Models for Type Ia supernovae and related astrophysical transients
NNoname manuscript No. (will be inserted by the editor)
Models for Type Ia supernovae andrelated astrophysical transients
Friedrich K. R¨opke · Stuart A. Sim
Received: date / Accepted: date
Abstract
We give an overview of recent efforts to model Type Ia super-novae and related astrophysical transients resulting from thermonuclearexplosions in white dwarfs. In particular we point out the challenges result-ing from the multi-physics multi-scale nature of the problem and discusspossible numerical approaches to meet them in hydrodynamical explosionsimulations and radiative transfer modeling. We give examples of how thesemethods are applied to several explosion scenarios that have been proposedto explain distinct subsets or, in some cases, the majority of the observedevents. In case we comment on some of the successes and shortcoming ofthese scenarios and highlight important outstanding issues.
The theoretical description of Type Ia supernovae and related astrophys-ical transients as thermonuclear explosions of white dwarfs stars has seenrapid development over the past decade. Multidimensional hydrodynamicalsimulations of the explosion phase were conducted, and the results couldbe directly used as input for radiative transfer simulations that derive syn-thetic observables from such models in a consistent way. This allowed toconnect modern supernova theory directly to astronomical observations andfacilitated a way to validate modeling assumptions by comparison with as-tronomical data.
F. K. R¨opkeZentrum f¨ur Astronomie der Universit¨at Heidelberg, Philosophenweg 12, 69120Heidelberg, Germanyand
Heidelberger Institut f¨ur Theoretische Studien, Schloss-Wolfsbrunnenweg 35, 69118Heidelberg, GermanyE-mail: [email protected]. A. SimSchool of Mathematics and Physics, Queen’s University Belfast, Belfast BT7 1NN,UK a r X i v : . [ a s t r o - ph . S R ] M a y The result of theoretical efforts is a consistent theoretical modelingpipeline for thermonuclear explosions in white dwarf stars. It starts outfrom a model of the progenitor and extends over multidimensional hydro-dynamical simulations of the explosion phase. Nucleosynthesis processes init are usually determined in a post-processing step. This gives a multidi-mensional picture of the structure (in particular the density, the velocityand the chemical composition) of the ejecta cloud, that serves as an inputto radiative transfer calculations. These, in turn, allow to derive syntheticobservables.In the two parts of this article, we discuss two main ingredients to thismodeling pipeline: hydrodynamics simulations of the explosion phase to-gether with nucleosynthesis calculations, and the radiative transfer in theejecta. the ignition itself may take centuries or happen dynamically. Thus, ignitionis a marginal case that may be addressed in the framework of hydrody-namical simulations, at least as much as time scales are concerned. Theexplosion itself proceeds in the transonic regime and is certainly accessibleto such a numerical treatment.The time scales on which most observables form are much longer –days, weeks, or months. Because the supernova ejecta are in homologousexpansion by then (hydrodynamical effects are frozen out) and the radiationfield is dynamically unimportant (at least to zeroth order; see Woosley et al.2007 for a discussion of the effect of Ni decay on the density and velocityprofiles), this can be treated in a modeling approach that is separated fromthe hydrodynamical simulations of the explosion phase and uses their resultsonly to define the background state of the expanding ejecta (see Sect. 3).The spatial scale problem in thermonuclear supernova models is no lesschallenging. Due to the extreme temperature sensitivity of the involved nu-clear reactions, burning is confined to the hottest regions and propagatesin thin fronts. Typically, these have widths far below the millimeter scale.This scale is extremely small compared with that of the exploding whitedwarfs (with radii of a few thousand kilometers). Seen from the large globalscales, it is well-justified to approximate combustion waves as sharp discon-tinuities separating the fuel from the ash material. In this discontinuityapproximation, jump conditions over the combustion front can be estab-lished according to the laws of fluid dynamics. They distinguish betweentwo modes of propagation for the combustion front: subsonic deflagration and supersonic detonation .Both deflagrations and detonations are subject to multidimensional hy-drodynamic instabilities (for a recent review see R¨opke 2017). While forthe latter case, it is generally assumed that the effects on the overall ex-plosion process are weak, deflagration burning is most likely dominated –and as a consequence significantly boosted – by the interaction with suchinstabilities. If ignited near the center of the white dwarf star, a defla-gration becomes turbulent. This is an implication of buoyancy instabilitybetween the central hot and light ashes and the dense and cold unburntfuel ahead of the flame. As a result, in the non-linear regime of the Landau-Rayleigh-Taylor instability, bubbles of burning material rise towards thestellar surface (but see Hristov et al. 2017). The flame front is located attheir interfaces. Outside of the bubbles, cold unburnt material sinks downtowards the center of the white dwarf. This leads to shear motions at theflame. Typical Reynolds numbers are as high as 10 and consequently a turbulent energy cascade forms. At the largest scales, kinetic energy is in-jected by large-scale turbulent eddies, that subsequently decay to smallerscales constituting the inertial range, in which kinetic energy is transportedfrom the large to the small scales without energy loss. Only at the micro-scopic Kolmogorov scale, the turbulent energy is finally converted to heatby viscous effects.On a wide sub-range of that turbulent cascade, the deflagration flameinteracts with turbulent eddies (see R¨opke and Schmidt 2009 and R¨opke2017 for discussions of turbulent deflagrations in SNe Ia). The effect of this interaction depends on whether turbulence corrugates the flame structureonly on large scales, or whether it penetrates the internal flame structureand modifies the microphysical transport in it. The first case, which cor-responds to the so-called flamelet regime of turbulent combustion, appliesto most of the explosion period. Here, the flame front is stretched out andwrinkled so that its surface area is greatly enlarged.Only at the latest times, when the star has expanded significantly andthe burning densities are low, the flame structure broadens. With the expan-sion, turbulence gradually freezes out, but if the prevailing turbulent inten-sities are still high, a modification of the flame structure is expected. It hasbeen suggested (e.g. Khokhlov et al. 1997, Lisewski et al. 2000a, R¨opke et al.2007, Woosley 2007, Schmidt et al. 2010, Poludnenko et al. 2011) that in thisregime transitions of the flame propagation mode from subsonic deflagra-tion to supersonic detonation are possible. Such deflagration-to-detonationtransitions (DDTs) are observed in terrestrial chemical combustion, butthere they are mostly associated with obstacles or walls of the combus-tion vessel. The existence of unconfined DDTs, as would be required inthe astrophysical context, remains unproven. Sufficiently strong turbulentmixing inside a broad flame structure is proposed to lead to conditions inwhich a detonation wave can form via the Zel’dovich gradient mechanism(Zel’dovich et al. 1970).In addition to these uncertainties in the flame propagation mechanism,the problem of the initial conditions poses a fundamental challenge to mod-eling thermonuclear supernova explosions. As to now, progenitor systemsof Type Ia supernovae are not observationally established. Although the as-tronomical identification of a progenitor would help to constrain potentialscenarios, it would not completely solve all problems of initial conditionsfor explosion simulations. The progenitor structure and the ignition pro-cess are not directly accessible to observations and have to be modeled.As discussed above, the timescales dominating the pre-ignition evolutionphases cannot easily be addressed in multidimensional simulations. The re-sulting uncertainty in the initial conditions is a fundamental obstacle toexplosion modeling. The equations of hydrodynamics forming the basis forthe description of the explosion processes are hyperbolic partial differen-tial equations. Thus they pose initial value problems. The choice of theinitial conditions therefore has a strong impact on the numerical solution(or even determines it). One should thus avoid to draw conclusions fromthermonuclear supernova simulations that are dominated by an arbitraryof the initial conditions.2.2 Numerical implementationSeveral approaches have been taken by different groups to meet the chal-lenges laid out above and perform simulations of thermonuclear supernovaexplosions. An overview of modeling the combustion physics is given inR¨opke (2017). Here, we will focus on one particular choice.The impracticality to resolve the tiny internal structure of combustionwaves in full-star supernova explosion simulations requires to model their propagation in parametrized approaches. The physical structure is either ar-tificially broadened so that it can be represented on the computational grid(Khokhlov 1995, Vladimirova et al. 2006, Calder et al. 2007), or it is com-pletely ignored and the combustion front is treated as a sharp discontinuityseparating the fuel from the ashes. An appropriate technique to achievethis (at least up to the spread in hydrodynamical quantities introducedby the numerical Riemann solver) is the so-called level-set scheme (Os-her and Sethian 1988, Reinecke et al. 1999). In this front-tracking method,the combustion wave is associated with the zero level-set of a signed dis-tance function G . Its motion is due to advection of the G -field and dueto burning. This is captured by an appropriate “level-set equation”. Whilethe advection part can be determined from the underlying hydrodynamicsscheme, the advancement due to burning is not consistently treated in thediscontinuity approximation. It is a parameter of the model that has tobe determined externally. For laminar deflagration flames, for instance, itcan be derived from resolved one-dimensional simulations (e.g. Timmes andWoosley 1992). For detonations, the Chapman-Jouguet case is a reasonableapproximation at low fuel densities. At higher densities, however, nuclearstatistical equilibrium establishes behind the detonation front and reactionsare partially endothermic. This gives rise to detonations of “pathological”type, that have to be studied in off-line simulations (Sharpe 1999).The fundamental importance of hydrodynamical instabilities for thepropagation of deflagrations requires special modeling approaches. As dis-cussed in Sect. 2.1, the interaction of the flame front with self-generatedturbulence boosts the burning efficiency. Because only the largest scalesof the turbulent cascade are resolved, the effect of flame surface enlarge-ment due to interaction with turbulent eddies on smaller scales has to becompensated by imposing an effective turbulent burning speed on the scaleof numerical resolution. This effective turbulent flame propagation veloc-ity replaces the laminar flame speed in the level-set equation. Accordingto Damk¨ohler (1940), it scales with the turbulent velocity fluctuations onthe considered length scale. Because of numerical dissipation, these are dif-ficult to determine close to the resolution of the computational grid, andtherefore turbulent subgrid-scale models are employed to determine them(see e.g. Niemeyer and Hillebrandt 1995, Schmidt et al. 2006, R¨opke andSchmidt 2009, Ciaraldi-Schoolmann et al. 2009, Hicks and Rosner 2013,Jackson et al. 2014, R¨opke 2017 for a discussion of approaches used inSN Ia explosion models). It is one of the important achievements of multi-dimensional simulations to capture the effect of turbulent flame accelerationin a self-consistent way.Another challenge is the modeling of nuclear reactions that take placein and behind the combustion wave. Two major obstacles have to be over-come in this context. The first is that many reactions are involved in theburning and an extended nuclear network is necessary to predict the syn-thesis of all involved isotopes. Solving the full network concurrently withthe hydrodynamic simulation requires substantial computational effort, inparticular in three-dimensional setups. While this is a practical challenge,the second is more fundamental. If combustion waves are represented as discontinuities, their internal structure and details of the reactions are notcaptured. Artificially broadened combustion waves face the problem thatthe length scales on which the species conversion and energy release pro-ceed physically are not resolved. They are also challenged by the numericaleffort of an extended nuclear network. For this reason, reduced nuclearnetworks are usually employed in the hydrodynamic explosion simulations,that follow only a few representative species (accounting, for instance, forunburnt fuel material, intermediate mass elements, and nuclear statisticalequilibrium compositions). The primary goal of the description of nuclearreactions in the hydrodynamic explosion simulations is to model the en-ergy release driving the dynamics. With reduced networks and artificiallybroadened combustion waves, it is possible to approximate the energy re-lease to a sufficient accuracy. In models with very few representative speciesand/or discontinuity descriptions of the combustion waves, the energy re-lease cannot be consistently reproduced and has to be calibrated. This iseither done on the basis of one-dimensional resolved flame simulations or inan iterative procedure involving a sequence of explosion models and nucle-osynthesis post-processing step. Such post-processing is also necessary toachieve the secondary goal of modeling the burning processes: the deter-mination of detailed nucleosynthetic yields and the chemical structure ofejected material in thermonuclear supernova explosions. The key idea is toplace virtual particles (so-called tracers) in the material of the explodingwhite dwarf star so that each represents a certain fraction of the total mass.These tracer particles are then passively advected with the flow of the ex-ploding material and record the thermodynamic trajectories representativefor the fraction of mass they follow. This data is then used as input to apost-processing step that reconstructs the details of the nuclear reactionsbased on an extended nuclear reaction network (see, e.g., Travaglio et al.2004).The detailed hydrodynamic and chemical structure of the ejected mate-rial is part of a modeling pipeline that follows the supernova event from theprogenitor structure over hydrodynamical explosion simulations and nucle-osynthetic post-processing to the formation of observables that can then becompared to astronomical data. It is input to multidimensional radiativetransfer calculations that will be discussed in Sect. 3.2.3 Requirements for a viable explosion scenarioA fundamental goal of modeling thermonuclear explosions in white dwarfsis to reproduce the characteristic spectral features of Type Ia supernovae.The lack of hydrogen and helium is characteristic for this class of objects.Moreover, spectral features indicate the presence of substantial amounts ofiron group and intermediate-mass elements. This is prototypical for burningcarbon-oxygen white dwarf matter. Irrespective of the combustion wave be-ing a deflagration or a detonation, the released energy and the compositionof the ash depends on the fuel density ahead of the front.At the highest densities, as encountered in the cores of massive whitedwarf stars, the ash temperatures become high enough to establish nuclear statistical equilibrium (NSE) conditions. Freeze-out from NSE occurs whenthe ejecta expand and iron group nuclei are formed. At lower fuel densities,burning is incomplete and intermediate-mass elements (Si, S, Ca, etc.) aresynthesized. At even lower densities, carbon burns to oxygen, and belowa certain threshold, burning ceases and unprocessed carbon-oxygen whitedwarf material is left behind.The fact that intermediate-mass elements are seen in the spectra impliesthat a substantial amount of the stellar material must be processed atsufficiently low densities ( ρ fuel (cid:46) g cm − ) to enable incomplete burning.The burning front therefore must either (1) pre-expand a Chandrasekhar-mass WD, which requires a sub-sonic flame propagation mode, (2) proceedas a detonation in a pre-expanded Chandrasekhar-mass WD in a delayeddetonation scenario, or (3) form a detonation in a sub-Chandrasekhar massWD. We will discuss these possibilities in Sect. 4. considered in any method that aims to predict synthetic spectra. Severalcontemporary radiative transfer approaches embed the assumption of high-velocity gradients in the form of the Sobolev approximation (see e.g. Sobolev1960, Lamers and Casinelli 1999)). This approach makes it relatively easy totake into account very large numbers of bound-bound transitions at modestcomputational cost (either directly, e.g. Mazzali and Lucy 1993, or via anexpansion opacity formalism, e.g. Karp et al. 1977, Blinnikov et al. 1998).However, the Sobolev approximation does have limitations, particularly inrelation to the treatment of overlapping lines (e.g. Baron et al. 1996)),which becomes increasingly common at short wavelengths. Consequently,the most sophisticated radiative transfer codes avoid this approximationand treat individual line profiles in detail.Reasonably accurate radiative transfer simulations also depend stronglyon the calculation of the temperatures and ionization/excitation conditionsin the ejecta. Local thermodynamic equilibrium (LTE) is often adopted as afirst estimate but departures from equilibrium have important consequencesand non-LTE effects become increasingly important with time as the ejectaexpand. As illustrated by Dessart et al. (2014), accurate synthetic observ-ables depend on describing a range of complicated microphysics, whose roleevolves with time. By the latest epochs commonly observed for thermonu-clear supernovae ( ∼ −
300 days post explosion), the ejecta are very farfrom LTE: at these ”nebular” epochs, ionization and heating controlled bythe ongoing injection of non-thermal particles (from radioactive decay) andcooling dominated by multiplets of forbidden lines, predominantly of theiron group elements (for a recent review of the modeling of nebular spectra,see Jerkstrand 2017). Consequently, modeling of spectra at these epochs iscritically dependent on the microphysics and quality/quantity of the atomicdata (radiative and collisional) available for the necessary ions.3.2 Implementation and application to modern explosion modelsGiven the competing demands on implementations and computational re-sources (i.e. need to address complicated/time-dependent microphysics inexpanding, inhomogeneous 3D ejecta models), most published studies todate have made several necessary approximations. Currently, perhaps themost important trade-off made in relation to the study of hydrodynam-ical explosion models is that between simplified microphysics and multi-dimensional effects in the ejecta. For example, while several of the MonteCarlo radiative transfer codes (e.g. SEDONA, Kasen et al. 2006, or AR-TIS, Kromer and Sim 2009) can compute orientation-dependent syntheticobservables for fully 3D ejecta models, these codes use the Sobolev approx-imation and either use LTE or relatively simple nLTE approximations. Incontrast, sophisticated 1D codes can avoid the Sobolev approximation andtreat many levels of many ions in full nLTE for SNe Ia explosion models(see, e.g., H¨oflich et al. 1998, Baron et al. 2006, Blondin et al. 2013). progenitor model/initial guess hydrodynamic multi-Dexplosion simulation nucleosynthesispostprocessing radiative transfersimulation comparison withobservational data
Fig.
3D modeling pipeline
To meet the requirements for a viable explosion scenario discussed in Sect. 2.3,different modes of ignition and flame propagation are necessary, dependingon whether the exploding star is close to the Chandrasekhar-mass or belowthat limit. The first case constitutes so-called Chandrasekhar-mass explo-sion models and the second sub-Chandrasekhar mass models. We explorethese model classes below. They are addressed with our modeling pipelineshown in Fig. 1.4.1 Chandrasekhar-mass white dwarf explosion modelsApproaching the Chandrasekhar mass, the density in the core of a whitedwarf increases steadily. This will lead to the ignition of carbon fusionin the so-called intermediate thermopycnonuclear regime (Gasques et al.2005), i.e. under conditions where the density has an appreciable effect onthe reaction rate. Initially, energy losses due to neutrinos formed in plasmondecays and electron-nucleus bremsstrahlung cool the stellar center. Ignitionoccurs when the central density reaches high enough values so that neutrinolosses become insufficient to balance the energy production due to carbonburning.This does, however, not yet trigger the explosion process. It rather leadsto a century of “simmering”, in which convective motions efficiently trans-port the energy generated in the stellar center outward. The fluid motionsare highly turbulent. On this background, a hotspot finally develops, out ofwhich a deflagration wave is formed by thermonuclear runaway. Simulatingthe entire simmering phase is virtually impossible, because a century can-not be bridged and the spatial resolution is far from resolving turbulence.Nonetheless, ignition simulations have been performed (H¨oflich and Stein2002, Kuhlen et al. 2006, Zingale et al. 2009, Nonaka et al. 2012). A three-dimensional simulation following the last hours until the first thermonuclearrunaway occurs at a radius of ∼
50 km off-center is presented by Nonakaet al. (2012). The results indicate that a second runaway at a different lo-cation shortly after the first is unlikely. Thus, to current best knowledge,the deflagration will form in a single region off-center of the WD star. Fig. Simulation of a deflagration (orange/white contour) in a Chandrasekhar- mass white dwarf star (blue color).
The pure deflagration scenario follows possibility (1) described in Sect. 2.3.After formation of the deflagration wave near the center, it propagatestowards the surface, subject to buoyancy instability. Multidimensional sim-ulations clearly show the formation of “mushroom-shaped” bubbles givingrise to a complex morphology of the flame front (see Fig. 2 for an illustra-tion). The flame strongly accelerates due to the interaction with turbulence.The scaling behavior of turbulent motions in this situation was unclear for along time. Based on highly resolved three-dimensional simulations, however,it could be shown that at small length scales and for most of the burningturbulence is isotropic and follows Kolmogorov scaling (Zingale et al. 2005,Ciaraldi-Schoolmann et al. 2009).The strength of the deflagration and the overall outcome of the explosionphase fundamentally depend on the initial conditions. Several parametersare expected to vary in nature from event to event, such as the centraldensity and the chemical composition of the exploding white dwarf star.Other parameters are unknown or subject to uncertainties in the numericalmodeling. A parameter of models of the explosion phase is the ignitiongeometry. The length scales of the actual flame formation cannot be resolvedin multidimensional simulations. Therefore, the effect is usually mimickedby placing a number of flame kernels near the stellar center. Although thisdoes not necessarily capture the ignition physics, it is a way of defining awell-posed initial setup. For a single sphere, the Rayleigh-Taylor instabilityis seeded by random numerical noise and imprinting a certain spectrum of resolved perturbations on the flame geometry ensures convergence of themodel. The simulations of Fink et al. (2014) showed that the initial flameshape has a tremendous impact on the strength of the burning. In thisstudy, a sequence of models was presented with varying numbers of ignitionsparks, that were randomly placed in the central region of the white dwarfstar. Sparse ignitions naturally lead to aspherical flame geometry evolution,whereas an on average isotropic flame propagation is only possible withdense ignitions. In the former case, only a small fraction of the star isburned. Due to buoyancy, the flame quickly rises towards the surface leavingthe far side of the star unaffected. The energy liberated in this process leadsto the ejection of parts of the stellar material, and a bound remnant is leftbehind. In contrast, a complete disruption of the white dwarf star is possiblefor dense isotropic ignition configurations. In all cases, however, the ejectastructure is well-mixed on large scales due to the flame instabilities. Evenwith dense ignition configurations, the production of Ni hardly exceeds0 . M (cid:12) .The relatively low Ni production means that pure deflagration modelsgenerally fail to account for the observed brightness of the majority ofSNe Ia (the predicted peak luminosity of such models is too low). However,the Ni masses are consistent with those required for some low-luminosityevents, of which several sub-classes have now been identified (Taubenberger2017). In particular, the range of Ni masses predicted in pure deflagrationmodels (Jordan et al. 2012b, Fink et al. 2014) is roughly consistent with theobserved range of brightness spanned by the Type Iax supernovae (Foleyet al. 2013).The potential identification of Type Iax supernovae with pure defla-grations is supported by comparison of synthetic spectra and light curvesto observations. For example, Kromer et al. (2013) compared model pre-dictions for one of the pure deflagration models of Fink et al. (2014) fora range of photospheric-phase epochs to SN2005hk (Phillips et al. 2007),a well-observed member of the SNe Iax class. They found fair agreementin both the strengths and shapes of spectral features across a range ofphases in the optical and infrared, and also good correspondence betweenthe model and observed colors around maximum lights. However, some im-portant discrepancies do remain. In particular, the model light curves evolvetoo quickly, most notably in the post-maximum decline phase in the redderbands (Kromer et al. 2013). This systematic discrepancy is also apparentin the comparison of a different model from the Fink et al. (2014) sampleto a fainter member of the SNe Ia class, 2015H (Magee et al. 2016). Severalstudies have also drawn attention to evidence that the ejecta of SNe Iaxare not fully mixed (Stritzinger et al. 2015, Barna et al. 2017), which isdifficult to reconcile with a turbulent deflagration model. In addition, it re-mains unclear whether pure deflagration models can account for the lowestluminosity members of the SNe Iax class, such as SN2008ha (Foley et al.2009), which requires less than 0 . M (cid:12) of Ni. Such a low mass of Nimight be achieved under conditions whereby burning in the deflagrationis occurs only in a limited central region of the WD, for example due to an exotic composition (Kromer et al. 2015) – however, it remains to bedemonstrated whether this can be realized in nature.One outstanding, but noteworthy, feature of the comparison of SNe Iaxand pure deflagration models is potential role of the residual material fromthe WD, that was still bound at the end of the explosion phase (in e.g. themodels compared to observations mentioned above, ∼ M (cid:12) or more of themass of the initial WD remains bound at the end of the explosion simula-tion). Some of the Ni synthesized in the explosion remains in this material(Kromer et al. 2013, Fink et al. 2014), meaning that it will experience on-going energy injection which will plausible drive further expulsion of mass(Foley et al. 2016). Further study, both of physical conditions in the resid-ual material (Shen et al. 2017) and of late-phase observations of SNe Iax(Foley et al. 2016) are needed to explore this topic in more detail.
Enhancing the Ni production and the explosion energy to values necessaryto reproduce normal SNe Ia is not possible by simply increasing the numberof ignition kernels or tuning the initial parameters of the exploding whitedwarf. A fundamental change in the burning mode is required – a transitionfrom the initial deflagration, that is necessary to pre-expand the material,to a subsequent detonation. This scenario follows possibility (2) describedin Sect. 2.3 This is the idea of the class of delayed detonation models . Thekey question in these is obviously if and how a detonation is triggered in alate burning stage.Several mechanisms have been proposed for initiating detonations indelayed detonation models. A spontaneous deflagration-to-detonation tran-sition (DDT) may be caused by intrinsic processes in the burning wave.Whether or not such DDTs occur in thermonuclear supernova explosionsremains uncertain. Some necessary conditions were laid out in the studiesof (Lisewski et al. 2000b, Woosley 2007, Woosley et al. 2009, 2011). Twoother mechanisms, the gravitationally confined detonation (GCD, Plewaet al. 2004) and the pulsational delayed detonation (PDD) mechanisms(e.g., Bravo and Garc´ıa-Senz 2006), rely on weak initial deflagration stagesthat fail to gravitationally unbind the white dwarf star.The mechanism for GCD assumes that a one-sided ignition leads to anasymmetric deflagration that is too weak to gravitationally unbind the WDstar. The deflagration ash breaks out of the star’s surface and sweeps aroundit to collide in the antipode. Clearly, a successful ignition of a detonationin this collision favors stronger impact which in turn implies a weak defla-gration phase. The resulting detonation then burns the bound core of theobject. With weak deflagrations, it will not be very expanded and thus par-ticularly bright events with high masses of synthesized Ni are expected. Abound white dwarf resulting from a weak deflagration will pulsate. Thesepulsations may aid the formation of a detonation (Jordan et al. 2012a).Both the GCD and the PDD scenarios share the characteristic feature thatthe products of high-density deflagration will be located in the outer part of the ejected material at high velocities. This is in conflict with observations(Seitenzahl et al. 2016).Also for the classical (DDT) delayed detonation scenario, several prob-lems persist. It has been suggested as a model for the bulk of normal SNe Ia.This requires them to reproduce individual supernova observations. Stud-ies based on 1D DDT models have generally been fairly successful in thisregard (e.g. H¨oflich et al. 1998, Blondin et al. 2013). Radiative transfersimulations based on multi-dimensional simulations of DDTs (e.g., in 2DKasen et al. 2009, Blondin et al. 2011, or in 3D Seitenzahl et al. 2013, Simet al. 2013) have also generally found that some DDT models can providea fairly good (albeit far from perfect) match to many properties of thelight curves spectra, and indeed spectropolarimetry (Bulla et al. 2016) ofindividual SNe Ia.In addition, if DDT models are responsible for the full population, ob-served trends between characteristic features should be reproduced. Themodel explosions should be able to cover the range of brightnesses observedfrom normal SNe Ia. This requires a nickel mass production in the rangefrom below 0 . M (cid:12) to 0 . M (cid:12) , see e.g. Scalzo et al. (2014). Delayed detona-tions face a fundamental challenge here. Generally, stronger deflagrationslead to increased expansion before the detonation phase sets in (R¨opke andNiemeyer 2007, Mazzali et al. 2007). Consequently, the detonation runs overlower-density material and produces less Ni. Therefore, the faintest mod-els are expected to be those with the strongest deflagration. This was testedin multi-spot ignition setups that allow to vary the deflagration strengthsignificantly. The strongest deflagrations produce (cid:38) . M (cid:12) of Ni and inthe subsequent detonation little is added to this amount. This means thatmulti-spot ignitions with many, on average isotropically distributed kernelsare required to reach the fainter end of the distribution of normal SNe Ia.These, however seem unlikely to be realized in nature (Nonaka et al. 2012).Furthermore, when such models are invoked, they appear to fail to fullyproduce the relatively rapid light curve evolution that is observed to co-incide with low luminosity (i.e. the light curve width-luminosity relation:see e.g. Sim et al. (2013)). The second problem in this context is that thebrightness distribution of normal events is observed to peak at explosionsproducing ∼ . M (cid:12) of Ni. It is not obvious why the initial parameterssuch as flame ignition geometry, central density and chemical compositionof the progenitor white dwarf star, or the initiation mechanism of the det-onation, should favor a configuration producing this amount of radioactivenickel.We note that, although the 3D simulations of Sim et al. (2013) have diffi-culties reproducing the observed width-luminosity relation with a faster de-cline of the B -band light curve for fainter events, it may be possible to con-struct such a relation in Chandrasekhar-mass white dwarf star explosions(Kasen and Woosley 2007). Recent studies (Blondin et al. 2017, Goldsteinand Kasen 2018), however, increasingly indicate that sub-Chandrasekharmass explosions are required to capture the full range of the observed rela-tion. (cid:12) is required to produce an explosion with bright-ness characteristic of the most common SNe Ia (see Sim et al. 2010, Shenet al. 2017). In addition, the low densities in sub-Chandrasekhar mass whitedwarfs means that their detonation yields significant quantities of interme-diate mass elements, allowing for a relatively good match to observed spec-tra across a broad range of explosion luminosity. One important challengefor this class of simple (toy) sub-Chandrasekhar mass models is that theystruggle to sustain sufficiently slow light curve evolution to account for thebrighter end of the SNe Ia distribution. However, the variation in ejectamass amongst sub-Chandrasekhar mass models does naturally suggest alink between brightness and light curve evolution and simulations have fa-vored sub-Chandrasekhar models for relatively faint explosions (see, e.g.,Blondin et al. 2017).The problem with sub-Chandrasekhar mass explosion models of the sortdescribed above is that the physical ignition of detonations in such objectsdoes not intrinsically arise from an evolutionary process (such as the ac-cretion of mass towards the Chandrasekhar limit). It has to be caused bysome vigorous event. Two possibilities are commonly discussed.One way to ignite a detonation in a carbon-oxygen white dwarf star isthat it accretes helium-rich material from a companion. Due to instabili-ties in the accretion process or once the accreted shell has grown massiveenough, a detonation triggers in the He material. It propagates the carbon-oxygen core and drives a shock wave into it. This shock wave may triggera secondary detonation in carbon-oxygen rich material – constituting theso-called double detonation explosion scenario (see Fink et al. 2010, Molland Woosley 2013 for recent multidimensional supernova simulations fol-lowing this paradigm). It is conceivable that this occurs when the shockhits the outer edge of the core (“edge-lit double detonation”) or due to aspherical collimation of the inwards propagating oblique shock wave nearits center. The latter case was shown to robustly lead to detonations ofthe core by a geometric amplification effect (Fink et al. 2007). It has to beemphasized, however, that many of the published models simply assume a primary detonation of the He shell. The ignition process is very hard to resolve numerically and the success of the scenario hinges on it to occurin reality. Synthetic spectra have been computed from double detonationmodels for a range of conditions (e.g. Nugent et al. 1997, Kromer et al.2010, Woosley and Kasen 2011), with results that depend significantly onthe assumed structure and mass of the helium layer at explosion. Indeed,it generally appears to be the case that the influence of the outer ejectalayers (rich in helium and/or helium-detonation ash) is mostly detrimen-tal to the agreement of the models with observations: if substantial heliumshells are invoked this leads to effectively suppressing the characteristic fea-tures of intermediate mass elements and to very dramatic line blanketingeffects associated with heavy elements that are synthesized in the heliumdetonation. Thus, for such double detonation models to be viable, very lowmass helium shells must be invoked: whether such low masses of helium canreally be ignited and/or sustain a detonation is a topic of active study (e.g.Shen and Bildsten 2007,?, 2014).An alternative scenario is that of violent mergers (Pakmor et al. 2010,2011, 2012). In contrast to the classical merger paradigm, the explosion hap-pens before the two white dwarfs have formed a single object. In the inspiralprocess, the lighter of the pair is disrupted and its material plunges into themore massive primary that is only weakly affected by tidal forces. This im-pact may trigger a detonation of the primary – a sub-Chandrasekhar masswhite dwarf. A recent update of the model (Pakmor et al. 2013) suggeststhat the ignition of the detonation is triggered even earlier in the inspiralprocess when He rich material (that always exists in low quantities on topof a carbon-oxygen white dwarf) is accreted from the secondary to the pri-mary. This rapid accretion process leads to hydrodynamical instabilities inthe He-layer on the primary and triggers a detonation in this shell. Sim-ilar to the double detonation scenario, the actual supernovae results froma secondary detonation of the core material. In contrast to that scenario,however, in the violent merger case the He shell is less massive and lessdense so that its imprint on the predicted observables is much reduced.It is remarkable that population synthesis studies predict a peak ofthe distribution of white dwarf mergers at primary masses that produce ∼ . M (cid:12) of Ni (Ruiter et al. 2013). Moreover, also the temporal evolu-tion of the luminosity function resulting from sub-Chandrasekhar mass WDdetonations seems to match the observations (Shen et al. 2017).
Thanks to advances in computing resources and numerical methods in re-cent years, we are now able to perform meaningful fully 3D explosion simu-lations for a range of progenitor scenarios that have been proposed for TypeIa supernovae. Combined with radiative transfer post-processing, which al-lows predictions to be made that can be directly compared to observations,such simulations are now playing a key role in driving our understand-ing of the nature and physics of thermonuclear supernovae. However, thestate of the art remains incomplete and far from satisfactory – numerous limitations persist. These include clearly posing initial conditions for explo-sion simulations in the context of particular progenitor modeling, properrepresentation of the dynamics and instabilities during the thermonuclearcombustion in full star models, and adequate description of the complex ra-diation processes responsible for spectrum formation in the evolving ejecta.The last decade has demonstrated that such multi-dimensional simulationsare possible. The goal for the future will be their development towards alevel of predictive power than allows for ever-improving quantitative testingby comparison to the increasing wealth of observational data. Acknowledgements
The work of FKR is supported by the Klaus Tschira Founda-tion and by the Collaborative Research Center SFB 881 “The Milky Way System”(subproject A10) of the German Research Foundation (DFG).
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