Type Ia Supernovae from Merging White Dwarfs II) Post-Merger Detonations
Cody Raskin, Daniel Kasen, Rainer Moll, Josiah Schwab, Stan Woosley
aa r X i v : . [ a s t r o - ph . H E ] D ec Draft version January 30, 2018
Preprint typeset using L A TEX style emulateapj v. 12/16/11
TYPE IA SUPERNOVAE FROM MERGING WHITE DWARFS II) POST-MERGER DETONATIONS
Cody Raskin , Daniel Kasen , Rainer Moll , Josiah Schwab , & Stan Woosley Draft version January 30, 2018
ABSTRACTMerging carbon-oxygen (CO) white dwarfs are a promising progenitor system for Type Ia supernovae(SN Ia), but the underlying physics and timing of the detonation are still debated. If an explosionoccurs after the secondary star is fully disrupted, the exploding primary will expand into a denseCO medium that may still have a disk-like structure. This interaction will decelerate and distortthe ejecta. Here we carry out multi-dimensional simulations of “tamped” SN Ia models, using bothparticle and grid-based codes to study the merger and explosion dynamics, and a radiative transfercode to calculate synthetic spectra and light curves. We find that post-merger explosions exhibit anhourglass-shaped asymmetry, leading to strong variations in the light curves with viewing angle. Thetwo most important factors affecting the outcome are the scale-height of the disk, which dependssensitively on the binary mass ratio, and the total Ni yield, which is governed by the central densityof the remnant core. The synthetic broadband light curves rise and decline very slowly, and thespectra generally look peculiar, with weak features from intermediate mass elements but relativelystrong carbon absorption. We also consider the effects of the viscous evolution of the remnant, andshow that a longer time delay between merger and explosion probably leads to larger Ni yields andmore symmetrical remnants. We discuss the relevance of this class of aspherical “tamped” SN Ia forexplaining the class of “super-Chandrasekhar” SN Ia.
Keywords: hydrodynamics – nuclear reactions, nucleosynthesis, abundances – supernovae: general –white dwarfs INTRODUCTION
There are currently two broad classes of models forSN Ia. In the first so-called “single degenerate” sce-nario, a CO white dwarf (WD) accretes material froma main sequence or post-main sequence companion(Whelan & Iben 1973; Nomoto 1982). As the WD growsand approaches the Chandraskehar mass ( M ch ), its cen-tral density increases, and a subsonic deflagration burn-ing front is ignited near the center, likely later tran-sitioning to a detonation (see Hillebrandt & Niemeyer2000, and references therein). Alternatively, a deto-nation may occur in a surface layer of accreted he-lium before the WD has reached M ch , setting up a sec-ondary detonation of the WD core and a subsequentSN Ia (Woosley & Weaver 1994; Livne & Arnett 1995;Fink et al. 2007, 2010; Ruiter et al. 2011; Sim et al.2012; Moll & Woosley 2013).The second, “double degenerate” scenario invokesthe dynamical merger of two WDs in a close binary(Iben & Tutukov 1984; Webbink 1984; Benz et al. 1990;Yoon et al. 2007; Pakmor et al. 2010). Since WDs aresupported by degeneracy pressure, their inverse mass-radius scaling relationship often leads to catastrophicmass loss once one of the pair of WDs overflows its Rochelobe. Marsh et al. (2004) have demonstrated that fora large parameter space of possible WD binary masses,mass transfer is unstable and leads to the complete dis-ruption of the less massive companion star. Nuclear Science Division, Lawrence Berkeley National Lab-oratory, Berkeley, CA, USA Departments of Physics and Astronomy, University of Cali-fornia, Berkeley, CA, USA Departments of Physics and Astronomy, University of Cali-fornia, Santa Cruz, CA, USA
When such a merger occurs, there are several pos-sible outcomes. On the one hand, the violent na-ture of the mass transfer itself may initiate a promptdetonation in the primary WD as material from thecompanion accretes onto its surface on a dynamicaltimescale. This outcome of such “peri-merger” detona-tions has been considered by Pakmor et al. (2010, 2011)and Guillochon et al. (2010) and is the subject of a com-panion paper (Moll et al. 2013). In such a scenario, thecompanion WD has only begun to be tidally disrupted,and remains largely intact at the time of detonation.On the other hand, the companion might be fully dis-rupted before a violent detonation takes place, form-ing a disk around the primary WD. Shear heatingand compression of the differentially rotating rem-nant may possibly set off a detonation in the primary(van Kerkwijk et al. 2010; Shen et al. 2012). If neitherof these outcomes result, the disk will evolve viscouslyto a more spherical geometry (Schwab et al. 2012), andas the degenerate core is compressed on a longer timescale, it may either explode or collapse to form a neutronstar (Saio & Nomoto 1985; Yoon et al. 2007; Shen et al.2012). Raskin & Kasen (2013) have considered howthese outcomes might be distinguished by looking forthe observational signatures of material that has beendynamically ejected during the merger process.WD cores that explode inside the envelope of a dis-rupted secondary star would resemble the “tamped”SN Ia models considered originally by Khokhlov et al.(1993) and Hoeflich & Khokhlov (1996). In those pa-rameterized 1D models, a near- M ch degenerate CO WDwas assumed to explode into a spherical medium of CO.The interaction of the SN ejecta with the CO mediumdecelerated the ejecta, piling up material into a dense Raskin, Kasen, Schwab, Moll, & Woosley 2013 shell. Because the surrounding medium was assumed tobe at relatively small radii ( r . cm) any directemission from the interaction would be difficult to de-tect. The restructuring of the SN ejecta has an impacton the light curves and spectra, however, and a simi-lar shell structure has been invoked to explain the 1–2week velocity plateau seen in some SN Ia (Quimby et al.2007). Fryer et al. (2010) also considered the case wherethe CO medium was more extended and ongoing inter-action influenced the brightness of the light curve.The tamped SN Ia models from WD merger haverecently gained more interest as a possible explana-tion for the class of so called “super- M ch ” SN Ia(Howell et al. 2006a; Hicken et al. 2007; Scalzo et al.2010; Taubenberger et al. 2011; Silverman et al. 2011).These rare events are nearly twice as bright as the mostcommon SN Ia, have broad light curves, and often showstrong carbon lines in their spectra. In addition, theejecta velocities, as measured from the Doppler shift ofthe SiII line, are often ∼
20% lower than those of normalSN Ia. In an empirical analysis of SN 2007if, Scalzo et al.(2010) inferred an ejecta masses of M = 2 . ± . M ⊙ and a Ni mass of M ni = 1 . ± . M ⊙ . These es-timates explicitly assumed a shell-like ejecta structuresuch as that in a tamped SN Ia model. Estimatesfor SN 2009dc were in a similar range, M & . M ⊙ and M ni ≈ . − . M ⊙ (Taubenberger et al. 2011;Silverman et al. 2011). To explain the strong observedcarbon lines, a relatively large mass of unburned CO isapparently required (Hachinger et al. 2012).These inferred properties of the super- M ch SN are dif-ficult to reconcile with CO WD explosions, either in thesingle or double degenerate scenario (Maeda & Iwamoto2009). In rare cases, the merger of two massive WDsmight have a total ejected mass exceeding 2.0 M ⊙ , butthe production of & . M ⊙ of Ni would still be dif-ficult. In simulations where a detonation is assumed tooccur early in the merger process, Ni is synthesizedonly in the primary WD, because the tidal distortion ofthe secondary WD usually reduces its density below thethreshold for burning to nuclear statistical equilibrium.This limits the total Ni produced in peri-merger det-onations to be near or less than the mass of the moremassive WD. If an explosion occurs post-merger, how-ever, the WD primary will be compressed by the coa-lescence with the secondary, and the central density ofthe merged remnant is likely to be higher, allowing for agreater Ni production.One limitation of most empirical analyses of the super- M ch SN has been the assumption of spherical symmetry.If the SN ejecta is asymmetric, the resulting emissionwill be anisotropic and, depending on the viewing angle,the Ni mass required to produce the observed peakluminosity could be reduced (Hillebrandt et al. 2007;Kasen & Plewa 2007). Simulations of peri-merger det-onations find that the ejecta is highly asymmetric due tothe interaction of the exploded primary with the nearlyintact secondary star. For cases with massive ( ∼ . M ⊙ )CO WD primaries, the light curves can approach the ob-served luminosities of the super- M ch events from someviewing angles (Moll et al. 2013). If, however, the deto-nation occurs at a later phase, the secondary will havebeen tidally disrupted into a disk. The resulting asym- metry and orientation effects will be distinctive and havenot been examined before.In this paper, we calculate the observational propertiesof multi-dimensional, tamped SN Ia from WD mergers,and consider, in particular, their relevance to super- M ch SN. A combination of lagrangian and grid-based hydro-dynamical tools is used to model the merger dynamicsand explosion, and a separate radiation transport code isemployed to generate angle-dependent light curves andspectra. In a companion paper (Moll et al. 2013), theresults of simulations of early-time detonations in thesesystems was described. Here we focus on the later stagesof the WD merger process where only a single degener-ate core remains, enshrouded by either a disk or a morespherical distribution of matter (Schwab et al. 2012).The synthesis of Ni is determined and the observableproperties calculated. Section 2 describes the numericalhydrodynamics methods, initial conditions, and mergerand detonation results. Section 3 reports the results ofthe radiation-transport calculations and Section 4 dis-cusses our results. MERGER SIMULATIONS & DETONATIONS
Smoothed particle hydrodynamics (SPH) codes areespecially useful for simulating compact object merg-ers since they conserve angular momentum explicitly.This is crucial for properly modeling the resultantmerger remnant configuration. Following the proce-dure of Raskin et al. (2012), we simulate several pairsof WD binaries whose masses are given in Table 1, using snsph (Fryer et al. 2006). We use the Helmholtz free-energy equation of state (EOS: Timmes & Arnett 1999;Timmes & Swesty 2000) for its applicability to a rangeof states from ideal gas to degenerate electron pressuresupport, and including photon pressure support.
Table 1
Simulated binary mass pairs and the q -parameter. All masses aresolar. m m q ( m /m )0p9-0p6 0.96 0.64 0.670p9-0p8 0.96 0.81 0.841p0-0p6 1.06 0.64 0.601p0-1p0 1.06 1.06 1.001p2-0p6 1.20 0.64 0.531p2-1p0 1.20 1.06 0.881p0-0p4 1.06 0.40 He 0.38 Each of our stars consists of 4 × particles of approxi-mately equal mass, arranged using weighted Voronoi tes-sellations ( wvt : Diehl et al. 2012). Such a configuration,sometimes called a Dirichlet tessellation, maximizes com-pactness for particles of varying size (smoothing length)and greatly reduces the time to relaxation over moreglass-like arrangements. Greater compactness also pre-vents entropy reversals that can sometimes result fromhigh entropy particles slipping through cracks in moreuniform particle arrangements, and does so without theneed for an unsuitably large artificial viscosity. Since in-teractions between particles in SPH codes are typicallydominated by the most massive particle in the interac-tion, constraining our particles to roughly equal massesensures that the interaction of the accretion flow from the ost-Merger Detonations ≈ × Kwith random temperature fluctuations of order ∼ K.Each star has a constant composition of 50% C and50% O (CO) by mass fraction, with the exception ofthe 0.40 M ⊙ star in simulation 1p0-0p4, which is initial-ized with 100% He. Our simulations employ the α -chain13-isotope nuclear burning network, aprox13 (Timmes1999), in order to capture any nuclear burning that mightoccur during the merger phase. This is important sincenuclear energy generation plays a part in setting the scaleheight of the remnant disk.Marsh et al. (2004) studied the dissipative forces in-volved in binary mass transfer for WDs and gave an-alytical estimates of the parameter space for stable andunstable mass transfer. In this case, unstable mass trans-fer refers to a binary system wherein, once mass transferbegins, the secondary star continues to lose mass at anaccelerated rate until its complete disruption on a dy-namical timescale. As seen in Figure 1, which repro-duces the analytical estimates of the stable and unstableparameter space, all of our simulations except for simula-tion 1p0-0p4 lie well within the regime of unstable masstransfer. 1p0-0p4 lies in the region between stability andinstability, and as will be discussed, this simulation alsoexhibits unstable mass transfer. Figure 1.
Stable and unstable mass transfer regimes for binaryWD mergers, reproduced from Marsh et al. (2004) with our simu-lations indicated by crosses. The circled cross, simulation 1p0-0p4,lies in the region of parameter space between always unstable andalways stable.
Additionally, Dan et al. (2011) have shown that ini-tial conditions which place the WDs too near each otheroften result in unrealistic accretion rates and can dis-tort the final remnant configuration. Too vigorous aflow can result in hotter disk temperatures, and thus alarger scale height. As in Raskin et al. (2012), we en-sure robust initial conditions by first relaxing the WDpair in their combined gravitational potential. We thenmove the stars nearer each other gradually until parti-cles residing in the tidal bulge of the less massive WDare just below the rotating-frame potential maximum be- tween the two stars. In this way, the merger simulationbegins with synchronized WDs that are already tidallydistorted (and not ringing), and with the less massiveWD just beginning to overflow its Roche lobe. Figure2 demonstrates this for the initial conditions of simu-lation 1p0-0p6, wherein the 0.64 M ⊙ WD has filled itsRoche lobe, but no single particle has yet traversed thepotential barrier between the two stars. In this case,Φ = Φ g − . r and is normalized to − Figure 2.
A rotating-frame, gravitational potential map for theinitial conditions of simulation 1p0-0p6. In the top frame, a densityslice of the 0.64 M ⊙ star is overlaid with potential contours, withthe first contour at the potential energy of the L1 point, wherethe potential has been normalized to unity. In the bottom frame,a representative subsample of particles is plotted against a traceof the system’s potential through the x -axis. As is evident, thestar has filled its Roche lobe, but no single particle has crossed thepotential barrier between the two stars. Initializing our simulations this way ensures that theresultant remnant disks after accretion will be concen-trated along the equator and not overly spherical as aresult of excessive heating during an unphysically vigor-ous mass-exchange. As will be seen in later sections, thescale height and equatorial concentration of the disk playa crucial role determining the angle-dependent spectra ofthese models.
Remnant Configurations
For all the merger simulations, including 1p0-0p4,there is an extended accretion phase that persists for
Raskin, Kasen, Schwab, Moll, & Woosley 2013 several orbits. Some material is lost from the companionand becomes unbound from the system during this phasethrough the L2 point ( M ej ∼ − M ⊙ ), as describedin Raskin & Kasen (2013). Eventually, the companionstar is completely disrupted and forms a hot accretiondisk around the primary. For simulation 1p0-1p0, smallasymmetries between the two stars were sufficient to un-bind one of the stars rather than both. This mechanismis described in more detail in Raskin et al. (2012). Theresult of each of these simulations is a mostly degeneratecore surrounded by a thermalized disk of C and O.The exception is simulation 1p0-0p4, where the disk iscomprised almost entirely of He with faint traces of Cand O lost from the surface of the primary and as aresult of some very mild burning during the merger. Allof the material that rotates as a solid body is assumedto be the core, with the rest of the material comprisinga hot disk envelope. The majority of the disk materialfollows a roughly r − profile as was found in Fryer et al.(2010). Figure 3 demonstrates that much of the disk issub-Keplerian as it is partially thermally supported. Figure 3.
Disk profiles for simulation 1p0-0p6. The upper paneldisplays the density profile of the disk, roughly reproducing the r − profile of Fryer et al. (2010), while the lower panel demonstratesthe sub-Keplerian rotation rate of the disk. In general, the scale heights of the disks are correlatedwith the q -values of the binary systems, with smaller q -values resulting in larger scale heights. For scale heightsof H ≈ − × cm, listed in Table 2, we ex-pect the viscous evolution of these disks to behave likethat of thick disks, with evolution times ∼
10 hours(van Kerkwijk et al. 2010). It is notable that simulation1p0-0p4 features a remnant disk with a scale height of H ≈ cm. The disruption of the companion in thissimulation was much more catastrophic than in the CO-CO mergers since small changes in mass lead to largerincreases in the radius of an already reduced mass com-panion. In addition, helium burning liberates consid-erably more energy in this simulation, thermalizing thedisk to a greater extent than in the CO-CO mergers.This large discrepancy in the scale heights between theCO-CO mergers and simulation 1p0-0p4 has an impor-tant impact on the variability of the spectra at different viewing angles of the SN produced inside these disks.This effect will be discussed further in § Table 2
Scale heights of the remnant disks of each of our mergersimulations. q H [10 cm]0p9-0p6 0.67 4.370p9-0p8 0.84 2.181p0-0p6 0.60 4.211p0-1p0 1.00 2.501p2-0p6 0.53 4.051p2-1p0 0.88 2.581p0-0p4 0.38 9.59 For the most part, little nuclear burning takes placeduring the merger process. Though the SPH simula-tions of many of the mergers do reach detonation condi-tions at the surface of the primary as prescribed by e.g.
Seitenzahl et al. (2009), these simulations, nevertheless,do not detonate self-consistently. There are a variety ofpossible explanations for this, some of which are code-dependent. Guillochon et al. (2010) have found that insimilar models with highly resolved accretion flows ofpure He, self-initiating surface detonations can some-times result. However, hydrodynamical simulations ofthe scale presented in this paper cannot accurately re-solve the detonation conditions inside the accretion flow.Moreover, it remains an open question as to how WDmergers or their remnants ignite in nature, and since weaim to quantify the effects of the remnant disk (or vis-cously evolved envelopes) on the light curves and spectraof SN Ia embedded within them, for this paper, we chosenot to insert artificial detonations at these early stagesin the merger simulations in order to explore the conse-quences.
Explosion Dynamics with snsph
Typically, conditions that lead to a self-consistent det-onation in an SPH calculation are stricter than those ofgrid codes, and surface detonations are notoriously diffi-cult to ignite. We do not expect the question of centralignition or surface ignition to drastically affect our con-clusions of how the SN shock interacts with the disk,since this detail mainly concerns the production of Niand not the detonation geometry, and as will be dis-cussed, even the Ni yield is not drastically altered. Forthese reasons, we artificially detonate our merger rem-nants via an instantaneous conversion of the unburnedremnant core to burned ash.During the merger phase of the calculation, we used aprox13 to capture any nucleosynthesis that may occurfrom sub-critical nuclear burning at the shock front ofthe accretion stream with the primary. Once a quasi-equilibrium was reached, the hydrodynamics calculationwas taken offline, and all material at densities > g cm − was instantaneously burned. In order to de-termine the composition of the incinerated material, aseries of 1D white dwarf detonations of various masseswas calculated offline using the kepler
1D implicit hy-drodynamics package (Weaver et al. 1978; Woosley et al.2002). The results were used to create a lookup table thatwas employed for the instantaneous conversion phase in ost-Merger Detonations snsph . The network in these kepler calculations wasadaptive in time (Rauscher et al. 2002), and for thesestudies, included between 380 to 564 isotopes coupleddirectly to the nuclear energy generation at each timestep. The lookup table produced this way provides apost-detonation ash abundance vector with solar metal-licity pollutants for a range of pre-shock densities of sub-Chandrasekhar WDs (Woosley & Kasen 2011). This ashcomposition strongly depends on the core density, andsince the variation in the total Ni yield between anytwo of our pre-computed models is less than 5% by mass,we interpolated between the results of kepler for var-ious initial core masses to approximate abundances fora range of central densities. Figure 4 shows a sampleash composition for a range of pre-shock densities froma kepler calculation of a 1.2 M ⊙ WD detonation.
Figure 4.
Ash composition as a function of pre-burned densityfor a 1.2 M ⊙ WD detonation as calculated by kepler . Only themost abundant isotopes are plotted.
We deposit the nuclear energy release (minus neutrinolosses) as thermal energy. This unbinds the star andleads to explosion shock that propagates through thedisk. We simulate this shock and expansion phase us-ing normal hydrodynamics, again coupled to aprox13 .The approach is based on the fact that the nucleosyn-thesis in a supersonic detonation is primarily a functionof the density of the fuel. However, our instantaneousdeposition of energy neglects the expansion of shockedmaterial that will occur while the detonation is propa-gating through the remnant star. In a later section, wecompare the results obtained this way with those com-puted self-consistently with a grid code, and show thatthe results are very similar.Using aprox13 for the hydrodynamical portion of thedetonation simulation after the instantaneous conversionof the core allows us to capture α -chain nuclear burningin the portions of the core below > g cm − and inthe disk as the SN shock overtakes it. Specifically, thisincludes triple- α burning in the helium disk of simula-tion 1p0-0p4. The hydrodynamical time step in all casesis limited to the smaller of either the Courant conditionor the burning timescale that results in at most a 30%change in internal energy. Unstable isotopes, such as Fe, which has a half-life of ≈ Ni, which has a half-life of roughly6 days, and so we do not track the energy deposition from the decay of these short-lived nuclei. snsph
Detonations
Figure 5 illustrates the time evolution of the shock asit encounters the remnant disk at the equator. Whileit expands relatively freely at the poles, the high den-sity disk material acts to slow the SN ejecta along theequator. The result is an hourglass shape and a strongglobal asymmetry that will influence the light curves andspectra.The isotopic stratification during homology is illus-trated in Figure 6. As expected, the iron-group elements(IGE) are concentrated in the central regions of the SNejecta, while intermediate-mass elements (IME) and COmake up the outer layers respectively. In the plane ofthe disk, both the IGE and IME are slowed due to theinteraction of the SN shock with the remnant disk. Wewill discuss the details of the light curves and spectra in § snsph simulations. Since the Ni yield depends onthe central density of the remnant core, the final productvaries with both the primary mass and the total mass.
Explosion Dynamics with castro
In order to ensure that the shock geometry and thedetails of the shock interaction with the disk are notspurious or method-dependent, we also employ an Eule-rian hydrodynamics code, castro (Almgren et al. 2010;Zhang et al. 2011) for simulations 0p9-0p8 and 1p0-0p6(henceforth referred to as 0p9-0p8c and 1p0-0p6c, respec-tively, when intended to refer to the castro counter-parts to the snsph simulations.). Since castro sim-ulations are computationally expensive as compared to snsph simulations, we restrict our exploration to thesetwo comparators. The equation of state remains theHelmholtz free-energy EOS, and the nuclear energy cal-culations are restricted to a 199-isotope lookup table.The final remnant stage of simulation 0p9-0p8 was in-terpolated onto a 3-level static mesh. Each level con-sists of a cube centered on the point of highest density.The domain sizes are (12 × cm) , (48 × cm) and(192 × cm) . The grid is resolved by 320 zones ateach level in the 1p0-0p6c model, and 256 zones at eachlevel in the 0p9-0p8c model. The innermost level wasdropped when the detonation reached its boundaries.Once the ejecta neared the boundary of the largest do-main, we mapped the data into a new domain twice aslarge, dropping the innermost level of refinement. Thisstep was repeated until the ejecta neared the boundariesof a domain of (3 . × cm) , at which time the ex-pansion is largely homologous, and the internal energy isonly a small fraction ( < . snsph simulations used for calculating themerging process, castro is unable to handle vacua. Thedensity of the ambient medium was set to 10 − g cm − at the beginning, and lowered by factors of 10 down to10 − g cm − during remaps into bigger domains (to fillthe volume not covered by the old domain). The totalmass of the ambient medium filling empty regions is al-ways smaller than 1 . × − M ⊙ .To further increase the numerical stability during theshock breakout and to contain detrimental effects on the Raskin, Kasen, Schwab, Moll, & Woosley 2013
Figure 5.
Time evolution of the shock geometry of simulation 0p9-0p8 as the detonation shock interacts with the remnant disk depictedwith slices in the x - z plane (each time-frame is square). Table 3
Isotope yields and kinetic energy for our snsph simulations.0p9-0p6 0p9-0p8 1p0-0p6 1p0-1p0 1p2-0p6 1p2-1p0 1p0-0p4 He 7 . × − . × − . × − . × − . × − . × − C 0.228 0.229 0.251 0.226 0.261 0.234 2 . × − O 0.315 0.384 0.318 0.452 0.291 0.376 3 . × − Ne 3 . × − . × − . × − . × − . × − . × − . × − Mg 1 . × − . × − . × − . × − . × − . × − . × − Si 0.155 0.215 0.107 0.257 0.0546 0.168 0.120 S 0.121 0.152 0.0811 0.154 0.0414 0.1033 0.115 Ar 2 . × − . × − . × − . × − . × − . × − . × − Ca 1 . × − . × − . × − . × − . × − . × − . × − Ti 4 . × − . × − . × − . × − . × − . × − . × − Cr 9 . × − . × − . × − . × − . × − . × − . × − Fe 1 . × − . × − . × − . × − . × − . × − . × − Ni 0.687 0.664 0.856 0.886 1.09 1.23 0.704KE [erg] 1 . × . × . × . × . × . × . × < A <
40 0.316 0.416 0.216 0.457 0.110 0.304 0.275 A ≥
44 0.716 0.692 0.894 0.926 1.16 1.30 0.734 time step, we employed a velocity cap of 3 × cm s − ≈ . c . A tally of the total kinetic energy subtracted thisway is smaller than 1 . × erg at the end of the simu-lation, which is insignificant compared to the remainingkinetic energy.In Table 4, we compare the two major hydrodynamicscodes used in this paper for simulating detonations, cas-tro and snsph . As snsph is a particle code, it does nothave a lower limit on the size of its interpolants - parti-cles have a fixed mass and their sizes are determined bytheir densities at any time step. Typical particle sizes inhigh density regions are roughly ( ∼ cm) , but canvary considerably. snsph also does not have a fixed do-main size - simulations can grow without bound as theinterpolants are lagrangian. castro Detonations
Figure 7 shows a snapshot of simulation 0p9-0p8c atthe time when the detonation is set off. There is hot ma-terial on the interface between the central post-mergerobject and the disk, indicated by red contours in the plot,but the temperatures are in general too low ( < K)to set off an autonomous detonation. The detonationis forcibly initiated by means of a spherical detonator(indicated by the yellow circle in Figure 7) of radius300 km, central temperature 1 . × K, and outer veloc-ity 8 × cm s − near the surface of the merged object(on the y = z = 0 line at x = 3, where the density isabout 5 × g cm − , see Figure 7). With the explodingstar being surrounded by a disk in every direction in theorbital plane, the ashes are accelerated most effectively ost-Merger Detonations Table 4
A comparison of the hydrodynamics methods, snsph and castro , used to simulate detonations. snsph castro
Initial Condition Binary System Post-Merger RemnantAlgorithm 3D SPH 3D Eulerian MeshInterpolants 4 × particles 3 × (320 /256 ) cellsSmallest Interpolant ( ∼ cm) (4 . × cm) Domain Size – (3 . × cm) EOS Helmholtz HelmholtzReaction Network aprox13 + lookup table lookup tableIgnition Location Global ( ρ > g cm − ) Edge-lit Figure 6.
Isotopic stratification in the supernova remnant of sim-ulation 0p9-0p8 during the homologous expansion phase. The blueregion indicates the highest concentrations of iron-group elements,while the red region demarcates the intermediate-mass element dis-tribution. The brightest areas in each region represent the highestconcentrations by mass-density. The rest of the domain in thisimage is almost entirely carbon+oxygen, as indicated in the lowerpanel which plots density (scaled by mass-fraction of each of thethree major isotope groups) as a function of velocity. along the polar axis.Simulation 1p0-0p6c was similarly detonated by meansof a spherical detonator near the surface of the remnantcore. The nucleosynthetic yield for the castro simula-tions are similar to that of the snsph result, with 0p9-0p8c resulting in ≈ .
72 M ⊙ of Ni and ≈ .
02 M ⊙ of sta-ble Fe and 1p0-0p6c resulting in ≈ .
80 M ⊙ of Ni and ≈ .
03 M ⊙ of stable Fe. Comparing the ejecta velocitiesand isotopic distribution as in Figure 8 for 1p0-0p6(c), wefind very close agreement between the snsph and cas-tro detonation results, despite the snsph detonationsbeing centrally ignited. The nucleosynthetic yields forour castro simulations are shown in Table 5. Ni Production and Viscous Evolution
We find that the Ni production in post-merger deto-nations is greater than one in which the WD primarydetonates during the merger itself (Moll et al. 2013).This is because the coalescence with the disrupted sec-
Figure 7.
Density in the orbital plane at the beginning of the det-onation simulation for 0p9-0p8 with castro . The yellow and redcontours, at 7 × K, shows the location of the manually planteddetonator in the merged object, and hot regions at the interfacewith the surrounding disk. The inset shows a larger region, with awhite square indicating the boundaries of the main plot.
Table 5
Isotope yields and kinetic energy for our castro simulations.1p0-0p6c 0p9-0p8c He 6 . × − . × − C 0 .
293 0 . O 0 .
373 0 . Ne 9 . × − . × − Mg 1 . × − . × − Si 0 .
100 0 . S 5 . × − . Ar 8 . × − . × − Ca 8 . × − . × − Ti 1 . × − . × − Cr 2 . × − . × − Fe 5 . × − . × − Fe 3 . × − . × − Ni 0 .
799 0 . . × . × < A <
40 0 .
175 0 . A ≥
44 0 .
837 0 . ondary WD leads to a compression of the WD primary,increasing its central density. We estimate the nucle-osynthetic yields from these results using the kepler -compiled lookup table. Figure 10 shows that this com-pression occurs quite rapidly a few 100 seconds afterRoche overflow, and increases the estimated Ni yield byabout 30% for simulation 1p0-1p0. In Moll et al. (2013),a peri-merger detonation of simulation 0p9-0p8 in cas-
Raskin, Kasen, Schwab, Moll, & Woosley 2013 (cid:1) (cid:2) (cid:4) (cid:5) (cid:6) (cid:7) r (cid:9) (cid:10) (cid:11)(cid:12)(cid:13)(cid:9)(cid:14) (cid:15) (cid:11)(cid:9)(cid:13)(cid:9)(cid:14) (cid:15) (cid:14)(cid:9)(cid:14) (cid:15) (cid:12)(cid:13)(cid:9)(cid:14) (cid:15) (cid:1) (cid:2) (cid:4) (cid:5) (cid:6) (cid:7) r (cid:9) (cid:10) (cid:11)(cid:12)(cid:13)(cid:9)(cid:14) (cid:15) (cid:11)(cid:9)(cid:13)(cid:9)(cid:14) (cid:15) (cid:14)(cid:9)(cid:14) (cid:15) (cid:12)(cid:13)(cid:9)(cid:14) (cid:15) (cid:1) (cid:16) (cid:4)(cid:5)(cid:6) (cid:7) r(cid:9) (cid:10) (cid:11)(cid:12).(cid:18)(cid:13)(cid:9)(cid:14) (cid:15) (cid:11)(cid:9).(cid:18)(cid:13)(cid:9)(cid:14) (cid:15) (cid:11)(cid:18)(cid:14)(cid:14)(cid:14) (cid:14) (cid:1) (cid:16) (cid:4)(cid:5)(cid:6) (cid:7) r(cid:9) (cid:10) (cid:18)(cid:14)(cid:14)(cid:14) (cid:9)(cid:14) (cid:15) (cid:9).(cid:18)(cid:13)(cid:9)(cid:14) (cid:15) (cid:12).(cid:18)(cid:13)(cid:9)(cid:14) (cid:15) Figure 8.
Comparison of the azimuthally averaged distributionsof Si and Ni during the homologous expansion phase for sim-ulations 1p0-0p6c (left sides of panels) and 1p0-0p6 (right sides).The two methods, castro and snsph , produce very similar results,despite the castro simulations featuring a surface detonation andthe snsph simulations featuring centrally ignited remnants. tro yields 0.58 M ⊙ of Ni, while the post-merger det-onation (included here as 0p9-0p8c) yields 0.72 M ⊙ , a24% increase.Following the merger, the structure of the remnantwill be further modified as the disk viscously evolves.We also wish to examine the effects of the viscous evo-lution in order to explore how the production of Niand Si changes as a function of the time between themerger and subsequent detonation. Schwab et al. (2012)recently simulated this phase using the staggered-meshcode zeus-mp2 (Hayes et al. 2006) (hereafter called sim-ply zeus ) in 2D. Here we employ the same methods fol-lowing the procedures laid out in Schwab et al. (2012).This code utilizes the Helmholtz EOS and approximatesmagnetic stresses via a shear stress term, T ij = ρν ( ∂ i v j + ∂ j v i ) , (1)where ν is the dynamic viscosity coefficient; ν = 3 × − c s Ω k , (2)with c s denoting the local sound speed and Ω k the Kep-lerian angular velocity. As in Schwab et al. (2012), onlythe azimuthal ( T rφ and T θφ ) components in the stresstensor were retained. The merged end-state of many of our snsph simula-tions are initialized in zeus on a spherical polar grid,with logarithmic grid spacing in the radial direction. Af-ter many viscous times ( ≈ s), the core density foreach simulation rises precipitously, with e.g. zues counterpartto snsph simulation 1p0-1p0) rising from 7 . × gcm − to 2 . × g cm − . The disks in these sim-ulations evolve to a more spherical shape, as shown inFigure 9. This more closely resembles the tamped con-figurations in the 1D models of Khokhlov et al. (1993)and Hoeflich & Khokhlov (1996). (cid:1) [ (cid:3) (cid:4) (cid:5) (cid:6) (cid:7)(cid:8) (cid:9) (cid:10)(cid:11)(cid:7)(cid:8) (cid:9) (cid:12)(cid:11)(cid:7)(cid:8) (cid:9) (cid:13)(cid:11)(cid:7)(cid:8) (cid:9) (cid:14)(cid:11)(cid:7)(cid:8) (cid:9) (cid:15)(cid:11)(cid:7)(cid:8) (cid:9) (cid:16)[(cid:3)(cid:4)(cid:5)(cid:6) (cid:7)(cid:8) (cid:9) (cid:10)(cid:11)(cid:7)(cid:8) (cid:9) (cid:12)(cid:11)(cid:7)(cid:8) (cid:9) (cid:13)(cid:11)(cid:7)(cid:8) (cid:9) (cid:14)(cid:11)(cid:7)(cid:8) (cid:9) (cid:15)(cid:11)(cid:7)(cid:8) (cid:9) Figure 9.
The density distribution in the x-y plane of simulation1p0-1p0 after 10 s. Viscous evolution has spherized the system,erasing the disk. We estimate the nucleosynthetic yields from these re-sults by interpolating the kepler table results to thefinal density profiles of the zeus simulations. For 1p0-1p0z, we find a ≈
30% increase in the Ni yield after 10 s, and a commensurate drop in Si production by morethan half (see table 6). As Figure 10 shows, dependingon the time to detonation, the Ni yield can vary byas much as 80%. Simulation 1p2-1p0z also exhibits theexpected decrease in Si, however the increase in Niis not quite as substantial, with more material formingstable Fe, as indicated in Table 6, which lists the Ni, Fe, and Si yields from each of our simulations. RADIATION-TRANSPORT
In order to synthesize light curves and spectra for ourdetonation models, we use the radiative transfer code sedona (Kasen et al. 2006). sedona uses a Monte Carlotechnique wherein photon packets are emitted in the SNejecta envelope which then scatter and absorb through-out a homologously expanding grid. The grid data isinterpolated from the snsph results; given the near per-fect axial symmetry of these models, the models wereazimuthally averaged and the radiation transport calcu-lations run in 2D. The source geometry for the photonpacket flux is first determined by energy deposition fromthe radioactive decay of Ni and Co and by any shock-heated gas. These packets then propagate throughout ost-Merger Detonations Figure 10.
The estimated Ni production for various timesthroughout the merger and viscous evolution of simulation 1p0-1p0(z). After the companion is completely disrupted, there is ashort period of pulsations in the primary core while it adjusts tothe new gravitational potential. This is reflected in the uncertaintyof the Ni yield at ≈
300 s.
Table 6
Isotope yields and kinetic energy for each of our simulations forwhich we initiated detonations, in addition to estimated yields(italics) from viscously evolved remnants. Simulations labeledwith a “c” are castro results, while those labeled with a “z” areresults from zeus . Ni Fe Si KE [erg]0p9-0p6 0.69 0.02 0.16 1.29 × × × × × × × × × the domain where scatterings and absorptions are calcu-lated from the opacities and emissivities of each cell theyencounter. Temperatures for each cell are calculated inan iterative way by fixing the thermal emission rate tothe calculated rates of photon packet absorption plus anysurplus energy from radioactive decay. All photon pack-ets that escape the system along certain lines of sight areused to construct synthetic light curves and spectra. Light Curves & Spectra
Figure 11 plots the near maximum light synthetic spec-tra for a sample of the models. The color coding showsthe orientation which varies from a viewing angle alongthe pole ( θ ≈ ◦ ) to one one along the equator ( θ ≈ ◦ ).The model ejecta exhibit a nearly perfect top/bottom re-flective symmetry, and so the spectra as seen from view-ing angles for 90 ◦ < θ < ◦ are essentially identical tothose with θ < ◦ .Several strong orientation effects can be seen in thesynthetic spectra of models 0p9-0p8, 1p0-1p0, and 1p2-1p0. First, the overall luminosity is higher from the equa-torial views, a result of the larger projected surface areaof the ejecta when viewed from these angles (see Fig-ure 6). Second, the Doppler shifts of most absorptionfeatures are lower for equatorial views, a result of thedeceleration of the ejecta by the surrounding disk. For model 0p9-0p8, the velocity (as measured from the min-imum of the SiII line with rest wavelength 6355 ˚A) is v ≈ ,
000 km s − for θ = 0 ◦ but only 9600 km s − for θ = 0 ◦ . Third, the IME absorption features are gener-ally weaker from the equatorial views. This is also dueto the deceleration of the ejecta, which narrowed the ve-locity range of IMEs above the photosphere. Fourth, thecontinuum is significantly bluer and the ultraviolet (UV)flux is much higher from viewing angles near θ ≈ ◦ .This is presumably due to the reduced line blanketingfrom iron group elements for this orientation. The diskinteraction slows the Ni ejecta to ∼ − in theequatorial regions, which is well below the photosphereat these epochs. In the absence of fast-moving iron, theline blanketing is weak and the light emerges in largepart at blue and UV wavelengths.The spectra of model 0p9-0p8 viewed near the pole( θ < ◦ ) are fairly similar to standard SNeIa (repre-sented here by SN2011fe), though the absorption featuresare slightly higher velocity than normal. However, for thelines of sight nearer the equator ( θ > ◦ ), the spectralack strong SiII and SII features, and more closely resem-ble that of the peculiar SN Ia 1991T (Filippenko et al.1992). However, unlike SN 1991T, strong CII absorp-tion features are seen in the model spectra near 6300 ˚Aand ≈ ⊙ companion starforming a more spherical configuration than in other sim-ulations (see Table 2). While iron absorption appearssomewhat faster than standard, the SiII line, thoughshallow, has a fairly normal velocity. CaII and SII linesare suppressed. No helium lines are visible in the the syn-thetic spectra, despite the large mass of unburned heliumremaining in the outer layers of the ejecta, however thismay be the result of the neglect of non-thermal excita-tion in these LTE radiative transfer calculations (Lucy1991).Figure 12 plots the synthetic broadband light curvesfor several of our simulations as viewed from either apolar or an equatorial angle. Table 7 also lists the B-band magnitudes at peak and the decline rate ∆m ( B )of the light curves at these two angles. Both the lightcurves rise times and decline rates are slow comparedto those of normal SN Ia. This reflects the relativelylong effective diffusion time in these models, which havea larger total ejecta mass and a larger fraction of un-burned material than standard M ch explosion models.The post-maximum B-band light curve is also stronglyinfluenced by line-blanketing, which affects the declinerate by progressively redistributing flux to longer wave-lengths (e.g., Kasen & Woosley 2007). From the equa-torial viewing angles, the low velocity of the iron groupelements results in reduced line blanketing, and hence aslower B-band decline rates from these orientations.Figure 13 plots the width-luminosity relation (i.e.,peak B-band magnitude vs. the decline rate ∆m ( B )) ofthe models. The peak B-band magnitude of the modelsvary by as much as 0.4 mag depending on the viewing an-0 Raskin, Kasen, Schwab, Moll, & Woosley 2013 gle. From angles near θ = 0 ◦ , the more massive models,in particular 1p2-1p0, predict a peak B-band magnitudeapproaching the brightness of the observed super- M ch events. However, the B-band decline rate of the modelsis smaller than that observed for specific cases such asSN 2003fg. As a whole, the decline rate of the modelsis typically too slow, given their peak brightnesses, whencompared to the observed Phillips relation. Table 7
Peak B-band magnitude and ∆m for each of our simulations forwhich we constructed light curves and spectra at θ = 14 ◦ / θ = 91 ◦ . The three SN whose spectra are shown inFigure 11 are also listed for comparison. (B)0p9-0p6 -18.9/-19.1 0.46/0.380p9-0p8 -18.7/-19.1 1.04/0.461p0-1p0 -19.1/-19.4 0.48/0.241p2-0p6 -19.4/-19.6 0.41/0.361p2-1p0 -19.3/-19.7 0.76/0.211p0-0p4 -19.1/-19.1 0.65/0.561991T -19.87 0.942003fg -19.94 0.92011fe -19.41 1.05 DISCUSSION
We have brought to bear a number of computationaltools to study the effects of various remnant configu-rations from double-degenerate mergers on a SN thatmight explode within them. First, we have shown that ageneric range of disk configurations can result from CO-CO mergers. We find that the disk scale height increaseswith the mass ratio ( q ) of the binary progenitor system.This scale height plays a critical role in the line of sightvariability of the resulting SN spectra. We also find thatmergers with low-mass helium companions can still oc-cupy a portion of parameter space where binary mergersare unstable. Moreover, such high- q mergers form muchmore spherical envelopes, all but erasing the variabilityof the observables with the line of sight.By comparing the results of centrally ignited modelsto surface detonators, we have found that the location ofignition in post-merger detonation has little affect on theSN structure as compared to the presence of an accretiondisk. Both our grid-based and particle-based detonationmodels produce very similar radioactive yields as well,with the grid-based detonations featuring only slightlygreater kinetic energies. This small disparity may onlybe the result of the different methods for igniting thedetonation as opposed to the location of the detonatoror the differences between particle-based and grid-basedhydrodynamics codes.Our synthetic light curves and spectra of most modelsposses strong viewing angle dependences as a result ofthe ejecta asymmetry imparted by the interaction withthe CO disk. The one exception is the He + CO WDmerger 1p0-0p4, where the remnant disk had a large scaleheight due to the high- q value. In general, observationsalong the equatorial region ( θ ≈ ◦ ) feature slower lineabsorptions, and brighter SNe with wider light curves.The sign of this trend conform to the Phillips Relation,but all of our models for which we have constructed lightcurves are much longer-lasting (have much smaller ∆m values) than standard SN Ia. The maximum light spectrawere peculiar from some viewing angles, showing weakIME absorptions and relatively strong CII absorptions.Tamped WD explosions such as those studied herehave often been invoked to explain the class of very lumi-nous, super- M ch SN Ia. In this context, our models havecertain interesting properties: the surrounding CO diskdecelerates the ejecta while remaining unburned, leadingto relatively lower IME absorption velocities, and strongCII lines when the event is viewed from the equatorialregions. In addition, the light curves are also brighterfrom these viewing angles, by as much as 40%, due tothe larger projected surface area of the hourglass shapedejecta. While these trends all have the right sense to ex-plain the super- M ch events, the particular models consid-ered here do not succeed in quantitatively reproducing allof the observed properties. In particular, even the mostmassive model we consider, 1p2-1p0, is slightly dimmer( ∼ . Ni masses inferred for some of thesuper- M ch events ( M ni ≈ . − . M ⊙ ) may thereforebe an overestimate of what is truly required to explainthe observed brightness.We found that for post-merger detonations have anenhanced Ni production compared to peri-merger det-onations, due to compression of the primary WD inthe merged remnant. Our viscous evolution simulationdemonstrate that additional compression occurs in thesubsequent hours. Thus, explosion occurring with longerdelay times after the merger event can result in ∼ Ni. However, longer delay timeswill also result in much more spherical explosions. If thesystem were to explode in this state, or at some subse-quent phase, the observables would be distinct from thoseconsidered here. In further studies, we will consider theimpact of these more evolved systems on the likelihoodof detonations and their ejecta structure and spectra.As the precise timing between WD mergers and sub-sequent SN Ia is still an open question, explorations ofall possible outcomes are important for both populatingand constraining the menagerie of transients. Some por-tion of parameter space in Nature may produce SNe likethose explored here. With the recent deluge of new andunusual transient observations from wide-field surveys,models like these may prove valuable for their classifica-tion.
ACKNOWLEDGMENTS
This research has been supported by the DOE HEPProgram under contract de-sc0010676; the National Sci-ence Foundation (AST 0909129 and AST-1109896) andthe NASA Theory Program (NNX09AK36G). DK is sup-ported by a Department of Energy Office of NuclearPhysics Early Career Award (de-sc0008067). RainerMoll acknowledges support by the Alexander von Hum-boldt Foundation through the Feodor Lynen ResearchFellowship program. JS is supported by an NSF Gradu-ate Research Fellowship. We thank John Bell and AnnAlmgren for their major roles in developing the CASTRO ost-Merger Detonations Figure 11.
Near maximum-light synthetic pectra from simulations 0p9-0p8, 1p0-1p0, 1p2-1p0, and 1p0-0p4 at several viewing anglesfrom 14 . ◦ < θ < . ◦ . High values of θ correspond to lines of sight through the accretion disk. Several prominent absorption featuresare indicated with gray bars, and comparison spectra for a standard SNIa (SN2011fe; Nugent et al. (2011)), a peculiar SNIa (SN1991T;Filippenko et al. (1992)) and a superluminous SNIa (SN2003fg; Howell et al. (2006b)) are plotted with offsets. code. This research used resources of the National En-ergy Research Scientific Computing Center, which is sup-ported by the Office of Science of the U.S. Departmentof Energy under Contract No. DE-AC02-05CH11231.This research used resources of the Oak Ridge Leader-ship Computing Facility at the Oak Ridge National Lab-oratory, which is supported by the Office of Science ofthe U.S. Department of Energy under Contract No. DE-AC05-00OR22725. We are grateful for computer timesupplied by the Advanced Computing Center at ArizonaState University, and to Frank Timmes for technical sup-port and for insightful discussions during the construc- tion of our nuclear network.REFERENCES Almgren, A. S., Beckner, V. E., Bell, J. B., et al. 2010, ApJ, 715,1221, 1221Benz, W., Cameron, A. G. W., Press, W. H., & Bowers, R. L.1990, ApJ, 348, 647, 647Dan, M., Rosswog, S., Guillochon, J., & Ramirez-Ruiz, E. 2011,ApJ, 737, 89, 89Diehl, S., Rockefeller, G., Fryer, C. L., Riethmiller, D., & Statler,T. S. 2012, ArXiv e-prints, arXiv:1211.0525Filippenko, A. V., Richmond, M. W., Matheson, T., et al. 1992,ApJ, 384, L15, L15 Raskin, Kasen, Schwab, Moll, & Woosley 2013
Figure 12.
Lightcurves in the U, B, V, and I bands from simulations 0p9-0p8, 1p0-1p0, 1p2-1p0, and 1p0-0p4 at θ = 14 ◦ (solid lines)and θ = 91 ◦ (dashed lines). o55o(cid:3)nttM(cid:7)(cid:8)ntoo(cid:7)(cid:9) (cid:10) (cid:11) (cid:12)(cid:13)(cid:13)(cid:12) (cid:14) (cid:15) T (cid:17) (cid:9) (cid:13) (cid:18) (cid:19)(cid:12) (cid:20)(cid:21) (cid:22) (cid:23) ntRt o5Rt (cid:26)(cid:27) o- Th(cid:23)i t tRn- tR- tR∆- o oRn- oR-
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