Inward propagation of nuclear-burning shells in merging C-O and He white dwarfs

We have investigated the consequences of merging double white dwarf systems by calculating evolutionary models of accreting white dwarfs. We have considered two cases: a massive C-O white dwarf of similar to 1 M-. accreting C-O mixture and a low-mass white dwarf with an initial mass of 0.4 M-. accreting matter composed mostly of helium. The accretion rate of the C-O white dwarf is assumed to be 1 x 10(-5) Mo-. yr(-1). After carbon burning is ignited at M-r similar to 1.04 M-., the flame propagates inward because of heat conduction. By inserting enough grid points to resolve the structure of the flame, we have obtained almost steady burning in most phases of evolution, but we have found a new phenomenon that the strength of flame sometimes oscillates because of a thermal instability in an early phase of the evolution. In calculating evolutionary models, we have occasionally employed a steady-burning approximation, in which the propagation speed of flame is given a priori. We have considered two extreme cases for the interior abundance of the massive white dwarf: X-c = 0.5 and X-c = 0.2. For X-c = 0.5, the flame becomes very weak at M-r similar to 0.4 M-. and the inward propagation stalls there. But a few thousand years later, the flame is reactivated because of contraction and propagates to the center. For X-x = 0.2, the flame propagates smoothly and reaches the center in similar to 1000 yr. In both cases the C-O mixture has been burned into an O-Ne-Mg mixture without causing an explosive phenomenon. For helium-accreting low-mass white dwarfs, we have considered accretion rates of 1 x 10(-7) and 1 x 10(-6) M-. yr(-1). After a fraction of M-. is accreted to the white dwarf, helium is ignited in the outer part and a shell flash occurs. Such a shell flash diminishes when about 10% of helium in the convective shell is burned into carbon and oxygen. The next shell hash occurs at a shell interior to the previously flashed shell. After less than 30 shell flashes, the helium ignition occurs at the center and steady burning begins. Thus, the merging produces a helium star that burns helium at the center. The mechanism of the propagation of the burning shell in this case is compressional heating during interpulse phases, in contrast to the case of the carbon-burning flame where the conduction drives the inward propagation of the flame. We infer that such a low-mass double white dwarf system could be a progenitor of the AM CVn stars.