3 Changes in the dominant mode in 1996

We observed GD 358 in August 1996 to provide simultaneous observations to compare to HST time resolved spectroscopy made on August 16. The 1996 data covers 10 days of the most remarkable amplitude behavior ever seen in a pulsating white dwarf. Observing this behavior is serendipitous, as individual observers and the WET have observed GD 358 off and on for 20 years without seeing this sort of behavior.

Figure 3 shows how the amplitude of the k=8 P=423 s mode changed with time during our observations in August 1996. The amplitude changes we saw in our optical data are unprecedented in the observations of pulsating white dwarf stars; no report has been made of such a large amplitude variation in such a short amount of time. Here we describe what we found in our data.

The lightcurves acquired in August 1996 are displayed in Fig. 4 and the Fourier transform for two lightcurves is shown in Fig. 5. Those in the first and second panels of Fig. 4 look very different from each other. The Fourier transform of the lightcurve from the first panel is similar to that from the 1994 WET data where we identified over 100 individual periodicities, while the Fourier transform of the second panel is dominated by only a single periodicity (Fig. 5); this represents a complete change in the mode structure, as well as the period of the dominant mode, in about one day!

In run an-0034, the k=8 P= 423 s mode's amplitude is 170 mma, which is the largest amplitude we have ever seen for this mode. To check for additional pulsation power (perhaps lower amplitude pulsations dwarfed by the 423 s mode power), we prewhitened the an-0034 lightcurve by the 423 s mode. Prewhitening subtracts a sinusoid with a specified amplitude, phase and period from the original lightcurve, and it helps us look for smaller amplitude pulsations by eliminating the alias pattern of the dominant pulsation mode from the Fourier transform. In Fig. 6, we show the Fourier transform of the an-0034 lightcurve both before and after prewhitening. We see now that GD 358 was indeed dominated entirely by a single mode at a different period from the dominant mode a day earlier. We refer to this event by the musical term "forte'', or more informally as the "whoopsie''.

Given the spectacular behavior of GD 358 in August 1996, we obtained follow-up observations in September 1996 and April 1997. Table 4 shows the journal of observations for the September 1996 and April 1997 data. The lightcurve and the power spectrum during these observations (Fig. 7) seem to have returned to the more or less normal state seen in the past (Fig. 2), not the unusually high amplitude state it was in in August 1996 (Fig. 5).

We show the lightcurves of GD 358 at three different times in Fig. 8. The middle panel shows the lightcurve when the amplitude of the 423 s mode was at its largest. The lightcurve looks almost sinusoidal, with the single 423 s mode in the power spectrum. The result of this is that we obtain similar values for the peak-to-peak semi-amplitudes and the FT amplitude of the 423 s mode; this implies that a single spherical harmonic is a good representation of the stellar pulsation at this time. The other two lightcurves, however, each containing several pulsation modes, are less sinusoidal. If the non-sinusoidal nature of a lightcurve comes from the fact that many modes are present simultaneously, then one would expect the shape of the lightcurve to be sinusoidal only when it is pulsating in a single mode. On the other hand, in the August 1996 sinusoidal lightcurve, the peak-to-peak light variation was about 44% of the star's average light in the optical. We would expect such a large light variation to introduce nonlinear effects into the lightcurve, even if the star is pulsating in a single mode, causing the lightcurve to look nonsinusoidal. Thus, the nearly sinusoidal shape of our lightcurves (Fig. 8) is a mystery, except for the theoretical models of Ising & Koester (2001), which predict sinusoidal shapes for large amplitude modes even with the nonlinear response of the envelope.

After the P=423 s mode reached its highest amplitude in run an-0034, the k=9 P=464 s mode started to grow and the 423 s became smaller, but there was still very little sign of the usually dominant k=17P=770 s mode. In Fig. 2, we present the Fourier amplitude spectra of the light curves obtained each year, 2000 on top, 1996, 1994, and 1900 on the bottom, on the same vertical scale. Note that the 1996 data set is low resolution, because of its smaller amount of data. It is clear that the periodicities change amplitude from one data set to the other. It is important to notice that the periodicities, when present, have similar frequencies over the years. The amplitudes change, and even subcomponents (different m values) may appear and disappear, but when they are present, they have basically the same frequencies (typically to within 1 $\mu$Hz).

Figure 7: GD 358 Fourier transform at four different times along with their spectral windows. The 1994 and 1997 Fourier transforms look similar (within the observed frequency resolution, that is). The September 1996 data look similar as well to these two data sets, but the highest amplitude modes have shorter frequencies (longer period). Obviously, the August 1996 Fourier transform looks very different from the other Fourier transforms.

In the September 1996 data, the Fourier transform shows that GD 358 is pulsating in periods similar to what we are familiar with from the WET data of 1990, although the highest peaks are at $1082~\mu$Hz, $2175~\mu$Hz and $2391~\mu$Hz. The very limited data set and the complex pulsating structure of the star makes interpretation of these peaks difficult (Fig. 7). It is not until the data taken in April 1997 when we observe the 770 s mode as the highest amplitude mode in the Fourier transform, as in 1990 and 1994. We do not have data to fill in the gap between September 1996 and April 1997 to see how the amplitude changed, but even by August 19th, the modes at k=15 and 18 were already starting to appear. The time scale which the star took to change from its normal multi-mode state to a single mode pulsator was very short, about one day. The reverse transition started one week after the event. An estimate of the total energy observed in pulsations is best obtained by measuring the peak-to-peak amplitudes in the light curves directly, instead of adding the total power from all the modes. For the largest amplitude run in 1996, an-0034, observed with the 2.1 m telescope at McDonald, we estimate a peak-to-peak semi-amplitude of 220 mma. For comparison, the measured Fourier amplitude for the k=8 mode for that run is 170 mma. For two runs at the same telescope in 2000, we obtain a peak-to-peak semi-amplitude of 120 mma. Again for comparison, the Fourier amplitudes of the large modes present are 30 mma, but there are several modes, and many combination peaks. As the observed pulsation energy is quadratic in the amplitudes and the frequencies, it corresponds to an increase of around 34% in the radiated energy by pulsations, from the amplitudes, plus a factor of 2.8 from the frequency. Just two days after the "forte'', the peak-to-peak amplitude decreased by a factor of 5, but during our observations a month later, it had already increased to its pre-"forte'' value. It is important to notice that the observed amplitude is not directly a measurement of the physical amplitude, as there are several factors that typically depend on $\ell$, including: geometrical cancellation, inclination effects, kinetic energies associated with the oscillatory mass motions, together with a term that depends on the frequency of pulsation squared. If we assume that the inclination angle of the pulsation axis to our line of sight does not change, and that the $\ell$ values of the dominant modes do not change, then it must be the $\ell$ distribution of the combination frequencies that changes and produces a difference in the peak-to-peak variations in the light curve, if the total energy is conserved. This is plausible, as relatively small variations in the amplitudes of the dominant periods can dramatically change the amplitudes of the linear combination frequencies, but not necessary.

Figure 8: GD 358 lightcurves over time. The shape of the lightcurve was sinusoidal when the amplitude was highest. The 1994 and September 1996 data exhibit similar pulse shapes and their corresponding power spectra also look similar (Fig. 7).