Multi-Dimensional Parameter Study of Double Detonation Type Ia Supernovae Originating from Thin-Helium-Shell White Dwarfs
Samuel J. Boos, Dean M. Townsley, Ken J. Shen, Spencer Caldwell, Broxton J. Miles
DDraft version February 1, 2021
Typeset using L A TEX twocolumn style in AASTeX63
Multi-Dimensional Parameter Study of Double Detonation Type Ia Supernovae Originating fromThin-Helium-Shell White Dwarfs
Samuel J. Boos, Dean M. Townsley, Ken J. Shen, Spencer Caldwell, and Broxton J. Miles Department of Physics & Astronomy, University of Alabama, Tuscaloosa, AL, USA Department of Astronomy and Theoretical Astrophysics Center, University of California, Berkeley, CA, USA Applied Research Associates, Inc. Raleigh, NC, USA (Received January 28, 2021; Revised xxx; Accepted xxx)
Submitted to ApJABSTRACTDespite the importance of Type Ia supernovae (SNe Ia) throughout astronomy, the precise progenitorsystems and explosion mechanisms that drive SNe Ia are still unknown. An explosion scenario that hasgained traction recently is the double detonation in which an accreted shell of helium detonates andtriggers a secondary detonation in the underlying white dwarf. Our research presents a number of highresolution, multi-dimensional, full star simulations of thin-helium-shell, sub-Chandrasekhar-mass whitedwarf progenitors that undergo a double detonation. We confirm the viability of the double detonationacross a range of helium shell parameter space as well as present bulk yields and ejecta profiles foreach progenitor. The yields obtained are generally consistent with previous works and indicate thelikelihood of producing observables that resemble SNe Ia. The dimensionality of our simulations allowus to examine features of the double detonation more closely, including the details of the off-centersecondary ignition and asymmetric ejecta. We find considerable differences in the high-velocity extentof post-detonation products across different lines of sight. The data from this work will be used togenerate predicted observables and may further support the viability of the double detonation scenarioas a SNe Ia channel as well as show how properties of the progenitor or viewing angle may influencetrends in observable characteristics.
Keywords: nuclear reactions, nucleosynthesis, abundances — supernovae: general INTRODUCTIONType Ia supernovae (SNe Ia) are some of the most lu-minous and consequential events in astronomy, but themystery regarding their exact origin remains. While thesource behind their extreme luminosity and character-istic light curve has been known for decades (Pankey1962; Colgate & McKee 1969), there has yet to be acomplete and precise explosion mechanism and progen-itor system model that can fully explain the origin andnature of SNe Ia. This can be disconcerting among thenumerous and diverse areas of research that rely on ob-servations and trends of SNe Ia. A more accurate de-piction of the origin of SNe Ia would serve to improvetheir contributions to many areas of research, includingnucleosynthesis, galaxy evolution, and cosmology.SNe Ia distinguish themselves from other supernovaetypes in the lack of H and He lines in their spectra, in-dicating a C-O white dwarf progenitor (see Maoz et al.2014, for a review). Additionally, SNe Ia have distinc- tive Si, Ca, and Fe spectroscopic features (Parrent et al.2014). A notable trend in the light curves of SNe Ia isthe Phillips relation in which more luminous SNe Ia de-cline more slowly from peak brightness (Phillips 1993).This unique characteristic allows the distance modulusof a SN Ia to be calculated from its decline rate andapparent peak magnitude. The general trend embodiedin the Phillips relation arises from the tendency of Fe-group elements, particularly radioactive Ni, to provideboth the energy source and the principal component ofopacity in the remnant ejecta. Thus a single parametersets the brightness, the diffusion time (Hoeflich et al.1996; Pinto & Eastman 2001), and the time for ion-ization transitions effecting color evolution (Kasen &Woosley 2007). However, such a deterministic trendimplies an underlying systematic variation of the ejectaprofile with Ni mass that itself requires a deeper ex-planation from the explosion mechanism (Woosley et al.2007). A comparable relation between the decline rate a r X i v : . [ a s t r o - ph . H E ] J a n Boos et al. and the intrinsic color of the SNe Ia is similarly criticalin correcting for extinction (Jha et al. 2007). As thefull mechanics that make SNe Ia into this nearly single-parameter family are unclear, further understanding ofthem is beneficial to the role of SNe Ia as a rung inthe cosmic distance ladder and in the determination ofthe Hubble constant (Riess et al. 2016, 2019), as well asevidence for the accelerating expansion of the universe(Riess et al. 1998; Perlmutter et al. 1999; Betoule et al.2014; Scolnic et al. 2018). Reproducing the Phillips re-lation, among other SN Ia traits, via simulation wouldserve to help determine the viability of the particularcandidate model as well as reducing uncertainties in thecalculation of SN Ia distances.While the precise model is still a mystery, there isstrong consensus that SNe Ia are the result of ther-monuclear explosions in C-O white dwarfs (WDs) dueto estimated energetic outputs and the aforementionedspectral features (Maoz et al. 2014; Seitenzahl & Towns-ley 2017). Theories vary in both progenitor system andexplosion mechanism. A promising explosion scenario,which we will discuss here, is the double detonation.In the double detonation, an accreted layer of He at thesurface of a C-O WD ignites and undergoes a thermonu-clear runaway. The secondary explosion in the core maybe triggered directly at the core-shell interface or whenthe inward propagating shock from the initial explosionconverges at some location within the core (Livne &Glasner 1991; Woosley & Weaver 1994; Fink et al. 2007;Fink et al. 2010; Shen & Bildsten 2014; see Townsleyet al. 2019 for a historical overview). Unlike a modelthat requires a primary core ignition, in this scenarioWDs with masses below the Chandrasekhar mass canlead to SNe Ia, increasing the range of theoretically vi-able progenitor systems. Ideally, the WD mass is thesingle parameter, so that the total Ni ejected and theoverall ejecta profile vary together to give, roughly, asingle sequence.A new, compelling motivation for double detonation isthe dynamically driven double degenerate double deto-nation (D ) model (Shen et al. 2018b). In the D model,the shell ignition is brought on by mass transfer from adegenerate companion (Guillochon et al. 2010). As thisscenario would occur in a tight binary, a supernova inthe primary WD would unbind the binary system andeject the companion at a speed close to its pre-explosionorbital velocity. Shen et al. (2018b) identified three pos-sible candidates for these “runaway” stars in the Gaiaastrometric survey, with one being loosely traced backto a supernova remnant.Much research on double detonations has examinedrelatively thick He layers, which produce results that did not match observations of normal SNe Ia very well(Woosley & Weaver 1994; Hoeflich & Khokhlov 1996;Nugent et al. 1997). Also, some work strongly sug-gested that a thin enough He shell would produce aspectroscopically normal SN Ia (Kromer et al. 2010;Woosley & Kasen 2011; Townsley et al. 2012; Mooreet al. 2013). It was then found that the inclusion of keyelements, particularly nitrogen, in the shell abundancesand nuclear reaction network eased the constraints onthe shell masses necessary to produce and host a det-onation (Shen & Moore 2014). Townsley et al. (2019)explored the double detonation scenario by conducting asingle, full-star simulation of a modestly enriched, thin-shell progenitor and found that these cases can indeedproduce spectroscopically normal SNe Ia. We follow upon that work here by exploring a broader set of progen-itors.In this work, we greatly extend the parameter spaceof the thin-shell SN Ia candidate from Townsley et al.(2019). We simulate double detonations in progeni-tors with a wider range of masses and He shell thick-nesses and abundances, generating isotopic yields and2-dimensional ejecta profiles to be used in the generationof predicted observables. The shell masses used in ourthin-shell progenitors are lower than in previous multi-dimensional studies, enabled by our use of an expandednuclear network. We discuss the details of our simu-lations in Section 2, including the progenitors, startingconditions, and methods used. Results of the simula-tions such as yields and profiles, as well as remarkabledetails of the explosion process and comparison to otherwork, are discussed in Section 3. Conclusions, in ad-dition to some discussion on consequential results, arehighlighted in Section 4. METHODSThis study builds on our previous work simulating su-pernovae and uses the astrophysical reactive fluid dy-namics software
FLASH and the nuclear reaction net-works from
MESA . In this section we detail both the cho-sen starting point of our simulation, that is the progen-itor WD and He ignition, as well as the methods usedto simulate the explosion and compute yields and pro-files. Many of these are as used in previous work, butwe also note some changes new to this work, specificallythe improved treatment of the domain-filling “fluff.”2.1.
Progenitors
The parameters of the ten C-O WD progenitors usedin this study are shown in Table 1. Progenitors vary intotal, core, and shell mass as well as He shell geometricthickness as determined by the temperature at the base ouble Detonation SNe Ia Table 1.
ProgenitorsM tot M shell ρ base T base Expansion Time( M (cid:12) ) ( M (cid:12) ) (10 g cm − ) (10 K) (s)0.85 0.033 2 3.0 21.41.00 0.016 2 3.5 97.11.10 0.008 2 3.5 97.21.02 0.021 2 5.0 31.31.02 0.021 a a Decreased O of the shell. They can be categorized into two shell baselayer density regimes: thin (2 and 3 × g cm − ) andthick (6 and 14 × g cm − ). The eight thin casesare the thin-shell candidate progenitors of SNe Ia at thefocus of this study. The two thick shell cases provide auseful comparison to the many other simulation studiesthat examined thick shell detonations (e.g. Fink et al.(2007); Kromer et al. (2010); Polin et al. (2019)).The composition of all but one progenitor very closelymatches that of the modestly enriched shell model fromTownsley et al. (2019). By mass, the core compositionis 0.4 C, 0.587 O, and 0.013 Ne. The shell compo-sition is 0.891 He, 0.05 C, 0.009 N, and 0.05 O.This moderate enrichment enhances the ignition in theshell (Shen & Moore 2014). One thin-shell progenitoruses an alternative shell composition with reduced Oof 0.015, with the difference made up in the He. Thismay be more representative of the enrichment based onmodest mixing between the He layer and the outer partof the core, which is made up of the C-rich ashes of thelate burning stages on the asymptotic giant branch. Theprogenitors are constructed to be hydrostatic with thesame equation of state used by
FLASH and thus necessi-tate no settling when initialized in
FLASH .The core of each progenitor has a uniform temperatureof 3 × K. The He layer has an adiabatic temperatureprofile intended to mock up what may be present dueto mixing induced by a dynamical mass transfer streamfrom the secondary (Guillochon et al. 2010). The choiceof base layer temperature is based on the local tem-perature runaway timescale determined from constant-volume self-heating calculations of material with com-position, density, and temperature corresponding to theshell base layer (see Figure 1). For each shell base den-sity, in most cases, the shell base temperature is chosen
Figure 1.
Runaway timescale contours for He shell mate-rial. The timescale was taken as the temperature runawaytime in constant density self-heating simulations. Circles in-dicate the He shell base parameters of the progenitors usedin this work. Some progenitors have the same shell base den-sity and temperature, but differing core and shell masses orcompositions, and overlap on this plot. such that the corresponding local runaway timescale istens of seconds. This is greater than the time it takesthe He denotation to propagate around the star, about2 seconds, so that ignition only occurs at our chosensite. Two cases have shorter runaway times, compara-ble to the propagation time around the star. These arethe case with a lower oxygen abundance and a standardcomposition case both with masses of 1 . M (cid:12) , whichhave a higher base temperature, 5 × K. This leadsto a geometrically thicker and therefore more massive Heshell for a given core mass and base density. The ques-tion of the appropriate temperature state of the shelland further exploration of how the ignition takes placeare left to separate work.2.2.
Nuclear reaction network and limiter
We utilize the same 55-species nuclear reaction net-work in our simulations as Townsley et al. (2019),designed to obtain accurate energy release for high-temperature He- and C-O-burning. The nuclear reac-tions are integrated during each step in each cell us-ing the constant temperature burning functionality pro-vided by the
MESA/net module version 9793.We use a burning limiter (Kushnir et al. 2013) sim-ilar to that discussed by Shen et al. (2018a) but withsome modifications to make the energy release rate in-
Boos et al. dependent of the timestep and to be more efficient. Theburning integration is limited so that the energy changeper mass in a single timestep | ∆ E| is no larger than c V T | ∆ ln T | max ∆ t hydro / ∆ t sound . Here c V is the mass-specific heat, | ∆ ln T | max is chosen to be 0.1, ∆ t hydro isthe hydrodynamic timestep, and ∆ t sound is the soundcrossing time of the smallest cell in the simulation. As aresult, when the limiter is active, the energy release rate, | ∆ E / ∆ t hydro | only depends on the value chosen for thelimit and the cell size, not the CFL or hydro timestep.Since ∆ t hydro is always smaller than ∆ t sound , and Shenet al. (2018a) limited the energy release during ∆ t hydro in a similar way, the value of | ∆ ln T | max is larger herefor the same energy release rate limit as in Shen et al.(2018a).Having an energy release rate that is independent oftimestep choice ameliorates one of the features of ourlimiter implementation criticized by Kushnir & Katz(2020). Their method also goes further to implementadditional limiter criteria, for example with some treat-ment of composition, that we do not pursue here. Webelieve that our current limiter implementation is suf-ficient for our desired accuracy under the methods uti-lized here. Temperature histories are post-processed toobtain the final yields and there is no explicit treatmentof detonation curvature, which is sensitive to the thick-ness of the reaction front.The energy limit, including calculation of c V , is eval-uated at the beginning of the timestep. The nuclearreactions are then integrated using a sub-stepped Bader-Deuflhard scheme until a time ∆ t hydro is reached or theenergy limit is overstepped. In the latter case, the fi-nal abundances for the step are determined by linearinterpolation between the last two sub-steps of the inte-gration, such that the energy release is the limit ∆ E .2.3. Domain fluff
The portion of the computational domain outside theinitial star is filled with low-density material. We referto this material, as is conventional, as “fluff.” While thismaterial is low enough density to be a negligible portionof the overall mass on the domain, some care in choosingits state is necessary in order to not adversely impactthe computation of the explosion of the star. Thereare two significant issues to be addressed in setting thestate, which are in tension with one another. The firstof these implies that the density should be chosen assmall as possible, while the second limits how small thedensity can be.The lowest possible density is desirable because, forany given fluff density, the outgoing ejecta will even-tually encounter enough fluff that a reverse shock will be created at some finite ejecta speed. Thus a higherdensity fluff limits the time to which a simulation canbe run without the reverse shock falling below some de-sired speed in the ejecta such that it affects a substantialamount of ejecta mass. For example, in previous work(Townsley et al. 2019), and the cases with masses of 0.85and 1 . M (cid:12) with shell base density 2 × g cm − here,a fluff density of 10 − g cm − limited the time of thesimulation to about 25 seconds, by which some regionsof slower eject in the southern hemisphere had a reverseshock below 20,000 km s − . (See Figure 2 in Townsleyet al. 2019.) This limitation has been present in manyof the simulations based on the setups introduced byTownsley et al. (2009).The second issue prevents the density from being cho-sen arbitrarily small. The sound speed in the fluff mustbe kept from being so large that the sound crossing timeof cells in the fluff region do not constrain the timestepto be smaller than it otherwise would be based on thestar’s interior. Adjacent blocks (regions of the grid thatmust have a uniform cell size, typically 8 × × (cid:112) T /ρ , this creates a lower limit on thefluff density.The sound speed constraint also creates an indirectlower limit on the temperature of the fluff near the sur-face of the initial hydrostatic star. The outer edge ofthe star is, by necessity, unresolved on the computa-tional mesh. For any stellar surface, the scale height,which is the length on which the density changes byorder unity, becomes very small compared to the over-all stellar size near the photosphere. This defines the“edge” of the star, where the density drops off to zero.Since this region is unresolved in our initial state, thelast cell containing stellar material (not just fluff) has afinite, and usually appreciable, pressure. With no res-olution limitation, this pressure would be balanced bygravity. However, the limited resolution does not allowthat. As a result, if the temperature of the fluff is chosentoo low so that the pressure in the fluff (which is almostentirely radiation pressure, due to the low density) ismuch lower than the pressure of the last zone of thestar, a shock will be created in the fluff. This shock willcause the temperature in the fluff to rise quite signifi-cantly due to the low density, leading to a sound speedthat is too high. As such, the minimum temperature ofthe fluff near the star’s surface, and therefore the mini-mum density as well, is determined by the resolution ofthe grid at the stellar surface. A finer spatial grid means ouble Detonation SNe Ia − in around 20 secondsor so for a typical supernova ejecta. This problem canbe avoided by introducing an outward down-gradient inboth the density and temperature together, maintaining T /ρ at a uniform value. By making this gradient phys-ically large in extent, any dynamics it introduces due tothe concomitant gradient in pressure is slow enough tonot be relevant to the simulation.We make choices of the fluff profile that, in testing,proved sufficient to keep the reverse shock outside thelocation marked as fluff using a composition-like massscalar for simulations of up to 100 seconds. As a re-sult, the velocity extent of the usable ejecta is now de-termined by the resolution of the grid at the star-fluffboundary. For this work we choose to start the decreasein the T and ρ of the fluff at the star-fluff boundary,decreasing in a log-linear fashion to a radius of 5 × cm, outside of which the density is 10 − g cm − andthe temperature 3 × K. As mentioned above, the T and ρ near the edge of the star is determined by thatnecessary to be similar to the pressure in the last zone ofthe star. Specifically, we set the temperature of the fluffat the edge of the star such that the radiation pressurein the fluff is equal to the gas pressure of the last zone ofthe star. The inner density is such that the sound speedis approximately constant throughout the fluff at about2 × cm s − .2.4. Post-processing for nucleosynthesis
To enable more accurate nuclear yields, a particle trac-ing post-processing technique is used as in Shen et al.(2018a) and Townsley et al. (2019) following completionof the primary hydrodynamic nuclear simulation. Atinitialization, approximately 100,000 Lagrangian parti-cles are distributed throughout the core and shell of thestar. In order to enable sufficient sampling of the shellashes where the ejecta vary the most, 20% of the parti-cles are assigned to the He shell. In addition to havingmore particles per mass in the shell, a region of the shellwithin 10 ◦ of the symmetry axis is chosen for even higherdensity of particles, with 20% of the shell particles des-ignated to this area.The position, velocity, temperature, and density isrecorded for every particle at each timestep in the sim-ulation. MESA ’s one zone burner with a 206-nuclide nu-clear network is used with the temperature and density history of each particle to determine the correspondingelemental evolution. Shen et al. (2018a) used the samepost-processing scheme for similar detonation simula-tions and found that nuclear yields between the mainsimulation and post-processing stages differ by upwardsof 10%, with energetics being consistent within 0.3%.Miles et al. (2019) showed that this method also givesgood agreement with results in which the unresolveddetonation structure is explicitly reconstructed. Com-bining the final position, velocity, and mass fraction ofeach particle with the final state of the simulation, wegenerate a 2-dimensional ejecta profile.2.5.
Simulations
FLASH version 4.3. The built-in nuclear energy deposition in
FLASH is replaced with reaction integration using
MESA described above in Section 2.2.Our simulations utilize an adaptive mesh in
FLASH .With this, regions with large temperature, density, orpressure gradients and energy generation are increasedin resolution. The maximum resolution of the simula-tions is typically 4 km. Regions outside the star are notrefined. The strategy utilized is as described in detail inTownsley et al. (2009). The sub-Chandrasekhar-massWD progenitors used here are about a factor of twolarger in radius than the Chandrasekhar-mass progeni-tors used in Townsley et al. (2009). Thus the resolutionof non-burning regions, needed to maintain hydrostaticbalance of the progenitor, is 32 instead of 16 km. Theexception to this is that full resolution is enforced at thecore-shell composition boundary. The location of thedetonation in the He shell is tracked and this enhancedresolution condition is relaxed after the detonation haspassed a given region. Material that expands beyondthe radius at which the initial stellar surface is locatedhas a minimum cell size of 64 km.Once the burning phases, both He and C detonations,of the explosion are complete, the grid is coarsened toa uniform maximum refinement level that coarsens asthe ejecta expands. The minimum cell size is chosenso that there are no more than 1024 cells between thecenter of the domain and the edge of the expanding shell.This yields several changes in maximum refinement levelbetween the end of the burning at about 3 seconds afterHe ignition and the end of the simulation at 100 seconds.To initiate the shell detonation phase, a circular iso-choric hotspot is placed on the core-shell interface. Thetemperature profile declines linearly from the center ofthe hotspot, with a maximum temperature of 1 . × K and the minimum temperature matching that of the
Boos et al.
Figure 2.
Temperature and density during a double detonation explosion simulation of a 1 . M (cid:12) C-O WD with a thin0 . M (cid:12) He shell. Temperature is shown as a color scale, while white and grey lines are logarithmically equally spaced densitycontours. White contours are spaced every decade and labeled for the three highest values in each frame, while grey contoursare only shown for log ρ = 5 . base layer of the ambient He shell. The hotspot sizesused range between 100 - 500 km and vary across theprogenitors as they were chosen to be the smallest nec-essary, within 50 km, to trigger a He shell detonation.Thinner He shells require larger hotspots to trigger adetonation.In order to properly capture the convergence of theinward traveling shock from the He shell detonation, re-finement of the shock region within the WD is forced un-til ignition is reached. In a single run of the progenitorwith the least massive shell, the maximum refinementwas temporarily improved to 2 km in order to observea core detonation due to the convergence of the inwardshock. The limiter was relaxed briefly during both ig-nition phases as the main goal of this study is to char-acterize the overall explosion outcome, leaving detailedstudy of the ignition processes to separate work.Most of our simulated explosions were allowed to runfor 100 seconds from primary ignition, the large majorityof which the ejecta spends in a generally homologousstate. Three of our runs utilized an earlier fluff schemethat only allowed the simulation to run to around 25seconds. There are no consequential differences betweenejecta profiles in velocity space due to these differing running times, only in the velocity extent of the usableejecta region due to the location of the reverse shock.The 2-dimensional ejecta profiles are constructed on auniform grid in velocity. We use a velocity cell size of500 ×
500 km s − . The scaling from the spatial grid inthe simulation to the velocity grid is determined by av-eraging the relation t exp = r/v r over all the final statesfor the tracer particles. Here v r is the radial compo-nent of the velocity of the material (as sampled by theparticle) and r is the radial location of each particle. t exp is the resulting expansion time, which is the timesince the homologous (i.e. Hubble-flow-like) expansionstarted assuming a time-independent velocity. These areshown for each simulation in Table 1 and are generallya few seconds shorter than the simulation time due tothe time required for the energy deposition prior to ejec-tion. Once t exp is determined, it can be used to map anyspatial location to an equivalent velocity location.The spatial density distribution from the hydrody-namic simulation is then mapped to the velocity gridand regridded (averaged or interpolated) onto the uni-form velocity grid. The abundances are then determinedby averaging the results of the individual tracer parti-cle nucleosynthesis using a cloud-in-cell method basedon the final velocity for each particle. Cells in which ouble Detonation SNe Ia Figure 3.
Zoomed-in frames showing the temperature anddensity during the secondary, core ignition in our simulationof a 0 . M (cid:12) WD with a thin He shell of 0 . M (cid:12) . Temper-ature and density are shown as a color scale and contours,respectively, as in Figure 2. The point of first ignition is lo-cated well away from the symmetry axis in the simulation,the left edge of the domain. Figure 4.
Same as the 2.66 s frame in Figure 3 but for Oabundance (mass fraction) instead of temperature, showingthe core ignition is clearly off the symmetry axis. the mass scalar marking the fluff (non-star) material inthe hydrodynamic simulation is greater than 1% or forwhich v r < r/t exp −
500 km s − , the latter being dueto the impact of the reverse shock, are excluded fromthe ejecta and filled with low density pure He. In theouter regions of the ejecta, some cells may not containany tracer particles. For these, the closest cells thatdo contain particles are determined and abundances areinterpolated where multiple cells are available or extrap-olated using a constant outward profile.Many species present at 100 s will decay by the timeof maximum light. Except as indicated when a spe-cific isotope is designated, we here show total elemen-tal abundances after 12 days of decay using rates in-cluded in the tables available in MESA . Both single- andtwo-step decay chains are computed analytically, whichis sufficient for the 206-nuclide set used for nucleosyn-thetic post-processing. The following elements are notdecayed: Ni, Co, Fe, Cr, and V. These are leftto be treated explicitly in any follow-up radiative trans-fer computations. We note that Mn, produced to theground state in our simulations, is left to decay. RESULTSWe have simulated double detonations in ten uniqueprogenitors and determined the yields and ejecta profilesfor each. This section contains a presentation of ourresults as well as a comparison to previous work anddiscussion of particular highlights.3.1.
Overview of simulation
We first give a brief overview of the explosion processthat occurs in a similar way in each simulation. Figure 2shows representative times from the double detonationexplosion simulation of a 1 . M (cid:12) C-O WD with a thin0 . M (cid:12) He shell. This figure also demonstrates theassumed azimuthal symmetry, such that the domain isa 2-dimensional plane whose left edge ( r = 0 here) isthe symmetry axis. Azimuthal symmetry is appropriateunder the assumption we have made here of a singleignition site. We will refer to the + z hemisphere, wherethe He ignition occurs, as the “northern” hemisphere,and the − z as the “southern”.In the first frame, the He burning can be observedgrowing from a small hotspot placed at the core-shellinterface. At 1.78 s, the He shell detonation is nearingcompletion and the inward-traveling shock in the south-ern hemisphere can be observed in the distortion of thecore density contours. This shock can be seen closer tothe central axis at 2.20 s after the shell He has beendepleted. The convergence of this shock and associatedsecondary ignition are detailed in Figures 3 and 4. By Boos et al.
Table 2.
Explosion Yields 1M tot ρ b, M shell LME Shell LME IME Shell IME HME (- Ni) Shell HME (- Ni) Ni Shell Ni( M (cid:12) ) ( M (cid:12) ) ( M (cid:12) ) ( M (cid:12) ) ( M (cid:12) ) ( M (cid:12) ) ( M (cid:12) ) ( M (cid:12) ) ( M (cid:12) ) ( M (cid:12) )0.85 2 0.033 0.213 0.023 0.499 9.7 × − × − × − × − × − × − × − × − × − × − × − × − × − × − b × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − a Shell base density (10 g cm − ) b Decreased O Dynamics of Secondary Ignition
Since our main goal is to obtain ejecta that can becompared to observed objects, we did not perform adetailed investigation of the secondary ignition. How-ever, it is worthwhile to comment on what we observein our simulation and how it relates to limitations dueto resolution and the imposed axisymmetry. We find, inat least some of our models, a core detonation ignitionthat takes place some distance from the symmetry axis(see Figures 3 and 4 for an example). This is in one ofour thin-shell models, with the resulting C detonationignition point around 7.5 × cm from the symmetryaxis. This feature confirms 1-dimensional work (Kush-nir et al. 2012; Shen & Bildsten 2014) on the expecteddynamics of the ignition, but bears further study in mul-tiple dimensions.In all cases, the trailing edge of the converging shockgrew to temperatures of over 10 K, but did not alwayslead to an obvious ignition before convergence along thesymmetry axis. While most prominent in the higher-mass shell cases, we find that a clearly off-center coreignition does not necessarily occur in all of our runs. Itis difficult to determine precisely if the ignition occursoff-center in the unclear cases due to spatial resolutionissues. Additionally, increasing the resolution in thesecases would only increase the effect that the symmetryaxis has on producing an ignition. The only progenitor that had a systematically uniquedetonation was our thickest-shell case ( M tot = 1 . M (cid:12) , M sh = 0 . M (cid:12) ). In it, the second detonation is triggeredwhere the converging shock meets the core-impactingshell material as the He detonation is completing andcolliding near the south pole, occurring slightly earlierand closer to the core-shell boundary than in our othernine runs. This may to be similar to the “scissor” mech-anism (coined by Gronow et al. 2020) seen in other thick-shell, multi-dimensional simulations (Garc´ıa-Senz et al.2018; Gronow et al. 2020). In Gronow et al. (2020),this alternative core ignition ultimately had little signif-icant effect on the calculated observables between sim-ilar models. While the precise location and mechanismbehind this secondary ignition is different to our otherruns, we still present the yields and ejecta from this caseas useful comparisons. We note that the aspects of thiscore ignition in the thick shell case are particularly sen-sitive to the use of a limiter and choice of resolution andthus leave detailed analysis to separate work.3.3. Yields
The main final yields for each simulation are shownin Table 2. Displayed are the post-processed yieldswith each isotope grouped into low- (Z ≤
10, LME),intermediate- (11 ≤ Z ≤
20, IME), and high-mass ele-ment (Z ≥
21, HME) groups. The Ni yields are sub-tracted from the HME and displayed on their own forclearer analysis. Additionally, yields for additional sig-nificant isotopes are shown in Table 3. Yields that areattributed to the original shell material are shown inaddition to the total yields for each element group andisotope.Core yields are directly correlated with core mass withmore massive progenitors generating more HME, includ- ouble Detonation SNe Ia T a b l e . E x p l o s i o n Y i e l d s M t o t ρ b , M s h e ll C Sh e ll C S i Sh e ll S i C a Sh e ll C a T i Sh e ll T i C r Sh e ll C r ( M (cid:12) )( g c m − )( M (cid:12) )( M (cid:12) )( M (cid:12) )( M (cid:12) )( M (cid:12) )( M (cid:12) )( M (cid:12) )( M (cid:12) )( M (cid:12) )( M (cid:12) )( M (cid:12) ) . . . . × − . . × − . . × − . × − . × − . × − . × − . . . × − . × − . . × − . . × − . × − . × − . × − . × − . . × − . × − . × − . . × − . . × − . × − . × − . × − . × − . . . × − . × − . . × − . . × − . × − . × − . × − . × − . . b . × − . × − . . × − . . × − . × − . × − . × − . × − . . . . × − . . × − . . . × − . × − . × − . × − . . . × − . × − . . × − . . × − . × − . × − . × − . × − . . . × − . × − . . × − . . × − . × − . × − . × − . × − . . . × − . × − . . × − . . × − . × − . × − . × − . × − . . . × − . × − . . × − . . × − . × − . × − . × − . × − Sh e ll b a s e d e n s i t y ( g c m − ) b D ec r e a s e d O ing Ni. This is a reflection of the more complete burn-ing expected in more massive WD detonations. Like-wise, the IME and LME yields generally decrease withcore mass. This can be observed in greater detail inTable 3 where the yields of the relatively lower-mass el-ements ( Si, Ca, and Ti) decrease with core mass.The range of Ni produced in all of our progenitorsare consistent with that of normal, observed SNe Ia(Stritzinger et al. 2006).There appears to be a relationship between shell massand total HME yields, even though very little HME isgenerated in the shell itself, in that more massive shellsresult in slightly more HME being produced in the core.This can be seen most clearly for the 1 . M (cid:12) totalmass progenitors with shell base densities of 2, 3, and6 × g cm − . These three progenitors have HME (in-cluding Ni) core yields that increase with shell basedensity even though the core masses actually decreaseslightly (up to 0 . M (cid:12) ) as shell base density increases.This effect does not appear to be extremely consequen-tial as the HME yields vary at most by about 0 . M (cid:12) .Yields from the shell material are determined mostlyby the combination of the shell density, giving thestrength of the detonation, and the mass within the low-est scale height of the shell. The geometric thickness ofthe shell determines the curvature which also can af-fect the yields in a way that is less dominant than thedensity (Moore et al. 2013). Thus, more complete burn-ing occurs and more HMEs are produced in the shellfor progenitors with higher shell base densities. Addi-tionally, progenitors with more massive cores (and thushigher curvature at the shell) will see weaker burning.This is reflected in our yields. For example, HME and Ni yields, for any choice of total mass (0.85, 1.00, or1 . M (cid:12) ), are higher for denser shells. This trend istrue when weighting for shell mass as well. With re-gards to the effects of curvature, this is most easily seenwhen examining the yields for a choice of shell base den-sity across core masses. For example, the abundance of Ni is higher for the 0 . M (cid:12) progenitor than that of1.00 and 1 . M (cid:12) for both the 2 and 3 × g cm − shell. Both of these trends are consistent throughoutthe yields except for a few special cases, including the Cr produced in the 2 × g cm − , 1 . M (cid:12) case. Thisdiscrepancy is explained in Section 3.5.3.4. Abundance and Ejecta Profiles
Boos et al.
Figure 5.
Post-processed and decayed ejecta abundances (mass fraction) for a selection of our models for some consequentialelemental/isotopic. The top three models represent the thin-shell regime while the bottom two are thick-shell models. Asignificant amount of radioactive shell material is made in the thick shell models only. ouble Detonation SNe Ia Figure 6. tot = 1.00, M sh = 0.021 thin-shell model at threeangles, +45 ◦ , 0 ◦ , and − ◦ , relative to the equator. Boos et al.
Figure 7. × g cm − ) but varying core mass. ouble Detonation SNe Ia Figure 8. Boos et al.
As observed in Figures 5 and 6, the outer regions ofthe abundance profiles are very dependent on the polarangle. Specifically, material generated closer in polarangle to the ignition pole is later found at higher veloc-ities in the ejecta. For example, Figure 6 shows the Tipeak at approximately 21,000, 17,000, and 14,000 kms − for 1-dimensional ejecta profiles at +45 ◦ , 0 ◦ , and − ◦ from the equatorial plane, respectively. Addition-ally, the abundance of the burned products and theirvelocity widths varies with polar angle as well. This canalso be observed in the Ti peaks in Figure 6 where Tipeak abundance and band velocity width increases withpolar angle.The origin of the asymmetric ejecta comes from thesystematically off-centered secondary ignitions in oursimulations. The secondary ignition takes place approx-imately halfway between the core and surface which re-sults in a stronger detonation towards the original ig-nition point. Thus, material generated along any givenradius in the unperturbed progenitor will have a polarangle dependence in its corresponding ejecta distribu-tion. The asymmetric nature of our ejecta has been seenbefore in previous multi-dimensional studies of doubledetonations (Fink et al. 2010; Garc´ıa-Senz et al. 2018;Gronow et al. 2020) and is characteristic of similarlyignited double detonations.In addition to the bulk yields, the velocity extents ofthe burned products in the core are clearly influencedby the choice of progenitor with larger cores resulting infaster outer core and shell ashes. This can be observedin the 1-dimensional abundance profiles at the equatorfor three separate runs in Figure 7. Additionally, wefind no obvious visible effect from shell thickness on theinner region of the ejecta. For example, the distributionof burned core material is mostly unchanged for the pro-genitors in Figure 8 where the core masses are consistentwithin 0 . M (cid:12) .The final tick on the mass coordinate axis in Figure 8indicate the velocity at which the enclosed mass is equalto the original core mass. Notably, this tick is not di-rectly consistent with the position of the products orig-inating from the densest part of the He layer as mightbe typical in a 1-dimensional or otherwise symmetricsimulation (see Figure 3 in Woosley & Kasen (2011) orFigure 1 in Polin et al. (2019)). This is for two rea-sons: the aforementioned asymmetry of the ejecta andsome degree of mixing between core- and shell-burnedmaterial. Inspection of the evolution of abundance pro-files and the core-shell interface shows that this mixingoccurs mostly during the late burning and early ejectastage. Kelvin-Helmholtz (shear) instabilities are gener-ated along the surface of the core as the initial detona- tion proceeds along the shell. The velocity imparted tothe shell by the detonation causes shell material to flowalong the surface of the core, so that some core materialmixes with the bottom of the shell. Additional mixingis also introduced near the end of the secondary deto-nation as it travels back through the shell at an obliqueangle.As seen in the yields, the properties of the shell havea significant influence on the shell ejecta profiles withthicker shells producing more higher-massed elements.This is most clear in examination of the core-shell inter-faces in Figure 8 which presents ejecta profiles of progen-itors with similar core masses but varying shell masses.For the thinnest progenitor in Figure 8, the Ca levelsare low at the interface in addition to very little HME.In the thicker shells, more Ca is produced in additionto the presence of Ti and radioactive isotopes. Thickershells also result in faster outermost shell-burned ejecta.This can be observed in Figure 7 where the outermostburned ejecta velocity, roughly associated with the Sipeak, increases with shell thickness.Two strong Si peaks are apparent in the ejecta of eachof our runs. These originate from the outer parts of boththe core and shell where burning is less complete (e.g.little to no Ni in these regions). The exact propertiesof these peaks vary across the progenitors but are gener-ally consistent. The Si abundance at these peaks reachupwards of 0.5 and 0.1 for the inner and outer peaksrespectively. The velocity where these peaks occur isalso highly dependent on the polar angle. This is bestobserved across the frames in Figure 6, where the ve-locities of the Si peaks drop as the polar angle increasesaway from the original He ignition point. At any givenpolar angle, these peaks are generally found at highervelocities for progenitors with more massive cores. Ad-ditionally, the outer Si peak velocity, along with that ofCa, increases with shell thickness and mass.Extra material and burning can be seen within a fewdegrees along the positive and negative symmetry axisin Figure 5. This is mostly due to the colliding He shelldetonation with the symmetry axis and is consideredto be a likely exaggerated but unavoidable effect in 2-dimensional simulations. 3D double detonation simula-tions from Gronow et al. (2020) show similar but lesspronounced radial protrusions along the ignition axis,but it is unclear whether this is attributable to the Hedetonation or the outgoing core detonation shock.We note that the outermost edge ejecta of the thickestshell in Figure 8 does not quite return to the initial Heshell composition as in all the other presented ejecta.This is related to the artifacts apparent for the samerun in Figure 5 and is a result of poor shell particle ouble Detonation SNe Ia
Reprocessing of shell material
For some of the very densest shell material, there aretwo distinct burning stages. The first burning stage isthe typical shell detonation previously described. Thesecond burning stage occurs when the shock from thecore detonation travels outward through the He shellashes. For most of the shell material, this shock triggerslittle to no burning. The densest material at the bottomof the shell, however, is reprocessed by this shock andburning is extended.This two-phase burning process can be observed inFigure 9 which presents the abundance evolution of anindividual tracer particle that began just above the core-shell interface. The interface is at a density of 2 × gcm − , while this tracer started at a density of 1 . × g cm − and at a polar angle of 58 ◦ . The two burn-ing stages occur at 0.4 and 3.0 s and correspond withthe He detonation and the remnant shock from the coredetonation respectively. In this case, shell material thatoriginally burned to mostly S winds up as predominantlyCa and unburned He with significant traces (abundance > M tot =1 . M (cid:12) , M sh = 0 . M (cid:12) case in Figure 5. There isa compact region of elevated Cr abundance in thenorthern region of the shell ashes. Progenitors withmore massive cores have more significant reprocessingdue to the stronger corresponding outgoing shock. Weattribute the dependence on polar angle to both the de-gree of alignment of the outgoing shock with the radialstellar density gradient, causing more shock strengthen-ing (Miles et al. 2019), as well as the more planar nature
Figure 9.
Abundance (mass fraction) and state evolutionof a single tracer that began the simulation in the shell nearthe core-shell interface in the 1 . M (cid:12) , 0 . M (cid:12) run. Twodistinct burning stages occur at 0.4 and 3.0 s, correspondingwith the He detonation and the remnant shock from the coredetonation respectively. of the secondary detonation shock near the polar region(see panels 4 and 5 in Figure 2).3.6. Comparison to our previous work
For comparison to the single simulation from Townsleyet al. (2019), we examine the 1 . M (cid:12) progenitor fromour work that used our standard He shell abundances.The progenitors are identical between the two simula-tions, but Townsley et al. (2019) did not implement aburning limiter and used tracer particle sampling thatwas more coarse. One notable difference in the outcomesof these two simulations is a modest difference in total Ni yield. Our simulation generates 0 . M (cid:12) , about10%, less Ni than that of Townsley et al. (2019).We believe this difference may be attributed to theuse of a limiter in our study. Other yields are slightlydifferent but are still relatively consistent compared tothe differences between simulations in this work. Basedon the work by Miles et al. (2019), we consider the dif-ferences between this work and that of Townsley et al.(2019) to be consistent with those expected from the dif-fering treatments of the under-resolved detonations. Useof non-limited burning is expected to slightly overpre-dict the completeness of burning, while use of a limiterslightly underpredicts the same. We consider the yieldsusing a limiter to be closer to the correct yields and suf-ficient for current purposes. More accurate yields would6
Boos et al. require more advanced techniques for treating the unre-solved, curved detonation front in the presence of den-sity gradients as discussed in Miles et al. (2019). Thechoice of progenitor is clearly the dominant factor in in-fluencing the final abundances among the cases we studyhere. 3.7.
Comparison to other work
Few studies, especially those in multiple dimensions,reach comparably low shell masses of our progenitorsdue to less-complete nuclear networks preventing theHe shell detonation and as such, direct comparisonsare limited. The bulk yields in our runs of 1 . M (cid:12) and above progenitors are roughly consistent with pre-vious 1-dimensional simulations of detonations in sub-Chandrasekhar WDs of similar masses, both with Heshells (Woosley & Kasen 2011; Polin et al. 2019) andwithout (Sim et al. 2010; Shen et al. 2018a; Miles et al.2019; Kushnir et al. 2020). The Ni yields for the0 . M (cid:12) runs in this study disagree with results of sim-ilar progenitors in some previous studies, producingabout twice as much as that in Polin et al. (2019) andSim et al. (2010). They are, however, more consistentwith yields from Shen et al. (2018a) and Miles et al.(2019) that simulated core ignitions in bare WDs. Bulk Ni yields from our 1 . M (cid:12) models are roughly con-sistent with the 3-dimensional study of Gronow et al.(2020), but are notably higher than that from Garc´ıa-Senz et al. (2018). (See Shen et al. 2018a for a detailedcomparison of yields from sub-Chandrasekhar simula-tions by various authors, including some that disagree.)With regards to HME produced in the shell, our mod-els produce much less than that from Polin et al. (2019)for the very thinnest shell models. For example, the1 . M (cid:12) , 0 . M (cid:12) shell model from Polin et al. (2019)produced over three orders of magnitudes more Ni inthe shell than comparable models from this work. Re-sults for the thick shell yields are more consistent be-tween our work and Polin et al. (2019), in addition tothose from Woosley & Kasen (2011). The thickest shellmodel from this work produces over a magnitude more Ni in the shell than comparable, albeit rotating, pro-genitors in Garc´ıa-Senz et al. (2018).Again, direct comparisons of ejecta profiles are diffi-cult due to a lack of variety of profiles shown in worksfrom previous multi-dimensional and (comparably) thin-shell double detonation studies. We compare our thick-est shell model (0 . M (cid:12) core, 0 . M (cid:12) shell) to model9C (0 . M (cid:12) core, 0 . M (cid:12) shell) from Woosley & Kasen(2011). In our thick shell model, Si can be foundstarting at around 10,000 km s − at the southern pole.The extent of the Si ejecta layers reach upwards of 15- 18,000 km s − , depending on polar angle. In model 9Cfrom Woosley & Kasen (2011), Si is prevalent (abun-dance > − . Ad-ditionally, Ni can be found in our thick shell modelejecta between 11,000 and 26,000 km s − , again depen-dent on polar angle. In the Woosley & Kasen (2011)1-dimensional counterpart, Ni is prevalent between10,000 and 23,000 km s − . As discussed in section 3.4,there are also some differences attributable to mixing atthe core-shell boundary. DISCUSSION AND CONCLUSIONS4.1.
High-velocity material
Regardless of total progenitor mass, very little HMEsor radioactive material are generated in the shell of ourthin-shell models. Rather, the shell ashes are heavilydominated by IMEs, especially Si and Ca. This is con-sistent with the high velocity features observed in SNeIa (Childress et al. 2014; Maguire et al. 2014; Silver-man et al. 2015) given that the material generated inthe shell is found at the highest velocities. The lack ofHMEs generated in the shell region of our models bol-sters the potential for the double detonation scenariocontributing to observed SNe Ia. While radiative trans-fer calculations must be completed to directly compareto observation, the yields and profiles of our thin-shellmodels indicate a range of viable progenitors provideda sufficiently-low shell base density.4.2.
Si velocities and line of sight
If SNe Ia in nature produce asymmetric ejecta as inour models, line of sight to an observed event wouldalmost certainly have an effect on the observables asseen in the singular case of Townsley et al. (2019).Zhang et al. (2020) examined Si ii absorption veloci-ties among observed SNe Ia and showed that the distri-bution is bimodal. This may be due to separate popu-lations (symmetric and asymmetric) in SNe Ia and/orthe nature of asymmetric ejecta itself. Zhang et al.(2020) conducted statistical simulations using line ofsight-dependent Si ii velocities parameterized by the ve-locity range and critical polar angle beyond which thevelocity remains relatively constant. This parametriza-tion is able to fit the line velocity results from our previ-ous work in Townsley et al. (2019) (see Figure 4 in Zhanget al. 2020). It was found that the observed distributioncannot be explained with only the distribution impliedby the 1 . M (cid:12) progenitor and the assumption of a ran-dom line-of-sight. Either a separate, less-asymmetric,population or an angular distribution with high Si ve-locities more constrained to the polar region than foundby Townsley et al. (2019) was required. ouble Detonation SNe Ia ii line velocities from most of our mod-els to resemble the line of sight-dependent model usedby Zhang et al. (2020) to represent those results. Whencomparing the Si ii λ − acrossviewing angles. We expect that the spectral line veloc-ity will be connected to the velocity where the outercore-produced Si peaks, for any given viewing angle.From the comparison of equatorial profiles of modelsat a single mass with different He shell masses in Fig-ure 8, the extent of the core-produced Si does not de-pend strongly on the He shell thickness. However, asseen in both the 2-dimensional ejecta (Figure 5) and thecomparison among equatorial profiles with different WDmasses (Figure 7), the nature of the asymmetry is sen-sitive to the WD mass. The additional dependence onmass presents the interesting possibility that the overallexpected Si velocity distribution may not be just thatseen from the 1 . M (cid:12) progenitor. Precise analysis of theabsorption lines demands radiative transfer calculations,planned for future follow-up work.The outer, high-velocity portion of the ejecta presentseven more asymmetry than that in which the Si ii linesare primarily formed. These present possible opportuni-ties for spectral signatures indicative of specific aspectsof the He shell detonation. One particularly interest-ing aspect of the asymmetric ejecta is the line-of-sightdependence caused by the reprocessing described in Sec-tion 3.5. This leads to the prevalence of some material inthe northern hemisphere, like Ca, that would not other-wise be produced in some runs. If the double detonationscenario contributes to a number of observed SNe Ia, itis possible that this reprocessing may introduce a line-of-sight effect on the spectra and/or photometry that isunique to the double-detonation mechanism. This na-ture of this mechanism encourages further analysis andconsideration across a wider range of potential progeni-tor models with various shell characteristics.4.3. Conclusions
We have simulated double detonations in ten uniqueprogenitors across a range of core masses and shell thick-ness, focusing on the thin-shell candidate model. Wehave presented the results from these simulations inbulks yields and ejecta profiles. In summary, our mainconclusions are: • Yields of our thin-shell progenitors are similar toTownsley et al. (2019), indicating that spectro-scopic features of these progenitors are likely to be consistent with observed SNe Ia. Cases with dif-ferent WD masses but equal He shell base densitiesproduce a similar degree of HMEs in the shell. • HME yields, including Ni, increase with progen-itor mass and cover the range of expectations fornormal SNe Ia. • Shell yields are very sensitive to the choice of shellbase parameters, with shells with a base densitybetween 3 and 6 × g cm − starting to producesignificant amounts of elements that may lead tospectral abnormalities. This will constrain the al-lowed shell thickness. • The core is ignited, in at least some cases, somedistance away from the symmetry axis, confirm-ing 1-dimensional work (Kushnir et al. 2012; Shen& Bildsten 2014), an aspect that requires furtheranalysis in multi-dimensional studies. • Ejecta from our simulations are modestly but sig-nificantly asymmetrical, leading to a line-of-sightdependence of the ejecta velocity extent of certainelements, including Si and Ca. • Non-trivial burning may occur in the shell afterthe shell detonation is complete by the shock fromthe core detonation passing out into the partially-burned shell. Due to the varying angle of the out-going shock, this results in further asymmetriza-tion of the ejection. • Comparison of ejecta profiles to previous 1-dimensional studies reveals differences near thecore-shell boundary that are due to shear mixinginduced by the He detonation.Our simulated double detonations show that sub-Chandrasekhar WDs with a range of He shell parametersand WD masses are viable candidates for some portionof observed SNe Ia judging by the simulations yields andejecta structure. To further verify the likelihood of thecontribution of double detonations on the population ofobserved SNe Ia, radiative transfer calculations must beperformed using these ejecta in order to determine howthese models truly compare to reality.ACKNOWLEDGMENTSThis work was supported by the NASA AstrophysicsTheory Program (NNX17AG28G). Computations wereperformed on NASA’s Pleiades and institutional re-sources at the University of Alabama. We thank CarlaFr¨olich for supporting B.J.M.’s contribution to thiswork.8
Boos et al.
Portions of this work were supported by the UnitedStates Department of Energy, under an Early Ca-reer Award (Grant No. SC0010263), by the Office ofScience, Office of Nuclear Physics, award DE-FG02-02ER41216, awards DE-AC02-05CH11231 and DE-SC0017616 (D.K.), by SciDAC award DE-SC0018297(D.K.), by the Research Corporation for Science Ad- vancement under Cottrell Scholar Awards, and by theGordon and Betty Moore Foundation through GrantGBMF5076 (D.K.).
Software:
FLASH (Fryxell et al. 2000; Dubey et al.2009, 2013, 2014, flash.uchicago.edu),
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