Deflagration to Detonation Transition in Thermonuclear Supernovae

We derive the criteria for deflagration to detonation transition (DDT) in a Type Ia supernova. The theory is based on the two major assumptions: (i) detonation is triggered via the Zeldovich gradient mechanism inside a region of mixed fuel and products, (ii) the mixed region is produced by a turbulent mixing of fuel and products either inside an active deflagration front or during the global expansion and subsequent contraction of an exploding white dwarf. We determine the critical size of the mixed region required to initiate a detonation in a degenerate carbon-oxygen mixture. This critical length is much larger than the width of the reaction front of a Chapman-Jouguet detonation. However, at densities greater than simeq 5 x 10^6 g cm^-3, it is much smaller than the size of a white dwarf. We derive the critical turbulent intensity required to create the mixed region inside an active deflagration front in which a detonation can form. We conclude that the density rho_tr at which a detonation can form in a carbon-oxygem white dwarf is low, less than 2 - 5 x 10^7 g cm^-3, but greater than 5 x 10^6 g cm^-3.