperlimits to thesurfacetemperaturesof neutronstarssetby previousobservationsareindicatedby circles.

In the above curves, pion neutrinocoolings are not included.The dashedcurves(hIl) show themaximumeffect of thepresenceof pion condensateson coolings[22].Here weassumedthat n7~(pionnumberdensity)= n,, (neutronnumberdensity)andeq. (ha)wasused,with p,. = PN 3 x iO’

4g cm3,where p~.is the critical density abovewhich pions appear.Most likely, p,~lies within PN—1OPN.

According to a bestestimatecurrentlyavailable,p~.= ~~2PN (see e.g. [22]).

2.3.5. DiscussionsFromfig. 7 we may notethe following points: (i) thegapbetweentheupper(Ib) and the lower (Ia)

boundariesof acceptabletemperaturesis relatively largefor model A but itis small for model B, intheneutrinocooling regions(the left half); (ii) thedifferencebetweenthe low temperatureasymptoticline for (Ia)—(Ib) with no superfluidityandthat for thecase(ha) with superfluidity is small for model Awhile it is large for model B (the right lower region);(iii) for interestingneutrOnstarcandidatesamongstellarobjectswhich are about 300—20000 yearsold, the surfacetemperat~1reis predictedto be inthevicinity of a few million degreesto nearlyten million degreesin the absetliceof pions, while it canbe below a million degreesin thepresenceof pion condensates;and (iv) in the presenceof nucleonsuperfluidities(solid curves)a staris predictedto berelatively cold (~ 105°K)for radiopulsars(t ~years),unlessadditionalheatingmechanismsareoperating.The lastconclusion(iv) is consistentwithourearlierwork. The first two pointscanbe easilyunderstoodin the following way. We maynotethatthedifferencebetweeen(Ib) (the upperboundary)and (ha) (the lower boundary)is that theURCA andthe neutrino processeswith neutral currentsare neglectedin the case (Ib), and that the majorcontributionto cooling is theURCA processin (ha) while that is the crustbr~msstrahlungin (Ib). Wealso note that the URCA rates are much higher (—100 times at 1090I~)relative to the crustbremsstrahlungin model A, while that is muchsmaller (— comparableat iO~0K)for model B, duetothepresenceof the largeoutercrust in model B (— 5—11 km) while it is very thin (—0.5 km) in modelA. Thentheconclusion(i) is obvious.The point (ii) will follow naturally if wenotethat thesuperfluideffect is small in the later photon cooling stagesfor model A becausethe majorcontribution to thetotal energyfor this modelcomesfrom theneutronenergyof normal(non-sñperfluid)neutronsin thecentralcore(p~ hO’5g cm3) which occupiesa major portion of the star (—6.3km radius)while formodel B nucleonsuperfluiditieseffectively eliminatenucleonenergiesrelativeto the electronenergy(which is muchlessthannucleonenergiesin theabsenceof superfluidity).

In the density region (II) in table 2 (where p = iO’~—io’~g cm3), we assumedthat the criticalsuperfluidtemperaturesof neutronsand protonsare 8 x 108°Kand 2 x 101O0K, respectively.In theregion(I) (>iO’~g cm3),we cansafely saythat neutronsarenot in thesup~rfluidstate.Thestateofprotonsdependson thepartial densityof protonsPp which is model depende~it.In the region(I) it canbe from afew percentto almostcomparablewith theneutrondensity [8,37~.In the formercasetheprotongapwill be ans-wavegapand thecorrespondingcritical temperaturecanbe=2x lObo0K. Thenthe URCA processwill be effectively cut off for model A, too. If the latter applies (p

0 pp), theURCA cooling will dominate(in the absenceof pions), becauseprotonswill be in a normal state.After examiningtheearlierwork on compositions[8,37], we concludedthat for model A, p,, 0.ip,.and thus protons in the region(I) are most likely in the p-wave superfluid state.Thereforewe usedT~= 2x IV°K (with m~/m~= 1). We usedtheseassumptionsto obtainthes~hidcurve(lIa) for modelA (seetable2). We emphasisethat themajordifferencebetweenthesolid cUrve (Ila) of model A and


that of model B comesfrom thepresenceof this non-superfluidneutroncore in thecenterof model Aand theabsenceof suchacore in model B. The possibleimportanceof suchanormalneutroncore atthe center (on cooling problems)was recognisede.g. by Takatsuka[48]but it was quantitativelyconfirmedherefirst by comparingour model A andmodel B.

We calculatedthe case (hlb), too. For both model A and model B, these curves practicallycoincidedwith the curves(Ila) of the respectivemodels(thoughthey showedslight deviationsin theintermediateregion of t 104_106years).The reasonis clearwhen we note that proton superfluidityeffectively kills the URCA processby the time t a year (with Te 107°Kand T, 1O9°K)andthereafter.

The effect of our mostuncertainfactors,the effectivemassesof neutronsandprotons,is takenintoaccountin the following way. In ourcurves(ha), we used(m~/m~)= 0.8 and(m~Im~)= 1. It is mostunlikely that they cango above1 [24].They canbe significantly less than 1 [39,40]so that neutrinoluminosities which depend strongly on effective masses,namely the URCA processand nucleonbremsstrahlungprocessesinvolving neutral currents,will becomenegligible as comparedwith theotherprocesses,most importantly thecrustbremsstrahlung.Thatcasecorrespondsto our curves(hb)whereeq.(12b) was usedfor total luminosity. To seethemaximumpossibleeffect of effectivemasses,our calculationsarerepeatedwith (m~/m,.)= (m~/m~)= 1. Thatincreasesour luminositiesat mostbya factor of 2—3, but its effect on cooling curves was negligible. We may understandthis relativeinsensitivity of temperatureson various factors more easily if we note that surfacetemperaturedependson luminosity as L”4 (seeeq. (3b)), and that the significantreductionof temperaturesin thepresenceof pion condensatesis due to the increaseof luminosity by suchadrasticallylarge factoras— l0~!

To seethe effect of stellar masseswe haveextendedour calculationsto 0.5M®stars.The generaleffect of stellarmassis to decreasetemperatures(atgiven ages)with decreasingmasses.For model Athe effect is practicallythesameas our earlierresultsshown in fig. 3a. For instance,at theageof theCrab neutronstarthe surfacetemperatureis decreasedonly by a factorof 2. The decreaseis evenlessformodel B. This is becausethedensitiesinvolved in thePPS-TImodelof h.3M

0(modelB) andthatof O.5M®aresimilar. At t = 1000yearsthesurfacetemperatureis still abovetwo million degrees.It wasnot significantly decreasedevenwhenthemassis reducedto 0.2M0(alreadyanunrealisticallytoo lowmass).

The possibleeffectof generalrelativity on thermodynamicpropertiesofneutronstarswasstudiedbyVan Riper [49].The model he usedis similar to ourmodel A thoughthemassis somewhatlarger,withM l.4M0, R 7km, and pC = 6.2 x iO~g cm

3.The overall effect was to lower the cooling curves(surfacetemperaturesat given ages)by afactorof --2 in theneutrinocooling regions(thoughit wasincreasedby a factor of --.3 after — abillion years).Sincethis effect dependson theredshiftand thuson M/R, the temperaturedecreasesa little (by a factorof — 1.7) for our model A, while its effect isinsignificant for our model B.

It hasbeenpointedout thatphotonemissivitymaybe decreasedby a factorof — (2—3) or lessin thepresenceof strong magneticfields of H ~ 1012 gauss[50].However,its effect will not be serious,especially for young neutron stars (~hO 000 years), becausegenerally neutrino cooling is stilldominantfortheseobjects(fig. 7). Thepossibleeffectof heavyneutrinosisalsopointedout [51],butweestimatethat that effect alsois not importantdueto theL”4 dependenceof Te mentionedearlier.(See[25]for details.)

Our conclusionis: the most likely valuesof surfacetemperaturesof neutronstarswithout pionsshouldbe in the regionbetweenthesolid curves(A) and (B) in fig. 7. In the neutrinocooling regions


(the left sideof thecrosses)they shouldlie in theshadedregions,thoughthe lower boundariesmay befurther loweredby a factor of — 1.2—1.5 dueto the effectsof general relativistic thermodynamics,stellarmasses,magneticfields, etc. However,it is mostunlikely that the theoreticalcurvescan belowered significantly below — a million degreesboth for the Crab pulsarand the Vela pulsar,unlesspions arepresentin neutronstars.

At the moment we cannotthink of any other ways by which the above conclusion may besignificantly altered, unless possibly if we accepta possibility that these,pulsarsare quark starsinsteadof neutronstars.That is possibleif a phasetransition of neutronmatterto quarkmattertakesplace at densitiesof our currentinterest (--. h014_1016g cm3). According to Baym andotherssuch apossibility is highly remote[52]*4,though very recentlya fewothersexpressØddifferent opinions[53].However,evenif they arerealquarkstars,their generalproperties(suchasequationsof state,stellarmass,radius,density,etc.) seemto besimilar to thoseof neutronstars,accordingto [54],andwe seeno reasonwhy their propertiesrelated to cooling problems, such as lUminosities, etc., shouldsignificantly change.However,it is desirablethat the last statementbe qheckedthrough detailedcalculations.In any case,it may be worthwhile to consider,beforeclosing this section,pion coolingproblemssomewhatmorein detail.

First we point out that the dashedcurves (III) in fig. 7 are meantto give a rough idea of themaximumeffects of the presenceof pions. This is becausewe assumedthat p~,the density abovewhich pions appear,is PN 3 x 10”’ g cm3. While the exact value of p,,. is still uncertain,it is mostunlikely that it is much below ~ The best estimate currently av~ilablegives p,~ 2PN

6x hO’4gcm3 [22].Effects of thepresenceof pionsboth on theequationof stateand on specific heats(andhencethe

energycontent)are not includedin our calculations.Variousauthorsestimatedequationsof stateinthepresenceof pions [8,22]. The bestreview concerningthis matter is foutid in the latestpaperbyBaym and Pethick [8].The generaleffect was to soften theequationof st~teand henceto increasedensityand decreasemass.However,due to variousuncertainties,presentestimatesarepreliminarythough illustrative, as pointed out by the aboveauthors.The effect on the specific heatsmay be todecreasetheenergysomewhat,but on theotherhandsuperfluidgapsof protonsandneutronsbecomenegligible in the presenceof pion condensates[55] and in that sensethe energywill be increased.Inany case,theeffectof theabovefactorson cooling is not serious,ascompatedwith thedrasticeffecton pion neutrinoluminosity [25,55].

Sincethecentraldensityof model B is 4.2 x iO’~g cm3,thereshouldin fa~tbeno pionsin model Bif p.,. is indeed = 2PN. On the other hand for model A, if p.,’. = 2PN, there~should be no significantdeviation from the dashedcurve (III) shown in fig. 7 (becauseits density is much higher, 3.5 x1O’~g cm3). For this curve to be significantly raised(highertemperaturesat given ages),p.,. must behigherthan—2 x iO’~g cm3.In otherwordsdependingon theexactvalueof~p.,,., cooling curvescanbeanywherebetweenthecurves(III) and thecurves(Ila) of given models.

We may note that model A and model B representtwo extremeeffectsof equationsof state,theformerwith thehighestdensityandthe latterwith the lowestdensity(among“realistic” models)whenthestellarmassis fixed at 1.3M

0.Whenthemassis fixed at this value, densItiesof neutronstarswithother “realistic” equationsof statelie betweenthesetwo extremes.If nuclear theoriescan give afairly reliableestimateof p.,’., theexact valueof theobservedsurfacetemperaturemay give us fairlygood estimateof the correctequationof state.(p.,.

2PN may be alreadya fairly reliable estimate.)

*4 However,see theirrecentreview [8] for their latestopinion concerningthis matter.


2.3.6. ComparisonsWe next comparetheaboveresultswith thoseof otherrecentworkers.Malone,in his Ph.D. thesis,

studiedneutron star cooling problems [56].His temperaturesare generally lower than ours. Forinstance,at 1000 yearshis surfacetemperatureis about(1—2) million degrees.The majorreasonseemsto be incompletetreatmentof various neutrinoprocesses(e.g. neglectof suppressionof the URCAprocessdue to protonsuperfluidity), inadequatetreatmentof magneticfields, etc. Maxwell [24]alsoworked on neutron star cooling problems. His temperaturesqualitatively agreewith our model B,though they are somewhatlower than ours. At the age of the Crab neutron star, the surfacetemperatureis estimatedto bein thevicinity of —(2—3) x h06°K(markedas ~ in fig. 7). G. Brown [23]alsomadean analyticestimateof cooling equations.At theageof theCrabneutronstar,his estimategives Te 4.4x 1O”K, abouthalf way betweenour upperand lower limits.

We hada chanceto examinerathercarefullythework by Maxwell [24]andcameto theconclusionthat themajorreasonfor the factthat his estimatedvaluesarecloserto ourmodel B thanmodel A isthat he useda uniformdensity= nucleardensityin his model,which is closeto thecentraldensityofour model B (4.2x iO’~gcm3). He also usedthe Te~Tirelationgiven by us [11]for H = 1012 gauss(the curve (II) in fig. 2 shown in the presentpaper)while we usedH = 5 x 10’~gauss(seethecurve(IV) in fig. 2). That is, our surfacetemperaturesshould be higher than thoseof Maxwell at giveninternal temperatures(which control neutrinocoolings). At a first glance,this small differencein Hmay seeminsignificant. However,examiningfig. 2 carefullywe caneasilyseethat that is not so. Forinstance,at T,= 108°K(correspondingto typical very soft X-ray sources),Te is appreciablyhigherforthe case (IV) (H 5 x 10’~gauss) than for the case (II) (H 1012 gauss).The correspondingTe 1.3x 106°Kfor the former while Te 9x h05°Kfor the latter. More minor reasonsfor thediscrepancyare asfollows. (i) In a neutronstarof densities PN (ourmodel B andMaxwell’s model),the density gradientsnearthe surfaceare more gradual and deviation from a perfectly constantdensitymodel (which Maxwell used)becomesmore serious(than, say,our model A) (see[25]).Theneteffect is in the directionof overestimatingcooling rates. (ii) The reductionof thecrustneutrinobremsstrahlungratesdue to finite nuclearsize effects[25]was neglectedin Maxwell’s work.

Maxwell et al. emphasisedgeneralincreaseof various neutrinoluminositiesin theirnewequationsover earlierones[24,35], especiallyan increaseof thenucleonURCA luminosity by afactorof —50overthat given by BahcallandWolf [5].Sawyeron theotherhandpointedout [57]that this may beanoverestimateconsideringthe difficulty of handlingthe additionaltensortermswhich are includedinthenewURCA rates[35a].The whole questionis currentlyundercareful thoroughre-examinationbyMaxwell andFriman[58],but it is almostcertainthat thatwill not affect in any way our conclusionspresentedin this paper*S.We may notethat thenewpion cooling ratesderivedfor pion condensatesby Brown’s group [22]and usedby us [25]are aboutthreetimes theolder onesderivedby BahcailandWolf for free pions [5]*6, but this, too, did not give significantchangesto our olderpion coolingcurvesshown in [17].The reasoncanbe tracedagainto thevery slow dependenceof temperaturesonluminosities.Sawyerand Soni recentlypointedout the reductionof pion neutrinoluminositiesdue toabsorptionprocesses[59], but this will not affect our conclusionsbecausetheir effect is negligibleafter the first few hours (following the supernovaexplosion)when internal temperaturesbecome~hO9°K.

*STheir revisedpreprint,which cameout in January1979, confirmsthis statement.

*6 We shouldpointout that Kiguchiof Kyoto independentlyreacheda qualitatively similar conclusion[22].


3. Observational aspects

hn this sectionwe considerdetectabilityof neutronstars.By this we meandetectionof radiationemitted from the stellar surface. Therefore we are mainly concernedwith cooling and heatingproblemsof neutron stars.Among heating problems,we may feel that problemsof binary X-raypulsarsand X-rays burstersshould be included. Howeverboth of theseproblemsare currently soimportantthat they may deserveseparatereviews.Also this reportwill becometoo long if they areincluded.Thereforetheseproblemswill be only briefly discussedin section3.5.2.

According to the presenttheoriesof stellar evolution neutronstarsare~formedmost commonlythroughsupernovaexplosions(seee.g. [46,47]). While it is possiblethat the~4endup in different wayscurrenttheoriesseeno problemin forming neutronstarsin this way [47].Without additionalheatingmechanismsstarsjust keepcooling as discussedin detail in earlier sections~The wavelengthAmax atwhich a starradiatesblackbodyradiation most intenselyis relatedto the surfacetemperatureTe as

Amax(CIfl) = hc/(4.965lkTe) O.29h81Te(°K). (13)

When Te is —h05°Kto ,~~hOS0K,thatcomesin theX-rayregion. Thecorrespondingluminosity is foundfrom eq. (3b). It dependson thesizeof the emitting region.For a neutronstarof 10 km radiusit isfrom —1O~to —hOd’ erg sec~in the above temperatureranges. For soft X-rays of —0.3 keY,L h0~~ergsec’ 3L

0, while for typical X-raysof —3 keV, L iO~erg sec’ 3 x 103L

0, whereL0is the solar luminosity. In the following, we comparethesenumberswith the X-rays observedfromvarious supernovaremnantsandpulsars.

If supernovaremnantscontaina pulsar,wecansafely assumethepreseneeof a neutronstar [6,7].Thereforea mostpromisingway to testour theorieswill be to examinepulsars,especiallyvery youngpulsars.We thereforefirst considertheseobjects,namelytheCrabpulsarand theVela pulsar,afterabrief reviewof observationalmeansuseful for ourpresentpurpose.We then~examineothersupernovaremnantsbeforewediscussolderpulsars(~1O~years).This is becausefrom ~urtheoreticalpredictions(e.g. fig. 7) it is most likely that additionalheatingmechanismsarerequiredif surfaceradiationis to bedetectedfrom the latter.

3.1. Observationalmeans

In ordertoidentify surfaceradiationfromaneutronstar,wemustfirst establishthat it isapointsource.Furthermore,it is desirableto seethat it is at leasta thermalradiation.Strictl~’speaking,it shouldbeablackbodyradiation.However,in thepresenceof strongmagneticfields of H ~ 1012 gauss,somedegreeofdistortionoftheblackbody(spectral)shapeis expected[50].Wewishto pointoutherethatundersuchstrongmagneticfields, radiationfrom thestellarsurfacemay be at leastmil~Jlypulsed.Someauthorsdemonstratedthatsurfaceradiationfrom astronglymagnetisedneutronstarcanbeassharpasthat fromanobservedbinaryX-raypulsar[60].Theyassumedthatthestellarsurfaceis i~agaseousstate.Howeverthesurfaceof anisolatedneutronstarwhichwe arehoping to observeis morelikely in a liquid or solidstate*

7.Wenotethat thesurfacetemperatureof suchastaris E 107°K(seeth~following discussionsfordetails).The exactpropertiesof radiationsfrom “magneticmetal”or “magneticliquid” surfacesareyetto befoundout. Accordingto ourpreliminarystudies[61],it isquitepossiblethatsuchradiationsarealso

*lThjs is becauseatransitionfrom a gasto a “magneticliquid—solid phase”mostlikely occursat — l07°K,~sincethe magneticpotential is a fewkeV (34].


pulsed.Becauseof sucha possibility, it is highly desirableto checkperiodicitiesif point sourcesaredetected.

Attemptsto detectthermalradiation directly from the surfacewere carriedout in thepastmainlythrough lunaroccultationexperiments.We expecta suddendrop of intensity whena point sourceisocculted.By selectingonly the off-pulse portions of datafrom theseexperiments,failure to detectsuchdropscould setan upperlimit to the stellarsurfacetemperature[18,19].

The abovemethod, however,is limited to objectswhich happento lie very closeto the eclipticplane,such as the Crab pulsar. Fortunatelywe haveprospectsof identifying various other pointsourcesaswell in very nearfuture,throughthe HEAO-B andotherexperiments.

The HEAO-B, the Second High Energy Astronomical Observatory,launched successfully inNovember1978, is expectedto makepossiblemany advancesin our understandingof high energyphysics [62].It would be impossibleto discusshere all of theseexciting possibilities,and so weconfine our attentionto the part which will be directly related to our currentproblemsof interest.Thereare two instrumentson boardwhich will be especiallyuseful for our presentproblem,(1) thehigh resolutionimager(HRI) and (ii) the imagingproportionalcounter(IPC). The former (HRI) hasanextremelyhigh spatialresolution,with the effective resolutionof a few arc seconds.As it is, thereisno energyresolution.However,through the useof two filters someroughestimatesof spectrumispossible.Thetemporalresolutionis 7.81~s.Sensitivityis 1 countsec~per7 UFU(UHURU counts)(= I

count sec’ per 3 X 10’°erg cm2 sec’), in the interval of 0.1—4keV (Crab spectrum).The latter(IPC) hasa moderatespatial resolutionof a few arc minutes and a moderatespectral resolution.The temporal resolution is 63j.~s.Sensitivity is 1 count sec’ per UFU (= I count seC’ per 4 x

ergcm2sec~)in the interval of 0.1—4keV (Crab spectrum).Further details arefound in [63].In orderto appreciatethehigh spatialresolutionof theHRI, a few arcseconds,we may only note

that the size of the CrabNebula is a few arc minutes in X-rays. Although the HRI hasno energyresolution,thereare variouswaysof estimatingspectrumof point sourcesthroughcombinedusesoftheHRI, IPC,and someof spectrometerson borad, theOGC, BBFS,etc.[20,63,64].

From the energybandsin which theseinstrumentsare effective, it is clear that the HEAO-B isaimed at studiesof soft X-ray sources.As will be shownin detail in subsequentsections,no pointsourceof thermalX-rayshasbeendetectedwith effectivetemperatureshigherthan—5 X 106°K,fromany of isolated neutron star candidates(pulsarsand supernovaremnants).That does not excludepossibilities that the surfaceemits soft X-rays. It hasbeenhard to detectthem becauseluminositydecreasesasx T4 and the flux getstoo weak.However,theHEAO-B will haveasensitivitysomeiO~times greaterthan that availablein thepast.Thereforeit will be ideal for detectingandstudyingverysoft X-ray sources.Also from the very high time resolutionsof the HRI and IPC (microsecondsrange),we cancheckrapidperiodicitiesof orderof pulsarperiods(if present).

In anycase,the first stepwill beto examineall X-ray supernovaremnantsandyoungpulsars(listedin table 4) with the HRI on the HEAO-B. We should also examineisolatedolder pulsarsand radiosupernovaremnants.As will becomeclear shortly (sections3.4 and 3.5.1), the IPC will be moreappropriatefor theseolderobjects.hf no point sourceis detectedfrom anisolatedneutronstar, thatwill give anupperlimit to thesurfacetemperature.How far down that canbelowered dependson theindividual case. With less obscurationof the source and shorter distance, it can be loweredsignificantly. For instance,for theVelapulsarit can be loweredasfar downasto —5 x hO5°K,with theHEAO-B [20].

If we detectpoint sourcesin any of theseobjects,morework is desirable.For instance,it will beextremelyuseful if we canfurtherdetectregularperiodicities like thoseof pulsarsfrom sourcesin

Sachiko Tsuruta, Thermal properties anddelectability of neutron stars —1 263

Table 4Variouscharacteristicsof supernovaremnantsdetectedas X-ray sources,as describedin the text (sections3.2.1, 3.2.2 and 3.3). Notation is

explainedin thetext (section3.3).

Flux F(ergcm2sec’)

Age Distance Radius b Luminosity(years) d(kpc) R(pc) (0) 0.5—2keV 2—IOkeV L,(ergsec’) T(°K)

(a) Crabnebula 924 1.7—2 1.5 —5.8 I.6x I0~’ l.6x i0~~ —

Cas-A 306 3 2 —2.1 9.1 x l0~° 4.5xio~~ {1~<~Tycho 406 3 3.2—6 + 1.4 I.8x 10_b0 7x l0~ { ~

(b) 6 287.8 506 2.5 9—18 3 x l0~’ 8 x l0~

SN 1006 972 1.2 4.4 + 14.5 ~ 2x 10” {(2_4~iO~

RCW 86 1793 14 2 x I0~~ J2.5 x 106(SN 185?) (?) 2.5 1 6 x io~

Puppus-A 5000 1.2—2.2 10—20 —3.4 5.6x it)-’ I.3x 10’° 10” {(1.~3~(o~06

IC 443 6000 1.5—2.5 20 +3 6x 10” 4x 10” — I0~(c) Vela-X 13 000 0.5—1 20—40 —3 7 X l0’ 3 X 10” (2.5—4.3)X 10~

Cygnus loop 20 000 0.8 40 —8.6 9.2 x l0~° 8 x 10” (2—4) X 10~Lupus loop 0.5 20 + 15 2x 10” 10” (l—5)X 106

supernovaremnants(SNR), becausethenwe canbe surethat thereare neutronstarsin theseSNR.As notedbefore,this point canbe checkedwithoutdifficulties by theHRI and IPC. Spectralstudiesarehighly recommended,too, thoughthesemay be morecomplicated.UnØerfavourableconditions(less interstellarabsorption,less circumstelharcomponents,shorterdistanCe,etc.), fairly good esti-matesof spectramaybe possible.Forinstance,if thecircumstellarbackgro4ndis not serious,wemaybe justified to usethe IPC to getadequateestimateof the spectrum.This methodis perfectlyvalid,for instance,for near-byolderpulsarswith age~ i05 yearssince theyhardly haveany circumstellarmatter.It shouldbe adequatefor theVela pulsar,too, atleastasa first step~Evenin anunfavourablecase,we may at leastusethe filter spectrometer(BBFS) with theHRI, to g~ta roughestimateof thespectra[63].The aboveproceduresarealreadyincludedin theexistingpro~~osalsfor theHEAO-B, atleastin ageneralform, andthe resultsareexpectedto comeout within oneto two years.All sourceslisted in theproposals[62]arescheduledto be observedat leastoncewithin the first six monthsafterthe launch[20].

Beyond the HEAO-B, various exciting possibilitiesexist in the immediateandnearfuture,in theUSA, Europe,Japan,etc. Among major spaceprogramsin future which involve large scaleX-rayobservatoriessuchassatellitesandspaceshuttles,we arealreadyawareof theAXAF, the STIX, etc.of theUnitedStates,theCORSAanda seriesof theASTROsatellitesof J~pan,etc.,thoughmanyoftheseare still only at theproposalstages.Amongthese,theAXAF, theAth’ancedX-Ray Astronomi-cal Facilities, may turn out to be mostpromisingfor ourpresentpurpose,~becauseit is expectedtohavea large mirror of — twice thediameterof theHEAO-B mirror. It is h~pedto be launchedin themid 1980’s.

As we notedalready,what is desirablehereare instrumentsof very high sensitivitiesas well as


high spatialandspectralresolutions,in the low energybandof E3 keV. As to the first two points,theHEAO-B is alreadyexcellent.As notedalready,a very high quality spatialresolutionis not criticallyimportantexceptfor very youngsourcessuchasthe Crab(seesection3.4 for furtherdetails.)In anycase,wecanexpectanexcitingoutcomefromthevariousproposalsalreadyexistingandthosestill to beplannedforthis satellite(if it staysup for a few years).hf we cango furtherin thenext seriesof spaceprograms,what is most desirablefor our presentproblem will be instrumentsof better spectralresolutionswhile maintainingspatialresolutionsand sensitivitiesof comparablequalitiesas thoseoftheHEAO-B instruments,in the low energyband of ~3keV. A bestway to achievethis goalmay besimply to increasethe numberof filters on a mirror-HRI type device [65].That canbe done fairlyeasilywith the mirror in the AXAF. In that case,the sensitivity and spatial resolutionalso may besignificantly increased.

We canmaintaina moderatespectralresolutionby usingaproportionalcounter(PC).However,ifwe try to usea PC with a mirror, thespatialresolutionis limited to ~ 1 arcminutes.This is becausetheangularresolutionof a mirror-detectorsystemis restrictedby thegeometryof themirror and thepositional resolutionof thedetector,andthe latterof a PC is not very good. (This is why theangularresolutionof the IPC is ~ 1 arc minutes.)Onepossibleway to overcomethis problemmaybe to useamodulationcollimater (MC). This is becausethe angularresolution of a MC is determinedby thegeometryof the collimater alone, independentof the resolutionsof the detector.In principle, bycombining aMC anda PC,we canachievea moderatespectralresolutionwhile maintaininga spatialresolutionof — severalarcseconds[65].In that case,we canconstructa higher qualitydevice,if weusethe FTT insteadof an ordinaryMC. The VFT, theFourierTransformTelescope,is an improvedversionof a modulationcollimateroriginally inventedby M. Oda[66].It is alsocalled“the multipitchmodulation collimater” [67].This multi-pitch device is plannedto be used in the STIX, but it isdoubtful that the STIX may becomedirectly useful for our presentpurpose.(However, seesection3.5.1.)

If weadopta MC—PC combination,we may just aswell usetheSPCinsteadof aPC. The SPC,theScintillationProportionalCounter,hasa betterspectralresolutionthanan ordinaryPC[68].SincetheSPChasa very poorpositional resolution,this counteris not suitablefor a mirror. Therefore,if wechoosetheSPC as a detector,we haveto dependon a MC (not a mirror) for a spatialresolution.Abestdevice of this kind canbe constructedby combiningtheSPCand theFTF. Eventhoughit is stillin a planning stage,suchan instrumentis expectedto be on board the ASTRO-B. This satellite isprimarily aimed at studiesof harderX-rays (�2keV). However,without difficulties we canmodifysuchan instrumentto suit our studiesin the soft X-ray range.If sucha missionis realised(probablywith a spaceshuttle), it will becomepossibleto study evendetailedstructuresof thermalspectra—

deviationof the continuum from the blackbodyshapedue to strong magneticfields, gravitationalredshiftsof line emissions,etc.

For our presentpurpose(studiesof soft X-ray point sources),a most seriousdisadvantageof amodulationcollimator type instrument(as comparedwith a mirror type device) is the effect of thediffuse background,becausethenoise due to the lattergets very seriousin the soft X-ray band. Ontheotherhand,by theuseof a mirror we candistinguishbetterbetweena pointsourceandthediffusebackground.Whenwe takeinto considerationall of thesevariousfactors,we may concludethat forstudiesof point sources,a mirror typedevice will be generallybetter suited for soft X-rays while aMC type will be betterfor harderX-rays(~3keV).

Theseand othervariousnew observationalapproachesarecurrentlyundertesting stages.So far,we haveconsideredmainly largescaleprogramssuchasthoseinvolving satellitesand spaceshuttles.


However,if our purposeis only spectralstudiesof selectedpoint sourcesalreadydiscovered,it maybe more economicalto userockets. In any case,whenwe think of recentingeneousinventionsanddiscoveries([1, 4, 66], etc.), we feel that it may not be beyondusto achievein very nearfuture ourpresentgoal,— namely,to observeneutronstarsdirectly if their surfacetemperaturesare > — severaltimes 105°K.If we can observationallyprovethat young neutronstars(~a~few times iO~years)arecolderthanthat (Te~ 5 X 105°K),the impactof suchdiscoverieswill befar reaching.If suchradiationsare detected,furtherspectralstudies,etc., asdescribedabove,may giveval~iableinsight into variousimportantproblems.

In the following sections,we elaborateon the abovegeneralstatementsby consideringindividualinterestingobjectswherevariousneutronstar theoriesmay be tested.

3.2. Youngpulsars

3.2.1. The Crab pulsarThe Crabpulsar,NP0531,emits non-thermalradiationin the radio,optical, andX-ray—gammaray

regions. It is superposedonto the non-thermal(synchrotron)radiationfroth the surroundingnebula,which extendsfrom the radio to thegammaray regions.Both (the nebulaand thepulsar)arebelievedto be the remnantsof a supernovaexplosionrecordedin AD 1054. Theirdistanceis estimatedto be

1.7—2 kpc (kiloparsecs).Thenebulaemits —3 x 1O_8ergcm2sec’ in 2—~OkeY. With the distanced = 2 kpc, that correspondsto 1.5x io~~erg sec’. The time averagedpulsar flux is about8% in atypical X-ray band of 1—10keV, but it increasesto 20% at 20 keV, becau$ethe nebulaspectrumissteeperthanthepulsarspectrumin theseenergybands.At higherenergies,~however,the ratio of thepulsar to the nebulaflux staysnearly constant.More information about the Crab is found e.g. in[69,70].

We caneasily seethat observedpulsarradiationis not ordinarysurfaceradiationin the followingways.First of all, we expecthigherintensity if it is from thesurface.Forinstance,in a typical energybandof 1—10keV, we expect— iO~erg sec’,while theobservedpulsedflux is only — 1O~ergsec’.*SMoreover,thespectrumis non-thermal.

The upperlimit to the surfacetemperatureof the Crabneutronstarwasfirst set by theColumbiagroup[18]at 4.7X 106°Kthroughlunaroccultationexperiments.It is markedby thecircle (1) in fig. 7.It lies almostin themiddle of our theoreticallypredictedrangewithout pion coolings.A newerupperlimit setby Toor and Seward[19], Te = 3 X 1O6°K,is marked by the circle (2). It is still within thepossiblerangefor model B. However, it is alreadyoutsidethe possiblerangeset for model A. Wenotethat nucleartheoriesfavour modelA overmodel B. Thereforeif obserivationsas well as theoriesfavour the former morethanthe latter, we may concludethat thealreadyexisting upperlimit to theCrab neutronstar temperaturewill favour the presenceof pions, and i~that senseit is alreadyinteresting.

In fig. 7 thelowestboundaryof the theoreticallyestimatedsurfacetemperatureof theCrabneutronstar (the curve (ha) of model B at t = i03 years)is 2.7X 1O6°Kin the abse~ceof pions. Someminoreffects,suchasthe effectsof stellarmasses,magneticfields and relativistk~thermodynamics,arenotincludedin theaboveestimate.Ourfurtherstudiesshowthat theabovelithit will not go downbelow

~8 Eventhoughemissivitymay bedecreasedunderstrongmagneticfields [50],it will notbe decreasedOsorethanby a factorof --2—3,with

H 5 x 1012 gauss.


two million degreeswhen theseextraeffectsare included. (See [25]for details.) Therefore,if theobservedupperlimit is furtherloweredto — 106°K,that may stronglysupportthepresenceof pions inthis neutronstar (section2.3.5 for further details). As shown in appendix 1, that may be possiblethrough theHEAO-B. Furtherdecreaseto ~ l06°K will be very difficult, however,dueto the strongsynchrotronbackground[20,65].

If an additionalpoint sourceis detectedat the site of thepulsarNP 0532, as very soft X-rays of0.2—0.6keY, we should first see if the observedintensity is consistentwith the surfaceradiation

(including corrections for interstellar absorption,magnetic effects, etc.). Then it is desirable toestimateits spectrum.That may not be very easymainly becauseof relatively strong synchrotronradiationsboth from thepulsarand thenebulawhich continueto theselow energybands.However,somesuccessmay be possiblein nearfuture (seeappendix2). It maybe useful to checkperiodicities,too, and thatcanbe donewith the HRI in theHEAO-B. This is becausethesurfaceradiationmay bepulsed,aspointedoutearlier.In that case,theadditionalpulsedradiationmay be quite different fromthe alreadyobservednon-thermalpulsedradiation, possibly broaderand out of phase,though theperiodwill be morelikely thesame.This is becausethealreadyobservedpulsarradiationmaybe dueto sharpbeamingmechanismsin the magnetosphere[71]or beyond[72],and the direction of thebeamingmaybe quite differentfrom thedirectionof themagneticaxis. Ontheotherhand,thesurfaceradiation, if pulsed,should be pulsedalongthe magneticaxis. This is becausephotonscan escapemoreeasily along themagneticaxis thanperpendicularto it [11,12, 16]. It is alsobecausethepolarregionsmay be heatedby infalling positrons[71]and they canbe hotterthanthe rest of the stellarsurface*9. The resultsof our preliminary studiesof the effect of magneticfields on the mannerofsurfaceradiationwere givenin [73].In similar studiesby someotherauthors[60]thesurfacematterwasassumedto be in agaseousstate,while it is mostlikely that theCrabneutronstarhasa liquid orsolid surface[25].It is needlessto saythat furtherstudiesin this directionaredesirable,andarenow inprogress[61].In any case,if this additionalpulsedcomponentis detectedin the soft X-ray region,comparisonof the two pulsedcomponentsmay give us useful informationabout thepulsarradiationmechanism,aswell as surfacepropertiesof magneticneutronstars.

If this additional point sourceturns out to be due to thermalradiationfrom the surfaceof theneutronstarwith Te— 106°K,the most likely explanationwill be thepresenceof pions in this neutronstar. In that case,the observedvalue of Te may give useful insight into the equationof state,andhencenuclearpotentialsandstronginteractions,as explainedearlier in section2.3.5.

3.2.2. The VelapulsarThe Vela X supernovaremnant,an extendedsource of — 20—40pc (parsecs)is estimatedto be

roughly — lO~yearsold and at a distanceof —500—1000pc. It emits X-rays with peakluminosity of—3 x 10” ergsec’ in the energybandof 0.1—2keV. Seetable4. It fits with theblastwavemodel ofexpandingsupernovashells,in which shockheatedplasmain the interstellarmedium emits thermalradiation[74].A moredetaileddescriptionof this object is founde.g. in [69,70, 75].

The Vela pulsar,PSR0833—45,doesnot lie in the centreof the remnant,but thereis little doubt

*9 Sincetemperaturesnearand on the polar hot spotsmaynot be uniform, that may alsocausea small degreeof distortion of the spectral

shape(from theblackbodyshape),in addition to themagneticdistortion.However, we expectthat suchdistortions arenot serious(at leastitshould be still thermal).(See[25]for details.)


abouttheassociationof thesetwo objects(e.g. theapproximateagreementbetweenthe “X-ray age”for theVela X and the “spin down age” of PSR0833-45[75]).Pulsedemissionshavebeendetectedandconfirmedin the radio,optical and thegammaraybands(of ~ 35 MeY). In theX-ray band,therehavebeenoccasionalandsometimesconflicting reportsof detectionsof a discretesource(sometimespulsed)atthe site of PSR0833—45[69,70, 75]. The presentstatus,however,~ssomewhatconfused.Inany case,noneof thesereportsis conclusive.It is highly desirablethat thispoint will be clarified innear future. Fortunately,the region of the sky where the Vela pulsarlies ~smuch clearerthantheCrabregion,and this goal shouldbe easilyobtainablethroughtheHEAO-B andotherobservationsinthe immediatefuture (seesection3.1).For instance,if blackbodyradiationis emittedfrom a neutronstarwith Te = 106°K,thecorrespondingluminosity is — iO” erg sec’.If thestar is at a distanceof theYela pulsar, we should detectfluxes of — l0’°ergcm2sec’. From our earlier discussions(seesection3.1),it is clearthat this is easilywithin the detectablerangeof the IIEAO-B instrumentsHRIand IPC. In fact, due to the far lessamountof material to block the source,the upper limit to thesurfacetemperaturecanbe loweredasfar downas to —5x l05°K[20]in thecaseof theYelaneutronstar. The aboveconclusionwill not be significantly changedwhen magneticeffectson emissivity,spectralshape,etc.,areincluded(seesection2.3.5 andref. [25]for details).

From fig. 7, we note that 5 x 1050K is far below the lowest temperatureswhich are theoreticallypossiblein theabsenceof pions. If we dependon conventionalphysicswhith we know of, it is mostunlikely that the aboveconclusioncanbe altered(seediscussionsin section2.3.5). Therefore,if wedetectno point sourceas far down asto this thresholdvaluewhich canbe reachedby the HEAO-B,the impact of suchan outcomeon fundamentalphysics (nuclear,particle and high energyphysics)may be beyondour imagination.(Also, see *10•)

Dueto the lack of detectionof X-raysstrongerthan—5x l0~~ergsect,thesurfacetemperatureoftheVela neutronstar is most likely below —5x 106°K.If a point sourceof soft X-rays is observedatthe Vela pulsar site, the outcomemay be equally valuable,but first of all that hasto be confirmed.Unfortunately, earlier reports of detectionsof such a discrete source at PSR0833are not yetconfirmed,andmoreovertheyareconflicting amongthemselvesasto the intensity,natureof radiation(pulsedor not), etc. [70,75]. We may note that the finest spatialresolution~’achievedin theseearlierreportswas — (1/3)°[76],largerthanthe sizeof thenon-thermalX-ray sourdein theCrabnebula,andthat thesize of theYela X is muchlarger,~ 20pc. Therefore,this point(whetherit is in fact a smallerextendedsourceor aneutronstar)canbe easilycheckedby theHRI.

If a point sourceof soft X-rays is detectedand confirmedat the site of the Vela pulsar, furthercarefulwork is highly desirable,— examinationof the spectrum,periodicity~etc.,as describedearlier(sections2.3.5, 3.1, 3.2.1). We shouldpointout that wewill havea muchbetterchanceof doing so andthus deciding whetherthe observedradiation is indeed due to thermal radiation from the stellarsurface,in the presentcase(the Vela pulsar) than the previousone (the Crab pulsar), due to itslocation— closerandclearer(lesscircumstellarand interstellarmatterto obscurethesource),andalsodue to the absenceof a strongnon-thermalX-raypulsarcomponent.Here,we may be ableto usetheHRI and OGS togetherto obtain the spectrumif the observedtemperatureis — l06°Kor higher,butthe IPC also shouldbe adequate,at leastasa first step.

In any case,consideringthe various points notedabove,we should em~hasisebeforeclosingthissectionthat theYela pulsarwill definitelybe far morepromisingthantheCrabpulsar.

~ Sucha low upperlimit will alsoforce into cornerssomeof heatingtheories.Seesections3.4 and3.5.


3.3. Other supernovaremnants

In this sectionwe considersupernovaremnants(SNR) which were identified with galactic X-raysources.Most of them arelisted in table 4 togetherwith the two alreadyconsideredin section3.2.Theyare dividedinto threegroups,(a) theCrab-like,(b) theyoungerand (c) theolderSNR. The group(a) is unique. The group (b) includes relatively young SNR (~twothousand years) whoseidentificationswith historicalsupernovaeventsarefairly reliableor very likely. Thegroup(c) containsolder SNR whose agesare derived from observations(expansionvelocity and size) and supernovatheory.Most of them are identifiedwith radioand optical remnants.Theyare listed in theorder ofincreasingage.The estimatedage,distanced (kpc), radius of the sourceR (pc), latitude b (degree),measuredflux F (ergcm2sec~)in two energy bands(0.5—2 keY) and (2—10keV), estimatedtotalX-ray luminosity L~(erg sec~),and temperatureof theemittingthermalplasmaT(°K),are listed foreachsource(wheneverknown). L~is takenover the (2—10keY) bandfor the youngerremnants(b)andover the (0.1—2 keY) bandfor theoldersources(c). The table waspreparedfrom [69,70, 75], andwe refer to thesereferencesfor furtherdetails.

First we note that they are all relatively closeto us (~3 kpc) and that theyare nearthe galacticplane (with most of them at hi ~6°,though two are nearly at — + 15°).Although they are quiteheterogeneous,the following statementmay be roughly valid; theyoung SNR (b) are hotter(~107°K)and less luminous(— i034—i035ergsec’),while theolderSNR arecooler(~ 10~°K)and more luminous(— 10”— 1036ergsect).We may say that they areall relatively weakX-ray sourcesascomparedwithtypical binary X-ray sources(L~ l0~~ergsec’ orhigher).We generallyobtain betterspectralfits formanyof youngersourcesif we acceptthe two componenttheory.That is, the X-ray emitting bodyconsistsof two components,a hotterplasmaanda coolerplasma.Temperaturesof both componentsaregivenin the last column for sourcesin thegroup(b). Thesegeneralcharacteristicsareconsistentwith the~blastwavemodel in which thermalradiationcomesfrom shockheatedplasmain interstellarmedium.According to this model,X-rays arecausedby outgoingshockwavesin theolder remnants(c), but in theyoungerremnants(b) thehottercomponentis due to theoutgoingshocksandthecoolercomponentis due to the reverseshock[74].Detailedmappingshavebeenmadeavailable for manyofthesesources,and their extendednatureis without question.We caneasilycheckthevalidity of theabovemodel throughfurtherspatialstudiesof theseSNR with the HRI and IPC.

Among thesesupernovaremnants,definite associationswith pulsarsareacceptedonly for two, theVala X—PSR0833pair and the Crab nebula—NP0531 pair. A possibleassociationof IC443 with theradiopulsar PSRO611 was suggested.Their ageconsiderationsmay makethat associationsomewhatuncertain.That is, thepredictedageof 1C443 from observationsand supernovatheory is 6000—13000years,while theageof PSRO611 estimatedfrom its spin-downratesis 90000 years.However,due tothenatureof suchtheories,especially the latterestimate,we suggestthat it may be worthwhile tostudy this source,too. Eventhough the sourceis ratherweak, it is locatedin the region of the skywhere density of X-ray sourcesis relatively low. Careful studies with the HRI and IPC arerecommended.

No pulsarsareso far found in any otherSNR. It is quite possiblethat supernovaeventsendup indifferent ways,as total disruptionswith no compactremnants,formation of black holes insteadofneutronstars,etc.(seee.g. [46]).On theotherhand,we will faceno conflict if mostof theX-raySNRpossesspulsarsand yetpulsarshavebeenso far found in only two of them. First of all, beamingoftheobservedradiopulsarsis so sharpthat we canseethem only whenthey are favourablysituated(so that thepulsarbeamswill enterourfield of view). Secondly,thepulsedradiationis so weak(much

Sachiko Tsuruta, Theimal properties and detectability of neutron stars —1 269

lessthanthe totalenergywhich becomesavailablethroughthe slowing down) that thesourcehastobe very closeto us.For instance,if theCrabpulsarwere locatedas far asat thedistanceof SN 1604(Kepler’s supernova),it would havebeentoo weakto be detected.Note that most of SNR listed intable 4 are further away thanthe Crab and the Vela remnants.Therefore,it may be worthwhile totentatively assumethat there are neutron stars in other supernovaremnants,too, besidesthoseconsideredin section3.2, and examinethem carefully.

Let us considerthosegroupedas(b). First of all, it is clear that themajorportion of theobservedX-raysarenot from the surfaceof neutronstars.The observedintensitiesaretoo low. Theyareonly

iOM~iO3sergsec~in the energyband of (1—10keY), while if they weredue to blackbodyradiationfrom the surface,they should be ~ iO~ergsec~.Theoreticallyestimated:surfacetemperaturesofstarsof this agegroup (—300—2000years) are in the range of —2x 106—107°Kwith correspondingluminositiesof — iO~—iO~ergsec’. Lack of observedX-rays strongerthan —10” ergsec~meansthat Te ~ 3 x 106°K.That makesmodel A highly unlikely unlesspions are present,while model B(without pions) is still consistentwith observations.If the two componentinterpretationis valid inthesesources,the higher energy componentis too hot (~107°K)to be coitisistentwith the surfaceradiationpicture. The coolercomponentis still too strongfor youngerremnantssuchas Cas A andTycho (still ~5x 106°K).As elaboratedin detail in section2.3.5, the aboveconclusionis still validwhen magneticeffects,etc.,are takeninto account.However,the situation is more complicatedforsomeof theoldermembersof this group,especiallyfor SN 1006 andSN 185. For thesetwo remnants,temperaturesof thecoolercomponentsare —2 x 106°Kand —2.5 x l06°K,respectively,with observedluminositiesof order of —2 x l0~~erg sec~.For both of them, we cannotexclude a possibility thatthey arethermalradiationfrom neutronstarsurfaceswith Te 2 X 106°K.At the momentwe cannottell whether theselow temperaturecomponentsare due to the reverse shocks,the stellar surfaceradiationsor somethingelse.However,that canbe checkedeasilywith theHEAO-B. For instance,iftheyarefrom reverseshocks,the emitting region shouldbe extendingover a volume far largerthanthespatialresolutionwhich canbe achievedby theHRI. We may havereasonsto be notvery hopefulaboutCasA and Tycho,sincewe expectmorecircumstellarmaterial aroundyoungerremnantsandsincetheyare more distant thanthe Crab and the Vela remnants(their ga1~cticlatitudesare small,too*fl. Chanceswill be muchbetter for the last two older SNR amongthegroup (b), especiallySN1006. If thermal radiationfrom the neutron star is superimposedonto the radiationfrom the coolercomponentnebulaplasma,both with two million degrees,it may not be too hard to distinguishbetweenthe two components,throughcareful mappingsof the sourceby the HRI. FortunatelySN1006 hashigh galactic latitude (+ 14.5)and it is estimatedto be closerthan the Crab. It is also freefrom sucha strongnon-thermalcomponentasL~ iO~erg sec’ which theCrabneutronstarhastocopewith. Therefore,eventhoughSN 1006 andtheCrabSNRareapproximatelyof thesameage,theupper limit to the surfacetemperatureof the neutronstar in SN 1006 can be lowered significantlybelow a million degrees(which is expectedto be impossiblefor theCrab). tueto thesereasons,SN1006 may turn out to be more interestingthan the Crab pulsar.That is especiallytrue if regularperiodicity is discoveredfor this X-ray source,for thenit will be definitethat it hasaneutronstar. Inany case,it is highly desirableto study this object very carefully, andpay evenmoreattentionthanwhat will be givento the Crabpulsar,especiallyif aperiodicity is detected.

Nextwe considertheolderSNR groupedas(c) in table4. In theabsenceof pions, theorypredictsthe surfacetemperatureto lie at — 106_5x 106°K(seefig. 7). The correspOndingluminosity lies at

~ We may howeverhavea betterchancewith them thanwith theCrab, dueto theabsenceof thestrongsynchrotronbackground.


l033_1036ergsec’. Thesevaluesof temperaturesand intensitiesare roughly comparablewith theobservedvaluesfor this agegroup,thoughthesituationheredependshighly on individualcases.Mostof them werealreadyfound to be extendedsources.From observedfluxes, thesurfacetemperaturesof theneutronstarsshould be ESx 106°Kif they arepresent.If theseremnantspossessneutronstarsradiatingfrom the surfacesat ~5x l06°K,thechanceof distinguishingbetweenthesecomponentsandthe nebulacomponentsmay be better for some sourcessuchas IC443 and Lupus Loop where theobservedfluxesareweaker,while it may be harderfor strongersourcessuchasPuppusA andCygnusLoop. On theotherhand,theyareall very extendedobjects,spreadingover some20—80pcdiameters.Therefore,careful spatial studiesof theseremnantswith the IPC and HRI may be sufficient toidentify point thermal sources.Once such point sourcesare found, the IPC may be sufficient forspectralstudies,asa first stepin any case,becausewe expectlesscircumstellarmatterto obscurethepoint sourcesin theseolder remnants.

Among major historical SNR, Kepler’s supernovawas not includedin theabovelist (table 4). NoX-rays have been detectedfrom this SNR yet. However,this remnantis much further away thanthoselisted in table 4, and it is locatedin the part of the sky which is ratherconfusing when seenthrough the X-rays. Due to the greatly increasedsensitivities,it may be worthwhile to checkthisobject also by the HEAO-B. If point sourcesare detectedfrom any of theseremnants,studiesofpossibleperiodicities,as well as careful spectralstudies,will be very useful asin thepreviouscases(section3.2).

3.4. Radiopulsars— Heatingproblems

Radio pulsars(pulsarswhich emit radio pulsesonly) are estimatedto be ~ l0~yearsold [6,10].From fig. 7, it is clear that they shouldbe colderthan —l05°K,in the presenceof magneticfields of~ 10~~gaussand with superfluidnucleons(that is, we mustexcludethe H = 0 curve and curves(Ia)and (Ib)). Therefore, if we detect thermalradiationsfrGm thesepulsars,it is most likely that someheatingmechanismsareat work [15,77].In fig. 7, theupperlimit to thesurfacetemperatureobtainedthrough theUV observations[78]is shownby thedownwardpointed arrowsmarked(UV). The limitwhich canbe setthroughtheHEAO-B dependson the individual sources.For near-bypulsars,it cango as far down as to —(2—3) X 105°K[64,771.

There are various ways in which pulsarscan be heated.Greensteininvestigatedheating due tofrictions betweensuperfluidneutronsand chargedparticlesin theouter crust [15].According to hisestimates,arotating neutronstarshouldbe kept typically at — severaltimes l05°K.Also, accordingtothe pulsar theory of the Ruderman’sgroup [71],infalling positrons will keepthe polar regionsof apulsarheatedto a certaindegree:the faster it rotatesthe hotterthe polar capswill be. For typicalradiopulsars,theestimatedtemperatureis of theorderof — l05—l06°K[77].

If the surfaceradiationis detectedfrom youngerradio pulsarsof ~ 106 years,both of the abovemechanismsmaybe working. If suchradiationis detectedfrom olderpulsarsof ~ 106 years,that maysupportthe friction theory[15],for thepositronheatingis expectedto be insufficientfor them.On theother hand, if thermal X-rays are not detectedfrom any of all pulsars,with the expectedsurfacetemperatures~(2—3)x l05°K,that maybe embarrassingfor all of thesetheories.Especially,theabovepulsarmodel predictsthat thepolarregionsof theVela pulsarwill beheatedto ~l06°K.Therefore,ifno radiation is detectedfrom this pulsar down to —5 x I o~°K,which canbe easilycheckedby theHEAO-B, theabovepulsartheorymay be in trouble.

Sachiko Tsuruta, Thermal properties and detectability ofneutron stars —1 271

We may note that theseheatingmechanismswill be at work only when the star was originallycolderthan the temperatureto which it canbe heated.We cansafelyassumethat all observedradiopulsarsexcept one are isolated neutron stars (no other detectablecompanionsin the vicinity).Therefore,the IPC is quite adequate.Sincewe alreadyknowthat only a point sourceexistsatthesiteof a radio pulsar,theextremelyhigh resolutionof theHRI is not required.We note that the spectralresolutionis muchbetterfor the IPC thanfor theHRI.

3.5. Othermajorheatingmechanisms— Accretion

In theprevioussection,we consideredpossibleheatingof pulsarsdue to frictions betweenthe twocomponentsin rotatingneutronstarsandaccretionof positronsto thepolarcaps.The formermay becloselyrelatedto otherobservedcharacteristicsof pulsars,suchasglitches,speed-upsandothermoreminor fluctuationsin periodsandfrequencies.A most recentaccountof theseproblemswasgiven byGreenstein[15] and we refer to that paperfor details. The latter problem,that of accretion,is soimportant that it may requirea separatereview. Therefore,we give below only a brief summary.Besidesthese,someauthorsconsidered“noise” [15,79], plasticity [80],etc., as possiblesourcesofheating. Heatingthrough plastic flows will not be importantfor neutronstarswe are consideringinthis report,older than — 100 years,becauseby that time the outerheavyion layersof the starwilldefinitelybein a rigid solid state.“Noise” maykeepolderpulsarsat ~‘105°K.Wereferto [15,77,79]forfurtherdetails*’2.

3.5.1. Accretionof interstellar matterAn isolatedneutronstarmay accreteinterstellarmatterif thepulsar-likeejectionof particlesdoes

not takeplace.The rateof accretionin this caseis expectedto be ~10h1 g sec~’[81].A crudeestimateof themaximumheatingby accretionmaybe madeby assumingthat all theki~ieticenergyreleasedbythe infalling particlesis radiateddirectly as blackbodyradiationfrom theboiUbardedarea.Then,forsphericalaccretion,the surfacemay be keptat somewhatabove—105°K.Fo~a strongly magnetisedneutron star with H ~ 1012 gauss,accretionmay heatup the polar region$ to —106°K,though itdependson the accretionrate, the extentof the bombardedarea,etc. Radiationfrom suchisolatedneutronstarsso far has been too weak to be detected—l029—l033ergsec~without correction forinterstellarabsorptions—intheenergybandof —0.03—0.5keY. However,it maybe within thecapacityof the HEAO-B and other similar programsin nearfuture, to searchfor such objects— isolatedneutronstarswhich are not conventionalradio pulsarswith various reasons.Theremay be neutronstarswhosemagneticfields areintrinsically weakor thosewhich are so old that magneticfields havedissipated.For them,we expectsphericalaccretionand the temperaturesto ~vhichsuchaccretionofinterstellarmatter can raise them will be probablytoo low (less than a few times l0~°K).(Thecorrespondingsurfaceradiationwill be too weak, ~1029ergsec1,exceptpcssibly for someof theclosestones.)However,it is quite possiblethat a stronglymagnetisedneutron~starsomehowdoesnotact asa pulsar,due to its old age, too slow rotation,etc.*’3. In that case,th~starmay be heatedto

l06°Kor somewhathigherthroughaccretionof ordinaryinterstellarmatteralone.If so,many of the

*12 A neutronstarcan beheatedthroughvarious phasetransitions(e.g. solidification of theouterheavyion layers,transitionsto superfluid

states,a pionphase,quark andvarioushyperonstates,etc.). However, theseeffects areall relatively minor [l~, 25].*13 In fact,sometheoriespredict thepresenceof manysuch apparently“dead” neutronstarsall over(seee.g. [82]).

272 Sachiko Tsuruta. Thermal properties and detectability of neutron stars — I

closerstarsshouldbe easilyobservableby detectorson theHEAO-B andsomeothersin nearfuture.(Seesection3.1.)The troubleis, we do notknow wheretheywill be. In this sense,generalsystematicsurveysof theX-raysky by instrumentsof extremelyhigh sensitivitiesin soft X-raywindows,suchastheHRI and IPC on the HEAO-B, may prove to be very fruitful. Someof thesestarsmay be at ornearvery old SNR (suchasNorth Polar Star). Someothers may be at the site of X-ray novae,andtransientX-ray and gammaray sources[77].Their radiation may be pulsed if they are stronglymagnetised,becausein that casethe radiationwill be emittedfrom the polarhot spots.Therefore,ifsuchpoint sourcesof very weak and soft X-rays are detected,it will be useful if we check theirpossibleperiodicitiesandstudy theirspectra.As beforesuchperiodicitiescanbecheckedby theHRIor IPC. If the absenceof conventionalradiopulsarsis due to slow rotationsof thestars,theperiodsmay be substantiallylonger than thoseof conventionalpulsars(>a few seconds).The pulse shapealso may be quite different from thoseof conventionalpulsars(possiblybroaderand more like sinewaves).Wemay bejustified to useinstrumentsof less spatialresolutions(thantheHRI) butof muchbetterspectral resolution(thanthe HRI), e.g. the IPC, a combinationof the FI’T and PC, etc.(seesection3.1),becausetheseobjectsareexpectedto havenegligible amountof circumstellarmatter.

Accordingto someof binary evolution theories(see e.g. [82]),an old radio SNR may not retainaneutronstarremnantbecausethesupernovaexplosionwhich was responsiblefor the extendedradioremnanthasdisruptedthe binary systemand the neutron star hasescapedthe system.However,itmay be worthwhile to include in generalX-ray surveysradio SNR andtheir neighborhoods,becausetheescapingneutronstarsmay be still in the vicinity of theoriginal siteof the binary system.

If soft X-rays are detectedfrom ordinary radio pulsars,most likely they are not due to theaccretionof interstellarmatter,becausemost likely thepulsarmechanismwhich requiresoutflowingplasmafrom thepolarcapswill preventsuchaccretion.In that case,the positronheatingin thepulsarmodel developedby Ruderman’sgroup [71],the frictional heating,noise [15,79], etc., which werediscussedpreviously, may be more likely. Similarly, this heatingmechanismby accretionof inter-stellarmatterwill not be valid for binary X-ray sourcesandyoungerSNR. In the formercasematterfrom thebinary companionand in the latter casethecircumstellarmatterejectedfrom thesupernovaexplosionareexpectedto dominatethephysicalsituation.

3.5.2. Binary X-ray sourcesand burstersAnotherimportant casewhereheatingis dueto accretionis whenthematteris suppliedthroughthe

masslossof thebinary companion.Here,a typical accretionrateis — 1016g sector higher [81].Thestellar surfacein this casecan be heatedto severalmillion degreesfor sphericalaccretionand to

or higher for polar accretionwith strong H [77].Due to the resulting higher temperaturesinvolved, variousinterestingnewphenomenacan takeplace.

First of all, the rateof accumulationof theaccretingmatter in this caseis sufficiently high so thatnuclear fusion becomesimportant [83,84]. Recently, such nuclear fusion suddenly receivedmuchattention in connectionwith the “X-ray bursters”.This is becausethrough accumulationof obser-vational data it hasbecomeincreasinglylikely that unstablenuclearreactionsin neutronstarsareresponsiblefor at leastone typeof the “burster” phenomenon[85].

In ordinary binary X-ray pulsars,the observedX-rays may be emitted nearthe polar capsofstronglymagnetisedrotatingneutronstars[60,86, 87]. However,accordingto theabovepicture of an“X-ray burster”,theobservedX-raysareemittedas“bursts” from thewholesurfaceof a neutronstar(possiblywith weakor no magneticfields). It wasproposedthat unstablenuclearreactionsin neutron


starsmay manifestthemselvesasnovaeand transcientX-ray sourcesaswell [85].A detailedaccountof thesepossibilitiesmaybe given elsewhere[77].

In connectionwith the aboveproblems,aninterestingpoint is the role of heatconductionwithin astar. Conductivity in neutron stars is expectedto be extremely high [11, 16, 50, 88]. From ourpreliminary studies,it seemsthat aneutronstarin abinary systemcanbe heatedsufficiently throughnuclearfusion causedby sphericalaccretionof matter from the companionstar, so that it can emitvery soft thermalX-rays from thewhole stellarsurfacewith Te l05—afewtimes l09°K.This processmay be responsiblefor the steadyweakX-ray sourcesdetectedfrom someof the “X-ray bursters”.ThesituationismorecomplicatedforbinaryX-raypulsars.Herethesurfacetemperaturedependsonhowefficiently heat can be conducteddownward and thus how much fraction of the nuclear energyreleasednear the polar capscan be carried downward,through conduction,to heatthe whole star.Only preliminaryresultsof studiesof the role of heatconductionhasbeengiven in [73],for suchpolaraccretiononto astronglymagnetisedneutronstar. A moredetailedaccountIS: expectedto be given in[61].

Thermalconductivity in the presenceof magneticfields is still the most,unknown factor in theabove problem. However, it is possible that the neutron star in such a binary system is heatedsufficiently throughnuclearfusion so that it will becomea soft X-ray emitter,dependingon differentassumptionswe makefor the effect of magneticfields on conductivity [61].Turning the problemaround,therefore, the lack of detectionof thermalX-rays at very low energy bandsmay give ususeful information on the upper limit to the effect of magnetic fields on thermal conductivity.Therefore, we highly recommendthat binary X-ray pulsarsand X-ray burstersalso be checkedthroughgeneralsky surveysby theHEAO-B, AXAF andotherfuturesatellit~experimentswhich canachievehigh sensitivitiesin very soft X-raywindows.

3.6. Conclusion

Herewe summariseour conclusions.Theseare:(i) According to the “standard” (no pions) cooling theories,predictedsurfacetemperaturesof

youngerneutronstars(of ~ afew times iO~yearsof age)arein the rangeof — 106—l07°K.Includedinthis grouparetheCrabpulsarand theVela pulsar.TheCrabneutronstarmay facevariousproblems(seesection3.2.1),while theVela pulsarwill probablybe themost rewardingone(section3.2.2).It ishighly recommendedthat periodicities be checkedif point sourcesof soft X-rays are detectedinsupernovaremnants(SNR), becausesuchradiationsmay be pulsed(section3.1). If pulsar-likeregularperiodicitiesaredetected,suchSNR within this agerangeshould also be includedin this group, (i).Among these,some,e.g. SN 1006, may turn out to be very promising.The abovetemperaturerangecanbe easilyexamined,fortunately,throughthe HEAO-B, the AXAF, etc., fOr mostof theseobjects.The startingpoint will beto searchfor point sourcesat the site of theseobj~ctsthroughthe HRI ontheHEAO-B. (That is alreadyincludedin theexisting HEAO-B proposals[6~].)If surfaceradiationsare detectedin this range, the exact observedtemperaturesand further sj~ectralstudieswill mostlikely prove to be invaluablefor our understandingof various importantproblems,suchasmagneticand other propertiesof densematter,pulsar radiationmechanisms,as we’l as other fundamentalproblemsin nuclear,particleandhigh energyphysics.We may notethat for~someof oldermembersof this groupsuchasthe Vela pulsar,the IPC on theHEAO-B may be adeq~iatefor further spectralstudies,while for some of youngerones,e.g., the Crab pulsar,we may requiremore sophisticatedinstrumentssuchas acombinationof many filters with a largemirror—HRI t~pedevice.

274 Sachiko Tsuruta, Thermal properties and detectability ofneutron stars — I

If it is observationallyproved that the surfaceof neutronstarsin this agegroup is colder than106°K,the outcome may be equally exciting. In that case,thepion coolingswill bestsupport the

observations,unless someotherequally adequateexplanationcanbe found.The presenceof quarksmay be anotherpossibility, though we feel at the moment that that explanationmay not haveanequallyadequatesuccess(asthepion coolings).

(ii) Older neutronstars(~ i0~years)are theoreticallypredictedto be very cold (E105°K)unlesssomeheating mechanismsare at work. If rotation is not too slow, olderpulsarsmay be heatedto

105—106°Kor a little higherby various mechanisms,— accretionof positions,frictions, “noise”, etc.(seesection3.4). If surfaceradiationsof this degreearedetectedfrom someof theseobjects,that maysupportsomeof heatingtheories.If suchpoint sourcesaredetectedat the sitesof old SNR of this agegroup,searchfor regularperiodicitiesmay be rewardingas before,for detectionof suchperiodicitieswill confirm thepresenceof neutronstars.

Apparently “dead” neutronstarsmay be heatedto luminosities of — 1029_1034ergsec~with thesurfacetemperaturesof —105—106°Kthrough accretionof interstellarmatter.Someclosermembersofthis classmay be detectableby instrumentsof highly increasedsensitivities,suchas those in theHEAO-B, theAXAF, etc.Fortheseobjects(all in thegroup(ii)), the IPC typedeviceshouldbequiteadequatebecausetheeffect of circumstellarmatteris none or negligible.

Acknowledgements

Since I startedwriting this reportmorethana yearago, this paperhasbeengreatlybenefitedfrommy staysat Max-Planck-Institut,Tokyo University, Kyoto University, MontanaState University,AspenCenterfor Physics,andUniversity of Cambridge,andalso from shortervisits to variousotherinstitutions,including NORDITA, SUNY at Stony Brook, HarvardUniversity, ColumbiaUniversity,NagoyaUniversity,andUniversityof Chicago.I wish to thankmy colleaguesin theseinstitutionsfortheir hospitality, help and encouragement.I wish to thank the participantsof the workshopsandconferencesat NORDITA (Copenhagen),Aspen (Colorado), Hachioji (Tokyo), as well as variouscolloquiaandseminarsin manyof theaboveinstitutions,for their stimulatingandhelpful discussions.Thanks are due especially to the following personsfor their assistancein various ways, for thecompletionofthis report,— helpfulsuggestions,constructivecriticisms,hospitality,encouragement,etc.:G.E.Brown,A.Bohr,H. Bethe,0. Maxwell, D. Pines,G. Baym,C. Pethick,W. Weise,P. Gorenstein,F.Seward,R. Giacconi,M. Ruderman,R. Novick, D. Helfand,R. Kippenhahn,G. Börner,J. Trumper,W.Hildebrandt,H.C. Thomas,A. Fabian,M. Rees,K. Makishima,M. Oda,S. Shibazaki,C. Hayashi,S.Ichimaru,S. Hayakawa,H. Sato,K. Sato,N. Itoh, T. Daishido,G. Caughlan,R. Swenson,R. Sawyer,J.Wright, M. Aizenman,D. Schramm,G. Field,R. McCray,etc.,andthosewhosehelpfulcommunicationsareacknowledgedin thereferences.I acknowledgewith thanksspecialfinancialsupportsfrom YamadaFoundation, the National ScienceFoundation (of the USA), Max-Planck-Institut, Kyoto University,Tokyo University,JóchiUniversity,NORDITA, andAspenCenterfor Physics,whichhelpedtheaboveactivities.

Appendix 1

A crude estimatecan be madeas follows [65].In the interval 0.1—3keY, the observednebulasynchrotronradiation strengthis —2 x iO~erg sec~(with d = 2 kpc). Assumethat the radius of the


CrabnebulaX-ray sourceis —2 arcminutesand that it is a uniformdisk. With thespatialresolutionoftheHRI=4 arcseconds,theCrabnebulacan bedivided into — iO~pixels.Therefore,the synchrotronstrengthperonepixel ~ 2 x l0~~ergsec~.If the total L~,from theneutronstarsurfaceis 4 this~ the detectionis impossible (regardlessof the interstellarabsorption).If we assumethat thelimit of the observability(of the surfaceradiation component)is ~ ~ we get L~9(min)4 x 10” ergsec~.This correspondsto Te 10

6°K.

Appendix 2

If a point sourceof thermalX-rays is detectedin a young SNR (supernovaremnants)suchastheCrab,we needan instrumentof high spatial andspectralresolution.TherefOre,an instrumentof theIPC type with a moderatespatial resolution (— a few arc minutes) will not be useful. The MC typedevicewill notbeadequateeither,becausewe expecttheobservedX-rays to be very soft. We havetodependon a device with a largemirror suchasthe HRI in the HEAO-B (see section3.1). A mostpromisingwaymay be to simply usemanyfilters on aHRI type mirror on theAXAF, sinceits mirroris expectedto be very large (seesection3.1). For detailedspectralstudiesof theseyoung objects,especiallythe Crab,we may haveto go beyondthe HEAO-B becauseit hasonly two filters and theHRI—OGCcombinationwill causesourceconfusionswhichwill beparticulaitlyseriousin thepresenceof strongsynchrotronbackground[65].

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Thermal properties and detectability of neutron stars - I cooling and heating of neutron stars