A&A 431, 635–645 (2005) Astronomy DOI: 10.1051/0004-6361:20041859 & c ESO 2005 Astrophysics Full-star type Ia supernova explosion models F. K. Röpke and W. Hillebrandt Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, 85741 Garching, Germany e-mail: [fritz;wfh]@mpa-garching.mpg.de Received 18 August 2004 / Accepted 20 October 2004 Abstract. We present full-star simulations of type Ia supernova explosions on the basis of the standard Chandrasekhar-mass deflagration model. Most simulations so far considered only one spatial octant and assumed mirror symmetry to the other octants. Two full-star models are evolved to homologous expansion and compared with previous single-octant simulations. Therefrom we analyze the effect of abolishing the artificial symmetry constraint on the evolution of the flame surface. It turns out that the development of asymmetries depends on the chosen initial flame configuration. Such asymmetries of the explosion process could possibly contribute to the observed polarization of some type Ia supernova spectra. Key words. stars: supernovae: general – hydrodynamics – methods: numerical 1. Introduction A solution to the problem is found in the concept of tur- bulent combustion. Realizing that the astrophysical scenario Recent observations have shown that spectra of – at least some is characterized by Reynolds numbers as high as 1014, one – type Ia supernovae (SNe Ia) show significant polarization expects the development of turbulent patterns in the flow, as (e.g. Howell et al. 2001; Wang et al. 2003). The origin of this soon as instabilities induce them. Such instabilities are in- effect is still unclear, but some possible mechanisms have been herent in the propagation of deflagration flames. In the situa- proposed (Wang et al. 2003; Kasen et al. 2003, 2004). Besides tion of thermonuclear supernovae the primary instability is the peculiar spatial structures of the remnant and interaction of the Rayleigh-Taylor instability. The resulting turbulent motions ejecta with the companion star, anisotropies of the explosion will interact with the flame wrinkling and stretching its sur- process itself could contribute to the observed polarization. face. Since the fuel consumption rate of deflagration flames is determined by the flame surface area, turbulence will sig- This idea fits well into the currently favored astrophysi- nificantly accelerate the flame propagation. Three-dimensional cal model of SNe Ia. These events are assigned to thermonu- SN Ia models (e.g. Hillebrandt et al. 2000) taking this effect / clear explosions of carbon oxygen white dwarf (WD) stars in into account have been shown to lead to robust explosions and binary systems (for a review of SN Ia explosion models see reproduce many features of SNe Ia (e.g. Reinecke et al. 2002b; Hillebrandt & Niemeyer 2000). From non-degenerate compan- Gamezo et al. 2003). ions they accrete matter until they approach the Chandrasekhar There have been speculations on the possibility of a later mass. After some thousand years of convective carbon burning transition from the deflagration to a detonation, but a viable thermonuclear reactions in the center of the star form a flame, mechanism to trigger this transition in SNe Ia explosions could i.e. a small spatial region undergoes thermonuclear runaway. not be identified yet (Niemeyer 1999). This flame can travel outward in two modes. If it is mediated by In any case, the deflagration stage requires thorough in- shock waves, the propagation proceeds at supersonic velocities vestigation especially in the context of asymmetries of the ex- leading to so-called detonations, while a mediation due to heat plosion process for it is intrinsically anisotropic. Due to the conduction of the degenerate electrons leads to subsonic defla- mentioned buoyancy instabilities the large scale features of the grations. A prompt detonation could be excluded as a model for flame show rising “bubbles” of hot and light ashes and in be- SNe Ia since it failed to produce the observed composition of tween downdrafts of cold and dense fuel. It is still an open the ashes (Arnett 1969). It burns the entire star at high densities question, how strong the resulting asymmetries may become. producing mainly iron group elements. This issue will be addressed in the present study. Consequently a mechanism had to be developed wherein It is clear that in order to explore asymmetries of the explo- the star can expand prior to incineration. This is provided if sion process it is mandatory to simulate the full star. Most of the flame propagation starts out as a deflagration (e.g. Nomoto the three-dimensional simulations so far have been restricted to et al. 1976). That flame propagation mode, however, is much one spatial octant assuming mirror symmetry to the other oc- too slow to explain SN Ia explosions. tants. Obviously, this approach imposes an artificial symmetry Article published by EDP Sciences and available at http://www.aanda.org or http://dx.doi.org/10.1051/0004-6361:20041859 636 F. K. Röpke and W. Hillebrandt: Full-star type Ia supernova explosion models constraint on the flame evolution and the goal of our work is In this regime, turbulent eddies wrinkle and stretch the to study the effects that result from removing this symmetry. flame but do not penetrate its internal structure. The internal There has been only one previous simulation of the deflagra- flame structure (which extends to less than a centimeter) as well tion phase in the full star (Calder et al. 2004), but it failed to as the smallest scales of flame interaction with turbulence are explode. The reason for this is presumably that it was started tiny compared with the dimensions of a WD star (∼108 cm). It with very artificial conditions for the initial flame (a perfect is therefore well justified to treat the resulting “turbulent flame sphere). Nevertheless, the merit of this model is that it points brush” as a discontinuity (the so-called flame front) separating to the importance of the initial flame configuration in the con- the fuel from the ashes, as long as the small-scale effects are text of asymmetries of the explosion process. appropriately modeled. The flame ignition is difficult to address in numerical sim- Two approaches are taken to ensure that. First, known ef- ulations and afflicted with large uncertainties (Woosley et al. fects have to be taken into account. This applies mainly to the 2004). We will explore the impact of initial flame configura- interaction of the flame with turbulence on unresolved scales. tions on the explosion models in a parametrized way. As a Wrinkling will increase its surface and therefore enhance the first step, in this study we investigate anisotropies in the flame burning rate. The shortcoming in resolving the flame surface evolution resulting from simulations covering the full star in- can be compensated by assigning an increased turbulent burn- stead of only one octant and from moderately asymmetric ini- ing velocity to the flame front reproducing the correct fuel tial flame configurations. This allows comparison with previous consumption rate. It turns out that in the flamelet regime the simulations. Detailed studies of multi-spot ignition scenarios turbulent burning velocity is proportional to the turbulent ve- and strongly dipolar ignited models are subject to forthcoming locity fluctuations and decouples completely from the laminar publications. burning speed determined by the microphysics. Second, one may suspect that on unresolved scales new physical effects oc- 2. Implementation cur (like “active turbulent combustion”, Niemeyer & Woosley 1997) that affect the flame propagation on large scales. This has 2.1. Flame model to be studied in separate small-scale models focusing on a nar- The stratification of the hot and light ashes below cold and row window in scale space. Simulations of the cellular burning dense fuel leads to a buoyancy instability known as the regime (Röpke et al. 2003, 2004a,b; Bell et al. 2004) led to the Rayleigh-Taylor instability. In its non-linear stage, bubbles of conclusion that the resulting increased burning velocity can be ff burning material rise into the fuel with a velocity (Davies & “renormalized” with little e ect on the large scale flame propa- Taylor 1950) gation in most stages of the SN Ia explosion. At least from this regime drastic effects are unlikely. v ∝ g LAt, (1) Due to the expansion of the WD, the density at which burn- where g and L denote the gravitational acceleration and the ing takes place decreases steadily. This is accompanied by a length scale of the bubble, respectively. The Atwood number gradual broadening of flame width. At some point turbulent ff At characterizes the density contrast between the interior and eddies can a ect the flame’s internal structure and burning en- the exterior of the bubbles. At the interfaces of the burning ters the so-called distributed regime. Since this is expected to bubbles shear flows give rise to a secondary instability, the happen at very late stages, it may not contribute much to the Kelvin-Helmholtz instability. It generates turbulent velocity production of iron group nuclei. Nevertheless, it could signifi- fluctuations at this (integral) length scale L, which decay to cantly increase the abundances of intermediate mass elements smaller scales in a turbulent cascade over a so-called iner- and hence the total energy release with consequences for mod- tial range and are dissipated at the Kolmogorov length. For eling of lightcurves and spectra. This regime is not yet acces- ff the velocity fluctuations v at some scale l in the inertial range sible in our code but current e orts aim at including it in fu- Kolmogorov-scaling yields ture simulations. In this respect the results presented here are preliminary. l 1/3 v(l) = v(L) · (2) To summarize the turbulent flame propagation in thermonu- L clear supernova explosions we note that large-scale features are driven by buoyancy according to Eq.