RadiativeHeatingandCooling incircumstellarenvelopes Dipl.-Phys.PeterWoitke ausberlin von zurerlangungdesakademischengrades dertechnischenuniversitatberlin VomFachbereich04(Physik) DoktorderNaturwissenschaften(Dr.rer.nat.) genehmigtedissertation Berlin1997 D83

Promotionsausschu Vorsitzender:Prof.Dr.D.Zimmermann Berichter: TagdermundlichenPrufung:18.06.1997 Prof.Dr.E.Sedlmayr Priv.Doz.Dr.J.P.Kaufmann

HatmaninnereRuhe,dannentstehtGelassenheit. Weiman,womaninnehaltenmu,entstehtgeistigeFestigkeit. HatmanGelassenheit,dannentstehtbesonnenesNachdenken. GibtesgeistigeFestigkeit,dannentstehtinnereRuhe. GibtesbesonnenesNachdenken,sokommtdasGelingen. (Konfuzius,BuchderRiten)

Zusammenfassung KleineFestkorperpartikelvoneinerGroebiszuetwa1mbildenaufdenersten Blickeineunbedeutende,eherstorendeKomponentederMaterieinunseremKosmos.Genauerbetrachtetkommtdiesen"Staubteilchen\jedocheinegrundsatzliche Bedeutungzu.AufgrundihrergroenWirkungsquerschnittefurdieWechselwirkung mitverantwortlichfurdenkosmischenkreislaufunddiechemischeevolutionder scheneigenschaftendesgasesinderinterstellarenmaterieundsindohnezweifel Universums.Siebeeinussenwesentlichdiedynamischen,thermischenundchemi- mitlichtpragensieinganzentscheidenderweisedaserscheinungsbilddesheutigen Materie.MankannohneUbertreibungsagen,daesohnedieExistenzderStaubteilchenwederdieErde,nochdenMenschen,javielleichtnichteinmaldieSonnmodynamischeBedingungen.NebenhohenDichtensindinsbesondereniedrige,aber gebenwurde. DieBildungdieserStaubteilchenausderGasphaseerfordertrelativspezischether- nichtzuniedrigetemperaturenunterhalbdersublimationstemperaturdesbetrachtetenfestkorpermaterialserforderlich.diesevoraussetzungistabsolutzwingend. demzufolgegeltendiemassivenwindedieserobjektealshauptproduktionsstatten soliegendiesevorallemindenzirkumstellarenhullenvonkuhlenriesensternenvor; FragtmannachderExistenzsolcherBedingungeninastrophysikalischenObjekten, desstaubesimuniversum.beiriesensternenmiteektivtemperaturenunterhalb vonetwa3000kistdermechanismusderstaubbildungunddesmassenverlustes nichtzuletztdurchdiearbeitenderberlinerarbeitsgruppevonprof.dr.sedlmayr durchabsorptions{undstreuprozessedenimpulsdesstrahlungsfeldesteilweise MolekulzumFestkorperzuermoglichen.DieentstehendenStaubteilchennehmen erreichtdasgastemperaturen,dieniedriggenugsind,umdenphasenubergangvom hinreichendverstanden:beigenugendgroenradialenabstandenvomstern aufundgebendiesendurchstoeandasgasweiter.dieserimpulseintragtreibt denstellarenwind. NebendiesenSternengibteseineReihevonweiterenstaubbildendenObjekten. InsbesondereexistierteinezahlenmaigeherunbedeutendeKlassevonRCoronae deckenkonnen,sodadieserfurdasbloeaugefurmonateoderjahrevomhimmel zurbildungvonriesigenstaubwolken,diedengesamtensternvorubergehendver- diesenobjektenkommtesinunvorhersagbarenzeitlichenabstandenimmerwieder Borealis(RCB)Sternen,diesichnichtrechtindasobigeBildeinordnenlassen.Bei zuverschwindenscheint1.beobachtungenlegennahe,daderstaubbildungsproze undnovae{explosionenvor. 1Ahnliche,wennauchnichtderartspektakulareBeobachtungenliegenfurWolf{Rayet{Sterne i

ii beidiesensternenineinerentfernungvonnureinigenwenigensternradienstattndenmu,obwohldiercb{sterneeektivtemperaturenvonetwa7000kbesitzen, diealsoheierundvielleuchtkraftigeralsdiesonnesind.dievorliegendearbeit ZUSAMMENFASSUNG nimmtdiesebeobachtungsergebnisseernst. dungindernahediesersternemitdenublichentheorienerklaren?setztmandie durchdiercb{sterneaufeineharteprobegestelltwerden:latsichdiestaubbilfernungenvomsternsehrhohetemperaturen,sodadiestaubbildungstheorien GangigeMethodenzurTemperaturbestimmungergebeninsogeringenradialenEnt- KannesinderNahevonheienSternenzuthermodynamischenBedingungenkom- Temperaturenalserwartetherrschen. GultigkeitderTheorienvoraus,somussenentwederdieBeobachtungenfalschsein, men,diestaubbildungsprozessezulassen?angeregtdurchdiesefragestellungunter- oderesmussenindernahediesersterne zumindestzeitweilig vielniedrigere suchtdievorliegendearbeitdenthermischenzustanddunnergaseunterdemein- uvonstellarenstrahlungsfeldern.eshandeltsichhierbeizunachstumallgemeine (nichtrcb{spezische),grundlegendestudien.einemethodezurzeitabhangigen TemperaturbestimmungvonGaseninzirkumstellarenHullenwirdentwickelt,die ModellrechnungeninzukunftigenArbeitenverwendetwerdenkann. vonvornhereinsokonzipiertist,dasiealselementarerbestandteilvonkomplexeren DasthermodynamischeKonzeptdieserMethodeberuhtaufeinernon{LTEBeschreibungdesGases,inderjedocheineGeschichtsabhangigkeitderKonzentrationendenetischesGleichgewicht(\steadystate")vorausgesetzt.Eswirdgezeigt,dadiesgangevonAtomenundeinfachgeladenenIonen,Vibrations{undRotationsuber- DiefolgendenradiativeProzessewerdenindieserArbeitberucksichtigt:Linienuber- vonh2,gebunden{frei{ubergangevonatomenausdemgrundzustandund(im gangevonpolarendiatomischenbzw.linearenmolekulen,quadrupol{ubergange MolekuleundderBesetzungsdichtenvernachlassigtwird.Stattdessenwirdeinki- AnnahmeeinegewohnlichethermodynamischeBeschreibungdesGaseszulat. sorptionsprozesseprozeitaufnimmtbzw.durchemissionsprozesseverliert.diera- diativenheiz{undkuhlratenbildensomitdiegrundlagezurthermodynamischen ModellierungdesGases. radiativenheiz{undkuhlraten,d.h.diewarmemengen,diedasgasdurchabziationsprozesseundfrei{frei{ubergange.dieseprozesseergebenindersummedie FallevonWassersto)ausangeregtenelektronischenNiveaus,fernerPhotodisso- dasichdieradiativenheiz{undkuhlratenausgleichen(strahlungsgleichgewicht). renhullenvonrcb{sternenbestimmt.diesezustandezeichnensichdadurchaus, ZunachstwerdendiestabilenGleichgewichtszustandedesGasesindenzirkumstella- DreiAnwendungenderentwickeltenMethodewerdenvorgestellt: Eswirdjedochfestgestellt,dadasStrahlungsgleichgewichteinezwarnotwendige, Gasesdarstellt.UntergegebenenDruck{undStrahlungsfeldbedingungenkonnen mehrerelosungenexistieren,d.h.eineraumlichekoexistenzvonheien,atomaren abernichthinreichendebedingungzurberechnungdesthermischenzustanddes

Phasennebenkalten,molekularenPhasenerscheintprinzipiellmoglich("thermische ZUSAMMENFASSUNG iii Bifurkationen\). DerRelaxationsprozedesGaseszumStrahlungsgleichgewichtwirdindenzirkum- StowellenureinhinreichenddichtesGasinderLageist,denStrahlungsgleichgewichtszustandnacheinigerZeitwiederzuerreichen.BeiTeilchendichten<108cm tendesgaseshinterstowellendiskutiert,diedurcheinepulsationdeszentralensternsverursachtwerden.esergibtsich,danachderpassageeinersolchestellarenhullenvonc{sternenuntersucht.hierbeiwirdinsbesonderedasverhal- SchlielichwirddaszeitabhangigethermischeVerhaltendesGasesindenzirkum- diesersternhullenverliert. StrahlungsgleichgewichtesihrebestimmendeBedeutungfurdieTemperaturstruktur verhaltsichdasgaszunehmendadiabatisch,sodaschlielichdiebedingungdes einembestimmtendichtebereichkanndabeidasgasdurcheinen2{stufen{proze, riodischesituationstudiert,inderdasgasindernahedessternsfortlaufenddurch Stowellenerhitztundkomprimiertwird,undinderZwischenzeitreexpandiert.In stellarenhullenvonpulsierendenrcb{sternengenaueruntersucht.eswirdeinepe- bestehendausradiativerkuhlunggefolgtvonadiabatischerexpansion,temperaturenerreichen,dieweitunterhalbderstrahlungsgleichgewichtstemperaturliegen. SchonbeiradialenAbstandenvonetwa1:5 3RtretenhierbeizeitweiligTemperaturenunterhalbvon1500Kauf,abhangigvonderStowellengeschwindigkeit.Diese DievorliegendeArbeitenthaltsomitgrundlegendeErkenntnisseuberdasthermodynamischeVerhaltenderGaseinzirkumstellarenHullen.Neue,alternativeWege VerdunklungsereignissedieserObjekteauslostwerdenkonnten. NahederRCB{SternedurchStowellenverursachtwird,wodurchdiespektakularen ArbeitstelltdaherdieHypotheseauf,dadieKondensationvonRuteilcheninder zurstaubbildungwerdenaufgezeigt.

iv

Abstract presentincircumstellarenvelopes. radiationelds.onthebasisofasteady{statenon{ltedescription,theradiative Thisthesisinvestigatesthethermalstateofdilutedgasesbeingexposedtostellar Thefollowingradiativeprocessesareexamined:linetransitionsofneutralandsingly ionizedatoms,vibrationalandrotationaltransitionsofpolardiatomicandlinear heatingandcoolingratesofthegasaredetermined,consideringthetypicaldensities Athermodynamicdescriptionofthegasisdevelopedwhichallowsforatime{ photodissociationandfree{freetransitions. theelectronicgroundstatesand(incaseofhydrogen)fromexcitedelectroniclevels, molecules,respectively,quadrupoletransitionsofh2,bound{freetransitionsfrom dependentdeterminationofthetemperaturestructureinthecircumstellarenvelopes stellarenvelopesofrcoronaeborealis(rcb)stars.itisfoundthatthecondition modelcalculations.threeapplicationsofthisdescriptionarepresented: First,thestableradiativeequilibriumstatesofthegasarecalculatedforthecircum- ofcoolandwarmstarsandcanbeincludedintomorecomplex,e.g.hydrodynamic, andradiationeld.thus,aspatialcoexistenceofhot,atomicandcool,molecular gas.morethanonetemperaturesolutionmayexistforxedconditionsofpressure ofradiativeequilibriumisnotsucientinordertodeterminethetemperatureofthe Second,therelaxationprocesstowardsradiativeequilibriumisstudiedinthecircumstellarenvelopesofC{stars.Thecharacterofthethermalrelaxationbehind propagatingshockwaves,whicharecausedbyapulsationofthecentralstar,is phasesisprincipallyconceivable(\thermalbifurcations"). discussed.itisfoundthatthegasmustbesucientlydenseinordertobecapable toreestablishradiativeequilibriumafterthepassageofsuchshocks.fordensities nallytheconditionofradiativeequilibriumloosesitssignicanceconcerningthe determinationofthetemperaturestructure. <108cm 3,thebehaviorofthegasbecomesmoreandmoreadiabatic,sothat Third,thetime{dependentbehaviorofthegasinthecircumstellarenvelopesof thegasisfoundtoundergoatwo{stepcoolingprocess,consistingofradiativecoolingathightemperaturesfollowedbyadiabaticexpansionatlowtemperatures.in atmospheresisdeveloped,wherethegasisperiodicallyheatedandcompressedby shockwavesandre{expandsbetweentheshocks.withinadistinctdensityinterval pulsatingrcbstarsisinvestigatedmoredetailed.amodelforshocklevitated atradialdistancesassmallas1:5 3R,despiteofthehigheectivetemperaturesof peraturesbelow1500k(farbelowthevaluesexpectedfromradiativeequilibrium) thiscaseaconsiderablesupercoolingofthegasoccurs,temporarilyproducingtem- closetorcbstarsiscausedbyshockwaves,whichmighttriggerthespectacular thesestars.thus,thisthesisstatesthehypothesisthattheonsetofdustformation declineevents. v

Contents Zusammenfassung Abstract vi ListofFigures ListofSymbols xiv x 1Introduction ListofTables xv 1.1The ModelsofDustFormation:::::::::::::::::::::::: Importance of for 1.2CriticalCommentsontheUsualMethodofTemperatureDetermination2 1.3ThePuzzleofDustFormationaroundRCBStars::::::::::: 1 1.5AimandStructureofthisWork:::::::::::::::::::::13 1.4RequirementsforDierentApproachesofTemperatureDetermination11 6 2TheThermodynamicConcept 2.1FirstLawofThermodynamicsandEquationofState:::::::::17 2.2LTEandNon{LTE::::::::::::::::::::::::::::18 17 3RadiativeHeatingandCooling 2.3SteadyState::::::::::::::::::::::::::::::::20 3.1Bound{BoundTransitions::::::::::::::::::::::::24 3.1.1EscapeProbabilityMethodforanN{Level{Systemwithout 23 3.1.1.2DiscussionoftheApplicabilityofSobolevTheory::28 3.1.1.1NumericalIterationScheme::::::::::::::27 Continuum::::::::::::::::::::::::::::24 3.1.3RotationalTransitionsofLinearPolarMolecules::::::::34 3.1.2LinesofAtomsandIons:::::::::::::::::::::32 3.1.1.3AnExemplaryTwo{Level{Atom:::::::::::29 vi

3.1.4VibrationalTransitionsofDiatomicPolarMolecules::::::38 3.1.3.2Fast,ApproximateMethod::::::::::::::36 3.1.3.1RotationalHeatingandCoolingbyCO::::::::36 3.1.4.1VibrationalHeatingandCoolingbyCO:::::::40 3.2Bound{FreeTransitions:::::::::::::::::::::::::44 3.1.5QuadrupoleTransitionsofH2::::::::::::::::::42 3.1.4.2Fast,ApproximateMethod::::::::::::::41 3.2.1TheRateEquationsforanN{LevelSystemwithContinuum:44 3.2.3OtherNeutralAtoms:::::::::::::::::::::::52 3.2.2TheH{Atom:::::::::::::::::::::::::::48 3.2.1.1Fast,ApproximateMethod::::::::::::::46 3.3PhotodissociationandRadiativeAssociation::::::::::::::53 3.4Free{FreeTransitions:::::::::::::::::::::::::::58 3.5OverviewoftheConsideredRadiativeProcesses::::::::::::59 3.3.1TheH Heating/CoolingRate::::::::::::::::::55 4TheCalculationoftheEquationofState 3.6FurtherHeatingandCoolingProcesses:::::::::::::::::61 4.2CalculationoftheInternalEnergy::::::::::::::::::::67 4.1CalculationoftheParticleConcentrations:::::::::::::::65 5ThermalBifurcationsintheCircumstellarEnvelopesofRCBStars71 5.1TheModel:::::::::::::::::::::::::::::::::72 5.1.1DenitionoftheRadiativeEquilibriumGasTemperature:::72 5.2Results:::::::::::::::::::::::::::::::::::73 5.1.3ApproximationoftheRadiationField::::::::::::::73 5.1.2ElementAbundances:::::::::::::::::::::::72 5.2.1DegreeofIonization:::::::::::::::::::::::73 5.2.4RadiativeEquilibriumTemperatureSolutions:::::::::77 5.2.3RadiativeHeatingandCoolingRates::::::::::::::75 5.2.2Chemistry:::::::::::::::::::::::::::::74 6RadiativeCoolingTimeScalesintheCircumstellarEnvelopesof 5.3Discussion:::::::::::::::::::::::::::::::::83 C{Stars 6.1TheModel:::::::::::::::::::::::::::::::::86 85 vii

6.1.3ApproximationoftheRadiationField::::::::::::::87 6.1.2ElementAbundances:::::::::::::::::::::::87 6.1.1DenitionoftheRadiativeCoolingTimeScale:::::::::86 6.2Results:::::::::::::::::::::::::::::::::::88 6.2.1CompositionoftheGas:::::::::::::::::::::88 6.1.4LocalVelocityGradient:::::::::::::::::::::87 6.2.2InternalEnergy::::::::::::::::::::::::::90 6.2.3TheRadiativeCoolingTimeScaleandtheRoleoftheVarious 6.2.4DependenceontheRadiationField:::::::::::::::94 6.2.5DependenceontheVelocityGradient::::::::::::::94 HeatingandCoolingProcesses:::::::::::::::::91 6.2.6ComparisontoAnalyticalHeating/CoolingFunctions:::::94 6.2.6.3ResultsoftheComparison:::::::::::::::95 6.2.6.2LTEHeating/CoolingFunction::::::::::::95 6.2.6.1Bowen'sHeating/CoolingFunction::::::::::95 7Shock{InducedCondensationaroundRCBStars 6.3Discussion:::::::::::::::::::::::::::::::::99 6.2.7TheTransitionfromIsothermaltoAdiabaticShocks:::::98 7.1TheModel:AFixed,PeriodicallyShockedFluidElementinaConstantRadiationField:::::::::::::::::::::::::::101 7.1.2Re{ExpansionPhases:::::::::::::::::::::::104 7.1.1ShockTransitions:::::::::::::::::::::::::103 7.1.3Thermodynamics:::::::::::::::::::::::::105 7.1.4TheModelingProcedure:::::::::::::::::::::106 7.2Results:::::::::::::::::::::::::::::::::::109 7.1.6ExaminedRangeofParameters:::::::::::::::::107 7.1.5OverviewofIntroducedParameters:::::::::::::::107 7.2.1CyclicVariationsinthePeriodicallyShockedFluidElements:109 7.2.4PreconditionsforCarbonNucleation::::::::::::::115 7.2.3DependenceonShockVelocity::::::::::::::::::115 7.2.2DependenceonDensity::::::::::::::::::::::113 7.3Discussion:::::::::::::::::::::::::::::::::119 7.3.1AdvantagesoftheModel:::::::::::::::::::::119 7.2.5DependenceonRadialDistance:::::::::::::::::118 7.3.3InterpretationsofObservationswithRegardtotheModel::121 7.3.2Criticism::::::::::::::::::::::::::::::120 viii

ACurrentStatusofRCBResearch 8Conclusions 127 125 A.1GeneralObservations:::::::::::::::::::::::::::127 A.2ObservationsDuringtheDeclineEvents::::::::::::::::129 A.3Models:::::::::::::::::::::::::::::::::::132 A.3.1HistoricalModels:::::::::::::::::::::::::132 A.3.2ModelCalculations::::::::::::::::::::::::132 References A.3.3EmpiricalModels:::::::::::::::::::::::::134 Danksagung 136 Lebenslauf 145 146 ix

ListofSymbols symboldescription Aul Einsteincoecientforspontaneousemission unit page Blu BulB frequencyintegratedplanckfunction rotationalconstant ergs 1cm 2 B Einsteincoecientforabsorption Einsteincoecientforstimulatedemission Planckfunction ergs 1cm 2Hz 1 erg 1cm2s 1 35 3 Cul D0mol Ea Clu ratecoecientforcollisionalexcitation ratecoecientforcollisionalde{excitation activationenergyofachemicalreaction 25 54 2 Eel Eion Etrans Ediss Erot totaldissociationpotentialenergy totalelectronicexcitationenergy totalionicpotentialenergy Evib totalrotationalexcitationenergy Eul totaltranslationalenergy totalvibrationalexcitationenergy energydierencebetweenupperandlowerstate 67 fg Iinc J() freeenthalpyofformationatstandardpressure incidentcontinuousintensityfromdirection p ergs 1cm 2Hz 1str 1 54 Jcont J meanspectralintensity frequencyintegratedmeanintensity 25 ul rotationalquantumnumber continuousmeanintensityatlinecenter 35 23 Jul J lineaveragedcontinuousmeanintensity ergs 1cm 2Hz 1 ul formingpervolumeandpersecond nucleationrate,i.e.thenumberofseedparticles solidangle cm 3s 1 115 temperature{independentratecoecientforcollisionalde{excitation cm3s 1 32 epe simpliedmeanescapeprobability totalnetradiativeheatingfunction ergs 1cm 3 { 25 26 meanescapeprobability Qbf bqrad bqdust ofdustgrains totalnetradiativeheatingratepermass totalnetheatingratepermassduetopresence ergs 1g 1 Qrad netradiativeheatingfunctionduetobound{free netradiativeheatingfunctionduetofree{free 22 transitions ergs 1cm 3 45 x 58

Qvib Qchem symboldescription Qrot rotationalnetheatingfunction vibrationalnetheatingfunction unit page Rrad chemicalreaction netradiativeheatingfunctionofaphoto{ ergs 1cm 3 37 Rij stellarradius 41 S totalratecoecientfortransitioni!j 54 SLul supersaturationratioofthegaswithrespectto linesourcefunction graphite ergs 1cm 2Hz 1 { 115 24 254 Si(Tg) TM Sahafunctionofleveli temperature cm 3 Trot vibrationaltransitionmoment [cgs] 45 Tbb rotationalexcitationtemperature vibrationalexcitationtemperature blackbodytemperature 36 392 Te Tg Trad TRE eectivestellartemperature uniquekinetictemperatureofthegas radiationtemperature 17 4 VWg radiativeequilibriumtemperature specicvolume1= cm3g 1 K 22 ZII Zrot dilutionfactor partitionfunctionofansinglyionizedatom 45 Zvib amu rotationalpartitionfunction vibrationalpartitionfunction atomicmassunitmc12=12 g 36 3 i bi i(tg) atm standardatmosphericpressure1:013106 photo{recombinationcoecienttoleveli departurecoecientfromltebi=ni=ni dyncm 2 cm 3s 1 39 c ratecoecientforcollisionalionizationfromlevel i cm3s 1 46 e speedoflight cms 1 50 45 jul El internalenergyofthegas elementabundancebynumber gasemissioncoecient ergs 1cm 3Hz 1str 1 ergg 1,18 25 frot gl;gu dimensionlessmeanintensityatul 66 ul numberofrotationaldegreesoffreedom statisticalweightsoflowerandupperlevel ratecoecientforcollisionalde{excitation cm3s 1 { 26 67 322 hhhiabs hhiem Planck'sconstant enthalpypermassunith=e+p= h=(2) ergg 1 ergs 25 k meanabsorbedphotonenergy 70 kf meanemittedphotonenergy 39 kr Boltzmannconstant ratecoecientofaforwardchemicalreaction ergk 1 47 B ratecoecientofareversechemicalreaction gasabsorptioncoecient depends 54 J mel Planckmeanabsorptioncoecient intensitymeanabsorptioncoecient wavelength massofneutralatomofelementel cm 1 3 xi g 662

symboldescription me mred;i electronmass unit page D eredspeciesandcollisionpartneri reducedmassforcollisionsbetweentheconsid- cos g{ 45 n dipolemomentofamolecule [cgs] ni nl particledensityinlte,chemicalequilibrium 25 nu populationofleveli 35 levelpopulationofthelowerlevel 50 nel levelpopulationoftheupperlevel 24 n<h> at II totalneutralatomparticledensity singlyionizedatomparticledensity 25 n<he> totaldensityofh-nucleiinallionic,atomicand molecularformsh=(pelelmel)) 66 ncr totalheliumparticledensityinatomicorionized 29 nthick population criticaldensity(n<h>{value)forthermal 74 ne eects criticaldensity(n<h>{value)foropticaldepths electrondensity 31 ithr nmol nsp totalparticledensityofmoleculemol totalparticledensityofonespecies cm 3 32 ul 36! thresholdfrequencyforphotoionization!mol linecenterfrequency eigenfrequencyoftheharmonicoscillator 45 39 25 2 pj j-thvibrationaleigenfrequencyofamolecule gaspressure Hz 67 ul(;) p psat prolefunctionoftheconsideredtransition standardpressure vaporpressureofneutralatomsoverthebulk material dyncm 2 Hz 1 115 22 54 sel r radialdistancetothecenterofthestar massdensityofthegas gcm 3 25 f() mol mentel stoichiometriccoecientofmoleculemolforele- StefanBoltzmannconstant ergcm 2K 4 66 4 0 bf i() photodissociationcrosssection totalcrosssectionforrotationalde{excitation bound{freeabsorptioncrosssectionfromleveli cm2 54 35 45 3 cool es ul meansobolevopticaldepth radiativeheating/coolingtimescale s anglebetweentheconsideredrayandtheradial characteristic direction 26 transitions temperature of vibrational K 25 86 v1 dv dl vibrationalquantumnumber hydrodynamicgasvelocity terminalwindvelocity localmeanvelocitygradient kms 1 cms 1 { 39 25 xii 28 26

ListofFigures 1.1Possibleradiativeequilibriumtemperaturesoverdilutionfactorina 1.3TemperaturestructureinhydrodynamicmodelsusingLTE{cooling:12 1.2SketchofanRCrBdeclineevent.:::::::::::::::::::: Planck{typeradiationeld:::::::::::::::::::::::: 75 1.4Temperaturestructureinhydrodynamicmodelsusing2{cooling:::12 3.2Temperaturedependenceofthelinecoolingrate::::::::::::30 3.1Thecoolingratepermassofanexemplarytwo{level{atom::::::30 2.1Thepoolsanduxesofenergyinthegas::::::::::::::::19 3.5VibrationalcoolingrateandexcitationtemperatureofCO::::::40 3.4RotationalcoolingrateandexcitationtemperatureofCO:::::::37 3.3Dependencyofthelinecoolingrateontheradiationeld:::::::30 3.6ThequadrupolecoolingrateofH2::::::::::::::::::::43 3.8Thebound{freeplusbound{boundcoolingrateofhydrogeninthe 3.7Thebound{freeplusbound{boundcoolingrateofhydrogeninthe casewithcontinuousradiationeld:::::::::::::::::::49 casewithoutcontinuousradiationeld:::::::::::::::::49 3.10Thebound{freeandfree{freecoolingratesofH withoutcontinuous 3.9Detailsofthehydrogencoolingrate::::::::::::::::::51 3.11Thebound{freeandfree{freecoolingratesofH withcontinuous 5.1ElementabundancesofRCoronaeBorealis:::::::::::::::72 3.12Overviewoftheconsideredheatingandcoolingprocesses:::::::60 radiationeld:::::::::::::::::::::::::::::::57 5.2Heating/coolingratesasfunctionofthegastemperature:::::::77 5.3ThermalbifurcationsinRCBenvelopesforp=102and100dyncm 2 5.4ThermalbifurcationsinRCBenvelopesforp=10 2and10 4dyncm 280 5.5ThermalbifurcationsinRCBenvelopesforp=10 6and10 8dyncm 881 79 6.1Thecomposition,theinternalenergyandthenetheatingfunctionof thegasasfunctionoftemperatureanddensity:::::::::::::89 xiii

6.4RadiativecoolingtimescalesforC{starenvelopesinthecaseJ= 6.3MostecientcoolingprocessreferringtoFig.6.2:::::::::::92 6.2RadiativecoolingtimescalesforC{starenvelopesforthecaseJ=0:92 6.6Radiativecoolingtimescalescalculatedfromtheanalyticalheating/coolingfunctionproposedbyBowen(1988)::::::::::::96 6.5MostecientcoolingprocessreferringtoFig.6.4:::::::::::93 B(3000K)::::::::::::::::::::::::::::::::93 7.1Ballistictrajectoriesofxeduidelementsintheenvelopeofapulsatingstar:::::::::::::::::::::::::::::::::102 6.7RadiativecoolingtimescalecalculatedfromLTE:::::::::::96 7.2Schematicdescriptionofthethermodynamicprocessesoccurringina 7.3Timevariationsofthethermodynamicstatevariablesinaxed,periodicallyshocked,circumstellaruidelement:::::::::::::110 xeduidelementofthecseofapulsatingstar:::::::::::106 7.4Detailsoftheperiodictimevariations::::::::::::::::::112 7.5Cyclicvariationsofdensityandtemperatureinxeduidelements:113 7.6Minimumgastemperaturesandthepossibilityofcarbonnucleation tooccurataradialdistanceofr=2r:::::::::::::::::117 xiv

ListofTables 1.1Observationalandtheoreticalconstraintsonthenucleationdistance 3.2Vibrationalandrotationalheatingandcooling:consideredspecies 3.1Atomiclineheatingandcooling:consideredspeciesandtransitions:33 inrcbenvelopes:::::::::::::::::::::::::::::10 3.3Bound{freeheatingandcooling:consideredspeciesandatomicdata:53 3.4Overviewoffurtherheatingandcoolingprocesses:::::::::::62 andmoleculardata::::::::::::::::::::::::::::35 5.1AbundantmoleculesinthecircumstellarenvelopesofRCBstars:::74 5.2Importantheating/coolingprocessesforRCBabundances:::::::76 4.1Moleculardataforthedeterminationoftheinternalenergy::::::69 7.1Resultsoftheshock{inducedcondensationmodelasfunctionofradial distanceandshockvelocity::::::::::::::::::::::::118 xv

xvi

Chapter1 Introduction 1.1 ModelsofDustFormation TheImportanceofTemperatureDeterminationfor Comparedtotheusualorganizationalformsofmatterinspacelikestarsandthe interstellarmedium(ism),thereareremarkableandexceptionalthermodynamic starissucientlycoolthatitsradiationelddoesnotionizethesurroundingcse, tomassloss stillhigherbyordersofmagnitudethanintheism.ifthecentral byordersofmagnitudethanintheinteriorandtheatmospheresofstars,but due conditionsinextendedcircumstellarenvelopes(cses).here,thedensitiesarelower energybarriersduringtheirformation. temperaturescanexist,whichareononehandlowenoughtoensurethestability ofcomplexchemicalstructures,butontheotherhandhighenoughtobridgethe Therefore,thecircumstellarenvelopesofcoolstarsareacosmiclaboratory,where space.thehighdensitiescombinedwithlow,butnottoolow,temperaturesprovide largeamountsofcomplexchemicalandphysicalprocessescanoccur.theseprocessesareoffundamentalimportancefortheevolutionofmatter,especiallyfor almostidealpreconditionsforcondensationprocesses. thetransitionfrommoleculestodustgrains,i.e.theprimaryformationofsolidsin nomenalikenovaeandsupernovae,shockwavesinthemostdensepartsoftheism BesidestheCSEsofcoolstarsonlyafewclassesofastrophysicalobjectsareknown (probablyconnectedwithstarformation)orcometimpactsonplanets.therefore, whichshowsimilarthermodynamicconditions.thesearetherareexplosivepheistenceofdustparticlesisofgreatestimportancefortheappearanceofthepresent particles(dustgrains)inspace.theseparticlesarecarriedintotheismbystellar windsandnallycanbeobservedeverywhereintheuniverse.thisubiquitousex- thecsesofcoolstarsaresupposedtobethemainproductionsitesofsmallsolid nallytotheformationofseedanddustparticles,showsuchastrongtemperature Thechemicalreactions,whichsuccessivelyleadtoincreasingcomplexityingasesand lastbutnotleast,alsofortheexistenceoflife,includingmankind. universe,forthecirculationofmatter,fortheformationofstarsandplanetsand, smalltemperaturewindowofafewhundreddegreesbelowthesublimationtemper- dependencethatevenslighttemperaturedeviationscanchangetheformationrates byordersofmagnitude.therefore,eectivenucleationisgenerallyrestrictedtoa 1

atureofthesolidmaterial,whichistobeconsidered1.thusitisimmediatelyclear 2 CHAPTER1.INTRODUCTION ontheproperdeterminationofthetemperatureinthemedium.questionableor thattheresultsoftheoreticalmodelcalculationsofdustformationcriticallydepend insucientmethodsfortemperaturedeterminationcaneasilyinducesevereerrors intheresultsofsuchcalculations. Forthemodelingandtheunderstandingofdustformationfromthegas Howprofoundisourknowledgeofthetruethermodynamicstateofthegases(especiallytheirtemperature)inastrophysicalobjects?Inexceptionalcasesadirect thegasisabsolutelyrequired. phase,themostpreciseinformationaboutthethermodynamicstateof determinationofthetemperaturestraticationfromobservationsoftheobjects mightbepossible.however,theoreticalmethodsareusuallyrequiredwhichmay stillsuerfromlargeintrinsicuncertainties(seenextsection).astonishingly,most expenditureforthetheoreticaltemperaturedeterminationoftenseemstobeinadequatecomparedtothedetailedtreatmentofthechemistryanddustformation modelcalculationsconcerningdustformationincoolstellarenvelopesuserather simpleandnotveryreliablemethodsfortemperaturedetermination.atleast,the Thesequestionsgetevenmoreimportant,whenthegaselementsaresubjectto processesinsuchinvestigations. Howreliablearethoseresults?Inthiswaynewpotentialsites(e.g.closetohot dynamicprocesseswhichdirectlyaecttheinternalenergyofthegas.forinstance, inshockwaves,intheheatingbymagneto{acousticwavesorduringfastexpansions accompaniedbyadiabaticcooling.whichtemperaturedoesoneuseinsuchcases? 1.2 stars)fordustformationnotpreviouslyconsideredmightbediscovered. ThetemperaturestructureinextendedCSEsisusuallycalculatedbymeansof CriticalCommentsontheUsualMethodofTemperatureDetermination thesolutionofaradiativetransfer.assumingthatsolelyradiativeprocessesare equilibrium(re),wherethetotalamountofabsorbedradiativeenergyislocally balancedbythetotalamountofemittedradiativeenergyeverywhereintheenvelope. importantfortheheatingandcoolingofthegas,thegaswillrelaxtoradiative ThemeaningofthephysicalquantitiesisexplainedintheListofSymbolsonpagex. Incaseoflocalthermodynamicequilibrium(LTE),theemissivitycanbeeliminated RE: 4ZJd=4Zd (1.1) bymeansofkirchho'slaw=b(t): 1Thisstatementreferstoboththe\classical"andkinetictreatmentoftheproblem. LTE: ZJd=ZB(TRE)d (1.2)

1.2THEUSUALMETHODOFTEMPERATUREDETERMINATION Inordertoarriveatashortnotation,appropriatemeansoftheabsorptioncoecient 3 thetemperatureofthegasinradiativeequilibriumcanbeexpressedby (JandB)canbedenedsuchthatEq.(1.2)simpliestoJJ=BB(TRE)and AsfarastheassumptionsofREandLTEareappropriate,Eq.(1.3)togetherwith TRE4=J asolutionoftheradiativetransferdeterminesthepropertemperaturestructurein BJ: cases,wherethegasissucientlycoolformoleculeformation.consideringthe ofthekeyproblemsinastrophysicsandaverydiculttask,especiallyinthose calculationofthegasabsorptioncoecients.thiscalculation,however,isone theconsideredastrophysicalobject.theuncertaintyofthismethodlieswithinthe andbound{freetransitionsintheuvmaybeimportant.inprinciple,allthese rangingfromthenearirtothemicro{wavespectralregion.furthermoreelectronic gaswithasolarelementalcomposition,overabillionoflinetransitionsareknown H2Omolecule,forexample,whichisoneofthemostabundantspeciesinacool Therefore,duetothelackofknowledgeoftheexactfrequencydependencyof, shiftsduetohydrodynamicvelocitiesandselfshieldingcomeintoplay. questionsconcerningtheindividuallineprolesandtheeectscausedbydoppler transitionmustbetakenintoaccountforapropercalculationof.additional thesimplifyingassumptionj=bisoftenmade.inthiscasewehavej=b,which givestheblackbodytemperature grey: T4 const.j=bishenceforthcalledthe\quasi{greyassumption".inthefollowing, Alternatively,Eq.(1.4)canbederiveddirectlyfromEq.(1.2)byassuming= bb=j: theintrinsicuncertaintyofthisapproximationfortheresultingtemperaturestructureisexplored.first,eq.(1.4)obviouslyhasalwaysexactlyonetemperaturesolutionforagivenj,whereaseq.(1.3)mayhavetwoormorestablesolutions,because thefractionj=bcanbetemperature{dependentitself.thereby,thequasi{grey assumptionignoresthepossibilityofthermalbifurcations,whichwewillencounter matedbyconsideringtheextremecaseofa{function laterinthiswork. Themaximumeectcausedbytruefrequency{dependentabsorptioncanbeesti- radiativeprocesswhichdominatestheheatingandcoolingofthegasandwhichhas Thiscaseisnotasarticialasitmightrstappear,becausethereisoftenonespecial =0( 0): (1.5) acertaincharacteristicwavelength.weconsidertheeectsinadilutedplanckeld oftype J=WB(Trad); (1.6)

whichisausefulapproximationoftheradiationeldifthegaselementmainly 4 CHAPTER1.INTRODUCTION receiveslightfromadistantblackbodysource.inasphericallysymmetric,optically thincsewithtrad=tewendthedilutionfactortobe Allain(1996)hasshownthatevenforanopticallythickCSEEq.(1.6)stillprovides W(r)=120@1 s1 R2 areasonablettotheresultsoffrequency{dependentradiativetransfercalculations r21a: (1.7) (Winters1994).Ingeneral,theparameterTradissmallerthanTeandWislarger comparedtoeq.(1.7).fortheassumedwendwb0(trad)=b0(tre)or whichcanbecomparedtotheblackbodytemperaturegivenbyeq.(1.4) TRE=h0 k,ln 1+1Wexph0 ktrad 1!; (1.8) Figure1.1depictstheresultsforsomearbitrarilychosencentralwavelengths0= c=0.first,iftheradiationeldisnotdiluted(w=1),theretemperaturealways Tbb=W1=4Trad: (1.9) star)thefrequencydependencyofismeaninglessfortheresultingtemperature, equalstrad.inthiscase(typicalforthedeepatmosphereandtheinteriorofthe whichmakesthetemperaturedeterminationveryreliable. Fartheroutintheenvelope,however,whereW<1,awidespreadofpossible temperaturesolutionsexistdependingonthecentralwavelength(notethelogarithmicscalingofthetemperatureaxis).thepossiblesolutionsliebetweentheuv{ limith0maxfktrad;ktg(tuv=trad)andtheir{limith0minfktrad;ktg (TIR=WTrad).TheblackbodytemperatureTbb(cf.Eq.1.9)isjustonesolutionin thefrequencydependencyof.forexampleatr=2r,theresultis620k,ifthe interactionbetweenmatterandradiationeldmainlytakesplaceat0=10m,but Thetheoreticallydeterminedtemperaturestructureisthereforeverysensiblefor between,concerningaspecialtypeoffrequencydependencyof. is1920k,if0=1m: Ifwecomparethesevaluestothesmalltemperaturewindow,whereecientdust condensationmaytakeplace,itisobviousthatanuncertaintyconcerningthefrequencydependencyofasconsideredabovecaneasilychangethetemperaturesto r=2r: 0=(1:::10)m)TRE=(1270650)K valueswellaboveorwellbelowpossiblecondensation,respectively.let'sgenerously thegasphasemayoccur.thecorrespondingradiusintervalsrcondintheoptically assumetcond=(1100200)kforthetemperature,whereecientnucleationfrom thinlimitarethengivenby Tcond=(1100200)K: 0=10m)rcond=(1:1:::1:35)R 0=1m grey )rcond=(11:5:::135)r )rcond=(2:7:::5:5)r

1.2THEUSUALMETHODOFTEMPERATUREDETERMINATION 5 Figure1.1:RadiativeequilibriumtemperaturesoverdilutionfactorWinaPlanck{type betweentheuvandtheir-limit,wherethecentralwavelength(seelabelsonsolidcurves) accordingtoeq.(1.7).theshadedregionindicatestherangeofpossibleretemperatures radiationeldwithtrad=3000k(cf.eq.1.6)accordingtoa{function{typegasabsorptioncoecient.theradiusaxisbelongstotheopticalthinlimit(pureradialdilution) issmallandlarge,respectively.thedashedlineshowstheblackbodytemperature.

TheseestimatesclearlyindicatethattheassumptionJ=Bhasadecisiveinuence 6 CHAPTER1.INTRODUCTION Alotofsimplifyingorevenunphysicalassumptionshavebeenmadeinthissection, ofdustformationinthecircumstellarenvelopesofcoolstars. sothatthecalculatednumbersaremeaningless.however,whatisimportantisthe onthecalculatedtemperaturestructurewithsevereconsequencesforthemodeling becomequestionable,whichisanothertopicofthiswork.moredetailedstudieson by1000k.theresultsareevenlessreliable,iftheassumptionsofreandlte cleartrendsintheresults.uncertaintiesconcerningthefrequencydependencyof theimportantheatingandcoolingprocessesarerequiredtotackletheproblemof caneasilychangetheresultsofthetheoreticallydeterminedtemperaturestructure tantandwhicharethecorrespondingrates?rememberingthestrongtemperature dependencyofnucleationfromthegasphase,suchinvestigationscanleadtoanew anddistincttheoreticalviewondustformationincses. theoreticaltemperaturedeterminationincses.whichspectralregionsareimpor- AnexampletotheaboveconclusioncanpossiblybefoundintheCSEsofRCoronae Borealisstars. 1.3 ThePuzzleofDustFormationaroundRCBStars knowntobeabout6thmagnitude,hadsuddenlydisappearedfromthesky.thestar remainedinvisibleforseveralmonthsandnallyrecoveredslowly.inthefollowing the25thbrighteststarinthenorthernconstellationcoronaeborealis,previously Morethantwocenturiesago,theGermanastronomerEduardPigottdiscoveredthat hisarticleonthisremarkablestarexactly200yearsago(pigott1797),whichestablishedanewclassofobjects theclassofirregularvariablestars.rcoronae years,pigottobservedsimilardisappearancesatirregularintervals.hepublished withinafewweeks,andtheeye{catchingshapeofthelightcurve,alwaysattracted muchinterestandfascinationintheastrophysicalcommunity.theuniquenessand (RCB)typedeclineeventswithdecreasesinvisualbrightnessofupto8magnitudes Borealisbecameitsrstmember.Sincethen,theunpredictableRCoronaeBorealis distinctivenessofthisextremetypeofvariabilityincontrasttothebroadvarietyof morecompleteobservationaldatahavebeencollectedcoveringthercbdecline Overtheyears,muchobservationaleortshavebeenundertakenandmoreand oneuniquephysicalmechanismwhichtriggersalltheevents. stellarparametersamongthercbstarsimmediatelysuggeststhattheremustbe Inspiteofthecompletenessandthequalityofobservationaldata,ourunderstanding observationsisoutlinedinappendixa. prehensivedatasetforthisextremetypeofstellarvariability.asummaryofthe events:photometry,spectroscopyandpolarimetry.theseobservationsformacom- ofthephysicalprocessescausingthercbdeclineeventsisstillratherpoor.since Loreta's(1934)andO'Keefe's(1939)basicsuggestionthatthedeclineeventsare causedbythesuddenoccurrenceofdustsomewhereinthelineofsighttowardsthe

1.3.THEPUZZLEOFDUSTFORMATIONAROUNDRCBSTARS 7 chromosphere? R CrB star stellar pulsation R cond =? ~7000 K ~10 4 L shock waves zone of possible nucleation t 0 t 1 dust growth & cloud formation & cloud acceleration t 2 radial dilution decline event visual brightness interpretation.inthiscase,thestellarparameters,thepulsationofthestar,theoccurrenceofshockwavesanddustcloudsandtheshapeofthelight-curvearesupportedby Figure1.2:Sketchofthephysicalprocesses,thegeometryandthetimeevolutionofan RCBdeclineevent.Avisualizationlikethisisalwaysamixtureofobservationalfactsand t 0 t 1 t 2 time ofthiswork. observations.thegeometryofthescenarioandthenucleationzonerefertothehypothesis

observer,theprogressduringthelastsixdecadesconcerningaphysicalexplanation 8 CHAPTER1.INTRODUCTION ofthephenomenonhasbeenslow.themostfavorablepicturetodayisthatclouds thestellarlight.inthelatephasesofadecline,thedustcloudmovesoutwardand Ifthedustcloudformsinthelineofsightitsuccessivelyeclipsesthestarandblocks radiallyacceleratedbyradiationpressureinrandomdirectionsawayfromthestar. ofcarbondustoccasionallyformfromthegasphaseneartothestar,whicharethen dispersesduetoradialdilution.thestarslowlyreturnstonormallight.thisoverall actuallycondensesisnowgenerallyaccepted. theearlyphasesofadecline,thephysicsandchemistryofthedeclinephase,the picture(sketchedinfig.1.2)andthefactthatitissomeformofcarbondustwhich thecircumstellarenvelopeetc.,arestillcontroversial.especiallysoarethephysical Allfurtherdetails,however,e.g.thedistanceofthedustcloudstothestarin survivalofthedustcloudsclosetothestar,thedynamicbehaviorofdustcloudsin reasonsfortheoccasionalonsetofdustformationandfortheformationofdust cloudsratherthansphericaldustshells.thus,thewholephenomenonisstillwaiting foraconvincingexplanation. OnekeytowardsabetterunderstandingoftheRCBdeclineeventsisgivenbythe radialdistancetothestellarphotosphere,wherefreshcarbondustcondensesfrom thegasphase.bymeansofareliabledeterminationofthisquantity,manyofthe proposedmodelsandscenarioscouldberuledoutimmediately.thedistanceto stellarradius,anangularresolutionof510 4arc-secondswouldberequired. RCrBisabout1000pcandinordertodetectadustcloudwithadiameterofone controversialquantitycannotbeobserveddirectly.however,thereexistsomeindirectobservationalcluesonthenucleationdistances,indicatingthatdustformation occursratherclosetothephotosphere: Temporalevolutionofemissionlines:Duringatypicaldeclineevent,arich Therefore,accordingtothepresentstateofobservationaltechniquesthishighly quentlycoverstheregionsresponsibleforthelineemissions.inthiscase,thedust (seeappendixa).itisnaturaltosuggestthattheexpandingdustcloudsubse- temporalevolutionofthreedistinguishableclassesofemissionlinescanbeobserved \chromospheric"emissionlinespectrumappears,similartoasolareclipse.aspecial cloudmustformbelowtheseregions.additionally,theemissionlinesareapparently Correspondingobservationalestimatesclaimanucleationdistanceof1:5 2R lesspolarizedthanthecontinuum(seeappendixa),whichsupportsthisscenario. absorptionlineshavebeendetectedjustinthebeginningofadecline(seeappendixa).sincesuchblueshiftsareonlyseeninthecontextofdeclineevents, (Claytonetal:1992). Dustaccelerationtimescale:Inmanycasesstronglyblueshifted(>200kms 1) onlypossibleratherclosetothestar,wheretheradiationuxissucientlyintense Accelerationstovelocitiesofafewhundredkm/swithinafewweeks,however,are (Whitneyetal:1993),yieldingnucleationdistancesof4 6R(Goeres1996). radiationpressureondustseemstoberesponsiblefortheaccelerationofthegas.

1.3.THEPUZZLEOFDUSTFORMATIONAROUNDRCBSTARS Declinetimescale:Theinitialdeclinephasetypicallylastsafewweeks.Ifthe 9 changesofthebrightnessandthespectraobservedduringthisphasearecausedby stellardisk,thedustcloudmustbelocatedclosetothestar(claytonetal:1992). anopticallythickdustcloud,radiallyexpandingandsubsequentlyobscuringthe Onlyinthiscase,thetangentialprojectionsofthemeasuredradialvelocitiescanbe aslargeasonestellarradiusperweek.fromthisargument,feast(1997)estimates Dustdilutiontimescale:Therecoveryphaseinlatedeclineissupposedtobe aconsequenceofthedustcloudmovingawayfromthestarataconstantvelocity theinitialradialdistanceofthedustcloudtobe2r. whileradiallydiluting.bysimultaneousmeasurementoftheexpansionvelocity(via absorptionlineblueshifts)andthegradientoflightincrease,anabsolutedistanceof thecloudcanbederived,compatiblewithnucleationdistancesof4 7R(Goeres thedeclinescanbeusedtoestimatetheangularcoverageofasingledustcloud 1996). IRuxconstancy:ThefactthattheIRuxesonlyshowminorchangesduring cloudmusthaveatleastthesizeofthestellardiskinordertooccultthestar,leads (Forrestetal:1972).Thesemi{coneangletogetherwiththeconditionthatsucha providesanestimateforthenucleationdistanceof2:5 6R(Goeresetal:1996). toaminimumdistanceofthedustcloudsatthebeginningofthedeclines,which (e.g.pugach1977,lawsonetal:1992).forsuchacorrelation,aphysicalconnection betweenthephotosphereofthestarandthecondensationzonewithaconstanttime individualobjectsthatthedeclineeventsalwaysbeginatxedpulsationphases Pulsationphasecorrelation:Thereissomeobservationalevidenceforcertain delayisrequired.thecloserthecondensationzonetothestar,themoreplausible thistypeofconnectionappears.thecharacterofthisargumentisonlyqualitative. theclearcommontendencyoftheobservationalndingsisthatdustcondensation Althoughatleastoneweaklinkcanbefoundineverychainoftheabovearguments, inrcbenvelopesinfactoccursfairlyclosetothephotosphereofthestar. Accordingtoallknowntheories(classicalnucleationtheory,chemicalpathwaycalculationsorthemodelingofchemicalreactionnetworks)theformationofasolid seemtocontradictthebasicsofdustformationtheory. Thesmallnucleationdistancesderivedfromobservationsatrstsight conditionspresentincsesforallhightemperaturecondensates(includinggraphite eredsolidmaterial.thisyieldslowertemperaturesthan2000kunderthedensity bodydemandstemperatureswellbelowthesublimationtemperatureoftheconsid- andsic).consideringthetypicalchemicalconditionspresentintheenvelopesof onlypresentoutsideabout11rforte=7000k.furthermore,ifweaskforthe InstandardmodelsforCSEs(cf.lastsection),suchtemperatureconditionsare (Goeres&Sedlmayr1992). RCBstars,temperaturesbelow1500Kareinevitablyrequiredforcarbonnucleation

Table1.1:ObservationalandtheoreticalconstraintsonthenucleationdistanceinRCB 10 CHAPTER1.INTRODUCTION envelopes.distancesaregiveninunitsofstellarradii. temporalevolutionandpolarimetryofchromospheric observations theory dustaccelerationtimescale emissionlines 1:5 2 declinetimescale dustdilutiontimescale 4 6 IRuxconstancy 2:5 6 4 7 2 sucientlylowgastemperaturefornucleation >11 pulsationphasecorrelation (close) sucientlylowdusttemperatureforduststability>20 minimumdistancerequiredtoassurethestabilityofsmallcarbondustparticlesin anopticallythinstellarradiationeld,theresultisabout20rforte=7000k (Fadeyev1988).InthecaseofhotRCBstarswitheectivetemperaturesupto TosummarizeTable1.1,twocontradictorypointsofviewcanbedistinguished.On formationappearstobeevenmoreserious. 20000K,onederivesdistancesaslargeas50Randsotheproblemofnearbydust onehand,theobservationalastronomersarguefordustformationneartothestaron thebasisofseveralsupporting,independentscienticndings.ontheotherhand, ofthepresentunderstandingofthercbtypedeclineevents.onceoneacceptsthat theoreticalmodelsfordustformationpredictlargenucleationdistancesasaconsequenceofthermodynamicconstraints.thisobviousconictbetweentheoryand dustformationoccursclosetothestar,asindicatedbyobservations,thereareonly observationtracesthroughthewholeliteratureandconstitutesthecentralproblem (i)therearefundamentalerrorsinthecurrentdustformationtheory.carbondust canbeformedfromthegasphasealreadyattemperaturesof3500 5000Kas twowaysoutofthisdilemma: presentat2raccordingtothestandardmodelsofcses. thecontextofmorecarefulmethodsforthetemperaturedetermination,takinginto conventionaltheoriesondustformationareapplicable,buthavetobediscussedin orytorcbenvelopesconcerningthetemperaturedeterminationofthegas.the (ii)thereisamistakeinthepreviousapplicationsofstandarddustformationthe- accountthedynamicconditionsinthecsesofrcbstars. AllRCBstarsmeasuredthusfarseemtobepulsatingvariables(Lawson&Kilkenny 1996).Thepulsationperiodsareoftheorderof40daysandtheradialvelocity variationsatthephotosphererangefromabout3kms 1to20kms 1(cf.AppendixA).Thestellarpulsationcreatesshockwaves,whichfurthersteepenupin

theatmosphereandpropagateintothecse(e.g.bowen1988,fleischeretal:1992). 1.4REQUIREMENTSFORDIFFERENTAPPROACHES 11 strongdeviationsfromreinthegasphase.if,however,oneofthethreeusual orlessperiodicalcompressionandre{expansionofthegas,bothofwhichmaycause timeagain.theshocksdissipatemechanicalenergyandfurthermoreinitiateamore Consequently,axeduidelementintheenvelopeishitbyshockwavestimeand doned,theradialrangeofnucleationdistancesprescribedbyobservationscould easilyopenup. assumptionsfortemperaturedetermination(re,lte,greygasopacities)isaban- 1.4 turedetermination RequirementsforDierentApproachesofTempera- Themostpromisingscienticmethodtogaininsightintothecomplexprocesses equationswhichareintegratedintime.besideshydro-andthermodynamics,these gas,wherethephysicalinteractionsareformulatedintermsofordinarydierential tions.thebasisofthesemodelsisahydro-andthermodynamicdescriptionofthe ofastrophysicalobjectsisthemodelingbyfulltime{dependentcomputersimula- modelsmayincluderadiativetransfer,chemistryanddustformation,accordingto putermodelscanhelptobuildupahigherlevelofcompletenessinscience. amongtheseprocessescanbecomethemaintopicofexamination.therefore,com- thechosendegreeofapproximation.thereby,allthenecessaryphysicalandchemicalprocessescanbeinvestigatedsimultaneously,sothatjustthecomplexinterplay However,theresultsofthisrathernewmethodofscienticcomputingcannotbe toabstractandtosimplify.inthiscontext,the\best"descriptionofaphysical processunderinvestigationisnotnecessarilythemostaccurateanddetailedone betterthanourphysicalunderstandingofthebasicprocessesinvolvedandourability processwhileusingtheleastamountofresources.suchadescriptionmustbe butadescription,whichcorrectlydescribesthemostimportantfeaturesofthe Mirasandlong{periodvariablesontheAGBandRCBstars,CepheidsandRVTauri ConcerningthetemperaturedeterminationintheCSEsofpulsatingstars(suchas partoftheinvestigationsofastrophysicalobjects. sucientlysimpleinordertobeincludedinmorecomplexmodelcalculationsas cannotbeobtainedbymeansofradiativetransfercalculationsalone. pulsationcallsforatime{dependenttreatmentofthethermodynamics.asargued above,strongdeviationsfromremayoccurandthegastemperaturestructure starsneartotheinstabilitystrip)thepresenceofshockwavescausedbythestellar transfer,usingthebasicassumptionsofre,lteandgreygasopacitiesasdescribedinsect.1.2,whichdeterminestheinstantaneousre{temperaturestructureculatedasfollows.bowen(1988)rstcarriesoutafrequencyintegratedradiative Fleischeretal:1992,Feuchtingeretal:1993),thegastemperaturestructureiscal- Accordingtothepresentstateofapproximationinsuchmodels(Bowen1988, Secondly,hecalculatesthecurrentgastemperaturebyassumingalocalrelaxation

12 CHAPTER1.INTRODUCTION Figure1.3:TemperaturestructureadoptedfromFeuchtingeretal:(1993). logradius[cm] Figure1.4:AdoptedfromBowen(1988) case,assumingthatthegasinstantaneouslyrelaxestoreeverywhereintheenvelope.applyingradiationhydrodynamics,feuchtingeretal:(1993)useanapproach, towardreataniterate.fleischeretal:(1992)considertheisothermallimiting Qrad,isanimportantingredient.QradvanishesinREandotherwisedeterminesthe timescaleforrelaxationtowardre. namicsinthesemodels,thetotalnetheatingrateduetoradiativegainsandlosses, whichissimilartobowen's,butmoreconsistent.inordertotreatthethermody- ConcerningthecalculationofQradinthemodelscitedabove,crucialassumptions havebeenmade(lteor,incontrast,qrad/2),yieldingsimpleanalyticalexpressionsforqrad.butdependingonwhichoftheseassumptionsisapplied,thperaturesusuallyveryclosetothere{temperaturestructureexceptforsomethin gastemperaturestructureturnsouttobequitedierent.intheltecase,the gastemperaturepeaksatthelocationsoftheshockfronts(cf.fig.1.3).incontrast,atsmalldensitiesintheqrad/2case,ratherbroadregionsofenhanced gastemperaturesbehindtheshocksareproduced,almostentirelydecoupledfrom there{temperaturestructure(cf.fig.1.4).theseresultsaretypicalexamplesfor ofthesemodelcalculations,e.g.themasslossrate,andeventhemodelstability ter,respectively.theresultinggastemperaturestructureaectsallotherresults shockwavesofpredominantly\isothermal"orpredominantly\adiabatic"charac- (Wood1979).Animportantfeedbackbetweenthegastemperaturestructureand thedynamicsoftheseenvelopesisgivenbythecondensationofseedparticlesfrom thegasphase(nucleation),whichisverysensitivetothegastemperature.ittriggers calculatedradiativeheating/coolingratesareveryecient,resultingingastem- ConsideringthedeterminationofQradinotherastrophysicalenvironmentsextensivemodelcalculationshavebeenmadeforstationaryplane-parallelshocks,e.g.in duetoradiationpressureondustgrains(fleischeretal:1991). thefurtherevolutionofthedustcomponentandhencetheaccelerationofthegas

theinterstellarmedium,whereallphysicalpropertiesarepurelydeterminedbythe 1.5.AIMANDSTRUCTUREOFTHISWORK 13 distancefromtheshockfront,butarenotexplicitlytime{dependent(hollenbach& McKee1979,Fox&Wood1985,Hollenbach&McKee1989,Gillet&Lafon1989, Neufeld&Hollenbach1994).Thissituationallowsforaveryaccuratephysicaldescription,includingnon{LTEionization,non{equilibriumchemistryandradiative modelsforpulsatingstarsforessentiallytworeasons.first,theshocksintheenvelopesofpulsatingstarsarenotstationary(e.g.axeduidelementwillstartto transfer.however,thisschemecannotbeeasilyappliedtothetime{dependent re{expandafterithasbeencompressedbyanpropagatingshockasopposedtothe stationarysituation)andsecond,thedetaileddescriptiongiveninthepaperscited models,atleastatthepresentstateofcomputerspeed. aboveismuchtooelaboratetobeincludedwithintime{dependenthydrodynamic radiativetrapping.ontheotherhand,itmustbesucientlysimpletobeincluded accountimportantfeaturessuchasthenon{ltepopulationofexcitedstatesor mustbephysicallybasedontherelevantheatingandcoolingprocesses,takinginto Thus,thereisagreatneedforarealisticcalculationofQrad.Ontheonehand,it intime{dependenthydrodynamicmodels. Thebasicaimofthisworkistogainmoretheoreticalinsightsonthetemperature 1.5 structureofcircumstellarenvelopes,especiallythoseofpulsatingstars,whereshock AimandStructureofthisWork oreticaltemperaturedeterminationinagivenradiationeld radiationtransfer wavespropagatethroughtheenvelopes.thisworkfocusesontheproblemofthe- calculationsareexplicitlynotconsideredandarenotperformed. sityrangetypicalforcses.however,thereisdetailedknowledgeavailableatboth 1014cm 3.Precedingstudiesintheliteratureareusuallynotapplicableinthisden- Forthispurpose,theradiativeheatingandcoolingofthegasincircumstellarenvelopesisinvestigatedfromtheverybeginning,examiningdensitiesfrom105to extremesofthisdensityinterval.forlargedensities,extensivecalculationsofgas thenetradiativeheatingrate.atlowdensities,theimportantradiativeheating absorptioncoecientsinstellaratmospheresexistwhich,incaseoflte,determine andcoolingprocessesareknownfromstudiesofinterstellarcloudsandinterstellar shockwaves.thisworkderivesadvantagesfrombothandintendstoclosethegap dynamicmodelcalculations.thepossibilityofafastandproperinclusionofthe Duetothetime{dependentconditionspresentinCSEsofpulsatingstars,thedeterminationofthetemperaturestraticationmustinvolvetime{dependenthydro- betweenthesedensitylimits. calculatedheatingandcoolingratesintohydrodynamicmodelsisanessentialconstraintfortheseinvestigations.itistheaimofthisworktolaythefoundationsfonamicmodelcalculations. amorereliabletreatmentofthetime{dependentthermodynamicsinsuchhydrody-

Althoughdustformationisrarelydiscussedinthisworkexplicitly,theworkisguided 14 CHAPTER1.INTRODUCTION bythecertaintythattheformationofsolidsincsesrequireslargedensitiesand veryspecialtemperatureconditions.thequestion,whichalwaysstandsbehindthe investigationsandisthebasicmotivationforthisworkis: Theworkisorganizedasfollows: Whereintheenvelopesuchthermodynamicconditionsmayoccur? LTE. thermodynamics.thelevelofapproximationforthisworkisxedandtheinternal Chapter2describesthebasicconceptforthetreatmentofthetime{dependent Chapter3containsthecalculationsofthevariousheatingandcoolingratesconsideringarbitraryradiationelds.Computationalmethodsaredevelopedwhich energyofthegasisdenedaccordingtothebasicassumptionofsteady{statenon{ andofh2;linetransitionsofneutralatomsandions;bound{freetransitions;free{ processesisinvestigated:rotationalandro{vibrationaltransitionsofpolarmolecules Duetothewidetemperaturerangetobeconsidered,avarietyofdierentradiative includetheimportanteectsofnon{lteandofopticalthicknessinspectrallines. freetransitionsandphotochemicalreactions.specialattentionispaidtowhichkind completesthetheoreticalpartofthiswork. eredsofar,butmightbeoffurtherinterestfortheheatingandcoolingofthegas, correspondingrates.ashortlistofradiativeprocesses,whichhavenotbeenconsid- ofatomicandmoleculardatahastobeknownforareliabledeterminationofthe ofthegasaredeterminedfromtheelementabundancesi,themassdensityofthe technicaldetailsofthecalculationofthevariousparticleconcentrationsandthe internalenergyofthegasareexplained.theparticleconcentrationsandthestate Chapter4outlinessomecommonfeaturesforthefollowingapplications.The gas,itstemperaturetg,andthecontinuousradiationeldj. Inthefollowingchapters,threeapplicationsofthetheoreticalmethodsarepresented: riummaynotbeunique,butcanhavetwoormorestabletemperaturesolutions. tionsinthecsesofrcbstars.itisshownthattheconditionofradiativeequilib- These\thermalbifurcations",inprinciple,allowforaspatialcoexistenceofhotand Chapter5examinesthetopologyoftheradiativeequilibriumtemperaturesolu- coolphasesinthecircumstellarenvelope. sityandtemperature.theimportanceofthedierentheating/coolingprocesses foracarbon{enrichedgastypicalforc{starsarecalculatedasfunctionofgasden- responsetopropagatingshockwaves.forthispurpose,radiativecoolingtimescales Chapter6investigatestherelaxationtowardsradiativeequilibrium,especiallyin isdiscussedandthemostecientprocessinthevariousdensityandtemperature

1.5.AIMANDSTRUCTUREOFTHISWORK regionsisdetermined.theresultsofthecoolingtimescalesarecomparedtothose 15 derivedfromformerlyappliedanalyticalheating/coolingfunctionsinpreviousresearch.thecharacterofthethermalrelaxationofthegasafterthepassageofshock wavesisdiscussed,providingnewfueltothecontroversyaboutwhethertheshocks incsesbehavepredominantly\isothermally"or\adiabatically". Chapter7againconsidersRCBstars.Aphysicalmechanismispresented,which envelopesofpulsatingrcbstars.amodelforxeduidelements,whichare periodicallyhitbystrongshockwavesproducedbythestellarpulsation,isdeveloped maybeessentialfortheoccasionalonsetofdustformationinthecircumstellar elements,areinvestigated.accordingtothismodel,thepreconditionsforeective carbonnucleationmaybetemporarilypresentquiteneartothephotosphereofa andthethermalenergybalance,thechemistryandthenucleationinsuchuid pulsatingrcbstar,despitetheirhigheectivetemperatures.thus,thisworkmight bridgethegapbetweenobservationsandtheoryconcerningrcbstarsasoutlined insect.1.3. Chapter8summarizestheresultsandpresentstheconclusionsofthiswork. thefascinatingclassofrcbstarsandsummarizespreviousmodels.thisappendix AppendixAgivesanoverviewofthecurrentstatusofobservationalknowledgeon providesanimportantbackgroundfortheinvestigationsinchapter5and7.

16

Chapter2 Thischapterintendstostatethebasicassumptionsofthisworkandtoclarifythe TheThermodynamicConcept distributionofthegasatrestisassumedtobegivenbyauniquemaxwelliandistribution,characterizedbyasinglekinetictemperature,whichishenceforthcalledthe Thecentralphysicalquantityofthisworkisthetemperatureofthegas.Thevelocity meaningofsometermsfrequentlyused. whichdistributethetotaltranslationalenergypresentamongthegasparticles.the sponsiblefortherelaxationtowardsthemaxwelliandistributionareelasticcollisions dierentkindsofparticles(e.g.electronsandatoms)areneglected.theprocessesre- \gastemperature"anddenotedbytg.dierencesinthekinetictemperaturesof correspondingrelaxationtimescaleisassumedtobeconsiderablyshorterthanany othertimescaleinherentinthephysicalsystemunderinvestigation.withregard tothisrelaxation,themostcriticalprocessistheequalizationofthetranslational theconclusionthattheexistenceofauniquemaxwelliancansafelybeassumed. Mihalas(1984,seep.29andp.387foramorecomprehensivediscussion)arriveat energiesbetweenlightandheavyparticles,becauseoftheinecientenergytransferratesofsuchcollisions.forconditionsinstellaratmospheres,mihalas&weibeterminationbasedontherstlawofthermodynamics Theaimofthisworkistodevelopatime{dependentmethodfortemperaturede- 2.1 FirstLawofThermodynamicsandEquationofState ofheattransferedtothegasq(countedpositiveforgains)plustheworkdoneto Equation(2.1)statesthatthechangeofinternalenergydEisgivenbytheamount de=q+w: thegasw(countedpositivewhenthesurroundingsdeliverworktothegas). Howarethegastemperatureandtheinternalenergyrelatedtoeachother?Theanswerofthisquestionsseemstobetrivial(givenbythewell{knowncaloricequationof state),butinfactdeservessomefurtherdiscussionfordilutedgasesunderastrophysicalconditions.besidesthetranslationaldegreesoffreedom,arealgas{consisting ofneutralatoms,electrons,ionsandmolecules{hasadditionalpossibilitiestostore energy,whicharehenceforthcalled\thepoolsofenergy".thepopulationofexcited 17

electronic,vibrationalandrotationalstatesrepresentsuchpools.furthermore,energyisstoredinpotentialformaccordingtothebindingforcesbetweenelectrons andatoms(ionizationpotential)andbetweentheconstitutingatomsofmolecules 18 CHAPTER2.THETHERMODYNAMICCONCEPT (dissociationpotential).consequently,theinternalenergyofthegasisdenedas ThedetailsoftheevaluationofthevariousenergytermsarestatedinChapter4 e=1etrans+eion+ediss+eel+evib+erot (Eq.4.1{4.6).IncaseofLocalThermodynamicEquilibrium(LTE),therelationship (2.2) degreeofionization,theconcentrationofthemoleculesandthepopulationofthe betweenthegastemperaturetgandtheinternalenergyeiswell{dened.the excitedlevelscanbedeterminedbymeansofsahaequations,thelawofmassaction andboltzmanndistributions,respectively.allenergytermsineq.(2.2)canbe ofanysuitablesetoftwolocalstatevariables. calculatedstraightforwardly,yieldinge=e(;tg)or,moregenerally,easfunction cessesalterthestateofthegasinvariousways.figure2.1sketchesthissituation. 2.2 ConsideringthedilutedgasesinCSEs,however,LTEisnotvalid.Radiativepro- LTEandNon{LTE Thegasisrepresentedbythebiggreybox,containingtheinternalpoolsofenergy. Thegasinteractswiththeradiationeldviatheexchangeofphotonsandwiththe arrowonthel.h.sinthegureandthegasmayexchangeheatwithitssurroundings freedomofthegas.workcanbedonetothegasw= pdvasindicatedbythe dustcomponent,e.g.viainelasticcollisions.atthesametime,internalprocesses (notshown).examplesforthesuchprocessesareheatconduction,viscousprocesses (greyarrows)redistributethetotalinternalenergyamongthevariousdegreesof andshockdissipation. TheradiativeprocessesgenerallydrivethegasawayfromLTE1,whereastheinternalprocesses(processesnotinvolvingphotonsordustgrains,thatis)drivethizationfollowedbyradiativerecombinationortheabsorptionofaphotonfollowed thetranslationalenergypoolisindirect,sotospeak,becauseatransmittingstate isinvolved.usually,atwo{stepprocessisrequired,forexampleacollisionalion- gastowardlte.ingeneral,theenergytransportbetweentheradiationeldand Thus,anon{LTEtreatmentforthedilutedgasesinCSEsisrequired.Balancing stateofthegascannotbecalculatedbythermodynamicconsiderations. bycollisionalde{excitation2.thesetransmittingstates(e.g.theexcitedelectronic thegainsandlossesbyallcollisional,radiativeandchemicalprocesses,thechange statesoftheatoms)areconsiderablyaectedbytheradiationeldandhence,the areusuallylessimportant,e.g.free{freeemission(bremsstrahlung). 1ExceptforthecasethattheradiationeldexactlyequalsaPlanckianofthegastemperature. 2Directlinksbetweentheradiationledandthetranslationalenergypooldoalsoexist,but

2.2.LTEANDNON{LTE 19 ionization electronic excitation radiation p dv translation vibrational excitation dust dissociation rotational excitation nation;bound{bound,free{free,vibrationalandrotationalemission/absorption; photodissociation/radiativeassociation.blackdashedarrowsshowenergyexchangeratesbetweendustandgas(viacollisionsandsurfacechemicalreactions). orparticlecreation/destruction,respectively.dottedgreyarrowsshowadditional Fullgreyarrowsrepresent\internal"energyuxesviacollisional(de-)excitation uxesviaphotons.fromtoptobottom:photoionization/radiativerecombi- Figure2.1:Thepoolsanduxesofenergy.Blackfullarrowsindicateenergy statesviachemicalreactions. examplesofinternalenergyuxesnotexplicitlyconsideredinthiswork,suchas pumpingbyuorescence,ro{vibrationalpumpingortheexcitationofvibrational

oftheparticledensitynjiofchemicalspeciesiinquantumstatejcanbeexpressed 20 CHAPTER2.THETHERMODYNAMICCONCEPT by(e.g.mihalas&weibelmihalas1984,p.389) d(nji=) dt =X fklg6=fijg nlkrl!j k!i njix DiusionprocessesareneglectedinEq.(2.3)andd=dtdenotestheLagrangian fklg6=fijg Rj!l i!k: (2.3) coecientrj!l derivativewithrespecttotime,consideringacomoving-movingframe.therate processeswhichdestroyaparticleofspeciesiinstatejandcreateaparticleof specieskinstatel.forsimplecollisional,absorptionoremissionprocesseswehave i=k.forchemicalprocesses(i6=k),thequantumindicesjandlusuallyrefertothe i!k[hz]denotesthetotalrateofallcollisional,radiativeandchemical groundstate.theratecoecientsmaycontainfurtherparticledensities(collision partnersorchemicalreactants)ortheradiationeldj,dependingonwhichtype ofprocessisconsidered.theratecoecientswillbequantiedinchapter3. 2.3 Ifthegasisexposedtostaticouterconditions(radiationeld,propertiesofthe dustcomponent,volumeetc.),thegaswillrelaxtowardsasteadystate3,i.e.the SteadyState concentrationsofallspeciesinallquantumstatesbecometime{independent d(nji=) calculatedbysolvingthecoupledalgebraicequationsoftypeeq.(2.3)forallspecies Incaseofasteadystatethel.h.sofEq.(2.3)vanishes.Theparticledensitiescanbe dt =0: (2.4) andstatesunderinvestigation.theresultsaretime{independentandcanformally beexpressedby Therefore,inadditiontothetwolocalstatevariablessucientinLTE,theradiation Consequently,thecaloricequationofstate(2.2)ofthegaswritesase=e(;Tg;J). nji=nji(;tg;j); (2.5) eldoccursasadditionalexternalparameterforthedeterminationoftheparticle densitiesandtheequationofstate4;5.theresultingsteadystateofcoursediers Jdisappears. internal(e.g.collisional)processesdominate,lteisvalidandthedependencyon fromlteingeneral.inthelimitingcaseoflargedensities,however,wherethe ofthesteadystate,whichisthemeanlocalvelocitygradientdv 3Othertermsusedintheliteratureare\kineticequilibrium"or\statisticalequilibrium". However,theinuenceofthisparameterissmall. thecalculationoftheratecoecientsforbound{boundtransitionsconcerningopticallythicklines. 5TheinuenceofthedustcomponentonthestateofthegasisneglectedinEq.(2.5) 4InChapter3itwillbestatedthatanotherexternalparameterentersintothedetermination dl.thisparameterisinvolvedin

2.3.STEADYSTATE 21 Theconditionofastaticenvironmentcanberelaxedtosomeextendconcerning thosetime{dependentsituationswherethechangesoftheouterconditionsoccur slowly.inthiscasethegasrapidlyaccommodatestothevaryingenvironment.this workassumesthatthisaccommodationinfactoccursinstantaneously: Theinternalrelaxationofthegastowardsteadystateisassumedtooccur onashorttimescalecomparedtothechangesoftheouterconditions. Inthiscase,acaloricequationofstateexists,asstatedabove,althoughthegasis notinlte.athermodynamicdescriptionofthegasisappropriate6.ofcourse,the 6Consideringtheexcitedelectronicstatesofatomsandions,therelaxationtimescaleisgivenby theradiativelifetimesofthelevelsatsmalldensitiesandisevenshorteratlargedensities.allowed electronictransitionshaveradiativelifetimesoftypically10 8s,butactuallytheslowestratewithin thelevelsystemdecidesupontherelaxationtimetowardacompletesteadystate,whichcanbe aslargeas102sforthemeta{stablelevelsincludedinthiswork.theradiativelifetimesofthe vibrationalandrotationallevelsofpolardiatomicmoleculesarefoundtobetypically10 2 101s. Thesetimescalesaretobecomparedtotypicalhydrodynamictimescales,whichfortheCSEs ofpulsatingstarsareapproximatelygivenbyonepulsationperiod,whichisabout107sformiras and106sforrcbstars.hence,therelaxationoftheelectronic,vibrationalandrotationalstates canbeassumedtobefast. Consideringtherelaxationtowardsionizationequilibrium(balancebetweenionizationandrecombinationrates),thecorrespondingtimescalesareverydierent.Ifphotoionizationdominates, e.g.consideringthecasej=b(7000k),therelaxationtimescalesarefoundtobe10 1 101s forallatomsunderinvestigation,independentofdensityandtemperature.ifcollisionalionization dominates,therelaxationtimescalestronglydependsonthedensity.inthecasej=b(3000k) therelaxationlasts104satadensityof1013cm 3,106sat109cm 3and109sat105cm 3, concerninglowtemperatures.ifthetemperatureishighenoughtocauseconsiderablecollisional ionization,therelaxationtimescalebecomesmuchshorter.hence,therelaxationofthedegree ofionizationisfastinthosecases,whereisitimportantforthecalculationoftheradiativeheating/coolingrates,butcanexceedthehydrodynamictimescalesotherwise. Thechemicalprocessesusuallyintroducethelargesttimescalestothegas(disregardingdust formationprocesses).thechemicalrelaxationtimescalescanindeedexceedhydrodynamictime scalesincses(e.g.becketal:1992).however,thechemicalprocessesthemselvesdonotprovide rstorderradiativeheatingandcoolingrates themostimportantheatingandcoolingprocesses usuallyinvolvethedegreesoffreedomdiscussedabove,whichcanbeassumedtobepopulated accordingtoasteadystate. Thus,onehastoconcludethattherelaxationofthegasinCSEstowardsitssteadysteadymay notalwaysbecomplete.afulltime{dependentnon{lteapproachwouldberequiredforamore accuratemodelingofthegas.inthiscase,therstlawofthermodynamicshastobeappliedtothe translationalenergyalonee=etrans=.acaloricequationofstatedoesnotexistintheusualsense orbecomesobsolete.onehastodeterminetheinternalenergydistributionprocessesinthiscase, e.g.howmuchtranslationalenergyisconsumedorliberatedduringachemicalreaction.atime{ dependenttreatmentofionizationandchemistrywithinhydrodynamicmodelsseemstobeoutof question,atleastatthecurrentstateofcomputerspeed.thiswouldintroducealargenumberof new(sti)dierentialequationstotheusualsetofhydrodynamicequationstobesolved,which requiresmuchmorecomputationaleorts.however,thepossibilitytoincludetheresultsofthis workintosuchcalculationsisregardedasessential,concerningfutureinvestigations.therefore, theassumptionofasteadystateismadenevertheless.itrepresentsanappropriatecompromise betweenaccuracyandexpense:itaccountsforthemostimportantnon{lteeects,butkeeps thingssimpleenoughforathermodynamicdescriptioncompatibletohydrodynamiccalculations. Forafurtherdiscussionofthistopic,seeMihalas&WeibelMihalas(1984,p.386{396)

particledensitiescanonlybedetermined,iftheradiationeldisknown;thatis, 22 CHAPTER2.THETHERMODYNAMICCONCEPT eventuallymodiedbydustinthecse,whichisalsoconsideredas\external". strongspectrallines.theradiationeldismainlyprovidedbyexternalsources, thecircumstellarenvelopesareopticallythinbydenition,maybeexceptforsome thereareasmanynewexternalparametersasarerequiredtospecifyj.however, much,responsiblefortheradiationeld.therefore,thenon{lteradiativetransfer calculationsconcerningthedustcomponent.thegasitselfisnot,oratleastnot Therefore,itseemsappropriatetoprescribetheradiationeldinCSEs(e.g.bya problemdecouplesincses(incontrasttothesituationinstellaratmospheres)and radiallydilutedphotosphericradiationeld)ortousetheresultsofradiativetransfer thethermodynamicbehaviorofthegascanbestudiedintheproposedway. TherstlawofthermodynamicsEq.(2.1)isfurtherspecializedinthefollowing. Accordingtothedenitionoftheinternalenergy(cf.Eq.2.2),theinternalenergy transferratesdonotcauseanyheatingorcooling,sincetheyonlytransferenergy eldandthedust,thatis)tothegas.letbqraddenotethetotalnetheatingrate givenbythesumofenergyuxesfromtheremainingexternalpools(theradiation fromoneinternalpooltoanother(cf.fig.2.1).thenetheatingratebq=q=dtis ofdustgrains.disregardingotherheating/coolingmechanismsasheatconduction, convectionandviscousprocesses,whichareusuallynegligibleatthelowdensities perunitmassandtimeduetoradiativeprocesses,whichisgiventheamountof netabsorbedphotonenergy,andbqdustthetotalnetheatingcausedbythepresence incses,therstlawofthermodynamicswrites pisthegaspressureandv=1=thespecicvolume.thegastemperaturecanbe de dt= pdv regardedasanimplicitresultofthesolutionofeq.(2.6),inferredfromthecaloric dt+bqrad+bqdust; (2.6) stateofthegasisdetermined.thenetradiativeheatingratebqradisakindofuseful equationofstate. byproductofthesecalculations. ThemaintaskofthefollowingChapter3willbetoquantifyalltheimportant Asaconsequenceofthesteady{stateassumption,theinternalenergyandthenet internalandexternalrates,asfaraspossible.bymeansoftheserates,thesteady parametricspecicationofj.thus,itisguaranteedthattheproposedtime{ tabulatebqradandeasfunctionoftwostatevariables,sayandtg,andasuitable arereadilyavailableinhydrodynamicmodelcalculations.itishencepossibleto heatingfunctionareentirelydeterminedbymeansoflocalphysicalquantities,which modelcalculationswithregardtofutureinvestigations. dependentmethodoftemperaturedeterminationcanbeincludedinhydrodynamic

Chapter3 RadiativeHeatingandCooling requiresaquantitativeanalysisofallradiativeprocessesoccurringintheconsidered istheinvestigationofthemostimportantheatingandcoolingprocesses,mainly uidelement adicultandprincipallyinnitetask.whatisfeasible,however, Thedeterminationofthetotalradiativeheating/coolingrateofthegasinCSEs relyingontheexperienceofprecedingstudies. Fromstellaratmospherecalculationsitisgenerallyknown,thatbound{freetran- onlyasecond{order{eectinthiscontext1.inpredominantlyneutralstellaratmospheres,thebound{freetransitionsofh areimportant.belowabout3500ksorptioncoecient(e.g.unsold1968)andhencefortheheatingandcoolingofthe gas.theadditionalconsiderationoflinetransitions(\lineblanketed"models)is sitionusuallyaretheprimarycausefortheshapeandthemagnitudeoftheab- theirelectronicbands,vibrationalandrotationalspectra,especiallythosemolecules moleculesenterintocompetitionandsoondominatetheabsorptioncoecientby Concerninginterstellarconditions,Hollenbach&McKee(1979,1989)pointedout thatforbiddenne{structurelines,meta-stabletransitionsandsomelow{lyingper- withpermanentdipolemoment(jrgensen1994). tenprovidethedominantcoolingmechanismforashockednon{moleculargas.if mittedlinetransitionsofthevariousneutralandsinglyionizedmetalatomsof- present,polarmoleculescontributebytheirlargeamountofallowedvibrationaland rotationallinetransitions.bound{freetransitions(mainlyofhydrogen),lyand HareimportantattemperaturesT>8000K,especiallyforlargedensities. ing/coolingratesforcses.ageneraltheoreticaldescriptionmustbedeveloped coolingprocessesmentionedaboveinordertounifythepictureofimportantheat- whichisapplicabletoboth,stellaratmospheresaswellasinterstellardensityconditions. Reviewingtheseexperiences,itisimportanttotackleatleastalltheheatingand processisalwaysdiscussedsimultaneously,whichisdierentfromotherapproaches concerninginterstellarmatter(e.g.spitzer1978),wheretheheatingandcooling Inthefollowingthenetheatingrateofoneparticularradiativeprocessanditsreverse spectrallines theeectsoflinesinamovingmediummaybelargerbyordersofmagnitude ratesareusuallydiscussedstrictlyapart. 1Thisstatementreferstoastaticatmosphere,whereselfshieldingdiminishestheinuenceof 23

24 3.1 Bound{BoundTransitions CHAPTER3.RADIATIVEHEATINGANDCOOLING systemofboundstatesispresented.themethodisapplicabletolinetransitionsof methodforthecalculationoftheheating/coolingrateofanarbitraryn{level{ Themostbasicformofinteractionbetweenmatterandradiationeldisgivenby atomsandions,tovibrationalandrotationaltransitionsofmoleculesandalsoto theabsorptionandemissionoflinephotons.inthissection,ageneraltheoretical quadrupoletransitionsofh2.ithasthefollowingfeatures: a)thecalculationofthelevelpopulationisperformedundertheassumptionof atomictransitions. thedensityconditionspresentincircumstellarenvelopes,especiallyforallowed steady-statenon{lte.itisabsolutelynecessarytoconsidernon{lteunder b)comparedtointerstellarconditions,thehighdensitiesencounteredincircumstellarenvelopesmaycauselargeopticaldepthsinthelines,whichsignicantly changetheheating/coolingratesduetoradiativetrapping.theseeectsare c)sincepropagatingshockwavesmaybepresentinthecircumstellarenvelopes ofpulsatingstars,largevelocitygradientsoccur.incontrasttosteady,plane{ tackledbyapplyinganescapeprobabilitymethod. incaseofsphericalsymmetry. asoutlinedbyhollenbach&mckee(1979).thisworkusesthesobolevtheory tryoftheowrequiresadierentmethodtocalculatetheescapeprobabilities parallelshocks(e.g.intheism),theexplicittime{dependenceandthegeome- d)lineabsorptioniscompletelytakenintoaccount.theintensecontinuous radiationeldincircumstellarenvelopeschangesthecoolingratessignicantly 3.1.1EscapeProbabilityMethodforanN{Level{SystemwithoutContinuum andcaninfactleadtonetlineheating incontrasttointerstellarconditions. isthetotalrateofenergywhichistransferredto/fromtheradiationeldvialine AnatomicormolecularN{levelsystemisconsidered.Thequantitytobedetermined statisticalequationsaregivenby emission/absorption.thistotalenergytransferrateiscalculatedintwosteps. First,thelevelpopulationsniarecalculatedbymeansofthestatisticalequations (\steadystatenon{lte")andsecondly,theenergytransferrateisdetermined.the andcanbesolvedtogetherwiththeequationfortheconservationofthetotalparticle nixj6=irij=xj6=injrji; (3.1) densityoftheconsideredspeciesnsp=pini.theratecoecientsaredenedby

3.1.BOUND{BOUNDTRANSITIONS Rlu=BluJul+Clu; Rul=Aul+BulJul+Cul (3.2) (3.3) 25 whereuandllabelanupperandlowerlevel,respectively.theratecoecients forstimulatedemissionbulandabsorptionblucanbecalculatedfromthosefor excitationculbyapplyingadetailedbalancerelation: spontaneousemissionaulbyapplyingtheeinsteinrelations.similarly,therate coecientsforcollisionalexcitationclucanbecalculatedfromthoseforcollisional Blu=gu Bul= 2h3ulAul glbul c2 (3.4) Clu=gu glculexp( Eul=kTg) (3.5) Thefrequencyintegratedmeanintensity Jul=1 (3.6) asknown),butismodiedbylineemissionandabsorptionintheconsideredresonanceregionitself,whichbecomesimportantforopticallythicklines.anexact isnotsolelygivenbytheincidentcontinuousintensitiesiinc 4ZZul(;)I()dd ()(whichareregarded (3.7) diativetransfercalculationsinthemovingmedium,whichgoesfarbeyondthescope ofthiswork.fortunately,thereareapproximateescapeprobabilitytechniquesavailable,whichcanaccountforthemostimportantresonanceeects.thisworkusesthe solutionofthisproblemcanonlybeachievedbyfrequencydependentnon{ltera- foradetaileddescriptionseewoitke1992)2: Sobolevapproximationinthecaseofsphericalsymmetry(e.g.Puls&Hummer1988, Jcont ul Jul=Pe =1 4ZIinc uljcont ul+(1 Pe ul()d ul)slul (3.9) (3.8) SLul=2h3ul c2 gunl glnu 1! 1 Pe ul=1 4Z1 exp S ul() ul()d (3.11) (3.10) dvk dl()=1 2vr+2@v S ul()=c3aul 83ul gu glnl nu!dvk dl() 1 @r (3.13) (3.12) ignoringradiationtransfereectsintheconsideredresonanceregionitself. Pe continuousmeanintensityatlinecenterfrequencycausedbyincidentradiation ulisthemeanescapeprobabilityandslulthelinesourcefunction.jcont 2AdiscussionoftheapplicabilityofSobolevtheorytotheshockedenvelopesofpulsatingstars ulisthe isgivenonp.28.

26 Jcont ul,inprinciple,resultsfromthecalculationofacontinuousradiativetransfer CHAPTER3.RADIATIVEHEATINGANDCOOLING (withouttheconsideredline).forsimplicity,theincidentintensitiesareassumedto ineq.(3.11):anappropriatemeanvelocitygradientddv inordertoavoidtheelaborateandtime{consumingintegrationoverthesolidangle beisotropicineq.(3.8)3.s dvk dlthevelocitygradientonaconsideredray.thefollowingapproximationisused ulistheso{calledsobolevopticaldepthofthelineand probabilitiesarecalculatedaccordingto *dv dl+=13@v @r+232x 1=2 0 1vr dleisdenedandtheescape x0=1+max(0; @v @r.vr) (3.14) es ul=c3aul 83ul gu glnl nu!*dv dl+ 1 (3.16) (3.15) epe ul=1 exp es es ul() andreachesamaximumvalueof33%aroundpe0:5.dierentowgeometriesand TheerrorofthisprocedurevanishesforthetwoimportantcasesPe!0andPe!1 ; (3.17) forddv caseswithvanishingvelocitygradientscanbetackledbyusingdierentexpressions toeliminatetheunknownlinesourcefunctionsslul.itisstraightforwardtoshow Forthenumericalsolutionofthestatisticalequations(3.1)itisveryadvantageous dleineq.(3.14)assummarizedinneufeld&kaufman(1993). that nuaul+(nubul nlblu)jul=nuaulepe =AulePeul 1 Jcont SLul! nu(1+jul) nlgu ul gljul! (3.19) (3.18) combinedintothefollowingsetofeectiveratecoecients localcontinuousradiationeld.bymeansofeq.(3.19)allaboveequationscanbe wherejul=c2=(2h3ul)jcont erul=aulepe ulisadimensionlessquantitywhichcharacterizesthe erlu=gu gl AulePe ul(1+jul)+cul uljul+culexp Eul ktg!: (3.21) (3.20) ofrij,wherethelinesourcefunctionsdonotappearanymore. ThelevelpopulationscannowbecalculatedbysolvingEq.(3.1)witheRijinstead eectoftheincidentintensitiesisproportionaltotheescapeprobabilityinthatparticulardirection, wherethevelocitygradientissmallerorlargerthanthemeanvelocitygradient,respectively:the toeq.(3.8),iftheconsidereduidelementmainlyreceivesthelightfromaparticulardirection, 3Strictlyspeaking,theinuenceofincidentcontinuousradiationincreasesordecreasescompared butthe(isotropic)re-emissionisproportionaltothemeanescapeprobability.

3.1.BOUND{BOUNDTRANSITIONS Finally,afterhavingdeterminedthelevelpopulations,thenetheatingratecan 27 readilybecalculatedeitherfrom Qcoll=XlXu>lEulCul nu nlgu glexp Eul ktg! orbymultiplyingtheradiativerateineq.(3.18)by Eulandsummingupthe (3.22) contributionsfromalltransitions: Qrad=XlXu>lEulnuAulePe ul Jcont SLul 1! =XlXu>lEulAulePe ul nlgu gljul nu(1+jul)! ul (3.24) (3.23) thegainsandlossesfromthetranslationalandtheradiativepool(theonlyconsideredsourceterms)balanceeachother(cf.fig.2.1).thisequalitydemonstratesthe Bothexpressionsareequivalentandmustyieldthesameresult(Qcoll=Qrad),since physicalmeaningofthebasicassumptionofsteadystate:afastrelaxationofthe isassured.equation(3.23)showsthemodicationsfromtheusualexpression degreeofexcitationoftheconsideredspeciesisassumedsuch,thatqcoll=qrad PEulnuAulcausedbyopticalthicknessandincidentcontinuousradiation(note howeverthatthelevelpopulationsarealsoaected). probabilitiesdependonthelevelpopulations.inmostcases,adirect{iteration Thesolutionofthestatisticalequationsstillrequiresaniteration,sincetheescape 3.1.1.1NumericalIterationScheme eld{conditions,wherex(it)meansquantityxatiterationstepit: convergesrapidly,buttherearealsocasesinwhichthisprocedurefails.thefollowingscheme,whichmaybecalledadecelerated{iteration,convergesforall consideredmodelatomsunderallconsidereddensity{,temperature{andradiation 1.PuteS 2.CalculateeRul(it)andeRlu(it)accordingtoEqs.(3.20)and(3.21). 3.Determinenj(it)fromthestatisticalequations(3.1). ul(0)=0;epe ul(0)=1;qrad(0)=1099. 4.CalculateQcoll(it)andQrad(it)accordingtoEqs.(3.22)and(3.24). 8.CalculateePe 7.PuteS 5.Dene=j1 Qrad(it)=Qrad(it 1)j. 6.CalculateeS ul(it)=es ulaccordingtoeq.(3.16). 9.Gobacktostep2unless<10 10. ul(it)fromeq.(3.17). ul(it 1)+[eS ul es ul(it 1)]exp( maxf0;(it 30)=10g). 10.TakeQradasnalresult,ifthepopulationisclosetoLTE otherwise almostequallylargenumbers.) relyonqcoll(toavoidtheerrorsproducedbythesubtractionoflarge,

28 Thetotallineheating/coolingratedependsonthefollowingphysicalparameters: CHAPTER3.RADIATIVEHEATINGANDCOOLING Theparticledensityoftheconsideredspeciesnsp,theparticledensitiesofthecollisionpartners,thegastemperatureTg,thecontinuousbackgroundradiationeld Thefollowingatomicormoleculardataarerequired:thestatisticalweightsgiand energieseioftheconsideredlevels,theeinsteincoecientsforspontaneousemission Jcont ulandthelocalmeanvelocitygradientddv Aulandtheratecoecientsforcollisionalde{excitationCul(Tg)(wheretheparticle dle. toarbitraryconditionsofdensity,temperatureandradiationeldandcanbeapplied ofthetotallineheating/coolingrateofanarbitraryn{level{system.itisapplicable Thepresentedmethodisanuniversalandrapidlyconvergingtoolforthecalculation densitiesofthecollisionpartnersenterintothecalculation). makessense. 3.1.1.2DiscussionoftheApplicabilityofSobolevTheory toavarietyofowgeometries,asfarastheinvolvedescapeprobabilityconcept TheapplicationofSobolevtheorytotheshockedenvelopesofpulsatingstarsrequires somecriticalremarks: 1)Sobolevtheoryisapplicableonlyincaseoflargevelocitygradients,where thesizesoftheresonanceregions(whereanemittedlinephotoncanstillbe re-absorbed)aresmallcomparedtotypicalscaleheightsoftheenvelope.in caseofthermalbroadening,thisconditioncanbewrittenas Ddv vthmax(dln dle Regardingtheresultsoftime{dependentmodelsfortheenvelopesoflong{ dr;dlnt dr;dlnnsp periodicvariablestars(bowen1988,fleischer1994),thisconditionseemsto dr;:::): (3.25) bejustevenfullled.thethermalvelocitiesareafewkms 1,themean velocitygradient(cf.eq.3.14)istypically5to50kms 1=Randthescale frontsinthepost{shockregions,wherethetemperaturegradientscanbefairly height(ther.h.s.ofeq.3.25)istypically1r.problemscanoccurveryclose tothestar,wherethescaleheightcanbemuchsmaller,andclosetoshock 2)Duetothestrictdivisionbetweenconsideredlineandcontinuum,lineoverlaps large. areanintrinsicproblemofsobolevtheory.thesobolevtheoryrequiresthat theemittedlinephotonsofoneparticulartransitioncannotbere-absorbedby anyotherlinetransitionanywhereelseintheenvelope: v1c< l.h.s.ofeq.3.26)isabout10 4forAGB{stars,whichisusuallymuchsmaller Themaximumrelativeshiftofthelinesduetohydrodynamicvelocities(the (3.26)

3.1.BOUND{BOUNDTRANSITIONS thantherelativespacingoftheconsideredspectrallines(ther.h.s.ofeq.3.26, 29 whereisthefrequencydierenceoftwoconsideredlines).thiscondition wherether.h.s.ofeq.(3.26)isgivenby2hb=h!210 3(cf.Sect.3.1.4). Buteveninthiscase,condition(3.26)remainsvalid.Problemscanoccurin becomesmoreseriousforthevibrationalbandsofdiatomicpolarmolecules, 3)Theproblemofnon{monotonicvelocitygradientsinthesaw{toothlikevelocityeldsofCSEsofpulsatingstarscoupledwiththequestionofthedierence betweenlocalandglobalescapeisignored.thereby,theradiativeheating electronicbandsofmolecules,wherethespacingoftheindividuallinesiseven morenarroworinverynarrowspacedatomicmultiplets. 4)Closetothelocationofavelocitydiscontinuity(causedbyashockfront),most ofe.g.onepost{shocklayerbylineemissionsfromthepost{shockregionof anothershockwaveisneglectedinthiswork. coolingofotherwiseopticallythicklinesincreases,whichmaybeimportant justforthehotemittingpost{shockregionsdirectlybehindshockwaves. Therefore,largerescapeprobabilitiescanoccurinthiscaseandtheradiative ofthedisturbingabsorberaremissinginthedirectionsacrossthediscontinuity. Theadvantagesofthepresentedescapeprobabilitymethod,however,clearlyoutweightheseshort{comings.Aslongasnobetterandcomparablesimplemethodsare available,thesobolevtheoryisjusttheappropriatecompromisebetweensimplicity andaccuracyofthephysicaldescription.usingthistheory,theresultsoftheline heating/coolingratesareentirelydeterminedbylocalphysicalproperties(whichare availableinhydrodynamicmodels),stillincludingthemostimportantlinetransfer completely,whichwouldinducemuchlargererrors. 3.1.1.3AnExemplaryTwo{Level{Atom eects.theonlyrealalternativewouldbetoignoreopticaldeptheectsofthelines Inordertodemonstratethebasicfeaturesofthelineheating/coolingfunctions,an exemplarytwo{level{atomisexamined.thefollowing(typical)atomicparameters andphysicalconditionsareconsidered: g1=g0;e=k=10000k;a10=10 2Hz;C10=n<H>10 10Hzcm3 Theresultinglineheating/coolingratepermassQrad=asfunctionofthetotal nh:nh2:nhe:nsp=1:0:0:1:10 4;*dv dl+=20kms 1 hydrogenparticledensityn<h>isdepictedinfig.3.1forthecaseofnegligiblecontinuousradiationeldjcont=0.theguredemonstratesthefundamentaldensity{ 500R sitiesncrandnthick,whicharedenedbelow: dependingontherelationbetweenthedensityofthegasn<h>andtwocriticalden- dependenceofthetwo{levelcoolingrate.threedierentcasescanbedistinguished,

30 CHAPTER3.RADIATIVEHEATINGANDCOOLING I II III Figure3.1:Thecoolingrate(fulllines,leftaxis)andtheexcitationtemperature, denedbyktexc=eul=ln((gunl)=(glnu)),inunitsoftg(dashedlines,rightaxis) ofanexemplarytwo{level{atominthecaseofnegligiblecontinuousradiationeld Jcont =0. Figure3.2:Thetemperaturedependence 1010cm 3andJcont ofthelinecoolingratepermassforn<h>= catepointsalreadydepictedinfig.3.1. =0.Fullcirclesindi-Figure3.3:Thedependencyoftheline coolingratepermassontheradiationeld forn<h>=1010cm 3andTg=2000K.

3.1.BOUND{BOUNDTRANSITIONS I.n<H>>nthick:Thelineisopticallythickandthecoolingrateislimitedby 31 thelocalsurroundings.lteisvalid.inthelimitingcasen<h>!1the escapeprobabilityscalesaspe!1=s/ 1.Thereby,Qradbecomesdensity{ radiativetrapping,whereonlyafractionoftheemittedlinephotonscanescape II.ncr<n<H><nthick:Theatomisthermallypopulated(LTE)andthelineis independent. III.n<H><ncr:Theatomispopulatedsub{thermally(non{LTE)andthecooling opticallythin.thecoolingrateissimplygivenbythethermalrateofemitted rateislimitedbytherateofenergytransferredfromthegasviacollisions, photons,leadingtoqrad/. ncrdenotestheusualcriticaldensityforthermalpopulationandnthickcorresponds isfollowedbyspontaneousemission. whichyieldsqrad/2.inthelimitingcasen<h>!0,eachexcitingcollision tos=1.thecriticaldensitiesaredenedby nthick=8 ncr= C10=n<H> A10g0 A10 g110 c3*dv dl+n<h> nsp : (3.28) (3.27) ase.g.incaseofthelargeeinsteincoecientsofallowedtransitions,thecooling Insomecases,thetwocriticaldensitieswilloverlap(ncr>nthick).Ifthishappens, ratedirectlychangesfromtheqrad/2totheqrad=constbehavior. hence,forecientlinecooling4.afurtherincreaseofthetemperaturedoesnot temperatureofabouttg>e=5kisrequiredforecientcollisionalexcitationand, Figure3.2showsthe\freezingout"oftheconcerneddegreeoffreedom.Aminimum increasethecoolingratemuch.however,sinceatthesametimeothercoolinglines temperature{dependenceisuniqueforalldensities:theqrad(n<h>)-curveissimply enterintocompetitionandbecomemuchmoreecient,theconsideredspectral shiftedup-anddownwardsinfig.3.1accordingtothattemperature{dependence. densityandtemperatureconditionsinordertobeanecientcoolant.thedepicted linegetslessimportantincomparison.hence,aspectrallinerequiresveryspecial heatinginthecaseofintensecontinuousradiationeldsasshowninfig.3.3.this Linetransitionscancausenetcoolingofthegas,butcaninfactalsocausenet resultisstraightforward,butfundamentallydierentfromtheexperiencewithinterstellarmatter,wherethecontinuousradiationeldmaybeneglectedandwhereline transitionsgenerallycauseradiativecooling.equation(3.23)expressesthelinear dependencyshowninfig.3.3. Allthediscusseddependenciesofthetwo{levellineheating/coolingfunctionare concerningamulti{level{atom. outlinedinthiswork. quitegeneralandapproximatelyapplyalsototheotherheating/coolingmechanisms 4However,theconsideredlinecanstillbeinterestingforradiativeheatingasfarasEl<kTg

32 3.1.2LinesofAtomsandIons CHAPTER3.RADIATIVEHEATINGANDCOOLING envelopes,theselectionofspeciesandlinesiscrucial.theselectiondependsonthe consideredelementalabundancesandontheconsidereddensityandtemperature conditions.asarguedabove,oneshouldespeciallyincludeavarietyoflineswith Forageneraldiscussionoftheimportanceoflineheatingandcoolingincircumstellar atomicdata(especiallythecollisionrates)canbeproblematic.theselectionoflines inthisworkismainlybasedontheexperienceofhollenbach&mckee(1989).since theirworkrelatestointerstellarconditions,mainlythefew,low{lyinglevelsofthe dierenteulvalues(dierentspectralregions).furthermore,theavailabilityof Incaseoflargerdensities,wherethepopulationisgenerallyclosertoLTE,more moreabundantatomsandionsaretakenintoaccount. linesenterintocompetitionandthechosenselectionmaybeinsucient.especially tobeimportantsolelyatsmalldensities,atroublesomeexpansionofthelinelist oftransitionstobeconsidered).however,sincespectrallineshavegenerallyproven important(whichimmediatelycausestroublesduetotherapidlyincreasingnumber forhighgastemperatures,eventhepopulationofveryhigh{lyinglevelsmaybecome dominateatlargerdensitiesanyway. Therefore,onlyafewfurtherlines,whichsatisfytheconditions wouldberatherfruitless,becausethebound{freeheatingandcoolingrateswill largeelementalabundance, neutralorsinglyionized, lowexcitationenergyel, havebeenadditionallyincluded,especiallyfrommendoza(1983)andthereferences collisionaldataavailable largeaul,dierenteulvaluesand therein.thecompletenessofthemodelatomsisanothernecessarypreconditionfor non{lteinvestigations.if,forexample,atransitionwiththeprincipalquantum numberjump9!5looksinteresting,allthe55transitionsuptolevels9shouldbe takenintoaccount.table3.1summarizestheselectionofspeciesandlinetransitions inthiswork,comprising15speciesand85lines.thelistincludesmostoftheexisting lowerthan10 4)andwithneutralHatoms: withfreeelectrons(whichareusuallydominantunlessthedegreeofionizationis Theratesforcollisionalde-excitationareassumedtobegivenbythecollisionrates low{lyingenergylevelsoftheconsideredspecies. Cul(Tg)=neeul(Tg)+nHHul(Tg) xul(tg)=xultg=trefxul (3.29) ThecollisionalratesareoftenrepresentedasEq.(3.30),sothatforonecollisionrate usuallytwoparameters(ul,andul)aretobecollectedforelectronsandh{atoms. ThereareoftendierenttsfordierenttemperatureregimesTref.

3.1.BOUND{BOUNDTRANSITIONS Table3.1:Atomiclineheatingandcooling:consideredspeciesandtransitions33 H Levels(1) ul[m](2) n=1,n=2,n=3 0.1215,0.1025,0.656 Ref. He+n=1,n=2,n=3,n=4,n=50.0304,0.0256,0.0243,0.0237, 11S,23S,21S,23P0,21P0 1.56,1.08,0.887,3.56,2.06,4.88 0.0626,0.0601,0.0591,0.0584, 5 0.164,0.122,0.109, 4 C 3P0,3P1,3P2 0.469,0.321,1.01 609.2,229.9,369.0 6 C+ 3P,1D2,1S0 0.984,0.462,0.873 N 2P1=2,2P3=2 2P,4P 157.7 0.233 N+ 4S3=2,2D,2P 0.520,0.347,1.04 O 0.656,0.306,0.576 O+ 3P2,3P1,3P0 3P,1D2,1S0 63.1,44.2,145.6 0.633,0.297,0.558 4S3=2,2D5=2,2D3=2 0.373,0.372,508 Si+ 3P0,3P1,3P2 3P,1D2,1S0 129.6,44.8,68.4 1.62,0.653,1.10 S 2P1=2,2P3=2 2P,4P 3P2,3P1,3P0 34.8 0.224 25.2,17.4,56.6 S+ 3P,1D2,1S0 4S3=2,2D3=2,2D5=2,2P1=2,2P3=20.673,0.672,0.408,0.407,314.5, 1.10,0.459,0.773 1.034,1.029,1.037,1.032,213.2 5D4,5D3,5D2 24.0,14.2,34.2 4 Fe+ 5D4,5F5,5F4 6D9=2,6D7=2,6D5=2 1.44,1.36,22.3 26.0,15.0,35.4 (1):Levelsarelistedintheorderofenergy(rstlevel=groundlevel).Levelswithout 6D9=2,4F9=2,4F7=2,4D7=2 5.34,4.12,1.26,17.9,1.64,1.80 (2):Orderoftransitions:1!0fortwo-level-atoms,1!0,2!0,2!1forthree-levelatoms,1!0,2!0,3!0,2!1,3!1,3!2forfour-level-atoms,1!0,2!0, lowerindexaremultipletswhicharetreatedassinglelevels. 3 5:Luttermoser&Johnson(1992) 4:Mendoza(1983)andreferencestherein,Hul=10 12cm3s 1isassumed 3:Hollenbach&McKee(1989)andreferencestherein 3!0,4!0,2!1,3!1,4!1,3!2,4!2,4!3forve-level-atoms. 6:EinsteincoecientsfromMihalas(1978),collisionalde-excitationratesfrom Mihalas&Stone(1968)

34 Inconclusion,theselectionoflineshasbeenperformedinviewoftheimportancefor CHAPTER3.RADIATIVEHEATINGANDCOOLING (usuallyclosetothephotosphereofthestar)wherethedensityislarge.incontrast, however,refertoanopticaldepth1,i.e.toaparticularshellofthecse wouldprobablysuggesttoconsiderthoselines,whichcanbeseen.theselines, thegas notfromtheobservationalpointofview.atrstsight,anastronomer ceedsaccordingtothemethodsoutlinedinsect.3.1.1.eachrowintable3.1 thelineslistedintable3.1maynotevenoccurinthestellarspectra. sultsroughlyareasuperpositionofseveraltwo{level{typefunctionsasdepicted Thecalculationofthevariouslineheating/coolingfunctionsstraightforwardlypro- morecomplex,sincethepopulationofthelowerlevelchangesandtheupperlevel infig.3.1.thebehavioroflineswithlargerexcitationenergy,however,isusually istherebyconsideredasclosedmulti{level{systemwithprowni=nsp.there- canbepumpedbyanothertransitionetc.inarealphysicalsituation,theconcentrationsofthecarriersofthelinesnsp=n<h>additionallydependonthetemperature, thedensityandtheradiationeld.thesameoccursfortheelectronconcentration, whichisofcrucialimportanceforthecollisionrates. theradiativeenergyexchange.especiallythero{vibrationaltransitionsofabundantpolarmoleculeshaveproventobeimportantunderinterstellarconditions Assoonasmoleculesbecomeabundantinthegasphase,theyusuallydominate (e.g.neufeld&kaufman1993),intheatmospheresofcoolstarsandeveninthe mentofmoleculesinradiativetransferarisesfromthelargenumberoflinetransitions outeratmosphereofthesun(e.g.ayres1981).thegeneralproblemofthetreat- tobeconsidered.fornon{lteinvestigations,ahugeamountofmoleculardatahas tobecollected(individualradiativelifetimes,collisionratesetc.).thisprocedureis 3.1.3RotationalTransitionsofLinearPolarMolecules troscopicdetails,butinthetotaleectofmoleculesfortheradiativeheatingand nately,therearesomeapproximateanalyticalexpressionsavailableforcertaintypes ofmolecules(e.g.diatomicmolecules).sincewearenotinterestedinanyspec- onlyfeasibleforaveryfewwell{knownmoleculesandsubsetsoftransitions.fortu- coolingofthegas,theseanalyticalapproximationsarejustappropriate. Concerningtherotationaltransitionsoflinearpolarmolecules,thebasicmodelofa rigidrotatorprovidesthestatisticalweightsgjandenergiesejofthelevels.the EinsteincoecientsforthealloweddipoletransitionswithselectionruleJ!J 1 (forspontaneousemission)canbederivedfromtherotationalconstantbandthe dipolemomentd(chin&weaver1984).theratesforcollisionalde-excitationcul areadoptedfromhollenbach&mckee(1979) AJ!J 1=644 EJ=J(J+1)hB gj=2j+1 3hJ!J 1 (3.32) (3.31) c 32D2J+1 J (3.33)

3.1.BOUND{BOUNDTRANSITIONS Cul=0glhB ktgexp El ktgxinivth;i (3.34) 35 malvelocityand0isthetotalcollisionalcrosssection,whichisusuallyestimated wherej!j 1=2JBisthefrequencyofthetransition,vth;ithemostprobablether- vth;i=q8ktg=mred;i; (3.35) momentd,whichcanbetakenfromvariousmoleculardatatables,e.g.landolt{ arethetotalcollisionalcrosssection0,therotationalconstantbandthedipole Themoleculardatarequiredforthecalculationoftherotationalheating/coolingrate tobe10 15cm2. sideredmoleculesinthiswork. Bornstein(Hellwege1982).Table3.2summarizesthesemoleculardataofthecon- Table3.2:Vibrationalandrotationalheatingandcooling:considered Species(3)[K]1=10[Hz]B[MHz]D[D]0[cm2] speciesandmoleculardata OH CO 3084 5134 34.4 15.9 556141 57636 0.1098 1.667 HCN C2H CH 4113 115 425473 43675 44316 2.985 1.46 0.8 C2N SiC 2939 1830 {(1) 2.3 { 56694 11863 20298 1.45 0.6 1.7 SiO SiN SiS 1638 1769 1058 21882 21788 9077 3.098 1.73 2.3 (1):Thevibrationalheating/coolingfunctionofthismoleculecannotbe CS 1830 10(2) 24496 1.957 treatedaccordingtosect.3.1.4,sinceitisnotdiatomic. 10 15 (2):Estimated.Thecorrespondingnetvibrationalheatingfunction,however,isnotsignicantlyaectedbythechoiceofthisparameter (cf.sect.3.1.4). consideredinthiswork.inthecasec>o,wateritisalmostabsent (3):Sinceonlyapplicationsforcarbon{enrichedcasesaremade,H2Oisnot Thecalculationoftherotationalheating/coolingfunctioncanbeperformedsimilarlytothelastparagraph.Insteadofsolvingthestatisticalequations(3.1)forall fromthegasphase. consideredrotationallevelpopulationsnj,however(whichwouldalsobepossible, buttooelaborateforourpurpose),iusethefollowingapproximatemethodproposed

36 bykrugeretal:(1994).therotationalstatesareassumedtobepopulatedaccording CHAPTER3.RADIATIVEHEATINGANDCOOLING toaboltzmann{distributionwithayetunknownrotationalexcitationtemperature Trot: Zrot=kTrot nj=nmolgj hb Zrotexp EJ ktrot (3.37) (3.36) overtherotationalstatesineq.(3.22)byintegrals,itcanbeshownthatthetotal rateofcollisionalenergytransfersimpliesto BymeansofEqs.(3.31),(3.32),(3.34),(3.36)and(3.37)andbyreplacingthesums Therotationaltemperatureisfoundbyiteration,untilthebothresultsforQcoll Qcoll=0nmolk(Trot Tg)Xinivth;i: (3.38) intheindividuallines,istherebycarriedoutovertherstjmax=(7ktrot=hb)1=2 (typically102)rotationalstatesbypl;u!pjmax heatingrateaccordingtoeq.(3.24),whichproperlyincludestheopticaldeptheects andqradfromeqs.(3.38)and(3.24)areequal.theevaluationoftheradiativenet 97%ofthetotalthermalemissionrateintheopticallythinlimit. J=1(u=J;l=J 1),yieldingabout 3.1.3.1RotationalHeatingandCoolingbyCO Forexample,therotationalheatingcoolingfunctionofCOisbrieydiscussed,which outlinedintable3.2andthefollowingphysicalconditionsareconsidered: isofspecialimportanceduetoitslargeabundance.themoleculardataofcoare Figure3.4depictstheresultsforthecaseJcont nh:nh2:nhe:nco=0:1:0:2:10 3;*dv =0.Thedensity{dependenceofthe dl+=20kms 1=500R (withthecriticaldensitiesncr106cm 3andnthick109cm 3forCO,cf.Eqs.3.41 and3.42below).duetotheincreasingpopulationofthehigherrotationallevels andthesmallerradiativelifetimesoftheselevels,however,therotationalcooling rotationalcoolingfunctionisgenerallysimilartoatwo{level{typecoolingfunction 3.1.3.2Fast,ApproximateMethod functionscalesasqrot/t2g,whichisdierentfromatwo{level{typecoolingfunction. upperresultscanbeused,ifthecontinuousradiationeldinthemicro{wavespectral (e.g.forthemodelcalculationsinchapter7).insuchcases,thefollowingttothe calculationoftherotationalheating/coolingfunctionsmaybetootimeexpensive Forcertainapplications,eventherathersimplemethoddescribedaboveforthe

3.1.BOUND{BOUNDTRANSITIONS 37 Jcont Figure3.4:RotationalcoolingrateandexcitationtemperatureofCOincase regiontslikejcont =0.Arrowsindicatethetrendforincreasinggastemperature. WB(Trad)W2kTrad(=c)2(Rayleigh{Jeansapproximation): Qrot;LTE=102442DB2 Qrot=Qrot;LTE n<h>+1+n<h> ncr nthick! 1 3c3h2 nmolk2tgwtrad Tg (3.39) ncr=102442db2ktg 3h2c30vth (3.40) nthick=0:08ktg 2DB*dv dl+n<h> nmol ; (3.42) (3.41) centrationsofthecollisionpartners.consideringtypicalastrophysicalrelevant molecules,thecriticaldensitiesforthermalpopulationoftherotationalstatesncr wherevth=n 1 rangebetweenabout105cm 3(e.g.SiS)and108cm 3(e.g.HCN). <H>Pinivth;iisthemeanthermalvelocitywithrespecttothecon- beanalyticallyderivedfromeq.(3.23).equation(3.40)expressesthedependencies atthecriticaldensities,<10%elsewhere).qrot;lteistherotationalheating/cooling functionincaseoflte(trot=tg)andvanishingopticaldepths(epe Equation(3.39)isaveryusefultformulawithacceptableaccuracy(error<35% oftherotationalheating/coolingfunctionuponthetemperatureandtheradiation ul=1),whichcan

38 eld.astherotationalfrequenciesarelocatedinthemicro{wavespectralregion CHAPTER3.RADIATIVEHEATINGANDCOOLING (cf.their{limitinfig.1.1).theoppositecaseismuchmoreprobable:therotationaltransitionswillalmostalwayscausenetradiativecooling.accordingto occursincasewtrad>tgwhichseemsunlikelytooccurincircumstellarenvelopes (h1!0=k=2hb=k5:5kforco),radiativeheatingviarotationalpumpingsolely thecomparativelyweaktemperature{dependence,rotationalheating/coolingisespeciallysignicantatlowgastemperatures.therelevanceofaconsideredmolecule scalesasnmol2db2,whichisimportantforthechoiceofthemoleculestobetaken intoaccount. ing/coolingmechanismforthegas.thevibrationalspectraofpolyatomicmolecules Theallowedvibrationaltransitionsofpolarmoleculesalsoprovideaneectiveheat- 3.1.4VibrationalTransitionsofDiatomicPolarMolecules arealreadyrathercomplex,sothatnoclosedanalyticalexpressionsforthemeanradiativelifetimesandthecollisionratesareknown.therefore,thisworkrestrictsto thevibrationaltransitionsofdiatomicpolarmoleculeswithselectionrulesv!v 1 heating/coolingbypolyatomicmoleculesprobablyisasecond{order{eect5.the themostabundantpolarmoleculesinthegasphase(e.g.co).thevibrational correspondingwavelengthstypicallyrangefrom4mto12m6. ;J!J1(forspontaneousemission).Fortunately,thesemoleculesareusually rotatorisapplied,whichissucientforthepurposeofthiswork. Forthelevelenergies,themostsimplemodelofaharmonicoscillatorandarigid J!J+1=v644 Ev;J=h!v+12+J(J+1)hB gv;j=gj (3.43) J!J+1!3(TM)2J+1 (\P{branch") (3.45) (3.44) Av!v 1 J!J 1=v644 3h v!v 1 J!J 1 c C10= 1 exp( =Tg)XiniexpBi AiT 1=3 ktg=1atm 2J+1 g(\r{branch") (3.47) (3.46) Ai=1:1610 3mred;i =h!=k 1amu1=24=3 (3.49) (3.48) Cvv0=(v v0)c10exp Bi=18:42+0:015Aimred;i (v v0 1)1:5=Tg 1amu1=4 1+1:5=Tg! (3.51) (3.50) 5AnexceptionistheH2Omoleculeincaseofanoxygenrichelementalcompositionofthegas. 6Notethatovertonetransitionsarenotconsideredhere(cf.discussioninSect.3.6).

3.1.BOUND{BOUNDTRANSITIONS visthevibrationalquantumnumber,!theeigenfrequencyoftheharmonicoscillator 39 andtmitstransitionmoment,whichisrelatedtothemeanradiativelifetimeof analyticalrepresentationoftherateofcollisionalde-excitationoftherstvibrational therstexcitedvibrationalstatevia1=10=a1!0 representationoftheeinsteincoecientsisadoptedfromnuth&donn(1981).the statec10istakenfrommillikan&white(1964).thelandau{tellercoecients J!J+1+A1!0 J!J 1.Theanalytical accordingtoeqs.(3.49)and(3.50).thecorrespondingcollisionalcrosssections systems"(diatomicmoleculeplusatomordiatomicmoleculeascollisionpartner) forvibrationalde-excitationaremuchlessthanthegeometriccrosssectionsofthe AiandBiaretobedeterminedbyexperimentsorcanbeestimatedfor\simple moleculesandshowastrongtemperature{dependence.thecollisionalde-excitation Equations(3.43){(3.51)formausefulsetofapproximateanalyticalexpressionsfor \surprisalanalysis"(elitzur1983). ratesforhigherquantumnumbersv>v0accordingtoeq.(3.51)areestimatedby therequiredmoleculardataintermsofafewbasicquantities,whicharetheeigenfrequency!,therotationalconstantbandthetransitionmomenttm(orthemean availableonlyforafewwell-knownmolecules(fromlaboratoryexperimentsorab caneasilybeobtainedfromvariousmoleculardatatables,whereasthelatteris lifetimeoftherstexcitedvibrationalstate10,respectively).thersttwodata Ofcourse,moreaccurateEinsteincoecientsandcollisionaldatacanbeusedfor to100hz.theobviousadvantageoftheanalyticalexpressionsaboveistheirbroad applicabilitytodiatomicpolarmolecules.thedisadvantageisthemodestaccuracy. initoquantummechanicalcalculations).typicalvaluesfor1=10rangefromabout1 ingtoboltzmanndistributions: Asinthelastsection,thero{vibrationalstatesareassumedtobepopulatedaccord- particularmolecules,ifavailable. Zvib= nv;j=nmolgj 1 exp( h!=ktvib) ZvibZrotexp 1 vh! ktvib J(J+1)hB ktrot! (3.53) (3.52) againfoundbyiteration,untiltheresultsforqradandqcollderivedfromeq.(3.24) andeq.(3.22)areequal.equation(3.22)istherebyevaluatedsolelyforthevitationalheating/coolingfunctionandthevibrationalexcitationtemperaturetvibis Therotationaltemperatureisconsideredasknownfromthecalculationofthero- l=fv 1;J1g),whichyields98%ofthetotalthermalemissionrateinthe rate.equation(3.24)isevaluatedaccordingtopl;u!pvmax brationalstatesandrestrictedtotherstvmax=1+6ktvib=h!vibrationallev- elspl;u!pvmax v=1pv 1 opticallythinlimit. v0=0(u=v;l=v0),yieldingabout99%ofthetotalcollisional v=1pjmax J=0(u=fv;Jg;

40 CHAPTER3.RADIATIVEHEATINGANDCOOLING Figure3.5:VibrationalcoolingrateandexcitationtemperatureofCOincase 3.1.4.1VibrationalHeatingandCoolingbyCO Jcont =0. tion.incaseofco,moreaccuratecollisionaldataareavailable:the1!0rate andtheconsideredvelocitygradientandgasabundancesaregiveninthelastsec- ofthecomoleculeiscalculated.themoleculardataforcoaregivenintable3.2 Inordertoillustratetheoutlinedprocedure,thevibrationalheating/coolingfunction seehollenbach&mckee1989).landau{tellercoecientshavebeenexplicitlymeasuredfortheco{hecollisions(millikan&white1964). Figure3.5depictstheresultsforthecaseJcont &Hollenbach1994)andforcollisionswithH2moleculesfromRosenbergetal:(1971, coecientsforcollisionswithhatomsaretakenfromglassgold(1993,seeneufeld essentiallyisatwo{level{typecoolingfunctionandconsequentlyshowsallthefeaturesdiscussedinsect.3.1.1.3.thehighervibrationallevelsv2areusually =0.Thevibrationalcoolingrate notverysignicant.accordingtothelargeeinsteincoecientsofthevibrational transitions,themaximumpossibleemissionrateintheopticallythinltecase Qrad/2totheQrad=constcaseataboutn0cr1011:5cm 3forCO.Thebasicslope sitions.consequently,thevibrationalcoolingfunctiondirectlychangesfromthe isneverrealized,becausetheemissionislimitedeitherbyinsucientcollisional pumpingorbyradiativetrapping,whichisthetypicalbehaviorofallowedtran- ofthetemperature{dependenceisthesameasdepictedinfig.3.2,althoughfor

3.1.BOUND{BOUNDTRANSITIONS temperaturestg>,thehighervibrationallevelscausesomemodications.the 41 vibrationalband. tedphotonsarespreadamongthenestructureofthep{andr{branchofthe sensibilityofthevibrationalheating/coolingtoopticalthickness,sincetheemit- maindierencetoanordinarytwo{level{typecoolingfunctionarisesfromtheweak 3.1.4.2Fast,ApproximateMethod assumedthatthebackgroundradiationeldisconstantoverthevibrationalband andequalsjcont formulaisdesigned,whichcanbeappliedintime{criticalmodelcalculations.itis Similartotherotationalheating/coolingfunctionintheprevioussection,afastt 1!0:Qvib=Qvib;LTE Qvib;LTE=nmol n<h>+1+n<h> ncr nthick! 1 exp(=tg) 1 h! B1!0(Tg) 1! Jcont (3.54) ncr=n<h> 10C10 (3.55) nthick=26:410*dv dl+ktg hb h! hc!3n<h> nmol (3.56) n0cr=pncrnthick (3.58) (3.57) occursincasejcont elsewhere.thedependencyofthevibrationalheating/coolingfunctionontheradiationeldisexpressedbyeq.(3.55),oncemoreindicatingthatradiativeheating Theaccuracyofformula(3.54)isabout35%atn<H>n0crandbetterthan10% exchangerateincaseoflte(tvib=tg)andnegligibleopticaldepths(epe Althoughthismaximumpossiblerateisusuallynotrealized(seeabove),itscales theresultsasformulatedineq.(3.54).asfarasncr>nthicksisvalid,thevibrational >Bandradiativecoolingotherwise.Qvib;LTEistheenergy heating/coolingrateisalmostentirelyindependentfromthemeanlifetime10.this ul=1). ingtypicalvaluesfor10andc10fordiatomicpolarmoleculesandgastemperatures diatomicpolarmolecules,forwhichthe10{valuesarenotexactlyknown.consider- 500 2000K,thecriticaldensitiesforthermalexcitationofthevibrationalstates allowsforthedeterminationofthevibrationalheating/coolingratesalsoofthose ncrareoftheorder1010 1017cm 3,whichduetoradiativetrappingareusually signicantlyreduced(ncr!n0cr)byupto4ordersofmagnitude7.consideringthe extremelytemperature{dependentandthevibrationalenergiesofthereactantsmaybeinvolved. rotationalstates).chemicalreactionsmightbeaectedbytheseeects,sincemanyreactionsare polarmoleculescanbeexpectedincircumstellarenvelopes(incontrasttothepopulationofthe 7Consequently,strongnon{LTEeectsconcerningthepopulationofthevibrationalstatesof presentedbycherchneetal:(1992). grainsintheseenvelopes.arstapproachtohandlereactantsofdierenttemperatureshasbeen Thissituationmayhavesevereconsequencesforthechemistryandalsoforthenucleationofdust

42 mostlyrealizedcaseqvib/2,theimportanceofamolecularspeciesunderexaminationconcerningitscontributiontothetotalheating/coolingofthegasscalesas CHAPTER3.RADIATIVEHEATINGANDCOOLING nmolh!c10. betweenthegasandtheradiationeld,becausethesemoleculeshaveextremely 3.1.5QuadrupoleTransitionsofH2 Unpolarmoleculesmayusuallybeneglectedconsideringthetotalenergyexchange smallradiativetransitionprobabilities.theh2molecule,however,maybesucientlyabundantinordertocompensateforthis.itsro{vibrationalquadrupole Theradiativeheating/coolingfunctionofH2iscalculatedanalogouslytoSect.3.1.4. transitionsareknowntobesignicantinwarminterstellarcloudsandarelocated roughlybetween1mand25m. Sect.3.1.4.Thelevelenergiesarederivedfromthespectroscopicconstants tionprobabilitiesmustbeused,whichmeansamuchlargerexpensecomparedto Sincenoanalyticalexpressionsareavailable,anextensivelistofindividualtransi- E(v;J)=hcwev+12 wexev+122+bvj(j+1) DeJ2(J+1)2 we=4401:2;xe=121:33;bv=60:853 3:062v+12;De=0:0471[cm 1] (3.59) asgivenbyhuber&herzberg(1979)andtheeinsteincoecientsforspontaneous emissionofthe(forbidden)ro{vibrationalquadrupoletransitionsv!v0;j! vibrationaltransitions). Thecollisionalvibrationalde-excitationrates1!0forH{atomsandH2{molecules fj 2;J;J+2garetakenfromTurneretal:(1977),wherealltransitionswithv5 areadoptedfromlepp&shull(1983)andreferencestherein.thoseforhe{atoms andj20aretakenintoaccount(comprising114purerotationaland898ro{ Eq.(3.51).Thecollisional(de{)excitationoftherotationalstatesisnotconsidered collisionalratesforthehighervibrationalstatesareagainestimatedaccordingto areestimatedaccordingtoeq.(3.47)withahe=145:5andbhe=20:77.the approximationisreliable,unlessthegasdensityislowerthan105cm 3. indetailhere instead,therotationaltemperatureofh2isassumedtoequal thegastemperature.accordingtothecalculationsoflepp&shull(1983),this IftheopticaldepthsinthelinesareneglectedasassumedintheworkofLepp&Shull (1983),theirresultscanbereproducedwithinamaximumfactorof2(generally 105cm 3,provingthatthepresentedmethodincludingtheintroductionofexcitationtemperaturesworksproperly.Figure3.6showstheresultsoncemoreforthe casejcont =0,nH:nH2:nHe=0:1:0:2andDdv muchbetter)forallconsideredgastemperaturesandfordensitieslargerthan opticallythick.thevibrationalcoolingrateismoreimportantforhightemperatures(tg>1000k)andhighdensities,whereitexceedstherotationalcoolingrate densitiesunlessn<h>>1012cm 3,whereeventhequadrupoletransitionsbecome totheassumptiontrot=tg,therotationalcoolingrateisproportionaltoforall dle=20kms 1=500R.According

3.1.BOUND{BOUNDTRANSITIONS 43 coolingrate(thinfulllines),therotationalcoolingrate(dottedlines,leftaxis) Jcont Figure3.6:Thetotalquadrupolecoolingrate(thickfulllines),thevibrational andthevibrationalexcitationtemperature(dashedlines,rightaxis)ofh2incase byaboutoneorderofmagnitude.thecriticaldensityforthermalpopulationofthe =0. vibrationalstatesoftheh2moleculestronglydependsonthegastemperatureand rangesfrom106to1010cm 3. ThetotalcontributionofH2heatingandcoolingroughlystaysproportionaltothe typicalforforbiddenlines.incomparisontootherheating/coolingrates,which anaccuracyofaboutoneorderofmagnitude).thisbehaviorisanaturalconsequencefromthelowtransitionprobabilitiesofthequadrupoletransitions,andis gasdensityoverthewholeconsidereddensityrangeofcircumstellarenvelopes(with decreaseasqrad/2forsmalldensities,theh2quadrupoleheating/coolingisespeciallysignicantatlowdensity(e.g.interstellar)conditions.

44 3.2 Bound{FreeTransitions CHAPTER3.RADIATIVEHEATINGANDCOOLING Bound{freetransitions(photoionisationandradiativerecombination)generallyprovideimportantheatingandcoolingratesassoonasconsiderablefractionalionization i)astronguvradiationeldispresentwhichcausesboth,photoionisation andnetradiativeheatingofthegas.thiscaseisgenerallyrealizedinthe ispresentinthegas,whichoccursinthefollowingtwocases: ii)thegasisdenseandhot,sothatcollisionalionizationcausesconsiderable overwhelmingpartoftheism(exceptforthedenseandshieldedmolecular fractionalionization.hightemperatures(>8000kforhydrogen)areusuallyrequiredforeectivecollisionalionizationwhich,followedbyradiative clouds),wheretheinterstellaruvradiationeldinteractswiththegas. dependontherelationbetweenthegastemperatureandthepresentuvradiationeld,seebelow).thecompetitiveprocessesofcollisionalionization recombination,preferablycausesnetcoolingofthegas(thedetails,however, andthree{bodyrecombinationarefurthermoreresponsibleforkeepingtheionizationequilibriumclosetolteinstellaratmospheres.inreturn,thelarge ConsideringthephysicalconditionsinCSEs,largefractionalionizationespecially radiativeheatingandcoolingofthegasase.g.intheatmospheresofhotstars. bound{freeopacitiesinthecaseofltecontroltheradiativetransferandthe beexpected. TheconditionsinthepredominantlyneutralCSEsofcool(e.g.AGB)starsdo alreadysucientlyintensetocauseconsiderablebound{freeradiativeheatingcan occursaroundwarmandhotstars,wherethephotosphericuvradiationeldis generallynotfavorlargebound{freeheating/coolingrates.thereare,however, importantexceptionsfromthisrule:first,ifchromosphericactivityispresent, interstellaruvradiationeldcanpenetrateintotheconsideredlayerofthecse,it radiativeheatingbybound{freetransitionsofthegascanbeeective.second,ifthe followedbyradiativerecombinationcanbeanimportantcoolingprocesses. thehotpost{shockgaslayersinthecsesofpulsatingstars,collisionalionization willcauseconsiderablefractionalionizationandradiativeheating.third,concerning 3.2.1TheRateEquationsforanN{LevelSystemwithContinuum ined.besidesthebound{boundprocessesdiscussedbefore,theprocessesofpho- (denotedby\ii")fortherstionizedstateoftheconsideredspeciesisexamtoionisation,radiativerecombination,collisionalexcitationandthree{bodyrecombinationaretakenintoaccount.analogouslytosect.3.1.1thelevelpopulations AlevelsystemconsistingofNboundelectronicstatesplusoneadditionallevel n1;n2;:::;nn;niiarederivedfromthestatisticalequations(3.1),assumingthatthe netproductionratesofallconsideredstatesvanish(steady{statenon{lte).this

3.2.BOUND{FREETRANSITIONS assumptionismorerestrictiveinthissection,becausethetimescaleforrelaxation 45 circumstances(e.g.inthecaseoflowfractionalionization)andmightexceedother, e.g.hydrodynamictimescales.furthermore,thelevelsystemisassumedtobe paredtothetimescalesforrelaxationoftheexcitedboundstatesundercertain towardsionizationequilibrium(niipiriii=piniriii)canbemuchlargercom- \closed"inthesensethatotherprocesses,whichmightprovideadditionalsource termsfortheparticledensitiesoftheneutralandsinglyionizedatoms(e.g.chemical Theratecoecientsforthebound{boundtransitionsaregivenbyEqs.(3.20)and reactions,chargeexchangereactions)areneglected. (3.21),whereasthoseforthebound{freetransitionsareformulatedaccordingto Mihalas(1978): RiII=41 ithrj Zhbf RIIi= ne i()d+nei(tg) (3.60) Si(Tg) 0 B@41 ithr Z 22 Si(Tg)=2ZII gi(2mektg)1:5 c2+j h!exp h ktgbf i()d+nei(tg)1ca(3.61) ilabelsaboundstate,bf h3 exp i ktg! i()isthebound{freeabsorptioncrosssectionandi(tg) (3.62) theratecoecientforcollisionalionization(weonlyconsidercollisionswithelectrons Thetotalradiativeheating/coolingfunctionofsuchamulti{levelsystemcomprisescontributionsfrombound{boundtransitions,whicharecalculatedaccording toeq.(3.24)andfrombound{freetransitions: Qbf rad=4nxi=11 ithr ZniJ niine Si(Tg)2h3 partitionfunctionoftheionizedstate. i=hthecorrespondingthresholdfrequency,si(tg)thesahafunctionandziithe here).iistheenergydierencebetweenthei-thlevelandthecontinuum,ithr= Itisimportanttonotethattheevaluationoftheradiativeheating/coolingratedependsonthedenitionoftheinternalenergy.Inthiswork,thetotalabsorbed/emitted photonenergyiscalculatedandtheionizationenergiesioccuraspotentialsinthe thiscase,nointernalionizationpotentialsareconsidered,butanadditionalfactor internalenergy(cf.eq.2.2).concerningotherpublications,theradiativeheating andcoolingratesoccasionallyrefertothepooloftranslationalenergyalone.in c2+jexp h ktg!bf i()d (3.63) functionzii,onlybf (h i)=happearsineq.(3.63),describingthegainorlossofpuretranslational energy. Besidesthedataforthelevelenergiesi,thestatisticalweightsgiandthepartition i()andi(tg)arerequiredforthecalculationofthebound{free

46 radiativeheating/coolingfunctions.theso{calledphoto{recombinationcoecients CHAPTER3.RADIATIVEHEATINGANDCOOLING Fortheactualsolutionoftheoutlinedsystemofequations,allintegralsareevaluated relationsforbound{freetransitions,whicharealreadyincludedineqs.(3.61,3.63). areprincipallynotneeded8,sincetheycanbededucedfromtheeinstein{milne boundtransitions,wherethesameprocedureasoutlinedinsect.3.1.1.1isapplied. includingniirequiresan(inner)iterationoftheescapeprobabilitiesofthebound{ Thesystemofequationsiswell{denedforgiventotalparticledensitynsp=nII+ numerically.thesolutionofthestatisticalequations(3.1)forthelevelpopulations Pni,giventemperatureTg,givenradiationeldJandgivenelectrondensityne. conservationne=pnii,comprisingallionsunderconsideration,whichinreturn Another(outer)iterationisnecessarytoachievethephysicalconditionofcharge yieldstheelectrondensity. Accordingtotheoutlinedequations,thedegreeofionizationofthegasandthe bound{freeheating/coolingratesarecalculatedsimultaneously.opticaldepthseffectsarenotincludedconcerningthebound{freetransitions incontrasttothe bound{boundtransitionsdiscussedbefore,whereitwaspossibletoapplysobolev theory9.thisproblemcouldonlybehandledbymeansofnon{local(non{lte) assumethegastobeopticallythininthecontinuumandputj=jcont theradiativeheatingandcoolingofsinglegaselements,weignoretheseeects, radiativetransfercalculations.sincethebasicapproachofthisworkistodetermine,wherejcont 3.2.1.1Fast,ApproximateMethod isthecontinuousbackgroundradiationeld. cientsas ing/coolingfunctionscanbederivedbyintroducingthephoto{recombinationcoef- Ausefulandquiteilluminatingformforthegeneralbound{freeratesandheat- i(t)=4 Si(T)1 ithr22 Z ThesecondpartofEq.(3.64)providesacommontformula,wheretheparameters c2exp h ktbf i()daitbi (3.64) aiandbiareoccasionallystatedintheliterature.asfarasstimulatedbound{free emissioncanbeneglected(whichusuallyisaveryaccurateapproximationinthe UV10),therecombinationratescanbere-writtenas RIIi=nei(Tg)+ Si(Tg)i(Tg): n2e lations,seebelow. 8Thephoto{recombinationcoecients,however,areveryusefulforquick,approximatecalcu- (3.65) canbelocatedintheopticalorevenirspectralregion. drivethegastowardslte{ionizationalreadyatcomparativelysmallergasdensities. 10Note,however,thatthecorrespondingwavelengthsofrecombinationstohighlyexcitedstates 9Suchopticaldeptheectsareexpectedtoreducethebound{freeheating/coolingratesand

3.2.BOUND{FREETRANSITIONS IfthecontinuousradiationeldtslikeJWB(Trad)W2h3=c2exp( h ktrad) 47 thephoto{recombinationcoecient: (Wienapproximation),alsotheionizationratescanpartlybeexpressedintermsof thenetbound{freeheatingratethenreducesto Bydeterminingthederivative@i=@(1=kT)fromEq.(3.64),itcanbeshownthat RiII=WSi(Trad)i(Trad)+nei(Tg) (3.66) Qbf rad=nxi=1niwsi(trad)i(trad)hhiabs hhiem i i=i+(1:5+bi)ktg; =i+(1:5+bi)ktrad i niinei(tg)hhiem i (3.68) (3.67) Equation(3.67)isexactasfarastheupperconditionsarevalidandthederivative wherehhiabs=emisthemeanabsorbedandemittedphotonenergy,respectively. (3.69) dzii=dtcanbeneglected.thebigtechnicaladvantageofeqs.(3.65),(3.66)and parameters(aiandbi)havetobeknownforeachconsideredbound{freetransition. (3.67)isthatnointegralsoccurandthatinsteadofafunction(bf tonsequalsthenumberoffree{boundemittedphotons(asinthecaseofnegligible Equation(3.67)demonstratesthatevenifthenumberofbound{freeabsorbedpho- i())onlytwo collisionalionization),thenetrateoftransferredenergydoesusuallynotvanish,in contrasttoallbound{bound{typetransitionsdiscussedintheprevioussection.the energy.inthecaseofthermodynamicequilibrium,however,wherej=b(tg)and ni=niine=si,thenetbound{freeradiativeheatingrateaccordingtoeq.(3.63)or reasonlieswithintheintegrationovertheabsorbed/emittedphotonspectrum,since accordingtoeq.(3.67)isindeedzero asdemandedbydetailedbalance. themeanabsorbedphotonenergyusuallydiersfromthemeanemittedphoton Themostsimplecaseoccurs,ifsolelythegroundstateoftheneutralatomisconsideredandifthecollisionalionizationratesareneglected.Fromtheconditionof steadystaten1r1ii=niirii1itfollowsthatinthiscasethenetheating/coolingrate simpliesto (independentofthevalueofthedilutionfactorw),whichcorrespondstotheuv{ Inthiscase,radiativeheatingoccursforTrad>Tgandradiativecoolingotherwise Qbf rad=niine1(tg)1:5+b1k(trad Tg): (3.70) limitdepictedinfig.1.1.

48 3.2.2TheH{AtomCHAPTER3.RADIATIVEHEATINGANDCOOLING However,thehigh{lyingenergylevelsofhydrogenmakeitalmostinaccessiblefor thetotaldegreeofionizationandtheradiativeheatingandcoolingofthegas. Accordingtoitsoverwhelmingabundance,hydrogenisalwaysimportantforboth, collisionalexcitationandcollisionalionizationatlowertemperatures,sothatthe andphysicalconditionsareconsidered: oftherstthreeboundlevelsandtheionizedstate11.thefollowingdataofhydrogen Fordemonstration,apurehydrogenplasmaisexaminedinthefollowing,consisting importanceofhydrogenismainlyrestrictedtohightemperatures. n=13:598ev=n2;gn=2n2;zii=1;bf n<h>=n1+n2+n3+nii;ne=nii;*dv dl+=20kms 1=500R n()=2:815(+29)gnii n53 whichareoftheorderofunity.thecollisionalionizationratecoecientsn(tg)are takenfromluttermoser&johnson(1992)andthereferencestherein.thehydrogen abbreviationx(y)=x10yincgs{units.gniiarethebound{freegauntfactors, Thebound{freeabsorptioncrosssectionsaretakenfromMihalas(1978)withthe bound{freetransitionsii!1(lyman{continuum),ii!2(balmer{continuum)and II!3(Paschen{continuum)arecalculatedbymeansoftheexactequationsgivenin Sect.3.2.1.Thetreatmentofthehydrogenbound{boundtransitions2!1(Ly), Figures3.7and3.8showtheresultingtotal(bound{freeplusbound{bound)cooling 3!1(H)and3!2(H)hasbeendescribedinSect.3.1. ratesofhydrogen.twoguresarepresentedhere,sincethedegreeofionizationand plotsbefore. eld,whichischosentobezerointherstandtoequalaplanckianof3000kin thesecondgure.notethescalingofthey{axiswhichisdierentfromtheother theheating/coolingratesstronglydependonthecontinuousback{groundradiation heatingandcoolingisfoundtobeunimportantfortg<6000k.howeverforhigher gastemperatures,hydrogencoolingsoonbecomesecientandnallyhydrogen Comparedtothemagnitudeoftheheating/coolingratesdiscussedsofar,hydrogen providesthedominantcoolingrateofthegasattemperaturesabove8000k. Thistemperature{dependencyisaconsequenceofthehigh{lyingenergylevelsof temperatures. ThetotalhydrogencoolingrateisfoundtoscaleroughlyasQ/2,whichisan indicatorforstrongnon{lteeectsinthelevelpopulations12. hydrogen,whichcanbecollisionallyexcitedorionizedsolelyinthecaseofhighgas ofhydrogentotheelectrondensity. stellarenvironmentsfordescribingaccuratelyboththeemergenthspectrumandthecontribution 11Luttermoseretal:(1989)haveshownthatathree{levelmodelforhydrogenissucientincool Sect.3.1.Concerningthebound{freeheating/coolingfunctions,wehavenIIneSiniinLTE, whichaccordingtoeq.(3.63)alsoimpliesq/asfarashydrogenismostlyneutral. 12LTEwithoutopticaldeptheectsimpliesQ/forspectrallinesasalreadydiscussedin

3.2.BOUND{FREETRANSITIONS 49 Figure3.7:Thetotal(bound{freeplusbound{bound)hydrogencoolingrate (fulllines,leftaxis)andthedegreeofionization(dashedlines,rightaxis)inthe casewithoutcontinuousradiationeld. Figure3.8:SameasFig.3.7,butwithanunderlyingcontinuousradiationeld.

50 ThedegreeofionizationinFig.3.7showsastep{likebehavior.Thisisaneect CHAPTER3.RADIATIVEHEATINGANDCOOLING density,lyandforlargerdensitiesalsohbecomeopticallythick.consequently, causedbythevaryingopticaldepthsofthehydrogenlines:withincreasinggas theeectiveradiativebound{boundratesaccordingtoeqs.(3.20)and(3.21)becomenegligiblecomparedwiththecollisionalrates,forcingtheupperlevelofthdersofmagnitude,leadingtosuccessivelyenhancedelectronconcentrationsfrom consideredtransitiontoachievethermalpopulationwithrespecttothelowerlevel. Therefore,thecollisionalionizationratefromthatupperlevelisincreasedbyor- therighttotheleftinfig.3.7.infig.3.8,thisbehaviorissmearedout,sincethe ratesforphotoionisationenterintocompetition. ThehydrogennetcoolingratesinthecaseJ=B(3000K)depictedinFig.3.8 arefoundtobelargerthaninthecasej=0,sincephotoionisationproduces considerablyhigherelectronconcentrations,providingmorecollisionpartners13. Furtherdetailsconcerningtherelativecontributionsofthedierenttransitionsand thelevelpopulationsarepresentedinfig.3.9,consideringthecasetg=8000kand J=0.Theb{factorsfordeparturesfromLTEarecalculatedas ne= bi=ni=ni=nisi(tg)=n2 0:5+q0:25+n<H>(1=S1+1=S2+1=S3) e (3.72) (3.71) Hydrogenbound{freecoolingisfoundtodominateinhotdensemedia,whereas particulardensity,whichdependsontheconsideredgastemperatureandvelocity withincreasinggasdensity.thetransitionbetweenthesetwocasesoccursata emissioninhydrogenlinesdominatesthecoolingofahotthingas,whichisa straightforwardconsequenceoftheincreasingopticaldepthsinthehydrogenlines gradient.infig.3.9,thistransitiondensityisaboutn<h>=1010cm 3.Thebumps onthetotalcoolingratesdepictedinfig.3.7correspondtothesetransitions. TheLyman{continuumalwaysprovidesthemostimportantbound{freecooling outbyhollenbach&mckee(1989)andneufeld&hollenbach(1994),althoughly Lyisusuallythemostimportanthydrogencoolinglineashasalreadybeenpointed rate.therelativecontributionsoftheothercontinuawithrespecttothelyman{ isopticallythickforallconsidereddensities(epe continuumscaleas1=n3,whichcanbeanalyticallyderivedfromeq.(3.63). gradient).however,forlargedensities,hbecomesmoreecientthanly,because thehtransition(3!2)doesnotinvolvethegroundleveland,hence,isnotsomuch aectedbyopticalthickness.therefore,thereisinfactasmalldensity{interval, 210:5=n<H>forthechosenvelocity wherehisthemostecientcoolingprocess,alreadymoreimportantthanly areconstant,anincreaseofthebackgroundcontinuousradiationeldalwaysimpliesreducednet changeoftheconcentrationsofthecollisionpartners.iftheconcentrationofthecollisionpartners andstillmoreimportantthanbound{freetransitions(aroundn<h>=1010cm 3in coolingratesandnallycausesnetradiativeheating. 13Thisisdierentfromallcoolingratesdiscussedsofar,sincewehavenotyetconsidereda

3.2.BOUND{FREETRANSITIONS 51 Figure3.9:DetailsforthecaseTg=8000KandJ=0.Upperpanel:relativecontributionsofthedierentbound{boundandbound{freetransitions. Fig.3.9).Lyisalsoalwaysopticallythick(similartoLy)andhencealwaysmuch Lowerpanel:b{factorsforthehydrogenlevels. lessimportantthanh. ThelowerpanelofFig.3.9showsthegradualchangefromalmostLTE{ionization andlte{population(bn1)atlargedensitiestopronouncednon{lteconditionsat stateispopulatedhyper{thermally,whichisanimportantresultforthecsesofcool (providedthattrad<tgforaplanckeldj=b(trad)).consequently,theground bii{factorindicatesthatthedegreeofionizationofhydrogenisalwayssub{thermal smalldensities,causedbythedecreasingrelevanceofthecollisionalprocesses.the stars,sinceitkeepsthegaspredominantlyneutralalsoatfairlyhightemperatures Thepopulationsoftheexcitedhydrogenlevelsarecompletelydecoupledatsmall andlowdensities,wherehydrogenwouldbestronglyionizedaccordingtolte. thestronglydecreasingescapeprobabilitiesofthelinesphotons.ltepopulation densitiesandarethermallycoupledtothegroundstateforlargedensities,causedby ionization,however,extremelylargedensitiesarerequired(e.g.n<h>>1016cm 3 fortg=10000k),wherethethree{body{recombinationratesbecomerelevant. tothehigherexcitedlevels,nallyalsofortheionizedstate.forcompletelte{ isestablishedindirectionofincreasinggasdensitiessuccessivelyfromthelower

52 ThesituationinthecaseJ=B(3000K)isquitedierent.Here,thenetbound{free CHAPTER3.RADIATIVEHEATINGANDCOOLING coolingratesdominateovernetbound{boundcoolingratesunlessthegasdensityis smallerthan107cm 3forallconsideredtemperatures.Negativeandpositivenet contributionsfromthedierenttransitionsmayoccuratthesametime,although eective,whereasatsmalldensitieslydominates. thesumofallcontributionsalwaysresultsinanetcooling. Tosummarize,hydrogenismainlyanimportanthigh{temperaturecoolant,approximatelycontributingasQ/2.ForlargedensitiestheLyman{continuumismost 3.2.3OtherNeutralAtoms Concerningotheratomsthanhydrogen,solelytheelectronicgroundstatesofthe neutralatomsareconsideredinthisworkforpracticalreasons.furthermore,since recombinationratesandthebound{freeheating/coolingfunctions.theapproxiproximatemethodoutlinedonp.46canbeusedforthecalculationofthephoto{ allbound{freetransitionsfromthegroundstatesarelocatedintheuv,theap- theapplicationoftheapproximatemethodisratheraccurateandverypractical, sationcrosssectionsofthevariousmetalatomsfromschmutzler(1987).therefore, ableagreementwiththosecalculatedfromeq.(3.61),whenapplyingthephotoioni- matephoto{recombinationratesderivedfromeq.(3.65)arefoundtoshowreason- Theratesofcollisionalionizationaredeterminedfromtheanalyticalexpression givenbyallen(1973) sinceitavoidsthetime{consumingnumericalfrequencyintegrations. whereo1isthenumberofopticalelectronsoftheneutralatom.table3.3summarizes 1(Tg)=1:1( 8)o1qTg1=eV 2exp ktg!; 1 (3.73) thedatausedforthedeterminationofthethebound{freeheating/coolingrates. bound{freetransitionsareoncemoresummarized.themethodsareusedforthe andthecalculationofthebound{freeheatingandcoolingrates: determinationoftheelectronconcentration,theconcentrationsofthevariousions Attheendofthissection,theimportantfeaturesofthedevelopedmethodsforthe Ionizationequilibrium(steadystatenon{LTE)isassumedtodeterminethe photoionisationand{recombination,collisionalionizationandthree{body{ particledensitiesoftheconsideredatoms,ionsandelectrons.theratesof Acoupleofsimplifyingassumptionsareusedforotheratomsthanhydrogen. Hydrogenistreatedmoreaccurately,includingofthersttwoexcitedlevels. recombinationaretakenintoaccountforeachatom. Bound-freeopticaldepthseectsareignored.

3.3.PHOTODISSOCIATIONANDRADIATIVEASSOCIATION Table3.3:Bound{freeheatingandcooling:consideredspeciesandatomicdata53 Species1[eV](1)g(1) He 24.587 Z(1;3) II 2.06(-10)-0.67 a(2) 1 b(2) 1 o(4) 1 NC 11.260 14.534 4 9 1.43(-10)-0.61 1.30(-10)-0.62 Mg OS 13.618 10.360 4 1.24(-10)-0.63 1.40(-10)-0.63 3 Si 7.646 8.151 19 26 3.70(-10)-0.86 1.50(-10)-0.64 4 (1):Allen(1973). Na Fe 7.870 5.139 25 2 30 1 1.50(-10)-0.65 1.40(-10)-0.69 (2):Beck(1993)andreferencestherein. 2 (3):Forsimplicity,thepartitionfunctionZIIisapproximatedbythe 1 (4):Thenumberof\optical"electronsisthenumberofelectronsin thelastoccupiedquantumstate statisticalweightofthegroundstateoftheionizedatom. densitiescanbereproduced. changefromalmostlteionizationatlargedensitiestonon{lteionizationatsmall Inconclusion,theoutlinedmethodsareapproximate,butsimpleandapplicableto thewiderangeofdensityconditionspresentincircumstellarenvelopes.thegradual 3.3 Radiativegainsandlossesofthegascanalsobecausedbychemicalreactions. Accordingtothedenitionoftheinternalenergyinthiswork(cf.Fig.2.1),solely PhotodissociationandRadiativeAssociation (photodissociationorradiativeassociation)contributetotheradiativeheatingor thosereactionswhichareaccompaniedbyanabsorptionoremissionofaphoton from3to8ev(exceptionsco:11.1evandn2:9.9ev),whichalreadygivesa cooling,respectively14. Thedissociationpotentialsofthemoleculesofastrophysicalinteresttypicallyrange about10ev,thedissociationenergiesaresubstantiallysmaller.thus,asfarashard rstimpressionoftheconcernedwavelengthregionoftheradiativeprocessesunderinvestigation.comparedtotypicalmolecularionizationpotentialsofmorethan energiesandmightbeconsideredasadditionalsourcetermsforthesepools,butdonotdirectly heating/coolingofthegaseveniftheyare\exothermic"or\endothermic".suchreactionsonly convertdissociationpotentialenergiesintotranslational,ro{vibrationalandelectronicexcitation 14Puregasphasereactions,whichdonotinvolvephotons,donotcontributetotheradiative aectthetotalinternalenergyofthegas.

54 UVradiationisabsent,photodissociationisexpectedtobemoreecientforthe CHAPTER3.RADIATIVEHEATINGANDCOOLING heatingofthegasthanphotoionisationinthemoleculardomainofcircumstellarenvelopes,evenifthecorrespondingphotocrosssectionsaresmaller.inthefollowing, aphoto{chemicalreactionofprototype isconsidered,whereaandblabelanatom,ion,moleculeorelectronandabthe AB+hkf *)kra+b (3.74) mostlyinitiatedbyabsorptioninelectronicbands(e.g.thelyman{andwerner{ andreversereaction,respectively.irrespectiveofthefactthatphotodissociationis correspondingcompositespecies.kfandkraretheratecoecientsoftheforward bandsofh2),whichareinprinciplenarrow{spacedbound{boundtransitions,the photodissociationcrosssectionsf()areassumedtobegiveninaquasi{continuous FromthedetailedbalanceconsiderationnABkfJ=B=nAnBkritisfound way kf=4zj hf()d: (3.75) kr(t)=nab nanbt4zb(t) hf()datexp Ea witheabeingtheactivationenergy.comingbacktotherstpart,nxistheparticle ThesecondpartofEq.(3.76)istheusualArrheniuslawforthebackwardreaction kt: bedeterminedbymeansofthelawofmassactionfromthecorrespondingfree enthalpyofformationatstandardpressurep densityofspeciesxinchemicalequilibrium,i.e.therstfractionineq.(3.76)can fg (T)=fG AB(T) fg A(T) fg B(T)<0 nanbt=kt nab p exp fg (T) kt (3.78) (3.77) Thecontributionofthephotochemicalreaction(3.74)tothetotalnetheatingfunctionofthegasisgivenby nxaretheactualparticledensities,whichinthiscontexthavetobedeterminedfrom Qchem rad=4z0@nabj nanbnab nanbtgb(tg)1af()d: (3.79) implythattheinvolvedmoleculehasbetoabundant). thosereactions,whichhavethelargestnetphoto{rates(thisdoesnotnecessarily thesteady{statesolutionofachemicalreactionnetwork.especiallyinterestingare dierentspeciestobeconsideredandthepooravailabilityofappropriatemolecular Thegeneralproblemofthetreatmentoftheseprocessesisthelargeamountof

3.3.PHOTODISSOCIATIONANDRADIATIVEASSOCIATION dataf()15.therefore,thesofaroutlinedequationsdonotlookverypromising, 55 approximatemethodforthebound{freetransitionsdiscussedinthelastsection.we assumethecontinuousradiationeldtotlikejwb(trad)andconsiderthe Thisproblemcanbeavoidedbythefollowingconsideration,analogouslytothe sincetheycansolelybeappliedtoaveryfew,well{knownmolecules(e.g.h2,co). casemaxfktg;ktradgd0ab<h,applyingwien'slawasbefore.bydierentiating Eq.(3.76)withrespectto1=kT,thefollowingexpressioncanbederived: hhiabs= fg (Trad)+Ea+( 1)kTrad Qchem rad=nabwkr(trad)nanb nabtrad hhiabs nanbkr(tg)hhiem hhiem= fg (Tg)+Ea+( 1)kTg (3.82) (3.81) (3.80) havetobeknown.theconsiderationsndthemeanabsorbedandemittedphoton sinceonlythegibbsenergiesfg andthearrheniuscoecientsa,andea photo{reactioncaneasilybeappliedtotheresultsofchemicalreactionnetworks, ComparedtoEq.(3.79),thisexpressionforthenetheatingrateofaconsidered energiestobeoforder fg +Ea ktd0ab+ea,i.e.themoleculedissociation energyplustheactivationenergyoftheradiativeassociationreaction.thenet associationforthethermalbalanceofthegasincses,the(steadystate)resultsof chemicalreactionnetworkcalculationsarerequiredprovidingthevariousconcentrationsofthespeciesunderexamination.suchinvestigationsgobeyondthescopeof thisworkandmustbelefttofutureinvestigations.however,animportantexample heatingratevanishesinthecaseofthermodynamicequilibrium,asexpected. Foracomprehensivediscussionoftheimportanceofphotodissociationandradiative isconsideredinthefollowingsection. 3.3.1TheH Heating/CoolingRate ThenegativehydrogenionH showsexceptionallylargephoto{ratesincircumstellar atmospheresofwarmstarsasthesun.therefore,itisimportanttoconsiderthe envelopes(beck1993).itsbound{freeandfree{freetransitionsarefurthermore TheconcentrationofH incircumstellarenvelopesismostlycontrolledbythefollowingtworeactions(becketal:1992): H +h H +H kf;2 kf;1 *)kr;2 *)kr;1 H2+e : H+e (3.83) radiativeheatingandcoolingbyh inmoredetails. well{knowntobethemostsignicantcontributortothegasopacityinthestellar minestheratesandtheheating/coolingfunction diculttomeasureinlaboratoryexperiments. 15ConsideringthefrequencyintegrationinEq.(3.79),thesoftendoff()almostentirelydeter- (3.84)

56 Thus,theconcentrationofH insteadystate(\kineticequilibrium")isalways CHAPTER3.RADIATIVEHEATINGANDCOOLING proportionaltotheelectronconcentration: Reaction3.83(\H bound{free")istheprocesstobeconsideredfortheradiative nh =nenhkr;1+nh2kr;2 heatingandcoolingofthegas.theaboveoutlinedmethodsarestraightforwardly kf;1+nhkf;2 (3.85) ation.thereactionratecoecientskf;1andkr;1arecalculatedbymeansofthe \exact"eqs.(3.75)and(3.76),wherealabelsthehatom,btheelectronandab andradiativerecombinationratherthanasphotodissociationandradiativeassoci- applied,althoughthebound{freetransitionsofh areclassiedasphotoionisation thenegativeionh.thebound{freeabsorptioncrosssectionofh isinterpolated fromtablesgivenbywishart(1979).stimulatedrecombinationsaretreatedinlte, forsimplicity.theratecoecientsofthesecondreaction(3.84)aretakenas kf;2=1:35( 9)cm3s 1(Schmetekopfetal:1967) kr;2=nh nh nh2nekf;2: (3.87) (3.86) ionizationandphotoionisationofmetalatomswithlowionizationpotentials(na, ThenH densityiscalculatedafterwardsfromeq.(3.85)16.accordingly,collisional ne,nhandnh2aredeterminedbymeansofthemethodsoutlinedinchapter4. Consideringagasofsolarelementalabundances,therequiredparticledensities Mg,Fe,...)areimportantlow{temperatureelectrondonatorsandprovideelectron concentrationsofatleast10 5forTg>5000K,leadingtoconsiderableH particle andthefree-freeheating/coolingrateisdeterminedby18 densities17. TheradiativeheatingandcoolingbyH comprisesbound{freeandfree-freecontributions.thebound{freeheating/coolingrateiscalculatedaccordingtoeq.(3.79), Qrad(H )=Qbf Qrad(H )=4nHneZJ B(Tg)()d; rad(h )+Qrad(H ) (3.88) Stilley&Callaway(1970): wherethefree{freecrosssection()isttedonthedipolelengthcalculationsof (3.89) ()= 1:3727( 25) +4:3748( 10) 2:5993( 7)=Tg 2!1 exp( h 17Incontrast,theconsiderationofapurehydrogenplasmaleadstoasystematicunderestimation 16Thisisanapproximateprocedure,sinceitneglectsthefeedbackontheformerparticledensities. ktg)cm5 oftheh concentrationandheating/coolingrates:theresultingelectronconcentrationsare solelyrefertothethermalmotionofthegas. smallerinthiscase,especiallyaroundtg6000k,justwheretheheatingandcoolingofh turns outtobemostsignicant. 18Kirchho'slaw=B(Tg)isapplicabletofree{freetransitionsalsoinnon{LTE,sincethey

3.3.PHOTODISSOCIATIONANDRADIATIVEASSOCIATION 57 dashedlines,rightaxis)ofh inthecasewithoutcontinuousradiationeld. free{freecoolingrate(shortdashedlines,leftaxis)andtheconcentration(long Figure3.10:Thetotal(bound{free+free{free)coolingrate(fulllines),the Figure3.11:SameasFig.3.10,butwithunderlyingcontinuousradiationeld.

58 Figures3.10and3.11depicttheresultsforthetwocasesJ=0andJ=B(3000K). CHAPTER3.RADIATIVEHEATINGANDCOOLING foradensemedium,e.g.instellaratmospheres.thestep{likebehavioroftheh leastq/2.therefore,theradiativeheatingandcoolingofh isonlyimportant Inbothcases,theradiativecoolingrateofH scalesasq/nhne,whichimpliesat concentrationandtotalcoolingratesfortg=10000kandtg=8000kinfig.3.10 inthegures),hydrogenismostlylockedinh2,andtheconcentrationandthe metalswithlowionizationpotentials.forevenlowergastemperatures(notshown whereasforlowergastemperaturestheelectronconcentrationiscontrolledbythe correspondtothestep{likedegreeofionizationofhydrogen(cf.figs.3.7and3.8), heating/coolingratesofh rapidlyvanish. considerablysmallerduetothelargephotoionisationrateskf;1.however,sincethe radiativecoolingratesofh arerelatedtonhneandnottonh,thecoolingrates ThecalculatedH concentrationsforthecasej=b(3000k)(cf.fig.3.11)are elds(notshown),thebound{freetransitionseectivelydestroythenegativeion, belessimportantthanbound{freeheating/cooling.forevenmoreintenseradiation dierentelectronconcentrations.free{freeheating/coolingofh isalwaysfoundto remainsimilar.thedierencesbetweenfigs.3.10and3.11aremainlycausedbythe chemistry(atlargedensities)quicklyrestorestheh ions. temperatures,wheretheproductofelectronandatomichydrogendensityislarge. Inconclusion,H ismainlyanimportantcoolantforlargedensitiesandmedium sothatradiativeheatingbyh israrelyfoundtobesignicant only,ifanactive 3.4 Ifthegasisalmostfullyionizedandthedensityislarge(n<H>>1011cm 3),freefreeemission(Bremsstrahlung)becomesaneectivecoolingprocess.Weusethe Free{FreeTransitions Allen(1973)andinclude,forconsistency,thereverseprocessoffree{freeabsorption bymeansoftherelation=b(tg) ordinaryexpressionforfree{freeemissionforapartiallysinglyionizedgasgivenin (Tg)=5:44( 39)n2e qtgexp h ktg Qrad=4Z(Tg) B(Tg) 1!d: J (3.91) (3.90) diativeheating,iftheincidentradiationeldmainlyconsistsofirphotons,where otherradiativeheatingprocessesbecomeimpossible.however,thegasmustbe quently,free{freetransitionsprincipallyprovideoneoflastingpossibilitiesforra- Free{freetransitionsalwaysconcernthewholeelectromagneticspectrum.Conse- considerablyionized(qrad/n2e)forsuchheating.

3.5.OVERVIEWOFTHECONSIDEREDRADIATIVEPROCESSES OverviewoftheConsideredRadiativeProcesses 59 providearstimpressionoftheirneteectandtheirimportancefortheheating cretion.theshownspectralpositionsofthevariousradiativeprocessesalready Figure3.5summarizestheradiativeheating/coolingfunctionsconsideredinthis andcoolingofthegas.ingeneral,radiativeheatingoccursinthecasej>b(tg) work.towhatextendthisselectioniscomplete,mustbelefttothereader'sdis- andcoolingintheoppositecase19.forthedepictedcaseofadilutedplanck{type thermalrelaxationofthegastowardsradiativeequilibrium,thegastemperature radiationeld,theradiativeprocessesatshortwavelengths(bound{freetransitions, (vibrationalandrotationaltransitions,spectrallines)forcooling.consideringthe willtuneinsuchaway,thatthesegainsandlossesbalanceeachother.notethat spectrallines)areresponsibleforradiativeheatingandthoseatlongwavelengths theformationofmoleculesinthegasintensiestheinteractionbetweenmatterand quentlyleadingtolowerradiativeequilibriumtemperatures(thiseectcaninfact causethermalbifurcationsinthegasasdiscussedinchapter6). radiationeldatlongwavelengths,thusreinforcingradiativecoolingandconse- moreplacedtogether: Aspartofthissummary,theimportantfeaturesofthedevelopedmethodsareonce Allconsideredradiativeprocessesaretreatedinnon{LTE.Thenon{LTEdescriptionofthemoleculesisrestrictedtoindividualvibrationalandrotational Allcorrespondingreverseprocessesaretakenintoaccount,relyingondetailed state. excitationtemperatures.theheating/coolingratesarecalculatedinsteady{ atureandtheradiationeld.inthecasej=b(tg)(asinthermodynamic balanceconsiderations.consequently,eachconsideredpairofforwardand equilibrium),alldiscussednetheating/coolingratesvanish(qrad=0). caseactuallyoccursdependsuponaspecicrelationbetweenthegastemper- reverseprocesscanappearasbothnetradiativeheatingandcooling.which Theheating/coolingratesareformulatedforarbitraryradiationeldsJ.EspeciallysimpleexpressionsarederivedfordilutedPlanckeldsoftypeJ= Opticaldepthseectsareincludedforallbound{boundtypetransitionsand neglectedotherwise. WB(Trad)20. sourcefunctions(consideringe.g.atwo{level{atom)generallysatisfyj><s><b(tg). asfarastheconsideredheating/coolingprocessesmerelyrefertosuchregion. 19Theserelationsarenotexact,butusuallycorrectalsoinnon{LTE,sincethecorresponding 20ThetparametersWandTradcanbedierentfordierentspectralregions(e.g.UVandIR),

60 CHAPTER3.RADIATIVEHEATINGANDCOOLING H2ro{vib.quadrupole:ca.1 25m Balmercontinuum: Paschencontinuum: H bound{free: <0:365m free{freetransitions: allwavelengths <0:821m <1:65m Figure3.12:Overviewoftheconsideredheatingandcoolingprocesses.Thefull lineshowsanassumedcontinuousmeanintensityjaccordingtoeq.(1.6)with diativecoolingatlongwavelengths,respectively.thelowerpanelindicatesthe betweenmatterandradiationeld,favoringradiativeheatingatshortandra- W=0:029(r=3Rforpureradialdilution)andTrad=3000K.Thedashedline wavelengthregionsoftheconsideredheating/coolingprocesses. istheplanckfunctionfortg=1500k.thearrowsindicatetheenergyexchange

3.6.FURTHERHEATINGANDCOOLINGPROCESSES FurtherHeatingandCoolingProcesses 61 Thetheoreticalpartofthisworkceaseswithsomeremarksonthoseheatingand Ofcourse,theproperinclusionofallradiativeprocessesisprincipallydesirable.The constructionofsuchanutmost\complete"set,however,isalonglastingprocess andcannotbecarriedoutbyonesinglework.theselectionofheating/cooling coolingprocesseswhichhavenotbeentakenintoaccount. numberofprocesses(morespecies,morelines,etc.)ofthealreadyconsideredtypes ofprocesses,andsecond,totakeintoaccountfurthertypesofprocesses. Apurequantitativeextensionwillprobablynotleadtosubstantialchangescomparedtotheforthcomingresultsofthiswork,sincethemostpromisingcandidatesof eachconsideredtypeofprocessarealreadyincluded.whatmaybecrucial,however, aretheadditionaltypesofprocessesnotconsideredsofar,whichmightproveto functionsofthisworkmaybeextendedintwoways:first,toincludealarger theseobstaclesshallbexedinthissection. them,moreorlessserious,specicobstaclesoccurredwhichpreventasimplequantitativediscussionforthetimebeing.togetherwithsomevaluationsandremarks, beimportantundercertaincircumstances.astheauthorstartedtostudyafewof Table3.4listssomeinterestingcandidatesofdierenttypesofprocesses,theexpectedspectralregionofabsorbed/emittedphotons,theexpectedeectforthegasments,however,arenecessaryconcerningtheenergygainsandlossescausedbythe givenforreasonsofcompletenessandtoopenthediscussion.someadditionalcom- andsomecomments.extensiveexplanationsarenotincluded,astable3.4ismainly thefacedobstaclesforthedeterminationofthecorrespondingheating/coolingrates (e.g.fleischeretal:1992)andcannotbedeterminedbyanysteady{stateconsiderations.therefore,theheatingandcoolingprocessescausedbythepresenceofdusponent(hereespeciallythetotaldustsurface)shouldbetreatedtime{dependently Duetothelargetimescalesinvolvedinthedustformationprocess,thedustcom- presenceofdustgrains: Consequently,energytransferratesdirectlyoccurbycollisions.Thecorrespondingratescaneasilybeaddedtothetotalnetradiativeheatingrateofthegas,if Dustgrainsprovideasimilarexternalpoolastheradiationeld(seeFig.2.1). arenotexplicitlyincludedinthiswork. perature,driftvelocities,etc.)areavailable.theratesforthermalaccommoda- tion(energytransferviainelasticgas{dustcollisions)aregiven,forexample,in theappropriateinformationsaboutthedustcomponent(totalsurface,dusttemsionswithmovingdustgrainscausedbyradiationpressure)ingoldreich&scoville (1976)orinKrugeretal:(1994). Morediculttodeterminearetheheating/coolingratescausedbysurfacechemical Burke&Hollenbach(1983)andthosefordriftheating(energytransferviagascolli- reactions,whichprincipallyexchangealltypesofgasinternalenergies(especially dissociationandionizationpotentials)withthedustcomponent.verydetailed knowledgeaboutthesereactionsisrequired.

62 Table3.4:Overviewoffurtherheatingandcoolingprocessesnotexplicitlyconsideredin CHAPTER3.RADIATIVEHEATINGANDCOOLING processthiswork. spectral photodissociation severalev numberoftrans., (),concurring region (estimated) importantforwhat? obstacles comments electronic reactionchannelsand (2) molecules trans.of severalev rad.heatingin moleculardomain, maybeimportant counterbalanceforrot. andvib.cooling numberoftrans.,rad. lifetimes,coll.rates, lineoverlaps probablystrong non{lteeects: vib.trans. andsmallcoll.rates shortrad.lifetimes of (typically0:01 1s) polyatomic molecules 3 20m heatingandcoolingselectionrules,rad. lifetimes(analytical expressions),coll. rates unpolarpolyatomic vib. moleculesmaybecome overtone polarduring trans. 1 5m heatingandcoolingrad.lifetimes (analytical expressions) A{coes.aboutone magnitudesmaller thanforgroundtone vibration,(1) rot.trans. ofnon{linear molecules 20 1000GHz lifetimes(analytical expressions) selectionrules,rad. trans. bf.trans. fromexcited (1) states severalev heatingandcooling, indirect (de{)excitationof boundlevels atoms (),completemodel trans.of bf.and. (2) negative ions, molecules several0.1ev heatingandcooling(),concurring reactionchannelsand rates (2) chemical reactions gasphase depends electronicandvib. levelsofreaction products,probably followedbyemission, indirectexcitationofreactionrates, i.e.cooling reactants reactionheats,energy distributionamong thevariousdegreesof freedomonthe dicult,cooling approximately proceedsonchemical bf.=bound{free,.=free{free,rad.=radiative,rot.=rotational,vib.=vibrational,a{ timescale coes.=einsteincoecientsforspontaneousemission(inverseoftheradiativelifetimes),coll. rates=collisionalde-excitationratecoecients,trans.=transitions,()=corresponding photocrosssections. (2)=seethesimple,approximatemethodsproposedinthiswork (1)=polyatomicmolecules(exceptforH2O)aregenerallylessabundantthandiatomic molecules(e.g.co)incses

3.6.FURTHERHEATINGANDCOOLINGPROCESSES Table3.4continuedfrompage62 63 process spectral region (estimated) importantforwhat? obstacles comments Raman scattering UVand optical moleculargasesby rad.heatingof inelasticscattering absolutecrosssections (Stokesand Anti-stokes)of individualmolecules, e.g.ofh2 forramanscatteringtoarbitrary wavelengths,however, crosssectionsare unpolarmoleculesand appliestopolarand dust: thermal accommodation dust:drift heatingandcooling accordingto small heating { temperaturedierence betweengasanddust ecientmechanism, dust:surface seekrugeretal:(1994) reactions { energies(heatingand dissociationpotential gainorlossof excitedreactants cooling),desorptionofreactionrates, (heating) reactionproducts reactionheats,energy distributionamong thevariousdegreesof freedomofthe dicult

64

Chapter4 TheCalculationoftheEquationofState physicsandthethermodynamicdescriptionofthegas.havingoncedetermined Thecalculationoftheequationofstateprovidesthebasiclinkbetweenthemicro- themicrophysicalquantities(theparticledensities)asfunctionofasuitableset goingbackintothedetailsofmicrophysics. modelingofthegascanbeperformedonahigher,thermodynamiclevelwithout propertiesofthegascanbedeterminedbymeansofstatisticalmethods.thus,the ofthermodynamicstatevariables(e.g.temperatureanddensity),allmacroscopic sidersasteadystate.consequently,twoadditionalexternalparametersenterinto and.aspointedoutinchapter2,thisworkdoesnotrelyonlte,butcon- ThischapterdescribestheassumptionsandthenumericaltechniquesusedtodeterminetheparticledensitiesandtheinternalenergyofthegasasfunctionofTg theusualthermodynamicdescription:theradiationeldjandthemeanvelocity gradientddv aresummarizedinthisseparatechapter. dle.sincethetechniquesarethesameforallfollowingapplications,they 4.1 Thebasisforthecalculationoftheparticleconcentrationsaretheelementabundances.Inthiswork,amixtureoftheelementsH,He,C,N,O,Na,Mg,Si,SandFe CalculationoftheParticleConcentrations isconsidered.sincedierenttypesofstarswithdierentabundancesareconsidered ions,electronsandmolecules: intheforthcomingapplications(c{stars,rcbstars),theassumedabundancesare statedseparately(cf.sect.5.1.2andsect.6.1.2).thefollowingbasicassumptions aremadeinordertocalculatethevariousparticledensitiesoftheneutralatoms, Neutralandsinglyionizedatomsaretakenintoconsideration.Theratiosbetweentheparticledensitiesofionsandneutralatomsarecalculatedbymeans (steadystatenon{lte),asdescribedinchapter3. toionization,-recombination,collisionalionizationand3-bodyrecombination ofthestatisticalequationseq.(3.1),takingintoaccounttheratesofpho- Forsimplicity,theratiosbetweentheparticledensitiesofmoleculesandneutralatomsarecalculatedaccordingtochemicalequilibrium1.Negativeions calculatingthesteadystatesolution(\kineticequilibrium")ofacompleteandreliablechemical 1Thisisofcourseasimplifyingassumption.Animprovementofthemodelmaybeachievedby 65

66 aretreatedlikemolecules(exceptforh,cf.sect.3.3.1).thechemistry CHAPTER4.THECALCULATIONOFTHEEQUATIONOFSTATE comprises130species(gail&sedlmayr1986),wheresomelargerpurecarbon moleculeshaveadditionallybeenincludedusingthethermo{chemicaldata {iterationtechniques,untiltheconservationofchargeandelementsisassured.the followingschemeisapplied,wherethequantities,tg,jandddv TheparticledensitiesarenallyfoundbymeansofnestedNewton{Raphsonand fromgoeres&sedlmayr(1992). 1. Estimatetheelectrondensityneandallneutralatomdensitiesinthe electronicgroundstatenel 0. dlearegiven: 3. 2. Performaninner{iterationforeachatomicspeciesinordertosolvethe coupledequationsforthelevelpopulationsandtheescapeprobabilities pendontg,jandne. Calculatethebound{freeandfree{boundratesRiIIandRIIi,whichde- 3a. (comparesect.3.1.1.1),i.e.: 3b. Tg,J,theescapeprobabilitiesePe partners. DeterminethelevelpopulationsandtheionparticledensitiesnEl Calculatethebound{boundrateseRulandeRlu,whichdependon ulandthedensitiesofthecollision 3c. fromthestatisticalequations(3.1). populationsandthelocalvelocitygradientddv CalculatetheescapeprobabilitiesePe ulwhichdependonthelevel dle. II 3d. 4. calequilibriumaccordingtothetotalneutralatomdensitiesnel Calculatetheparticledensitiesofthemoleculesnmolbyassumingchemi- Gobacktostep3aunlesstheprocedurehasconverged. 5.Calculatethecurrenterrorsofchargeandelementconservation,i.e. gastemperaturetg. atandthe ~F(ne;nH0;:::;nFe 0)= 0 B@ PElElmEl nhat nhii PmolnmolsHmol H ne PElnEl II+Pmolnmolsemol 1CA 6.PerformoneNewton{Raphsoniterationstep,i.e.solveD~F~n=~Ffor PElElmEl nfe Fe at nfe. II PmolnmolsFe thecorrections~nandput~n!~n ~n,wherethecomponentsofthe mol 7.Gobacktostep2unlessallfurthercorrectionsbecomesmall= vector~nareshownastheargumentof~fintheupperequation. elementabundances,mostoftheimportantreactionchannelsprobablyinvolvetheabundantpure reactionratenetwork,which,however,goesbeyondthescopeofthiswork.concerningthercb j=e;h;:::;fefnj=njg<10 10. max carbonmolecules,whichareallradicalsandwhosereactionratesareonlypoorlyknown.

4.2.CALCULATIONOFTHEINTERNALENERGY Thesuccessfulconvergenceofthisiterationschemecriticallydependsonthequality 67 theelectrondensity)maydependontheescapeprobabilities,asdemonstratedfor areincluded.inthiscase,thedegreeofionizationofaconsideredatom(andhence {iteration(step3 3d)isnecessary,ifbound{freetransitionsfromexcitedlevels ofstep1,i.e.therstestimateoftheelectronandneutralatomdensities.theinner hydrogeninsect.3.2.2.ifnosuchbound{freetransitionsareconsidered(asforall havingsolvedtheaboveiterationscheme.especiallyallexcitedstatesofionscan becalculatedafterwards,sinceonlytherstionizationstageistakenintoaccount. otherelementsthanhydrogeninthiswork),thesystemofequationsdecouplesand Thedescribedmethodyieldsallparticlesdensities,includingtheconsideredlevel thepopulationsofthemulti{levelatomswithoutcontinuumcanbedeterminedafter populations,asfunctionofthemassdensity,thegastemperaturetg,thecontinuousbackgroundradiationeldjandthevelocitygradientddv 4.2 CalculationoftheInternalEnergy dle. ingoncedeterminedtheparticledensitiesasoutlinedabove,theevaluationofthe internalenergyiscomparablesimpleandnotverytime{consuming. Fordynamicconsiderations,theproperdeterminationoftheinternalenergyisas Accordingtothedenitionoftheinternalenergyinthiswork(cf.Chapter2),the importantasthedeterminationoftheradiativeheatingandcoolingrates2.hav- electronic,vibrationalandrotationalexcitationenergies.thedierenttermsare internalenergycomprisesoftranslational,ionizationanddissociationpotentialand calculatedasfollows: Etrans=32nkTg Eion=XElnEl IIEl II+XElnEl IIIEl II+El III+::: (4.1) Ediss= XmolnmolD0mol (4.3) (4.2) Evib=XmolnmolXj Eel=Xi;jnijEij exph!mol gjh!mol jj (4.4) Erot=Xmolfmol 2nmolkTmol rot ktmol vib 1 : (4.6) (4.5) considere=fkt=(2)atthesametime.thedenitionsoftheinternalenergyandtheradiative heatingandcoolingratesrefertoeachother.forexample,theinternalenergyinthemolecular domainturnsouttobenegativeinthiswork. 2Warning:Oneshouldnottaketheradiativeheatingandcoolingratesoutofthisworkand

nisthetotalgasparticledensity(atoms+ions+electrons+molecules).ii=iii 68 CHAPTER4.THECALCULATIONOFTHEEQUATIONOFSTATE (measuredfromthevibrationalgroundstate),i.e.theenergyrequiredtototally dissociatethemoleculeintoitsconstitutingatomsat0k.bydenition,neutral forreasonsofcompleteness).d0molisthetotaldissociationpotentialofamolecule istheionizationpotentialoftherst/secondionizationstage(thelatteronlygiven atomshavezeropotentialenergies. nijistheparticledensityofspeciesiinthej-thexcitedelectronicstateandeij largerotationaltemperatures,whichissucientinthiscontext.frotisthenumber thecorrespondingenergydierencetoitselectronicgroundstate.!mol eigenfrequencyofamoleculeandgjthecorrespondingdegeneracy.equation(4.5) assumesindependent,harmonicoscillators.equation(4.6)istheclassicallimitfor j isthej-th ofrotationaldegreesoffreedom(2forlinearmolecules,3otherwise). Asfaraspossible,thevibrationalandrotationalexcitationtemperaturesTrotand Forthecalculationofthedissociationpotential,vibrationalandrotationalexcitation areassumedtoequalthegastemperature. Forthosemoleculeswhicharenotconsideredtherein,theexcitationtemperatures TvibarecalculatedbymeansofthemethodsoutlinedintheSects.3.1.3and3.1.4. mentabundances.additionaldatafortheionizationpotentialsel theselectedmoleculesandsummarizesthenecessarymoleculardata.theselection comprisesthemostabundantmoleculesinbothcases,c{starandrcbstarele- energies,onlytheabundantmoleculesmustbetakenintoaccount.table4.1lists electroniclevelsei;jcanbefoundinthetables3.1and3.3. theinternalenergyprovidethecaloricandthermalequationsofstateintheform Insummary,theoutlinedmethodsforthecalculationoftheparticledensitiesand IIandtheexcited p=p;tg;j;ddv e=e;tg;j;ddv dle=1(etrans+eion+ediss+eel+evib+erot) ne+xel(nel at+nel II)+Xmolnmol!kTg: (4.7) Asimilarexpressioncanbewrittenforthetotalradiativenetheatingrateofthe (4.8) densities,whichdependondensityandtemperature,theradiationeldandthe gas,whichiscalculatedaccordingtochapter3asfunctionofthevariousparticle velocitygradient Equations(4.7)to(4.9)denethethermodynamicsystemwhichisexaminedinthefollowingpartsofthiswork. dle: (4.9) Qrad=Qrad;Tg;J;Ddv independentstatevariablesissucienttodeterminethethermodynamicstateand TogetherwiththetwoexternalparametersJandDdv butotherusefulchoicescanbee.g.(p;),(p;tg)or(p;h),dependingontheproblem. henceallgasproperties.equations(4.7)to(4.9)areformulatedintermsof(;tg), dle,anysuitablesetoftwo

4.2.CALCULATIONOFTHEINTERNALENERGY 69 moleculed0mol[ev](1)!mol[1/cm]and(degeneracy)(2) Table4.1:Moleculardataforthedeterminationoftheinternalenergy. CO H2 11.11 4.48 4158.5(1) 2143.2(1) frot C2H2 CH 12.07 3.46 2732.8(1) 1920.0(1),640.0(2),3220.0(1) CH4 16.78 3373.7(1),1973.8(1),3281.9(1),611.6(2),729.3(2) 16.99 2916.5(1),1534.0(2),3018.7(3),1306.0(3) C3 6.15 1828.0(1) C4 13.70 1224.5(1),63.1(2),2040.0(1) C5 19.49 26.78 1568.0(1) 350.0(1),450.0(1),1088.0(1),1103.0(1),1431.0(1), 2220.0(1),2344.0(1) 112.0(2),222.0(2),648.0(2),863.0(1),1632.0(1), 3 C7 38.70 710.0(2),1206.0(1),1745.0(1),2132.0(1), 73.0(2),157.0(2),240.0(2),598.0(2),631.0(1), C10 59.30 2281.0(1),2376.0(1) 184.0(2),253.0(2),419.0(1),497.0(2),555.0(2), 2 N2 9.90 1118.0(2),1522.0(2),1971.0(2),2013.0(2) 568.0(2),577.0(1),661.0(1),690.0(2),946.0(1), 2330.0(1) 3 C2N2 CN 13.90 7.72 2042.4(1) 1924.0(1),324.0(2),1050.8(1) HCN 21.32 2330.0(1),846.0(1),2158.0(1),503.0(2),234.0(2) SiC2 13.09 2096.3(1),713.5(2),3311.5(1) Si2C 13.05 1742.0(1),837.0(1),186.0(1) SiO 11.10 4.58 670.0(1),275.0(2),1600.0(1) 983.0(1) 3 SiS 8.23 6.38 1229.6(1) 744.5(1) (1):ThetotaldissociationpotentialenergycanbedeterminedfromtheJANAFtables CS 7.35 1272.2(1) (Chaseetal:1985)accordingtoD0mol=fH0(mol) PElsEl molfh0(el)at0k. 2 (2):ValuesfordiatomicmoleculesaretakenfromHuber&Herzberg(1979)accordingto ForthelargercarbonmoleculesC7andC10,thedissociationpotentialsaretaken fromabinitioquantummechanicalcalculations(\scaledbindingenergies"from!=!e 2!exe.ValuesforpolyatomicmoleculesfromChaseetal:(1985).Values Raghavachari&Binkley1987). therelationfvib=3n 5=3N 6forlinearandnon{linearmolecules,respectively. forc4,c5,c7andc10fromraghavachari&binkley(1987).thereadermayverify

Inpractice,suchdependenciesarecodedbynumericalinversion.Onecomputer 70 CHAPTER4.THECALCULATIONOFTHEEQUATIONOFSTATE valuesforandtgwhichyield(p;h)bynewton{raphsoniteration. above,yieldingthevaluesofallstatevariablesasfunctionof(;tg).ife.g.a formulationin(p;h)isneeded,anothercomputerroutinendsthecorresponding routinecarriesoutthedeterminationofthermodynamicstateofthegasasstated

Chapter5 ThermalBifurcationsinthe Asarstapplicationofthethermodynamicdescriptiondevelopedinthiswork,the CircumstellarEnvelopesofRCBStars topologyoftheradiativeequilibriumsolutionsisinvestigated. Radiativeequilibrium(RE)isdenedastheequalityofradiativegainsandlosses. anylong{termphysicallyrealizedsolutionmustbethermallystable,thecondition criterionforthethermalstabilityofgasesunderastrophysicalconditions1.since ductionandheatingbymagneto-acousticwavesorcosmicrays),reisthemain Supposingthatotherheatingandcoolingprocessesarenegligible(asheatcon- ofreprovidesthebasicequationforthedeterminationofthegastemperaturein thecaseofstaticconditions. However,asshowninthischapter,theconditionofREmaynotbeunique,but Muchmore1986),inlatetypestars(Kneer1983)andintheinterstellarmedium occurintheoutersolaratmosphere(ayres1981;muchmore&ulmschneider1985; canhavetwoormorestabletemperaturesolutions.thesemultiplesolutionsare (e.g.biermannetal:1972). commonlycalled\thermalbifurcations".thermalbifurcationsarewell{knownto mightberelatedtotheformationofdustintheseenvelopes,whichcausesthespectacularrcb{typedeclineevents(cf.sect.1.3andappendixa)tionofwhetherornotlow{temperaturesolutionsalreadyexistatsmallradialdis- ThischapterinvestigatesthecircumstellarenvelopesofRCBstars.Here,thequestancestothestarisofspecialscienticinterest.Theoccurrenceofsuchsolutions Themainintentionofthischapteristodemonstratethatthermalbifurcations,in Thetemperaturesofthesephasescaneasilydierbyseveralthousandsofdegrees. principle,canleadtodierent,coexistingphasesofthegasinpressureequilibrium. ThephenomenonofthermalbifurcationsisexpectedtooccurfrequentlyinallpartiallymoleculargasespressionalongaRE{trajectory,dp 1Asecondarycriterionforthermalstabilityisthatthegasmustoerresistanceagainstcom- dre>0.otherwise,thegasisunstableagainstcollapsing 71

72 TheModel CHAPTER5.THERMALBIFURCATIONSINRCBSTARS 5.1.1DenitionoftheRadiativeEquilibriumGasTemperature valuesofthegaspressurep,theradiationeldjandthemeanlocalvelocity eachotheraredetermined.theregastemperaturestre gradientddv Inthefollowingthegastemperatureswhereradiativeheatingandcoolingbalance dleaccordingto garecalculatedforgiven Equation(5.1)isanimplicitdenitionoftheREgastemperature,whichmayof Q radp;tre g;j;ddv coursebenon{unique.forstabilityonehastorequirethatthederivativeofqrad dle=0: evenmoreradiationandwillheatupfurther. intemperaturetgwillincreasenetheating,thatisthegaselementwillabsorb withrespecttotemperatureissmallerthanzero.otherwise,asmallenhancement @Qrad @TgTg=TRE g (<0, Thebifurcationpoints,wheresolutionsappearordisappear,satisfyEq.(5.1)and >0,unstableRE (5.2) havezerorstderivatives.theyonlyexistforcertainvaluesoftheotherparameters, e.g.forspecialradiationelds. TheelementabundancesoftheprototypestarRCoronaeBorealisareconsidered, 5.1.2ElementAbundances adoptingthevaluesfromcottrell&lambert(1982)2.rcbsarechemicallypeculiarstars,showingstronghydrogendeciencyandconsiderablecarbonenrichment (cf.appendixa).mgandneareassumedtohavethesolarabundancesgivenby workforallmodelswithregardtorcbstars. Allen(1973).Figure5.1summarizesthechoiceoftheelementabundancesinthis 2OtherRCBstarsshowconsiderable,individualdeviationsfromtheseabundances,especially Figure5.1:AssumedelementabundancesofRCoronaeBorealis forh:heandc:n:o(lambert&rao1994). 6 Na H S Si Al Fe Mg Ne 5 4 N 3 O C 2 1 He 0 log ε

5.2.RESULTS 5.1.3ApproximationoftheRadiationField 73 eldisused.theradiationeldisttedbyaradiallydilutedblackbodyeldof determinationoftheparticledensitiesandthecalculationoftheradiativeheating andcoolingrates.inthischapter,atwo{parameterapproximationoftheradiation Theradiationeldisanimportantingredientforthemodel,enteringintoboththe theeectivetemperatureteofthecentralstar estareneglected,i.e.thecseisassumedtobeopticallythin.teissettobe Absorptionbetweentheouteredgeofthephotosphereandthelocationofinter- J(r)=121 q1 R2=r2B(Te): (5.3) approximationreasonablytsthestellarspectrumintheopticalandirregionwith 7000K,whichisarepresentativevalueforthisclassofstars(cf.AppendixA).The amaximumdeviationofafactor1:5,butleadstosomewhattoohighintensities continuousuv{emissionsfromshockedgaslayersinthecircumstellarenvelopeand bethedominantsourceforradiationatallwavelengths chromosphericemissions, depthsinthestellaratmosphere.furthermore,thestellarphotosphereisassumedto for<300nm(asplundetal:1997),whichisaconsequenceofthelargeuvoptical regions,respectively,ascomparedtoeq.(5.3). nored.sucheectswouldenhancethemeanintensitiesintheuvandirspectral alsoir{emissionsfromextendedcircumstellardust{shells(cf.appendixa)areig- ofthercbapplicationsarestatedrst.inthefollowing,thetypicalfeaturesforthe 5.2 BeforestudyingthestructureoftheRE{solutions,someofthemicrophysicalresults Results ionizationandthechemistryofthegasaresummarizedandtheroleofthevarious heating/coolingprocessesisdiscussed.regardingtheabundances(cf.fig.5.1),the resultscandieralotfromthoseofahydrogen{richgaswithnearlysolarabundances asencounteredintheinterstellarmediumorforexampleintheatmospheresof AGB{stars. 5.2.1DegreeofIonization Fractionalionizationusuallyturnsouttobelarge,irrespectiveofthegastemperature.Thisisaconsequenceofthelargeratesofphotoionizationaccordingtothe 30000K,wheretheratesofcollisionalionizationcomeintoplay.Consequently, thedegreeofionizationequalsalmost1fortg>30000kandisapproximatelygiven menthelium,however,ismostlyneutralunlessthegastemperatureislargerthan assumedradiationeldwithitsstronguvintensities.themostabundantele- carbonismainlypresentintheformofmoleculesandtheelectronsareprovidedby bythec/he{ratioatlowergastemperatures.atverylowtemperaturestg<1200k,

74 otherelements,mainlysiandmg.accordingtothesomewhatpoortoftheuv CHAPTER5.THERMALBIFURCATIONSINRCBSTARS partofthestellarspectrum,theseresultsarestillpreliminary3. Thecharacteroftheresultschangesforverylargedensities(n<He>>1014cm 3 tionrates.consequently,thedegreeofionizationsmoothlyreachestheresultsof eciencyofthevariouscollisionalprocesses,especiallythethree{bodyrecombinamosphericlayersofthestar,thestateofthegasisclosetolteduetothelarge or>10 9gcm 3orp>100dyncm 2).Forsuchdensities,aspresentintheat- Saha{ionizationwithincreasingdensities. 5.2.2Chemistry Moleculesbecomeabundantinthegasphaseapproximatelybelowadividinglinein thegastemperature/density{planereachingfromtg4000katn<he>=1014cm 3 moleculestooccurarecoandn2. totg1500katn<he>=105cm 3.Withdecreasingtemperature,therst Elementsmostabundantmolecules(2) purecc2,c3,c4,c5,c7, Table5.1:AbundantmoleculesinthecircumstellarenvelopesofRCBstars(1) abundantmolecules(3) C/N O C10(monocyclicring),... N2,CN,C2N,C2N2 CO NCN,C4N2 NO,O2,CO2,C2O Si/S Mg H SiC2,Si2C,SiC,SiO,SiS,CS C2H,HCN,H2,C2H2 SiN,Si2,Si3,Si2N,SO,SN,S2 CH,OH,HN,HS,SiH,CH4,C2H4 (1):resultingfromequilibriumchemistrybasedontheelementabundancesgivenin Fe (atomic) MgO,MgN,MgS,MgH Fig.5.1fortherangen<He>=106:::1012cm 3andTg=800:::5000K FeO (3):moleculeswithmax (2):moleculeswithnmol=n<C>>10 6somewhereinthe(n<He>;Tg){plane Table5.1reviewsthemoreabundantmoleculesforthehydrogen{decientand moleculeiscomposedof. Elfnmol=n<El>g>10 10,whereElincludesallelementsthe carbon{richelementcompositionconsideredhere.thechemistryisdividedintothe followingsubgroups.themostabundantgroupcontainsthepurecarbonmolecules chemistry.oxygenismostlyblockedbytheformationofcoandconsequentlyall SeeGoeres&Sedlmayr(1992)formoredetailedinformationconcerningthecarbon gastemperaturetheconcentrationsofthemorecomplexcarbonmoleculesincrease. withsmallchains,whichareallradicals,andmonocyclicrings.withdecreasing othermoleculescontainingoxygenarenotabundant.especiallyh2oispractically forrcbstarsinfutureinvestigations. 3Asubstantialimprovementofthemodelmaybeachievedbyusingadetailedmodelspectrum

5.2.RESULTS absentfromthegasphase.thenextgrouparecompoundsformedoutofnitrogen 75 andcarbon.themostimportantnitrogenmolecule,however,isn2.furthermore, (exceptforsio).ironandmagnesiumbearingmoleculesareunimportant. formedoutoftheseelementsandtheabundantandunblockedelementscandn thereareseveralabundantsilicon,sulphurandhydrogenbearingmolecules,all 5.2.3RadiativeHeatingandCoolingRates Thequestionofimportantcontributorstotheheatingandcoolingofsuchspecial Ddv variousradiativeprocessesforatypicalchoiceoftheparameterste,r=rand case.table5.2summarizestheresultsofthisworkconcerningtheroleofthe gashastobeinvestigatedcarefully.noprecedingstudiesareavailableforthis therstlineofeachpanelintable5.2)increaseswithincreasinggastemperature bymanyordersofmagnitudeandmoderatelydecreaseswithdecreasingdensity. Theimportanceoftheindividualheatingandcoolingprocessesstronglydepends dle.theabsolutevalueofthetotalradiativeheating/coolingrate(asgivenin ontemperatureanddensity.usuallyonespecialradiativeprocessdominatesin acertaintemperature/densityregime.allbasicradiativeprocessesmaycause heatingorcoolingandchangethesignatdierenttemperatures,whichdependon theelectrondensity)averycomplexpictureappears,whichshowsthefollowing boththecarriersoftheheating/coolingratesandthecollisionpartners(especially oftheprocess(cf.fig.3.5).togetherwiththestronglyvaryingconcentrationof therelationbetweenjandthesourcefunctionatthecharacteristicwavelength features: Free{freeheating/coolingisimportantforlargedensities(n<He>>1012cm 3). Bound{freetransitions,mainlyofHeandC,providethemostimportant Theheating/coolingratesoflinetransitionscoverthewholetemperature/den- heating/coolingprocessatlargedensities(n<he>>1011cm 3),whereallthe bound{boundtypetransitionsareopticallythick. temperatures.themostimportantcontributorsarehe+,c+,n+,s+and ingofthegasfornottoolargedensities(n<he><1011cm 3)andnottoolow sity{planeandaregenerallyimportant.theydominatetheheatingandcool- Assoonaspolarmoleculesbecomeabundantinthegasphase,theirlargenumberofallowedtransitions(vibrationalandrotational)dominatestheradiative Fe+,becauseofthehighfractionalionizationinthemodel(cf.Sect.5.2.1). insect.5.2.2.coplaystheoverwhelmingroleconcerningtheheatingandcoolingofthegasbymolecules,sinceitisthemostabundantpolarmoleculeby heatingandcoolingofthegas.thishappensbelowthedividinglinedescribed andsis.forlargerdensities(n<he>>1010cm 3)thevibrationaltransitions areimportant,whereasforsmallerdensitiesthepurerotationaltransitionsare approximatelytwoordersofmagnitude.furtherimportantmoleculesarecs moresignicant.

76 CHAPTER5.THERMALBIFURCATIONSINRCBSTARS Table5.2:Importantheating/coolingprocessesforRCBabundancesasfunctionoftemperatureTganddensityn<He>(1);(2).ParametersarechosenasTe=7000K,r=2Rand Ddv dle=10kms 1=R(3). 50000K He-bf 1014cm 3 1:0(14) 1012cm 3 1010cm 3 108cm 3 106cm 3 C-bf He-bf C-bf 1:0(12) HeII HeI He-bf 1:2(11) CII HeII HeI 3:3(9) CII HeII HeI 4:3(7) 20000K He-bf C-bf O-bf 5:3(10) He-bf C-bf O-bf 5:7(8) CII HeI SiII 8:1(8) CII HeI SiII 1:7(7) CII OII NII 3:5(5) 10000K C-bf O-bf 1:2(8) CII C-bf SiII 4:8(6) CII SiII NII 2:3(7) CII NII SII 4:7(5) CII NII OII 2:6(4) 6000K +C-bf +O-bf 1:9(6) +C-bf +O-bf 2:0(5) CII NII SiII 3:2(5) NII CII SII 1:7(4) NII OII FeII 2:7(3) 3000K +C-bf +2:1(7) CO-vib +C-bf 4:6(5) +CII+C-bf +O-bf +1:8(5) +CII+NII +SII +4:9(3) +SII+NII FeII +4:7(2) 1500K +Fe-bf +Si-bf+Mg-bf +7:3(4) CS-vib CO-vib+SiII 6:2(4) +CII+C-bf +SiII +5:6(4) CO-rot+CII +SII 7:2(3) CO-rot+SII +NII 5:1(3) 800K +Mg-bf +Fe-bf+Na-bf +1:8(4) +CO-vib +SiS-vib+SiII +2:6(4) +CO-vib +SiS-vib CO-rot +4:9(2) CO-rot+FeII HCN-rot 1:5(3) CO-rot+FeII +He-bf 1:0(3) 400K +Mg-bf +Fe-bf+Na-bf +2:7(4) +CO-vib+SiII +1:1(4) +SiS-vib+SiII +FeII +7:6(2) +FeII+CO-rot +SiII +4:0(2) +CO-rot+FeII +SiII +2:6(2) Eachpanelofthetablehastwoentries: (1)Therstlineistheresultingtotalnetradiativeheatingratepermassofthegas (2)Alistofthethreemostecientheating/coolingprocessesisstatedbelowinorder ofdecreasingabsolutenetrates: =cooling,+=heating,i=linesofneutral atom,ii=linesofionizedatom,=free{free,bf=bound{free,vib=vibrational, Qrad=[ergs 1g 1],whereX(Y)meansX10Y. (3)R=73Risassumedinthiscontext. rot=rotationaltransitions.

5.2.RESULTS 77 10 3dyncm 2(n<He>109:::21010cm 3),Te=7000K,r=3RandDdv Figure5.2:Heating/coolingratesasfunctionofthegastemperatureforp= 10kms 1=R.Thethickfulllineshowsthetotalnetheatingrate.Theother dashedanddottedlinesdepictthefree-freerateq,thetotalbound{freerate Qbf(allatoms/ions),thetotallineheating/coolingrateQLines(allatomsand dle= solutions. ions),thetotalvibrationalrateqvib(allmolecules)andthetotalrotationalrate Qrot(allmolecules).Thecirclesdenotestableradiativeequilibriumtemperature 5.2.4RadiativeEquilibriumTemperatureSolutions Thesolutionsoftheradiativeequilibriumproblemarerelatedtothechangesofsign Iwillbrieyexplorethereasonsforthesechangesofsigninthefollowing. ofthetotalnetradiativeheatingfunctionqradasafunctionofthegastemperature. Theheating/coolingratesasfunctionsofthegastemperatureareshowninFig.5.2 forasamplechoiceoftheparameters.thesumsoftheratesofallkindsofprocesses changethesignatdierenttemperatures.fortheparameterschoseninfig.5.2, (free-free,bound-free,lines,vibrationalandrotationaltransitions)aredepicted.for onends:bound{free7000k,spectrallines4800k,free-free615k,vibraingthedirectiontolowergastemperatures,thedierentprocessessubsequently sucientlyhightemperatures,allradiativeprocessescausenetcooling.consider-

78 tional575kandrotational200k.finally,forsucientlylowtemperatures,all CHAPTER5.THERMALBIFURCATIONSINRCBSTARS ishenceforthcalledthe\high{temperaturesolution".forthehightemperaturesin problem.consideringthedirectiontolowtemperatures,therstsolutiontooccur radiativeprocessescausenetheating. Fig.5.2,thelinetransitionsprovidethedominantheating/coolingprocess.Conse- Thus,therealwaysexistsatleastonestablesolutionfortheradiativeequilibrium ture,wherethetotallineheating/coolingrateqlineschangesitssign. quently,thehigh{temperaturesolution(4830k)isusuallyclosetothetempera- Thehigh{temperaturesolutionreferstoapredominantlymolecule{free, ThechangeofsignofQLinesiscausedbythetemperature{dependentcompeting QLinesforsmallandQbfforlargedensities,respectively. partiallyionizedgas.thetemperatureisxedbythechangeofsignof bination,andcollisionalionizationfollowedbyradiativerecombination4.ifmolecule citationfollowedbylineemission.thechangeofsignofqbfiscausedbythe competingprocessesofphotoionizationfollowedbycollisional(three{body)recom- processesoflineabsorptionfollowedbycollisionalde-excitation,andcollisionalex- betheonlysolutionandtheradiativeequilibriumproblemwouldbeunique. However,oncethegashasreachedasucientlylowtemperature,moleculesbecome formationwasnotpossibleinthegasphase,thehigh{temperaturesolutionwould locatedintheirandmicrowavespectralregion,entersintocompetitionwiththe abundant.theirlargenumberofallowedvibrationalandrotationaltransitions, otheratomictransitionswhichsubstantiallyincreasestheeciencyoftheinteraction temperaturesarerequiredtocauseachangeofsignofthemolecularheating/cooling faintnessofthecentralstaratthesewavelengths,assketchedinfig.3.5.muchlower betweenthegasandtheradiationeldatlongwavelengths.thereby,theappearance functions. ofmoleculescausesreinforcedcoolingforthepresentbecauseofthecomparable temperaturesolutionsresultfromanequilibriumbetweenatomicheating Theadditionaltemperaturesolutionsarecausedbythepresenceof andmolecularcooling.thelow{temperaturesolutionsarecausedbya molecules.twotypesofstablesolutionsarefound.themedium{ ForexampleinFig.5.2,onendsasecondstablesolutionat1900K,where theradiativeheatingbylinesandbound{freetransitionsisbalancedbyvibrational changeofsignofthedominantmolecularheating/coolingfunction. cooling.ataboutthethirdstablesolution(565k)thevibrationalheating/cooling superpositionofthenumeroustransitions. functionchangesitssign.additionalunstablesolutionsexistat2290kand 1440K. 4Strictlyspeaking,evenQbfandQLinesmaychangethesignmorethanonce,becauseofthe

5.2.RESULTS 79 p=10+2dyncm 2 p=100dyncm 2 Figure5.3:ThermalbifurcationsinRCBenvelopesforp=102dyncm 2(upper panel,n<he>1014:::51015cm 3)and100dyncm 2(lowerpanel,n<He> unstablesolutions,respectively.theradiusaxisbelongstotheopticalthinlimit 1012:::51013cm 3).TheradiativeequilibriumtemperaturesolutionsTRE shownversusdilutionfactorwinaplanck{typeradiationeldwithte=7000k andforhdv dli=10kms 1=R.Fullanddottedblacklinesindicatestableand gare bodytemperaturetbbarethesameasshownandexplainedinfig.1.1. (pureradialdilution)accordingtoeq.(1.7).theuv{andir-limitandtheblack

80 CHAPTER5.THERMALBIFURCATIONSINRCBSTARS p=10 2dyncm 2 p=10 4dyncm 2 1010:::51011cm 3)and10 4dyncm 2(lowerpanel,n<He>108:::5109cm 3). Figure5.4:SameasFig.5.3,butforp=10 2dyncm 2(upperpanel,n<He>

5.2.RESULTS 81 p=10 6dyncm 2 p=10 8dyncm 2 Figure5.5:SameasFig.5.3,butforp=10 6dyncm 2(upperpanel,n<He> 106:::5107cm 3)and10 8dyncm 2(lowerpanel,n<He>104:::5105cm 3).

82 ThegeneraltopologyoftheradiativeequilibriumsolutionsisdepictedintheFigs.5.3 CHAPTER5.THERMALBIFURCATIONSINRCBSTARS dilutionaccordingtoeq.(1.7),butitsmeaningismoregeneral.wcharacterizes Thisfactorisrelatedtoadistinctradialdistanceinthecaseofpuregeometric to5.5.thetemperaturesolutionsareshownasfunctionofthedilutionfactorw. thestructureofthebifurcations.w=1togetherwithreimpliescompletethermodynamicequilibrium(te)accordingtotheconceptofthiswork:inthecase wherew=1theradiationeldisanon{dilutedplanck{eldj=b(trad)andthe thedeparturefromanequilibriumand,hence,isanappropriatevariabletostudy isdirectlybalancedbyitscorrespondingreverseprocess,whichcharacterizeste. (cf.chapter3),thisworkaccuratelydescribesthisbehavior. Sinceallreverseprocessesareincludedbymeansofdetailedbalanceconsiderations onlysolutionofreisgivenbytre g=trad.every(collisionalorphoto-)process AllcalculatedREtemperaturesolutionsarelocatedbetweentheIR{limit(TIR= Thermalbifurcationsarefoundtooccurunderthefollowingconditions: WTe)andtheUV{limit(TUV=Te),nicelyconrmingthesimpleresultsofSect.1.2 wherelteanda{typegasabsorptioncoecienthavebeenconsidered. 1)Ahigh{temperaturestablesolutionmustbepossible,i.e.aradiativeequilibriumstateofthegasmainlyconsistingofatomsandions5. 2)W<0:1(r>1:5R)isrequiredtomakepossibleamolecule{rich,low{ 3)p<1dyncm 2(n<He><1013cm 3)isrequiredtolimittheinuenceofthe temperaturesolutionasmotivatedbytheir{limit. notproduceadditionalsolutions. lowtemperatures(cf.table5.2)and,consequently,moleculeformationdoes rates.fortoolargedensities,qbfdominatestheheatingandcoolingevenfor bound{freeheating/coolingratescomparedtothemolecularheating/cooling and810kforp=10 2dyncm 2andW=0:05)candierbyseveralthousands dilutionfactor.thestabletemperaturesolutions(e.g.5220k,2000k,1220k simultaneoustemperaturesolutionsmayexist,dependingonthepressureandthe Undertheseconditions,thegasisalwaysfoundtobeatleastbi{stable.Upto4 ofdegrees,usuallyyieldingonehigh{temperature,atomicsolutionandoneormore whichcanbeseenbycomparisonofthefigs.5.3to5.5.thegeneraltendencyis low{temperature,molecularsolutions. AnotherresultofthemodelisthattheREgastemperaturesaredensity{dependent, thatathingastendstobecoolerthanadensegas,consideringthesamebranch comparedtothelatter,yieldinglowerretemperaturesaccordingtofig.1.1. ofsolution.thisiscausedbytheincreasingimportanceofspectrallinesandrotationaltransitionscomparedtobound{freeandvibrationaltransitionsfordecreasing density,respectively.theformertransitionshavelongercharacteristicwavelengths entersintothemolecularregimeanddisappears.onlyonelow{temperaturesolutionremainsin thiscase. W<0:01inFig.5.5.Inthiscasethe\high"{temperaturesolutiondropsbelow2000K,whereit 5Aviolationofthiscriterionoccursatsmallpressuresp<10 6dyncm 2andlargedilutions

5.3.DISCUSSION Discussion 83 mainlyconsistingofatomsandions.bothphasesareinradiativeequilibriumand ThecircumstellarenvelopesofRCBstarsshowamulti{stablecharacter.Coolgas phases,mainlyconsistingofmolecules,canprincipallycoexistbesideshotphases, reinforcedradiativecooling6.thegasthencoolsdowntomuchlowertemperatures, Themulti{stablecharacterofthegascausesakindof\coolingtrap".Oncethe gashasreachedasucientlylowtemperature,moleculesareformedwhichcause inpressurebalancewitheachother. untiltheheatingandcoolingbymoleculesaloneproducesanothersolutionofthe Thermalbifurcationsarefoundtooccurinalargerangeofexaminedparameters, concerningboththeradialdistancetothestarandthegaspressure.thesendings radiativeequilibriumproblemandstabilizesthelowtemperature. thermalbifurcationsareexpectedtooccurmainlyinthecsesofwarmstarswith indicatethattheoccurrenceofthermalbifurcationsisnotrestrictedtothecsesof Te>4500K,wheretheatomic,high{temperaturesolutionstillexists(cf.criterion1 RCBstars,butisacommonphenomenoninpartiallymoleculargases.However,the oftheitemlistonthepreviouspage). ConcerningtheCSEsofcoolstars(asC{andM{starsontheAGB),theradiative equilibriumgastemperaturesareexpectedtobemuchlowerthantheblack{body quently,thesolutionsoftheradiativeequilibriumproblemshouldbesimilarto temperatures.thegasinthesecircumstellarenvelopesismolecule{rich.conse- inthischapterisnotappropriateforthiscaseandtheresultscanbedierent. enshroudedandhenceopticallythick.theapproximationoftheradiationeldused Nevertheless,theconsequencesofthemulti{stablecharacterofthegasreachfar, thelow{temperaturesolutionsdiscussedabove.however,theseenvelopesaredust{ asf.kneer(1983)wroteinviewofthisinstability:\iconcludethatrestellar atmosphereswithte=5800kmaynotexist,inprinciple".iwouldnotgothat far,butconsiderforexampleagaselementwhichslowlymovesoutwardsinacse Thegastemperatureslowlydecreaseswithincreasingradialdistancedowntoabout consistsofatomsandionsaslongasthehigh{temperaturesolutionisrealized. oftheelementshallbeslow,sothatreremainsvalid.thegaselementmainly withatemperaturestructuresimilartothatdepictedinfig.5.5.themotion 2000K,untilsuddenly,atabout4RinFig.5.5,acertainamountofmoleculeshas beenformed,justsucienttodestabilizetheradiativeequilibrium.thegasthen quicklycoolsdowntowardsthesecond,low{temperaturesolutionat200k.the nalchemicalcompositionandtheamountofdustformedinthegaselementwill cruciallydependontherelationbetweenthechemicalandthecoolingtimescale duringthistransition.intheend,thechemistryfreezesoutanddustformation generaltheoreticalviewofthechemistryandthedustformationprocessesinstellar becomesimpossibleagain.ifthisscenarioprovestobetrueitwouldchangeour 6Howsuchasucientlylowtemperaturecanbereached,isleftopenforthepresent.

84 envelopesquitedramatically.othertopicsrelatedtothemulti{stablecharacterof CHAPTER5.THERMALBIFURCATIONSINRCBSTARS thegascouldbeinhomogeneities,cloudformationorahysteresis{likebehaviorof ityofreunderdynamicconditions.forexample,thelow{temperaturesolutions calculateradiativecoolingtimescales,whichgiveanimpressionontheapplicabil- thegasinthecsesofpulsatingstars. caneasilybedestabilizedbyadiabaticheating/coolingrates,whichdiminishesthe TheresultsofthischapterrefertotheassumptionofstaticRE.Chapter6will meaningofthelow{temperatureresultsofthischaptertosomeextent. theradiationeld.eachradiativeprocesswhichisadditionallytakenintoaccount dependsonthedetailsof(i)thechemistry,(ii)theheating/coolingfunctionsand(iii) areliabledeterminationofthegastemperatureisdicult.inthestaticcaseitreally Incontrasttothegeneralnding,thatthermalbifurcationsshouldoccurprincipally, theresultsofchapter7,aswillbediscussedtherein. maychangetheresultsfortre gsubstantially.thisisfundamentallydierentfrom

Chapter6 RadiativeCoolingTimeScalesinthe Thesecondapplicationofthethermodynamicmethodsdevelopedinthisworkinvestigatestherelaxationtowardsradiativeequilibrium.Agaselementinnon{RE isconsidered.theelement,beinghotterorcoolerthaninre,willconsequently radiateawayexcessinternalenergy(radiativecooling)orgainradiativeenergyby CircumstellarEnvelopesofC{Stars Thecharacterofthethermalbehaviorofthegasunderdynamicalconditionscan whichisdenedbelow. netabsorption(radiativeheating),respectively.thekeyquantitywhichdescribes theeciencyofthisrelaxationisthetimescaleforradiativecoolingorheating, thegasquasiinstantaneouslyrelaxestowardsreand,consequently,thecondition consideredprocess.iftheradiativecoolingtimescaleisshorterthantheothers, timescale")withtheotherhydrodynamicorchemicaltimescalesinvolvedinthe bediscussedbycomparingthistimescale(henceforthcalledthe\radiativecooling andmustbecalculatedtime{dependently. thantheothers,thetemperatureofthegasdependsonthehistoryoftheprocess ofrecanbeusedtodeterminethegastemperature.ifitiscomparableorlarger ConcerningthechemistryandthedustformationintheCSEsofpulsatingstars,the temperatureunderdynamicconditionsisinvestigated. Inthefollowing,theapplicabilityofREforthedeterminationofthegas characterofthethermalrelaxationofthegasinresponsetopropagatingshockwaves pulsatingstarsarehitbyshockwavestimeandtimeagain.theshocksdissipate whichsteepenuptoshockwavesintheatmosphereandpropagateintothecse isofspecialimportance.thepulsationintheinteriorofthestarproduceswaves, mechanicalenergyandheatupthegastoconsiderablyhightemperatures.thegas mustbeabletoradiateawaythisexcessinternalenergybeforethenextshockhits (e.g.bowen1988,fleischeretal:1992).thus,thegaselementsintheenvelopesof formation.infact,fromobservations,justtheoppositeconclusionscanbedrawn. chemicalanddustformationprocesses. Followingthisconsideration,onewouldexpectthestellarpulsationtohinderdust theelement.otherwise,itwillneverbecomesucientlycooltoallowforcomplex Manyofthedust{formingobjectsareknowntobepulsatingstars.Moreover,a 85

strongcorrelationbetweentheoccurrenceofanirexcess(indicatingdustformation) 86 CHAPTER6.RADIATIVECOOLINGTIMESCALESINC{STARS byobservations. stars(jura1986),i.e.stellarpulsationfavorsdustformation.therefore,anecient relaxationoftheshock{heatedgasincircumstellarenvelopesseemstobeconrmed andalightvariability(indicatingstellarpulsation)canbeobservedforlatetype ThischapterconsiderstheCSEsofpulsatingC{stars.ItpicksupthecontroversialquestionofwhethertheshocksintheseCSEsbehavepredominantly\isothermally"or\adiabatically"(moreinformationsaboutthiscontroversycanbefoundin Sect.1.4).Aclaricationofthisquestionisanimportantsteptowardstheprincipal understandingofdustformationinthecsesofpulsatingstars. 6.1.1DenitionoftheRadiativeCoolingTimeScale TheModel Anarbitraryphysicalquantityyshallbeconsidered.Thetimeevolutionofyis assumedtobegivenbytherstorderordinarydierentialequation Theequilibriumvaluesofthephysicalquantityyareimplicitlydenedbyf(y)=0. dy dt=f(y): (6.1) timerequiredfortherelaxationist=y y Ifyistorelaxtowardequilibrium,i.e.y(t+t)y,therstorderestimateofthe FirstorderTaylorexpansionofEq.(6.1)intimeyieldsy(t+t)=y(t)+tf(y). Theradiativecoolingtimescaleisdenedanalogously,consideringpureradiative heating/coolingaccordingtode f(y): (6.2) cool;tg;j;ddv dle=e;tre dt=bqradwithbqrad=qrad= g;j;ddv bqrad;tg;j;ddv dle e;tg;j;ddv dle gisoneretemperaturesolutionasdenedinchapter5.apartfromtheproblemofremulti{stability,tre : (6.3) thermodynamicquantitiesandtg,thecontinuousbackgroundradiationeldj andthelocalvelocitygradientddv theentiredensity{andtemperature{rangeencounteredintheshockedenvelopesof gandtherebycoolarecompletelydeterminedbythe pulsatingc{stars. dle.inthefollowing,theaimistocalculatecoolfor

6.1.THEMODEL 6.1.2ElementAbundances 87 TheelementcompositionofC{starsisassumedtobesolarexceptforcarbon.The byc=o=1:7,whichaccordingtofrantsman&eglitis(1988)isarepresentative fromallen(1973).carbonisassumedtobeoverabundantwithrespecttooxygen TheabundanceofHeisassumedtobeHe=H=0:1andtheabundanceofMgistaken solarabundancesareadoptedfromlambert&rao(1994)andreferencestherein. valueforc{stars. FortheapplicationswithregardtoC{starsthemeanback{groundintensitiesare assumedtobegivenbyannon{dilutedplanckeld(w=1),thatis 6.1.3ApproximationoftheRadiationField Thisisdoneforthreereasons.First,theCSEsofC{starsaresupposedtobedust enshroudedandhencenotopticallythin.eq.(6.4)representsthelimitingcaseofan J=B(Trad): (6.4) opticallythickcse.second,theassumptionconsiderablysimpliestheevaluation ofthecoolingtimescaleasdenedabove.accordingtoeq.(6.4)thereisalways latterofcourserequiresmoredetailedknowledgeaboutj.theparametertradis todeterminecoolandnottondthespecictemperaturesolutionsofre.the well{denedaccordingtoeq.(6.3).third,thecalculationsareperformedinorder exactlyone(trivial)retemperaturesolutiongivenbytre g=trad.thereby,coolis assumedtovarybetween0and3000kforc{starenvelopes,considering3000kas remarkablysimilar(cf.sect.6.2.4).therefore,thechoiceoftheradiationeldis arepresentativevaluefortheeectivetemperaturesofthesestars.evenforthe notcrucialforthedeterminationofcool.chromosphericemissionsandcontinuous extremecasesj=0andj=b(3000k),theresultsforthecoolingtimescalesare emissionsfromshockedgaslayersinthecseareagainignored. 6.1.4LocalVelocityGradient meanvelocitygradientddv theshockedenvelopesofcoolpulsatingstars(e.g.fig.1ofwintersetal:1994),the Regardingthetypicalsaw{tooth{likevelocitystructuresinmodelcalculationsfor fronts).therefore,thisparameterisxedandsettoddv throughoutthewholeconsideredcircumstellarshell(exceptfortheverythinshock characteristicvalueoftheorderofv1=rvaryingbyaboutoneorderofmagnitude dle,asdenedbyeq.(3.14),hasmoreorlessacertain inuenceofthisparameterissmall(cf.sect.6.2.5). dle=20kms 1=500R.The

6.2 88 Results CHAPTER6.RADIATIVECOOLINGTIMESCALESINC{STARS Beforediscussingtheresultsfortheradiativecoolingtimescales,someofthemicrophysicalresultsshallbestatedrst:thecompositionofthegas(degreeofionization 6.2.1CompositionoftheGas andchemistry),theinternalenergyandtheroleofthedierentradiativeprocesses. thegureshowscontourlinesoftheconcentrationsofh2ande inthetemperature/density{plane,indicatingwhetherthegasispredominantlymolecular,atomic ThecompositionofthegasisroughlydepictedinFig.6.1.Theupperpanelof orionized.thetwoextremecasesj=0andj=b(3000k)areconsideredonthe leftandrighthandsideoffig.6.1,respectively. Theresultingelectrondensityisveryimportantforthecalculationoftheradiative collisionpartner,alsoonthebound{boundcollisionrates.thedegreeofionization heating/coolingrates.ithasadecisiveinuenceonthebound{freeratesand,as processeslinearlydependonne(cf.eq.3.60and3.61),thedensity{dependence ofthegasisfoundtostronglydependontheradiationeld. cancelsoutandthecontourlinesarehorizontallinesonthelefthandsideoffig.6.1. whichismainlybalancedbyradiativerecombination.sincetheratesoftheboth InthecaseTrad=0,fractionalionizationissolelycausedbycollisionalionization, Thedeviationsfromstraightlinesathightemperaturesarecausedbycollisional thanadirectcollisionalexcitationh+e!h++2e (cf.sect.3.2.2).fractional ionizationisfoundtobenegligible(<10 5)forTg<5000KinthecaseJ=0. excitationfromexcitedstatesofhydrogen.forlargedensitiesatwo{stepcollisional processh+e!h+e andh+e!h++2e turnsouttobemoreecient importantthancollisionalionizationfortg<5000k.sincethephotoionizationrates izationpotentials(si,mg,fe,na)additionallyproducesfreeelectronsandismore aredensity{independent,buttheradiativerecombinationratesdodependonthe ForTrad=3000K(righthandside)photoionizationofmetalatomswithlowiontainedforlowtemperatures,dependingonthedensity1.Thus,fractionalionization isfoundtobemuchlargerthaninlteatlowtemperaturesforthisradiationeld. density,thecontourlinesareapproximatelyverticallinesfortg<5000konthe righthandsideoffig.6.1.adegreeofionizationaslargeas10 3to10 5isre- radiativerecombinationratesintheentiretemperature/density{planeunderin- Thethree{bodyrecombinationratesarefoundtobenegligiblecomparedtothe tionunlessthetemperatureisaslargeas>10000k.onlyforverylargedensities, (n<he>>1016cm 3)thecalculatedfractionalionizationofthegasapproachesLTE. vestigation.consequently,lte(saha){ionizationisneverachieved.forexam- ple,thegasremainspredominantlyneutralfortheradiationeldsunderexamina- andgraindrift,forexample. 1Suchdegreeofionizationisexpectedtocauseconsiderableeectswithregardtograincharge

6.2.RESULTS 89 Figure6.1:Thecomposition,theinternalenergyandthenetheatingfunctionofthegas (fulllines).themiddlediagramsshowthetotalinternalenergyofthegaseona linearscalerangingfromabout 1:4(12)to+1:3(13)erg/g.Thezero{lineisadditionally asfunctionoftemperatureanddensity.theupperdiagramsarecontourplotsoftheh2{ concentrationlog(nh2=n<h>)(dottedlines)andtheelectronconcentrationlog(ne=n<h>) shownasadashedlineandthesignofeisindicated.thelowerdiagramsshowthe AllcalculationaremadeforDdv considersthecasej=0,whereastherightcolumnconsidersthecasej=b(3000k). QradispositivebelowandnegativeabovethedashedTg=3000K{line.Theleftcolumn totalnetheatingfunctionofthegaslogjqrad=j[ergg 1s 1].Ontherighthandside, dle=20kms 1=500R.

Thechemicalcompositionofthegasiscalculatedbyassumingchemicalequilibrium 90 CHAPTER6.RADIATIVECOOLINGTIMESCALESINC{STARS theconcentrationsofthemoleculesarefoundtobeverysimilarcomparedtotheresultsofpreviousworksusingchemicalequilibrium.theresultsoftheapplicationof withrespecttotheneutralatomdensitiesinthiswork(cf.chapter4).consequently, entneutralatomdensitiesduetotheupperresultsconcerningtheionizationwhich (e.g.gail&sedlmayr1988).somemodicationsarecausedbythesomewhatdier- yieldsomewhatdierentmoleculeconcentrations(e.g.lesssiliconbearingmolecules chemicalequilibriumtoc{starsenvelopeshavethoroughlybeendescribedelsewhere linesintheupperpaneloffig.6.1. aresmall.theh2{concentration,forexample,isdepictedbythedottedcontour ifsiisstronglyionizedasinthecasetrad=3000k).however,thesemodications Thedeterminationoftheinternalenergyofthegasisanimportantingredientfor 6.2.2InternalEnergy dependenttreatmentofthermodynamics,forexampleinhydrodynamicmodels.it thecalculationofthecoolingtimescale.moreover,itisessentialforanytime{ providesthebasiclinkfromtheenergycontent,whichismodiedbyradiative heatingandcooling,tothetemperatureofthegas. ular,atomicandionizedstateofthegas.theregimesaredividedbyconsiderable Threedierentregimescanbedistinguishedwhichrefertothepredominantlymolec- plane.theinternalenergydiersalotfromthatofanidealgas(e=fkt=(2)). ThemiddlediagramsofFig.6.1depictcontourlinesofeinthetemperature/density{ gaswhilethetemperatureonlychangesgradually.withinonephase,theinternal energybarriersinbetween.inodertoovercomesuchabarrier(toperformaphase transition),aconsiderableamountofenergyistobeaddedortoberemovedfromthe tionenergy,arefoundtosignicantlycontributetothetotalinternalenergyofthe Allcomponentsoftheinternalenergy(cf.Eq.2.2),exceptfortheelectronicexcita- anidealgas.theinternalenergyalwaysincreasesmonotonicallywithtemperature. energyapproximatelydependslinearlyonthetemperature,closetothebehaviorof energyisnegativeinthemolecularregime).erotisabout0:57etransassoonash2 forhightemperatureswhereitreachesabout4etransat20000k.edissdominates theinternalenergyatlowtemperatures,about 59Etransat500K(theinternal gas,atleastinaparticularregionoftemperatureanddensity.eionisimportant thepopulationofthevibrationalstatesofthemoleculesisstronglyaectedbythe radiationeld.itsmaximumcontributionisfoundtobe0:07etransfortrad=0 and1:2etransfortrad=3000k.eelisfoundtobenegligible(<310 4Etrans). ismoreabundantthanh.thecontributionofevibdependsontrad,indicatingthat density.thedependenceontheradiationeldissignicant.theionicpotential Theinternalenergyofthegasisnotcompletelydeterminedbytemperatureand principallyalsopresent,butnegligible. sponsibleforthesedependences.thedependenceofeonthevelocitygradientis energyeionandthevibrationalexcitationenergyofmoleculesevibaremainlyre-

6.2.RESULTS 6.2.3TheRadiativeCoolingTimeScaleandtheRoleoftheVarious 91 Trad=0)andinFig.6.4(forTrad=3000K).Thedashedarrowswillbediscussedlater TheradiativecoolingtimescalesasfunctionofandTgaredepictedinFig.6.2(for HeatingandCoolingProcesses to10yearsforawarmandthingas.thecoolingtimescalestronglydepends Typicalvaluesforcoolarefoundtorangefrom10 2sforahotanddensegas radiativeheatingfunctionbqradisadditionallyshowninthelowerpaneloffig.6.1. (insect.6.2.7)andarenotofinterestforthepresent.forcompleteness,thenet onboththegastemperatureandthegasdensity.thetemperature{dependenceis foundtocomprise9ordersofmagnitudeatlargeand3ordersofmagnitudeatsmall isalsostrong,8ordersofmagnitudeathighand3ordersofmagnitudeatlower densities,consideringgastemperaturesof500:::20000k.thedensity{dependence gastemperaturesrequiredfortheecientexcitationoftheupperstates,bynon{ functions,whichareaectedbythevaryingparticleconcentrations,byminimum Thesedependencesresultfromasuperpositionofthedierentheatingandcooling temperatures,consideringdensitiesof104:::1014cm 3. LTEeectsandbyradiativetrapping.Ingeneral,adensegasheatsandcools toprovideareasonablettotheresults.thereareevencases,whereadensegas n<h>>1011cm 3.Inthisregion,allimportantradiativeheatingandcoolingrates moreecientlythanathingas.however,asimpleapproachlikeqrad/2fails areofbound{boundtransitiontypeandthetotalrateisstronglyreducedbythe heatsandcoolslessecientlythanathingas.thisoccursfortg<4000kand lowcontinuousradiationeldsandhighdensities,theheatingandcoolingofthe areuncertaininthisregion.forsucientlylowgastemperatures(tg<1500k), ofthestrengthbutofthenumberoflinestakenintoaccount.therefore,theresults largeopticaldepthsinthelines.theheating/coolingeciencyisnomoreaquestion gasispossiblycontrolledmostlybythepresenceofdustgrains(e.g.viathermal accommodation),sincedustformationcantakeplaceecientlyinthisregion. followingpictureappears: Themosteectiveheating/coolingprocessesarestatedintheFigs.6.3and6.5.The Forhightemperatures(Tg>6000:::10000K),hydrogencoolingdominates. tiesbound{freecoolingofhydrogenturnsouttobemoreimportantduetothe largeopticaldepthinthehydrogenlines.free{freecoolingisalsoimportant ForsmalldensitiescoolingbyLyandHisecient,whereasforlargedensi- Forintermediatetemperatures,thereisazoneofconsiderablesmallerheating/coolingrates,i.e.largercoolingtimescales.Inthiszone,thetemperature isalreadytoolowinordertoexcitetheh{atoms,butstilltoohighforcon- CIIandOII.Atverylargedensities(n<H>>1011:::1012cm 3),bound{free transitionsofh dominatetheradiativeheatingandcooling. ofneutralandsinglyionizedmetalatoms:ci,oi,sii,fei,feiiandalsosi, forhightemperaturesandlargedensities. siderablemoleculeconcentrations.theremainingradiativeprocessesarelines

92 CHAPTER6.RADIATIVECOOLINGTIMESCALESINC{STARS Figure6.2:Contourlinesoftheradiativecoolingtimescales(fulllines).The (isobar)coolingtrackwithamaximumradiativecoolingtimescaleofoneyear digitsonthecurveslabellogcool[days].thedashedarrowindicatesthecritical onthetrack.parameters:trad=0andddv dle=20kms 1=500R. H bf H I H bf C I Si I O I C I Fe I Fe II O I CO vib Figure6.3:MostecientcoolingprocessreferringtoFig.6.2(schematically, C 2 H rot rot=rotational,vib=vibrational,i=linesofneutralatom,ii=linesofionized HCN SiO vib H SiS vib CO rot atom,bf=bound{free). 2

6.2.RESULTS 93 established.notethatthecoolingtimescaleremainspositiveandsteady,al- thoughthenetradiativeheatingfunctionqradchangesitssignattg=3000k. Figure6.4:Contourlinesoflogcool[days]asinFig.6.2,butforTrad=3000K. ThecriticalcoolingtrackendsatTg=Trad,whereradiativeequilibriumisre- H bf CII H I H bf O I C I CO vib CO rot C 2 H rot Figure6.5:Mostecientheating/coolingprocessreferringtoFig.6.4. H 2 vib H 2 vib HCN rot H SiS vib 2 CS rot rot CO rot Fe II Fe II

94Forlowtemperatures(Tg<2000:::4000K,assoonasCOisabundant), CHAPTER6.RADIATIVECOOLINGTIMESCALESINC{STARS CNareimportant.Forsmalldensitiesrotationalheating/coolingdominates, rotationaltransitionsofco,sis,hcn,c2h,cs,h2andalsoofsioand polarmoleculesdominatetheradiativeheatingandcooling.vibrationaland vibrationalstates(cf.sect.3.1). whereasforlargedensitiesthevibrationalheating/coolingismoreimportant, asexpectedfromthelargercriticaldensitiesforthethermalpopulationofthe ThecoolingtimescaleasfunctionofJisexpectedtovarybetweenthevaluesshown inthefigs.6.2and6.4,supposedthat0jb(3000k)forc{starenvelopes. 6.2.4DependenceontheRadiationField Theradiativecoolingtimescaleisfoundtobeonlymarginallyaectedbythechoice oftheradiationeld.themaximumdeviationbetweentheextremestrad=0and Trad=3000Kisfoundtobe2.4dex,whichoccursintheregioncontrolledbyH Forexample,atTg<3000KinFig.6.4,theradiativeprocesseschangefromcooling smaller.thestandarddeviationisfoundtobe0.6dex.thisisasurprisingresult. atlargedensitiesandwarmtemperatures.however,theusualdeviationismuch eciencyofradiativeheatingandcoolingisaninherentfeatureofthegas,mainly controlledbytemperatureanddensity. eciencythanthoseresponsibleforcooling.butthisturnsouttobewrong.the toheating.onewouldexpecttheprocessesresponsibleforheatingtohaveadierent assumedvalueof20kms 1=500R.SignicantdierencesareonlyfoundforTg< Thecalculationshavebeenrepeatedwiththeonetenthandtentimestheusually 6.2.5DependenceontheVelocityGradient careanywayasstatedinsect.6.2.3.qradnevervariesmorethanlinearlywithddv 4000Kandn<H>>1011cm 3,wheretheevaluationofQradhastobetakenwith Forsmallerdensitiesorhighertemperaturesthedependenceismuchsmaller. dle. Inthefollowing,theresultsofthisworkarecomparedtopreviousanalyticalap- 6.2.6ComparisontoAnalyticalHeating/CoolingFunctions temperaturestructures. proachestodeterminetheradiativeheatingandcoolingratesincircumstellaren- encesinthehydrodynamicmodelcalculationsconcerningforexampletheresulting velopes.aspointedoutinsect.1.4,theseapproachesleadtoconsiderabledier-

6.2.RESULTS 6.2.6.1Bowen'sHeating/CoolingFunction 95 bybowen(1988): Thefollowinganalyticexpressionofthenetradiativeheatingratehasbeenproposed BowenstrictlyassumesQrad/2throughoutthecircumstellarenvelope.Theheating/coolingprocesseswhichbehaveinsuchawayarelimitedbythecollisionalenergy 2C0(Trad Tg) (6.5) bqrad=3k onemightcalleq.(6.5)thestrictnon{lteheating/coolingrate.atemperature{ independentcoolingtimescaleisfurthermoreassumed.theparameterc0,reecting transferasinthelimitingcaseofsmalldensities(cf.sect.3.1.1.3).consequently, rateiscalculatedasdescribedbybowen(1988),althoughseveralassumptionsare turescoolingbyemissioninlyisadditionallytakenintoaccount.thish{cooling involvedhere,whichwithregardtothisworkseemtobequestionable,asforexampletheassumptionofaconstant(density{independent)escapeprobabilityfor Ly. theradiativecoolingtimescale,ischosentobe10 5gscm 3.Forhighertempera- 6.2.6.2LTEHeating/CoolingFunction bymeansoftheassumptionoflteb"=bb(tg): thegas,thefollowingexpressionforthenetradiativeheatingratecanbeobtained StartingfromtheexactexpressionbQrad=4RbJd 4Rb"d,wherebisthe trueabsorptioncrosssectionpermassandb"isthespectralemissivitypermassof bqrad=4bj(;tg)t4 rad bb(;tg)t4 g tionpermassofthegas.isthestefanboltzmannconstant.thisanalyticalform bj(;tg)istheintensity{meanandbb(;tg)theplanck{meanabsorptioncrosssec- (6.6) ofthenetradiativeheatingratehasbeenusedbyfeuchtingeretal:(1993),assuming eralrequiresverylargedensities(cf.chapter3).furthermore,ifoneuses'swhich collisionallypopulatedlevelsoftheconsideredatomsandmolecules,whichingen- Qrad/results.UsingEq.(6.6)meanstoassumethatallradiativeprocessesreferto aconstantgreygasabsorptioncrosssection.asfarasbisdensity{independent, callythin2.duetothelackofplanck{meanopacities,rosselandmeanopacitiesare usedinthefollowingforbothopacitiesineq.(6.6).therosselandmeanopacities examplethenumerouslinesofmolecules),theincludedlinesareassumedtobeopti- havebeencalculatedbyopacitysamplingmethodswithrespecttospectrallines(for 6.2.6.3ResultsoftheComparison br(;tg)areinterpolatedfromtablesprovidedbyscholz(scholz&tsuji1984). thelteheating/coolingfunctionaredepictedinthefigs.6.6and6.7,respectively. TheresultingradiativecoolingtimescalesaccordingtoBowen'sandaccordingto 2Infact,thesetwoassumptionscontradicteachother.

96 CHAPTER6.RADIATIVECOOLINGTIMESCALESINC{STARS ing/coolingfunctionproposedbybowen(1988).trad=3000kiscon- sidered. Figure6.6:logcool[days]calculatedfromtheanalyticalheat- Figure6.7:logcool[days]asintheuppergure,butcalculatedfrom thelte{heating/coolingfunction.nocriticalcoolingtrackexistshere, year. sincetheradiativecoolingtimescaleisalways(much)shorterthanone

6.2.RESULTS Inthiscontext,e=3kTg=(2)with=1:27amuisassumedforbothapproaches 97 muchagreementwiththeresultsobtainedinthiswork.bowen'sheating/cooling functionyieldsastrongdensitydependence(cool/ 1,verticalcontourlines), NeitherthecoolingtimescalesderivedfromBowen'snorfromtheLTErateshow underdiscussion.theseresultscanbecomparedtofig.6.4. ing/coolingratescalculatedinthiswork,bowen'srateusuallygivesmuchsmaller values(uptoafactorof10 6inthelow{density,low{temperatureregime),whereas coolingtimescales(roughlyhorizontalcontourlines).comparedtotheheat- whereasthelteheating/coolingfunctionyieldsmoreorlessdensity{independent density,high{temperatureregime).thebestthatcanbesaidisthatthecooling twoanalyticalformulae. timescalescalculatedinthisworkusuallyliebetweenthevaluesderivedfromthe thelterateusuallygivesmuchlargervalues(uptoafactorof106inthelow{ coolingmoredetailed,andsincehydrogencoolingisdominantathightemperatures paredtotheresultsofthiswork.asbowen'sratetosomeextenttreatshydrogen 8000K)Bowen'srategivesaboutthesameslopeandorderofmagnitudecom- Someroughagreementsarefound,nevertheless.Forhightemperatures(Tg> andcoolingforhightemperatures,anintermediateminimumforthepredominantly accordingtothiswork,thisagreementwastobeexpected.thelterateproduces ciencyinthemolecularregimeatlowtemperatures.bestagreementwiththelte asimilartemperature{dependenceasfoundinthiswork:veryeectiveheating atomicphaseatwarmtemperaturesandare{increaseoftheheating/coolinge- coolingtimescaleisfoundonthelefthandsideofthediagramsatlargedensities probably,itisbecause detailedagreementisnotachieved,notevenfortheselargedensities.thisdisagreementmightbecausedbymissingradiativeprocessesinthiswork.however,more (yieldingsimilarcoolingtimescaleswithinabout2ordersofmagnitude).however, ii)accordingtothiswork,thegasisstillfarfrombeinginlteatn<h>= i)rosselandmeansofhavebeenusedinsteadofplanckmeans, Insummary,bothanalyticalheating/coolingfunctionsyieldpooragreementwith iii)thelteheating/coolingfunctionneglectsopticaldepthsinthelines. and 1014cm 3(especiallywithregardtothedegreeofionization,cf.Fig.3.9), theresultsofthiswork.bowen'srateseemstounderestimateandthelterate stressesthenecessitytousemoredetailedmodelcalculationsfortheradiativeheatingandcooling.theproposedanalyticalfunctionsareinsucienttodescribethe seemstooverestimatetheheating/coolingeciencybyordersofmagnitude.this radiativeheatingandcoolinginthecircumstellarenvelopesofcoolstars.

6.2.7TheTransitionfromIsothermaltoAdiabaticShocks 98 CHAPTER6.RADIATIVECOOLINGTIMESCALESINC{STARS discussionofthecharacterofthethermalrelaxationbehindpropagatingshockwaves inthecircumstellarenvelopesofpulsatingstars. Thecalculatedradiativecoolingtimescalesofthisworkallowforaquantitative consistingofh+10%he).afterthepassageoftheshock,thegasradiatesawaythe Agaselementbeinghitbyastrongshockwaveofvelocityvsisalmostinstantaneouslyheateduptohightemperatures(11500K[vs=20kms 1]2foranidealgas excessinternalenergydissipatedbytheshock,i.e.itrelaxestore,inprinciple. P.Furthermore,thepropagatingshockwavesinitiateaconsiderablecompression However,consideringthemoreorlessperiodicallyshockedenvelopesofpulsating wavehitstheelement.thistimeisgivenbyaboutonestellarpulsationalperiod andre-expansionofthegasaccompaniedbyconsiderableadiabaticheating/cooling. stars,onlyalimitedtimeforthisrelaxationisavailable,beforethenextshock ThetimescaleoftheseprocessesisalsogivenbyP.Thus,therelationbetweenthe radiativecoolingtimescalecoolandthestellarpulsationalperiodpdeterminesthe characterofthethermalbehaviorofthegasinthecsesofpulsatingstars. IfcoolismuchsmallerthanP,thegasquicklyrelaxestoREbehindtheshocks. coolp,isothermalshocks Theadiabaticheating/coolingratesaremeaningless.REisestablishedintheoverwhelmingpartsofthecircumstellarenvelope,exceptforsomethintemperature temperaturestructureofthecsecanbecalculatedbyassumingre. peaksatthelocationoftheshockfronts(cf.fig.1.3).apartfromthesepeaks,the cool>p,adiabaticshocks (6.7) coolingadiabaticallyinthemeantime.thetemperaturestructuremustbecalculatedtime{dependently. Inanalogytothesituationinstationaryshocks(e.g.Neufeld&Hollenbach1994),an isobarcoolingtrackinthetemperature/density{planeisconsideredinthefollowing. coolingtimescalecoolonthetrack.theisobarcoolingtrackwithcool1yronthe track(henceforthcalledthe\criticalcoolingtrack")isdepictedinfig.6.2,fig.6.4 andalsoinfig.6.6asdashedgreyarrow,consideringoneyearasatypicalperiodof Thetotalcoolingtimealongsuchatrackisroughlygivenbythemaximumradiative sequently,thegassubsequentlyheatsupduetotheshocks(cf.fig.1.4),roughly radiateawaytheenergydissipatedbyoneshockwithinonepulsationalperiod.con- If,however,coolexceedsP,a\chromospheric"situationresults.Thegascannot isanestimateforthedividinglinebetweentheshocksofpredominantlyisothermal andpredominantlyadiabaticcharacter.gaselementswhichareshockedtotheleft meanparticlemassduetophasetransitions(cf.fig.6.1).thecriticalcoolingtrack pulsatingc{stars.thedeviationsfromastraightlinearecausedbychangesofthe ofthecriticalcoolingtrackscanreestablishrebeforethenextshockarrives thosewhichareshockedtotherightofthecriticalcoolingtrackwillbehitbythe nextshockbeforerecanbeachieved.

6.3.DISCUSSION Accordingtotheresultsofthiswork,atransitionofthecharacteroftheshockwaves 99 AroundTg5000Kthecoolinggaselementspendsmostofitstotalcoolingtime. frompredominantlyisothermaltopredominantlyadiabaticwithdecreasingdensity. istobeexpectedtooccuraroundpost{shockdensitiesof106:::108cm 3,changing Thecoolingtimescaleinthistemperatureregionisfoundtovaryby4 5orders ofmagnitudefortheentirerangeofconsidereddensities,whichis10ordersof nalanswertothesequestionscanonlybeobtainedbymeansoftime{dependent forisothermalshocks,densitiesaslargeas1010:::1012cm 3wouldberequired.A magnitude.therefore,asharptransitionisnotexpectedtooccur,ratheragradual hydrodynamicmodelcalculations. changeoverabroadrangeofdensities.forexample,ifcool<0:01pwasdemanded readyoccursat1011cm 3.TheLTEratepredictstheshockstobecloseto AccordingtoBowen'srate,thetransitionfromisothermaltoadiabaticshocksal- theisothermallimitingcaseforalldensities.thisexplainsthedierencesbetween themodelcalculationsofbowen(1988)andfeuchtingeretal:(1993)concerningthe 6.3 resultingtemperaturestructures. namicsinthemodelcalculationsforcoolstellarenvelopes,especiallyinthecase Theresultsofthischapterstronglysuggesttoincludetime{dependentthermody- Discussion previouslyusedarenotsucientinthiscontext. Atimescalediscussioncanbeperformedinordertoclarifywhetherornottheconditionofradiativeequilibrium(RE)canbeusedtodeterminethegastemperature. Bycomparingtheradiativecoolingtimescalecool,asdepictedintheFigs.6.2and culationoftherelevantheatingandcoolingrates.simpleanalyticalexpressions ofpulsatingstars.thebasisforthethermodynamicdescriptionisarealisticcal- 6.4,withtheothertimescalescontrollingthephysicalprocessunderconsideration, itcanbedecidedwhetherthetemperaturesmaybecalculatedfromradiativetransfercalculations(assumingre),orwhether,forinstance,asimpleadiabaticcooling lawismoreappropriate. Thegeneraltendencyoftheresultsobtainedinthisworkisthatthecondition determinationonthebasisofreisjustied.however,ingeneral,time{dependent tobeoftheorderofdays,sothatreisprobablyestablishedandatemperature Forthelargedensitiesclosetothestartheradiativecoolingtimescalesarefound ofrecanonlypartlybeusedinordertodeterminethetemperatureofthegas. beconsidered,themorequestionablethedeterminationofthetemperatureonthe eectsasadiabaticcoolingcanthroughoutbeimportant.thelowerthedensityto basisofrebecomes.forinstance,atn<h>107cm 3inthewarmatomicphase, theradiativecoolingtimescaleapproachestheorderofoneyear,alreadyclosetothe Concerningtheshockedenvelopesofpulsatingstars,thethermodynamicsshouldbe expansiontimescaleinstationarywindmodelsforc{stars(e.g.krugeretal:1994).

treatedtime{dependentlyandifonlyfortheexistenceoftheshockwaves.additionally,apartfromthelocationsoftheshockfronts,strongdeviationsfromreare 100 CHAPTER6.RADIATIVECOOLINGTIMESCALESINC{STARS toapproximatelyadiabatic. expectedtooccurroughlyatdensitiesn<h><108cm 3,connectedwiththegradual transitionofthecharacteroftheshocks,changingfromapproximatelyisothermal heatingandcoolingratescalculatedinthisworkintothetime{dependenthydrodynamicmodelcalculationscanbeachievedbytabulatingqradandtheinternal temperature{dependenceofthechemicalandthenucleationprocesses,pronounced energyeasfunctionsof,tgandfurtherparameterscharacterizingthelocalcontinuousradiationeldandthelocalmeanvelocitygradient.accordingtothestrong eectsareconceivable3.anothertopicwhichmightberelatedtotheresultsofthis formationprocessesareaectedistobeinvestigated.theproperinclusionofthe peraturestructuresofcoolstellarenvelopes.howfarthechemicalandthedust Time{dependentthermodynamiceectscancausesubstantialchangesinthetem- workistheformationofchromospheres. dynamicmodelingmayhavewithregardtodustformation. 3Thefollowingchapterwilldemonstratewhatsevereconsequencesatime{dependentthermo-

Chapter7 RCBStars Shock{InducedCondensationaround modynamicprocess.themostcomplexlevelincludedintheworkisachieved: Thethirdandlastapplicationinthisworkstudiesadistincttime{dependentther- ThecircumstellarenvelopesofpulsatingRCBstarsareconsidered.Athermodynamicdescriptionforxeduidelementswhichareperiodicallyhitbyshockwaves isdeveloped.astheshockscompressthegas,itre{expandsinthemeantimewhich time{dependentnon{lte(inthesteadystateapproximation)andnon{re. causesconsiderableadiabaticcooling.theinternalenergybalance,thetemperature requiredforsuchnucleation.thecalculationsprovideahypothesisforthephysical elementsareinvestigated.specialattentionispaidtotheminimumradialdistance causeofthespectacularrcbdeclineeventswhicharesupposedtobecausedby ofthegasandthepossibilityforeectivecarbonnucleationtooccurinsuchuid AppendixA. andthescienticmeaningofthesestudiesarefurtherdescribedinsect.1.3andin dustformationclosetotheserelativelyhotstars.theastronomicalbackground 7.1 mentinaconstantradiationfield TheModel:AFixed,PeriodicallyShockedFluidEle- Achosenuidelementinthecircumstellarenvelopeofapulsatingstaristime followsaroughlyballistictrajectory,whilecooling,chemicallyrelaxingandre{ Fleischeretal:1992,Feuchtingeretal:1993).Betweentheshocks,theuidelement Theshocksaccelerate,heat,chemicallyalterandcompressthegas(Bowen1988, andtimeagainhitbypropagatingshockwavescausedbythestellarpulsation. playofdierentphysicalprocessestakesplace:hydrodynamics,thermodynamics, expanding(gillet&lafon1983,bowen1988). Thus,theCSEsofpulsatingstarsareastrophysicalsiteswhereacomplexinter- thermodynamicconsequencesofthehydrodynamicsituationofperiodicallyshocked circumstellarshockwavesisaverychallengingworkwhichgoesfarbeyondthescope ofthisthesis.instead,themodelcalculationspresentedinthischapterstudythe chemistryanddustformations.acompletemodelingoftheseprocessesoccurringin gas.asimplegas{boxdescriptionsuitableforthethermodynamicinvestigationsis 101

r r(t) developedforthissituationwhich,accordingtoobservations,isapparentlycommon 102 CHAPTER7.SHOCK{INDUCEDCONDENSATIONINRCBSTARS amongthedust{formingobjects(jura1986). Thecircumstellarenvelopeisassumedtooscillateinaperiodicmanner1.Furthermore,masslossisneglected.Masslossleadstoanadditionaloutwardmovemencussedinthischapter,whichisthetemporalsupercoolingofthegasbelowitsRE{ Therefore,theneglectionofmasslosssystematicallyunderestimatestheeectdis- accompaniedbyadditionalexpansion,whichcausesadditionaladiabaticcooling. temperatureduringthephasesofre{expansion.accordingtotheseassumptions, theuidelementexactlyreturnstoitsstartingpoint(cf.fig.7.1)andallhydrodynamicandthermodynamicquantitiesvaryperiodicallyintimewiththepulsation asbeingtypicalfortheenvelopesofpulsatingstars. periodofthestar.wewillconcentrateonthismostsimplecase,whichisconsidered v s.t Figure7.1:Lagrangiantrajectories(schematically,upperpanel)anddistance r gravitationaldeceleration.there{expansionofthegasbetweentheshocksis inthelaboratoryframe.onaverage,theaccelerationbyshocksisbalancedby cumstellarenvelopeofapulsatingstarwithoutmassloss.vsistheshockvelocity betweentwoneighboringuidelements(lowerpanel)intheshock{levitatedcir- t causedbythephase{shiftoftheballistictrajectoriesaccordingtothegravitation TheperiodicsituationassketchedinFig.7.1canbedividedintotwophases:the ofthestar. shocktransitionandthere{expansionofthegas.bothprocessesareexaminedin thefollowinginorderndanappropriateprescriptionoftheperiodicboundary envelopeispossibleevenifthestellarpulsationisperfectlyperiodic. conditions,whichthegaselementsareexposedto. 1Fleischeretal:(1995)havepointedoutthatamulti-periodicorevenchaoticoscillationofthe

7.1.1ShockTransitions 7.1.THEMODEL 103 TheshocktransitionsaretreatedbyapplyingtheRankine{Hugoniotrelations(e.g. Landau{Lifschitz1959).Theshockfrontisconsideredasinnitesimallythinandthe magneticeldsandvanishingcontributionsoftheradiativeuxareassumed: actualtransitionprocessasinstantaneous.thejumpconditionsforaplane{parallel ofmass,momentumandenergyinacomovingframeoftheshockfront.negligible perpendicularshock(~v?tothefront)aregivenbytheequationsoftheconservation p1+1v21=p2+2v22 12v21+h1=12v22+h2 1v1=2v2 (7.1) Equations(7.1)relatethehydrodynamicandthermodynamicpropertiesoftheupstreamow(index1,\pre{shock")tothoseofthedownstreamow(index2,\post{ shock").h=e+p=istheenthalpyandetheinternalenergypermassofthegas. Togetherwiththeequationofstate(cf.Chapter4)thepost{shockquantitiescan theequationofstateisapplicableagain.thistimeisassumedtobesmallcompared passageoftheshockwave,whenthegashasjustrelaxedtoitssteadystate,sothat Strictlyspeaking,theso{denedpost{shockstatereferstoadenitetimeafterthe becalculatedfromthepre{shockthermodynamicstateandtheshockvelocityv1. ofequations(7.1)requiresaniteration.thefollowingsimpleiterationschemeis topulsationperiodpandtheradiativecoolingtimescale. appliedwhichisfoundtoconvergereliably: Duetothenon{trivialequationofstateinvolved,theactualsolutionofthesystem 1)Startwithacompressionratiooffour(v2=v1=4). 2)Puth2=h1+(v21 v2)=2andp2=p1+1v1(v1 v2). 4)Dene=j1 (2v2)=(1v1)j. 3)Calculatethepost{shockdensityaccordingtotheequation 5)Performoneiterationstepbyv2!0:1v2+0:9v11=2. ofstateintheform2=2p2;h2;j;ddv dle. TheradiationeldJ,thevelocitygradientDdv 6)Gobacktostep2unless<10 8. beequalonbothsidesofthefront(cf.sect.7.1.6). Theseparametersaresettoxedvaluesduringthecalculationsandareassumedto additionalparametersforthecalculationoftheequationofstate(cf.chapter4). dleandtheelementabundancesiare Thecalculatedcompressionratios2=1forstrongshocks(v1pre{shocksound Accordingtothetheoreticaldescriptionoutlined,thegasiscompletelydissociated bythedissociationandionizationpotentialenergytermsintheequationofstate. valuesrangefrom5to9,dependingontheshockvelocity.thiseectiscaused speed)arefoundtobelargerthanthemaximumvalueof4foranidealgas.typical

andpartiallyionizedbystrongshocks.sincethedissipatedshockenergyispartly 104 CHAPTER7.SHOCK{INDUCEDCONDENSATIONINRCBSTARS consumedinordertobreakthechemicalbondsandtoionizetheatoms,thepost{ valuesare20000kto70000kforshockvelocities20kms 1to50kms 1.Thepost{ massivehydrogen{decientgas. shockgastemperaturesarefoundtobelowercomparedtoanidealgas.typical shocktemperaturesarehighercomparedtoahydrogen{richgasbecauseofthemore Betweentheshocks,thechangeoftheinternalenergyofthegasiscalculatedviathe 7.1.2Re{ExpansionPhases rstlawofthermodynamics(cf.sect.7.1.3).inordertondanappropriatedescriptionofthere{expansionprocess,asuitablestatevariableischosen,whoseexplicit cannotbylimitedtosingleuidelementconsiderations.thefollowingapproachis consistentphysicaldescriptionwouldincludetime{dependenthydrodynamicsand time{dependencecanbeprescribed.ofcourse,thisisanapproximateprocedure.a adaptedtotheexperienceofhydrodynamicmodelcalculations. p1isthepre{shockandp2thepost{shockgaspressure(cf.eq.7.1),modthe p 1=(t)=p 1= 2 +p 1= 1 p 1= 2 tmodp modulofunctionandtheadiabaticindexofthegas,whichisassumedtobe5/3 P (7.2) ismotivatedasfollows: inthiscontext.themainideaofthisapproachistoassumethatthegaspressure (i)inthelimiting(adiabatic)caseofnegligibleradiativeheating/cooling,where monotonicallydecreasesbetweentheshockswithapower{lawintime.theapproach pv=const,thevolumevarieslikeasaw{toothfunctionintime V1isthepre{shockvolumeandVmintheminimumvolumeofaxeduidelement V(t)=Vmin+(V1 Vmin)tMODP duringoneperiodiccycle.vminequalsthepost{shockvolumev2intheadiabatic P : (7.3) Inthiscase,theLagrangiantrajectoriesr(t)aresecond{orderpolynomialsandthe readilyobtainedifthegasactuallybehavespurelyballisticassketchedinfig.7.1. dependenthydrodynamicmodelcalculations(e.g.bowen1988,seehisfig.4).itis case.equation(7.3)providesagoodttothevolumevariationsfoundintime{ theabsoluteradialdistance,theenclosedvolumev/r2risproportionaltor and,therefore,alsoasaw{toothfunction.ingeneral,takingintoaccountradiativeheatingandcooling,thecalculatedvolumevariationdoesnotdiermuchfrom Eq.(7.3)asdemonstratedinFig.7.3.ThemaindierenceisthatinthiscaseVmin distancebetweentwoneighboringuidelementsrvarieslikeasaw{toothfunctionsintime.supposedthattheamplitudeofradialmotionissmallcomparedto issmallerthanv2,whichwillbecomemoreclearinthenextparagraph.

(ii)accordingtoeq.(7.2)thegaspressurevariesonatimescaleofp.hence,fast 7.1.THEMODEL 105 thepassageofashockwave,wherethegasishotandcoolsveryeciently.this theuidelementcompressesbyfastcooling,whichespeciallyoccursshortlyafter radiativecoolingwithcoolpautomaticallyproceedsisobaricly.consequently, aboutafactor4isfollowedbyasubsequentpost{shockcompression,whichamounts 1979and1989,Neufeld&Hollenbach1994),wheretheinitialshockcompressionof uptoafactor100.thereasonforthisbehavioristhattheowissubsonicbehind matcheswellwiththeresultsofstationaryshockmodels(e.g.hollenbach&mckee thefront,sothatpressurebalancecanestablish.thecitedcalculationsshowthat astationaryow.therefore,againconsideringtheperiodicshocks,vminingeneral compressionbyradiativecooling2.thepost{shockcoolingusuallyproceedssofast, doesnotcoincidewithv2,butissubstantiallysmallerduetosubsequentpost{shock p=constisvalidwithin25%accuracyinthewholepost{shockregioninthecaseof (iii)accordingtotheassumptionthatthepressurevariationismonotonicallydecreasingbetweentheshocks,theamplitudeofpressurevariationisgivenbythejump thateq.(7.3)isstillagoodapproximationoftheresultingoverallvolumevariation. conditionseq.(7.1).thereisnoneedtointroduceadditionalfreeparametersin ordertodescribetheamplitudeofthecyclicvariationscausedbytheshocks.especiallyvminisaresultofthecalculations.ifthevolumewaschosentobeprescribed, Theresultsarenotmuchaectedbytheassumedslopeofthepressuretime{ dependence.additionalcalculationshavebeencarriedoutwithdierentvalues forandalsocalculationswherethevolumewaschosentobeprescribed(using onefreeparameter,thatisvmin,wouldbeadditionallyrequired. onlytheexistenceofaperiodicalperturbationoftheuidelementanditsamplitude (characterizedbytheshockvelocity)areapparentlyimportant. Eq.(7.3)withVminasadditionalfreeparameter).Theresultsareverysimilar 7.1.3Thermodynamics Thetimeevolutionoftheinternalenergyeoftheconsidereduidelementduring dynamics there{expansionphasesisstraightforwardlycalculatedviatherstlawofthermo- thespecicenthalpyh=e+p=andtosolveinstead Sincethegaspressureischosentobeprescribed,itismoreconvenienttoconsider de dt= pdv dt+bqrad: (7.4) Equation(7.5)issolvedbyimplicitnumericalintegrationwithadaptivestepsize dh dt=+vdp control.thekeyforthiscalculationisthedeterminationofthestateofthegasand dt+bqrad: thepressuregradientprovidesanon{negligiblehydrodynamicforce. 2Consequently,thetrajectoriesareinfactnotpurelyballistic.Inthehotpost{shockregions

thenetradiativeheating/coolingrateasfunctionofp,h,jandddv 106 CHAPTER7.SHOCK{INDUCEDCONDENSATIONINRCBSTARS calculations. andqradateveryinstantoftime,sothateq.(7.5)becomesanordinarydierential equation.thegastemperaturetgasfunctionoftimeisanimplicitresultofthese dlewhichyieldsv issketchedinfig.7.2.auidelementinthecseofapulsating(rcb)staris Aschematicdescriptionofthethermodynamicprocessesandthemodelingprocedure 7.1.4TheModelingProcedure considered.inphase1theelementishitbyapropagatingshockwave,whereitis heatedandcompressed(!v2).duringphase2itcoolsdownandfurthercompresses duetofast,approximatelyisobaricradiativecooling(!vmin).accordingtothe periodicityintheseenvelopes,thegaselementnallyre-expandsduringtherestof eachperiodicalcycleinphase3(!v1).thesethreephasesrepeatperiodically. thermodynamic processes model 1) shock solution of the transition jump conditions } 2) post shock cooling solution of dh dp ^ = V + Qrad dt dt for given p=p(t) 3) re expansion Figure7.2:Thethreeperiodicallyrepeatingphasesofshocktransition,post{ : heating heating(inphase3).thetheoreticaldescriptionoftheprocessesisoutlinedon shockcoolingandre{expansionforauidelementinthecircumstellarenvelope : radiative / cooling therighthandside ofapulsatingstar.wiggledarrowsindicatenetradiativecooling(inphase2)or : adiabatic cooling

Themodelsimulatestheseprocessesbysolvingtheshockjumpconditionsatthe 7.1.THEMODEL 107 mentbecomeperiodically.usually3to25periodsarerequiredinordertoachieve instantsoftimewheretheshockwaveshitthegaselementt2f0;1p;2p;:::gand bycalculatingtherstlawofthermodynamicsinthemeantime.thecalculations arecontinueduntilthevariationsofthethermodynamicquantitiesinthegasele- periodicity.indetail,thecalculationsproceedasfollows: 1)ChooseaxedradiationeldJ,axedvelocitygradientDdv 2)Ateveryfullperiod,solvethejumpconditionsEq.(7.1)forthepost{ arbitraryinitialenthalpyh1. shockvelocityv1andatxedpre{shockgaspressurep1.startwithan dle,axed 3)Considerthetimevariationofthegaspressureduringtheforthcoming shockstate(p2;h2). 4)Gobacktostep2,unlessallvariationshavebecomeperiodic. thalpyaccordingtoeq.(7.5),yielding(p1;h01)attheendofthisperiod. periodasexplicitlygivenaccordingtoeq.(7.2)andcalculatetheen- Thenal(periodic)resultsofthemodeldependonthefollowingparameters: 7.1.5OverviewofIntroducedParameters Twoparametersforthedescriptionofthebackgroundcontinuousradiationeld(Teandr=R,cf.Eq.5.3). Parametersforthecompositionandtheoveralldensityoftheconsidered Twoparametersforthestrengthandthefrequencyofthepropagating shockwaves(v1andp). Twoadditionalparameterswhoseeectsontheresultsaresmall,that gaselement(iandp1). isthelocalmeanvelocitygradientddv explicitpressuretime{dependenceduringre{expansion(cf.eq.7.2). dleandthepowerindexforthe 7.1.6ExaminedRangeofParameters Teandr=R:TheeectivetemperatureofthecentralRCBstarisassumedto be7000kthroughoutthischapter,whichapparentlyisarepresentativevaluefor beconstantduringthecalculations.incontrast,asignicantradialmotionofthe uidelement(assketchedinfig.7.1)isassumedtobesmallcomparedtor,so thatr=r,forsimplicity,isxed.thereby,themeanintensityjisassumedto thisclassofstars(lambert&rao1994).thevariationoftheradialpositionofthe areconsidered. furthercomplicationofthemodelofminorimportance.radialdistancesof1:5 5R uidelementwouldimplyanadditionaltime{dependenceofj,whichisregardedas

P:Thepulsationperiodisassumedtobe44days,whichisthevaluesuggestedby 108 CHAPTER7.SHOCK{INDUCEDCONDENSATIONINRCBSTARS Fernieetal:(1972)forRCrB.SinceotherRCBstarsshowverysimilarvaluesforP (cf.appendixa)thisparameterisalsoxedforthecalculations. willfurthermoredependontheconsideredradialdistance.time{dependentmodels v1:theshockvelocityisuncertain,maybedierentfordierentrcbstarsand developsomewherebelowthephotosphere,wherethevelocityvariationisusuallya fewkms 1.Theshocksconsiderablysteepenupaccordingtotheexponentialdensity 1992,Wintersetal:1994,Feuchtingeretal:1993)indicatethattheshocksbeginto forthecircumstellarenvelopesoflong{periodvariables(bowen1988,fleischeretal: gradientintheouteratmosphereandsoonreachshockvelocitiesof30kms 1(the shockvelocitycanapproximatelybeidentiedwiththeamplitudeofvelocityjumps occurringinthesemodels).atlargerradialdistances,thedensitygradientbecomes orevenalmostzero,dependingonthemodel.itisunclear,whethertheseresults smallerandtheshockvelocityusuallytendstodecreaseagain,leadingto10kms 1 canbeadoptedtorcbstars.largephotosphericvelocityvariationsof20km 1 (implyingshockvelocitiesof40kms 1)havebeenobservedforRYSgr,whichis thestrongestknownpulsatingrcbstar(cf.appendixa).thesemeasurements strongershockscanbeexpectedinthecse3.however,rysgrisanexceptional refertothelineformationregion,i.e.tothephotosphereofthestar.considerably case.otherrcbstarsshowradialvelocityvariationsofabout5to10kms 1, butnolinesplittingwhichisanindicatorforshockactivityinthephotosphere (AppendixA).ShockactivityintheCSEisprobablynotdirectlydetectable,at leastnotatmaximumlight(apartfromthedeclineevents),whenthestaristoo bright.therefore,thequestionsoftheexistenceofshockwavesinthecsesof RCBstarsandtheirvelocitiescannotbedecidedbyobservationsyet.Thiswork fromtheory.shockvelocitiesof20 50kms 1areconsidered. presupposesthepresenceofshockwavesinrcbstarenvelopes,since(i)rcbstars showconsiderableradialvelocityvariationsatthephotosphereand(ii)evensmall i:theelementalabundancesoftheprototypestarrcrbareadoptedfromcottrell& amplitudewavesareknowntosteepenuptoconsiderableshockwavesinthecse Lambert(1982),cf.Fig.5.1. increasingradialdistance,theactualdensitystructureofthecircumstellarenvelopes p1:thepre{shockpressureoftheuidelementsisvariedindependentlyofr=r. Althoughthemeangaspressurecanbeexpectedtomonotonicallydecreasewith comparedtoagbstarenvelopes.forshock{levitatedcsesassketchedinfig.7.1,v1gep ThegravitationalforceatthephotosphereofanRCBstarisabout30timeslargerthanthatof isagoodapproximation.geisthegravitationaldecelerationcorrectedforradiativeacceleration. 3ThefollowingtheoreticalconsiderationpointstolargershockvelocitiesinRCBstarenvelopes aboutafactorof10smaller,yieldingabout3timeslargershockvelocitiescomparedtoagbstars. If10kms 1isconsideredasatypicalvalueforAGBstars,valuesofabout30kms 1arededuced anagbstar(roughlyassumingequalstellarmassesandluminosities)andthepulsationperiod fortheenvelopesofrcbstars.

7.2.RESULTS 109 ofrcbstarsisnotknown.pre{shockgaspressuresof10 7 10+1dyncm 2are considered. Themeanlocalvelocitygradientisnocrucialparametertothemodelandisassumed tobegivenbyddv dle=v1=r.thestellarradiusisassumedtober=73rinthis context(fernie1982).thepowerindexfortheprescriptionofthetime{dependence ofthegaspressureissetto=5=3. 7.2 Results 7.2.1CyclicVariationsinthePeriodicallyShockedFluidElements Anexampleoftheresultsforthecyclicvariationsofthethermodynamicstate variablesinxed,periodicallyshocked,circumstellaruidelementsisdepictedin Fig.7.3.Therst3periodsafterperiodicityhasbeenachievedareplottedona lineartimescale.thepost{shockgastemperatureisfoundtobe24000kinthe consideredcase,whichisoutofthedepictedrange.duringtherst1:5%ofthe period(16hours)thegasecientlycoolsdownto10000k,whichcausesfurther compression.theshockcompressionfactoris6:2andthepost{shockcompression factoris2:0.anattempttodepictthedierencebetweentheshockandthepost{ shockcompressionismadeinthemiddlepanel,butonthislinearscalethetotal (shock+post{shock)compressionphaseappearslikeasingle,almostinstantaneous process.thegasapproximatelyre{expandsadiabaticallyafterthiscompression.a saw{toothlikebehaviorofthevolumeresults.thetemperaturereachestre gafter about10%oftheperiodandisclearlybelowtre g afterwards.hereandinthe remainderofthischaptertre gdenotestherststable\high{temperature"solution ofradiativeequilibriumasdenedinchapter5.tre g isdensity{dependentand hencenotconstant. Infact,thevalueofTRE gispracticallymeaninglessforthegasinthedepictedcase ofp1=1:610 5dyncm 2,whichcorrespondstoadensityvariationofn<He>= 6107:::7108cm 3.Thetime{dependenttemperatureofthegasisessentially determinedbytheshocktransitionandtheeciencyoftheradiativecoolingathigh{ temperaturesduringthepost{shockcoolingphase.thesetwophasesdeterminethe starttemperatureandthetotal(de{)compressionfactorfortheforthcomingphase ofre{expansionwhichproceedsapproximatelyadiabatically.reisneverrealized andcannotbeusedtodeterminethetemperatureofthegas. FurtherdetailsareshowninFig.7.4,wherethesamesettingoftheparameters isinvestigatedexceptforalargershockvelocityof50kms 1.Inthiscase,the post{shocktemperatureisfoundtobe64000kandtheshockandpost{shock compressionfactorsare8:4and11,respectively.allradiativeprocessescause netcoolingbehindtheshockandsincethecoolingtimescaleisasshortasinitially 100s,theuidelementveryquicklycoolsdownduetoradiativelosses.Within therst0:3%oftheperiod(3hours),thetemperaturedropsto8000k.

110 CHAPTER7.SHOCK{INDUCEDCONDENSATIONINRCBSTARS plottedfordistancer=2r,shockvelocityv1=20kms 1andpre{shockpressure specicvolumeandthegastemperature(calculated,middleandlowerpanel)is Figure7.3:Timevariationsinaxed,periodicallyshocked,circumstellaruid p1=1:610 5dyncm 2.Thedottedlinesinthemiddlepanelindicatethepre{ elementofanrcbstar.thegaspressure(assumed,upperpanel)andthe depictstheradiativeequilibriumgastemperature. shock,post{shockandminimumvolume.thedashedlineinthelowerpanel

7.2.RESULTS Duringthisphase,whichisplottedonalogarithmictimescaleinFig.7.4,the 111 panel.thetemperaturereachestre uidelementcompressesascanbeseenfromtheincreasingdensityintheupper theremainingtimeoftheperiod,theuidelementre-expandsbyatotalfactorof 92.Thisre-expansioncausesintenseadiabaticcoolingasindicatedbythecooling rateqadb=vdp=dtinfig.7.4,whichistheconcurringratefordh=dtineq.(7.5). gafter1%oftheperiod(10hours).during heatingfunctionqradchangesitssign(notethetwofoldlogarithmicy{axisinthe Consequently,thegastemperaturedecreasesbelowTRE Thedecisivepointforthethermalbehaviorofthisuidelementisreachednow.The lowerpaneloffig.7.4). gandthetotalnetradiative ofthegasissucientinordertoovercomethismaximum.iftheanswerisno,the adiabaticcoolingofthegasiscompensatedbynetradiativeheating,thecoolingof atatemperatureof3000k.thequestioniswhetherornottheadiabaticcooling pointisrelatedtotherstintermediatemaximumofqrad(tg)depictedinfig.5.2 Tg<TRE timeoftheperiod(atlowergastemperaturesjqradjisusuallysmallercomparedto thegasisstoppedandthere{expansionproceedsmoreorlessisothermally(with therstmaximum),andthere-expansionapproximatelyproceedsadiabatically. g).iftheanswerisyes,theadiabaticratedominatesduringtheremaining Thecharacterofthere{expansionprocess,beingeitherisothermaloradiinantlyneutral,atomicphaseofthegas(causedbylineandbound{free transitions)priortomoleculeformation4. abatic,isdecidedbytheeciencyoftheradiativeheatinginthepredom- Inthegure,theadiabaticcoolingexceedsthenetradiativeheatingrate(jQadbj> heating/coolingfunctionsenterintocompetition,andsoonbecomemoreimportant lineheating/coolingratehasbeendominating,nowthevibrationalandrotational coolenoughinordertoallowforconsiderablemoleculeformation.whilesofarthe jbqradj)and,thus,thecoolingofthegascontinues.subsequently,thegasbecomes thanqlines.sincethemolecularratescausenetradiativecoolingforthepresent, Qradagainchangesitssignandtheadiabaticcoolingofthegasisnallyeven process. duringthere{expansion,whichisplottedonalinearscaleinfig.7.4.consequently, supportedbynetradiativecooling.however,qadbremainsthemostimportantrate Thus,thegastemperatureislowerthaninradiativeequilibriumalmostallthetime. there{expansionwhichtakesabout99%oftheperiodapproximatelyisanadiabatic Thegastemperaturenallyreachesaminimumvalueof780Kandisbelow1500K discussedinthesect.7.2.4. forabout60%oftheperiodatdensitiesn<he>=4108:::1:5108cm 3.These arethermodynamicconditionsfavorableforeectivecarbonnucleation,aswillbe excessinternalenergydissipatedbyashock,cf.sect.7.2.2. 4Thisstatementreferstocaseswherethegasrstofalliscapabletoquicklyradiateawaythe

112 CHAPTER7.SHOCK{INDUCEDCONDENSATIONINRCBSTARS Figure7.4:Detailsofthetimevariationsinaxed,periodicallyshocked,circumstellar uidelementofanrcbstar.thex{axisisbrokeninthisplot.therst2%ofthe theheatingandcoolingrates,respectively.thethickfulllineshowsthetotalnetradiative andthetotalheliumparticledensity(dottedline).themiddleandlowerpaneldepict Theupperpanelshowsthegastemperature(fullline),theREtemperature(dashedline) periodaredepictedonalogarithmicscale,whereastheother98%areplottedlinearly. rateandtheotherthindottedanddashedlinesdepictpartialrates,thesesarethenet thethickdottedline.parameters:r=2r,v1=50kms 1andp1=1:610 5dyncm 2. sumsoftheradiativegains/lossescausedbytheindicatedtransitiontype(i.e.allbound{ freetransitions,allvibrationaltransitionsetc.).theadiabaticcoolingrateisdepictedby free{free,bound{free,atomicline,vibrationalandrotationalrates.thelatterratesarethe

7.2.RESULTS 113 Figure7.5:Cyclicvariationsofdensityandtemperatureinperiodicallyshocked uidelementsatr=2r.theelementsdierbydierentvaluesofthepre{shock v1=20kms 1andv1=50kms 1,respectively.Theshortdashedlinesindicate theshocktransitions.thelongdashedlineshowstheradiativeequilibriumgas gaspressure.thegrayandblackcyclesdepicttheresultsforshockvelocities 7.2.2DependenceonDensity temperature. pressurep1,whichxesthemeandensityofthegasduringtheperiodicvariations. Theresultsdiscussedsofarhavebeencalculatedforaparticularpre{shockgas shocksisquitedierentforotherdensities.thisdensity{dependenceisdepicted sity(cf.chapter6),thethermalbehaviorofthegasinresponsetotheperiodic infig.7.5,wheretheperiodicallyrepeatingthermodynamicprocessesappearas Sincetheeciencyoftheradiativeheating/coolingisstronglyaectedbytheden- counterclockwisecycles. withthepost{shockstate(uppercorner),thegaselementreachesre(leftcorner) aratedandanalmosttriangularcycleresults(seel.h.s.ofthegure).beginning Concerningverylargedensities,thethreephasessketchedinFig.7.2arewellsep-

within1%oftheperiodduetoecient,approximatelyisobaricradiativecooling.theslightdeparturefromastraightlineonthiscoolingtrackisrelatedto 114 CHAPTER7.SHOCK{INDUCEDCONDENSATIONINRCBSTARS expansion.therefore,thegastemperaturestaysclosetotre of2.sincethecouplingtoreisstrongatthesedensities,theadiabaticcooling therecombinationofhe,wherethemeangasparticleweightchangesbyafactor ratesarenegligiblecomparedtotheradiativeratesinthesubsequentphaseofre{ caseofisothermalshocks.forabout99%oftheperiod,theelementiscloseto uppercorneragain,etc.thetriangularcyclesaretypicalresultsforthelimiting leadingtotherightcorner.finally,theuidelementisshockedandjumpstothe gduringthisprocess duringtheoverwhelmingpartoftheperiod.however,thisprocedureisonlyfeasibleforlargedensities.roughlyspeaking,reisareliablecriterionfortemperature determinationfordensitiesn<he>>1011cm 3. RE.Therefore,theconditionofREcanbeusedtodeterminethegastemperature consequently,thegasneverapproachesre.onthecontrary,acorona{likesituation energydissipatedbyoneshockcannotberadiatedawayduringoneperiodand, results,wherethegasisheateduptoextremelyhightemperaturesduetotheenergy Ontheotherextreme,consideringthecaseofverysmalldensities,theexcessinternal dissipationofwaves.forexample,inthecycleonther.h.s.forv1=50kms 1,thegas Oncethegashasrecombined,itscoolingtimescalebecomesmuchlargerthanthe ispredominantlyionizedandalwayshotterthan20000k.thisbehavioristypical period. forthelimitingcaseofadiabaticshocks.theresultingperiodictracks(seer.h.s.of Concerninganintermediaterangeofdensities,theradiativeenergyexchangeis Fig.7.5)consistofshocktransition,Herecombinationandadiabaticre{expansion. downfarbelowtre coolingratesduringthephasesofre{expansion.inthiscasetheuidelementcools shocks,butisnottooecientinordertobemaintainbalancewiththeadiabatic ecientenoughinordertocauseafastrelaxationofthegastowardsreafterthe ofmoleculedissociationenergy. (mainlycoandc3),wherethefurthercoolingofthegasisdelayedbytheliberation adiabatictracksatthelowerpartoffig.7.5arecausedbymoleculeformation g asdiscussedinsect.7.2.1.thesmallkinksonthealmost Asupercoolingofthegasoccurswithinadistinctdensity{intervalcaused byatwo{stepprocessofradiativecoolingathightemperaturesfollowed dependentcooling/heatingeciencyofthegas.regardingthebroadspectrumof Thedensity{dependencedepictedinFig.7.5isanaturalconsequenceofthedensity{ byadiabaticcoolingatlowtemperatures. ofapulsatingstarthedensityisjustappropriateforthiseect. densitiesencounteredincses,itseemsinevitablethatsomewhereintheenvelope

7.2.RESULTS 7.2.3DependenceonShockVelocity 115 arelargerforstrongshocks,implyinglargeradiabaticcoolingratesduringthephases mightbesurprising,butitisactuallystraightforward.thetotalcompressionratios butdoalsoallowforlowerminimumtemperatures.atrstsight,thisdependence Largershockvelocitiesv1producehighermaximumtemperaturesbehindtheshocks, oftheperturbation,causingbothup{anddownwarddeviationsfromre. ofre{expansion.theshockvelocitycanberegardedasameasurefortheamplitude 7.2.4PreconditionsforCarbonNucleation Inthefollowing,thepossibilityofeectivecarbonnucleationtotakeplaceinthese counteredincircumstellarenvelopes,thesizeofthecriticalclusterusuallyisassmall periodicallyshockeduidelementsisinvestigated5.consideringthedensitiesen- temperatureofmacroscopicdustparticles(the\dusttemperature").accordingto as10atoms.therefore,thechemicalreactionsinvolvedintheformationprocessof suchseedsareassumedtobecontrolledbythegastemperatureratherthanthere assumedradiationeld,macroscopicgrains(strictlyspeaking,graphitegrainsinthesmallparticle limitofmietheory)denitelyevaporateatthesmallradialdistancesunderinvestigation,because theirinternaltemperaturesaremuchtoohigh(fadeyev1988).incontrast,largemoleculesmight 5Theformationofmacroscopicdustgrainsisnotdiscussedinthiswork.Accordingtothe bestableprovidedthattheiropticalanduvabsorptionpropertiesarecomparablesmaller. onceadustcloudhasformed,theradiationowsaroundtheopticallythickregionandnewdust easilyincreasetheabsorptioncoecientofthegas/dustmixturebyafactorof105.therefore, tocausealocalreductionofthedusttemperature.thephasetransitionfromgastodustcan Absorptionbythedustitselfisapromisingcandidateinordertoblockotheradiationeldand Apparently,theformationofdustclosetothestarmustbeaccompaniedbysomekindofshielding. particlesmaycondenseandgrowintheshadowofthiscloud,whereasthegrainsattheinneredge ofthecloudtowardsthestarswillevaporate.aquasistablesituationmightbeconceivablewhere star.sphericaldustformationinadistinctradiallayercausesanincreaseofthedusttemperatures inthelayerswithintheshellviaback{warmingandhasalmostnoeectonthedusttemperature Incontrast,theformationofasphericaldustshellseemstobeabsolutelyimpossibleclosetothe thedustcloudsurvivesthestrongradiationeldviaself{shieldinginadynamicalsense. ratherthandustshellformationincaseswherethegasissucientlydenseandcoolfornucleation intheouterlayers,becausetheradiationuxisnotblocked,butjusttransmitted. Thus,aninstabilitycausedbydustformationpossiblyexistswhichfavorsdustcloudformation butastrongradiationeldhinderstheseedparticlestogrowfurther. starenvelopes.inordertoclarifythesequestions,atleast2dmodelcalculationsarerequired, Dicultquestionsareraisedbytheseconsiderations,whichmaybeimportantnotonlyforthe dustformationinrcbstarsbutforanyharshradiationeldenvironments,e.g.inwolf{rayet dustgrains.dustformationclosetothestarinanycasemustproceedviathisrststep. theformationofseedparticlesandleaveasidetheproblemofthethermalstabilityofmacroscopic ontherstnecessarystepconcerningthetransitionfromthegasphasetodustparticles,whichis challengingproblemwhichgoesfarbeyondthescopeofthiswork.therefore,iwillconcentrate whichmustincluderadiativetransferandtime{dependentdustformation/destruction avery

thisassumption,thesupersaturationratiosiscalculatedas 116 CHAPTER7.SHOCK{INDUCEDCONDENSATIONINRCBSTARS S=nCkTg isthevaporpressureofcarbonatomsoverthebulkmaterial(graphite)atgas wherencistheparticledensityofneutralcarbonatomsinthegasphaseandpsat psat(tg); (7.6) Figure7.6showsthisconditionandgivesanoverviewofallresultsconcerningthe temperature.anecessaryconditionforcarbonnucleationtotakeplaceiss>1. periodicallyshockeduidelementsatr=2r.theminimumgastemperature occurringinoneperiodiccycleisdepictedasafunctionofthemeanheliumparticle densityduringthecyclewhichisdenedas Figure7.6demonstratesthattheconditionsappropriateforeectivecarbonnucleationaretemporarilypresentintheperiodicallyshockeduidelements,concerning 0n<He>dt: (7.7) n<he>=1pzp adistinctdensity{intervalbracketedbyn<he>=107:3cm 3and109:3cm 3.Incontrast,dustformationisthermodynamicallyimpossibleatr=2R,ifthetemperature ofthegasisgivenbytre persecond,iscalculatedbyapplyingclassicalnucleationtheory(gailetal:1984). ThenucleationrateJ,whichisthenumberofseedparticlesformingpervolumeand librium(cf.chapter5). g,whichisthehigh{temperaturesolutionofradiativeequiestingdensity{interval,nucleationratesofj=n<he>=10 13:5:::10 15:5s 1occur, ThenucleationratesareplottedascontourlinesinFig.7.6.Consideringtheinter- ecientcarbonnucleationmaytakeplace. Thetotalgrowthtimeforaseedparticletoreachamacroscopicsize,say0:01m, elsfortheenvelopesoflong{periodvariables(fleischeretal:1992),indicatingthat whicharelargevaluescomparedtothoseexperiencedfromtime{dependentmod- thatdustgrowthispossible(cf.footnote5): ofcarbonlockedinco).thedusttemperatureisassumedtobesucientlylowso intoaccountallthermallyimpingingcarbonbearingspecies(exceptfortheamount canbeestimatedbyconsideringthethreedimensionalgrowthbyaccretion,taking V0=4=3a30isthemonomervolume(a0=1:2810 8cmforgraphite,Gailetal:1984), tot gr= V0(n<C> n<o>)vth 0:01m (7.8) sidered)andthestickingprobability.assuming=1,alowerlimitiscalculated fortheactualtotalgrowthtime. vth=qktg=(2mc)thethermalvelocity(whichisabitsmallerifmoleculesarecon- isoftheorderofafewweeks(feast1986).inanycase,thetotalgrowthtimemust regionshouldnotexceedthetimescaleoftheinitialdropofthelight{curve,which InordertocauseaRCBdeclineevent,thetotalgrowthtimeinthedustforming

7.2.RESULTS 117 meangasdensityduringthecyclefortwodierentshockvelocitiesasindicated. theminimumtemperaturesoccurringinoneperiodiccycleasfunctionofthe tooccurataradialdistanceofr=2r.thefulllinesandpointsdepict Figure7.6:Minimumgastemperaturesandthepossibilityofcarbonnucleation oftheguresketchesthecondensationregime,i.e.theregionoffavorable thermodynamicconditionsforcarbonnucleation.onlybelowthes=1{limit, thegasissupersaturatedwithrespecttographite.contourlinesofthelogarithm Thedashedlineistheradiativeequilibriumgastemperature.Thelowerpart onepulsationperiod. oftheclassicalnucleationratej[cm 3s 1]areplotted.Ontherightedge,the growthtimeforaseedparticletoreachthemacroscopicsizeof0:01mexceeds

notexceedthepulsationperiodofthestar.thisconditionisadditionallyshown 118 CHAPTER7.SHOCK{INDUCEDCONDENSATIONINRCBSTARS thegasoccurs. thisconditionisjustfullledwithinthedensity{interval,wherethesupercoolingof infig.7.6,constitutinganabsolutelowerlimitforthedensityinthedustforming region,whichcanberesponsibleforanrcbdeclineevent.asdepictedinfig.7.6, Thepredictionsofthemodelconcerningtheminimumradialdistancerequiredfor ecientcarbonnucleationareofspecialinterest.previousmodelingofdustformationinthecsesofrcbstarshassueredfromthenecessitytoconsiderratherlargtionstellusthatdustformationprobablyoccursmuchclosertothestar(cf.sect.1.3 Thedependenceoftheresultsofthismodelontheparameterr=Risdepictedin andappendixa). radialdistancesinordertoobtainsucientlylowtemperatures,whereasobserva- 7.2.5DependenceonRadialDistance Table7.1,wheretheminimumoftheTg;min{curve(cf.Fig.7.6)isstatedintherst rowandtheintervalofmeanheliumparticledensitieswithtg;min(n<he>)<1500k isstatedinthesecondrow. Table7.1:Resultsasfunctionofradialdistanceandshockvelocity. r=1:5r v1=20kms 1 2200K v1=50kms 1 r=2:0r 1950K (0:2:::6:5)108cm 3 1060K r=3:0r(3:7:::13)108cm 3 1300K { (0:2:::18)108cm 3 710K r=5:0r(2:3:::77)108cm 3(0:2:::650)108cm 3 950K (0:2:::69)108cm 3 500K 200K ismuchlessdistinctivethanexperiencedfromre.achangeoftheshockvelocity, thestar,theeasierlowtemperaturesappropriateforcarbonnucleationareachievable.however,intheexaminedcaseoftime{dependentnon{re,thisdependence Thegeneraltendencyoftheresultsisasexpected:thelargertheradialdistanceto forexample,caneasilycauseverydierentconditions.consideringthe20kms 1 shocks,gastemperatureslowerthan1500kareproducedforr>3r,whereasconcerningthe50kms 1shocks,evenlowergastemperaturesoccurforallconsidered radialdistances. interval,centeredaroundafew108cm 3inallconsideredcases.Theparticular temporalsupercoolingofthegasbehindshockwavesisonlypossiblewithinaspecial, Comparedtotheinuenceofr=R,thedensity{dependenceisveryselective.The narrowdensity{interval.thedeeperthetemperature{minimum,thewiderthis

7.3.DISCUSSION density{rangeisingoodagreementwiththeestimatespresentedbygoeres(1992) 119 forthedensityofthedustformingregionsintheenvelopesofrcbstars. Therefore,theinuenceofr=Rontheresultsislesspronouncedthantheinuence atradialdistancesassmallas1:5 3R.Strictlyspeaking,thisstatementrefersto notverydistinct.theessentialoutcomeofthismodelitthatshockwavesare principallycapabletoproducelowtemperaturesappropriateforcarbonnucleation ofv1andn<he>.thepredictedvaluesforthecondensationdistancearehence heatingandcompressionfollowedbyre{expansionissupposedtobeaninevitable, theinvestigatedcaseofperiodicshocks.however,sincethebasicprocessesofshock straightforwardconsequenceofcircumstellarshockwaves,iconclude: everasucientlystrongshockwaveencountersthosepartsofthecir- cumstellarenvelope,wherethegasdensityisjustappropriateforthe two{stepcoolingprocessdescribedinsect.7.2.2. Favorablethermodynamicconditionsforcarbonnucleationoccur,when- pulsatingrcbstarshasbeeninvestigated.thecomplexinterplaybetweenshock{ 7.3 ThethermodynamicbehaviorofperiodicallyshockeduidelementsintheCSEsof Discussion gasdensityandthegastemperature,dependingontheshockvelocity. foundtooccurinthissituation,comprising1 2ordersofmagnitudesforboththe Largetime{variationsofthethermodynamicconditionsinxeduidelementsare heating,radiativeheatingandcoolingandadiabaticcoolinghasbeenexamined. Thecalculationsprovideahypothesisforthephysicalcauseoftheonsetofdust formationclosetoapulsatingrcbstar,connectedwiththequestionofthetrigger conditionsforcarbonnucleationarefoundtobetemporarilypresentclosetothe star,despiteofthehigheectivetemperaturesofthercbstars.thefollowing ofthercb{typedeclineevents.asaconsequenceofthepresenceofshockwaves, twobasicconditionsarerequiredinordertoallowforeectivecondensationclose thegasisusuallynotinre.inthetime{dependentnon{resituation,favorable toarbitraryshockwaves,nomatterhowtheshockwascreated. shockvelocitymustbelargerthanabout20kms 1.Theresultscanbegeneralized tothestar.thegasdensitymustbebracketedbyabout108 109cm 3andthe overviewofpreviouslypublishedmodelscanbefoundinappendixa.ashort discussionofthepossiblelinkstoobservationscompletesthischapter. hand,andthemainpointsofcriticismontheotherhand.forcomparison,an InthefollowingIwillbrieysummarizetheadvantagesofthismodelontheone 7.3.1AdvantagesoftheModel 1)Theobviousattractionofthemodelisthatdustformationclosetothestar isexplainedfromphysics.themodelpredictsthattemperaturesaslowas

1201000Kcanbepresentatradialdistancesassmallas2R.Accordingtothe CHAPTER7.SHOCK{INDUCEDCONDENSATIONINRCBSTARS 2)Thecondensationdistancesarefoundtobeassmallas1:5 3R,inagreementwiththevaluesinferredfromobservations(e.g.Claytonetal:1992, factmeansfavorableconditionsforcarbonnucleation. calculations,theseconditionslastformorethanhalfoftheperiodwhichin cf.sect.1.3).incontrast,noneofthepublishedmodelscanexplainthis 3)Thenarrowdensity{intervalnecessaryforthetwo{stepcoolingprocesscausing fundamentalfeatureofthedustformationinrcbenvelopesinaquantitative thelowtemperaturesagreeswithpreviousestimatesofthedensityinthenucleationzoneofrcbstars(goeres1992).thisagreementisnotself{evident. way. icalpropertyofthegas,whichistheradiativeheating/coolingeciency.this eciencydecreaseswithdecreasinggasdensityduetoincreasingnon{lte Thedensity{dependenceofthemodeliscausedbyacompletelydierentphys- 4)Thedependencyofthemodelontheeectivetemperatureofthestarissmall. PreliminarytestcalculationswithTe=5000 9000Kyieldsimilarresults eects. phenomenonisreportedforavarietyofstarscomprisingeectivetemperatures occurat1:5rforashockvelocityof50kms 1.Thisinsensibilityofthe modelwithrespecttoteapparentlyagreeswithobservations,sincethercb asdepictedabove.evenforte=9000k,gastemperaturesbelow1000k pronouncedte{dependence. of4000 20000K.Incontrast,allotherproposedphysicalmodelsexhibita 7.3.2Criticism 1)ShockactivityinthephotosphereofRCBstarsisonlyconrmedforone absorptionlinesplitting(cf.appendixa).therefore,thepresenceofshock wavesinrcbenvelopesisgenerallydoubted. velocityvariations,butnoshockactivityinthephotosphereasinferredfrom exceptionalobject,whichisrysgr.otherrcbstarsshowconsiderableradial circumstellarenvelopearerequired.adirectobservationofcircumstellarshock wavesisverydicult,duetocontrasteectswithregardtothebrightstar.a tothephotosphereofthestar,whereasinformationsabouttheconditionsinthe Comments:Thebasicproblemoftheaboveargumentisthattheobservationsrefer chancetoobservecircumstellarpropertiesmaybepresentduringtheearlyphases ofthedeclineevents.atthepresentstateofobservations,nopreciseinformations aboutcircumstellarshockactivityhavebeendeduced,atleastonecannotruleout dependingonthephotosphericdensitygradient.justinthosecases,wherethe knowntobecapabletosteepenuptoconsiderablystrongshockwavesinthecse, theory,evensmallamplitude(subsonic)wavesinthephotosphereofthestarare thepossibilitythatshockwavesareinfactpresentinallrcbstarsenvelopes.from initialradialpulsationissmall,thedensitygradientturnsouttobelarge,which ampliesthewaves.

7.3.DISCUSSION 2)RCBstarsshowsimilardeclineeventswithrespecttodeclinefrequency,time 121 scalesanddeclineamplitude,regardlessoftheirspecialpulsationproperties (e.g.theradialvelocityamplitudes).therefore,acausalconnectionbetween pulsationanddustformationseemssuspect. cumstellarshockwaves,sothatsomecorrelationsareexpected.fromobservations, substantial.inanycase,thepulsationofthestarshouldberesponsibleforthecir- thecommentsontheupperpointmayberepeated,butinfactthiscriticismismore Comments:Thisisthemostseriousobjectiontotheproposedmodel.Ofcourse, evidenceispoorconcerningthiscorrelation(cf.appendixa). ofthestar(cf.appendixa)wouldcontradicttheaboveargument,butobservational theoftenclaimedcorrelationbetweenthebeginofadeclineandthepulsationphase 3)Themodelatrstsightseemstosuggestdustshellformationratherthan dustcloudformation,asfarasasphericallysymmetricpulsationofthestaris considered.dustshellformation,however,canberuledoutfromobservations alwaysonlyaveryfew probablyjustone dustcloudperpulsationperiod clouds(cf.footnote5)isnotveryconvincing,atleastcannotexplainwhy (cf.appendixa).feast(1997)arguesthattheproposedinstabilitywhich survivesthedust{destroyingradiationeld. mightberesponsiblefora\fragmentation"oftheformingdustshellintodust Comments:Thisiscertainlyaweakpointofthemodel.However,themodelin becomesopticallythickduetodustformation,thebasicassumptionofopticalthinnessbreaksdownandtheradiationeldmustbecalculatedbymeansofthesolution whichinanycasemustbethestartingpointofthedeclineevents.oncethegas factdoesnotmakeanypredictionsofwhathappensaftertheonsetofnucleation. Itonlyintendstoshowhowtheonsetofnucleationispossibleneartothestar, ofa(non{local)radiativetransfer.itisprincipallynotpossibletomodeltheformationofacloudwithouttakingintoaccounttheimportantphysicalinteractiontency.atrivialwayoutistheprescriptionofanon{sphericalsituationpriorto dustformation.onecould,forexample,considerasuperpositionofnon{spherical inamore{dimensionalway.allmodelspublishedsofarsuerfromthisinconsis- presentedthermodynamicmethodsareofcourseapplicabletosuchprescribedsituations.however,inmypersonalopinion,suchassumptionsdonotreallyexplain anything.whatisnecessaryisthemodelingofthephysicalprocessofcloudforma- inhomogeneities.inbothcases,theconditionsfordustformationaredierentin neighboringuidelements,whichmightleadtotheformationofdustclouds.the shockwavesduetonon{radialpulsations,oronecouldprescribetheexistenceof tothestarisseriousandisnotrestrictedtotheproposedmodel. placepriortodustformation,orwhetherthecauseofcloudformationisrelatedto theprocessofdustformationitself.theproblemofthesurvivalofthedustclose tionfromapreviouslyhomogeneoussituation,nomatterwhetherthisprocesstakes 7.3.3InterpretationsofObservationswithRegardtotheModel 1)IftheRCBdeclineeventsareinfactcausedbyshock{inducedcondensation, density.thiscouldtosomeextentexplaintheprincipalsimilarityofthelight{ similarinitialthermodynamicconditionsofthegaswouldbepresentinthe nucleationzoneatthebeginningofalldeclines,concerningforinstancethegas

122curvesconcerninglightamplitudesandtimescalesinvolved,irrespectiveofthe CHAPTER7.SHOCK{INDUCEDCONDENSATIONINRCBSTARS widerangeofstellarpropertiesofrcbstarssuchaseectivetemperature, elementabundancesandpulsationproperties.themodelsuggeststhatthe 2)Accordingtothemodel,aformingdustcloudinthelineofsightwouldalwaysbelocatedbehindashockwave,whosehotpost{shockregionmightbe responsibleforsome\chromospheric"lineemissionsasobservedduringthe fartherout,theshockencounterslessandlessdensepartsofthecircumstellar earlydeclines.ifthentheentirecomplexofshockwaveanddustcloudmoves envelope,probablycausingafadingofthelineemissionsasthedeclinepro- andapartfromthestar. declineeventsarecausedbyadistinctphysicalprocessofthegasindependent aboutv1=2.theshock{inducedemissionlinesareexpectedtobesharpand outwards,leavingthepost{shockgaswithanoutwarddirectedvelocityof 10kms 1agreeswiththeproposedscenario,becausetheshockpropagates gresses.additionally,theobservedblue{shiftoftheemissionlinesoftypically unpolarized,inagreementwithobservations,becausethegasemitsundisturbedinfrontofthedust.therefore,themodelseemstogenerallyagree staticchromosphere. emissionslines,ifinterpretedasshockactivityratherthanasactivityfroma withthespectralpropertiesandthetime{evolutionoftheobservednarrow 3)Theproposedmechanismconstitutesacausalconnectionbetweenshockwaves ofthestaratmaximumlight(e.g.lawsonetal:1992),supposedthatthesmall theobservedcorrelationbetweenthebeginofadeclineandthepulsationphase rangeofthegasdensitiesnecessaryforthismechanismissolelypresentata anddustformationincircumstellarenvelopes,whichmightnaturallyexplain Insummary,theproposedmodelprovidesasolutiontothecentralproblemhowdust takeaparticulartimetoreachthisdistance. particularradialdistancetothestar,andsupposedthattheshocksalways condensationmayoccurclosetothestar,butiscertainlynotcapabletoprovidea fromobservations:temperatures,densities,radialdistancesandtimescales.it yieldsabouttherightconditionsfordustformationasinferredfromtheoryand completeexplanationofthepuzzleofthercbdeclineeventssofar.morecomplex apparentlybridgesagapbetweenthetheoryofdustformationontheonehand,and modelcalculationsarerequiredinordertoachievethisaim.thepresentmodel toincludethedevelopedthermodynamicmethodsintomorecomplexcalculations asakindofstartingpoint. bythecontroversyaboutthecondensationdistances.therefore,itseemspromising theobservationsofrcbstarsontheotherhand(cf.sect.1.3),whichismanifested Otherresults,whichareworthtobementioned,areasfollows.Thelow{temperature solutionsofrefoundinchapter5areneverfoundtoberealizedortohaveany eectontheresultsintheperiodicallyshockedsituation.oncetheadiabaticcooling theyaredenitelystrongerthantheremainingheating/coolingratesaroundthe ratesaresucientlystronginordertodestabilizethehigh{temperaturesolution,

7.3.DISCUSSION low{temperaturesolutions.similarly,thedetailsofthechemistry,theradiative 123 heatingandcoolingratesandthespectralpeculiaritiesofthebackgroundradiation thedensity{intervalappropriateforthetwo{stepcoolingprocessshiftsabit.hence, elddonotcauseprincipalchangesinthecyclicthermodynamicprocesses.what incontrasttotheresultsofthethermalbifurcationsdiscussedinchapter5,the resultsofthischapterhaveamuchmoregeneralmeaning. happens,forexample,ifanotherimportantheating/coolingrateisincluded,isthat eectivetemperature,thepulsationpropertiesofthestarandtheelementalabundances(e.g.thehdeciency)mightprovideanexplanationforthefact,thatthlarenvelopesofpulsatingstars notonlyinrcbenvelopes.theinuenceofthe Inprinciple,theeectdiscussedinthisworkisexpectedtooccurinallcircumstel- stellarparametersmustbefurtherinvestigated.especiallythedependencesonthe includingradiativetransferandatime{dependenttreatmentofthedustcomplex elingofthecircumstellarenvelopesofrcbstars.higherdimensionalcalculations dynamicmodelcalculationsarerequiredinordertoallowforamorerealisticmod- RCBphenomenonisrestrictedtoaspecialclassofobjects.Time{dependenthydro- dustgrainsclosetothestar,closelyrelatedtotheself{shieldingindustclouds. wouldberequiredinordertomodeltheformationanddestructionofmacroscopic

124

Chapter8 investigated.radiativeheatingandcoolingrateshavebeencalculatedconsideringthetypical(p;t){rangeandtheradiationeldspresentinthecircumstellar Conclusions Thethermalstateofdilutedgasesbeingsubjecttostellarradiationeldshasbeen temperatureofgasesunderstaticaswellasdynamicconditions.asanimportant Thesestudiesintendtolaythefoundationsfortheoreticalmethodstodeterminethe envelopes(cses)ofcoolandwarmstars. Theresultsofthisworkshowthatanon{LTEtreatmentoftheatomsandmolecules ingredient,suchmethodsmustbepartofanyfundamentalmodelingofcses,especiallywithregardtothesimulationofthechemicalanddustformationprocesses, isessentialinordertocalculatetheeciencyoftheradiativeheatingandcooling whichareknowntobestronglytemperature{dependent. processesincses.thepossibilitytoincludethecalculatedheatingandcooling ratesintomorecomplexcalculations(e.g.time{dependenthydrodynamicmodels) isalsoregardedasessential.apropercoupling,however,canbeachievedonlyif thebasicassumptionsarecompatible.therefore,acompromisemethodhasbeen proposedwherethestateofthegasiscalculatedbymeansoftheassumptionofa macroscopicpropertiesofthegasdonotdependonhistoryandcanbecalculated depthseectsofspectrallinesinsobolevapproximation.ontheotherhand,all populationoftheexcitedelectronic,vibrationalandrotationalstatesandoptical steadystate.ononehand,thismethodaccountsfornon{lteionization,non{lte models. Thus,athermodynamicdescriptionhasbeendeveloped,wherethestateofthegasis asfunctionoflocal,instantaneousphysicalquantities,whichareavailableinsuch gradientddv twoexternalparameterswhicharetheradiationeldjandthelocalmeanvelocity determinedbytwoindependentstatevariables,e.g.andtg(asusualinlte),plus processesbymeansofdetailedbalanceconsiderations. fulltime{dependentnon{lteapproach.itincludeslteasalimitingcase,which occursatlargedensities.thelatterisachievedbystrictlyincludingallreverse dle.themethodgoesonestepbeyondlte,butdoesnotrepresenta First,thetopologyofthesolutionsofradiativeequilibrium(RE)hasbeenexamined, Threeapplicationsofthismethodhavebeenpresented: ormorestabletemperaturesolutions.twodierenttypesofsolutionshavebeen consideringthecsesofrcoronaeborealis(rcb)stars.theresultsshowthat theconditionofre,i.e.theequalityofradiativegainsandlosses,canhavetwo identied:high{temperature,predominantlyatomicstatesandlow{temperature, 125

predominantlymolecularstates.themolecule{richstatesarefoundtobesubstantiallycoolerthanablackbodyinre.thisresultisstraightforward,inferredfrom 126 CHAPTER8.CONCLUSIONS sequencesforthechemistryandthedustformationintheseenvelopes.concerning thelargesensitivityofthemoleculesintheinfraredspectralregion.itisexpected thatthisresultisvalidincoolstellarenvelopesaswell,possiblywithimportantcon- additionallycomeintoplaywhichmeansthatinprincipleaspatialcoexistenceof thecsesofwarmstarswithte>4500k,thehigh{temperature,atomicsolutions iedforthecaseofc{starsenvelopes.comparisontotheothertimescalesinvolved Second,thetimescalesofradiativerelaxationprocessestowardsREhavebeenstud- inpressurebalancewitheachother(\thermalbifurcations"). high{temperatureandlow{temperaturegasphasesisconceivable,bothinreand scaleismuchshorterthantheothers,thecharacterofthethermodynamicprocessis oftemperaturedeterminationwhicharebasedonre.iftheradiativecoolingtime approximatelyisothermalandthetemperaturecanbecalculatedbymeansofre.in intheprocesstobemodeledyieldsacriterionfortheapplicabilityofthemethods theoppositecasethegasbehavesmoreorlessadiabatically.thethermalrelaxation creaseoftheeciencyoftheradiativeheatingandcoolingprocesses.consequently, ofsuchrelaxation.withdeceasingdensity,increasingnon{lteeectsleadtoade- densityhasbeenidentiedtobethekeyquantitywhichdecidesuponthecharacter ofthegasbehindcircumstellarshockwaveshasbeendiscussedaccordingly.thegas agradualchangeofthenatureoftheshocksisexpectedtooccuraround108cm 3, changingfrompredominantlyisothermaltopredominantlyadiabatic.theseresults toshock{levitatedatmospheresofpulsatingstars.largetime{variationsofthe stronglysuggesttoincludetime{dependentmethodsfortemperaturedetermination thermodynamicconditionsarefoundtooccurinsuchuidelements,comprising Third,amodelforperiodicallyshockeduidelementshasbeendeveloped,applicable intothemodelsoftheenvelopesofpulsatingstars. aftertheheatingandcompressionbyashockwave,thegasrstradiatesawaythe WithregardtoRCBstars,thefollowingeecthascometolight.Incertaincases 1 2ordersofmagnitudesinthestatevariables,dependentontheshockvelocity. excessinternalenergydissipatedbyashockwaveandthenre{expandsadiabatically. 1:5 3R,despiteofthehigheectivetemperaturesofthesestars.Suchconditions temporarilyoccurforshockvelocities20 50kms 1atradialdistancesassmallas Thistwo{stepcoolingprocesscanproducetemperaturessubstantiallylowerthanin REwithinadistinctdensityinterval.Temperaturesaslowas1000Karefoundto arefavorableforcarbonnucleation.thus,thepresentworkstatesthehypothesis larenvelopeshavebeenundertaken,providingnewinsightsandnewideasonthe Inconclusion,basicstudiesofthethermodynamicbehaviorofgasesincircumstel- mighttriggerthespectacularrcb{typedeclineevents. thattheonsetofdustcondensationclosetothestariscausedbyshockwaves,which processesleadingtodustformation.

AppendixA ThisappendixintendstogiveabriefoverviewonthecurrentstatusofRCBresearch, CurrentStatusofRCBResearch providinganimportantbackgroundfortheinvestigationsinchapter5and7.since thisclassofstarsshowssomanyinterestingaspectsinvariouselds,onlythetopics andintherecentreviewsoflambert&rao(1994)andclayton(1996). ThereadercanndfurtherinformationsinthediplomathesisofS.Friedrich(1995) whichprovidecluesonthedustformationandthedeclineeventsaresummarized. A.1 Classication:TheclassofRCBstarstodaycomprises32knownobjectsinour Galaxy(Lambert&Rao1994).Thisnumbervariesasaconsequenceofrecentobservationsandtheclassicationofsomeobjectsisstillunderdiscussion(Clayton 1996).Acertainvericationrequiresatleastthesuccessfulobservationofonede- GeneralObservations V854Cen,wasnotdiscoveredbefore19861.Themaincriterionforclassicationis ofrcbstarsisundoubtfullymuchlarger,probablybetween200and1000inthe Galaxy(Lawsonetal:1990).Forexample,thethirdbrightestRCBstaronthesky, clineevent,whichisquiteadicultobservationaltask.therefore,thetruenumber alsotheoccurrenceofsmall{scalevisualbrightnessvariations.rcbstarsaresingle theoccurrenceofthercb{typedeclineevents(cf.a.2:lightcurves).additional starsoftypicalspectraltypef{gibwithabsolutebrightnessesmv= 4to 5 varyinghydrogendeciencyamongtheobjectsof(logh=he= 0:3to 7:2),and criteriaareacarbonoverabundance(c=he=0:01to0:1)andaclearbutstrongly alsoextremevaluesfromabout4000kforwxcraandsapsuptoabout20000k (Feast1979),suggestingluminositiesofaboutL104L.Thestellarmassescanbe fordycenandv348sgroccur.comparedtothislargespreadofeectivetemperatures,thedeclineeventsoftheindividualrcbstarsshowaremarkablesimilarity (involvingdustformation),whichisapparentlynotverysensitivetotheeective intheirdeclinelightcurves,e.g.thedeclineamplitudesandtimescalesinvolved. temperatureofthestar. Therefore,auniquephysicalmechanismseemstoberesponsibleforalltheevents determinedbypulsationalmodels,yielding0:8 0:9M(Wei1987).Theeective temperaturestypicallyarete=(70001000)k(lambert&rao1994).however, 1Thisisbecausethisobjectisindeclinemostofthetime. 127

128 Pulsations:Besidesthedeclineevents,allthoroughlyobservedRCBstarsshow AppendixA 0:1to0:4magforperiodsoftypically40days(Feast1990;Lawsonetal:1992; variationshaveadditionallybeendeducedfromthedoppler{shiftofphotospheric Lawson&Kilkenny1996).ForaconsiderablenumberofRCBstars,radialvelocity moreorlessperiodicalvisualbrightnessvariationswithamplitudesrangingfrom Thereareseveralcluesthatthepulsationshaveadirectfeedbackonthedynamicsandthechemistryoftheouteratmospheresofthesestars.ThemostextremceptionalcaseofRYSgr.Thevariationsofvelocityandbrightnessusuallyoccurin phase,suggestingthercbstarstobeirregularradialpulsators(lawsonetal:1990). absorptionlines,yieldingabout5to10kms 1andabout20kms 1fortheex- propagationofashockwavethroughtheatmosphereofthestar(lawson1986, Lawsonetal:1991).Furthermore,besidessomepermanentemissionlinesprobably pulsatorrysgrshowsphase{correlatedline{splitting,whichisinterpretedasthe ofchromosphericcharacter(e.g.cii1335a),thereoccurphase{correlatedemission featuresintheuv,whichmightbecausedbyshockheating.theequivalentwidths ofelectronicabsorptionbandsofc2(swan)andcn(violet),associatedwiththe MassLoss:ThequestionofwhetherornottheRCBstars besidestheoccasional caseofrcrb(claytonetal:1995). outeratmosphere,showaclearcorrelationwiththepulsationalphaseofthestarin thedeclines(cf.a.2:spectroscopy)areblue{shiftedbytypically10kms 1,which masslosseventsduetodustcloudformation undergoanunderlyingpermanent However,theseeectscanhardlybedistinguishedfromthedustandgasclouds, accordingtofeast(1990,1996)suggestsapermanentradiation{drivenmassloss. massloss,isundercontroversialdiscussion.thenarrowemissionlinesseenduring whichapparentlyarepresentoutofthelineofsightatanytimenearthestar RCBenvelopes,aclaricationofthisquestionwouldbedecisivelyimportant.In (cf.a.1:dustshells).forthetheoreticalunderstandingofthedustformationin amassivestellarwind,whichisnotdrivenbythedustitself,thedustformation mightbeasecondaryprocessandcouldoccurratherdistantfromthestar.in contrast,ifthereisnomassivewind,thedensitiesaretoolowanddustcannotform dustcloudformationeventsrangefrom10 6Myr 1(Feast1986)to10 7Myr 1 atlargedistances.estimatesforthemeanmasslossratesinferredfromthesumof 2 25m,irrespectivewhetherthestarisindeclineornot.Theexcesscanbetted (Claytonetal:1992). DustShells:TheinfraredphotometryofRCBstarsshowaclearexcessatabout byblackbody{curvesofcharacteristictemperatures600{900k(kilkenny&whittet 1984).Thisthermalemissionisobviouslycausedbythetotalamountofdustinthe Sge(Jurcsik1996),showingamoreorlesspermanentIRexcessafterthosedeclines,butnoexcess ofthelineofsight2;3.themeanradialdistancesofthese\dustshells"areestimated vicinityofthestar,whichhasbeenformedduringtheformerdeclineeventsorout before. 3Sincethesedustcloudshaveproventobeopticallythininlatedecline,itwouldactuallybe 2ThisinterpretationisstronglysupportedfromtheobservationoftheveryrstdeclinesofFG moreappropriatetottheexcessby(1=)b(t).

tobe10{90stellarradii(walker1985).recently,feastetal:(1997)publishedextensivelong{terminfraredphotometrydatafor12rcbstarsandconcludedthat A.2.OBSERVATIONSDURINGTHEDECLINEEVENTS 129 mass,increases.thisisanargumentinfavoroftheformationofhotdustcloseto themeandusttemperatureisincreasing,ifthel{ux,representingthetotaldust thereisevidenceforaspreadindusttemperaturesineachrcbshell,wherethe thestar.sincethelimitingvalueof1500kisconstantforallrcbstarsobserved, hottestcomponentsarealwayslimitedbyabout1500k4.feast(1997)arguesthat thecondensationtemperatureofcarbonpossiblyisthecontrollingfactorinallrcb envelopes.furthermore,anotherexcessinthefarir(60{100m)canbeobserved forseveralobjects(e.g.rcrbandsutau),whichpointstodistant,fossildustand StellarEvolution:ThepopularityoftheRCBstarsisalsocausedbytheirmysteriousorigin.SincetheRCBstarsaresorare,theymustbeeitheramanifestation Gillettetal:(1986),thelinearsizeoftheshellofRCrBisabout18arcmin(8pc)5. gasshells,probablyconnectedwiththeformerevolutionofthestar.accordingto infactatopicofrecentresearchonstellarevolution.themergingoftwowhite Twomajorevolutionaryscenarioshavebeenworkedoutduringthelast15years (Iben1983,Schonberner1986,Renzini1990),indicatingthattheRCBstarsare ofapeculiarsidepathofstellarevolution,oracommon,butrapidlyevolvingstage. dwarfs(doubledegenerate(dd)scenario)andthere{birthofanpost{agbstar Thesemodelsmakedierentpredictionsaboutthesurfaceelementalcomposition, agreementwithobservationsisstillratherpoor.lastbutnotleast,theveryfast thelifetimesofrcbstarsandtheirspatialdistributioninthegalaxy.however, asaconsequenceofalastthermalpulse(finalheliumshellflash(ff)scenario). evolutionoffgsge(kipper1996,jurcsik1993)acrossthehr{diagramduringthe lastcenturysuggeststhatthebirthofanewrcbstarhasactuallybeenobserved. LightCurves:Theindividuallightcurvesofthespectaculardeclineeventsare quitedierentinappearanceconcerningboththedierenteventsandthedierent A.2 ObservationsDuringtheDeclineEvents stars.nevertheless,theireye{catchingshapeissotypicalthattheyessentiallydene thenalrecoveryphasethereusuallyisaphaseoflow{levelchaoticlightvariation, fromdeepdeclinesusuallytakesmonthsoryears.betweentheinitialdecreaseand thevisualbrightnessoftypically3 6magwithinafewweeks,whereastherecovery thisclassofobjects.thelightcurvesofthercbstarsstartwithasuddendropin superimposingeachother,areoftenobserved,suggestingmultipledustformation events. lastingbetweenzeroandseveralyears(goeres&sedlmayr1992).multipleminima, colorvariations.theinitialdecreaseinlightalwaysappearsslightlyreddened.the ColorVariations:ThedeclineeventsofRCBstarsareaccompaniedbycomplex 4Ingoodagreementwiththecarboncondensationtemperature,cf.A.3:Goeres&Sedlmayr 5Ashugeastheangularsizeofthemoon.

130 lightthenmaybecomebluish(a\blue"declineaccordingtocottrelletal:1990)or AppendixA remainsreddened(a\red"decline).asthedeclineprogresses,astrongreddeningoc- (Alexanderetal:1972,Cottrelletal:1990).ThenallightincreaseapparentlyproceedsonauniquelineforallRCBstarswithslope+5intheV=(B V){diagramcurs,untilthelightnallyincreasesagainandthestarslowlyreachesitsusualbrightnessandcolor.ThesevariationsresultintypicalloopsintheV=(B V){diagram whichisanunusuallylargevaluecomparedtointerstellarreddening(cottrell1996), providingcluesonthenatureofthedust.thecauseofthebluinginearlydecline hasbeenproposedtobeanadditional,radialextended,hotterlightsourcethanthe staritself(pugach1991,claytonetal:1992),emittingat3900 5700Amainlyin cloudradiiduringtheinitialformationand/orvaryingdistancesfromthelineof formoflineemission(asplund1995).this\chromosphere"initiallyisnot,orisat sightmightproduceredorbluedeclines. declinesmaybecausedbydierentcloudgeometriesduringthedeclines.varying leastmuchless,eclipsedbythedustcloud.thedierencebetweenredandblue havesmalleramplitudescomparedtotheopticalregion.asarule,theamplitudes IRObservations:Thelightvariationsintheinfraredregionsduringthedeclines decreasewithincreasingwavelengthandvanishesataboutthel{band(=3:6m), wouldbeexpectedifthedustwaspresentinasphericalsymmetricfashion.this wherethelightisalreadydominatedbythermaldustemission(e.g.feastetal:1997). Noanti{correlationbetweentheopticalandIRbrightnesshasbeenfound,which isthemainobservationalargumentfordustcloudformationratherthandustshell thestar. formation(forrestetal:1972).furthermore,thedustmassproducedinonedecline ExtinctionoftheDustParticles:ThepossibilitytoobservetheRCBstars apparentlyissmallcomparedtothetotalmassofthedustpresentinthevicinityof twice,uncoveredandcoveredbydust,allowsforadirectdeterminationoftheextinctioncurveofthematerialresponsibleforthedeclineevents.theresultsclearly aswellasthespecialappearanceofthisfeatureisdiscussedinvariouspublications caseofrcbstars(e.g.hechtetal:1984).thegeneralshapeoftheextinctioncurve positionofthewell{knowngraphite\bump"at2200aisshiftedto2400 2500Ain indicatethecarbonaceouscharacterofthedustmaterial(/1=).however,the Maron1989,Wright1989,Drilling&Schonberner1989,Hecht1991,Jeery1995, concerningthenatureofthedustgrainsinrcbenvelopes(e.g.holmetal:1987, fullerenes).theonlyclearuniquetendencyinthesepapersseemstobetheunusual bon,\onion{like"structures,amorphouscarboncorescoveredbygraphitemantles, hydrogenatthesurfaceorunusuallatticeormicroscopicstructures(\glassy"car- Zubko1996).Manyinterpretationsarepossible.Unusualsizedistributions,no smallradiiofthedustparticles,typically50atomaximumvaluesofabout600a. Spectroscopy:Sofar,onlytwodeclineeventsofRCBstarshavebeencompletely monitoredasfunctionoftimebyopticalspectroscopy:the1967declineofrysgr (Alexanderetal:1972)andthe1988declineofRCrB(Cottrelletal:1990).However,fragmentaryspectraldataisavailableforseveraleventsofthethreebrightest

CurrentStatusofRCBResearch RCBstarsRCrB,RYSgrandV854Cen,coveringcertainphasesofthedeclines 131 Asplund1995).Thespectraindicateaspecialtimeevolution.Untilthebeginningof (e.g.lambertetal:1990,lawsonetal:1992,claytonetal:1992,rao&lambert1993, distinguishbetweenthreecomponents,namede1,e2andbl.mostoftheemission 1992)6.Astheintensityofthephotospheric(absorptionline)spectrumdecreases,a rich\chromospheric"emissionlinespectrumcomestolight.alexanderetal:(1972) adecline,nospectralchangeshavebeenreportedsofar(cottrelletal:1990,lawson (8eV)ofneutralorsinglyionizedmetalatoms,whichareblue-shiftedbytypically 10kms 1anddisappearaftersomeweeks.AsmallernumberofnarrowE2{emission linesoflowexcitationenergies(3ev),mainlymultiplettsofsciiandtiiiremain linebelongtotheclassofnarrow(50kms 1)E1{linesofhighexcitationenergy visiblefor50 150days.Asthedeclineprogresses,theopticalspectrummainly consistsofvebroad(100 200kms 1),unshiftedBL{emissionlines:CaIIH&K, \chromospheric",althoughtheydonotlooklikethechromosphericemissionsofany absorptionlinespectrumre{appearsandsoondominatesthelightfromtheremainingemissionlines.thephysicalnatureoftheemissionlinesisusuallydescribedby NaIDandalineat3888A,probablyHeI(Feast1975).Finallythephotospheric absorptioncomponentscanbeobserved,especiallyblue{shiftedfeatureswithtypical velocitiesof200kms 1towardstheobserver.Thesecomponentsaresupposedto TheBL{linesusuallyshowamulti{componentstructure.Dierentemissionand otherstars(clayton1996). radiationpressure.hence,thesevelocitiescanbeassociatedwiththevelocitiesof thedustclouds. originatefromthegasdraggedalongwiththedustcloudsbeingacceleratedby region,havebeenreported.incontrast,theemissionlinesremainmoreorlessunpolarized(whitneyetal:1992).thephysicaleectcausingthepolarizationismainly allowforimportantconclusions.first,thedustisdistributednon{sphericallyand thescatteringoflightatthesurfacesofdustgrains7.therefore,theseobservations Polarization:Thelightduringthedeclinesgenerallyisstronglypolarized(Serkowski&Kruszewski1969,Coyne&Shawl1973,Standfordetal:1988,Emov1990).In thecontinuum,degreesofpolarizationupto14%,especiallyinthebluespectral ratherclosetothephotosphere. dustseemstoformbelowthelineemissionregions,suggestingdustformationoccurs forthelineemissions,atleastmuchlessthanthephotosphere.consequently,the second,thedustcloudcausingthedeclinedoesnoteclipsetheregionsresponsible declinestooccuratparticularpulsationalphasesofthestar(e.g.lawsonetal:1992). acausalconnectionbetweenthepulsationsandthedeclineevents.foratleast FurtherObservations:Severalobservationalcluescanbefoundwhichpointto Furthermore,themultipledropsofthelightcurveinthebeginningofthedecline twoobjects(rysgrandv854cen)thereissomeevidencefortheonsetofthe canbemade. 6Tocatchastarjustbeforeadecline,however,needsaveryluckymoment,sincenopredictions 7Possiblyatthesurfacesofotherdistantdustcloudsoutofthelineofsight.

132 eventsseemtooccurattimeintervalscorrespondingtotheperiodofthestar(feast AppendixA Finally,long{timevariationsofthedecline{activityhavebeenreported,actingon timescalesofafewthousandyears(menzies1986,feast1990).accordingtothe meantimebetweenthedeclines,isincreasingwithincreasinghydrogenabundance. 1996).AccordingtoJurcsik(1996),thedecline{activity,i.e.theinverseofthe personalopinionoftheauthor,theobservationsreviewedinthislastparagraphare lessstrikingthanthoseoutlinedbefore,stillleavingenoughroomforinterpretation. proachestomodelthercbdeclineeventshavebeencarriedoutsofar.themodest Comparedtothenumberandthequalityofobservations,onlyafewtheoreticalap- A.3 Models theoreticalapproachesandsolutions.aconsistent,physicaldescriptionoftheproblemobviouslymustcontain(i)adetailedcalculationofthermodynamics,chemistry activityofthetheoreticiansispossiblycausedbythecomplexityoftheprocesses anddustformation,(ii)asolutionofradiativetransferand(iii)amodelingofthe andthesomewhattroublesomegeometryinvolved.theseobstaclespreventsimple hydrodynamicsforthedust{enrichedgas.duetothecloudygeometryallthese keyproblemsofthedeclines,e.g.onthetriggerforthesuddenonsetofdustformation.ontheotherhand,severalempiricalmodelsprescribetheexistence,the Presently,thereareononehandafewtheoreticalworkswhichfocusoncertain Noneofthepublishedmodels includingthiswork satisfytheserequirements. investigationshavetobeworkedoutinmorethanonespatialdimension. consequencesandargueinfavororagainstcertainscenarios. geometryandthemovementofthedustinfrontofthestar,calculatetheobservable tionoccasionallyoccursinamassive,sphericalstellaroutow,whichcausesthe Loreta(1934)andO'Keefe(1939):Loreta(1934)assumesthatdustforma- A.3.1HistoricalModels declines.o'keefe(1939)agreeswithloreta'shypothesis,butproposesthatthe dustformsinejectedblobsofgas,similartosolarprotuberances.thesolidmatter proposedtocondenseisbelievedtobe\principallycarbon".bothmodelsassume thatdustformationtakesplaceratherdistantfromthestar,wherethetemperature modynamicconsiderations,o'keefederivescondensationtemperaturesof1360k, islowenoughtoallowforthephasetransition.basedonsomefundamentalther- declines,theprogresssincethenhasapparentlybeenratherslow. statements,whichuptodateprovidethebasicideafortheexplanationofthercb densities107cm 3anddistancesofabout8stellarradii.Reviewingtheseearly Wdowiak(1975):Giantconvectioncellsareproposedtobepresentatthesurfaces A.3.2ModelCalculations ofrcbstars.scalingtheobservationsofthegranulationandthesuper{granulation

CurrentStatusofRCBResearch ofthesuntogiantstardimensions,wdowiakarguesforconsiderablylowertemperaturesovercertainrestrictedareasofthestar.followinghisideas,thisfavorsdust thispictureandarguedthateventhesemi{regularvisuallightvariationsmightbe formationovertheseareas,followedbydustcloudejection.feast(1996)tookover 133 onlyqualitatively,asnocalculationsofthesuper{granulationhavebeencarriedout sphere,especiallyforhotrcbstars,aswdowiakstatedhimself.hisargumentis fewthousanddegreeslessmaynotbesucientfordustformationinthephoto- causedbythiseectratherthanbystellarpulsation.problemsremainasevena andtheformationofdusthasnotbeencalculated. Fadeyev(1983{1988):Y.Fadeyevwastherstwhoappliedclassicalnucleation theory,basedonthebulkmaterialdataforgraphite,tothecircumstellarenvelopes isprescribedast=t(r),andaradiallyexpandinguidelementisfollowedstarting ofrcbstars8.inhislatestmostadvancedwork(fadeyev1988),thetemperature atthesonicpointwithagiveninitialvelocity.accordingtotheassumptionsofan opticallythinradiationeld,greygasopacities,radiativeequilibriuminthegasand gasduetoradiationpressureondustgrainsyieldsabout150days.inhisearlier starandits(prescribed)masslossratearefound.theaccelerationtimescaleofthe includingthegreenhouseeectforamorphouscarbon,dustformationispossible works,atemporalenhancementofthegasdensitycausedbypropagatingshock outsideofabout20r.strongcorrelationswiththeeectivetemperatureofthe wavesarealsoconsidered.themodelprincipallyhasdicultiestoexplain(i)the dustformationinrcbstarsofdierenteectivetemperatures,(ii)thepre{existence ofamassivestellarwindand(iii)theoccurrenceofhighvelocityfeaturessoonafter thebeginningofadecline. ingconditionsinrcbenvelopes.thechemistryisdominatedbyamixtureofpure carbonmoleculesinaninertheliumgas,similarlytorecentlaboratoryexperiments thoroughlyinvestigatedthecarbonchemistryandthenucleationundertheprevail- Goeres&Sedlmayr(1992):Goeres(1992)andGoeres&Sedlmayr(1992)have concerningtheformationof\buckyballs"(c60).however,themainchemicalpathwaytotheformationofsootparticlesinvolvessmallcarbonchains,monocyclicrings (PACs).Fullerenesareproposedtoformasminorby{productsofthispathway.Gas andlargerdehydrogenized,curved,butnotclosedpolyaromaticcarbonmolecules temperaturesroughlybelow1500kareinevitablynecessaryforcarbonnucleation. ThemaingrowthspeciesistheabundantC3radical.Moleculedriftisproposedto triggerthefurthergrowthtolargermolecules.thedeclinesarecausedbydensity enhancementsduetosuperimposingshockwaveswhichoriginatefromnon{radial pulsations.thedescendingandtheascendingbranchesofthelightcurveareexplainedbyhydrostaticdustgrowthandradialdilutionataconstantoutowvelocity, thesameprincipalproblemsasabove. Asplund&Gustafsson(1996):Gustafsson&Asplund(1996)haveworkedout detailedatmospherecalculationsforhydrogendecientstars(static,planeparalrespectively.thegastemperatureisprescribedasinthemodeloffadeyev,causing 8...andpublishedhisresults.

134 lel,lte),usingaccuratelycalculated,line{blanketedabsorptioncoecients,which AppendixA heliumionizationzone(atross10).intheselayers,theradiativeacceleration surfacesofthestarsarebelowtheso{denededdingtonlimit( =grad=ggrav=1). However,radiativeinstabilitiesarepresentinthedeeperphotosphericlayersatthe yieldgoodagreementwiththeobservedspectra.accordingtothesemodels,the exceedsthegravitationaldeceleration >1which,accordingtotheirmodels,is layersareunstableagainstcompressionandoutwardaccelerationofgasblobs.as balancedbypressureinversions.asplund&gustafsson(1996)recognizethatsuch theyputforwardthemselves,thereasonforadeclineeventisproposedtobethe accelerationofsuchagasblobinthedeepphotosphereofthestar,followedbya supersonicinjectionthroughtheatmosphere,radiativecoolinganddustformation. Thus,thecauseoftheRCBdeclinesmightbefoundintheradialatmosphericstructureofthestaritself.Themodelseemstobepromising,butsofartheinvestigations Themodeldoesnotexplainthereasonfordustformationclosetothestar,itonly ofblobinjectionhavenotbeenperformed.dustformationhasnotbeencalculated. arerestrictedtohydrostaticconsiderations.hydrodynamicalmodelsfortheprocess A.3.3EmpiricalModels providesthenecessarydensityconditions. Humphreys&Ney(1974):Asecondarycoolstarwithanopticallythickdust envelopecausesthedeclineevents.suchbinarymodelshaveprincipalproblemsto Wingetal:(1972)andGrinin(1988):Orbitingdustcloudsfromtimetotime noobservationalevidenceforbinaryrcbstarshavebeenreportedsofar. explaintheasymmetryandthetruerandomnessofthelightcurves.furthermore, Keplerorbits. shouldbedrivenawayfromthestarduetoradiationpressureratherthandoing obscurethestar theproblemsarethesameasabove.moreover,thedustclouds approachhasvariedabit,butthemainidearemainstheexpansionofadustcloudof modelforthedustcloudevolutionwhichcausesthedeclines.overtheyears,the Pugach(1984{1994):Pugachandco{workershavedevelopedacomprehensive initiallyinnitesimalsmall,sphericaldustcloudwithagaussiandensityprole forthedierentcolorshavebeenperformedforthefollowinggeometry.amassive, homologouslyexpands(v=r=t)ataxedplacewithacertainosetfromthe constantmassataxedlocationinfrontofthestar.radiativetransfercalculations onwavelength.pugachshowedwithhiswork,thattheshapeofthelightcurve, additionallyemittedradiation,whichisnotaectedbythedustcloudbutdepends lineofsightinfrontofthestar.themodelintroducesthreeparameters:the andthecolorvariations,canbereproducedbythisscenario.nohydrodynamical totaldustmass,theosetfromthelineofsightandtheintensityofscatteredor of5 5001020g(Pugach&Koval'chuk1994).Theshortcomingofthismodelisof movementofthecloudisneeded,nodustformationmustbeconsidered.itcanallbe explainedbypuregeometry.estimatesforthetotaldustcloudmassesyieldvalues

CurrentStatusofRCBResearch coursethatitdoesnotreallyexplainanything.theexistenceofthedustcloudis 135 Emov(1990):Spontaneouschangesoftheabsorptionpropertiesofapre{existing prescribedandthereasonforthehomologouscloudexpansionremainsmysterious. particles.themodelcantosomeextentdescribetheshapeofthelightcurveandthe ofsuchspecialdustisnotprovided. colorvariations,butareasonforthespontaneouschangesaswellastheexistence dustshellcausethedeclines,e.g.viaspontaneousalignmentofnon{sphericaldust FurtherModels:Manyfurthercommentsandestimatesarestatedintheliterature,e.g.Feast(1986,1996,1997),Alexanderetal:(1972),Forrestetal:(1971sultsinviewofsomead{hocassumedscenarios.Therefore,theydonotappear 1972).However,thesepublicationsmainlypresentobservationsanddiscussthere- asextra\models"inthisappendix.nevertheless,importantconclusionscanbe drawnfromtheseconsiderations.thestandardmodelinthesepublicationsclearly istheformationofdustcloudsneartothestar,followedbyradialexpansionand dilution.fromtheirobservationsofrcrb,forrestetal:(1972)concludedthatthe sizesofthedustcloudsandthedeclineactivitiesofrcbstarsareinagreementwith theformationofonedustcloudperpulsationalperiod. dustcloudcausingthe1972declineonlycoveredabout3%ofthesolidangle(correspondingtoasemiconeangleofabout20).accordingtofeast(1986),theangular

References Alexander,J.B.,Andrews,P.J.,Catchpole,R.M.,Feast,M.W., AspectroscopicandphotometricstudyofthepulsatingRCoronaeBorealis LloydEvans,T.,Menzies,J.W.,Wisse,P.N.J.,Wisse,M.(1972). Allain,T.(1996).Photodestructionandgrowthofinterstellarpolycyclicaromatichydrocarbons.Dissertation,TechnischeUniversitatBerlin,FRG. typevariablerysagittarii.mnras158,305{360. Asplund,M.(1995).SpectroscopyofRYSgrduringthe1993minimum. Allen,C.W.(1973).AstrophysicalQuantities.London:TheAthlonePress. Asplund,M.,Gustafsson,B.(1996).ArethedeclinesofRCoronaeBorealis A&A294,763{772. Asplund,M.,Gustafsson,B.,Kiselman,D.,Eriksson,K.(1997).Line- (Hrsg.),Hydrogen-DecientStars,pp.39{42.ASPConf.Ser.96. starscausedbysuper-eddingtonluminosities?inc.s.jeeryundu.heber Ayres,T.R.(1981).Thermalbifurcationinthesolarouteratmosphere. decientcarbonstars.a&a318(2),521{534. blanketedmodelatmospheresforrcornonaeborealisstarsandhydrogen- Beck,H.K.B.(1993).Ionization,ChemistryandDustFormationintheOut- ApJ244,1064{1071. Beck,H.K.B.,Gail,H.-P.,Henkel,R.,Sedlmayr,E.(1992).Chemistry owsofclassicalnovaeandredgiants.dissertation,technischeuniversitat, Berlin,FRG. Biermann,P.,Kippenhahn,R.,Tscharnuter,W.,Yorke,H.(1972). incircumstellarshells.i.chromosphericradiationeldsanddustformation inopticallythinshellsofm-giants.a&a265,626{642. Bowen,G.H.(1988).Dynamicalmodellingoflong-periodvariablestaratmospheres.ApJ329,299{317. PhaseTransitionintheInterstellarMedium.A&A19,113{122. Burke,J.R.,Hollenbach,D.J.(1983).TheGas{GrainInteractioninthe ChaseJr.,M.W.,Davies,C.A.,DowneyJr.,J.R.,Frurip,D.J.,Mc- 234. InterstellarMedium:ThermalAccommodationandTrapping.ApJ265,223{ InJ.Phys.Chem.Ref.Dat.,Vol.14Suppl.1.NationalBureauofStandards. Donald,R.A.,Syverud,A.N.(1985).JANAFThermochemicalTables. 136

Chin,G.,Weaver,H.A.(1984).VibrationalandrotationalexcitationofCO Cherchneff,I.,Barker,J.R.,Tielens,A.G.G.M.(1992).Polycyclic 269{287. AromaticHydrocarbonformationincarbon-richstellarenvelopes.ApJ401, Clayton,G.C.(1996).TheRCoronaeBorealisStars.PASP108,225{1996. Clayton,G.C.,Whitney,B.A.,Meade,M.R.,Babler,B.,Bjork- incomets:nonequilibriumcalculations.apj285,858{869. Clayton,G.C.,Whitney,B.A.,Stanford,S.A.,Drilling,J.S. man,k.s.,nordsieck,k.h.(1995).long-termspectroscopicandpo- larimetricmonitoringofrcoronaeborealisnearmaximumlight.pasp107, (1992).ObservationsofRCoronaeBorealisstarsindecline:Empiricalargumentsfordustformationnearthestellarsurface.ApJ397,652{663. 416{426. Cottrell,P.L.(1996).RCoronaeBorealisstars:currentstatusoftheobservationaldata.InC.S.JeeryundU.Heber(Hrsg.),Hydrogen-Decient CoronaeBorealisandXXCamelopardalis.ApJ261,595{611. Stars,pp.13{25.ASPConf.Ser.96. Cottrell,P.L.,Lambert,D.L.(1982).ThechemicalcompositionofR Coyne,G.V.,Shawl,S.J.(1973).PolarimetryofRCoronaeBorealisat Cottrell,P.L.,Lawson,W.A.,Buchhorn,M.(1990).The1988decline ofrcoronaeborealis.mnras244,149{167. Drilling,J.S.,Schoenberner,D.(1989).Onthenatureofnewlyformed visuallightminimum.apj186,961{966. Efimov,Y.S.(1990).RCrBinthebrightnessminimumof1988/1989.SvA34, 247{254. dustaroundthehydrogen-decientstarv348sagittarii.apj343,l45{l48. Elitzur,M.(1983).Onvibrationalexcitationofinterstellarmolecules.ApJ266, Fadeyev,Y.A.(1983).GraphitegrainformationintheatmospheresofRCoronaeBorealisstars.Ap&SS95,357{368. 609{613. Fadeyev,Y.A.(1986).TheoryofdustformationinRCoronaeBorealisstars. Fadeyev,Y.A.(1988).CarbondustformationinRCoronaeBorealisstars. Dordrecht,pp.441{451.D.ReidelPub.Comp. InK.Hungeretal.(Hrsg.),HydrogenDecientStarsandRelatedObjects, Feast,M.W.(1975).TheRCoronaeBorealistypevariables.InV.Sherwoodet al.(hrsg.),vaiablestarsandstellarevolution,pp.129.iausymp.67. MNRAS233,65{78. Feast,M.W.(1979)..InF.M.Batesonetal.(Hrsg.),ChangingTrendsin VariableStarReseach,pp.246.IAUColl.46.

Feast,M.W.(1990).TheperiodicitiesofRCoronaeBorealisstarsandtheir Feast,M.W.(1986).TheRCrBstarsandtheircircumstellarmaterial.In drecht,pp.151{161.d.reidelpub.comp. K.Hungeretal.(Hrsg.),HydrogenDecientStarsandRelatedObjects,Dor- Feast,M.W.(1996a).SomegeneralproblemsconcerningRCBstars.InC.S. shells.aspconf.ser.11,538{548. Feast,M.W.(1996b).Thepulsation,temperaturesandmetallicitiesofMira JeeryundU.Heber(Hrsg.),Hydrogen-DecientStars,pp.3{11.ASPConf. andsemiregularvariablesindierentstellarsystems.mnras278,11{21. Ser.96. Feast,M.W.(1997).TheRCoronaeBorealisstars{II.Furtherinferences Feast,M.W.,Cartner,B.S.,Roberts,G.,Catchpole,R.M.(1997). fromtheinfrareddata.mnras285(2),339{357. Fernie,J.D.(1982).RCoronaeBorealisnearmaximumlight.PASP94,172{ tions.mnras285(2),317{338. TheRCoronaeBorealisstars{I.Infraredphotometryandlong-termvaria- Fernie,J.D.,Sherwood,V.,DuPuy,D.L.(1972).Aphotometricstudy 176. Feuchtinger,M.U.,Dorfi,E.A.,Hofner,S.(1993).Radiationhydrodynamicsinatmospheresoflong{periodvariables.A&A273,513{523. ofselectedrcoronaeborealisvariables.apj172,383{390. Fleischer,A.J.,Gauger,A.,Sedlmayr,E.(1991).Generationofshocks Fleischer,A.J.,Gauger,A.,Sedlmayr,E.(1992).CircumstellarDust shellsaroundlong-periodvariables.i.dynamicalmodelsofc-starsincluding dustformation,growthandevaporation.a&a266,321{339. byradiationpressureonnewlyformedcircumstellardust.a&a242,l1{l4. Forrest,W.J.,Gillett,F.C.,Stein,W.A.(1971).VariabilityofradiationfromcircumstellargrainssurroundingRCoronaeBorealis.ApJ170, L29{L31. Fleischer,A.J.,Gauger,A.,Sedlmayr,E.(1995).CircumstellarDust mechanismcausedbydustformation.a&a297,543{555. shellsaroundlong-periodvariables.iii.instabilityduetoanexterior{ Forrest,W.J.,Gillett,F.C.,Stein,W.A.(1972).InfraredmeasurementsofRCoronaeBorealisthroughits1972marchejuneminimum.ApJ178icalmodels.ApJ297,455{475. L129{L132. Fox,M.W.,Wood,R.P.(1985).ShockwavesinMiravariables.II.Theoret- Frantsman,Y.L.,Eglitis,I.E.(1988).TheC/O{ratioinNstars: observationsandtheory.sva14,l109{l111.

Gail,H.-P.,Sedlmayr,E.(1986).Theprimarycondensationprocessfordust Friedrich,S.(1995).DiePhysikderRCrB-Sterne:Phanomenologieundihre KonsequenzenfurdieModellierung.Diplomarbeit,TechnischeUniversitat, Berlin,FRG. Gail,H.-P.,Sedlmayr,E.(1988).Dustformationinstellarwinds.IV.Heteromolecularcarbongrainformationandgrowth.A&A206,153{168. aroundlatem-typestars.a&a166,225{236. Gillet,D.;Lafon,J.-P.J.,David,P.(1989).Radiativeshocksinatomic Gillett,F.C.,Backman,D.E.,Beichman,C.,Neugebauer,G.(1986). andmolecularstellaratmospheres.iii.theshockwavevelocityproblemin ApJ310,842{852. IRASobservationsofRCoronaeBorealis-detectionandstudyofafossilshell. Mirastars.A&A220,185{196. Goeres,A.(1996).ChemistryandThermodynamicsoftheNucleationinRCrB Goeres,A.(1992).StaubbildungindenHullenvonKohlenstosternen:RCoronaeBorealis.Dissertation,Techn.Univ.Berlin,Berlin. Goeres,A.,Sedlmayr,E.(1992).TheenvelopesofRCoronaeBorealisstars. I.Aphysicalmodelofthedeclineeventsduetodustformation.A&A265, pp.69{81.aspconf.ser.96. starshells.inc.s.jeeryundu.heber(hrsg.),hydrogen-decientstars, Goldreich,P.,Scoville,N.(1976).OH-IRStars.I.Physicalpropertiesof 216{236. Grinin,V.P.(1988).Ontheblueemissionvisibleduringdeepminimaofyoung circumstellarenvelopes.apj205,144{154. Gustafsson,B.,Asplund,M.(1996).Modelatmospheresforcoolhydrogendecientcarbonstars.InC.S.JeeryundU.Heber(Hrsg.),Hydrogen{ irregularvariables.svalett.14,27{28. DecientStars,pp.27{38.ASPConf.Ser.96. Hecht,J.H.(1991).ThenatureofdustaroundRCoronaeBorealisstars: Hecht,J.H.,Holm,A.V.,Donn,B.,Wu,C.-C.(1984).Thedustaround isolatedamorphouscarbonorgraphitefractals?apj367,635{640. Hellwege,K.-H.(1982).Landolt-Bornstein:Numericaldataandfunctional Springer-Verlag. relationshipsinscienceandtechnology,bandii/14ofnewseries.berlin: RCoronaeBorealistypestars.ApJ280,228{234. Hollenbach,D.,McKee,F.(1989).Moleculeformationandinfraredemissioninfastinterstellarshocks.III.ResultsforJshocksinmolecularclouds. ApJ342,306{336. Hollenbach,D.,McKee,F.(1979).Moleculeformationandinfraredemission infastinterstellarshocks.i.physicalprocesses.apjs41,555{592.

Huber,K.P.,Herzberg,G.(1979).MolecularSpectraandMolecularStruc- Holm,A.V.,Wu,C.-C.,Hecht,J.,Donn,B.(1987).ThefadingofR ture,bandiv.constantsofdiatomicmolecules.vannostrandreinholdcom- pany. CoronaeBorealis.PASP99,497{508. Humphreys,R.M.,Ney,E.P.(1974).InfraredStarsinbinarysystems. Iben,I.J.,Kaler,J.B.,Truran,J.W.,Rezini,A.(1983).Ontheevolutionofthosenucleiofplanetarynebulaethatexperienceanalheliumshell ApJ190,339{347. Jeffery,C.S.(1995).Theultravioletpropertiesofcoolmaterialejectedby ash.apj264,605{612. Jrgensen,U.G.(Hrsg.)(1994).IAUColloquium146:MoleculesintheStellar Environment,Berlin.SpringerVerlag. hydrogen-decientstars.a&a299,135{143. Jurcsik,J.(1996).OntheInter-fadesPeriodsofRCrBTypeVariables.InC.S. Jura,M.(1986).TheRoleofdustinmasslossfromlate-typestars.Irishastr. JeeryundU.Heber(Hrsg.),Hydrogen-DecientStars,pp.96.ASPConf. J.17,322{330. Kilkenny,D.,Whittet,D.C.B.(1984).Infraredphotometryandbroadband Ser.96. Kneer,F.(1983).ApossibleexplanationoftheWilson-Bappurelationandthe chromosherictemperatureriseinlate-typestars.apj128,311{317. uxdistributionsofsouthernrcoronaeborealisstars.mnras208,25{33. Kruger,D.,Gauger,A.,Sedlmayr,E.(1994).Two{uidmodelsforstationarydust{drivenwinds.I.Momentumandenergybalance.A&A290,573{ 589. Lambert,D.L.,KameswaraRao,N.(1994).TheRCoronaeBorealisStars Lambert,D.L.,Rao,N.K.,Giridhar,S.(1990).HighresolutionspectroscopyofRCoronaeBorealisduringthe1988{1989minimum.JA&A11, {AFewMereFacts.JA&A15,47{67. Lawson,W.A.(1986).RYSgr:pulsationrelatedphenomenon.InK.Hungeret 475{490. Lawson,W.A.(1992).SpectroscopyoftheRCoronaeBorealisstarV854Cen 215.D.ReidelPub.Comp. al.(hrsg.),hydrogendecientstarsandrelatedobjects,dordrecht,pp.211{ Lawson,W.A.,Cottrell,P.L.,Clark,M.(1991).RadialvelocityvariationsoftheRCoronaeBorealisstarRYSgr.MNRAS251,687{692. throughadeclineonset.mnras258,33{36.

Lawson,W.A.,Cottrell,P.L.,Gilmore,A.C.,Kilmartin,P.M. Lawson,W.A.,Cottrell,P.L.,Kilmartin,P.M.,Gilmore,A.C. (1992).PredictingmasslosseventsinRCoronaeBorealis:declinesofV854 Cen.MNRAS256,339{348. Lawson,W.A.,Kilkenny,D.(1996).Theobservationalcharacterization (1990).Thephotometriccharacteristicsofcoolhydrogen-decientcarbon U.Heber(Hrsg.),Hydrogen-DecientStars,pp.349{360.ASPConf.Ser.96. ofhydrogen-decientcarbonstarsaspulsatingstars.inc.s.jeeryund stars.mnras247,91{117. Loreta,E.(1934).NotasullestellevariabiliRCoronidi.Astron.Nachr.254, Lepp,S.,Shull,M.(1983).TheKineticTheoryofH2Dissociation.ApJ270, 578{582. Luttermoser,D.G.,Johnson,H.R.(1992).Ionizationandexcitationin coolgiantstars,i.hydrogenandhelium.apj388,579{594. 151. Maron,N.(1989).PropertiesofthecircumstellargrainsinRCoronaeBorealis. Luttermoser,D.G.,Johnson,H.R.,Avrett,E.H.,Loeser,R.(1989). Ap&SS161,201{207. Chromosphericstructureofcoolcarbonstars.ApJ345,543{553. Mendoza,C.(1983).Compilationoftransitionprobabilities,electronexcitation Menzies,J.W.(1986).RYSgr:Canthetimeofthenextminimumbepredicted? PlanetryNebulae,Dordrecht,pp.154{172.D.ReidelPublishingCompany. ratecoecientsandphotoionizationcrosssections.ind.r.flower(hrsg.), Mihalas,D.(1978).StellarAtmospheres(2nded.).SanFrancisco:W.H.FreemanandCompany. Dordrecht,pp.207{210.D.ReidelPub.Comp. InK.Hungeretal.(Hrsg.),HydrogenDecientStarsandRelatedObjects, Mihalas,D.,WeibelMihalas,B.(1984).FoundationsofRadiationHydrodynamics.OxfordUniversityPress. Millikan,R.C.,White,D.R.(1964).Systematicsofvibrationalrelaxation. Muchmore,D.(1986).Non-uniquesolutionstothestellaratmosphereproblem. A&A155,172{174. J.Chem.Phys.39(4),3209{3213. Neufeld,D.A.,Hollenbach,D.J.(1994).Densemolecularshocksand Muchmore,D.,Ulmschneider,P.(1985).EectsofCOmoleculesonthe accretionontoprotostellardisks.apj428,170{185. outersolaratmosphere:atime{dependentapproach.apj142,393{400. Neufeld,D.A.,Kaufman,M.J.(1993).Radiativecoolingofwarmmolecular gas.apj418,263{272.

O'Keefe,J.A.(1939).RemarksonLoreta'sHypothesisconcerningRCoronae Nuth,A.N.,Donn,B.(1981).Vibrationaldisequilibriuminlowpressure Borealis.ApJ90,294{300. clouds.apj247,925{935. Pigott,E.(1797)..Philos.Trans.R.Soc.London1,133. Pugach,A.F.(1977).OntheconnectionbetweenpulsationsofRYSgrand Pugach,A.F.(1984).AmodeloftheRCoronaeBorealisphenomenon.SvA28, 1{3. thetotallightdeclines.inform.bull.variablestars1277(iaucommun.27), Pugach,A.F.(1990).InterpretationofphotometricobservationsofRCoronae Borealis.Lightcurves.SvA34,646{649. 288{292. Pugach,A.F.(1992).InterpretationofphotometricobservationsofRCoronae Pugach,A.F.(1991).InterpretationofphotometricobservationsofRCoronae Borealis.Anoncentraleclipsebyaninhomogeneouscloud.SvA36,612{618. Borealis.Colorfeatures.SvA35,61{65. Pugach,A.F.,Koval'chuk,G.U.(1994).InterpretationofRCoronaeBo- Pugach,A.F.,Skarzhevskii,V.O.(1993).Interpretationofphotometric ports38,219{224. realisphotometricobervations:thesyntheticlightcurve.astronomyre- Puls,J.,Hummer,D.J.(1988).TheSobolevapproximationforthelineforce observationsofrcoronaeborealis.approximationtables.astronomyreports37,169{175. Raghavachari,K.,Binkley,J.S.(1987).Structure,stabilityandfragmentationofsmallcarbonclusters.J.Chem.Phys.87(4),2191{2197tinuumopacity.A&A191,87{98. andlinesourcefunctioninaspherically{symmetricalstellarwindwithcon- Rao,N.K.,Lambert,D.L.(1993).HighresolutionspectroscopyoftheR Renzini,A.(1990).EvolutionaryscenariosforRCrBstars.ASPConf.Ser.11, CoronaeBorealisstar,V854Centauri,duringadeepminimum.AJ105,1915{ 549{556. 1926. Schmetekopf,A.L.,Fehsenfeld,F.C.,Ferguson,E.E.(1967).LaboratorymeasurementoftherateconstantforH +H!H2+e.ApJ148, Schmutzler,E.(1987).ZumthermischenZustanddunnerPlasmenunterdem EinubeliebigerPhotonenspektren.Dissertation,UniversitatBonn,Bonn, L155{L156. FRG.

Scholz,M.,Tsuji,T.(1984).Theeectsofsphericalextensionuponthephotosphericstructureandspectrumofredgiants:comparisonofMandCstars. Schonberner,D.(1986).EvolutionarystatusandoriginofextremelyhydrogenlatedObjects,Dordrecht,pp.471{480.D.ReidelPub.Comdecientstars.InK.Hungeretal.(Hrsg.),HydrogenDecientStarsandRe- A&A130,11{18. Serkowski,K.,Kruszewski,A.(1969).ChangesinpolarizationoftheRCrB Spizer,L.(1978).PhysicalProcessesintheInterstellarMedium.New{York:J. starrysgr.apj155,l15. Stanford,S.A.,CLayton,G.C.,Meade,M.R.,Nordsieck,K.H. (1988).RCoronaeBorealisdustejections:Apreferredplane?ApJ325,L9{ answhitney,b.a.,murison,m.a.,nook,m.a.,anderson,c.m. Wiley&Sons. Stilley,J.L.,Callaway,J.(1970).Free-freeabsorptioncoecientofthe negativehydrogenion.apj160,245{260. L12. Turner,J.,Kirby-Docken,K.,Dalgarno,A.(1977).TheQuadrupole Unsold,A.(1968).PhysikderSternatmospharen(2.Auageed.).Berlin,Heidelberg:Springer. 281{292. Vibration{RotationTransitionProbabilitiesofMolecularHydrogen.ApJS35, Walker,H.J.(1985).IRASphotometryofdustshellsaroundhydrogendecientstars.A&A152,58{62. Wdowiak,T.J.(1975).Coarsephotosphericconvectionandtheejectionofdust Whitney,B.A.,Balm,S.B.,Clayton,G.C.(1993).Dustformationin byrcoronaeborealis.apj198,l139{l140. Whitney,B.A.,Clayton,G.C.,Schulte-Ladbeck,R.E.,Meade, ASPConf.Ser.No.45. RCBstars.InD.Sasselov(Hrsg.),LuminousHighLatitudeStars,pp.115{122. Wing,R.F.,Baumert,J.H.,Strom,S.E.,Strom,K.M.(1972).Infrared tothegeometryofthedustandemission-lineregion.aj103,1652{1657. photometrieofrcrbduringitsrecentdecline.pasp84,646{647. M.R.(1992).SpectropolarimetryofV854Centauriatminimumlight:Clues Winters,J.M.,Fleischer,A.J.,Gauger,A.,Sedlmayr,E.(1994). Winters,J.M.(1994).Internalstructureandopticalappearanceofcircumstellardustshellsaroundcoolcarbongiants.Dissertation,TechnischeUniversitat, CircumstellarDustshellsaroundLong-periodVariables.II.Theoretical Berlin,FRG. Wishart,A.W.(1979).Thebound{freephoto-detachmentcross-sectionofH. MNRAS187,59p{60p. lightcurvesofc{stars.a&a290,623{633.

Wood,P.R.(1979).PulsationandmasslossinMiravariables.ApJ227,220{ Woitke,P.(1992).StaubbildunginderSupernova1987A.Diplomarbeit,TechnischeUniversitat,Berlin,FRG. Wright,E.L.(1989).FractaldustgrainsaroundRCoronaeBorealisstars. 231. Zubko,V.G.(1996).OntheinterpretationoftheextinctioncurvesofRCB stars.mnrasinpress,. ApJ346,L89{L91.

MeinenDank::: IchdankeihmfurdieFreiheit,dieermiraufmeinemwissenschaftlichenWegeinraumte, :::mochteichzunachstherrnprof.dr.sedlmayraussprechen.vonseinerunverwechelbarenartzudenkenhabeichundwerdeichhoentlichauchnochinzukunftviellernen. deresmirdurchunburokratischemanahmenermoglichte,andasinstitutfurastonomie undastrophysikzuruckzukehrenundhiermeinedissertationzubeenden. WeiterhindankeichHerrnPriv.Doz.Dr.KaufmannfurdieErstellungdesZweitgutachtens FurdieToleranz,dieHilfbereitschaftunddiefruchtbarenDiskussionenbeiderErstellung sowieherrnprof.dr.zimmermann,dersichbereiterklarthat,denprufungsvorsitzzu unddasvertrauen,dasermirtrotzzwischenzeitlicherdierenzenschenkte.erwares, michbeiholgerbeck,christianehelling,janmartinwintersundbeipetercottrell,die derarbeitmochteichallenmitgliederndesinstitutesdanken,insbesonderebedankeich ubernehmen. DanielKrugerhatganzwesentlichbeidernaturwissenschaftlichenKonzeptiondieserAr- unentgeltlichearbeitandenrechnerndesinstitutesdiesearbeiterstmoglichmachte. zurseitegestandenhaben.weiterhindankeichuwebolick,derdurchseinegrotenteils mirbeiderkorrekturderarbeitunddererledigungderprufungsformalitatentatkraftig chieren... unddeinekorrekturenbesondersbedanken vielleichtkannichmichdafurbaldrevanbeitmitgewirkt.diegrundideeerwuchsausseinerdiplomarbeit,aufdiewiederumandreasgaugereinuhatte.ichmochtemichbeidir,daniel,furdiezahlreichendiskussionen AchimGoeresdankeichvonHerzenfurdieBeratungnichtnurinfachlichenFragen.Neben meinearbeitodervielmehrmeinepersonzuunterstutzen.ichwerdenievergessen,wie hergab,ummeinendarauffolgendenkurzvortrageinzuleiten. einerartinnererseelenverwandschaftfuhleichbeiihmstetsdasaufrichtigebestreben, MeininnigsterDankgehortjedochDietrichEwert,vondessenHartnackigkeit,schierunendlicherEnergieundliebevollerZielstrebigkeitichnochviellernenwerde.IneinerZeit, ermuntert,meinearbeitfortzusetzen. eraufderinternationalentagunginbamberg(1995)seineneingeladenenvortragdafur alsichmitderastrophysikinnerlichfastschonabgeschlossenhatte,hatermichtaglich

Lebenslauf PeterWoitke PersonlicheDaten Alter:32Jahre Geburtsort:Berlin{Spandau Familienstand:ledig Anschrift:Schwendyweg6,13587Berlin Sept.1977{Dez.1983 Sept.1971{Juli1977 SchuleundStudium Apr.1984{Okt.1992 Astrid{Lindgren{GrundschuleinBerlin{Staaken Freiherr{vom{Stein{GymnasiuminBerlin{Spandau StudiumderPhysikanderTechnischenUniversitatBerlin ThemaderDiplomarbeit:"StaubbildunginderSupernova Jan.1984{Marz.1984 StudienbegleitendeTatigkeiten: IndustriepraktikumbeiderFirmaSiemens 1987A\,AbschlualsDiplom{Physiker Apr.1989{Marz.1992 Apr.1987{Feb.1989 KurseundPraktikainBionikundEvolutionsstrategie TutorimphysikalischenGrundpraktikum(Projektlabor) abnov.1992 furastronomieundastrophysikdertuberlinbei BeruicherWerdegang TatigkeitalswissenschaftlicherMitarbeiteramInstitut BetreuungvonSeminarvortragenundDiplomarbeiten ArbeitundMitarbeitanwissenschaftlichenPublikationen TeilnahmeaninternationalenTagungen(z.B.St.Louis,1994) Prof.Dr.E.SedlmayrmitdenAufgabenbereichen: Okt.1996{Jan.1997 derfirmabbjservisggmbh TatigkeitalsNetzwerk{AdministratorundInformatikerbei Sprachen: BesondereKenntnisse EDV: (Schulkenntnisse) Englisch(sicherinWortundSchrift),Franzosisch diverseerfahrungenmitpcs,workstationsundgrorechnernunterdos,windowsundunix.computersprachen: Programmen(Paradox5.0)sowiemitStandardsoftware{ C,C++,Fortran,GFA{Basic,PascalundAssembler.ErfahrungenmitPC{NetzwerkenunterNovel3.12,Datenbank{ Sport: ProduktenwieMS-WordundExcel. imvolleyball TatigkeitalsTrainervonHerren-undDamenmannschaften Berlin,den31.Juni1997