INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS A MATHEMATICAL AND GENERAL

J.Phys.A:Math.Gen.39(2006)4411–4419

JOURNALOFPHYSICSA:MATHEMATICALANDGENERAL

doi:10.1088/0305-4470/39/17/S16

Denseplasmasinastrophysics:fromgiantplanetstoneutronstars

GChabrier1,DSaumon2andAYPotekhin3

123

EcoleNormaleSup´erieuredeLyon,CRAL(UMR5574CNRS),FranceLosAlamosNationalLaboratory,NM87545,USAIoffePhysico-TechnicalInstitute,StPetersburg,Russia

Received14August2005,infinalform22November2005Published7April2006

Onlineatstacks.iop.org/JPhysA/39/4411

Abstract

Webrieflyexaminethepropertiesofthedenseplasmascharacteristicoftheinteriorofgiantplanetsandoftheatmospheresofneutronstars.Specialattentionisdevotedtotheequationofstateofhydrogenandheliumathighdensityandtotheeffectofmagneticfieldsonthepropertiesofdensematter.PACSnumbers:05.70.−a,64.10.+h,96.30.Kf,96.30.Mh,97.60.Jd(Somefiguresinthisarticleareincolouronlyintheelectronicversion)

1.Introduction

Anaccuratedeterminationofthethermodynamicpropertiesofmatterunderextremeconditionsoftemperatureanddensityisrequiredforacorrectdescriptionofthemechanicalandthermalpropertiesofmanydenseastrophysicalbodies,includinggiantplanets,low-massstars(i.e.,starssmallerthantheSun)andso-calledcompactstars(whitedwarfs,browndwarfsandneutronstars).Theseobjectsarecomposeddominantlyofion–electronplasmas,whereionsarestronglycorrelatedandelectronsarestronglyorpartiallydegenerate:theclassicalCoulombparameter󰀄i=(Zie)2/kBTaiislargeandtheelectrondensityparameter󰀇󰀅coupling󰀆1/3

islessthanunity(herea0=h¯2/(mee2)denotestheelectronicBohrradius,rs=aia0Zi

ai=(3/4πni)1/3themeaninter-ionicdistance,Zitheionchargenumber,andnitheionnumberdensity).Thecorrectdescriptionofthestructureandcoolingoftheseastrophysicalbodiesthusrequirestheknowledgeoftheequationofstate(EOS)andthetransportpropertiesofsuchdenseplasmas.Inthisshortreview,wefocusonthetwoextremesofthisrangeofastrophysicalobjectsintermsofmatterdensity:Jovianplanetsandneutronstars.Aswillbeshowninthenextsections,modernexperimentsandobservationsprovidestringentconstraintsonthethermodynamicpropertiesofdensematterunderthephysicalconditionscharacteristicoftheseobjects.

0305-4470/06/174411+09$30.00©2006IOPPublishingLtdPrintedintheUK

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2.TheequationofstateofhydrogenandthestructureofJovianplanets2.1.Hydrogenpressuredissociationandionization

JupiterandSaturnarecomposedofabout70%–97%bymassofhydrogenandhelium.TemperaturesandpressuresrangefromT=170KandT=136KattheP=1barsurface,respectively,atthesurface,toT>8000K,P>10Mbaratthecentre.AtpressuresaroundP∼1–3Mbar,correspondingtoabout80%and60%ofJupiter’sandSaturn’sradius(measuredfromthecentre),respectively,hydrogenundergoesatransitionfromaninsulatingmolecularphasetoaconductingionizedplasma.Thedescriptionofthistransition,describedasthepressureionizationormetallizationofhydrogen,hasremainedachallengingproblemsincethepioneeringworkofWignerandHuntington[1].Muchexperimentalworkhasbeendevotedtothisproblem,buttheresultsremainsomewhatinconclusive.Severalhigh-pressureshockwaveexperimentshavebeenconductedinordertoprobetheEOSofdeuterium,theisotopeofhydrogen,intheregimeofpressureionization.Gasgunshockcompressionexperimentsweregenerallylimitedtopressuresbelow1Mbar[2],probingonlythedomainofmolecularhydrogen.Newtechniquesincludelaser-drivenshockwaveexperiments[3,5],pulse-powercompressionexperiments[6]andconvergentsphericalshockwaveexperiments[7,8]andcanachievepressuresupto5Mbarinfluiddeuteriumathightemperature,exploringforthefirsttimetheregimeofpressuredissociationandionization.TheserecentexperimentsgivedifferentresultsatP󰀁1Mbar,however,andthiscontroversyneedstobesettledbeforearobustcomparisonbetweenexperimentandtheorycanbemadeintheverydomainofhydrogenpressureionization.

Onthetheoreticalfront,alotofefforthasbeendevotedtodescribingthepressureionizationofhydrogen.TheEOScommonlyusedformodellingJovianplanetinteriorsistheSaumon–Chabrier–VanHorn(SCVH)EOS[9–11]whichincludesadetaileddescriptionofthepartialionizationregime.ThisEOSreproducestheHugoniotdataofNellisetal[2]butyieldstemperaturesabout30%higherthanthegasreshockdata,indicatinginsufficientD2dissociation[12].Aslightlyrevisedversion[13]recoversthegasgunreshocktemperaturedataaswellasthelaser-drivenshockwaveresults[3],withamaximumcompressionfactorofρ/ρ0󰀁6,whereρ0=0.17gcm−3istheinitialdensityofliquiddeuteriumat20K.Ontheotherhand,theearlierSESAMEEOS[14],basedonasimilarformalism,predictsasmallercompressionfactor,withρ/ρ0󰀁4,ingeneralagreementwithalltheotherrecentshockwaveexperiments.AbinitioapproachesforthedescriptionofdensehydrogenincludepathintegralMonteCarlo(PIMC)[15–17]andquantummoleculardynamics(QMD)simulations.Thelattercombinemoleculardynamics(MD)anddensityfunctionaltheory(DFT)totakeintoaccountthequantumnatureoftheelectrons[18–21].TherelevanceofearlierMD-DFTcalculationswasquestionedonthebasisthatthesesimulationswereunabletoreproducedatafromgasgunexperiments[18].Thisproblemhasbeensolvedwithmoreaccuratesimulations[19–21].Althoughanabinitioapproachismoresatisfactorythanthephenomenologicalapproachbasedoneffectivepotentials,inpracticethesesimulationsalsorelyonapproximations,suchasthehandlingoftheso-calledsignproblemfortheantisymmetrizationofthefermionwavefunctions,orthecalculationoftheelectronfunctionaldensityitself(inparticulartheexchangeandcorrelationeffects),ortheuseofeffectivepseudo-potentialsofrestrictedvalidity,additiontofinitesizeeffects.Moreover,thesesimulationsaretoocomputationallyintensiveforthecalculationofanEOScoveringseveralordersofmagnitudeindensityandtemperature,asnecessaryforthedescriptionofthestructureandevolutionofastrophysicalbodies.

Figure1comparesexperimentalandtheoreticalHugoniotsintheP–ρandP–Tplanes.Thedisagreementbetweenthelaser-drivenexperimentsandtheothertechniquesisillustratedintheP–ρdiagram.WhereastheSCVHEOSachievesamaximumcompressionsimilartothe

Denseplasmasinastrophysics4413

Figure1.Experimentalshock(P,ρ,T)dataandtheoreticalHugoniotsofdeuterium.Sourcesofdataaregasgunetal[2,12],Zmachine[6],NOVA[3,4]andCSSW[7,23].CurvesshowHugoniotscomputedfromtheEOSsofSCVH[11],SESAME[14],PIMC[15]andMD-DFT[20].

laser-drivendata,alltheothermodelspredictcompressionfactorsintheP–ρplaneinagreementwiththemorerecentdata.TheMD-DFTresults,however,predicttemperaturesforthesecondshocksignificantlylargerthantheexperimentalresults[12].Eventhoughtheexperimentaldouble-shocktemperaturemaybeunderestimatedduetounquantifiedthermalconductionintothewindowuponshockreflection,andthusrepresentsalowerlimitonthereshocktemperatures,thedisagreementintheT–Vplaneissignificant.Asnotedpreviously,thedegreeofmoleculardissociationhasasignificantinfluenceonthethermodynamicpropertiesofthefluidandinsufficientdissociationinthesimulationsmayresultinoverestimatesofthetemperature.IthasbeensuggestedthattheLDA/GGAapproximationsusedinMD-DFTunderestimatethedissociationenergyofD2[22].Thiswouldleadtoevenlessdissociation.ThefactthatcompressionalongtheexperimentalHugoniotremainssmallthussuggestscompensatingeffectsinthecaseofhydrogen.Morerecent,improvedsimulations[21],however,seemtopartlysolvethisdiscrepancyandtoproducereshocktemperaturesinbetteragreementwiththeexperimentalresults.PeakcompressioninthemodernMD-DFTsimulationsoccursinthe∼0.2–0.5Mbarrangearoundadissociationfractionof∼50%.

Thedifferencesinthebehaviourofhydrogenathighdensityandtemperatureillustratedbythevariousresultsdisplayedinfigure1bearimportantconsequencesforthestructureandevolutionofourJovianplanets.Thesedifferencesmustbecorrectlyunderstoodbeforethedescriptionofhydrogenpressuredissociationandionizationstandsonfirmgrounds.AsnotedbyBoriskovetal[23],alltherecentexperimentsagreequitewellintermsoftheshockspeedusversustheparticlevelocityup,almostwithintheirrespectiveerrorbars.Errorbarsanddifferencesin(us,up)areamplifiedinaP–ρdiagrambyafactorof(ρ/ρ0−1).Thesearechallengingexperimentsasthedifferencesseeninpanel1offigure1arisefromdifferencesinusandupoflessthan3%.

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Figure2.AdiabatsforhydrogeninP–ρandP–Tplanes.Thecurveslabelled‘S’and‘J’showtheSCVH-interpolatedEOSadiabatsofSaturnandJupiter,determinedbyT=136KandT=170KatP=1bar,respectively.Thefirstandsecond-shockHugoniotscalculatedwiththeSESAMEEOSareshownbytheheavysolidlinelabelled‘H’.Thelightsolidcurve(right-handscale)showsthedifferencebetweenJupiteradiabatscalculatedwiththeSESAMEEOSrelativetotheSCVH-interpolatedEOS.

2.2.TheinteriorsofJupiterandSaturn

TherapidrotationofJovianplanetsinducesanon-sphericalgravitationalfieldthatcanbeexpandedinLegendrepolynomialsPn(cosθ):

󰀂󰀁󰀉n∞󰀈󰀃ReqGM

V(r,θ)=−JnPn(cosθ),(1)1−

rrn=1whereMandReqdenote,respectively,theplanetmassandequatorialradius,andtheJnare

thegravitationalmoments:

󰀄1

Jn=−r󰀂nPn(cosθ)ρ(r󰀂,θ)d3r󰀂.(2)nMReqVBecauseofnorth-southsymmetry,themomentsofoddorderarenull.Thefirstthree

non-vanishingmoments,J2,J4andJ6,havebeenmeasuredwithhighaccuracyforbothplanetsduringspacecraftfly-bymissions.Combinedwiththeplanetmass,radiusandrotationperiod,theseprovideintegralconstraintsonthedensityprofileoftheplanet,ρ(r,θ),tobecomparedwiththecorrespondingvaluesfromastructuremodelobtainedforaself-gravitatingandrotatingfluidbodyinhydrostaticequilibrium.TheEOSprovidestheP(ρ)relationneededtoclosethesystemofequations.ThestructureoftheH/Heenvelopesofgiantplanetsisfixedbythespecificentropydeterminedfromobservationsattheirsurface.Theveryhighefficiencyofconvectionintheinterioroftheseobjectsleadstonearlyadiabaticinteriorprofiles.ThestructureoftheplanetisthusdeterminedbythechoiceofthehydrogenEOSandtoalesserextentbytheheliumEOSusedinthemodels.AdetailedstudyoftheinfluenceoftheEOSofhydrogenonthestructureandevolutionofJupiterandSaturnhasbeenconductedrecently[24].Fortunately,someshockwaveexperimentsoverlapJupiter’sandSaturn’sadiabats.Figure2displaysJupiter(J)andSaturn(S)adiabatsforhydrogencalculatedwiththeSCVHEOSandthefirstandsecondshockHugoniotscalculatedwiththeSESAMEEOSandillustratestherelativedifferencesindensitybetweenJupiteradiabatscomputedwiththesetwoEOSs.AsdemonstratedbySaumonandGuillot[24],thesmall(󰀃5%)differenceonthe(P,ρ)relationalongtheadiabatbetweenthetwoEOSs,representativeofthetwosetsofexperimentalresults,islargeenoughtoaffectappreciablytheinteriorstructureofthemodels.NotethattheSESAMED2Hugoniotatlowdensityissomewhatstifferthanthegasgunexperiments[2]anddoesnotrecovertheidealD2gasentropyatlowtemperatureanddensity.Nomodel

Denseplasmasinastrophysics4415

ofJupitercouldbeobtainedwiththisEOS[24].AslightlymodifiedSESAMEEOS,whichdoesrecovertheH2entropyatlowtemperatureanddensity,yieldsJupitermodelswithaverysmallcoremass,Mcore∼1M⊕(M⊕isthemassoftheEarth)andamassMZ∼33M⊕ofheavyelements(Z>2)mixedintheH/Heenvelope.TheSCVHEOSyieldsmodelswithMcore=0–4M⊕andMZ∼15–26M⊕.ModelsofSaturnarelesssensitivetotheEOSdifferences,sinceonly∼70%ofitsmassliesatP>1Mbar,comparedto91%forJupiter.ModelscomputedwiththeSCVHandthemodifiedSESAMEEOShaveMcore=10–21M⊕andMZ=1–6M⊕and4–8M⊕,respectively.Asseeninfigure2,thetemperaturealongtheadiabatismoresensitivetothechoiceoftheEOS.Thisaffectsthethermalenergycontentoftheplanetandthusitscoolingrateandevolution.Equationsofstatewhichareadjustedtofitthedeuteriumreshocktemperaturemeasurements[25]leadtomodelsthattake∼3GyrforJupitertocooltoitspresentstate.Evenwhenconsideringuncertaintiesinthemodels,orconsideringthepossibilityofaH/Hephaseseparation,suchashortcoolingageisunlikelytobereconciledwiththeageofthesolarsystem.Thisastrophysicalconstraintsuggeststhatthereshocktemperaturedataaretoolow.

2.3.Heliumequationofstateandtheplasmaphasetransition

Theplanetinteriormodelsarealsoaffected,toalesserextent,bytheuncertaintiesoftheheliumEOS.AmodelEOSforheliumathighdensity,coveringtheregimeofpressureionization,hasbeendevelopedrecentlybyWinisdoerfferandChabrier[26].ThisEOS,basedoneffectiveinteractionpotentialsbetweenHe,He+,He++ande−species,reproducesadequatelyexperimentalHugoniotandsoundspeedmeasurementsupto∼1Mbar.Inthismodel,pressureionizationispredictedtooccurdirectlyfromHetoHe++.Becauseoftheuncertaintiesinthetreatmentoftheinteractionsathighdensity,however,thepredictedionizationdensityrangesfromafewto∼10gcm−3.Comparisonofthemodelpredictionswithavailablemeasurementsofelectricalconductivityofheliumathighdensity[27,28]isunderway.

Thepressureionizationandmetallizationofhydrogenhavebeenpredictedtooccurthroughafirst-orderphasetransition,theso-calledplasmaphasetransition(PPT)[1,29–31,10,42].NearlyallofthesePPTcalculationsarebasedonchemicalEOSmodels.SuchmodelsarebasedonaHelmholtzfreeenergythatincludescontributionsfrom(1)neutralparticles(atomsandmolecules),(2)afullyionizedplasmaand(3)usuallyacouplingbetweenthetwo.Itiswellknownthatrealisticfullyionizedplasmamodelsbecomethermodynamicallyunstableatlowtemperaturesandmoderatedensities.ThisisanalogoustothebehaviourofexpandedmetalsatT=0thatdisplayaregionwheredP/dρ<0andevenP<0[32].Thisbehaviourofthefullyionizedplasmamodelisformallyafirst-orderphasetransitionandreflectstheformationofboundstatesintherealsystem.Inotherwords,thechemicalmodelshaveafirst-orderphasetransitionbuiltinfromtheonset,andthisphasetransitioncoincides,notsurprisingly,withtheregimeofpressureionization.ThisrepresentsacommonflawinthistypeofmodelsanditfollowsthattheirpredictionofaPPTinhydrogenisnotcredible.Thisisfurthersupportedbyadetailedstudyoftwoofthesemodels[10,42].Ontheotherhand,recentabinitiosimulationsfindasharp(6±2%)volumediscontinuityatconstantpressure[21,33]ordP/dT<0atconstantvolume[43–45],afeatureconsistentwiththeexistenceofafirst-orderphasetransition.Atthesametime,thepaircorrelationfunctionexhibitsadrasticchangefromamoleculartoanatomicstatewithametalliccharacter(finitedensityofelectronicstatesattheFermilevel).Thesetransitionsarefoundtooccurinthe∼0.5–1.25Mbarand∼1500–3000Ktemperaturerange.Whiletheseresultsaresuggestive,asystematicexplorationofthispartofthephasediagramremainstobedone.Notethata

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first-orderstructuraltransitionforH2atT=0ispredictedtooccuratapressureP󰀁4.0Mbar,fromDFTcalculationsbasedonexactexchangecalculations[22].ThereissofarnopublishedexperimentalevidenceforthePPTbutitcannotyetberuledout.Giventhedifficultyofmodellingthisregionofthephasediagramofhydrogen,onlyexperimentscanultimatelyestablishwhetheraPPTexistsornot.

3.Densematterinstrongmagneticfields:neutronstarstructureandcooling

Neutronstars(NS)consistofacoreofnucleonssurroundedbyanenvelopeofnucleiandelectronsformingaCoulombplasma.Coolingratesofthesestarsaredeterminedbytheheatcapacityandneutrinoemissionprocessesintheircoresandbyheattransportintheenvelopes.Fortheneutrinoemission,mostimportantaretheso-calleddirectUrca(Durca)processes(beta-decayandbeta-capture)andmodifiedUrca(Murca)processes(thesamebutwithparticipationofanadditionalnucleon,whichhelpstofulfilmomentumconservation).TheMurcaprocessesarelessefficient,buttheyworkineverysufficientlyhotNS.Incontrast,themostefficientDurcaprocessesoperateonlyiftheprotonfractioninthecoreislargeenough(otherwisethemomentumconservationconditionfordegeneratenucleonscannotbesatisfied).SomemodelsofnuclearmatterpredictthataNSwithrelativelyhighmassshouldhaveasufficientprotonfractionatthestellarcentrefortheDurcaprocessestooccur.Suchstarsshouldcoolfaster,whichopensapossibilityoftestingtheEOSofsuperdensematterthroughobservations.Thecoolingratesarealsostronglyaffectedbynucleonsuperfluidity(see[34]forreviewandreferences).

MostNSshavemagneticfieldsB∼1011–1013G,whereassome(so-calledmagnetars)arethoughttohavefieldsashighas∼1014–1015G.ThephotosphereofaNSischaracterizedbytemperaturesTs󰀁105–107K(dependingontheagetandmassMofthestar)anddensitiesρ󰀁10−2–104gcm−3(dependingonTandB).TraditionallytheNScrustisassumedtobecomposedofiron.However,theouterlayers,includingtheatmosphere,canbecomposedoflightelements(H,He,C)accretedontopoftheironlayer.ThereforethedeterminationofthetemperatureprofilesandemittedspectraofNSsrequiresanaccuratedescriptionoftheformationofboundstatesandpressureionizationoftheseelementsinastrongmagneticfield.

Thequantum-mechanicalpropertiesoffreechargedparticlesandboundspecies(hydrogenatomsandmolecules)arestronglymodifiedbythemagneticfield,whichtherebyaffectsthethermodynamicpropertiesoftheplasma[35,36].ThetransversemotionofelectronsinamagneticfieldisquantizedintoLandaulevels.TheenergyofthenthLandaulevelofthe√2

¯ωcninthenon-electron(withouttherestenergy)ismec(1+2bn−1),whichbecomesh

¯eB/mec=11.577B12keV,istheelectroncyclotronenergy,relativisticlimit,whereh¯ωc=h

2

b=h¯ωc/mec=B12/44.14isthefieldstrengthintherelativisticunits,andB12=B/(1012G)isatypicalmagnetic-fieldscaleforNSconditions.Theatomicunitforthemagnetic-field

¯e)×(e2/a0)=2.35109G.Itisconvenientstrengthissetbyh¯ωc=e2/a0,i.e.,B0=(mec/h󰀇×2

todefineadimensionlessmagnetic-fieldstrengthγ=B/B0=bαf,whereαfisthefinestructureconstant.

Forγ󰀎1,asencounteredinNSs,theground-stateatomicandmolecularbindingenergiesincreaseas∼ln2γ.TheHatominastrongmagneticfieldiscompressedinthetransversedirectionstotheradius∼am,where

am=(h¯c/eB)1/2=γ−1/2a0=2.56×10−10B12

−1/2

cm(3)

isthequantummagneticlength,whichbecomesthenaturallengthunit.Theincreaseofbinding

energiesanddecreaseofsizesleadtoasignificantincreaseofthefractionofnon-ionizedatomsintheplasmaatthephotosphericdensities(whicharehigherforstrongermagnetic

Denseplasmasinastrophysics4417

Figure3.Effectivesurfacetemperature(asseenbyadistantobserver,Ts∞)versusNSagetforassumedNSmassM=1.3and1.5solarmasses.ThedotswitherrorbarsshowestimatesofNSagesandeffectivetemperaturesfromvariousobservations;thedotswitharrowsindicateobservationalupperlimits.Left:coolingofNSswithdifferentrelativemasses󰀅M/Mofaccreted(H–He–C)matter(valuesoflog󰀅M/Mareindicatednearthecurves).Solidcurvesrefertonon-accreted(Fe)ironenvelopeofthestar.Right:coolingofNSswithironenvelopefordifferentmagneticfieldstrengths(logBinGauss).

fields).Forexample,atT=106KandB=1013G,thetypicaldensityisρ∼1gcm−3,andthereare>1%ofatomsintheHatmosphere.Becauseofthealignmentoftheelectronspinsantiparalleltothefield,twoatomsintheirgroundstate(m=0)donotbindtogether,becauseofthePauliexclusionprinciple.OneofthetwoHatomshastobeexcitedinthem=−1statetoformthegroundstateoftheH2molecule[36].Anotherimportanteffectisthatthermalmotionofatomsacrossthefieldstronglymodifiestheirbindingenergiesandradiativetransitionrates.Asshownin[37–39],theallowanceforpartialionizationandthermalmotioniscrucialforneutron-staratmospheremodelling.

3/2

AslongasT󰀍h¯ωc/kB=1.343×108B12Kandρ󰀍ρB≈7.1×103B12gcm−3,theelectroncyclotronenergyh¯ωcexceedsboththethermalenergykBTandtheelectronFermienergykBTF,sothatthefieldisstronglyquantizing(e.g.,[35]).Inthiscase,typicalfortheNSphotospheres,theelectronspinsarealignedantiparalleltothefield.TheelectronFermienergydecreases;thereforetheonsetofdegeneracyisshiftedtohigherdensities(slightlybelowρB).Protonmotionisalsoquantizedbythemagneticfield,butthecorrespondingcyclotronenergy

¯ωcme/mp.ismuchsmaller,h¯ωcp=h

Amodelwhichdescribesthethermodynamicsofaninteracting(H2,H,H+,e−)plasmainastrongmagneticfieldwasconstructedin[37].Onthebaseofthismodel,theEOSformagnetizedHatmospheresofNSs,aswellastheiropacities,wereexploredandtabulatedin[38,39].

LandauquantizationofelectronorbitsaffectsnotonlytheEOSandtheradiativeopacities,butalsotheheatconductioninthesurfacelayers(see[35]andreferencestherein).TheEOSofstronglymagnetized,partiallyionizedhydrogenplasmaaswellastheelectronconductivitiesandradiativeopacitiesinneutronstarmagnetizedenvelopeswereusedin[40]tocalculatethethermalstructureandcoolingofsuperfluidNSswithaccretedenvelopesinthepresenceofstrongdipolemagneticfields.In[40](seealso[41]andreferencestherein),theeffectof

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neutronsuperfluidityintheNSinnercrustwasalsoexamined.Theaccountoftheeffectsofaccretedmatter,magneticfieldandneutronsuperfluidityalterstheNScoolingsignificantly.

Figure3displaystheoreticalcoolingcurvesofNSswith(lowercurves,M=1.5M󰀊)orwithout(uppercurves,M=1.3M󰀊)Durcaprocessesinthecore,withorwithoutaccretedenvelopes,andwithmagneticfieldofdifferentstrengths,comparedtotheestimatesoftheeffectivetemperatureobtainedfromobservations(see[34]forreferences).Asseeninthefigure,thepresenceofalight-element(accreted)envelopeincreasesTsattheearlycoolingstage(t󰀂105yr),andasaresultthethermalenergybecomesexhaustedsooner.Themagnetar-likemagneticfieldB󰀁1014Gactsinasimilarway,whereasaweakerfieldalmostdoesnotaffectthecooling.

Forsimplicity,infigure3weneglecttheeffectsofsuperfluidity.Theirdiscussioncanbefoundin[34,40,41].4.Conclusions

Inthisbriefreview,weconsideredthedescriptionofthethermodynamicpropertiesofdenseCoulombmatterintwospecificastrophysicalcontexts,Jovianplanetsandneutronstars.Thedescriptionofthepressureionizationofhydrogenandotherelements,aswellasthepresenceofstrongmagneticfields,playsanimportantroleindeterminingthemechanicalandthermalpropertiesandtheevolutionoftheseobjects.Modelsincludingthesecomplexeffectscansuccessfullyexplainavarietyofobservations.Ontheotherhand,modernexperimentsand/orobservationscanenableustodiscriminatebetweenvariousEOSmodelsinplanetinteriorsandleadtoabetterdeterminationofmassesofaccretedenvelopes,surfacemagneticfieldsandeventuallytheEOSofsuperdensematterinneutronstars.Acknowledgments

WethankDGYakovlevforprovidinguswithhiscompilationofobservationaldataandperformingcoolingcalculationswithourphysicsinputforfigure3.TheworkofGCwaspartiallysupportedbytheCNRSFrench–RussiangrantPICS3202.TheworkofAYPwaspartiallysupportedbytheRLSSgrant1115.2003.2andtheRFBRgrants05-02-16245,03-07-90200and05-02-22003.TheworkofDSwassupportedinpartbytheUnitedStatesDepartmentofEnergyundercontractW-7405-ENG-36.References

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