4 The population of AM CVn stars

We used the population synthesis program SeBa, as described in detail in Portegies Zwart & Verbunt (1996), Portegies Zwart & Yungelson (1998) and Nelemans et al. (2001b to model the progenitor populations. We follow model A of Nelemans et al. (2001b), which has an IMF after Miller & Scalo (1979) and flat initial distributions over the mass ratio of the components and the logarithm of the orbital separation and a thermal eccentricity distribution. We assume an initial binary fraction of 50% and that the star formation decreases exponentially with time, which is different from other studies of close double white dwarfs that assume a constant star formation rate. The mass transfer between a giant and a main-sequence star of comparable mass is treated with an "angular momentum formalism'' which does not result in a strong spiral-in (Nelemans et al. 2000).

  
4.1 The total population

We generate the population of close double white dwarfs and helium stars with white dwarf companions and select the AM CVn star progenitors according to the criteria for the formation of the AM CVn stars as described above. We calculate the birthrate of AM CVn stars, and evolve every system according to the recipe described in Sect. 2, to obtain the total number of systems currently present in the Galaxy (Table 1) and their distribution over orbital periods and mass loss rates (Fig. 5).

The absence of an effective coupling between the accretor spin and the orbital motion (model I) reduces the current birth rate AM CVn stars from the white dwarf family by two orders of magnitude as compared to the case of effective coupling (model II). The fraction of close double white dwarfs which fill their Roche lobes and continue their evolution as AM CVn stars is 21% in model II but only 0.2% in model I (see also Fig. 1). In model I the population of AM CVn stars is totally dominated by the helium star family. In model II where tidal coupling is efficient both families have a comparable contribution to the population. Increasing the mass of the critical layer for ELD from 0.15 ${M}_{\odot }$ to 0.3 ${M}_{\odot }$ almost doubles the current birth rate of the systems which are able to enter the semi-degenerate branch of the evolution. In the latter case almost all helium star binaries that transfer matter to a white dwarf in a stable way eventually become AM CVn systems (see Fig. 3).

In Fig. 5 we show the total current population of AM CVn systems in the Galaxy in our model. The evolutionary paths of both families are indicated with the curves (see also Fig. 4). Table 1 gives the total number of systems currently present in the Galaxy. The evolution of the systems decelerates with time and as a result the vast majority of the systems has orbital periods larger than one hour. The evolutionary tracks for the two families do not converge, since the mass loss of the helium stars prevents their descendants from recovering thermal equilibrium in the lifetime of the Galactic disk (see Sect. 3.3).

The minimum donor mass attainable within the lifetime of the Galactic disk is $\sim$ $0.005~\mbox{${M}_{\odot}$ }$ for the descendants of the helium white dwarfs and $\sim$ $0.007~\mbox{${M}_{\odot}$ }$ for the descendants of the helium stars. This is still far from the limit of $\sim$ $0.001~\mbox{${M}_{\odot}$ }$ where the electrostatic forces in their interiors will start to dominate the gravitational force, the mass-radius relation will become $R \propto M^{1/3}$ (Zapolsky & Salpeter 1969), and the mass transfer will cease.

In Table 1 we give the local space density of AM CVn systems estimated from their total number and the Galactic distribution of stars, for which we adopt

$$\rho(R, z) = \rho_{\rm0} \; {\rm e}^{-R/H} \; \mbox{sech}(z/h)^2 \quad \mbox{pc}^{-3}$$ (9)

as in Nelemans et al. (2001b). Here H = 2.5 kpc (Sacktt 1997) and h = 200 pc, neglecting the age and mass dependence of h. These estimates can not be compared directly to the space density, estimated from the observations: 3   10^-6 pc^-3 (Warner 1995). In our model the space density is dominated by the long-period, dim systems, while Warner's estimate is based on the observed systems which are relatively bright. For a comparison of the observed and predicted populations we have to consider selection effects.

 

 

Table 1: Birth rate and number of AM CVn systems in the Galaxy. The first column gives the model name (Sect. 3.4) followed by the current Galactic birthrate ($\nu$ in yr^-1), the total number of systems in the Galaxy (#) and the number of observable systems with V < 15 (# obs). The last column ($\sigma$ in pc^-3) gives the local space density of AM CVn stars for each model. Due to selection effects the number of observable systems is quite uncertain (see Sect. 4.2)

	white dwarf family			He-star family
Mod.	$\nu$	#	# obs	$\nu$	#	# obs	$\sigma$
	10-3	107		10-3	107		10-4
I	0.04	0.02	1	0.9	1.8	32	0.4
II	4.7	4.9	54	1.6	3.1	62	1.7

  
4.2 Observational selection effects: From the total population to the observable population

The known systems are typically discovered as faint blue stars (and identified with DB white dwarfs), as high proper motion stars, or as highly variable stars (see for the history of detection of most of these stars Ulla 1994; Warner 1995). The observed systems thus do not have the statistical properties of a magnitude limited sample.

Moreover, the luminosity of AM CVn stars comes mainly from the disk in most cases. Despite the fact that several helium disk models are available (e.g. Smak 1983; Cannizzo 1984; Tsugawa & Osaki 1997; El-Khoury & Wickramasinghe 2000) there is no easy way to estimate magnitude of the disk. Therefore, we compute the visual magnitude of the systems from very simple assumptions, to get a notion of the effect of observational selection upon the sample of interacting white dwarfs.

The luminosity provided by accretion is

$$L_{\rm acc} \approx 0.5 \; G \; M \dot{m} \; \left( \frac{1}{R} - \frac{1}{R_{\rm L1}} \right).$$ (10)

Here R is the radius of the accretor and $R_{\rm L1}$ is the distance of the first Lagrangian point to the centre of mass of the accretor. We use an "average'' temperature of the disk (see Wade 1984), which may be then obtained from $L = S \sigma T^4$, where S is the total surface of the disk:

$$S = 2 \pi (R_{\rm out}^2 - R_{\rm WD}^2).$$ (11)

We use $R_{\rm out} = 0.7 R_{\rm L1}$. The visual magnitude of the binary is then computed from the effective temperature and the bolometric correction (taken from Kuiper 1938), assuming that the disk is a black body. This allows us to construct a magnitude limited sample by estimating the fraction of the Galactic volume in which any system in our theoretical sample may be observed as it evolves.

 

 

Table 2: Orbital periods, visual magnitudes and theoretical mass estimates for known and candidate AM CVn stars

Name	Period	mv	m	m	Ref.
	${\rm s}$		(ZS)	(TF)
AM CVn	1028.7	14.1-14.2	0.033	0.114	1
HP Lib	1119	13.6	0.030	0.099	2
CR Boo	1471.3	13.0-18.0	0.021	0.062	3
V803 Cen	1611	13.2-17.4	0.019	0.054	2
CP Eri	1724	16.5-19.7	0.017	0.048	2
GP Com	2970	15.7-16.0	0.008	0.019	2
RX J1914+24	569	>19.7	0.068	-	4
KL Dra		16.8-20			5

Theoretical mass estimates (in ${M}_{\odot }$) obtained from Eq. (5) are labelled by ZS, estimates from Eq. (8) by TF.
References: (1) Ptterson et al. (1993), (2) Warner (1995), (3) Provencal et al. (1997), (4) Cropper et al. (1998), (5) Schwartz (1998).

Figure 6: Magnitude limited sample ( $V_{\rm lim} = 15$) of the theoretical population of AM CVn stars for model I (left) and model II (right). The grey scale gives the number of systems, like in Fig. 5 but now on a linear scale (upper branch for the helium star family; lower branch for the white dwarf family). The selection criteria are described in Sect. 4.2. The periods of the observed systems (Table 2) are indicated with the vertical dotted lines. The stability limits for the helium accretion disk according to Tsugawa & Osaki (1997) are plotted as the solid slanted and dashed horizontal lines. Between these lines the disk is expected to be unstable. The upper dashed line is for an accretor mass of 0.5  ${M}_{\odot }$, the lower for a 1.0  ${M}_{\odot }$ accretor. The rulers at the top indicate the theoretical relation between the period and the mass for the mass-radius relations of the two AM CVn star families given by Eqs. (5) and (8)

We derive $P - \dot{M}$ diagrams for both models, similar to the ones for the total population, but now for the "observable'' population, which we limit by V = 15. Changing $V_{\rm lim}$ doesn't change the character of graphs, since only the nearby systems are visible. The expected number of observable systems for the two families of progenitors is given in Table 1 and shown in Fig. 6. The observable sample comprises only one star for every million AM CVn stars that exists in the Galaxy. A large number of AM CVn stars may be found among very faint white dwarfs which are expected to be of the non-DA variety due to the fact, that the accreted material is helium or a carbon-oxygen mixture.

In the "inefficient'' model I about one in 30 observed systems is from the white dwarf family. This is a considerably higher fraction than in the total AM CVn population where it is only one out of 100 systems. In the "efficient'' model II, the white dwarf family comprises $\sim$60% of the total population and $\sim$50% of the "observable'' one. The ratio of the total number of systems of the white dwarf family in models I and II is not proportional to the ratio of their current birthrates. This reflects the star formation history and the fact that the progenitors of the donors in model I are low mass stars that live long before they form a white dwarf. In model I the fraction of the observable systems which belong to the white dwarf family is higher than the fraction of the total number of systems that belong to this family. This is caused by the fact that the accretors in these systems are more massive (see Fig. 1), thus smaller, giving rise to higher accretion luminosities.

To compare our model with the observations, we list the orbital periods and the observed magnitude ranges for the known and candidate AM CVn stars in Table 2. For AM CVn we give $P_{\rm orb}$ as inferred by Patterson et al. (1993) and confirmed as a result of a large photometry campaign ( Skilman et al. 1999) and a spectroscopic study (Nelemans et al. 2001a). For the remaining systems we follow the original determinations or Warner (1995). Most AM CVn stars show multiple periods, but these are close together and do not influence our qualitative analysis. KL Dra is identified as an AM CVn type star by its spectrum (Jha et al. 1998), but still awaits determination of its period. The periods of the observed AM CVn stars are shown in Fig. 6 as the vertical dotted lines. The period of RX J1914+24 is not plotted because this system was discovered as an X-ray source and it is optically much fainter than the limit used here.

Figures 5 and 6 show that the uncertainty in both models and observational selection effects make it hard to argue which systems belong to which family. According to model I the descendants of close double white dwarfs are very rare. However, in that case one might not expect two observed systems at short periods (AM CVn and HP Lib). In both models I and II, systems with long periods (like GP Com) are more likely to descend from the helium star family. In the spectrum of GP Com, however, Marsh et al. (1991) found evidence for hydrogen burning ashes in the disk, but no traces of helium burning, viz. very low carbon and oxygen abundances. It is not likely that any progenitor of the helium star family completely skipped helium burning. More probably, this system belongs to the white dwarf family.

Most systems in the "observable'' model population have orbital periods similar to the periods of the observed AM CVn stars that show large brightness variations; thus most modelled systems are expected to be variable. These brightness variations have been interpreted as a result of a thermal instability of helium disks (Smak 1983). In Fig. 6 we show the thermal stability limits for helium accretion disks as derived by Tsugawa & Osaki (1997): above the solid line the disks are expected to be hot and stable; below the horizontal dashed lines the disks are cool and stable and in between the disks are unstable. Note that the vast majority of the total Galactic model population (Fig. 5) is expected to have cool stable disks according to the thermal instability model, preventing them from being detected by their variability.

The period distributions of the "observable'' population in our models agree quite well with the observed population of AM CVn stars. Better modelling of the selection effects is, however, necessary.

4.3 Individual systems

Table 2 gives theoretical estimates of the masses of the donor stars in the observed AM CVn stars, derived from the relation between the orbital period and the mass of the donor (see Sect. 3.4 and Fig. 6).

AM CVn stars may be subject to tidal instability due to which the disk becomes eccentric and starts precessing. Such instabilities are used to explain the superhump phenomenon in dwarf novae (Whitehurst 1988).

For AM CVn and CR Boo the observed 1051.2 s (Provencal et al. 1998) and 1492.8 s ( Provencal et al. 1997) periodicities are interpreted as superhump periods. Following Warner (1995) we compute the mass ratio of the binary system using the orbital period ( $P_{\rm orb}$) and the superhump period ($P_{\rm s}$) via:

$$\frac {P_{\rm s}}{P_{\rm s} - P_{\rm orb}} \approx 3.73 \, \frac{1+q}{q}\cdot$$ (12)

This results is q = 0.087 and 0.057 for AM CVn and CR Boo respectively. Assuming that they belong to the white dwarf family their accretor masses are $M=0.38\,\mbox{${M}_{\odot}$ }$ and $M=0.37 \,\mbox{${M}_{\odot}$ }$. These values are at the lower end of the predicted distribution. If we apply the semi-degenerate mass-radius relation, the estimated masses of the accretors are high, even close to the Chandrasekhar mass for AM CV. The formation of systems with high-mass accretors has a low probability (see Fig. 3), which suggests that either Eq. (12) is not applicable for helium disks or alternatively that these binaries do not belong to the helium star family.

Maybe the most intriguing system is RX J1914.4+245; detected by ROSAT (Moch et al. 1996) and classified as an intermediate polar, because its X-ray flux is modulated with a 569 s period, typical for the spin periods of the white dwarfs in intermediate polars. Cropper et al. (1998) and Ramsay et al. (2000) suggest that it is a double degenerate polar with an orbital period equal to the spin period of the accreting white dwarf. The mass transfer rate in this system, inferred from its period ( $\dot{m} \approx 1.8~ 10^{-8} \, \mbox{${M}_{\odot}$ }$ yr^-1) is consistent with the value deduced from the ROSAT PSPC data (Cropper et al. 1998) if the distance is $\sim$100 pc.

Even though polars have no disk, the coupling between the accretor and donor is efficient due to the strong magnetic field of the accretor. We therefore anticipate that Eq. (4) applies without the correction introduced by Eq. (7). It may well be that magnetic systems in which the coupling is maintained by a magnetic field form the majority of stable AM CVn systems of the white dwarf family. We do not expect this system to belong to the helium star family, since its period is below the typical period minimum for the majority of the binaries in this family.

RX J0439.8-809 may be a Large Magellanic Cloud relative of the Galactic AM CVn systems. This system was also first detected by ROSAT (Greiner et al. 1994). Available X-ray, UV- and optical data suggest, that the binary may consist of two degenerate stars and have an orbital period < 35 min (van Teeseling et al. 1997,1999). RX J1914.4+245 and RX J0439.8-809 show that it is possible to detect optically faint AM CVn stars in supersoft X-rays, especially in other galaxies. The possibility of supersoft X-rays emission by AM CVn stars was discussed by Tutukov & Yungelson (1996). There are two probable sources for the emission: the accreted helium may burn stationary at the surface of the white dwarf if $\dot{m} \sim 10^{-6} \, \mbox{${M}_{\odot}$ }$yr^-1 and/or the accretion disk may be sufficiently hot in the same range of accretion rates. However, the required high accretion rate makes such supersolf X-ray sources short-living (see Fig. 4) and, therefore, not numerous. Note that AM CVn, CR Boo, V803 Cen, CP Eri and GP Com are also weak X-ray sources (Ulla 1995).

The most recently found suspected AM CVn star, KL Dra, is also variable. Therefore we expect it to lie in the same period range as CR Boo, V803 Cen and CP Eri. Taking the limits for stability as given by Tsugawa & Osaki (1997) we expect the orbital period to be between 20 and 50 min (Fig. 6).