2006 WHOLE EARTH TELESCOPE OBSERVATIONS OF GD358: A NEW LOOK AT THE PROTOTYPE DBV

Before we can look in detail at the 2006 XCOV25 FT, we must investigate GD358's amplitude and frequency stability over the entire timebase. GD358 is known for large scale changes in amplitudes and small but not insignificant frequency variations on a variety of timescales (K03). Amplitude and/or frequency variations produce artifacts in FTs, greatly complicating any analysis. We divided the data set into three chunks spanning ≈180 hr and calculated the FT of each chunk (Figure 2). For the dominant peak, the frequencies are consistent to within measurement error and the amplitudes remain stable to within 3σ. The differences between each FT are explained by variation in window structure and resolution from chunk to chunk. We are confident that GD358 was fairly stable over the length of XCOV25.

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To select the statistically significant peaks in the Fourier transform, we adopt the criterion that a peak have an amplitude at least 4.0 times greater than the average noise level in the given frequency range. This leads to a 99.9% probability that the peak is a real signal present in the data, and is not due to noise (Scargle 1982; Breger et al. 1993, K03, for example). Here, "noise" is defined as the average amplitude after prewhitening by the dominant frequencies in the FT, and is frequency dependent. This is a conservative estimate, as it is impossible to ensure that all of the "real" frequencies are removed when calculating the noise level. This is most certainly the case in GD358, where the FT contains myriad combination frequencies. Figure 3 displays the average amplitude, specified as the square root of the average power after prewhitening by 50 frequencies, as a function of frequency, for 2006, and previous WET runs on GD358 in 2000, 1994, and 1990.

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The amplitude limit we select is very important in determining "real'signals. To confirm our uncertainty estimates, we performed several Monte Carlo simulations using the routine provided in Period04, software devoted to the statistical analysis of time series photometry (Lenz & Breger 2005). The Monte Carlo routine generates a set of light curves using the original times, the fitted frequencies and amplitudes, and added Gaussian noise. A least-squares fit is performed on each simulated light curve, with the distribution of fit parameters giving the uncertainties. Our Monte Carlo results are consistent with our average amplitude noise estimates (Table 2), confirming our use of the average amplitude after prewhitening.

For both Fourier analysis and multiple least squares fitting, we use the program Period04. The basic method involves identifying the largest amplitude resolved peak in the FT, subtracting that sinusoid from the original light curve, recomputing the FT, examining the residuals, and repeating the process. This technique is fraught with peril, especially in multiperiodic stars, where it is possible for overlapping spectral windows to conspire to produce alias amplitudes larger than real signals. Elimination of this alias issue was the driving motivation behind the development of the WET, whose goal is to obtain nearly continuous coverage over a long time baseline. Our current data set on GD 358 does contain gaps, but we have minimized the alias problem.

To illustrate the procedure we followed, let us examine the region of dominant power at 1234 μHz (Figure 4). Comparison of this region with the spectral window demonstrates that most of the signal is concentrated at 1234 μHz. We fit the data with a sine wave to determine frequency, amplitude, and phase, and subtract the result from the original light curve. The second panel of Figure 4 shows the FT prewhitened by this frequency. Careful examination reveals two residual peaks (arrows) that are clearly not components of the spectral window. We next subtract a simultaneous fit of the 1234 μHz frequency and these two additional frequencies, with the results displayed in panel 3 of Figure 4. At this point, we must proceed with extreme caution. The remaining peaks, which correspond closely with aliases in the spectral window, are significant and cannot be ignored. We are faced with two possibilities: (1) these peaks represent real signals, and (2) amplitude modulation is present. While we have ruled out large modulations in frequency/amplitude during XCOV25, small scale amplitude modulation may be intrinsic to GD358, or artificially present in the data set. An examination of data from individual sites reveals that although the frequencies from each site are the same within the statistical uncertainties, two observatories, one using a CCD and one a PMT, report consistently lower amplitudes, probably due to beating effects. Removing the suspect observations from the data set does not alter the prewhitening results for the 1234 μHz region. As an additional test, we closely examined the FTs from Figure 2. Prewhitening the 1234 μHz regions of each chunk consistently produces the same five peaks. Finally, we note that one of these peaks appears in combination with other modes (Table 3). Therefore, we believe our first possibility is most probable, all five peaks are real and are listed in Table 2.

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We employed this procedure to identify frequencies satisfying our criteria of amplitudes 4σ above the average noise. The large amplitude power at 1740 μHz is particularly complex. Unlike our example at 1234 μHz, this mode contains three peaks with amplitudes over 10 mma and contains as many as nine frequencies (Figure 5). Figure 6 demonstrates the prewhitening results for modes at 2154 and 2362 μHz.

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Our final identifications result from a simultaneous nonlinear least squares fit of 130 frequencies, amplitudes, and phases. We adopt the l = 1 mode identifications for GD358 established in W94. We emphasize, however, that k, representing the number of nodes in the radial component of displacement from the center to the stellar surface, cannot be observationally determined and may not correspond precisely to the values given here. Tables 2 lists 27 identified independent frequencies. Table 3 presents significant combination frequencies. In the interest of space, we do not list all of the combination frequencies.

The 2006 XCOV25 data set illustrates GD358's continuing tendency for changing the distribution of amplitudes among its excited modes. Using the k identifications established in W94, we detect power at k = 21, 19, 18, 17, 15, 14, 12, 11, 9, and 8. The principal frequency is at 1234 μHz (k = 18) with an amplitude of 24 mma. The previously dominant k = 17 and 15 modes (K03) are greatly diminished in amplitude (1.4 and 5.6 mma, respectively), and we do not detect the k = 16 or k = 13 modes reported by W94 and K03. We cannot identify the suspected l = 2 mode at 1255.4 μHz or k = 7 at 2675.5 μHz noted by K03). K03 suggest that k = 7 may have been excited to visibility via resonant coupling with k = 17 and 16. Since neither has significant amplitude in 2006 (Figure 1), it logically follows that this mode would not be detected. Perhaps the greatest surprise from XCOV25 is the appearance of prominent power at the predicted value for k = 12, a region of the FT previously devoid of significant peaks.

Figure 7 presents a "snapshot" of multiplet structure in the XCOV25 FT. The only modes exhibiting structure with splittings in agreement with previous observations are k = 9 and 8. We find average multiplet splittings of 3.83 and 3.75 μHz, respectively. The dominant k = 18 is a quintuplet with an average splitting of 5.6 μHz. The 5.6 μHz splitting also appears in the k = 15 and k = 12 modes. The k = 15 mode contains five peaks, only three of which are split by 5.6 μHz. The k = 12 mode could contain as many as nine components, if we relax our 4.0 σ detection criteria slightly. A possible interpretation for k = 12 is two, perhaps three, overlapping triplets, each with ≈5.6 μHz splitting. An l = 2 mode is predicted at ≈1745μHz, so a second possible interpretation is an overlap of l = 1 and l = 2 modes. We point out that the other high-k modes (17, 16, 14, and 13) reported by W94 to have frequency splittings of ≈6 μHz are either not detected here, or do not have sufficient amplitude to investigate multiplet structure.

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Our goal for XCOV25 is the identification/confirmation of l and m values for modes in GD358's 2006 light curve. This information is required to fit our high signal-to-noise lightcurve using Montgomery's technique. However, the 2006 multiplet structure proves to be puzzling. In the limit of slow rotation, we expect each mode to be split into 2l + 1 components. Except for k = 9 and k = 8, we do not find the distinct triplets attributable to l = 1. Indeed, k = 18 is a quintuplet at first glance suggesting l = 2, and k = 15 and 12 are both complex but lacking equal splittings. We cannot confirm the l or m designations of W94 and K03 based on multiplet structure alone. Fortunately, the identification of the spherical degree of GD358's pulsation modes as l = 1 does not rely solely on multiplet structure. Table 4 presents the mean period spacing for XCOV25, as well as WET runs from 1990, 1991, 1994, and 2000. The average period spacing, which is dependent on stellar mass, is 38.6 s. If GD358's modes are consecutive l = 1, the period spacing corresponds to a stellar mass of ≈0.6 M_o, in agreement with spectroscopic results (Beauchamp et al. 1999). If they are consecutive l = 2 modes, the derived stellar mass becomes ≈0.2 M_o, making GD358 one of the lowest mass field white dwarfs known and incompatible with the spectroscopic log g. In addition, the distance of 42 ± 3pc derived from GD358 models assuming l = 1 is consistent with the trigonometric distance of 36 ± 4pc. The distance derived derived from models assuming l = 2 is ≈75 pc (Bradley & Winget 1994). Is it possible that we have an amalgam of l = 1 and 2 modes? l = 2 modes are predicted to be quintuplets, fitting with our results for k = 18. However, we expect the ratio of rotational splittings between l = 1 and l = 2 to be R_1,2 = 0.60 (Winget et al. 1991). Assuming k = 8 and 9 are l = 1 as indicated by their triplet structure, we expect an l = 2 splitting of 6.3 μHz, much larger than the observed 5.6 μHz splitting. We have no easy explanation for the 5.6 μHz splitting, and hesitate to identify k = 18 as an l = 2 mode solely on the basis of a quintuplet. The k = 12 and k = 15 modes contain the mysterious 5.6 μHz splitting, but the splittings are unequally spaced. The "best fit" pulsation models do predict l = 2 modes at 1250 μHz (near k = 18) and 1745 μHz (near k = 12) (Metcalfe et al. 2003). The 1250 μHz mode is not detected in this data set. Some of the complex peaks at k = 12 could correspond to an underlying l = 2 mode, but the splittings do not support this conclusion. Finally, optical and UV radial velocity variations have been used to determine the l values for several modes in GD358, including k = 18. Kotak et al. (2003) identified radial velocities corresponding to k = 18, 17, 15, 9, and 8 in their time-resolved optical spectra. They determined that these modes all share the same l value, which is probably l = 1. Castanheira et al. (2005) use HST UV time resolved spectroscopy to determine that k = 9 and 8 are best explained as l = 1.

While we are fairly certain that the majority of GD358's modes are l = 1, except for k = 9 and 8, we cannot directly provide m identifications for the observed multiplet components or confirm the m identifications of W94 and K03. For this data set, we must resort to indirect means. In the next section, we examine the combination frequencies, and, limiting ourselves to l = 1, explore their potential for pinning down the multiplet m components in GD358.

GD358's 2006 FT contains a rich distribution of combination frequencies, from differences to simple harmonics to fourth order combinations. Combinations peaks are typically observed in the FTs of large amplitude pulsators (Dolez et al. 2006; Yeates et al. 2005; Thompson et al. 2003; Handler et al. 2002, W94). Their frequencies correspond to integer multiples of a single frequency (harmonics), combinations (both sums and differences) of different components of a given multiplet, or combinations (again both sums and differences) of components of different modes (cross combinations). The general consensus on the origin of these combinations dictates that they are not independent modes, but are the result of nonlinear mixing induced by the convection zone in the outer layers of the star (Brickhill 1992; Wu 2001; Ising & Koester 2001). The convection zone varies its depth in response to the pulsations, distorting and delaying the original sinusoidal variations, and redistributing some power into combination frequencies. While combinations themselves do not provide additional direct information about the interior structure of the star, they are sensitive to mode geometry, making them potential tools for identifying l and/or m values and orientation for each mode.

Wu (2001) provides a theoretical overview of combinations and their interpretation. Handler et al. (2002) and Yeates et al. (2005) apply Wu's approach to several ZZ Ceti (DA) stars. Yeates et al. (2005) focus on the dimensionless ratio R_c first introduced by van Kerkwijk et al. (2000). The theoretical value of R_c is given as $R_c = \it {F(\omega _i,\omega _j,\tau _{\rm c_0},2\beta +\gamma)}\it {G(l,m,\theta)}$. The first term, F, includes the frequencies of the parent modes (ω_i, ω_j), as well as properties inherent to the star, such as the depth of the stellar convection zone at equilibrium ($\tau _{{\rm c}_0}$) and the sensitivity of the convection zone to changes in temperature (β, γ). The second term, G, includes geometric factors such as the l and m values of the parent modes, and the inclination angle of the pulsation axis. Both Wu (2001) and Yeates et al. (2005) present theoretical predictions for various combinations of l and m values.

The observed value of R_c is defined as the ratio between the observed amplitude of a combination and the product of the observed amplitudes of its parent modes, including a correction factor incorporating the bolometric correction and an estimate for the convective exponent. The convective exponent appears in theoretical estimations of the thermal response timescale of the convection zone. From Montgomery (2008), we estimate the bolometric flux correction for GD358 to be 0.35 and the convective exponent to be 25. For harmonic combinations, this produces a corrective factor of 0.22. Cross combinations include an additional factor of 2, resulting in a correction of 0.44. With this formalism, and limiting ourselves to the previous identification of the principal modes as l = 1, and with the smorgasbord of combination frequencies present in GD358, it should be possible to determine m values of the parent modes, to provide limits on the inclination angle, and to further study convection in white dwarfs. However, interpretation of the combination frequencies is not as straightforward as we would hope.

The analysis of R_c for harmonic combinations is simplified by the fact that we are certain that the combination contains a single parent, and so single values for l and m. In the 2006 data set, we detect the dominant 1234 μHz mode's first, second, and third harmonics, and place upper amplitude limits of ≈0.3 mma on higher orders. The first harmonic, at 2468.282 μHz with an amplitude of 5.1 mma, is the 12th highest amplitude peak in the FT and the second highest amplitude combination frequency observed.

The second largest mode and most complicated multiplet in the 2006 FT is k = 12. Focusing on its three large amplitude components (Table 3), we find the first harmonic of 1736 μHz, near 3472 μHz with an amplitude of 1.5 mma. Interestingly, the largest peak near 3472 μHz is actually a sum of k = 12 components (see Table 3), not a simple harmonic, a behavior distinctly different from k = 18. We place an upper limit of 0.5 mma for a 1st harmonic of 1741.67 μHz, 0.6 mma for 1749.08 μHz, and upper limits of 0.3 mma for all higher orders for all other components of k = 12.

For the remaining modes in Table 2, we detect the first harmonic of 1039 μHz (k = 21) with an amplitude of 0.7 mma. For k = 19, 17, 15, 14, and 11, we place upper limits of 0.5 mma for their first harmonics. The 2154 μHz peak, identified as k = 9, m = 0 (k = 9⁰) in W94, has a first harmonic with an amplitude of 1.0 mma. It is surprising that we do not detect a harmonic of the larger amplitude k = 9⁺¹ component. We place upper limits of 0.4 mma for harmonics of the k = 9^±1 components. Finally, the 2359 μHz component of k = 8, identified as k = 8⁻¹ in W94, has a first harmonic at 4719.483 μHz (2.1 mma). We place upper limits of 0.4 mma for harmonics of the k = 8⁰ and k = 8⁺¹ components. We note again that the larger amplitude k = 8⁺¹ does not have a significant harmonic.

We begin by plotting the observed values of R_c for first order harmonics in Figure 8. If no harmonic is detected, we present upper limits. Since we do not have m identifications for most modes but we know that m must be the same for harmonic parents, we plot theoretical predictions for l = 1, m = 0, 0 combinations for inclinations of 0, 20, 40, and 60° (solid lines) and l = 1, m = 1, 1 and m = −1, − 1 (dotted line). R_c values for m = 1, 1 and m = −1, − 1 combinations do not depend on inclination. Surprisingly, the observed R_c values and upper limits for the high-k modes are lower than the theoretical predictions. We could argue that, under the theoretical assumptions of Wu (2001), the detected high-k multiplet components are not consistent with m = 0.

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From the k = 9 harmonics, we find behavior closer to theoretical predictions. The k = 9⁰ harmonic is consistent with an inclination of between 0 and 40°, while the k = 9^±1 components are in the direction of theoretical predictions. However, for k = 8, the k = 8⁻¹ harmonic has a larger value of R_c than the k = 8⁺¹ upper limit, something that is difficult to explain using current theory and simple viewing arguments. The upper limit of R_c for k = 8⁰ is 29, and is not shown in Figure 8.

The most eye-catching example of cross combination peaks is found near 3000 μHz (Figure 1), corresponding to linear combinations between the two largest multiplets, k = 18 and k = 12. Our resolution and sensitivity are sufficient to reveal nine combinations, with amplitudes ranging from 3.4 mma to 1.1 mma. In several cases, multiple parent identifications are possible for each mode. For the following discussion, we include identifications for detected combinations involving at least one large amplitude parent, with most including the 1234 μHz mode in combination with components of k = 12 (Table 3). The 1234 μHz mode also produces large amplitude combinations (4.2 and 3.8 mma) with the 1039 (k = 21) and 1173 (k = 19) μHz modes.

Figure 9 presents observed R_c values and upper limits for first order cross combinations including the dominant 1234 μHz mode as one parent. We include the same theoretical predictions as in Figure 8, adding predictions for l = 1, m = +1, − 1 combinations for inclinations of 20, 40, and 60° (dotted lines), and m = 0, ± 1 combinations (dot/dash line). The R_c values for combinations of 1234 μHz with the high-k modes 21, 19, 15, and one component of k = 12 (1739 μHz) are consistent with either m = 0, 0 combinations with an inclination of roughly 40° or m = +1, − 1 combinations with an inclination closer to 50°. Recalling that the harmonic combinations (Figure 8) argue that none of the high-k modes are m = 0, the second choice of m = +1, − 1 combinations seems more likely. This further implies that 1234 μHz is either m = +1 or m = −1, and the other modes are all the remaining value.

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The relative values of R_c among the 3000 μHz combinations (k = 18+12), while independent of most model parameters, do depend on inclination, and should reflect their respective projection in our line of sight. We should find R_c values corresponding to combinations of m = 0, ± 1, m = 0, 0, m = −1, + 1, and m = same. However, even incorporating our tenuous identification of 1234 μHz as m = −1, given the number of multiplet components and large uncertainties in the inclination angle, our analysis based on Wu (2001) predictions produces no clear identification.

Our analysis of GD358's combination frequencies is complex. We find a very tenuous identification for the 1234 μHz mode of m = ±1, requiring the remaining high-k modes' largest component to be the remaining value. We find no clear m identifications for the components of k = 12. This mode is clearly more complex than the simple multiplet structure expected in the limit of slow rotation.

There are multiple reasons why GD358's combination frequencies defy simple interpretation. First, this star simply has too many principal modes simultaneously excited to large amplitude. The cross-talk between them is large, and the perturbative treatment of Wu (2001) may not be valid. Second, both Wu (2001) and Yeates et al. (2005) assume the intrinsic amplitudes are the same for every component in a multiplet, and this may simply not be the case for GD358. Third, the theoretical value of R_c relies on correct identification of l and m, although in simpler cases, as in Yeates et al. (2005), it can be used as a mode identification technique. While we are fairly certain that GD358's pulsations are l = 1, the assumption of a classical triplet split by rotation is not valid for GD358's high-k modes. Finally, we raise a possibility suggested by the k = 9 and 8 components and their harmonics: to first order, the m = 0 components sample the radial direction, and the m = ±1 components sample azimuthal direction. Components of the same mode have the same inclination, so cancellation effects should be the same. The k = 9 and 8 multiplet amplitudes and presence/absence of harmonics argue that something is interrupting the azimuthal symmetry of GD358. It is possible that the pulsations do not all share the same inclination, in which case they would not combine as expected by Wu (2001). The oblique pulsator model is well known, especially in the case of roAp stars. For example, Bigot & Dziembowski (2002) show that the main mode of HR 3831, a pulsating roAp star, departs from alignment with the magnetic axis. GD358's combinations are trying to tell us something important about the pulsation geometry, but we do not yet understand how to interpret it.

While the general frequencies of the high-k multiplets agree with the identifications of W94 and K03, the 2006 multiplets themselves are complex, and cannot be explained by simply invoking rotation using standard theory and simple geometric viewing arguments. Except for the k = 9 and k = 8 modes, we do not find triplet structure as reported in W94 and cannot provide m identifications. The facts that the multiplets do change and the 2006 combinations do not conform to theoretical predictions leads us to expand our investigation to include archival observations of GD358. In the following, we will examine the structure of GD358's multiplets in detail over timescales of years.

In 1996 June, GD358's light curve (third panel, Figure 16) was typically nonlinear, with a dominant frequency of 1297 μHz corresponding to k = 17, and a peak to peak intensity variation of ≈15%. The k = 9 and 8 modes were present with amplitudes of 8 and 9 mma, respectively. On August 10 (suh-0055, second panel Figure 16), the light curve was again typical, with a 20% peak to peak intensity variation, and main frequencies consistent with previously observed high-k modes. The k = 9 and 8 modes had upper amplitude limits of 8 mma. On August 12, 27 hr later, GD358 exhibited completely altered pulsation characteristics (an-0034, top panel Figure 16). The lightcurve was remarkably sinusoidal, with an increased peak to peak intensity variation of ≈50%. The k = 8 mode dominated the lightcurve with a amplitude of 180 mma, the highest ever observed for GD358. The k = 9 mode was also present, with an amplitude of ≈20 mma. The high-k modes, dominant 27 hr prior, had upper detection limits of 7 mma. Over the next 24 hr, the k = 8 mode began to decrease in amplitude (Figure 17). Between Aug 13 and 14, k = 9 grew in amplitude as k = 8 decreased, reaching a maximum of ≈57 mma, and then began to decrease as the k = 8 mode continued to dissipate. By August 16, a mere 96 hr after the sforzando began, GD358 was an atypically low amplitude pulsator, with power detected in the k = 9 and 8 modes at amplitudes of ≈7 mma, and upper limits of less than 5 mma for the high-k modes. By September 10, less than a month after the start of the sforzando, the high-k modes had reappeared, with the dominant mode at ≈1085 μHz at an amplitude of 29 mma.

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The only observations we have during the maximum sforzando are the PMT photometry (Table 5) observations. Differential photometry does not measure standard stars. However, we can calculate the ratio between the average counts from GD358 and its comparison star, assuming the same comparison star was used each night, and accounting for factors such as extinction and sky variation. In the following, we calculate ratios based on sky subtracted counts obtained within ±1.5 hr of the zenith. We focus on the McDonald 2.1 m runs an-0034, an-0036, an-0038, and an-0040. The ratio between GD358 and the comparison star decreased from 1.23 during the height of the sforzando (an-0034, Aug 12) to 1.01 on Aug 15 (an-0040). We also examined observations from Mt. Suhora (Table 5). Suh-0055 was obtained ≈24 hr before the sforzando, and the ratio between GD358 and the comparison star was 0.11. Suh-0056 took place 18 hr after the sforzando maximum but before the pulsations had completely decayed. For suh-0056, the ratio between GD358 and the comparison star increased to 0.16. For the remaining Mt. Suhora observations, obtained during the following days, the ratio between GD358 and the comparison star decreased to 0.12. Observation logbooks at Mt. Suhora demonstrate that the same comparison star, but not the same one used at McDonald, was used each night (Zola 2008). While we cannot pin down any exact numbers due to the nature of differential photometry, the observations of GD358 are consistent with a flux increase in the observed bandpass during the maximum sforzando.

The onset of convection goes hand in hand with the initiation of pulsation in both hydrogen and helium white dwarfs (Brickhill 1991; Wu 2000; Montgomery 2005). A primary goal of XCOV25 is to use GD358's nonlinear lightcurve to characterize its convection zone, improving the empirical underpinnings of convective heat transport theory. Our asteroseismological investigation points to evidence that something in the star's outer surface layers is influencing its pulsation modes. The convection zone lies in the outermost regions of GD358, so we are led to consider the relationship between the star's convection zone and its photometric behavior, both typical and during the sforzando.

In general, theoretical studies of convective energy transport are based on the mixing length theory (MLT) (Bohm-Vitense 1958). Originally intended to depict turbulent flows in engineering situations, MLT enjoys success in describing stellar convection (Li & Yang 2007). It remains, however, an incomplete model with unresolved problems. The actual mixing length is not provided by the theory itself, but is defined as (αH_p), where H_p is the pressure scale height, and α is an adjustable variable. This adjustable parameter undermines the ability of theoretical models to make useful predictions. Advances in convection theory include stellar turbulent convection (Canuto & Mazzitelli 1991; Canuto et al. 1996), which establishes a full range of turbulent eddy sizes. Work is currently underway employing helioseismology to compare MLT and turbulent convection models (Li & Yang 2007). Our work with convective light curve fitting will also provide important tests for convection theory.

Physical conditions in white dwarf atmospheres differ by orders of magnitude from those in envelopes of main sequence stars like the Sun. For GD358, models indicate that the convection zone is narrow, with a turnover time of ≈1 second and a thermal relaxation timescale of ≈300 s (Montgomery 2008). The pressure scale height is small, limiting the vertical height of the convective elements. Montgomery & Kupka (2004), employing an extension of stellar turbulent convection, calculate that 40–60% of GD358's flux is carried by convection, depending on the adopted effective temperature.

Convective lightcurve fitting is based on the assumption that the convection zone, by varying its depth in response to the pulsations, is responsible for the nonlinearities typically observed in GD358's lightcurves (Montgomery 2005). During the sforzando, within an interval of 27 hr, GD358's lightcurve became sinusoidal, arguing that the convection zone was inhibited, at least as sampled by the k = 8 mode. For example, if this large amplitude mode was the m = 0 member of the triplet, then its brightness variations would be predominantly at the poles and not the equator. Thus, all that is required to produce a sinusoidal lightcurve is a mechanism to inhibit convection near the poles.

How could convection in any star be decreased on such short timescales? The obvious method, but not necessarily the most physical, is to raise GD358's effective temperature, forcing it to the blue edge of the instability strip, and decreasing convection. Such a temperature increase has the observable consequence of increasing the stellar flux. A flux increase of 20% in our effective photometric bandpass would produce an equivalent change in the bolometric flux of ≈40%, corresponding to a temperature change of ≈2200 K. The sforzando photometry is consistent with such an increase in flux, and Weidner & Koester (2003) find that T_eff ≈ 27, 000 K is required to simulate GD358's light curve during the maximum sforzando. Castanheira et al. (2005) use FOS spectroscopy of GD358 obtained on 1996 August 16 to determine T_eff = 23900 ± 1100 K, a number not inconsistent with previous observations (Beauchamp et al. 1999). However, the FOS spectra were acquired after the pulsations had greatly decreased in amplitude (Figures 16 and 17). It is possible that GD358 had cooled by this point. Recent theoretical evidence also indicates that simply "turning off" convection would have minimal observable effects on GD358's spectrum (Koester 2008).

The connection between convection and pulsation during the sforzando may also depend on the origin of the event. Was the sforzando a short-term change, such as a collision, or a more global event intrinsic to GD358?  The growth and dissipation of modes during the sforzando could be connected to pulsation growth timescales, or may be governed by a completely different mechanism. Pulsation theory predicts that growth rates for high-k modes are much larger than for low-k modes. This simply means that high-k modes are easier to excite and have growth times (and decay times) of days rather than years (Goldreich & Wu 1999). This makes sense for GD358, where we typically find the high-k modes to be much more unstable in amplitude over time than k = 9 and 8 (Figure 10). Figure 17 plots the amplitudes for the k = 8, k = 9, and "high-k" modes during the sforzando. We note that the "high-k" points do not correspond to any particular k value, but give the highest amplitude (or upper limit) between 1000 and 1500 μHz. We fit the observed amplitudes for k = 8 and the high ks with a simple exponential decay model $A(t) = A_{0}e^{-t/\tau _d}+c$, where A(t) is the amplitude at time t and τ_d is the damping constant. Our fits give a damping timescale of τ_d = 1.5 ± 0.2 days for both the k = 8 mode and the high ks. The k = 9 mode, which actually grew in amplitude later, as k = 8 decayed, has an upper limit of $\tau _{9^d}<1.1$ days. We only have a few points measuring the growth timescale (τ_g) of k = 8 and 9, so we place an upper limits for k = 8 of $\tau _{8^g}\le 1.2$ days. For k = 9, we find an upper limit of $\tau _{9^g}\le 2.2$ days. These timescales are all roughly equivalent, are all much longer than the dynamical timescale, and not distributed as predicted by theoretical pulsation growth rates. The results for k = 9 and 8 are much shorter than the expected pulsation damping timescale of about two months (Montgomery 2008a), arguing that the source of the sforzando was capable of interacting with the pulsations to produce changes on faster timescales than theoretical growth rates.

Our purpose in entering this discussion of the sforzando was to provide insight into connections between convection and pulsation in GD358. We find evidence that convection diminished during the sforzando. The decreased convection was accompanied by the rapid disappearance of GD358's high-k modes and the transfer of energy to the k = 8 and 9 modes. The photometry is consistent with a temperature increase, which would inhibit convection. The sudden transfer of energy from the high-k modes to k = 9 and 8 argues that a mechanical (radial or azimuthal) or thermal structural change altered GD358's outer layers, modifying mode selection. We find that the growth/damping timescales during the sforzando for the k = 9 and 8 modes do not agree with the expectations of pulsation theory, and may be governed by a completely different mechanism.

Magnetic fields have long been known to be capable of inhibiting convection. For example, localized fields associated with sunspots in the Sun are observed to inhibit convection in the surrounding photosphere (Biermann 1941; Houdek et al. 2001). Winget (2008) estimates that the magnetic decay time at the base of the convection zone is ≈3 days. In the next section, we investigate the possible implications of magnetic fields on GD358 and its pulsations.

Our current understanding of magnetic fields in stars like the Sun invokes the dynamo mechanism, which is believed to function in the narrow region where rotation changes from the latitudinal rotation of the outer convective layers to spherical rotation in the radiative zone (Morin et al. 2008). The solar magnetic field consists of small, rapidly evolving magnetic elements displaying large scale organization, with a cycle time of 22 years during which the polarity of the field switches. Localized fields, with strengths hundreds of times stronger than the global field, exert non-negligible forces on charged particles, influencing convective motion. Solar pulsations are observed to vary in frequency with the solar cycle in these regions (Schunker & Cally 2006). Although the exact mechanism is uncertain, the frequency shifts are interpreted as representing change in the Sun's internal structure or driving (Woodward & Noyes 1985; Kuhn 1998; Houdek et al. 2001).

A pulsating white dwarf represents an extreme environment. The atmosphere and the convection zone are thin, the surface gravity orders of magnitude higher, and differential rotation may or may not play a role (Kawaler et al. 1999). The magnetic field in a typical white dwarf is confined to the nondegenerate outer layers. The field may dominate near the surface, but deeper in the atmosphere, gas pressure will prevail. Drawing an analogy with the Sun, a surface magnetic field in GD358 probably has an associated cycle time similar to the solar cycle. Hansen et al. (1977) and Jones et al. (1989) discuss this topic from a theoretical perspective, predicting cycle times ranging from two to six years.

What are the observable consequences of a magnetic field on white dwarf pulsations? Jones et al. (1989) find that, in the presence of a weak field, defined as strong enough to perturb but not dominate mass motions, multiplet frequencies are increased with respect to the central mode in a manner proportional to m². The m = 0 mode is also shifted, unlike the case for rotation. Each g-mode pulsation samples the magnetic field at a different depth. A logical expectation is that, for low-k modes with reflection points below the convection zone, we can treat the magnetic field as a perturbation, while for high-k modes such a treatment is not valid.

Our best candidates to investigate the influence of a magnetic field on multiplet structure are GD358's k = 9 and 8 modes. These two modes are consistent with rotationally split triplets and we can examine their multiplet structure in detail over 16 years of observations (Figures 18 and 19). From 1990 to 2006, k = 9 had an average multiplet width of 7.7 ± 0.1 μHz, while k = 8 is similar, at 7.5 ± 0.1 μHz. Figures 18 and 19 show that the complete multiplet width and the splittings with respect to the central mode wander up to 0.5 μHz from the average values. Both Figures 18 and 19 reveal a dramatic change during the sforzando. In June 1996, both multiplets increased in total width by ≈1μHz. Prior to August 1996, the retrogradesplitting (0, −1) for k = 8 was consistently smaller by ≈0.2 μHz than the prograde splitting (0, +1). After 1996 August, the retrograde splitting became consistently larger by ≈0.5 μHz than the prograde splitting and has remained so through 2007, at the resolution of our observations. The case is not so straightforward for k = 9, but this mode clearly shows a large change in 1996, especially for the (0, −1) splitting (Figure 18). In the previous section, we questioned whether the sforzando was a short-term event or a global, enduring change. This analysis of k = 9 and 8 points toward a long-term change in GD358's resonance cavity. The changes we observe in the multiplet structure could be explained by a change in magnetic fields.

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A magnetic field also introduces an additional symmetry axis. Our analysis of GD358's harmonics and combination frequencies hints that something is disrupting its azimuthal symmetry. It is conceivable that high-k modes would align more or less with the magnetic field, while the low-k modes align with the rotation axis. Wu (2001) & Yeates et al. (2005) do not explore the possibility of multiple axes or the effects of a magnetic field on nonlinear mixing by the convection zone. Future work could explain why GD358's modes do not combine as expected by simple theory.

The presence of the k = 12 mode offers a final hint of a possible surface magnetic field. The second largest mode in the 2006 FT, k = 12 appears in a region that is normally devoid of significant power. If, for this mode, the magnetic field can no longer be treated as a perturbation, we would expect a multiplet with (2l + 1)² components, as opposed to the three expected from rotation (Unno et al. 1979). For l = 1, this corresponds to nine magnetic components. We do find six peaks above our 4σ limit, with three additional peaks slightly just our criteria. Using the magnetic model of Unno et al. (1979), we are able to match the frequencies of the k = 12 mode components, but not the amplitudes (Figure 20).

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