A currently favored model for Type Ia supernovae consists of a carbon-oxygen (CO) white dwarf ( 0.6–1.0 M), surrounded by a thick layer of helium ( 0.2–0.3 M), which explodes as a consequence of successive detonations in the helium layer and the CO core. Previous studies, carried out in one and two dimensions, have shown that this model is capable of providing light curves and late-time spectra in agreement with observations, though the peak light spectrum may be problematic. These same studies also highlighted a key uncertainty in the model. When properly considered in three dimensions, will the helium detonation actually succeed in igniting a corresponding detonation in the carbon core? In this paper we follow the hydrodynamic evolution of a representative case calculated in three dimensions using the smoothed particle (SPH) approach to multi-dimensional hydrodynamical modeling. Several fine zoned simulations are also carried out in one dimension to elucidate shock hydrodynamics that cannot be resolved in a calculation that carries the whole star. Consistent with the previous results by Benz (1997) and Livne & Arnett (1995), our calculations show that the initial stages of helium ignition strongly influence the development of the explosion. In particular, the altitude above the core boundary at which the first hot spots appear will determine the character of detonation in the core. This altitude is sensitive to the carbon mass fraction in the CO core and to the pre-explosive mixing between the CO core and helium layer. We also find, for a given helium layer and CO core mass, that the number and geometrical distribution of these hot spots influences the evolution of the explosion and the nucleosynthetic yield. A model in which the ignition begins at five distinct points produces more intermediate mass elements than another model in which the ignition commences at a single point. Nevertheless, given that a successful double detonation occurs, the energetics and gross features of the explosion are not very different from what is seen in one- or two-dimensional simulations.

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