3 The nature of the mass donor: Two formation scenarios

   
3.1 Close double white dwarfs as AM CVn progenitors

From the spectra of AM CVn stars it is inferred that the transferred material consists mainly of helium. Faulkner et al. (1972) suggested two possibilities for the helium rich donor in AM CVn: (i) a zero-age helium star with a mass of 0.4 - 0.5 ${M}_{\odot }$ and (ii) a low-mass degenerate helium white dwarf. The first possibility can be excluded because the helium star would dominate the spectrum and cause the accretor to have large radial velocity variations, none of which is observed. Thus they concluded that AM CVn stars are interacting double white dwarfs. Their direct progenitors may be detached close double white dwarfs which are brought into contact by loss of angular momentum due to GWR within the lifetime of the Galactic disk (for which we take 10 Gyr). The less massive white dwarf fills its Roche lobe first and an AM CVn star is born, if the stars do not merge (see Sect. 3.2). We discussed the formation of such double white dwarfs in Nelemans et al. (2001b).

To calculate the stability of the mass transfer and the evolution of the AM CVn system, one needs to know the mass-radius relation for white dwarfs. This depends on the temperature, chemical composition, thickness of the envelope etc. of the white dwarf. However, Pnei et al. (2000) have shown that after cooling for several 100Myr the mass-radius relation for low-mass helium white dwarfs approaches the relation for zero-temperature spheres. As most white dwarfs that may form AM CVn stars are at least several 100Myr old at the moment of contact (Tutukov & Yungelson 1996), we apply the mass-radius relation for cold spheres derived by Zapolsky & Salpter (1969), as corrected by Rappaport & Joss (1984). For helium white dwarfs with masses between 0.002 and 0.45  ${M}_{\odot }$, it can be approximated to within 3% by (in solar units)

$$R_{\rm ZS} \approx 0.0106 - 0.0064 \; \ln M_{\rm WD} + 0.0015 M_{\rm WD}^2.$$ (5)

We apply the same equation for the radii of CO white dwarfs, since the dependence on chemical composition is negligible in the range of interest.

   
3.2 Stability of the mass transfer between white dwarfs

In Fig. 1 we show the limiting mass ratio for dynamically stable mass transfer (Eq. (4)) as the upper solid line, with $\zeta (m)$ derived from Eq. (5). The initial mass-transfer rates, as given by Eq. (3), can be higher than the Eddington limit of the accretor (Tutukov & Yungelson 1979). The matter that cannot be accreted is lost from the system, taking along some angular momentum. The binary system may remain stable even though it loses extra angular momentum. However heating of the transferred material, may cause it to expand and form a common envelope in which the two white dwarfs most likely merge (Hang & Webbing 1999). Therefore, we impose the additional restriction to have the initial mass transfer rate lower than the Eddington accretion limit for the companion ($\sim$10 $^{-5}\,\mbox{${M}_{\odot}$ }$ yr^-1). This changes the limiting mass ratio below which AM CVn stars can be formed to the lower solid line in Fig. 1. In this figure we over-plotted our model distribution of the current birthrate of AM CVn stars that form from close binary white dwarfs (see Sect. 4).

In the derivation of the Eq. (3) it is implicitly assumed that the secondary rotates synchronously with the orbital revolution and that the angular momentum which is drained from the secondary is restored to the orbital motion via tidal interaction between the accretion disk and the donor star (see, e.g. Verbunt & Rappaport 1988, and references therein).

However, the orbital separation when Roche lobe overflow starts is only about 0.1 $R_{\odot}$ and the formation of the accretion disk is not obvious; the matter that leaves the vicinity of the first Lagrangian point initially follows a ballistic trajectory, passing the accreting star at a minimum distance of $\sim$10% of the binary separation (Lubow & Shu 1975), i.e. at a distance comparable to the radius of a white dwarf ($\sim$ $0.01~R_{\odot}$). So, the accretion stream may well hit the surface of the accretor directly instead of forming an accretion disk around it (Webbink 1984).

The minimum distance at which the accretion stream passes the accretor is computed by Lubow & Shu (1975) (their $\widetilde{\omega}_{\rm min}$), which we fit with

 

$$\frac{r_{\rm min}}{a} \approx 0.04948 \; - \; 0.03815 \; \log (q)$$ (6)

$$+ \; 0.04752 \; \log^2 (q) \; - \; 0.006973 \log^3 (q).$$

The value of q at which the radius of the accretor equals $r_{\rm min}$ is presented as a dotted line in Fig. 1; above this line the accretion stream hits the white dwarfs' surface directly and no accretion disk is formed.

In absence of the disk the angular momentum of the stream is converted into spin of the accretor and mechanisms other than disk-orbit interaction are required to transport the angular momentum of the donor back to the orbit. The small separation between the two stars may result in tidal coupling between the accretor and the donor which is in synchronous rotation with the orbital period. The efficiency of this process is uncertain (Smarr & Blandford 1976; Campbell 1984), but if tidal coupling between accretor and donor is efficient the stability limit for mass transfer is the same as in the presence of the disk (Eq. (4)). In the most extreme case all the angular momentum carried with the accretion stream is lost from the binary system. The lost angular momentum can be approximated by the angular momentum of the ring that would be formed in the case of a point-mass accretor: $\dot{J}_{\dot{m}} = \dot{m} \sqrt{G M a r_{\rm h}}$, where $r_{\rm h}$ is the radius of the ring in units of a. This sink of angular momentum leads to an additional term $-\sqrt{(1+q) r_{\rm h}}$in the brackets in Eq. (3). As a result the condition for dynamically stable mass transfer becomes more rigorous:

$$q < \frac{5}{6} + \frac{\zeta(m)}{2} -\sqrt{(1+q) r_{\rm h}}.$$ (7)

This limit (with $r_{\rm h}$ given by Verbunt & Rappaport (1988) and again the additional restriction of a mass transfer rate below the Eddington limit) is shown in Fig. 1 as the dashed line.

Figure 1 shows that, with our assumptions, none of the AM CVn binaries which descend from double white dwarfs (which we will call the white dwarf family) forms a disk at the onset of mass transfer. After about 10⁷ yr, when the donor mass has decreased below 0.05 ${M}_{\odot }$ (see Fig. 4) and the orbit has become wider a disk will form.

Figure 1: Stability limits for mass transfer in close double white dwarfs. Above the upper solid curve the mass transfer is dynamically unstable. Below the lower solid line the systems have mass transfer rates below the Eddington limit in the case of an efficient tidal coupling between the accretor and the orbital motion. If the coupling is inefficient, this limit shifts down to the dashed line. Above the dotted line the stream hits the companion directly at the onset of the mass transfer and no accretion disk forms. As the evolution proceeds (parallel to the arrow) a disk eventually forms. The gray shades give the current model birthrate distribution of AM CVn stars that form from close binary white dwarfs. The shading is scaled as a fraction of the maximum birthrate per bin, which is 1.4   10-4 yr-1

Only one of the currently known 14 close double white dwarfs possibly is an AM CVn progenitor; WD1704+481A has $P_{\rm orb} = 3.48$hr, $m = 0.39 \pm 0.05\,\mbox{${M}_{\odot}$ }$ and $M = 0.56 \pm 0.05\,\mbox{${M}_{\odot}$ }$ (Maxted et al. 2000). It is close to the limit for dynamical stability, but because the initial mass transfer rate is expected to be super-Eddington it may merge.

  
3.3 Binaries with low mass helium stars as AM CVn progenitors: A semi-degenerate mass donor

Another way to form a helium transferring binary in the right period range was first outlined by Savonije et al. (1986). They envisioned a neutron star accretor, but the scenario for an AM CVn star, with a white dwarf accretor, is essentially the same (Iben & Tutukov 1991). One starts with a low-mass, non-degenerate helium burning star, a remnant of so-called case B mass transfer, with a white dwarf companion. If the components are close enough, loss of angular momentum via GWR may result in Roche lobe overflow before helium exhaustion in the stellar core. Mass transfer is stable if the ratio of the mass of the helium star (donor) to the white dwarf (accretor) is smaller than $\sim$1.2 (Tutukov & Fedorova 1989 Ergma & Fedorova 1990). When the mass of the helium star decreases below $\sim$ $0.2\,\mbox{${M}_{\odot}$ }$, core helium burning stops and the star becomes semi-degenerate. This causes the exponent in the mass-radius relation to become negative and, as a consequence, mass transfer causes the orbital period to increase. The minimum period is $\sim$10min. With strongly decreasing mass transfer rate the donor mass drops below 0.01  ${M}_{\odot }$ in a few Gyr, while the period of the system increases up to $\sim$1hr; in the right range to explain the AM CVn stars. The luminosity of the donor drops below $10^{-4}\,\mbox{${R}_{\odot}$ }$ and its effective temperature to several thousand K. We will call the AM CVn stars that formed in this way the helium star family. Note that in this scenario a disk will always form because the orbit is rather wide at the onset of the mass transfer. The equations for efficient coupling thus hold.

The progenitors of these helium stars have masses in the range 2.3-5 ${M}_{\odot }$. The importance of this scenario is enhanced by the long lifetimes of the helium stars: $t_{\rm He} \approx 10^{7.15}~M^{-3.7}_{\rm He}$yr (Iben & Tutukov 1985), comparable to the main-sequence lifetime of their progenitors, so that there is enough time to lose angular momentum by gravitational wave radiation and start mass transfer before the helium burning stops.

Our simulation of the population of helium stars with white dwarf companions, suggests that at the moment they get into contact the majority of the helium stars are at the very beginning of core helium burning. This is illustrated in Fig. 2. Having this in mind, we approximate the mass-radius relation for semi-degenerate stars by a power-law fit to the results of computations of Fedorova (1989) for a 0.5 ${M}_{\odot }$ star, which filled its Roche lobe shortly after the beginning of core helium burning (their model 1.1). For the semi-degenerate part of the track we obtain (in solar units):

$$R_{\rm TF} \approx 0.043 \; m^{-0.062}.$$ (8)

Trial computations with the relation $R \approx 0.029 \, m^{-0.19}$ from the model of Savonije et al. (1986) which had $Y_{\rm c} = 0.26$ at the onset of the mass transfer reveals a rather weak dependence of our results on the mass-radius relation.

As noticed by Savonije et al. Savonije et al. (1986), severe mass loss in the phase before the period minimum, increases the thermal time-scale of the donor beyond the age of the Galactic disk and thus prevents the donor from becoming fully degenerate and keeps it semi-degenerate.

Another effect of the severe mass loss is the quenching of the helium burning in the core. Tutukov & Fedorova (1989) show that during the mass transfer the central helium content hardly changes (especially for low mass helium stars). Therefore, despite the formation of an outer convective zone, which penetrates inward to regions where helium burning took place, one would expect that in the majority of the systems the transferred material is helium-rich down to very low donor masses. However, donors with He-exhausted cores at the onset of mass transfer ( ) may in the course of their evolution start to transfer matter consisting of a carbon-oxygen mixture (see Fig. 3 in Ergma & Fedorova 1990, who used the same evolutionary code as Tutukov & Fedorova).

Figure 2: Cumulative distribution of the ratios of the helium burning time that occurred before the mass transfer started and the total helium burning time for the model systems

Figure 3: Distribution of helium stars with white dwarf companions that currently start stable mass transfer. In the systems to the right of the dotted and dashed lines the white dwarfs accrete at least 0.3 ${M}_{\odot }$ and 0.15  ${M}_{\odot }$ before ELD, respectively, at high accretion rates. These lines show two choices of mass limits above which, with our assumption, the binaries are disrupted by edge-lit detonation before they become AM CVn systems. The systems in the top left corner are binaries with helium white dwarf accretors

Figure 4: Examples of the evolution of AM CVn systems. The left panel shows the evolution of the orbital period as function of the mass of the secondary (donor) star. The right panel shows the change in the mass transfer rate during the evolution. The solid and dashed lines are for white dwarf donor stars of initially 0.25  ${M}_{\odot }$ transferring matter to a primary of initial mass of 0.4 and 0.6  ${M}_{\odot }$, respectively, assuming efficient coupling between the accretor spin and the orbital motion. The dash-dotted and dotted line are for a helium star donor, starting when the helium star becomes semi-degenerate (with a mass of 0.2 ${M}_{\odot }$). Primaries are again 0.4 and 0.6 ${M}_{\odot }$. The numbers along the lines indicate the logarithm of the time in years since the beginning of the mass transfer

Figure 3 shows the population of low-mass helium stars with white dwarf companions which currently start mass transfer, (i.e. have ), derived by means of population synthesis. It is possible that only a fraction of them evolves into AM CVn stars. Upon Roche lobe overflow, before the period minimum, most helium donors lose mass at an almost constant rate close to $3 \,~ \,10^{-8}\,\mbox{${ M}_{\odot}$ }$yr^-1. Accretion of He at such rates by a carbon-oxygen (CO) white dwarf may trigger a detonation in the layer of the accumulated matter (Taam 1980). This may further cause the detonation of the underlying CO dwarf, so-called edge-lit detonation, ELD (Livne 1990; Livne & Glasner 1991; Woosley & Weaver 1994; Livne & Arnett 1995). The conditions for ELD to occur: the mass of the white dwarf, the range of accretion rates, the mass of the accumulated layer, etc. are still actively debated.

However, examples computed by Limongni & Tornambè (1991) and Woosley & Weaver (1994) show that if yr^-1 the helium layer inevitably detonates if and . Therefore, as one of the extreme cases, we reject all systems which satisfy these limits from the sample of progenitors of AM CVn stars, assuming that they will be disrupted before the helium star enters the semi-degenerate stage. As another extreme, we assume that only 0.15  ${M}_{\odot }$ has to be accreted for ELD (as a compromise between results of Limongi & Tornambè 1991; Woosley & Weaver 1994). The relevant cut-offs are shown in Fig. 3.

For CO white dwarf accretors less massive than 0.6 ${M}_{\odot }$ we assume that accretion of helium results in "flashes'' in which the He layer is ejected or lost via a common envelope formed due to expansion of the layer. Such events may be repetitive.

Limongi & Tornambè (1991) show that for accretion rates below $10^{-8}\,\mbox{${M}_{\odot}$ }$yr^-1 more than $\sim$ $0.4~\mbox{${M}_{\odot}$ }$ helium has to be accreted before detonation. For systems in which the donor becomes semi-degenerate and the mass accretion rates are low we limit the accumulation of He only by adopting the Chandrasekhar mass as a maximum to the total mass of the accreting white dwarf.

  
3.4 Summary: Two extreme models for AM CVn progenitors

We recognise two possibilities for each family of potential AM CVn systems, which are: efficient or non-efficient tidal coupling between the accretor and the orbital motion in the white dwarf family, and two limits for the disruption of the accretors by ELD in the helium star family. We compute the populations for every possible solution and combine them into two models: model I, in which there is no tidal coupling and ELD is efficient in the destruction of potential progenitor systems (an "inefficient'' scenario for forming AM CVn systems) and an "efficient'' model II, in which there is an effective tidal coupling and ELD is efficient only in systems with the most massive donors. However, we give the birth rates and number of the objects for the four different solutions separately in Table 1.

Figure 5: The current Galactic population of AM CVn systems of the two families for models I (left) and II (right). The grey scale indicates the logarithm of the number of systems. The upper branch is the helium star family; the lower branch the white dwarf family. The lines show the orbital period and mass transfer rate evolution and correspond to the lines in Fig. 4

Figure 4 presents two examples of the evolution of the orbital period and the mass transfer rate for both families of AM CVn systems. Initially the mass transfer rate is very high but within a few million years it drops below 10^-8  ${M}_{\odot }$ yr^-1. In the same time interval the orbital period increases from a few minutes, in the case of the white dwarf family, or from a little over 10 min, for the helium star family, to a few thousand seconds. The semi-degenerate donor systems have lower mass transfer rates and larger periods for the same donor mass due to their larger radii. The fact that the period is independent of the accretor mass (M) is a consequence of Eq. (2) and Keplers 3rd law leading to $P \propto (R^3/m)^{1/2}$.