Astron. Astrophys. 322, L9-L12 (1997)

2. The model

We consider a C-O white dwarf accreting matter from a companion with solar chemical composition (X=0.7, Y=0.28, Z=0.02). For the outcome of the evolution, we only consider the Chandrasekhar-mass SN Ia explosion of the white dwarf with an initial mass less than . The companion star is assumed to be lobe-filling, and mass transfer occurs through Roche-lobe overflow. Detached systems like symbiotic stars, in which the companion star of the white dwarf loses mass via a wind, are not taken into account, due to the deficiency of massive white dwarfs among such systems, and hence the low frequency ( yr^-1) of SNe Ia (Yungelson et al. 1995).

The system can be specified by three parameters: the initial white dwarf mass , the initial donor mass and the orbital period at the beginning of mass transfer. We have followed the evolution of these binary systems for various sets of these parameters, using an updated version of the evolution code developed by Eggleton (1971).

Whether the white dwarf can grow in mass is determined by the "accumulation ratio" , the fraction of the accreted hydrogen that converts into elements heavier than helium. The value of is related to mass loss from the white dwarf during hydrogen and helium burning, i.e., , where and are the fraction of the transferred mass accumulated during hydrogen and helium burning, respectively.

It has been shown (e.g. Nomoto et al. 1979; Fujimoto 1982) that steady hydrogen burning on the white dwarf surface can occur when , where is the mass transfer rate, yr^-1 and (note that even in this stable burning situation, not all of the accreted hydrogen is available for burning, because some of it can be lost via a radiatively driven wind). As , hydrogen burning is unstable and occurs in flashes, in which part of the white dwarf envelope may be ejected. For the hydrogen accumulation ratio , we have taken the following assumptions: (1) if , we adopt the strong wind solution by Hachisu et al. (1996), which allows burning of hydrogen into helium at a rate limited to , the excess material being blown off in the wind, i.e., ; (2) if , we fit the data of the accumulation ratio calculated by Prialnik & Kovetz (1995) for hydrogen burning and flashes (for each combination of and ) with the following conditions: (i) when declines below yr^-1, a strong nova explosion is assumed to occur and no mass accumulates on the surface of the white dwarf, that is ; (ii) when , .

The steady hydrogen burning converts hydrogen into helium on top of the C-O white dwarf, and increases the mass of the helium layer gradually. When its mass reaches a certain value, helium ignites. Helium shell burning is unstable if yr^-1, and a flash grows, during which part of the envelope mass is blown off (Kato, Saio & Hachisu 1989). For the helium accumulation ratio , we use the results given by Kato et al. (1989).

The evolution of the system is driven by the nuclear evolution of the donor star, and the change of the orbital angular momentum of the system mainly caused by wind mass loss from the white dwarf, which is assumed to leave the system carrying the specific orbital angular momentum of the white dwarf (the mass loss in the donor's wind is supposed to be negligible, but its effect on the change of , i.e., magnetic braking, is included). In each time step, we numerically calculate the response of the donor radius and the Roche-lobe radius, the difference of which determines the mass transfer rate, to mass loss from the donor, and mass and angular momentum loss from the binary respectively (a semi-analytical method was adopted by Hachisu et al. 1996 and by Di Stefano and Nelson 1996). The wind decreases both the total mass and the orbital angular momentum of the binary, leading to increase and decrease of the orbital separation, respectively. Thus the stability of mass transfer and the secular evolution of the binary sensitively depends on the values of , and the mass ratio. Generally we have two kinds of products: (1) grows to 1.4 , producing a SN Ia; (2) the calculations are stopped before reaches because exceeds yr^-1 at which there is no wind solution, or declines below yr^-1, at which a strong nova explosion is assumed to occur, or for a low-mass () donor, the helium core mass reaches 0.46 .

© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998

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