A promising model for normal Type Ia supernova (SN Ia) explosions are delayed detonations of Chandrasekhar-mass white dwarfs, in which the burning starts out as a subsonic deflagration and turns at a later phase of the explosion into a supersonic det
onation. The mechanism of the underlying deflagration-to-detonation transition (DDT) is unknown in detail, but necessary conditions have been determined recently. The region of detonation initiation cannot be spatially resolved in multi-dimensional full-star simulations of the explosion. We develop a subgrid-scale (SGS) model for DDTs in thermonuclear supernova simulations that is consistent with the currently known constraints. The probability for a DDT to occur is calculated from the distribution of turbulent velocities measured on the grid scale in the vicinity of the flame and the fractal flame surface area that satisfies further physical constraints, such as fuel fraction and fuel density. The implementation of our DDT criterion provides a solid basis for simulations of thermonuclear supernova explosions in the delayed detonation scenario. It accounts for the currently known necessary conditions for the transition and avoids the inclusion of resolution-dependent quantities in the model. The functionality of our DDT criterion is demonstrated on the example of one three-dimensional thermonuclear supernova explosion simulation.
Delayed detonations of Chandrasekhar-mass white dwarfs (WDs) have been very successful in explaining the spectra, light curves, and the width-luminosity relation of spectroscopically normal Type Ia supernovae (SNe Ia). The ignition of the thermonucle
ar deflagration flame at the end of the convective carbon simmering phase in the core of the WD is still not well understood and much about the ignition kernel distribution remains unknown. Furthermore, the central density at the time of ignition depends on the still uncertain screened carbon fusion reaction rates, the accretion history and cooling time of the progenitor, and the composition. We present the results of twelve high-resolution three-dimensional delayed detonation SN Ia explosion simulations that employ a new criterion to trigger the deflagration to detonation transition (DDT). All simulations trigger our DDT criterion and the resulting delayed detonations unbind the star. We find a trend of increasing iron group element (IGE) production with increasing central density for bright, faint, and intermediate SNe. The total 56Ni yield, however, remains more or less constant, even though increased electron captures at high density result in a decreasing 56Ni mass fraction of the IGE material. We attribute this to an approximate balance of 56Ni producing and destroying effects. The deflagrations that were ignited at higher density initially have a faster growth rate of subgrid-scale turbulence. Hence, the effective flame speed increases faster, which triggers the DDT criterion earlier, at a time when the central density of the expanded star is higher. This leads to an overall increase of IGE production, which off-sets the percental reduction of 56Ni due to neutronization.
The delayed detonation model describes the observational properties of the majority of type Ia supernovae very well. Using numerical data from a three-dimensional deflagration model for type Ia supernovae, the intermittency of the turbulent velocity
field and its implications on the probability of a deflagration-to-detonation (DDT) transition are investigated. From structure functions of the turbulent velocity fluctuations, we determine intermittency parameters based on the log-normal and the log-Poisson models. On the other hand, the analysis of the turbulent velocity fluctuations in the vicinity of the flame front by Roepke suggests a much higher probability of large velocity fluctuations on the grid scale in comparison to the log-normal intermittency model. Following Pan et al., we computed probability density functions for a DDT for the different distributions. Assuming that a DDT can occur in the stirred flame regime, as proposed by Woosley et al., the log-normal model would imply a delayed detonation between 0.7 and 0.8 seconds after the beginning of the deflagration phase for the multi-spot ignition scenario used in the simulation. However, the probability drops to virtually zero if a DDT is further constrained by the requirement that the turbulent velocity fluctuations reach about 500 km/s. Under this condition, delayed detonations are only possible if the distribution of the velocity fluctuations is not log-normal. From our calculations follows that the distribution obtained by Roepke allow for multiple DDTs around 0.8 seconds after ignition at a transition density close to 1x10^7 g/cm^3.
We analyze the statistical properties of the turbulent velocity field in the deflagration model for Type Ia supernovae. In particular, we consider the question of whether turbulence is isotropic and consistent with the Kolmogorov theory at small leng
th scales. Using numerical data from a high-resolution simulation of a thermonuclear supernova explosion, spectra of the turbulence energy and velocity structure functions are computed. We show that the turbulent velocity field is isotropic at small length scales and follows a scaling law that is consistent with the Kolmogorov theory until most of the nuclear fuel is burned. At length scales greater than a certain characteristic scale, turbulence becomes anisotropic. Here, the radial velocity fluctuations follow the scaling law of the Rayleigh-Taylor instability, whereas the angular component still obeys Kolmogorov scaling. In the late phase of the explosion, this characteristic scale drops below the numerical resolution of the simulation. The analysis confirms that a subgrid-scale model for the unresolved turbulence energy is required for the consistent calculation of the flame speed in deflagration models of Type Ia supernovae, and that the assumption of isotropy on these scales is appropriate.
