198 9ApJ. . .346. .847C -1 5 (Lundqvist andFransson1988). SN1980Kwasfolloweddown evidence forpulsaractivity (Bandiera,Pacini,andSalvati dence forpulsaractivityinsupernovae hasbeenlacking.Radio (Bandiera, Pacini,andSalvati1984;Pacini1987;Michel, emission fromsupernovae,which hasbeentakenaspossible Kennel, andFowler1987).However, clearobservationalevi- vational effectsofapulsarnebulahavebeenemphasized assumed thattheyarerapidlyrotatingpulsarsandtheobser- M yrfor24s. putation wasroughlyequivalenttoamassaccretionrateof 14 nosity manyordersofmagnitudehigherthanthis.Theircom- photon luminosityattheEddingtonlimitandaneutrinolumi- considered indetail,theinitialconditionsformasswere a 1984), canbemodeledindetail bycircumstellarinteraction static power-lawdensitydistribution.Theseauthorsfound a While thephysicalprocessesclosetoneutronstarwere accretion ofupto10“Mmatterontoaneutron star. star. Zel’dovich,Ivanova,andNadezhin(1972)computed the enough todrivetheobjectovermasslimitforaneutron found thatthetotalaccretedmasswasconsiderable,probably pressure gradientdrivesmattertowardtheneutronstar.He (1971) emphasizedthatneutrinolossesinthevicinityof neutron stardecreasethepressureinthatregionso matter fromthesupernovatowardneutronstar.Colgate form insupernovae,therewerecalculationsofthefallback 0 0 The AstrophysicalJournal,346:847-859,1989November15 © 1989.TheAmericanAstronomicalSociety.Allrightsreserved.PrintedinU.S.A. More recentworkonneutronstarsinsupernovaehas Soon aftertherecognitionthatneutronstarsarelikelyto 12 38-1 _4 42-1 56 Subject headings:stars:accretion—neutronsupernovae accreting envelopeislargerthanthemagneticpressureofa10Gneutronstarfield. rotation duringthefallbackprocess.Throughoutearlyaccretionphases,pressureatbaseof so thatitsenergycanberadiatedawayinaboutayear.Thistimescalewouldlengthenedbyeffectsof reduce orevenreversetheinflowoutsideshockedenvelope.Theenvelopemassissmallatanage>0.5yr, nosity attheEddingtonlimitforaheavyelementgas,3.5x10ergss,isexpected.Thisluminositycan begin todiffuseoutfromtheshockedregion.Whileanenvelopeispresentaroundneutronstar,alumi- age of7months,theshockradiusreachesradiationtrappingininflowsothatcan by ashockwaveisreleasedneutrinolosses.Whentheaccretionratedropstoabout3x10Matan solutions inwhichgravitationalenergygeneratedbysubsonicinfallthroughanextendedenvelopebounded trapped intheflowandneutrinolossesareimportantclosetoneutronstarsurface.Wefindsteadystate the largeaccretionratesoccurringduringearlyphases(3x10"to310Myr),radiationis more smaller.ThetimescaleforthisaccretioninSN1987Aisafewhours.Atanageofabout10days, radiative diffusionbecomesimportantandthepressureforcesarenolongerabletolimitaccretion.For this phasemaybeabout0.1M;formorenormalTypeIIsupernovae,themassafactorof100or ginally boundgasfallsbackoraspressureforceslimittheaccretion.TheeffectofNiheatingonatimescale of daysistoreducetheaccretionrate.ForparametersrelevantSN1987A,totalmassfallbackduring peaks whenthereverseshockwavereachesneutronstarandsubsequentlysteeplydeclineseitherasmar- trate onfallbackafterareverseshockwavemakesitswaytothecenterofsupernova.Therate clusion. Fallbackduringthisearlyphasedependsontheunknownexplosionmechanism,sohereweconcen- explosion islikelytounbindsurroundingcorematerial,althoughearlyneutrinolossescouldaffectthiscon- 0 0 0 © American Astronomical Society • Provided by the NASA Data System Accretion ofmatterontoaremnantneutronstaratthecentersupernovaisinvestigated.Theinitial I. INTRODUCTION NEUTRON STARACCRETIONINASUPERNOVA Department ofAstronomy,UniversityVirginia Received 1989January20;acceptedMay22 Roger A.Chevalier ABSTRACT 847 46 39- 37_1 Colgate’s arguments,Michel (1988) estimatedthefallbackrate mass infall.Whilepressureforces playedanimportantrolein magnetic effectsmayberesponsible forturningaroundthe for aneutronstarandhesuggested thatneutrinoheatingand back couldincreasethetotal massovertheupperlimit phenomenon. Colgate(1988) reemphasized thepointthatfall- , andtherehasbeen renewedinterestinthefallback study ofneutronstaraccretion,ifitdoestakeplacein this energy ofonlyabout8x10ergs. “injection” periods.A0.5speriodcorrespondstoarotational phase,itisplausiblethattheyarebornwiththeir (Narayan 1987;EmmeringandChevalier1989).Whileneutron many pulsarsare“injected”withperiodsof0.5sormore early times.Recentstudiesofpulsarstatisticshaveshown that stars couldbebornrapidlyrotatingandspindowninanon- cases isthattheneutronstarrelativelyslowlyrotatingfrom energy inputfromapulsar(Chevalier1987).TheHalumi- nosity inthiscaseisabout3x10ergss^ slower declineover4yr(Rupenetal1987),whichsuggests compatible withradioactiveenergyinput,SN1986Jshoweda complete wavelengthcoverage.WhileSN1980Kand steady luminositysourceinSN1987A,wherewehavemore yr (UomotoandKirshner1986)withoutshowinganyevidence of steadypulsarenergyinput.Stronglimitscannowbesetona to anHaluminosityof2x10ergssatagenearly 1987A showeddeclinesoveraperiodofalmost2yrthatare The proximityofSN1987Ashouldmakepossibleadetailed A possiblereasonfortheapparentlow-energyinputinmany 198 9ApJ. . .346. .847C w 6 5 where aand/?isdimensionlessconstant. because the samedimensionsasEandK.Thisispossibleifs =2 below itsequilibriumvalueso thatthereissomeinitialinfall. is thenadimensionlessconstant.Theinitialconditionsoutside For smallervaluesofathe neglectoftheinfallbecomes In anactualsituation,thecore pressuremaydropsomewhat pressure the explosionaretakentobehydrostaticequilibriumwith the GM, whereGisthegravitationalconstant,canbeexpressed in increasingly accurate. resulting shockwaveexpandsasapowerlawintime K givenbyp=Kr~,wherewisaconstant(Sedov1959).The explosion inapower-lawdensityprofilealreadydetermines effects ofacentralmassonanexplosion.Theproblem two parameters,theenergy£,andinitialdensityparameter case wheretheflowisself-similarinordertoestimate computations. Inasimplifiedtreatmenthere,weexamine explosion wouldinvolvenumericalworkincludingequationof Colgate (1971)hasalreadynotedthedifficultieswithnumerical over averylargerangeintemperature,density,andradius. state, radiativetransport,neutrinolosses,andhydrodynamics mathematical detailsmaywishtobeginwiththissection. Throughout thispaper,weuseastandardneutronstarmassof discussed in§IV.Effectssuchasrotation,neutronstarvelo- period in§III.Theaccretionofmatterontheneutronstaris imations aremadesothatsemianalyticmethodscanbeused. electron scatteringEddingtonlimit,whichis may formaroundtheneutronstar.Pinto,Woosley,and which alsosuggestsfuturework.Thereaderuninterestedin major conclusionsofthispaperaresummarizedin§VII, cities, andmagneticfieldsarebrieflytreatedin§V.InVI,we M =1.4M©andradiusr10cm. novae andmentionthepossibilityofacentralblackhole.The discuss therelevanceofresultstodifferentkindssuper- The explosionphaseistreatedin§IIandthelaterfallback the fallbackphenomenon,althoughanumberofapprox- mass (setto1.4M©),cisthespeedoflight,andkopacity. thetical pulsarwouldbesmotheredbyinfallforthefirst0.14yr on theassumptionofballisticmotion.Hefoundthatahypo- where Gisthegravitationalconstant,Mneutronstar and thatsubsequentlyadiskwithmassoforder10“M© Ensman (1988)notedtheplausibilityofalateluminosityat 848 ns A centralmass,M,canbeaddedtotheflowifparameter A fullcalculationoftheneutronstaraccretionduring Our purposehereistogiveamoredetaileddescriptionof ^Edd “' 4tüGMc © American Astronomical Society • Provided by the NASA Astrophysics Data System a K R= At s 3 = 3.5x10 II. THEEXPLOSIONPHASE a = r —~— GMK GMK 21 E 0.2 cmg 1/(5-s) 2 /(5-s) , t ergs s (2.1) (2.3) (2.2) CHEVALIER yi46)l3 2 2 1/3 are erroneous. tions ofCarrusetalarenotisentropic andwebelievethatthey at thecontactdiscontinuity where U=rj.They1.5solu- y =1.5thepostshockflowis expected tobeisentropic.Thisis perty appliestothecentral region oftheshockedflow.For true inoursolutionsandthepressure anddensityremainfinite entropy attheshockfront;wenotethatp/poct'~. For behavior canbeobtainedbyexaminingtheevolutionof the this point,p-►constantandp-+oo.Someinsightinto this radius atwhichthereisacontactdiscontinuitywithU=rj. At y =1.5and5/3forarangeofvalues^.Weagree with their resultsfory=5/3.Thesolutionsextendintoafinite solution tothedifferentialequationsisfound. time, butcanbedeterminedfromenergyconservationonce a y >1.5,theentropydropsto0atearlytimesandthat pro- For agivensituation,theconstantßisnotknownahead of (rj =1)are Taking C=KA,theboundaryconditionsatshockfront remains constantduringtheexplosionat where theprimeimpliesd/drj.Thecaseofinterestinthis shock expansionrate,theMachnumberofshock, section hasa=fands2,and~A(ßE/K).Forthestated where Cisaconstant,leadingtotheequations where visthevelocityandyadiabaticindex.Theself- similar variablesaretakentobe f Carrus etal(1951)havealreadysolvedthisproblemfor The fluidequationstobesolvedare [2(a —1)+(yl)as]Pa(Urj)P Ct as 2<,1>,s 1 =^’pQ(ri), v =aAf-'Uit]),pCa,4r‘~P(>7), a(U -r,)U'+(a-l)U+^F-^=0, 2 (u -f/)Q'+airsfi=o, (y -l)^+2’l)y^ -J- +v—y dp p dt drp 2 t;- (y +l)^_(2y^—y1) z T: +7rrH—j"=0> dv 1dpGM dt drpr dp dv2pv„ dt drr 2 e/# 2 (y -1MT+2 2 (y -hl)«^ 3 ya* dp , dt dr yo—p;— P(U-ri) _ Q^ Q' =0, Vol. 346 (2.12) (2.11) (2.10) (2.9) (2.8) (2.7) (2.6) (2.5) (2.4) 198 9ApJ. . .346. .847C _1 2 2 mass, pocrand-►constant atthecenter.Valuesofa,ßk, for thiscase.For00.017,p-►ooalongwith p.Thecriticalvalueofkcorre- the left-handsideleadstorelation Taking thevalueofconstantfromconditionsat the depend ona.Theconstantkisdefinedinthesamewayas where S=(9k—4)/(3/c2)andkbareconstantsthat imately hydrostaticatmosphere.Theasymptoticconditionsin 0.035, 0.017,and0.0060.Fora>0,theflownowextendsto for theshockedgasinasupernova.Lidov(1957)hasanalyzed zero energy.They=4/3solutionsareofspecialinterest some analyticprogresswiththey=4/3solution.Manipula- the centercanbeexpressedas r =0wherei;-►andpooinordertomaintainanapprox- because thisadiabaticindexislikelytoberoughlyappropriate because thematerialbeingaddedtoshockedregionhas flow. Thiscaseisnotofinteresthere. contact discontinuityarethusnotcomplete.Theymustbe energy oftheflowisconstant.Theself-similarsolutionsto incorporated intotheblasthasnegativeenergy,yettotal flows extendtor=0whenacentralmassisnotpresent.We equal tothetotalinitialmassintor=0.Thefactthatflows medium. Webelievethisisincorrect;thetotalshockedmass No. 2,1989 shock frontandgoingtothecentralasymptoticconditions on tion oftheself-similarequationsleadstorelation Lidov (1957).AsdiscussedbyLidov,itispossibletomake sponds to(47ua)hereandheillustratesresultsfora=0.060, this caseandweagreewithhisresults.Hisparameterßcorre- addition ofenergy.Fory>4/3,thepreshockmatterthatis believe thatthisisbecausethey>4/3flowinfactrequires extend onlytoafiniteradiusispeculiarbecausetheblastwave radius oftheshockedflowwithafiniteininitialr~ tant aspectoftheseflows.Carrusetal.(1951)identifytheinner found forJï>1or(x/ß<4/3y. internal energycontentofthegas.Self-similarsolutionscanbe immediate postshockgasvelocitybeinggreaterthanthegravi- tational escapevelocity.However,thiscriterionneglectsthe 9 1 t t t 1 2 This propertydoesnotapplytothey=4/3solutions The factthatthesolutionsendatafiniteradiusisanimpor- Carrus etal.(1951)findalowerlimitonJtbasedthe © American Astronomical Society • Provided by the NASA Astrophysics Data System 3 _7( 16(1 —2fc)7<4’ (2 -3kf(8a^d+60^ 2/32/3Si p =bKA-t-*(r/Aty,(2.14) p “(S,+l)r’ p GM v =kr/t,(2.13) NEUTRON STARACCRETIONINASUPERNOVA <2i6) (2.15) 6 _1 9 23-151 -1 23 -2 with theenvelopedrivesareverse shockwavetothecenter. (1988) alsosuggeststhatthehotradiationbubblecreated by early thattheconditionsdependonexplosionmechanism. is shorterthantheagefori<0.1s.Thistimesufficiently equation (2.19),weestimatethatthelosstimeoutsideof10 cm central gas.However,neutrinolossesareonlyimportant at if neutrinolossescausealossofpressuresupportforthe some ofwhichisboundtothe neutronstar.Thisisthephaseof The neutronstaristhensurrounded byhigherdensitygas, times, thedetailsdependon explosionmechanism. While theremaybeasmallamountofmassfallbackatlater neutrino heatingisabletoejectallthemassoutsidecore. Wilson findthatallthematteroutsideofaradius150km, or trino heatingoccursonatimescaleofabout0.5s.Bethe and loss ratesofSchinderetal.(1987)andthestructuregiven by very earlytimesintheblastwaveevolution.Usingneutrino g is3GM/rfory=4/3andthus3timesthemagnitudeof onto theneutronstarbecause thecollisiononmantlegas a totalcoremassof1.665M,isejectedfromthecore.Colgate For example,inthemodelofBetheandWilson(1985),neu- the gravitationalenergy.Thegasisunbound. the gravitationalfieldandcangiveanoutwardacceleration. the rarefactionwave,internalenergycandoworkagainst From equation(2.19)itcanbeseenthattheinternalenergy evolves towardfreeexpansionwithv=r/t.Thetimescalefor This valueisveryclosetothecriticalcorresponding this tooccurisroughlythedoublingtimeforradius.Near rarefaction wavemovesbackintotheshockedgasand where tisins.The2Mpointataradiusof3x10cm,and then K =10gcmandFergs,thecentralconditionsare the timethatshockfronttakestoreachthisradiusis3.5s. k =fand,fordefiniteness,wetakethecriticalcase.For Since theshockfrontthenencounterslowerdensitymaterial,a gem. WithM=1.4fortheneutronstarmass,wehave in thetransitiontoHelayer.TheprofilefityieldsK»10 imoto (1988),itappearsthatapocrdensityprofileisan 0 region oftheinitialoxygenlayer.Thedensityprofilesteepens adequate fittothe2Moutsidedensecore.Thisis SN 1987A.InthemodelofShigeyama,Nomoto,andHash- 0 0 0 4-/33 However, thereisanotherchance formattertofallback As emphasizedbyColgate(1971),thisconclusionischanged i; =^,p13*10igcm(2.19) We nowapplytheseconsiderationstothecentralregionof 0.1 0.02920.2920.4120.380 0.08 0.02290.2860.4290.194 0.0621 0.01750.2810.4440.0 0.02 0.005370.2690.482-0.604 0.04 0.01100.2750.464-0.286 0.01 0.002650.2650.491-0.791 0 0.0.2620.500-1.000 a =0.019Í„-—rr)(TTTi“)'•(2.18) a ßkS í1 Properties ofy=4/3BlastWaves 2351 VlO gcmVVlOergs/ TABLE l 849 198 9ApJ. . .346. .847C y 3 56 39 _1 56 56 3 3 where visthevelocityrange of theexpandinggasandkis are theenergyinputfrom reverseshockwaveandthe at theendofthissection.At early times,thedominanteffects opacity. Theeventualeffectsof radiativediffusionarediscussed energy input.Itisusefultouses=p/pasaparameter (Nomoto etal1988;Woosley,Pinto,andWeaver1988).The there wasasufficientlymassivehydrogenenvelopefor for SN1987Aasareference,althoughthediscussionshouldbe 850 here. evolution examinedbyColgate(1988)andinthenextsection for radiativetransfereffectsis with y=4/3(radiationpressuredominant)andthetimescale phases oftheevolution.Atearlytimesgasisopticallythick because theentropyremainsapproximatelyconstant over mixing ofthecorematerialmentionedabovespreads center, theenergyinputfromradioactivitycanleadtoexpan- cm “,wheretisareferencetimesincetheexplosionandp gas, asappearstoberequiredbyobservationsofSN1987A responsible formixingcoregasandperhapssomeenvelope reverse shockwavetobedrivenbackthroughthecoreof applicable toawiderangeofevents. and thusonthehydrodynamicevolutionofexplosion. the conditionsinmediumsurroundingneutronstar the lateraccretionrate. total accretedmass,whichisearly,butwouldreduce this additionalexpansionisimportant,itdoesnotaffectthe radioactive matterovergaswitharelativelylargetotalenergy This hasbeenreferredtoasthe“Nibubble.”However, the densityatthattime.IfallofNiisconcentratedto the coredensityisroughlyuniformwithi^1x10gs expansion intheabsenceofacentralpointmass.Modelsfor time scaleforthereverseshocktopassthroughcoreis the coregasatapostexplosionphase.Itmayalsobe star (Woosley1988;Shigeyama,Nomoto,andHashimoto ticular supernovaunderconsideration.Here,wetakemodels These conditionsdependonthedetailedpropertiesofpar- for SN1987A,thekineticenergyofejectaischangedby with amaximumvelocityof2000kms,asmaybethecase reduced. If0.07M©ofNiismixedthrough4material so thatthedynamicaleffectofradioactiveenergyinputis sion andsubstantiallyreducethedensityinNiregion. reach thestellarsurface,whichisabout2hr,or7x10s,for about thesameastimethatoutershockfronttakesto 4%. Inthefollowing,weassumethatpiremainsconstant.If SN 1987A(Woosley1988;Shigeyamaetal.1988)showthat the reverseshockedgasishigh,andreturnstouniform SN 1987A(Shigeyamaetal.1988).Theradiationpressurein 1988). Thishastheeffectofraisingdensityandentropy e a An importantaspectofSN1987Aisthatitlikely The internalenergyofthegasismodifiedbyradioactive Infall ontoacentralneutronstarinsupernovadependson © American Astronomical Society • Provided by the NASA Astrophysics Data System ~ 110day 2i vKptiy a 2 0.2 cmg 3c III. LATERFALL-BACK 1 K 2000 kms' 2 rr1 v 1/2 93 10 gcms* 1/2 CHEVALIER (3.1) 5 3915 56 56 8-1 -1/32 1315 56 2 —i/tr 56 -1 56 9/2 2 3281 417 Thus theaccretionrateshouldbegivenbyBondiexpres- equation (3.2),wefindthatradioactiveheatinglimitstheaccre- heating hasincreasedtheinternalenergyofcoregas.Using tion rateatanageof6x10s(7days). and thenafteratimeoforderT=8.8dayswhenradioactive sion foraperiodafterthereverseshockhastraversedcore, For Ç=1,pi10,ands6.3xwehavet1.7 characteristic lengthscalefortheaccretionphenomenonis mass inastaticmediumwastreatedbyBondi(1952).The present analysisnolongerapplies. increase insbutthisoccursonatimescaleT=114days,so increase insduetoNidecaysisthen inversely proportionaltotheuncertainamountofmixing).The be expressedas x =ÇGM/c^.TheBondianalysisdoesnotallowforexpan- central mass,andthecorrespondingaccretiontimescaleis and isthegassoundspeedatalargedistancefrom that radiativetransfereffectsbecomeimportantandthe The energyinputfromCodecayscancauseafurther in 4M©ofmatter,=5.6x10ergsgs(Kis g cms.Iftheradioactiveenergyinputrateperis estimate thatafterthereverseshockfront,pjpt—6.3x10 sion arerelativelysmall.ThecriticaltimeatwhichT=tcan and weestimatethatprovidedTT.ForNi,=8.8daysandthevalueofdependson that thepressureeffectscan be neglectedsothattheflowis drops belowM.Toanalyze this phaseoftheflow,weassume expression assumesthaty=4/3;for4/3,À0.707.Using is thedensityatalargedistancefrommass.Thesecond where Àisadimensionlessconstantthatdependsonyand p^ sion canbecomesignificant so thatthemassaccretionrate steeply withtime:Moct~ . tive heatingincreasessast , themassaccretionratedrops the reverseshock(7x10s)is2.2gs“or350 M© the parametersforSN1987A,accretionrateattime of r aec er e a x 10sandfor=—1.1,wehavet8.6s. 1 a a e r B s s B e erc The Bondi(1952)accretionrateisgivenby The problemofaccretiongaswithafinitepressureonto As notedabove,earlyinthe evolution theeffectsofexpan- 13X7_/ Sr =UpataV^!~1-1MgCmS~.(3.3) £213 y - =+\(Pat)-^T[Tt)]•(3.2) P Pi3 yr n Af =^^s\~2~JPooC00 2 B • „JGM\ Tc2 5.77 \3/(CGM)' 3 Í4\(pt)sf =a 2 (GM)(ptyi _ a ç3/2 3/2 Vol. 346 (3.4) (3.5) 198 9ApJ. . .346. .847C 2 1 2 define flowing butisboundtotheneutronstar;and(3)flow into threeregions:(1)theflowisinfalling;(2)out- dt =—GM/r.Thechangeindensityofthegascanbefound and thesolutionisnowgivenby (t >i),thepointwherei;=0isgivenby at r.Wethenhave which isthemaximumextentreachedbyagaselementstarting outflowing andunbound.Forregions12,itisusefulto algebraically lengthy.Attimet>,theflowcanbedivided by calculatingthedifferentialexpansionofgaselement. r toandithasadensityp.Theradialmotionofanelement No. 2,1989 In region3,itisusefultodefine applies toregion1andthetop2.Atlatetimes where inplacesthattherearetwosigns,thebottomsign of gasisthemsimplyfoundfromthesolutiondr/ assume thatthegasisuniformlyexpandingwithvelocityv= to thescaleofflow.Forinitialconditionsattimei,we the pressureeffectshavealengthscalethatissmallcompared ballistic. Therationaleforthisassumptionisthatwhent>r, 0 0 0 0 0 0 a The calculationoftheflowisstraightforward,althoughit © American Astronomical Society • Provided by the NASA Astrophysics Data System 2 \2GMtrj’ 0 2 1 2 Vr 2GMtJ’ (1 îi-V 0 _ /SGMt\ NEUTRON STARACCRETIONINASUPERNOVA 1/3 1/2 1/2 (3.14) (3.13) (3.12) (3.6) 2/3 3r -5/3 1/2 1/2 with Matthetransitiontime r.Settingt=z,wehave central mass. but wecanextrapolateback totcheckhowitcompares dependence thatcharacterizesmostlengthscalesrelated to a It isnoteworthythattheexpansionratefasterthan i by findingtheradius,r,atwhichPi=p^;itis flow isreducedandmassconserveddespitethehigherden- mass deceleratestheflowsothatvolumeoccupiedby the inconsistent withmassconservation.However,thecentral exact profiletowithin20%.Thefactthatthederiveddensity is sities. Thesizeoftheregionincreaseddensitycanbefound The densityatrisgivenby and atlargeradii,thedensitytendstowardp^=Potto/t)-F° dimensional analysis.Theinteriordensityisgivenby dence ofMwasalreadyfoundbyMichel(1988)essentially that ismarginallyboundtotheneutronstar.Theidepen- The relevantvalueofphereistheinitialdensityatpoint t =10iand100i,thedensityprofilePi+p^reproduces flow ati>with form, theactualvelocityanddensityprofilesatagiventimeare free expansion.Fori=10iand100i,thevelocityiswithin is v=+(2GM/r)andoutsideofr,thevelocitytendstoward free-fall velocityv=—(2GM/r).Atr,theoutward quite simple.Insideofr,thevelocityrapidlyapproaches late times. b a0 this pointdividesregions2and3.Wenotethatrjr=1.77at marginally boundtotheneutronstarexpandsas where pisagaingivenbyequation(3.11).Thepointthat 0 10% ofrjtby1.8r.Theinteriorregiontendstowardasteady t e f 0 >p intheabsenceofacentralmassmightappearto be o 0 o c e z z e 0 The valueofMyderivedhere onlystrictlyappliestoí>» While thedetailedsolutiontakesacomplicatedparametric 0 dr 2 dr 3 M =1-6Ç b 2,31/3-5/ =%n(2GM)pioir.(3.20) Pl0 2 3,9l2/98/9 r, =(^JTt*(2GM)tö•(3.22) 5/6 5/3/3 Pe 3/2[35/2(2GMí¿)]’ M =§7r(2GM)pioi~•(3.19) r+ f0 1/2 v =(2GM)(-+' 5 (Pa 4)4?’ (GM) S^GMt^Po 1/2 4 '1 l\ M =5.5Í f 5 (Paftsf (GM) (3.23) (3.21) (3.17) (3.15) (118) 851 198 9ApJ. . .346. .847C 4 4 3 n+1 34 present model,ß=1atatime where t=icr(Begelman 1978);whenQ>1,theaccretion rate ismodifiedbydiffusion effects.Inthecontextof flow whenriscomparabletotheBondiradius,=GM/c^. matter atsmallradiileadsto(Begelman1978) inflow timeforaflowwiththeopticaldepthdominated by inward convectedluminositydefinesatrappingradius,r,for The radiuswheretheoutwarddiffusionluminosityequals with timesothattheconvectedluminositysteadilydecreases. effects oftheaccretionprocess.However,bothManddrop here, 3Mc^isinitially>L,sothattherearenoexternal flux. Forasteadystateinflow,theinwardconvectedlumi- nosity, L.However,aradialinflowgivesconvectedinward from theoutersurface.Thetimescaleforthistooccuristensof More exactly,therelevantparameteris Radiative diffusioncanbeexpectedtoinfluenceaBondi-type the radiationinflow.Equatingdiffusiontimeto large radii(Begelman1978).Fortheparametersofinterest nosity atlargeradiusis3Mc^,wherethesoundspeed nosity inthefluidframeisessentiallyEddingtonlumi- nova boundaryandweconcentrateonithere. tant somewhatearlierthanthelossofenergyfromsuper- this processoccursonrelativelysmallscales,itbecomesimpor- arises inwardduetoadiabaticcompressionuponinfall.Since diffusion tendstoflattentheradiationpressureprofilewhich days (eq.[3.1]andFig.10ofWoosley1988).Onasmallscale, gates infromtheouterpartsofsupernovaasenergyislost large andsmallradii.Atradii,adiffusionwavepropa- it indicatesthattheaccretionprocesscaninvolveasignificant In ouranalysispressureforceslimittheaccretionrate,asina reimploded inatimeof5x10sandthattheremainingmass the effectsofradiativediffusioncanmodifyflowatboth Bondi flow,whileinColgate’sanalysisthepressureforcesdrive of thestarisreimplodedonatimescalethat5timeslonger. the estimatebyColgate(1988),whofindsthatabout1M©is amount ofmass.However,themassestimateissmallerthan at timer=1.7x10s.Thismassestimateisquitecrude,but lowered byallowingforthetransitiontoexpansionphase equation (3.5),and=7x10sforthetimeofreverse transition timeTisthatthereaslightsteepeninginthe rate acrossthetransitionpoint.Thechangeingoing the Bondirate,Mwhichassumesadiabaticflow.Eventually a largeraccretionrate. shock wave,wehaveM=0.15.Thisestimateisslightly have theaccretedmassM=Bt¡~/(n—1).Taking dependence ofMifsisconstant. so thatthereisapproximatecontinuityofthemassaccretion 852 n =3/2,B1.26x10fortheSN1987Aparametersin the totalaccretedmassisatearliesttime,i.We bB trB tr Edd Edd a a B acc0 acc e f 56 n In aradiation-dominatedflow,theradiativediffusionlumi- We foundthatheatingbyNidrivestheaccretiontoward If weexpressMasBt~,notethatsincen>1,mostof © American Astronomical Society • Provided by the NASA Astrophysics Data System t d 2 5 _ 4cs"J "376 (pJ^KGMji e = r t s/2 c 3 Tue» 4nc * Mk (3.25) (3.24) (3.26) CHEVALIER 56 56 93-21 56 2 n the timeduringwhichradioactiveheatingisimportant so brief.Thesubsequentaccretionrateisgivenby(eq.[3.19]) t =i.Forourparameters,thistimeisaboutequalbecause estimated byequatingMwithinequation(3.19) flattening ofthepressureprofilebyradiativediffusion; inner densityprofilesteepensinordertocompensateforthe envelope aroundtheneutronstarmovesoutsidetrapping The valueofiisuncertain,butitcanbeseenthatthe there isnottimeforthemtohaveasignificantinfluencehere. so thatMremainsfairlyconstantwithtime.Theabovevalue result canbeadecreaseinthesoundspeedatsonicpoint. The accretionratecanincreaseduringthisphasebecausethe important oncetheshockfrontboundinganaccretion opposing theinflow.Thiseffectisexpectedtobeparticularly not takeintoaccounttheeffectofradiationpressurein the flowinballisticphase.However,equation(3.28)does about 30days.Bythistime,radiativediffusionprobablykeeps become significantcomparedtoNiheatingatanageof supernova expansion.Thetimeatwhichthisoccurscanbe of Mcanbereducedbytheeffectsthermalpressure,but For thecasestudiedhere,sisincreasingduetoNiheating frame luminosityatlargeradius. (Pat* =1x10gscmandk0.2),wefindt« radius andisdiscussedinthenextsection. M isratherinsensitivetot.TheeffectsofCoheating luminosity becomescomparabletoLinestimatingtherest is brief.Attimet,M^AtiGMcI(kc^\sothattheconvected so wehave Using equation(3.2)forsandothertypicalparameters 8.4 days.Thus,thesecondphaseofadiabaticBondiaccretion falls backontothecentralneutronstar.Inthissection, we important (Zel’dovich,Ivanova,andNadezhin1972),webegin examine thepile-upofgasonneutronstarsurface. index y.TheflowisthendefinedbythetwoparametersB and by examiningasimplecaseofadiabaticaccretion.Theaccre- Although anumberofphysicalprocessesarelikelyto be the neutronstarradiusr,butitshouldapproximate the shocked flowisassumedtobeadiabaticwithan tion isassumedtobecoldwitharateofM=Bt~and the taken tobe as equations(2.4)-(2.6),andtheself-similarvariablesare now actual flowonscalesr>.Thefluidequationsarethesame 0d df 0 GM andisexpectedtobeself-similar.Thisapproachneglects e d fQ Edd d e ns ns 4 The accretionrateMeventuallybecomeslimitedbythe M =1.2xlO" After i,theaccretionrateisgivenbyQM(Begelman1978), In theprevioussection,wecalculatedrateatwhichmass d f dB n1/3213P Li{r,) l3/ ~(2GM)t’- ’ V =a(2GM)t~U(ti),P = ^P(r,), (2G)l/3 1 IV. NEUTRONSTARACCRETION r Br"- M =3.14 d 93 10 gscm“ Patl 3l (Pa t)KS' ae X GMct .-2 D^-n-5/3 1 _Jo_\ 10 days/ M yr 0 -1/3 . (3.28) Vol. 346 (3.27) 198 9ApJ. . .346. .847C 2 mass fluxtoapproach0atr=0. for particularvaluesofnandy.Theshockboundarycondi- where r\isaneigenvaluetobedeterminedfromthesolution (aßt}) terminequation(2.9)isreplacedbyl/(2arç).Theposi- equations (2.8H2.10)withs=3(n+l)/2,exceptthatthea/ where againa=f.Theself-similarequationsarethesameas No. 2,1989 cal solution.Thenumericalintegrationsaresimplifiedby following asymptoticforms: tions aregivenby tion oftheaccretionshockfrontisgivenby and abareconstantstobedeterminedfromthenumeri- where a guessforrjanddeterminingthevalueofthatallows A solutionisfoundbyintegratinginfromtheshockfrontwith gas initiallyovershootsitsequilibriumpositionandthengrad- layers tendtowardastaticstatewith cussed intheprevioussection.Itcanbeseenthatinner Table 2givestheconstantsformassaccretionratesdis- dividing outtheasymptoticpower-lawdependenceofQandP. ually relaxes. It isofinterestthatthevaluesbarepositive.Thepostshock However, fory=4/3,thesolutionofequation(4.9)isx 1 or x >1.Fory=5/3,thisissatisfiedforalls1,orn—5/3. s 1x s t 2 3 2 5/3 44/30.260955246 1.27x10“14.0 5/3 40.278744235 8.84xKT42.7 1.5 3.754/30.239653125 1.45xKT10.1 Analysis oftheself-similarequationsforsmallrjgives P = The solutiongiveninequations(4.6)-(4.10)requiresthat n syr¡ b 8(s +1)' s x ~ .J.b^s—x2)j~| Properties oftheAdiabaticAccretion Solutions <_1)/2sGA © American Astronomical Society • Provided by the NASA Astrophysics Data System p =aB(2GM)"r,-*.(4.10) 1/ PM =2+r —l{w') 21/ s ±{s+8[s2-2(sl)/y]} Q{r,s) = (^t)’ 212 R =YiilGMŸ'h,(4.2) s x M«- U =bri+ (x -IX«x+2) l TABLE 2 x-2Xs+ + NEUTRON STARACCRETIONINASUPERNOVA p (s+l)r ]• (4.8) (4.9) (4.5) (4.7) (4.6) (4.4) (4.3) 1 1/32/30/3 34 35 1,32,3 3-4355/1 x9 10 4 285/31 4 03 4 parameters thoughttobeapplicableSN1987A.Atimerange where ristheradiusinunits of10cmandTradisthe 0.26(2GM)i «8.7x10Í4cm,sothattheself-similar flatter thanr~.Atatimegreater10s,theenvelope hydrostatic structureisdescribed by(eq.[4.10]) ation pressuredominates(y =4/3)and51.0x10,the tion. Fory=4/3,theshockradiusisgivenbyr = rj isaconstant<1.Thisenvelopehasdensityprofilethat is to buildupanenvelopewitharadius~r¡(2GM)twhere given aboveattimetandremainsthatlevelfora of the reverseshockfront,Mprobablybuildsuptovalue by thereverseshockwaveisturnedaround.Afterarrival of g scmandt=10s,wehaveM1.0xi given bythefallbackvalueM(eq.[3.19]).Forpíq=110 (1977), allowafinitemassaccretionrateasr-►0. solution appliestoradiii>9x10cm.Assumingthatradi- structure isexpectedtoapproachthen=5/3self-similarsolu- order tbeforedropping.Constantmassaccretionisexpected sion mechanismandonthewaythatinwardflowcreated accretion historyisuncertainbecauseitdependsontheexplo- n =5/3beginningatatimeoforder10s.Thepreviousmass or M=2.2x10Í4gs“,whereiisthetimeinunitsof During mostofthistimeperiod,themassaccretionrateis about 10dayswhenradioactiveheatingbecomesimportant. of interestisfromthetimereverseshock,about10s,to The differencebetweenthesolutionsisthatof boundary condition,thesolutiontookformdescribedhere. value ofr¡approachedthecriticalrequiredbyourinner equations withr¡lessthanthecriticalvalueofrjledtosolu- form. Inthepresentwork,wefoundthatintegrationof from thatgivenhereandthebasicsolutionstakeadifferent the shockfront.However,smallradiusexpansionof equivalent. Theonlyminordifferenceinvolvesthedensityat with adensitydistributionpccr~.Ifœisidentified Sakashita andYokosawa,likethoseofKazhdanLutskii tions oftheformgivenbySakashitaandYokosawa,butas solution givenbySakashitaandYokosawa(1974)isdifferent those usedhereandthedimensionlesssimilarityequationsare (9k +2s)/(3k2).Itcanbeseenthatm>ssothepres- where kandaarepositiveconstants,n=2GMm Sakashita andYokosawa(1974)ofinitiallycoldinfallgas sure anddensityincreasewithtime. equations (4.6H4.10)toapply.Forn<|,thecharacterof 10 s.Theaccretionenvelopeisthusdeterminedbyaflowwith limits isclosertothesolutiongivenin§II.Wehave 3(n H-l)/2,thetwodimensionalparametersareequivalentto solution changesandthenatureofsmallradiusasymptotic s —1sothat>2(orn^)isrequiredforthesolutionin l sh s s9 0 0 f0 0 4 s s l1 12 4-3 125- p =0.0095r^gcm, 3.6xlO^dynescm, We nowconsiderconditionsintheaccretionenvelopefor The similaritysolutionsareclosetothecasestudiedby pi32i3 ~ p\ptj’ 125 T =6.1xlOVi’K, (4.14) rad p “(m+l)r’ p GM v =—/ci-, m ( r\- t (4.13) (4.12) (4.11) 853 198 9ApJ. . .346. .847C n -40 7610-3 -1 35-_1 -3_1924 50 -43 -2 5 3 -3 56 61 14 write €„=FpwithF3.1x 10incgsunitsandn=2.5. some inaccuracy,theygreatly simplifythesolution.Wecan Also, relativisticeffectsareneglected andheatingofthegasby good approximationthrough mostoftheenvelope,butthat nated, withy=4/3throughout.Itcanbeshownthatthis isa Another approximationisthattheenveloperadiationdomi- where itisshockedandtheenergylostbyneutrinos.In falling accretingmatterextendstotheneutronstarsurface, the pairpressurebecomessignificant closetotheneutronstar. approximate eisthisrangeby(T/10*K)ergscm s. and 10-ergscmsatK(Schinderetal1987). We ergs cmsatT=10K,IO K, (1987). Asimpleformforispossiblebecausethetem- (2.6) withthed/dtterms=0andaneutrinolossterm, cooling timetobethisshortrequiresaminimumaccretionrate absence ofcooling,thetimescaleforshockwavetomove extended adiabaticenvelope.Attheotherextreme,free- important neartheneutronstarsurfaceforanyplausibleadia- neutron starsurfaceinourmodel,theneutrinocoolingtimeis pressure ocr.Ifweextendanrprofilebacktothe inflowing regionclosetotheneutronstarisconvectivewith found thatthepressureprofileissteeperthanr.In pair emissivitydominatestheneutrinolosses(seealsoColgate the energylossperunitvolumeestimatedfromSchinderet al out oneneutronstarradiusisabout10"s.Fortheneutrino batic structureandthatthesewillleadtothecollapseof (1972) withapproximatelyconstantM,theradiationpressure expected toincreasetowardthesurfaceofneutronstar.In cooling isunimportantintheoutersteeppower-lawpartof peratures anddensitiesneartheneutronstararesuchthat the . Zytkow (1977)foraneutronstarwithmassiveenvelope,the is proportionaltorintheextendedenvelopethatbuildsup time-dependent modelofZel’dovich,Ivanova,andNadezhin the adiabaticmodelwithconstantMdiscussedabove,we adiabatic envelope.However,theradiationpressureis for y=5/3. trino coolingratesofSchinderetal(1987),wefindthatthis transfer canleadtoabreakdownofthisassumption(Colgate where pandaretheradiationthermalpressures,rhis radiation-dominated. Wenotethat Ni decaysisnegligible.While ourapproximationsintroduce state, inflowingenvelopeofshockedgasispresentaroundthe neutron star,butwithsignificantneutrinocoolingatthebase here areexpectedtoleadanextendedenvelopearoundthe around theneutronstar.InsteadymodelofThorneand outside oftheshockfront.Neutrinocoolingandradiative shows thattheenvelopestructureisnotsubstantiallychanged heavy elementgas,andkisBoltzmann’sconstant.Table2 the meanparticleweighttakentobe2amuforanionized radiation temperature.Infact,theenvelopeisonlymarginally of theenvelope.Wenextinvestigatemodelsinwhichasteady of about10Myr“.Themassaccretionratesinterest 854 1971). Inthisregime,eisindependentofdensityand10 1971 ;Zel’dovich,Ivanova,andNadezhin1972).Usingtheneu- n —(y —1)€„ontheright-handsideofequation(2.6),where€ is <0.1 snearthesurface.Itappearsthatneutrinolossesare radth 0 n n The steadystateequationsarethesameas(2.4)- Up tothispointwehaveconsideredflowsthatareadiabatic © American Astronomical Society • Provided by the NASA Astrophysics Data System Prad^ ñip Pth ~pkT 1.5r 0.25 11 > (4.15) CHEVALIER -4 where thesubscriptnsdenotesvalueatneutron star. envelope closesttotheneutronstar.Approximatingthisheight which theneutrinocoolingoccursin“scaleheight”of the Using equations(4.17)and(4.18) forthepressureleadsto tion gives by r/4foranrpressureprofileenvelope,energyconserva- 6%. Thestrongdependenceof€„onpleadstoasituation in about 6%,pby8%,andoverestimatesthemagnitudeof v by Over muchofthevolume,equation(4.17)underestimatesp by conditions, itisstraightforwardtoshowthat where thesubscriptshreferstovalueatshockfront.We is adiabaticovermostofthevolume.Inaddition,flow important closetotheneutronstarsothatpostshockflow from analyticalconsiderations.Theneutrinolossesareonly equal totheneutronstarradius.Table3givesvaluesofr losses. Thevalueofrisvarieduntilthestagnationradius note that vdv/dr terminequation(2.5)canbeneglected.Underthese highly subsonicexceptclosetotheshockwave,sothat obtained forvariousvaluesofM. ceeds inwarduntiltheinflowstagnatesbecauseofneutrino conditions areappliedatthatpoint.Theintegrationthenpro- where For y=4/3,wecanapproximatethestructureby the shockradius,r,ischosenandstrongy=4/3 flow ischaracterizedbythemassaccretionrateM.Avaluefor pressure andtobefallingatthefree-fallvelocitysothat The accretinggasatlargeradiiisassumedtohaveanegligible neutron starsurface,whichisassumedheretoactlikeawall. of thesteadystateequationsfromshockfrontto sh sh sh -1/(y-1)y/(y p ocr,~ The basicfeaturesoftheenvelopestructurecanbederived A solutionfortheenvelopestructureinvolvesanintegration V P2V,nr Psh —7po?lPin'. ° 47rrt;“(shJ rsh4/15 1/5 8hin V3/(7tF)(GAf) Shock RadiiforSteadyStateEnvelopes 0.01 8 1 ... 100 « 4.0x10—cm .(4.20) r 1/23/55 Y77r2\ r»M~ s p,417) /2 ’•-*(£) *•'”'•"(4)' M /2GMV J -1 (M© yr) M 04 TABLE 3 \M yr7 / MV- e (3_2y)/(y_1) and t;ocr.(4.16) GMM 9 8 7 2.2 x10 3.6 x10 5.6 x10 — in (cm) Vol. 346 (4.19) (4.18) 198 9ApJ. . .346. .847C 9 -1 No. 2,1989 studied thestabilityofaplanar shockfrontwithradiative Imamura (1982)andImamura, Wolff,andDurisen(1984)have it. Animportantquestioniswhether thisoscillatoryprocessis envelope hascollapsed,continued massaccretioncanrebuild stable sothatitrelaxestothe steadystateflow.Chevalierand the coresofmassivestarsnear theendoftheirlives.Once losses. Thissameprocessleadstotheacceleratedevolution of sure atthebaseofenvelope,whichincreasesneutrino tational contractionoftheenvelope,whichincreasespres- a runawayprocessinthattheneutrinolossesleadtogravi- independent ofradius,wheremisagaintakentobe2amu.The hin (1972;seealsoBisnovatyi-KoganandLamzin1984),this is the envelope.AsdiscussedbyZeFdovich,Ivanova,andNadez- tually, neutrinolossesareexpectedtoinitiatethecollapse of neutron starsteadilyincreasethroughthecalculation.Even- neutrino lossrateandthetemperaturedensitynear the neutrino losses.Finally,theinfalltimescale,r/y,isindependent surface are the Mrangeofinterest.Theconditionsnearneutronstar assumption ofradiationpressuredominanceisverygoodover thermal pressuretoradiationis that wereinitiallymadeinderivingthemodel.Theratioof that wouldbereleasedinasteadystateflow.However, the expected steadystatevalue,andtheneutrinolossesremain described in§III. This timescaleislessthantheforMtochangeover of radiusovermostthevolume(seeeq.[4.17]),anditis and areappropriateforthepairemissivitydominanceof sentation oftheshockradius. Comparison withTable3showsthatthisisanexcellentrepre- small comparedtothegravitationalenergy extends outto3x10cm,considerablylargerthanthe of steadystatestructuresasthemassaccretionratechanges range ofinterest.Thustheenvelopecanevolvethroughaseries M© yroveratimeof24s.Duringthistime,theshockradius remains open.Thetime-dependentcalculationsofZel’dovich, approaches thesteadystateinatime-dependentsituation influence ofneutrinolosses.Thequestionhowtheflow plausible structurefortheneutronstarenvelopesunder Ivanova, andNadezhin(1972)followamassfluxofabout14 We arenowinapositiontochecksomeoftheassumptions We believethatthesteadystatesolutionsgivemost 46 Ëgrav =1.2X10| lo 73 262 T &2.2x10(—;n,(4.22) p =1.2x10———gem“, p =5.6x10—rdynescm © American Astronomical Society • Provided by the NASA Astrophysics Data System Prad~ mp~ P thhl_ x 04 = 2.4 01 04 VM yrV ( M\- 0 \M yr7 \M yrV ( M\ / M\- q q 1 0.021 M© yr M M© yr' 1 NEUTRON STARACCRETIONINASUPERNOVA M© yr M M s . 1 ergs s(4.24) (4.23) (4.21) for aparticularvalueofM can befoundbyintegratingthe trino lossesisreduced.Thetotal massintheenvelope,M, reduce thepressureinenvelope sothattheeffectofneu- does dropsharply,ararefaction waveisexpectedtomoveinto the envelopefromvicinity oftheshockfront.Thismay neutron starisneededtodetermine whetherthisoccurs.IfM study ofthethermalstructuregassurrounding the have ahigheropacity,anoutflowcanbesetup.Adetailed the formerfree-fallregion.Ifouterlayersarecooler and that matterisnolongeracceleratedtowardtheneutronstar in sure approximatelycancelsoutthegravitationalattraction so the opacityisconstantthroughoutregion,photonpres- layers cankeeptheluminositynearL. months forSN1987A.Atlatertimes,whenr>,therate of envelope. Fromequation(3.28),thisoccursatanageofabout 7 important. Theratioofinflowtimescale,i,todiffusion in determiningwhentheeffectsofradiativediffusionbecome a dramaticeffectontheevolutionofaccretionprocess. If than L.However,thereleaseofenergyfromdeeper gravitational energyreleaseneartheshockfrontbecomes less scale, i,isgivenby the trappingradius,r(eq.[3.24]).Thisradiusalsoplaysarole radii andisclosetoLonlywhenrapproximatelyequal where equations(4.17)and(4.18)havebeenusedforthepost- ponent oftherestframeluminositygivenby frame mustbeapproximatelyattheEddingtonlimitforradi- For M<,theradiationisabletoescapefromshocked shock properties.Theconvectedluminosityissmallestatlarge ation pressuresupport,L.Thereisalsoaconvectedcom- Under theseconditions,thephotonluminosityinfluid hydrostatic equilibriumwithadominantradiationpressure. discussed above,theneutronstarenvelopeisinapproximate equation (3.24)leadstoacriticalvalueforM transfer probablyonlyplaysaroleinthelaterevolution.As further study. approaches Lonlyforr«.Settingequaltoin structure, thephotonopticaldepthsaresolargethatradiative gravitational field.Stabilityundertheseconditionsrequires case hastheadditionalfactorsofsphericalgeometryanda shock fronttorwhile<.However,theluminosity shows thattherecanbesomediffusionofradiationfromthe diffusion intotheinflowinggasoutsideshockfrontbecause rapidly withpressure,asisthecasehere.However,present the shockfront,ittravelseasilytolargerradii.Blondin(1986) once theradiationisabletodiffusefirstscaleheightoutside the largestradius,i.e.,atr.Thisisalsorelevantradiusfor structure isstableforcasesinwhichthelosstermincreases In theshockedenvelope,diffusioneffectsoccurfirstnear cooling tothistypeofoverstability.Theresultisthattheshock env Edd shtr in Edd diff tr Edd c Edd Eddshtr trsh sh The escapeoftheradiationatEddingtonlimitcanhave While neutrinolossesplayacrucialroleintheenvelope A>28xl01 - =.~Vcn,*g-)"eF-(4“) 2 ¿conv =-I6nrpv«7M,(4.25) 5/7 í KV idiff ‘ tin _4nrc 855 198 9ApJ. . .346. .847C 2 1 418 IMrll may havesignificantangularmomentum. mantle materialintothecentral region.Thisoutermaterial Colgate 1988).However,thereverse shockwavecanmixouter with modelswithoutrotation(Burrows1988).Theseconsider- of matterthatisinitiallyclose totheneutronstarcore(seealso ations indicatethatrotationdoes notplayaroleinthefallback Also, theobservedneutrinoburstfromSN1987Aisconsistent there issomeevidenceforaslowinitialrotationrateofpulsars. gas totheneutronstarsurface.Itwasmentionedin§I that retion processbypreventingthedirectinfallofaccreting elements arelikelytooccur. velocity carriesthecompactobjecttoaregionwherelighter the ambientdensityisuniform,conditionsareindependent cantly modified.Thegassoundspeeddoesdropwithtime,but note thatifacompositiongradientispresent,theneutron star modify thehydrodynamicresultsof§§IIIandIV.However, we of positionwithinthecore,asinHubbleflow.Ifthereisa star tendstocomovewithgasexpandingatitsvelocityand,if the explosiontendstowarduniformexpansion.Theneutron pulsar velocity.Thustheinitialaccretionrateisnotsignifi- modifying theneutronstaraccretionprocess. the soundspeedofgasissomewhatlargerthanatypical ity relativetothesurroundingmantlegashaspotentialfor metry intheinitialexplosionmechanism.Aneutronstarveloc- side oftheneutronstar. radial densitygradient,theaccretionratemaybelargeronone that thepulsarsreceivetheirvelocitiesasaresultofanasym- to betenable(e.g.,DeweyandCordes1987),soitisplausible The runawaystarhypothesisforthevelocitiesdoesnotappear velocity isassociatedwiththeneutronstarformationprocess. population cannothavesuchalargevelocitydispersion,the order of200kms“(e.g.,Lyne1987).Sincetheirprogenitor prolonged. tional effectsareimportant(see§V),theaccretionperiodis radiation effectsdonoteffectivelycutofftheinfallorifrota- in aboutayear.Thus,significantaccretionontotheneutron can leadtotheaccretionofenvelopeontoneutronstar important inthisphase,accretionattheEddingtonlimitrate for theremnantenvelope.Evenifneutrinolossesarenot star canceaseafewyearsaftertheexplosion.However,if tant, ataboutM=3x10“yr“, If theaccretionceaseswhenradiativeeffectsbecomeimpor- density (eq.[4.17])overtheenvelope: 856 0eny If adiskcanformaroundthe neutronstar,theaccreted Angular momentumoftheinfallinggascanchangeacc- We concludethatatypicalpulsarvelocitydoesnotstrongly When thereverseshockwavereachesexplosioncenter, areknowntohavetypicalspacevelocitiesonthe Menv1/2 (2GM) © American Astronomical Society • Provided by the NASA Astrophysics Data System V. ASYMMETRIESANDPULSAREFFECTS 7 «4.5 X10~|1-0.068In a) NeutronStarVelocities Mq yr 2r w* w) si (^\ ]n M b) Rotation M o (4.27) CHEVALIER 41 12 1/2 pulsator inSN1987A.Ifitis notpresent,thetotalluminosity § IV,itispossibleforradiation pressuretoaccomplishthisby observations bringsintoquestion therealityofacentralrapid an ageof2yr. infall ratescalculatedhere.According tothecalculationsin the centralneutronstarrequiresatruncationofhighmass bility (Wangetal1989).Ineithercase,directobservation of Chevalier (1989).Apulsatingneutronstarisanotherpossi- the implicationsofpresenttheoryaregiveninWoosley and et al1989).Ifthesourcewasarapidlyrotatingneutron star, rapid pulsationsfromSN1987Aon1989January18(Kristian The presentworkwascompletedpriortotheobservation of to observetheeffectsofneutronstaraccretioninasupernova. surface sothatthemagneticfieldcanplayarole. effects haveachancetocutofftheaccretionthatneutron tion mightalsoreducetheinfallpressureatneutronstar occur within2yrofthetimeexplosion(see§IV).Rota- is entirelysuppressedbytheaccretion.Itonlyafterradiative star magneticfieldcanbegintoaffectthedynamics.Thismay here (M^3x10“Myr“).Thismeansthatpulsaractivity infall (eq.[4.22]),itisclearthatthepressuredominatesat 1987A becausethisobjectpromisestogiveusourbestchance the neutronstarsurfaceformassaccretionratesofinterest When thispressureiscomparedtotheofspherical calculations areneededtoexploretheseissues. matter mayaccumulateinadisksothatthesteadyaccretion rounding medium,thepressureofamagneticdipolefieldis with magneticfieldsofB«10G.Intheabsenceasur- rate ontheneutronstarsurfaceisreduced.Two-dimensional sate forthegravitationalenergyrelease.Atsametime, probably neededfortheneutrinolossestobeablecompen- found in§IVthattheneutrinolossesclosetoneutronstar tion, thecentralpressureisreducedandalargervalueofr surface playacrucialroleintheenvelopestructure.Withrota- envelope becauseitprovidessupportforthegas.We period isincreased. § I).Ifapulsarmagneticfieldinfluencestheflow,estimated the initialpulsarperiodsderivedfromstatisticalstudies(see M =0.1,thisinitialperiodisconsiderablysmallerthan where /isthemomentofinertianeutronstar.For 0 tion periodis momentum ofthismatterisM(rGM),wherethe angular momentumoftheneutronstar,starrota- accreted mass.Ifthisangularmomentumisthedominant velocity. IntheNewtonianapproximation,totalangular matter isaddedtotheneutronstarorbitingatitsKeplerian sh acc0 accns The lackofconfirmation the 1989January18pulsation The numericalestimatespresentedinthispaperareforSN Most pulsarsarethoughttoberotatingneutronstarsborn Rotation canhaveasubstantialeffectontheneutronstar , i-.0x(£)''dynescm-=. 4 P =452mS;(51} KoTm;)(l0gcm)- c) PulsarEffects VI. SUPERNOVAE Vol. 346 (5.2) 198 9ApJ. . .346. .847C 3 12 38_1 3 56 32 type flow(see§III)islessthanisothattheBondiaccretion The densityandpressureatthistimethusscaleasR~;the with differentinitialradii,R.Undertheseconditions,similar sound speedisindependentofR.ThetimetosetupaBondi- the reverseshockfronttopasscenter,i,scalesasR. shock fronttopassthroughtheenvelope,andthustimefor velocities areexpectedinvariouscases.Thetimefortheouter We considerexplosionswithsimilarenergiesandmasses,but however, someapproximatescalingrelationscanbederived. Type IIsupernovaprobablyhasaninitialradiusafactor>10 with arelativelysmallradius,about3x10cm(e.g.,Woosley envelopes. SN1987Awasanunusualcasebecauseitexploded (eq. 3.4),thetransitiontimetoaballisticphase,isincreased by studied forthiscasewithdetailedhydrodynamiccalculations; times larger.Thepostreverseshockwaveaccretionshouldbe neutron starswithdifferentproperties. mass withessentiallynolighterelementsfrommixingand wave, andthefallbackisonlythatfromexplosionphase. mass loss;theexplosionsoftheseobjectsmayberelatedto class ofmassivestarsisexpectedtoundergoextensiveenvelope cause theformationofareverseshockwaveinmantle,so explanations are(a)theluminosityofcentralsourcecutoff expected todrivetheaccretion totheballisticcase.Thus, models donotenteranearlyballisticphase.Radioactive the twotypesofmassivestarsupernovaemaygiveriseto a smallerlikelihoodofsignificantangularmomentum.Thus, Compared totheothercase,accretionmayinvolvelittle Type lbsupernovae.Inthiscase,thereisnoreverseshock that thegeneralscenariopresentedhereapplies.However,a The deathsofstarswithinitialmassesgreaterthanabout8 In general,theenvelopeisprobablysufficientlymassiveto to betheexplosionsofmassivestarswithhydrogenenvelopes. remnants areformed.NormalTypeIIsupernovaethought although theremaybesomemassrangeinwhichblackhole able tosupernovaeinwhichaneutronstarremnantisformed. below). roughly sphericalaccretionatalargerate(see the accretionandluminositysubsequentlydied limited accretion.Iftheluminositylimitislower,possible L =3.5x10ergss,asexpectedforradiationpressure- pulsar isbelieved tohavebeenpresentis SN 1054,whichis pulsar activityinobserved supernovae. Oneeventwherea central pulsarmayceasewithin ayearoftheexplosion. late timeaccretionrateisgiven byequation(3.24)beforeradi- heating canfurtherincreasesanddecreasetheaccretion rate, R, whichcanbeafactorof10ormore.Thusthelarge-radius We notethattheinitialvalueofsisocR,so of t typical TypeIIsupernovaisreducedfromthe0.1Mdeduced rate, M,appliesattimei.Thetotalaccretedmassis oc M areexpectedtoleadtheformationofacompactobject, out (asabove),or(b)thereisacentralblackholeundergoing show whetherthereisasteadyluminositysourceatlevelof source. During1989,thedeclineofCoenergyinputwill No. 2,1989 ative effectscutofftheinflow. Theabilityoftheinfalltoburya as inSN1987A.However,radiativediffusionisalways above byafactorof100ormore. M(t)t ocR~Rcc.Thustheaccretedmassfor a of thesupernovaplacesconstraintsonnaturecentral 1988; Shigeyama,Nomoto,andHashimoto1988).Atypical rev rev Edd e e c 0 Brevacc q rcyTcy We nowturntoTypeIIsupernovaexplosionswithmassive The considerationsinthispapershouldbegenerallyapplic- The largerradiuscanalsochangethelateraccretionrate. As notedin§I,thereislittle unambiguousevidencefor © American Astronomical Society • Provided by the NASA Astrophysics Data System NEUTRON STARACCRETIONINASUPERNOVA 2 6 56 56 56 energy giventotheinnercore massissufficienttoprovidea positive bindingenergy.Neutrino lossesmaychangethiscon- inside theshockfront.Inan adiabatic explosion,theinternal mated bystudyingself-similar blastwavesinamediumwithan parameters suchthatthenucleon densityisfairlyconstant in theinnercoreofprogenitor ofSN1987A,whichhas pressure sothatequation(3.28)maycontinuetodescribe the low, theouterpartsofinflowarenotaffectedbyradiation r~ densityprofile.Thiscondition maybeapproximatelymet the gravitationalenergyisgraduallyreleasedinsteadofbeing accretion rateforaperiodofyears.Thissituationischanged if dissipative processesintheaccretinggas.Ifluminosity is swallowed bytheblackhole. rotation leadstodiskformationaroundtheblackholeso that siderably lessthanL.Theexactluminositydependson the the vicinityoftrappingradius,butitislikelytobecon- hole accretion,therewillbeaphotonluminositygeneratedin that asolidsurfaceispresentatradiusof10cm.Withblack presented in§IVarenolongerapplicablebecausetheyassume is expectedtobesmallsothatneutronstarcollapseat (Arnett 1988;Woosley1988).InanormalTypeIIsupernova, time isunlikely. the accretedmassafterreverseshockwavepassesthrough formation scenariotoexplaintheclaimedneutrinoburstsfrom the MontBlancneutrinoburstwithneutronstarformation However, thereappeartobeseveredifficultiesinidentifying reverse shockwave,whentheNiisexpectedtomixwith Woosley 1988)isnotvalidforaccretionafterthepassageof ejection ofsubstantialNiimplieslittlemassfallback(e.g., accreted ontothecentralobjectafewhoursafterexplosion. gained fromfollowingsupernovaetolatetimes. infall modelspresentedhere.Itisclearthattheremuchtobe SN 1987Aseparatedby4.7hr(e.g.,Hillebrandtetal.1987). outer corelayers.Thereissomeinterestinadelayedblackhole neutron startocollapseablackhole.Theargumentthatthe neutron starmass,thismaybeenoughmasstocausethe Depending onthenuclearequationofstateandinitial from thevicinityofneutronstarinfirstmonthsafter the explosion,whichisclosetolowertimelimitfor This proposalrequiresthatthepulsarenergybeabletoescape central pulsarprovidedtheluminosityforlatelightcurve. expected intheseobjects.Chevalier(1977)proposedthatthe explosions ofstarswithinitialmasseslessthanabout15M, (e.g., Nomoto1987).Thisisprobablyageneralpropertyofthe likely thatessentiallynoNiwassynthesizedinthisexplosion moving envelopegaspredictedbythismodelhasnotyetbeen in asupernova.Thefallbackratecalculations§IIIareinde- so thatlatelightcurvespoweredbyradioactivityarenot envelope wasinitiallypresentandthatthemixtureofH controversial. Chevalier(1977)proposedthatahydrogen .WehavefoundthatfortheparametersofSN pendent ofwhetherthecentralobjectisaneutronstaror observed. SN1054wasobservedforover600days,butitis shock wavepassingthroughthemantlegas.However,fast- He observedintheCrabNebulaisresultofreverse associated withtheCrabpulsar.Thenatureofthiseventisstill 1987A, about0.1Mofmatter(withinafactorfew)is Edd 0 0 The resultsofthispapercanbesummarizedasfollows: If acentralblackholeispresent,theaccretionsolutions Finally, wecommentonthepossiblepresenceofablackhole 1. Theeffectofacentralmassonanexplosioncanbeesti- VII. SUMMARY 857 198 9ApJ. . .346. .847C -431 _41 3(n+1)/2 5/3 - -15 .1987,Natare,329,611. Colgate, S.A.1971,Ap.J.,163,221. Chevalier, R.A.,andImamura,J.N. 1982, Ap.J.,261,543. Chevalier, R.A.1977,inSupernovae, ed.D.N.Schramm(Dordrecht:Reidel), Carrus, P.A.,Fox,Haas,F.,and Kopal,Z.1951,Ap.J.,113,1. Arnett, W.D.1988,Ap.J.,331,377. Burrows, A.1988,Ap.J.,334,891. Bondi, H.1952,M.N.R.A.S.,112,195. Bisnovatyi-Kogan, G.S.,andLamzin,S.A.1984,SovietAstr.—AJ,28,187. Bethe, H.A.,andWilson,J.R.1985,Ap.J.,295,14. Begelman, M.C.1978,M.N.R.A.S.,184,53. Bandiera, R.,Pacini,F.,andSalvati,M.1984,Ap.J.,285,134. Blondín, J.M.1986,Ap.J.,308,755. much smallerthanthetotalaccretedmass. mass accretionrate.Theintheextendedenvelope is mass accretionratesofinteresthere(10to10Myr). and areverseshockwavepropagatesbacktothecenter.Some batic structure. Neutrino lossesareexpectedtoleadthecollapseofanadia- envelope structurehasashortcoolingtimescaleatitsbase. with time.Wefindthatfortypicalparameters,anyplausible is about10Myr;thisresultnotstronglydependent envelope progenitor,theaccretedmassmaybeafactorof100 implies thatmostofthemassaccretionoccursattime expansion effectsbecomedominant,weapproximatethefall- These envelopesinvolvearadiallyextended,approximately tion envelopeisshortcomparedtothetimeforchangesin the to theneutronstarsurface.Theflowtimethroughaccre- adiabatic regionandawithstrongneutrinolossesclose density profileissteeperandthepressureincrease envelope isbuiltwithdensityocr~;forn<^,the later timewhenthesupernovadensityislower. drops asi“. back byballisticmotion.Inthiscase,themassaccretionrate formula. Underthesecircumstances,adiabaticexpansionleads is theexpansiontimescale,accretiongivenbyBondi pressure-limited accretionisshortcomparedtotheage,which core materialisdeceleratedbyitsinteractionwiththeenvelope (Bethe andWilson1985).Here,weconcentrateonthelater explosion mechanismiscrucialfordeterminingtheflow elusion, buttheyareonlyimportantatearlytimeswhenthe 858 varies asi".Fory=4/3andn>1/3,anapproximatelystatic tion envelopesonneutronstarsifthemassaccretionrate on thesupernovatype. source canbeneglected.Theaccretionrateatanageofayear the lateraccretiontobeballistic,ifluminosityofcentral smaller becausethereverseshockwavereachescenterata M. ForatypicalTypeIIsupernovawithanextended appropriate toSN1987A,thetotalaccretedmassisabout0.1 the reverseshockfrontreachescenter.Forparameters steepens thedeclineratewhilecoolinglowersit.Once to amassaccretionratedroppingasi.Heatingofthegas bound totheneutronstar.Ifcharacteristictimefor accretion phase. properties. Iftheexplosionoccursbyneutrinoheating,thereis of themassnearcentralneutronstarisexpectedtoremain a clearbifurcationbetweentheneutronstarandejectedmass 0 o 0 P- 53. 4. Thediffusionofradiationfromthecentralregioncauses 6. Steadystateaccretionenvelopesareplausibleforthe 2. Insupernovaexplosionswithamassiveenvelope,the 3. Thesteepdeclineofthemassaccretionratewithtime 5. Self-similarsolutionsarecalculatedforadiabaticaccre- © American Astronomical Society • Provided by the NASA Astrophysics Data System CHEVALIER REFERENCES 4-1 evolution ofaneutronstarinsupernova. Kristian, J.A.,etal.1989,Nature,338,234. Imamura, J.N.,Wolff,M.T.,andDurisen, R.H.1984,Ap.J.,276,667. bear ontheseproblems.However,itisexpectedthatfuture process. Atthepresenttime,therearefewobservationswhich Lyne, A.G.1987,inIAUSymposium 125,TheOriginandEvolutionofNeutron Lundqvist, P.,andFransson,C.1988, Astr.Ap.,192,221. Lidov, M.L.1957,SovietAstr.—AJ, 1,588. Kazhdan, Ya.M.,andLutskii,A.E.1977, Astrofizika,13,535. Hillebrandt, W.,Höflich,P.,Kafka,Müller,E.,Schmidt,H.U.,andTruran, Emmering, R.T.,andChevalier,A.1989,Ap.J.,345,931. Dewey, R.J.,andCordes,J.M.1987,Ap.321,780. Colgate, S.A.1988,inSupernova1987AtheLargeMagellanicCloud,ed. M. observations ofSN1987Awillclarifysomeaspectstheearly study inordertodeterminewhetheritcancutoffthefallback tions. Also,thestabilityofthesesolutionsrequiresstudy. Finally, theescapeofradiationfromaccretionregionneeds correspondence fromS.Sakashita.Thisresearchwas sup- lated withmoreaccurateequationofstateandcoolingfunc- fallback afterthepassageofreverseshockwavecanbe ported inpartbyNSFgrantsAST-8615555andAST-8818362. separated fromtheaccretiononneutronstarsurface. treated bynumericalhydrodynamicsandcaninitiallybe processes requiremoredetailedexamination.Theproblemof ciency ofthegenerationaccretionluminosityisprobably son, andE.Myraaregratefullyacknowledged,asishelpful Discussions withS.Baibus,P.Becker,R.Blandford,C.Frans- analytic estimatesoftherelevantphysicalprocesses.These ties ofpulsars.Thepresentpaperhasconcentratedonsemi- the possibilityofblackholeformation,andinitialproper- born neutronstar,thecompositiononastarsurface, greatly reducedinthiscase. envelopes. Unlessrotationaleffectsareimportant,theeffi- Models ofsteadystateaccretionenvelopesneedtobecalcu- from anumberofpointsview:theobservabilitynewly absence ofasolidsurfacedoesnotallowforshockedaccretion little effectontheestimatedfallbackrates.However, plausible magneticpressureattheneutronstarsurface.A dropped toarelativelylowvalue. and canonlyhaveaneffectwhenthemassaccretionratehas typical pulsarmagneticfieldcannotreversetheinitialinflow ing medium.Rotationcandelaytheaccretionontoneutron times, thepressureinaccretionflowislargerthan star surfacebutitseffectsaredifficulttoestimate.Atearly small effectbecausetheneutronstarisinauniformlyexpand- We estimatethatatypicalneutronstarvelocityhasrelatively envelope. Thisoccursatanageof7months.Beforethistime, the effectsofacentralneutronstararehidden. inflow. Atanaccretionratebelowabout3x10“Myr, the photonscanbegintodiffuseoutfromshocked Eddington limit.Initially,thisluminosityistrappedbythe photon luminosityinthefluidframeisapproximatelyat 0 Stars, ed.D.J.HelfandandH.Huang (Dordrecht:Reidel),p.23. J. W.1987,Astr.Ap.,180,L20. p. 341. Kafatos andG.Michalitsianos(Cambridge:CambridgeUniversityPress), This workwasstimulatedbyaconversationwithS.Colgate. Accretion ontoaneutronstarinsupernovaisimportant 9. Thepresenceofablackholeinsteadneutronstarhas 8. Otherfactorscanaffecttheneutronstaraccretionrate. 7. Sincetheaccretionenvelopesareradiationsupported, Vol. 346 198 9ApJ. . .346. .847C No. 2,1989 Schinder, P.J.,Schramm,D.N.,Wiita,Margolis,S.H.,andTubbs,L. Sakashita, S.,andYokosawa,M.1974,Ap.SpaceSei.,31,251. Rupen, M.P.,vanGorkom,J.H.,Knapp,G.R.,Gunn,E.,andSchneider, Nomoto, K.,Shigeyama,T.,Kumagai,S.,andHashimoto,M.1988,Proc.Astr. Nomoto, K.1987inIAUSymposium125,TheOriginandEvolutionofNeutron Narayan, R.1987,Ap.J.,319,162. Pinto, P.A.,Woosley,S.E.,andEnsman,L.1988,Ap.J.{Letters),331,L101. Pacini, F.1987,inProc.ESOWorkshoponSN1987A,ed.I.J.Danziger Michel, F.C,Kennel,C.F.,andFowler,W.A.1987,Science,238,938. Michel, F.C.1988,Nature,333,644. Roger A.Chevalier:DepartmentofAstronomy,UniversityVirginia,P.O.Box3818,Charlottesville,VA22903 (Garching: ESO),p.565. D. P.1987,A.J.,94,61. Soc. Australia,7,490. Stars, ed.D.J.HelfandandH.Huang(Dordrecht:Reidel),p.281. 1987, Ap.J.,313,531. © American Astronomical Society NEUTRON STARACCRETIONINASUPERNOVA Provided bythe NASA Astrophysics Data System ZeFdovich, Ya.B.,Ivanova,L.N.,andNadezhin,D.K.1972,SovietAstr.—AJ, Woosley, S.E.,Pinto,P.,andWeaver,T.A.1988,Proc.Astr.Soc.Australia,7, Woosley, S.E.,andChevalier,R.A.1989,Nature,338,321. Woosley, S.E.1988,Ap.J.,330,218. Thorne, K.S.,andZytkow,A.N.1977,Ap.J.,212,832. Wang, Q.,Chen,K.,Hamilton,T.T.,Ruderman,M.,andShaham,J.1989, Uomoto, A.,andKirshner,R.P.1986,Ap.J.,308,685. Shigeyama, T.,Nomoto,K.,andHashimoto,M.1988,Astr.Ap.,196,141. Sedov, L.I.1959,SimilarityandDimensionalMethodsinMechanics(New Nature, 338,319. 355. York: AcademicPress). 16,209. 859