4 Main periodicities in the 2000 WET data

4.1 Assumptions and ground rules used in this work

We observed GD 358 as the primary target in May-June of 2000 to provide another "snapshot'' of the behavior of GD 358 with minimal alias problems. For the period May 23rd to June 8th, this run provided coverage ($\sim$80%) that was intermediate between the 1994 run (86% coverage) and the 1990 run (with 69% coverage). The 2000 WET run had several objectives: 1) look for additional modes besides the known $\ell=1$ k=8 through 18 modes; 2) investigate the multiplet splitting structure of the pulsation modes; 3) look for amplitude changes of the known modes; 4) determine the structure of the "combination peaks'', including the maximum order seen; and 5) provide simultaneous observations for HST time resolved spectroscopy.

Before we can start interpreting the peaks in the FT, we need to select an amplitude limit for what constitutes a "real'' peak versus a "noise'' peak. Kepler (1993) and Schwarzenberg-Czerny (1991, 1999), following Scargle (1982), demonstrated that non-equally spaced data sets and multiperiodic light curves, as all the Whole Earth Telescope data sets are, do not have a normal noise distribution, because the residuals are correlated. The probability that a peak in the Fourier transform has a 1/1000 chance of being due to noise, not a real signal, for our large frequency range of interest, requires at least peaks above $4\langle {{\rm Amp}}\rangle$, where the average amplitude $\langle {{\rm Amp}}\rangle$ is the square root of the power average (see also Breger et al. 1993 and Kuschnig et al. 1997 for a similar estimative).

 

 

Table 5: Average amplitude of datasets, from 1000 to 3000 $\mu$Hz.

Year	${{\rm BCT_{start}}}$	$\langle {{\rm Amp}}\rangle$
	(days)	(mma)
1990	244 8031.771867	0.62
1994	244 9475.001705	0.58
1996	245 0307.617884	1.44
2000	245 1702.402508	0.29

Table 5 shows that the noise, represented by $\langle {{\rm Amp}}\rangle$, for the 2000 data set is the smallest to date, allowing us to detect smaller amplitude peaks. Several peaks in the multi-frequency fits are below the $4\langle {{\rm Amp}}\rangle$ limit and therefore should be considered only as upper limits to the components.

The present mode identification follows that of the 1990 data set, published by Winget et al. (1994). They represent the pulsations in terms of spherical harmonics $Y_{\ell,m}$, with each eigenmode described by three quantum numbers: the radial overtone number k, the degree $\ell$, also called the angular momentum quantum number, and the azimuthal number m, with $2\ell+1$ possible values, from $-\ell$ to $+\ell$. For a perfectly spherical star, all $(2\ell+1)$ eigenmodes with the same values of k and $\ell$ should have the same frequency, but rotation causes each eigenmode to have a frequency also dependent on m. Magnetic fields also lift the m degeneracy. The assigned radial order k value are the outcome of a comparison with model calculations presented in Bradley & Winget (1994), and are consistent with the observed mass and parallax, as discussed in their paper. Vuille et al. (2000) determinations followed the above ones. In the upper part of Fig. 1, we placed a mark for equally spaced periods (correct in the asymptotic limit), using the 38.9 s spacing derived by Vuille et al., starting with the k=17 mode. The observed period spacings in the FT are very close to equal, consistent with previous observations.

 

 

Table 6: Our multisinusoidal fit to the main periodicities in 1990.

k	Frequency	Amplitude	$T_{{\rm max}}$
	$\rm (\mu Hz)$	(mma)	(s)
18	$1233.408 \pm 0.017$	$5.05 \pm 0.13$	$731 \pm 7$
17+	$1291.282 \pm 0.018$	$5.04 \pm 0.14$	$395 \pm 6$
17	$1297.590 \pm 0.006$	$14.60 \pm 0.14$	$24 \pm 2$
17-	$1303.994 \pm 0.019$	$4.71 \pm 0.14$	$411 \pm 7$
16+	$1355.664 \pm 0.106$	$0.87 \pm 0.14$	$471 \pm 37$
16	$1361.709 \pm 0.040$	$2.21 \pm 0.14$	$18 \pm 14$
16-	$1368.568 \pm 0.031$	$2.96 \pm 0.14$	$627 \pm 11$
15+	$1420.932 \pm 0.010$	$9.32 \pm 0.14$	$416 \pm 3$
15	$1427.402 \pm 0.005$	$19.24 \pm 0.14$	$425 \pm 2$
15-	$1433.853 \pm 0.011$	$7.90 \pm 0.14$	$88 \pm 4$
14+	$1513.023 \pm 0.017$	$5.23 \pm 0.14$	$123 \pm 5$
14	$1518.991 \pm 0.009$	$9.71 \pm 0.14$	$270 \pm 3$
14-	$1525.873 \pm 0.016$	$5.35 \pm 0.13$	$459 \pm 5$
13+	$1611.671 \pm 0.016$	$5.70 \pm 0.14$	$140 \pm 5$
13	$1617.297 \pm 0.017$	$5.28 \pm 0.14$	$508 \pm 5$
13-	$1623.644 \pm 0.019$	$4.70 \pm 0.14$	$407 \pm 5$
12	$1733.850 \pm 0.163$	$0.53 \pm 0.13$	$474 \pm 44$
11+	$1840.022 \pm 0.136$	$0.65 \pm 0.14$	$41 \pm 35$
11	$1846.247 \pm 0.135$	$0.66 \pm 0.14$	$504 \pm 34$
11-	$1852.099 \pm 0.093$	$0.94 \pm 0.14$	$132 \pm 24$
10+	$1994.240 \pm 1.071$	$\leq 0.14$
10	$1998.919 \pm 0.060$	$1.50 \pm 0.14$	$25 \pm 14$
10-	$2007.992 \pm 0.117$	$0.84 \pm 0.14$	$7 \pm 26$
9+	$2150.430 \pm 0.048$	$1.932 \pm 0.13$	$174 \pm 10$
9	$2154.052 \pm 0.020$	$4.59 \pm 0.13$	$336 \pm 4$
9-	$2157.834 \pm 0.032$	$2.81 \pm 0.13$	$400 \pm 7$
8+	$2358.975 \pm 0.016$	$5.68 \pm 0.13$	$120 \pm 3$
8	$2362.588 \pm 0.016$	$5.77 \pm 0.13$	$422 \pm 3$
8-	$2366.418 \pm 0.017$	$5.34 \pm 0.13$	$268 \pm 3$

 

 

Table 7: Main periodicities in 1994.

k	Frequency	Amplitude	$T_{{\rm max}}$
	$\rm (\mu Hz)$	(mma)	(s)
18+	$1228.712 \pm 0.022$	$2.77 \pm 0.13$	$252.6 \pm 12.0$
18	$1235.493 \pm 0.005$	$12.94 \pm 0.13$	$170.9 \pm 2.6$
18-	$1242.364 \pm 0.016$	$3.66 \pm 0.13$	$62.4 \pm 9.0$
17+	$1291.093 \pm 0.010$	$6.17 \pm 0.13$	$250.3 \pm 5.1$
17	$1297.741 \pm 0.003$	$22.11 \pm 0.13$	$37.7 \pm 1.4$
17-	$1304.459 \pm 0.010$	$6.25 \pm 0.13$	$615.0 \pm 5.0$
16+	$1355.388 \pm 0.035$	$1.70 \pm 0.13$	$167.0 \pm 17.6$
16	$1362.298 \pm 0.060$	<0.89
16-	$1368.541 \pm 0.031$	$1.92 \pm 0.13$	$322.3 \pm 15.5$
15+	$1419.650 \pm 0.003$	$18.37 \pm 0.13$	$46.4 \pm 1.6$
15	$1426.408 \pm 0.004$	$15.55 \pm 0.13$	$239.7 \pm 1.8$
15a	$1430.879 \pm 0.006$	$10.61 \pm 0.13$	$187.9 \pm 2.7$
15-	$1433.169 \pm 0.014$	$4.46 \pm 0.13$	$104.1 \pm 6.5$
14	$1519.903 \pm 0.028$	$1.09 \pm 0.13$	$485.2 \pm 20.0$
13+	$1611.357 \pm 0.012$	$5.02 \pm 0.13$	$466.0 \pm 5.0$
13	$1617.474 \pm 0.009$	$3.46 \pm 0.13$	$183.3 \pm 1.1$
13-	$1624.568 \pm 0.010$	$6.07 \pm 0.13$	$101.0 \pm 4.1$
12	$1746.766 \pm 0.064$	$0.93 \pm 0.13$	$414.2 \pm 25.0$
11	$1863.004 \pm 0.184$	<0.71
10	$2027.325 \pm 0.457$	<0.46
9+	$2150.504 \pm 0.019$	$3.15 \pm 0.13$	$346.7 \pm 6.1$
9	$2154.124 \pm 0.013$	$4.76 \pm 0.13$	$12.3 \pm 4.0$
9-	$2157.841 \pm 0.022$	$2.69 \pm 0.13$	$144.2 \pm 7.1$
8+	$2358.883 \pm 0.013$	$4.50 \pm 0.13$	$398.2 \pm 3.8$
8	$2362.636 \pm 0.006$	$9.25 \pm 0.13$	$274.6 \pm 1.8$
8-	$2366.508 \pm 0.007$	$4.22 \pm 0.13$	$169.7 \pm 3.8$

 

 

Table 8: Main modes in 1996.

k	Frequency	Amplitude	$T_{{\rm max}}$
	$\rm (\mu Hz)$	(mma)	(s)
19	$1172.66 \pm 0.15$	$2.5 \pm 0.6$	$17.4 \pm 47.5$
18	$1253.65 \pm 1.01$	< 2.1
17+	$1291.13 \pm 0.11$	$4.3 \pm 0.6$	$653.6 \pm 28.1$
170	$1295.38 \pm 0.17$	$2.6 \pm 0.6$	$139.7 \pm 43.6$
17-	$1304.68 \pm 0.11$	$4.8 \pm 0.6$	$346.3 \pm 25.7$
16+	$1355.21 \pm 2.02$	< 1.9
160	$1362.55 \pm 0.15$	$2.7 \pm 0.6$	$172.2 \pm 41.2$
16-	$1379.64 \pm 0.16$	$2.7 \pm 0.6$	$378.6 \pm 41.0$
150	$1427.47 \pm 0.92$	< 2.2
15-	$1434.36 \pm 0.18$	$2.2 \pm 0.6$	$154.5 \pm 47.1$
14	$1520.58 \pm 0.18$	$2.0 \pm 0.6$	$398.3 \pm 46.6$
13+	$1611.60 \pm 0.18$	$2.1 \pm 0.6$	$461.8 \pm 42.7$
130	$1617.51 \pm 0.35$	$1.1 \pm 0.6$	$183.5 \pm 79.9$
13-	$1619.63 \pm 0.84$	< 2.2
12	$1736.10 \pm 0.34$	$1.1 \pm 0.6$	$323.6 \pm 75.2$
11	$1862.93 \pm 0.39$	$0.9 \pm 0.6$	$58.8 \pm 78.7$
10	$2027.41 \pm 0.26$	$1.4 \pm 0.6$	$471.9 \pm 47.9$
9+	$2149.97 \pm 0.07$	$5.6 \pm 0.6$	$69.8 \pm 11.8$
90	$2153.84 \pm 0.05$	$7.6 \pm 0.6$	$155.2 \pm 8.5$
9-	$2157.89 \pm 0.04$	$9.1 \pm 0.6$	$426.7 \pm 7.2$
8+	$2358.63 \pm 0.03$	$12.6 \pm 0.6$	$125.3 \pm 4.7$
80	$2362.50 \pm 0.02$	$23.2 \pm 0.6$	$104.8 \pm 2.5$
8-	$2365.98 \pm 0.02$	$22.2 \pm 0.6$	$142.9 \pm 2.6$

4.2 Nonlinear least squares results from 1990, 1994, and 1996

For a more self-consistent comparison, we took the data from the 1990, 1994 WET runs and the August 1996 run and derived the periods of the dominant modes via a nonlinear least squares fit. In Tables 6, 7 and 8 we present the results of a non-linear simultaneous least squares fit of 23 to 29 sinusoids, representing the main periodicities, to the 1990, 1994 and 1996 data sets. We use the nomenclature k^a, for example 15^-, to represent a subcomponent with m=-1 of the k=15 mode in these tables. The difference in the frequencies reported in this paper compared to the previous ones is due to our use of the simultaneous non-linear least-squares frequency fitting rather than using the Fourier Transform frequencies.

We note that both the Fourier analysis and multi-sinusoidal fit assume the signal is composed of sinusoids with constant amplitudes, which is clearly violated in the 1996 data set. The changing amplitudes introduce spurious peaks in the Fourier transform. This will not affect the frequency of the modes, but the inferred amplitude will be a poor match to the (non-sinusoidal) light curve amplitude.

In Table 5 we present the average amplitude of the data sets, from 1000 to 3000 $\mu$Hz, after the main periodicities, all above $4\langle {{\rm Amp}}\rangle$, have been subtracted. For the 2000 data set, the initial $\langle {{\rm Amp}}\rangle$ for the frequency range from 0 to 10 000 $\mu$Hz, is 0.69 mma. For the high frequency range above 3000 $\mu$Hz, $\langle {{\rm Amp}}\rangle\simeq 0.2$ mma.

4.3 Mode analysis of the 2000 WET data

To provide the most accurate frequencies possible, we rely on a non-linear least squares fit of sinusoidal modes with guesses to the observed periods, since these better take into account contamination or slight frequency shifts due to aliasing. In Table 9 we present the results of a simultaneous non-linear least squares fit of 29 sinusoids, representing the main periodicities of the 2000 data set, simultaneously. All the phases have been measured with respect to the barycentric Julian coordinated date BCT 2 451 702.402 508.

 

 

Table 9: Main modes in 2000.

k	Frequency	Period	Amplitude	$T_{{\rm max}}$
	$\rm (\mu Hz)$	(s)	(mma)	(s)
20	$1110.960 \pm 0.017$	$900.122 \pm 0.014$	$2.04 \pm 0.08$	$870.19 \pm 8.61$
19	$1172.982 \pm 0.013$	$852.528 \pm 0.009$	$2.74 \pm 0.08$	$164.86 \pm 6.07$
$\ell=2$	$1255.400 \pm 0.002$	$796.556 \pm 0.002$	$14.86 \pm 0.08$	$747.02 \pm 1.05$
18	$1233.595 \pm 0.018$	$810.639 \pm 0.012$	$1.96 \pm 0.08$	$354.91 \pm 8.11$
17+	$1294.284 \pm 0.094$	$772.628 \pm 0.056$	$0.38 \pm 0.082$	$579.63 \pm 39.54$
17	$1296.599 \pm 0.001$	$771.248 \pm 0.001$	$29.16 \pm 0.08$	$247.81 \pm 0.52$
17-	$1301.653 \pm 0.053$	$768.254 \pm 0.031$	$0.68 \pm 0.08$	$314.52 \pm 22.52$
16	$1362.238 \pm 0.159$	$734.086 \pm 0.086$	$0.42 \pm 0.12$	$263.36 \pm 63.93$
16-	$1378.806 \pm 0.007$	$725.265 \pm 0.004$	$5.35 \pm 0.08$	$514.70 \pm 2.66$
15+	$1420.095 \pm 0.001$	$704.178 \pm 0.001$	$29.69 \pm 0.08$	$418.14 \pm 0.49$
15	$1428.090 \pm 0.052$	$700.236 \pm 0.025$	$0.70 \pm 0.08$	$217.29 \pm 19.78$
15-	$1432.211 \pm 0.036$	$698.221 \pm 0.018$	$1.08 \pm 0.089$	$355.58 \pm 13.30$
14	$1519.811 \pm 0.134$	$657.977 \pm 0.058$	$0.266 \pm 0.08$	$301.55 \pm 48.28$
13+	$1611.084 \pm 0.116$	$620.700 \pm 0.045$	$0.31 \pm 0.08$	$448.66 \pm 39.81$
13	$1617.633 \pm 0.174$	$618.187 \pm 0.066$	$0.21 \pm 0.08$	$448.19 \pm 58.97$
13-	$1625.170 \pm 0.235$	$615.320 \pm 0.089$	$0.15 \pm 0.08$	$299.24 \pm 79.32$
12	$1736.277 \pm 0.034$	$575.945 \pm 0.011$	$1.04 \pm 0.08$	$230.37 \pm 10.79$
11	$1862.871 \pm 0.042$	$536.806 \pm 0.012$	$0.84 \pm 0.08$	$503.66 \pm 12.43$
10	$2027.008 \pm 0.028$	$493.338 \pm 0.007$	$1.29 \pm 0.08$	$350.14 \pm 7.49$
9+	$2150.462 \pm 0.012$	$465.016 \pm 0.003$	$2.96 \pm 0.08$	$35.49 \pm 3.09$
9	$2154.021 \pm 0.007$	$464.248 \pm 0.001$	$5.34 \pm 0.08$	$252.38 \pm 1.71$
9-	$2157.731 \pm 0.014$	$463.450 \pm 0.003$	$2.57 \pm 0.08$	$174.68 \pm 3.53$
8+	$2359.119 \pm 0.006$	$423.887 \pm 0.001$	$5.57 \pm 0.08$	$166.60 \pm 1.49$
8	$2362.948 \pm 0.094$	$423.200 \pm 0.017$	$0.38 \pm 0.08$	$81.50 \pm 21.93$
8-	$2366.266 \pm 0.006$	$422.607 \pm 0.001$	$5.79 \pm 0.08$	$418.36 \pm 1.44$
$2\times 18$	$2510.761 \pm 0.021$	$398.286 \pm 0.003$	$1.70 \pm 0.08$	$334.47 \pm 4.58$
$2\times 17$	$2593.208 \pm 0.004$	$385.623 \pm 0.001$	$7.82 \pm 0.08$	$249.57 \pm 0.96$
7	$2675.487 \pm 0.004$	$373.764 \pm 0.001$	$8.49 \pm 0.08$	$193.30 \pm 0.86$
$2\times 15$	$2840.195 \pm 0.008$	$352.089 \pm 0.001$	$4.29 \pm 0.08$	$47.50 \pm 1.60$

Armed with the new frequencies in Table 9, we comment on regions of particular interest in the FT. First, we identify several newly detected modes at P=373.76 s, $f=2675.49~\mu$Hz, amp = 8.43 mma; P=852.52 s, $f=1172.99~\mu$Hz, amp = 2.74 mma; and P=900.13 s, $f=1110.95~\mu$Hz, amp = 2.03 mma. Based on the mode assignments of Bradley & Winget (1994) we identify these modes as k=7, 19, and 20. The mode identification is based on the proximity of the detected modes with those predicted by the models, or even the asymptotical period spacings, but also because of resonant mode coupling, i.e., a stable mode will be driven to visibility if a coupled mode falls near its frequency, as it happens for k=7, which is very close to the combination of k=17 and k=16; and k=20, which falls near the resonance of the 8^- and the $\ell=2$ mode at $1255~\mu$Hz (see next paragraph). It is important to note that these modes appear in combination peaks with other modes, as shown in Table 11. This reinforces our belief that these modes are physical modes, and not just erroneously identified combination peaks. We note that Bradley (2002) analyzed single site data taken over several years, and found peaks at 1172 or $1183~\mu$Hz in April 1985, May 1986, and June 1992 data, lending additional credence to the detection of the k=19 mode or its alias.

Figure 9: Peaks around k=18 in the 1990 (solid line) and 2000 (dashed line) transforms.

The first previously known region of interest surrounds the k=18 mode, which lies near $1233~\mu$Hz, according to previous observations. In the 2000 data, the largest amplitude peak in this region lies at $1255~\mu$Hz, which is over $20~\mu$Hz from the previous location. Given that other modes (especially the one at k=17) has shifted by less than $4~\mu$Hz, we are inclined to rule out the possibility that the k=18 mode shifted by $20~\mu$Hz. One possible solution is offered by seismological models of GD 358, which predict an $\ell=2$ mode near $1255~\mu$Hz. For example, the best ML2 fit to the 1990 data (from Metcalfe et al. 2002, Table 3), has an $\ell=2$ mode at $1252.6~\mu$Hz (P= 798.3 s). This would also be consistent with the larger number of subcomponents detected, although they may be caused only by amplitude changes during the observations. Figure 9 shows the region of the k=18 mode in the FT for the 1990 data set (solid) and the 2000 data set (dashed); it is consistent with the k=18 mode being the $1233~\mu$Hz for both data sets, and they even have similar amplitudes. While we avoided having to provide an explanation for why only the k=18 mode would shift by $20~\mu$Hz, we have introduced another problem, as geometrical cancellation for an $\ell=2$ mode introduces a factor of 0.26 in relation to unity for an $\ell=1$ mode. Thus, the identification of the $1255~\mu$Hz as an $\ell=2$ mode, which has a measured amplitude of 14.86 mma, implies a physical amplitude higher than that of the highest amplitude $\ell=1$ mode, around 30 mma.

Kotak et al. (2003), analyzing time-resolved spectra obtained at the Keck in 1998, show the velocity variations of the k=18 mode at $1233~\mu$Hz is similar to those for the k=15 and k=17 modes, concluding all modes are $\ell=1$. They did not detect a mode at $1255~\mu$Hz.

In Fig. 10 we show the pre-whitened results; pre-whitening was done by subtracting from the observed light curve a synthetic light curve constructed with a single sinusoid with frequency, amplitude and phase that minimizes the Fourier spectrum at the frequency of the highest peak. A new Fourier spectrum is calculated and the next dominant frequency is subtracted, repeating the procedure until no significant power is left. It is important to notice that with pre-whitening, the order of subtraction matters. As an example, in the 2000 data set, if we subtract the largest peak in the region of the k=18 mode, at 1255.41 $\mu$Hz, followed by the next highest peak at 1256.26 $\mu$Hz and the next at 1254.44 $\mu$Hz, we are left with a peak at only 1.3 mma at 1232.76 $\mu$Hz. But if instead we subtract only the 1255.41 $\mu$Hz followed by the peak left at 1233.24 $\mu$Hz, its amplitude is around 3.1 mma, i.e., larger. Pre-whitening assumes the frequencies are independent in the observed, finite, data set. If they were, the order of subtraction would not affect the result. Because the order of subtraction matters, the basic assumption of pre-whitening does not apply. We attempt to minimize this effect by noting that the frequencies change less than the amplitudes, and use the FT frequencies in a simultaneous non-linear least squares fit of all the eigenmode frequencies. But even the simultaneous non-linear least squares fit uses the values of the Fourier transform as starting points, and could converge to a local minimum of the variance instead of the global minimum.

The modes with periods between 770 and 518 s (k=17 through 13) are present in the 2000 data, though with different amplitudes than in previous years. Another striking feature of the peaks in 2000 is that one multiplet member of each mode has by far the largest amplitude, so that without data from previous WET runs, we would not know that the modes are rotationally split. The frequencies of these modes are stable to about $1~\mu$Hz or less with the exception of the 16^- mode, where the frequency jumped from about $1368.5~\mu$Hz in 1990 and 1994 to about $1379~\mu$Hz in 1996 and 2000 (see Fig. 11). Most of these frequency changes are larger than the formal frequency uncertainty from a given run (typically less than $0.05~\mu$Hz), so there is some process in GD 358 that causes the mode frequencies to "wobble'' from one run to the next. We speculate that this may be related to non-linear mode coupling effects. Whatever the origin of the frequency shifts, it renders these modes useless for studying evolutionary timescales through rates of period change.

The k=12 through 10 modes deserve separate mention because their amplitudes are always small; between 1990 and 2000, the largest amplitude peak was only 1.6 mma. The small amplitudes can make accurate frequency determinations difficult, and all three modes have frequency shifts of 13 to $33~\mu$Hz between the largest amplitude peaks in a given mode. The k=10 mode shows the largest change with the 1990 data showing the largest peaks at $1998.7~\mu$Hz and  $2008.2~\mu$Hz, while the 2000 data has one peak dominating the region at  $2027.0~\mu$Hz. An examination of the data in Bradley (2002) shows that the k=12 mode seems to consistently show a peak near 1733 to $1736~\mu$Hz, and that only the 1994 data has the peak shifted to  $1746.8~\mu$Hz, suggesting that 1994 data may have found an alias peak or that the $1736~\mu$Hz mode could be the m=+1 member and the $1746.8~\mu$Hz mode is the m=-1 member. The data in Bradley (2002) do not show convincing evidence for the k=11 or 10 modes, so we cannot say anything else about them.

It is interesting to note that the k=8 and k=9 modes are always seen as a triplet, with 3.58 $\mu$Hz separation for k=9, even in the 1996 data set. Our measured spacings are 3.54 and 3.69 $\mu$Hz, from m=-1 to m=0 and m=0 to m=1. The k=8 mode in 2000 shows an m=0 component below our statistical detection limit (A= 0.41 mma, when the local $\langle {{\rm Amp}} \rangle=0.29$ mma), but the m=1 and m=-1 modes remain separated by $2\times 3.58~\mu$Hz. All the higher k modes are seen as singlets in the 2000 data set. We also note that the k=8 and 9 modes have by far the most stable frequencies. The frequencies are always the same to within $0.3~\mu$Hz, and in some cases better than $0.1~\mu$Hz. However, the frequency shifts are large enough to mask any possible signs of evolutionary period change, as Fig. 12 shows. Thus, we are forced to conclude that GD 358 is not a stable enough "clock'' to discern evolutionary rates of period change.