5 Discussion

A population synthesis study for AM CVn stars (and related systems) was done by Tutukov & Yungelson (1996), who considered only an "efficient'' model. Their derived birthrate for the white dwarf family is 1.3   10^-2 yr^-1, a factor three higher than the value in our "efficient'' model. This difference can in part be explained by the different treatment of the mass transfer from a giant to a main sequence star of comparable mass (see Nlemans et al. 2000,2001b). Most close double white dwarfs in our model have a mass ratio close to unity for which stable mass transfer is impossible (Sect. 2), while in the model of Tutukov & Yungelson (1996) they predominantly have $q \sim 0.5$-0.7 which is more favourable for stable mass transfer. Our higher integrated star formation rate only partly compensates for the loss of stable systems.

Another difference is that Tutukov & Yungelson (1996) conclude that the helium star family (non-degenerate helium stars in their terminology) do not contribute significantly to the AM CVn population. This is a consequence of their assumption that these systems, after the period minimum, live only for 10⁸yr. In contrast, our calculations show that their evolution is limited only by the lifetime of the Galactic disk. Tutukov & Yungelson estimate the total number of AM CVn stars from the helium star family as (1.9-4.6) 10⁵ depending on the assumptions about the consequences of the accretion of helium. We find 2   10⁷ even when we let ELD destroy the systems which accrete only 0.15 ${M}_{\odot }$.

An additional complication is the possibility of the formation of a common envelope for systems where the accretion rate exceeds the rate of stationary helium burning at the surface of the accreting white dwarf ($\sim$ $10^{-6}\,\mbox{${M}_{\odot}$ }$ yr^-1). If such a common envelope forms the components of system may well merge. If it happens, it will occur directly after the Roche-lobe contact, when the highest accretion rate occurs. We do not consider this possibility in our model, because it involves too many additional (and unknown) parameters. Applying only the requirement that stable systems should accrete below the Eddington rate, we may overestimate the birthrate of AM CVn stars.

We did not discuss the Roche lobe overflow by low mass stars with almost exhausted hydrogen cores ( $X_{\rm c} \sim 0.01$) which may also result in the formation of helium transferring systems with orbital periods $\sim$10 min (Tutukov et al. 1987) because of its extremely low probability.

The prescription for ELD is related to the problem of SNIa progenitors. In model I almost all accretors in the helium star family with initial $M \geq 0.6$ ${M}_{\odot }$ "explode'' and the ELD rate is close to 0.001 yr^-1. If ELDs really produce SNeIa, they may contribute about 25% of their currently inferred Galactic rate. In model II 0.3 ${M}_{\odot }$ must be accreted prior to the explosion, and the ELD rate is only about $4 \, ~ \,10^{-4}$ yr^-1. Even in model I we find a much lower ELD rate than Tutukov & Yungelson (1996 who find 0.005). This is partly due to a lower birthrate, but also to the different treatment of the mass transfer from a giant to a main sequence star of comparable mass (Nelemans et al. 2000), which causes the accretors in our model mainly to have masses below 0.6 ${M}_{\odot }$. Such systems probably never experience ELD.

In model II the accretors in both families may accrete so much matter that they reach the Chandrasekhar mass. The rates for the white dwarf and helium star families for this process are 3   10^-6 and 5   10^-5 yr^-1.

In both families the accretors can be helium white dwarfs (see Figs. 1 and 3). It was shown by Nomoto & Sugimoto (1977) that accretion of helium onto helium white dwarfs with $\dot{m} = (1$- $4) \, ~ \, 10^{-8} \, \mbox{${M}_{\odot}$ }$ yr^-1 results either in a helium shell flash (at the upper limit of the accretion rates) or in central detonation which disrupts the white dwarf (for lower $\dot{m}$). The detonation occurs only when the mass of the accretor grows to $\sim$ $0.7\,\mbox{${M}_{\odot}$ }$. In our calculations this happens for the helium star family at a rate of $\sim$ 4   10^-6yr^-1. For the white dwarf family it happens only in model II, at a rate of $\sim$ 2   10^-6 yr^-1.