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

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4. Data analysis

Since we had access to telescopes with sizes from 0.6 to 3.6 m, the S/N of the data varied. In Fig. 2 we show pieces of the light curve observed with the 3.6 and the 0.6 m telescopes at Mauna Kea on successive nights in the same hour angle interval. In the upper panel the light curve from the 3.6 m telescope is essentially noise-free, but the curve is still irregular. The next panel shows the light curve observed with the 0.6 m telescope, and it does not look significantly different. In the 3rd panel we show for comparison the light curve of PG 1159-035, a rich DO-pulsator of the same magnitude as AM CVn, observed with a 0.75 m telescope (Winget et al. 1991). In this case the pulses have small amplitudes but even with a small telescope a regular pattern is seen in the light curve.


[FIGURE] Fig. 2. Part of the light curve of AM CVn observed with the CFHT 3.6 m telescope is shown in the upper panel. The second panel shows the light curve 24 hrs later at the same location (Hawaii) observed with a 0.6 m telescope. The lower panel shows the light curve of a rich low amplitude pulsator PG 1159-035 observed with a 0.75 m telescope. The lower (solid) curve in each panel is a synthetic light curve calculated from the frequencies, amplitudes and phases determined. The unit mmi is milli-modulation intensity

A comparison of these light curves tells us immediately that AM CVn cannot be interpreted as a system with regular, small amplitude linear pulsations which is observed for some g -mode pulsating white dwarfs. We may have nonlinear pulsations or a combination of pulsations and flickering due to mass transfer.

Fig. 3 presents the Fourier Transform (FT) of the entire WET-run. Note the change in y-scale to accommodate the dynamic range of each panel. We find a series of peaks at 1902, 2853, 3805, 4756 and 5708 µHz. These peaks are numerically related to a fundamental frequency [FORMULA]  µHz - which itself is not detected. The 989 µHz peak which was present with large amplitudes in 1978, 1982 and 1987 (Provencal et al. 1995), has an amplitude of only 1.2 mma in this data set. Next we will describe the FT in general terms, then in the following subsections discuss details.

[FIGURE] Fig. 3. Fourier Transform of the total data set. The ordinate shows relative power in units of µmp (modulation power which is (ma)2). The ordinate scales are different for each panel in order to accommodate the range observed in each frequency interval. The frequency is in µHz

4.1. Results of the Fourier Transform in general terms

When we interpret the FT of a light curve we must be aware of its limitations when used on data sets with gaps. If we assume that some real modulation frequencies are present, the resolution and alias pattern generated by each real peak in the FT is described by the spectral window, the pattern of peaks generated in a FT by a single sinusoid sampled exactly as the data (Nather et al. 1990). Our WET coverage was not complete, resulting in a spectral window containing small sidebands due to remaining periodic gaps in our data samples (Fig. 4). The temporal resolution of our FT, given as the inverse of the length of the campaign, is [FORMULA]  µHz.

[FIGURE] Fig. 4. The window function for the WET run on AM CVn. The sidebands are quite small and the resolution is about 1 µHz. The window function is in power and normalised to unity

The power spectrum's noise level begins to rise exponentially below 400 µHz, masking any real power in this part of the spectrum. Polynomial division forces the noise level to zero below 50 µHz, reducing our sensitivity to low frequency variations. The dominant peak is at 1903 µHz ([FORMULA]). We find additional peaks at each [FORMULA] higher frequency up to 5708 µHz. In addition to this series of harmonics, there are other smaller peaks, but no obvious pattern is found.

We tested the stability of the resolved power spectrum with a simple comparison between the first and the second half of the run (Fig. 5). We find that the main peaks are present in both parts, but the higher harmonics are somewhat stronger in the second part. However, the lower priority and less favourable sampling in the second part of the run degrades our window function, reintroducing alias features and increasing the noise level, possibly accounting for the increased amplitude of the higher harmonics.

[FIGURE] Fig. 5. Amplitude-FT of the first and second half of the run, with inserts showing the window functions in amplitude for each part

The most surprising and unexpected feature in the temporal spectrum is a sideband splitting of 20.8 µHz (or a period of 13.4 hrs) associated with some of the harmonics, always on the high frequency side.

4.2. Peaks in the 1000 µHz region

The question confronting us with AM CVn is the existence of a 951 µHz (1051 s) modulation. To settle this question we show an enlargement of the 1000 µHz region in Fig. 6. The only significant peaks in this region are at 972.3 µHz (1028.5 s), 988.8 (1011.4 s) and 1045 µHz (957 s), all with an amplitude about 1.2 mma. We have not seen the 1045 µHz power before, and the power previously reported at 976 µHz (Solheim et al. 1984) is not detected here. Most importantly, we conclude that the 951 µHz modulation is not detected in this data set.

[FIGURE] Fig. 6. Power spectrum of the region 900-1100 µHz with the frequencies corresponding to some of the interesting frequencies (951, 972, 976, and 1045 µHz) marked with arrows

It is interesting to note that the 972.3 µHz peak is 21 µHz from the undetected [FORMULA] at 951.3 µHz, the same sideband splitting as we find in the harmonics. We can therefore interpret it as a high frequency sideband to [FORMULA]. The 989 µHz peak has been observed at much larger amplitudes in the past, and is incommensurable with the set of numerically related frequencies. This modulation must have a different physical origin than the set of numerically related peaks. The 1045 µHz modulation has not been detected before and may either be spurious or a sideband to the 989 µHz peak, with a [FORMULA] Hz.

4.3. A search for constant frequency- and period splittings

Constant frequency and period spacings are frequently observed in the power spectra of classical hydrogen dwarf novae. Such frequencies could arise from beating with the orbital period or rotational period, g -mode pulsations or other causes. In a system with a disk we may not be able to observe the fine structure of the pulsation spectrum of the central white dwarf, but instead detect a pattern related to disk phenomena. To investigate this possibility we have searched the power spectrum for incidences of equal splittings with a method developed and described by Provencal (1994). This method does not require prior choices as inputs as in the KS (Kolmogorov-Smirnov) test. The method uses the spectral window to create a template containing two peaks separated by the frequency splitting of interest. This template is formed by adding, in both frequency and phase space, two spectral windows, one of which is offset by the frequency difference of interest with respect to the other. The template is then passed through the FT of the data set, and compared with the pattern of power surrounding every peak above an amplitude threshold decided upon by the user. The square of the difference between the template and the region around the actual peak is calculated, and summed for every peak above the amplitude limit. After each peak has been tested, a new template is created with a slightly different frequency splitting, and the process is repeated. The final product is an average difference for the entire FT, as function of the frequency splitting. If a certain splitting occurs in the FT a multiple of times, this will show up as a lower than average difference.

The result of the Provencal method used on the AM CVn data set is shown in Fig. 7. The two deep minima are aliases due to the data sampling, while the minimum at 21 µHz is the fine structure sidebands we have discovered. No additional splittings are detected.

[FIGURE] Fig. 7. Equal frequency splittings in the 1990 AM CVn FT. The deepest minima are window artefacts. The minima at 20.8 µHz is the fine structure associated with the 1902.5, 2853.8, and 3805 µHz frequencies (Provencal 1994). The bar at 47 µHz is a one sigma significance level

4.4. Periods and phases of significant peaks

To verify the amplitude and phase of the sidebands of the harmonically related peaks, we prewhitened the data set in the following way: We identified the largest amplitude peak in a band, and subtracted a sinusoid with the same amplitude and phase from the light curve, repeating the process until no obvious peaks remained in the FT. An example of the prewhitening process for the 1902.5 µHz band is shown in Fig. 8. The exact periods, amplitudes and phases for the largest peaks were determined by a non-linear least square fit (Kepler 1993). The result is given in Table 3. Although [FORMULA]  µHz is itself undetected, we have adopted a naming strategy based on its numerical relationship to the other peaks. We have named the sequence of harmonics of the fundamental 951.3 µHz frequency for [FORMULA], where n identifies the [FORMULA] th harmonic. For the high frequency sidebands we add a second index 0, 1, or 2 to identify the sideband number. We have chosen 951.3 µHz as a fundamental frequency, since most of the observed frequencies fall into a harmonic pattern of this frequency. This is different from the conclusions of Provencal et al. (1995) who designate [FORMULA] (1902.5 µHz) as the fundamental frequency. As a time reference we have used the start of the WET campaign: [FORMULA]  BJD. The phases given in the table refer to the first [FORMULA] (for zero phase) after [FORMULA].

[FIGURE] Fig. 8. The prewhitening process shown for the 1902.5 µHz region. The upper panels show the original FT, and the lower panels show the pattern left after successive subtraction of single sinusoids marked with arrows

[TABLE]

Table 3. Peaks detected or identified in the FT


We note that the "harmonic pattern" [FORMULA], [FORMULA], ..., [FORMULA] is very precise, with deviations from exact ratios of the order [FORMULA]. We have searched for, but not detected [FORMULA].

The average sideband difference is [FORMULA] µHz, corresponding to a period of [FORMULA]  hrs. Unfortunately, our observing and reduction technique, with subtraction of third order polynomials to remove long term variations, is not suited for detection of low frequency modulations in our temporal spectrum. However, we find that the amplitude of the strongest peak is modulated with this period, and will address this in Sect. 4.8 (Fig. 11).

We find no additional single peaks, except those given in Table 3. A band of excess power is seen between 1218 and 1248 µHz, and two peaks, each 1 mma are located at frequencies 1227 and 1307 µHz, but they seem to be unrelated to the set of harmonically related peaks in Table 3, and have a low signal to noise ratio. It should be noted that in an analysis of older data sets Provencal et al. (1995) have detected a power at [FORMULA] in some seasonal Fts.

The precise numerical relations between the frequencies [FORMULA] suggest that they are harmonics of one frequency ([FORMULA]), and the constancy of the frequency splitting over a factor 3 in frequency range argues for a geometrical relationship which preserves the frequency splitting over such a range. In the following we will discuss the stability of the significant peaks, and how they modulate the light curve.

4.5. The stability of the significant peaks

To study the coherency of the various peaks we have divided the light curve in 9 parts, each of which are long enough to resolve the 21 µHz structure. We have fitted each subset, using linear least squares, with all of the frequencies determined above, to produce O-C diagrams to detect phase variations (Kepler 1993; Solheim 1993b).

To reduce the effect of aliasing, each part of the data set was prewhitened with the periods, amplitudes and phases given in Table 3, leaving only one peak in each group at a time for linear sinusoidal fit.

Examples of the resulting amplitude and phase variations after prewhitening are shown in Fig. 9. With some exceptions discussed below, the amplitudes and phases for the stronger peaks are constant. We can conclude that we have coherent oscillations in all of the harmonic peaks except [FORMULA], [FORMULA] and [FORMULA], which may have separate origins.

[FIGURE] Fig. 9. Amplitude and phase variations for the strongest ([FORMULA]) group (1903 µHz) during the campaign. The top panels show amplitude variations and the lower panels show the phase variations. The abscissa unit is [FORMULA]  s, and the ordinate unit of the upper panels is mma (milli-modulation amplitude), while the ordinate unit of the lower panels is s

In the [FORMULA] group (Fig. 8) the amplitudes are almost constant for [FORMULA] and [FORMULA]. The amplitude of [FORMULA]  s is generally low (4 mma), but has a dramatic rise (to 14 mma) between day 6 and 9. This coincides with the rise in amplitude for [FORMULA], indicating that these two periods are harmonically related, i.e. we may interpret [FORMULA] as a pulse shape harmonic of [FORMULA]. The phase of [FORMULA] shows no variations. The phase of [FORMULA] shows a systematic drift in the first part of the run, then jumps back to average around day 9. The phase of [FORMULA] jumps between two levels P/2 apart, again an indication that [FORMULA] is a pulse shape harmonic of [FORMULA].

In the 350 s multiplet ([FORMULA]) we also observe a constant amplitude except for an "event" taking place between day 6 and 9, which is strongest for [FORMULA]. The phase is drifting during the run for [FORMULA] and [FORMULA].

In the remaining peaks ([FORMULA] - [FORMULA]) the event between day 6 and 9 is not observed. The amplitude of [FORMULA] is constant. The amplitude of [FORMULA]  s slowly decreases during the run, while the amplitude of [FORMULA] slowly increases, making a sudden jump at the end of the run. The amplitude of [FORMULA] is only significantly different from zero in the central part of the run. We find no significant phase variations except a phase shift of P/2 at the very end of the run for [FORMULA], indicating that this is a pulse shape harmonics of [FORMULA].

We conclude that the peaks [FORMULA] and [FORMULA] are stable in amplitude and phase, while the peaks [FORMULA], [FORMULA], [FORMULA], [FORMULA], and [FORMULA] experience systematic drifts, and the rest are phase variable or uncertain in our data set. The periods [FORMULA] and [FORMULA] may be interpreted as pulse shape harmonics, while the others may represent real physical periods or beat periods. It should be emphasised that we can make only weak conclusions concerning the stability of the low amplitude modulations, and that we only discuss stability on time scales longer than 13.4 hrs. A summary of the stability of the strongest peaks is given in the last column of Table 3.

4.6. The average modulation shapes

We can find additional hints of the origin of AM CVn's photometric variations from a study of the average modulation shape. The average modulation shape is formed by folding a light curve at a period of interest. It contains the same information present in the FT, but provides an additional tool to interpret harmonic structures. We determined the average modulation shape of the dominant frequencies in AM CVn, and show some of them in Fig. 10.


[FIGURE] Fig. 10. Pulse shape for 4 modulations. a [FORMULA] 951.3 µHz, b [FORMULA] µHz, c 988.8 µHz, and d [FORMULA] µHz. The ordinates are modulation intensity (mmi)

The average modulation profile at the unseen fundamental frequency 951.3 µHz (Fig. 10a) shows one narrow and one wide peak per period. This difference has led previous observers (see Solheim et al. (1984) for a review) to identify these as primary and secondary maxima. If the two humps were equal, we would observe only the first harmonics in the FT. The higher harmonics will make two consecutive peaks of [FORMULA] look different. All the observed harmonically related frequencies could be interpreted as pulse shape harmonics of the fundamental frequency. However, since the harmonics are phase stable to the limit of detection (possibly except [FORMULA]), and have relatively constant amplitudes (Solheim 1993b), they can also be due to modulation of physical features. An argument against the sequence being simple harmonics of a fundamental frequency, is our expectation that the frequency splitting in that case would increase proportional to the order of the harmonics, while we observe a constant splitting.

The profile of [FORMULA]  µHz (Fig. 10b) is almost saw-toothed and variable in shape, and requires higher order pulse shape harmonics to be explained. The lack of coherency for [FORMULA] and the amplitude correlation with [FORMULA] indicates that [FORMULA] (1944.4 µHz) is a pulse shape harmonics of [FORMULA]. If this frequency refers to a physical structure, it must have an irregular shape, and change on a short time scale. Higher order harmonics of the [FORMULA] pulse are not detected.

The sideband [FORMULA] and most of the remaining peaks investigated are all close to pure sinusoids. In Figs. 10c and 10d we show 988.8 µHz and [FORMULA]  µHz as examples. The coherency of the sidebands and their sinusoidal form indicate that they are aspect modulations. The 988.8 µHz modulation shows considerable seasonal variations (Provencal et al. 1995), arguing for the origin of this frequency in some feature which varies in size and shape. A lack of sidebands and the temporal instability for [FORMULA] argues for it being a pulse-shape harmonics for [FORMULA]. The same is the case for [FORMULA] which jumps [FORMULA] in phase, as expected from pulse-shape harmonics.

4.7. A 13.32 hrs period detected

Patterson et al. (1993) report modulation of AM CVn's absorption line profiles with a period of 13.38 h, which they interpret as the precession period of an elliptical disk in the AM CVn system. The disk precession may be the cause of the 20.8 µHz frequency splitting we observe.

To investigate the effect of the precession period on the amplitude of [FORMULA], we have fitted, using linear least squares, with a fixed period [FORMULA]  s (1902.5 µHz) to sections of the data about 4 cycles ([FORMULA]  minutes) in length, resulting in 176 amplitude determinations which we have folded with the period [FORMULA] and shown in Fig. 11, This method is similar to the technique employed by Kurtz et al. (1989) for detection of precession in magnetic Ap stars.


[FIGURE] Fig. 11. The modulation amplitude of the strongest modulation [FORMULA]  µHz versus the precession period [FORMULA].4. The pulsation amplitude has been calculated by fitting the period 525.618 s to sections of the data about 4 cycles ([FORMULA] minutes) long. The solid line shows a linear least square fit with a period 13.32 hrs, a mean amplitude 13.1 mma and a sinusoidal variation with [FORMULA]  mma

The large scatter in the diagram is due to the error in determination of amplitudes in the short light curve pieces. The upper envelope of the points shows the sinusoidal amplitude modulation expected from two unresolved nearby frequencies.

The period of the amplitude modulation and its time of maximum is determined by a non-linear least square analysis of the amplitude versus time distribution. We get as the period of amplitude modulation:

[EQUATION]

or

[EQUATION]

and

[EQUATION]

If we compare [FORMULA] with the phase of skewness variations determined by Patterson et al. (1993) for the broad absorption lines, we find their maximum value of skewness at [FORMULA] BJD, which is in phase with our maximum amplitude. This strengthens the idea that the 1902.5 µHz modulation and the line skewness are related.

If we compare the various modulation periods, we find:


[TABLE]

We conclude that these periods are all the same within the error of measurement and are most likely related to the same phenomenon.

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© European Southern Observatory (ESO) 1998

Online publication: March 30, 1998
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