Mass ejection in failed supernovae: equation of state and neutrino loss dependence
DD RAFT VERSION F EBRUARY
12, 2021Typeset using L A TEX twocolumn style in AASTeX63
Mass ejection in failed supernovae: equation of state and neutrino loss dependence M ARIO I VANOV AND R ODRIGO F ERN ´ ANDEZ Department of Physics, University of Alberta, Edmonton, AB T6G 2E1, Canada.
ABSTRACTA failed core-collapse supernova from a non-rotating progenitor can eject mass due to a weakening of gravityassociated to neutrino emission by the protoneutron star. This mechanism yields observable transients and setsan upper limit to the mass of the black hole (BH) remnant. Previous global simulations of this mechanismhave included neutrino losses parametrically, however, with direct implications for the ejecta mass and energy.Here we evolve the inner supernova core with a spherically-symmetric, general-relativistic neutrino radiation-hydrodynamic code until BH formation. We then use the result in a Newtonian code that follows the response ofthe outer layers of the star to the change in gravity and resolves the surface pressure scale height. We find that thedense-matter equation of state (EOS) can introduce a factor ∼ variation in gravitational mass lost to neutrinos,with a stiff EOS matching previous parametric results, and a soft EOS yielding lower ejecta masses and energiesby a factor of several. This difference is caused primarily by the longer time to BH formation in stiffer EOSs.With a soft EOS, our red and yellow supergiant progenitors fail to unbind mass if hydrogen recombinationenergy is not included. Using a linear ramp in time for mass-energy lost to neutrinos (with suitable parameters)yields a stellar response within ∼ of that obtained using the detailed history of neutrino losses. Our resultsimply quantitative but not qualitative modifications to previous predictions for shock breakout, plateau emission,and final BH masses from these events. Keywords: gravitation (661) – hydrodynamics (1963) – supernova neutrinos (1666) – shocks (2086) –black holes (162) – core-collapse supernovae (304) INTRODUCTIONCore-collapse supernova theory has focused for decadeson achieving successful explosions in order to account forthe majority of observed events (e.g., Janka et al. 2016; Bur-rows & Vartanyan 2021). Black hole (BH) formation infailed supernovae has received more recent attention, duein part to the need to explain the presupernova progenitorpopulation (e.g., Kochanek et al. 2008; Smartt et al. 2009)and the mass distribution of remnant BHs and neutron stars(NSs) (e.g., Raithel et al. 2018; Woosley et al. 2020). Theincreasing number of BH-BH binaries detected in gravita-tional waves by Advanced LIGO & Virgo (Abbott et al. 2019,2020) has further expanded our empirical knowledge of thecompact object mass distribution, with BH masses now ex-tending to the regime where pair-instability supernovae arerelevant (with GW190521; Abbott et al. 2020). On the the-oretical side, simulations of BH-forming supernovae – thatinclude some form of neutrino physics – have been carriedout for large numbers of progenitors in spherical symmetry(e.g., O’Connor & Ott 2011; Ugliano et al. 2012; Ertl et al.2016; Ebinger et al. 2019; Warren et al. 2020; Couch et al.2020), and more recently in three-dimensions for a handfulof cases (e.g., Chan et al. 2018; Kuroda et al. 2018; Walket al. 2020; Chan et al. 2020; Pan et al. 2020).Observational predictions for BH formation events arestrongly dependent on the degree of rotation in the progen-itor star. For stars with the right angular momentum pro- file, a neutrino-cooled disk can form outside the BH and along-gamma-ray burst with an associated supernova can beproduced (e.g., MacFadyen & Woosley 1999, see also Na-gataki 2018 for a review). Even if the disk forms at largerradii, other transients could also arise due to outflows fromthe disk and/or thermonuclear explosions (e.g., Bodenheimer& Woosley 1983; Woosley & Heger 2012; Kashiyama et al.2018; Zenati et al. 2020).In the absence of any significant rotation, mass ejectioncan still occur in a failed supernova if a protoneutron star isformed (i.e., for progenitor masses below the pair instabilitythreshold). Neutrino emission reduces the gravitational mass,generating an imbalance between the acceleration of gravityand the pressure gradient in the progenitor, driving an out-flow (Nadyozhin 1980). Mass ejection from this mechanismtherefore sets an upper limit on the mass of the BH, since anysubsequently formed accretion disk would eject additionalmass, further reducing that of the remnant BH. For red super-giants (RSGs), this mechanism can eject the weakly boundhydrogen envelope, generating an optical transient with peakluminosity ∼ erg s − and ∼ yr duration (Lovegrove &Woosley 2013). The failed supernova candidate N6946-BH1,associated with the disappearance of a RSG, showed a simi-lar year-long transient (Gerke et al. 2015; Adams et al. 2017;Basinger et al. 2020).This neutrino-induced mass ejection is not limited toRSGs, however. Fern´andez et al. (2018, hereafter F18)showed that blue supergiants (BSGs) and Wolf-Rayet (WR) a r X i v : . [ a s t r o - ph . H E ] F e b I VANOV & F
ERN ´ ANDEZ stars also eject mass through this mechanism, although theejecta masses, energies, and velocities differ significantlyfrom those in RSGs. This diversity of progenitors implieselectromagnetic signals that span a wide range in luminos-ity, duration, and peak wavelength, resulting in new typesof transients. Of particular recent interest are failed super-novae from BSGs as possible progenitors of fast blue opti-cal transients (e.g., Kashiyama & Quataert 2015; Marguttiet al. 2019) and very massive BH remnants from failed su-pernovae around the lower edge of the “pair instability gap”(e.g., Farmer et al. 2019; Mapelli et al. 2020; Marchant &Moriya 2020; Renzo et al. 2020).A key limitation of the global simulations of F18 is the useof an analytic model to describe the decrease in gravitationalmass with neutrino emission. While this parametric ap-proach, which follows that of Lovegrove & Woosley (2013),provides a computationally inexpensive input, it takes im-portant quantities (neutrino cooling timescale, binding en-ergy and maximum mass of the NS) as free parameters. Thephysics that sets these quantities is complex, depending onthe transport of neutrinos in the protoneutron star and on theequation of state (EOS) of dense matter.Here we improve upon the work of F18 by modeling theevolution of the inner core with the spherically symmetric,general-relativistic neutrino radiation-hydrodynamics code
GR1D (O’Connor & Ott 2010), which yields the history ofgravitational mass lost to neutrinos until BH formation givena dense-matter EOS and presupernova star. This physically-grounded gravitational mass decrease is then used, in lieu ofa parametric scheme, as an input to the same global hydrody-namic setup used by F18 to characterize the response of thestar to the change in gravity. These global simulations areperformed at spatial resolutions such that the surface pres-sure scale height is resolved in all progenitors, thereby alsoimproving upon the work of F18. We focus on understandingthe dependence of the ejecta properties on the EOS, on thespatial resolution of the simulation, on the need to know thedetailed neutrino emission history of the protoneutron star,and on the implications of these inputs for the electromag-netic signatures expected from these events.The structure of this paper is the following. Section 2describes our assumptions and the numerical method em-ployed, the presupernova progenitors used, and the range ofmodels evolved. Section 3 provides an overview of results forvarious progenitors, the effect of varying the EOS, the sen-sitivity to the history of gravitational mass loss, the effectsof spatial resolution, and the observational implications. Asummary and discussion follows in Section 4. METHODS2.1.
Physical Model and Approximations
Our aim is to compute the hydrodynamic response of thestar to the decrease in the gravitational mass from neutrinoemission after core bounce in the case where the supernovafails and a BH forms. We restrict ourselves to progenitors forwhich rotation is negligible, and ignore the effect of hydro-dynamic instabilities that operate within the first second after
Figure 1.
Schematic of our computational approach. For a givenpresupernova progenitor (§2.2) and dense-matter EOS, the evolu-tion of the inner core is followed with the general-relativistic, neu-trino radiation-hydrodynamics code
GR1D (§2.3), focusing on thegravitational mass lost to neutrinos ∆ M G ( t ) [eq. 5] at some transi-tion radius r in . The region outside r in is then evolved with FLASH ,accounting for the mass flowing supersonically into r in as well asthe change in gravity due to ∆ M G ( t ) (§2.4), and focusing on anymass ejected from the stellar surface. The majority of our modelsonly interpolate ∆ M G ( t ) from GR1D , while a smaller sample em-ploys an analytic approximation for this function or maps the initialcondition for
FLASH directly from
GR1D (Table 2). bounce, which have an important role in a successful explo-sion (e.g., Janka et al. 2016; here we only consider failures).We therefore carry out our analysis in spherical symmetry.We also ignore any circumstellar material that the star couldhave ejected prior to undergoing core-collapse. This materialcan modify the observational signatures of shock breakout(e.g., Chevalier & Irwin 2011; Katz et al. 2012; Haynie &Piro 2020). Our focus is the total energy and mass of anyejecta arising from the change in gravitational acceleration.In previous work (F18), we removed the inner stellar coreand computed the hydrodynamic response of the star usinganalytic prescriptions for the evolution of the gravitationalmass enclosed within this radius. This enclosed mass evolvesdue to material accreted through the inner radius and dueto neutrino emission during the protoneutron star phase (seealso Lovegrove & Woosley 2013). This modeling approachis acceptable because material that accretes through the innerboundary reaches supersonic speeds shortly after collapse,without any pressure feedback on the outer layers.Here we replace the analytic prescriptions for the in-ner core evolution used in F18 with a neutrino radiation-hydrodynamic calculation (Figure 1). Due to the complexityof the latter and the limitations in thermodynamic range setOS
AND NEUTRINO EFFECTS IN FAILED SN E Pre-supernova progenitors
The pre-supernova stars we explore are shown in Table 1.We adopt the same fiducial solar-metallicity progenitors as inF18: a 15 M (cid:12) RSG (denoted by R15), a 25 M (cid:12) BSG (B25),and a 40 M (cid:12) WR (W40). These models cover three differ-ent regimes in core compactness parameter (O’Connor & Ott2011; Sukhbold & Woosley 2014; Sukhbold et al. 2018) ξ . = 2 . r ( M = 2 . M (cid:12) ) / , (1)and in envelope compactness ξ env = ( M cc /M (cid:12) )( R cc /R (cid:12) ) , (2)where M cc and R cc are the mass and radius of the star atcore-collapse. The values of ξ . and ξ env are shown in Ta-ble 1 (see Pejcha & Thompson 2015; Ertl et al. 2016; M¨ulleret al. 2016; Murphy & Dolence 2017 for explodability crite-ria other than ξ . ).In addition to these baseline models, we consider two yel-low supergiants (YSGs), one of 22 M (cid:12) and solar metallicity(Y22), and another of 25 M (cid:12) and low-metallicity (Y25); amassive 80 M (cid:12) low-metallicity blue supergiant (B80); and a50 M (cid:12) solar-metallicity WR (W50). The YSGs are such thanone of { ξ . , ξ env } has a value similar to R15, while the othervaries. W50 has a higher core-compactness than W40, andB80 has high values of both core- and envelope compactness(in F18 no mass is ejected from this model). All of these pro-genitors are computed with the stellar evolution code MESA version 6794 (Paxton et al. 2011, 2013, 2015, 2018). Param-eters and physical choices are described in Fern´andez et al.(2018) (see also Fuller et al. 2015), and inlists are publiclyavailable .To connect with previous work, we also evolve models s20and s40 from Woosley & Heger (2007). These progenitorshave been used in BH formation simulations (e.g., O’Connor2015; Pan et al. 2018) and in a code-comparison study of 1Dcore-collapse supernova simulations (O’Connor et al. 2018),providing calibration values.2.3. Inner core evolution
The evolution of the inner stellar core from collapse un-til BH formation is modeled with the spherically-symmetric bitbucket.org/rafernan/bhsn mesa progenitors Table 1.
Presupernova progenitors used in this study. Columnsfrom left to right show model name, type of star (RSG: red super-giant, YSG: yellow supergiant, BSG: blue supergiant, WR: Wolf-Rayet), zero-age main sequence mass, initial metallicity, mass atcore-collapse, effective temperature at core-collapse, core compact-ness (eq. [1]), and envelope compactness (eq. [2]). Model S40 hasthe structure of a BSG but it is truncated below its photosphere.Model Type M zams Z M cc T eff ξ . ξ env ( M (cid:12) ) ( Z (cid:12) ) ( M (cid:12) ) ( K)R15 RSG 15 1.00 11 3 0.24 0.010B25 BSG 25 1.00 12 15 0.33 0.120W40 WR 40 1.00 10 260 0.37 27Y22 YSG 22 1.00 11 5 0.54 0.016Y25 YSG 25 0.01 23 4.6 0.25 0.024W50 WR 50 1.00 9 215 0.55 22B80 BSG 80 0.01 55 28 0.97 0.79S20 RSG 20 1.00 16 2.5 0.28 0.015S40 BSG 40 1.00 15 ... 0.54 1.3 neutrino radiation-hydrodynamics code
GR1D version 1(O’Connor & Ott 2010). The code solves the equations ofgeneral-relativistic hydrodynamics in spherical coordinatesusing the radial gauge, polar slicing metric (Romero et al.1996). The finite-volume hydrodynamics solver employs apiecewise-parabolic reconstruction, and performs a temporalupdate using a second-order Runge-Kutta scheme.During collapse, neutrino effects are modeled via a param-eterization of the electron fraction with density (Liebend¨orfer2005). After bounce, neutrino cooling in this version of GR1D is modeled with a gray leakage scheme for ν e , ¯ ν e and a composite heavy lepton neutrino ν x . The opacityhas contributions from charged-current absorption on nu-cleons and neutral-current scattering on nucleons and nu-clei. Emission accounts for charged-current reactions aswell as thermal emission from electron-positron pair anni-hilation and plasmon decay (we did not include nucleon-nucleon bremsstrahlung). The local effective emission rateis an interpolation between the diffusive and free emissionrates (Ruffert et al. 1996; Rosswog & Liebend¨orfer 2003).Neutrino heating due to charge current absorption is com-puted from the enclosed local luminosity obtained from theleakage scheme. The local outgoing luminosity is then cor-rected for the neutrino energy lost to heating.We consider three finite-temperature EOSs in our calcula-tions: SFHo (Steiner et al. 2013) as our default (soft) case,DD2 (Hempel et al. 2012) as a stiff variant, and LS220 (Lat-timer & Swesty 1991) as another reference case. These EOSsare commonly used in the core-collapse supernova and NS Available at stellarcollapse.org I VANOV & F
ERN ´ ANDEZ merger literature (e.g., Pan et al. 2018; Vincent et al. 2020),thus providing a connection to previous work (while DD2and SFHo are consistent with experimental constraints andunitary gas bounds on the symmetry energy and its densityderivative, LS220 is not; c.f. Tews et al. 2017). We ne-glect light clusters when computing opacities and emissiv-ities with the DD2 and SFHo EOSs, as these species havea sub-dominant effect on the post-bounce evolution (Yudinet al. 2019; Nagakura et al. 2019).In most models, the computational domain is discretizedwith a uniform grid of cells from the origin out to km,and logarithmic spacing for larger radii, for a total grid sizeof cells. In all RSG models and some YSGs and BSGs,we double the resolution in the uniform section of the grid( r < km) as these models take longer to reach BH for-mation and reach more compact shock radii. The maximumradius is set by a density close to the lowest value in the tabu-lated EOS ( × g cm − ), corresponding typically to sev-eral times cm, much smaller than the radius of the starat core-collapse. The dynamical time at the outer boundaryis typically ∼ s, much longer than the time to form a BHin most models, justifying our approximation of neglectingthe evolution of the outer stellar layers when evolving thecore with GR1D . Simulations are deemed to have formed aBH when the density increases rapidly with time to values ∼ − × g cm − (near the upper limit of the EOS ta-ble), at which point the simulation stops. In only one case(model R15 with the DD2 EOS) we fail to reach BH forma-tion within . s of evolution and the code crashes, althoughbased on the central value of the lapse function ( . ), themodel is close to BH formation.While a newer version of GR1D is available, whichtreats neutrinos with a multigroup moment (M1) scheme(O’Connor 2015) and therefore provides a more accuratemeasure of the gravitational mass lost to neutrinos, the con-vergence of the transport algorithm near BH formation ismore fragile than for the leakage scheme. Evolving the s40progenitor with the LS220 EOS, we obtain post-bounce timesto BH formation { . , . } s, maximum PNS baryonicmasses (at a density of g cm − ) { . , . } M (cid:12) , andmaximum PNS gravitational masses { . , . } M (cid:12) withthe leakage and M1 versions of GR1D , respectively. Thesemasses and BH formation times are the same as those re-ported in O’Connor (2015) for the leakage version, while forthe M1 version the masses are the same within but theBH formation time is about longer. These results are alsoconsistent (within ∼ ) with the 1D results of Pan et al.(2018) for the s40 progenitor using the LS220 EOS and adifferent code.2.4. Response of the star to neutrino losses
The hydrodynamic response of the outer layers of the starto the decrease in gravity due to neutrino emission is mod-eled in spherical symmetry with the Newtonian hydrody-namic code
FLASH version 3 (Fryxell et al. 2000; Dubeyet al. 2009), modified as described in Fern´andez (2012) andF18. The code solves the Euler equations in spherical co- ordinates with the dimensionally-split Piecewise ParabolicMethod (PPM; Colella & Woodward 1984, Fryxell et al.1989) and the
Helmholtz
EOS (Timmes & Swesty 2000).The computational domain spans a radial interval [ r in , r out ] that varies for different evolution modes and pro-genitors, as explained below. The boundary conditions areset to outflow at r = r in and r = r out . The mass flowing outthrough the inner radial boundary is kept track of as a scalarbaryonic mass M B , flash , such that total mass is conservedclose to machine precision (F18).The gravitational acceleration in FLASH g F is a sum ofthe contribution from the baryonic mass in the computationaldomain and the gravitational mass M G , flash inside r in , g F ( r, t ) = − Gr (cid:20) M G , flash + 4 π (cid:90) rr in ρ ( ζ, t ) ζ dζ (cid:21) ˆ r, (3)where ρ is the mass density. Initially, the gravitational mass M G , flash is either equal to the baryonic mass enclosed within r = r in in the presupernova progenitor, or otherwise mappedfrom GR1D , depending on the mode of evolution as describedbelow. This gravitational mass is subsequently updated fromtime t n to t n +1 by adding the change due to the baryonicmass flowing through the inner boundary over the time step,and correcting for the instantaneous difference between bary-onic and gravitational masses as computed by GR1D or froman analytic fit, M ( n +1)G , flash = M ( n )B , flash + 4 π (cid:90) t n +1 t n (cid:2) r ρ max( − v r , (cid:3) (cid:12)(cid:12) r in dt − ∆ M G ( t n +1 ) + ∆ M G ( t = 0) , (4)where ∆ M G ( t n ) = M B , gr1d ( t n ) − M G , gr1d ( t n ) , (5)is the instantaneous difference between the baryonic andgravitational masses enclosed by r = r in in GR1D ( M B , gr1d and M G , gr1d , respectively), and ∆ M G ( t = 0) is an initialEOS-dependent offset between these two masses in GR1D ∆ M G ( t = 0) (cid:39) π (cid:90) r in ρr dr (cid:20) GM G , gr1d ( r ) c r − e int c (cid:21) , (6)where e int is the specific internal energy, and wehave assumed non-relativistic collapse speeds as well as GM G , gr1d ( r ) /c (cid:28) r in deriving equation (6). For theSFHo and DD2 EOSs, the term in square brackets is ∼ − ,whereas for LS220 it is ∼ . , given an extra offset in e int (of order the nuclear binding energy per nucleon) neededto connect the nuclear EOS with a low-density continuation(O’Connor & Ott 2010). Figure 2 shows the evolution of ∆ M G at r in = 2 000 km in GR1D for our fiducial progeni-tors using all three EOSs considered in this study.Since at the onset of significant neutrino emission we have M G , flash (cid:39) M G , gr1d (cid:39) M B , gr1d , the formulation in equa-tion (4) preserves consistency in the mass evolution within FLASH , which is entirely baryonic, while also accountingOS
AND NEUTRINO EFFECTS IN FAILED SN E Figure 2.
Evolution of the difference ∆ M G (eq. [5]) between bary-onic and gravitational masses in GR1D , evaluated at r = 2 000 km,as a function of time from the onset of core-collapse. Top, middle,and bottom panels correspond to our fiducial presupernova modelsW40, B25, and R15, respectively (Table 1). For each case, the re-sult obtained with the DD2, SFHo, and LS220 EOS is shown, aslabeled. The gray vertical band shows the range of maximum shockradius time t stall obtained with the three EOSs ( . − . s forR15, . − . s for B25, and . − . s for W40, withthe largest values corresponding to the DD2 EOS). The larger valueof ∆ M G at t = 0 (eq. 6) for the LS220 EOS is due to an offsetin the internal energy needed to connect the high- and low-densityregimes of the EOS (O’Connor & Ott 2010). Table 2.
Evolution modes for
FLASH simulations. ∆ M G ( t ) initial conditionfrom GR1D analytic from
GR1D
Interpolation (1) yes no noAnalytic (2) no yes noRemap (3) yes no yes for the mass-energy lost to neutrinos via ∆ M G . When aBH forms at time t = t bh , the mass difference ∆ M G ( t bh ) becomes a constant in equation (4).The initial condition for FLASH and the evolution of theinner core ( r < r in ) are treated in three different ways, toassess the sensitivity of our results to the details of the innercore history (Table 2):1. Interpolation : by default, we initialize
FLASH withthe pre-collapse profile, and interpolate ∆ M G ( t ) at r = r in as a function of time from GR1D . This ap-proach provides a more realistic value for the grav-itational mass lost to neutrinos relative to the mod-els of F18, while starting from the same initial con-dition. Figure 2 shows the evolution the mass differ-ence ∆ M G ( t ) at r = 2 × cm for the three fidu-cial progenitors and EOSs used. The curves start froma very small value ∆ M G ( t = 0) and increase almostlinearly until BH formation. The inner radial boundaryfor these models is located at r in = 2 × cm as inF18 (approximately at the outer edge of the iron core),and ∆ M G ( t ) at that radius is interpolated for use in FLASH .2.
Analytic : given the overall simplicity of the function ∆ M G ( t ) , we evolve a second group of models byinitializing the domain in the same way as with In-terpolation mode, but now we parameterize the func-tion ∆ M G ( t ) as a linear ramp that turns on and offat specified times t stall and t bh , respectively, reachingthe same maximum value as the instantaneous function ∆ M G ( t ) . The aim is to quantify the degree to whichthe details of the gravitational mass loss history (as op-posed to just the final magnitude and overall timescale)influences the results. The inner boundary for thesemodels is also located at r in = 2 × cm.3. Remap : a third group of models are such that the initialcondition for
FLASH is mapped from
GR1D , in addi-tion to interpolating the mass difference ∆ M G ( t ) at r = r in as a function of time. Profiles of density,pressure, velocity, and composition are mapped at atime t stall when the shock reaches its maximum am-plitude, usually ∼
200 ms after the onset of collapse(Figure 2). The inner radius r in of the computationaldomain in FLASH is chosen such that (1) the shock ra-dius in
GR1D never exceeds it, and (2) the flow at thisradius is supersonic, so that there is no hydrodynamic I
VANOV & F
ERN ´ ANDEZ feedback to regions outside this transition. Figure 3shows a snapshot of the velocity profile in a
GR1D runof model W40 at the time of mapping into
FLASH . Forthis model, r in = 2 × cm and the time of mappingis ∼ ms after the onset of collapse. This is not ourdefault mode of evolution because inconsistencies be-tween general-relativistic and Newtonian evolution arenot entirely negligible at the level of precision neededto model this mass ejection mechanism.The computational domain is discretized with a logarith-mic grid using a resolution of cells per decade in ra-dius ( ∆ r/r (cid:39) . ) for RSG, YSG, and BSG, models, andtwice that value ( ∆ r/r (cid:39) . ) for WRs. At this reso-lution, the pressure scale heights at the surfaces of all pro-genitors are resolved. We also evolve models with a lowerresolution of cells per decade in radius to compare withthe results of F18, which used this value in their highest res-olution models.Like in F18, following BH formation, the inner radialboundary of the simulation is moved out by a factor of atspecific times, once the flow in the entire region [ r in , r in ] has become supersonic, so that the minimum hydrodynamictime step becomes larger ( ∆ r ∝ r in a logarithmic grid) andevolution of the shock up to the stellar surface and beyondcan be followed at a smaller computational cost.The outer radius of the computational domain in FLASH isset at r out = { × , × , × } cm for RSG/YSG,BSG, and WR progenitors, respectively, corresponding tofactors − times the stellar surface at core collapse.The domain outside the star is filled with a constant-densityambient medium in hydrostatic equilibrium, with the samecomposition as the stellar surface. The ambient densitiesare { − , − , × − } g cm − for RSG/YSG, BSG,and WR progenitors, respectively. These densities are lowenough that the ambient mass swept up by the shock is muchsmaller ( (cid:28) ) than the ejecta mass itself, with negligibleslowdown. While the mass in ambient for the RSG/YSGmodels could in principle reach ∼ . M (cid:12) at the maximumsimulation radii, we normally stop our simulations much ear-lier than that point. A floor of temperature at K is adoptedin all simulations, consistent with the low-temperature limitof the Helmholtz EOS in
FLASH (in practice this is whatsets the stopping time for RSG, YSG, and some BSG sim-ulations). The density floor is set times lower than theambient for RSGs, YSGs, and BSGs, and a factor of 5 lowerthan the ambient for WRs.2.5.
Models Evolved
All of our simulations are listed in Table 3. We adopt theSFHo EOS and interpolation of ∆ M G ( t ) from GR1D (Ta-ble 2) as our default setting.The three fiducial progenitors described in §2.2 (R15, B25,W40) are evolved using the three EOSs described in §2.3,with progenitor names appended the letters { S,L,D } whenusing the SFHo, LS220, or DD2 EOS, respectively, andending in { } in accordance to the inner core evolution Figure 3.
Velocity and sound speed as a function of radius inthe core-collapse simulation of the W40 progenitor carried out with
GR1D using the SFHo EOS. The time shown corresponds to that atwhich, when using remap evolution mode (Table 2), we map vari-ables into
FLASH for subsequent evolution. The vertical blue lineindicates the position of the inner radial boundary in
FLASH . modes listed in Table 2. For example, model R15S1 is theR15 progenitor evolved with the SFHo EOS in GR1D , and in-terpolating ∆ M G ( t ) from GR1D into
FLASH . We also evolvethe 3 fiducial progenitors varying the evolution mode of theinner core, using the SFHo EOS. The remaining progeni-tors are all evolved using the SFHo EOS and interpolationof ∆ M G ( t ) , with the exception of the B80 progenitor, whichwe evolve using the SFHo and DD2 EOS.Each progenitor is evolved at the maximum resolutionlisted in §2.4, while fiducial progenitors are also evolved atthe lower resolution used in F18, to compare results. Themaximum run time for each model is set either by the shockemerging from the star and reaching nearly constant total en-ergy, or otherwise when the temperature in the postshockflow reaches the floor value of K, at which point non-conservation of thermal energy ensues and results becomeunreliable. RESULTS3.1.
Overview of evolution for different progenitors
The inner core evolution in all of our models is qualita-tively the same. After core-collapse and bounce, the shockstalls and then gradually recedes on a timescale of ∼ safter bounce. At that time, the difference between baryonicand gravitational masses ∆ M G stops increasing, as shown inFigure 2.At a radius ∼ cm, where the local free-fall time t ff is comparable to the time to change the gravitational massvia neutrino cooling ( ∼ t bh ), an acoustic pulse forms whichOS AND NEUTRINO EFFECTS IN FAILED SN E Table 3.
Summary of results. Columns from left to right show model name, evolution mode (I: interpolation, A: analytic, R: remap, c.f. Table 2),EOS used in
GR1D , time to BH formation from the onset of core-collapse t bh , maximum gravitational mass lost to neutrinos ∆ M G ( t bh ) ,maximum kinetic energy of the acoustic pulse in the FLASH simulation E simk , max , mass ejected in the FLASH run M ej , total energy of ejecta in FLASH run E ej , and recombination energy E rec (eq. 9) for stars with extended hydrogen envelopes. Results are reported for FLASH runs attheir maximum resolution, see §3.4 for results at lower resolution. Model W40S1.7 has its inner radial boundary at r in = 2 × cm, insteadof the default r in = 2 × cm for interpolation mode.Model Mode EOS t bh (s) ∆ M G ( M (cid:12) ) E simk , max ( erg) M ej ( M (cid:12) ) E ej ( erg) E rec ( erg)R15S1 I SFHo 2.836 0.196 2.15 2.19 -0.119 0.398R15S2 A 2.06 2.16 -0.154 0.388R15S3 R 0.57 0.99 -0.268 0.179R15L1 I LS220 2.947 0.222 2.42 2.42 -0.103 0.440R15D1 DD2 > > M ej ( − M (cid:12) )B25S1 I SFHo 1.791 0.173 2.27 2.80 0.399B25S2 A 2.27 2.82 0.404B25S3 R 0.14 0.45 0.012B25L1 I LS220 1.864 0.198 2.55 3.18 0.593B25D1 DD2 2.895 0.261 5.40 5.45 1.76Y22S1 SFHo 0.931 0.139 0.55 5.39 -0.013 0.010Y25S1 2.542 0.175 3.18 15.5 -0.082 0.028 M ej ( − M (cid:12) )W40S1 I SFHo 1.535 0.157 1.40 1.44 0.067W40S2 A 1.36 1.36 0.063W40S3 R 0.06 0.01 < < moves outward with a Mach number of order unity (F18;Coughlin et al. 2018a). The subsequent evolution of thissonic pulse and its effect on the stellar envelope depends onthe type of stellar progenitor. Table 3 summarizes the resultsof our hydrodynamic simulations.For models in which we initialize the domain in FLASH byremapping fluid quantities from
GR1D at t = t stall (evolutionmode 3 in Table 2), there are discrepancies at the (cid:46) inthe fluid quantities (slightly lower density, lower internal en-ergy, differing infall velocity) within the outermost mappingradius ( ∼ cm) relative to the presupernova model. Thisresults in the remapped models undergoing a slighly fastercollapse than corresponding models that only make use of ∆ M G ( t ) to modify gravity (evolution modes 1 and 2), whichyields either a reduced kinetic energy of the shock and loweramount of mass ejected (models R15S3 and B25S3) or nomass ejected at all (model W40S3). Resolving this discrep-ancy requires a self-consistent treatment of the entire starusing a general-relativistic framework, which is beyond thescope of our study. In the rest of the paper we limit the dis-cussion to models that only interpolate ∆ M G ( t ) from GR1D or otherwise approximate it analytically.Figure 4 shows the evolution of the energy in the soundpulse as it travels outward through the envelope and becomesa running shock, for our three fiducial progenitors. Follow-ing F18, we define the pulse shell as the region limited by I
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Figure 4.
Left:
Evolution of the energy components in the acoustic pulse generated by neutrino-induced gravitational mass decrease for ourthree fiducial progenitors: R15 (top), B25 (middle), and W40 (bottom). Each panel shows the kinetic (red), internal (blue), gravitational(purple), and total energy (black) for models that interpolate ∆ M G ( t ) from GR1D and vary the EOS: DD2 (thin solid), SFHo (thick semitrans-parent), and LS220 (dashed). Only positive total energies are shown.
Right:
Evolution of the mass contained by the pulse. Curves correspondto the same models as in the left panel. the forward edge of the pulse/shock on its front, and by thestagnation point on its rear, which separates expanding fromaccreting flow. Figure 4 also shows that the mass in this pulsechanges with time, as it sweeps up material on its front andloses it to fallback accretion from its rear. In all cases, thekinetic energy in the pulse is initially a small fraction of thegravitational and internal energies in the shell. As the pulsepropagates out, the kinetic energy eventually becomes com-parable and/or exceeds the (more rapidly decreasing) thermal and gravitational energies as it emerges from the stellar sur-face. The final net energy of the outgoing shock is compara-ble to its initial kinetic energy.The propagation of weak shocks in gravitationally boundstellar envelopes does not conserve energy (Coughlin et al.2018b). Depending on the radial dependence of the stel-lar density profile and on the initial strength of the pulse,the shock can accelerate or decelerate, yielding behavior thatranges from a strong shock as it reaches the stellar surface toOS
AND NEUTRINO EFFECTS IN FAILED SN E ∆ v pulse imparted to amass shell by the change in gravity over a free-fall time t ff , ∆ v pulse ( r ) = ( G ∆ M G /r ) t ff (Coughlin et al. 2018a, F18).The kinetic energy in the pulse is ∆ E pulse (cid:39) M pulse ∆ v , (7)with M pulse (cid:39) πr H p , and H p being the pressure scaleheight. In terms of stellar quantities, we can write the maxi-mum kinetic energy that a stellar shell can have as (F18) E k , max (cid:39) . × (cid:16) α . (cid:17) (cid:18) H p /r . (cid:19) (cid:18) ∆ M G . M (cid:12) (cid:19) × (cid:18) × cm r (cid:19) erg , (8)where we have evaluated equation (7) at the location whereenergy extraction is maximal (point of acoustic pulse for-mation), and where α = d ln M ( r ) /d ln r . The maxi-mum kinetic energies obtained in the simulations (Figure 4,also shown in Table 3 as E simk , max ) are broadly consistentwith equation (8) given the characteristic gravitational masschanges ∆ M G ( t bh ) shown in Table 3.The mass ejected via this mechanism is set, to order ofmagnitude, by the exterior mass coordinate in the star atwhich the gravitational binding energy is comparable to theshock energy. RSGs, with weakly bound hydrogen en-velopes, can eject several solar masses of slowly-moving ma-terial (Lovegrove & Woosley 2013), whereas BSGs and WRstars eject much smaller masses at higher speeds (F18).Figure 5 shows the ejecta mass and energies as a func-tion of core compactness (eq. 1) and envelope compactness(eq. 2), for all of our simulations that interpolate ∆ M G ( t ) from GR1D runs that use the SFHo EOS. The inverse rela-tion between ejected mass and envelope compactness is ev-ident: given shock energies of characteristic magnitude of ∼ erg, the mass ejected is inversely proportional to thesurface gravity of the star. The inverse dependence of theejecta energy with core compactness can be understood fromthe monotonic decrease in ∆ M G ( t bh ) with core compact-ness (Figure 6). Higher compactness is associated with ashorter time to BH formation (O’Connor & Ott 2011; daSilva Schneider et al. 2020), which results in less gravita-tional mass lost to neutrinos. While the ejecta energy isweakly correlated with envelope compactness, the energy perunit mass (and hence velocity) of the ejecta is strongly depen-dent on the envelope compactness (Figure 6): stars that canunbind matter from their surfaces do so at speeds that scalewith the escape velocity at that location ( v esc ∝ ξ / ).To illustrate these trends with specific cases, we can com-pare models R15S1 and Y22S1, which have similar envelopecompactness (0.010 and 0.016 respectively), but the YSG has Figure 5.
Ejecta energy (top) and mass (bottom) as a function ofcore compactness ξ . (left; eq. 1) and envelope compactness ξ env (right; eq. 2) for all models that interpolate ∆ M G ( t ) from GR1D us-ing the SFHo EOS. The error bars indicate the change introduced,for our 3 fiducial progenitors (R15, B25, W40), by evolving the in-ner core with the DD2 EOS instead. Colored markers correspondto their respective progenitors: red circles for RSGs, blue squaresfor BSGs, black diamonds for WRs, yellow left triangles for YSGs,green right triangles for S20 and S40 models. Open and full sym-bols in the top row denote bound and unbound ejecta, respectively,with the exception of the R15 models, for which we have plottedboth R15S1 (open circle, bound) and R15D1 (full circle, unbound). more than double the core compactness of the RSG (0.54 ver-sus 0.24, respectively). Model R15S1 ejects about 2.2 M (cid:12) ,a factor of 40 more than the YSG ejecta (5.39 × − M (cid:12) ),which follows from the larger value of ∆ M G ( t bh ) for theRSG. In both cases, the ejecta is bound (see §3.2 for a discus-sion of hydrogen recombination energy). Likewise, modelsR15S1 and Y25S1 have similar core compactness (0.24 and0.25, respectively), but the YSG’s envelope compactness ismore than double that of the RSG (0.010 and 0.024, respec-tively). The Y25 model ejects . M (cid:12) , which is approx-imately 14 times less compared to the ∼ M (cid:12) from theRSG. In both cases the ejecta is bound, with the energy perunit mass being larger in the YSG. Model S20S1 has highercore and envelope compactness than model R15S1, and alsoejects bound mass.Model B80S1 has a higher core (0.97) and envelope com-pactness (0.79) than model B25S1 (0.33 and 0.12 respec-tively). While B25S1 ejects 2.8 × − M (cid:12) with energy × erg by the end of the simulation, model B80S1 gen-erates a shell with mass ∼ × − M (cid:12) that is bound( E ej ∼ − erg) by the time it approaches the stellar sur-0 I VANOV & F
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Figure 6.
Left:
Maximum gravitational mass lost to neutrino emission ∆ M G ( t bh ) as a function of core compactness ξ . . Right : Ejectaenergy per unit mass as a function of envelope compactness ξ env . Symbols and error bars have the same meaning as in Figure 5, with the extrahigh-compactness BSG on the left panel representing model B80S1, which does not eject mass. An upper limit is used for ∆ M G ( t bh ) frommodel R15D1 since it did not collapse to a BH in GR1D within . s of evolution. face and reaches the floor of temperature (after which timethe internal energy starts rising and the subsequent evolu-tion is unreliable). This M (cid:12) progenitor also fails whenusing parameterized neutrino mass loss in F18. ModelS40S1, in contrast, is a BSG with similar envelope compact-ness as B80S1 but smaller core compactness (0.54), ejectingmarginally bound mass in amounts comparable to the WRmodels, which have much higher envelope compactnesses.Model W50S1 has a higher core compactness thanmodel W40S1 (0.55 and 0.37 respectively), but alower envelope compactness (22 and 27, respectively).Model W50S1 ejects less mass and with lower en-ergy (5.4 × − M (cid:12) and 1.9 × erg) than W40S1(1.4 × − M (cid:12) and 6.7 × erg). While the energy per unitmass is comparable, it is higher in the progenitor with higherenvelope compactness (W40S1).A systematic uncertainty of ∼ in our results usingthe default evolution mode arises from our choice of r in =2 × cm. Model W40S1.7 is identical to W40S1 exceptfor the position of the inner radial boundary set at r in = 2 × cm. The resulting ejecta mass and energy are about higher in the model with the smaller inner radial boundary(see also F18 for further tests on the sensitivity of the choiceof boundary position).3.2. EOS Dependence
Table 3 shows that for the same stellar model and spatialresolution, using the stiffer DD2 EOS to evolve the inner coreresults in more mass ejected and with higher specific energy,by a factor of several, relative to using the softer SFHo orLS220 EOS. Equation (8) shows that the maximum kineticenergy of the shock is proportional to the square of the grav-itational mass lost to neutrinos ∆ M G . All models evolvedwith the DD2 EOS achieve higher values of ∆ M G ( t bh ) thantheir equivalent models evolved with the SFHo EOS by a fac-tor of up to two. Results obtained with the SFHo and LS220 EOSs are usually very similar to each other and often over-lap.The origin of this trend with EOS stiffness is illustrated inFigure 2: the stiffer DD2 EOS yields a longer time to BH for-mation and therefore results in more gravitational mass lostto neutrino emission than in the models that use the SFHoEOS. While there are some differences in the growth rateof ∆ M G , associated with the differing neutrino luminosi-ties, which in turn are most sensitive to the effective nucleonmasses in the EOS (Schneider et al. 2019), these luminos-ity differences are sub-dominant compared to those arisingfrom the time interval during which the PNS emits neutrinos,which is set primarily by the accretion rate and the maxi-mum mass at finite entropy ( M tov ) that the EOS can support,which depends on both cold and thermal pressure compo-nents (Hempel et al. 2012; da Silva Schneider et al. 2020).The same trend of increasing ejecta mass and energy with in-creasing M tov was found by Lovegrove & Woosley (2013)using a parameterized evolution of the inner core.Figure 5 shows the spread in ejected masses due to the EOS(represented as error bars) for our three fiducial progenitors.Aside from the magnitude of the gravitational mass lost and aminor change in the location of the acoustic pulse formation(radius at which t bh (cid:39) t ff ), the evolution of the shock as itpropagates into the envelope is qualitatively the same for allEOSs, as shown in Figure 4A key quantitative difference introduced by the EOS is thatwhen using SFHo or LS220, none of the RSG progenitorseject unbound mass (Table 3). The equation of state usedin FLASH assumes fully ionized nuclei, but it is known thathydrogen recombination can dominate the energetics uponsubsequent expansion of the shock in failed supernovae fromRSGs (Lovegrove & Woosley 2013). To estimate the impor-tance of this missing effect, we compute the recombinationenergy E rec that can be liberated if all the hydrogen con-OS AND NEUTRINO EFFECTS IN FAILED SN E Table 4.
Comparison of the ejecta masses and energies obtained inF18 (high-resolution “e HR” models, using parametric inner coreevolution) and in this work (interpolating ∆ M G ( t ) from GR1D us-ing either the SFHo [S1] or DD2 [D1] EOS and at the same resolu-tion as F18; c.f. Table 5), for our three fiducial progenitors (Table 1).Progenitor M ej ( M (cid:12) ) E ej (10 erg)[S1,D1] F18 [S1,D1] F18R15 [2.2, 3.4] 4.2 [-0.1, 0.5] 1.9B25 [0.03, 0.05] 0.05 [0.5, 1.8] 1.6W40 [1, 6] × − × − [0.06,0.31] 0.25 tained in the ejecta recombines to the ground state, E rec = M H m p χ H (9) M H = (cid:90) M ej X ( M ) dM, (10)with X the mass fraction of hydrogen in the ejecta, m p theproton mass, and χ H = 13 . eV. The resulting recombina-tion energies for RSGs and YSGs are shown in Table 3. Withthe exception of model s20 and the model that maps initialprofiles from GR1D (R15S3), the ejecta for all other RSGprogenitors should be unbound after including this contribu-tion. Regarding YSGs, model Y22S1 can potentially unbindits ejecta marginally when including hydrogen recombina-tion (e.g., with a more precise calculation than our rough es-timate), while for model Y25S1 recombination energy fallsshort by a factor 3 and unbinding is unlikely.The massive BSG progenitor B80 fails with both the SFHoand DD2 EOSs. While the latter yields a larger shock ki-netic energy by a factor ∼ , the qualitative result does notchange: as the shock approaches the stellar surface, it reachesthe floor of temperature and the internal energy begins ris-ing. Prior to that, the mass in the shell is (cid:46) − M (cid:12) andthe outward-moving material is gravitationally bound with E ej (cid:39) − erg. Further study of these failing models willrequire a low-temperature EOS consistent with that used toevolve the presupernova model.Comparing to the results of F18, which were obtained us-ing an analytic model for the inner core evolution and there-fore of ∆ M G ( t ) , we find that their results are in broad agree-ment with our models that use the DD2 EOS. In contrast,results obtained with the softer SFHo EOSs have energieslower by a factor of several compared to F18. A side-by-sidecomparison of results for the same fiducial progenitors and atthe same spatial resolution is shown in Table 4. The relationbetween the two sets of results follows from the fact that F18assumed M tov = 2 . M (cid:12) as an input to their parametric neu-trino scheme. The maximum NS mass at an entropy of k B per baryon correlates well with t bh when comparing amongdifferent EOSs (Hempel et al. 2012). The choice of F18 ismuch closer to the finite-entropy M tov for the DD2 EOS( . M (cid:12) at k B per baryon; M. Hempel, private commu- time [s] M G [ M ] t bh t stall W40 SFHo G R D A n a l y ti c Figure 7.
Evolution of ∆ M G as a function of time since core-collapse for the W40 progenitor. The result from GR1D using theSFHo EOS is shown in black, while our linear ramp parameteriza-tion is shown in red, with the star denoting the time of BH forma-tion. The vertical dashed blue line shows the time t stall at whichthe shock stalls, which we use to start the ramp function (same timeas in Figure 3). The vertical dashed red line shows the time of BHformation t bh . nication) than for the SFHo EOS ( . M (cid:12) at k B per baryon,Steiner et al. 2013).3.3. Simplified inner core evolution: linear ramp for ∆ M G To assess the sensitivity of mass ejection to the detailedhistory of neutrino emission by the protoneutron star beforeBH formation, we explore a set of models in which we pa-rameterize ∆ M G ( t ) as a linear ramp in time (Figure 7). Theinput parameters are the maximum value of ∆ M G , the timeof BH formation t bh , and a starting time, which we chooseto be that when the shock radius reaches its maximum value( t stall , same as that used when remapping the domain from GR1D ). These parameters are generally reported in (or other-wise are straightforward to obtain from) published studies ofBH formation in failed SNe.Figure 8 compares the evolution of the energy, mass,and velocities of the outgoing shell for models W40S1 andW40S2, with the former interpolating ∆ M G ( t ) from GR1D and the latter using the analytic ramp prescription. The shockproperties are very close to one another in both models, witha relative difference of in energy and in mass by the endof the simulation. A smaller relative difference in final en-ergy and mass (few percent) is found for the correspondingBSG models (B25S1-B25S2), while for the RSG the massejected is nearly the same whereas the negative ejecta energydiffers by 30% (Table 3).The low sensitivity to the detailed neutrino history can beunderstood from the fact that variations in neutrino emis-sion (as inferred from Figure 2) occur on timescales much2 I VANOV & F
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Figure 8.
Comparison between models W40S1 and W40S2, whichinterpolate ∆ M G ( t ) from GR1D or use an analytic ramp model(Figure 7), respectively. Top, middle, and bottom panels show theevolution of the energies, mass, and velocity of the sound pulse, aslabeled ( v shock and v trail are the velocities of the forward and rearend of the shell, respectively). The vertical dotted line indicates thetime at which the shock emerges from the stellar surface. shorter than t bh . Mass shells for which the local dynamicaltime t dyn is comparable to this neutrino variability timescaleare accreted into the BH before there is sufficient time to af-fect the emergence of the sound pulse at a location such that t dyn ∼ t bh . Marginally bound shocks (from RSGs) are moresensitive to small changes in neutrino emission history thanthose from progenitors that eject mass more robustly.3.4. Effect of spatial resolution
Table 3 reports the mass ejected and final shock energy forselected models at two spatial resolutions: the same as in F18( ∆ r/r = 4 . × − ) for comparison, and at 4-8 times finergrid spacing per decade in radius ( ∆ r/r = 0 . − . × − ).The highest resolution is used to resolve the surface pressurescale height of the W40 model with ∼ cells ( H p /R cc (cid:39) . × − ). For RSGs and BSGs, the surface pressure scale cm)0100020003000400050006000 v e l o c it y ( k m s − ) B25D1 R bo R cc c τ ( r ) shock ( r shock ) numbo WKbo
Figure 9.
Forward shock velocity v shock as a function of forwardshock radius r shock in model B25D1 at high resolution. The numer-ical shock breakout velocity v numbo is obtained by searching for theposition at which v shock = c/τ ( r ) , where τ ( r ) is the optical depthin the presupernova star. For reference, we also show the analyticvalue v WKbo from equation (11). The numerical shock velocity is notsmoothed for this model. height is resolved with { − , − } cells in our low-highresolution models, respectively.The ejecta masses and energies are essentially convergedwith resolution for the RSG and BSG models that use theDD2 EOS (R15D1 and B25D1). Their SFHo counterparts(R15S1 and B25S1), with weaker explosions, are somewhatsensitive to resolution, with changes of a few percent in theirejected mass and − in ejecta energies when increasingthe resolution by a factor 4. For the WR progenitor, on theother hand, resolving the surface pressure scale height evenwith a few cells results in an increase of in ejecta massand energy for model W40S1, with a − change in themore energetic model W40D1. We expect further increasesin resolution to behave in the same way as for RSG and BSGmodels. As discussed in §3.1, an additional ∼ uncer-tainty in ejecta mass and energy arises from the position ofthe inner radial boundary.As an additional diagnostic of our simulations, we examinethe velocity at shock breakout, and compare with analyticformulae (as reviewed in Waxman & Katz 2017). The latterpredict a shock breakout velocity v WKbo (cid:39) v ∗ × (cid:40) M . , v . ∗ , . R − . , ( BSG, WR )4 . M . , v . ∗ , . R − . , ( RSG ) , (11)where v ∗ = (cid:112) E ej /M ej , M ej , = M/ (10 M (cid:12) ) , v ∗ , . = v ∗ / (10 . cm s − ) , and R cc , = R cc / (10 cm ) , and wherewe have ignored the dependence on opacity and detailed stel-lar density profile dependence. The corresponding values of v WKbo are shown in Table 5 for selected models.The shock breakout velocity is measured from the
FLASH simulations as follows. We use the position of the forwardOS
AND NEUTRINO EFFECTS IN FAILED SN E Table 5.
Resolution dependence of key quantities for selected models. Columns from left to right show model name, EOS used in
GR1D , spatialresolution in the
FLASH run ( ∆ r/r = { . , . , . } × − corresponds to { , , } cells per decade in radius in a logarithmicgrid), mass ejected in the FLASH run M ej , total energy of ejecta in the FLASH run E ej , shock breakout velocity measured in the FLASH simulation, and analytic breakout velocity (eq. [11]).Model EOS ∆ r/r M ej E ej v numbo v WKbo ( − ) ( M (cid:12) ) ( erg) (km s − ) (km s − )R15S1 SFHo 4.5 2.20 -0.130 60 ...R15S1 1.1 2.19 -0.119 60 ...R15D1 DD2 4.5 3.38 0.498 80 30R15D1 1.1 3.37 0.489 80 30 M ej ( − M (cid:12) )B25S1 SFHo 4.5 2.90 0.491 900 600B25S1 1.1 2.80 0.399 1,000 500B25D1 DD2 4.5 5.46 1.77 1,300 900B25D1 1.1 5.45 1.76 1,300 900 M ej ( − M (cid:12) )W40S1 SFHo 4.5 1.32 0.059 9,000 9,000W40S1 0.6 1.44 0.067 17,000 10,000W40D1 DD2 4.5 6.15 0.308 10,000 13,000W40D1 0.6 6.30 0.326 20,000 13,000 shock r shock as a function of time to calculate the forwardshock velocity v shock using time-centered finite difference.When the resulting velocity curve is too noisiy, we smooththis function using a Savitsky-Golay filter to suppress fluc-tuations. We then obtain the shock breakout velocity by aniterative process: we start with the shock velocity at the stel-lar surface, and compute the shock breakout optical depth τ bo = c/v shock , which is then used to obtain the shock break-out radius R bo using the optical depth profile in the presuper-nova progenitor. The shock velocity at the newly obtainedposition r = R bo is then used to compute a new opticaldepth, which then yields another value for R bo . This pro-cess converges after a few iterations.Table 5 shows the numerical shock breakout velocities forselected models. Agreement with analytical values is goodto within a factor of . Figure 9 shows an example of theforward shock velocity profile and its relation to the analyticvalue for model B25D1.Given that our simulations resolve the surface pressurescale height in all models, we revisit the prediction of Cough-lin et al. (2018a) regarding the outward acceleration of thephotosphere ahead of the main shock. A photospheric shockis possible given that the entire envelope experiences a nearlyinstantaneous change in the acceleration of gravity, and re-gions near the surface of the star have a rapidly decreasingsound speed with radius. Figure 10 shows radial profiles of velocity at a time when the main shock is about to reach theoriginal pre-collapse surface at r = R cc .The RSG model is such that outward acceleration of mat-ter near the photosphere is very efficient, launching a seriesof surface shocks that reach ∼ R cc by the time the mainshock approaches the stellar surface. These surface shockscontain very little mass, and include ambient material. Partof this dynamics is likely to arise from the temperature be-ing near the floor value of the EOS ( K), which is higherthan the actual surface temperature of the RSG and thereforeimplies inconsitent thermodynamics (fully ionized ions as-sumed in
FLASH ) relative to the EOS used to construct thepresupernova model.A cleaner surface shock is visible in the BSG model, whichdisplays a quiescent ambient ahead of it. The magnitude ofthis surface shock is smaller by a factor of almost ∼ compared to the main shock, and it does not lead to a sig-nificant expansion of the stellar surface ahead of the arrivalof the main shock. For the WR model, the entire region in-side the photosphere moves outward at ∼ km s − and,while no independent shock is visible before the arrival ofthe much faster main shock, this photospheric expansion issignificantly faster than in the RSG and BSG models, as pre-dicted by Coughlin et al. (2018a). Like in the BSG model,however, the expansion of the star is minimal before the ar-rival of the main shock.4 I VANOV & F
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100 s R cc radius (cm)0100 2 4 6 8radius (10 cm)025 1 . . . . cm)0100 Figure 10.
Radial velocity profile of stellar matter at a time just before the main shock reaches the pre-collapse stellar surface (red vertical linelabeled R cc ), for selected models at their highest resolutions. The gas near the photosphere acquires positive velocity even before the arrival ofthe shock, with noticeable ‘surface’ shocks in the RSG and BSG models. The dashed grey line shows the velocity of ambient material in modelR15D1 (the main shock has a smaller velocity but carries more mass than the surface shock). Implications for electromagnetic counterparts
Non-rotating failed supernovae are expected to generateelectromagnetic emission in the form of shock breakout (Piro2013; Lovegrove et al. 2017) and plateau emission (Love-grove & Woosley 2013). The predicted plateau propertiesfor a RSG progenitor are consistent with those of the failedsupernova candidate N6946-BH1 (Gerke et al. 2015; Adamset al. 2017; Basinger et al. 2020). A wider range in emissionproperties is expected when accounting for WRs and BSGs(F18).Following F18, we estimate bolometric shock breakoutproperties in our models using a combination of analyticformulate and quantities measured from the
FLASH simu-lations. The shock breakout luminosity is estimated as (Piro2013) L bo (cid:39) E rad max( t lc , t diff ) , (12)where t lc = R cc /c is the light-crossing time over the stellarradius and t diff = ( R cc − R bo ) /v bo is the radiation diffu-sion (and shock crossing) time, with R bo the stellar radiusat which the optical depth is c/v bo . The radiation energy E rad is obtained for WRs using the formulae in Waxman &Katz (2017), while for BSGs and RSGs it is measured di-rectly from the simulation ( E rad = (cid:82) aT dV over the vol-ume between R bo and R cc at the time when r shock = R cc ).The shock breakout velocity is set to the analytic value inequation (11) except for model R15S1, which has E ej < ,in which case we use the value obtained from the simula-tion (Table 5). The temperature of the breakout emissionis estimated using the luminosity and stellar radius, L bo =4 πR σT .Table 6 shows the shock breakout emission estimatesfor our fiducial models employing the DD2 or SFHoEOSs. For reference, the pre-supernova luminosities are { . , . , . } × L (cid:12) for models R15, B25, and W40, re-spectively. In all cases, the shock breakout luminosity ex-ceeds the presupernova luminosity, with characteristic values ∼ , , and erg s − for RSG, BSG, and WR, re- spectively. The timescales range from days, hours, and sec-onds, and the emission is dominated by optical, UV, and X-rays, for RSGs, BSGs, and WRs, respectively. These esti-mates ignore the effects of an intervening wind (e.g., Cheva-lier & Irwin 2011; Katz et al. 2012; Haynie & Piro 2020),hence they are likely very rough, particularly for WRs. TheEOS introduces a variation of a factor of a few in luminosityand timescale, while the uncertainty in breakout velocity (Ta-ble 5) adds another factor of uncertainty. Overall, resuls arequalitatively the same as those from F18, as expected fromTable 4Plateau emission is estimated using the formulae of Kleiser& Kasen (2014) L pl = 1 . × E / , M − / , R / , κ − / . T / , erg s − (13) t pl = 120 E − / , M / , R / , κ / . T − / , d , (14)where L pl is the plateau luminosity and t pl the plateau du-ration, E ej , = E ej / (10 erg ) , R cc , = R cc / (500 R (cid:12) ) , κ . is the opacity in units of . cm g − , and T rec , is the recombination temperature at the ejecta photospherein units of K (see also Kasen & Woosley 2009). Weevaluate these quantities assuming a recomination temper-ature of K for RSGs and WRs, and
K for BSGsgiven the surface composition, as well as κ . = 1 (F18).The characteristic expansion velocity of the ejecta is v exp = (cid:112) E ej /M ej . Table 6 shows that these properties are againconsistent with those estimated in F18, with plateau emis-sion lasting for years, months, and days for RSGs, BSGs, andWRs, respectively. Luminosities are much fainter than nor-mal supernovae, with characteristic values , , and erg s − for RSGs, BSGs, and WRs, respectively. In thelatter case, plateau emission is fainter than the presupernovaluminosity.In the context of modern optical transient surveys (e.g.,Graham et al. 2019), the most promising signatures of non-rotating failed supernovae remain shock breakout in RSGsOS AND NEUTRINO EFFECTS IN FAILED SN E Table 6.
Estimates for the bolometric shock breakout and plateau emission for our fiducial progenitors, which interpolate ∆ M G ( t ) from GR1D and use the DD2 or SFHo EOS. Columns from left to right show: model name, shock breakout luminosity, breakout time (maximum betweendiffusion and light-crossing times), shock velocity at breakout v bo , effective temperature at breakout T bo , plateau luminosity, plateau duration,and final shock velocity v exp = (cid:112) E ej /M ej . Model R15S1 ejects bound material so no plateau properties are computed, and the breakoutvelocity is obtained from the simulation (Table 5); for all other models the breakout velocity is computed using equation (11). For reference,the pre-supernova luminosities are { . , . , . } × L (cid:12) for models R15, B25, and W40, respectively.Model L bo t bo v bo T bo L pl t pl v exp ( L (cid:12) ) (km s − ) ( K) ( L (cid:12) ) (d) (km s − )R15S1 . d * . ... ... ...R15D1 d
40 0 . B25S1
50 6 h
500 5 9 20 400
B25D1
300 3 h
900 8 20 20 600
W40S1
400 1 s ,
600 140 0 . , W40D1
600 1 s ,
000 150 0 . , and plateau emission in BSGs, both of which have durationsof the order of days and bolometric luminosities erg s − .Shock breakouts from BSGs are bright candidates for surveyswith hour-long cadences and UV capabilities.The emergence of fast blue optical transients (e.g., Droutet al. 2014) as an observational class has driven interest infailed supernovae from BSGs as possible progenitors (e.g.,Kashiyama & Quataert 2015). Although the plateau lumi-nosities by themselves would place these transients at thefaint end of the class (Suzuki et al. 2020), more power canbe extracted from the compact object by extended fallbackaccretion from the ejected shell (F18), which would resultin a brighter lightcurve (e.g., Dexter & Kasen 2013; Moriyaet al. 2019). Such a fallback scenario has been proposedas a possible driver of activity from AT 2018cow (Marguttiet al. 2019). Additional power and/or diversity of propertiescan arise if ejected shells intract with a dense circumstellarmedium (e.g., Tsuna et al. 2020).A dark collapse is still a possibility for some progenitors.Very small amounts of bound ejecta are predicted for ourYSGs and the most massive BSG in our sample. Transientsfrom these events might be faint enough that they are missedby surveys, and collapse is only found in searches for disap-pearing progenitors (e.g., Reynolds et al. 2015). SUMMARY AND DISCUSSIONWe have carried out global simulations of non-rotatingfailed supernovae in spherical symmetry, modeling the evo-lution of the inner supernova core ( r (cid:46) , km) with thegeneral-relativistic, neutrino radiation-hydrodynamics code GR1D . The resulting gravitational mass loss is then used ina Newtonian hydrodynamic simulation that follows the re-sponse of the outer layers of the star to the change in gravity.Relative to previous work by F18, we can now connect theEOS of dense matter to final ejecta masses and energies fromfailed supernovae and the associated electromagnetic emis-sion. We also employ much higher spatial resolution in theouter stellar layers than in previous work, thus addressing uncertainties in ejecta masses, energies, and velocities. Ourmain results are the following:1. – The ejecta masses and energies can vary by a factor ofseveral depending on the stiffness of the EOS. The dominanteffect that determines this dependency is the time to BH for-mation, which is longer for a stiffer EOS, resulting in moregravitational mass lost to neutrinos (Fig. 2) and therefore amore energetic shock (Fig. 4).Previous work by F18, which used a parametric ap-proach to the inner core evolution, is consistent with ourstiff EOS (DD2) results (Table 4) given the maximum NSmass ( . M (cid:12) ) used in their parametric scheme. Our resultsare also consistent with the trends with maximum NS massfound in the work of Lovegrove & Woosley (2013), whichalso used a parametric approach for the inner core evolution.2. – When using a soft EOS (SFHo), our RSG and YSGprogenitors fail to eject unbound mass (Table 3). Accountingfor the energy released by hydrogen recombination (not in-cluded in the FLASH
EOS) could provide sufficient thermalenergy to unbind the ejeta in most of these bound cases.3. – Predictions for shock breakout and plateau emission re-main largely unchanged relative to F18, with variations of afactor of a few in luminosity, duration, velocity, and effectivetemperature (Table 6). The most promising candidates foroptical transient surveys remain the shock breakout from anRSG and plateau emission from a BSG, both of which havedurations on the order of days and luminosities on the orderof erg s − . Detecting shock breakout from BSGs willrequire UV photometry on an hour-long cadence.4. – Using a linear ramp with time for the gravitational masslost to neutrinos (Fig. 7) yields ejecta masses and energieswithin ∼ of those obtained using the detailed history of ∆ M G ( t ) (Fig. 8). Variations in the neutrino luminosity ontimescales smaller than the BH formation time have a very6 I VANOV & F
ERN ´ ANDEZ limited impact on mass ejection properties.5. – Resolving the surface pressure scale height of the WRprogenitors with a few cells results in an increase of ∼ in ejecta masses and energies relative to the unresolved case(Table 5). Further increases in resolution have the most im-pact when mass ejection is weak, as inferred from our RSGand BSG models. Analytic estimates for the shock breakoutvelocity agree to within a factor of ∼ with our numericallymeasured values (Fig. 9).6. – With our fully resolved stellar surfaces, we searched forthe precursor shocks predicted by Coughlin et al. (2018a),finding clear evidence for it in the BSG progenitor (Fig. 10).A strong surface shock is visible in the RSG, but incon-sistencies in the thermodynamics ( T > K in the EOSused in
FLASH ) preclude a more definitive statement. TheWR progenitor does not show a clear surface shock whiledisplaying a noticeable positive velocity for the entire pho-tospheric region as the main shock approaches the stellarsurface. Expansion of the stellar surface prior to the arrivalof the main shock is only significant in our RSG models.The main uncertainty for mass ejection and electromag-netic signal prediction is the angular momentum distributionin the progenitor star. If the infalling stellar material can cir-cularize into an accretion disk, outflows from this disk caneject additional matter or even reverse the infall of stellar lay-ers still in the process of collapsing (e.g., Woosley & Heger2012; Quataert & Kasen 2012; Perna et al. 2014; Kashiyama& Quataert 2015; Kashiyama et al. 2018; Murguia-Berthieret al. 2020; Zenati et al. 2020). General considerations aboutthe angular momentum in presupernova envelopes suggestthat disk formation is ubiquitous (F18), and even in the ab-sence of significant rotation, convective eddies in RSG en-velopes can give rise to transient disk activity once they col-lapse (Quataert et al. 2019).An additional uncertainty for electromagnetic signal pre-diction is the amount of mass present in the circumstellarmedium of the progenitor. In addition to the strong line-driven winds in WR stars, enhanced mass loss prior to core-collapse is regularly inferred from supernova observations(e.g., Bruch et al. 2020). Theoretically, enhanced mass loss(above steady line-driven winds) is expected from binary in-teractions (e.g. Smith 2014) or from additional energy in-jection such was wave energy dissipation (Quataert & Sh-iode 2012; Quataert et al. 2016; Fuller 2017; Fuller & Ro2018). This enhanced mass loss can significantly modify the shock breakout signal (e.g., Chevalier & Irwin 2011; Katzet al. 2012; Haynie & Piro 2020).Besides the outer structure of the progenitor, more pre-cise estimates for the strength of the main shock drivenby gravitational mass loss can be obtained by using ageneral-relativistic treatment for the entire star, better neu-trino transport, and more realistic progenitor models. Multi-dimensional simulations of BH-forming supernovae can alsopredict additional features such as SASI-modulated neutrinoand/or gravitational wave activity (e.g., Pan et al. 2020, inparticular their non-rotating progenitor), which would com-plement the electromagnetic signal in diagnosing the physicsof stellar-mass BH formation, should such an event occur inour Galaxy. ACKNOWLEDGMENTSWe thank Coleman Dean, Steven Fahlman, Craig Heinke,Sharon Morsink, Mathieu Renzo, and Greg Sivakoff for help-ful discussions and/or comments on the manuscript. Wealso thank Matthias Hempel for information about the DD2EOS, and Alex Heger for information about the s20 ands40 progenitors. The anonymous referee provided con-structive comments that improved the manuscript. This re-search was supported by the National Sciences and Engi-neering Research Council (NSERC) of Canada through Dis-covery Grant RGPIN-2017-04286, and by the Faculty ofScience at the University of Alberta. The software usedin this work was in part developed by the U.S. Depart-ment of Energy (DOE) NNSA-ASC OASCR Flash Cen-ter at the University of Chicago. Data visualization wasdone in part using
VisIt graham , cedar , and b´eluga clus-ters. Software:
FLASH version 3 (Fryxell et al. 2000; Dubeyet al. 2009),
GR1D version 1 (O’Connor & Ott 2010),
Matplotlib (Hunter 2007),
MESA version 6794 (Pax-ton et al. 2011, 2013, 2015, 2018),
MESA SDK (Townsend2020),
NumPy (Harris et al. 2020),
VisIt (Childs et al.2012)REFERENCES
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