Astron. Astrophys. 332, 939-957 (1998)

5. Discussion

The intense WET observing campaign representing almost half the existing photometry of this system has produced a unique data set on AM CVn.

Given all the spectroscopic and photometric evidence, we have great difficulty in interpreting AM CVn as a single star, although it has many properties in common with single white dwarfs (Patterson et al. 1992; Provencal et al. 1995).

A model for the system must explain the following observations:

• the numerical relationship between the , , ..., and modulations
• the lack of power at µHz
• the 20.8 µHz sidebands associated with 4 of the harmonics ()
• the amplitude modulation of the  µHz peak in phase with the maximum absorption line skewness modulation observed by Patterson et al. (1993)
• the unrelated 988.8 µHz modulation
• the low amplitude peak at 972.3µHz which is  µHz ()
• that the modulations at , and maybe are pulse shape harmonics
• the long time phase stability of the 1902.5µHz modulation with s s^-1 (Provencal et al. 1995)

In addition, AM CVn's UV and optical spectra are complex, containing broad asymmetric features, P Cygni profiles of UV-resonance lines indicative of a wind, and a continuum spectrum which indicate a very hot central region, possibly containing a white dwarf (Solheim 1993c; Solheim & Sion 1994; Bard 1995).

5.1. The precessing disk model

Possible explanations for the system are discussed by Patterson et al. (1993) and Provencal et al. (1995). In the following we concentrate on one model which gives a simple explanation of nearly all the observations. It is a variation of the precessing disk model proposed for the AM CVn system by Patterson et al. (1993). They proposed that the 20.8 µHz modulation of the spectral lines is due to a prograde precessing apsidal line in an elliptical disk (), and that  µHz is the superhump frequency ().

We will accept the explanation for the identification of , but we find it more likely that is identified as the orbital frequency - because of the long term phase and amplitude stability of , and because this peak always has most power.

Numerical simulations show that superhump periods exist in systems where the mass ratio is smaller than a critical ratio of if the mass transfer rate is large enough (Whitehurst 1988; Osaki 1989; Lubow 1991). The key factor for development of an eccentric disk with observable superhumps is expansion of the disk beyond the critical radius for the 3:1 resonance between the orbital frequency and particle orbital frequencies. Ichikawa et al. (1993) show that such expansion can happen during a normal dwarf nova outburst. If the mass transfer rate is greater than , the system will go into a permanent superoutburst, and may show continuous super- humps (Osaki 1995, 1996). AM CVn's absorption line spectrum, flux distribution, and extreme mass ratio suggests a system in constant superoutburst, providing us with an excellent laboratory to study accretion and the superhump phenomenon. This has precedence: permanent superhumpers exist also among the hydrogen CVs. PG 0917 + 142 (Skillman & Patterson 1993) is a recent example, and we propose that the high mass transfer objects AM CVn and EC 15330-1403 are their helium CV counterparts. The absorption line profile modulation observed by Patterson et al. (1993) can be explained as due to the change of aspect angle with , which also modulates the amplitude of the 1902.5 µHz frequency in the light curve.

It should be noted that the generally adopted interpretation of the superhump phenomenon in terms of a precessing disk based on numerical results by Whitehurst, Lubow and Ichikawa etc. is by no means well established. These results were obtained with quasi-particle codes (SPH or otherwise) with limited spatial resolution and high intrinsic viscosity. Calculations with hydrodynamic finite different codes yield different results. The dominant term in the tidal potential has , i.e. two-fold azimuthal structure. Indeed, the results of hydrodynamic calculations using the full tidal (point mass) potential of the secondary star show a strong two-armed spiral tidal (shock)wave in the disk, corotating with the companion. The dissipation in this shock wave detorques the disk material in the outer regions whereby the disk gets an elliptical shape which is fixed in the binary frame (e.g., see Fig. 10 in Savonije et al. (1994)). Recently, such an tidal shock wave has in fact been observed in the CV IP Pegasi during outburst (Steeghs et al. 1997). In the SPH calculations no such clear spiral structure is visible, instead the disk becomes eccentric and starts to precess. Precessing disks could only be obtained in the hydro calculations if (artificially) only the (weaker) tidal component was applied and the precession disappeared once the full tide (including the dominant component) was applied to the disk (Heemskerk 1994). Therefore the precessing disk model is at least open to debate. However, in the case of AM CVn we cannot have an elliptical shape which is fixed in the binary frame. This would give a modulation as soon as the inclination is . However, observations of superhumps suggest indeed a slow precessional motion in the disk during its permanent superoutburst. If one accepts this model the following remarks are relevant:

If we in the binary frame have a prograde precessing disk, an observer in an inertial frame would observe as the superhump frequency:

or

When we identify as , we get  µHz or  s, which we have detected as a low amplitude coherent modulation (Table 3). We find that is in good agreement with the disk apse precession rate found by numerical simulations by Lubow (1991).

We observe beat between the binary orbital frequency and the prograde precession as . This is not consistent with the observed fact that all SU UMa stars show superhump periods which are slightly longer than the orbital periods. Warner (1995b, Table 3.3) shows that for the SU UMa stars the superhump period is % longer than the orbital period. In order to observe the prograde precession as a shorter superhump period, we have to observe features in the disk in a fixed frame relative to the observer, and this will give us valuable information in modelling the disk shape. We conclude that the AM CVn system displays a mixture of properties from the two competing theories:

Since most power is seen in the peak at frequency Hz it must indeed correspond with the expected two-armed tidal spiral wave rotating with the orbital angular speed which, presumably due to aspect variation, gives a photometric signal modulated with twice the orbital frequency. Hence Hz. The modulation at 3 times can be interpreted as the weaker tidal response of the disk which may arise when the disk has expanded in size. It may also be interpreted as a two-fold spiral rotating with 3/2 times , as expected from Lubow's non-linear wave interaction model. However, if the signal (at 5 ) is real, it is difficult to explain in Lubow's model. In his model would correspond to a two-fold spiral moving with 1.5 times the orbital angular speed. In combination with the observed two-fold tidal spiral () this could never give rise to time variation with . But in terms of direct tidal responses it could be the combined time-signal of a two-fold and a three-fold spiral wave both moving with the orbital angular speed through the disk.

The components with the above frequencies plus Hz could then be interpreted in terms of their interaction with a precessing (with ) eccentric mode, whereby Hz. However, the peaks with the above-mentioned main frequencies plus twice Hz remain unexplained. However, is probably a pulse-shape harmonic, while may be spurious or .

Simulations of disks in systems with small mass ratios demonstrate that the difference between and diminishes with declining mass ratio (Hirose & Osaki 1990). A relation between and has been established by Warner (1995b):

where for . When we identify and  s we get .

The secondaries in AM CVn systems have never been directly observed. Theoretical modelling has led to two possibilities concerning their nature. The first one assumes a completely degenerate secondary as proposed by Faulkner et al. (1972). Using , we find and . The second possibility is to accept the conclusions of Savonije et al. (1986) who demonstrate that the high mass transfer rate drives the secondary out of thermal equilibrium and keeps it semi-degenerate. In this case the mass radius relation (Savonije et al. 1986) gives and . The semi-degenerate solution gives a reasonable fit for all the AM CVn stars, while solutions for fully degenerate secondaries give too low mass for the primary in most cases (Warner 1995a). As radius of the accreting star we use the Hansen and Kawaler (1994) formula for a C-O white dwarf intermediate between non-relativistic and full-relativistic degeneracy, which gives  km.

We can also use the assumption that AM CVn is in permanent superoutburst and compare for the two choices of the stellar masses. We use the following equation from Smak (1983)

where is the minimum mass transfer rate for a constant high state, where , is the disk radius, and is the radius of the accreting white dwarf. As disk radius we use , which is the tidal radius of the disk (Hirose & Osaki 1989). For we get  g/s, and for we get  g/s. The best fit to the observed continuum disk spectrum based on stellar atmosphere models, gives  g/s (Bard 1995), which give us another argument for the high mass solution which we will adopt in the following.

If the maximum and minimum velocities determined from the absorption lines originate in the disk, we can determine limits for mass and inclination (Patterson et al. 1993; Warner 1995a). The only solution for is an inclination of degrees, which requires an inner disk radius of .

With  s and  s, we obtain a beat frequency of 21 µHz (Eq. (1)) which may be present as high frequency sidebands to all physical frequencies in the system. This gives us an added bonus - we can use the 21 µHz frequency splitting to identify disk related physical frequencies. We also expect to observe some sum and difference frequencies and pulse-shape harmonics as is commonly observed in systems with non-sinusoidal modulations.

For the modulation at the independent frequency of 988.8 µHz, we may challenge the origin proposed by Patterson et al. (1993). They suggest that the 988.8 µHz frequency is a beat frequency resulting from a 16.5 µHz (or period 16.8 hrs) retrograde precession of the line of nodes in a slightly tilted disk. This would result in low frequency sidebands or amplitude modulations with this frequency which we have searched for, but did not find in our data. This search may be more profitable when the amplitude of the 988.8 µHz modulation is high. From a study of absorption line variations Patterson et al. (1993) conclude that the tilt could be at most a few degrees.

Patterson et al.'s (1993) proposed identification of the 988.8 µHz modulation as a disk tilt frequency is based on their identification of 972.3 µHz as the orbital frequency. Our identification of 951.3 µHz as the orbital frequency and 972.3 µHz as the (prograde) superhump frequency, leads to a prediction of a lower frequency for a beat between the (retrograde) nodal regression frequency () and the orbital frequency. Such a modulation is not detected, which may mean that a tilt is small or not present.

The amplitude variable modulation at 988.8 µHz is left unexplained by the expected disk frequencies. We have two possible other explanations for this modulation: It could either be the frequency of rotation or a g -mode pulsation of the accreting white dwarf. If we observe it must be due to the variable aspect of the accretion column on one of the poles. For both explanations we would expect higher amplitudes in the far UV, which is observed in HST spectra (Solheim et al. 1997). Arguments against the frequency of rotation interpretation are the rapid amplitude changes observed, no X-ray modulation at this frequency, and no polarization detected. AM CVn shows soft X-ray luminosities far below what is derived for polar systems, but not totally inconsistent with X-ray luminosities of non-magnetic CV's (Ulla 1995).

The implications of explaining the 988.8 µHz as a g -mode pulsation will be further discussed in Sect. 5.

5.2. The missing power at explained by a two-fold azimuthal structure or two spiral arms

We now consider a long standing problem for AM CVn: Why do we not observe any power at the fundamental orbital frequency , which we do for the other permanent superhumper in the AM CVn family, EC 15330-1403?

Any explanation of this problem would involve a phenomenon that happens twice per orbital period, and looks exactly the same each time. One possibility proposed by Solheim et al. (1984) and worked out in more detail by Provencal et al. (1995) is an intermediate polar system where a magnetic field sweeps a hole in the disk and the two accreting poles are seen twice per orbital revolution. In this case, we expect a negative because the accretor will be spinning up (Solheim et al. 1984) or a positive if the magnetic field is coupled to the slower rotating outer part of the disk (Provencal et al. 1995). An argument in favour of this model is that a hole in the disk is required to explain the maximum and minimum velocities in the disk with the high mass for the primary object, as described in the previous section.

We find several arguments against this interpretation. First, a magnetic field is not observed, although the small field required may escape detection. In addition, we do not find the variable X-ray emission expected from accreting columns (Ulla 1995). ROSAT observations of AM CVn can be interpreted as consisting of two sources: A blackbody source consistent with a white dwarf of  K, assuming , or  K, assuming , and an extra hard (bremsstrahlung) component with a temperature of a few keV, where is the interstellar hydrogen column density in the direction of AM CVn (van Teeseling 1995). The hard component is probably related to the accretion column on the white dwarf, and the blackbody source is most likely the central white dwarf or its boundary layer.

We propose that a solution to the , or the "double hump" problem, is found in the noncircular shape of AM CVn's disk, combined with the , two-armed spiral. But to explain the we have to add a spiral or triangular shape in the outer part, and finally a precessing, elliptical wave to completely explain the observations. Schematically the system may look as in Fig. 12.

Fig. 12. Geometry of a disk with strong tidal response which gives rise to spiral arms, and weak response which gives rise to triangular shape in the outer parts. For AM CVn the 20.8 µHz sideband indicates in addition a prograde precessing eccentricity

A disk with two-fold azimuthal structure, or tidal bulges is equivalent to the Earth with lunar tides. The Earth tides are, on average, very well synchronized with the lunar orbit - but the amplitude, and to some degree the phase may vary from day to day because of local weather. Small variations in the mass transfer rate and a precessing elliptical disk may mimic local weather on earth, changing the amplitude and phase of the tidal variations on short time scales, creating differences between the predicted and observed light curve as demonstrated in Fig. 2.

The phase variability of 2 on several short time scales (Solheim et al. 1984) may be linked to small variations in the precession rate, arising from variations in or chaotic behaviour in the disk. Fig. 13 shows that the amplitude of is modulated with the precession frequency (). Again, we would expect the same amplitude modulation if the two frequencies are independent. A larger modulation amplitude implies a higher disk eccentricity, which occurs when the position of the secondary star coincides with the major axis of the elliptical disk, perhaps introducing increased mass transfer.

Fig. 13. The amplitude of the modulation at 988.8 µHz as observed in the period 1976-1993

Based on this model, the disk must have a small eccentricity, which precesses with the period , explaining the observed superhump period of  s. An eccentricity -0.2 is necessary to explain the 13.4 hr line-profile changes observed by Pattersen et al. (1993).

What can we expect for in this model? We expect to decrease through loss of angular momentum by general relativity, which is more than compensated for by the high mass transfer which leads to orbital expansion. Assuming conservative mass transfer, is predicted to be between 3.6 and  s s^-1 (Faulkner et al. 1972). However, a wind is observed in the system, and mass and angular momentum are not conserved. Therefore we expect a somewhat larger rate of period change than predicted by Faulkner et al. Provencal et al. (1995) determined a for the 525.6 s period of s s^-1, a factor of 50 too large to be explained solely by GR and conservative mass transfer. Either the system is losing additional angular momentum at a high rate or the observed is not a measure of orbital evolution.

We have now explained the modulations with the largest amplitudes in the temporal spectrum. The final question is why we do not detect itself? In high inclination disks we would expect occultations due to the secondary passing in front of the disk or the bright spot, producing modulations with the orbital frequency. As shown in Fig. 12, we find that, for our choice of masses and , no disk occultations will take place if - so disk occultations cannot occur in this case since our solution gives .

The tidal bulges and the two spiral arms are on average stationary features seen twice per orbital period. Changes in the mass transfer rate and the disk ellipticity may change the position of the tidal bulges, which we observe as irregularities or 'phase jitter' in the light curve and the superhump period.

From UBVRI observations of the light curve, Massacand & Solheim (1995) conclude that the modulation has equal amplitude in each band, which can be explained by the disk size and shape effects described above. This reinforces our conclusion that is a result of changing aspects of the disk.

The modulation has a slightly larger amplitude at longer wavelengths and may arise from the cooler, outer part of the disk (Massacand & Solheim 1995). Based on our conclusion we can now present a complete table of identification of the 4 independent frequencies in the system and their beats, harmonics, sums and differences as given in Table 4.

Table 4. Explanations of frequencies in AM CVn

5.3. The 988.8 µHz modulation could be a g -mode pulsation

The 988.8 µHz modulation has many properties common with observed g -mode pulsations. It varies in amplitude on many time scales - years to even hours, perhaps explained by beating of closely spaced modes or fine structure due to rotational splitting. It has a higher amplitude in the far UV (Solheim et al. 1997). For a single white dwarf pulsator this is a result of limb darkening, which is stronger in the far UV (Robinson et al. 1995). It is also within the possible range of periods expected for DOs (Bradley 1995) and perhaps DBs with hot accreted envelopes (Nitta 1996). In order to fit the UV observation the central star of AM CVn is more likely a DO with temperature 150-180,000 K (Bard 1995; Nymark 1997).

Identification of g -mode pulsations in AM CVn opens interesting possibilities for modelling the interior of the accretor. Pulsation modes may be amplified by parametric resonance with the orbital frequency of the secondary object. This may be the first hint of a cooler star disguised with a hotter envelope, and may explain the rapid amplitude variations. Since the resonance is not exact, and the pulsation may at times be out of phase with the orbital phase, the pulsations may be damped instead of driven. Over the years the observed amplitude has changed with a factor more than ten, from 1.2 mma in 1990 to 15 mma in 1982, which is a change from 10 to 100 mma if the disk were not present (Fig. 13). When the amplitude is largest we should expect the pulses to be highly driven and non sinusoidal, generating a series of higher harmonics of the pulsation frequency. In the WET run reported in this paper, such harmonics of 988.8 µHz were not observed, due to the small amplitude (1.2 mma) of the variation itself.

The modulational frequency of 988.8 µHz found in AM CVn is near the pulsation frequencies observed for the hottest DOV stars such as RXJ 2117+34 (  K) (Moskalik & Vauclair 1965; Werner et al. 1996). This supports other evidence for the classification of the central star as a DO.

The disk in AM CVn stars contributes the majority of light in the optical part of the spectrum, but the contribution from the hot accretor will be relatively stronger in the far UV, predicting higher pulsation amplitudes in this spectral region. Monitoring AM CVn's light curve in the far UV over several days, would be of great importance for settling the question of g -mode pulsations in AM CVn. With such monitoring it may also be possible to detect frequency sidebands due to rotation of the white dwarf.

© European Southern Observatory (ESO) 1998

Online publication: March 30, 1998
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