Table Of Content

Received;accepted;published PreprinttypesetusingLATEXstyleemulateapjv.5/2/11 PUSHINGCORE-COLLAPSESUPERNOVAETOEXPLOSIONSINSPHERICALSYMMETRYI: THEMODELANDTHECASEOFSN1987A A.Perego1,M.Hempel2,C.Fro¨hlich3,K.Ebinger2,M.Eichler2,J.Casanova3,M.Liebendo¨rfer2,F.-K.Thielemann2 (Dated:May26,2015) Received;accepted;published 5 ABSTRACT 1 Wereportonamethod,PUSH,forartificiallytriggeringcore-collapsesupernovaexplosionsofmassivestars 0 in spherical symmetry. We explore basic explosion properties and calibrate PUSH to reproduce SN 1987A 2 observables. Our simulationsare based on the GR hydrodynamicscode AGILE combinedwith the neutrino y transportschemeIDSAforelectronneutrinosandASLfortheheavyflavorneutrinos.Totriggerexplosionsin a theotherwisenon-explodingsimulations,thePUSHmethodincreasestheenergydepositioninthegainregion M proportionallyto the heavyflavor neutrinofluxes. We explorethe progenitorrange 18 –21M . Our studies ⊙ revealadistinctionbetweenhighcompactness(HC)(compactnessparameterξ >0.45)andlowcompactness 1.75 2 (LC)(ξ <0.45)progenitormodels,whereLCmodelstendtoexplodeearlier,withalowerexplosionenergy, 1.75 2 andwithalowerremnantmass.HCmodelsareneededtoobtainexplosionenergiesaround1Bethe,asobserved forSN1987A.However,allthemodelswithsufficientlyhighexplosionenergyoverproduce56Niandfallback ] isneededtoreproducetheobservednucleosynthesisyields. 57−58Niyieldsdependsensitivelyontheelectron R fractionand onthe locationofthe masscutwith respectto the shellstructureofthe progenitor. We identify S a progenitoranda suitable setof parametersthatfitthe explosionpropertiesofSN 1987Aassuming0.1M ⊙ . h of fallback. We predict a neutron star with a gravitational mass of 1.50 M⊙. We find correlations between p explosion properties and the compactness of the progenitor model in the explored mass range. However, a - morecompleteanalysiswillrequireexploringofalargersetofprogenitors. o Subjectheadings:hydrodynamics,nucleosynthesis,stars: neutron,supernovae: general,supernovae: individ- r t ual(SN1987A) s a [ 1. INTRODUCTION is required to address the following fundamental questions: 2 What are the conditions for explosive nucleosynthesis as a Core-collapse supernovae(CCSN) occur at the end of the v life of massive stars (M & 8 − 10M ). In these violent functionofprogenitorproperties?Whatistheconnectionbe- 5 ⊙ tweentheprogenitormodelandthecompactremnant? How events,thecoreofthestar gravitationallycollapsesandtrig- 4 do these aspects relate to the explosion dynamics and ener- gers a shock wave, leading to the supernovaexplosion. De- 8 getics? Thelackofreadilycalculablesupernovasimulations spite many decades of theoretical and numerical modeling, 2 withself-consistentexplosionsisaproblemformanyrelated the detailed explosion mechanism is not yet fully under- 0 fields, in particular for predicting nucleosynthetic yields of stood. Simulationsin sphericalsymmetryincludingdetailed 1. neutrino transportand generalrelativity fail to explode self- supernovae. Aswe willcontinuetoarguebelow,spherically 0 consistently,exceptforthelowest-masscore-collapseprogen- symmetric models of the explosion of massive stars are still 5 itors (Fischeretal. 2010; Hu¨depohletal. 2010). There are apragmaticmethodtostudylargenumbersofstellarprogen- 1 many ongoing efforts using multi-dimensional fluid dynam- itors, from the onset of the explosion up to several seconds : aftercorebounce. v ics,magneticfields,androtationtoaddressvariousremaining In the past, supernova nucleosynthesis predictions re- i open questions in core-collapse supernova theory (see, e.g., X lied on artificially triggered explosions, either using a pis- Janka (2012); Jankaetal. (2012); Burrows (2013)). Among ton (e.g. Woosley&Weaver 1995; Limongi&Chieffi 2006; r thosearealsotechnicalissues,forexampletheconsequences a Chieffi&Limongi 2013) or a thermal energy bomb (e.g., ofneutrinotransportapproximations,theconvergenceofsim- Thielemannetal. 1996; Umeda&Nomoto 2008). For the ulationresults, orthedependenceofthesimulationoutcome piston model, the motion of a mass point is specified along onthedimensionalityofthemodel.Awarenessofthisdepen- a ballistic trajectory. For the thermal energy bomb, explo- dence is especially important because not all investigations sionsaretriggeredbyaddingthermalenergytoamasszone. canbeperformedinacomputationallyveryexpensivethree- Inbothcases,additionalenergyisaddedtothesystemtotrig- dimensional model. While sophisticated multi-dimensional ger an explosion. In addition, the mass cut (bifurcation be- models are needed for an accurate investigation of the ex- tween the proto-neutron star (PNS) and the ejecta) and the plosionmechanism,theyarecurrentlytooexpensiveforsys- explosion energy are free parameters which have to be con- tematic studies that have to be based on a large number of progenitor models. But such a large number of simulations strained from the mass of the 56Ni ejecta. While these ap- proaches are appropriate for the outer layers, where the nu- carla [email protected] cleosynthesis mainly depends on the strength of the shock 1Institut fu¨rKernphysik, Technische Universita¨t Darmstadt, D-64289 wave, they are clearly incorrect for the innermost layers. Darmstadt,Germany There,theconditionsandthenucleosynthesisaredirectlyre- 2Departementfu¨rPhysik,Universita¨tBasel,CH-4056Basel,Switzer- lated to the physics of collapse and bounce, and to the de- land 3DepartmentofPhysics,NorthCarolinaStateUniversity,RaleighNC tails of the explosion mechanism. Besides the piston and 27695 thermal bomb methods, another widely used way to artifi- 2 Peregoetal. ciallytriggerexplosionsistheso-called“neutrinolight-bulb”. placetheinnermost1.1M ofthePNSbyaninnerboundary. ⊙ In this method, the PNS is excised and replaced with an The evolution of the neutrino boundary luminosity is based inner boundary condition which contains an analytical pre- onan analyticcoolingmodelof the PNS, whichdependson scription for the neutrino luminosities. The neutrino trans- a set of free parameters. These parametersare set by fitting port is replaced by neutrino absorption and emission terms observationalpropertiesof SN 1987Aforprogenitormasses in optically thin conditions. Suitable choices of the neu- around20M (seealsoErtletal.(2015)). ⊙ trinoluminositiesandenergiescantriggerneutrino-drivenex- Artificialsupernovaexplosionshavebeenobtainedbyother plosions(e.g.,Burrows&Goshy1993;Yamasaki&Yamada authors using a grey leakage scheme that includes neutrino 2005;Iwakamietal.2008,2009;Yamamotoetal.2013).The heatingviaaparametrizedcharged-currentabsorptionscheme light-bulb method has also been used to investigate mod- (O’Connor&Ott2010)insphericallysymmetricsimulations els with respect to their dimensionality. The transition from (O’Connor&Ott2011). spherical symmetry (1D) to axisymmetry (2D) delivers the Inthispaper,wereportonanewapproach,PUSH, forar- newdegreeoffreedomtobringcoldaccretingmatterdownto tificially triggering explosions of massive stars in spherical theneutrinosphereswhilematterinotherdirectionscandwell symmetry. In PUSH, we deposit a fraction of the luminos- longerinthegainregionandefficientlybeheatedbyneutrinos ity of the heavy flavor neutrinos emitted by the PNS in the (Herantetal. 1994). The standing accretion shock instabil- gain region to increase the neutrino heating efficiency. We ity (SASI, e.g., Blondinetal. 2003; Blondin&Mezzacappa ensure an accurate treatment of the electron fraction of the 2006; Schecketal. 2008; Iwakamietal. 2009; Fernańdez ejectathroughaspectralneutrinotransportschemeforν and e 2010;Guilet&Foglizzo2012)stronglycontributestothisef- ν¯ andadetailedevolutionofthePNS.Wecalibrateournew e fect in 2D light-bulb models and leads to strong polar os- methodbycomparingtheexplosionenergiesandnucleosyn- cillationsofexpansionduringtheunfoldingoftheexplosion thesisyieldsofdifferentprogenitorstarswithobservationsof (Murphy&Burrows2008).Itwasfirstexpectedthatthetrend SN1987A.Thismethodprovidesaframeworktostudymany towardasmallercriticalluminosityforsuccessfulexplosions importantaspectsofcore-collapsesupernovaeforlargesetsof willcontinueasonegoesfrom2Dtothree-dimensional(3D) progenitors: explosive supernova nucleosynthesis, neutron- models (Nordhausetal. 2010; Handyetal. 2014), but other star remnant masses, explosion energies, and other aspects studiespointedtowardthecontrary(Hankeetal.2012;Couch wherefullmulti-dimensionalsimulationsarestilltooexpen- 2013). One has to keep in mind, that a light bulb approach siveandtraditionalpistonorthermalbombmodelsdonotcap- mightnotincludethefullcouplingbetweentheaccretionrate ture allthe relevantphysics. With PUSH we can investigate andtheneutrinoluminosity. However,recentmodelsthatde- general tendencies and perform systematic parameter vari- rivetheneutrinoluminosityfromaconsistentevolutionofthe ations, providing complementary information to “ab-initio” neutronstarsupporttheresultthat2Dmodelsshowfasterex- multi-dimensionalsimulations. plosions than 3D models (Mu¨lleretal. 2012a; Bruennetal. The article is organized as follows: Section 2 describes 2013; Dolenceetal. 2013; Takiwakietal. 2014). Most im- oursimulationframework,thestellar progenitormodels, the portant for this work is a finding that is consistent with all newmethodPUSH,andourpost-processinganalysis. InSec- aboveinvestigations:In3Dthereisnopreferredaxis.The3D tion3,wepresentadetailedexplorationofthePUSHmethod degrees of freedom lead to a more efficient cascade of fluid and the results of fitting it to observablesof SN 1987A. We instabilitiestosmallerscales. Inspiteofvividfluidinstabil- alsoanalyzeaspectsofthesupernovadynamicsandprogeni- ities, the 3D models show in their overall evolution a more tordependency.InSection4,wediscussfurtherimplications pronouncedsphericity than the 2D models. Hence their av- ofourresultsandalsocomparewithotherworksfromthelit- erageconditionsresemblemorecloselytheshockexpansion erature. A summary is given and conclusions are drawn in thatwouldbeobtainedbyanexploding1Dmodel. Section5. Ina 1DmodelwithdetailedBoltzmannneutrinotransport twoothermethodstotriggerexplosionsusingneutrinoshave 2. METHODANDINPUT been used (Fro¨hlichetal. 2006; Fischeretal. 2010). These 2.1. Hydrodynamicsandneutrinotransport “absorption methods” aim at increasing the neutrino energy deposition in the heating region by mimicking the expected Wemakeuseofthegeneralrelativistichydrodynamicscode neteffectsofmulti-dimensionalsimulations. Inonecase,the AGILEinsphericalsymmetry(Liebendo¨rferetal.2001). For neutral-currentscatteringopacitiesonfreenucleonsareartifi- the stellar collapse, we apply the deleptonization scheme of ciallydecreasedtovaluesbetween0.1and0.7timestheorigi- Liebendo¨rfer (2005). For the neutrino transport, we em- nalvalues.Thisleadstoincreaseddiffusiveneutrinofluxesin ploy the Isotropic Diffusion Source Approximation (IDSA) regionsofveryhighdensity.Thenetresultsareafasterdelep- for the electron neutrinos ν and electron anti-neutrinos ν e e tonizationofthePNSandhigherneutrinoluminositiesinthe (Liebendo¨rferetal. 2009), and an Advanced Spectral Leak- heatingregion. Intheothercase,explosionsareenforcedby agescheme(ASL)fortheheavy-leptonflavorneutrinosν = x multiplyingthereactionratesforneutrinoabsorptiononfree ν ,ν ,ν ,ν (Peregoetal. 2014). We discretize the neutrino µ µ τ τ nucleonsby a constantfactor. To preserve detailed balance, energyusing20geometricallyincreasingenergybins, inthe theemissionratesalsohavetobemultipliedbythesamefac- range 3MeV ≤ E ≤ 300MeV. The neutrino reactions ν tor. Thisreducesthetimescaleforneutrinoheatingandagain included in the IDSA and ASL scheme are summarized in resultsinamoreefficientenergydepositionintheheatingre- Table 1. They represent the minimal set of the most rele- gion. However, the energy associated with these explosions vant weak processes in the post-bounce phase, particularly werealwaysweak. up to the onset of an explosion. Note that electron cap- Recently, Uglianoetal. (2012) have presented a more so- tures on heavy nuclei and neutrino scattering on electrons, phisticatedlight-bulbmethodtoexplodesphericallysymmet- which are relevant in the collapse phase (see, for exam- ricmodelsusingneutrinoenergydepositioninpost-shocklay- ple, Mezzacappa&Bruenn 1993c,b), are not included ex- ers. Theyuseanapproximate,greyneutrinotransportandre- plicitly in the form of reaction rates, but as part of the pa- Pushing1DCCSNetoexplosions:modelandSN1987A 3 rameterizeddeleptonizationscheme. IntheASLscheme,we dripline. TheHS(DD2)wasfirstintroducedinFischeretal. omit nucleon-nucleon bremsstrahlung, N + N ↔ N + N + (2014), where its characteristic properties were discussed ν + ν¯ , where N denotes any nucleon (see, for example, and general EOS effects in core-collapse supernova simu- x x Hannestad&Raffelt 1998; Bartletal. 2014). We have ob- lations were investigated. Fischeretal. (2014) showed that servedthatitsinclusionwouldoverestimateµandτneutrino the HS(DD2)EOSgivesa better agreementwith constraints luminositiesduringthe PNS coolingphase, due to the miss- fromnuclearexperimentsandastrophysicalobservationsthan ingneutrinothermalizationprovidedbyinelasticscatteringon the commonly used EOSs of Lattimer&Swesty (1991) and electronsandpositronsatthePNSsurface.However,wehave Shenetal. (1998). Furthermore, additional degrees of free- also tested thatthe omissionofthisprocessdoesnotsignifi- dom, such as various light nuclei and a statistical ensemble cantlychangetheµandτneutrinoluminositiespredictedby of heavy nuclei, are taken into account. The nucleonmean- theASLschemebeforetheexplosionsetsin. Thus,neglect- field potentials, which are used in the charged-currentrates, ingN-Nbremsstrahlungisonlyrelevantforthecoolingphase, havebeencalculatedconsistently(Hempel2014). Themax- whereitimprovestheoverallbehaviorwhencomparedtosim- imum mass of a cold neutron star for the HS(DD2) EOS is ulationsobtainedwithdetailedBoltzmannneutrinotransport 2.42M (Fischeretal.2014), whichiswellabovethelimits ⊙ (Fischeretal.2010,e.g.). fromDemorestetal.(2010)andAntoniadisetal.(2013). The equation of state (EOS) of The EOS employed in our simulations includes an exten- Hempel&Schaffner-Bielich (2010) (HS) that we are siontonon-NSEconditions. Inthenon-NSEregimethenu- usingincludesvariouslightnuclei,suchasalphas,deuterons clear composition is described by 25 representative nuclei or tritons (details below). However, the inclusion of all from neutrons and protons to iron-group nuclei. The cho- neutrino reactions for this detailed nuclear composition sen nuclei are the alpha-nuclei 4He, 12C, 16O, 20Ne, 24Mg, would be beyond the standard approach implemented in 28Si, 32S, 36Ar, 40Ca, 44Ti, 48Cr, 52Fe, 56Ni, complemented current supernova simulations, where only scattering on by 14N and the following asymmetric isotopes: 3He, 36S, alpha particles is typically included. To not completely 50Ti, 54Fe, 56Fe, 58Fe, 60Fe, 62Fe, and 62Ni. With these nu- neglect the contributions of the other light nuclei, we have clei it is possible to achieve a mapping of the abundances addedtheir mass fractionsto the unboundnucleons. This is from the progenitorcalculations onto our simulations which motivatedbytheirveryweakbindingenergiesand,therefore, is consistent with the provided electron fraction, i.e., man- bytheideathattheybehavesimilarlyastheunboundnucleon tainingchargeneutrality.AllthenuclearmassesM aretaken component. i from Audietal. (2003). To advect the nuclear composition In all our models we use 180 radial zones which include insidetheadaptivegrid,weimplementtheConsistentMulti- the progenitorstar up to the helium shell. This corresponds fluid Advection (CMA) method by Plewa&Mu¨ller (1999). toaradiusofR ≈ (1.3−1.5)×1010 cm. Withthissetup,we Forgivenabundances,thenon-NSEEOSiscalculatedbased modelthecollapse, bounce,andonsetoftheexplosion. The on the same underlying description used in the NSE regime grid of AGILE is adaptive, with more resolution where the (Hempel&Schaffner-Bielich 2010), but with the following gradientsofthethermodynamicvariablesaresteeper.Thus,in modifications:excitedstatesofnucleiareneglected,excluded thepost-bounceandexplosionphases,thesurfaceofthePNS volume effects are not taken into account, and the nucle- andtheshockarethebetterresolvedregions.Thesimulations onsaretreatedasnon-interactingMaxwell-Boltzmanngases. arerunforatotaltimeof5s, correspondingto& 4.6safter Suchaconsistentdescriptionofthenon-NSEandNSEphases corebounce. Atthistime, theshockhasnotyetreachedthe prevents spurious effects at the transitions between the two externaledgeofourcomputationaldomain. regimes. OutsideofNSE,anapproximateα-networkisusedtofol- lowthechangesincomposition.ExplosiveHelium-,Carbon-, Table1 Relevantneutrinoreactions. Neon-,andOxygen-burningarecurrentlyimplementedinthe simulation.Notethatthethermalenergygenerationbythenu- Reactions Treatment Reference e−+p↔n+νe IDSA a clearreactionsisfullyincorporatedviathedetailednon-NSE e++n↔p+νe IDSA a treatment. We donothavetocalculateexplicitlyanyenergy N+ν↔N+ν IDSA&ASL a liberation,butjustthechangesintheabundances.Withinour (A,Z)+ν↔(A,Z)+ν IDSA&ASL a relativistictreatmentoftheEOS(appliedbothinthenon-NSE e−+e+↔νµ,τ+νµ,τ ASL a,b andNSEregime)energyconservationmeansthatthespecific Note. — Nucleons are denoted by N. The nucleon charged current internalenergye isnotchangedbynuclearreactions. This rates are based on Bruenn (1985), but effects of mean-field interactions int (Reddyetal.1998;Robertsetal.2012;Mart´ınez-Pinedoetal.2012;Hempel is due to fact that eint includes the specific rest mass energy 2014)aretakenintoaccount. e , wheree isgivenbythe sum overthe massesof all mass mass References.— (a)Bruenn(1985);(b)Mezzacappa&Bruenn(1993a). nucleiweightedwiththeiryieldY = X/A, i i i e = YM . (1) mass i i 2.2. Equationofstateandnuclearreactions Xi For the high-density plasma in nuclear statistical equi- However,ifwedefinethespecificthermalenergyethas librium (NSE) the tabulated microphysical EOS HS(DD2) e =e −e , (2) is used. This supernova EOS is based on the model th int mass of Hempel&Schaffner-Bielich (2010). It uses the DD2 thenuclearreactionswilldecreasetherestmassenergy(i.e., parametrization for the nucleon interactions (Typeletal. increase the binding) and consequently increase the thermal 2010), the nuclear masses from Audietal. (2003), and the energy.Thistreatmentofthenon-NSEEOSisconsistentwith Finite RangeDroplet Model (Mo¨lleretal. 1995a). 8140nu- theconventionusedinallhigh-densityNSEEOSs. clei are included in total, up to Z = 136 and to the neutron Due to limitationsof our approximateα-network, and be- 4 Peregoetal. causewedonotincludeanyquasi-statisticalequilibriumde- 1010 HCmodels scription, we apply some parameterizedburningfor temper- 108 LCmodels atures between 0.3 and 0.4 MeV. A temperature dependent burningtimescale is introduced, which gradually transforms 106 the initial non-NSE composition towards NSE. For temper- 3] 104 1010 aretuacrehsesofa 0co.4mMpoesVitioanndcloabseovteo,NthSeEnaonnd-NitSsEtheprhmasoedyanlwamayics -gcm 102 108 [ 1 pHrSo(pDerDti2e)sEbOecSo.mEveenvetrhyousgimhitlhaertwtootphheasNesSEarepbhaasseedoofntthhee nsity 10-2 106 smaamine,idnupeuttophthyesilcism,sitmedalsle,tbuotfunnuacvleoiidcaobnlseiddeiffreedreinncneosnc-aNnSrEe-. De 10-4 1040 1 2 3 To assure a smoothtransition forall conditions, we have in- 10-6 troduced a transition region as an additional means of ther- 10-8 modynamicstability. We have chosen a parameterizationin 10-10 termsoftemperatureandimplementalineartransitioninthe 0 5 10 15 20 temperatureintervalfrom0.40MeVto0.44MeV.Wecheck EnclosedMass[M ] that the basic thermodynamicstability conditionds/dT > 0 Figure1. Density profiles as function of ZAMS mass for the progenitor isalwaysfulfilled. modelsincludedinthisstudy(18.0M⊙to21.0M⊙). HCmodelsareshown inred,LCmodelsareshowninblue. Theverticallineintheinsetislocated 2.3. Initialmodels at1.75M⊙andindicatesthatmassatwhichthecompactnessparameterξ1.75 isdetermined(seeEquation3). For this study, we use solar-metallicity, non-rotating stellar models from the stellar evolution code KEPLER 1.2 (Woosleyetal. 2002). Our set includes 16 pre-explosion onsetofcollapse modelswithzero-agemainsequence(ZAMS)massbetween bounce 1.0 18.0M and21.0M inincrementsof0.2M . Thesemodels ⊙ ⊙ ⊙ havebeenselectedto haveZAMSmassaround20M , sim- ⊙ ilartotheprogenitorofSN1987A(e.g.,Podsiadlowskietal. 0.8 2007).WelabelthemodelsbytheirZAMSmass.InFigure1, [] thedensityprofilesofthe progenitormodelsareshown. For 75 0.6 each of them the compactness parameter ξ is defined fol- 1. M lowing O’Connor&Ott (2011) by the ratio of a given mass 0.4 MandtheradiusR(M)whichenclosesthismass: ξ ≡ M/M⊙ . (3) 0.2 M R(M)/1000km 0.0 Typically, either ξ1.75 or ξ2.5 are used. The compactnesscan 17.5 18.0 18.5 19.0 19.5 20.0 20.5 21.0 21.5 be computed at the onset of collapse or at bounce, as sug- M [M ] ZAMS gested by O’Connor&Ott (2011). For our progenitors, the difference in the compactness parameter between these two Figure2. Compactness ξ1.75 as function of ZAMS mass for our pre- explosionmodelsattheonsetofcollapse(greencrosses)andatbounce(ma- momentsisnotsignificantforourdiscussions.Thus,forsim- gentapluses). plicity,inthefollowingwewilluseξ computedattheonset 1.75 of the collapse. The progenitor models considered here fall into two distinct families of compactness: low compactness core.Theaccretionluminositydependsnotonlyontheaccre- (ξ <0.45;LCmodels)andhighcompactness(ξ >0.45; tion rate but also on the evolutionof the mass and radius of 1.75 1.75 HCmodels),seeTable2. Figure2showsthecompactnessas the PNS, which is treated accuratelyand self-consistentlyin functionofZAMSmassfortheprogenitorsofthisstudy.The ourmodels. non-monotonousbehavior is a result of the evolution before In order to trigger explosions in the otherwise non- collapse. The mass rangebetween 19 and21 M is particu- explodingsphericallysymmetricsimulations, we relyonthe ⊙ larly prone to variations of the compactness. For a detailed delayed neutrino-driven mechanism, which was first pro- discussionofthebehaviorofthecompactnessasfunctionof posed by Bethe&Wilson (1985). Despite the lack of con- ZAMSmassseeSukhbold&Woosley(2014). sensus and convergence of numerical results between different groups, recent multi-dimensional simulations of CC- 2.4. ThePUSHmethod SNehaveshownthatconvection,turbulenceandSASIinthe shockedlayersincrease theefficiencyat whichν andν¯ are 2.4.1. Rationale e e absorbed inside the gain region, compared with spherically ThegoalofPUSHistoprovideacomputationallyefficient symmetricmodels(see,forexample,Janka&Mueller1996; frameworktoexplodemassivestarsinsphericalsymmetryto Nordhausetal.2010;Hankeetal.2012,2013;Dolenceetal. study multiple aspects of core-collapse supernovae. The us- 2013;Couch2013;Melsonetal.2015). Thiseffect,together age of a spectral transport scheme to compute the ν and ν¯ with the simultaneous increase in time that a fluid particle e e luminositiesprovidesa more accurateevolution of Y of the spendsinsidethegainregion(e.g.,Murphy&Burrows2008; e innermostejecta, whichisa crucialaspectfornucleosynthe- Handyetal. 2014), provides more favorable conditions for sis. Theneutrinoluminositiesincludetheaccretioncontribu- the development of an explosion. Moreover, according to tion,aswellastheluminositycomingfromPNSmantleand multi-dimensionalexplosionmodels,theshockrevivalisfol- Pushing1DCCSNetoexplosions:modelandSN1987A 5 Table2 Progenitorproperties MZAMS ξ1.75 ξ2.5 Mprog MFe MCO MHe Menv (M⊙) atcollapse atbounce atcollapse atbounce (M⊙) (M⊙) (M⊙) (M⊙) (M⊙) 18.2 0.37 0.380 0.173 0.173 14.58 1.399 4.174 5.395 9.186 18.6 0.365 0.375 0.170 0.170 14.85 1.407 4.317 5.540 9.313 18.8 0.357 0.366 0.166 0.166 15.05 1.399 4.390 5.613 9.435 19.6 0.282 0.288 0.118 0.117 13.37 1.461 4.959 6.243 7.125 19.8 0.334 0.341 0.135 0.135 14.54 1.438 4.867 6.112 8.428 20.0 0.283 0.287 0.125 0.125 14.73 1.456 4.960 6.215 8.517 20.2 0.238 0.241 0.104 0.104 14.47 1.458 5.069 6.342 8.125 18.0 0.463 0.485 0.199 0.199 14.50 1.384 4.104 5.314 9.187 18.4 0.634 0.741 0.185 0.185 14.82 1.490 4.238 5.459 9.366 19.0 0.607 0.715 0.191 0.191 15.03 1.580 4.461 5.693 9.341 19.2 0.633 0.737 0.191 0.192 15.08 1.481 4.545 5.760 9.325 19.4 0.501 0.535 0.185 0.185 15.22 1.367 4.626 5.860 9.365 20.4 0.532 0.594 0.192 0.192 14.81 1.500 5.106 6.376 8.433 20.6 0.742 0.95 0.278 0.279 14.03 1.540 5.260 6.579 7.450 20.8 0.726 0.904 0.271 0.272 14.34 1.528 5.296 6.609 7.735 21.0 0.654 0.764 0.211 0.212 13.00 1.454 5.571 6.969 6.026 Note.—ZAMSmass,compactnessξ1.75andξ2.5attheonsetofcollapseandatbounce,totalprogenitormassatcollapse(Mprog),massoftheironcore(MFe), carbon-oxygencore(MCO),andheliumcore(MHe),andmassofthehydrogen-richenvelope(Menv)atcollapse,foralltheprogenitormodelsincludedinthis study.Thetoppartofthetableincludesthelow-compactnessprogenitors(LC;ξ1.75<0.4atcollapse),thebottompartincludesthehigh-compactnessprogenitors (HC;ξ1.75>0.45atcollapse). lowed by a phase where continued accretion and shock ex- andthecoolingoftheformingcompactobject. Asshownby pansioncoexistoveratimescale of& 1s(e.g.,Schecketal. O’Connor&Ott (2013) in broad progenitor studies, during 2006;Marek&Janka2009;Bruennetal.2014;Melsonetal. theaccretionphasethatprecedestheshockrevival,theprop- 2015). During this phase, matter accreted through low- ertiesoftheν spectralfluxescorrelatesignificantlywiththe x entropy downflows onto the PNS continues to power an ac- propertiesofν ’sandν¯ ’s. Unliketheelectron(anti-)neutrino e e cretionluminosity.There-ejectionofafractionofthismatter luminosities, that in spherically symmetric models decrease by neutrino heating accelerates the shock and increases the suddenlyoncetheshockhasbeenrevived,ν luminositiesare x explosionenergy. Thelengthofthisphase,theexactamount onlymarginallyaffectedbythedevelopmentofanexplosion. ofinjectedenergy,anditsdepositionratearestilluncertain. ThisallowsPUSHtocontinueinjectingenergyinsidetheex- Inspired by the increase of the net neutrino heating that a panding shock for a few hundreds of milliseconds after the fluidelementexperiencesduetotheabovementionedmulti- explosionhassetin. Moreover,sincethisenergyinjectionis dimensionaleffects,PUSHprovidesamoreefficientneutrino providedbytheν fluxes,itchangessignificantlybetweendif- x energydeposition inside the gain region in spherically sym- ferentprogenitorsandcorrelateswiththeν andν¯ accretion e e metricmodels. However,unlikeothermethodsthatuseelec- luminosities(atleast,duringtheaccretionphase). tron flavor neutrinos to trigger artificial 1D explosions (see Section 1), in PUSH we deposit a fraction of the luminos- 2.4.2. Implementation ity of the heavy flavor neutrinos (ν ’s) behind the shock to x Theadditionalenergydeposition,that representsthe main ultimatelyprovidesuccessfulexplosionconditions. Thisad- feature of PUSH, is achieved by introducinga local heating ditionalenergydepositioniscalibratedbycomparingtheex- term,Q+ (t,R)(energyperunitmassandtime),givenby plosionenergiesandnucleosynthesisyieldsobtainedfromour push progenitorsample with observationsof SN 1987A. This en- ∞ sures that our artificially increased heating efficiency has an Q+ (t,r)=4G(t) q+ (r,E)dE, (4) push Z push empiricalfoundation. Thus, we can make predictionsin the 0 senseofaneffectivemodel. where Despitethe factthatν ’scontributeonlymarginallyto the energy deposition insidex the gain region in self-consistent q+ (r,E)≡σ 1 E 2 1 dLνx F(r,E), (5) models (see, for example, Bethe&Wilson 1985) and that push 0 4m m c2! 4πr2 dE ! b e they only show a weak dependence on the temporal varia- tionoftheaccretionrate(see,forexample,Liebendo¨rferetal. with 2 2004), their usage presents a number of advantages for our 4G2 m c2 purposes. They represent one of the largest energy reser- σ = F e ≈1.759×10−44cm2 (6) voirs available, but they do not directly change the electron 0 π((cid:16)~c)4 (cid:17) fractionY (unlikeelectronflavor neutrinos). Thisallowsus e beingthetypicalneutrinocross-section,m ≈1.674×10−24g to trigger an explosion in 1D simulations without modify- b an average baryon mass, and (dL /dE)/(4πr2) the spectral ingνe andν¯e luminositiesnorchangingchargedcurrentreac- energyfluxforanysingleν neutrνinxospecieswithenergyE. tions. The ν luminositiesare calculated consistently within x x Notethatallfourheavyneutrinoflavorsaretreatedidentically ourmodel. Theyincludedynamicalfeedbackfromtheaccre- bytheASL scheme,andcontributeequallyto Q+ (seethe tion history, progenitor properties of each individual model, push factor4appearinginEquation(4)). 6 Peregoetal. ThetermF(r,E)inEquation(5)definesthespatiallocation µandτneutrinoluminosityasenergyreservoirsuggest whereQ+ (t,r)isactive: k &1. push push • t sets the time at which PUSH starts to act. We re- 0 if ds/dr>0 or e˙ <0 on F(r,E)= νe,νe , late t to the time when deviations from spherically ( exp(−τνe(r,E)) otherwise symmoentricbehaviorappearinmulti-dimensionalmod- (7) els. Matter convection in the gain region sets in once where τ denotes the (radial) optical depth of the electron neutrinoνse, s is the matter entropy and e˙ the net specific theadvectiontimescaleτadvandtheconvectivegrowth energyrateduetoelectronneutrinosandνae,nν¯eti-neutrinos. The time scale τconv satisfy τadv/τconv & 3 (Foglizzoetal. 2006). For all the modelswe have explored,thishap- two criteria aboveare a crucialingredientin our description pensaroundt = 0.06−0.08s. Intheaboveestimates, of triggering CCSN explosions: PUSH is only active where τ = M˙ /M , where M˙ istheaccretionrate electron-neutrinosareheating(e˙ >0)andwhereneutrino- adv shock gain shock drivenconvectioncanoccur(ds/νed,νre<0). at the shock and Mgain the mass in the gain region, andτ = f−1 , where f istheBrunt-Vaïsa¨lafre- ThetermG(t)inEquation(4)determinesthetemporalbe- conv B−V B−V haviourofQ+ (t,r). Itsexpressionreads quency. Consideringthatτconv ∼ 4−5ms, we expect push t ∼ 0.08 − 0.10s, in agreement with recent multi- on dimensional simulations (see, for example, the 1D- 0 t≤t on 2D comparison of the shock position in Bruennetal.  (t−ton)/trise ton <t≤ton+trise 2013). G(t)=kpush× 1(toff +trise−t)/trise ttoonff+<ttris≤e <tofft+≤ttroisffe , • trise defines the time scale over which G(t) in-  0 t>toff +trise (8) tlcharreegaestsiemtsemfrusoclmtai-ledziemtrhoeantstoicohnkaaprluaschpt.eerritzuWersbeatthciooennsgnrebocwetttwthrieseeonfwttihhthee anditissketchedinFigure3. Notethatthroughoutthearticle shock radius (R ) and the gain radius (R ) (e.g. we always measure the time relative to bounce, if not noted shock gain Janka&Mueller1996). Foglizzoetal.(2006)showed otherwise. that convection in the gain region can be significantly The cumulative energy deposited by PUSH, E , can be push stabilized by advection, especially if τ /τ only calculatedfromtheenergydepositionratedE /dtas adv conv push marginally exceeds 3, and that the growth rate of the Epush(t)=Z t dEdptush! dt′ =Z t Z Q+pushρdVdt′. Ofanstethstegortohweirnhgamndo,daeilsowdiemrilnimishitedto.Ttrihseusi,strriesepr≫eseτncotnevd. ton ton  Vgain  (9) bytheoverturntimescale,τoverturn,definedas whereVgainisthevolumeofthegainregion.Boththesequan- τ ∼ π(Rshock−Rgain), (11) titieshavetobedistinguishedfromthecorrespondingenergy overturn hvi gain andenergyrateobtainedbyIDSA: wherehvi istheaveragefluidvelocityinsidethegain gain t dE t region. In our simulations, we have found τ ≈ E (t)= idsa dt′ = e˙ ρdV dt. (10) overturn idsa Z dt ! Z Z νe,ν¯e  0.05s around and after ton. In case of a contract- 0 0  Vgain  i0n.2g−sh0o.c3ks,aSfAteSrIbsoaurneceal(sHoaenxkpeeectteadl.t2o0d1e3v)e.lHopenacreo,uwnde assume0.05s.t .(0.30s−t ). rise on • t setsthetimeafterwhichPUSHstartstobeswitched off off. We expect neutrino driven explosions to develop fort . 1s duetothe fastdecreaseofthe luminosities during the first seconds after core bounce. Hence, we fix t = 1 s. PUSH is not switched off suddenly at off the onsetof the explosion, but ratherstarts decreasing naturally even before 1 s after core bounce due to the decreasingneutrinoluminositiesandduetotherarefac- tionofthegainregionabovethePNS.Thesubsequent injection of energy by neutrinos in the accelerating Figure3. ThefunctionG(t)determinesthetemporalbehavioroftheheating shockisqualitativelyconsistentwithmulti-dimensional duetoPUSH.Thequantitytonisrobustlysetbymulti-dimensionalmodels. simulations,whereaccretionandexplosioncancoexist Weconsideravalueof80msinourcalculationsandavalueof120msfor duringtheearlystagesoftheshockexpansion.Thede- testing. triseandkpusharesetbyourcalibrationprocedure,spanningarange from50msto250ms,andfrom0(PUSHoff)to∼4, respectively. Since crease of dEpush/dt on a time scale of a few hundreds weassumethattheexplosiontakes placewithinthefirstsecondaftercore of milliseconds after the explosion has been launched bounce,weusetoff =1s. makesourresultslargelyindependentofthechoiceof t forexplosionshappeningnottooclosetoit. ThedefinitionofG(t) introducesa setof(potentially)free off parameters: Whiletonisrelativelywellconstrainedandtoff isrobustlyset, t and especially k are still undefined. We will discuss rise push • k is a global multiplication factor that controls di- theirimpactonthemodelandontheexplosionpropertiesin push rectly theamountof extraheatingprovidedbyPUSH. detail in Sections 3.1.1-3.1.3. Ultimately, we will fix them Thechoicesofσ asreferencecross-sectionandofthe usingacalibrationprocuredetailedinSection3.4. 0 Pushing1DCCSNetoexplosions:modelandSN1987A 7 2.5. Post-processinganalysis To compute the explosion energy, we assume that the to- talenergyoftheejectawithrest-massessubtractedeventually For the analysis of our results we determine several key convertsintokineticenergyoftheexpandingsupernovarem- quantitiesforeachsimulation. Thesequantitiesareobtained nant at t ≫ t . The quantity e includes the rest mass fromapost-processingapproach.Wedistinguishbetweenthe expl total contributionviae , seeEquation(2). Instead,ifwewantto explosion properties, such as the explosion time, the mass int calculatetheexplosionenergy,we havetoconsiderthe ther- cut,or the explosion energy, and the nucleosynthesis yields. mal energy e . Therefore, we define the specific explosion The former are calculated from the hydrodynamicsprofiles. th energyas The latter are obtainedfromdetailed nuclearnetworkcalcu- e =e +e +e , (14) lationsforextrapolatedtrajectories. expl th kin grav and the time- and mass-dependent explosion energy for the 2.5.1. Accretionratesandexplosionproperties fixedmassdomainbetweenm andMas 0 For the accretion process, we distinguish between the ac- m0 cretion rate at the shock front, M˙ = dM(R )/dt, and Hexpl(m0,t)=− eexpl(m,t)dm. (15) shock shock Z theaccretionrateonthePNS,M˙ =dM(R )/dt. Inthese M PNS PNS expressions, M(R) is the baryonicmass enclosed in a radius This can be interpreted as the total energy of this region in R,R istheshockradius,andR isthePNSradiusthat a non-relativistic EOS approach, where rest masses are not shock PNS satisfiestheconditionρ(R )=1011gcm−3. included. PNS We consider the explosion time t as the time when the Theactualexplosionenergy(stilltime-dependent)isgiven expl shockreaches500km,measuredwithrespecttocorebounce by (cf. Uglianoetal. (2012)). In all our models, the velocity Eexpl(t)= Hexpl(mcut(t),t). (16) ofmatterattheshockfronthasturnedpositiveatthatradius i.e.,forthematterabovethemasscutm (t). cut and the explosion has been irreversibly launched. There is To identify the mass cut, we consider the expression sug- no unique definition of t in the literature and some other expl gestedbyBruenninFischeretal.(2010): studies(cf.Janka&Mueller(1996);Handyetal.(2014))use differentdefinitions,e.g.,thetimewhentheexplosionenergy m (t)=m max(H (m,t)) , (17) cut expl increases above 1048 erg. However, we do not expect that (cid:16) (cid:17) thedifferentdefinitionsgivequalitativelydifferentexplosion wherethemaximumisevaluatedoutsidethehomologouscore times. (m & 0.6M⊙),whichhaslargepositivevaluesofthespecific For the following discussion, we will use the total energy explosion energy eexpl once the PNS has formed due to the ofthematterbetweenagivenmassshellm uptothestellar high compression. In the outer stellar envelope, before the 0 surface: passage of the shock wave, eexpl is dominated by the negative gravitationalcontribution. However, it is positive in the m0 neutrino-heated region and in the shocked region above it. E (m ,t)=− e (m,t)dm. (12) total 0 ZM total Hence,theabovedefinitionofmcutlocatesessentiallythetran- sitionfromgravitationallyunboundtoboundlayers.Thefinal Mistheenclosedbaryonicmassatthesurfaceofthestarand masscutisobtainedfort=t . m0 isabaryonicmasscoordinate(0 ≤ m0 ≤ M). etotal isthe Ourfinalsimulationtimetfinal&4.6sisalwaysmuchlarger final specifictotalenergy,givenby thanthe explosiontime and, as we will show later, it allows E (t) to saturate. Thus, we consider E (t = t ) as the e =e +e +e , (13) expl expl final total int kin grav ultimateexplosionenergyofourmodels. Inthefollowing,if i.e.,thesumofthe(relativistic)internal,kinetic,andgravita- weuseEexplwithoutthetimeasargument,wemeanthisfinal tionalspecificenergies. Forallthesequantitieswemakeuse explosionenergy. ofthegeneral-relativisticexpressionsinthelaboratoryframe 2.5.2. Nucleosynthesisyields (Fischeretal. 2010). The integral in Equation (12) includes both the portion of the star evolved in the hydrodynamical Topredictthecompositionoftheejecta,weperformnucle- simulationandtheouterlayers,whichareconsideredassta- osynthesis calculations using the full nuclear network Win- tionaryprofilesfromtheprogenitorstructure. net (Winteleretal. 2012). We include isotopes up to 211Eu Theexplosionenergyemergesfromdifferentphysicalcon- covering the neutron-deficient as well as the neutron-rich tributions (see, for example, the appendix of Schecketal. side of the valley of β-stability. The reaction rates are the (2006) and the discussion in Uglianoetal. (2012)). In our same as in Winteleretal. (2012). They are based on ex- model,wearetakingintoaccount: (i)thetotalenergyofthe perimentally known rates where available and predictions neutrino-heatedmatter that causes the shock revival; (ii) the otherwise. The n-, p-, and alpha-captures are taken from nuclearenergyreleasedbytherecombinationofnucleonsand Rauscher&Thielemann (2000), who used known nuclear alphaparticlesintoheavynucleiatthetransitiontonon-NSE masses where available and the Finite RangeDroplet Model conditions;(iii)thetotalenergyassociatedwiththeneutrino- (Mo¨lleretal. 1995b) for unstable nuclei far from stability. drivenwinddevelopingaftertheexplosionuptotheendofthe Theβ-decayratesarefromthenucleardatabaseNuDat2§. simulation; (iv)the energyreleasedbythe explosivenuclear We divide the ejecta into different mass elements of burningintheshock-heatedejecta;and(v)thetotal(negative) 10−3 M⊙ each and follow the trajectory of each individual energy of the outer stellar layers (also called the “overbur- mass element. As we are mainly interested in the amounts den”).Wearepresentlynottakingintoaccoutthevariationof of 56Ni, 57Ni, 58Ni, and 44Ti, we only consider the 340 in- theejectaenergyduetotheappearanceoflate-timefallback. nermostmasselementsabovethemasscut,correspondingto Thisisjustifiedaslongasthefallbackrepresentsonlyasmall fractionofthetotalejectedmass. §http://www.nndc.bnl.gov/nudat2/ 8 Peregoetal. a totalmass of 0.34M . The contributionof the outermass behaviorbetweenlowandhighcompactnessmodelsisseen. ⊙ elementstotheproductionofthosenucleiisnegligible. The LC models (ξ < 0.4) result in slightly weaker and 1.75 For t < t , we use the temperature and density evolu- fasterexplosions,withlessvariabilityintheexplosionenergy final tion from the hydrodynamicalsimulations as inputs for our andintheexplosiontimefordifferentvaluesofk .Evenfor push network. Foreachmasselementwestartthenucleosynthesis relativelylargevaluesofk ,theexplosionenergiesremain push post-processingwhenthetemperaturedropsbelow10GK,us- below1 Bethe (1 Bethe, abbreviatedas 1 B, is equivalentto ingtheNSEabundances(determinedbythecurrentelectron 1051 erg). On the other hand, the HC models(ξ > 0.45) 1.75 fractionYe)astheinitialcomposition.Formasselementsthat explodestrongerandlater,withalargervariationintheexplo- neverreach10GKwestartatthemomentofbounceanduse sion properties. In this case, for highenoughvaluesof k push theabundancesfromtheapproximateα-networkatthispoint (& 3.0), explosion energies of & 1 Bethe can be obtained. as the initial composition. Note that for all tracers the fur- TheHCmodelsalsoleadtoalargervariabilityoftheremnant therevolutionofYe inthenucleosynthesispost-processingis masses,eventhoughthiseffectislesspronouncedthanforthe determinedinsidetheWinnetnetwork. explosiontimeorenergy.Forthevaluesofk usedhere,we push Attheendofthesimulations,i.e.t = tfinal,thetemperature obtain(baryonic)remnantmassesfromapproximately1.4to anddensityoftheinnerzonesarestillsufficientlyhighfornu- 1.9M . ThedifferencesofLCandHCmodelswillbeinves- ⊙ clearreactionstooccur(T ≈1GKandρ≈2.5×103gcm−3). tigatedfurtherinSection3.3. Therefore,weextrapolatetheradius,densityandtemperature There are three models with 0.37 . ξ . 0.50 (cor- 1.75 uptotend =100susing: responding to ZAMS masses of 18.0 (HC), 18.2 (LC), and 19.4M (HC))whichdonotfollowthegeneraltrend.Inpar- r(t)=r +tv (18) ⊙ final final ticular,wefindthethresholdvalueofk forsuccessfulex- push t −3 plosionstobehigherforthesemodels. Acommonfeatureof ρ(t)=ρ (19) final thesethreemodelsisthattheyhavethelowestFe-coremassof t ! final allthemodelsinoursampleandthehighestcentraldensities T(t)=T[s ,ρ(t),Y (t)], (20) final e attheonsetofcollapse. where r is the radial position, v the radial velocity, ρ the Thechoiceoftrise doesnotaffecttheobservedtrendswith density, T the temperature, s the entropy per baryon, and kpush: similar behaviors are also seen for 50ms . trise . Y the electron fraction of the mass zone. The tempera- 250ms. e tureiscalculatedateachtimestepusingtheequationofstate 3.1.2. t on of Timmes&Swesty (2000). The prescription in Equations Totestthesensitivityofourmethodtotheparametert ,we (18)–(20)correspondstoafreeexpansionforthedensityand on computemodelswithk = 2.0andt = 0.15sforavery anadiabaticexpansionforthetemperature(see,forexample, push rise largeonsetparameter, t = 120ms. We comparethe corre- Korobkinetal.(2012)). on spondingresultswith the onesobtainedfort = 80ms. As on 3. FITTINGANDRESULTS expected,theshockrevivalhappensslightlylater(withatem- poralshiftof∼30ms),theexplosionenergiesaresmaller(by To test the PUSH method, we perform a large number of ∼ 0.05B) andthe remnantmasses are marginallylarger(by runswherewevarythefreeparametersandexploretheirim- 0.08M ). However,all the qualitativebehavioursdescribed pact on the explosion properties. We also analyze in detail ⊙ above,aswellasthedistinctionbetweenhighandlowcom- thebasicfeaturesofthesimulationsandoftheexplosionsin pactnessmodels,donotshowanydependenceont . Inthe connectionwiththepropertiesoftheprogenitorstar. Finally, on following,wewillalwaysassumet =80ms. we fitthe freeparametersin the PUSH methodto reproduce on observedpropertiesofSN1987Aforaprogenitorstarinthe 3.1.3. k &t push rise range18-21M . ⊙ In Sections 3.1.1 and 3.1.2, we have investigated the de- 3.1. Generaleffectsoffreeparametervariations pendencyofthemodelonthesingleparameterskpushandton. Now,weexploretheroleoft incombinationwithk .For 3.1.1. k rise push push this,weapproximatelyfixtheexplosionenergytothecanon- Theparameterwiththemostintuitiveandstrongestimpact ical value of ∼ 1 B for the high compactness models (cor- ontheexplosionisk . Itsvaluedirectlyaffectstheamount responding,forexample,tothepreviouslyexaminedmodels push of extra heating which is provided by PUSH. As expected, with k = 3.0 and t = 150ms), and investigate which push rise larger values of k (assuming all other parameters to be other combinationsof k and t result in the desired ex- push push rise fixed) result in the explosion being more energetic and oc- plosion energy. We restrict our explorations to a sub-set of curringearlier. Inaddition,afasterexplosionimpliesalower progenitor models (18.0 M , 18.6 M , 19.2 M , 19.4 M , ⊙ ⊙ ⊙ ⊙ remnant mass, as there is less time for the accretion to add 19.8 M , 20.0 M , 20.2 M and 20.6 M ) that spans the ⊙ ⊙ ⊙ ⊙ masstotheformingPNS. ξ -range of all 16 progenitors. Figure 5 summarizes the 1.75 Beyondthesegeneraltrendswithk ,thedetailedbehav- explosionenergies,explosiontimes, andremnantmassesfor push iordependsalsoonthecompactnessoftheprogenitor.Forall variouscombinationsof k and t forprogenitorsof dif- push rise 16progenitormodelsinthe18-21M ZAMSmassrange,we ferentcompactness. Therequiredconstraintcanbeobtained ⊙ have explored several PUSH models, varying k between byseveralcombinationsof parameters, whichlie ona curve push 0.0 and 4.0 in increments of 0.5 but fixing t = 80 ms and in the k -t plane. As a general result, a longer t re- on push rise rise t = 150 ms. For k 6 1, none of the models explode quiresalargerk toobtainthesameexplosionenergy.This rise push push and for k = 1.5 only the lowest compactnessmodelsex- canbeunderstoodfromthedifferentrolesofthetwoparame- push plode. Figure 4 shows the explosion energy, the explosion ters: whilek setsthemaximumefficiencyatwhichPUSH push timeandthe(baryonic)remnantmassasfunctionofthepro- depositsenergyfromthereservoirrepresentedbytheν lu- µ,τ genitor compactness for k = 1.5,2.0,3.0,4.0. A distinct minosity, t sets the time scale over which the mechanism push rise Pushing1DCCSNetoexplosions:modelandSN1987A 9 3.5 the 18.0 M progenitor model, but with different combina- k =1.5 ⊙ push tions of t and k . Runs with larger parameter values 3.0 k =2.0 rise push push require PUSH to deposit more energy (see (E + E ) at push idsa 2.5 kpush=3.0 t ≈ texpl), and the correspondingdeposition rates are shifted g] kpush=4.0 towards later times. Moreover, for increasing values of trise, er 2.0 the explosion time t becomes larger, but the interval be- expl 51 tween (t +t ) and t decreases. Despite the significant 0 1.5 on rise expl 1 variation of k between different runs, the peak values of [ push expl 1.0 d(Epush+Eidsa)/dt at the onsetof the shockrevivalthatpre- E ceedstheexplosionareverysimilarinallcases. 0.5 3.5 0.0 k =2.2,t =50ms push rise -0.5 3.0 kpush=2.5,trise=100ms 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 k =3.0,t =150ms 2.5 push rise [] 1.0 1.75 erg] 2.0 kkpush==33..58,,ttrise==220500mmss k =1.5 push rise 0.9 push 51 k =2.0 0 1.5 push 1 0.8 k =3.0 [ 0.7 kpush=4.0 expl 1.0 push E 0.6 0.5 s] [ pl 0.5 0.0 x e t 0.4 -0.5 0.3 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 [] 0.2 1.75 1.0 0.1 0.9 kpush=2.2,trise=50ms k =2.5,t =100ms 0.0 push rise 0.8 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 k =3.0,t =150ms push rise [] 0.7 k =3.5,t =200ms 1.75 push rise 2.4 kpush=1.5 s] 0.6 kpush=3.8,trise=250ms [ 2.2 kkppuusshh==23..00 texpl 00..45 k =4.0 2.0 push 0.3 ] M 0.2 1.8 [ m 0.1 e Mr 1.6 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.4 [] 1.75 1.2 2.4 k =2.2,t =50ms push rise k =2.5,t =100ms 1.0 2.2 push rise 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 kpush=3.0,trise=150ms [] k =3.5,t =200ms 1.75 2.0 push rise Figure4. Explosionenergies(top),explosiontimes(middle),and(baryonic) ] kpush=3.8,trise=250ms M remnant mass (bottom) as function ofcompactness for kpush 1.5, 2.0, 3.0, [ 1.8 and4.0, andfixedtrise = 0.15sforallprogenitor models included inthis m studay(ZAMSmassbetween18.0and21.0M⊙).Non-explodingmodelsare Mre 1.6 indicatedwithEexpl = −0.5Binthetoppanelandareomittedintheother panels. 1.4 reaches this maximum. Together, they control the slope of 1.2 G(t)intherisingphase(seeFigure3). Amodelwithalonger 1.0 risetimereachesitsmaximumefficiencylater,atwhichtime 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 theluminositieshavealreadydecreasedandapartoftheab- [] 1.75 sorbed energy has been advected on the PNS or re-emitted Figure5. Explosionenergies(top),explosiontimes(middle),and(baryonic) in the form of neutrinos. To compensate for these effects, remnantmass(bottom)asfunctionofcompactnessforpairsofkpushandtrise, a larger kpush is required for a longer trise. This is seen in andfortheprogenitormodelswithZAMSmass18.0,18.6,19.2,19.4,19.8, Figure 6, where we plot the cumulative neutrino contribu- 20.0,20.2,and20.6M⊙. tion (E + E ) and its time derivative for four runs of push idsa 10 Peregoetal. 30 solid: d(Epush+Eidsa)/dt 7 ExplosionpropertiTeasbfoler3tworeferenceruns dashed: E +E 25 push idsa long vertical: t + t 6 Quantity HC LC on rise -1s] 20 short vertical: texpl 5 Zξ1A.7M5 S (M(-)⊙) 01.693.27 02.02.803 B B] ton (ms) 80 ergy rate [ 1105 trise= 100 ms, kpush= 2.5 34 Energy [ ttkMrepixsurpeeslmhn (((Mmm(-)⊙ss))) 13.70173135.0012.40669 n t = 150 ms, k = 3.0 E rise push 2 Eexpl(tfinal) (B) 1.36 0.57 trise= 200 ms, kpush= 3.5 Epush(toff+trise) (B) 3.51 1.08 5 1 Eidsa(toff+trise) (B) 2.76 1.01 Eidsa(tfinal) (B) 4.10 2.11 0 0 Note.—ThesetworunsareusedtocomparetheHCandLCsamples. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Time post bounce [s] Figure6. Temporalevolutionofthetotalneutrinoenergycontributioninside theexplosionenergy.Onlytheenergydepositedbyneutrinos thegainregion(Epush+Eidsa)(solidlines)andofitstimederivative(dashed in the region above the final mass cut will eventually con- lines), forfourrunswiththesameZAMSprogenitor mass(18.0M⊙),but tributetotheexplosionenergy. differentcombinationsofPUSHparameterstriseandkpush.Foreachrun,the verticallinescorrespondtot = ton+trise (long,dashed)andtotexpl (short, Toidentifythisrelevant neutrinocontribution,in Figure7 dot-dashed). we show the time evolution of the integrated net neutrino energy deposition E (mfin,t) within the domain above the ν cut 3.1.4. toff fixed mass mficunt = mcut(tfinal). We choose mficunt to include all EventhoughPUSH is active up to t +t & 1s, its en- therelevantenergycontributionstothe explosionenergy,up off rise ergydepositionreducesprogressivelyonatimescaleofafew to the end of our simulations. Despite the significant dif- 100 ms after the explosion has set in (see Figure 6). This ferences in magnitudes, the two models show overall simi- showsexplicitlythatthevalueoft doesnothaveimportant lar evolutions. If we compare E (mfin,t) at late times with off ν cut consequencesin oursimulations, atleastaslongaswe have (Epush(toff +trise)+Eidsa(tfinal))fromTable3,weseethatitis typicalexplosiontimeswellbelowonesecond.Theobserved significantly smaller. About two thirds of the energy origi- decreaseofthePUSHenergydepositionrateafterthelaunch nallydepositedinthegainregionareadvectedontothePNS oftheexplosionwillbeexplainedinSection3.3. andhencedonotcontributetotheexplosionenergy. In addition to the neutrino energy deposition, in Fig- 3.2. Contributionstotheexplosionenergy ure 7 we also show the variation of the total energy for the domain abovemfin, i.e., ∆E (mfin,t) = E (mfin,t)− In the following, we discuss the contributions to and the cut total cut total cut sourcesof the explosionenergy, i.e., we investigatehow the Etotal(mficunt,tinitial), where tinitial is the time when we start our explosionenergyis generated. Thisisdonein severalsteps: simulation from the stage of the progenitor star. The varia- first,wehaveacloserlookattheneutrinoenergydeposition. tionofthetotalenergycanbeseparatedintothenetneutrino Thenweshowhowitrelatestotheincreaseofthetotalenergy contributionandthe mechanicalworkatthe innerboundary, oftheejectedlayers,andfinallyhowthisincreaseofthetotal ∆Etotal = Eν + Emech. We note that in our general relativis- energytransformsintotheexplosionenergy.Forthisanalysis, ticapproachthevariationofthegravitationalmassduetothe we havechosenthe 19.2and20.0M ZAMSmassprogeni- intenseneutrinoemissionfromthePNSisconsistentlytaken ⊙ tormodelsasrepresentativesofthe HCandLCsamples, re- intoaccout.ItisvisibleinFigure7,thatthenetdepositionby spectively.Weconsidertheirexplodingmodelsobtainedwith neutrinosmakesupthelargestpartofthechangeofthetotal ton = 80ms, trise = 150ms, andkpush = 3.0. A summaryof energy. The transfer of mechanicalenergy Emech is negative theexplosionpropertiescanbefoundinTable3. becauseoftheexpansionworkperformedbytheinnerbound- Thetableshowsthatforbothmodelsneutrinosarerequired aryduringthecollapseandthePNSshrinking. Howeveritis todepositanetcumulativeenergy(Epush+Eidsa)muchlarger significantlysmallerinmagnitudethanEν. thanE torevivetheshockandtoleadtoanexplosionthat Next, we investigate the connection between the varia- expl matchestheexpectedenergetics. Forthetworeferenceruns, tion of the total energy and the explosion energy. In Fig- whenthe PUSH contributionis switched off(t = t +t ), ure 7, we show the variation of the explosion energy above off rise thecumulativedepositedenergyis∼4timeslargerthanE . the fixed mass mfin, i.e. ∆H (mfin,t) = H (mfin,t) − expl cut expl cut expl cut ThiscanalsobeinferredfromFigure6forotherruns.Thatra- H (mfin,t ), together with the relative variation of the expl cut initial tioincreasesfurtherupto∼5.5att=tfinal,duetotheneutrino time-dependent explosion energy, ∆Eexpl(t) = Eexpl(t) − energydepositionhappeningatthesurfaceofthePNSwhich H (mfin,t ). It is obvious from Equations (2), (12), generatestheν-drivenwind. AccordingtoEquations(9)and expl cut initial and (15) that the difference between ∆H (mfin,t) and (10),E andE arethetotalenergieswhicharedeposited expl cut push idsa ∆E (mfin,t) is givenby the variationof the integratedrest inthe(time-dependent)gainregion.Thisneutrinoenergyde- total cut positionincreasestheinternalenergyofthematterflowingin massenergy,∆Hexpl(mficunt,t)=∆Etotal(mficunt,t)−∆Emass(mficunt,t). thatregion. However,since theadvectiontimescale is much In Figure 7, −∆E (mfin,t) can thus be identified as the mass cut shorter than the explosion timescale, a large fraction of this difference between the long-thin and the short-thick dashed energyisadvectedontothePNSsurfacebytheaccretingmass lines. We find that the overall rest mass contribution to the beforetheexplosionsetsin,andhencedoesnotcontributeto finalexplosionenergyis positive, butmuchsmaller thanthe