1 Introduction

AM CVn stars are helium-rich faint blue objects that exhibit variability on time-scales of $\sim$1000 s. Smak (1967) discovered that the prototype of the class, AM CVn (=HZ 29), shows photometric variability with a period $\sim$18 min and suggested that it is a binary. Immediately Paczynski (1967) realised that it could be a semi-detached pair of degenerate dwarfs in which mass transfer is driven by loss of angular momentum due to gravitational wave radiation. After flickering, typical for cataclysmic binary systems, was found in AM CVn by Warner & Robinson (1972), this model was used to explain this system by Faulkner et al. (1972). There are currently 8 AM CVn candidates (Table 2) which have been studied photometrically in great detail (for reviews see Ulla 1994; Warner 1995; Solheim 1995). AM CVn stars also attracted attention as possible sources of gravitational waves (Hils & Bender 2000, and references therein).

After the introduction of the concept of common envelope evolution for the formation of cataclysmic variables and X-ray binaries (Paczynski 1976), the formation of close double white dwarfs through two of such phases was anticipated by Tutukov & Yungelson (1979, 1981). The emission of gravitational waves would subsequently bring the two white dwarfs into a semi-detached phase. Nather et al. (1981) independently suggested this scenario for the formation of AM CVn itself. In an alternative scenario the white dwarf donor is replaced by a helium star that becomes semi-degenerate during the mass transfer (Iben & Tutukov 1991).

Throughout this paper we use the term AM CVn for binaries in which a white dwarf accretes from an another white dwarf or from a semi-degenerate helium star, irrespective how they would be classified observationally.

This paper continues our study on the formation and evolution of the Galactic population of close double white dwarfs (Nelemans et al. 2000, 2001b). Here we study the population that becomes semi-detached and transfers mass in a stable way. In addition we examine the alternative case where the donor is a semi-degenerate helium star. In Sect. 2 we outline the evolution of binaries driven by gravitational wave radiation. We review the models for the formation of AM CVn stars and discuss the stability of the mass transfer in Sect. 3. The results of our population synthesis and a comparison with observations are presented in Sect. 4. The differences with previous studies are discussed in Sect. 5 after which the conclusions follow.