198OApJ. . .238 . .991G © 1980.TheAmericanAstronomicalSociety.Allrightsreserved.PrintedinU.S.A. The AstrophysicalJournal,238:991-997,1980June15 has employednumericalcodesforsphericallysym- remaining physics,suchasnuclearreactions,energy dynamics permitsadetailedtreatmentofthe metric hydrodynamics.Thisapproachtothehydro- transport, neutrinoandtheequationof details ofthethermodynamicsandconsiderasimple state. However,itcanbeinstructivetodisregardthe al. 1971;Arnett1977ft;Barkat1977),starsmassive hydrodynamics. the inputphysicsandconsequentbehavioris be obtainedanalytically,therelationshipbetween analytic equationofstate.Manyresultsmaythen very similarprecollapseconfiguration.Thisconsistsof enough toignitecarbonnonexplosivelyallevolvea supernova progenitors(seePaczynski1971;Ikeuchiet more transparent,anditispossibletoinvestigate a hotdegenerateironcoreofsome1-3M,with metry) thatismoredifficulttoexaminewithnumerical 21453 andAST76-80801A02). when themeanadiabaticindex,ÿ=\yPdVßyPdV, approximated byan«=3polytrope(VanRiper1978; overlying burningshells.Thecorecanberoughly behavior (suchasdeparturesfromsphericalsym- drops below4/3.Thisresultsfromsomecombination Nadyozhin 1977).Itbecomesdynamicallyunstable pressure andinitiatecollapse, theincreasingdensity and generalrelativity.After theseeffectsreducethe of inverseßdecay,endothermicdissociationiron, soon causesthecoretobecome opaquetoneutrinosso 0 v 1 Most workonstellarcorecollapseinsupernovae According torecentworkontheevolutionof SupportedinpartbytheNational ScienceFoundation(AST78- © American Astronomical Society • Provided by theNASA Astrophysics Data System pressure atagivendensityisreducedbyupto3%fromthevalueformarginallystablestaticcore. homologously collapsingconfigurations.Homologouscollapseoftheentirecoreispossibleif stellar coresduringtheearlyphasebeforenucleardensityisreached.Wefindafamilyofexact nonradial modesarecalculated,anditisfoundthatallessentiallystable. varies asthe3/2powerofreducedpressureatonsetcollapse.Linearperturbations these homologouslycollapsingsolutionsareseparableinspaceandtime.Loworderradial Subject headings:stars:collapsed—interiorssupernovae For agreaterpressurereduction,aninnercorecancollapsehomologously,themassofwhich = 3,correspondingtoy4/3.Suchpolytropesprovideareasonableapproximationcollapsing We investigatethecollapseofnonrotatinggassphereswithapolytropicequationstate:n I. INTRODUCTION Division ofGeologicalandPlanetarySciences,CaliforniaInstituteTechnology 1 HOMOLOGOUSLY COLLAPSINGSTELLARCORES W. K.KelloggRadiationLaboratory,CaliforniaInstituteofTechnology Received 1979September26;acceptedDecember5 Stephen V.Weber Peter Goldreich ABSTRACT AND 991 the equationofstate.Bythiswemean trope wouldbeareasonablysimpleapproximationto that thesubsequentcollapseisnearlyadiabatic(Arnett where Pisthetotalpressure,pdensity,y=4/3 1917a; Betheetal.1979;SaenzandShapiro1979). equation ofstateisreasonablefortheprecollapsecore constant intimealonganygivenstreamline,butnot the adiabaticindex,andkisaconstantbothinspace and intime.“Adiabatic”normallymeansthatkis necessarily constantfromstreamlinetostreamline. will rangeaboveandbelowthis.Thepressurere- y is4/3attheonsetofcollapse,althoughlocalvalue since thecoreresemblesan«=3polytrope.Themean space forahydrostaticconfiguration,andyisreplaced using asmallerk:forcollapsingcoresthanthehydro- duction attheonsetofcollapsecanbemodeledby equal 1+1/«,norisitnecessarilyconstant. by 1+\¡n,withnthepolytropicindex.Foranormal “Polytropic” normallymeansthatkisconstantin static cores.Thestatementthatcorecollapseisadia- polytrope, y=dlogP/dpdoesnotnecessarily The equationofstatewillbecomestifferatnuclear that theequationofstateisassimple(1). batic meansinthiscasethatitconservesentropy,not densities, anessentialfeatureforcausingacorebounce proposed equationofstate shouldbereasonableat and forallowinganeutronstar toform.However,the meant onlytodescribethe earlystagesofcollapse densities lowerthannuclear density.Thismodelis before nucleardensityisreached. These resultsindicatethatan«=3,y4/3poly- The numericalmodelsindicatethattheproposed P =KP\(1) 198OApJ. . .238 . .991G 1/3 is independentofdensity.Putanotherway,the characteristic masscorrespondingtotheJeanslength mations ofthestar.Therefore,centercore the samepowerofradiusforhomologousdefor- 992 Jeans mass,regardlessofthedensityincrease.This gravitational energyandthethermalscaleas complicated equationofstatewouldnotgiveidentical could exist.Indeed,homologouscollapseoftheinner collapse, sincethecenteralonewouldbelessthan should notrunawayfromtheouterpartsduring equation (3)canbeintegrated equation, andPoisson’sequation: radius, sothathomologousexactsolutionsforcollaps- Van RiperandArnett1978;1978).Amore models (Nadyozhin1977;EpsteinArnett1977¿* potential; Gisthegravitationalconstant;andh Here, uisthefluidvelocity;0gravitational equation ofstateisnottoodifferentfromthatgivenby ing coreswouldnotbeexpected.However,ifthetrue part ofthecorehasbeenobservedinnumerical suggests thatahomologouslycollapsingsolution obtained fromastreamfunction:u=Sv.(Notethat heat function{h=)dP/p4jcpforourequationof scaling ofgravitationalandthermalenergywithcore an arbitraryconstantcan beaddedtov.)Then, state). Iftheflowisvorticityfree,velocitymaybe equation (1),thedeparturefromhomologyshouldbe is essentialifsupernovaearetobepowerfulsourcesof cally symmetricsystems.Growthofnonradialmodes that nogravitationalradiationisemittedbyspheri- radial modes,isanimportantquestion.Onereason small. § III,andthenormalmodesarediscussedinIV. cores. SincetheJeansmassisindependentofdensityin vestigate thestabilityofhomologouslycollapsing perturbations ofthoseconfigurationsarederivedin ties ofthesesolutions.Theequationsdescribinglinear collapsing solutionsdoesexistanddiscusstheproper- This suggeststhatthecollapsewillbestable. gravitational radiation.Itisrelativelyeasytoin- our model,thereisnotendencytowardfragmentation. Section Vpresentsasummaryandconclusions. One nicefeatureofthisequationstateisthatthe We beginwiththeequationofcontinuity,Euler’s The stabilityofcorecollapse,especiallyfornon- In §II,wedeterminethatafamilyofhomologous 2 ^ +V(i|ii|)(VXu)u\h\(t)=0,(3) © American Astronomical Society • Provided by theNASA Astrophysics Data System II. HOMOLOGOUSLYCOLLAPSINGCORES 2 - +i|V*;|/*(/>=0. (5) dv 2 Ô V(j) —4nGp=0.(4) £ +\.(pu)=0,(2) GOLDREICH ANDWEBER which isseparableinrandt.Theexpressionontheleft where wehavealsosubstitutedthestreamfunction The constantofintegrationhasbeenincorporatedinto the timedependenceofmostgeneralexactly The generalsolutiontothisdifferentialequationgives and alsogivesanonlineardifferentialequationfora{t). pression dependsontimeonly,sobothhavebeen is afunctionofradiusonly,whilethemiddleex- the potential. The constantofintegration,C,determinesthecon- homologous solution. equated toaconstant.Thispairofequationsimplies in termsofthecentraldensity, into thecontinuityequation.Next,wescaledensity corresponding totheJeanslength(seeEddington once, yieldingtheenergyintegral where thesubscriptcindicatesvalueat/?=0.We star isinfinite.Equation(14) canbeintegratedagainif traction (orexpansion)velocity whentheradiusof and thepotentialintermsofsquarecentral sound speed, scale theradialcoordinatefrom(dimensioned)Rto profile doesnotevolve.Euler’sequationyields are transformedbythistimedependentscalingto (dimensionless) r=R/a(t).Thedifferentialequations 1926). Thescalefactoristimedependentinourcase: the trivialrelation/=0,whichstatesthatdensity or u=àrsothatthecontinuityequationreducesto C =0andA>0: i21l3 2 p dt a(t) =(yP/py/(nyGp)p-(^^J,(6) The differentialequationfora(t)canbeintegrated It isconventionalforpolytropestousearadialscale c To seekahomologoussolution,wesetv=Xßaär 12 2 2 + a(\v—âr)-\p/pVz;=0,(7) ^ -r-\+ja~\Sv\h>(12) a =(6A)1/3(i+io)2/3 (15) (^J - 3 2 =Pc/=(^YV a~V(¡) —AnGp0,(9) 2 \¡/ =^Xr-2>f,(13) 31/2 31/6 /k\ /k\ 12 \ inG\ nG 1 /2 vy ^ nG )a 3 / Vol. 238 ,/remainsfiniteatallr.Physically, cally, thislimitexistsbecausef(r)becomestangentto solution ispossible,2æ0.00654376.Mathemati- the limitingvalueof2isreachedwhensurface polytropes, for(lefttoright)A=0, 0.002,0.004,andX. m W m 13 For A=0,thestarhaszerobindingenergyandis We useequation(13)toeliminateij/inPoisson’s Numerical solutionsofequation(16)forseveral There isamaximumvalueof2forwhichphysical Fig. 1.—Densityprofilef(r)cep ofhomologously-collapsing © American Astronomical Society • Provided by theNASA Astrophysics Data System d 2 L±(2l\ r r dr\dr) 3 + /=¿- COLLAPSING STELLARCORES (16) 3 3 2 312 can beperformed,yielding we write density, p.Inordertoseethatp/p=2when, and thecoremass,M,isexpressedintermsofmean after equation(12)isusedtoevaluatetheacceleration, and useequation(16)toevaluate/.Theintegralthen while p!pdecreasesfrom0.01846to0.006544.The configuration isonlyneutrallystableandhighly the coreisinfreefall.Theconditionù(R)= product rp/pisrelativelyinsensitiveto2,increasing 2, rincreasesfromapproximately6.897to9.889, centrally condensed. core. Thereasonforthissmallrangeisthatthestatic and 2determines/c,therangeofcorrespondstoa seen tobesatisfiedwhendf(r)/dr=0. Thus, theconditionforfreefallatsurfacecanbe homologously collapsingconfigurationfortheentire pressure, bynomorethan2.9%,andstillfinda range of2.9%injc.Consequently,ifwebeginwitha only byafactor1.0449inthisinterval.SpecifyingM 267o)- Althoughtheentirecorecannotcollapseho- novae issubstantiallylarger(Betheetal[1979]obtain static (2=0)core,wecanreducek,andthusthe mologously whenthepressurereductionisthislarge,a pressure duetothematerialfurtherout(whichis less massiveinnercorecandoso,whiletheremainder — GM/Rbecomesp/p=Àindimensionlessunits, nearly infreefall)isnegligible.Themassoftheinner of thecoreisleftbehind.The2=homologous cm structure oftheinnercoretoextentthat solution shouldbeagoodapproximationtothe core willbe where Mandkarethemassvalueoffor core massesagreewiththepredictionofequation(17) marginally stableinitialstaticcore. c inner coreisingoodagreementwiththeresultsofVan s c ms difference isonly6%ofthetotalcoremassandmay to withinafewpercentforpressurereductionsofup starting froman«=3polytrope,withy1.32.His s = 4ßnr(plp^kIuG).As2increasesfromzeroto finds thatthemassofhomologous innercoreis not bewelldetermined.Note, however,thatVanRiper prediction whenthepressurereductionis707,butthis Riper (1978),whonumericallyintegratescollapses, 50%. Hisinnercoremassis307largerthanour predictions maybeinerror ifydifferssubstantially from 4/3. sensitive tothevalueofy, so thatourquantitative Sc m 0 0 0 The coremassforthecollapsingpolytropesisM The pressurereductioninitiatingcollapseinsuper- This predictionforthemassofhomologous 312 p/p =(4nr/3)J4nrfdr, cs p/l>c =¿-(3/r)df(r)/dr. s M= 1.0449Í—J,(17) 1C0 3/2 / K\ 993 198OApJ. . .238 . .991G 994 collapsing configurationsdescribedinthepreceding g(t) =exp as section. Theperturbeddynamicalvariablesarewritten The functiong(t)givesthetimedependenceof scaled perturbations,andisassumedtohavetheform, where We havechosentoworkwithperturbationsofthe ment vectorasismoreconventional(see,e.g., stream functionratherthantoworkwiththedisplace- especially inanacceleratedcoordinatesystem. algebraically simplertoworkwithscalarvariables, Eddington 1926;LedouxandWalraven1958).Itis collapse oftheunperturbedconfigurationhascancel- It canbeseenthatthetimedependencedueto (18)-(20) todefinetheperturbations,weobtain is theprincipalrestoringforce.For/7-modes, displacement vectorwhichisprimarilyhorizontal. value problem. They producelittlepressureperturbation,andgravity led outofthedynamicalequations,leavinganeigen- radius vector.The/-modeisintermediateinfrequency the dominantrestoringforce.Cowlingalsoidentifiesa displacement vectorisprimarilyradial,andpressure oscillation ofasphericalstar.The#-modeshave for /7-modesincreasesasthe numberofmodesin- ponent ofthedisplacementvectorandpressure creases, whilethefrequency decreaseswithincreasing between the/7-modesand #-modes.Thefrequency perturbation arebothconstantinsignalonganygiven fundamental or/-mode,forwhichtheradialcom- mode, butshallregarditas thelowestorder/?-mode. order for#-modes.Weshall notdistinguishthe/- Consider linearperturbationsofthehomologously Linearizing equations(7)-(9),andusing Cowling (1941)classifiedthemodesofnonradial © American Astronomical Society • Provided by theNASA Astrophysics Data System 3 Vf 4>(r, f)=~Í—ja^[iAo+iMrMO]•(20) p(r, o=(—Ja-/[l+(rM0],(19) Pl III. LINEARPERTURBATIONEQUATIONS 12 2 ÍP +O^)']^!+fpiiK=0,(24) 2 Wv +3\\ogf-\vp=0,(23) 12 1l v{r,t) ={aàr+^v(r)g{t),(18) r1 t =(±nGpy(forallX), ffc 1,2 31/2 3/2 = (9k/2)t(forXÏ0).(22) l 4 //c\ ( K\ t dt' ff 23 91/2 V^ +3/=0.(25) Pl = expOi/^),A0, (2/A)P i0(21) = Í? 2 GOLDREICH ANDWEBER 2 1/2 1/2 1/2q3/2 r 1/2 2 2 -(i +1) 2 x \f^i2dV =0. #-modes further.Forthe/7-modes,equation(24)can convective modes.Thesehavep=0,p^r)if/^r) is assumedinourmodel,the#-modesarejustneutral 0, Viv+3Vlog/'Vi^=0.Weshallnotdiscussthe be usedtoeliminateAssumingthattheangular where =pwandm—pip+(A/2)].Thus,we homogeneous differentialequations harmonics, weobtaintwosecond-orderordinary dependence oftheperturbationsisgivenbyspherical v have p=(A/8)±(A/8—m),andg{i)becomes exp (±/ra//)forA=0;tcca,q—\/6± to differenteigenvalues^mareorthogonal: real andthateigenfunctions£i(f*),Ç()corresponding (1/36 —2m/9A)forA^0.Thedisplacementvector equations (26)and(27)numerically,usingthemethod for thelowestthreemodes/=0,1,2and00. <*; =ôr/risrelatedtovrby^Vw. we havenotdonesobecausethisispossibleinthe tions toasinglesecond-orderdifferentialequation,but static A=0casehavebeencomputedbefore. (26) and(27)areregularityconditions.Regularityat ism =—3A,wr,\l/6\_r(df/dr) +/]—3Ar.This Schwarzschild (1941)hasevaluatedradialmodes,and convenient linearlyindependentconditioncanbe condition servesonlytodeterminethearbitrarycon- infinity isinsuredbyocratr=r[i.e.,¿/i/^/dr the entirecore.Consequently, itperturbsthepartic- reasonable agreementwithours. including itexactly;however,hiseigenvalueisstillin our resultsareinsatisfactoryagreementwithhis.For/ specified. stant inthedefinitionofstreamfunction,andany ular homologouslycollapsing solutionwehaveused mode correspondstoahomologous perturbationof approximates thepotentialperturbationratherthan value forthelowest/=2modeA0.He general case.Cowling(1941)hasobtainedtheeigen- Regularity ofwatthesurfacerequires3(df/dr)(dw/dr) r dr\ 2 2 ^ --fw+3-y-il/mw=0, f d(w\/(/+1Kdfdw, < AareplottedinFigure2.Someofthemodesfor i s + (/WiA*^9].For/=0,thefourthboundary — i/ji+mw=0atr.For/0,regularityof = A0,itispossibletoreducethesystemofequa- 2r1 2 m s Cowling (1941)showedthatfory=1+1/«,which We havefoundeigenvaluesandeigenfunctionsof 21 r It caneasilybeprovedthattheeigenvalues,m,are The boundaryconditionsfordifferentialequations The lowestradialmodecan be foundanalytically.It r dr 1 d(y\/(/-hi) v2 dr IV. NORMALMODES r dr 2 i¡/ -h3fi¡/—mw=0, 1 Vol. 238 (26) (27) 198OApJ. . .238 . .991G l2/3 213 1 2/31 1/2 for ourunperturbedconfigurationintothegeneral this modeisproducedsimplybysetting^0in perturbations dogrowintimefortheformercase,but gives atimedependenceofeithert~or¿.The homologously-collapsing solution.Thevalueofm equation (15).Thiscanbeseenby No. 3,1980 Thus, thisshouldnotbeconsideredagrowingmode.It has m=—X,givingatimedependenceforwandil/of is onlyanindicationthatonehaschosenincon- venient originoftimefortheunperturbedconfigu- unimportant asa—►0. ration. ThetmodeisalinearizationoftheC^0 ing ourexpectationthatthevalueofCbecomes amplitude ofthismodegoestozeroas>0,confirm- nonzero collapsevelocityinthelimita—>oo.The —►(25/8)/l asA—>A.Thereasonforthisisnot neutral displacementoraconstanttranslationvelocity uniform displacementoftheentirestar.Thissolution solution ofequation(14);thatis,thewitha modes with{bottomtotop)0-2nodes. SR =ar\wcct°ort.Thesecasescorrespondtoa difficult tounderstand.First, ifweemploytheWKBJ above, theeigenvalueapproachessamelimitm of theentirestar.Theeigenfunctionsarew=r, t~ ort.Morerelevantisthetimedependenceof approximation forshortwavelength perturbations,we k cc(P/p)“oc/.When both/anditsderivative find thattheradialwavenumber, k,variesas go tozeroatthesurface, aninfinitenumberof 1 m = 3(df/dr)—Xr. The lowest/=1modeisatrivialsolution,giving Fig. 2.—Eigenvalues(m)asfunctionsof2for/=0,1,2, For allmodesexceptthetwospecialcasesdiscussed © American Astronomical Society • Provided by theNASA Astrophysics Data System lR{t +t)-R(t)]IR{t) 0 l2> x R{t)tIR(t)a:t-lt=r\ Q COLLAPSING STELLARCORES 2 q12 5/4 wavelengths willbefoundinanyneighborhoodofr. for Ar=r—«l,/moc.Thissolutionis essential singularityatAr=0.ForX<,f(Ar)will w ocAr,q——5/2±(25/42m/X).Weseethat More rigorously,wecanfindanasymptoticsolution becomes imaginaryform/X>25/%,producingan centrated tothesurfaceas2—►X.Thebehaviorof eigenfunctions asX^>XisillustratedbyFigure3, become linearratherthanoscillatoryinthatregime. become linearforArsufficientlysmall,andvv(Ar)will for asequenceofvaluesX.Thevariationthe which showsthe/=2eigenfunctionswithtwomodes of modes,m-+25/$Xandthemodesbecomecon- Consequently, foreigenfunctionswithafinitenumber character. ModesfordifferentXnottoocloseto eigenfunctions arecharacteristicintheirqualitative and {toptobottom)A=0,0.0065, 0.0065435.Theverticalscaleis and 0-3nodes,scaledasinFigure3.Theseparticular cies areisolated. s generally bemuchlessthanthecentralpressure.A limiting solution.Forarealstellarcoretherewillbe artifact ofthespecialmathematicalcharacterthis tude variestoomuchtopermitaworthwhileplotof amplitude ofw(r)predictedbyWKBJtheory, This featureissomewhatobscuredinthefiguresby concentrated nearthesurface,asin/?-modesgenerally. for theirbehaviorasr->0.Thepowerinthesemodesis look muchthesame,asdomodesfordifferent/,except normal modesareregularandthattheeigenfrequen- finite soundspeedatthesurfacewillinsurethat s w(r) ocr“y,hasbeenremovedbecausetheampli- arbitrary. Notehowthenodesconcentrate towardthesurfaceas some finitepressureatthesurface,althoughthiswill w(r) byitself. scaling. m m m m m 5/4 The singularbehavioratXshouldberegardedasan Fig. 3.—Scaledeigenfunctionsr/(r) w(r) for1=2,twonodes, Figure 4illustrateseigenfunctionsfor2=0,/2, m 01 23456789 10 995 198OApJ. . .238 . .991G q1/2 1/64 23 996 discussed earlier)areoscillatory.Theeigenvalues,m, modes isthatall(exceptingthetwospecialcases time dependenceoft,q=—1/6±(2/3)z.Itis are alwaysgreaterthan25A/8.Thislimitingcasegivesa sufficient form/X>1/8toobtainacomplexeigen- modes doesincreaseduringthecollapseas frequency. Theamplitudeoftheoscillationsforall compression. Amplificationofthissortshouldnotbe to adiabaticamplificationofsoundwavesdue insures thatperturbationswillnotattainhighampli- described asinstability.Also,thesmallexponent ¿~ xa“.Thisincreaseinamplitudecorresponds radius isreducedbyafactoroforder10“,per- collapse ofastellarcoretoneutronstar,inwhichthe the collapseisbyaverylargefactor.Evenin tude duringcollapseiftheyareinitiallysmall,unless turbations willbeamplifiedonlybyafactorof3-6. 0-3 nodes.Theverticalscaleisarbitrary.Modesofdifferent/and/, are qualitativelysimilartothese. can explaintheformationofahomologousinnercore, Barkat, Z.1977,inSupernovae,ed. D.N.Schramm(Dordrecht: Bethe, H.A.,Brown,G.E.,Applegate, J.,andLattimer,J.M.1979, .19776,Ap.J.Suppl.,35,145. Arnett, W.D.1977a,Ap.J.,218, 815. 5/4 The mostimportantcharacteristicofthenormal Fig. 4.—Scaledeigenfunctionsr/(r)w(r)for/=2,A0,and Our polytropicmodelsforcollapsingstellarcores Nucl. Phys.,A324,487. Reidel), p.131. © American Astronomical Society • Provided by theNASA Astrophysics Data System V. CONCLUSIONS GOLDREICH ANDWEBER REFERENCES -1/4 which hasbeenobservedinnumericalmodels.The mass oftheinnercoreisgivenbymodel:it collapse (tobeprecise,itisupto4.57olargerthanthe essentially theChandrasekharmasscorrespondingto core shouldbesimilartothatofastationaryn=3 the reducedentropyofadiabaticphase polytrope, butwithsomewhatmoreextendedouter Chandrasekhar mass).Thedensityprofileoftheinner layers. course. Hunter(1962)discussesthestabilityofper- contrasted withthepressure-freecase,inwhichper- amplitude ofthefractionaldisplacementincreasingas sense thatperturbationsareoscillatory,withthe turbations growmonotonicallyandrapidly(Lynden- pressure canstabilizesphericalcollapseisfamiliar,of Æ asthecoreradiusdecreases.Thismaybe turbations ofhomogeneousgassphereswithy=4/3. Bell 1964;Lin,Mestel,andShu1965).Thefactthat In thatcase,theunperturbedcollapseiseffectively wavelengths. Lynden-Bell’s(1979)statementbasedon pressure-free becausethereisnopressuregradient,and perturbations arestableonlyforsufficientlyshort the homogeneouscase,thatshapeofaspherewill be stableonlyiftheinwardaccelerationof limiting casewhenthesurfacecollapsesinfreefall. apply toourmodels.Ourspheresarestableeveninthe boundary islessthan2/5ofthefreefallvalue,doesnot culated numericallybyShapiro(1977)andSaenz than freefall. However, thecollapseofinteriorisalwaysslower that departuresfromsphericalsymmetryandthe ploying homogeneousspheroidsorellipsoids,predict s tant willremainverynearlysphericallysymmetric,and We concurthatcoresinwhichrotationisnotimpor- only afterthebounceofcoreatnucleardensities. emission ofgravitationalradiationbecomeimportant Shapiro (1978,1979).Thesesimplifiedmodels,em- core bouncecanbeunderstoodintermsofthis diation, beforethecorebounce. therefore willnotemitsignificantgravitationalra- drodynamics ofthebounceitselfandproduction relatively simpleanalyticmodel.Theinterestinghy- Weber wassupportedbyaChaimWeizmannpostdoc- for excellentassistancewiththecomputing.StephenV. Frank Shu.WealsoacknowledgeGlenHermannsfeldt a reflectedshockmustbeapproachedinanotherway. toral fellowship.ThisworkwassupportedbyNSF grants AST78-21453andAST76-80801A02. Cowling, T.G.1941,M.N.R.A.S., 101,367. Eddington, A.S.1926,TheInternal ConstitutionoftheStars Hunter, C.1962,Ap.J.,136,594. Epstein, R.I.1977,inSupernovae, ed. D.N.Schramm(Dordrecht: This homologouslycollapsingcoreisstableinthe Nonspherically symmetriccollapseshavebeencal- We aregratefulforcommentsbyDougKeeleyand The hydrodynamicsofcorecollapseupuntilthe (Cambridge: CambridgeUniversity Press). Reidel), p.183. Vol. 238 198OApJ. . .238 . .991G Pasadena, CA91125 Peter Goldreich:DivisionofGeologicalandPlanetarySciences,170-25,CaliforniaInstituteTechnology, Stephen V.Weber:TheoreticalAstrophysics,130-33,CaliforniaInstituteofTechnology,Pasadena,CA91125 calculations givesignificantlylargerhomologouscoremassesthantheanalyticformulaifk/k<0.2.Thiscan core collapseisexpectedtobewellwithintheregimeinwhichourresultsshouldvalid. A homogeneouspressure-freespherecollapseshomologously,andapressuregradient-freeconfigurationis dynamically equivalenttoonethatispressure-free.Therefore,thenearlyhomogeneouscentralpartof numerical calculation(±2%),providedthepressurereductioninitiatingcollapseisnottoolarge.The .1979,Observatory,99,89. Ikeuchi, S.,Nakazawa,K.,Murai,T.,andHayashi,C.1971,Prog. Paczynski, B.E.1971,ActaAstr.,21,271. No. 3,1980 initial pressurereductionisverylarge,thehomologouscoremustformfrommaterialthatinitiallynearly gradients canproduceanappreciableeffect. initial configurationwillcollapsenearlyhomologously,regardlessoftheequationstate,untilsmall be explainedeasily.Thegradientsofdensityandpressuregotozeroatthecenterinitialconfiguration. factor thanarealcorecollapsewouldprovide.Fortunately,theactualpressurereductionwhichinitiatesstellar homogeneous, sotheasymptoticsolutionmaynotbereacheduntildensityhasincreasedbyalarger homologous coremassesagreewithouranalyticresult(eq.[17])towithintheaccuracyclaimedfor obtained bynumericalintegrationofcollapses,inwhichtheequationstate(1)isemployed.The Nadyozhin, D.K.1977,Ap.SpaceSei.,51,283. Lynden-Bell, D.1964,Ap.J.,139,1195. Ledoux, P.,andWalraven,T.H.1958,HandbuchderPhysik, Lin, C.C.,Mestel,L.,andShu,F.H.1965,Ap.J.,142,1431. 0 Note addedinproof.—VanRiper(1980,privatecommunication)hasprovideduswithhomologouscoremasses 353. Our analyticresultsshouldalwaysbevalidasymptoticallyasthedensitygoestoinfinity.However,if Theor. Phys.,46,1713. © American Astronomical Society • Provided by theNASA Astrophysics Data System COLLAPSING STELLARCORES .1979,Ap.J.,229,1107. Saenz, R.A.,andShapiro,S.L.1978,Ap.J.,221,286. Van Riper,K.A.,andArnett,W.D.1978,Ap.J.{Letters),225, Van Riper,K.A.1978,Ap.J.,221,304. Shapiro, S.L.1977,Ap.J.,214,566. Schwarzschild, M.1941,Ap.J.,94,245. Wilkinson, J.H.1955,TheAlgebraicEigenvalueProblem(Oxford: Clarendon Press). L129. 997