3. Results and discussion
An example of the calculated evolution (, , d) is plotted in Fig. 1. In this case the white dwarf can grow to to trigger a SN Ia. Despite of an initial mass ratio of 2.5, the mass transfer is stabilized by the strong wind, and maintains a rate around yr-1 during the whole evolution, i.e., nearly above the steady burning region. A large fraction () of the transferred matter is blown off in the wind. The orbital period decreases to d and then increases before the white dwarf mass reaches .
A possible candidate that shows a similar evolutionary path as in Fig. 1 is the transient supersoft X-ray source RX J0513.9-6951 in the Large Magellanic Cloud (Schaeidt et al. 1993). Spectroscopy and photometry observations suggest that this 0.76 d source may be a massive () white dwarf accreting at a high ( yr-1) rate from a more massive () companion star (Alcock et al. 1996; Southwell et al. 1996). The bipolar outflows with high speed ( km s-1) (Cowley et al. 1996; Southwell et al. 1996) are clear evidence of strong wind during high accretion case. The final evolution of this source will probably lead to a SN Ia or accretion-induced collapse (Kahabka 1996).
Figure 2 presents another example of the evolution for , , and d. The mass transfer rate, as in Fig. 1, is high enough to maintain stable hydrogen burning and weak helium flashes to increase the white dwarf mass. The orbital period increases as mass is transferred from the less massive donor to the white dwarf. Similar features can also be found in Fig. 2 in Hachisu et al. (1996).
We summarize the final outcome of our evolutionary calculations in Fig. 3, showing the distribution of the progenitor systems of SNe Ia in the diagram. For given white dwarf masses, Fig. 3 shows the lower and upper boundaries of the initial orbital period and the companion mass, within which the mass transfer is suitable for the white dwarf to grow to (filled and open circles correspond to and , respectively). It can be seen that there are two "islands", or two types of progenitor systems in the diagram. One is close binaries that contain a massive () donor with an initial orbital period of several tenth of a day to several days, and mass transfer occurring in Case A and Case B (see dotted curves in the figure). The initial white dwarf mass that may increase to in this case can actually be as low as . The other is low-mass () binaries with long orbital period (tens to hundreds of days), in which the donor is a red giant, i.e., in shell hydrogen burning phase, and the required initial mass of the white dwarf is , a bit larger than the value () suggested by Hachisu et al. (1996). This is because Hachisu et al. (1996) assumed that all hydrogen is converted into helium, and then to heavier elements when the mass transfer rate becomes yr-1 after the wind stops. In our calculations we combined the results of Prialnik & Kovetz (1995) and Kato et al. (1989) for the accumulation ratio, and found it generally , which makes it difficult for lower-mass white dwarfs to grow to . The results, however, are not conclusive due to the uncertainties in the accumulation ratio.
According to the standard model for SSS by van den Heuvel et al. (1992), one can see from Fig. 3 that not all SSS will evolve to SNe Ia: no SN Ia is produced when the initial donor mass lies between and . The mass transfer rates in such cases either rise beyond yr-1 or rapidly decrease below yr-1 before reaches . On the other hand, the hot white dwarfs in the SN Ia progenitor systems may not be observed during the wind phase due to the self-absorption of X-rays by the wind itself, until the mass transfer rate decreases below and the wind stops (Hachisu et al. 1996). However, the observations of RX J0513.9-6951 suggest that if is not much above yr-1, the white dwarf can still be seen as a supersoft X-ray source.
Using the method suggested by Iben & Tutukov (1984), one can estimate the birth rate of SNe Ia, based on Fig. 3, to be yr-1 in our Galaxy, roughly consistent with the observed rate. The two types of progenitor systems in Fig. 3 can also qualitatively account for SN Ia explosions in both spiral and elliptical galaxies.
We finally discuss the main uncertainties in our work. Firstly, only the Chandrasekhar-mass white dwarf model for SNe Ia is considered. Though it is more preferred by observations than the sub-Chandrasekhar-mass model, the latter can not be ruled out (cf. Branch et al. 1995). Secondly, the strong wind claimed by Hachisu et al (1996) is a breakthrough in the binary evolution theory. Its reliability, however, needs verification by more detailed theoretical calculations and observations. Thirdly, the accumulation ratio of the accreted matter is very difficult to estimate, due to the complicated situation on the surface of a white dwarf. It depends on the mass and temperature of the white dwarf, as well as the mass transfer rate. For hydrogen burning, Prialnik & Kovetz (1995) performed multi-dimensional calculations, but their results are still uncertain (MacDonald 1996). For helium burning, the only calculation of the accumulation ratio we can find in the literature is by Kato et al. (1989) for a white dwarf with the old opacities. It is worthwhile in this respect to emphasize the importance of careful study of the condition under which accreting white dwarfs can accumulate hydrogen and helium, the condition for hydrogen and helium explosions, and the accumulation efficiency during the steady and unsteady burning cases.
© European Southern Observatory (ESO) 1997
Online publication: June 5, 1998